Lesson 12.2: Advanced Locomotion Planning

Learning Objectives

After completing this lesson, you will be able to:

• Implement trajectory optimization techniques for locomotion planning
• Design locomotion strategies for uneven terrain navigation
• Develop reactive behaviors for real-time locomotion adjustments
• Integrate perception and planning for adaptive locomotion
• Evaluate and select appropriate planning algorithms for different scenarios

Introduction

Advanced locomotion planning goes beyond simple predefined gaits to enable humanoid robots to navigate complex environments with varying terrain, obstacles, and dynamic conditions. This lesson explores sophisticated planning techniques that allow robots to adapt their locomotion patterns in real-time based on environmental feedback, optimize their movements for efficiency and stability, and respond to unexpected situations during locomotion.

Trajectory Optimization for Locomotion

Mathematical Formulation

Trajectory optimization in humanoid locomotion involves finding the optimal sequence of states and controls that minimize a cost function while satisfying dynamic and kinematic constraints.

Optimal Control Problem:

min ∫[t0, tf] L(x(t), u(t), t) dt + Φ(x(tf))
subject to: ẋ(t) = f(x(t), u(t), t)
           x(t0) = x0
           g(x(t), u(t)) ≤ 0
           h(x(t), u(t)) = 0

Where:

• x(t): State vector (positions, velocities, CoM, etc.)
• u(t): Control vector (joint torques, desired positions)
• L: Running cost function
• Φ: Terminal cost function
• f: System dynamics
• g, h: Inequality and equality constraints

Cost Function Design

Stability Cost

J_stability = ∫ ||ZMP_ref(t) - ZMP_actual(t)||² dt

Energy Efficiency Cost

J_energy = ∫ ||τ(t)||² dt

Tracking Cost

J_tracking = ∫ ||x_desired(t) - x_actual(t)||² dt

Smoothness Cost

J_smoothness = ∫ ||u̇(t)||² dt

Optimization Approaches

Direct Methods

• Discretize the continuous problem into a nonlinear programming (NLP) problem
• Single shooting: Integrate dynamics forward in time
• Multiple shooting: Break trajectory into segments with continuity constraints
• Collocation: Approximate state and control trajectories using polynomials

Indirect Methods

• Use Pontryagin's Minimum Principle
• Solve boundary value problem
• Less common due to complexity of humanoid dynamics

Model Predictive Control (MPC)

• Receding horizon optimization
• Solve finite-time optimization repeatedly
• Feedback through state re-initialization
• Handles constraints explicitly

Multi-Body Trajectory Optimization

For humanoid robots, trajectory optimization must consider the full multi-body dynamics:

Whole-Body Optimization

• Optimizes all degrees of freedom simultaneously
• Ensures coordinated motion
• Computationally expensive
• Suitable for offline planning or simplified models

Hierarchical Optimization

• Optimizes different aspects in sequence
• Lower priority tasks executed when higher priority tasks allow
• More computationally efficient
• Common approach in real-time systems

Uneven Terrain Navigation

Terrain Classification and Mapping

Terrain Types

• Flat Ground: Level surfaces with consistent friction
• Rough Terrain: Small obstacles, irregular surfaces
• Stepped Terrain: Discrete height changes (stairs, curbs)
• Sloped Terrain: Inclined surfaces
• Compliant Terrain: Soft surfaces (sand, grass)

Perception Integration

• LIDAR: Provides accurate 3D terrain mapping
• Stereo Vision: Dense depth information
• RGB-D Cameras: Color and depth information
• Tactile Sensors: Ground contact information

Adaptive Gait Generation

Terrain-Aware Gait Parameters

• Adjust step height for obstacles
• Modify step length for stability on slopes
• Change walking speed for safety on uneven terrain
• Adapt foot orientation for surface normals

Online Gait Adaptation

• Real-time adjustment of gait parameters
• Uses terrain feedback to modify planned trajectories
• Maintains stability margins on varying terrain

Footstep Planning on Complex Terrain

3D Footstep Planning

• Plans foot placements in 3D space (x, y, z)
• Considers terrain height and orientation
• Ensures stable foot contacts

Sampling-Based Approaches

• RRT* for footstep sequence planning
• Probabilistic completeness for complex terrain
• Balances optimality with computation time

Optimization-Based Approaches

• Minimize energy consumption on terrain
• Maximize stability margins
• Consider multiple terrain constraints simultaneously

Stability Considerations on Uneven Terrain

Variable Support Polygon

• Support polygon changes with foot placement on uneven terrain
• ZMP must remain within variable polygon
• Requires dynamic adjustment of balance control

Friction Cone Constraints

• Different friction coefficients on various surfaces
• Affects maximum allowable forces
• Influences gait parameter selection

Reactive Behaviors in Locomotion

Real-Time Reaction Framework

Behavior Hierarchy

• Emergency Behaviors: Fall prevention, immediate response
• Stability Behaviors: Balance recovery, ZMP adjustment
• Navigation Behaviors: Obstacle avoidance, path following
• Efficiency Behaviors: Energy optimization, speed control

Disturbance Response

Push Recovery

• Detect external forces through IMU and force sensors
• Compute capture point to determine recovery strategy
• Execute appropriate recovery behavior (ankle, hip, stepping)

Unexpected Obstacle Response

• Detect obstacles in path using perception systems
• Modify footstep plan to avoid obstacles
• Adjust gait parameters for safe navigation

Multi-Modal Control Architecture

Discrete Event Systems

• Switch between different locomotion modes
• Events trigger mode transitions
• Modes: walking, standing, stair climbing, etc.

