In view of the structural characteristics of spherical 3-DOF parallel mechanism, which all the joints are rotating joints and the joint axes intersect at one point, the position of the component is expressed by quaternion by means of the corresponding relationship between the long arc on the sphere surface and quaternion algebra. The product of quaternions is used to describe the component position change caused by joint rotation, and the direction cosine of the joint axis is obtained through the corresponding relationship between the geometric addition of the long arc on the sphere surface and the multiplication of quaternions, thus the constraint equation of the robot is established. The constraint equation is proceed subtly by a clever variable, which reduced the data in the symbolic operation of MATLAB, solved the problem that the effective results could not be obtained when the operation exceeded the computer memory, and the positive closed equation of the robot is established. By using the influence coefficient method, the transfer relation between the angular velocity and angular acceleration of the robot active component and the corner cone on the end component is established. The examples show that, there are 8 kinematics solutions for the given position of the active component in the top pyramid group. And according to the example of the continuous motion of the active robot component, the changing course of angular velocity and angular acceleration in the position of the top pyramid is simulated.