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We present a tracking controller for mobile multi-robot systems based on dual quaternion pose representations applied to formations of robots in a leader-follower configuration, by using a cluster-space state approach. The proposed controller improves system performance with respect to previous works by reducing steady-state tracking errors. The performance is evaluated through experimental field tests with a formation of an unmanned ground vehicle (UGV) and an unmanned aerial vehicle (UAV), as well as a formation of two UAVs.

Concurrent position and attitude (pose) tracking of a rigid spacecraft is addressed when there are no (both linear and angular) velocity measurements, and only pose information is available. A second-order observer with a simple structure is designed to provide spacecraft velocity estimates. It consists of one part that mimics the six-degrees-of-freedom (6DOF) spacecraft kinematics and dynamics and a correction part driven by the error between pose measurements and estimates. The resultant velocity estimates are then utilized to construct a velocity-free pose tracking controller with a proportional-derivative plus feedforward form. All the theoretical designs and analyses are conducted in a compact dual-quaternion framework and incorporate hybrid control, homogeneous nonsmooth feedback, and Lyapunov theory. Rigorous proof shows that the observer alone and the combined observer-controller both achieve global finite-time convergence, which is an advantageous property compared with existing 6DOF velocity observers and velocity-free pose controllers. Numerical examples are presented to demonstrate the application and effectiveness of the proposed method.

Modern approaches for robot kinematics employ the product of exponentials formulation, represented using homogeneous transformation matrices. Quaternions over dual numbers are an established alternative representation; however, their use presents certain challenges: the dual quaternion exponential and logarithm contain a zero-angle singularity, and many common operations are less efficient using dual quaternions than with matrices. We present a new derivation of the dual quaternion exponential and logarithm that removes the singularity, we show an implicit representation of dual quaternions offers analytical and empirical efficiency advantages compared with both matrices and explicit dual quaternions, and we derive efficient dual quaternion forms of differential and inverse position kinematics. Analytically, implicit dual quaternions are more compact and require fewer arithmetic instructions for common operations, including chaining and exponentials. Empirically, we demonstrate a 30–40% speedup on forward kinematics and a 300–500% speedup on inverse position kinematics. This work relates dual quaternions with modern exponential coordinates and demonstrates that dual quaternions are a robust and efficient representation for robot kinematics.

Recently, continuum flexible robots have been designed for the use in diverse applications; including the exploration of confined static and dynamic environments. One of the challenging tasks for those robots is planning optimal trajectories due to, not only the redundant Degrees of Freedom (DOF) they own but also their compliant behaviour. In this paper, an Imitation-based Pose Planning (IbPP) approach is proposed to teach a two-section continuum robot the motion primitives that will facilitate achieving and generalizing for spatial point-to-point motion which involves both position and orientation goals encoded in a dual quaternion form. Two novel approaches are proposed in this research to intuitively generate the motion demonstrations that will be used in the proposed IbPP. Namely, a flexible input interface, acting as a twin robot, is designed to allow a human to demonstrate different motions for the robot end-effector. Alternatively, as a second approach, the Microsoft Kinect sensor is used to provide motion demonstrations faster via human arm movements. Based on the kinematic model of the two-section continuum robot, a Model Reference Adaptive Control (MRAC) algorithm is developed to achieve tracking the generated trajectory from the IbPP and to guarantee the robustness against the model uncertainties and external disturbances. Moreover, controller stability analysis is developed based on Lyapunov criteria. Finally, simulations are conducted for the two-section continuum robot to prove the ability of the proposed IbPP with the two proposed inputs to learn and generalize for spatial motions, which in future could be easily accommodated for robots with multiple sections. In addition, the proposed MRAC shows a significant performance towards following the required trajectory and rejecting the external disturbance.

21th IFAC World Congress, online, 11 Jul 2020 - 17 Jul 2020; IFAC-PapersOnLine 53(2), 9992-9999 (2020). doi:10.1016/j.ifacol.2020.12.2717 special issue: "21th IFAC World Congress / Edited by Rolf Findeisen, Sandra Hirche, Klaus Janschek, Martin Mönnigmann", Published by Elsevier, Frankfurt ; München [u.a.]

