移动机械臂是由移动平台和机械臂组成的超冗余系统，兼具机械臂的高灵巧性和移动平台的高灵活性，可以在复杂环境中灵活完成各种任务，在生活和工业中应用越来越广泛。把移动机械臂系统视为一个整体进行轨迹规划是高效执行复杂任务的前提，但这也使得轨迹规划问题变得更加的复杂。传统方法如轨迹优化或者采样策略，其计算复杂度限制了实时性，极大地影响了实际应用，因此研不需要极大计算量的超冗余系统的轨迹规划方法十分的重要。针对这一问题，本文基于模型预测路径积分控制框架，研究了轨迹规划的安全性和稳定性问题，提出了改进的模型预测路径积分控制策略，具体内容如下： (1)拓展模型预测路径积分控制框架：本文首先将移动平台与机械臂的运动学方程整合为统一模型，构建状态和控制输入联合优化框架，通过路径积分方法将轨迹规划问题转化为随机最优控制问题，并求解出概率分布形式。随后引入重要性采样策略，通过设计自适应采样分布函数，将高维轨迹搜索转化为带权重的轨迹采样拟合过程。最后进一步提出动态代价调整机制、控制平滑设计及解耦控制代价与温度系数策略，优化算法效率与轨迹质量。 (2)研究轨迹规划安全性与稳定性：针对轨迹规划过程中安全性无法保证的问题，本文引入控制障碍函数，研究倒数控制障碍函数和零点控制障碍函数，提出了安全临界控制器，通过二次规划实时修正控制输入，确保状态约束的硬性满足；针对轨迹规划过程中稳定性不足的问题，本文引入控制李雅普诺夫函数，并拓展到指数控制李雅普诺夫函数和快速指数控制李雅普诺夫函数，提出了指数稳定控制器，通过优化求解实时调整控制输入，加速轨迹收敛。 (3)提出改进的模型预测路径积分控制策略：本文将控制障碍函数与控制李雅普诺夫函数的约束条件整合至模型预测路径积分控制框架中，提出了安全稳定协同控制策略。控制障碍函数的硬约束通过二次规划显式求解，控制李雅普诺夫函数的软约束通过在代价函数中引入轨迹稳定性惩罚项。本文在仿真环境中对改进后的模型预测路径积分控制策略进行验证，仿真结果表明该策略可以很好地处理移动机械臂轨迹规划任务，并且在保证轨迹规划安全性的同时提高了轨迹规划的稳定性。
The mobile manipulator is a super-redundant system composed of a mobile platform and a manipulator. It possesses both the high dexterity of the manipulator and the high flexibility of the mobile platform, enabling flexible execution of various tasks in complex environments. It is increasingly widely applied in both life and industrial scenarios. Considering the mobile manipulator system as an integrated whole for trajectory planning is a prerequisite for efficient task execution, yet this also increases the complexity of trajectory planning. The computational complexity of traditional methods such as trajectory optimization or sampling strategies limits real-time performance and significantly impacts practical applications. Therefore, studying trajectory planning methods for superredundant systems that require minimal computational resources is of critical importance. To address this issue, this paper investigates the safety and stability of trajectory planning based on the model predictive path integral control framework and proposes an improved model predictive path integral control strategy. The specific contents are as follows: (1)Expanding the model predictive path integral control framework: This paper first integrates the kinematic equations of the mobile platform and manipulator into a unified model, constructs a joint optimization framework for state and control input, and transforms the trajectory planning problem into a stochastic optimal control problem through path integral methods, expressed in a probability distribution form. Subsequently, the importance sampling strategy is introduced. By designing an adaptive sampling distribution function, the high-dimensional trajectory search is transformed into a weighted trajectory sampling fitting process. Finally, dynamic cost adjustment mechanisms, control smoothing designs, and decoupling strategies for control cost and temperature coefficients are proposed to optimize algorithm efficiency and trajectory quality. (2)Studying on safety and stability of trajectory planning: To address the issue of unguaranteed safety during trajectory planning, this paper introduces the control barrier function. Inverse control barrier functions and zero control barrier functions are studied, leading to the proposal of a safety-critical controller. Real-time correction of control inputs via quadratic programming ensures the hard satisfaction of state constraints. For the problem of insufficient stability during trajectory planning, this paper introduces the control Lyapunov function, extending it to exponential control Lyapunov functions and fast exponential control Lyapunov functions. An exponential stability controller is proposed. Real-time adjustment of control inputs through optimization solutions accelerates trajectory convergence. (3) Proposing an improved model predictive path integral control strategy: This paper integrates the constraint conditions of the control barrier function and control Lyapunov function into the model predictive path integral control framework, proposing a safety-stability control strategy. Through quadratic programming the hard constraints of the control barrier function are explicitly solved and the soft constraints of the control Lyapunov function are addressed by introducing trajectory stability penalty terms into the cost function. The improved model predictive path integral control strategy is validated in a simulation environment. Simulation results demonstrate that the strategy effectively handles mobile manipulator trajectory planning tasks, improving trajectory planning stability while ensuring safety.
