Physics-Informed Identification of Tensegrity Prestress across Taut-Slack Transitions

Nomenclature

Structure and dynamics

$q$, $\dot{q}$, $\ddot{q}$ — nodal coordinates, velocities, accelerations ($\mathbb{R}^{3n}$, $n$ nodes)

$n_{\mathrm{free}}$, $n_{\mathrm{mem}}$ — free coordinates, members

$M$, $G$ — constant mass matrix, gravity vector

$X_k$, $Y_k$ — bar and cable member operators

$R(q)$, $\nabla R$ — constraint functions and their Jacobian

$\lambda$ — constraint multipliers

$\kappa$ — quadratic velocity terms, bar constraints

$f_s$, $f_d$, $f_{\mathrm{ext}}$ — cable elastic, damping, external forces (N)

$p$ — forcing amplitude (N)

$A$ — equilibrium matrix $s$, $m$ self-stress states, mechanisms

$T$, $V_g$, $V_s$ kinetic, gravitational, strain energy (J)

$E$, $W$ mechanical energy, work (J)

Members

$s_k$ member vector of cable $k$

$\bar{L}_k$ length of bar $k$ (m)

$L_k^{\mathrm{ref}}$, $L_{0k}$ length of cable $k$ in the reference configuration, and its unstressed rest length (m)

$\sigma_k$, $K_k$ force density (N/m) and stiffness (N/m) of cable $k$

$EA$, $c_k$ cable axial stiffness (N) and damper coefficient (N s/m)

$T_k$, $g_k$ cable tension (N) and slackness (m)

Learning

$\tau$, $T_{\mathrm{ref}}$ scaled time $t/T_{\mathrm{ref}}$ and reference time (s)

$q_\theta$, $\lambda_\phi$ displacement network, multiplier head

$c$, $s(\tau)$ record descriptors; hard-IC window

$r_{\mathrm{EOM}}$ equation-of-motion residual

$r_{\mathrm{bar}}$ bar-length constraint residual

$\mathcal{L}_{\mathrm{data}}$ sensor-data loss

$\mathcal{L}_{\mathrm{EOM}}$, $\mathcal{L}_{\mathrm{bar}}$ physics, constraint losses

$w_b$, $w_d$ loss weights

$\beta$ sharpness of the smoothed cable law (–)

Identifiability and assessment

$S(t)$ sensitivities of the sensor accelerations

$J$ Fisher information matrix

$s_{n,i}$ noise standard deviation, channel $i$

$S$, $\Gamma$ stress matrix, member grouping matrix

$\lambda_{\min}$ smallest eigenvalue (stability margin)

$U_c$ cable utilization (–)

$\hat{\sigma}$, $\hat{c}$, $\widehat{EA}$ identified prestress, damping, stiffness

Abbreviations

CRLB: Cramér–Rao lower bound; DAE: differential-algebraic equations; DOF: degree of freedom; EOM: equations of motion; LP: linear program; NCP: nonlinear complementarity problem; PINN: physics-informed neural network; RMSE: root-mean-square error.

Introduction

Tensegrities are self-stressed networks of bars in compression and cables in tension, in which self-stress, instead of moment-resisting joints, provides the structural integrity [1, 2, 3]. This self-stress governs the tangent stiffness, the natural frequencies, and the load paths, so a single geometry behaves differently at different prestress levels [4, 5]. This dependence makes tensegrities tunable and collapsible, since a tensegrity collapses when its prestress is removed. High prestress levels, however, may reduce the capacity of the system to sustain external loads. Prestress cannot be inferred from the geometry or measured without embedded sensors, and it may decrease during service through creep, anchorage slip, and thermal cycling; its loss is a critical damage mode [6, 7]. Because stiffness and frequency scale with the self-stress, this loss changes the structural behavior while the geometry stays visibly unchanged.

Forward analysis of tensegrities is well developed. Form-finding determines the shapes that sustain a self-stress [8], and dynamic formulations predict motion about a prestressed configuration [9, 10]. These analyses take the member parameters as input. Structural identification solves the opposite problem: it infers the model, or the state, that produced an observed response, for model updating, health monitoring, virtual sensing, and control [11, 12, 13].

For a tensegrity, identification is both more critical and more difficult than for a conventional frame. It is more critical because the governing quantity, the prestress, may decrease during service and is not directly observable from sparse structural measurements. It is more difficult for three reasons rooted in the mechanics. The prestress is distributed: once several independent self-stress states exist, it is no longer fixed by the geometry, and a change in the rest length of one cable redistributes force through the whole network. The cables are unilateral: they carry tension only and may slacken under strong excitation, which makes the dynamics nonsmooth. Finally, the instrumentation is sparse: measuring every member force is impractical, and the few measurements available must be combined with equilibrium, compatibility, the constitutive law, and member-wise admissibility.

The recovery of the prestress state and the member properties of a tensegrity from sparse, noisy dynamic records, with the tension-only cable law $T_k \ge0$, $g_k \ge0$, $T_k g_k = 0$ kept in the model, has received little attention. Here $T_k$ is the tension in cable $k$ and $g_k$ its slackness. It is also unclear which member parameters a given record can determine, and whether a taut–slack transition supplies information that a taut record lacks.

Earlier identification studies address parts of this problem. Prestress-dependent dynamic properties have been tracked on a grid [14], on a simplex across 13 measured prestress levels [15], and on a five-module active tensegrity across 11 levels [16]. Self-diagnosis locates unknown loads and damage [7, 17], cellular decomposition determines the minimal sensor configurations [18], and global equilibrium recovers several cable forces from measurements on a few members [19]. Modal updating and Bayesian filtering track prestress-sensitive frequencies and parameter drift on linearized models [20, 21, 22]. These studies are limited to the linearized taut regime. They rely on modal extraction, which loses sensitivity under sparse instrumentation and on structures whose modes are stiffened elastically rather than geometrically [23]. They also fix the damping beforehand or absorb it into the identified parameters, although measured tensegrity dissipation comes mainly from joint friction [24].

Physics-informed neural networks can avoid the linearization. A network approximates the response while the governing equations enter the training objective as a residual, and unknown physical parameters enter as trainable variables, so that response reconstruction and parameter identification become one inverse problem [25, 26, 27]. For structural systems the approach is established: it has been applied to inversion in solid mechanics [28], to structural parameter identification, to nonlinear and thermally damped systems, and to health assessment [29, 30, 31]. Bayesian variants quantify the uncertainty in the recovered quantities, and recent work estimates state and parameters jointly under incomplete measurement [32, 33]. All of these studies treat lumped systems divided by algebraic block or by degree of freedom; a tensegrity requires a division by physical member.

Physics-informed learning has been applied to tensegrities for force-density-informed prestress design [34], for dynamic response prediction [35], and in geometrically nonlinear, symmetry-reduced, and energy-informed forms [36, 37, 38]. All of these studies solve the forward problem on taut structures. The learning-based work closest to identification fits the parameters of the physics engine used to simulate tensegrity robots to their locomotion trajectories [39, 40], without a structural interpretation of the identified quantities or a statement of what a record can determine. Related fields have addressed parts of the nonsmooth problem. An active elastoplastic constraint has been used as an additional measurement equation [41], and the sensitivity to an event has been included in the Fisher information through saltation matrices, which map a small change in the state just before an event to the change just after it [42]. Complementarity conditions have been included in physics-informed losses for static contact [43], and networks for nonsmooth dynamics exist for low-dimensional friction oscillators [44].

The motivation for this study follows from this review. First, the identification methods applied to tensegrities are limited to the linearized taut regime and do not keep the tension-only cable law when cables slacken. Second, physics-informed methods have identified the parameters of lumped structural models, but for tensegrities they have been used only for forward analysis of taut structures. Third, the studies of nonsmooth identification reviewed above do not treat cable structures, and they do not quantify how much information a taut–slack transition contributes.

This study aims to address these points with a physics-informed network for the inverse problem. The equations of motion are written in constrained Cartesian coordinates [10]. In this form the mass matrix is constant, which stabilizes training, and the cable force densities enter the residual linearly, which gives closed-form residual sensitivities. A displacement network and a constraint-multiplier network are trained on that residual, with the force densities, the axial stiffness, and the damping coefficients treated as trainable parameters.

The study makes three contributions: an inverse formulation for tensegrity dynamics that keeps the tension-only cable law through taut–slack transitions; an identifiability analysis that shows, before inversion, which member parameters a record can determine and to what precision; and a demonstration of both on benchmark structures and on measured records. In the formulation, the tension-only condition is imposed through a smoothed (softplus) cable law, which is compared on identical records with two alternatives: a Fischer–Burmeister complementarity residual (a function that is zero exactly when the tension-only conditions hold) and collocation points concentrated near the transitions. The identifiability analysis uses Fisher information and Cramér–Rao bounds, and the linear entry of the force densities shows which members a record cannot determine. Where the parameter set is ill-conditioned, tying members into groups can restore a well-posed estimate, and those groups are the clustered cables already used for actuation [45, 46]. The demonstration uses simulated records of planar tensegrity frames built from one, two, three, and five Snelson X-modules, the largest of which forms a 16-cable two-span bridge, and of a three-dimensional triplex prism, with gravity and in a zero-gravity prestress series. It then uses measured records from three laboratory tests: the modal frequencies of a hexagonal tensegrity prism, chirp records of a biomimetic leg whose spring rates were measured independently, and impact records of a full-scale tensegrity simplex with a force transducer in each cable, at 13 prestress levels. No measured record in this study contains a slack cable. The treatment of taut–slack transitions is therefore validated on simulated records only, and the measured experiments validate the rebuilt models, the small-oscillation identification, and the time-domain residual.

The remainder of this study is organized as follows. § 2 presents the dynamics formulation, the benchmark structures, and the simulated datasets. § 3 develops the inverse formulation, the transition treatments, and the identifiability analysis. § 4 reports the demonstrations, § 5 discusses their implications and limits, and § 6 summarizes the findings and the scope.

Tensegrity Dynamics and Simulated Datasets

Constrained Lagrangian dynamics in Cartesian coordinates

The dynamics are written in redundant Cartesian nodal coordinates $q \in\mathbb{R}^{3n}$, following Goyal and Skelton [10] as implemented in the STEDY code [47]. Each rigid bar contributes consistent translational and rotational inertia to a mass matrix $M$ constant in $q$. Gravity enters as a constant generalized force $G$. Support conditions are linear constraints $Aq=b$ selecting the fixed coordinates. Each bar $k$ imposes the quadratic length constraint $q^{\mathsf{T}}X_k^{\mathsf{T}}X_kq=\bar{L}_k^2$, with $\bar{L}_k$ the bar length and $X_k$ the constant operator extracting the bar vector. The position-level constraints are collected as $R(q)=0$. The motion then satisfies the index-3 differential-algebraic system, which is integrated in its index-1 (acceleration-level) saddle-point form,

with $\lambda$ the constraint multipliers and $f_s$, $f_d$ the cable elastic and damping forces defined below. The kinetic energy is $T=\frac{1}{2}\dot{q}^{\mathsf{T}}M\dot{q}$. Differentiating the position-level constraints twice gives their velocity- and acceleration-level forms,

in which $\kappa$ collects the quadratic velocity terms of the bar constraints. Combining Eq. (1) with the acceleration-level condition gives the saddle-point system solved at every right-hand-side evaluation,

which returns the accelerations and the bar constraint forces together. An admissible initial state satisfies $R(q_0)=0$ and $\nabla R(q_0)\dot{q}_0=0$. This study uses the constant mass matrix and the linear entry of the cable force densities in $f_s$ throughout. Parameter sensitivities of the residual are therefore closed-form (§ 3.3).

The simulated records are obtained by integrating Eq. (1) through Eq. (3) with tight tolerances. The simulator reproduces the state derivatives of the STEDY code to a relative accuracy of $10^{-14}$, and its small-oscillation frequencies agree with those of an independently assembled prestressed truss tangent stiffness (Table 2). Its energy and constraint checks are described in § 2.5.

Check	Level
Residual against simulator state derivatives, slack included	$10^{-15}$
Residual at exact parameters, exact cable law	$10^{-15}$
Smoothed-cable-law peak force-density bias at $\beta= 800$	$< 1\%$
Simulator against classical-limit truss frequencies	$3 \times10^{-9}$
Frame generator against earlier X2 frequencies	$10^{-6}$
Flexible-bar residual against flexible-bar simulator	$10^{-9}$

Tension-only cable law and taut–slack transitions

Each cable $k$ is linear elastic and works only in tension. With $s_k=Y_kq$ the member vector, $L_{0k}$ the rest length, that is, the length of the unstressed cable, and $K_k$ the stiffness, the force density is

and the nodal cable force is $f_s=-\sum_k\sigma_kY_k^{\mathsf{T}}Y_kq$. Axial viscous dampers with coefficients $c_k$ act only while the cable is taut. Viscous cable damping is a modeling choice made for differentiability; measured tensegrity dissipation is dominated by joint friction [24], and since the damping coefficients are among the identified parameters, the consequence of that mismatch is shown on measured data in Section 4.1.2 and discussed in § 5. Equation (4) is the piecewise-smooth form of the complementarity law $T_k\geq0$, $g_k\geq0$, $T_kg_k=0$, in which $T_k=\sigma_k\lVert s_k\rVert$ is the tension in cable $k$ and $g_k=L_{0k}-\lVert s_k\rVert+T_k/K_k$ is its elastic slackness. That slackness vanishes exactly while the cable is taut and reduces to the geometric slack $L_{0k}-\lVert s_k\rVert$ while force-free. Classical forward solvers treat the taut–slack transitions with complementarity time-stepping [48, 49].

The prestress state is described by the cable force densities $\sigma_k$ at the reference configuration, and errors in the identified prestress are quoted throughout as force-density errors. Here $\sigma_k$ denotes a force density, the tension per unit length of the cable, and not a stress; the symbol $q$, often used for force densities in form-finding, is kept for the nodal coordinates of the dynamic formulation in § 2.1. Member properties follow the convention $K_k=EA/L_k^{\mathrm{ref}}+\sigma_k$ and $L_{0k}=L_k^{\mathrm{ref}}(1-\sigma_k/K_k)$, in which $L_k^{\mathrm{ref}}$ is the length of cable $k$ in the prestressed reference configuration, longer than its rest length $L_{0k}$. A given $\sigma$ then makes the reference configuration an exact static equilibrium. Admissible prestress states come from a linear program over force densities that balances gravity with all $\sigma_k\geq\sigma_{\min}$. The self-stress null-space modes of the equilibrium operator are computed for the identifiability analysis.

Benchmark structures and prestress series

Figure 1 presents the benchmark structures. The planar $X_n$ frames (Figure 1a) are staircases of Snelson X-modules, each the classical assembly of two crossed struts and four perimeter cables [1, 2]. Chaining $n$ modules gives $2(n+1)$ nodes, $2n$ bars (the module diagonals), and $3n+1$ cables. With the two end base nodes pinned and motion confined to the plane, the frame has $2n$ mechanical degrees of freedom (DOFs): its $4n$ free nodal coordinates less one length constraint for each of its $2n$ bars. The X1 frame ($n=1$, two DOFs) is the entry point for the identification experiments. Its base cable joins the two pinned nodes and therefore transmits no information to the dynamics. That member is provably unidentifiable, a built-in control for the identifiability analysis. The X2 frame (Figure 1b) has the geometry of the earlier simulations on which this study builds. Its node 3 is kinematically locked by its two bars to the pinned supports, and the structure and default forcing are mirror-symmetric. The evaluation metrics treat both properties explicitly. The code that generates the frames is verified against the earlier X2 implementation (Table 2). Scale-up studies use $n$ as the size parameter.

