Spacecraft Dynamics And Control Modeling Using Geometric Algebra

The coupling effect between spacecraft translational motion and rotational motion cannot be ignored in advanced space missions,such as spacecraft formation flying,proximity operations,and continuous thrust control.Therefore,the coupled spacecraft position and attitude modeling and control are increasingly being investigated.Among the combined spacecraft position and attitude motion modeling methods,the dual-quaternion-based method has become increasingly attractive in recent years.Dual quaternion is claimed to be computationally efficient,nonsingular,and compact.By using dual quaternions,the attitude and position motion equations can be modeled in the same form as the quaternion-based attitude motion equations.Moreover,the attitude and position controllers can be designed as a single control law with desirable properties similar to the quaternion-based attitude control law.Dual quaternions inherit the problems of quaternion.The dual-quaternion-based model describes the six-degree-of-freedom motion by using eight parameters.Therefore the normalization constraint problem is inevitable.The constraint also needs to be guaranteed by additional operations,thus possibly causing inconveniences.A pose representation should be explored that is complementary to dual quaternions just as the complementarity of MRP representation to the quaternions forspacecraft modeling and control applications.This desired representation should be able to provide a uniform model for the combined rotational and translational motions similar to the dual quaternion but should avoid the normalization constraint resulting from the dual-quaternion representation.The main contents are listed as follows:The twistor representation is introduced as a complementary representation to dual quaternions for the combined attitude and positionmotion parameterization.A twistor can be used as a complementary representation to dual quaternions for the modeling and control problem of the combined spacecraft attitude and position relative motion.Just as the dual quaternion,the twistors not only provide a uniform representation for the combined spacecraft rotational and translational motion but also avoid the normalization constraint resulting from the dual-quaternion representation.A novel approach towards the modeling and control problems of spacecraft relative translation and rotation by using twistors is presented.The proposed twistors-based controller ensures the global asymptotical stability of the relative motion states.The proposed controlleravoids the inconvenient computation of the logarithm of dual quaternion in the dual-quaternion-based controller,whereas the control performance is acceptable.The relative pose motion on the plane under the proposed twistor-based controller,the dual quaternion vector part based controller,and the dual quaternion logarithm based controller with the same control coefficient showed that: for the attitude motion part,when the initial angle of rotation is larger than ?,the state converges to the closer equilibrium point 2? instead of zero under the control of both controllers.For the position motion part,the trajectory of former two controller bends to the same side,whereas the trajectory of the latter controller bends to the other side.Moreover,the curvature of the trajectory under the former two controller is even,whereas that under the latter controller is either flat or flexural in different section.Another sliding mode controller is designed for the spacecraft relative translation and rotation control problem to overcome the uncertain of the mass parameter and the disturbance force and torque.The process of the controller design and the stability proving is implemented basing on the existing MRP-based attitude controller.The twistor-based sliding mode controller can ensure the global asymptotic stability of the close-loop system.The states converge fast with hardly overshoot or steady state error,and the system is robust to the uncertain of the mass parameter and the disturbance force and torque.A novel method using twistors is presented to solve the pose estimation problem of the combined spacecraft translational and rotational motion.Twistor representation allows the use of the unscented Kalman filter in the proposed pose estimation method to achieve a higher estimation accuracy compared with the method using dual-quaternion-based extended Kalman filter.It is difficult to design the UKF filter using dual quaternions because of the normalization constraint problem.To overcome this problem,twistor representation in the framework of dual numbers is introduced to describe the spacecraft local pose error,whereas dual quaternions are used to describe the absolute pose.The UKF schemes which take different measurements are designed based on the twistor representation.The proposed UKF schemes using twistors are compared with the dual-quaternion-based multiplicative extended Kalman filter through simulations.The large initial error causes a larger pose estimation error in the EKF than in the UKF.The generated results imply that the nonlinearity of the original observation model for the sensor output cannot be captured by the EKF method,whereas the UKF method can obtain the correct estimation result.The advantage of UKF is its use of second-order or higher-order approximations of nonlinear functions as basis,unlike EKF,which is essentially based on first-order approximation.Thus,as the system shows strongernonlinearity,the UKF exhibits more advantages than the EKF.Moreover,for the UKF method,different initial errors did not affect the position estimation error.The statistical results of the attitude errors in UKF are negligible because the attitude error curve significantly surges until the end of the simulation.On the basis of the results of the position estimation precision,constructing dual quaternion observation from original sensor observations may exhibit reduced estimation precision.The UKF method is known to demand higher computational cost than the EKF method.The amount of computation required for the covariance decomposition and sigma point propagation increases as the dimension of the vector increases.Our experience in the simulation shows that the presented UKF algorithm is slower than the EKF algorithm.A process noise covariance matrix and an observation noise covariance matrix instead of expanding the state vector can be exploited to reduce the computational load in UKF.As a result,the UKF algorithm can be less slow than the EKF algorithm.