A typical hand-eye robot system has two related kinematic loops, which proper functioning requires the accurate determination of robot-world (RW) and hand-eye (HY) transformations in them. Insofar as HY calibration can resolve only the issues related to HY transformation, this study is focused on the joint treatment of the RW&HY transformations, which process is referred to as simultaneous RW&HY calibration. An alternative linear programming is adopted for the construction of semi-convex objective functions via two algorithms. These involve the homogeneous matrix and dual quaternion parameterizations, respectively. Their feasibility was further tested using simulated and real experimental datasets and compared to the results obtained via two available methods (NL and LMI). The results obtained strongly suggest that the homogeneous matrix parametrization (ALP1) had better performance than the dual quaternion one (ALP2). Meanwhile, both of them yielded the good optimal calibration solutions instead of local minima ones. Therefore, both formulations provide new insights into the behavior and complexity of the simultaneous RW&HY calibration.

Tese (doutorado) — Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Elétrica, 2016. O objetivo principal deste trabalho é contribuir para a compreensão e para o avanço das técnicas de controle para robôs manipuladores no contexto de aprimorar o estado-da-arte em robótica. O presente trabalho fornece soluções para uma série de problemas relevantes e desafiadores com ênfase na precisão, robustez, otimização e flexibilidade do controle cinemático no espaço de tarefas. Estas características são essenciais para o desenvolvimento de ferramentas que permitam robôs executarem tarefas sofisticadas em cenários cada vez menos controlados. As contribuições são obtidas a partir da exploração da álgebra de quatérnios duais que é a abordagem escolhida para a modelagem e o controle da cinemática acoplada dos robôs manipuladores. A fim de defrontar a suscetibilidade de manipuladores com respeito a erros de modelagem, incertezas, distúrbios exógenos e a influência destes na pose do efetuador, adaptamos técnicas de controle H∞—válidas apenas para perturbações aditivas—para a álgebra de quatérnios duais. Assim, obtemos um vínculo intuitivo e elegante entre o desempenho sobre a trajetória do efetuador e diferentes fontes de incertezas e perturbações, ao mesmo tempo que atenuamos perturbações no sentido H∞ com o mínimo esforço de controle instantâneo. A partir da álgebra de quatérnios duais, também propomos novas estratégias de controle ótimo para manipuladores. Em contraste a técnicas clássicas, a otimização foca-se em variáveis do espaço de tarefas. Além de prevenir derrapagens e sobrepassos, esta nova estratégia é particularmente relevante ao lidarmos com os desafios inerentes a cenários que requerem maior precisão e segurança de interação em contraste com velocidade de convergência—é o caso, por exemplo, de robôs complacentes, múltiplos braços e interação homem-robô, onde a velocidade no espaço de tarefas é fator que influencia largamente a aceitação humana ao contato. A fim de aprimorar a reatividade do manipulador, introduzimos um novo critério que enriquece movimentos no espaço nulo. Utilizando controle chaveado, relaxamos determinadas especificações reduzindo o número de graus de liberdade controláveis, mas mantendo o efetuador dentro de subespaços desejados. Assim, o espaço nulo da Jacobiana é ampliado com mais graus de liberdade possibilitando a execução de tarefas adicionais. Aparte destes requisitos, visamos igualmente assegurar a manipulação adequada em toda a área de trabalho. Singularidades intrínsecas à cinemática são defrontadas tanto por meio de movimentos no espaço nulo—que mostramos ser válidos no formalismo de quatérnios duais—como através de uma nova estratégia de prevenção com base em critérios H∞. A solução é adequada para a variedade de quatérnios duais e regula explicitamente a compensação entre a exatidão e esforço de prevenção de singularidades inevitáveis. Finalmente, também abordarmos a obstrução topológica referente a estabilização global a partir de controladores contínuos—restrição válida para todas as representações de corpos rígidos. No caso de quatérnios duais unitários, a obstrução é representada pela problema de cobertura dupla inerente ao grupo. Neste trabalho, propomos uma lei de controle chaveada com histerese que assegura a estabilidade global robusta e assintótica no espaço de quatérnios duais. Uma extensa coleção de exemplos, simulações e experimentos são realizados para ilustrar a eficácia e relevância dos resultados. Ademais, e mais importante, os resultados propostos são validados por uma exploração matemática rigorosa a fim de garantir precisão e síntese adequada para aplicações práticas. Assim, o trabalho formaliza a teoria de controle na álgebra de quatérnios duais e aprimora o estado-da-arte de controle para manipuladores—contribuindo com o esforço contínuo para o desenvolvimento da robótica do futuro. In the light of the