Hybrid Control Systems

• Continuous dynamics within discrete modes
• Discrete transitions between modes
• Ensures stability during mode changes

Online Planning and Replanning

Receding Horizon Approach

• Plan trajectory over finite horizon
• Execute first portion of plan
• Replan with updated state information

Anytime Algorithms

• Provide best available solution at any time
• Improve solution as computation time allows
• Critical for real-time applications

Integration of Perception and Planning

Sensor Fusion for Locomotion

Multi-Sensor Integration

• Combine data from various sensors for robust perception
• Handle sensor failures gracefully
• Provide uncertainty estimates

State Estimation

• Estimate robot state (position, velocity, orientation)
• Integrate proprioceptive and exteroceptive sensors
• Handle sensor noise and delays

Perception-Action Coupling

Closed-Loop Planning

• Planning based on current perception
• Execution with feedback
• Continuous replanning as perception updates

Predictive Perception

• Anticipate future sensor readings
• Plan based on predicted environment
• Handle sensor latency

Advanced Planning Algorithms

Sampling-Based Motion Planning

RRT (Rapidly-exploring Random Tree Star)*

• Asymptotically optimal path planning
• Builds tree of possible locomotion sequences
• Balances exploration and optimization

PRM (Probabilistic Roadmap Star)*

• Precomputes roadmap of possible configurations
• Query-specific path optimization
• Efficient for repeated planning tasks

Optimization-Based Planning

Sequential Quadratic Programming (SQP)

• Solves nonlinear optimization iteratively
• Handles constraints effectively
• Suitable for trajectory optimization

Interior Point Methods

• Efficient for large-scale optimization
• Handles inequality constraints well
• Good convergence properties

Learning-Based Approaches

Reinforcement Learning for Locomotion

• Learn optimal locomotion policies through interaction
• Handle complex terrain without explicit modeling
• Require extensive training

Imitation Learning

• Learn from human demonstrations
• Transfer human locomotion strategies
• Reduce need for manual tuning

Implementation Challenges

Computational Complexity

Real-Time Requirements

• Locomotion planning at 100-200Hz for stability
• Balance optimization quality with computation time
• Use hierarchical approaches to manage complexity

Model Simplification

• Simplified dynamics models for real-time planning
• Balance accuracy with computational efficiency
• Validate simplified models against full dynamics

Robustness and Safety

Uncertainty Handling

• Account for model uncertainties
• Handle sensor noise and failures
• Ensure safe operation under uncertainty

Fallback Behaviors

• Default safe behaviors when planning fails
• Graceful degradation of capabilities
• Emergency stop procedures

Integration Challenges

Multi-System Coordination

• Coordinate planning with control systems
• Handle timing and communication issues
• Ensure consistency between systems

Calibration and Tuning

• System-specific parameter tuning
• Balance generalization with performance
• Maintain performance across different conditions

Performance Evaluation

Metrics for Advanced Planning

Planning Quality Metrics

• Solution optimality (cost function value)
• Constraint satisfaction
• Computational efficiency (planning time)

Locomotion Performance Metrics

• Stability margins during execution
• Tracking accuracy of planned trajectories
• Success rate in challenging environments

Adaptation Metrics

• Response time to environmental changes
• Recovery from disturbances
• Learning rate for new terrain types

Future Directions

Emerging Technologies

Deep Learning Integration

• Neural networks for terrain classification
• Learning-based planning heuristics
• End-to-end locomotion learning

Advanced Sensing

• Event-based cameras for high-speed perception
• Multi-modal sensing for robust environment understanding
• Wearable sensors for enhanced state estimation

Research Frontiers

Dynamic Terrain Adaptation

• Real-time learning of terrain properties
• Adaptive control strategies
• Predictive terrain modeling

Human-Robot Interaction in Locomotion

• Socially-aware navigation
• Collaborative locomotion
• Adaptive behavior for human comfort

Summary

Advanced locomotion planning for humanoid robots involves sophisticated techniques that enable navigation in complex and dynamic environments. Trajectory optimization provides mathematical frameworks for generating optimal locomotion patterns, while terrain adaptation techniques allow robots to navigate uneven surfaces. Reactive behaviors ensure robustness to disturbances and unexpected situations. The integration of perception and planning creates closed-loop systems that can adapt to changing conditions. Despite significant challenges in computational complexity and system integration, these advanced techniques are essential for achieving human-like locomotion capabilities in humanoid robots.

Further Reading

• Wensing, P.M., & Orin, D.E. "Generation of Dynamic Walking Gaits" - Advanced gait generation techniques
• Kuindersma, S., et al. "Optimally Time-Consistent Locomotion Planning" - Trajectory optimization for locomotion
• "Planning Algorithms" by LaValle - Comprehensive treatment of motion planning
• "Robotics: Systems and Algorithms" by Fox, Burgard, and Thrun - Perception-action integration