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Chapter 03: Whole-Body Optimization

Overview​

This chapter explores advanced optimization techniques for controlling humanoid robots as complete systems, considering all degrees of freedom, contact forces, and dynamic constraints simultaneously. This enables complex, dynamic, and efficient movements.

Learning Objectives​

  • Understand whole-body dynamics
  • Learn optimization-based control
  • Explore trajectory optimization
  • Master contact force optimization
  • Understand real-time implementation

Core Concepts​

1. Whole-Body Dynamics​

System Model:

  • All joints considered together
  • Coupled dynamics
  • Contact forces included
  • Full state space

Equations of Motion:

M(q)\ddot{q} + C(q, \dot{q}) + G(q) = \tau + J^T F

Where:

  • M: Mass matrix
  • C: Coriolis forces
  • G: Gravity
  • Ï„: Joint torques
  • J: Jacobian
  • F: Contact forces

Complexity:

  • High-dimensional (30+ DOF)
  • Nonlinear dynamics
  • Contact constraints
  • Real-time requirements

2. Optimization-Based Control​

Optimization Problem:

\min_{\tau, F} \quad Cost(q, \dot{q}, \tau, F)

Subject to:

  • Dynamics constraints
  • Contact constraints
  • Joint limits
  • Torque limits
  • Stability constraints

Cost Functions:

  • Energy minimization
  • Tracking error
  • Smoothness
  • Stability margin
  • Task completion

3. Trajectory Optimization​

Planning Horizon:

  • Short-term: Immediate actions
  • Medium-term: Motion sequences
  • Long-term: Task completion

Optimization Methods:

  • Direct Methods: Discretize then optimize
  • Indirect Methods: Optimal control theory
  • Sampling: RRT, PRM variants
  • Gradient-Based: Fast convergence

Constraints:

  • Dynamic feasibility
  • Contact constraints
  • Obstacle avoidance
  • Stability requirements

4. Contact Force Optimization​

Contact Modeling:

  • Friction cones
  • Contact points
  • Force distribution
  • Stability margins

Optimization:

  • Distribute forces optimally
  • Maintain stability
  • Minimize internal forces
  • Satisfy friction constraints

Applications:

  • Multi-contact locomotion
  • Manipulation with contacts
  • Pushing tasks
  • Climbing

5. Real-Time Implementation​

Computational Challenges:

  • Large optimization problems
  • Real-time constraints (less than 1ms)
  • Numerical stability
  • Solution quality

Approaches:

  • Model simplification
  • Efficient solvers
  • Parallel computation
  • Predictive control
  • Hierarchical optimization

Trade-offs:

  • Accuracy vs speed
  • Optimality vs feasibility
  • Complexity vs performance

Technical Deep Dive​

Optimization Pipeline:

Task Specification
    ↓
Trajectory Generation (Offline)
    ↓
Real-Time Optimization (Online)
    ↓
Torque Commands
    ↓
Execution

Real-World Application​

Dynamic Locomotion:

  • Running and jumping
  • Parkour movements
  • Complex terrain
  • Fast, efficient motion
  • Whole-body coordination

Hands-On Exercise​

Exercise: Design an optimization problem for:

  • A dynamic walking motion
  • Include all constraints
  • Define cost function
  • Discuss solution method

Summary​

Whole-body optimization enables:

  • Complex dynamic motions
  • Efficient energy use
  • Coordinated movements
  • Advanced capabilities
  • Natural-looking motion

References​

  • Whole-Body Control
  • Trajectory Optimization
  • Contact Force Control