ongoing effort to improve the state-of-the-art in robotics, this thesis’s main goal is to contribute to the understanding and advancement of control techniques for robot manipulators. In this context, the present work provides solutions to a number of relevant and challenging task-space kinematic control problems for robot manipulators in the sense of accuracy, robustness, optimality and flexibility. These are cornerstone features required to unfold tools for robots to execute sophisticated tasks deployed at increasingly less controlled scenarios. To support the theory presented herein, this thesis exploits the advantages of the dual quaternion algebra to model and control the robot coupled translation and rotation kinematics. First, to address the manipulator liability to modeling errors, uncertainties, exogenous disturbances, and their influence upon the robot orientation and translation kinematics, we adapt H∞ techniques— suitable solely for additive noises—to dual quaternion algebra. In this sense, we provide an intuitive and elegant connection between performance effects over the end-effector trajectory and different sources of uncertainties and disturbances while satisfying disturbance attenuation properties with minimum instantaneous control effort. Sharing the dual quaternion advantages, we also propose novel optimal control strategies in dual quaternion task-space. In contrast to standard optimal control techniques, herein, we focus on optimizing task-space variables. In addition to preventing drifts and overshoots, the analysis is particularly relevant to address the challenges for increasing accuracy in scenarios that require precise and safe interaction in contrast to convergence speed—for instance, compliant robots, multiple arms and human-robot interaction where literature shows that controlled speed at taskspace largely influences the human acceptance of the contact. Finally, to cope with reactiveness requirements, we introduce a new criterion to enrich the selfmotions of robot manipulators. Using a switching control technique, we relax particular task specifications in order to control fewer degrees of freedom while maintaining the end-effector within desired subspaces. As a consequence, the Jacobian nullspace is enlarged with more degrees of freedom to perform additional self-motion tasks. Departing from the requirements for robustness, optimality and reactiveness, this thesis also aims at ensuring proper manipulation throughout the workspace. Singularities instrisic to the kinematic modeling are addressed both by means of self-motion—which is shown to be valid within the dual quaternion formalism—and using a novel avoidance strategy based on the H∞ criteria. The novel solution is suitable for dual quaternion manifold and explicitly regulates the trade-off between exactness and effort of avoiding inescapable singularities. Finally, we also approach the topological obstruction to global stabilization using continuous feedback—inherent to all rigid body representations. In the case of unit dual quaternions, the obstruction is reflected by the unwinding phenomenon that stems from the group double cover. Herein, we propose a switching control law with hysteresis that ensures robust global asymptotic stability within dual quaternion space. An extensive collection of examples, simulations and experiments are performed to illustrate the effectiveness and relevance of our results. Most important, all the proposed results are supported by rigorous mathematical evidence to ensure accuracy and proper synthesis with practical applications. Therefore, we believe the approaches herein formalize the control theory within unit dual quaternions and improve the state-of-the-art in control of manipulators—contributing to the ongoing effort of unfolding tools for the future of robotics.

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 based on dual quaternions as well as the prime decomposition. The results show that the variable-DOF 5R2P mechanisms have a 1DOF spatial 5P2P motion mode and a 2DOF 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.

The Delta robot, widely used in fast pick-and-place applications with pure position control, is a parallel kinematic chain with three rotational inputs resulting in three pure translations at the end-effector. This paper proposes a complete task-space impedance control with inverse dynamics to give this robot compliant behavior, enabling it to be used in tasks involving physical interaction. For that purpose, the well-known usage of dual quaternion algebra for kinematics modeling is novelly integrated with a neural network to compose a compact representation for the forward kinematics function, that is singularity-free and suitable for real-time calculation. This network computes the forward kinematics more than 150 times faster than a numeric equation solving algorithm, with an average estimation error of less than 0.5 mm. The proposed algorithm is implemented in a rigid body simulator, and the performance of the complete system is analyzed.

