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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. [ABSTRACT FROM AUTHOR]

In order to solve the problem of noise interference in camera calibration and robot forward kinematics solution, this study proposes a dual quaternion hand-eye calibration algorithm based on twice opposition-learning and random differential variation (ODHPO). The hand-eye calibration equation was innovatively rewritten in the form of dual quaternion, and the F-norm minimization model of the rotation and translation error function was constructed for optimization. At the same time, the penalty function method is introduced to effectively transform the constrained optimization problem in hand-eye calibration into an unconstrained problem, which is further solved by the ODHPO algorithm for global optimization. Compared with the singular value decomposition algorithm based on the traditional dual quaternion method, the ODHPO algorithm performs better in global optimization ability and convergence stability. Through numerical simulation and real robot hand-eye calibration experiments, it is proved that the proposed algorithm is superior to the traditional dual quaternion (CDQ) algorithm, the classical algorithms Tsai method and Navy method in terms of solution accuracy, sensitivity to the number of pose transformations and stability, demonstrating its potential for application and practical significance in robot vision system. ● An improved hunter-prey optimization algorithm with twice opposition-learning and random difference variation is proposed. ● A dual quaternion hand-eye calibration algorithm based on improved hunter-prey optimization is proposed. ● The effectiveness of the improved algorithm is verified by 12 benchmark test functions ● The proposed calibration algorithm is superior to other traditional calibration algorithms [ABSTRACT FROM AUTHOR]

The dual quaternion plays an advantageous role in the field of robotic kinematics and dynamics due to its compact expression. It is well known that quaternion can be easily implemented as algebraic operations on vectors through matrix operators. Similarly, the dual matrix operators can convert dual quaternion operations to matrix operations. However, it has not received adequate attention, and thus there are very few applications at present. With the dual matrix operators of dual quaternion, this paper re-verifies the equivalency between the conjugate formula of unit dual quaternion and dual Euler–Rodrigues formula. To combine this equivalence with the homomorphic mapping of Lie groups, a theoretical correlation of the current hand-eye calibration methods is established. Motivated by the attempt to establish a more complete scheme for hand-eye calibration, a new simultaneous method based on the conjugate formula of dual quaternion is proposed. The method verifies that the rotation problem can be extended to the rigid-body transformation problem via the dual algebra. Finally, the simulation and experimental validations for the hand–eye calibration method are given. • The conjugate formula of unit dual quaternion is re-verified via the dual operations. • A new simultaneous method is defined for hand-eye calibration. • The theoretical correlation with the existing methods is established. [ABSTRACT FROM AUTHOR]

An integrated attitude-orbit controller based on extended state observer (ESO) is designed for spacecraft to settle down the unavailability of angular velocity and centroid velocity during on?orbit flight.First,the spacecraft’s integrated attitude?orbit kinematics and dynamics model is presented based on dual quaternion.Second,an extended state observer is designed for the case of velocity screw information missing,which takes the model parameter error and external disturbance into account,and the finite time convergence of the observation error is analyzed by using Lyapunov stability theorem.Then a fast terminal sliding mode controller for integrated attitude?orbit control is designed based on the above extended state observer,and its finite time convergence is analyzed.Finally,a numerical simulation is given to verify the effectiveness of the control system. [ABSTRACT FROM AUTHOR]

Whenrobots physically interact with the environment, compliant behaviors should be imposed to prevent damages to all entities involved in the interaction. Moreover, during physical interactions, appropriate pose controllers are usually based on the robot dynamics, in which the ill-conditioning of the joint-space inertia matrix may lead to poor performance or even instability. When the control is not precise, large interaction forces may appear due to disturbed end-effector poses, resulting in unsafe interactions. To overcome these problems, we propose a task-space admittance controller in which the inertia matrix conditioning is adapted online. To this end, the control architecture consists of an admittance controller in the outer loop, which changes the reference trajectory to the robot end-effector to achieve a desired compliant behavior; and an adaptive inertia matrix conditioning controller in the inner loop to track this trajectory and improve the closed-loop performance. We evaluated the proposed architecture on a KUKA LWR4+ robot and compared it, via rigorous statistical analyses, to an architecture in which the proposed inner motion controller was replaced by two widely used ones. The admittance controller with adaptive inertia conditioning presents better performance than with a controller based on the inverse dynamics with feedback linearization, and similar results when compared to the PID controller with gravity compensation in the inner loop. [ABSTRACT FROM AUTHOR]