The triplex prism (Figure 1c) is the three-dimensional benchmark: bottom triangle pinned, top triangle free, three bars, and nine cables, which leave six DOFs from nine free coordinates. Its construction reproduces two classical results, independent checks of the mechanics. The free-floating pure self-stress exists only at the twist angle of $30^\circ$, the value $90^\circ- 180^\circ/n_b$ for a prism with $n_b = 3$ bars. In that state, the force densities of the bars, the cross cables, and the triangle cables are in the ratio $-t : t : t/\sqrt{3}$, with $t$ the prestress magnitude. Under gravity with the base pinned, the members must also carry their own weight. The linear program of § 2.2, with every cable force density at least 1 N/m, then finds a prestress with all bars in compression for twist angles from just above $30^\circ$ to about $58^\circ$, for the bars of this study (radius 5 cm, density 500 kg/m$^3$). The lower bound is the self-stress twist, and the upper bound grows with the self-weight relative to the minimum force density. The benchmark is built at $45^\circ$, inside this range. The prestress of the gravity-loaded prism is fixed uniquely by equilibrium; the only free self-stress states involve the bottom cables, which do not affect the dynamics. The prism therefore poses an inverse problem with a unique answer. The $X_n$ frames, by contrast, have free self-stress states and are used for the studies at several prestress levels. The prestress series of the prism is generated on the zero-gravity $30^\circ$ configuration, where the force densities scale linearly with the prestress level.

Excitation suite, sensor protocols, and noise model

Each record is generated by forward simulation from rest in the prestressed equilibrium fixed by the linear program of Section § 2.3 ($q(0)=q_0$, $\dot{q}(0)=0$). A time-dependent force $f_{\mathrm{ext}}(t)$ is then applied at one or more coordinates while Eq. (1) is integrated forward. A harmonic record applies at coordinate $d$ the force

with amplitude $p$ (N) and frequency $f_0$ (Hz). Three further classes replace this scalar signal at the same driven coordinate: a half-sine impulse, a band-limited earthquake-like signal, and stationary Gaussian noise. The earthquake-like class is white noise, low-pass filtered, shaped by a trapezoidal envelope, and normalized to its peak. The repeated runs with multi-point excitation in Section § 4.3 superpose signals of the form of Eq. (5) at two to four coordinates with distinct amplitudes and frequencies.

The two random classes are generated from fixed seeds, purely so that every dataset can be regenerated exactly; the seeds play no methodological role. These dataset seeds are also distinct from the training seeds used later: the records are fixed once, while the identification is repeated from several random network initializations and noise realizations to quantify the scatter of the identified values (§ 4.8).

The simulated tests cover the X2 frame at three prestress levels under harmonic and impulse forcing, with earthquake-like forcing added at the lowest level, together with the gravity-loaded prism and the zero-gravity prism at three prestress levels. At each combination the forcing amplitude is set on both sides of the slackening threshold, the smallest amplitude at which the forward simulation’s event tracker reports a cable leaving tension. Below that threshold a record stays strictly taut; above it the records contain many taut–slack transitions.

Every record spans 5 s at a 0.01 s output step, integrated at relative and absolute tolerances of $10^{-8}$ with continuous tracking of taut–slack transitions. The stored outputs of each record are the positions, velocities, accelerations, cable tensions, per-cable taut flags, and the times of the taut–slack transitions.

The sensor data are the instrumented-coordinate accelerations of the integrated trajectory, with Gaussian noise added at 2% of each channel’s root-mean-square. The sensor protocols of the identification experiments subsample the free coordinates under the same 2% noise. The retained fraction ranges from four-ninths to two-thirds across the demonstrations of Section § 4; the X1 frame uses half. The remaining coordinates are held out for the posterior-predictive check of § 4.2.

Constraint and energy checks

Two verification checks are evaluated along every stored record. The first is the relative violation of the squared bar length; since a relative length error $\epsilon$ violates the squared length by $2\epsilon$ to leading order, a reported violation $v$ corresponds to a length error of $v/2$ and converts directly to a physical length. The second is the energy-balance error $|E(t)-W(t)-E(0)|$, with $E$ the total mechanical energy and $W$ the work integral of the external and damping forces; the balance holds across taut–slack transitions because a slack cable stores no strain energy.

On the records used for identification the worst figures, at the end of the record, are a length drift of 0.33 $\mu$m on the 1.414 m bar diagonal and a relative energy error of $5\times10^{-6}$. These bound the numerical error the integration carries into the inverse problem; they establish that the simulated records are far more accurate than the 2% sensor noise added to them. The same checks apply unchanged to trained networks, where they measure the physical consistency of reconstructed motions.

Methodology

Physics-informed inverse formulation

Network parameterization and hard constraints

A displacement network and a multiplier head share a trunk, drawn as the computational graph of Figure 2c. The trunk maps the scaled time $\tau=t/T_{\mathrm{ref}}$, embedded in a fixed Fourier feature basis, through a tanh multilayer perceptron;

velocities and accelerations follow by automatic differentiation of the outputs in time. Outputting the constraint multipliers alongside the displacements places the formulation in the mixed-variable, hard-constraint PINN class [50, 51].

The displacement head outputs only the free coordinates, and $q_\theta(\tau)=q_0+s(\tau)B\tilde{u}_\theta(\tau)$ imposes the fixed ones by construction: $B$ selects the free coordinates and the window $s(\tau)=(1-e^{-\tau/\tau_{\mathrm{IC}}})^2$ enforces $q(0)=q_0$ and $\dot{q}(0)=0$ exactly, so no initial-condition loss is needed.

The multiplier head outputs the full vector $\lambda_\phi(\tau)$ and is initialized at static equilibrium multipliers. In the synthetic benchmarks these are computed from the exact prestress state, a value available only because the benchmark is synthetic. That value is used only for the initialization and is never a trainable parameter. In an application, the same initialization would be computed from the nominal or design prestress, and an error in it would change only the starting point of the head. Every transition treatment uses this initialization. For multi-record training the trunk is additionally conditioned on record descriptors $c$.

Equation-of-motion residual with trainable member parameters

The physics residual evaluates Eq. (1) on the network outputs,

in which the force densities $\boldsymbol{\sigma}$, the axial stiffness, and the damper coefficients are trainable parameters, represented in logarithmic form to enforce positivity. The external forcing $f_{\mathrm{ext}}$ is a known input, as in a controlled excitation test with prescribed, recorded load. An unknown or ambient forcing would add its amplitude and phase as unknowns; that is left to future work. All structural operators in Eq. (6) are constant tensors precomputed from the incidence matrices, so the residual applies unchanged to any tensegrity topology. Because the residual is linear in $\boldsymbol{\sigma}$, the sensitivities $\partial r_{\mathrm{EOM}}/\partial\sigma_k$ are closed-form, and § 3.3 uses this property in the identifiability analysis. The residual implementation is verified by the checks collected in Table 2; training itself uses the smoothed cable law described below.

Throughout this study the exact parameters of a synthetic benchmark are used in three places only: to verify the residual implementation, to evaluate the reference Fisher analysis of § 3.3, and to score the recovered values after a run. The identification itself uses only the noisy sensor records, the known forcing, the geometry and supports, and the stated initial guesses.

The training objective combines three terms: a sensor-data loss on measured accelerations at instrumented coordinates, the mean squared residual of Eq. (6) on fresh collocation points resampled each step, and the squared bar-length constraint residual,

each normalized by its natural scale: the pooled mean-square measured acceleration over the instrumented channels, the squared forcing amplitude, and the squared bar length. The data normalizer, a single global response power, stays bounded away from zero for a forced record, so no denominator regularization is needed.

Because each term of Eq. (7) is divided by its own scale, the remaining weights are dimensionless, and data and physics are weighted equally. A sensitivity study in Section S2 of the Supplementary Material varies both weights on the X1 taut record. Reproducibility is flat to within about 2 percentage points across three decades of $w_b$ around the value used throughout ($w_b=100$), and worsens sharply only when the bar constraint is effectively unenforced. Changing the data-to-physics ratio between 0.3 and 3 changes the mean force-density error by about 1 percentage point; at 10, where the sensor data are effectively ignored, the error increases sharply. Under model-form error, down-weighting the suspect physics makes the identification slightly worse (6.0 against 5.8% at a ratio of 0.1), because the residual’s error is structured by the geometry and still informs the parameters.

A single Adam optimizer [52] updates the network weights and the physical parameters jointly, in two groups with learning rates of $2\times10^{-3}$ and $10^{-2}$, both cosine-annealed to zero over the run [53]. The physical parameters are represented in logarithmic form, which makes their updates multiplicative. One relative step size therefore serves force densities, an axial stiffness, and damper coefficients whose values differ by orders of magnitude (details in Section S2 of the Supplementary Material). No gradient clipping is applied and no per-structure tuning is performed: the same two rates are used for every result reported here.

Table 1 lists the benchmark structures; the full table of simulation, sensing, and training parameters, with the origin of each value, and the archival scheme that reproduces every run exactly are given in Section S2 of the Supplementary Material. The X1 demonstrations of § 4.2 use a four-layer trunk of width 64, 256 collocation points per step, and 25,000 steps; the scale-up grid of § 4.3 uses the same architecture at 12,000 steps.

Structure	Nodes	Bars	Cables	Mech. DOFs	Prestress source
X1 frame	4	2	4	2	LP, 25 N/m min. force density
X2 frame	6	4	7	4	LP, 25 N/m min. force density
X3 frame	8	6	10	6	LP, 25 N/m min. force density
X5 bridge (two-span)	12	10	16	8	LP, 25 N/m; intermediate support
Triplex prism (45°, gravity)	6	3	9	6	uniquely fixed by equilibrium
Prism series (30°, zero-g)	6	3	9	6	self-stress scale 1–15 N/m
Hexagonal prism (laboratory)	12	6	18	free–free	single self-stress mode

The values in these tables come from different sources. The member properties of the planar benchmarks, cables of radius 1 mm and modulus 70 MPa and rigid bars of radius 5 cm and density 500 kg/m$^3$, and their geometry are those of the earlier simulations on which this study builds; they were kept so that the frame generator could be verified against those simulations (Table 2). The excitation amplitudes are set on both sides of the slackening threshold of each structure (§ 2.4), and the sensor noise and the sensor count are varied in § 4.9. The method settings, namely the network size, the learning rates, the loss weights, and the smoothing sharpness $\beta$, were fixed once and used unchanged from the two-degree-of-freedom frame to the 16-cable bridge and the three-dimensional prism, over a 15-fold range of prestress and sensor noise from 2 to 10 %. The sensitivity of the results to the loss weights is reported in Section S2 of the Supplementary Material, and the bias introduced by $\beta$ is bounded in § 3.1.3.

Treatments of the taut–slack transition

The tension-only law of Eq. (4) is a complementarity condition, and a gradient-based optimizer must cross the taut–slack boundary repeatedly while training. A force law whose derivative jumps at that boundary produces discontinuous gradients there. The primary treatment therefore replaces the ramp in the force density with a smoothed (softplus) cable law,

and smooths the on–off switching of the damping by the matching sigmoid. Because the smoothed cable law is differentiable, the gradients that reach the optimizer remain continuous across the taut–slack boundary, where the exact law has a kink.

The smoothing introduces a bias, and the sharpness $\beta$ is chosen to keep it small. The bias is confined to a boundary layer of width $1/\beta$ in the strain coordinate, where the softplus exceeds the exact ramp by at most $\ln2/\beta$. With $\beta= 800$, the layer width is $1.3 \times10^{-3}$, the largest overshoot is $9 \times10^{-4}$, and the peak bias is below 1 % of the recovered force densities. The same value is used for all structures and records. Outside the boundary layer, the bias decays rapidly to zero.

Two alternatives are compared with the smoothed cable law on identical records in § 4.5. The complementarity treatment keeps the exact law. It removes the nonsmoothness from the residual by adding an output head for the non-negative cable tensions, and it links each tension to the elastic law through a Fischer–Burmeister complementarity residual, a function that is zero exactly when the tension-only conditions hold, adapting the static-contact construction of Sahin et al. [43] to the dynamic setting. The event-aware treatment keeps the smoothed cable law and concentrates half of each batch’s collocation points near the taut–slack transitions detected in the current solution. Both are specified in Section S1 of the Supplementary Material. All three treatments share the same trunk architecture, loss weights and normalizations, and optimizer settings; none was tuned separately.

Adequacy of the nodal member model

The formulation above is nodal: cables are massless axial force elements, bars are rigid links, and member motion follows the node motion. This representation is used for all identification results reported in this study. It does, however, discard the member’s own dynamics, and near slack this omission introduces a model-form error. A member-resolved alternative treats each member as a one-dimensional continuum coupled to its neighbors at the nodes.

The two descriptions agree except at very small cable tensions. Bending and tension terms balance at $T^{\star}=E_c I_c(\pi/L_c)^2$, about $5 \times10^{-4}$ N for the cables of this study, against working tensions of 10 to 130 N. While the cable carries load, the flexural term is therefore four to five orders of magnitude below the tension term. The taut-string idealization, however, degenerates exactly at slack: its transverse wave speed $\sqrt{T_c/\rho_c A_c}$ falls to zero, and so does its fundamental frequency. Restoring the flexural term removes the degeneracy and leaves a finite frequency there. The member-resolved model is therefore used only as a diagnostic, to show when the nodal idealization becomes inaccurate; the ratio $T^{\star}/T$ measures by how much.

Identifiability analysis

Although Eq. (6) is linear in the force densities, the map from the parameters to the sensor accelerations is nonlinear, because it passes through the solution of the equations of motion. The sensitivity of this map, $S(t) = \partial a_{\mathrm{sensors}}(t)/\partial\theta$, is therefore obtained by central finite differences of the verified forward model (relative step $10^{-3}$, floored at $10^{-6}$). The Fisher information matrix of the parameter set $\theta= (\sigma, c)$ then follows for a given record, sensor set, and noise level:

Here $\Sigma_n = \operatorname{diag}(s_{n,i}^{2})$ collects the per-channel noise variances, with $s_{n,i}$ the standard deviation of sensor channel $i$: 2 % of that channel’s root-mean-square. The two instrumented channels have comparable RMS, so these bounds coincide with the equal-variance form to the precision reported below. The same information matrix underlies parameter identifiability and optimal sensor placement in structural dynamics [54, 55].

The analysis is local to the reference parameters. After removing the near-null direction spanned by the inert base cable, the identifiable block of $J$ is only moderately conditioned: its condition number is about $1.9 \times10^{2}$ on the taut record and $5.6 \times10^{2}$ on the slack-transition record. Locality does not rule out practical non-uniqueness, in which a run fits the record closely and still returns wrong parameters. This study calls such a run a silent failure; § 4.3 and § 4.7 report one on the X2 taut record and one on the bridge.

Concentration of prestress information in mechanism modes

The numerical analysis above measures how much information a record carries. A modal argument shows where that information comes from, before any data are taken.

Linearized about an equilibrium configuration, the tangent stiffness separates into a material part and a geometric part,

in which the operators $G_k$ depend on the geometry alone, so $K_g$ is linear in the force densities. This is the same linearity that Eq. (6) exploits, seen in the modal domain. For a mode $\phi_i$ normalized so that $\phi_i^\mathsf{T}M\phi_i = 1$,

the sensitivity following from the standard eigenvalue derivative together with the linearity of $K_g$.

The two terms of Eq. (10) behave differently. An infinitesimal mechanism is a displacement pattern that the compatibility operator annihilates. Its members do not extend to first order, so $\phi_i^\mathsf{T}K_e\phi_i \simeq0$: the frequency is carried entirely by the geometric term. That term is homogeneous of degree one in $\sigma$. Scaling the self-stress by $\lambda$ therefore scales $\omega_i^2$ by exactly $\lambda$, and the relative sensitivity reaches its largest possible value,

A mode stiffened elastically is at the other end of the same expression: with $\phi_i^\mathsf{T}K_e\phi_i \gg\phi_i^\mathsf{T}K_g\phi_i$, the ratio is small and the frequency is nearly insensitive to prestress.

