Whole-Body Control (WBC)

WBC's core mechanism is a strict task hierarchy. Higher-priority tasks (e.g., maintaining balance) are satisfied exactly, while lower-priority tasks (e.g., arm posture) are achieved only in the null space of higher tasks. This is mathematically solved using projected inverse dynamics or null-space projections. For example, a humanoid will always prioritize keeping its Center of Mass (CoM) within the support polygon over reaching for an object.

• Critical Tasks: Balance, contact force constraints, joint limits.
• Secondary Tasks: End-effector tracking, posture control, gaze direction.
• Tertiary Tasks: Energy minimization, comfort posture.

WBC formulates control as a single, unified Quadratic Program (QP). This optimization simultaneously computes optimal joint torques, contact forces, and joint accelerations subject to a full set of dynamic constraints.

Key constraints explicitly modeled include:

• Actuator Torque Limits: τ_min ≤ τ ≤ τ_max
• Friction Cone Constraints: Ensures contact forces remain within a pyramid to prevent slipping.
• Contact Consistency: Forces are only applied at feet in contact with the ground.
• Dynamic Feasibility: Solutions obey the rigid-body dynamics equation: M(q)q̈ + C(q, q̇) + G(q) = Sᵀτ + J_cᵀF_c. This holistic approach prevents the internal conflicts common in decentralized controllers.

Unlike fixed-base manipulators, legged robots have a floating base (typically the pelvis or torso) with six unactuated degrees of freedom (3 translation, 3 rotation). WBC explicitly accounts for this by treating the base acceleration as a variable in the optimization. The solver computes the required ground reaction forces that, through the centroidal dynamics, generate the necessary base motion. This is fundamental for generating dynamic motions like jumping or recovering from a push, where the base must accelerate freely.

A key subtask within WBC is the regulation of the robot's centroidal momentum (the combined linear and angular momentum of its entire body about its Center of Mass). By controlling the rate of change of centroidal momentum—via the Centroidal Momentum Matrix—the controller directly influences the aggregate motion of the body. This provides a powerful, compact representation for balance. For instance, to counteract a rotational disturbance, WBC can command a specific rate of change of centroidal angular momentum by coordinating all limb movements.

WBC typically functions as a high-frequency (500-1000 Hz) local controller. It is fed desired task references (e.g., footstep locations, swing foot trajectories, CoM motion) from a slower, planning layer. This planner is often a Model Predictive Control (MPC) module based on a Reduced-Order Model (ROM) like the Linear Inverted Pendulum (LIP). The MPC handles long-horizon planning for stability, while WBC precisely executes these plans on the full-order robot model, compensating for dynamic effects the planner ignored.

The computational engine of modern WBC is a high-speed QP solver. The control problem is transformed into a standard QP form: minimize (1/2)xᵀQx + cᵀx subject to Ax = b, Cx ≤ d. Solvers like OSQP, qpOASES, or ProxQP compute solutions in milliseconds. This enables real-time control at the servo level. The cost function (Q) encodes task errors (e.g., foot tracking error), while the equality constraints (A, b) enforce strict priorities and dynamics, and inequality constraints (C, d) represent physical limits.

Centroidal dynamics governs the relationship between the net external wrenches (forces and moments) acting on a robot and the motion of its center of mass (CoM) and its centroidal angular momentum. It is a critical simplification used in WBC, as it allows planners to reason about the robot's overall motion without solving the full multi-body dynamics. In WBC, tasks like balance are often formulated as constraints or objectives on centroidal quantities, which are then distributed to individual joints.

The Quadratic Program (QP) formulation is the standard mathematical optimization problem at the heart of most WBC implementations. It provides a computationally efficient framework for resolving multiple, potentially conflicting tasks. The core structure is:

• Cost Function: A quadratic expression that penalizes deviations from desired tasks (e.g., foot tracking error, desired torso orientation).
• Linear Constraints: Hard limits that must be satisfied, such as:
  • Actuator torque limits
  • Friction cone constraints for foot contacts
  • Joint position and velocity limits
  • Dynamic consistency between limb motions and centroidal dynamics WBC solves this QP at a high rate (e.g., 1 kHz) to compute optimal joint torques or accelerations.

Model Predictive Control (MPC) is a complementary, higher-level controller often used in conjunction with WBC. While WBC operates at a fast rate to satisfy instantaneous constraints, MPC plans over a future time horizon (e.g., 0.5-1.0 seconds). MPC typically uses a simplified model (like centroidal dynamics) to optimize future footstep placements, CoM trajectories, and ground reaction forces. The optimal plan from MPC is then fed as a reference trajectory to the WBC layer, which handles the precise, whole-body execution subject to all physical constraints.

Floating base dynamics refers to the equations of motion for a legged robot whose base (e.g., torso) is not fixed to the world, possessing six unactuated degrees of freedom (three translation, three rotation). This is fundamental to WBC, as the controller must account for the coupling between limb movements and base motion. WBC frameworks explicitly incorporate the floating base dynamics to compute the forces and torques required to achieve desired base acceleration while accounting for the inertial effects of moving limbs.

Task-space formulation is the control paradigm where objectives are defined in operational spaces (Cartesian coordinates) rather than joint space. WBC is inherently a task-space controller. Common tasks include:

• End-effector control: Precisely positioning a foot or hand.
• Center of Mass (CoM) control: Regulating balance.
• Torso orientation control: Keeping the body level.
• Momentum control: Regulating centroidal angular momentum. WBC's hierarchy assigns priorities to these tasks, using null-space projections to execute lower-priority tasks in a way that does not interfere with higher-priority ones.

Inverse dynamics is the computation of the joint torques required to achieve a desired joint acceleration, given the robot's current state, mass properties, and applied forces. WBC commonly uses an inverse dynamics solver as its final step. After the QP optimization determines the optimal joint accelerations, contact forces, and possibly base acceleration, the inverse dynamics module computes the specific joint-level torques commands to send to the actuators. This separates the high-level optimization from the low-level dynamics computation.