Generative Evolutionary Design of Voxel-Based Soft Robots with Provable Optimality

I. INTRODUCTION

• MISCO targets VSR design challenges including combinatorially vast search spaces, costly evaluations, and a quality-diversity dilemma; multi-task design automation remains under-explored.Source: §I · p. 1
• MISCO’s MEC-VAE combines task conditioning, position awareness, and inter-voxel signaling, while shared parameters allow design experience to transfer across tasks.The framework also introduces adjustable sampling of suboptimal designs to balance exploitation and exploration.Source: §I · p. 2
• MISCO-A nearly consistently outperforms competitive baselines in optimization efficiency, while MISCO-B improves diversity without sacrificing much efficiency.The experiments span eight Evolution Gym locomotion and manipulation tasks of varying difficulty.Source: §I · p. 2
• MISCO is reported to nearly consistently outperform competitive baselines in optimization efficiency; its biodiversity-focused variant improves diversity without sacrificing much efficiency.Source: §I · p. 2

A. A brief introduction to VSRs

• The introduction describes two-dimensional Evolution Gym robots as 5 × 5 grids whose entries encode empty, rigid, soft, horizontal-actuator, or vertical-actuator voxels.Source: A. A brief introduction to VSRs · p. 3

B. Related work

• Earlier VSR methods include evolutionary search, generative encodings, differentiable methods, and curriculum strategies, but typically lack explicit distributions over promising designs or spatial interaction modeling.Source: B. Related work · pp. 3–4
• Prior work had not integrated NCA into distribution-based evolution, while deep-generative EDA theory and practical management of the quality-diversity trade-off remained open problems.Source: B. Related work · p. 4

A. Overview of MISCO

• MISCO alternates morphology proposal, control optimization and fitness evaluation, adjustable natural selection, and MEC-VAE model updates; the paper states almost-sure convergence to optimal distributions under mild conditions and a linear average convergence rate.Source: A. Overview of MISCO · p. 4

B. Robot proposal

• MEC-VAE’s decoder integrates task awareness, position awareness, and inter-voxel coordination to generate voxel-specific latent representations and sample complete robot designs.A shared output layer produces a material-type distribution at each position.Source: B. Robot proposal · p. 5, 4) Voxel sampling · p. 5

1) Task awareness:

• Each task has a learnable embedding mapped to a Gaussian prior over task-aware latent variables, allowing that task to sample varied morphological prototypes.Source: 1) Task awareness · p. 5

2) Position awareness:

• Learned embeddings of each voxel’s x- and y-coordinates encode its position in the material matrix.Source: 2) Position awareness · p. 5

3) Inter-voxel coordination:

• NCA updates voxel latents through repeated shared neural rules using neighboring states, coordinating local functionalities before the output layer samples voxel materials.The paper describes information exchange between adjacent voxels and iterates the updates for a predefined K steps.Source: 3) Inter-voxel coordination · p. 5, 4) Voxel sampling · p. 5

4) Voxel sampling:

• Voxel-specific task requirements are coordinated through message passing to decode cohesive morphologies, with shared parameters supporting design experience transfer across tasks.The authors state that this modular representation reduces-rank problem formulation and contributes to sample-efficient design.Source: 4) Voxel sampling · pp. 5–6

1) Control optimization:

• PPO trains a controller for each morphology from interaction trajectories, using advantage-based updates constrained by a KL-divergence term.The optimized controller's cumulative reward measures task fitness.Source: 1) Control optimization · p. 6

2) Construction of the global morphology-fitness pool:

• Evaluated morphologies and fitness scores accumulate in a global pool, enabling sample reuse across evolutionary stages and task environments.Fitness is the cumulative reward of a complete trajectory after policy optimization.Source: 2) Construction of the global… · p. 6

D. Natural selection

• A-B balancing selects top-ranked designs for exploitation and fitness-weighted random designs for diversity, with α controlling their relative proportions.Higher α favors top-ranked designs; lower α increases exploration of suboptimal designs.Source: D. Natural selection · p. 6

E. Model update

• MEC-VAE is updated by maximizing an ELBO that combines reconstruction loss and latent-distribution regularization, using reparameterized sampling to propagate gradients through the encoder.The encoder estimates task-latent distributions from morphology and task inputs; the ELBO provides a tractable lower bound when marginal likelihood is intractable.Source: E. Model update · p. 7

1) Asymptotic convergence to optimal solutions:

• Under mild conditions, MEC-VAE's elite set converges almost surely to a set containing an optimal design, while its learned distribution ultimately places nonzero probability only on optimal designs.The theorem allows multiple global optima and does not require a particular probability allocation among them.Source: 1) Asymptotic convergence to optimal… · pp. 7–8, F. Theoretical guarantees of convergence · p. 7

2) Linear convergence rate:

