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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.

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.]

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.

*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.

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 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 contribution presents a modeling technique for an aerial manipulator based on a quad-rotor vehicle provided with a robotic arm. Dual quaternions, which are a little explored but powerful mathematical tool, are proposed as an alternative to the classical Euler angles approach for the kinematic and dynamic modeling. A feedback control law based on dual quaternions is used to validate the proposed model, taking into account the external effects of the robotic limb. Numerical simulations and experiments validate the proposal, opening a path for future research.

A kinematic modeling method based on the dual quaternion is proposed. The method requires less calculation time comparing to the traditional approaches, e.g. the approach using the Denavit-Hartenberg formalism. The forward kinematic problem for 6-DOF manipulator is derived using quaternions. When solving the inverse kinematic problem, decomposition to the Paden-Kahan sub-problem using the dual quaternion is applied. The new sub-problem is formulated as a “rotation about two subsequent axes for a given amount” In general, with this method, the kinematic problem of multi-DOF manipulator is decomposed into several sub-problems. A 6-degree of freedom manipulator is presented as an example. The kinematics modeling and analysis verifies the correctness and feasibility of proposed method.

this paper attempts to compare operation modes of three one translational and two rotational (1T2R) zero-torsion parallel mechanisms (PMs), 3-PSP PM, 3-RPS PM and 3-SPR PM, using dual quaternions and geometric constraints. At first, the geometric constraints of their limbs are presented based on motion characteristics. Then, constraint equations of these mechanisms are obtained according to dual quaternions in combination with geometric constraints of their limbs. The operation mode analysis of these three mechanisms is thus carried out by prime decompositions. And the kinematic interpretation of their operation modes is presented. Finally, the axodes of 3-PSP PM, 3-RPS PM and 3-SPR PM are drawn to character and compare their operation modes. The results reveal motion characteristics of these mechanisms.

International audience; In this work we present the kinematic model of an anthropomorphic hand, consisting of fingers with coupled joints similar to humans, for forward and inverse kinematics computation. Additionaly, a strategy for relative pose control between the finger tips and thumb is also developed. The formulation uses geometrical analysis, and the kinematic model can easily be integrated with the kinematics of a robotic arm in order to benefit from the added redundancy. The forward and inverse kinematic formulations use unit dual-quaternion representation of screw based kinematics which ensures singularity free representation of orientation. The model hence obtained is implemented on an anthropomorphic hand for validation.

International audience; A novel approach to the kinematic coordination problem for dual-arm robots and for the definition of bimanual tasks is proposed, where both modelling and control aspects of the problem are handled using dual quaternion representation of pose and screw-based manipulator Jacobian. The proposed formulation is advantageous in terms of computation and storage efficiency compared to other formulations of dual-arm manipulator forward kinematics and Jacobian computation. Unit dual quaternion representation is also capable of avoiding representational singularities related to orientation. The coordination problem for dual-arm manipulation is addressed using relative Jacobian, and parametrized task definition is achieved using virtual mechanisms and task-specific Jacobians. The framework hence developed has been validated using a dual-arm Baxter robot for steering wheel rotation and bimanual pouring tasks, thus validating the position controller with both relative and global task-space motions. The proposed experiments use a visual sensing strategy to detect and follow the manipulated objects during the task.

Since vision-based sensors typically cannot directly measure the relative linear and angular velocities between two spacecraft, it is useful to develop attitude- and position-tracking controllers- namely, pose-tracking controllers-that do not require such measurements. Using dual quaternions and based on an existing attitude-only tracking controller, a pose-tracking controller that does not require relative linear- or angular-velocity measurements is developed in this paper. Compared to the existing literature, this velocity-free pose-tracking controller guarantees that the pose of the chaser spacecraft will converge to the desired pose independent of the initial state, even if the reference motion is not sufficiently exciting. In addition, the convergence region does not depend on the gains chosen by the user. The velocity-free controller is verified and compared with a velocity-feedback controller through two simulations. In particular, the proposed velocity-free controller is compared qualitatively and quantitatively with a velocity-feedback controller and an extended Kalman filter using a relatively realistic satellite proximity-operation scenario.