Eq. (11) is a property of the geometry and the mode shape, computable before any record is taken, and it accounts for observations that otherwise look unrelated. The rotational mode of the simplex of Małyszko and Rutkiewicz [15] is a mechanism, and its squared frequency is proportional to the self-stress scale in the present rebuilt model over a 12-fold range, the intercept of a linear fit being $3 \times10^{-6}$ of the mean, while the translational modes of the same structure are flat across all 13 measured prestress levels. Two full-scale structures measured at the same laboratory show the two extremes. The five-module active tensegrity shifts its lowest frequency by about 29 % over a 1.7-fold force change [16]. The footbridge moves about 1 % while its self-stress quadruples [23]. The ratio in Eq. (11) is near 1 for the first and near 0 for the second. The X-frames of this study, whose frequencies shift about 12 % over a 50-fold prestress change, lie between them.

The identification of § 3.1 is a time-domain residual inversion and does not use modes. Eqs. (10) and (11) therefore do not describe the identification method itself; they describe the structure of the information the record contains, which is what the Fisher matrix above then measures numerically. Also, a small geometric fraction limits what the frequencies reveal about prestress; a full acceleration record can reveal more. The transient formulation recovers the force densities of the X-frames where a modal method would have little to work with (§ 4.3).

The analysis is based on three structural facts: which members the free coordinates cannot detect, how precisely the remaining members can be determined, and how much information is lost during the slack intervals. First, members whose ends are both restrained transmit no force to the free coordinates. Neither node moves, so the column of $\partial f/\partial\sigma$ belonging to such a member is identically zero. The member’s column of the sensitivity matrix $S$, whose rows are the sensor samples and whose columns are the parameters, is then zero as well, and so are its row and column of $J$. These members span the null space of $J$. On the X1 frame the base cable’s sensitivity is about 11 orders of magnitude below the identifiable members’ on the taut record, and about 6 orders below on the slack-transition record. $J$ itself has exactly one near-null eigenvalue, matching exactly one inert member (Figure 3). A taut–slack transition cannot remove this null space. A transition changes which members carry force, thereby redistributing information among the directions that the equilibrium operator determines, but it adds no information to the part of that operator’s null space that remains in every configuration. A doubly restrained member is invisible to the free coordinates whether taut or slack.

A self-stress combination among members that do move is a weaker case. It produces no free-coordinate force at the reference configuration. The operator, however, depends on the configuration, and the combination leaves the null space as soon as the structure displaces. A record with sufficiently strong excitation therefore provides information about it. The distinction is testable before any data are taken: which members have both ends restrained is a property of the geometry and the supports alone. The intervention study of § 4.4 confirms this: richer excitation and taut–slack transitions recover the identifiable members, while more sensors and more records do not.

Applied to the 12 classic tensegrity forms assessed in Section S3 of the Supplementary Material, this argument determines 164 of 183 cables at first order: 17 of the 19 undetermined cables are individually invisible members spanning two restrained nodes, and the other two are first-order self-stress combinations on the X2 and X3 frames. Identifiability in these structures depends on the support conditions, not on their size or redundancy, and the remedy differs: an invisible member needs a support layout or an instrumentation plan, since no record of any length or amplitude will substitute, while a first-order combination needs only a record with larger excitation.

The same algebra shows how the null direction can be removed. Suppose the members are tied into groups, each group sharing one tension or one force density, so that $\sigma= \Gamma^{\mathsf{T}}g$. Here $g$ holds one value per group, and row $i$ of the grouping matrix $\Gamma$ holds the coefficients that map the value of group $i$ to its members. By the chain rule, the sensitivity to the group values is $S\Gamma^{\mathsf{T}}$, and the Fisher information becomes $\Gamma J\Gamma^{\mathsf{T}}$. This matrix keeps a null direction of $J$ only if that direction lies in the range of $\Gamma^{\mathsf{T}}$, that is, in the row space of $\Gamma$. The members with nonzero entries in the null direction form its support, and the null direction cannot lie in the row space once a member of its support shares a group with a member outside it.

On the X2 frame, grouping can be examined both for static equilibrium and for the dynamic identification, and the two cases differ. For static equilibrium, the equilibrium sensitivity $\partial F/\partial\sigma$ of the seven independent cables gives a Gram matrix with eigenvalues $\{0,1,1,1,2,2,3\}$. The zero eigenvalue makes this parameterization singular: an equal increase of the force densities in the two bottom-chord cables leaves the forces at the free coordinates unchanged. When three cables are tied into one routed group, the smallest eigenvalue rises from 0 to 1.0, and the parameterization is no longer singular.

The dynamic identification uses a different matrix, the Fisher information $J$, assembled from the acceleration sensitivities over the whole record. Under the taut X2 protocol, $J$ is not singular even for independent cables; its smallest eigenvalue is 76. The static null direction does not remain in the dynamic problem, because a change of force density alters the geometric stiffness even where it leaves the reference-configuration force balance unchanged. The acceleration sensitivity along that direction is 11% of that of the strongest member, which is six orders of magnitude above the finite-difference noise floor.

Grouping still improves the dynamic problem. With the same routed group, the smallest eigenvalue of $J$ rises from 76 to $2.8 \times10^{2}$, and at 2% sensor noise the relative Cramér–Rao bound of the worst grouped force density is 0.09%. Both analyses use the same structure and the same record and differ only in the parameterization, so the improvement comes from the parameterization alone. Grouping is the standard remedy for ill-conditioned updating [45, 56]. The tensegrity setting adds that the groups need not be chosen for numerical convenience: the clustered cables routed through frictionless pulleys for actuation [46] already share a tension by construction. Such a structure does not have the static singularity that the same geometry with independent cables brings to any equilibrium-based identification.

Second, the diagonal of the inverse of the identifiable block gives the Cramér–Rao lower bounds. At 2% sensor noise the best attainable standard deviations on the X1 frame are 0.013–0.030 N/m for the force densities and 0.0026 N s/m for the damper coefficient. The identification of § 4.2 achieves a single-run damping error within a few percent of that bound, while its force-density errors are within a factor of 20 of theirs. That gap comes from the bias of the optimization and the smoothing. The bound limits the scatter of repeated identifications, not the error of a single run. The eight-seed study of § 4.8 finds the achieved dispersion 2.7 to 8.3 times above it. The identification is therefore accurate but not statistically efficient.

Third, a slack interval removes the force information of the slack member, while the transition times themselves carry information about rest lengths. On the slack-transition record the bound for the most-loaded cable increases by a factor of about two relative to the taut record. That factor measures the information lost to the slack intervals. On this record, the central-difference sensitivities span transition instants where the acceleration is nonsmooth, and this bound is therefore approximate.

Baselines

Two classical methods are applied to the same records, the same noisy data, and the same initial parameter guesses as the physics-informed identification. The first is deterministic time-domain model updating by least-squares shooting: Levenberg–Marquardt [57, 58] over $(\log\sigma, \log c)$, with every residual evaluation a full forward simulation of the verified solver. The second is a joint state–parameter unscented Kalman filter [59, 60] on the augmented state of free coordinates, velocities, and log-parameters. It propagates that state by the same equations of motion under a fixed-step Runge–Kutta scheme with a parameter random walk. Symmetric-eigenvalue matrix square roots, covariance eigenvalue bounds, and state clipping stabilize the filter and keep the sigma-point propagation numerically stable. Both use the exact forward model, and the comparison therefore separates the identification strategy from modeling error. Classical modal updating is used where measured frequency-response data are the target [20].

Structural assessment of the identified model

The recovered force densities describe the prestress state of the structure, and they are the input to a downstream assessment that requires no further measurement: the tangent stiffness at the identified equilibrium classifies the self-stressed shape as super-stable, stable, or unstable, and the member forces yield strength and buckling utilizations for the cables and struts. The formulation, the classification conditions, the material library, and the energy-balance check that accompanies them are set out in § 9.5 of the Supplementary Material. The property that matters for identification, how the assessment responds to identification error, is reported in § 4.14.

Results

Verification of the physics residual

Before any identification, two checks confirm that the physics residual is implemented correctly. First, the residual must reproduce the equations of motion of the verified simulator, including configurations with slack cables. Second, with the exact parameters, the residual must be zero on the reference trajectories: exactly with the exact cable law, and up to the known smoothing bias with the smoothed cable law. Table 2 collects these and the other numerical checks in one place, and the precision figures are not repeated elsewhere. These checks separate errors in the physics from errors of the optimization in all the results that follow.

Demonstration on the X1 frame

The inverse experiments begin on the X1 frame at prestress level $\sigma_{\min} = 25$ N/m. Its exact force densities are $\boldsymbol{\sigma} = (25, 52.2, 25, 25)$ N/m for the base, top, and two vertical cables. The damper coefficient is $c = 1$ N s/m, and the axial stiffness is fixed at its exact value. The forcing acts on the vertical coordinate of one top node. The sensors record noisy accelerations, with $2\%$ noise, at the two coordinates of that node, which are half the free coordinates. Two records are inverted under the smoothed cable law: a taut record ($p = 10$ N, no taut–slack transitions) and a slack-transition record ($p = 40$ N, 20 taut–slack transitions).

In the taut regime, the three identifiable force densities are recovered to relative errors of 0.33, 0.37, and 1.0 %, and the damper coefficient to 0.27 % (Figure 4, top). Reconstruction at the two unmeasured coordinates errs by 1.3 % of the motion scale. Re-simulation with the recovered parameters reproduces an unmeasured acceleration to a relative root-mean-square error of 1.7 % (Figure 5). The slack-transition record gives similar force-density errors, 0.20 % for the top cable and 1.1 and 1.3 % for the verticals, through 20 taut–slack transitions with the smoothed cable law alone. Its damping error is larger, 6.2 %. Its reconstruction error is 0.8 % of the motion scale, and re-simulation reproduces the unmeasured acceleration to 2.1 % (Figure 5). Training histories for both records appear in Figure 4 (bottom), and multi-seed uncertainty statistics in § 4.8.

Of the figures just quoted, the reconstruction and re-simulation errors measure agreement with the data, while the parameter errors measure agreement with the exact values. Only the parameter agreement is evidence that the identification worked; a good response fit is necessary but not sufficient. A silent failure, defined in § 3.3, is a good fit with wrong parameters; § 4.4 shows one in which the record is fitted while the force-density error reaches 79.8 %. The misfit cannot separate the two cases. The identifiability analysis of § 3.3 is computed from the geometry and the sensor set before the record is inverted, and it shows which members can be determined and to what precision; the recovered values are then compared with that prediction. The Cramér–Rao bounds of § 4.8 give the precision attainable at the stated noise level. A recovery close to the bound is therefore at the limit of what the record allows; a recovery far above it is not, however small the misfit. Finally, the scatter across repeated runs from different training seeds, each a fresh network initialization and noise realization (§ 4.8), distinguishes a parameter that the record determines from one at which the optimizer happened to stop. The predictions at unmeasured coordinates quoted above are reported as a consistency check only.

The record inverted in this section was generated by the same forward model that the residual uses. The accuracies above might therefore reflect only self-consistency, a circularity that the inverse-problems literature calls an inverse crime. § 4.9 addresses it directly, with generators whose governing equations the residual cannot represent at any parameter value. There, lumping the cable mass at the nodes violates the massless-chord cable law and roughly doubles the taut force-density error, and replacing the rigid bars with elastic ones removes the algebraic constraints and changes the integration scheme. Sections 4.10 to 4.12 then use measured records only, and § 4.11 runs the residual itself on measured records from hardware built by another group.

The inert base cable behaves as theory requires in both regimes. Initialized at 10 N/m, it drifts without converging, to 21.5 N/m in the taut record and toward zero in the slack-transition record, while every other parameter converges. Its recovered value is therefore not an error of the method; the record does not determine it.

This structure is unfavorable for a modal method. The natural frequencies of the X-frames shift by only about 12 % over a 50-fold change in prestress level, which gives frequency-based updating little information. A full-scale tensegrity footbridge shows the same insensitivity: its natural frequencies move by about 1 % as its self-stress ratio quadruples [23], because its deformation modes are stiffened mainly by member extension. A structure measured in the same laboratory shows the opposite case. A five-module active tensegrity whose lowest mode is a prestress-stiffened mechanism shifts that frequency from about 2.6 to 3.3 Hz across 11 measured prestress levels, while its member forces rise by a factor of 1.7 [16]. Although both structures were measured with the same instrument in the same laboratory, their frequencies change by 1 and by 29 %. The difference lies in whether the observed mode is stiffened geometrically or elastically, as the modal argument of § 3.3 shows. The transient formulation avoids the problem: the force densities enter the equation-of-motion residual linearly, and they are recovered with errors near 1 %. The slack-transition record loses force information during its slack intervals and still identifies the members that remain active, consistent with the information argument of § 3.3.

Identification across the benchmark structures

The physics-informed identification is applied to the staircase frames and the three-dimensional prism at 2 % noise (Table 3). Every single-point row, the X1 rows included, uses 12,000 steps; the repeated two-point runs use 25,000. The demonstrations of § 4.2 ran for 25,000 steps, which explains why their errors are below the X1 rows here; § 4.5 isolates this effect of training length. The held-out ratio is the ratio of the sensor-data loss on the held-out 30 % of time samples to that on the training 70 %; a value near 1 means the fit generalizes. Sensors are placed at half the free coordinates on the single-point runs and repeated X2 runs, and at two-thirds (8 of 12) on the repeated X3 runs. Errors cover the identifiable members only. The base cable is inert, and excluded, only on the X1 frame (§ 3.3).

Structure	Record	Forcing	Transitions	max $\hat{\sigma}$ err (%)	$\hat{c}$ err (%)	recon. (%)	held-out ratio	Outcome
X1 frame	taut	1-pt	0	3.5	1.3	4.1	2.8	identified
X1 frame	slack-transition	1-pt	20	18.3	3.2	11.5	9.5	partial
X2 frame	taut	1-pt	0	79.8	7.0	5.3	4.6	silent failure
X2 frame	taut	2-pt	0	2.1	0.1	1.3	6.3	identified
X2 frame	slack-transition	1-pt	20	20.8	5.5	1.7	6.0	partial
X2 frame	slack-transition	2-pt	61	1.8	2.7	0.7	6.9	identified
X3 frame	taut	1-pt	0	99.7	95.0	$>100$	485.6	fit failure
X3 frame	taut	2-pt	0	$>100$	23.8	12.1	31.8	fit failure
X3 frame	slack-transition	1-pt	18	$>100$	30.0	36.7	92.6	fit failure
X3 frame	slack-transition	2-pt	21	2.9	0.9	7.8	4.7	identified
Triplex prism	taut	1-pt	0	0.6	3.2	0.8	2.2	identified
Triplex prism	slack-transition	1-pt	14	2.0	0.5	1.0	4.4	identified

The three-dimensional prism is identified as accurately as the X1 frame. Its force densities, fixed uniquely by gravity and equilibrium, are recovered from four measured channels of nine free coordinates with errors of 0.6 % on the taut record and 2.0 % on the slack-transition record; the damping errors are 3.2 and 0.5 %. On the X2 frame, stronger excitation improves the identification. The low-amplitude taut record leaves weakly excited members essentially unidentified, with a maximum error of 80 %, while the slack-transition record, with 20 taut–slack transitions, recovers them several times more accurately. Within the structures and excitation suites tested, the information gained from the stronger motion outweighed the information lost while individual cables were slack, consistent with the Fisher analysis of § 3.3. The method fails detectably on the X3 frame at this step count, with parameter errors approaching or exceeding 100 %.

