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Kinematic Feedback Control Using Dual Quaternions Kinematic Feedback Control Using Dual Quaternions Alexander Meyer Sjøberg1 and Olav Egeland1 Abstract— This paper presents results on kinematic con- the unwinding problem [2], [3] that appear when feedback trollers for the stabilization of rigid body displacements using is taken from the quaternion vector. Quaternions have been dual quaternions. The paper shows how certain results for used in attitude estimation [14], where an overview is found quaternion stabilization of rotation can be extended to dual quaternions stabilization of displacements. The paper presents in [4]. Dual quaternions have been used for control in [9] a relevant background material on screw motion and the where the controller of [1] was extended to dual quaternions, screw description of lines and twists. Moreover, results are and in [10] where an adaptive controller was presented. presented on the computation of the exponential functions for A controller based on a small-angle approximation of the dual quaternions for use in numerical integration. The paper logarithm of the dual quaternion was proposed in [11], [22]. presents and analyzes different controllers based on feedback from dual quaternions, where some of the controllers are known In [16], [13] kinematic control was proposed with feedback from the literature, and some are new. In particular, it is from the vector part of the dual quaternion, and the hybrid shown which controllers give screw motion, and it is discussed control method of [17] was used to avoid the unwinding how this will affect the performance of the controlled system problem. compared to other controllers that are not based on screw In this paper, a new kinematic controller with feedback motion. This analysis is supported by Lyapunov analysis. Also, certain passivity properties for dual quaternions are presented from a dual quaternion is presented based on controllers for as an extension to previously published results on passivity for quaternions [23], [6]. This controller can be implemented quaternions. to give a screw motion, or more direct translation. Hybrid control is not used as the focus is on the kinematic properties I. INTRODUCTION of the dual quaternion feedback, and how this relates to previ- Quaternions have been studied extensively in the control ous quaternion control methods. The controller does not have community for the last three decades for use in attitude problems with unwinding as there is an unstable equilibrium control and estimation on SO(3). More recently, also dual at p. Moreover, a solution for efficient computation of the quaternions have been studied for control and estimation on exponential function for a dual quaternion is presented for SE(3). Many of the properties of quaternions are transferred use in numerical time integration that eliminates the need to dual quaternions, and a geometric interpretation based on for normalization of the real term of the quaternions and a dual angle about a line can be seen as an extension of projection of the dual term. the quaternion interpretation in terms of a rotation about The paper is organized as follows: Sect. II presents back- a vector. Still, there are some notable differences between ground on quaternions, dual quaternions, screws, and screw quaternions on SO(3) and dual quaternions on SE(3) that motion. Sect. III presents different controllers based on dual have an impact on controller design that will be discussed in quaternion feedback, and an analysis of the properties of the this paper. In particular, a bi-invariant metric on SO(3) can controllers, and Lyapunov analysis and passivity properties. be described in terms of the rotation angle q, and Lyapunov Sect. IV presents the implementation aspects and results functions can be formulated based on the quaternion. In from simulations of the kinematic controllers, while Sect. V contrast to this, there is no bi-invariant metric on SE(3), and concludes this paper. distance measures will be scale-dependent [20]. Moreover, to formulate norms or Lyapunov functions on SE(3) from II. PRELIMINARIES dual quaternions, it is necessary to use inner products on the Euclidean representation of real and dual parts. A. Quaternions Quaternions were used for attitude control with feedback A quaternion q = h + s is written as the sum of a scalar from the quaternion vector in [23] using Lyapunov