Robot programming tools ranging from inverse kinematics (IK) to model predictive control (MPC) are most often described as constrained optimization problems. Even though there are currently many commercially-available second-order solvers, robotics literature recently focused on efficient implementations and improvements over these solvers for real-time robotic applications. However, most often, these implementations stay problem-specific and are not easy to access or implement, or do not exploit the geometric aspect of the robotics problems. In this work, we propose to solve these problems using a fast, easy-to-implement first-order method that fully exploits the geometric constraints via Euclidean projections, called Augmented Lagrangian Spectral Projected Gradient Descent (ALSPG). We show that 1. using projections instead of full constraints and gradients improves the performance of the solver and 2. ALSPG stays competitive to the standard second-order methods such as iLQR in the unconstrained case. We showcase these results with IK and motion planning problems on simulated examples and with an MPC problem on a 7-axis manipulator experiment.

翻译：机器人编程工具从逆运动学（IK）到模型预测控制（MPC），通常被描述为约束优化问题。尽管目前市场上存在许多商业二阶求解器，但机器人学领域的最新研究主要聚焦于针对实时机器人应用对这些求解器进行高效实现与改进。然而，这些实现大多局限于特定问题且难以获取或实现，亦未能充分利用机器人问题的几何特性。本文提出采用一种快速、易实现的一阶方法——增强拉格朗日谱投影梯度下降法（ALSPG）——通过欧几里得投影充分挖掘几何约束的潜力来解决此类问题。研究表明：(1) 使用投影替代完整约束与梯度能显著提升求解器性能；(2) 在无约束情况下，ALSPG与iLQR等标准二阶方法保持竞争力。我们通过仿真示例中的逆运动学与运动规划问题，以及七轴机械臂实验中的模型预测控制问题，验证了上述成果。