The failures are of two kinds, and only one is flagged by the fit. In a fit failure, the network cannot reproduce the measured motion at all, and the held-out ratio flags it without the exact parameters. Within the runs of this fixed-length, 2 %-noise grid with the smoothed cable law, the held-out ratio stays below 7 on every successful run and rises to 93 and 486 on the X3 runs. The threshold depends on the protocol, because the baseline of the ratio follows the noise level: the successful 10 %-noise inversion of § 4.9 has a ratio of 20. The ratio flags overfitting of the instrumented channels; a failure that does not fit even those channels appears instead as a large reconstruction error at a low ratio. The two signals are therefore complementary.

The second kind is silent. On the X2 taut record, the motion is fitted well, although not to the noise floor: the reconstruction error is 5.3 %, and the held-out ratio is 4.6, inside the band of successful runs. The parameter errors still reach 80 %. This case shows most clearly that prediction accuracy and parameter accuracy are different quantities: a model can reproduce the measured response closely while its force densities are badly wrong. For this reason, response fit is reported throughout as a consistency check and never as evidence of identification. The failure is reproducible: across six training seeds, the maximum force-density error is 81 $\pm1$ %. A Fisher analysis of that record at the exact parameters, following § 3.3, finds no local lack of information. The sensitivities of the seven members span only a factor of three, and every relative Cramér–Rao bound is below 0.5 %. The record therefore contains the information, but the finite-length optimization does not extract it: a distant parameter set reproduces the measured channels nearly as well. Neither the held-out ratio nor the local Fisher analysis detects that practical non-uniqueness.

Both kinds of failure respond to richer excitation, but differently on the two frames. On the X2 frame, two-point forcing, at two coordinates and two frequencies with the same sensors and noise, removes both single-point deficits. On the taut record, it removes the silent failure: the maximum error drops from 80 to 12.4 % in a run of the same 12,000 steps (Table 4), and to 2.1 % after 25,000 steps, with a damping error of 0.1 %. On the slack-transition record, it reduces the worst-member error from 21 to 1.8 %. On the X3 frame, richer excitation succeeds only with the slack-transition record, and only with the sensor set widened from 6 to 8 of the 12 free coordinates. That record recovers all 10 cables to within 2.9 % and the damping to 0.9 % through 21 taut–slack transitions at 25,000 steps; at 12,000 it has not yet converged. The low-amplitude taut record with the same two-point excitation remains a fit failure, flagged again at a held-out ratio of 32. At this training length, the richness of the excitation sets the failure boundary more than the size of the structure does, and strong excitation that produces taut–slack transitions is the most effective improvement.

Intervention	max $\hat{\sigma}$ err (%)	$\hat{c}$ err (%)
Baseline (one-point taut, four sensors)	79.8	7.0
+ sensors (all eight coordinates)	76.7	1.9
+ two-point off-resonance forcing	12.4	0.4
+ three pooled taut records	90.4	1.8
One-point slack-transition record	20.8	5.5
Combined (transitions, two-point, eight sensors)	9.2	3.3

The X3 frame needs a fuller explanation, because the method succeeds on the X1 and X2 frames but not on the X3 frame. The failure is not caused by missing information. The Fisher analysis of the X3 records with multi-point excitation gives worst relative Cramér–Rao bounds below 0.7 %, which shows that the records determine the parameters. Nor is it removed by a larger network or longer training alone: a five-layer trunk of width 160 trained for 60,000 steps on the gently forced taut record still overfits the instrumented channels, with a held-out ratio above 100 and a median parameter error near 100 %. Stronger excitation removes it, as the convergence with the slack-transition record above shows. At this scale, the boundary is therefore an optimization limit of gently forced records, which richer excitation moves and a larger network does not. The X2 taut record shows the same boundary at a smaller scale.

Removal of a silent failure

The X2 taut record is used here to test which intervention removes a silent failure. Table 4 starts from that record, with one-point forcing, four sensors, and a maximum force-density error of 79.8 %. The training length is held at 12,000 steps, and one factor is changed at a time. More sensors do not help: enlarging the sensor set from four of the eight free coordinates to all eight changes the force-density error only to 76.7 %. The failure is therefore not caused by a lack of sensors. More data of the same kind do not help either: three pooled taut records of the same one-point form give 90 %. Records that excite the same modes carry redundant information about the non-unique direction.

Excitation that exercises more of the structure removes the failure. Two-point forcing at two frequencies away from the resonances reduces the error to 12.4 %, a one-point slack-transition record reduces it to 20.8 %, and the two together with the full sensor set reduce it to 9.2 %. The damping coefficient, well excited throughout, is recovered to a few percent by every intervention. A silent failure caused by practical non-uniqueness is therefore removed by richer excitation. More sensors and more copies of the same record do not remove it. This agrees with the identifiability reading of § 3.3.

Comparison of transition treatments

Table 5 compares the three transition treatments on the same slack-transition records under one protocol, with four seeds at each of two training lengths. This separates the effect of the treatment from that of the training length.

Structure	Treatment	Steps	max $\hat{\sigma}$ err (%)	$\hat{c}$ err (%)	recon. (%)	transitions
X1	smoothed cable law	12,000	18.82	3.8	11.6	20
X1	Fischer–Burmeister NCP	12,000	26.97	42.6	15.5	20
X1	event-aware collocation	12,000	17.90	2.4	10.9	20
X1	smoothed cable law	25,000	0.77	6.5	0.5	20
X1	Fischer–Burmeister NCP	25,000	2.55	24.4	1.7	20
X1	event-aware collocation	25,000	0.82	6.3	0.6	20
X2	smoothed cable law	12,000	21.13	5.6	1.6	20
X2	Fischer–Burmeister NCP	12,000	100.82	23.3	8.4	20

The training length has the stronger influence. Every treatment improves by more than an order of magnitude between 12,000 and 25,000 steps. The median error of the smoothed cable law falls from 18.8 to 0.77 %, and that of event-aware collocation from 17.9 to 0.82 %. At the shorter length, none of the three has converged on this record, and the errors there measure progress toward convergence.

Among the treatments, the smoothed cable law and event-aware collocation are indistinguishable. Their medians differ by less than a percentage point at 12,000 steps and by 0.05 percentage points at 25,000, and their seed ranges overlap almost completely. Concentrating collocation points near the transitions therefore neither helps nor hurts once the smoothed cable law is in place. The complementarity treatment performs worst. Its median force-density error is 27.0 % at 12,000 steps and 2.55 % at 25,000, against 18.8 and 0.77 % for the smoothed cable law. Its damping error, 42.6 and 24.4 %, is several times that of the other two treatments. Its additional tension head adds unknowns that the record must determine, which is a likely reason for the slower convergence. The same ordering holds on the X2 frame, where the complementarity run exceeds 100 % while the smoothed cable law reaches 21.1 %.

These results support the smoothed cable law, which matches the best alternative at lower cost and has no auxiliary unknowns. Training long enough to converge matters more than the choice of transition treatment. Both comparisons cover all identifiable cables; on the X2 frame the largest error of the complementarity run is on the base cable, which is identifiable there. No treatment was tuned separately, and the comparison is reported as measured.

Prestress series on the prism

At its pure self-stress twist, the zero-gravity prism has a one-parameter series of prestress states, and the identification follows this series (Figure 6). At prestress levels 5 and 15 N/m, the identifiable force densities are recovered to about 2 %, including the ratio of $\sqrt{3}$ between the cross-cable and triangle-cable force densities. At the lowest level (1 N/m), two of the members at the smallest force density are not recovered, with errors of 32 and 60 %, while the rest stay inside the 10 % band.

The miss is systematic. Six training seeds at the same 12,000 steps miss the same two members in the same direction, with an average worst-member error of 66.1 %; the run shown in Figure 6 is the most favorable of the six.

The usual explanations at low prestress are a small taut margin and weak information. The taut margin is small enough that the record contains 45 taut–slack transitions. A Fisher analysis of the same excitation and sensing quantifies the loss of information: the worst relative Cramér–Rao bound grows 17-fold, from 0.047 % at 15 N/m to 0.804 % at 1 N/m, and the conditioning of the information matrix becomes worse.

Neither effect explains the size of the miss. The bound at level 1 is 0.80 %; a method that attains the bound could therefore recover these members with errors well below 1 %. The gap to the bound lies in the optimization. Doubling the training length to 25,000 steps halves the error, to 31.8 % on average over the same seeds. That is still about 40 times above the 0.80 % bound at this level. The level-1 result is therefore an optimization limit, not an identifiability limit. The information does decrease as the prestress approaches zero, and the prestress series quantifies this decrease. At level 1, however, the record still contains the information, and the inversion has not yet extracted it.

Extension to a two-span bridge

The largest structure in the study is an X5 planar X-frame with 16 cables and eight mechanical degrees of freedom, against 10 cables and six for the X3 frame. Under gravity, the plain X5 frame is statically unstable, because its long unbraced span has two sway modes with negative stiffness. Pinning one intermediate base node removes the instability and turns the frame into a two-span tensegrity bridge (Figure 7).

The bridge is identified from a taut record and a slack-transition record. Each is forced at four top coordinates and instrumented at nine of the 18 free coordinates. Two of the nine are on the top-chord node above the interior support. The two bars at that node connect it to pinned base nodes and hold it fixed, which is the nodal counterpart of the invisible cable of § 3.3. These two channels carry no signal, and seven channels remain active. The Fisher analysis on the seven active channels confirms that both records are informative. No null direction appears among the 16 force densities and the damping coefficient, and the worst relative Cramér–Rao bounds are 0.6 % on the taut record and 0.2 % on the slack-transition record.

Both central findings recur at this scale. The slack-transition record is the more informative: it recovers the damping to 0.9 % and the force densities to a median of 4.2 %, with 9 of the 16 cables within 5 %. The taut record fits the sensors equally well, with a held-out ratio of 2.7 against 3.3 for the slack-transition record, and the Fisher analysis shows that it is also informative. Even so, it recovers the force densities only to a median of 16 % and the damping to 5 %. That is a silent failure of the same type as on the X2 frame, at more than twice the number of cables, and the held-out ratio again does not detect it.

The best result also shows the expected effect of size. The worst cable on the slack-transition record reaches 29 %, against errors of a few percent on the smaller structures, most likely because the active sensors are spread across more unknowns. The method therefore extends to a bridge-scale structure, and both central findings hold there.

Uncertainty and classical baselines

Eight independently seeded repetitions of the taut X1 inversion quantify the scatter of the identification, each with a different noise realization and network initialization over 12,000 steps. The seed-to-seed standard deviations are 0.085, 0.081, and 0.11 $N/\mathrm{m}$ for the identifiable force densities and 0.018 $N\,\mathrm{s}/\mathrm{m}$ for the damping coefficient. All four are between 2.7 and 8.3 times the Cramér–Rao bounds of § 3.3 (Figure 3c). Bias dominates the force-density error: the seed means are 2–4 % from the exact values, each bias roughly an order of magnitude above its own seed standard deviation. This bias is attributed to the finite training length and the smoothed cable law; the measurement noise changes from seed to seed and does not produce a common bias. The damping seed mean is closer still, 0.7 % from exact, though its scatter of 1.8 % exceeds its small bias. The held-out ratio averages 2.8 across seeds, consistent with the successful-run band.

The classical identification methods extend the comparison. Levenberg–Marquardt shooting is started from the same records, noise, and distant initial guess, with all force densities at 10 $N/\mathrm{m}$ and the damping at 0.3 $N\,\mathrm{s}/\mathrm{m}$. It gives different results on the two records. On the low-amplitude taut record, it satisfies its convergence criteria at a spurious minimum: the identifiable force-density errors are 61–97 %, and the damping is overestimated 66-fold. The failure is evident without the exact parameters, since the final misfit is orders of magnitude above the noise floor. On the slack-transition record, the same method recovers the identifiable force densities with errors below 0.1 % and the damping within 0.2 %, and the inert member goes to zero.

The computational cost shows the same split by record. Shooting performs one full forward simulation per residual evaluation, and its cost therefore depends on the path of the search. On the taut record, the search moved into high-damping regions where the equations become stiff, and 68 simulations took about three hours. On the slack-transition record, 232 simulations took 83 seconds. The physics-informed identification takes 2819 and 4643 seconds on the two X1 records in its 25,000-step runs on the same hardware. On the slack-transition record, where shooting converges, shooting is both faster and more accurate than the physics-informed identification. On the taut record, the physics-informed identification is faster and more accurate. The advantage of the physics-informed identification is therefore its robustness from a distant initialization.

Repeated starts confirm that the taut failure depends on the starting point: of several distant restarts, at least one reaches the exact force densities to within 1 % while others converge to spurious minima.

The two recursive filters show the opposite pattern. The joint state–parameter unscented Kalman filter uses standard scaled sigma points on the augmented state and three passes over the record. It succeeds on the taut record, recovering the identifiable force densities to 3.1 % and the damping to 1.5 % from the same distant initialization. On the slack-transition record, its error grows to 183 %, and it overestimates the damping. The likely cause is that the unscented transform propagates sigma points across the taut–slack boundary and averages states on opposite sides of it. A particle filter built around an ensemble Kalman filter represents the parameters by a weighted set of samples instead of one covariance. From the distant initialization, it recovers neither record; its errors are 62 % on the taut record and above 400 % on the slack-transition record. At 2 % noise, the likelihood is so sharply peaked that one particle takes almost all the weight at every step. The effective sample size is then 1 out of 400, and the filter reduces to a greedy search. This may explain why recursive filters in the tensegrity literature are used on linearized models with informed priors [21].

Two of the three classical methods succeed in opposite regimes, and the particle filter fails in both. Shooting converges to a spurious minimum on the taut record and is exact on the slack-transition record; the unscented filter does the reverse. Only the physics-informed identification succeeded on both records from the distant initialization, recovering the identifiable force densities to 1.0 and 1.3 %. Classical updating with an exact forward model remains strongly competitive when the optimization problem is well conditioned, as shooting shows on the slack-transition record. In that record, the taut–slack transitions appear to make the optimization problem easier to solve. The same transitions that cause the unscented filter to fail let an independent gradient-based method reach the exact parameters, two orders of magnitude faster than on the taut record. The benefit of the transitions is therefore not specific to the physics-informed identification.

Table 6 compares the four methods in three respects: recovery accuracy, running cost, and how many quantities the user must choose before the method runs. A hyperparameter is any numeric or structural choice that is not a property of the structure or the record and not a convergence tolerance. The shared initialization of all force densities at 10 N/m and damping at 0.3 N s/m is counted once for every method that uses it. The physics-informed identification has by far the most hyperparameters of the four. Shooting needs two choices and outperforms it on the slack-transition record in both accuracy and wall time.

Method	Taut record max $\hat{\sigma}$ err. (%)	Taut record $\hat{c}$ err. (%)	Slack-transition record max $\hat{\sigma}$ err. (%)	Slack-transition record $\hat{c}$ err. (%)	Cost and tuning wall (s)	Cost and tuning hyper-parameters
Physics-informed identification	1.0	0.3	1.3	6.2	2819 / 4643	15
Levenberg–Marquardt shooting	97.4	6504	0.07	0.2	10 700 / 83	2
Unscented Kalman filter	3.1	1.5	182.7	133.5	63 / 47	11
Particle filter with EnKF	61.7	67.1	488.1	80.8	762 / 751	6

The unscented filter has the lowest cost and is accurate on the taut record. For a structure known to stay taut, it is therefore the appropriate choice. Shooting is the least tuned and the most accurate of all four on the record it can solve. The particle filter is less accurate and more costly in every comparison. The failure comes from the likelihood: at 2 % noise it is too narrow for 400 particles to represent. The physics-informed identification is robust in both regimes, at the cost of 15 hyperparameters and of wall times one to two orders of magnitude above those of the filters.

