Physical AI: Concepts, Techniques, and Challenges

I Overview and Foundations

”Pure logical thinking cannot yield us any knowledge of the empirical world; all knowledge of reality starts from experience and ends in it.”

— Albert Einstein

Introduction to Physical AI

For decades, artificial intelligence (AI) operated inside computers primarily, processing digital data and generating symbolic outputs without directly altering the physical world. Recent advances extend AI beyond this digital boundary by enabling it to perceive and understand physical environments, reason about them, and act within them. This emerging capability has given rise to physical AI, which studies how artificial systems shall interact with the physical world. Yet, what exactly is physical AI? What does its technical landscape look like, and what core challenges does it entail? And, what are typical application scenarios of physical AI? This chapter addresses these questions by tracing AI’s expansion from digital to physical space, defining physical AI and its boundaries with neighboring fields, and examining its core characteristics, system architectures, and application domains.

Background: A Paradigm Expansion of AI

Digital AI and Physical AI

Artificial intelligence has developed through several complementary traditions, from symbolic reasoning and knowledge representation to statistical learning and deep neural networks. Deep learning substantially advances the ability of computational systems to extract useful representations from complex data, with major improvements in speech recognition, visual recognition, and related information-processing tasks. These achievements establish representation learning as a central methodological foundation of contemporary AI (LeCun et al., 2015).

In this chapter, Digital AI refers to capabilities that operate directly on digital representations, such as text, images, software, and structured records. This definition emphasizes the system’s operational interface over the origin of its data. For example, while an image may depict a physical scene, recognizing its contents does not require the system to actually interact with that environment. Physical AI, in contrast, applies intelligence to perceive, influence, and adapt to physical processes through direct interaction. Definition 1.1 in § 1.1.2 further clarifies this fundamental distinction.

The distinction becomes apparent when comparing the description of an action with its execution. An AI system may correctly explain how to pick up a glass while remaining unable to estimate its mass, establish a stable grasp, detect incipient slip, or adjust contact forces. Successful manipulation depends on relationships among geometry, material properties, sensing, and actuation that cannot be assessed solely through the linguistic plausibility of a response. The relevant criterion is whether the system produces the intended physical outcome under the conditions encountered.

This distinction is related to the symbol grounding problem of how symbols can acquire meaning through a system’s own capacities rather than solely through interpretation by an external observer. The formulation by Harnad (1990) identifies the limitations of defining symbols only through other symbols and motivates their connection to nonsymbolic representations. For physical AI, the corresponding engineering problem is to connect concepts such as support, contact, and stability to observations and action consequences.

Grounding should not, however, be reduced to a categorical claim that digital models possess no physical knowledge. Models trained on descriptions, images, videos, or simulated trajectories can acquire useful regularities concerning the physical world. The unresolved issue is whether these regularities support reliable intervention across changes in objects, environments, and operating conditions. Semantic competence, predictive accuracy, and physical competence are related but distinct properties.

Figure 1.1 summarizes this expansion from digital tools and simulated agents to real-world single-agent systems, multi-agent collaboration, and infrastructure-level physical AI. Across these physical deployments, the unit of analysis extends from an isolated model to the integrated system of computation, sensing, actuation, communication, and environment. This perspective has important historical precedents. Brooks (1991) demonstrates an approach to robotics in which intelligent behavior arises through direct perception-action coupling, challenging the assumption that autonomous behavior must depend on a centralized symbolic description of the world. Contemporary physical AI extends this longstanding concern with situated action by incorporating learned representations, scalable training, and predictive models.

Core Characteristics

Physical AI operates under constraints that cannot be removed by increasing model capacity alone. Three characteristics are particularly important, namely, consequential interaction under uncertainty, perception of evolving physical states, and the integration of empirical learning with physical structure.

Consequential Interaction Under Uncertainty Physical systems are only partially observable. Object properties, external disturbances, and equipment conditions may be unknown or change during execution. Actions also affect the evidence available to the system. Moving a camera changes visibility, probing a surface reveals compliance, and applying force may alter an object’s configuration. Perception and action consequently form an interdependent process.

Physical interaction can incur recovery costs and irreversible consequences. A simulated episode can often be restarted from a saved state, whereas a spilled chemical, damaged component, or contaminated sample may require substantial intervention or may be impossible to restore. The operational distinction is that physical systems generally lack an exact, inexpensive reset mechanism. Planning must therefore consider the recoverability of future states as well as the likelihood of immediate task success.

Perception of Evolving Physical States Physical AI requires observations that are temporally and spatially meaningful. Vision can describe appearance and geometry, while force, touch, proprioception, temperature, pressure, and other measurements reveal complementary aspects of interaction. A manipulation system, for example, may need to distinguish visual contact from load-bearing contact and a completed motion from a successfully transferred object.

Although sensors typically produce sampled digital measurements, the processes they measure evolve in continuous time and may include discrete events such as impact, grasp release, or equipment switching. Reliable interpretation therefore depends on sampling rates, calibration, synchronization, coordinate frames, and measurement uncertainty. A correct estimate obtained too late may be unusable for control. More generally, computation and physical dynamics must be analyzed together because execution timing can affect system behavior (Lee, 2008).

Learning Informed by Physical Structure Data provide evidence about real operating conditions, including effects that are difficult to model analytically. Physical knowledge provides structure that can restrict implausible predictions and reduce the burden of learning. Conservation relationships, symmetries, geometric constraints, and governing equations can be incorporated through model architectures, training objectives, simulation, or constrained decision procedures. Physics-informed machine learning formalizes several ways of combining observations with mathematical descriptions of physical processes (Karniadakis et al., 2021).

The appropriate balance depends on the task. Explicit models may be effective when governing relationships are sufficiently understood, whereas learned components can represent uncertain interactions or compensate for model mismatch. Physical priors are themselves conditional on assumptions and operating regimes. For example, an inaccurate friction model or an inappropriate rigid-body approximation can produce systematic errors. Physical consistency therefore requires validation within the task’s operating regime, regardless of the model family.

Together, these characteristics shift evaluation toward closed-loop performance. Relevant measures include task completion, robustness to disturbance, response time, resource consumption, and recovery from failure. Prediction accuracy remains valuable, but its practical significance depends on how prediction errors propagate through decisions and physical execution.

Why Physical AI Is Needed and Increasingly Feasible

Physical AI is motivated by tasks that require systems to act on the physical world. Flexible manufacturing requires adaptation to variations in parts and processes. Transportation and energy systems require coordinated responses to changing demand and operating conditions. Scientific experimentation requires procedures to be adjusted in response to measurements. Furthermore, extreme environments, such as deep space and the deep sea, drive the need for local autonomy, as communication constraints and physical hazards severely limit continuous human intervention.

These demands do not imply that established automation is inadequate in every setting. Carefully engineered automation remains effective where tasks and operating conditions can be specified in advance. The additional research challenge is to preserve useful performance when relevant conditions change and exhaustive preprogramming becomes impractical. Physical AI addresses this challenge by integrating interactive learning with perception, inference, prediction, and adaptation.

Several technical developments support this direction. Sensing and local computation provide the practical interfaces for observing and influencing physical processes. Simulation supplies controlled environments for training and evaluation, while methods for transfer from simulation to reality reduce dependence on collecting every training example on physical hardware. For example, Tobin et al. (2017) demonstrate that varying simulated visual conditions can support transfer of a vision model to real robotic tasks. Such results establish a useful transfer mechanism, while leaving its effectiveness dependent on the relationship between training variation and deployment conditions.

