用李代数统一建模柔性机械臂，实现无离散化控制与稳定拓展。
Modular Lie Algebraic PDE Control of Multibody Flexible Manipulators
- 基于李代数将刚体运动与弹性变形统一为旋量形式。
- 无需模态截断，控制器保证所有关节误差指数收敛。
- 结构模块化，可扩展至任意长度多体系统，适合高精度控制场景。
本文提出一种针对任意链节数的串联柔性机械臂的子系统自适应控制框架，其中每段连杆的弹性形变偏微分方程（PDE）在整个控制设计中保持连续，不进行空间离散或模态截断。所有动力学量——刚体运动、弹性形变及连杆间约束力——均以固定于本体的旋量（twists）和螺力（wrenches）统一表示，嵌入se3李代数结构。通过将应变型形变PDE代入动力学方程，消去分布式的弹性加速度，得到仅由本体旋量加速度和形变场驱动的可控动力学形式。各子系统期望旋量轨迹通过考虑偏转补偿的逆运动学生成。名义控制器经逐子系统李雅普诺夫函数证明可实现旋量误差指数衰减。自适应修改用在线估计参数替代精确物理参数，引入投影律更新，并加入参数估计误差项。在所有连杆求和后，交互功率项因牛顿第三定律与se3*se3上的自然功率配对框架不变性而完全抵消，从而在名义与自适应控制器下均保证所有旋量误差指数收敛且弹性变形有界。该螺旋理论结构使交互项抵消精确，稳定性证明具有模块化与任意长度可扩展性。框架在三维运动的双连杆柔性机械臂上进行了数值验证。
原文摘要 · Abstract (English)
This paper presents a subsystem-based adaptive control framework for serial flexible manipulators with an arbitrary number of links, in which the elastic deformation PDE of each link is carried through the entire control design without spatial discretization or modal truncation. All dynamic quantities -- rigid-body motion, elastic deformation, and inter-link constraint forces -- are expressed uniformly as body-fixed twists and wrenches within the se3 Lie-algebraic structure. A controllable form of the per-link dynamics is derived by substituting the strain-based deformation PDE into the dynamic equation, eliminating distributed elastic acceleration and yielding a model governed by the body-fixed twist acceleration and deformation field. Desired subsystem twist trajectories are generated via a deflection-compensating inverse kinematics procedure. A nominal per-link controller is proven to produce exponential twist error decay via a per-subsystem Lyapunov function. An adaptive modification replaces exact physical parameters with online estimates governed by a projection-based law, augmenting with a parameter estimation error term. Upon summing over all links, the interaction power terms telescope to zero by Newton's third law and the frame invariance of the natural power pairing on se3*se*(3), establishing exponential convergence of all twist errors and bounded elastic deformation under both nominal and adaptive controllers. The screw-theoretic structure renders interaction term cancellation exact, making the stability certificate modular and scalable to chains of arbitrary length. The framework is validated numerically on a two-link flexible manipulator in three-dimensional motion.
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