Pooling records reduces the noise- and record-specific bias of a single inversion. Three taut records at forcing amplitudes of 6, 10, and 14 N are inverted jointly by a single network conditioned on the record descriptor, with shared physical parameters. They recover the identifiable force densities to 0.01–0.5 % and the damping to 0.2 %, roughly halving the best single-record force-density errors. Sharing parameters across inexpensive repeated tests is well suited to monitoring practice, where many short records are easier to obtain than one long one.

Robustness to noise, model-form error, and adverse sensing

Three robustness studies on the X1 frame are summarized here; their tables and full discussion are given in § 9.6 of the Supplementary Material.

Accuracy decreases moderately with noise: a fivefold increase in noise increases the force-density error by a factor of 1.8, and removing one of the two measured channels increases it by less than that. Identifying the axial stiffness together with the force densities shows the trade-off that the Fisher analysis predicts. On the taut record, the stiffness is recovered to 7.4 %, but the force-density errors grow to 35 %; on the slack-transition record, the joint identification fails, and the held-out ratio flags the failure.

The simulated records above are generated and inverted with the same forward model, which is an inverse crime (§ 4.2). Records from generators that the residual cannot represent, with the cable mass lumped at the nodes or with elastic bars integrated by a different solver, at most double the taut force-density error of the X1 frame, to 7.0 %. A 5 % perturbation of the mass matrix raises the error on the prism from 0.6 to 3.5 %. On the slack-transition record, every generator stays within the seed scatter of the matched case.

Under adverse sensing, the force-density error grows from 2.8 % in the baseline case to between 4.7 and 12.2 % for a single channel, 8 % noise, or a quarter of the samples in intermittent blocks, and to 25.0 % when the harmonic forcing is replaced by an impulse. All four conditions together give no larger error than the impulse alone. On a single adverse record, the excitation therefore limits the accuracy more than the quality of the measurement does.

Validation on measured data

No cable slackens in the measured experiments of this and the next two sections. These experiments validate the rebuilt models, the small-oscillation identification, and, on the leg of § 4.11, the time-domain residual on measured records. The treatment of taut–slack transitions remains validated on simulated records only.

A published experiment provides measured frequencies against which the formulation is tested. The public dataset accompanying Huang et al. [20] provides full-field motion-capture displacement histories of all 12 nodes of a hexagonal tensegrity prism (six bars, 18 cables, modeled free–free). It also provides a driving-point frequency response with four identified peaks at 7.285, 9.688, 10.430, and 11.396 Hz, and the complete parametric model. The prism was rebuilt here from the published parameters: rigid bars of measured mass 42.3 g, cable stiffness $EA = 379$ N, and the design prestress of 10 N in the two ring-cable groups. The structure carries a single self-stress mode, and scaling it to the ring value reproduces the published vertical-cable tensions of 11.88 N.

The rebuilt model is verified in the modal domain. All eight published theoretical frequencies (5.39–13.16 Hz) are reproduced to within 1.0–2.0 %, including both degenerate pairs. The small uniform overprediction is consistent with the neglected string mass (2 % of the total). Matching these frequencies requires the transverse geometric stiffness of the compressed bars in the linearization: a cables-only stiffness assembly overestimates this structure’s mechanism-dominated frequencies by tens of percent. The classical-limit check of § 2.1 was extended to include this term.

Identification proceeds in the modal domain (Figure 8b). A single prestress scale fitted to the four measured peaks returns a ring tension of 10.6 N, 6 % above the design value, with a root-mean-square frequency residual of 0.33 Hz. The peaks are matched to the model’s distinct frequency lines by order; the dataset’s mode shapes are not reconstructed here. One peak is questionable: at 9.688 Hz it is nearer the mode-6 line than the mode-4 pair it is assigned to. Even so, the order-preserving assignment fits all four peaks with the 0.33 Hz residual, while pairing 9.688 Hz with mode 6 gives a residual of 0.92 Hz. That alternative still returns a ring tension of 8.6 N, near the design value. The conclusion therefore does not depend on the assignment: the prestress can be identified from the frequencies near the design value, while the frequencies are nearly insensitive to $EA$.

No single prestress level makes all four residuals zero: the measured-to-model frequency ratios at the design prestress span 0.99–1.08. The source publication’s own theoretical values show the same inconsistent offsets from its measurements, which points to unmodeled effects such as joint masses and distributed string inertia. Adding the cable stiffness as a second parameter reduces the residual by only 9 % while increasing $EA$ by an implausible factor of 18. The mechanism-dominated modes the test excites are sensitive to prestress and nearly insensitive to $EA$. This is an experimental example of the flat directions predicted by the Fisher analysis of § 3.3.

The dataset also distributes the raw motion-capture records, and this study examined them to decide whether a time-domain inversion was possible. Three dwell records are provided, each at a single excitation frequency, sampled at 90 Hz over 61 to 163 s. They are not suitable for a time-domain inversion. No force channel is recorded, and the drive point is not documented. A residual inversion would therefore have the amplitude, the phase, and the position of the load as further unknowns. The motion is also too small to slacken a cable. The largest change in any inter-node distance across the three records is 7.6 mm, against 14.4 mm of prestretch on the shortest cable at the design prestress. This comparison does not require the markers to be registered to the model. In addition, forming accelerations by differentiating the displacements twice reduces the signal-to-noise ratio in the excitation band from between 8.7 and 42 to between 0.19 and 2.4, with only 8.6 to 12.4 samples per cycle; the ratio is below 1 in two of the three records. Narrowband filtering about the drive frequency restores the ratio, but it also linearizes the record. The modal comparison above is therefore the comparison that this dataset supports.

Identification against an independently measured spring rate

The prism test of § 4.10 compares an identified prestress with a design value. A second published experiment provides a measured reference, and its raw records allow the time-domain residual of § 3.1 to be run on measured data, which no other experiment in this study permits.

Wang et al. [61] report a planar biomimetic leg: a fixed pelvis, a femur on a cable-constrained hip that acts as a revolute joint, and a tibia carried on an antiparallelogram knee, with compliance from four extension springs. Five spring configurations are tested. Every quantity used here is published: node coordinates and body properties, spring rates measured on a dedicated spring rig, and frequencies identified from chirp tests. The 18 raw chirp records behind those tables were also supplied by the authors, who granted their use for this study with citation and acknowledgment.

The rebuilt model is checked before it is inverted. Built from the published tables alone, with no fitted constant, it reproduces the authors’ own computed frequencies to 0.015 % root-mean-square over 10 values, and it reproduces the eight frequencies of their gravity-ablation table, a physical regime not used to build the model, to 0.094 %. Against their measurements, the same model differs by 3.2 %, which this study cannot reduce, because the authors’ own computation differs by 3.19 % from the same measurements.

The knee spring rate is the identification target. This choice is made before any inversion, by the identifiability analysis of § 3.3 applied to the springless Case E. Gravity supplies most of the restoring stiffness of the first mode: the hip springs contribute only 17 to 23 % of it, while the knee springs contribute 56 to 70 % of the restoring stiffness of the second mode. A frequency error enters an identified spring rate amplified by 2/s, with s the share of the restoring stiffness that the spring supplies. The amplification is about 3 at the knee and about 10 at the hip. At the hip, the source publication’s own computed-to-measured frequency gap of 3.25 to 4.89 % already corresponds to spring-rate errors of 37 to 47 %. The hip rate therefore cannot be identified reliably from this experiment. It is included below only as a nuisance parameter, with its statistics given in Section S4, and the knee provides the test.

When identified from the measured frequency pair of each case, the knee scale agrees with the rig-measured rate to 7.6 % root-mean-square, with a worst case of 12.6 % (Figure 9b, Table 7). The rig rate is measured independently of this identification. When identified record by record instead of from the published pairs, the knee scale is within 8 % in three of the four cases, with record-to-record spreads of 4 to 6 percentage points.

case	rig rate (N/m)	identified scale	error (%)
A	188.0	0.957	−4.3
B	188.0	1.030	+3.0
C	60.6	0.874	−12.6
D	60.6	0.934	−6.6
root-mean-square error			7.6

The records also allow the time-domain residual of § 3.1 to be run on measured data: the drive is known, the response is measured, and the stiffness and damping are unknown. The fit is taken on the smoothed response envelope rather than point by point, because a small frequency error accumulates into a large phase error across a resonance passage. Six unknowns are identified per record: two stiffness scales, two joint dampers, the chirp start, and one drive amplitude; the hip scale is among them only so that it is set by the record. The procedure is first tested on a simulated record with known parameters and 2 % noise, where it recovers all six unknowns to within 5 % at an envelope error of 0.0026 (details in § 9.4).

On the measured records, the knee is identified to about the precision of the rig measurement. Pooled over the 15 records with springs, the knee scale is 1.001, with a standard deviation of 0.183. The mean agrees with the rig-measured rate to 0.1 %, and no case departs by more than 8 %. The nuisance parameter behaves as the screening predicts; § 9.4 reports it in full. The envelope errors range from 0.045 to 0.093, against 0.0026 on the simulated test, which shows that model form limits these fits.

The leg is not self-stressed, and its springs never slacken. It therefore tests the small-oscillation formulation and the time-domain inverse against a measured reference, but not the treatment of taut–slack transitions. The experiment shows that the method can recover an identifiable member property from measured records, to 0.1 % in the mean and 8 % at worst, with the identification target selected by the screening before any inversion.

Prestress identification against measured cable forces

The prism test of § 4.10 compares an identified prestress with a design value. A second experiment allows a stronger comparison. The structure is self-stressed, its prestress is measured directly by force transducers built into the cables, and it is tested at 13 independent prestress levels.

Małyszko and Rutkiewicz [15] report on a full-scale steel tensegrity simplex that is 1.186 m tall and consists of three struts and nine cables, with the lower triangle fixed and the upper triangle free (Figure 10). A force transducer was built into each of the nine cables; the transducers were individually calibrated and re-zeroed at the beginning of each prestress level by slackening the turnbuckles until the module carried no internal force. To obtain the dynamic response, the structure was struck with an instrumented hammer, and the response was recorded by three triaxial accelerometers mounted at the free nodes. The experimental data used in this study were collected by co-authors Małyszko and Rutkiewicz. The dataset comprises one strike at each of the 13 prestress levels, all at the same node and azimuth, so that the prestress level is the only variable across the set.

Prior to inversion, the rebuilt model is verified against the results of the source publication. Table 3 of Małyszko and Rutkiewicz [15] reports the base-cable force twice, once as measured and once as calculated from its Equation (5). The calculated column is an equilibrium statement about the published geometry, which this study can reproduce independently. Evaluating the single self-stress mode of the simplex on the published coordinates gives a base-to-cross force ratio of 0.2091. The calculated column gives 0.2079: this is the mean, over the 13 levels, of the ratio of the calculated base-cable force to the measured cross-cable force, and the per-level values range from 0.2070 to 0.2091. The two ratios differ by 0.6 %. The two formulations give the same static equilibrium before any measured record is used.

The rotational mode is isolated by projecting the three nodal accelerations onto the tangential directions of the upper triangle, that is, onto its rigid-body rotation about the vertical axis. This projection removes the flexural modes by construction and does not rely on their separation in frequency. This matters at the lowest prestress levels, where the rotational mode falls to 3.4 Hz. With this projection (Figure 10b, Figure 11d), the 13 records give the first frequency within 1.19 % root-mean-square of the values published in Table 4 of the 2020 study, with a worst case of 2.47 %. This study therefore extracts the same frequencies from the records as the 2020 study, and any later disagreement concerns the model.

The identifiability argument presented in § 3.3 can now be tested directly. Eq. (11) indicates that for a mode carrying no elastic strain energy, the squared frequency is proportional to the prestress, which yields a log-slope of $d \ln f / d \ln F = 1/2$; for an elastically stiffened mode, this log-slope approaches zero. The 2020 study identified every mode it could resolve at every level. Grouping these modes by family gives exponents of +0.41 for the rotational mechanism mode and +0.00 and +0.01 for the two flexural families below 20 Hz (Figure 11b). Under the same excitation and the same 14-fold change of force, one mode of the structure is more than 100 times as sensitive to prestress as another. The rotational family near 62 Hz is also flat, at −0.01. It turns the upper triangle but also stretches members, and Eq. (11) classifies a mode by the member stretching it produces. A second strike series at azimuth 330°, recorded by the same authors at all 13 levels and processed directly in this study, gives the same result. It reproduces the first frequency of the first series to 0.5 % on average, and its flexural peaks have exponents of +0.003 and +0.014, against +0.39 for the mechanism mode.

The mechanism law itself follows directly from the records, without a model. Fitting the squared first frequency against the transducer force across the 13 levels gives

with a residual scatter of 1.2 % of the mean. Figure 11a compares this line with the rebuilt model, and the two differ in two separate ways. First, the model line passes through the origin, as it must for an ideal pin-jointed simplex, while the measurement keeps a rotational frequency of 2.56 Hz at zero prestress. This frequency comes from a stiffness that the joints supply and the idealization omits. The Discussion of the 2020 publication states the same expectation, that the first frequency should be zero without prestress. The discrepancy therefore stems from the shared idealization failing to capture the physical hardware, not from a conflict between the two numerical models. The intercept also keeps the sensitivity in Figure 11b below the ideal value. The fitted mechanism exponent is +0.405 ± 0.012 at the 95 % level, significantly below the 1/2 of Eq. (11). Eq. (12) explains this shortfall: with $A = 0.0239$ and $B = 6.56$ its two constants, the local slope $d \ln f / d \ln F = AF/2(AF + B)$ rises from 0.245 at level 1 to 0.464 at level 13.

Second, the slopes differ. This is the more informative of the two differences, and the source publication supplies what is needed to explain it. In Table 2 of Małyszko and Rutkiewicz [15], the cable mass is reported as $0.516\ \mathrm{kg}\ \mathrm{m}^{-1}$ and categorized as non-physical: it is a reduced value for a consistent mass matrix, and it already includes the transducers and turnbuckles, distributed along each cable. The authors of the 2020 study confirm how it was formed: the weighed total less six joint masses of 0.897 kg and three struts at the published linear density, spread over the total cable length. Taken at face value, this mass determines the share of the cable mass that moves, with nothing to fit. The three lower horizontal cables join fixed nodes and contribute nothing. The three cross cables run from a fixed end to a free one, and a consistent mass matrix puts a third of each at the free end. The upper horizontal cables rotate with the top triangle, but along a chord whose mean squared radius is exactly half that of the nodes; half of their mass therefore counts when referred to a node. With the 25 g accelerometer mass stated in the 2020 publication, this gives a calculated mass of 0.342 kg at each free node.

The convention is no longer the only option, because the authors of the 2020 study have since weighed every element the module contains: rod, sensor, nuts, pads, plate, thimbles, clamps and shackles, together with the stated composition order of every cable. Only the turnbuckles, which were lost after the experiment, were never weighed directly. However, the difference between the weighed total of 16.205 kg and the remaining components gives 234 g for each turnbuckle, compared with the 260 g recalled by the authors of the 2020 study.

The struts need one further statement, because their mass is not uniform along the rod. The published 1.990 kg m$^{-1}$ is the linear density of the tube and of the threaded end sections alike. Each weighed strut is 224 g heavier than this density implies, and the surplus is carried at the two welded connections between the tube and the end sections. The threaded sections are 16 cm long; each connection is therefore 13.75 cm from its node and carries 112 g, and the model places these masses at those points. Placing every weighed part where the photographs and the composition orders put it gives 1.166 kg moving with each free node.

