Project Scope: Robot Morphology Optimization

This page records the morphology representation and optimization method currently included in the project. The broader technical survey is used as background; this document identifies the methods that are part of the initial implementation.

Included Methods

Area	Selected method	Project status
Morphology representation	Direct encoding	Included
Direct encoding variant A	Morphology tree	Included
Direct encoding variant B	Connection matrices \(C_a\) and \(C_s\)	Included
Morphology optimization	Genetic Algorithm (GA)	Included

Direct Morphology Encoding

The project represents each candidate robot morphology explicitly using one of two alternative direct encodings. The morphology tree and the connection matrices are not combined into a single representation.

Variant A: Morphology Tree

\[ \Phi_{tree} = G \]

where \(G = (V, E)\) is the morphology tree, with modules or components represented by vertices \(V\) and parent-child structural relationships represented by edges \(E\).

Variant B: Connection Matrices

\[ \Phi_{matrix} = (C_a, C_s) \]

where:

• \(C_a\) represents the actuator or active connection relationships between modules.
• \(C_s\) represents the structural or supporting connection relationships between modules.

Both direct encoding variants are intended to make the physical structure explicit and easy to decode into a robot model. Genetic operators can modify the selected representation directly, subject to validity constraints such as a connected morphology and valid module connections.

The exact contents and dimensions of \(C_a\) and \(C_s\) depend on the available module set and connection interface definitions. Their interpretation will be kept consistent with the robot model and simulator used by the project.

Genetic Algorithm Optimization

The genetic algorithm searches over a population of morphologies encoded with the selected representation variant. Each generation follows the general process below:

1. Initialize a population of valid morphology encodings.
2. Decode each individual into a robot model.
3. Evaluate the morphology using the selected task and objective function.
4. Select higher-performing individuals as parents.
5. Create offspring through crossover and mutation of the selected direct encoding.
6. Repair or reject invalid offspring.
7. Repeat until the stopping criterion is reached.

For an individual morphology \(\Phi_i\), the optimization objective is expressed as:

\[ \Phi^* = \underset{\Phi \in \mathcal{F}}{\operatorname{argmax}}\; F(\Phi) \]

where \(F(\Phi)\) is the task fitness and \(\mathcal{F}\) is the set of valid morphology encodings.

The fitness function may later combine task performance with engineering criteria such as energy consumption, morphology complexity, stability, or hardware cost. The initial implementation should keep the objective explicit so that these terms can be added without changing the representation format.

Included Optimization Components

The initial implementation covers:

• Encoding and decoding the selected direct representation.
• Supporting the morphology-tree and connection-matrix variants independently.
• Generating an initial population of valid candidates.
• Applying selection, crossover, and mutation.
• Evaluating candidate fitness.
• Preserving the best candidate through elitism, where appropriate.
• Enforcing morphology validity after genetic operations.

Currently Out of Scope

The following methods are described in the survey but are not part of the current project scope:

• Indirect encodings such as CPPNs, L-Systems, and cellular encodings.
• Graph grammars and language-model-based morphology generation.
• Deep reinforcement learning for morphology generation.
• Differentiable simulation and gradient-based morphology optimization.
• Diffusion-based morphology generation.
• Bayesian optimization and surrogate-model-based search.
• Joint brain-body optimization beyond the controller or evaluation mechanism required to score a morphology.

These methods can be considered later as alternative representations or optimization baselines. They should not be treated as required components of the initial direct-encoding genetic algorithm implementation.