Least-squares programming is a popular tool in robotics due to its simplicity and availability of open-source solvers. However, certain problems like sparse programming in the 0- or 1-norm for time-optimal control are not equivalently solvable. In this work we propose a non-linear hierarchical least-squares programming (NL-HLSP) for time-optimal control of non-linear discrete dynamic systems. We use a continuous approximation of the heaviside step function with an additional term that avoids vanishing gradients. We use a simple discretization method by keeping states and controls piece-wise constant between discretization steps. This way we obtain a comparatively easily implementable NL-HLSP in contrast to direct transcription approaches of optimal control. We show that the NL-HLSP indeed recovers the discrete time-optimal control in the limit for resting goal points. We confirm the results in simulation for linear and non-linear control scenarios.

翻译：最小二乘规划因其简洁性及开源求解器的可用性，在机器人学中广泛应用。然而，诸如时间最优控制中基于0-范数或1-范数的稀疏规划问题，无法通过等效方式求解。本文提出一种面向非线性离散动态系统时间最优控制的非线性分层最小二乘规划（NL-HLSP）方法。我们采用赫维赛德阶跃函数的连续近似形式，并引入额外项以避免梯度消失问题。通过保持状态与控制量在离散化步长间分段恒定，采用简单离散化方法，从而获得相较于最优控制直接转录方法更易实现的NL-HLSP形式。理论证明表明，对于静止目标点情形，该NL-HLSP方法确实能恢复离散时间最优控制。在线性与非线性控制场景的仿真结果验证了方法的有效性。