With the weighed masses, the model requires the prestress acting in the structure to be 1.384 times the value that the cross-cable transducers report. With the distributed-mass convention, it requires 1.373. The two mass models, built from different information, agree to 0.8 %. Table 3 of the 2020 publication gives 1.410 ± 0.027 for the same quantity, from two columns that need no model at all: the measured horizontal-cable force against the value its Equation (5) requires. The dynamic and static estimates differ by 1.9 %, which is one standard deviation of that measurement’s scatter. The two estimates are independent, and neither was fitted to the other. They share only the measured force. The remaining difference therefore concerns the transducer calibration or the force distribution. Figure 11a shows that comparison directly: evaluated at the prestress obtained from static equilibrium, rather than at the transducer reading, the model line lies just below the measured line across the whole range, separated by 6.6 Hz$^2$ at zero force and 4.9 Hz$^2$ at the highest level. This remaining separation is the joint stiffness that the pin-jointed idealization omits, and it is the only constant that the identification fits.

The identification is then compared with the transducers. The slope comes from the rebuilt model carrying the hardware mass just described, with nothing fitted. The force factor comes from Table 3 of Małyszko and Rutkiewicz [15] by static equilibrium alone. Only the joint stiffness is taken from the frequencies, because a pin-jointed model cannot supply it; it is fitted leave-one-out, so that no level informs its own prediction, and all other quantities come from the model (Table 8, Figure 11c, Figure 11d).

level	measured (N)	±	predicted (N)	error (N)	error (%)
1	263	27	253	−10	−3.76
2	551	37	568	+17	+3.02
3	896	40	958	+62	+6.89
4	1167	38	1212	+45	+3.86
5	1434	35	1487	+53	+3.72
6	1695	31	1684	−11	−0.66
7	1879	29	1873	−6	−0.30
8	2164	28	2145	−19	−0.88
9	2612	35	2568	−44	−1.67
10	2928	42	2891	−37	−1.25
11	3093	43	3121	+28	+0.92
12	3316	45	3269	−47	−1.42
13	3584	40	3552	−32	−0.90

The mean absolute error is 32 N, which is inside the 36 N transducer scatter reported in the 2020 publication. The worst error is 62 N, and the root-mean-square error is 2.9 %. For comparison, fitting both constants to the frequencies, without using the model, gives 28 N and 6.5 %. This is better in newtons but has twice the relative error, because the free fit misses the lowest level by 22 %, against 4 % for the model-based prediction. Fitting no constant, which leaves the structure without joint stiffness, gives 214 N and 29 %.

The fitted constant has a physical meaning: it is the rotational stiffness of the joints. Without it, the model line passes through the origin, and the 2020 publication states in its Discussion that an unprestressed simplex should have no first frequency at all. The force factor is taken from the source publication, not derived here. The comparison therefore inherits whatever lies behind it, including the effect on the force distribution of the 32.5 mm eccentricity between the strut axis and the cross-cable attachment, which this study does not resolve. Finally, the excitation is a light impact about a taut equilibrium. No cable slackens, and this experiment does not test the treatment of taut–slack transitions (§ 3.1.3).

The records also contain the damping of the rotational mode, which the source publication does not report. This study examined it but does not report values. The log-decrement value depends on the chosen window, because the decay rate drifts by about a quarter within a single ring-down. The values are also between 0.56 and 0.82 times those obtained in the 2020 study from the same experiment, a disagreement larger than any effect that could be claimed. The limitation of the viscous damping law is argued below from modal strain energy, which does not depend on these estimates.

An identifiability limit on damping

The same structure shows a limit of the damping model, which follows from the geometry alone. Damping enters the formulation of § 3.1 as an axial viscous coefficient in each cable. Its effect on a mode therefore scales with how much that mode extends the cables. Per unit of modal kinetic energy, the rotational mode of this structure changes the cable lengths about 2900 times less than the modes above it. It is an infinitesimal mechanism, stiffened by prestress rather than by extension, and axial damping is nearly insensitive to it. A single cable coefficient fitted to a damping ratio of a few percent in the rotational mode would predict damping ratios of order $10^{6}$ in the modes above it. A structure whose modal damping is of the same order across these modes therefore cannot be represented by axial dashpots with any coefficient. Such damping points instead to hysteretic or joint-friction dissipation.

This is a limitation of the damping model, not of the identification, and it supports the identifiability argument of § 3.3. The property that makes the prestress-carrying mode informative about prestress, namely its lack of elastic strain energy, is the same property that makes it uninformative about any extension-based damping. Prestress and damping are recoverable from different parts of the same measurement, and should not be quoted with a shared confidence. On records simulated with the axial law, the coefficient is recovered correctly; only a real structure reveals the limitation.

In addition, the simplex identification reads one frequency from each record and does not run the time-domain residual of § 3.1. It therefore tests the small-oscillation identification against a measured reference; the residual itself is tested on measured records with the leg of § 4.11.

Stability, strength, and energy of the identified structure

The assessment turns a set of force densities into a stability margin and member utilizations. With the exact force densities, it classifies every free-standing polyhedral tensegrity among the classic tensegrity forms as super-stable at its own self-stress, and the planar benchmark frames as merely stable under gravity. Under excitation, it tracks the instantaneous stability class of the prism through the same taut–slack transitions that the identification uses. The full demonstration, including the member utilization curves, the energy balance through taut–slack transitions, and the bar-stiffness sensitivity study, is given in § 9.5 of the Supplementary Material.

This section asks how the assessment responds to identification error. Table 9 answers this by substituting the identified force densities for the exact ones. The outcome follows the identifiability structure of § 3.3. The large force-density errors concentrate in the directions that the record cannot determine. These directions, such as the cable joining the two pinned nodes, carry no free-coordinate stiffness, and they therefore hardly change the stability margin. On the X1 frame, an 87 % maximum force-density error leaves the margin off by 0.04 %. The identified model therefore predicts stability well even where individual force densities are poorly recovered. The strength utilizations scale with the governing member force and follow the error on that member, which lies in the well-identified directions.

Recovered model	max. $\sigma$ error	margin error	$U_c$ error
X1, taut	14 %	0.12 %	0.3 %
X1, slack-transition	87 %	0.04 %	0.2 %
X2, multi-record	90 %	6.0 %	22 %

Discussion

This study provides the inverse counterpart to the forward tensegrity networks of Xu et al. [35] and Qin and Chen [37]. Here, the residual that those studies solve for the motion is used with the member parameters as unknowns. The constrained Cartesian formulation makes this identification tractable, because its mass matrix is constant and the force densities enter it linearly. Compared with the differentiable physics engines used in robotics [39, 40], the present formulation aims at the structural interpretation of the identified quantities. It adds an identifiability analysis with Cramér–Rao bounds, and it works at the sensor counts and noise levels of structural monitoring instead of motion capture.

Within the structures and excitation suites tested, taut–slack transitions acted less as an obstacle to identification than as a source of information. Records with repeated taut–slack transitions were inverted with errors of a few percent on the X1 frame and the prism. Where low-amplitude forcing leaves members weakly excited, as on the X2 frame, the strong slack-transition record recovered those members several times more accurately, to a worst-member error of 21 %. The gain from strong excitation outweighed the information lost while individual cables carried no force. On the already well-excited X1 frame and prism, the two records give comparable force-density accuracy, with neither record better for every parameter.

The classical baseline and the X3 frame show the same pattern. Shooting converges to a spurious minimum on the low-amplitude taut record but recovers the slack-transition record almost exactly (§ 4.8). Two-point excitation with taut–slack transitions succeeds on the X3 frame; the low-amplitude taut record with the same two-point excitation still fails (Table 3). Within the range tested, the excitation sets the limit of the single-record formulation. The prestress level sets a different limit. At the low-prestress end of the prism series, the information decreases 17-fold, and the identification reaches an optimization limit before the information runs out (§ 4.6).

The simplest transition treatment was the most reliable of the three: the smoothed cable law was at least as accurate as the alternatives on the X1 and X2 comparisons. The complementarity treatment slowed convergence, event-aware collocation neither helped nor hurt, and neither overtook the smoothed cable law on the structures tested. No treatment was tuned separately.

The failure diagnostics have known limits. Fit failures raise the held-out ratio one to two orders of magnitude above the successful-run band. The silent failure of the gently forced X2 record is well fitted yet wrong by up to 80 %, and neither the held-out ratio nor a local Fisher analysis at the exact parameters detects it; the Cramér–Rao bounds even show that the record is informative. Detecting such practical non-uniqueness without the exact parameters, through multi-seed scatter for instance, remains open. Richer excitation removes the failure itself.

The Cramér–Rao bounds give the scatter that a record permits, and the eight-seed study gives the scatter achieved, which is 2.7 to 8.3 times that floor. With eight seeds, this ratio is only indicative. Neither measure captures the dominant error. The seed means of the force densities are 2 to 4 % from the exact values, an order of magnitude more than the seed scatter. The error therefore comes mainly from bias. An interval built from the spread of repeated fits, whether through dropout, an ensemble, or a variational posterior over the weights, would not capture this bias and could be narrow where the estimate is least accurate. This study reports bias and scatter separately for that reason.

All evidence for identification through taut–slack transitions is simulated, on small to moderate structures with two to eight mechanical degrees of freedom and up to 16 cables. Damping is modeled as viscous and assigned to the cables, while measured tensegrity decay is dominated by joint friction [24]. Axial stiffness is fixed except in the joint identification of § 4.9, where it traded error against the force densities as the Fisher analysis predicts. The rigid-bar model with lumped inertia also omits the member internal displacements that shape the high-frequency response [62]. Identification is thus confined to the low-frequency, mechanism-dominated modes the benchmarks emphasize.

The most consequential of these limits is the massless-chord cable model, in which a cable carries force along its chord and has no mass of its own. A continuum cable supports one axial and two transverse fields; the error of neglecting them depends on how far those modes lie from the mechanism modes. On the benchmarks the cable axial fundamentals lie near 47 Hz and the transverse fundamentals between 10 and 36 Hz, one to two orders above mechanism modes near 1.4 Hz. The bar axial modes stand three orders above. Transverse cable motion is therefore the first neglected effect that a wider frequency band would capture. Restoring it takes more than the taut-string model. Its transverse wave speed $\sqrt{T_k/\rho A}$ vanishes with the tension, and the taut-string model therefore degenerates in the slack regime that this study addresses. The flexural rigidity, which the taut-string idealization discards, remains at zero tension. With it, the member behaves as a beam at zero tension, a well-posed model with finite frequencies. The two mode families only below a crossover tension $T^* = EI(\pi/L)^2$ of order $10^{-3}$ N, against working tensions of tens of newtons. A transverse model that remains valid at slack would therefore add no error while the cable is taut, and it would be a natural basis for identifying cable mass and bending together with the prestress.

Section 4.10 is modal and taut: it validates the structural model and prestress against a physical test, but not the time-domain residual inversion, because that dataset contains no forcing record. Sections 4.11 and 4.12 add two independent experiments, a biomimetic leg from another group with springs measured on a separate rig, and the simplex of Małyszko and Rutkiewicz with load cells in its cables. The leg experiment provides the most direct evidence for the time-domain residual, because its raw records were made available and they allow the time-domain residual itself to be run on measured data, with the drive known and the response measured. It recovers the knee spring to 0.1 % in the mean over 15 records, on an identification target the identifiability diagnostic selects in advance. The simplex provides the second measured validation. From 13 impact records at 13 measured prestress levels, the prestress is identified with a mean absolute error of 32 N, inside the 36 N scatter of the transducers. The hardware mass in this model comes from the elements weighed by the authors of the 2020 study, and a single joint-stiffness constant is fitted. None of the three experiments tests the treatment of taut–slack transitions, because each is excited about a taut prestressed equilibrium and no cable slackens. That validation requires a tensegrity test in which a cable slackens under measured excitation while the cable forces are recorded, and it is the next experimental step of this work.

The damping law is the one place where the measured data exceed what the formulation can represent. This study keeps the viscous law. The whole formulation and every simulated result are built on the axial coefficient of § 3.1, and on records generated by that law it is recovered correctly. The simplex shows that it cannot represent the dissipation of a real self-stressed structure. § 4.12 gives the structural reason. The mechanism mode stiffened by the prestress barely extends any member, and extension-based damping is therefore nearly insensitive to it. Yet the measured damping is near-uniform across modes whose strain energies differ by three orders of magnitude. A hysteretic or joint-friction law would be the natural extension. It changes only the member force law and leaves the inverse formulation as it is, with the constant mass matrix and the linear force-density structure. Making that change here would alter every result in § 4 without benefit for the taut–slack question that this study addresses; the change is therefore left to future work. The finding itself is reported in full because it bears on what any identification of damping in a mechanism-dominated tensegrity can claim.

Conclusions

This study formulated physics-informed inverse identification for tensegrity structural dynamics and examined it on benchmark structures, against classical identification methods, and on measured records from laboratory structures. The residual is the constrained Cartesian equation of motion. Its constant mass matrix and the linear entry of the force densities make the identification trainable and the residual sensitivities closed-form. The tension-only cable law is retained through a smoothed cable law, and a Fisher-information analysis shows, before inversion, which member parameters a record can determine and to what precision.

Four findings summarize what was learned. First, within the cases tested, the excitation, more than the structure size or the sensor count, set the limit of a single-record identification. Strong records with taut–slack transitions recovered members that low-amplitude taut records left undetermined, adverse sensing increased the error less than weak excitation did, and the failures that occurred were removed by richer excitation, not by more sensors, more data of the same kind, or larger networks. Within the structures and excitation suites tested, the taut–slack transitions supplied more information than the slack intervals removed.

Second, a good response fit is not evidence of identification. One benchmark record was fitted closely while its parameter errors approached 80 %, and the same silent failure recurred at bridge scale. The held-out ratio flags fit failures; neither it nor a local Fisher analysis detects silent failures, and their detection without the exact parameters remains open. Identification claims in this study are therefore based on the identifiability analysis and the parameter comparisons, never on fit quality.

Third, the benefit of taut–slack transitions is not specific to the physics-informed identification. Classical methods given the same records succeeded in opposite regimes: shooting converged to a spurious minimum on the taut record and was nearly exact on the slack-transition record, and the unscented filter did the reverse. The physics-informed identification succeeded on both from the same distant initialization, at a higher computational and tuning cost.

Fourth, the limits of a record can be predicted in advance. A member restrained at both ends is invisible at any amplitude, taut or slack; a first-order self-stress combination is recovered by a record with larger excitation; and grouping members that share a tension can remove the remaining degeneracy. The recovered prestress supports a stability and strength assessment whose margins are far less sensitive to identification error than the individual force densities, because the poorly recovered directions are the ones carrying no free-coordinate stiffness.

All evidence for identification through taut–slack transitions is simulated, on structures of two to eight mechanical degrees of freedom. The measured validations are taut: one modal, one time-domain, and one at 13 prestress states with a single fitted hardware constant. Damping is modeled as viscous and axial. The measured data from a self-stressed structure show that this law cannot represent joint-dominated dissipation, and prestress and damping should therefore not be quoted with a shared confidence. Cables are modeled as massless force elements; the error of this model was measured by deliberate model-form mismatch, and a refinement for the slack regime is outlined in § 3.2. Extension to measured records with taut–slack transitions, to a hysteretic or friction damping law, and to larger structures is ongoing work.

Data Availability

The simulated datasets and the code that generates them are available from the authors. The biomimetic-leg measurements analyzed in § 4.11, comprising 18 inertial-measurement-unit chirp records together with the measured spring and component properties, were provided by Y. Wang and colleagues and are used here with their permission; requests for those records should be directed to them. The simplex measurements analyzed in § 4.12 combine the published tables of Malyszko and Rutkiewicz [15], which is open access, with impact-hammer records, element masses and joint geometry held by the co-authors L. Malyszko and A. Rutkiewicz; requests for those records should be directed to them.