Learned physical models provide another foundation. Graph network simulators have demonstrated the ability to represent interacting particles and predict dynamics across fluids, rigid solids, and deformable materials within evaluated settings. These models illustrate how structured learning can support physical prediction beyond a single fixed configuration (Sanchez-Gonzalez et al., 2020). For autonomous action, their potential value lies in evaluating candidate interventions before execution, provided that prediction errors and operating limits are understood.

Large pretrained models also offer reusable semantic knowledge. For example, RT-2 shows that vision-language pretraining can be combined with robot demonstrations to support robotic action and improve generalization in the reported experiments (Zitkovich et al., 2023). This provides evidence that capabilities developed through digital learning can contribute to physical interaction when connected to appropriate action data and interfaces.

These developments make it increasingly feasible to combine digital capabilities, physical models, and control mechanisms to study how systems acquire transferable physical skills while maintaining measurable performance within specified operating conditions.

Physical AI: Definition and Boundaries

A Broad Definition of Physical AI

At its most general, physical AI concerns the capacity of an artificial system to interact with the physical world. This monograph uses physical AI for this artificial-system scope. The term physical intelligence is retained for general capacities, bodily mechanisms, and source-specific usage. The following definition makes the notion precise.

This definition identifies a functional scope rather than a particular architecture. Representations may be explicit, implicit, or distributed across several components. They need not constitute a complete internal simulator, and action selection need not depend on symbolic reasoning. What matters is whether the integrated system can use information about physical conditions to guide and revise its behavior.

Four implications clarify the definition. First, physical grounding requires observations to be connected to relevant quantities, events, or conditions in the world. Second, the system must have a path for intervention, whether through direct actuation or commands to controlled equipment. Third, feedback must inform subsequent behavior, allowing discrepancies between expected and observed outcomes to affect execution. Fourth, claims of intelligent capability should be supported by evidence of task-relevant inference, prediction, planning, or adaptation, rather than inferred merely from the presence of sensors and actuators.

These criteria define a continuum of capability. A fixed feedback regulator and a system that adapts across changing tasks may share essential control principles, yet differ substantially in the breadth of conditions they can handle. The distinction between automation and Physical AI is therefore most useful when expressed through demonstrated capabilities and operating assumptions, rather than an absolute technological boundary.

Terminological care is also necessary because physical intelligence has an established, more specific usage in mechanics and robotics. Sitti (2021) uses the term to describe functions such as sensing, actuation, memory, and computation physically encoded in an agent’s body. The broader definition of Physical AI adopted here includes such contributions while also encompassing computational and distributed systems that interact with physical processes. It should be understood as this chapter’s organizing framework, rather than a claim that the literature has converged on one definition.

Finally, key components should be distinguished from a complete system. A learned simulator, perception model, or action planner may provide an essential Physical AI capability. In isolation, however, it does not satisfy the system-level requirement for closed-loop physical intervention. This distinction allows purely digital modules to be recognized as enabling technologies without attributing autonomous physical competence to them prematurely.

Physical AI vs. Embodied AI

Embodied AI posits that artificial intelligence requires a body to house the system and drive its active sensorimotor relationship with the environment (Pfeifer and Iida, 2004). The concept of embodiment focuses on how this morphology, along with sensing and action capabilities, directly shapes environmental interactions and guides behavior. Importantly, embodiment is by no means limited to humanoid or biologically inspired designs. For physically deployed agents, research further highlights how mechanical structures and material properties can intrinsically support adaptive behavior, thereby reducing the control burden on a centralized controller (Pfeifer et al., 2007).

Nor does embodied AI necessarily require a deployed physical robot. Simulated agents are widely used to study navigation, interaction, and learning under embodied constraints (Shridhar et al., 2021). Habitat, for example, was introduced as a platform for training and evaluating virtual embodied agents in three-dimensional environments (Savva et al., 2019). Embodied AI therefore includes both simulated and physically deployed agents.

Within this chapter’s framework, physically deployed embodied AI constitutes a major realization of physical AI. Its characteristic analytical focus is the agent and its particular sensing and action capabilities. Physical AI adopts an additional system-level perspective, encompassing arrangements in which sensing, decision-making, and actuation are distributed across infrastructure. Figure 1.3 summarizes these distinctions and their overlap.

Consider an adaptive traffic-management system that estimates network conditions, changes signal timings, and evaluates the resulting traffic flows. Its interaction with the world is mediated by distributed detectors, communication links, and signal controllers rather than a single mobile body. Similar arrangements can be considered for factory coordination or power-system operation, provided that the system actually closes the loop between observations, decisions, and physical consequences. These are illustrative applications of the proposed definition, not claims that every traffic controller, factory scheduler, or grid-management platform possesses advanced intelligence.

Scientific equipment provides a concrete research example. Degrave and colleagues demonstrated deep reinforcement learning for magnetic control of plasma configurations in the TCV tokamak. The controller interacts with a physical process through the machine’s measurement and actuation infrastructure (Degrave et al., 2022). Under the present framework, this qualifies as Physical AI even though the system lacks the conventional form of a mobile or manipulative robot.

Such systems are not independent of physical embodiment in the broadest sense, because sensors, actuators, and material processes remain indispensable. Their distinguishing feature is that the physical interface is distributed across an installation or network. Within this monograph’s framework, embodied AI emphasizes how an agent’s body shapes sensing and action, while physical

AI emphasizes closed-loop intervention in real physical processes, including through distributed infrastructure. The two overlap in physically deployed embodied systems.

Physical AI vs. Cyber-Physical Systems and Digital Twins

Physical AI intersects closely with cyber-physical systems (CPS), but the concepts describe different aspects of a system. CPS concerns the integration of computation with physical processes, including the consequences of timing, concurrency, and interaction between software and its physical environment (Lee, 2008). Physical AI concerns the capabilities exercised through that integration, including interpreting conditions, anticipating consequences, choosing interventions, and adapting behavior.

A CPS may operate effectively through fixed control logic without possessing broad inference or adaptation capabilities. Conversely, a physically intelligent system depends on the engineering properties of its computational and physical infrastructure, including dependable sensing, timely communication, and executable control. Physical AI can therefore be understood as a capability layer within many CPS architectures. Describing it as a possible development toward greater autonomy is useful, provided that this does not imply that CPS research excludes advanced cognition or that learning automatically improves an engineered system.

Digital twins occupy a different position. They provide digital representations associated with identifiable physical counterparts, with connections that support updating and comparison between physical and digital states. The literature identifies multiple characteristics and implementation choices, making it important to specify how a proposed twin is connected to its physical counterpart (Jones et al., 2020).

Within a Physical AI architecture, a predictive digital twin can support state estimation, diagnosis, and evaluation of alternative interventions. It may help estimate how a production line would respond to a schedule change or how equipment behavior would change under another operating condition. Such uses require more than a visually accurate replica. The representation must capture the variables and dynamics relevant to the decision.

A digital twin does not necessarily select or execute actions autonomously, and a Physical AI system need not maintain a comprehensive twin. A compact learned dynamics model may be sufficient for a particular control task. Likewise, a generic world model becomes a digital twin only when it is meaningfully associated with a specific physical counterpart and maintained through an appropriate update relationship. Counterfactual predictions remain conditional on the validity of the model and its assumptions.

