Inverse Dynamics

Inverse dynamics is the computational core of feedforward control. Given a desired joint trajectory (position, velocity, acceleration), it calculates the precise actuator torques needed to follow that path, assuming a perfect model. This feedforward signal is combined with a feedback controller (like a PID) to compensate for disturbances and model inaccuracies, forming a computed-torque controller. This is essential for high-precision tasks in robotic arms, legged locomotion, and exoskeletons.

Inverse dynamics provides the mathematical framework for dynamic parameter estimation. The equations of motion can be rearranged into a linear-in-parameters form: Y(q, q̇, q̈) * Φ = τ, where Y is the regressor matrix (built from known kinematics), Φ is the vector of unknown dynamic parameters (masses, inertias, friction coefficients), and τ is the measured torque. By collecting data from excitation trajectories and solving this linear system, the unknown parameters Φ can be identified, directly calibrating the simulation model.

Inverse dynamics is used to quantitatively validate physics simulators. The process involves:

• Running a real robot (or high-fidelity reference simulator) through a motion and recording joint positions, velocities, and measured actuator torques.
• Importing the same motion (q, q̇, q̈) into the simulator.
• Using the simulator's internal model to compute the predicted torques via inverse dynamics.
• Comparing predicted vs. measured torques using metrics like Normalized Mean Absolute Error (NMAE). Large discrepancies indicate model fidelity issues, such as incorrect inertia tensors or unmodeled friction.

For robots equipped with joint torque sensors or force-torque sensors at the end-effector, inverse dynamics enables residual force analysis. The difference between the measured sensor reading and the predicted force/torque from the inverse dynamics model is the residual. Large, unexpected residuals can indicate:

• External contacts or collisions not in the planned motion.
• Payload changes (picking up/dropping an object).
• Model errors or actuator faults. This is critical for collision detection, adaptive control, and failure diagnosis in collaborative and safety-critical robots.

Inverse dynamics is a key constraint in trajectory optimization problems. When planning an optimal path (e.g., minimum time or energy), the optimizer must ensure the trajectory is dynamically feasible. The inverse dynamics equations translate the candidate acceleration profile into required torques, which are then checked against the robot's actuator limits (peak and continuous torque). This ensures the planned motion can be physically executed by the hardware. This is used in motion planning for humanoids, drones, and manipulators performing dynamic tasks.

For tasks requiring interaction with the environment (e.g., polishing, assembly, peg-in-hole), hybrid control is used. Inverse dynamics separates the control problem into subspaces:

• In the position-controlled subspace, it computes torques to track a desired motion.
• In the force-controlled subspace, it computes torques to achieve a desired contact force. The total command is the sum. This requires an accurate dynamic model to decouple the dynamics in each subspace, preventing force commands from causing unwanted motion and vice-versa. This is foundational for compliant manipulation.

Forward dynamics is the complementary computation to inverse dynamics. Given a system's current state (positions, velocities) and the applied forces or torques at its actuators, forward dynamics calculates the resulting acceleration and subsequent motion trajectory.

• Core Relationship: Forms a dual with inverse dynamics. Inverse dynamics calculates τ = f(q, q̇, q̈) (torques from motion), while forward dynamics calculates q̈ = f(q, q̇, τ) (motion from torques).
• Primary Use: Used within physics simulators to propagate the state of a system forward in time, and for simulation-based training of control policies.
• Example: In a robot arm simulation, forward dynamics uses the motor torques commanded by a controller to compute how the arm's joints will move in the next time step.

System identification is the process of constructing mathematical models of a dynamic system from measured input-output data. For inverse dynamics to be accurate, the underlying dynamic model (e.g., mass, inertia, friction parameters) must be correct.

• Direct Link: The accuracy of inverse dynamics calculations is entirely dependent on the fidelity of the identified dynamic model. Errors in identified parameters lead to erroneous torque computations.
• Process: Involves designing excitation trajectories, collecting data from the real system, and using estimation algorithms to fit model parameters.
• Goal: To minimize the reality gap by ensuring the simulation's physics model matches the real robot's behavior.

A dynamic regressor is a key mathematical formulation that enables parameter estimation for system identification. It expresses the inverse dynamics equation in a linear-in-parameters form.

• Formulation: The inverse dynamics torque τ is expressed as τ = Y(q, q̇, q̈) * π, where Y is the regressor matrix (composed of known functions of state and acceleration) and π is a vector of the unknown base dynamic parameters (like inertial moments, mass, friction coefficients).
• Utility: This linear form allows the use of efficient linear least-squares techniques to estimate π from collected (τ, q, q̇, q̈) data.
• Prerequisite: Essential for data-driven calibration of a robot's model before deploying model-based controllers that rely on inverse dynamics.

Feedforward control is a control strategy that uses a model of the system to proactively apply the forces needed to achieve a desired trajectory. It is the primary application of inverse dynamics in robotics.

• Mechanism: The controller uses an inverse dynamics model to compute the exact torques τ_ff required to follow a planned trajectory (q_d, q̇_d, q̈_d). These torques are commanded directly to the actuators.
• Advantage: Compensates for the system's known dynamics (like inertia and gravity) ahead of time, reducing the burden on the feedback controller (e.g., PID) which only has to correct for small disturbances and modeling errors.
• Architecture: Often used in a feedforward + feedback structure: τ_total = τ_ff(q_d, q̇_d, q̈_d) + τ_fb(error).

Computational Torque Control is a specific, model-based control law that explicitly uses inverse dynamics for nonlinear feedback linearization. It is a canonical application of inverse dynamics in robotics.

• Control Law: τ = M(q) * [q̈_d + K_v*(q̇_d - q̇) + K_p*(q_d - q)] + C(q, q̇)q̇ + g(q) Where M, C, and g are the inertia, Coriolis, and gravity matrices from the dynamic model.
• Function: The term M(q) * q̈_d + C(q, q̇)q̇ + g(q) is the inverse dynamics computation. The remaining M(q)*[K_v*ė + K_p*e] is a linear PD feedback law acting on a linearized system.
• Result: Effectively cancels the system's nonlinear dynamics and replaces them with an easy-to-tune linear error dynamics, enabling precise trajectory tracking.

Gravity compensation is a simplified, static form of inverse dynamics. It calculates only the actuator torques required to counteract gravitational forces acting on the robot, holding it in place or allowing it to move as if in zero gravity.

• Calculation: τ_gravity = g(q), where g(q) is the gravity torque vector from the robot's dynamic model. This is a subset of the full inverse dynamics calculation (which also includes inertial and Coriolis terms).
• Primary Use: A fundamental safety and functionality feature. It allows a robot to be backdrivable and easy for humans to guide (in zero-g mode), and forms the baseline for many compliant controllers.
• Example: Before executing a precise motion, a robotic arm first activates gravity compensation to prevent it from sagging due to its own weight.