In this study, we obtain triplets from quaternions. First, we obtain triplets from real quaternions. Then, as an application of this, we obtain dual triplets from the dual quaternions. Quaternions, in many areas, it allows ease in calculations and geometric representation. Quaternions are four dimensions. The triplets are in three dimensions. When we express quaternions with triplets, our study is conducted even easier. Quaternions are very important in the display of rotational movements. Dual quaternions are important in the expression of screw movements. Reducing movements from four dimensions to three dimensions makes our study easier. This simplicity is achieved by obtaining triplets from quaternions. [ABSTRACT FROM AUTHOR]

A multi-robot collaboration system can complete more complex tasks than a single robot system. Ensuring the calibration accuracy between robots in the system is a prerequisite for the effective inter-robot cooperation. This paper presents a dual-robot system for orthopedic surgeries, where the relationships between hand-eye, flange-tool, and robot-robot need to be calibrated. This calibration problem can be summarized to the solution of the matrix equation of AXB=YCZ. A combined solution is proposed to solve the unknown parameters in the equation of AXB=YCZ, which consists of the dual quaternion closed-form method and the iterative method based on Levenberg–Marquardt (LM) algorithm. The closed-form method is used to quickly obtain the initial value for the iterative method so as to increase the convergence speed and calibration accuracy of the iterative method. Simulation and experimental analyses are carried out to verify the accuracy and effectiveness of the proposed method.

The 6D Pose estimation is a crux in many applications, such as visual perception, autonomous navigation, and spacecraft motion. For robotic grasping, the cluttered and self-occlusion scenarios bring new challenges to the this field. Currently, society uses CNNs to solve this problem. The CNN models will suffer high uncertainty caused by the environmental factors and the object itself. These models usually maintain a Gaussian distribution, which is not suitable for the underlying manifold structure of the pose. Many works decouple rotation from the translation and quantify rotational uncertainty. Only a few works pay attention to the uncertainty of the 6D pose. This work proposes a distribution that can capture the uncertainty of the 6D pose parameterized by the dual quaternions, meanwhile, the proposed distribution takes the periodic nature of the underlying structure into account. The presented results include the normalization constant computation and parameter estimation techniques of the distribution. This work shows the benefits of the proposed distribution, which provides a more realistic explanation for the uncertainty in the 6D pose and eliminates the drawback inherited from the planar rigid motion.

In this paper, we develop a new method of analytical design and control of the spatial motion of a rigid body (in particular, a spacecraft considered as a rigid body) in a nonlinear dynamic formulation using dual quaternions (Clifford biquaternions). The control provides the asymptotic stability in general of any selected programmed motion in the inertial coordinate system and the desired dynamics of the controlled motion of the body. To build control laws, new biquaternion differential equations of perturbed spatial motion of a rigid body are proposed, in which unnormalized biquaternions of finite displacements, biquaternions of angular and linear velocities of the body and accelerations with nonzero dual scalar parts are used. The concept of solving the inverse problems of dynamics, the feedback control principle, and the approach based on the reduction of the equations of perturbed body motion to linear stationary differential forms of the selected structure invariant with respect to any selected programmed motion due to the corresponding choice of dual nonlinear feedbacks in the proposed biquaternion control laws is presented. Analytical solutions of biquaternion differential equations are designed to describe the dynamics of controlling the spatial body motion using the proposed biquaternion control laws. The properties and patterns of such control are analyzed. [ABSTRACT FROM AUTHOR]

It is well known that the automorphism group Aut(H) of the algebra of real quaternions H consists entirely of inner automorphisms iq : p → q ·p·q-1 for invertible q ∈ H and is isomorphic to the group of rotations SO(3). Hence, H has only inner derivations D = ad(x), x ∈ H. See [4] for derivations of various types of quaternions over the reals. Unlike real quaternions, the algebra Hd of dual quaternions has no nontrivial inner derivation. Inspired from almost inner derivations for Lie algebras, which were first introduced in [3] in their study of spectral geometry, we introduce coset invariant derivations for dual quaternion algebra being a derivation that simply keeps every dual quaternion in its coset space. We begin with finding conditions for a linear map on Hd become a derivation and show that the dual quaternion algebra Hd consists of only central derivations. We also show how a coset invariant central derivation of Hd is closely related with its spectrum. [ABSTRACT FROM AUTHOR]

