A trajectory tracking control algorithm based on Cartesian space force compensation is proposed to address the issues of low control accuracy and poor dynamic performance of trajectory tracking control for robotic arms with a serial elastic actuator(SEA). The algorithm aims to improve the accuracy of Cartesian space trajectory tracking. Firstly, the SEA and the 6-degree-of-freedom(6-DOF) robotic arm system model are introduced, and a position-velocity-torque mixed control algorithm based on neural network model predictive control(NNMPC) is designed. Subsequently, based on this joint controller, a Cartesian space force compensation control algorithm for flexible joint manipulators is designed. This algorithm calculates the force compensation value in Cartesian space based on the tracking error, combines it with a PID controller, converts it into a target torque compensation value in the joint space, and compensates it to the joint controller to achieve high-precision Cartesian space trajectory tracking. Cartesian space force compensation can effectively counteract external disturbances and system uncertainties, reduce tracking errors, and enable the manipulator to more accurately track the target trajectory. Finally, the effectiveness and superiority of the proposed trajectory tracking controller are validated through simulation and experimental trials. The experimental outcomes demonstrate that the controller achieves an overall accuracy of 1.86 mm, marking an enhancement of 2.91 mm and 1.77 mm over the uncompensated trajectory tracking and position-based compensation trajectory tracking control algorithms, respectively.