Dual quaternion-based inverse kinematics of the general spatial 7R mechanism

Abstract

The theory of dual quaternion and its use in serial mechanisms are described in this paper. A closed-form solution to the inverse kinematic analysis of the general 7-link 7R mechanism is presented. Dixon's resultant is used and the input-output equation is expressed in the form of a 6 × 6 determinant equated to zero, and the formulae to determine other angular displacements are expressed in the closed form. Numerical example confirms these theoretical results. The whole process is very simple and easy to program, which supplies a new method for the real use of the 7R mechanism. Downloaded fromThe inverse kinematics analysis of the 7R mechanism is that s i , α i , a i , and θ 7 are known parameters and Fig. 2 The general 7-link 7R mechanism

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

“…First, note thatq e = ±1 andω B = 0 are, in fact, the equilibrium conditions for the closed-loop system formed by (12), (9), and (13). Then, consider the following candidate Lyapunov function for the equilibrium pointq e = +1 and ω B = 0 (or equivalently, (ω B ) s = 0) motivated by Eq.…”

“…Consider the rigid body kinematic and dynamic equations (12) and (9). Let the input dual force be defined by the feedback control laŵ (15) whereẑ is the output of the following LTI systemẋ p = A x p + B q e ,ẑ = (CA) x p + (CB) q e , where (A, B, C) is a minimal realization of a strictly positive real transfer matrix C sp (s) with B a full rank matrix.…”

“…Furthermore, note thatq e = ±1,ω B = 0, andx sp = 0 is the equilibrium condition for the closed-loop system formed by (12), (9), (16), and (15). Consider the candidate Lyapunov function motivated by Eq.…”

“…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].…”

Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections

“…In the same year, Pujol [15,42] investigated the relation between the composition of rotations and the product of quaternions [43] and related the work to Cayley's early contribution [11,44] through Euler-Rodrigues parameters. Following various studies, the use of the Euler-Rodrigues formula and of the Euler-Rodrigues-parameters formulated unit-quaternion has been extended to a broad range of research topics including vector parameterization of rotations [33,40,45], rational motions [46][47][48][49], motion generation [50][51][52] and planning [53], kinematic mapping [54,55], orientation [56] and attitude estimation [57][58][59], mechanics [60], constraint analysis [61,62], reconfiguration [63,64], mechanism analysis [65][66][67] and synthesis [68,69], sensing [70] and computer graphics [71][72][73] and vision [74].…”