Abstract Rigid transformation including rotation and translation can be elegantly represented by a unit dual quaternion. Thus, a non-differential model of the Helmert transformation (3D seven-parameter similarity transformation) is established based on unit dual quaternion. This paper presents a rigid iterative algorithm of the Helmert transformation using dual quaternion. One small rotation angle Helmert transformation (actual case) and one big rotation angle Helmert transformation (simulative case) are studied. The investigation indicates the presented dual quaternion algorithm (QDA) has an excellent or fast convergence property. If an accurate initial value of scale is provided, e.g., by the solutions no. 2 and 3 of Závoti and Kalmár (Acta Geod Geophys 51:245–256, 2016) in the case that the weights are identical, QDA needs one iteration to obtain the correct result of transformation parameters; in other words, it can be regarded as an analytical algorithm. For other situations, QDA requires two iterations to recover the transformation parameters no matter how big the rotation angles are and how biased the initial value of scale is. Additionally, QDA is capable to deal with point-wise weight transformation which is more rational than those algorithms which simply take identical weights into account or do not consider the weight difference among control points. From the perspective of transformation accuracy, QDA is comparable to the classic Procrustes algorithm (Grafarend and Awange in J Geod 77:66–76, 2003) and orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015. https://doi.org/10.1186/s40623-015-0263-6).

To simplify the kinematics expression and to increase the motion response speed for manipulators,a kinematics modeling method is proposed based on dual quaternion. First,to reduce the computational amount,the general equation of the forward kinematics of the manipulator is presented using dual quaternion.Then,to solve the inverse kinematics problem of the assembled joint variables of manipulator,the sub-problems are classified into two groups of uniaxial and biaxialox,this classification method includes the existing sub-problem classification,which is beneficial to the joint variable grouping solution. Finally,taking a 6 R series manipulator as an example,the forward kinematics and inverse kinematics modeling and simulation analysis based on the dual quaternion are carried out. The results show that this method is effective,comparing with the spinor method,its calculation time of forward kinematics is reduced by about 30%,the geometric meaning of inverse kinematics is more clear. This method can be applied to other sub-problems solution of the manipulator configuration.

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 develops a relative state estimation method for two spacecraft based on monocular vision measurement, where the leader spacecraft of three-dimensional shape is observed by a calibrated camera fixed on the follower spacecraft. The two-dimensional image of the leader spacecraft with multiple geometric features is obtained by the camera. Multiple geometric features including points, lines and circles are described under dual number algebra, and the observation models are proposed based on the geometric projection relationships of these features. By the proposed method, the geometric features appeared on the leader spacecraft can be employed to estimate the relative state between the two spacecraft. Another contribution of this paper is to develop a six-degree-of-freedom relative motion dynamics using dual number. The dynamics model describes the relative motion between arbitrary points on the spacecraft, and both the kinematically and dynamically coupling effects are considered. Based on the derived process model and observation model, an extended Kalman filter is presented to estimate the relative state between the two spacecraft. In addition, an unscented Kalman filter is also developed to avoid complicated calculation of derivations. Finally, numerical simulations are performed to evaluate the proposed EKF and UKF approaches based on multiple geometric feature measurement. Simulation results verify the effectiveness of the proposed method, and show that by using EKF or UKF technique, the relative translation and rotation estimation errors converge rapidly with the desired high estimation accuracy. Comparison simulations are made with the situations using less feature measurement, which demonstrate that employing multiple feature measurement provides superior estimation performance than methods with less feature measurement. [ABSTRACT FROM AUTHOR]

