Quaternions and Dual Quaternions: Singularity-Free Multirobot Formation Control

Abstract

Cluster space control is a method of multirobot formation keeping that considers a group of robots to be a single entity, defining state variables to represent characteristics of the group, such as position, orientation, and shape. This technique, however, suffers from singularities when a minimal state representation is used. This paper presents three alternative implementations of this control approach that eliminate singularities through changes in the control architecture or through redundant formation definitions. These proposed solutions rely on quaternions, dual quaternions, and control implementations that produce singularity-free trajectories while maintaining a cluster level abstraction that allows for simple This work has been sponsored through USAIT Grant W911NF-14-1-0008, ITBACyT 2013-17, and Agencia Nacional de Promoción Científica y Tecnológica, FONCYT PICT 2014-2055.

Pose consensus based on dual quaternion algebra with application to decentralized formation control of mobile manipulators

“…An extension to consider time-varying formations is also devised. This result is more general than the previous ones found in the literature that also focus on the formation control of systems composed of rigid bodies [28,32,33,29] as our approach: (i) is decentralized in the sense that only neighbor information is needed by each agent, in contrast to the necessity of obtaining global information such as the state variables of a shape or of a leader as in [32,33]; and (ii) is also able to deal with general directed graph topologies, in contrast to the requirement of imposing some specific graph topologies such as undirected graphs [29] and rooted trees [28]; 4. On the application side, whole-body control and consensus protocols are used to propose a strategy that allows decentralized formation control of the end-effectors of mobile manipulators whose kinematic models are given directly in the algebra of dual quaternions; 5.…”

“…Each agent's desired relative pose to the center of formation is represented by the rigid motion given by the dual quaternion δ i ∈ S. The dual quaternion representing the relation from the inertial frame to the center of formation, i.e. the pose of group formation, is represented by x c ∈ S. This framework for defining the relation has parallels with Cluster Space Control in [32] where the relative poses of the agents are defined by means of relative transformations given by dual quaternions.…”

“…In networks composed of multiple robotic manipulators, described as rigid-body agents, the agents can be modeled with dual quaternions [31] and consensus theory can be used to analyze or design distributed control laws. Some advantages of using quaternions and dual quaternions in formation control are shown by Mas and Kitts [32] in the framework of Cluster Space Control, by defining each relative position of the agents by means of relative transformations given by dual quaternions. An application on formation of unmanned aerial vehicles is shown in [33].…”

Robust H∞ kinematic control of manipulator robots using dual quaternion algebra

“…Thanks to those advantages, there has been an increasing interest in the study of kinematic representation and control in dual quaternion space. Those works comprise rigid motion stabilization, tracking, and multiple body coordination (Han et al, 2008;Wang et al, 2012;Wang and Yu, 2013;Mas and Kitts, 2017), and kinematic control of manipulators with single and multiple arms and humanrobot interaction (Adorno et al, 2010;Figueredo et al, 2013;Adorno et al, 2015).…”

Modeling and Control for an Underwater Vehicle using Dual Quaternions.

“…One way to model the behavior of vehicle movement is by using Euler's angles [1,15],this models analyze different hydrodynamic parameters, including works where the unknown parameters of mass / inertia and damping are estimated, in order to be able to compensate them [16,17], but a problem that this type of model is the singularity that exists in Euler's angles, because of this, a model was developed based on quaternions to eliminate singularities exist in the Euler angles using matrices 6 × 6 [18], another way to model dynamics and kinematics in a more compact way that includes translation and rotation is to use dual quaternions. [19,20,2,21,18], this type of model can represent the behavior of the vehicle in a more compact and simple way, consequently, the computer process is less, in addition to eliminating the singularities of the Euler angles.…”