# Kinematics and Statics of the Gough-Stewart Platform
###### Abstract  
This paper presents an algorithm for the kinematics and statics analysis of a Gough-Stewart platform. Through defining the velocity screw, the relative angular and linear velocities of a single rigid body can be expressed as a single vector. The velocity screw equations of various mechanisms are deduced in detail, the forward and inverse kinematics of a parallel mechanism can be solved through the velocity screw equation. Similarly, the definition of the force screw allows all constraint forces and torques of a single rigid body to be expressed using a single vector, and the static screw equation can be used to solve the forward and inverse statics of a parallel mechanism in one coordinate system. The advantage of this approach is that kinematics and statics modeling are unified in screw coordinates because the kinematic parameters in screw form can be directly employed in statics modeling. The results of the kinematics and statics analysis of the Gough-Stewart platform validate this method. This algorithm is easy to compute and program with high efficiency, and it can also be applied to any other spatial, complex multi-rigid-body systems. kinematics; statics; screw; numerical algorithm +
Footnote †: journal: _Sci._  
0000-0002-3300-33000-3300-3300-3300-3300-3300-3300-3300-3300-  
and equilibrium states that govern their operation can be investigated. A systematic study of statics offers the groundwork for the subsequent study of their dynamic behavior and validates the potential in various engineering applications. The general law of the balance of the spatial force system is established based on the virtual displacement principle. The main research methods include the virtual displacement principle [17; 18] and the method of influence coefficient [19]. The statics of over-constrained and parallel mechanisms could be solved based on compliant mechanisms [20; 21; 22]. Screw theory is a convenient method to solve the kinematics and statics of parallel mechanisms. The expressions for the kinematics and statics could be simplified using screw theory, and some scholars have applied screw theory to solve the kinematics [23; 24] and statics [25; 26] in parallel mechanisms. Traditional static modeling methods, which are established in the Cartesian coordinate system, suffer from an unclear mechanism relationship using the dismantling method, and the vector method will introduce multiple variables, while the conventional methods start with displacement, and then solve the velocities and accelerations through first- and second-order numerical interpolation, respectively, which leads to high computational costs. This paper proposes a kinematics and statics analysis method to analyze a Gough-Stewart platform in screw coordinates. Through defining the velocity screw, the relative angular and linear velocities of a point in a single rigid body can be combined into a dual vector with six components. The velocities of all joints can be derived in one direct step through the velocity screw equation. Then, the displacements and accelerations can be carried out separately through a first-order numerical integration and a first-order numerical differential interpolation, individually and simultaneously. Therefore, compared with the conventional modeling methods using displacements as global variables, the second-order numerical interpolation can be avoided in the proposed method in this paper, which eliminates complex algebraic manipulation and leads to more stable and accurate results. The velocity screw equation could be established to calculate the absolute position and absolute posture of all the joints and any point in a single rigid body. Through defining the force screw, both constraint forces and constraint torque at all joints could be expressed in screw form. Both forces and torques are considered in the equilibrium of a rigid body, and the force screw equation is established to express the resultant action of a force system in an absolute coordinate system.