This paper proposes an innovated approach to model an electromagnetic force which is an ideal control force without plume and light pollution for optical imaging system. The main scope is modeling the 3-axis coupled electromagnetic force using dual quaternion for its advantages in describing spiral motion with translation and rotation motion simultaneously. The derivation procedure relies on a far field model of the electromagnetic force, and also, the force is re-formulized. Final equations show the dual quaternion dynamics can express the relative motion caused by the electromagnetic force acting on the position and attitude integrally. A new mission concept of diffraction imaging system in GEO is discussed as application scenario of the new dynamics model. Two cases with different mass were simulated to verify the new model, and the results display the electromagnetic force is significant for a small mass spacecraft in GEO. • The electromagnetic force is expressed by dual quaternion. • A novel dynamics model is derived to describe GEO electromagnetic collocated satellite. • The motion of diffraction imaging system is modelled using dual quaternion. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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• An algebraic synthesis method is proposed to form novel multi-loop linkages for rotation scaling. • Multi-loop planar, spherical, and spatial linkages can be constructed using the proposed method. • The generated linkages can exactly transmit angular motion with prescribed transmission ratios. • Exact angular motion transmission between parallel, intersecting, or skew axes is achieved. • Transmission characteristics of the multi-loop linkages are proved by kinematics analysis. This paper deals with the algebraic synthesis and input-output analysis of one degree-of-freedom (1-DOF) multi-loop planar, spherical, and spatial linkages for increasing or decreasing angular motion with a desired constant transmission ratio between two adjacent parallel, intersecting, and skew axes respectively. Based on the rotation scaling model and motion polynomials over dual quaternions, an algebraic synthesis method including three procedures is proposed to exactly construct novel multi-loop linkages owning arbitrarily prescribed constant transmission ratios and input/output axes. To illustrate this method, several 1-DOF multi-loop planar, spherical, and spatial linkages for rotation scaling are synthesized by designating various input and output axes as well as transmission ratios. Taking some multi-loop planar, spherical, and spatial linkages as examples, input-output analysis is carried out to verify their transmission characteristics. The results demonstrate that the generated 1-DOF multi-loop linkages can indeed transmit motion with prescribed constant transmission ratios and input/output axes. This work provides a framework for further investigation on mechanisms performing specified tasks for motion transmission. [ABSTRACT FROM AUTHOR]

This paper presents a new approach that uses rational Bézier motions to generate 5-axis tool path for sculptured surface machining (finish cut) with a flat-end cutter. By using dual quaternion to represent a spatial displacement, the representation of kinematic motions for the cutter bottom circle of the flat-end cutter is formulated. Based on that, a new approach for tool path generation using rational Bézier cutter motions is described, in which key issues such as interference avoidance and surface accuracy requirement are addressed. First, a set of cutter contact points on an iso-parametric curve of the designed surface is obtained based on a given fitting tolerance. The associated cutter locations (CLs) are then obtained by finding the suitable cutter orientations that avoid any gouging. The conversion from the CLs to dual quaternion representation is carried out and the rational Bézier dual quaternion curve for cutter motion is generated. The entire tool path is therefore established based on the cutter undergoing the rational Bézier motion. Next, the whole tool path is checked to find (1) if there is any interference between the cutter and the designed surface, and (2) whether the deviation between the surface generated by the cutter motion and the designed surface is larger than the given tolerance. The problematic CLs, which cause either gouging or accuracy problem, are then modified. The process of tool path checking → CLs modification → tool path regeneration continues until the whole tool path is interference-free and satisfies the accuracy requirement. Furthermore, a more accurate representation of the effective cutting shape is proposed, which is used to evaluate the scallop height between adjacent tool paths. A method for constructing the adjacent tool path has been developed by considering the allowable scallop height. Finally, computer implementation and an illustrative example are presented to demonstrate the efficacy of the approach. [ABSTRACT FROM AUTHOR]

Due to the high complexity of geo-spatial entities and the limited field of view of LiDAR equipment, pairwise registration is a necessary step for integrating point clouds from neighbouring LiDAR stations. Considering that accurate extraction of point features is often difficult without the use of man-made reflectors, and the initial approximate values for the unknown transformation parameters must be estimated in advance to ensure the correct operation of those iterative methods, a closed-form solution to linear feature-based registration of point clouds is proposed in this study. Plücker coordinates are used to represent the linear features in three-dimensional space, whereas dual quaternions are employed to represent the spatial transformation. Based on the theory of least squares, an error norm (objective function) is first constructed by assuming that each pair of corresponding linear features is equivalent after registration. Then, by applying the extreme value analysis to the objective function, detailed derivations of the closed-form solution to the proposed linear feature-based registration method are given step by step. Finally, experimental tests are conducted on a real dataset. The derived experimental result demonstrates the feasibility of the proposed solution: By using eigenvalue decomposition to replace the linearization of the objective function, the proposed solution does not require any initial estimates of the unknown transformation parameters, which assures the stability of the registration method.

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

This study investigates the tracking control problem of helical microrobots (HMRs) in complicated blood environments. The integrated relative motion model of HMRs is established by resorting to the dual quaternion method, which can describe the coupling effect between the rotational and translational motions. Subsequently, an original apparent weight compensator (AWC) is designed to alleviate the adverse effects of the HMR sinking and drifting owing to its own weight and buoyancy. An adaptive sliding mode control based on the developed AWC (AWC-ASMC) is constructed to guarantee the fast convergence of the relative motion tracking errors in the presence of model uncertainties and unknown perturbations. The chattering phenomenon of the classical SMC is significantly reduced using the developed control strategy. Furthermore, the stability of the closed-loop system under the constructed control framework is demonstrated by the Lyapunov theory. Finally, numerical simulations are performed to demonstrate the validity and superiority of the developed control scheme., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2023 ISA. Published by Elsevier Ltd. All rights reserved.)

