The task of inducing, via continuous static state-feedback control, an asymptotically stable heteroclinic orbit in a nonlinear control system is considered in this paper. The main motivation comes from the problem of ensuring convergence to a so-called point-to-point maneuver in an underactuated mechanical system. Namely, to a smooth curve in its state–control space which is consistent with the system dynamics and connects two (linearly) stabilizable equilibrium points. The proposed method uses a particular parameterization, together with a state projection onto the maneuver as to combine two linearization techniques for this purpose: the Jacobian linearization at the equilibria on the boundaries and a transverse linearization along the orbit. This allows for the computation of stabilizing control gains offline by solving a semidefinite programming problem. The resulting nonlinear controller, which simultaneously asymptotically stabilizes both the orbit and the final equilibrium, is time-invariant, locally Lipschitz continuous, requires no switching, and has a familiar feedforward plus feedback–like structure. The method is also complemented by synchronization function–based arguments for planning such maneuvers for mechanical systems with one degree of underactuation. Numerical simulations of the non-prehensile manipulation task of a ball rolling between two points upon the “butterfly” robot demonstrates the efficacy of the synthesis.

本文考虑了通过连续静态反馈控制在非线性控制系统中引入渐近稳定异宿轨道的任务。主要动机来自确保收敛到欠驱动机械系统中所谓的点对点机动的问题。即，其状态控制空间中的一条平滑曲线与系统动力学一致并连接两个（线性）可稳定平衡点。所提出的方法使用特定的参数化，连同状态投影到机动上，以为此目的结合两种线性化技术：边界平衡处的雅可比线性化和沿轨道的横向线性化。这允许通过解决半定规划问题来离线计算稳定控制增益。由此产生的非线性控制器同时渐近稳定轨道和最终平衡，是时不变的，局部 Lipschitz 连续的，不需要切换，并且具有熟悉的前馈加反馈结构。该方法还辅以基于同步函数的参数，用于为具有一定程度欠驱动的机械系统规划此类机动。球在“蝴蝶”机器人两点之间滚动的非抓取操作任务的数值模拟证明了合成的有效性。局部 Lipschitz 连续，不需要切换，并且具有熟悉的前馈加反馈结构。该方法还辅以基于同步函数的参数，用于为具有一定程度欠驱动的机械系统规划此类机动。球在“蝴蝶”机器人两点之间滚动的非抓取操作任务的数值模拟证明了合成的有效性。局部 Lipschitz 连续，不需要切换，并且具有熟悉的前馈加反馈结构。该方法还辅以基于同步函数的参数，用于为具有一定程度欠驱动的机械系统规划此类机动。球在“蝴蝶”机器人两点之间滚动的非抓取操作任务的数值模拟证明了合成的有效性。