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 引用本文: 傅景礼, 陆晓丹, 项春. 爬壁机器人系统的Noether对称性和守恒量. 力学学报, 2022, 54(6): 1680-1693.
Fu Jingli, Lu Xiaodan, Xiang Chun. Noether symmetries and conserved quantities of wall climbing robot system. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(6): 1680-1693.
 Citation: Fu Jingli, Lu Xiaodan, Xiang Chun. Noether symmetries and conserved quantities of wall climbing robot system. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(6): 1680-1693.

## NOETHER SYMMETRIES AND CONSERVED QUANTITIES OF WALL CLIMBING ROBOT SYSTEM

• 摘要: 爬壁机器人的运动是一种模仿壁虎爬行的运动, 爬壁机器人的运动可分解为四肢带动身体的运动, 先前的研究都是基于牛顿力学的方法. 本文采用Lagrange 力学的方法建立爬壁机器人系统的运动方程, 并运用Lie群分析方法建立该系统的Noether对称性理论, 得出爬壁机器人的运动规律. 首先, 给出非完整爬壁机器人系统的动能、势能和Lagrange函数以及所受的非完整约束, 从而建立了非完整爬壁机器人系统的Lagrange方程; 其次, 引入关于时间和广义坐标的无限小变换, 提出了非完整爬壁机器人系统的Hamilton作用量和Hamilton作用量的基本变分公式; 第三, 给出爬壁机器人系统 Noether对称性变换和广义准对称变换的定义, 判据和存在的Noether守恒量, 并提出了非保守完整系统和非保守非完整爬壁机器人系统的Noether定理; 最后, 以圆锥面上爬壁机器人为例, 对给出的守恒量直接进行积分给出圆锥面上爬壁机器人整体运动的精确解和四肢运动的数值解, 发现了该爬壁机器人的运动规律, 很好地验证了非完整爬壁机器人系统的Noether对称性理论. 本文的研究为Lie群分析方法应用于其他复杂的机器人系统以及柔性机器人系统的对称性求解提出了一种新的对称性求解方法.

Abstract: The wall climbing robot's motion is a kind of imitation gecko's crawling motion. The wall climbing robot's motion can be divided into four limbs driving the body's movement. The previous research is based on Newton's mechanics. In this paper, Lagrange mechanics method is used to establish the motion equation of the wall climbing robot system, and the Noether symmetry theory of the system is established by using the Lie group analysis method, and the motion law of the wall climbing robot is obtained. Firstly, the kinetic energy, potential energy, Lagrange functions and nonholonomic constraints of nonholonomic wall climbing robot system are given, and the Lagrange equation of nonholonomic wall climbing robot system is established. Secondly, by introducing infinitesimal transformation of time and generalized coordinates, the basic variational formulas of Hamilton action and Hamilton action of nonholonomic wall climbing robot system are proposed. Thirdly, the wall climbing robot system is given The definition, criterion and existing Noether conserved quantity of Noether symmetry transformation and generalized quasi symmetry transformation are introduced. The Noether theorem of non conservative holonomic system and non conservative nonholonomic wall climbing robot system is proposed. Finally, taking the wall climbing robot on the conic surface as an example, the given conserved quantity is directly integrated, and the exact solution of the whole motion of the wall climbing robot on the conical surface and the motion of the limbs are given The numerical results show that the motion law of the wall climbing robot is found and the Noether symmetry theory of the nonholonomic wall climbing robot system is well verified. This paper proposes a new symmetry solution method for Lie group analysis method applied to other complex robot systems and flexible robot systems.

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