Geometrically optimal gaits: a data-driven approach

Abstract

The study of optimal motion of animals or robots often involves seeking optimality over a space of cyclic shape changes, or gaits, specified using a large number of parameters. We show a data-driven method for computing the gradient of a cost functional with respect to a large number of gait parameters by employing geometric properties of the dynamics to efficiently construct a local model of the system, and then using this model to rapidly compute the gradients. Our modeling step specifically applies to systems governed by connection-like models from geometric mechanics, which encompass a number of high-friction regimes. We demonstrate using our method for optimizing gaits under noisy, experiment-like conditions by simulating planar multi-segment serpent-like swimmers in a low Reynolds number (viscous friction) environment. Our optimization results recover known results for 3-segment swimmers with a 66 dimensional gait parameterization, and extend to optimizing the motion of a 9 segment swimmer with a 264 dimensional gait space, using only 30 simulation trials of 30 gait cycles each. The data-driven geometric gait optimization approach we present is designed to operate on noisy, stochastically perturbed dynamics—as noisy and variable as experimental data—and efficiently optimize a large number of parameters. We believe this approach has the potential to significantly advance our ability to optimize robot gaits with hardware in the loop and to study the optimality of animal gaits with respect to hypothesized cost functions.

Appendix A: Regression for estimating the cost metric

The regression for computation of the metric $\mathcal {M}$ is centered about $\theta$ and $\dot{\theta }$, similarly to the construction of the regression for the connection $\mathbf {A}$. The metric approximation now takes the form:

$$\begin{aligned} \dot{s}^2_n \sim ({\dot{\theta }_n + {\dot{\delta }}_n}^\mathsf {T}) \left[ \mathcal {M}+ \left( \frac{\partial \mathcal {M}}{\partial {r}_j}\right) \delta ^j_n \right] (\dot{\theta }_n + {\dot{\delta }}_n) \end{aligned}$$

(23)

leading to the regression:

$$\begin{aligned}&\begin{bmatrix} \dot{s}^2_{_1} \\ \vdots \\ \dot{s}^2_{_N} \end{bmatrix} = \begin{bmatrix} 1,&{\dot{\delta }}_1,&{\dot{\delta }}_{_1} \hat{\otimes }\, {\dot{\delta }}_{_1},&\delta _1,&{\delta }_{_1} \otimes \, {\dot{\delta }}_{_1},&{\delta }_{_1} \otimes \, {\dot{\delta }}_{_1} \hat{\otimes } {\dot{\delta }}_{_1}\\ \vdots&\vdots&\vdots&\vdots&\vdots \\ 1,&{\dot{\delta }}_N,&{\dot{\delta }}_{_N} \hat{\otimes }\, {\dot{\delta }}_{_N},&\delta _N,&{\delta }_{_N} \otimes \, {\dot{\delta }}_{_N},&{\delta }_{_N} \otimes \, {\dot{\delta }}_{_N} \hat{\otimes } {\dot{\delta }}_{_N} \end{bmatrix} \cdot {R}^\mathsf {T}, \end{aligned}$$

(24)

$$\begin{aligned}&R = \begin{bmatrix} \widehat{\mathcal {M}_{i,j}\dot{\theta }^i\dot{\theta }^j},&\widehat{\mathcal {M}_{i,j} \dot{\theta }^i},&\widehat{\mathcal {M}_{i,j}},&\widehat{\frac{\partial \mathcal {M}_{i,j}}{\partial {r}_k} \dot{\theta }^i\dot{\theta }^j},&\widehat{\frac{\partial \mathcal {M}_{i,j}}{\partial {r}_k} \dot{\theta }^j},&\widehat{\frac{\partial \mathcal {M}_{i,j}}{\partial {r}_k}} \end{bmatrix}\nonumber \\ \end{aligned}$$

(25)

Here the modified exterior product $\hat{\otimes }$ includes only the upper triangular elements, i.e., $x\hat{\otimes }\,y = [\ldots , x_i y_j, \ldots ]$ s.t. $i\le j$. Following (14), at each m we solved for $1+d+\left( {\begin{array}{c}d\\ 2\end{array}}\right) +d+d^2+d\left( {\begin{array}{c}d\\ 2\end{array}}\right) \approx \frac{1}{2}d^3$ unknowns to construct our model of the metric.