Inverse Dynamics

Inverse dynamics is formulated using the equations of motion for a multi-body system. For a robot with floating base dynamics, this is typically expressed as:

M(q) * v̇ + C(q, v) * v + g(q) = Sᵀ * τ + Jᶜ(q)ᵀ * f

Where:

• M(q) is the mass/inertia matrix.
• C(q, v) represents Coriolis and centrifugal forces.
• g(q) is the gravitational force vector.
• S is the selection matrix for actuated joints.
• τ are the joint torques to be solved for (the output).
• Jᶜ(q) is the contact Jacobian matrix.
• f are the external contact forces (e.g., Ground Reaction Forces). Given a desired joint acceleration v̇, the solver computes the necessary torques τ.

The primary output of inverse dynamics is a feedforward torque command. This is the predicted force required to achieve the planned acceleration, assuming a perfect model of the robot's dynamics. In practice, this feedforward term is combined with a feedback controller (like a PD loop) to correct for model inaccuracies and disturbances. This combination is central to high-performance control architectures like computed torque control.

The accuracy of the inverse dynamics solution is critically dependent on the fidelity of the robot's dynamic model. Key model parameters that must be known or identified include:

• Link masses and centers of mass
• Inertia tensors for each body
• Kinematic parameters (link lengths, joint axes)
• Friction coefficients at the joints Errors in these parameters lead to inaccurate torque predictions, forcing the feedback controller to work harder and potentially degrading performance.

For complex robots like humanoids, the basic inverse dynamics equation is often insufficient due to contact constraints and task priorities. It is commonly reformulated as a Quadratic Program (QP). This optimization framework finds the optimal torques τ and contact forces f that:

1. Minimize a cost function (e.g., torque squared, deviation from desired acceleration).
2. Satisfy equality constraints (the equations of motion).
3. Satisfy inequality constraints (e.g., friction cones, torque limits, ZMP within support polygon). This QP-based inverse dynamics is the computational core of Whole-Body Control (WBC).

Inverse dynamics operates in the middle layer of a standard locomotion control stack:

1. High-Level Planner: Generates a desired Center of Mass (CoM) trajectory and footstep plan over a long horizon (seconds).
2. Inverse Dynamics / WBC: Receives the short-horizon (milliseconds) motion plan and solves for the instantaneous joint torques and contact forces needed to track it, respecting all physical constraints.
3. Low-Level Joint Control: The computed torque commands are sent to the joint-level actuators (often with local PID loops). This positioning allows it to bridge strategic planning with precise, constraint-aware actuation.

It is crucial to distinguish inverse dynamics from the related concept of Inverse Kinematics (IK).

• Inverse Kinematics solves for joint positions q given a desired end-effector pose (position & orientation). It is a purely geometric problem.
• Inverse Dynamics solves for joint torques/forces τ given a desired joint acceleration v̇. It is a dynamic problem involving mass, inertia, and forces. In a typical pipeline, IK provides desired joint angles for a footstep; a trajectory generator creates smooth joint velocity and acceleration profiles; then inverse dynamics computes the torques to realize those accelerations.

The complementary calculation to inverse dynamics. Forward dynamics computes the resulting acceleration of a robot given its current state (positions, velocities) and the applied joint torques or forces. It answers the question: 'What motion will these forces produce?' This is the fundamental simulation step used in physics engines and for predicting future states in Model Predictive Control (MPC).

A simplified but crucial model that describes the relationship between the net external wrench (combined force and moment) acting on the robot and the motion of its Center of Mass (CoM) and its centroidal angular momentum. It treats the entire multi-body robot as a single rigid body with its mass concentrated at the CoM. This abstraction is vital for high-level balance and motion planning, as it allows planners to reason about overall robot motion without solving the full multi-body dynamics at every step.

A hierarchical control framework that uses inverse dynamics as its core solver. WBC formulates locomotion as a Quadratic Program (QP) that coordinates all the robot's degrees of freedom to execute multiple, potentially conflicting tasks simultaneously (e.g., foot trajectory tracking, torso orientation, CoM acceleration) while strictly respecting physical constraints like joint limits, torque bounds, and friction cones. The output of the QP is the optimal set of joint torques.

The standard mathematical optimization structure for solving inverse dynamics and whole-body control in real-time. The problem is defined with:

• A quadratic cost function to minimize (e.g., error from desired tasks, effort).
• Linear equality constraints (e.g., the equations of motion from inverse dynamics).
• Linear inequality constraints (e.g., friction cone constraints, torque limits). Efficient, specialized QP solvers allow robots to solve this optimization at control frequencies of 500-1000 Hz.

The specific formulation of the equations of motion for a legged robot, whose base (torso) is not fixed to the world but 'floats' in 6D space (3D position + 3D orientation). This introduces six unactuated degrees of freedom. The dynamics are typically derived using Recursive Newton-Euler Algorithm (RNEA) or Lagrangian formulation. Inverse dynamics for a floating-base system calculates the joint torques and the unactuated base acceleration resulting from contacts and applied forces.

The fundamental physical quantity that inverse dynamics ultimately resolves. The GRF is the force vector (with normal and friction components) exerted by the ground on the robot's foot. In locomotion planning, desired GRFs are first computed from high-level models (like the Linear Inverted Pendulum) to achieve a certain CoM acceleration. Inverse dynamics then solves for the joint torques and contact forces that realize those desired GRFs while satisfying the full multi-body dynamics.