This paper investigates kinematics and statics analysis of a 3-UPU robot in screw coordinates. According to the definition of a twist, both the angular velocity of a rigid body and the linear velocity of a point on it are expressed in screw components. We therefore establish the twist equation (TE) to calculate the position and posture of each joint. This equation can be applied directly to analyze the statics. According to the definition of a wrench, both the force and torque of the planar linkage are expressed in wrench components. As a result, we establish wrench equations (WE) to express the resultant action of a force system in one coordinate frame. With the definition of twist and wrench, we can associate the kinematics with statics by unit screws. Compared with the traditional Cartesian coordinate method, the WE is free from complex algebraic manipulation and convenient to obtain wrench matrix by Plücker coordinates of each wrench. Numerical calculation is applied to solve the kinematics which can conveniently obtain the position and posture of each joint in absolute coordinate. Six unknown variables can be solved with each equation of wrench which contain 3 forces and 3 torques. The kinematics and static analysis of the 3-UPU robot validate this method.