Control input constraints especially velocity saturations make a gap between theoretical control laws and applications on real vehicles. This paper presents a first attempt to solve the set-point control problem for the unit dual quaternion (UDQ) based kinematic system subject to input constraints. It strictly proves that the UDQ-logarithm based proportional control law can still make this system stable by using Lyapunov method. The control performance of the system is illustrated and verified by numerical simulations.

This paper presents a framework based on dual quaternions for obtaining the forward and inverse kinematic models for a 6-dof (degrees of freedom) UR5 cobot. A forward kinematics model based on dual quaternions is developed for the cobot. Dual quaternion differential kinematics involving Jacobian transpose and damped least squares methods along with pose error feedback are used for determining the inverse kinematics model. Implementation of inverse kinematics methods for a given trajectory shows that Jacobian transpose method is faster, but gives more jerky motions and is less immune to singularity compared to damped least squares.

Kinematics studies the motion of bodies without consideration of the forces or moments that cause the motion. Robot kinematics refers the analytical study of the motion of a robot manipulator. Formulating the suitable kinematics models for a robot mechanism is very crucial for analyzing the behaviour of industrial manipulators. There are mainly two different spaces used in kinematics modelling of manipulators namely, Cartesian space and Quaternion space. The transformation between two Cartesian coordinate systems can be decomposed into a rotation and a translation. There are many ways to represent rotation, including the following: Euler angles, Gibbs vector, Cayley-Klein parameters, Pauli spin matrices, axis and angle, orthonormal matrices, and Hamilton 's quaternions. Of these representations, homogenous transformations based on 4x4 real matrices (orthonormal matrices) have been used most often in robotics. Denavit & Hartenberg (1955) showed that a general transformation between two joints requires four parameters. These parameters known as the Denavit-Hartenberg (DH) parameters have become the standard for describing robot kinematics. Although quaternions constitute an elegant representation for rotation, they have not been used as much as homogenous transformations by the robotics community. Dual quaternion can present rotation and translation in a compact form of transformation vector, simultaneously. While the orientation of a body is represented nine elements in homogenous transformations, the dual quaternions reduce the number of elements to four. It offers considerable advantage in terms of computational robustness and storage efficiency for dealing with the kinematics of robot chains (Funda et al., 1990). The robot kinematics can be divided into forward kinematics and inverse kinematics. Forward kinematics problem is straightforward and there is no complexity deriving the equations. Hence, there is always a forward kinematics solution of a manipulator. Inverse kinematics is a much more difficult problem than forward kinematics. The solution of the inverse kinematics problem is computationally expansive and generally takes a very long time in the real time control of manipulators. Singularities and nonlinearities that make the

Dual quaternion is the parameter that can describe position and attitude in a unified form. It is extension of dual number and quaternion; so, mathematical and parameter characteristics of those are holding. The attitude and orbital kinematics model can be concurrently integrated in a simplified and intuitive formulation with dual quaternion. From these merits, dual quaternion-based relative navigation for spacecraft proximity operation is investigated in this paper. Especially, the case is considered which interested points are not coincident with center of mass of spacecraft. At this time, it is obvious that position is affected by rotational motion of rigid body. With conventional decoupled relative kinematics, this attitude–position coupling effect cannot be described at once. However, the new dual quaternion-based relative kinematics can represent coupled orbital motion in a simple form. Motivated by these backgrounds, in this paper, to estimate relative position and attitude states using vision sensor, dual quaternion-based relative kinematics is formulated. To reduce the number of state variables, error dual quaternion is applied, which makes the number of state variable be six instead of eight, by using parameter constraints of dual quaternion. Extended Kalman filter and unscented Kalman filter are adopted to realize relative navigation. Since the necessity of velocity information in dual quaternion kinematics, two types system models are constructed—velocity propagation and velocity measurement model. In addition, the six line-of-sight measurements are used to update the dual quaternion parameter. Simulation results show that the kinematics model is accurately derived, states are well estimated, and statistics of estimation errors are consistent with estimated error covariance.

