Modeling and Control for an Underwater Vehicle using Dual Quaternions.

In this work, the analysis of the dynamic general model of an unmanned underwater vehicle (UUV) based on dual quaternions is presented, then the general dynamic model is reduced to a specific vehicle of 4 DoF, this model eliminates the singularities that exist with the representation of the Euler angle and that the model is more compact than others proposed in the literature [1],[2]. To demonstrate the applicability of the model, three controller strategies are proposed for tracking a trajectory, the first controller is a PD + G, under unknown disturbances it produces a considerable tracking error, the second is an adaptive controller that estimates unknown hydrodynamic parameters, and the third is a robust controller for unknown disturbances and parameter uncertainties. The closed-loop system stability analysis for each controller is based on Lyapunov’s theory, a set of numerical simulations is performedto show the behavior of the vehicle with the proposed controllers. The efficiency of the controllers is shown in Table 2 where it is deduced that the adaptive controller has a better performance. The graphics show that the robust controller has little error tracking and the computational cost is lower.

Structure Synthesis and Reconfiguration Analysis of Variable-DOF Single-Loop Mechanisms With Prismatic Joints Using Dual Quaternions

This paper deals with the structure synthesis and reconfiguration analysis of variable-DOF (variable degree-of-freedom) single-loop mechanisms with prismatic joints based on a unified tool - the dual quaternion. According to motion polynomials over dual quaternions, an algebraic method is presented to synthesize variable-DOF single-loop 5R2P mechanisms (R and P denote revolute and prismatic joints respectively), which are composed of the Bennett and RPRP mechanisms. Using this approach, variable-DOF single-loop RRPRPRR and RRPRRPR mechanisms are constructed by joints obtained from the factorization of motion polynomials. Then reconfiguration analysis of these variable-DOF single-loop mechanisms is performed in light of the kinematic mapping and the prime decomposition. The results show that the variable-DOF 5R2P mechanisms have a 1-DOF spatial 5P2P motion mode and a 2-DOF Bennett-RPRP motion mode. Furthermore, the variable-DOF 5R2P mechanisms have two transition configurations, from which the mechanisms can switch among their two motion modes.

Application of slice regularity to functions of a dual-quaternionic variable

In this paper, we present the algebraic properties of dual quaternions and define a slice regularity of a dual quaternionic function. Since the product of dual quaternions is non-commutative, slice regularity is derived in two ways. Thereafter, we propose the Cauchy-Riemann equations and a power series corresponding to dual quaternions.

A Closed-Form Solution to Planar Feature-Based Registration of LiDAR Point Clouds

Since pairwise registration is a necessary step for the seamless fusion of point clouds from neighboring stations, a closed-form solution to planar feature-based registration of LiDAR (Light Detection and Ranging) point clouds is proposed in this paper. Based on the Plücker coordinate-based representation of linear features in three-dimensional space, a quad tuple-based representation of planar features is introduced, which makes it possible to directly determine the difference between any two planar features. Dual quaternions are employed to represent spatial transformation and operations between dual quaternions and the quad tuple-based representation of planar features are given, with which an error norm is constructed. Based on L2-norm-minimization, detailed derivations of the proposed solution are explained step by step. Two experiments were designed in which simulated data and real data were both used to verify the correctness and the feasibility of the proposed solution. With the simulated data, the calculated registration results were consistent with the pre-established parameters, which verifies the correctness of the presented solution. With the real data, the calculated registration results were consistent with the results calculated by iterative methods. Conclusions can be drawn from the two experiments: (1) The proposed solution does not require any initial estimates of the unknown parameters in advance, which assures the stability and robustness of the solution; (2) Using dual quaternions to represent spatial transformation greatly reduces the additional constraints in the estimation process.

Dual-Quaternion Analytic LQR Control Design for Spacecraft Proximity Operations

Proximity operations offer aggregate capability for a spacecraft operating in close proximity to another spacecraft, to perform on-orbit satellite servicing, or to a space object to perform debris removal. To utilize a spacecraft performing such advanced maneuvering operations and perceiving of the relative motion of a foreign spacecraft, these trajectories must be modeled accurately based on the coupled translational and rotational dynamics models. This paper presents work towards exploiting the dual-quaternion representations of spacecraft relative dynamics for proximity operations and developing a sub-optimal control law for efficient and robust maneuvers. A linearized model using dual-quaternions for the proximity operation was obtained, and its stability was verified using Monte Carlo simulations for the linear quadratic regulator solution. A sub-optimal control law using generalized higher order feedback gains in dual-quaternion form was developed based on small error approximations for the proximity operation and also verified through Monte Carlo simulations. Necessary information needed to understand the theory behind the use of the dual-quaternion is also overviewed within this paper, including the validity of using the dual-quaternions against their Cartesian or quaternion equivalents.

Immersion and Invariance Adaptive Control for Spacecraft Pose Tracking via Dual Quaternions

This paper addresses the simultaneous attitude and position tracking of a target spacecraft in the presence of general unknown bounded disturbances in the framework of dual quaternions, which provides a concise and integrated description of the coupled rotational and translational motions. By virtue of the newly introduced dual direction cosine matrix, the dimension of the dual quaternion-based relative motion dynamics written in vector/matrix form can be lowered to six. Treating the disturbances as unknown parameters, a modular adaptive pose tracking control scheme composed of two separately designed parts is then derived. One part is the adaptive disturbance estimator designed based on the immersion and invariance theory. Driven by the disturbance estimation errors, it can realize exponential convergence of the estimations and has the nice “parameter lock” property, which can hardly be expected in the conventional certainty equivalent adaptive controllers. The other part is a proportional-derivative-like pose tracking controller where the estimated disturbances are directly used. The closed-loop stability of the relative motion system under different kinds of disturbances is proven by Lyapunov stability analysis. Simulations and comparisons with two previous dual quaternion-based controllers demonstrate the novel features and performance improvements of the proposed control scheme.