These distinctions establish complementary roles. CPS describes the integration of computational and physical processes. Embodied AI examines intelligence through situated sensorimotor interaction. Digital twins provide representations tied to physical counterparts. Physical AI, in the framework developed here, identifies the capacity to combine perception, physical understanding, decision-making, and feedback into effective intervention. This system-level perspective provides the basis for the architectural elements discussed in the following sections.

Core Elements of Physical AI

Definition 1.1 characterizes Physical AI as the capacity to perceive physical states, maintain task-relevant representations, and execute interventions through feedback. In practice, this capacity takes the form of a persistent closed loop. Information processing and decision-making must be translated into physical actions, and subsequent behavior must be revised according to observed outcomes. Physical AI therefore operates through a cycle of perception, modeling, prediction,

decision-making, execution, and feedback, rather than through a one-shot mapping from input to output.

As § 1.1.2 emphasizes, Physical AI is not synonymous with robotic intelligence. Whether a system exhibits it depends on its ability to close the loop between perception and action, not on its morphology. The elements discussed in this section apply equally to autonomous vehicles, industrial equipment, automated experimentation platforms, intelligent buildings, and energy infrastructures. In such systems, perception, computation, decision-making, and execution are often distributed across devices, edge nodes, cloud platforms, and physical facilities rather than concentrated within a single device.

Operating in a continuously changing and uncertain world also means that actions are constrained by dynamics, geometry, time, energy, materials, and safety requirements, and that their consequences are not always reversible, as § 1.1.1 discusses. A physically intelligent system must therefore answer not only “What should be done?” but also “Can it be done under the current conditions?”, “How can it be done safely?”, and “Does the execution produce the intended outcome?” Organized around these questions, Physical AI comprises four closely related elements, including physical state perception and estimation, physical world models, physical decision-making together with planning and control, and closed-loop feedback and multiscale adaptation. They correspond respectively to perception, representation, intervention, and feedback in the definition. The first three together constitute the physically intelligent agent, while the fourth connects the agent with the physical world into a continuously running loop. The relations among these elements, together with the data, hardware, and physical world on which they depend, are illustrated in Figure 1.4.

Physical State Perception and Estimation

Physical AI begins with information about the environment, task-relevant objects, and the system itself. Vision conveys identity, pose, geometry, and motion. Force and tactile sensing reveal contact, friction, and material properties (Dahiya et al., 2010), while acoustic sensing detects collisions and vibrations. Proprioception measures the system’s own configuration and load, and industrial systems add process variables such as temperature, pressure, and current. Chapter 9 treats the sensors themselves. The concern here is turning their signals into an estimate of physical state.

Measurements are incomplete, noisy, delayed, and often misaligned across modalities, so calibration, multimodal fusion, and state estimation are needed to obtain a temporally consistent state with explicit uncertainty (Kalman, 1960a; Baltrušaitis et al., 2019). Perception also supports active information acquisition. When observations are insufficient for a decision, the system can change its viewpoint, reposition sensors, or probe objects to estimate mass, stiffness, or friction, folding information acquisition into the decision loop (Bajcsy, 1988).

Physical World Models: Representation and Prediction

State estimates are organized into a world model, which describes the environment, the actions available to the system, and how physical states evolve over time (Ha and Schmidhuber, 2018; LeCun, 2022). It has two interdependent aspects. Representation captures what the model holds about the current world, and prediction anticipates what the world becomes after a given action.

Physical representations cover objects, poses, geometry, and spatial relations, together with velocities, masses, contacts, forces, material properties, affordances, and task constraints. A cup is at once a shape with a pose and an object that can be grasped, holds liquid, and spills when tilted. Representations may be explicit (maps, object lists, equations of motion, relational graphs), implicit (learned from data), or hybrid, combining physical priors with learning to retain interpretability while correcting for unmodeled effects. An effective world model is updated with each new observation and distinguishes directly observed, estimated, and unobserved states.

Prediction estimates the consequences of a particular action, covering trajectories, contact outcomes, equipment conditions, task success, and safety hazards over time scales from milliseconds to hours. Predictions may come from physical equations and simulators (Raissi et al., 2019) or from transition patterns learned from data (Sanchez-Gonzalez et al., 2020). Practical systems combine the two and revise predictions and their uncertainty online. In scenario simulation, the decision process submits candidate actions and the world model returns predicted outcomes with their uncertainty, enabling counterfactual comparison before anything is executed. This simulation runs online, unlike the physics engines and generative environments of Chapter 4, which are used offline to produce training data and test policies.

Physical Decision-Making, Planning, and Control

Physical decision-making selects actions from task objectives, the current state, predicted outcomes, and real-world constraints, usually through a hierarchy (Kaelbling and Lozano-Pérez, 2011; Sutton et al., 1999). At the top, the system sets goals and constraints and, when several physical agents share a task, allocates and coordinates subtasks among them. Foundation models for robot manipulation reason about the task and scene, decompose goals into executable steps, and produce action policies. Foundation models for whole-body motor control provide reusable feedback policies for executing movement goals and correcting deviations (Zeng et al., 2025; 2026). The levels differ mainly in time scale, from minutes or hours for task allocation to seconds for policies and milliseconds for motor commands. The boundary separating task-level reasoning and action-policy generation from motor control and feedback correction varies across systems.

Execution takes many forms, from joints and end effectors to steering and braking, instrument scheduling, or valves, pumps, and switches. An action is thus best understood as an observable intervention that changes a physical object, environment, or process. Such interventions must be feasible, safe, and timely. A logically valid plan can still fail because of collisions, actuator limits, or latency, so safety checks precede execution, and monitoring and emergency stops stay active throughout.

Closed-Loop Feedback and Multiscale Adaptation

Physical AI revises behavior according to the observed consequences of action. After each command, the system perceives again, checks whether the intended result is achieved, and updates its state estimate together with its world model and plan. When outcomes deviate from predictions, it identifies the cause and compensates, retries, replans, or terminates safely.

The loop resides in no single module but in the connections among the elements. Perception supplies the state, the world model answers action queries from the decision process, commands act on the world through actuators, and sensors return the consequences to perception Figure 1.4. The loop thus closes through the physical world rather than inside the agent. Fast reactive control, such as tightening a grip when an object begins to slip, may bypass the world model and respond directly to the current state.

Feedback operates at three nested scales. The fastest loop connects motor control and feedback correction with the actuators and sensors, adjusting position, velocity, or force in real time. The task-level loop connects task-level reasoning and action-policy generation with the world model and revises plans as the task progresses. The slowest passes through the offline data and simulation infrastructure, where deployment data drive continual learning (Parisi et al., 2019) and form a data flywheel. In multi-agent and infrastructure-scale systems these loops may be distributed across devices that coordinate toward a common objective, and humans may join them by providing goals, demonstrations, supervision, and intervention.

Physical AI therefore lies not in producing a plausible action or plan, but in sustaining this cycle of observation, prediction, action, verification, and revision under real physical constraints, whether the system is a single robot or distributed infrastructure.

The Role of Physical AI in the Evolution of AI

Section 1.1 contrasts digital intelligence, whose objects of computation and evaluation are recorded content, with the Physical AI defined in Definition 1.1. This subsection develops three claims about what that contrast means for the trajectory of AI. Physical AI is the stage at which artificial intelligence moves from understanding and generating digital information to perceiving, predicting, and changing physical processes. It builds on digital intelligence rather than replacing it and provides one of the important paths toward artificial general intelligence (AGI).