Aiming at hand-eye calibration problem in robotic hand-eye vision system, the hand-eye conversion relationship between end-effector and camera of manipulator was studied, the method of solving homogeneous matrix in hand-eye calibration was summarized, and a vision model based on motion posture was set up. The improved dual quaternion approach was used for solving separately the rotation parts and the translation of the hand-eye homogeneous transformation matrix. In order to obtain the optimum solution of the translation vector, the infinite norm minimization was introduced during the solving process of translation vector, the local optimal solution of translation vector was obtained by iteration. Simulations based on the considered optimal translation vector algorithm were performed in Matlab, and the classic; solving using dual quaternion and linear decoupling estimation algorithm were also implemented for comparison. The results indicate that the proposed method is better than traditional methods in the precision of calibration, robustness of noise and sensitivity and stability of motion posture numbers. It can meet the accuracy requirement of hand-eye calibration. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the factorizations of monic, bounded motion polynomials. We prove existence of factorizations, possibly after multiplication with a real polynomial Q , and provide an algorithm for computing Q and corresponding factorizations. The algorithm gives a much lower bound on the degree of the polynomial factor than the mere existence theorem. [ABSTRACT FROM AUTHOR]

Computationally efficient inverse dynamics is crucial to the real-time application of parallel robots. This paper provides a computationally more efficient solution to the inverse dynamics of a class of six-DOF parallel robots based on the dual quaternion approach under the principle of virtual power. A unit dual quaternion is selected as the generalized coordinates of the system. The equations of motion are then constructed by the principle of virtual power. The dual quaternion constraints are eliminated by the null space formulation to obtain the inverse dynamic solution. It is revealed that in the new solution, the Jacobian matrices and the orthogonal complement matrix are all linear with respect to the generalized coordinates. Additionally, the positions, velocities and accelerations of all bodies are quadratic with respect to the generalized coordinates, velocities and accelerations. Such succinct expressions render the new solution computationally more efficient. The execution time of the dual quaternion approach and the traditional one are compared under the same condition by two different six-DOF parallel robots: 6-UPS and 6-PUS. The results show that the new solution can save the computational cost by 43.45% and 38.45% respectively for the two robots, illustrating the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Abstract We discuss existence of factorizations with linear factors for (left) polynomials over certain associative real involutive algebras, most notably over Clifford algebras. Because of their relevance to kinematics and mechanism science, we put particular emphasis on factorization results for quaternion, dual quaternion and split quaternion polynomials. A general algorithm ensures existence of a factorization for generic polynomials over division rings but we also consider factorizations for non-division rings. We explain the current state of the art, present some new results and provide examples and counter examples. [ABSTRACT FROM AUTHOR]

The rigid motion involving both rotation and translation in the 3D space can be simultaneously described by a unit dual quaternion. Considering this excellent property, the paper constructs the Helmert transformation (seven-parameter similarity transformation) model based on a unit dual quaternion and then presents a rigid iterative algorithm of Helmert transformation using a unit dual quaternion. Because of the singularity of the coefficient matrix of the normal equation, the nine parameter (including one scale factor and eight parameters of a dual quaternion) Helmert transformation model is reduced into five parameter (including one scale factor and four parameters of a unit quaternion which can represent the rotation matrix) Helmert transformation one. Besides, a good start estimate of parameter is required for the iterative algorithm, hence another algorithm employed to compute the initial value of parameter is put forward. The numerical experiments involving a case of small rotation angles i.e. geodetic coordinate transformation and a case of big rotation angles i.e. the registration of LIDAR points are studied. The results show the presented algorithms in this paper are correct and valid for the two cases, disregarding the rotation angles are big or small. And the accuracy of computed parameter is comparable to the classic Procrustes algorithm from Grafarend and Awange (J Geod 77:66-76, 2003), the orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015), and the algorithm from Wang et al. (J Photogramm Remote Sens 94:63-69, 2014). [ABSTRACT FROM AUTHOR]