Purpose The purpose of this paper is to propose an error model for serial robot kinematic calibration based on dual quaternions. Design/methodology/approach The dual quaternions are the combination of dual-number theory and quaternion algebra, which means that they can represent spatial transformation. The dual quaternions can represent the screw displacement in a compact and efficient way, so that they are used for the kinematic analysis of serial robot. The error model proposed in this paper is derived from the forward kinematic equations via using dual quaternion algebra. The full pose measurements are considered to apply the error model to the serial robot by using Leica Geosystems Absolute Tracker (AT960) and tracker machine control (T-MAC) probe. Findings Two kinematic-parameter identification algorithms are derived from the proposed error model based on dual quaternions, and they can be used for serial robot calibration. The error model uses Denavit–Hartenberg (DH) notation in the kinematic analysis, so that it gives the intuitive geometrical meaning of the kinematic parameters. The absolute tracker system can measure the position and orientation of the end-effector (EE) simultaneously via using T-MAC. Originality/value The error model formulated by dual quaternion algebra contains all the basic geometrical parameters of serial robot during the kinematic calibration process. The vector of dual quaternion error can be used as an indicator to represent the trend of error change of robot’s EE between the nominal value and the actual value. The accuracy of the EE is improved after nearly 20 measurements in the experiment conduct on robot SDA5F. The simulation and experiment verify the effectiveness of the error model and the calibration algorithms.

This paper presents a new solution for inverse kinematics (IK) by using dual quaternions (DQ) as operators and combining them with the Paden-Kahan subproblem to determine the analytical solution. Kinematics under the structure of screw theory makes the process mathematically efficient and gives geometric meaning. Both derived results and simulation outgrowths, when compared with the generally used numerical solution and matrix-based analytical solution, show that our method is faster and does not suffer from the numerical instability caused by being near to a singular configuration. The method has been proven to have the advantages of reducing the computational load and promoting precision, and it can provide multiple choices for IK.

The avoidance of singularities is critical to design and control of parallel robots. This paper aims at efficient determination of the maximal singularity-free joint space and workspace of a class of six-DOF parallel robots with six kinematic chains of same type. We represent the singularity-free joint space by a 6-cube and determine it firstly. The singularity-free workspace is generated by continuous motion of all active joints in the singularity-free joint space. As a result, the boundary of the workspace can be obtained with simultaneous consideration of position and orientation of the mobile platform. The size relation between the maximal singularity-free joint space and workspace is discussed. To efficiently determine the singularity-free joint space and workspace, we propose dual quaternion-based Jacobian matrices and construct an efficient algorithm. The algorithm detects singularities in a given joint space and simultaneously calculates its corresponding workspace. The computational costs of the proposed algorithm and the traditional one are compared using a 6-U P S parallel robot, leading to 9 seconds and 458 seconds respectively. Finally, both the maximal singularity-free joint space and workspace of a 6- P US parallel robot are determined to further demonstrate the effectiveness of the new approach.

Using dual quaternions, the closure equations of a kinematic loop can be expressed as a system of multiaffine equations. In this paper, this property is leveraged to introduce a branch-and-prune method specially tailored for solving such systems of equations. The new method is objectively simpler (in the sense that it is easier to understand and to implement) than previous approaches relying on general techniques such as interval Newton methods or methods based on Bernstein polynomials or linear relaxations. Moreover, it relies on two basic operations — linear interpolation and projection onto coordinate planes— that can be efficiently computed. The generality of the proposed method is evaluated on position analysis problems with 0-, 1-, and 2-dimensional solution sets, including the inverse kinematics of serial robots and the forward kinematics of parallel ones. The results obtained on these problems show that the efficiency of the method compares favorably to state-of-the-art alternatives., This work was partially supported by the Spanish Ministry of Economy and Competitiveness through the project DPI2017-88282-P.

*Yüca, Gülsüm ( Aksaray, Yazar ), In this study, we are interested in the way quaternions to represent 3D and 4D rotations in Lorentzian space. We give a new method for obtaining a rotation matrix in Lorentzian space with the help of a unit quaternion. Furthermore, we prove that rotation matrices correspond to a quaternion leave invariant the same axis in Euclidean and Lorentzian space. Then, we introduce a semi-orthogonal matrix representation of a quaternion curve in 4D space. Moreover, we provide applications and draw their figures to explore visual representations. Finally, due to the importance of the dual space in kinematics, robotics, and other areas related, we carry this work into their dual spaces by using a dual quaternion.