New Capabilities Beyond Digital Intelligence The first extension concerns what the system perceives. Digital intelligence works on symbols and pixels that come from prior collection, curation, and encoding. Physical AI deals with physical states directly (Section 1.1), including the pose and velocity of objects, forces and friction at contact surfaces, pressure and temperature in fluids, and the composition and structure of materials. These states are continuous and evolve over time. They can only be partially observed through noisy sensors, and vision, touch, force, and other modalities must be fused within a common spatiotemporal frame. The object of perception thus shifts from previously recorded content to the current state of a world that keeps changing.

The second extension concerns what the system produces. Because physical systems lack an inexpensive reset (Section 1.1), a grasp, an injection, or a turn of the steering wheel changes the subsequent state of the system and often cannot be undone. Physical AI must therefore predict, before acting, how the world will evolve if a given action is taken. This prediction is interventional in nature and depends on causal relations and dynamics, not merely on statistical correlation (Schölkopf et al., 2021).

Building on Digital Intelligence These new capabilities do not make Physical AI a replacement for digital intelligence. The common-sense and domain knowledge accumulated in foundation models is the basis on which Physical AI understands and plans tasks (Ichter et al., 2023). In turn, Physical AI grounds that knowledge in measurable changes of state and tests it through real interaction. A large language model can search the literature, propose a hypothesis, and write an experimental protocol, but whether the protocol succeeds depends on accurate pipetting, stable temperature control, and correct sample handling. Only when digital reasoning is closed with physical execution can AI take part in the full process from posing a question and running an experiment to drawing a conclusion.

Section 1.1 explains that Physical AI enlarges the unit of analysis from the model to the system, and this enlargement has concrete consequences for how capability is attributed. The same policy may perform very differently on robot arms of different precision or in workshops with different lighting, and task success also depends on engineering conditions such as sensor calibration, fixture design, and safety interlocks. The shift also changes how general capability should be understood and measured, which leads to the third claim.

A Path Toward AGI Benchmarks in digital environments are prone to overfitting, and it is hard to rule out memorization caused by contamination of the training data. The physical world is open, continuous, and impossible to enumerate in advance, and whether a task has been completed is decided directly by physical outcomes. Moravec’s paradox points in the same direction, highlighting that perceptual and motor skills that humans find effortless are among the hardest for machines to acquire (Moravec, 1988). Physical AI therefore provides a demanding test of generality along four axes, including generalization to new objects and scenes as well as new task combinations, transfer across embodiments and tasks as well as from simulation to reality (Tobin et al., 2017; OpenAI et al., 2020), robustness to disturbances and sensor degradation as well as environmental drift, and long-term autonomy over days or months with little human intervention. Because mistakes in the physical world are often irreversible, evaluation cannot rest on task success rate alone. Real-time performance, sample efficiency, energy consumption, risk controllability, and recoverability after failure must be considered together, and they frequently trade off against one another. Physical AI is not the only such path, as mathematical reasoning, scientific discovery, and social interaction advance general capability from other directions, but it is the one in which capability is checked against physical reality.

Locating Physical AI on the Trajectory of AI Taking the degree of coupling between AI and the physical world as the axis, the position of Physical AI can be stated more concretely. At the loosely coupled end, AI acts as a digital tool that assists humans with design, analysis, and decision-making but leaves execution entirely to them. Next, simulated agents perceive and act in simulators or digital twins (Jones et al., 2020), where trial and error is cheap and safe but limited by the gap between simulation and reality. Physical AI proper begins when AI closes the loop in the real world. It starts with single-agent systems, such as a robot, vehicle, drone, or autonomous laboratory platform, where most current embodied intelligence research is concentrated. It then extends to multi-agent collaboration, in which groups of agents accomplish tasks that no single agent can, and finally to infrastructure-level Physical AI, in which distributed sensing, computing, and actuation resources embedded in cities, power grids, factories, and research facilities are scheduled and governed through common protocols. Section 1.5.6 returns to these three stages as the recurring physical compositions of deployed systems. In a network of autonomous laboratories spread across institutions, for example, the question is no longer only whether an experiment can be completed, but also how resources are allocated, how processes are audited, and how data ownership and reproducibility are established. These stages coexist and support one another rather than replacing each other. Digital tools and simulation continue to supply knowledge, data, and safety validation to physical systems. Physical AI thus occupies the tightly coupled end of this axis, where the value of AI is realized as a change in physical processes rather than only as the production of information. The application landscape in Section 1.5 makes this coupling concrete across industries, and Chapter 10 examines Physical AGI as a long-term research goal.

The Application Landscape of Physical AI

Physical AI connects the interpretation of physical state with decisions that change it, and its applications span individual robots, autonomous vehicles, instrument networks, factories, cities, and scientific facilities. These settings differ in scale by orders of magnitude, yet the questions that determine whether a system is useful remain largely the same, concerning what the task requires, what the environment allows, what resources are available, and which decisions may be taken without a human in the loop. Whether a particular configuration amounts to a deployable system is ultimately reflected in the cost of the valid work it produces. The subsections that follow first state the closed loop that every application instantiates, together with the four dimensions along which applications are described, namely task, environment, resources, and autonomy. They then survey four domains, from industrial operation through everyday services and public infrastructure to scientific inquiry, and finally extract the system archetypes that recur across those domains.

The Closed Loop and the Distinction from Conventional Automation

Every application of Physical AI instantiates one closed loop as shown in Figure 1.5. Physical perception acquires signals from the environment through sensors. State understanding assembles these signals into estimates of the objects, agents, and processes present, together with the representations and world models of how they evolve. Prediction and imagination extrapolate the estimated state forward and evaluate candidate actions, whether through learned dynamics or explicit physical simulation. Decision making and planning select and sequence actions under task and process constraints. Physical action executes the selected behavior through control and actuation, and the resulting environmental feedback becomes the input to the next cycle of perception. These stages give Definition 1.1 its operational form. Section 1.3 groups them into four elements comprising physical state perception and estimation, world modeling for representation and prediction, decision-making with planning and control, and closed-loop feedback with multiscale adaptation. The loop as a whole, rather than any single stage, defines an application. A system that perceives without acting merely observes its environment, and one that acts without state understanding merely executes a prescribed motion.

Conventional automation already incorporates feedback control, so the loop itself is not new (Åström and Murray, 2008). What distinguishes Physical AI is how the loop is sustained when conditions depart from those anticipated at design time, through adapting to changing conditions, interpreting unfamiliar tasks, and transferring knowledge across operations. Classical control holds a process near a setpoint under modeled disturbances. Physical AI re-estimates state, re-plans, and recovers when the task or the environment changes. The distinction is graded rather than binary. Applications occupy a spectrum from fixed procedures with local adaptation, such as an industrial manipulator re-grasping a shifted part, to open-ended task interpretation, in which a system must infer what is wanted from instruction, demonstration, or context (Ichter et al., 2023).

The position of an application on this spectrum is determined by the task rather than by the industry in which it appears. An application can be characterized along four dimensions, namely task, environment, resources, and autonomy. Resources include sensing, actuation, computation, energy, and human support. Autonomy specifies which decisions are delegated and when human participation is required (Parasuraman et al., 2000). These dimensions distinguish requirements that industry categories tend to merge. Within one factory, moving sealed containers and manipulating deformable materials impose different demands on contact control and recovery, and a laboratory may coordinate fixed instruments without a single mobile body. System composition and decision authority therefore require separate descriptions.

