In this paper, we propose a nonlinear adaptive position and attitude tracking controller for satellite proximity operations. This controller requires no information about the mass and inertia matrix of the satellite, and takes into account the gravitational force, the gravitygradient torque, the perturbing force due to Earth's oblateness, and other constant-but otherwise unknown-disturbance forces and torques. We give sufficient conditions on the reference motion for mass and inertia matrix identification. The controller is shown to be almost globally asymptotically stable and can handle large error angles and displacements. Unit dual quaternions are used to simultaneously represent the absolute and relative attitude and position of the satellites, resulting in a compact controller representation.

In this paper, we suggest a new representation for the combined translational and rotational dynamic equations of motion of a rigid body in terms of dual quaternions. We show that with this representation it is relatively straightforward to extend existing attitude controllers based on quaternions to combined position and attitude controllers based on dual quaternions. We show this by developing setpoint nonlinear controllers for the position and attitude of a rigid body with and without linear and angular velocity feedback based on existing attitude-only controllers with and without angular velocity feedback. The combined position and attitude velocity-free controller exploits the passivity of the rigid body dynamics and can be used when no linear and angular velocity measurements are available. I. INTRODUCTIONDual quaternions are built on, and are an extension of, classical quaternions. They provide a compact way to represent not only the attitude but also the position of a rigid body. They have been successfully applied to inertial navigation [1], rigid body control [2], [3], [4], [5], [6], [7], spacecraft formation flying [8], inverse kinematic analysis [9], computer vision [10], [11] and animation [12]. It has been argued that dual quaternions are the most compact and efficient way to simultaneously express the translation and rotation of robotic kinematic chains [13], [14]. Moreover, it has been shown that combined position and attitude control laws based on dual quaternions automatically take into account the natural coupling between the rotational and translational motion [5], [6]. Additionally, dual quaternions allow combined position and attitude control laws to be written compactly as a single control law.However, the property that makes dual quaternions most attractive and useful is that, as it will be shown, the combined translational and rotational kinematic and dynamic equations of motion written in terms of dual quaternions have the same form as the translational kinematic and dynamic equations of motion written in terms of quaternions.In this paper, we demonstrate, and take advantage of, this analogy between quaternions and dual quaternions to develop a combined position and attitude setpoint controller that does not require linear and angular velocity measurements from an existing attitude setpoint controller that does not require angular velocity measurements [15], [16].

An orientation estimation algorithm is presented. This algorithm is based on the Extended Kalman Filter, and uses quaternions as the orientation descriptor. For the filter update, we use measurements from an Inertial Measurement Unit (IMU). The IMU consists in a triaxial angular rate sensor, and an also triaxial accelerometer. Quaternions describing orientations live in the unit sphere of R 4. Knowing that this space is a manifold, we can apply some basic concepts regarding these mathematical objects, and an algorithm that reminds the also called "Multiplicative Extended Kalman Filter" arises in a natural way. The algorithm is tested in a simulated experiment, and in a real one.

Based on the highly successful quaternion multiplicative extended Kalman filter for spacecraft attitude estimation using unit quaternions, this paper proposes a dual quaternion multiplicative extended Kalman filter for spacecraft pose (i.e., attitude and position) and linear and angular velocity estimation using unit dual quaternions. By using the concept of error unit dual quaternion, defined analogously to the concept of error unit quaternion in the quaternion multiplicative extended Kalman filter, this paper proposes, as far as the authors know, the first multiplicative extended Kalman filter for pose estimation. The state estimate of the dual quaternion multiplicative extended Kalman filter can directly be used by recently proposed pose controllers based on dual quaternions, without any additional conversions, thus providing an elegant solution to the output dynamic compensation problem of the full six degree-offreedom motion of a rigid body. Three formulations of the dual quaternion multiplicative extended Kalman filter are presented. The first takes continuous-time linear and angular velocity measurements with noise and bias and discretetime pose measurements with noise. The second takes only discrete-time pose measurements with noise and hence is suitable for satellite proximity operation scenarios where the chaser satellite has only access to measurements of the relative pose, but requires the relative linear and angular velocities for control. The third formulation takes continuous-time angular velocity and linear acceleration measurements with noise and bias and discrete-time pose measurements with noise. The proposed dual quaternion multiplicative extended Kalman filter is compared with two alternative extended Kalman filter formulations on a five degree-of-freedom air-bearing platform and through extensive Monte Carlo simulations.

This paper takes advantage of a new, recently proposed representation of the combined translational and rotational dynamic equations of motion of a rigid body in terms of dual quaternions. We show that combined position and attitude tracking controllers based on dual quaternions can be developed with relatively low effort from existing attitude-only tracking controllers based on quaternions. We show this by developing an almost globally asymptotically stable nonlinear controller capable of simultaneously following time-varying position and attitude profiles without linear and angular velocity feedback based on an existing attitude-only tracking controller without angular velocity feedback.