Acknowledgments

The authors thank Yihang Wang for supplying the biomimetic-leg records and for answering questions about the excitation.

Supplementary Material

This supplement contains the implementation details and extended analyses that support the main text: the alternative treatments of the taut–slack transition, the optimizer and loss-weight details with the full parameter table, the assessment of the classic tensegrity forms, the exclusion analysis and hip statistics behind the biomimetic-leg reconstruction, the structural-assessment formulation with its demonstration on exact force densities, and the robustness studies on noise, model-form error, and adverse sensing. Section, table, and equation numbers of the form S1, S2 refer to this supplement; unprefixed numbers refer to the main text.

Alternative treatments of the taut–slack transition

The main text trains with the smoothed cable law and compares two alternatives with it on identical records. Their construction is as follows.

The complementarity treatment keeps the exact law and moves the nonsmoothness out of the residual, adapting the static-contact construction of Sahin et al. [43] to the dynamic setting. An auxiliary tension head outputs a non-negative tension $T_k(\tau)$ for every cable, and a softplus output layer enforces the positivity. A nominal tension scales this output. It is the 10 N/m initialization force density times the mean member length of the reference geometry, and it therefore carries no information about the prestress being identified. The cable term of the equation-of-motion residual uses the force density $T_k/\lVert s_k\rVert$. A Fischer–Burmeister residual then ties the tension to the elastic law,

with $g_k$ the elastic slackness defined in the main text. Tension and slackness are normalized by member scales before they enter the loss, and the loss gains the term $10\,r_{\mathrm{NCP}}^2$. The zero set of $\phi_{\mathrm{FB}}$ is exactly the tension-only cable law: $r_{\mathrm{NCP},k}=0$ if and only if $T_k\ge0$, $g_k\ge0$, and $T_kg_k=0$. The cable is then either taut and elastic or slack and force-free. The function $\phi_{\mathrm{FB}}$ is smooth everywhere except at $T_k=g_k=0$, where the cable is just taut and carries no force. This reduces the nonsmooth set from the whole taut–slack boundary to one point per cable.

The event-aware treatment keeps the smoothed cable law and concentrates the collocation points where its bias occurs. Every 500 steps, the current solution is checked for taut–slack transitions. Half of each batch’s collocation points are then drawn from narrow Gaussian windows around the detected transition times; the other half remain uniform. Apart from the tension head and the complementarity term, the three treatments share the same trunk architecture, loss weights and normalizations, and optimizer settings.

Optimizer, loss weights, and run archival

The larger learning rate on the physical parameter group can be used because those parameters are represented in logarithmic form. The size of an Adam step is set by the normalized gradient, not by the gradient magnitude. A step in $\log\sigma_k$ is therefore multiplicative in $\sigma_k$, and one relative step size serves force densities, an axial stiffness, and damper coefficients whose numerical values differ by orders of magnitude.

Table S1 varies the loss weights on the X1 taut record. Below $w_b\simeq10$, the bar-length constraint is not enforced and every quantity degrades together, reaching 29.5% at $w_b=1$. Above this value, the recovery is flat to within about 2 percentage points across three further decades, and the value used throughout, $w_b=100$, lies in the middle of this plateau. Between $w_{\mathrm{eom}}=0.3$ and 3, the mean force-density error changes by about 1 percentage point; at $w_{\mathrm{eom}}=10$, where the sensor data are effectively ignored, the error increases sharply.

	force density (%)		damping (%)
bar-constraint weight, matched generator
$w_b$	max	mean
1	29.50	23.11	16.59
10	7.42	5.99	0.62
100†	3.25	2.69	1.45
1000	3.86	3.18	2.57
10000	2.14	1.69	3.49
data against physics, matched generator
$w_{\mathrm{eom}}$	max	mean
0.1	4.74	3.54	8.43
0.3	4.77	3.95	1.59
1†	3.25	2.69	1.45
3	3.13	2.59	1.53
10	9.79	8.04	2.50
the same, with a model-form error present
$w_{\mathrm{eom}}$	max	mean
0.1	7.71	6.01	8.90
0.3	7.73	6.38	6.45
1†	6.84	5.78	2.73
3	6.85	5.72	2.31
10	11.87	9.86	1.30

The third block of the table carries a record with a model-form error the residual cannot represent. Down-weighting the residual to $w_{\mathrm{eom}}=0.1$ makes the identification worse, 6.0 against 5.8%, and the best value is at or slightly above equal weighting. The residual remains informative about the parameters even when it contains a model-form error, because this error is structured by the geometry and is not random. Reducing the weight of the residual discards this structure together with the error. The penalty for keeping the equal weighting used throughout is 0.06 percentage points.

Every identification run through the experiment engine has an archived per-run configuration file that reproduces it exactly. The robustness, baseline, and multi-record studies come from dedicated scripts that fix their protocols in code and record the varied factors with each result. Table S2 lists the simulation, sensing, and training parameters used throughout the main text.

Parameter	Value	Source
Members, synthetic benchmarks
Cable radius, modulus	1 mm, 70 MPa ($EA = 220$ N)	benchmark definition
Cable damping $c_k$	1.0 N s/m, taut cables only	benchmark definition
Bar radius, density	5 cm, 500 kg/m$^3$	benchmark definition
Bar treatment	rigid (length constraint)	§ 2.1
Hexagonal prism	$EA = 379$ N, bar mass 42.3 g	Huang et al. [20]
Prestress
Planar frames, bridge	LP, 25 N/m minimum force density	§ 2.3
Triplex prism, gravity	uniquely fixed by equilibrium	§ 2.3
Prism series, zero gravity	self-stress scale 1–15 N/m	§ 4.6
Free-standing polyhedra	own self-stress mode, sign-checked	§ 4.14
Members, structural assessment
Bar modulus, yield	206 GPa, 300 MPa (Q345 steel)	Ma et al. [63]
Cable effective modulus	76 GPa (steel rope)	Ma et al. [63]
Cable allowable stress	600 MPa, about half rope strength	European Committee for Standardization [64], ASTM International [65]
Section, buckling factor	circular $I = \pi R^4/4$, $K = 1$	Carpentieri et al. [66]
Rank tolerance	$10^{-6}$ relative singular value	§ 4.14
Excitation and records
Classes	harmonic, half-sine impulse, broadband	§ 2.4
Fixed protocol	1.0 Hz, one coordinate (two, prism)	§ 2.4
Multi-point excitation	two coordinates, 1.0 and 1.7 Hz	§ 4.4
Amplitude range	3–60 N, spanning both regimes	§ 2.4
Record length, output step	5 s, 0.01 s	§ 2.4
Reference integration	RK45, tolerance $10^{-10}$	§ 2.5
Sensing
Measured signal	nodal accelerations	§ 2.4
Instrumented coordinates	half to two-thirds of free coordinates; one channel in the adverse-sensing study	§ 4.9
Noise model	zero-mean Gaussian, scaled to per-channel RMS	§ 2.4
Noise level	2 % nominal; 5, 8, 10 % in the robustness studies	§ 4.9
Sparse, intermittent record	25 % of samples, three dropout windows	§ 4.9
Network
Trunk	4 layers $\times$ 64 units, tanh	§ 3.1
Input features	8 Fourier frequencies in scaled time	§ 3.1
Heads	displacement, constraint multipliers	§ 3.1
Hard initial condition	$s(\tau) = (1 - e^{-\tau/0.08})^2$	§ 3.1
Displacement scale	0.1 m	§ 3.1
Smoothing sharpness $\beta$	800	§ 3.1.3
Optimization
Optimizer	Adam, cosine-annealed	§ 3.1
Learning rates	$2 \times10^{-3}$ network, $10^{-2}$ parameters	§ 3.1
Loss	$\mathcal{L}_{\mathrm{data}} + \mathcal{L}_{\mathrm{EOM}} + 100\mathcal{L}_{\mathrm{bar}}$	§ 3.1
Collocation points	256 fresh points per step	§ 3.1
Training length	25,000 steps; 12,000 in the grid studies	§ 3.1
Data split	70 train / 30 held out	§ 3.1
Parameter initialization	$\sigma_k = 10$ N/m, $c = 0.3$ N s/m	§ 3.1
Seeds	0–8, network and noise realization	§ 4.8

Assessment across the classic tensegrity forms

The stability classification of the main text is based on the same rank count that governs identifiability. This section applies it beyond the identification benchmarks, to forms that are assessed with their exact force densities and are never identified.

The rank of the equilibrium matrix $A$ fixes the number of self-stress states $s = n_{\mathrm{mem}} - \operatorname{rank} A$ and infinitesimal mechanisms $m = n_{\mathrm{free}} - \operatorname{rank} A$, related by Maxwell’s rule $m - s = n_{\mathrm{free}} - n_{\mathrm{mem}}$ [4, 5]. Table S3 collects these counts and the class for the planar identification benchmarks. Ranks use a relative singular-value tolerance of $10^{-6}$, which accommodates the six-figure coordinates of the published polyhedral geometries. The octahedral forms satisfy their self-stress to machine precision and the rest to about $10^{-7}$. The table also covers three classic forms assessed with the same formulation (Figure S1): the tensegrity prisms $T_3$, $T_4$, $T_6$, the expanded octahedron (the six-strut tensegrity icosahedron) [67, 68], and the truncated-tetrahedron tensegrity [69]. These forms are free-standing and are therefore minimally grounded here. Their prestress is set from their own self-stress mode. Their counts come from the equilibrium matrix on all coordinates, since no reaction absorbs equilibrium at a grounding node. The supported frames and prisms are counted on their free coordinates. The self-stress count is precisely the dimension of the identifiable prestress space, and the mechanisms are what that prestress must brace. The rank deficiency that admits a self-stress thus also makes the prestress identifiable and stability-critical. The planar frames carry no mechanism and are merely stable under gravity, because their stress matrices are indefinite (Figure S2). Every free-standing polyhedral tensegrity in the table, by contrast, is super-stable at its own self-stress. Its margin falls as the number of struts grows (28.2 for $T_3$ to 8.4 for $T_6$) and as the mechanism count rises. The truncated tetrahedron is the extreme case: seven mechanisms braced by a single self-stress state. Gravity load moves the frames off the super-stability boundary.

Structure	Free	Mem.	Rank $A$	$s$	$m$	Resolvable	Class	$\lambda_{\min}$
X1 frame	4	6	4	2	0	3/4	stable	148.6
X2 frame	8	11	8	3	0	6/7	stable	211.6
X3 frame	12	16	12	4	0	9/10	stable	45.4
$T_3$ prism	9	12	8	4	1	6/9	super-stable	28.2
$T_4$ prism	12	16	11	5	1	8/12	super-stable	17.7
$T_6$ prism	18	24	17	7	1	12/18	super-stable	8.4
expanded octahedron	30	30	29	1	1	24/24	super-stable	2.0
octahedral cell	30	30	28	2	2	24/24	super-stable	2.5
dome	21	18	17	1	4	12/12	super-stable	17.4
truncated tetrahedron	30	24	23	1	7	18/18	super-stable	1.3
class-1 cell	30	30	29	1	1	24/24	super-stable	24.0
two-stage tower	18	27	18	9	0	18/21	super-stable	22.3

The column of determined cables gives the identifiability result of the main text. Of the 183 cables of the 12 structures, the free coordinates determine 164 at first order. Of the 19 undetermined cables, 17 are individually invisible, each spanning two restrained nodes. The remaining two are combinations of members: the X2 and X3 frames each carry a self-stress direction that produces no free-coordinate force at the reference configuration; no single cable of either is invisible, but one combination is undetermined at first order. The pattern is exact on the tensegrity prisms. There, the $n$ cables of the pinned base polygon join restrained nodes only and are undetermined, while every other member is determined; 3, 4, and 6 of the 9, 12, and 18 cables of $T_3$, $T_4$, and $T_6$ are therefore undetermined. In the free-standing polyhedra every cable is determined, since minimal grounding leaves no pair of fully restrained nodes for a cable to span.

Biomimetic-leg reconstruction details

Three candidate origins of the model-to-measurement frequency gap of the leg of Wang et al. [61] are excluded quantitatively. First, the gap is not measurement scatter. The springless case, repeated three times, has a standard deviation of 0.0006 Hz on $f_1$, and its 0.0155 Hz distance from the computed value is 26 standard errors. Second, the gap is not an artifact of the chirp, which would bias a peak in the direction of the frequency change. Driving the leg’s own mass and stiffness matrices with their protocol, a linear chirp from 0.01 to 4 Hz at 0.0327 Hz s$^{-1}$, and picking peaks from the response spectrum as they do displaces the peak by at most 0.8 % for modal damping between 1 and 5 %. Third, the gap is not an error in the reported masses or dimensions. No entry of their Tables D.5 or D.6 has enough influence: every log-sensitivity of $f_1$ is below 0.21 in magnitude. The most influential entry would need a 10 % error to move $f_1$ by 2 %, and it would increase the error in $f_2$.

The missing restoring stiffness has two parts: one that remains with no springs installed, measured by Case E, and one that grows with the installed hip springs. When the cases are ordered by the stiffness installed at the hip, the $f_1$ gap is 2.3, 3.4, 3.7, 5.1, and 5.1 % for Cases E, A, D, B, and C. Fitting a residual rotational stiffness at the two joints to Case E alone returns 0.107 N m rad$^{-1}$ at the hip and 0.011 N m rad$^{-1}$ at the knee, about 10 % of the gravitational restoring stiffness of the first mode. If a single multiplicative factor on the hip rate is also allowed, all eight frequencies of Cases A to D are reproduced within 1.2 % with two constants, and the factor is 1.27. This result is recorded as a diagnosis and not adopted as a correction, because it is fitted to the frequencies that it then reproduces.

The hip statistics behind the exclusion are as follows. When both scales are identified from the published frequency pairs, contrary to the screening, the hip rate exceeds the rig rate by 49 to 68 % across Cases A to D, with a root-mean-square of 56.5 %. Multiplying each case’s frequency gap by its amplification $2/s$ predicts 37 to 47 %, and the prediction and the outcome correlate at 0.89 across the cases. The overshoot is therefore the amplified model-form gap and not an artifact of the optimizer. Identified record by record, the hip scale is +29 ± 38, +62 ± 24, +185 ± 141 and +70 ± 8 % for Cases A to D, and in the time-domain envelope fits it pools to 0.931 ± 0.336 over the 15 records with springs. This value is centered on the rig value, but its coefficient of variation is 36 %, against 18 % for the knee, which is too imprecise to report as a result. The screening in the main text summarizes these statistics, and they are reported here in full.

The simulated test of the time-domain envelope identification, a record with known parameters and 2 % noise, returns the hip scale to −4.4 %, the knee scale to +0.1 %, both dampers to within 0.5 %, and the drive amplitude to +0.6 %, at an envelope error of 0.0026.

The measured records also give two quantities that the source publication does not report. The damping ratio of the second mode is 0.64 ± 0.24 % of critical across all 18 records, measured on the free ring-down after the chirp has passed the resonance; that of the first mode is near 0.4 %, with much wider scatter. Every record also requires a drive amplitude larger than the commanded stroke, by a pooled factor of 1.183 ± 0.137, and an independent comparison of swing amplitudes gives 1.21. Because two independent routes require the same increase of about 20 %, its likely source is the test stage.

Structural assessment of the identified model

This section contains the structural-assessment formulation of the main text and its demonstration with exact force densities. The propagation of identification error to the assessment remains in the main text, since it is a property of the identification.