From Automation to Autonomous Industrial Operation

Industrial production, logistics, and agriculture involve transportation, manipulation, inspection, and process regulation. Physical AI offers value when variability makes fixed procedures costly to maintain. Parts arrive in unfamiliar configurations, crops vary in maturity, or equipment behavior changes with wear. The underlying pattern is common to these sectors. Fixed, engineered procedures are replaced by state estimation and adapted planning, while the process constraints within which the adaptation must operate remain fixed.

Flexible Manufacturing and Lights-Out Production In an adaptive production system, visual and tactile observations update estimates of object and process state, while planning and control adjust operations within process constraints (Lee et al., 2015; Monostori et al., 2016). Variability enters as small batches, frequent product changeovers, and parts presented in unfixtured configurations, and the value is realized when changeover no longer requires re-engineering (Van Brussel et al., 1998). The benefit is measured by product quality, changeover time, resource consumption, and the effort of setting up a new task, while task variability and the resources required for changeover are the binding constraints. The limit of this development is the lights-out factory, in which no standing human presence remains on the floor. Lights-out operation does not eliminate human work. It moves that work to exception handling, supervision, and maintenance, and the cost of these activities becomes a primary economic variable rather than an overhead (Bainbridge, 1983).

Warehousing, Logistics, and Supply-Chain Orchestration Warehouses and distribution networks concentrate transportation and manipulation at high repetition rates, including automated storage and retrieval, mobile-robot fleets, sortation, palletizing and depalletizing, and parcel handling (Wurman et al., 2008). The distinguishing problem is not the execution of any single motion but orchestration, which involves allocating physical work across many machines as demand fluctuates, replanning after disruption, and keeping physical flows consistent with the inventory records and delivery commitments carried by information systems (Gerkey and Matarić, 2004). Local motion competence and network-level coordination are separate achievements. An operation can automate the former while the latter remains manual, and the gap between the two is closed in practice either by consistent state estimates or by buffer stock.

Agriculture, Forestry, Fishery, and Animal Husbandry Agricultural settings concentrate the constraints of Physical AI, including biological variability, unstructured outdoor terrain, weather-dependent timing, and long deployment cycles. Operations under development span the production sequence, including inspection and patrol, seeding and planting, mechanical weeding, fertilization, plant protection, harvesting, and grading and sorting of produce (Bac et al., 2014), together with livestock monitoring and aquaculture operations (Berckmans, 2017; Forel et al., 2018). Value appears in two forms, namely precision, in which seed, chemical, and water inputs are reduced by acting at the resolution of individual plants or animals, and timeliness, in which operations are performed within short agronomic windows. The acceptance criteria are correspondingly biological and seasonal rather than volumetric, such as fruit harvested without bruising at the correct maturity, weeding achieved with a specified reduction in chemical use, or illness detected early enough for intervention.

Energy, Mining, and Resource Systems Energy and mineral systems are high-risk and large-scale, including generation equipment, transmission corridors, pipelines, refineries, and open-pit and underground mines. Applications combine inspection of assets (such as turbine blades, transmission lines, and pipe corridors) (Nguyen et al., 2018), autonomous haulage and drilling (Wang and Huang, 2017; Marshall et al., 2008), and process regulation distributed across many devices. Equipment behavior changes with wear, which motivates continuously updated estimates of degradation in place of fixed maintenance schedules (Jardine et al., 2006). Limited access and the high consequence of failure shift the economics toward predictive operation and remote supervision. In these systems, safety cases rather than throughput determine which functions may be automated, and the allocation of authority between site automation and remote operation centers is itself a design decision.

Daily Services: From Dedicated Equipment to Embodied Companions

Service applications place Physical AI in environments that are built for people rather than for machines. At one end are dedicated devices (single-purpose appliances that perform one task under human setup), and at the other are personalized embodied companions intended to share unstructured space with untrained users. Consumer-grade embodied products, from desktop companions to inexpensive legged and humanoid platforms, make this trajectory visible at retail prices, although the distance between demonstrations of interactive behavior and dependable household service remains substantial (Srinivasa et al., 2012).

Human participation is an explicit design variable in these applications. Supervision, shared control, and selective delegation assign different responsibilities to users, operators, and machines. In human-facing services, decision authority and timely assistance matter alongside completion rates, so evaluation includes service quality, accessibility, and the burden placed on users and caregivers (Hoffman, 2019; Nikolaidis et al., 2016).

Home and Community Services A household assistant must accommodate changing layouts and object locations, because neither residents nor their belongings remain arranged to suit a stored map (Wisspeintner et al., 2009). Cleaning, tidying, laundry, and meal preparation combine manipulation with continual re-estimation of where things are and what state they are in. Community services extend the same capabilities into semi-public space, including delivery to doors and lockers, patrol and monitoring, and check-ins on residents who live alone. In these settings, acceptance is determined not by peak capability but by the frequency and manner in which the system requests help. If a robot completes 90% of its tasks unaided but leaves the remaining 10% unfinished in mid-task, it shifts the burden to its users rather than removing it (Forlizzi and DiSalvo, 2006).

Medical Rehabilitation and Eldercare A rehabilitation system must respect clinically established limits on force, range, and intensity while still challenging the patient enough for therapeutic effect. Applications include shared-control therapy devices that adjust assistance to measured performance, hospital logistics and delivery, continuous monitoring of activity, and assistance with activities of daily living (Kyrarini et al., 2021). Clinical deployment adds requirements that are foreign to industrial settings. Interventions must demonstrate safety and efficacy under the conventions of clinical study (Lo et al., 2010; Esquenazi et al., 2012), responsibility for decisions must be traceable to accountable parties, and eldercare in particular must weigh privacy and dignity alongside capability (Broadbent et al., 2009). In this domain, Physical AI cannot be separated from its regulatory and evidentiary context.

Commercial and Professional Services Retail, hospitality, professional cleaning, and food preparation occupy a middle ground, with spaces more structured than homes, because they are designed around service processes, yet occupied by the public and by staff whose cooperation cannot be assumed. Tasks include shelf monitoring and restocking, floor care, reception and guidance, and food handling to hygienic standards. Interaction with customers and employees is part of the task itself rather than a distraction from it, which places interface design and predictable, legible behavior on the critical path alongside manipulation competence. The economics are those of semi-structured spaces. Repetition is higher than in domestic service, while the tolerance for bespoke installation is lower than in industry.

Public Infrastructure: From Single-Vehicle Autonomy to City-Scale Coordination

Transportation, energy networks, and public infrastructure extend the concerns of the preceding sections across larger spatial and temporal scales. The unit of analysis shifts from one machine to populations of devices whose local actions must combine into system-level outcomes.

Intelligent Transportation Vehicles and drones combine local perception and control with routing and coordination (Yurtsever et al., 2020; Murray and Chu, 2015). For road vehicles, the difficult problems are increasingly systemic rather than vehicular, including mixed traffic in which autonomous and human drivers share conventions, fleet scheduling together with charging and cleaning for shared robotaxi operation, and the interfaces between vehicle autonomy and traffic management (Stern et al., 2018; Varaiya, 2013). For low-altitude aircraft, coordination is the founding problem. Airspace organization, geofencing, and conflict resolution for large numbers of small vehicles must exist before individual autonomy is useful at scale (Prevot et al., 2016). In both cases, the binding constraint is the coordination of many local planners toward network-level service quality.

