This study proposes an efficient Newton-type method for the optimal control of switched systems under a given mode sequence. A mesh-refinement-based approach is utilized to discretize continuous-time optimal control problems (OCPs) using the direct multiple-shooting method to formulate a nonlinear program (NLP), which guarantees the local convergence of a Newton-type method. A dedicated structure-exploiting algorithm (Riccati recursion) is proposed that efficiently performs a Newton-type method for the NLP because its sparsity structure is different from a standard OCP. The proposed method computes each Newton step with linear time-complexity for the total number of discretization grids as the standard Riccati recursion algorithm. Additionally, it can always solve the Karush-Kuhn-Tucker (KKT) systems arising in the Newton-type method if the solution is sufficiently close to a local minimum. Conversely, general quadratic programming (QP) solvers cannot accomplish this because the Hessian matrix is inherently indefinite. Moreover, a modification on the reduced Hessian matrix is proposed using the nature of the Riccati recursion algorithm as the dynamic programming for a QP subproblem to enhance the convergence. A numerical comparison is conducted with off-the-shelf NLP solvers, which demonstrates that the proposed method is up to two orders of magnitude faster. Whole-body optimal control of quadrupedal gaits is also demonstrated and shows that the proposed method can achieve the whole-body model predictive control (MPC) of robotic systems with rigid contacts.
翻译:本研究提出了一种高效的牛顿型方法,用于在给定模式序列下优化控制交换系统。 一种基于网状精密的方法, 用来将连续时间最佳控制问题(OCPs)分解为连续最佳控制问题(OCPs), 使用直接的多射法来制定非线性程序(NLP), 保证牛顿型方法的本地趋同。 提议了一种专门的结构开发算法( Riccati recuration ), 以便有效地为NLP 执行牛顿型方法, 因为它的宽度结构结构不同于标准的 OCP 。 此外, 拟议的方法将牛顿型电网的总数分解为直线性, 以直线性、时间兼容性、 最优化控制( OCPs), 将降低的离线性、 离线性、 最精确的电流化电流化电路( QPralislational) 系统进行修改后, 将显示快速的NPral- translational 矩阵, 将显示为Sloveal- slational- 递解算法。