# Screw Dynamics of a Multibody System by a
## 1 Introduction  
At present, parallel manipulators are widely used in machine tools because of their high stiffness and loading capacity, fewer driving branches, and easy control. With the application of parallel robots in machine tools, the dynamics analysis of a parallel manipulator becomes increasingly important. The mobility characteristics of the Schoenflies motion include three independent translations and one rotation around a fixed axis. The parallel mechanism that performs the Schoenflies motion can be used in special manipulation. For example, it can be used for product grabbing, assembly, and the processing production line. However, a Schoenflies parallel manipulator has a structure of multiple closed loops, which leads to a very complicated dynamics model. Inverse dynamics is utilized to calculate the driving force as an input to control the motion required by the end effector. Compared with their series counterparts with perfect dynamics modeling methods, the dynamics of parallel mechanisms is still a problem [1]. Therefore, this paper aims to propose an effective dynamics method for such multi-rigid-body systems. In the field of robot dynamics, research on the dynamics modeling of parallel robots has made significant progress. The main method includes the Newton-Euler equation, the Euler-Lagrange equation, the Kane method, and the virtual work principle [2]. The Newton-Euler equation is widely used to establish the dynamics model of a robot. Considering the inertial force and moment, the Newton equations and Euler equations of all rigid bodies can be obtained to solve the closed-loop dynamics of systems such as the Stewart platform [3], a self-reconfiguring chain [4], and multilink cable-driven hyper-redundant robots [5]. The Newton-Euler equations take all joint forces, including the torques and forces, and input and output into account, which is one major advantage of this method. However, a large number of algebraic and differential equations increase the calculation cost for the Newton-Euler method. From the perspective of analytical mechanics, some scholars utilized the Lagrange method to investigate the dynamics of manipulators [6; 7; 8]. The Lagrange equation of the first type takes the Lagrange multiplier and treats the constraint as an additional equation [9]. The Lagrange equation of the second type allows one to establish the ordinary differential equations of a mechanical system and obtain the solution of the equation by using a numerical algorithm [10]. For example, the inverse dynamics of the Stewart platform were derived by using the Lagrange method [11; 12]. [PERSON] et al. [13] proposed a modular analysis method for the dynamics of hybrid robots and demonstrated its application under a software framework. The Kane method takes the generalized rate as an independent variable and can eliminate constrained forces/moments [14]. [PERSON] et al. [15] derived a closed-form dynamics formula for a spherical parallel manipulator by using the Kane method. The virtual work principle provides a universal method for solving the dynamics problems of complex mechanical systems [16]. In the dynamic simulation and control of robots, these methods have higher computational efficiency [17; 18]. However, when conducting inverse dynamics and structure design, constraints need to be easily calculated. When analyzing the inverse dynamics of robots, these methods all have their individual limitations. In addition, the natural orthogonal complement method [19; 20], the particle dynamics and a recursive approach [21], a parallel Hamiltonian formulation [22], and a symbolic-numerical method [23] are adopted to establish the dynamics models of closed-loop systems. The existing methods for the dynamics of complicated multi-closed-loop systems often face severe difficulties [2; 24; 25]. In the aforementioned research, seldom were the dynamics discussed in the coordinates of a momentum screw.