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 $\ell_0$- or $\ell_1$-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：最小二乘规划因其实用性和开源求解器的易获取性，成为机器人学中的常用工具。然而，诸如$\ell_0$-或$\ell_1$-范数稀疏规划等时间最优控制问题，却无法通过等价方式求解。本文提出一种非线性分层最小二乘规划（NL-HLSP）方法，用于非线性离散动态系统的时间最优控制。我们采用赫维赛德阶跃函数的连续近似，并引入一项避免梯度消失的附加项。通过保持状态与控制量在离散化步长间分段常值的方式，采用简单离散化方法。相较于最优控制的直接转换方法，我们由此获得一种更易实现的NL-HLSP。研究表明，在静止目标点条件下，该NL-HLSP方法确实可恢复离散时间最优控制解。我们通过线性和非线性控制场景的仿真验证了上述结果。