Multiple impacts of 2D (planar) open-loop robotic systems composed of n elastic links and revolute joints are studied in this paper. The dynamic equations of motion for such systems are derived by the Gibbs-Appell recursive algorithm, while the regularized method is employed to model the impact-contact mechanism. The Timoshenko beam theory is used to model the transverse vibrations of the links. Also, both the structural damping and air damping are considered to enhance the modeling accuracy. The system joints are assumed to be frictionless and slack-free, but friction force is included for the links colliding with the ground. The n-flexible-link system considered goes through a flight phase and an impact phase during its motion. In the impact phase, new equations of motion are derived by including the terms caused by the viscoelastic forces in the system’s differential equations. Owing to the extremely short acting time of the impact force, the related differential equations can be solved only via special treatment, i.e. by detecting the exact moment of impact. To this end, entering or leaving the impact phase is analyzed and controlled with high precision by a special computational algorithm presented in this work. To demonstrate the efficacy and precision of the algorithm developed, computer simulations are conducted to study the dynamic behavior of a 3-link robotic mechanism. To investigate the effect of mode shape on the elastic deformation of links, four different mode shapes are used in the simulations and their results are compared.

中文翻译：

---

多个柔性链接中的摩擦冲击接触

二维（平面）开环机器人系统的多重影响n本文研究了弹性连杆和旋转接头。此类系统的动态运动方程由 Gibbs-Appell 递归算法推导，而正则化方法用于模拟冲击接触机制。Timoshenko 梁理论用于模拟连杆的横向振动。此外，结构阻尼和空气阻尼都被认为可以提高建模精度。假设系统关节是无摩擦和无松弛的，但摩擦力包括与地面碰撞的连杆。这n- 所考虑的柔性连杆系统在其运动过程中经历了飞行阶段和冲击阶段。在冲击阶段，通过在系统的微分方程中包含由粘弹性力引起的项，推导出新的运动方程。由于冲击力的作用时间极短，相关的微分方程只能通过特殊处理，即通过检测准确的冲击时刻来求解。为此，通过这项工作中提出的特殊计算算法，可以高精度地分析和控制进入或离开冲击阶段。为了证明所开发算法的有效性和精度，进行了计算机模拟来研究三连杆机器人机构的动态行为。为了研究振型对连杆弹性变形的影响，