The recovered force densities describe the prestress state of the structure. This prestress determines two safety-critical properties: the geometric stability of the self-stressed shape and the strength demand on its members. Both follow from the identified prestress without further measurement. The time-domain identification keeps the rigid-bar model; the assessment adds the members’ finite axial rigidity and their strength limits. The two agree on the mechanism-dominated response that sets the margin, as verified below.

Let $A$ be the equilibrium matrix mapping member force densities to nodal forces, $D = \operatorname{diag}(E_m A_m/\ell_m)$ the member axial stiffnesses, and $S = \sum_m \sigma_m C_m$ the stress matrix assembled from the force densities $\sigma_m$. The tangent stiffness is the Hessian of the total potential energy,

the sum of a material stiffness $K_M$ and the geometric prestress stiffness $K_G$; this is the standard tangent stiffness of the force-density method [70]. For the rigid bars of this study, the geometric stiffness is the constraint-Hessian stress matrix rather than the elastic-truss softening, and the transverse softening that governs a stocky member enters the buckling check below instead. Restricted to the free coordinates, the configuration is stable when the reduced $K_t$ is positive definite, and its smallest eigenvalue $\lambda_{\min}$ measures the margin.

Where the structure carries infinitesimal mechanisms, displacements $u_m$ in the null space of $A^{\mathsf{T}}$, stability depends on the prestress alone. The configuration is then prestress-stable when $u_m^{\mathsf{T}} S u_m > 0$ for every mechanism, equivalently when the reduced stress matrix $U_m^{\mathsf{T}} S U_m$ is positive definite [4, 5]. A configuration is super-stable when the stress matrix $S$ is itself positive semidefinite on the free coordinates [70]. It is then stable for any material stiffness and any positive scaling of the prestress. Without super-stability, stability also depends on the material stiffness. A third, weaker sufficient condition tests only the directions the material stiffness cannot resist. With $U_0$ a basis for the null space of $K_M$, the configuration is prestress-stable when

a test projected onto the mechanisms, which is not needed when $K_M$ is already definite [70]. The three classes, super-stable, stable, and unstable, follow from the signs of the two margins $\lambda_{\min}(K_t)$ and $\lambda_{\min}(K_G)$,

where each sign is taken against a tolerance scaled by $\lVert K_G\rVert$, because this norm sets the size of the mechanism-mode eigenvalues on which the classification depends.

Member strength is checked against the same force densities. A cable carries $T_m = \sigma_m \ell_m$ and a strut the balancing compression $C_m$. Each is measured against its capacity by a utilization,

with a circular section $I_m = \pi R_m^4/4$ and the pinned-pinned factor $K = 1$ appropriate to cable-terminated nodes [66]. A member reaches its limit at unit utilization. Geometric instability, indicated by $\lambda_{\min}$, and material failure, indicated by $U_c$ and $U_b$, are thus separated and reported together.

Member properties follow the tensegrity material library of Ma et al. [63]: Q345 structural-steel bars with $E_b = 206$ GPa, and wire-rope cables with an effective $E_c = 76$ GPa. The bar yield is taken as 300 MPa, a design value about 13% below the 345 MPa nominal that names the grade. The cable modulus is that of multi-strand wire rope. It is well below solid steel because the helical lay of the strands adds a geometric compliance in series with the wire material. It should not be read as the modulus of a spiral strand, which retains a larger fraction of the solid-steel value. The cable is checked against an allowable $\sigma_{\mathrm{all}} = 600$ MPa, about half the 1200 MPa breaking strength assumed by the source library. That allowable stress is well inside the range of standard products: spiral ropes for structural applications are specified in EN 12385-10 [64], and seven-wire prestressing strand under ASTM A416 carries a specified minimum tensile strength of 1860 MPa in grade 270 [65]. The utilization therefore already carries a factor of at least two on the material.

The model also tracks the structure’s energy through each excitation: the kinetic energy $T$, the gravitational potential $V_g$, and the cable strain energy $V_s$ are evaluated at every step. Their sum obeys the exact balance $\frac{\mathrm{d}}{\mathrm{d}t}(T + V_g + V_s) = (\mathbf{f}_{\mathrm{ext}} + \mathbf{f}_{\mathrm{damp}}) \cdot\dot{\mathbf{q}}$. The balance also serves as an integration check and shows the strain energy stored and released at the taut–slack transitions.

The assessment of this section turns a set of force densities into a stability margin and member utilizations. It is used in two different ways. The formulation, and the behavior of the classes across structures and configurations, are demonstrated with the exact force densities; they therefore show properties of the structure, not of one inversion. Only the propagation table of the main text substitutes the identified force densities for the exact ones, and it shows how much the identification error changes the assessment.

The lowest natural frequency from the tangent stiffness of Eq. (14) agrees with the constrained modal analysis of the rigid-bar dynamics to within 0.2%: 1.410 against 1.413 Hz on the X2 frame. The elastic-member assessment and the rigid-bar model therefore share the mechanism-dominated response. The assessment also admits a rigid or a flexible bar model. On the steel members, the two give stability margins within 0.5% of each other. The rigid-bar assumption behind the identification therefore carries into the assessment with negligible error. The strength utilizations depend only on the member forces and are identical between the two.

The triplex prism has one infinitesimal mechanism, and its reduced stress matrix is positive, with a margin of 28.2. The prestress alone therefore braces the mechanism, which is the defining prestress-stability property of a tensegrity [70]. The mechanism-projected condition agrees. The geometric stiffness restricted to the null space of the material stiffness has smallest eigenvalue 28.2 on the prism and stays positive on every free-standing structure assessed. On the planar frames the material stiffness is already definite and the test is not needed.

Classifying the reduced stiffness and stress matrix places each configuration in one of the three classes (Figure S3). The benchmarks fall into two of them. The triplex prism is the only benchmark with an infinitesimal mechanism, and its stability therefore depends on the prestress alone; it is also the only one that becomes unstable when the prestress is reversed. The planar frames carry no mechanism, and their material stiffness holds them whatever the prestress does; only the definiteness of the geometric stiffness changes, and it marks the super-stable boundary.

The class depends on the configuration as well as the topology. The triplex prism at its identified self-stress is super-stable in the sense of this section, and so is the X2 frame at the steel design prestress used below. Under gravity load, the frames instead carry an indefinite stress matrix and are merely stable. Scaling and reversing the prism prestress takes it through these classes: the mechanism loses its bracing as the prestress approaches zero and becomes unstable once the prestress reverses. On the X2 frame, a non-optimal self-stress makes the stress matrix indefinite, leaving a merely stable configuration held by the material stiffness. The identified load-balancing self-stress lies on the super-stability boundary.

The classification uses the same rank count as the identifiability analysis. Section S3 applies it to the classic tensegrity forms and gives the counts, classes, margins, and determined cables of all 12 structures.

The class is not only a property of the resting shape. The triplex prism is used for this demonstration for the same reason it is used for the prestress series: a frame without mechanisms cannot show it. Repeating the experiment on the X2 frame leaves its margin above 43 throughout, because the material stiffness holds the shape whatever the taut–slack state of the cables. Its class therefore does not change. Under excitation, the instantaneous configuration of the prism changes class as cables change between taut and slack (Figure S3, bottom). A slack cable drops out of the stress matrix, and the frozen shape passes from super-stable through merely stable to momentarily unstable once enough bracing is lost. Damping restores the super-stable equilibrium as the motion decays. The stability class thus tracks the same taut–slack transitions that the identification uses.

The X2 frame is rebuilt with the steel members of the material library at its design prestress, a maximum cable tension of 3 kN. There the reduced tangent stiffness is positive definite and the self-stressed shape is super-stable. The strongest cable carries half its allowable ($U_c = 0.50$), and the most compressed strut a small fraction of its buckling load

($U_b = 0.03$). Raising the prestress traces the utilization curves of Figure S3. The cable reaches its allowable ($U_c = 1$) at a maximum tension of 6 kN while the stability margin stays large and the strut stays far from buckling. On this frame, material strength, not geometric instability, governs the safe prestress.

Figure S4 traces the energy balance of this section on the X2 frame under resonant excitation that slackens the cables. The balance is a property of the constrained integration, and it is therefore reported on the reference planar benchmark only. The check is evaluated on every record behind the results above. The strain energy rises and falls as cables load and unload, and the mechanical energy tracks the work input. The balance residual stays below $10^{-4}\,\mathrm{J}$ through the taut–slack transitions.

The flexible-bar model also lets the identification carry the bar axial stiffness as an unknown. With the residual extended to elastic bars and $EA_{\mathrm{bar}}$ added to the trainable parameters, the extended residual is verified against the flexible-bar simulator (the verification table of the main text). A sensitivity analysis nonetheless shows that records of this bandwidth carry little information about the bar stiffness. Bar axial deformation is a high-frequency mode, decoupled from the mechanism motion whose frequencies the prestress sets. The low-frequency, mechanism-dominated record therefore carries little information about $EA_{\mathrm{bar}}$. Band-limiting the response below the bar axial modes removes about 70% of its sensitivity at the soft bar stiffness tested. The remainder, the bar’s quasi-static compliance, scales as $1/EA_{\mathrm{bar}}$. The stiff limit itself is too stiff to integrate and was not computed. Identifying the bar stiffness would require exciting the bar axial modes directly. The bar force, by contrast, is fixed by the self-stress and is identified alongside the cable force densities. It is the same separation of an identifiable force from an unidentifiable stiffness seen in the cable stiffness–prestress trade-off.

Robustness studies

This section gives the three robustness studies on the X1 frame that § 4.9 of the main text summarizes.

Noise, sensor count, and joint stiffness identification

Accuracy decreases moderately with noise: a fivefold increase in noise increases the force-density error by a factor of 1.8. Table S4 varies sensor noise and the sensor count, one factor at a time, on the X1 taut record. Every entry is a mean over seeds of the same 12,000-step protocol, and the spread is the scatter over seeds. The damping error is the more sensitive of the two and by far the more scattered; at the higher noise levels, its standard deviation approaches its mean. Removing one of the two measured channels leaves a single accelerometer coordinate for a two-degree-of-freedom structure, and the inversion still identifies the force densities and the damping to within a few percent. Halving the instrumentation therefore increases the error less than raising the noise fivefold.

Condition	Seeds	max $\hat{\sigma}$ err (%)	$\hat{c}$ err (%)
2 % noise, 2 sensors	9	3.60 $\pm$ 0.28	1.58 $\pm$ 0.93
5 % noise, 2 sensors	4	4.81 $\pm$ 0.76	4.10 $\pm$ 2.10
10 % noise, 2 sensors	4	6.60 $\pm$ 0.79	4.28 $\pm$ 3.53
2 % noise, 1 sensor	4	4.26 $\pm$ 0.67	3.57 $\pm$ 2.95

Identifying the axial stiffness as well shows the trade-off that the Fisher analysis predicts. With $EA$, $\sigma$, and $c$ identified jointly on the taut record, the stiffness is recovered to 7.4% but the force-density errors grow to 35%: the two parameter groups exchange error along their correlated direction. On the slack-transition record, the joint identification fails with an 83 % stiffness error, and the held-out ratio again flags the failure. Reliable stiffness identification therefore needs either records rich enough to separate the groups or an independent stiffness measurement. The joint formulation makes this ambiguity visible.

Model-form error

The simulated records of the main text are generated and inverted with the same forward model, which is an inverse crime (§ 4.2 of the main text). Four generators test whether the accuracies hold under model error, with the identification left unchanged throughout. The first perturbs the mass matrix by up to 5 % per coordinate, which represents unmodeled joint masses and fittings but leaves the model form intact. The second and third change the model form. The second lumps half of each cable’s mass at its end nodes, which makes the massless-chord cable law of the residual wrong at every parameter value. The third replaces the rigid bars with elastic ones, which removes the bar-length constraints, and it is integrated by an implicit multistep solver instead of the explicit scheme used for the matched records. Neither the equations nor their discretization are shared. The fourth tightens the generator tolerance from $10^{-10}$ to $10^{-13}$, which changes the discretization only.

The mass perturbation leaves the X1 taut recovery essentially unchanged, at 3.0 % against 3.5 % for the matched case. On the prism, it raises the force-density error from 0.6 to 3.5 % and the damping error from 3.2 to 4.2 %. Table S5 reports the rest. The lumped cable mass is only 1.2 % of the structural mass, but it changes the acceleration record by 8.6 %, four times the sensor noise, and doubles the taut force-density error to 7.0 %. Elastic bars increase the error by a factor of 1.5. Lumping the cable mass at the nodes is the more severe of the two standard choices. A consistent distributed-chord mass matrix moves a third of that mass into coupling between the end nodes; it changes the record by 5.7 % instead of 8.6 % and gives an error of 5.9 % instead of 7.0 %. Both increases are small, and the errors stay below 10 %. On the slack-transition record, every generator stays within the seed scatter of the matched case, and the model-form errors tested therefore do not limit that inversion. The tolerance row is unchanged to two significant figures, which shows that the integration accuracy does not affect the results. Within these tests, the accuracies reported in the main text therefore also hold for generators that the residual cannot represent.

Generator	Taut record		Slack-transition record
	$\sigma$	$c$	$\sigma$	$c$
matched	$3.5 \pm0.2$	1.1	$19.7 \pm2.1$	4.6
cable mass	$7.0 \pm0.2$	2.1	$21.0 \pm2.8$	5.7
flexible bars	$5.3 \pm0.4$	1.9	$17.0 \pm1.9$	6.0
tolerance $10^{-13}$	$3.5 \pm0.2$	1.1	$19.3 \pm1.6$	5.6

Adverse sensing

The identifications of the main text use favorable conditions: two acceleration channels, 2 % noise, harmonic forcing, and a dense record. Field monitoring rarely provides all four. The X1 record was therefore regenerated and inverted under four adverse conditions drawn from practice, first one at a time and then all together. The first is a single acceleration channel, the sparse sensor placement forced when a full structure cannot be instrumented [55, 71]. The second is 8 % measurement noise, the upper end of what vibration studies report [72]. The third keeps only a quarter of the time samples, in intermittent blocks with three sensor-off dropout windows; aging and telemetry faults make such data loss common [73, 74]. The fourth replaces the harmonic forcing with an impulse (impact) excitation [75]. Physics-informed methods have been used for identification from such sparse and noisy records [26, 76].

Turning each factor on alone separates their contributions (Table S6). The baseline case recovers the identifiable force densities to 2.8 %. Quadrupling the noise and halving the sensor count increase the error only by factors of 1.6 and 1.7. Discarding three quarters of the samples in intermittent blocks increases it by a factor of 4.3. Replacing the harmonic forcing with an impulse increases it by a factor of 8.8, the largest of the four. An impulse supplies energy only once, after which the structure vibrates freely. The cables therefore spend most of the record in free decay instead of being driven through their working range.

Condition	Sensors	Noise	Samples	Forcing	$\sigma$ error
baseline	2	2 %	500	harmonic	2.8 %
high noise	2	8 %	500	harmonic	4.7 %
single sensor	1	2 %	500	harmonic	4.9 %
sparse, dropout	2	2 %	106	harmonic	12.2 %
impulse	2	2 %	500	impulse	25.0 %
all four	1	8 %	120	impulse	24.9 $\pm$ 4.2 %

With all four together, the error is no worse than with the impulse alone, 24.9 % against 25.0 %. The fit remains close throughout, which indicates that the lost accuracy comes from reduced information and not from a failure to converge. The damping is the exception. Across three seeds of the combined condition, its error is 101 % with a scatter of $\pm$73 %, consistent with monitoring experience that damping estimates degrade first [72]. On a single adverse record, the excitation is the limiting factor: the attainable accuracy depends more on what the excitation makes observable than on how well the response is measured.