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The Robotic locomotion community is interested in optimal gaits for control. Based on the optimization criterion, however, there could be a number of possible optimal gaits. For example, the optimal gait for maximizing displacement with respect to cost is quite different from the maximum displacement optimal gait. Beyond these two general optimal gaits, we believe that the optimal gait should deal with various situations for high-resolution of motion planning, e.g., steering the robot or moving in “baby steps.” As the step size or steering ratio increases or decreases, the optimal gaits will slightly vary by the geometric relationship and they will form the families of gaits. In this paper, we explored the geometrical framework across these optimal gaits having different step sizes in the family via the Lagrange multiplier method. Based on the structure, we suggest an optimal locus generator that solves all related optimal gaits in the family instead of optimizing each gait respectively. By applying the optimal locus generator to two simplified swimmers in drag-dominated environments, we verify the behavior of the optimal locus generator. 

The complex dynamics of agile robotic legged locomotion requires motion planning to intelligently adjust footstep locations. Often, bipedal footstep and motion planning use mathematically simple models such as the linear inverted pendulum, instead of dynamically-rich models that do not have closed-form solutions. We propose a real-time optimization method to plan for dynamical models that do not have closed form solutions and experience irrecoverable failure. Our method uses a data-driven approximation of the step-to-step dynamics and of a failure margin function. This failure margin function is an oriented distance function in state-action space where it describes the signed distance to success or failure. The motion planning problem is formed as a nonlinear program with constraints that enforce the approximated forward dynamics and the validity of state-action pairs. For illustration, this method is applied to create a planner for an actuated spring-loaded inverted pendulum model. In an ablation study, the failure margin constraints decreased the number of invalid solutions by between 24 and 47 percentage points across different objectives and horizon lengths. While we demonstrate the method on a canonical model of locomotion, we also discuss how this can be applied to data-driven models and full-order robot models. 

In this work, we propose a method to generate reduced-order model reference trajectories for general classes of highly dynamic maneuvers for bipedal robots for use in sim-to-real reinforcement learning. Our approach is to utilize a single rigid-body model (SRBM) to optimize libraries of trajectories offline to be used as expert references that guide learning by regularizing behaviors when incorporated in the reward function of a learned policy. This method translates the model's dynamically rich rotational and translational behavior to a full-order robot model and successfully transfers to real hardware. The SRBM's simplicity allows for fast iteration and refinement of behaviors, while the robustness of learning-based controllers allows for highly dynamic motions to be transferred to hardware. Within this work we introduce a set of transferability constraints that amend the SRBM dynamics to actual bipedal robot hardware, our framework for creating optimal trajectories for a variety of highly dynamic maneuvers as well as our approach to integrating reference trajectories for a high-speed running reinforcement learning policy. We validate our methods on the bipedal robot Cassie on which we were successfully able to demonstrate highly dynamic grounded running gaits up to 3.0 m/s. 

A key problem in robotic locomotion is in finding optimal shape changes to effectively displace systems through the world. Variational techniques for gait optimization require estimates of body displacement per gait cycle; however, these estimates introduce error due to unincluded high order terms. In this paper, we formulate existing estimates for displacement, and describe the contribution of low order terms to these estimates. We additionally describe the magnitude of higher (third) order effects, and identify that choice of body coordinate, gait diameter, and starting phase influence these effects. We demonstrate that variation of such parameters on two example systems (the differential drive car and Purcell swimmer) effectively manages third order contributions.