Hyper-Dual Quaternions, HDQ, are first introduced in this paper. Utilizing the Hyper-Dual Numbers, HDN, Dual Quaternions, DQ, and hyper-dual angle, the general expression of HDQ is obtained. Rigid single- and multi-body kinematics such as in a serial robot kinematics, are then expressed in HDQ form. Taking advantage of the dual numbers’ “automatic differentiation” feature, HDQ encompasses both body pose (translational and rotational) and body velocities in a compact form with no need for further pose differentiation.

With the development of space science and technology, the requirements of spacecraft are increasing, and it is significant to research the multi-spacecraft system model. Based on the dual quatemion theory, this paper deduces the coupling kinematics equation of multi-spacecraft, and points out the coupling effect of relative position and attitude. After that, the simulation analysis of 6 kinds of multi-spacecraft is carried out based on the established multi-spacecraft position and orbit integration model, in which the six cases are 5, 15, 25, 35, 45 and 55 satellites respectively. Finally, the correctness and practicability of the model are verified by comparing between Matlab and STK simulation results.

Dual quaternions (DQ) provide a compact representation of position and attitude. They have been extensively used in fixed-base robotics due to their numerous computational advantages. This paper aims to develop the dynamic equations of motion of a robotic arm on a free-flying spacecraft using a Newton-Euler approach in a DQ modeling framework. A pose-stabilization maneuver for the end-effector is found via Differential Dynamic Programming (DDP). The results of this simulation are provided.

Accepted manuscript version. Published version available at https://doi.org/10.23919/ACC.2018.8431719. In this paper dual-quaternions are used to model a fully actuated rigid-body. A backstepping-controller that solves the trajectory tracking problem is derived and proved to provide uniform asymptotical stabilization of the error dynamics. Numerical simulations are provided where the controller is compared to existing dual-quaternion tracking controllers and it is shown to have similar performance.

This paper presents results on kinematic controllers for the stabilization of rigid body displacements using dual quaternions. The paper shows how certain results for quaternion stabilization of rotation can be extended to dual quaternions stabilization of displacements. The paper presents a relevant background material on screw motion and the screw description of lines and twists. Moreover, results are presented on the computation of the exponential functions for dual quaternions for use in numerical integration. The paper presents and analyzes different controllers based on feedback from dual quaternions, where some of the controllers are known from the literature, and some are new. In particular, it is shown which controllers give screw motion, and it is discussed how this will affect the performance of the controlled system compared to other controllers that are not based on screw motion. This analysis is supported by Lyapunov analysis. Also, certain passivity properties for dual quaternions are presented as an extension to previously published results on passivity for quaternions. © 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

This article addresses the inverse kinematic problem of the general 6R serial robot via unit dual quaternion and Dixon resultant. Firstly, we show quaternion and dual-quaternion. Secondly, by the quaternion and dual quaternion theory, unit dual quaternion kinematics equations of 6R robots are derived. Thirdly, the solution process was constructed via elimination method and Dixon resultant; According to Dixon resultant, we get a 6th-order determinant which is equal to zero, and we can get a 16th-order polynomial in single variable from the determinant. All roots are obtained by a series of solving processes. Finally, the numerical example verifies the correctness of the solution process.

This paper revises and extends the problem of robust singularity and joint limits avoidance to the cooperative task-space using unit dual quaternion framework — ensuring singularity-free coupled representation of the cooperative space. The research is paramount to cooperative control applications within flexible manufacturing systems and poorly structured environments where robustness and reactiveness play a significant role. Singularity-robust techniques are proposed to control cooperative task primitives while dynamically avoiding kinematic singularities for redundant and non-redundant cooperative task trajectories. A task self-motion, which uses the nullspace of the cooperative task, is also exploited to avoid joint limits or singularities and to define task-priority controllers. Simulated and experimental results illustrate the effectiveness of the proposed singularity-robust and task-priority primitive controllers and the usefulness of singularity-robust solutions and joint limits avoidance in the cooperative task-space.

