The Sequential Linear Quadratic (SLQ) algorithm is a continuous-time variant of the well-known Differential Dynamic Programming (DDP) technique with a Gauss-Newton Hessian approximation. This family of methods has gained popularity in the robotics community due to its efficiency in solving complex trajectory optimization problems. However, one major drawback of DDP-based formulations is their inability to properly incorporate path constraints. In this paper, we address this issue by devising a constrained SLQ algorithm that handles a mixture of constraints with a previously implemented projection technique and a new augmented-Lagrangian approach. By providing an appropriate multiplier update law, and by solving a single inner and outer loop iteration, we are able to retrieve suboptimal solutions at rates suitable for real-time model-predictive control applications. We particularly focus on the inequality-constrained case, where three augmented-Lagrangian penalty functions are introduced, along with their corresponding multiplier update rules. These are then benchmarked against a relaxed log-barrier formulation in a cart-pole swing up example, an obstacle-avoidance task, and an object-pushing task with a quadrupedal mobile manipulator.

中文翻译：

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基于连续时间DDP的模型预测控制中的约束处理

顺序线性二次（SLQ）算法是众所周知的差分动态规划（DDP）技术的连续时间变体，具有高斯-牛顿黑森（Gauss-Newton Hessian）近似。由于其解决复杂轨迹优化问题的效率，该系列方法已在机器人技术界获得普及。但是，基于DDP的配方的一个主要缺点是它们无法适当地包含路径约束。在本文中，我们通过设计一种约束SLQ算法来解决此问题，该算法使用以前实现的投影技术和新的增强拉格朗日方法来处理混合约束。通过提供适当的乘数更新定律，并解决单个内循环和外循环迭代，我们能够以适合实时模型预测控制应用的速率检索次优解决方案。我们特别关注不等式约束的情况，其中引入了三个增广的拉格朗日罚函数及其对应的乘数更新规则。然后，以四杆移动摆臂示例中的宽松对数障碍公式，避障任务和四足式移动机械手的推物体任务为基准。