Optimal behaviours of a system to perform a specific task can be achieved by leveraging the coupling between trajectory optimization, stabilization, and design optimization. This approach is particularly advantageous for underactuated systems, which are systems that have fewer actuators than degrees of freedom and thus require for more elaborate control systems. This paper proposes a novel co-design algorithm, namely Robust Trajectory Control with Design optimization (RTC-D). An inner optimization layer (RTC) simultaneously performs direct transcription (DIRTRAN) to find a nominal trajectory while computing optimal hyperparameters for a stabilizing time-varying linear quadratic regulator (TVLQR). RTC-D augments RTC with a design optimization layer, maximizing the system's robustness through a time-varying Lyapunov-based region of attraction (ROA) analysis. This analysis provides a formal guarantee of stability for a set of off-nominal states. The proposed algorithm has been tested on two different underactuated systems: the torque-limited simple pendulum and the cart-pole. Extensive simulations of off-nominal initial conditions demonstrate improved robustness, while real-system experiments show increased insensitivity to torque disturbances.

翻译：为完成特定任务，系统可通过利用轨迹优化、稳定性和设计优化之间的耦合来实现最优行为。这一方法对欠驱动系统（即执行器数量少于自由度的系统，因此需要更复杂的控制系统）尤为有利。本文提出一种新型协同设计算法——鲁棒轨迹控制与设计优化（Robust Trajectory Control with Design optimization, RTC-D）。其内层优化层（RTC）同步执行直接转录（DIRTRAN）以寻找标称轨迹，同时为稳定化时变线性二次型调节器（TVLQR）计算最优超参数。RTC-D通过设计优化层增强RTC，基于时变李雅普诺夫吸引域（ROA）分析最大化系统鲁棒性。该分析为偏离标称状态的一组状态提供稳定性形式化保证。所提算法已在两种不同欠驱动系统（力矩受限单摆和小车倒立摆）上完成测试。对非标称初始条件的大量仿真证明了鲁棒性提升，而真实系统实验表明其对力矩扰动的灵敏性降低。