Passive Dynamic Walker
Abstract
This project explores passive dynamic walking mechanisms through mathematical modeling and simulation. Starting with a basic compass gait model and progressing to a more complex kneeded walker, we analyze the dynamics and stability of these walking systems. Our study demonstrates how simple mechanical principles can achieve stable bipedal locomotion without active control.
Model Analysis
1. Compass Gait Model
Key Features:
- Two-link inverted pendulum structure
- Alternating stance and swing phases
- Simple heel-strike transition model
Dynamic Analysis:
Swing Phase
- Pendulum-like motion of the swing leg
- Conservation of angular momentum
- Gravity-driven natural dynamics
Heel Strike Phase
- Instantaneous collision model
- Momentum conservation principles
The compass gait model demonstrates how passive dynamics alone can create a stable walking pattern. During the swing phase, the motion follows natural pendulum dynamics, while the heel strike phase involves an instantaneous transfer of momentum that helps maintain the walking cycle.
2. Kneeded Walker Model
Advanced Features:
- Three-link mechanism with knees
- Multiple phase transitions
- Knee-strike and heel-strike dynamics
- More human-like walking motion
Knee Strike Dynamics:
Pre-Strike Phase
- Independent thigh and shank motion
- Two-link pendulum dynamics per leg
- Multiple degrees of freedom
Post-Strike Phase
- Knee locking mechanism
- Combined leg segment dynamics
- Reduced degrees of freedom
Knee Strike Transition
The knee strike represents a critical transition in the walking cycle. When the knee strikes, the system undergoes an instantaneous change where:
- Angular momentum is redistributed between segments
- The system transitions from 3 links to 2 links
- Impact forces are modeled using impulse-momentum principles
- Energy is partially conserved through the strike
Simulation Results
Our simulations demonstrate stable walking patterns for both models. The Kneeded Walker shows more natural gait characteristics with its knee joints, while the Compass Gait provides efficient basic locomotion. Phase portraits and trajectory analysis confirm the stability of both systems.
Key Equations
Dynamics Model:
The equation represents the core dynamics of both walker models, where M(q) is the inertia matrix, C(q,ḣq) captures Coriolis forces, G(q) represents gravitational effects, and τ denotes external torques.