Optimal control is a popular approach to synthesize highly dynamic motion. Commonly, $L_2$ regularization is used on the control inputs in order to minimize energy used and to ensure smoothness of the control inputs. However, for some systems, such as satellites, the control needs to be applied in sparse bursts due to how the propulsion system operates. In this paper, we study approaches to induce sparsity in optimal control solutions -- namely via smooth $L_1$ and Huber regularization penalties. We apply these loss terms to state-of-the-art DDP-based solvers to create a family of sparsity-inducing optimal control methods. We analyze and compare the effect of the different losses on inducing sparsity, their numerical conditioning, their impact on convergence, and discuss hyperparameter settings. We demonstrate our method in simulation and hardware experiments on canonical dynamics systems, control of satellites, and the NASA Valkyrie humanoid robot. We provide an implementation of our method and all examples for reproducibility on GitHub.

Parametric statistical models that are implicitly defined in terms of a stochastic data generating process are used in a wide range of scientific disciplines because they enable accurate modeling. However, learning the parameters from observed data is generally very difficult because their likelihood function is typically intractable. Likelihood-free Bayesian inference methods have been proposed which include the frameworks of approximate Bayesian computation (ABC), synthetic likelihood, and its recent generalization that performs likelihood-free inference by ratio estimation (LFIRE). A major difficulty in all these methods is choosing summary statistics that reduce the dimensionality of the data to facilitate inference. While several methods for choosing summary statistics have been proposed for ABC, the literature for synthetic likelihood and LFIRE is very thin to date. We here address this gap in the literature, focusing on the important special case of time-series models. We show that convolutional neural networks trained to predict the input parameters from the data provide suitable summary statistics for LFIRE. On a wide range of time-series models, a single neural network architecture produced equally or more accurate posteriors than alternative methods.