• ACR measures average convergence across generations: ACR = 1 means the optimum has been found, while ACR < 0 indicates worse performance than the initial population.The paper characterizes a positive limiting ACR as linear convergence.Source: 2) Linear convergence rate · p. 8
• If transitions from non-optimal populations to a ρ-promising region have a positive lower-bounded probability over κ generations, MEC-VAE's average convergence rate has a positive asymptotic lower bound.This is the paper's condition for linear convergence of ACR.Source: 2) Linear convergence rate · p. 8

A. Experiment settings:

• The benchmark covers eight EvoGym task instances spanning locomotion and manipulation at varying difficulty levels.The compared methods include GA, BO, CPPN-NEAT, RoboGAN, and MorphVAE.Source: A. Experiment settings · p. 8
• The stated MISCO procedure generates task-specific robots, optimizes controllers to evaluate fitness, then updates MEC-VAE using selected designs and variational inference.The pseudocode identifies A-B balancing as part of the model-update stage.Source: A. Experiment settings · p. 9
• Methods are evaluated until 1,000 robot designs have been assessed, with population size 25 and results averaged over three independent runs.The authors also report comprehensive ablation studies in Supplementary Material F.Source: A. Experiment settings · p. 9

1) Best fitness:

• MISCO-A achieves the best evolutionary outcome on five of eight tasks and remains strongly competitive on the other three.No baseline excels across all tasks; GA performs poorly on some, including Climber-v0.Source: 1) Best fitness · p. 9, Figure 3 · p. 10
• At 1,000 evaluations, MISCO-A ranks among the top three on every task and has the highest average rank, with larger margins on Climber-v0 and PlatformJumper-v0.Where MISCO-A does not rank first, such as DownStepper-v0, the best baseline has only a marginal advantage.Source: 1) Best fitness · p. 9
• MorphVAE performs commendably but underperforms MISCO in most cases, which the authors connect to the importance of problem-oriented design.The passage attributes MorphVAE's commendable performance to its use of variational autoencoders.Source: 1) Best fitness · p. 10
• MISCO-B maintains balanced performance across tasks and is suited to diversity-prioritizing settings, trading some search efficiency for exploration.It can slightly underperform some baselines in a few cases.Source: 1) Best fitness · pp. 10–11

2) Fitness distribution:

• MISCO-A places more probability density in high-fitness regions, especially on DownStepper-v0, Catcher-v0, Pusher-v0, and Carrier-v0, while MISCO-B’s distributions are superior or on par with MISCO-A and baselines.Panel c plots probability density against fitness.Source: 2) Fitness distribution · p. 11, Figure 5 · p. 12

3) Cumulative fitness time average (CFTA) :

• MISCO-A’s CFTA rises steeply and consistently across tasks, outpacing baselines as it exploits and refines top-ranking designs.The authors attribute this pattern partly to MEC-VAE tracking the distribution of high-performing robots for more targeted trial and error.Source: 3) Cumulative fitness time average… · p. 11
• At 1,000 evaluations, both MISCO variants show clear advantages in total average fitness; MISCO-B initially lags but later trends upward and surpasses baselines.Fig. 5(b) compares average fitness across evaluated designs and average task ranks.Source: 3) Cumulative fitness time average… · p. 11, Figure 5 · p. 12

D. Analysis of morphological diversity

• Diversity is measured among designs exceeding the pooled 95th-percentile fitness threshold, using MDS to combine per-position material variability with pairwise morphological distinctiveness.For 5 × 5 VSRs with five material types, the two MDS factors are equally weighted.Source: D. Analysis of morphological diversity · pp. 11–12
• MISCO-B achieves the highest average diversity rank, while both MISCO variants produce diverse designs; the authors attribute MISCO’s balance to MEC-VAE fitting elite-sample distributions.The reported gain over MorphVAE is attributed to the tailored architecture’s capacity to account for VSR interaction dynamics.Source: D. Analysis of morphological diversity · pp. 12–13, Table I · p. 13

V. DISCUSSION

• The discussion reports that adding suboptimal morphologies stimulates exploration and leads to leading morphological diversity without significantly sacrificing optimization efficiency.The paper also reports theoretical convergence analysis and more efficient optimization than baselines, particularly on complex tasks, alongside stable evolutionary trajectories.Source: §V · p. 13
• The authors identify model interpretability through inter-voxel connection strengths as a future research direction.They also propose exploring modular generative representations for both design and control.Source: §V · p. 13

Funding declaration

• The work was supported by the listed Chinese research institutions, foundations, and public computing resources.The declaration names the National Natural Science Foundation of China and other institutional support.Source: FUNDING DECLARATION · p. 13

Proofs of Theorems 1 and 2

• With probability 1, MISCO obtains an optimal solution as generations tend to infinity, and the population distribution converges to the optimal distribution.Once an optimum is found, it remains in elite samples; the proof also shows the probability of non-optimal designs tends to zero.Source: FUNDING DECLARATION · p. 14
• After an optimal design is found, selection pressure contracts the probability of every non-optimal design.The proof states that for each non-optimal x, Cτ+1(x) ≤ d · pτ(x) for some d < 1.Source: FUNDING DECLARATION · p. 14