analysis part h and a vector part s [7], [12]. Conjugation is given and in [6] using passivity. In [21] feedback was taken from by q∗ = h − s , and multiplication with a scalar l gives the Rodrigues vector and the modified Rodrigues vector. The lq = lh + lss . Let q = h + s and q = h + s be two need for velocity feedback was eliminated in [15], [1]. More 1 1 1 2 2 2 quaternions. Then the sum is q + q = h + h + s + s , recent work [17] has used hybrid control as a solution to 1 2 1 2 1 2 and the quaternion product is q1 ◦ q2 = h1h2 − s 1 · s 2 + *The research presented in this paper was funded by the Norwegian Re- h1s 2 + h2s 1 + s 1 × s 2. The norm of a quaternion is given ∗ search Council under Project Number 237896, SFI Offshore Mechatronics. by jjqjj2 = q ◦ q = h2 + s · s . 1The authors are with the Department of Mechanical and Indus- A unit quaternion q has unit norm which means that trial Engineering, Norwegian University of Science and Technology 2 2 (NTNU), 7491 Trondheim, Norway falexander.m.sjoberg, kqk = h + s · s = 1. A unit quaternion can be used to [email protected] describe a rotation (k;q) by an angle q about a unit vector k by letting q q q = h + s ; h = cos ; s = k sin 2 2 Then if R = I + sinqk× + (1 − cosq)k×k× 2 SO(3) is the associated rotation matrix, the rotation of a general vector u can be achieved with the two alternative expressions RRuu = q ◦ u ◦q∗. It is noted that q and −q describes the same rotation. The vector e = 2hss = k sinq will also be used. B. Kinematic Differential Equation of a Unit Quaternion Let q = qab be the quaternion describing the rotation from a frame a to a frame b. Then the kinematic differential equation is 1 1 Fig. 1. The displacement t shown as the sum of the translation u generated q˙ = q ◦ w b = w a ◦ q (1) by the rotation q about the screw axis kˆ, and the translation dkk along the 2 2 the screw axis. b b where w = w ab is the angular velocity of frame b with the same line. An illustration is shown in Fig. 1. This can a b respect to frame a in the coordinates of b, and w = q ◦w ◦ be described as a motion by a dual angle qˆ = q + ed about q∗ is the same vector in the coordinates of a. the line kˆ [18]. The displacement can be represented by C. Dual Vectors, Screws and Lines R (I − R)c + dkk T = T 2 SE(3) (3) A dual real number is written aˆ = a + ea0 where a and 0 1 0 a are real numbers and e is the dual unit which satisfies R k 2 where R is the rotation matrix corresponding to q and k, e 6= 0 and e = 0 [18]. A real-valued function f (a) can be c = k ×k0 and d = k ·t. If the motion from frame a to frame extended to a function of a dual number where b is done as a screw motion about a fixed line kˆ, then w = q˙k, d f (a) 0 and the velocity v of a point fixed in b that is on the screw f (aˆ ) = f (a) + e a (2) kˆ da ˆ ˙ axis k will be vkˆ = dkk. It is noted that E. Dual Quaternions 0 0 sinaˆ = sina + ea cosa; cosaˆ = cosa − ea sina The dual quaternion can also be written qˆ = q +eq0 where 0 ∗ A dual 3D vector is given by uˆ = u +eu0 where u and u0 are q and q are quaternions. The conjugate is given by qˆ = ∗ 0∗ 0 0 3D vectors. Computations on dual numbers and dual vectors q + eq . Let qˆ 1 = q1 + eq1 and qˆ 2 = q2 + eq2 be two dual are performed as operations on polynomials in the dual unit quaternions. Then the quaternion product is 0 0 e. For two dual vectors uˆ 1 = u1 +eu1 and uˆ 2 = u2 +eu2 this q q q q q q0 q q0 0 0 qˆ 1 ◦ qˆ 2 = q1 ◦ q2 + e(q1 ◦ q2 + q2 ◦ q1) gives uˆ 1 + uˆ 2 = u1 + u2 + e(u1 + u2), uˆ · uˆ 2 = u1 · u2 + e(u1 · 0 0 0 0 2 ∗ 2 0∗ 0 ∗ u2 + u1 · u2), and uˆ × uˆ 2 = u1 × u2 + e(u1 × u2 + u1 × u2). The norm is kqˆk = qˆ ◦ qˆ = kqk + e(q ◦ q + q ◦ q ).A A screw is a dual vector which satisfies the screw trans- dual quaternion is called a dual unit quaternion if kqˆk = 1, a 0a formation law. Let sˆa = s + esa be the screw referenced to which means that q is a unit quaternion and frame a and given in the coordinates of a. Then the same q ◦ q0∗ + q0 ◦ q∗ = 2(hh0 + s · s 0) = 0 (4) screw referenced to a frame b and given in the coordinates b 0b 0b a 0a a a of b will be sˆb = s + esb where sb = Rb(sa +t × s ). A displacement of a vector can be expressed as RRuu +t = A line can be described by the screw kˆ = k + ek0, where qˆ ◦ u ◦ qˆ ∗, while the screw transformation from frame a to 0 ∗ k is the unit vector along the line, and k = c × k is the frame b can written [5] sˆa = qˆ ◦ sˆb ◦ qˆ .
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