Control

Motion planning decides where the robot should go. Control is what makes the robot actually go there, in a world where the model is slightly wrong, the payload shifted, and the floor is never quite flat. The tool is feedback: measure where the robot is, compare it to where it should be, and correct. That loop runs at every level of the stack, from the motor current loops deep inside an actuator to the whole-body torque controller of a humanoid, and it keeps running underneath whatever learned policy sits on top.

The field's own estimate of its workhorse is blunt: more than 95% of all industrial control problems are solved by PID control, many of them without the D Åström 2008↓Feedback Systems: An Introduction for Scientists and EngineersKarl Johan Åström, Richard M. Murray, Princeton University Press, 2008. Everything in this module descends from that observation or reacts against it.

The loop is below, on the smallest problem that still shows everything: an inverted pendulum with a payload bolted off-center. Press Run, then drag the proportional gain down. Somewhere just under 9.81 in the slider's units the pole stops coming back up, and no setting of the other two gains rescues it.

PID: three ways to use an error

Close the loop on a scalar error $e(t) = r(t) - y(t)$, the difference between the reference and the measured output. The PID controller turns that error into an actuator command with three terms:

Åström and Murray's reading of the three terms is the one to keep: the proportional term reacts to the present error, the integral term accounts for the past, and the derivative term is a linear extrapolation into the future Åström 2008↓Feedback Systems: An Introduction for Scientists and EngineersKarl Johan Åström, Richard M. Murray, Princeton University Press, 2008. Proportional action alone always leaves a steady-state error, because a nonzero error is what generates the corrective torque; that is exactly what the integral term exists to erase. Derivative action adds damping, predicting where the error is heading, at the cost of amplifying measurement noise.

Tuning those three gains is its own craft. The Ziegler-Nichols rules of 1942, drive the loop into oscillation with proportional gain alone, then back off from the critical gain and period by fixed fractions, are still printed in every textbook Ziegler 1942↓Optimum Settings for Automatic ControllersJ. G. Ziegler, N. B. Nichols, Trans. ASME, 1942. The failure modes are just as canonical: integral windup, where a saturated actuator lets the integrator accumulate error it can never repay, and derivative kick, where a setpoint step spikes the derivative term.

The lab below is that loop made physical. The plant is the canonical one from underactuated control, a torque-driven inverted pendulum Tedrake 2024↓Underactuated RoboticsRuss Tedrake, MIT course notes, 2024:

with $\theta$ measured from upright and $\tau_{bias}$ a constant disturbance from a payload bolted off-center. Linearized about the upright, the loop is stable only when $K_p > m g \ell$, which is the 9.81 threshold the lab at the top of this module is built around.

State space and LQR

PID acts on one error. State feedback acts on the whole state. Write the plant in state-space form, $\dot{x} = Ax + Bu$, and close the loop with $u = -Kx$: every state gets its own gain, and pole placement becomes a linear-algebra exercise Åström 2008↓Feedback Systems: An Introduction for Scientists and EngineersKarl Johan Åström, Richard M. Murray, Princeton University Press, 2008. The natural question is which $K$ is best, and Kalman's 1960 formulation made that precise Kalman 1960↓Contributions to the Theory of Optimal ControlR. E. Kalman, Bol. Soc. Mat. Mexicana, 1960: pick the gain that minimizes a quadratic cost trading state error against control effort,

The optimal gain is $K = R^{-1} B^{\top} P$, where $P$ solves the algebraic Riccati equation:

That is the linear quadratic regulator. Two design knobs, $Q$ and $R$, replace per-gain intuition with a statement of what you care about: penalize angle deviation heavily and the controller gets stiff; penalize torque and it gets gentle. And the connection back is direct. Linearize the pendulum above, choose a diagonal $Q$ weighting angle and rate, and the LQR gain is a PD controller. The sliders you just moved were a hand-tuned version of what Riccati solves in one shot Tedrake 2024↓Underactuated RoboticsRuss Tedrake, MIT course notes, 2024.