Unmanned networked multirobot systems have the potential to accomplish complex field tasks with minimum human intervention. Motion coordination of vehicles that operate in different domains (land, sea, air) is one of the problems that need to be addressed to achieve such a goal. This article presents a representation method based on dual quaternions for leader- follower formation control architectures. This representation offers the most compact and computationally efficient screw transformation formalism and can be used to describe rigid body motions because they simultaneously describe positions and orientations with only eight parameters. A controller in dual quaternion formation space is proposed and analyzed. Computer simulation results and experimental tests applied to the task of escorting an UGV with UAVs are shown to verify the functionality of the proposed system.

One challenge in dual arm robotic field is the description of cooperative tasks of the two arms. Traditional methods with translational vector and rotational matrix are complex, especially for dual arm cooperative tasks. In this paper, using the ability of dual quaternion based full dual position control strategies, it is possible to present the cooperative kinematic control problem in a compact and lack of singularity manner. Simulations is carried out on ABB YuMi dual arm robot to show the effectiveness of the approach.

In this paper, a comparative study of two methods are presented to determine the forward kinematics and Jacobian matrix for any serial manipulator, based on product of exponentials formula and unit dual quaternion algebra. This comparison is qualitative and particularly quantitative taken into account computational costs and memory requirements that are very critical in the manipulator control and real-time algorithms. For the first time the computational cost in determining the Jacobian matrix by using these two algorithms is compared. The two algorithms are applied to the 7-DOF Kuka Lightweight Robot using MATLAB and compared.

A hybrid bimodal controller for rigid body pose stabilization within the group of unit norm dual-quaternions is proposed in this paper. Using two binary logic state variables, this hysteresis-based controller represents a middle term solution between the memoryless discontinuous controller and the fixed-width hysteretic one. The proposed strategy is novel within the dual-quaternions framework and addresses three common difficulties that appears in the literature of pose and attitude stabilization: global stability, robustness against chattering and against unwinding. The efficacy and performance of the proposed controller are illustrated with numerical examples.

This paper proposes a fully decentralized and recursive approach to online identification of unknown kinematic and dynamic parameters for cooperative manipulation of a rigid body based on commonly used local measurements. To the best of our knowledge, this is the first paper addressing the identification problem for 3D rigid body cooperative manipulation, though the approach proposed here applies to the 2D case as well. In this work, we derive truly linear observation models for kinematic and dynamic unknowns whose state-dependent uncertainties can be exactly evaluated. Dynamic consensus in different coordinates and a filter for dual quaternion are developed with which the identification problem can be solved in a distributed way. It can be seen that in our approach all unknowns to be identified are time-invariant constants. Finally, we provide numerical simulation results to illustrate the efficacy of our approach indicating that it can be used for online identification and adaptive control of rigid body cooperative manipulation., Comment: 8 pages

Formulation of proper and efficient algorithms for robot kinematics is essential for the analysis and design of serial manipulators. Kinematic modeling of manipulators is most often performed in Cartesian space. However, due to disadvantages of most widely used mathematical constructs for description of orientation such as Euler angles and rotational matrices, a need for unambiguous, compact, singularity free, computationally efficient method for representing rotational information is imposed. As a solution, unit quaternions are proposed and kinematic modeling in dual quaternion space arose. In this paper, an overview of spatial descriptions and transformations that can be applied together within these spaces in order to solve kinematic problems is presented. Special emphasis is on a different mathematical formalisms used to represent attitude of a rigid body such as rotation matrix, Euler angles, axis-angle representation, unit quaternions, and their mutual relation. Benefits of kinematic modeling in quaternion space are presented. New direct kinematics algorithm in dual quaternion space pertaining to a particular manipulator is given. These constructs and algorithms are demonstrated on the human centrifuge as 3 DoF robot manipulator.

