Simultaneous position and attitude control without linear and angular velocity feedback using dual quaternions

Abstract

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].

Pose-Tracking Controller for Satellites with Time-Varying Inertia

“…As shown in Ref. [9], the term vec q In other words, a spacecraft typically produces control forces and moments by changing its mass and inertia matrix, for example, by using thrusters and control moment gyros, respectively. In that case, the requiredṀ B can be calculated by solving the following equation: …”

“…(9) and (10), it follows thatV =−(q * B/D (q s 6) are estimates of the gravitational force, gravity-gradient torque, perturbing force due to Earth's oblateness, and dual disturbance force calculated using the estimated mass and inertia matrix. These terms can be thought of as an approximate cancellation of these forces and torques.…”

“…This interesting property has been recently used in Refs. [5,7,9,10] to extend existing attitude-only controllers with some desired properties into position and attitude controllers with equivalent desired properties. This was achieved by, essentially, replacing quaternions by dual quaternions in the attitude-only control laws and corresponding Lyapunov functions.…”

Extended Kalman Filter for Spacecraft Pose Estimation Using Dual Quaternions

“…Note that, whereas the relation between r B B∕I and r I B∕I is quadratic in q B∕I , q B∕I;d is related linearly in q B∕I with r B B∕I and r I B∕I . A unit dual quaternion is defined as a dual quaternion that belongs to the set [23] H u d fq ∈ H d : q · q qq q q 1g fq ∈ H d : q r · q r 1 and q r · q d 0g (22) From this constraint, assuming that −180 < ϕ < 180 deg, the scalar parts of the real and dual parts of a unit dual quaternion can be computed from their respective vector parts from…”

Dual quaternion based IMU and vision fusion framework for mobile augmented reality

“…Tracking would be more accurate if we can capture the translation due to rotational motion. In [9] authors talk about a combined position and attitude tracking controller based on dual quaternion, so that single control law can be used for both. [10] uses an EKF with dual quaternion based model for integrated orbit and attitude description of spacecraft motion.…”