SingleLeggedWalker

Code for evolving Continuous Time Recurrent Neural Networks (CTRNNs) to perform locomotion using a single-legged walker body — a minimal platform for studying sensorimotor integration in embodied cognition.

The base single-legged walker simulation was developed by Dr. Randall Beer. It has been extended here to support full sensorimotor configurations.

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Overview

Sensorimotor integration — the bidirectional coupling between sensation and motor control — plays a central role in embodied cognition. This project investigates how neurons self-organize when sensory and motor connections are not topologically constrained. By exhaustively exploring all possible sensorimotor configurations of a minimal CTRNN-controlled walker, we can ask: which connections are necessary for locomotion, and which improve it?

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The Walker Body

The walker has two functional parts — the leg and the foot — each equipped with both an effector and a sensor:

Component	Symbol	Role
Foot effector	E_ft	Lifts / plants the foot
Forward effector	E_f	Swings leg forward
Backward effector	E_b	Swings leg backward
Foot sensor	S_ft	Detects whether foot is planted
Angle sensor	S_a	Reports current leg angle

Sensory inputs are defined as:

$$o_{S_a} = w_a \cdot \frac{\phi}{\phi_{\max}}$$

$$o_{S_{ft}} = \begin{cases} 0 & \text{if } o_{E_{ft}} = 0 \ w_{ft} & \text{if } o_{E_{ft}} = 1 \end{cases}$$

The CTRNN neuron update equation is extended to incorporate sensory feedback:

$$\dot{o}_i = \frac{1}{\tau_i} o_i(1-o_i)\left(\sum_{j=1}^{N} w_{ji}o_j - \sigma^{-1}(o_i) + \theta_i + I_i + w_{ai}o_{S_a} + w_{fti}o_{S_{ft}}\right)$$

where sensory weight signs are constrained: $w_{ai}, w_{fti} \in {-1, 0, 1}$.

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Network Architectures

Two neural controller architectures are explored:

1-Neuron CTRNN

A single recurrently connected neuron drives all three effectors and receives input from both sensors. All 3⁵ = 243 possible sensorimotor configurations are exhaustively evaluated.

2-Neuron CTRNN

Two mutually connected neurons, each with its own connections to effectors and sensors. The combined effector output is capped to the range [-1, 1] to prevent "cheating" via additive output amplification.

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Evolutionary Search

Parameters are optimized using a real-valued evolutionary algorithm (TSearch):

Parameter	1-Neuron	2-Neuron
Generations	10,000	1,000
Population size	5,000	500
Mutation variance	0.2	0.2
Elitism fraction	0.05	0.05
Time constants τ	[1.0, 10.0]	[1.0, 10.0]
Weights w, biases θ	[-8.0, 8.0]	[-8.0, 8.0]
Sensory weights	[1.0, 8.0]	[1.0, 8.0]
Step size	0.01	0.01
Run duration	220	220

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Results

1-Neuron

• Only 27 of 243 configurations achieved forward motion.
• A positive foot effector connection (+E_ft) was necessary, along with either a positive forward or negative backward effector connection.
• A negative foot sensor and positive angle sensor connection produced the greatest fitness gains.
• Optimal configuration: +E_ft, +E_f, -E_b, -S_ft, +S_a

The optimal 1-neuron walker achieves oscillatory locomotion through the mechanical snapback mechanism rather than neural oscillation — demonstrating that body–environment coupling alone can generate rhythmic behavior.

2-Neuron

The best 2-neuron walker used two neurons with complementary connection patterns:

• Neuron 1: +E_ft, +E_f, -E_b, +S_ft, +S_a
• Neuron 2: +E_ft, -E_f, +E_b, -S_ft, -S_a

The two neurons appear to act antagonistically, with one neuron driving stance-phase behavior and the other driving swing-phase behavior.

Generalized Observation

For an N-neuron network to produce forward locomotion, the following conditions must hold across neurons:

$$\sum_{i=1}^{N} w_{ift} \geq 0 \quad \wedge \quad \sum_{i=1}^{N} w_{if} > \sum_{i=1}^{N} w_{ib}$$

Project Structure

SingleLeggedWalker/
├── main.cpp          # Entry point: sets up and runs evolutionary search
├── LeggedAgent.cpp   # Walker body simulation (physics + sensors/effectors)
├── LeggedAgent.h
├── CTRNN.cpp         # Continuous Time Recurrent Neural Network
├── CTRNN.h
├── TSearch.cpp       # Evolutionary search algorithm
├── TSearch.h
├── VectorMatrix.h    # Vector/matrix utilities
├── random.cpp        # Random number generation
├── random.h
└── Makefile

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Building & Running

make
./main

Output is written to standard out. Redirect to a file as needed:

./main > results.dat

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Background & References

This work builds on a lineage of minimally cognitive walker experiments using CTRNNs:

• Beer, R. D. (2010). Fitness space structure of a neuromechanical system.
• Poulet, J. F., & Hedwig, B. (2003). A corollary discharge mechanism modulates central auditory processing in singing crickets.

Previous models studied motor integration with limited or no sensory feedback, and when sensors were included, they were restricted to angle input with hand-coded phase-switching logic. This project is the first to exhaustively evaluate the full space of sensorimotor configurations for this walker body.

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License

MIT — see LICENSE for details.