The present work focuses on the development of a methodology that can be used to solve the inverse kinematics of 6R robot manipulators with Euclidean wrist. Dual quaternions are used as screw motion operators to perform line transformations to the joint axes. The transformations are performed after each joint angle is found. This process accommodates the robot pose on each step from the initial pose to the final pose when all the joint angles have been found. In each step, the appropriate sub problems of Paden-Kahan are used to find the joint angles. The main contribution of the presented methodology is that it is a more general solution than previous approaches. Additionally, an extension of the second sub problem of Paden-Kahan is derived for parallel axes. The proposed algorithm can be used to solve the inverse kinematics problem of most of the 6R robot manipulator configurations, with or without shoulder offset, and with or without intersecting axes of the first and second joint.

International audience; This work presents the modeling and control of an aerial manipulator based on a quadrotor with a robotic arm using the dual quaternion approach. First the kinematic model of the complete system is obtained using dual quaternions to represent multiple rotations and translations, then the dynamic model is developed via the Newton-Euler formalism. A position control law is used to stabilize the vehicle around a desired position and orientation using a smooth trajectory. The closed-loop system is then numerically simulated to corroborate stability. In addition, experimental flight tests were performed for validation.

Foldable structures, a new kind of space structures developed in recent decades, can be deployed gradually to a working configuration and also can be folded for transportation, thus have potentially broad application prospects in the fields of human life, military, aerospace, building structures, and so on. Combined with the technology of origami folding, foldable structures derive more diversified models, and the foldable structures in cylindrical shape are mainly studied in this paper. Some researchers use the theory of quaternion representing spatial fixed-point rotation and construct the rotating vector model to obtain the quaternion rotation sequence method (QRS method) analyzing origami, but the method is very limited and not suitable for the cylindrical foldable structures. In order to solve the problem, a new method is developed, which combines the QRS method and the dual quaternion method. After analyzing the folding angle via the QRS method for multivertex crease system and calculating the coordinates of all vertices via the dual quaternion, the rigid foldability can be checked. Finally, two examples are carried out to confirm validity and versatility of the method.

From the point of view of classical mechanics, deriving the equations of motion for systems of coupled rigid bodies is regarded as a straightforward procedure: once a suitable set of generalized coordinates and reference frames have been chosen, what remains is to either apply Lagrange’s equations or Newton and Euler’s equations to obtain the differential equations of motion. As the complexity of multibody system increases, the need for more elegant formulation of the equation of motion becomes an issue of paramount importance. Our primary focus is on the kinematic analysis of rigid bodies and serial manipulators (robotic systems) using simultaneously, both homogeneous transformations (4x4) matrices and Dual Quaternions, for the sake of results comparisons (cost,complexity,storage capacity etc.) . This paper has been done mainly for educational and peadagogical purposes, hoping that the scientific community will finally adopt and use Dual Quaternions at least when dealing with multibody systems and specially robotics. DOI: http://dx.doi.org/10.11591/ijra.v1i1.275

Solid object rotations or rotations of unit real vectors are often used in robotic moves. Yet, due to the complexity of calculations in kinematics analysis, quaternions began to be used in robotic moves. Quaternions help us to calculate rotations in reel space, and rotation and translation in dual space. Furthermore, we can easily define a screw or an axis through dual vectors. In recent years, there has been efficient studies in mechanics analysis by using dual quaternions and screw moves. In this study, we transferred the well known sequential and modular methods for the rotations being formed in real spherical wrist moves to dual space. Hence, while we rotate the unit vectors only in real space, now we can rotate screws .Hence, while we rotate the unit vectors only in real space, now we can rotate screws or the axis in the dual spherical mechanisms in dual kinematics. Through this study, by being switched to dual spherical mechanism the first time, we will see that we can carry out what we do with unit vectors in real space as well as in dual space. Furthermore, while the orbits drawn by the joint moves are curves in reel spherical wrist moves, the orbits drawn by the joint moves are ruled surfaces in dual spherical wrist moves. We think that this situation would establish a base for different mechanics studies.

