The convergent evolution of animals and robots makes it possible to find principles of locomotion without contingent dependence. (A) A maned wolf and a frogfish. Both species use the ground for locomotion in their environments. (B) Schematic illustration showing the symmetry and modularity of the maned wolf and frogfish bodies. The modules (blue outlined in red) are the joining connected body parts that move together during locomotion. The symmetry plane (green) indicates the bilateral symmetry of the body. Modularity and symmetry are present in phylogenetically distant species. Therefore, they are candidates for being necessary features (principles) of directed locomotion on the ground. (C) Convergent evolution of organisms with no common ancestor (i.e., animals and robots) and just one common behavior (i.e., locomotion) can be used to find principles of this behavior. It allows differentiating a necessary feature for enhanced locomotion from features resulting from the other animal’s functions or contingent on Earth’s evolutionary history. (D) The green and blue-filled circles represent the shape features of terrestrial and aquatic animals with directed locomotion on the ground. The green, orange, and blue unfilled circles represent, respectively, the shape features of robots with directed locomotion on the ground subject to 9.81m/s2 (Earth’s gravity), 3.721 (Mars’s gravity), and 0.1 (gravity plus buoyancy inside the water) acceleration towards the floor. The red intersection at the center represents the convergent features that indicate principles expected to be valid in different gravitational environments and organisms as different as animals and robots. (E) A voxel is the unit constituting the robots. The voxels are connected in different configurations (defining the body shape) and oscillate their volume independently (defining the body control), forming a mass-spring oscillating system. (F) Example of a Voxel-based Soft Robot. The maximum dimension that a robot can occupy is the 43 voxel’s space. (G) We simulate the robots in a physical environment that evaluates effects like gravity, friction, and floor stiffness in the mass-spring system of voxels. (H) The generation cycle is the unit of the robots in silico evolutionary process. The phenotype evaluation is the average speed of the robot in its environment calculated in the 30s of simulation (directed locomotion ability). The best robots are selected, and their genotype (CPPN’s networks) are mutated to produce a new generation of robots that will constitute the next initial population. (I) In each environment (water, mars, and earth), an evolutionary process of 1500 generation cycles results in a sample of robots with different bodies (shape and control) and performances. The shape and control features of the robots are analyzed looking for principles.

An intermediate number of modules sparsely connected is necessary for achieving higher directed locomotion performance. (A) The modules are groups of connected and synchronized voxels of a robot. Using the modules, we created two topological representations of the robot (TP1 and TP2). TP1 has information about the mean phase (represented by the color) and the position of each module. The size of each module is relative to the number of voxels it contains. Modules that touch the ground are represented as triangles, while those that do not are circles. TP2 representation shows the modules and their connections without adding any further information. (B) Robot’s modularity examples with the number of modules and the connectivity measure (AvgDegree). (C) Illustration of the hypothesized impact of the number of modules on the directed locomotion ability. The robots with the 20% top directed locomotion ability (red area) are the ones that maximize both the body’s degree of freedom and its coordination capacity by having an intermediate number of modules. (D) The results of the three environments confirm the hypothesis: the best robots (100-80% fitness layer in red) typically have an intermediate number of modules (more than one but less than 17). The module number range increases as the fitness layers worsen, representing the broad search. (E) Illustration of the hypothesized impact of the module’s connectivity (measured by the network average degree defined in Equation 1) on the directed locomotion ability. The robots with the 20% top directed locomotion ability (red area) maximize both the body’s degree of freedom and coordination capacity by having sparsely connected modules. (F) The results of the three environments confirm the hypothesis that robots with 20% better directed locomotion ability (100-80% fitness layer) typically have an average degree between 1.2 and 4 (sparsely connected). Robots with modules densely connected (more than four connections per module on average) do not have so good performances.

