On the Hardware Feasibility of Nonlinear Trajectory Optimization for Legged Locomotion based on a Simplified Dynamics

Simplified models are useful to increase the computational efficiency of a motion planning algorithm, but their lack of accuracy have to be managed. We propose two feasibility constraints to be included in a Single Rigid Body Dynamics-based trajectory optimizer in order to obtain robust motions in challenging terrain. The first one finds an approximate relationship between joint-torque limits and admissible contact forces, without requiring the joint positions. The second one proposes a leg model to prevent leg collision with the environment. Such constraints have been included in a simplified nonlinear non-convex trajectory optimization problem. We demonstrate the feasibility of the resulting motion plans both in simulation and on the Hydraulically actuated Quadruped (HyQ) robot, considering experiments on an irregular terrain

Real-time trajectory adaptation for quadrupedal locomotion using deep reinforcement learning

We present a control architecture for real-time adaptation and tracking of trajectories generated using a terrain-aware trajectory optimization solver. This approach enables us to circumvent the computationally exhaustive task of online trajectory optimization, and further introduces a control solution robust to systems modeled with approximated dynamics. We train a policy using deep reinforcement learning (RL) to introduce additive deviations to a reference trajectory in order to generate a feedback-based trajectory tracking system for a quadrupedal robot. We train this policy across a multitude of simulated terrains and ensure its generality by introducing training methods that avoid overfitting and convergence towards local optima. Additionally, in order to capture terrain information, we include a latent representation of the height maps in the observation space of the RL environment as a form of exteroceptive feedback. We test the performance of our trained policy by tracking the corrected set points using a model-based whole-body controller and compare it with the tracking behavior obtained without the corrective feedback in several simulation environments. We also show successful transfer of our training approach to the real physical system and further present cogent arguments in support of our framework

Rapid stability margin estimation for contact-rich locomotion

The efficient evaluation the dynamic stability of legged robots on non-coplanar terrains is important when developing motion planning and control policies. The inference time of this measure has a strong influence on how fast a robot can react to unexpected events, plan its future footsteps or its body trajectory. Existing approaches suitable for real-time decision making are either limited to flat ground or to quasi-static locomotion. Furthermore, joint-space feasibility constraints are usually not considered in receding-horizon planning as their high dimensionality prohibits this. In this paper we propose the usage of a stability criterion for dynamic locomotion on rough terrain based on the Feasible Region (FR) and the Instantaneous Capture Point (ICP) and we leverage a Neural Network (NN) to quickly estimate it. We show that our network achieves satisfactory accuracy with respect to its analytical counterpart with a speed up of three orders-of-magnitude. It also enables the evaluation of the stability margin's gradient. We demonstrate this learned stability margin in two diverse applications - Reinforcement Learning (RL) and nonlinear Trajectory Optimization (TO) for legged robots. We demonstrate on a full-sized quadruped robot that the network enables the computation of physically-realizable Center of Mass (CoM) trajectories and foothold locations satisfying friction constraints and joint-torque limits in a receding-horizon fashion and on non-coplanar terrains

Receding-horizon perceptive trajectory optimization for dynamic legged locomotion with learned initialization

To dynamically traverse challenging terrain, legged robots need to continually perceive and reason about upcoming features, adjust the locations and timings of future footfalls and leverage momentum strategically. We present a pipeline that enables flexibly-parametrized trajectories for perceptive and dynamic quadruped locomotion to be optimized in an online, receding-horizon manner. The initial guess passed to the optimizer affects the computation needed to achieve convergence and the quality of the solution. We consider two methods for generating good guesses. The first is a heuristic initializer which provides a simple guess and requires significant optimization but is nonetheless suitable for adaptation to upcoming terrain. We demonstrate experiments using the ANYmal C quadruped, with fully onboard sensing and computation, to cross obstacles at moderate speeds using this technique. Our second approach uses latent-mode trajectory regression (LMTR) to imitate expert data—while avoiding invalid interpolations between distinct behaviors—such that minimal optimization is needed. This enables high-speed motions that make more expansive use of the robot’s capabilities. We demonstrate it on flat ground with the real robot and provide numerical trials that progress toward deployment on terrain. These results illustrate a paradigm for advancing beyond short-horizon dynamic reactions, toward the type of intuitive and adaptive locomotion planning exhibited by animals and humans