In this paper we shall discuss connections between some old problems of elementary geometry from the beginning of the 20-th century and geometry of parallel manipulators. Beside of this we would like to demonstrate how dual quaternions give a natural tool for solution of problems connected with spatial motions. In order to give this demonstration a short excursion to the theory of kinematic spaces is necessary. As an application we show, how one problem of such kind can be solved by using computer algebra. At the end we explain how methods of approximate parameterization yield a natural tool for the visualization of trajectories of self-motions of parallel manipulators.

The most common approach used in modeling humanoid robots is to restrict the motion in two different planes, namely the frontal and sagittal plane, thereby using trigonometric expressions and Euler-angles for its description. In this paper some problems of trajectory interpolation and singularity of motion representation are discussed and the dual quaternion representation is used as an alternative to avoid these problems and to obtain the models ensuring a general spatial analysis of the motion trajectory. The aim of this work is to present a different method to model kinematics of humanoid robots avoiding the restriction to the frontal and sagittal planes. A spatial trajectory is therefore assumed and the joint angles are calculated to move the robot in human like motion. To achieve the goal an algebraic geometry approach based on dual quaternions is used to obtain the inverse kinematics expressions that are then polynomial functions. Simulations are performed to evaluate the model numerically.

This work addresses the task-space design problem of a linear-quadratic optimal tracking controller for robotic manipulators using the unit dual quaternion formalism. The efficiency, compactness, and lack of singularity of the representation render the unit dual quaternion a suitable framework for simultaneously describing the attitude and the position of the end-effector. Motivated by the advantages of this kinematic description, we propose a new task-space linear-quadratic optimal tracking controller in order to find an optimal trajectory for the end-effector, providing a tool to balance more conveniently the end-effector error and its task-space velocity. This is possible because the kinematic control problem using the dual quaternion transformation invariant error can be reduced to an affine time-varying system. The proposed optimal tracking controller allows the compensation of trajectory induced disturbances, as well as other modeled additive disturbances and known bias. Simulation results with different design parameters provide a performance overview, in comparison with standard kinematic controllers with and without a feed-forward term, for tracking a desired reference.

This paper presents general modal dual Quaternion (DQ) kinematics for multisection continuum arms. DQ's produce higher accuracy than homogeneous transformation matrices (HTM) when transformed to modal shape functions (MSF) of similar order and are numerically stable. Thus, the model is compact, more accurate and computationally efficient than the modal kinematics proposed by the author based on HTM's. Also, DQ kinematics does not suffer from singularity related limitations of Euler angle based inverse orientation kinematics. Recursive schemes for deriving DQ's and DQ Jacobians are also presented and can be extended arbitrarily. Both modal HTM and modal DQ kinematics are then applied to solve illustrative spatial inverse position and orientation tracking problems. Based on the results, this paper quantitatively compares both methods and highlights the advantages of modal DQ kinematics. The proposed DQ kinematics are easily extensible to variable length multisection continuum arm with general actuator configurations.

This paper presents a study on the higher-order motion of point-lines embedded on rigid bodies. The mathematic treatment of the paper is based on dual quaternion algebra and differential geometry of line trajectories, which facilitate a concise and unified description of the material in this paper. Due to the unified treatment, the results are directly applicable to line motion as well. The transformation of a point-line between positions is expressed as a unit dual quaternion referred to as the point-line displacement operator depicting a pure translation along the point-line followed by a screw displacement about their common normal. The derivatives of the point-line displacement operator characterize the point-line motion to various orders with a set of characteristic numbers. A set of associated rigid body motions is obtained by applying an instantaneous rotation about the point-line. It shows that the ISA trihedrons of the associated rigid motions can be simply depicted with a set of ∞2 cylindroids. It also presents for a rigid body motion, the locus of lines and point-lines with common rotation or translation characteristics about the line axes. Lines embedded in a rigid body with uniform screw motion are presented. For a general rigid body motion, one may find lines generating up to the third order uniform screw motion about these lines.