The catch is in the word "linearized". LQR's guarantee is local to the operating point; swinging the pole up from hanging takes a different controller entirely (energy shaping), with LQR catching it at the top Tedrake 2024↓Underactuated RoboticsRuss Tedrake, MIT course notes, 2024. Real systems run a gain schedule: different $K$ for different regions of the state space.

MPC: re-solve the plan at every step

Model-predictive controlmodel predictive controlControl by constant re-planning: at each control step, optimize a short sequence of future actions against an explicit dynamics model, execute only the first action, and solve again from the freshly measured state. Because the plan is recomputed online, model error is rejected by feedback at every step instead of being frozen into a policy's weights, so a model-predictive controller shows less of the frozen-policy sim-to-real failure mode, though a wrong dynamics model still costs it closed-loop performance. Long assumed too slow for a robot's full dynamics, the method reached whole-body scale in 2026, when Zhang and colleagues solved iLQR against MuJoCo's physics fast enough to close the loop in real time on real hardware. takes the optimal-control idea online. Instead of precomputing a gain, solve a short-horizon optimal control problem at every control step, from the current measured state, subject to the constraints that actually bind:

Then apply only the first input, measure the new state, and solve again. The receding horizon is what makes MPC robust: model error is not baked into a fixed gain but re-corrected at every step, because each new solve starts from reality. The constraints are the other half of the value: torque limits, joint limits, and friction cones enter the optimization directly rather than as saturation afterthoughts. Making that rigorous took decades; the modern stability theory, terminal costs and constraint sets that guarantee the receding-horizon loop inherits the infinite-horizon optimum, was consolidated by Mayne and collaborators in 2000 Mayne 2000↓Constrained Model Predictive Control: Stability and OptimalityD. Q. Mayne, James B. Rawlings, Christopher V. Rao et al., Automatica, 2000.

The industrial lineage is older than the theory. MPC grew out of power-plant and petroleum-refinery practice in the 1970s and 1980s, where slow, constrained, multivariable plants were exactly what PID could not handle Qin 2003↓A Survey of Industrial Model Predictive Control TechnologyS. Joe Qin, Thomas A. Badgwell, Control Engineering Practice, 2003; by 2003 a vendor survey counted thousands of installed applications, still concentrated in refining and petrochemicals Qin 2003↓A Survey of Industrial Model Predictive Control TechnologyS. Joe Qin, Thomas A. Badgwell, Control Engineering Practice, 2003. What changed since is compute. Di Carlo and colleagues put convex MPC on the MIT Cheetah 3 by simplifying the robot to a single rigid body with contact forces as decision variables: prediction horizons up to half a second, each QP solved in under a millisecond, replanned 20-30 times per second, one set of gains for every gait from standing to gallop Di Carlo 2018↓Dynamic Locomotion in the MIT Cheetah 3 Through Convex Model-Predictive ControlJared Di Carlo, Patrick M. Wensing, Benjamin Katz et al., IEEE/RSJ IROS, 2018. And in 2026, Zhang and collaborators showed the simplification is no longer forced: plain iLQR with MuJoCo dynamics and finite-difference derivatives runs whole-body MPC in real time on quadrupeds and a full-size humanoid, transferring to hardware with few sim-to-real accommodations Zhang 2026↓Whole-Body Model-Predictive Control of Legged Robots with MuJoCoJohn Z. Zhang, Taylor A. Howell, Zeji Yi et al., ICRA 2026. The standing assumption that full-order MPC is too slow for legged robots no longer holds, and the debate between MPC and learned policies has its own module in reward design and the MPC debate.

