# Dynamics of a 3-UPS-UPU-S Parallel Mechanism
## 1 Introduction  
Compared with its serial counterparts, a parallel mechanism provides higher load-bearing capability and can be precisely controlled with a quick response and adjustment [1]. Hence, parallel mechanisms have been commonly applied in equipment and mechanisms with large scalar and mission requirements to be heavy duty, which requires higher dynamic performance, higher structural stability and robustness and also accurate positioning and lower complexity in control. For parallel mechanisms with serial kinematic subchains, the motion characteristics, such as degrees of freedom, workspace and singularities, are fundamental to the design of the kinematic subchains [2; 3; 4]. These motion parameters can be suggested for the optimization of the design and dynamic properties of the parallel mechanisms [5]. Some researchers have disclosed the influence of the arrangement of the joints. Especially for the universal joints, the motion of the moving platform of the parallel mechanism is relevant to the layout of the joints and the direction of the rotary axes [6]. Then, the kinematic analysis is the basis for gaining the kinematic information of the structure and its controlling and correcting posture [7]. Many researchers have proposed various methods for analyzing the kinematics of parallel mechanisms [8; 9; 10; 11]. As one of those common methods, kinematics and as well as dynamics of mechanism can be studied systematically through the theory of screw. Screw theory is known as a computational, supremely efficient approach to analyze the kinematics, statics and dynamics of mechanisms [12]. The velocities, accelerations of each movable joints on sub chains and any point on moving platform can be obtained, which form the groundwork for further dynamic modelling [13]. Dynamic analysis is one of the kernels of mechanism control [14]. Methods of dynamic modelling have been widely studied, in that it is necessary for determining the forces and moment of joints and actuators. Traditionally, the dynamic analysis is carried out via two kinds of methods: methods based on vectors, such as the Newton-Euler method [15] and on energy, such as methods using Lagrangian equations [16; 17; 18; 19]. The dynamic modellingof parallel mechanisms is often complex and difficult, because it is based on the kinematic information of the mechanisms and dynamic parameters [20]. Conventional dynamics analysis through the Newton-Euler method relies on motion equations of each part of the mechanism. Describing the dynamics with Lagrangian equations is practical in solving the inverse dynamics, but it also requires as much large computation time as the Newton-Euler method does. [PERSON] and [PERSON] obtained the position information of a \(3-\) PSS parallel mechanism through a Jacobian matrix and established the dynamic model using Lagrangian equations [21]. Through the principle of virtual work, the dynamics is expressed by building the energy equations in a generalized coordinate framework and the complexity of analyzing the inverse dynamics can be reduced [22; 23; 24; 25]. Some other methods have also been explored and applied in dynamic analysis such as the recursive matrix method [26], the influence coefficient method, which is separate to the velocity and acceleration variables [27] and the Kane equation [28; 29; 30]. [PERSON] et al. established a dynamic model of the Stewart platform through a combination of Kane equation and the principle of virtual works [31]. Considering that screw theory has already been extensively used in dynamic analysis, based on the kinematic formulations through screw theory, dynamics modelling of mechanisms can be established more efficiently [32; 33; 34]. [PERSON] solved the inverse dynamics of a translational parallel mechanism via screw theory and the virtual work principle [35]. The existing methods of kinematic analysis often start from displacement and the second order numerical differential interpolation is required to calculate the acceleration.