# Intermediate Encoding Layers for the Generative Design of 2D Soft Robot Actuators: A Comparison of CPPN’s, L-Systems and Random Generation

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methods and Materials

#### 2.1. Test Case

#### 2.2. Encoding Methods

#### 2.2.1. Random Units

#### 2.2.2. Lindenmayer Systems

#### 2.2.3. Compositional Pattern Producing Networks

## 3. Results and Analysis

- 1.
- High constraint energy, high internal energy—Units that do not deform to the desired shape and require significant work to maintain that shape.
- 2.
- High constraint energy, low internal energy—Units that do not deform to the desired shape but do not require much work to maintain that shape.
- 3.
- Low constraint energy, low internal energy—Units that deform to the desired shape and do not require much work to maintain that shape.
- 4.
- Low constraint energy, high internal energy—Units that deform to the desired shape but require significant work to maintain that shape.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A schematic representation of the 2D grid domain for this work. We show a $5\times 5$ grid of regular square elements on the left. The outer edge elements form an exclusion zone that enforces a fully contained internal cavity for all candidate units. Elements in the exclusion zone are always material filled. In the central $3\times 3$ region, any element can be either a void or filled with material. On the right, we show how a single unit can be nested into a regular grid of similar units as a larger meta-unit.

**Figure 2.**Three simple deformation patterns considered. Each of the three shows an initially regular square unit (solid black line) deforming on inflation (blue dashed line). The top left shows single-axis elongation, the top right shows two-axis elongation and the bottom shows “shear”. A, B, C and D indicate the corners of the initial square unit.

**Figure 3.**

**Right**: A representative 5 × 5 unit implemented as a FE model, with prescribed displacement on all four edges (blue, green, red and yellow arrows).

**Left**: A representative 15 × 15 unit implemented as a FE model, with a single point fixed in translation and rotation (bottom left corner, pink arrows), and internal pressure (green arrows).

**Figure 4.**

**Left**: The re-configurable mould used to manufacture sample units. The interior of the mould has mounts for square blocks placed at the coordinates of voids.

**Right**: The test fixture for pressurising sample units. The unit is centred over the inflation port and placed between two thick perspex plates.

**Figure 5.**Photograph of a pressurised silicone sample scaled and overlayed with the results of a FE simulation of the same unit at the same pressure.

**Figure 6.**A graphical representation of the axiom, the first and second iteration of the L-System described in Table 1.

**Figure 7.**L-System interpretation of symmetry axioms according to Table 2. Axes of symmetry are indicated with dotted lines. (

**a**) Horizontal rotation. (

**b**) Vertical rotation. (

**c**) Horizontal and vertical rotation. (

**d**) Diagonal rotation. (

**e**) Negative diagonal rotation. (

**f**) Diagonal and negative diagonal rotation. (

**g**) Horizontal reflection. (

**h**) Vertical reflection. (

**i**) Horizontal and vertical reflection. (

**j**) Diagonal reflection. (

**k**) Negative diagonal reflection. (

**l**) Diagonal and negative diagonal reflection.

**Figure 8.**Simple example patterns generated using a CPPN encoding layer interpreted into the 2D Boolean domain used in this paper.

**Figure 9.**Histograms showing the distribution of unit constraint energy for 1000 units generated by a GA using either random, L-system and CPPN intermediate encoding layer. (

**a**) Random. (

**b**) L-System. (

**c**) CPPN.

**Figure 10.**Histograms showing the distribution of unit internal energy for 1000 units generated by a GA using either random, L-system and CPPN intermediate encoding layers. (

**a**) Random. (

**b**) L-System. (

**c**) CPPN.

**Figure 11.**Comparison of 1000 generated units in terms of internal and constraint energy for random, L-system and CPPN intermediate encoding layers. (

**a**) Random. (

**b**) L-System. (

**c**) CPPN.

**Figure 12.**Representative high-performing units generated with Random, L-system and CPPN intermediate encoding layers. (

**a**) Random. (

**b**) L-System. (

**c**) CPPN.

**Table 1.**Example L-System construction for a simple pattern in a grid domain. Starting with the axiom “F”, at each iteration, “F” is replaced by “F[−fF][+fF]”. Constants “[, ], +, −, and f” are not replaced in successive iterations. Including an interpretation layer, we can traverse the domain and remove material iteratively. “F” represents the removal of material at the current grid coordinate, “f” represents movement in the direction of the current travel direction, “[, ]” represents storing the current and retrieving the current grid position and travel direction from a last in, first out stack and “+, −” represent rotation of the current travel direction by an angle of ±45${}^{\circ}$.

Variables: | F |

Constants: | [, ], +, −, f |

Axiom: | F |

Production Rules: | F → F[−fF][+fF] |

Symmetry Axis | Rotational Axiom | Reflective Axiom |
---|---|---|

Horizontal | [F]++++[F] | [F]++++(F) |

Vertical | − −[F]++++[F] | − −[F]++++(F) |

Horizontal and vertical | [F]++[F]++[F]++[F] | [F]++(F)++[F]++(F) |

Diagonal | +[F]++++[F] | +[F]++++(F) |

Negative diagonal | −[F]++++[F] | −[F]++++(F) |

Diagonal and negative diagonal | +[F]++[F]++[F]++[F] | +[F]++(F)++[F]++(F) |

**Table 3.**Parameters and ranges available to the GA when designing using L-system intermediate encoding layer.

Parameter | Min | Max |
---|---|---|

Num Axioms | 1 | Number of predefined axioms |

Number of rules | 1 | Total domain |

Rule length | 2 | 5 |

Number of iterations | 1 | 5 |

**Table 4.**Parameters and ranges available to the GA when designing using CPPN intermediate encoding layer.

Parameter | Min | Max |
---|---|---|

Num of Hidden layers | 2 | 10 |

Size of first layer | 2 | 32 |

Element removal threshold | 0 | 100 |

**Table 5.**Performance comparison of units generated using Random, L-system and CPPN intermediary encoding layers in a GA. The table shows the function evaluations required to converge to a stable solution from the same starting point with the same GA settings.

Method | Function Evaluations | Standard Deviation |
---|---|---|

Random | 945 | 128 |

L-System | 398 | 198 |

CPPN | 650 | 54 |

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**MDPI and ACS Style**

Venter, M.P.; Conradie, N.T.
Intermediate Encoding Layers for the Generative Design of 2D Soft Robot Actuators: A Comparison of CPPN’s, L-Systems and Random Generation. *Math. Comput. Appl.* **2023**, *28*, 68.
https://doi.org/10.3390/mca28030068

**AMA Style**

Venter MP, Conradie NT.
Intermediate Encoding Layers for the Generative Design of 2D Soft Robot Actuators: A Comparison of CPPN’s, L-Systems and Random Generation. *Mathematical and Computational Applications*. 2023; 28(3):68.
https://doi.org/10.3390/mca28030068

**Chicago/Turabian Style**

Venter, Martin Philip, and Naudé Thomas Conradie.
2023. "Intermediate Encoding Layers for the Generative Design of 2D Soft Robot Actuators: A Comparison of CPPN’s, L-Systems and Random Generation" *Mathematical and Computational Applications* 28, no. 3: 68.
https://doi.org/10.3390/mca28030068