Whole-body QP: the bottom of the stack

Below the planner and the MPC layer sits the fastest loop in a legged or humanoid robot: the whole-body controller, which turns desired accelerations and contact forces into joint torques hundreds to thousands of times per second. The intellectual root is Khatib's operational-space formulation from 1987, which writes the dynamics where the task lives, at the end-effector, rather than in joint space Khatib 1987↓A Unified Approach for Motion and Force Control of Robot Manipulators: The Operational Space FormulationOussama Khatib, IEEE J. Robotics and Automation, 1987:

with $\Lambda$ the task-space inertia and $\mu$, $p$ the Coriolis and gravity terms, mapped back to joints through the JacobianJacobianThe derivative of the forward-kinematics map: the matrix relating joint velocities to end-effector velocity at the current configuration, with one column per joint. It changes with configuration and can lose rank at singularities, where covering some task-space direction would demand unbounded joint speeds. The same matrix maps a wrench at the end effector back to the joint torques that balance it through its transpose, which makes it the working object of both velocity-level control and statics.. Sentis and Khatib extended this into a hierarchy of tasks, so a humanoid can respect balance and contact constraints while lower-priority tasks, a reaching arm, a head gaze, fill the remaining degrees of freedomdegrees of freedomThe number of independent coordinates needed to specify a mechanism's configuration. An arm's degree-of-freedom count is its number of independent joints, so a 7-DoF arm places its end effector with one coordinate to spare beyond the six a rigid pose needs, and that redundancy is what lets the elbow reconfigure while the hand stays put. More degrees of freedom buy dexterity and obstacle avoidance at the price of a larger control problem. Sentis 2005↓Synthesis of Whole-Body Behaviors through Hierarchical Control of Behavioral PrimitivesLuis Sentis, Oussama Khatib, Int. J. Humanoid Robotics, 2005.

The modern implementation collapses that hierarchy into one quadratic program per control tick:

The decision variables are joint accelerations, torques, and contact forces at once; the equality constraint is the rigid-body dynamics itself; $\mathcal{F}$ is the friction cones that keep feet from slipping. Every task, hold the torso upright, track the swing foot, keep the head level, is a weighted row in the objective. This is whole-body controlwhole-body controlTreating all of a robot's actuated degrees of freedom, legs, torso, arms, and hands, as one coupled control problem instead of stacking an arm controller on top of a separate locomotion controller. The model-based version optimizes against the full rigid-body dynamics at once: Zhang and colleagues ran whole-body model-predictive control in real time on hardware with plain iLQR and MuJoCo dynamics, across dynamic quadruped locomotion and full-sized humanoid walking. The learned version reframes the same problem as tracking a retargeted human motion, the route H2O took for real-time whole-body teleoperation, with reinforcement learning supplying the balance and contact feasibility that raw retargeting loses. as it ships on humanoids today, and the humanoid whole-body control module covers how learned policies relate to it.

Impedance control: making contact a design variable

Everything above regulates motion in free space. The half it leaves out is contact, and contact changes the objective itself. A stiff position controller is the right answer where the world is exactly where the model says it is, and the wrong one the moment it is not: drive a rigid arm into a surface 2 mm nearer than modeled, and the position error no longer measures a tracking mistake, it measures stored energy in a collision. The stiffness-versus-compliance tradeoff is therefore a control objective, not a gain choice. You decide, as part of specifying the task, whether the arm should push back like a spring or yield like a suspension, and the controller implements that decision at the contact.