Urban Operations and Public Facilities Buildings, power systems, and water networks distribute sensing and actuation across many devices with no mobile body at all, an arrangement treated as a distinct system archetype in § 1.1.5.6. Model predictive control of building climate, demand response across a power network, and leakage detection in a water distribution network illustrate the range (Afram and Janabi-Sharifi, 2014; Siano, 2014; Chan et al., 2018). Communication capacity and consistent state estimates constrain coordination between local actions and system-wide outcomes. Value is measured by the reliability and continuity of service, energy and resource efficiency, and responsiveness to faults, observed over the scales of cities and seasons, which places a premium on sustained and validated operation rather than on peak performance.

Disaster Response and Public Safety Extreme environments impose additional restrictions. Disaster response combines difficult access, limited energy, and delayed or unavailable human intervention, making recovery mechanisms and bounded local autonomy particularly important. Search and rescue in collapsed structures, firefighting and hazardous-material intervention, and post-disaster inspection all require systems that decide and recover locally within a pre-authorized envelope and degrade safely when communication fails (Murphy, 2014; Delmerico et al., 2019; Tranzatto et al., 2022). Public-safety applications sharpen the question of authority more than any other setting in this chapter. The conditions under which a machine may act, and the parties who are answerable for its actions, must be specified before capability is deployed rather than after.

Scientific Inquiry: From Automated Instruments to Autonomous Laboratories

Scientific applications use physical action to acquire evidence and guide inquiry. An autonomous laboratory selects experiments, operates instruments, measures outcomes, and revises subsequent choices (King et al., 2009; Burger et al., 2020; MacLeod et al., 2020; Dai et al., 2024). Scientific value depends on measurement validity, sample traceability, and reproducible interpretation, alongside experimental throughput.

Automated and Self-Driven Laboratories The A-Lab illustrates this organization within a defined inorganic-materials synthesis workflow, integrating robotic operations, computational information, automated characterization, and active learning (Szymanski et al., 2023). Its subsequent correction underscores the need to validate material identification and novelty claims separately from successful workflow execution (Szymanski et al., 2026). Workflow autonomy (the reliable execution of planned operations) and epistemic autonomy (the production of claims that survive scrutiny) are different achievements, and evaluations in this field tend to measure the former while the value lies chiefly in the latter. Self-driven laboratory programs accordingly treat the validity chain from sample to published claim as part of the system to be engineered.

Physical Simulation and Digital Twins In scientific inquiry, simulation and digital twins, which § 1.1.2.3 distinguishes from generic world models, serve as instruments for experiment design, parameter estimation, and controlled testing. Validation takes two forms. For policy training, it concerns whether simulated variation supports performance in reality. For an operational digital twin, it concerns the accuracy, timeliness, and uncertainty of the estimates used in decisions. The two uses impose different obligations. A training environment must cover the variations a policy will meet, while an operational twin must declare how far ahead, and with what confidence, its estimates may be trusted.

Human-Machine Collaborative Scientific Cognition Autonomous laboratories change the division of cognitive labor in science rather than removing the scientist. In deployed systems, objective setting, prioritization among hypotheses, and the acceptance of evidence remain human responsibilities, while machines contribute breadth of search, uniformity of execution, and complete procedural memory. The interpretability of proposed experiments conditions trust. A suggestion whose rationale can be examined can also be disputed, and such dispute is one of the mechanisms by which science corrects itself. The productive design question, familiar from § 1.1.5.1, is which parts of inquiry to delegate. Within a research program, the answer to this question is itself something to be established experimentally.

Aerospace and Space Exploration Space exploration combines scientific objectives with the most constrained communication, energy, and mass budgets in this chapter. Planetary rovers schedule a sol of activity within command latency (Jet Propulsion Laboratory, 2021), on-orbit systems service and assemble spacecraft, and in-situ instruments analyze samples that cannot be returned. Underwater operations share the pattern of difficult access, limited energy, and delayed or absent intervention (Wynn et al., 2014). Both settings inherit bounded local autonomy as their organizing principle of deciding locally within a pre-authorized envelope, recording faithfully, and deferring the remainder to later analysis. This is the same principle identified for disaster response in § 1.1.5.4, applied where the return of data is slower still.

System Archetypes across Industries

Beneath the domain labels above, deployed systems assume a small number of recurring physical compositions, together with two modes that cut across them. Three compositions locate sensing and actuation together with computation in one body, in many bodies, or in the facility itself, while the two modes place either a human operator or a simulated world inside the loop. These compositions retrace the progression from single-agent systems through multi-agent collaboration to infrastructure-level Physical AI described in § 1.1.4. Taken together, the five archetypes span the domains surveyed above, each of which can be read through the four dimensions of § 1.1.5.1. Figure 1.5 shows both views together.

Single Embodied Agents The first composition concentrates sensing, actuation, computation, and energy in a single mobile body, as in the household assistant (Fu et al., 2025), the inspection quadruped (Miki et al., 2022), the planetary rover, the delivery drone (Loquercio et al., 2021). Its defining property is that resources and autonomy are co-located and bounded by one morphology, which in turn fixes the reachable task set. Most public attention attaches to this archetype, although the surveys above show it to be one composition among three.

Multi-Agent and Swarm Systems The second composition distributes work across many cooperating bodies, as in warehouse fleets, agricultural machine teams, drone formations, cooperative transport (Dorigo et al., 2021). Capability emerges from task allocation, communication, and consensus rather than from any single member, and the fleet thereby acquires properties that its individual members lack, such as tolerance of the failure of one member (Werfel et al., 2014; Tolstaya et al., 2020; Zhu et al., 2024). Assessment correspondingly shifts to fleet-level quantities such as throughput, availability, and the reliability of the coordination mechanism itself.

Environment-Scale Installations The third composition embeds sensing and actuation in the facility itself, as in automated warehouses, lights-out factories, autonomous laboratories (Gao et al., 2025), intelligent buildings, instrumented grids. No mobile body need exist. The laboratory described in § 1.1.5.1, which coordinates fixed instruments without any mobile body, belongs to this composition. Engineering attention shifts from the agility of a body to the interfaces of a facility, including how materials, information, and people enter and leave, and the system boundary becomes an explicit design decision rather than the surface of a robot.

Capability-Augmenting Tools and Collaborative Systems The first of the two cross-cutting modes extends a single person’s capabilities through intelligent hand tools, exoskeletons, augmented-reality task guidance, and collaborative robot cells (de Looze et al., 2016; Matheson et al., 2019). The human retains decision making, while Physical AI supplies perception, guidance, and force at the point of work. Autonomy is deliberately low, and integration with the human is the design center rather than a constraint. Because responsibility remains with the operator, this archetype provides the economic baseline against which more autonomous alternatives are judged (International Organization for Standardization, 2025).

Virtual-Real Interaction Systems The other cross-cutting mode spans physical and simulated environments through sim-to-real policy training (Tobin et al., 2017), operational digital twins, teleoperation with haptic and mixed-reality interfaces (Hokayem and Spong, 2006), and hardware-in-the-loop test benches. The virtual channel may support development, remain part of operation, or serve both roles. Action may originate from a human operator, a learned policy, or a simulator, and its physical effect depends on how the resulting command is executed. The validation questions of § 1.1.5.5, concerning whether simulated variation covers reality and how far estimates may be trusted, thereby become structural properties of the system rather than external checks.

