Nonlinear Optimal Control for Autonomous Space Robot Rendezvous with a Tumbling Non-cooperative Target Based on Synchronization Spinning Linear Approaching Strategy

The autonomous rendezvous and capturing programs demonstrate there is a great demand for robust and effective autonomous rendezvous control algorithm. Some demonstrations of Guidance, Navigation and Control in this field are currently undertaking in Aerospace Flight Dynamics Laboratory at Northwestern Polytechnical University, which is a top university in spacecraft dynamics and control in China. Due to fuel exhaust or attitude control system failure, it is common for the defunct satellites to tumble along the body‘s maximum moment axis as time goes on. They may pose potential risks to the normally operating satellites if not removed. However, the passive rotating satellite presents a bigger challenge for the powered space robot to approach with whatever in terms of ensuring safety or reducing fuel cost. This is the motivation of the research described in this paper. Approaching a tumbling target satellite in previous research work only addressed the translational motion challenges. Even though some studies considered rotational motion, these efforts were limited to simplified case of two dimensions. Furthermore, the previous studies failed to present an approaching strategy to ensure safety according to the rotating characteristic of the target. This paper greatly extends the previous research work by full states modeling of the relative motion, approaching strategy designing to avoid collision and optimal controllers derivation which minimize the control error, fuel consumption and in the meanwhile, reducing the onboard calculation to make it more suitable for real-time situation. Unlike previous study which focused on orbital motion only, a 6 DoF model was developed in this paper to formulate the coupled orbit and attitude dynamics of the controlled space robot and the passive tumbling satellite. TH equations were used which were suitable to describe satellites operating in an elliptic orbit rather than CW. And in terms of relative rotation, relative attitude dynamics were derived based on quaternions which could avoid the singularity caused by euler angles. In particular, perturbation induced coupling and dynamic coupling stem from the coupled model were presented to evaluate the coupling effect. When the manipulator equipped on the space robot begins to capture the target, the tumbling target will give rise to attitude disturbance. Therefore, it is quite necessary to design an safety approaching strategy to ensure the success of the proximity operations. Synchronization spinning linear approaching strategy allows the space robot to measure the relative position and attitude based on the visual instruments before capturing. To begin with, the space robot would move to a certain distance right above the maximum moment axes and stay till the relative navigation information is obtained. Then reorienting its attitude and matching its angular velocity to that of the target by enforcing control torque. After that, the distance is decreased along linear path through orbit control force given by small thrusters equipped on the robot’s body. By matching the angular velocity of the two vehicles and gradually decreasing the distance, it would lessen the attitude disturbance to the base of the robot and prevent attitude stumble from happening. Due to the highly nonlinear property rooted in this model, conventional linear control methods are unsuitable. Therefore, the control of space robot to approach a tumbling target was formulated as a unified optimal control problem. The State Dependent Riccati Equations(SDRE) method has been applied after transforming the dynamic equations into the state dependent coefficient form. Since the SDRE method needs to solve the Riccati equation repetitively at every integration step, it may cost an expensive computational efforts if the system order is higher. Therefore, the problem was then solved within the frame of θ − D method, which made a great improvement to the SDRE method in terms of the amount of calculations. The advantages of the new full-state modeling and effectiveness of the approaching strategy were demonstrated through numerical simulation by Matlab, and both the controllers are capable to control the relative orbit and attitude synchronizely. Even though θ − D controller got a slightly lower control accuracy, the computational time needed is a small fraction of the one needed by the SDRE approach. In sum, the θ − D is more suitable for implementation in real-time onboard guidance and control. Fig. 1 Approaching trajectory Fig. 2 Attitude tracking of the chaser in the orbital plane