Hogan's impedance controlimpedance controlHogan's 1985 formulation of compliant contact control: instead of commanding a position or a force, the controller programs the relationship between them, making the manipulator behave at the contact like a mass-spring-damper whose stiffness and damping the programmer chooses. The position error drives a force through the desired mechanical impedance, so the arm yields when it meets the environment and pushes back only as much as the programmed spring says. The port-based argument underneath is a duality: at a contact you may command motion or force, the two conjugate variables of the interaction port, but not both independently, and impedance control takes the motion as input and force as output. from 1985 states the idea: stop commanding position or force and command the relationship between them instead, so the manipulator presents itself to the environment as a programmable mass-spring-damper whose stiffness and damping you choose Hogan 1985↓Impedance Control: An Approach to Manipulation: Part I—TheoryNeville Hogan, ASME J. Dynamic Systems, Measurement, and Control, 1985. The argument underneath is a port argument. A contact is an interaction port whose conjugate variables are force and velocity, and a duality governs it: you may command motion or force at a contact, but not both independently, because the environment fixes the other one. Salisbury's active stiffness control, five years earlier, was already the special case: set the apparent Cartesian stiffness matrix per task axis Salisbury 1980↓Active Stiffness Control of a Manipulator in Cartesian CoordinatesJ. Salisbury, 19th IEEE Conf. Decision and Control, 1980. Admittance controladmittance controlThe dual of impedance control: force is measured and motion is commanded. A wrist force-torque sensor reads the contact force, the admittance law converts it into a position or velocity correction, and the underlying motion controller executes it. It is the natural choice on a stiff, geared, non-backdrivable arm, where the position loop is excellent but no torque command channel exists: admittance control synthesizes compliance in software on top of a position-controlled machine, at the cost of the measurement and the loop delay. Ott, Mukherjee and Nakamura's unified formulation shows the two schemes as the two causal assignments of the same interaction port. is the dual assignment: measure force with a wrist sensor, command motion. Each has a natural home. A low-inertia, backdrivable direct-drive arm realizes an impedance directly, because torque commands are its native language; a stiff geared arm with a wrist force-torque sensor runs admittance control in software on top of its excellent position loop. Ott, Mukherjee and Nakamura's unified formulation shows the two are the same port with the causality flipped, and how to switch between them without losing passivity Ott 2010↓Unified Impedance and Admittance ControlChristian Ott, Ranjan Mukherjee, Yoshihiko Nakamura, IEEE Int. Conf. Robotics and Automation, 2010.

Where the task geometry allows it, the older and still dominant scheme is Mason's and Raibert and Craig's hybrid force/position controlhybrid force/position controlRaibert and Craig's decomposition of a contact task into orthogonal subspaces: in a task frame chosen from the geometry of the contact, force is controlled along the axes constrained by the environment and position is controlled along the free axes, never both on the same axis. Polishing a surface controls force normal to it and tracks a path tangent to it; sliding a block in a slot controls force pressing into the slot walls and position along the slot. Mason's predecessor analysis established how to pick the constrained and free directions from contact geometry, which is what makes the decomposition a property of the task rather than a tuning choice.: decompose the contact into a task frame from its geometry, control force along the constrained axes and position along the free ones, never both on the same axis Mason 1981↓Compliance and Force Control for Computer Controlled ManipulatorsMatthew Mason, IEEE Trans. Systems, Man, and Cybernetics, 1981 Raibert 1981↓Hybrid Position/Force Control of ManipulatorsM. H. Raibert, J. J. Craig, ASME J. Dynamic Systems, Measurement, and Control, 1981. Polishing controls force normal to the surface and tracks a path tangent to it. The decomposition is a property of the task, which is both its strength, nothing is guessed, and its limit: someone must supply the contact geometry in advance.

