An efficient optimal planning and control framework for quadrupedal locomotion

Abstract

Abstract-In this paper, we present an efficient Dynamic Programing framework for optimal planning and control of legged robots. First we formulate this problem as an optimal control problem for switched systems. Then we propose a multi-level optimization approach to find the optimal switching times and the optimal continuous control inputs. Through this scheme, the decomposed optimization can potentially be done more efficiently than the combined approach. Finally, we present a continuous-time constrained LQR algorithm which simultaneously optimizes the feedforward and feedback controller with O(n) time-complexity. In order to validate our approach, we show the performance of our framework on a quadrupedal robot. We choose the Center of Mass dynamics and the full kinematic formulation as the switched system model where the switching times as well as the contact forces and the joint velocities are optimized for different locomotion tasks such as gap crossing, walking and trotting.

Online walking motion and foothold optimization for quadruped locomotion

“…This high level input is the actual relevant information, as it is often of secondary importance with which footsteps the robot reaches this goal. The only reason the footsteps follow the direction of the goal, is because the system knows it needs to move the body there (8), but has to maintain stability (6) while staying inside the kinematic range of the legs (7). By not fixing the footsteps beforehand, we allow the optimizer to modify the base of support as needed to perform the desired motion.…”

“…The following describes the constraints (6), (7), (8) and cost (3) necessary to generate the walking motions.…”

“…transformation of a continuous time Optimal Control (OC) problem into a discrete mathematical optimization problem and solved with off-the-shelf solvers ( [1]- [7]). Other approaches solve the OC problem directly through efficient solvers based on Dynamic Programing [8], [9]. These formulations are powerful, since they can deal with nonlinear dynamics and constraints and directly produce the optimal inputs τ to the system.…”

Hybrid direct collocation and control in the constraint-consistent subspace for dynamic legged robot locomotion

“…According to (21), there is no change of momentum in the null space of the constraints, or similarly, all the change of momentum has to be perpendicular to the null space of the constraints. However, and as in (10), this system of equations does not provide a unique solution.…”

“…Xi and Remy [20] applied direct collocation in a planar model using soft contact models. A series of approaches based on Differential Dynamic Programming (DDP) have presented alternatives to direct methods, where computational efficiency is the main motivation [10], [21].…”