The archetypes compose rather than compete with one another. An autonomous laboratory, for example, is an environment-scale installation that contains single agents, is trained through virtual-real interaction, is maintained by humans with augmenting tools, and is coordinated as a multi-agent system when replicated across sites. The physical composition fixes much of the resource profile of an application before any algorithm is selected, and the two modes then determine where authority resides and how much of the loop runs in simulation. For this reason, the four dimensions of § 1.1.5.1 are best assessed per archetype rather than per industry.

Summary and Organization of the Monograph

This chapter introduces Physical AI as a system-level capacity to perceive physical states, maintain task-relevant representations of physical relationships and action consequences, and select and execute interventions through feedback under physical, operational, and computational constraints. It contrasts this capacity with digital intelligence, makes the contrast precise in Definition 1.1, and marks its boundaries against embodied AI, cyber-physical systems, and digital twins. It then decomposes the capacity into four interdependent elements (§ 1.1.3), locates it on the axis of coupling between AI and the physical world (§ 1.1.4), and shows how deployed applications instantiate one closed loop under four assessment dimensions, across three physical compositions and two cross-cutting modes (§ 1.1.5). A single criterion runs through these perspectives. What a system can do is determined less by any individual model than by the integrated loop of perception, prediction, decision, action, and feedback, and by the physical and operational conditions under which the loop is sustained.

The remainder of this monograph develops that loop. Chapter 2 assembles the technical foundations shared by all four elements, including mathematical descriptions of the physical world, control theory and autonomous systems, embodied cognition, and learning and reasoning. The chapters that follow then take up the elements in order. Chapter 3 covers perception and state estimation. Chapters 4 and 5 cover physical data, simulation environments, and world models as resources that support prediction. Chapters 6 and 7 cover the foundation models and motor-control systems that realize decision-making and execution. Chapter 8 develops the definition, organization, task execution, and coordination of physical agents. Chapter 9 covers the hardware and systems that must sustain the loop reliably over long-term operation. The final chapters return to the questions left open here concerning how far the path toward general Physical AI extends (Chapter 10), how capabilities are evaluated and governed (Chapter 11), and what development strategies and longer-term outlook follow from them (Chapter 12).

The definitions and distinctions established in this chapter provide the vocabulary for all that follows. Whether a system amounts to Physical AI, where it sits between fixed automation and open-ended autonomy, and how much of the loop runs in simulation or under human authority are judged not by stated architecture but by demonstrated capability under specified conditions.

Technical Foundations

Building on the framework introduced in Chapter 1, this chapter first establishes foundations of Physical AI, including the mathematical modeling, control theory, cognitive AI, and usage of machine learning. These four points are organized around the closed physical interaction loop: observations inform state estimates, models and goals guide action, physical contact generates feedback, and learning updates behavior. Particularly, Section 2.1 introduces mathematical descriptions of geometry, dynamics, state, and uncertainty, which provide a shared language for later discussion. Section 2.2 explains how the core feedback control is used to build a robust and safe embodied system. On top of the control theory, Section 2.3 examines how to enable embodiment with perception, skill, and physical grounding of language and multimodal representations. Finally, Section 2.4 moves towards the modern machine learning-based methods. This closed-loop formulation provides the shared theoretical backbone for the perception, modeling, control, hardware, and safety chapters that follow.

Mathematical Descriptions of the Physical World

Physical AI requires an internal representation of the physical world that enables an agent to perceive, predict, and interact with its environment. Unlike purely symbolic systems, embodied agents must reason about continuous geometry, physical dynamics, uncertainty, and interaction constraints. Mathematical descriptions provide the foundation for this process by defining how physical entities are represented, how their states evolve over time, and how observations relate to underlying physical reality.

Classical robotics has developed rigorous mathematical formulations for geometry, motion, and dynamics, while modern embodied intelligence extends these concepts toward learned representations of complex environments. Together, geometric representations, physical dynamics, and predictive state descriptions form the foundation for physical world modeling.

Geometry and Spatial Representation

Euclidean Space and Rigid-Body Transformations The physical world is naturally described in three-dimensional (3D) Euclidean space. The position of an object can be represented as a point $\mathbf{x}_{\mathrm{pos}} \in\mathbb{R}^{3}$, while the complete pose of an object requires both translation and rotation.

A rigid transformation is represented as

where $\mathbf{R} \in\mathrm{SO}(3)$ is a rotation matrix in the special orthogonal group in three dimensions and $\mathbf{t} \in\mathbb{R}^{3}$ represents translation. The rotation matrix satisfies

Although rotation matrices provide an intuitive geometric interpretation, they contain redundant parameters. Unit quaternions provide a compact representation of orientation and avoid singularities associated with Euler angles.

For robotics and embodied systems, rigid transformations in the special Euclidean group $\mathrm{SE}(3)$ are commonly represented as

The corresponding Lie algebra $\mathfrak{se}(3)$ provides a local representation of transformations and is widely used in state estimation, simultaneous localization and mapping (SLAM), and motion optimization (Solà, 2017; Forster et al., 2017).

Explicit Geometric Representations A physical environment can be represented using explicit geometric structures that directly describe spatial entities. Point clouds represent surfaces through discrete sample sets of the form

Point clouds preserve metric geometry and naturally arise from sensors such as light detection and ranging (LiDAR) devices and depth cameras. They are widely used for localization, mapping, manipulation, and 3D scene understanding.

Mesh representations describe surfaces using vertices, edges, and faces. Compared with point clouds, meshes explicitly encode surface connectivity and are therefore important for simulation, collision detection, and physical interaction.

Implicit Representations of Physical Environments Instead of explicitly storing geometric elements, implicit representations describe the environment through continuous functions.

A signed distance function (SDF) represents geometry as the zero-level set of a continuous function:

SDFs provide useful geometric properties such as surface distance and occupancy, making them suitable for collision checking, motion planning, and reconstruction.

Neural implicit representations extend this idea by parameterizing continuous scene functions with neural networks. Neural radiance fields (NeRFs) represent a scene through a continuous mapping from spatial coordinates and viewing directions to density and appearance as

where $\sigma_{\mathrm{rad}}$ represents density and $\mathbf{c}_{\mathrm{rgb}}$ represents color. Such representations enable high-quality reconstruction and novel-view synthesis, providing a foundation for simulation-ready environments (Mildenhall et al., 2020).

Topology and Connectivity Geometric representations describe where objects exist, but embodied agents must additionally understand how spaces and objects are connected. Topological representations abstract environments into nodes and edges, where nodes represent meaningful states or locations and edges represent feasible transitions.

For navigation, topology captures reachability and connectivity of environments. For manipulation, object topology describes structural relationships among parts and possible interaction paths. Combining geometry and topology allows agents to reason not only about spatial configuration but also about possible actions.

Physical Dynamics and Interaction Modeling

Kinematics Kinematics describes how the configuration of a physical system changes without considering the forces causing the motion. For articulated robots, forward kinematics describes the relationship between joint space and task space as

where $\mathbf{q}$ represents joint variables and $\mathbf{T} \in\mathrm{SE}(3)$ represents the end-effector pose.

The inverse kinematics problem estimates joint configurations required to achieve a desired pose:

where $f^{-1}(\{\mathbf{T}\})$ denotes the set of joint configurations that reach the desired pose.

Inverse kinematics is fundamental for manipulation, locomotion, and motion planning, but can be challenging due to redundancy, constraints, and multiple feasible solutions (Khalil and Dombre, 2004).