The hardware consequence reaches everything the learned stack wants to do. A position-controlled, non-backdrivable geared arm cannot regulate contact force at all: it has no force channel, so the contact force is set by the position error, the gear ratio, and whatever the surface does, which is to say by nothing the controller commanded. A learned policy that outputs joint positions on that hardware cannot perform the contact-rich tasks the dexterity module argues are the frontier, however good the training data; no dataset creates a force channel the actuator does not have. The alternatives: torque-controlled arms, whose joint torques are commanded directly, and series elastic actuationseries elastic actuatorAn actuator with a deliberately soft spring placed between the gear train and the load, proposed by Pratt and Williamson in 1995. The spring is a sensor as much as a compliance: its deflection measures the transmitted force cheaply and robustly, the force loop closes on the spring rather than on a fragile wrist sensor, and the spring mechanically low-passes impact forces so a collision loads the drivetrain far below what a rigid joint would see. The cost is bandwidth and position accuracy, which is why the design dominates in legged robots and collaborative arms, where force fidelity and impact tolerance matter more than stiffness., which puts a deliberate spring between the gear and the load so the drivetrain has a mechanical force sensor and impact filter Pratt 1995↓Series Elastic ActuatorsGill Pratt, Matthew Williamson, IEEE/RSJ Int. Conf. Intelligent Robots and Systems, 1995. Cartesian impedance controllers of the Albu-Schäffer line made torque-controlled flexible-joint arms practical for fine contact work Ott 2003↓Decoupling Based Cartesian Impedance Control of Flexible Joint RobotsChristian Ott, Alin Albu-Schäffer, Andreas Kugi et al., IEEE Int. Conf. Robotics and Automation, 2003, and the shipped evidence is vendor-level: Franka's interface exposes a 1 kHz impedance layer as the product's core feature Franka Robotics 2026↓Franka Control Interface DocumentationFranka Robotics, Franka Robotics, as of 2026-08-20, while Universal Robots, the archetype of the geared collaborative arm, implements force mode as an admittance loop in software Robots 2025↓URScript: Dynamic Force ControlUniversal Robots, Universal Robots, as of 2026-08-20, 2025. Backdrivability is an actuator decision made long before any controller or policy runs, and backdrivabilitybackdrivabilityThe ease with which an external force can move a joint with the actuators passive: a direct-drive or quasi-direct-drive arm is backdrivable, because pushing on the link turns the motor with little resistance, while a high-ratio geared arm is not, because the gears and the controller's position loop hold the joint rigid against the push. It decides which compliance scheme fits the hardware: impedance control assumes the arm can be commanded as a force source, which suits a low-inertia backdrivable design, while a non-backdrivable geared arm needs admittance control built on a force sensor, or a series elastic element to restore a mechanical force channel. is the property that decides which of these schemes the hardware can even attempt.

The lab below puts that claim in your hands. Command a penetration depth, program the stiffness and damping of the contact impedance, and watch the peak force move against the object's crush limit and a transient contact-force limit stated from published biomechanical measurements Han 2024↓Evaluation of force pain thresholds to ensure collision safety in worker-robot collaborative operationsD. Han, M. Y. Park, J. Choi et al., Frontiers in Robotics and AI, 2024 rather than the paywalled ISO table ISO 2016↓ISO/TS 15066:2016, Robots and robotic devices — Collaborative robotsISO, ISO Technical Specification (public catalogue entry), 2016. Then select the position-controlled arm and watch the sliders grey out: there is nothing to program, because the position loop has already decided the force.

The policy-to-controller contract

A learned action is a controller request, not a motor command by default. Specify the request type, expected update rate, horizon, frame, saturation behavior and the lower loop that interprets it. A Cartesian delta sent to a position controller has different contact behavior from the same delta interpreted as an impedance target. A torque output transfers still more responsibility to the policy and demands a faster, better observed loop.

Keep deadline handling and hard bounds outside the model. On every tick, validate observation age, action shape, finite values, joint and workspace bounds, then define what the controller holds or stops when inference is late. Measure command-to-actuation latency rather than model latency alone. The action-space guide helps choose the interface; this control hierarchy determines whether that choice can be executed safely and at the claimed rate.

Where this meets the learned stack

Classical control did not lose to learning; it became the floor learning stands on. Every actuator in a learned-policy demo still closes a current loop, usually PI, at kilohertz. The interesting boundary is one level up. MPC has no sim-to-real gap in the reinforcement-learning sense, because it re-solves against the measured state every step rather than baking a model into weights; the sim-to-real transfer problem is largely a problem of open-loop policies. The honest 2026 picture is hybrid: Boston Dynamics added reinforcement learning to Spot's existing MPC stack to handle variability, rather than replacing it Dynamics 2024↓Starting on the Right Foot with Reinforcement LearningBoston Dynamics, 2024, and the classical floor a learning paper must clear now includes an iLQR controller the authors kept deliberately simple, an easy-to-reproduce hardware baseline by design Zhang 2026↓Whole-Body Model-Predictive Control of Legged Robots with MuJoCoJohn Z. Zhang, Taylor A. Howell, Zeji Yi et al., ICRA 2026.

The stack you have now seen end to end, kinematics to locate the body, planning to pick the motion, control to execute it, is the half of robotics that learning has absorbed least. State estimation, the module after this one, closes the loop: none of these controllers know where the robot is without it.