Empirical evidence of structures shared across tasks

• Elite robot morphologies from three algorithms were analyzed across tasks using 450 samples per task, with t-SNE visualizations indicating shared structures and potential cross-task synergies.The samples came from the top 5% of 1,000 evaluations per task in GA, BO, and CPPN-NEAT experiments, each repeated three times.Source: FUNDING DECLARATION · p. 15
• Perplexity values were all below 1 and mostly near 1, indicating both task separation and overlapping morphological features.Walker-v0 and DownStepper-v0 were particularly separated; the authors identify task awareness as important despite shared structures.Source: FUNDING DECLARATION · p. 15
• MEC-VAE’s 128 task-latent dimensions use low KL divergence for shared information and high divergence for task-specific variation.The analysis interprets this pattern as balancing shared representations with task-level specialization.Source: FUNDING DECLARATION · p. 15

A. Climber-v0

• Climber-v0 asks the robot to climb as high as possible through a flat, vertical channel, and is classified as medium difficulty.Its reward favors movement in the positive y-direction.Source: A. Climber-v0 · pp. 15–16
• Figure 6 presents cross-task design overlap, morphology perplexity, and dimensionwise average KL divergence for the task latent variables.Its bubble plot covers 128 dimensions; bubble size and color intensity encode divergence magnitude.Source: Figure 6 · p. 16
• Figure 7 illustrates various tasks in the EvoGym environment.The caption does not specify individual task encodings or outcomes.Source: Figure 7 · p. 16

B. DownStepper-v0

• DownStepper-v0 requires a robot to descend stairs of varying lengths and is classified as easy.The task observation includes the robot’s velocity, orientation, contact information, and terrain-height information.Source: B. DownStepper-v0 · p. 16
• Movement in the positive x-direction is rewarded in DownStepper-v0.The robot also receives a one-time reward of 2 for reaching the terrain’s end.Source: B. DownStepper-v0 · p. 16
• Rotating more than 90 degrees from the original orientation triggers a one-time penalty of −3 and resets the environment.Source: B. DownStepper-v0 · p. 16

C. Carrier-v0

• Carrier-v0 is an easy task in which the robot catches a box above it and carries it as far as possible, with a penalty for dropping the box below a threshold height.The robot and box are rewarded for moving in the positive x-direction.Source: C. Carrier-v0 · p. 16

D. Pusher-v0

• Pusher-v0 is an easy task in which the robot pushes a box initialized in front of it, rewarding positive x movement and penalizing separation in that direction.Source: D. Pusher-v0 · pp. 16–17

E. Catcher-v0

• Catcher-v0 is a hard task in which the robot catches a fast-moving, rotating box, with reward for moving toward it and a penalty for dropping it below a threshold height.Source: E. Catcher-v0 · p. 17

F. PlatformJumper-v0

• PlatformJumper-v0 traverses floating platforms at different heights, rewards positive x-motion, and applies a one-time -3 penalty for rotating over 90 degrees or falling.Falling triggers an environment reset.Source: F. PlatformJumper-v0 · p. 17

G. Flipper-v0

• Flipper-v0 asks the robot to perform as many counter-clockwise flips as possible on flat terrain, with reward for counter-clockwise rotation.Source: G. Flipper-v0 · p. 17

H. Jumper-v0

• Jumper-v0 asks the robot to jump as high as possible in place; its reward favors positive y-motion and penalizes x-motion.Source: H. Jumper-v0 · p. 17
• Across almost all eight tasks, removing individual components degrades performance; on Flipper-v0, an ablated MISCO-B beats the complete model but not MISCO-A.The exception does not overturn MISCO-A's higher performance in that task.Source: SUPPLEMENTARY MATERIAL E · p. 18
• MISCO-A reaches optimal solutions more efficiently, while MISCO-B emphasizes exploration and diversity, making the variants alternatives for different task priorities.MISCO-B's selection randomness is controlled by a tunable hyperparameter.Source: SUPPLEMENTARY MATERIAL E · p. 18

• MEC-VAE training uses 50 robot samples per update, with updates per generation increasing linearly from 50 to 250; NCA uses 30 coordination steps.Source: IMPLEMENTATION DETAILS · p. 18
• PPO optimizes controllers and evaluates robot fitness, while the actor, critic, and MEC-VAE MLPs each use two 64-unit Tanh hidden layers.The variance-generating MLP applies an exponential output activation to ensure non-negativity.Source: IMPLEMENTATION DETAILS · pp. 18–19
• Experiments ran on an Ubuntu 22.04 server with Intel Xeon processors and four NVIDIA Tesla RTX GPUs; code is available for reproducibility.Source: IMPLEMENTATION DETAILS · p. 18
• Supplementary Tables III–V list the VSR design problem, model hyperparameters, and MLP architectures, respectively.Source: IMPLEMENTATION DETAILS · p. 19, Table III · p. 19, Table IV · p. 19