The design of strapdown inertial navigation system (INS) algorithms based on dual quaternions is addressed. Dual quaternion is a most concise and efficient mathematical tool to represent rotation and translation simultaneously, i.e., the general displacement of a rigid body. The principle of strapdown inertial navigation is represented using the tool of dual quaternion. It is shown that the principle can be expressed by three continuous kinematic equations in dual quaternion. These equations take the same form as the attitude quaternion rate equation. Subsequently, one new numerical integration algorithm is structured to solve the three kinematic equations, utilizing the traditional two-speed approach originally developed in attitude integration. The duality between the coning and sculling corrections, raised in the recent literature, can be essentially explained by splitting the new algorithm into the corresponding rotational and translational parts. The superiority of the new algorithm over conventional ones in accuracy is analytically derived. A variety of simulations are carried out to support the analytic results. The numerical results agree well with the analyses. The new algorithm turns out to be a better choice than any conventional algorithm for high-precision navigation systems and high-maneuver applications. Several guidelines in choosing a suitable navigation algorithm are also provided.

Three methods for the formulation of the kinematic equations of robots with rigid links are presented in this paper. The first and most common method in the robotics community is based on 4/spl times/4 homogeneous matrix transformation, the second one is based on Lie algebra, and the third one on screw theory expressed via dual quaternions algebra. These three methods are compared in this paper for their use in the kinematic analysis of robot arms. The basic theory and the transformation operators, upon which every method is based, are referenced. Three analytic algorithms are presented for the solution of the direct kinematic problem corresponding to each method, and the geometric significance of the transformation operators and parameters is explained. Finally, a comparative study on the computation and storage requirements for the three methods is worked out.

This paper addresses the H∞ robust control problem for robot manipulators using unit dual quaternion representation, which allows an utter description of the end-effector transformation without decoupling rotational and translational dynamics. We propose three different H∞ control criteria that ensure asymptotic convergence, whereas reducing the influence of disturbances upon the system stability. Also, with a new metric of dual quaternion error in SE(3) we prove independence from robot coordinate changes. Simulation results highlight the importance and effectiveness of the proposed approach in terms of performance, robustness, and energy efficiency.

The manipulation in singular regions promotes an instantaneous reduction in mechanism mobility, which can result in some disturbances in the trajectory tracking. In proximity to the singularities, small velocities of the end-effectors generate high speeds in the joints due to the gradual reduction of the mobility. The phenomenon of kinematic singularity promotes a instantaneous instability in torque profile of the redundant robotic systems by the transformation of secondary joints to primary joints. The disturbances of the underwater environment intensify the effects of the singularities because the hydrodynamic strongly oppose to torque variations. This work presents a methodology for using dual quaternions in the posture feedback of a Underwater Vehicle-Manipulator System (UVMS) using the Davies method, which avoids kinematic singularities and ensures the optimal torque profiles.

In this paper, we compare three inverse kinematic formulation methods for the serial industrial robot manipulators. All formulation methods are based on screw theory. Screw theory is an effective way to establish a global description of rigid body and avoids singularities due to the use of the local coordinates. In these three formulation methods, the first one is based on quaternion algebra, the second one is based on dual‐quaternions, and the last one that is called exponential mapping method is based on matrix algebra. Compared with the matrix algebra, quaternion algebra based solutions are more computationally efficient and they need less storage area. The method which is based on dual‐quaternion gives the most compact and computationally efficient solution. Paden‐Kahan sub‐problems are used to derive inverse kinematic solutions. 6‐DOF industrial robot manipulator's forward and inverse kinematic equations are derived using these formulation methods. Simulation and experimental results are given.