Strandbeest Leg Optimization Using Genetic Algorithms

Source: https://en.wikipedia.org/wiki/Leg_mechanism

Technical Documentation

Overview

This document describes the implementation of a genetic algorithm (GA) designed to optimize the leg mechanism of Theo Jansen's Strandbeest. The algorithm aims to find optimal proportions for the 11-bar linkage system that creates the characteristic walking motion of these kinetic sculptures.

Problem Definition

The Strandbeest leg mechanism consists of 11 connected bars that form a walking linkage. The optimization challenge is to determine the optimal lengths of these bars to achieve:

1. A nearly horizontal motion during the ground contact phase
2. Appropriate foot clearance during the return phase
3. Mechanical feasibility and stability

Algorithm Implementation

Chromosome Structure

Each solution is represented as a list of 11 floating-point numbers corresponding to the lengths of the bars in the mechanism. The chromosome structure is:

lengths = [crank_length, bar_2_length, ..., bar_11_length]

Fitness Function

The fitness function evaluates three key aspects:

1. Ground Flatness (50% weight)

   • Measures deviation from horizontal during ground contact phase
   • Lower deviation scores higher
   • Critical for smooth walking motion

2. Step Height (30% weight)

   • Evaluates foot clearance during return phase
   • Penalizes both insufficient and excessive height
   • Optimal range: 15-25% of mechanism height

3. Mechanical Complexity (20% weight)

   • Penalizes impractical bar length ratios
   • Considers manufacturing feasibility
   • Promotes robust mechanical design

Genetic Operators

Selection

• Tournament selection with size 5
• Provides good balance between exploration and exploitation
• Maintains genetic diversity while favoring better solutions

Crossover

• Single-point crossover
• Crossover point randomly selected
• Preserves successful sub-structures of solutions

Mutation

• Gaussian mutation with 10% mutation rate
• Maximum mutation magnitude: ±10% of current value
• Helps escape local optima and maintains diversity

Evolution Strategy

• Population size: 100 individuals
• Elitism: Top 10% preserved
• Termination: 1000 generations or convergence
• Convergence defined as < 1% improvement over 50 generations

Performance Metrics

The algorithm tracks several metrics during optimization:

1. Best fitness per generation
2. Population diversity
3. Convergence rate
4. Constraint violations

Validation

Solutions are validated through:

1. Kinematic simulation
2. Physical feasibility checks
3. Stability analysis

Implementation Notes

Critical Considerations

1. All length measurements are in millimeters
2. Minimum bar length: 10mm
3. Maximum bar length: 100mm
4. Angular resolution: 50 points per cycle

Optimization Parameters

• Population size: 100
• Generations: 1000
• Mutation rate: 0.1
• Tournament size: 5
• Elite count: 10

Results Analysis

The optimization typically produces results with:

1. Ground contact deviation < 5mm
2. Step height between 15-25% of mechanism height
3. All bar lengths within manufacturable ranges

Future Improvements

Potential enhancements include:

1. Multi-objective optimization
2. Dynamic mutation rates
3. Advanced crossover methods
4. Real-time visualization
5. Parallel evaluation

References

1. Jansen, Theo. "The Great Pretender." 010 Publishers, 2007
2. Holland, John H. "Adaptation in Natural and Artificial Systems"
3. Goldberg, David E. "Genetic Algorithms in Search, Optimization and Machine Learning"