Abstract To solve the shortcomings of the traditional monocular vision measurement algorithm, which needs to be solved iteratively, this paper proposes a method based on binocular vision for measuring the relative pose of moving rigid bodies. In this paper, the coordinates of the feature points of the follower rigid body coordinate system are obtained by using the method of least squares. At the same time, combined with the dual quaternion algorithm, the leader and follower rigid bodies' coordinate systems are uniformly described, and the model takes the minimum sum of squared errors as the objective function and solves the pose parameters by the dual quaternion method. The simulation results show that the dual quaternion algorithm not only overcomes the shortcomings of the traditional quaternion and translation vector split description coordinate system transformation but also has strong stability; the relative position error is less than 0.05 m, and the relative attitude error is less than 0.2°, which can meet the measurement accuracy requirements of the relative pose of the moving rigid body. [ABSTRACT FROM AUTHOR]

Highlights • Space manipulators can act as actuators for spacecraft due to the dynamic coupling. • Pose control forces provides equality constraints for trajectory planning problem. • Convex optimization method found the globally optimal solution efficiently. • Balance manipulator compensates for coupling and environmental disturbance. Abstract This paper presents a trajectory planning algorithm for a space robot with dual-manipulators. Here one manipulator of the space robot captures a target, and another manipulator is free. In this case, this study uses one manipulator as the mission manipulator to capture the target, and another as the balance manipulator aiming at the compensation of the pose disturbance. For this method, a novel trajectory planning algorithm applied to the balance manipulator is presented. The trajectory planning problem is transformed into series of problems of the optimal state solution, and then the iterative algorithms for the trajectory planning are designed. In the iterative algorithms, the bias force on the spacecraft base caused by the balance manipulator is used as the compensation force. Then, to calculate the expected compensation force and torque, a pose control law for the spacecraft base is introduced. The expected compensation force and torque provide equality constraints for optimization problems, which implies that the trajectory planning algorithm compensates for not only the disturbance generated by the manipulator's motion, but also environmental disturbances. This is because the expected compensation force and torque depend on the pose change of the spacecraft base rather than the type of the disturbance. Numerical simulation was carried out to analyze the proposed trajectory planning method. It was observed that the method greatly reduces the disturbance of Manipulator A on the spacecraft base. These results validated the effectiveness of the proposed method for the trajectory planning to make the spacecraft base disturbance up to minimum. [ABSTRACT FROM AUTHOR]

In 2016, Yüuce and Torunbalcı Aydın [18] defined dual Fibonacci quaternions. In this paper, we defined the dual third-order Jacobsthal quaternions and dual third-order Jacobsthal-Lucas quaternions. Also, we investigated the relations between the dual third-order Jacobsthal quaternions and third-order Jacobsthal numbers. Furthermore, we gave some their quadratic properties, the summations, the Binet's formulas and Cassini-like identities for these quaternions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Rigid motion computation or estimation is a cornerstone in numerous fields. Attitude computation can be achieved by integrating the angular velocity measured by gyroscopes, the accuracy of which is crucially important for the dead-reckoning inertial navigation. The state-of-the-art attitude algorithms have unexceptionally relied on the simplified differential equation of the rotation vector to obtain the attitude. This paper proposes a functional iteration technique with the Rodrigues vector (named the RodFIter method) to analytically reconstruct the attitude from gyroscope measurements. The RodFIter method is provably exact in reconstructing the incremental attitude as long as the angular velocity is exact. Notably, the Rodrigues vector is analytically obtained and can be used to update the attitude over the considered time interval. The proposed method gives birth to an ultimate attitude algorithm scheme that can be naturally extended to the general rigid motion computation. It is extensively evaluated under the attitude coning motion and compares favorably in accuracy with the mainstream attitude algorithms. This paper is believed having eliminated the long-standing theoretical barrier in exact motion integration from inertial measurements. [ABSTRACT FROM AUTHOR]

SUMMARY: We provide necessary and sufficient conditions for all projective transformations of the projectivized dual quaternion model of rigid body displacements that are induced by coordinate changes in moving and/or fixed frame. These transformations fix the quadrics of a pencil and preserve the two families of rulings of an exceptional three-dimensional quadric. Moreover, we fully characterize the constraint varieties of dyads with revolute and prismatic joints in the dual quaternion model. The constraint variety of a dyad with two revolute joints is a regular ruled quadric in a three-space that contains a “null quadrilateral.” If a revolute joint is replaced by a prismatic joint, this quadrilateral collapses into a pair of conjugate complex null lines and a real line but these properties are not sufficient to characterize such dyads. We provide a complete characterization by introducing a new invariant, the “Study fibre projectivity,” and we present examples that demonstrate its potential to explain hitherto not sufficiently well-understood phenomena. [ABSTRACT FROM AUTHOR]