Conventional control of robotic manipulators requires prior knowledge of their kinematic structure. Model-learning controllers have the advantage of being able to control robots without requiring a complete kinematic model and work well in less structured environments. Our recently proposed Encoderless controller has shown promising ability to control a manipulator without requiring any prior kinematic model whatsoever. However, this controller is only limited to position control, leaving orientation control unsolved. The research presented in this paper extends the state-of-the-art kinematic-model-free controller to handle orientation control to manipulate a robotic arm without requiring any prior model of the robot or any joint angle information during control. This paper presents a novel method to simultaneously control the position and orientation of a robot&rsquo, s end effector using locally weighted dual quaternions. The proposed novel controller is also scaled up to control three-degrees-of-freedom robots.

This paper deals with the construction and reconfiguration analysis of a spatial mechanism composed of four circular translation (G) joints. Two links connected by a G joint, which can be in different forms such as a planar parallelogram, translate along a circular trajectory with respect to each other. A spatial 4G mechanism, which is composed of four G joints, usually has one degree-of-freedom (DOF). First, a 2-DOF spatial 4G mechanism is constructed. Then, a novel variable-DOF spatial 4G mechanism is constructed starting from the 2-DOF 4G mechanism using the approach based on screw theory. Finally, the reconfiguration analysis is carried out in the configuration space using dual quaternions and tools from algebraic geometry. The analysis shows that the variable-DOF spatial 4G mechanism has one 2-DOF motion mode and one to two 1-DOF motion modes and reveals how the 4G mechanism can switch among these motion modes. By removing one link from two adjacent G joints each and two links from each of the remaining two G joints, we can obtain a queer-rectangle and a queer-parallelogram, which are the generalization of the queer-square or derivative queer-square in the literature. The approach in this paper can be extended to the analysis of other types of coupled mechanisms using cables and gears and multi-mode spatial mechanisms involving G joints.

This objective of this paper is to develop a dual quaternion based method for estimating target volumes in radiation therapy for head and neck cancer. Inaccuracies in radiation targeting are responsible for incidental exposure of healthy adjacent tissues, causing significant morbidity and mortality. This paper focuses on inaccuracies incurred when a tumor is displaced during treatment. To address this problem, the clinical target must be expanded to cover the region through which the tumor might move. The resulting expanded target is known as the Planning Target Volume (PTV). In the current practice, the rotational components of displacements are neglected, producing planning target volumes that either miss the true target motion or are larger than needed to cover the target path. By using the dual quaternion based kinematic formulation, this paper represents and captures both translational and rotational inaccuracies. It then presents a framework for calculating the PTV swept out by the target as it shifts within its range of translations and rotations.

The parameterization of a rigid-body motion can be done using multiple algebraic entities. A very important criterion when choosing a parameterization method is the number of algebraic equations and variables. Recently, orthogonal dual tensors and dual quaternion proved to be a complete tool for computing rigid body displacement and motion parameters. The present research is focused on developing new methods for recovering kinematic data when the state of features attached to a body during a rigid displacement is available. The proof of concept is sustained by computational solutions both for the singularity-free extraction of a dual quaternion from feature-based representation of motion and for the recovery algorithms of the dual quaternion.

This article presents a numerical method for the inverse kinematics of serial chains by utilizing dual quaternion formulation of the robot kinematics within a converging paths algorithm. The method is inspired by iterative techniques such as FABRIK, however adding information on the kinematics of the chain to be solved. The method has been tested with 2R, planar 3R and spatial 3R robots. Future work includes optimizing the method to compare with other fast numeric algorithms for inverse kinematics.

The paper proposes a mathematical method for redefining motion parameterizations based on the joint-space representation of parallel robots. The study parameters of SE(3) are used to describe the robot kinematic chains, but, rather than directly analyzing the mobile platform motion, the joint-space of the mechanism is studied by eliminating the Study parameters. From the loop equations of the joint-space characterization, new parameterizations are defined, which enable the placement of a mobile frame on any mechanical element within the parallel robot. A case study is presented for a medical parallel robotic system in which the joint-space characterization is achieved and based on a new defined parameterization, the kinematics for displacement, velocities, and accelerations are studied. A numerical simulation is presented for the derived kinematic models, showing how the medical robot guides the medical tool (ultrasound probe) on an imposed trajectory.

