Crocoddyl: An Efficient and Versatile Framework for Multi-Contact Optimal Control

Abstract

We introduce Crocoddyl (Contact RObot COntrol by Differential DYnamic Library), an open-source framework tailored for efficient multi-contact optimal control. Crocoddyl efficiently computes the state trajectory and the control policy for a given predefined sequence of contacts. Its efficiency is due to the use of sparse analytical derivatives, exploitation of the problem structure, and data sharing. It employs differential geometry to properly describe the state of any geometrical system, e.g. floating-base systems. We have unified dynamics, costs, and constraints into a single concept -action -for greater efficiency and easy prototyping. Additionally, we propose a novel multipleshooting method called Feasibility-prone Differential Dynamic Programming (FDDP). Our novel method shows a greater globalization strategy compared to classical Differential Dynamic Programming (DDP) algorithms, and it has similar numerical behavior to state-of-the-art multiple-shooting methods. However, our method does not increase the computational complexity typically encountered by adding extra variables to describe the gaps in the dynamics. Concretely, we propose two modifications to the classical DDP algorithm. First, the backward pass accepts infeasible state-control trajectories. Second, the rollout keeps the gaps open during the early "exploratory" iterations (as expected in multiple-shooting methods). We showcase the performance of our framework using different tasks. With our method, we can compute highly-dynamic maneuvers for legged robots (e.g. jumping, front-flip) in the order of milliseconds.

Impedance Optimization for Uncertain Contact Interactions Through Risk Sensitive Optimal Control

“…The DDP controller used as a baseline in the presented experiments is from [29]. The kino-dynamic optimizer described in [31] is used to generate reference trajectories around which both iterative controllers DDP and Risk Sensitive are initialized.…”

“…In particular, we use a nonlinear iterative risk sensitive optimal control formulation [28], [24], [26] which enables to explicitly take into account the distribution of uncertainty while being numerically efficient. The algorithm is similar to what can be obtained using DDP [18], [29] while reasoning about the higher order statistics of the problem. Consider the following dynamics expressed as a nonlinear stochastic difference equation…”

“…where represents the difference between the two states x and x with the proper Lie group operations [33]. The propagation of the dynamics can be handled similarly using the ⊕ operation [29]. The gaussian noise ε ∼ N (0, Σ) is then defined on the same space as the estimation error, and the covariance can be propagated through the adjoint transformation while the state prediction follows the relative group composition operation described earlier in (4).…”

Nth Order Analytical Time Derivatives of Inverse Dynamics in Recursive and Closed Forms

“…A closed form of the EOM is obtained after replacing the system twist by (21). Alternatively, first the kinematic relation (21) and then the coefficient matrices in (30) are evaluated for a given state q, q. The generalized gravity forces are given as…”

“…These derivatives are useful in optimal control of legged robots (e.g. differential dynamic programming in Crocoddyl framework [21]) and their computational design & optimization [18].…”