The inverse kinematic step-by-step solution method based on Euler's angle was proposed to address the problem of incomplete or no analytical solution for the coaxial 3RRR spherical parallel mechanism (CSPM), which was the end-effector of the spherical wrist rehabilitation robot. The CSPM posture Euler angle can be decomposed into two sub-postures rotating around Z-axis and X, Y-axis based on the characteristics of the co-axial spherical parallel mechanism. The set of inverse kinematic solutions for the sub-postures rotating around X-axis and Y-axis was solved. The smaller value in the set of inverse kinematics solutions for each joint was selected and added to the angle of rotation around the Z-axis as the CSPM inverse kinematics solution. The correctness of the proposed method was verified by using CSPM forward kinematics. The actual attitude space of the wrist rehabilitation device was solved by using the proposed method with the constraints of no linkage collision point and no singularity configuration based on the real wrist motion range. The proposed inverse kinematics solution method was interconverted with unit quaternion in the actual posture space, and unit quaternion interpolation was applied to CSPM motion planning. The theoretical calculation results and experimental results were smooth trajectory curves, and the maximum value of both errors didn’t exceed 2.5°.