In this paper, a dual quaternion-based methodology for computing the forward and inverse kinematic models for a serial manipulator is presented. A dual quaternion-based forward kinematics model is developed for the Kuka LBR IIWA 7 R800 cobot. An inverse kinematics model is developed that uses dual quaternion differential kinematics and includes Jacobian transpose and damped least squares methods for determining Jacobian pseudo-inverse. Implementation of these methods on a given trajectory shows that, compared to damped least squares, the Jacobian transpose method is faster, but is less immune to singularity and gives more jerky motions.

This paper explores an accurate and complete spacecraft six-degree-of-freedom coupled relative motion model using the dual quaternion representation. Based on this technique, we build a scheme capable of describing both kinematic and dynamic coupling effects on the spacecraft relative translational motion through a further combination with the chaser’s precise absolute translational and rotational dynamics. This new model generalizes the existing nonlinear spacecraft relative translational model to include both the kinematic coupling effect due to the displacements of selected feature points relative to the spacecraft centers of mass and the dynamic coupling effect induced by the gravity gradient torque and the orbital perturbations. Several numerical simulations are implemented to validate the feasibility of the proposed model, for analyzing both the kinematic and dynamic coupling effects, on the relative translational motion of two arbitrary feature points in either Keplerian or J2 perturbed orbits. The results are further compared against the J2 perturbation effect., J.C. thanks the support of the Chinese Scholarship Council. This study was funded by the Ministerio de Economia y Competividad (Grant Numbers PGC2018-100928-B-100 and MTM2016-80117-P), the Catalan government (Grant Numbers 2017SGR-1049 and 2017SGR-1374) and the National Natural Science Foundation of China (Grant Number 11572248).

Two novel dual quaternion Kalman filters for pose estimation in a satellite relative navigation problem are developed in this work. The satellites are assumed to be equipped with a vision navigation system that provides batches of intersatellite lines-of-sight. The measurements of the inertial angular velocities and the relative translation velocity are corrupted by additive biases and white noises and transferred over an intersatellite link for processing. Novel techniques of the dual quaternion nonlinear constraints are proposed: 1) a normalization and projection step along with a partial reset, and 2) pseudo-measurement updates. Extensive Monte-Carlo simulation results, along with a detailed numerical study of the observability Gramians, illustrate the relative advantage of the proposed dual quaternion filters over various state-of-the-art pose filters.

Kinematics analysis studies the relative motions, such as, first of all, the displacement in space of the end effector of a given robot, and thus its velocity and acceleration, associated with the links of the given robot that is usually designed so that it can position its end-effector with a three degree-of-freedom of translation and three degree-of-freedom of orientation within its workspace. This chapter presents mainly, on the light of both main concepts; the first being the screw motion or/ and dual quaternions kinematics while the second concerns the classical ‘Denavit and Hartenberg parameters method’ the direct kinematics of a planar manipulator. First of all, examples of basic solid movements such as rotations, translations, their combinations and general screw motions are studied using both (4x4) matrices rigid body transformations and dual quaternions so that the reader could compare and note the similarity of the results obtained using one or the other method. Both dual quaternions technique as well as its counterpart the classical ‘Denavit and Hartenberg parameters method’ are finally applied to a three degree of freedom (RRR) planar manipulator. Finally, we and the reader, can observe that the two methods confirm exactly one another by giving us the same results for each of the examples and applications considered, while noting that the fastest, simplest more straightforward and easiest to apply method, is undoubtedly the one using dual quaternions. As a result this work may as well act as a beginners guide to the practicality of using dual-quaternions to represent the rotations and translations ie: or any rigid motion in character-based hierarchies. We must emphasize the fact that the use of Matlab software and quaternions and / or dual quaternions in the processing of 3D rotations and/or screw movements is and will always be the most efficient, fast and accurate first choice. Dual quaternion direct kinematics method could be generalised, in the future, to more complicated spatial and/ or industrial robots as well as to articulated and multibody systems

