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Week 7: Modern Control (MPC & WBC)

Why PID Fail in Humanoids​

PID controls each joint individually. But in a humanoid, moving the left arm shifts the Center of Mass (CoM), which might necessitate moving the right leg to maintain balance. The joints are coupled.

Model Predictive Control (MPC)​

MPC solves an optimization problem at every time step.

  1. Look Ahead: Predict what will happen for the next $T$ seconds if I apply inputs $U$.
  2. Optimize: Find the best $U$ to minimize cost (e.g., "stay upright", "use minimal energy").
  3. Act: Apply the first input $u_0$.
  4. Repeat.

Convex MPC​

For legged robots, we often simplify the robot to a single mass (Potato Model). The dynamics become linear, and the optimization becomes a Quadratic Program (QP), which can be solved in milliseconds.

Whole-Body Control (WBC)​

WBC runs at high frequency (1kHz) and ensures that the tasks requested by the planner/MPC are executed while respecting physical constraints:

  • Friction cones (don't slip).
  • Joint limits.
  • Torque limits.

Hierarchy of Tasks​

WBC allows strict prioritization:

  1. Priority 0: Don't fall over.
  2. Priority 1: Track the hand trajectory.
  3. Priority 2: Look at the target.

If Priority 1 conflicts with Priority 0, Priority 0 wins.

Lab: QP Solvers in Python​

We can use cvxpy to solve a simple QP relevant to force distribution.

import cvxpy as cp
import numpy as np

# Problem: Support a 10kg weight on 2 legs.
# Minimize effort (forces squared).
# Constraint: F1 + F2 = mg

F = cp.Variable(2) # Forces on [Left Leg, Right Leg]
mg = 10.0 * 9.81

# Objective: Minimize F1^2 + F2^2
cost = cp.sum_squares(F)

# Constraints
constraints = [
    cp.sum(F) == mg,  # Support the weight
    F >= 0            # Legs can only push, not pull
]

prob = cp.Problem(cp.Minimize(cost), constraints)
prob.solve()

print("Force Distribution:", F.value)
# Result should be [49.05, 49.05] (Equal distribution)