Contact Dynamics, Hybrid Systems, and Locomotion in Robotics

Recap of Previous Lectures

• Previous Topics: LQR with quaternions for quadrotors
  • Applied in convex MPC, drag&drop, and other tasks.
  • Mentioned no Euler angles in drone projects.
• Upcoming Topics: Contact dynamics, hybrid systems, legged robots, and locomotion.
  • Applications in various engineering fields including aerospace and chemical engineering.

Introduction to Contact Dynamics

Key Examples

• Bouncing Ball
  • In Air: Smooth dynamics, governed by classic ODEs.
  • On Impact: Discontinuous dynamics, jump discontinuities in velocity.
• Concepts
  • Jump Discontinuities: Velocity has a discontinuous change upon impact.
  • Non-Smooth Dynamics: Cannot be described purely by traditional ODEs due to discontinuities at the points of impact.

Solutions for Modeling Contact Dynamics

• Event-Based (Hybrid) Formulation

  • Smooth ODEs in between contacts with guard functions to detect impacts.
  • Guard Function: Condition to detect contact events.
  • Jump Map: Specifies state changes at the moment of contact.
  • Pros and Cons: Common in legged locomotion, easy to integrate, but requires pre-specified mode sequence.

• Time-Stepping (Contact Implicit) Methods

  • Converts dynamics and contact conditions into constraint optimization problems.
  • Solves for contact forces jointly with states.
  • Contact Forces as Inequality Constraints
  • Pros and Cons: No need for pre-specified mode sequence but results in harder optimization problems.

Dynamics Inside Hybrid Systems

Mathematical Formulation

• Smooth Vector Field (Continuous Dynamics)
• Guard Function: Detects contact (e.g., ball hitting the floor).
• Jump Map: Specifies changes in states (e.g., zeroing out vertical velocity).
• Integration: High-order integrators possible (e.g., RK4).
• Example: Bouncing ball with inelastic and elastic collisions.

Implementing Hybrid Dynamics

• Pseudo-Code Workflow
  • Integrate using a high-order method (e.g., RK4).
  • Evaluate guard function to detect impact events.
  • Apply Jump Map on state when guard condition is met.
  • Continue integration.

Simulation Example

• Falling Brick Simulation
  • Illustrates inelastic and elastic behaviors.
  • Showcased through a bouncing ball which either goes plop or bounces.

Hybrid Trajectory Optimization for Legged Robots

Simple Example: One-Legged Hopper

• Model: Two-point mass system (body and foot) with prismatic joint.
• State & Control Variables
  • State: Position and velocity of both masses.
  • Controls: Force and torque between body and foot.

Dynamics and Constraints

• Flexible and Fixed Time Formulations
  • Mode Sequences: Alternating groups of stance and flight phases.
  • Continuous Dynamics: Different dynamics for stance and flight.
  • Jump Maps: Apply changes in state at contact transitions (e.g., zeroing the foot velocity on ground contact).

Implementing Trajectory Optimization

• Cost Function: Quadratic costs for state and control.
• Initial Guess: Helps with solver convergence, plausible but not necessarily physically accurate.
• Constraints: Includes contact constraints (e.g., feet must not penetrate ground).
  • Example: Foot’s Z-coordinate in contact modes should be zero.

Simulation Results

• Visualized hopping motion for one-legged Hopper.
• Shows stance, flight, and transitions between modes.
• Demonstrated on a simplified one-legged robot but can be extended to complex systems like bipeds and quadrupeds.

Practical Challenges and Q/A

• Issue with Unplanned Contacts: Killer for robots; leads to unwanted behaviors.
• Initial Guess Importance: Critical for solver performance; wrong guesses lead to poor solutions.
• Adaptation in Hardware: Real-time control and disturbance rejection are significantly harder than offline trajectory planning.
• Deviation from Constraints: Solver might fail if initial or boundary conditions are too inconsistent.

Conclusion

• Hybrid system modeling and optimization is a powerful method in robotics.
• Application expands beyond robotics into control and physics-based optimization in various fields.
• The practical implementation requires careful consideration of dynamics, initial guesses, and solver techniques.