Based on the dual-quaternion description, a smooth six-degree-of-freedom observer is proposed to estimate the incorporating linear (translational) and angular velocity, called the dual-angular velocity, for rigid bodies. To establish the observer, some important properties of dual vectors and dual quaternions are presented and proved, additionally, the kinematics of dual-transformation matrices is deduced, and the transition relationship between dual quaternions and dual transformation matrices is subsequently analyzed. An important feature of the observer is that all estimation states are ensured to be $\mathcal {C}^{\infty }$ continuous, and estimation errors are shown to exhibit asymptotic convergence if the signals to be estimated are bounded. Furthermore, to achieve tracking control objectives, the proposed observer is combined with an independently designed proportional-derivative-like feedback control law (using full-state feedback), and a special Lyapunov “strictification” process is employed to ensure a separation property between the observer and the controller, which further guarantees almost global asymptotic stability of the closed-loop tracking error dynamics. Numerical simulation results for a prototypical spacecraft pose tracking mission application are presented to illustrate the effectiveness and robustness of the proposed method.

This paper is about the construction and reconfiguration analysis of a novel variable-DOF (or kinematotropic) single-loop 7R spatial mechanism, which is composed of seven R (revolute) joints. Firstly, the novel variable-DOF single-loop 7R spatial mechanism is constructed from a general variable-DOF single-loop 7R spatial mechanism and a plane symmetric Bennett joint 6R mechanism for circular translation. The reconfiguration analysis is then carried out in the configuration space by solving a set of kinematic loop equations based on dual quaternions and the natural exponential function substitution using tools from algebraic geometry. The analysis shows that the variable-DOF single-loop 7R spatial mechanism has five motion modes, including a 2-DOF planar 5R mode, two 1-DOF spatial 6R modes, and two 1-DOF spatial 7R modes and can transit between the 2-DOF planar 5R mode and each of the other motion modes through two transition configurations. There are two transition configurations from which the mechanism can switch among its four 1-DOF motion modes.

This article proposes a new algorithm of quaternion and dual quaternion in matrix form. It applies quaternion in special cases of rotated plane, transforming the sine and cosine of the rotation angle into matrix form, then exporting flat quaternions base in two matrix form. It establishes serial 6R manipulator kinematic equations in the form of quaternion matrix. Then five variables are eliminated through linear elimination and application of lexicographic Groebner base. Thus, upper bound of the degree of the equation is determined, which is 16. In this way, a 16-degree equation with single variable is obtained without any extraneous root. This is the first time that quaternion matrix modeling has been used in 6R robot inverse kinematics analysis.

When the robot is required to execute a certain task in the world coordinate system (WCS), it is necessary to find the coordinate transformation between the robot base coordinate system (RBCS) and WCS to enable the high precision motion planning. This paper presents a simple and accurate method that allows a simultaneous computation of the coordinate transformations (i.e., rotation and translation) from WCS to RBCS. Based on the dual quaternion, the robot kinematic model and formulas for calculating the transformation are derived, which allow calculating the rotation and translation simultaneously. Taking the unit dual quaternion as a constraint, the Lagrangian multiplier method is employed to obtain the optimum transformation. Both simulation and experiment results show that higher calibration precision is obtained. The proposed approach has certain reference value and guiding sense for other calibration problems.

This paper proposes a robust dual-quaternion based H ∞ task-space kinematic controller for robot manipulators . To address the manipulator liability to modeling errors, uncertainties, exogenous disturbances , and their influence upon the kinematics of the end-effector pose, we adapt H ∞ techniques – suitable only for additive noises – to unit dual quaternions. The noise to error attenuation within the H ∞ framework has the additional advantage of casting aside requirements concerning noise distributions, which are significantly hard to characterize within the group of rigid body transformations. Using dual quaternion algebra, we provide a connection between performance effects over the end-effector trajectory and different sources of uncertainties and disturbances while satisfying attenuation requirements with minimum instantaneous control effort. The result is an easy-to-implement closed form H ∞ control design criterion. The performance of the proposed strategy is evaluated within different realistic simulated scenarios and validated through real experiments.