Dynamics and Equations of Motion Dynamics describes how physical states evolve under forces and control inputs. For articulated systems, robot dynamics can be expressed as

where $\mathbf{M}$ represents inertia, $\mathbf{C}$ represents velocity-dependent effects, $\mathbf{g}$ represents gravity, and $\boldsymbol{\tau}$ represents actuator torque. The external wrench $\mathbf{w}_{\mathrm{ext}} \in\mathbb{R}^{6}$ stacks force and moment. The geometric Jacobian $\mathbf{J}(\mathbf{q}) \in\mathbb{R}^{6 \times n}$ maps the $n$ joint rates to linear and angular velocity at the wrench reference point, expressed in the same coordinate frame.

The Newton-Euler formulation provides an efficient recursive computation of rigid-body dynamics, while the Lagrangian formulation derives motion equations from energy functions:

These formulations provide the mathematical basis for simulation, control, and physical reasoning (Featherstone, 2008).

Contact Mechanics and Interaction Physical AI requires reasoning about interaction between objects. Unlike free-space motion, contact introduces constraints involving collision, friction, and force transmission.

The Coulomb friction model relates tangential and normal forces through

where $\mu$ denotes the friction coefficient.

Contact dynamics are often formulated as constrained optimization problems involving non-penetration constraints and friction cones. Accurate contact modeling is essential for grasping, manipulation, locomotion, and tool use (Posa et al., 2014; Stewart and Trinkle, 1996).

State Representation and Predictive Modeling

State-Space Representation A physical system is commonly described through a state variable $\mathbf{x}_{t}$ that summarizes the information necessary to predict future evolution. The state transition process can be represented as

where $\mathbf{u}_{t}$ represents actions and $\mathbf{w}_{t}$ represents disturbances.

The observation process connects physical states with sensory measurements through

This formulation provides the mathematical foundation for perception, estimation, planning, and control.

Latent States and Compact Physical Representations In realistic environments, the complete physical state is rarely directly observable. Sensors provide incomplete and noisy measurements, while variables such as object properties, hidden geometry, and contact states may remain unknown.

Therefore, modern Physical AI systems often learn latent state representations of the form

where $\mathbf{o}_{t}$ represents observations and $\mathbf{z}_{t}$ represents a compact latent description of the environment.

A useful latent representation should preserve task-relevant physical information while discarding irrelevant variations. Such representations provide the foundation for predictive learning and world models discussed in § 2.1.3 and Chapter § 4.3.

Uncertainty and Predictive Representations Physical environments are inherently uncertain due to incomplete observations, stochastic dynamics, and unknown parameters. Therefore, physical state representations should incorporate uncertainty:

rather than assuming exact knowledge of the state.

Probabilistic state estimation methods, including Kalman filtering and Bayesian inference, provide classical solutions for uncertainty estimation. Modern learning-based approaches extend these ideas by learning predictive representations directly from large-scale sensory experience.

These mathematical descriptions provide the bridge between explicit physical models and learned world models. Classical representations define the structure of the physical world, while learned representations enable scalable prediction and reasoning in complex environments.

Control Theory and Autonomous Systems

The mathematical descriptions in the previous section provide models of the physical world, and control theory in this section explains how an embodied agent can use measurements and actions to regulate its interaction with that world. Embodied intelligence requires continuous exchange of energy and information with the physical world. A robot must therefore not only infer an appropriate action but also realize it through actuators subject to finite torque, uncertain contact, delayed sensing, and disturbances that cannot be enumerated in advance. Control theory provides the formal language for this interaction. It asks how measurements should modify action, how models may be used without being trusted unconditionally, and under what conditions the resulting closed loop remains stable, accurate, and safe.

This section examines closed-loop autonomous operation, in which sensing, planning, and control repeatedly respond to the consequences of action. Motor-current and joint controllers react within microseconds to milliseconds. Whole-body controllers coordinate forces over tens of milliseconds, and task planners and behavior executives operate over seconds or minutes. Learning systems may update models over an entire deployment. Although these layers differ in representation and method, each compares an intended condition with an observed one and acts on the discrepancy. Feedback therefore connects classical control, optimization, planning, and learning-based autonomy rather than belonging to any one of them. The following subsections introduce feedback, control methods, and hierarchical autonomy. They also analyze stability, robustness, adaptation, and safety.

Foundations of Cybernetics: Feedback and Steady State

A feedback control system uses measurements of a physical system’s behavior to adjust its control inputs toward a desired target, which forms a feedback loop of measurement and correction. This section first introduces this feedback loop, and then discusses its transient and steady-state behavior, to assess how effectively a system approaches and maintains that target.

Negative Feedback Principle Wiener (1948) introduces cybernetics as a framework for studying control and communication in systems whose actions affect the conditions they subsequently observe. In this framework, a plant receives a control input $u(t)$, evolves under its own dynamics and external disturbances, and produces an output $y(t)$. For output tracking, a controller compares the measured output with a reference $r(t)$ and uses the tracking error to determine the next control command. The error, and the control input in the simplest case of proportional control with gain $K_{\mathrm{P}}$, are defined as

where the proportional law is the simplest instance of feedback. Integral and derivative action extend this relation in the proportional-integral-derivative controller introduced below.

Controllers may also use an estimate of the underlying state when the relevant variables are not directly measured. The loop is called negative feedback when the controller acts so as to reduce the measured deviation, as when a thermostat increases heating below its set point or a balancing vehicle generates torque opposing body tilt. Feedback is most valuable when the model is incomplete. If an actuator delivers less torque than commanded, or a manipulator grasps an object of unknown mass, a precomputed input cannot detect the resulting error. Feedback uses observations of the changed motion to compensate within the limits of sensing, actuation, and closed-loop bandwidth. Feedback thereby converts certain forms of uncertainty into observable error and permits repeatable behavior from imperfect components.

Negative feedback is not synonymous with stability. A correction whose sign is appropriate at low frequency may destabilize the loop once sensing, computation, or actuation introduces sufficient delay. Excessive gain has the same effect, and strong reaction to high-frequency measurement noise may excite unmodeled flexible modes. Feedback design must therefore shape the magnitude and phase of the loop response, not merely reverse the sign of the error (Åström and Murray (2008)). The signals entering the loop must also be distinguished. A reference describes intended behavior, a disturbance acts on the plant from outside, measurement noise corrupts the information available to the controller, and model uncertainty denotes the discrepancy between representation and mechanism. These signals can produce similar visible effects yet require different remedies, so an embodied controller is a negotiated balance among tracking, disturbance rejection, noise attenuation, actuator effort, and robustness. Feedback further depends on what is measured. Because the full state is rarely available, practical loops incorporate an estimator that fuses a dynamic model with sensor streams. Estimation and control become coupled processes in which actions determine which future observations become available.

Open-Loop and Closed-Loop Control Two basic control arrangements are open-loop and closed-loop control, illustrated in Figure 2.1 (Franklin et al. (2019); Dorf and Bishop (2017); Åström and Murray (2008)). An open-loop controller selects its input without measurements of the controlled outcome during execution. Such control is appropriate when the plant and environment are sufficiently repeatable, when feedback is unavailable, or when the maneuver is too fast for a sensor-to-actuator loop, but it cannot correct errors that become observable only after execution begins. Closed-loop control uses observations to revise action, and its advantage is made precise by the sensitivity functions. For a single-input, single-output linear system in the standard negative unity-feedback configuration with plant $P(s)$ and controller $C(s)$, the loop transfer function $L(s) = P(s)C(s)$ yields