A highly symmetric shape and symmetry breaking in the dynamics are necessary for achieving higher displacements. (A) Illustration example of voxels with shape and control symmetry in the y-axis. (B) Examples of robot’s shape and control symmetry measures in the x-axis (X), y-axis (Y), and their mean value (XY). (C) Illustration of the hypothesized impact of XY shape symmetry on the directed locomotion ability. A poor body weight balance increases the body’s chance of falling or making a curved path. The green area allows bilateral symmetry in the body (100% symmetry at least on one axis). (D) In the results of the three environments, the robots with 20% better directed locomotion ability (100-80% fitness layer in red) typically have a shape symmetry higher than 0.5. The lower fitness layer contains a higher range of symmetry values. (E) Illustration of the hypothesized impact of XY control symmetry on the directed locomotion ability. The body synchronicity between its parts is higher with increased XY control symmetry. However, locomotion depends upon instability (break of symmetry) in the direction of the movement. Thus, the robots with top 20% performance (red) maximize synchronicity but necessarily keep a break of total control symmetry in the body. (F) In the results of the three environments, the robots with 20% better performance (red) have a peak shifted to higher values of symmetry and a break of total control symmetry (XY control symmetry<1). (G) The bilateral symmetry is the type of shape symmetry most frequent in the top robots (100-80% fitness layer). Biradial symmetry is also present. (G) The bilateral symmetry is also the most frequent type of control symmetry in the top robots (100-80% fitness layer). Biradial control symmetry is not present in this layer.

Robots from different gravitational environments require distinct features for good locomotion performance. (A) Will the best robots in water be the best when transferred to mars or earth? (B) Transference protocol. We optimize the best 50 body shapes (just shapes without control) of each original environment in the new environment. In each new gravitational environment, the fixed shapes could test different controls of their body shape and keep the best. (C) Illustration of a hypothetical transference outcome in which the quality of transference (the ability of directed locomotion in the new environment) will depend on the gravitational difference between the new and original environments. (D) The transferred robots’ average new environment performance (directed locomotion ability). Robots originally from water (blue) cannot acquire a performance as good as the best robots originally from mars (orange) and earth (green) in the mars and earth new environment. Robots originally from mars and earth have worse locomotion when transferred to water and cannot move as well as a robot originally from water.

The relative volume of the feet affects robot performance when it goes to higher gravitational environments. (A) The voxels belonging to modules touching the ground during the robot’s movement are called feet voxels. (B) The distribution of feet voxels proportion in the body differs between the environments (*p < 0.01, ANOVA) and tends to smaller values when gravity increases. (C) In the environmental transitions that gravity increases - water to mars (dark blue), water to earth (light blue), and mars to earth (red) - the robots with a small proportion of feet voxels maintain their performance better (smaller difference between the new and the original environment). Spearman correlation coefficients of r = -0.39 (water to mars), r = -0.43 (water to earth), and r = -0.32 (mars to earth), all with p < 1e-08.

The results of the three experiments (63 CPPN, 83 CPPN, 43 DE) confirm the results of 43 CPPN that the best robots (100-80% fitness layer in red) typically have an intermediate number of modules and average degree compared to the other layers. Thus, the principles found seem to hold for different genotypes and sizes of robots.

The results of the three experiments (63 CPPN, 83 CPPN, 43 DE) confirm the results of 43 CPPN that the best robots (100-80% and 80-60% fitness layers) restricts to a small amplitude of values compared to the worst layers (40-20% and 20-0%). We did a bootstrap of the sample for each environment and each experiment, with N = 10000. We plotted the difference of the extreme values (amplitude) for each layer distribution. The points are the mean value of the distribution of amplitude values after N=10000 bootstrapping.

Alternative limbs configuration for bilateral quadrupeds.

The three experiments (63 CPPN, 83 CPPN, 43 DE) confirm the results of 43 CPPN that the best robots (100-80% fitness layer in red) typically are shifted to higher shape and control symmetry values compared to the other layers. Besides, in all the cases, there is a break of total symmetry in the control symmetry. Thus, the principles found are true even using different genotypes and robot sizes.