Sample Efficient Learning of Body-Environment Interaction of an Under-Actuated System

TL;DR

Locomotion in highly dissipative environments can be captured by motility maps within geometric mechanics, linking body velocity $v_b$ to shape changes via the Reconstruction Equation $v_b = A(r)\dot r + \mathbb{I}^{-1}(r)p$. The paper assesses four learning strategies—TLS, TLS+SUDS, GMR, and GMR+SUDS—and introduces Augmented Gaussian Branching Regression (A-GBR) to achieve sample-efficient motility-map estimates for underactuated systems. Using a 4-flipper, 3-segment granular swimmer, the study shows that GMR-based methods generally outperform TLS when data are plentiful, and incorporating SUDS improves data efficiency across regimes, with GMR+SUDS often delivering the best accuracy given sufficient data. These findings highlight how model expressiveness and data availability jointly shape predictive performance, informing the design of robust, adaptable locomotion strategies for robots operating in complex environments.

Abstract

Geometric mechanics provides valuable insights into how biological and robotic systems use changes in shape to move by mechanically interacting with their environment. In high-friction environments it provides that the entire interaction is captured by the ``motility map''. Here we compare methods for learning the motility map from motion tracking data of a physical robot created specifically to test these methods by having under-actuated degrees of freedom and a hard to model interaction with its substrate. We compared four modeling approaches in terms of their ability to predict body velocity from shape change within the same gait, across gaits, and across speeds. Our results show a trade-off between simpler methods which are superior on small training datasets, and more sophisticated methods, which are superior when more training data is available.

Figures (6)

• Figure 1: 4-flipper, 3-segment swimmer. Picture of the swimmer in the granular medium (left), and schematic showing markers and angles (right). If the robot axes were all perfect hinge joints, the shape of the robot was fully characterized by the two motor angles $\phi_1,\phi_2$, and the flippers angles $\alpha_{1,2},\beta_{1,2}$. We measured the swimmer’s pose using 11 markers: 4 on the tail link --- two on its top face (red, green dots), one raised on a spoke, and one lowered and attached to the back edge (teal, orange half-circles); 3 on the head link -- two on the top face (yellow, gray), and one raised (magenta); one each on each of the flippers (gunmetal blue, cyan, purple, and brown).
• Figure 2: Commanded and measured theoretical gaits. We plotted the commanded $(\phi_1, \phi_2)$ values (solid lines) and the measured angles (dots, same color). For scale, we included a circle of radius 1 radian (dashed black). Mild diamond and square (D-,S-) built by: $\phi_1=\cos(\theta)\pm\cos(3\theta)/9,\phi_2=\sin(\theta)\mp\sin(3\theta)/9$, and extreme diamond and square (D+,S+) built by: $\phi_1=\cos(\theta)\pm\cos(3\theta)/4,\phi_2=\sin(\theta)\mp\sin(3\theta)/4$
• Figure 3: Prediction error by frequency similarity and model type. We plotted box plots (default matplotlib parameters) of each category: same frequency error distributions (right), and different frequency error distributions (left).
• Figure 4: Prediction error by gait similarity and model type. We plotted box plots (default matplotlib parameters) of log RMS error distributions each category: same gait (right), near gait (middle), and far gait (left).
• Figure 5: Prediction error by combined frequency similarity and gait similarity, by model type. We plotted box plots (default matplotlib parameters) of log RMS error distributions each category.