Locomotion Selects for Symmetry:
Evolutionary Co-Design of Legged Robots on an Icosahedral Body

Jiong Lin1 · Yue Yao1 · Hod Lipson1

1 Creative Machines Lab, Columbia University

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Design space
Environment

One recorded 20 s rollout per clip, replayed through MuJoCo forward kinematics · abstract space = the original study’s clips (Genesis, position-controlled servos, capsule bodies) · fabricable space = the hw17 build (v50 printed parts, XM430 motor model, domain randomisation) trained in icos_v2, on flat ground and on the quantised 30 mm rough field

The 18 morphologies

Each body in both design spaces. Left: the abstract-space clip from the original study (Genesis, position-controlled servos). Right: the same morphology as the hw17 build trained in this work, on flat ground and on the 30 mm rough field. Numbers are the 20 s displacement of the recorded clip. Click a thumbnail to load it in the viewer.

Figures

The figures of the abstract, with self-contained captions for readers who arrive here without the paper. Plotting scripts are in vis/.

The same tripod candidate against the 15 reflection planes of the icosahedron
Fig. 1 — How symmetry is measured. The robot body is a regular icosahedron; any of its 20 faces can carry a leg. A regular icosahedron has exactly 15 mirror planes, so a body's symmetry can be checked plane by plane. Each panel shows the same three-legged candidate against one of the 15 planes (the green ring is the plane). A leg face is drawn blue when the reflection maps it onto another leg face and red when its mirror image lands on an empty face; grey faces carry no leg, and the digit on a face is that leg's number of joints. Two scores are written above each panel: F counts leg faces whose mirror image is not a leg face (placement mismatch), and J repeats the count at the level of joints and hinge-axis patterns (motor-level mismatch). A plane with F = J = 0 is an exact mirror plane of the body — here P7. The population curves in Fig. 3 use these scores averaged over all 15 planes.
Abstract-simulator candidates, realistic-simulator champions, and the champions walking on rough terrain
Fig. 2 — The two simulators and the evolved bodies. Top row: four candidates from one evolving population in the abstract simulator, where legs are capsule links driven by ideal position servos and every body starts from a straight-leg pose. Middle row, left to right: the champions evolved in the realistic simulator, rendered from the fabricable CAD (3D-printed leg modules and Dynamixel XM430-W350 servos) in their solved standing pose — the three-legged tripod (4,4,4), which has a mirror plane; the four-legged (3,3,3,3), which has none; and the (4,4,2,2) and (3,3,2,2,2) champions. The numbers are joints per leg. Bottom row: the tripod (left pair) and the (3,3,3,3) champion (right pair) walking on 30 mm quantized-noise terrain with policies trained on that terrain — two frames of each rollout.
Symmetry of the elites per generation in six evolutionary runs, against drift baselines
Fig. 3 — Does selection make populations more symmetric? Each panel tracks the symmetry of the best candidates of every generation in one evolutionary run; left column the face-placement score F, right column the motor-level score J, both averaged over the 15 planes, lower means more symmetric. Dots are individual candidates (thinned for legibility), the solid line is a linear fit over all candidates with its 95 % confidence band, and the dashed line is the drift baseline: the symmetry the same evolutionary operators produce under random fitness, i.e. without any selection for walking, matched to the species present in the set. Red: abstract simulator, mixed competition — one shared population of 32 over 95 generations (top-16 elites, then only its tripods). Green: abstract simulator, speciated evolution — seven leg topologies evolving side by side for 40 generations (top-10 elites, then the tripod species). Blue: realistic simulator — the tripod and the (3,3,3,3) topology re-evolved for 50 and 48 generations (top-8 elites). In the abstract simulator every fitted slope is negative and the elites end below drift; in the realistic simulator the elites sit only slightly below drift and the slopes are flat.
Displacement of every trained candidate per generation in mixed competition
Fig. S2 — Fitness in mixed competition. Every candidate trained in the abstract-simulator mixed-competition run, plotted by generation and coloured by leg count (green triangles 3 legs, orange squares 4, blue diamonds 5, red hexagons 6). Faded markers are newcomers screened with a 10-million-step training budget; solid markers were trained to 50 million steps. The dashed line is the best elite so far and the solid grey line the mean of the elite set: the best body reaches 9.1 m by generation 4, 10.9 m by generation 9 and 11.5 m at generation 46, and three-legged bodies occupy the top of the distribution from the first generations.
Share of elite slots per topology and per leg count over generations
Fig. S3 — The tripod takes over. Share of the 16 elite slots held by each leg topology (top) and by each leg count (bottom) in the same mixed-competition run. The first generation contains all seven topologies; five- and six-legged bodies leave the elite set within two generations, the four-legged topologies (4,4,2,2) and (4,3,3,2) hold a shrinking share until generation 20–30, and from generation 31 every elite slot is a tripod (4,4,4). This takeover is why the mixed-competition symmetry curves in Fig. 3 are compared with a drift baseline that follows the species mix of the elite set.