linkage-evolution

Evolved 4-Bar Linkage Mechanism

Evolving physical hardware using AI — A 3D-printable mechanical linkage evolved to trace a figure-8 pattern, demonstrating that evolutionary algorithms can discover optimized physical mechanisms.

Why This Matters

This is real hardware evolution. Unlike optimizing software algorithms, this project evolves a physical mechanism that you can:

1. 3D print the parts
2. Assemble with standard hardware
3. Crank by hand and watch it trace the evolved curve
4. Verify that evolution found a better solution than human design

The evolved linkage achieves 25% better fitness than the baseline, tracing a figure-8 pattern more accurately with optimized dimensions that evolution discovered — including the golden ratio (φ ≈ 1.618) for the tracer position!

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The Evolution Results

Before vs After

Metric	Baseline	Evolved	Improvement
Fitness Score	15.08	18.87	+25%
Path Accuracy	0.268	0.198	-26% error
Hausdorff Distance	0.458	0.439	-4%

Fitness Progression

Evolution ran for 15 generations, with steady improvement throughout:

Parameter Changes

Evolution discovered significant changes to all 6 parameters:

Key Discovery: Evolution independently found the golden ratio (φ ≈ 1.618) for the tracer X position! This mathematical constant appears throughout nature and engineering — and the AI discovered it through pure optimization.

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How 4-Bar Linkages Work

A 4-bar linkage is one of the simplest mechanisms that converts rotational motion into complex paths:

                    Tracer Point (traces the curve)
                         ●
                        /
    Crank (L2)    Coupler (L3)
        ●-----------●-----------●
       /                         \
      / θ (input)                  \ Follower (L4)
     ●                              ●
   Fixed                          Fixed
   Pivot A                        Pivot B
     |<-------- Ground (L1) ------>|

Components:

• Ground (L1): Fixed distance between the two pivot points
• Crank (L2): Input link that rotates (connected to a motor or handle)
• Coupler (L3): Connecting link that moves freely
• Follower (L4): Output link that rocks back and forth
• Tracer Point: A point on the coupler that traces the "coupler curve"

By varying the 4 link lengths and the tracer point position (6 parameters total), you can generate an infinite variety of curves — but finding the right parameters for a specific target shape is extremely difficult.

Motion Sequence

Here's how the evolved linkage traces its path as the crank rotates:

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3D Printable Design

The evolved linkage is ready to manufacture:

Evolved Dimensions

Part	Length	Material
Ground (L1)	89.6 mm	Base plate
Crank (L2)	24.6 mm	Red link
Coupler (L3)	86.2 mm	Green link
Follower (L4)	55.6 mm	Blue link

STL Files

Ready-to-print files in stl_figure8/:

• ground_plate.stl — Base with fixed pivot mounts
• crank.stl — Input link (attach to motor or handle)
• coupler.stl — With tracer point marker
• follower.stl — Output rocker link
• assembly_guide.txt — Full assembly instructions

Hardware Needed

• 4× M5 bolts (pivot joints)
• 4× M5 nuts
• 8× M5 washers
• Optional: Small bearings for smoother motion

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Running the Evolution

Prerequisites

cd showcase/linkage-evolution
python3 -m venv .venv
source .venv/bin/activate
pip install -e ../../sdk claude-agent-sdk matplotlib

Evaluate Existing Solutions

# Baseline linkage
python3 evaluate.py baseline.py --target=figure8

# Evolved champion
python3 evaluate.py evolved_figure8.py --target=figure8

Evolve for a New Target

Edit evolve_config.json to change the target shape:

{
  "evaluation": {
    "test_command": "python3 evaluate.py {solution} --json --target=heart"
  }
}

Available targets: heart, figure8, circle, ellipse, star, line, letter_d

Then run evolution:

/evolve "Evolve 4-bar linkage to trace a heart shape" --config=evolve_config.json

Export STL Files

python3 export_stl.py evolved_figure8.py output_directory/

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Project Files

linkage-evolution/
├── README.md                 # This file
├── linkage.py               # 4-bar linkage kinematics simulator
├── targets.py               # Target shape definitions
├── evaluate.py              # Fitness evaluation (path accuracy)
├── visualize.py             # Matplotlib visualization
├── export_stl.py            # 3D printing STL generator
├── generate_readme_images.py # Generate documentation images
│
├── baseline.py              # Starting linkage (fitness: 15.08)
├── evolved_figure8.py       # Evolved champion (fitness: 18.87)
├── evolve_config.json       # Evolution configuration
│
├── images/                  # Documentation images
│   ├── linkage_diagram.png
│   ├── evolution_comparison.png
│   ├── fitness_progression.png
│   ├── 3d_assembly.png
│   ├── parameter_evolution.png
│   └── motion_sequence.png
│
└── stl_figure8/             # 3D-printable parts
    ├── ground_plate.stl
    ├── crank.stl
    ├── coupler.stl
    ├── follower.stl
    └── assembly_guide.txt

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The Science Behind It

Why Linkage Optimization is Hard

Finding optimal linkage parameters is a non-convex optimization problem:

• The fitness landscape has many local optima
• Small parameter changes can drastically alter the traced curve
• The Grashof condition constrains which linkages can complete full rotations
• The relationship between parameters and output shape is highly nonlinear

Traditional approaches use:

• Trial and error with lookup tables
• Numerical optimization (often gets stuck in local optima)
• Synthesis algorithms (limited to specific curve families)

Why Evolution Works

Evolutionary algorithms excel at this problem because:

1. Population diversity — Explores multiple regions simultaneously
2. Crossover — Combines good traits from different solutions
3. Incremental improvement — Each generation builds on successes
4. No gradient required — Works despite discontinuous fitness landscape

The LLM-based evolution adds:

• Semantic understanding — Can reason about why a change might help
• Creative mutations — Tries non-obvious parameter combinations
• Knowledge transfer — Applies concepts like the golden ratio

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Historical Context

4-bar linkages have been studied for over 150 years:

• 1864: Peaucellier-Lipkin linkage — First mechanism to trace a perfect straight line
• 1876: Chebyshev's work on linkage synthesis
• 1950s: Freudenstein's analytical methods
• Today: Still an active area of research in mechanism design

This project demonstrates that AI can now contribute to this classical field, discovering optimized mechanisms through evolutionary search.

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Future Directions

• Evolve for more complex target shapes
• Multi-objective optimization (accuracy vs. mechanical advantage)
• 6-bar linkages for more complex curves
• Add physical constraints (material stress, joint angles)
• Integrate with CAD software for direct manufacturing

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Citation

If you use this project, please cite:

@misc{linkage-evolution,
  title={Evolved 4-Bar Linkage Mechanism},
  author={Agentic Evolve},
  year={2025},
  url={https://github.com/anthropics/agentic-evolve/showcase/linkage-evolution}
}

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Built with Agentic Evolve — LLM-powered evolutionary algorithm discovery