Dual quaternion-based relative navigation for position-attitude coupling motion of spacecraft

Abstract

Relative navigation of spacecraft is one of the indispensable research tasks for space missions to utilize multiple satellites or to approach target object. In general, it is possible to obtain the navigation solution based on the GPS. However, the relative position and attitude have to only be estimated based on the equations of motions and sensor measurements in the case of the deep space mission or the proximity operation mission. That is, the accuracy of the navigation solution is determined by those two elements defined for the target object or point. In other worlds, true states may be different if the orbit information of the relative object or the precise target position information on the object is not given in the mission of the proximity operation. In particular, conventional equations describing relative motion of two spacecraft cause a navigation error due to the position-attitude coupling motion in which the attitude change of the spacecraft affects the relative position to the target.

In this dissertation, dual quaternion-based relative navigation is proposed for spacecraft that takes into consideration the position-attitude coupling motion that may occur at arbitrary points other than the center of mass of the spacecraft. Dual quaternion is a parameter defined by combination of dual number and quaternion. It represents the attitude and position in a unified form and it can be used to express the position-attitude coupling equation model easily and intuitively. Using dual quaternion-based kinematics equations for vision-based relative navigation, more effective navigation solutions can be obtained in position-attitude coupling situations. In addition, nonlinearity analysis of the parameter can show that though the linearized filter can give good results because the error occurring in the linearization process of the kinematics is not large. This dissertation is summarized into three parts.

First, it is verified that the dual quaternion-based kinematics can describe the position-attitude coupling motion for spacecraft in relative orbit motion. For a relative navigation problem having a target point on the rigid body of a relative object, the distance on the object between center of mass and the target point becomes relatively significant, as a spacecraft gets closer to the other. The relative navigation of spacecraft is generally composed of equations of motion in which the position and attitude are independent based on the center of mass of the two spacecraft. This causes navigation errors to the target point rather than the center of mass, and therefore, additional computation must be required to correct the error. Alternatively, the dual quaternion simultaneously represents position and attitude information, and its kinematics includes translational motion that varies with the rotational motion of the body. Consequently, the dual quaternion-based kinematics, which is simply defined by a unified form, can effectively reflect the position-attitude coupling problem.

Second, the dual quaternion-based extended and unscented Kalman filter are applied to the problem of estimating the relative position and attitude of two spacecraft considering position-attitude coupling. The vision-based relative navigation system using beacons and a position sensing diode sensor is constructed, and the six line-of-sight vectors obtained from the image sensor are used as the measurement value. In the implementation of an extended Kalman filter, by deriving an error dual quaternion that utilizes two parameter constrains, the covariance calculation gain can be obtained with six parameters instead of eight parameters. For both filter methods of EKF and UKF, two system models, relative velocity state propagation and measurement models, are constructed and simulated under various conditions. Simulation results show that in all cases the error of the state variable converges within the allowable range, and in particular, it can effectively solve the problem of position-attitude coupling in the relative velocity measurement model.

Lastly, nonlinearity of the dual quaternion-based kinematics is analyzed. The nonlinearity index of the system shows the degree of error of the model that may occur in the propagation process of the linearization model of the nonlinear system. The larger the nonlinearity index means that there is difference between the linearized and nonlinear model, and the navigation error obtained through the linearized filter becomes larger. The quaternion-based equation of motion is the most linearized model among the various rotation parameters, and the linearized model of dual quaternion-based kinematics derived by a similar method of quaternion has a low nonlinearity index. This indicates that the dual quaternion-based kinematics can have an acceptable navigation solution even if a linearized filter is used. A relative navigation method for position-attitude coupling problem using the dual quaternion-based kinematics is suggested in this dissertation. While previous equations of motion considered position-attitude independently, a dual quaternion-based kinematics is computed the positions affected by the attitude change simultaneously. This yields a navigation solution effectively for any point other than the center of mass of the relative object, and results are better than the previous method in case of the relative angular and linear velocity information are given. Especially, it can be applied to the case where the relative orbit or target point information is not clear, so that it may be used more helpful in practical missions.