EI、Scopus 收录

 引用本文: 陈渊钊, 朱健成, 吴文军, 杜晓康, 章定国. 浮动坐标系下大变形柔性梁的变形梯度单元. 力学学报, 待出版.
Chen Yuanzhao, Zhu Jiancheng, Wu Wenjun, Du Xiaokang, Zhang Dingguo. Deformation gradient element for flexible beams with large deformation in a floating frame reference. Chinese Journal of Theoretical and Applied Mechanics, in press.
 Citation: Chen Yuanzhao, Zhu Jiancheng, Wu Wenjun, Du Xiaokang, Zhang Dingguo. Deformation gradient element for flexible beams with large deformation in a floating frame reference. Chinese Journal of Theoretical and Applied Mechanics, in press.

## DEFORMATION GRADIENT ELEMENT FOR FLEXIBLE BEAMS WITH LARGE DEFORMATION IN A FLOATING FRAME REFERENCE

• 摘要: 机械臂振动控制通常采用浮动坐标系法建立动力学模型, 以确保关节运动和振动抑制有合适控制变量. 但浮动坐标法受到小变形的限制, 求解大变形动力学问题的精度不高, 而常用的大变形建模理论, 如绝对节点坐标法可以精确求解机械臂的大转动和大变形问题, 却未能提供合适的振动控制变量. 为此, 本文提出了一种新的大变形场离散方案—变形梯度单元, 该单元采用节点变形及其梯度作为节点坐标, 无任何小变形假设, 能准确转化为节点的绝对节点坐标及其梯度, 可视之为绝对节点坐标法的等效单元, 能精准描述柔性体的大变形运动. 研究表明: 1)变形梯度单元求解梁结构的大变形静力学问题时, 其结果与绝对节点坐标法缩减梁单元的完全相同, 求解效率相当; 2)本文模型与绝对节点坐标法处理大变形动力学问题的解几乎完全相同, 可说明本文模型的正确性. 通过广义α法求解时, 由于本文模型和绝对节点坐标法的动力学方程形式上的差异, 两种方法的动力学数值解存在非常微小的差异; 3)对于大变形机械臂的关节运动控制, 基于小变形假设的一次近似刚柔耦合模型极易给出错误甚至发散的预测结果; 4)本文模型自然区分了关节的大转动和梁的大变形, 结合PD控制策略对系统施加关节驱动力矩和若干个横向控制力, 此时系统李雅普诺夫函数对时间的导数为非正定, 系统能量随着时间的递增会逐渐衰减, 系统稳定, 且横向控制力越多, 机械臂残余振动抑制效果越佳.

Abstract: The dynamic modeling of robotic arm vibration control often adopts the floating frame reference method to ensure appropriate control variables for joint motion and vibration suppression. However, the floating frame reference method is limited by small deformations and lacks precision in solving large deformation dynamic problems. The commonly used modeling theories for large deformation problems, such as the Absolute Nodal Coordinate Formulation (ANCF) which can accurately solve large rotation and large deformation problems in robotic arms, fail to provide suitable control variables for vibration control. To address this issue, this paper proposes a novel discretization scheme for large deformation fields - the deformation gradient element. This element employs nodal deformation and its gradient as nodal coordinates without any small deformation assumptions, and can be accurately transformed into absolute nodal coordinates and their gradients. It can be viewed as an equivalent element of ANCF, capable of precisely describing the large deformation motion of flexible bodies. The following conclusions are drawn:1) When solving large deformation static problems in beam structures, the results obtained using the deformation gradient element are identical to those using the reduced beam element of ANCF, with comparable computational efficiency. 2) The solutions of the proposed model and ANCF for large deformation dynamic problems are almost identical, validating the correctness of the proposed model. When solved using the generalized-α method, the slight differences in dynamic solutions between the two methods stem from the different forms of their dynamic equations. 3) For motion control of robotic arms undergoing large deformations, the first-order approximation rigid-flexible coupling model based on small deformation assumptions is prone to yielding erroneous or even divergent predictions. 4) The proposed model naturally distinguishes between large rotations of joints and large deformations of beams. By combining PD control strategies to apply joint driving torques and multiple transverse control forces to the system, the derivative of the system's Lyapunov function with respect to time is non-positive, ensuring that system energy gradually decays over time and the system remains stable. Moreover, the more transverse control forces applied, the better the suppression of residual vibrations in the robotic arm.

/

• 分享
• 用微信扫码二维码

分享至好友和朋友圈