Beyond Inverted Pendulums: Task-optimal Simple Models of Legged Locomotion

TL;DR

This work proposes a model optimization algorithm that automatically synthesizes reduced-order models, optimal with respect to a user-specified distribution of tasks and corresponding cost functions, and shows in simulation that the optimal ROM reduces the cost of Cassie's joint torques and increases its walking speed.

Abstract

Reduced-order models (ROM) are popular in online motion planning due to their simplicity. A good ROM for control captures critical task-relevant aspects of the full dynamics while remaining low dimensional. However, planning within the reduced-order space unavoidably constrains the full model, and hence we sacrifice the full potential of the robot. In the community of legged locomotion, this has lead to a search for better model extensions, but many of these extensions require human intuition, and there has not existed a principled way of evaluating the model performance and discovering new models. In this work, we propose a model optimization algorithm that automatically synthesizes reduced-order models, optimal with respect to a user-specified distribution of tasks and corresponding cost functions. To demonstrate our work, we optimized models for a bipedal robot Cassie. We show in simulation that the optimal ROM reduces the cost of Cassie's joint torques by up to 23% and increases its walking speed by up to 54%. We also show hardware result that the real robot walks on flat ground with 10% lower torque cost. All videos and code can be found at https://sites.google.com/view/ymchen/research/optimal-rom.

Figures (12)

• Figure 1: An outline of the synthesis and deployment of optimal reduced-order models (ROM). Offline, given a full-order model and a distribution of tasks, we optimize a new model that is effective over the task space (Section \ref{['sec:approaches']}). Online, we generate new plans for the reduced-order model and track these trajectories on the true, full-order system (Section \ref{['sec:mpc']}). This diagram also shows the bipedal robot Cassie (in the rightmost box) and its full model. Cassie has five motors on each leg -- three located at the hip, one at the knee and one at the toe. Additionally, there are 2 leaf springs in each leg, and the spring joints are visualized by $q_{16}$ to $q_{19}$ in the figure. The springs are a part of the closed-loop linkages of the legs. We model these linkages with distance constraints, so there are no rods visualized in the model.
• Figure 2: Relationship of the full-order and reduced-order models. The generalized positions $q$ and $y$ satisfy the embedding function $r$ for all time, and the evolution of the velocities $\dot{q}$ and $\dot{y}$ respects the dynamics $f$ and $g$, respectively.
• Figure 3: The linear inverted pendulum (LIP) model. It is a point mass model of which height is restricted in a plane. The point mass and the origin of this model correspond to the center of mass and the stance foot of the robot, respectively. In the examples of this paper, we initialize the reduced-order model to the LIP model during model optimization.
• Figure 4: The averaged cost of the sampled tasks of each model optimization iteration in Examples 1 to 5. Costs are normalized by the cost associated with the full-order model (i.e. the cost of full model trajectory optimization without any reduced-order model embedding). Therefore, the costs cannot go below 1. The costs at iteration 1 represent the averaged costs for the robots with the embedded initial reduced-order models, LIP. Note that the empirical average does not strictly decrease, as tasks are randomly sampled and are of varying difficulty.
• Figure 5: The diagram of the model predictive control (MPC) introduced in Section \ref{['sec:mpc']}. The MPC is composed of the controller process and the planner process, and it contains a time-based finite state machine which outputs either left or right support state. This finite state machine determines the contact sequence of the high-level planner and the contact mode of the low-level model-based controller. The high-level planner solves for the desired reduced-order model trajectories and swing foot stepping locations, given tasks (commands) and the finite state. For reduced-order models without body orientation (e.g. CoM model without moment of inertia), we send the turning rate command to the controller process instead of planner process. Inside the controller process, the regularization trajectories are used to fill out the joint redundancy of the robot. These regularization trajectories are derived from simple heuristics such as maintaining a horizontal attitude of the pelvis body, having the swing foot parallel to the contact surface, and aligning the hip yaw angle with the desired heading angle. All desired trajectories are sent to the Operational Space Controller (OSC) which is a quadratic-programming based inverse-dynamics controller sentis2005controlwensing2013generation.

Theorems & Definitions (5)

• Remark 1
• Proposition 1: Differentiability Condition jin2021safe
• Theorem 1: Envelope Theorem riley2012essential
• Corollary 1
• proof