In order to design a practical 4-DOF robot arm, in this paper, firstly, its model was created and assembled in SOLIDWORKS; and then the model was imported to Simulink for simulation. Finally, the DH parameter method was used to construct the robot kinematics model, and he homogeneous transformation matrix between the robot arm structure coordinate system was used to obtain the positive motion and inverse kinematics solutions. The Robots Toolbox under MATLAB was used for simulation; and the third, fifth, and seventh-order polynomial algorithm was used for trajectory planning in the joint space coordinate system of the manipulator; and the fitting curve of the angle, angular velocity and angular acceleration of each joint were obtained. The results show that among three methods, the seventh-order polynomial algorithm shows the best. The robot arm’s grasping operation trajectory is smooth, and the speed, acceleration, and jerk of each joint trajectory remain continuous, which can meet the actual application requirements and provide corresponding support for the robot arm’s target grasping motion planning.