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The authors consider current problems of the modern theory of dynamic systems on manifolds, which are actively developing. A brief review of such trends in the theory of dynamic systems is given. The results of the algebra of dual numbers, quaternionic algebras, biquaternions (dual quaternions), and their application to the analysis of infinitesimal neighborhoods and infinitesimal deformations of manifolds (schemes) are presented. The theory of differential-algebraic equations over the field of real numbers and their dynamics, as well as elements of trajectory optimization of respective dynamic systems, are outlined. On the basis of connection in bundles, the theory of differential-algebraic equations is extended to algebraic manifolds and schemes over arbitrary fields and schemes, respectively. [ABSTRACT FROM AUTHOR]

An equidistant double NURBS tool paths (EDNTP) fairing method satisfying the given fitting accuracy is proposed for the linear tool path in work coordinate system (WCS). By utilizing dual-quaternion to describe five-axis tool posture, the linear tool path in WCS is fitted with dual-quaternion B-spline vector function in order to obtain the equidistant double NURBS curves P and Q describing the coordinate points of tool tip and tool axis, respectively. Then, due to the different parameters of the two curves, a parameter synchronization interpolation model based on EDNTP is presented to realize synchronous motion of the two NURBS curve parameters during the interpolation process. The three-axis machining NURBS curve interpolation algorithm is utilized to interpolate the tool tip points NURBS curve of EDNTP for five-axis machining. In terms of parameter synchronization model, interpolation algorithm can be mapped to NURBS curve of tool axis points. Simulation and experimental results show that the proposed algorithm can generate G continuous EDNTP satisfying the given fitting accuracy and make the tool interpolate synchronously with the constant cutter length along the EDNTP during the interpolation process and achieve high machining precision which is mainly concentrated on the 10 mm series. [ABSTRACT FROM AUTHOR]

This paper exploits screw theory expressed via unit dual quaternion representation and its algebra to formulate both the forward (position+velocity) kinematics and pose control of an n -dof robot arm in an efficient way. Efficiency is in less computer memory usage, in fast computation of the equations, in singularity-free representation of task space, in robustness to numerical errors, and in compactness of the representations. The formulation is simple, intuitive and straightforward to implement. We validated this formulation experimentally on a 7 dof robot arm. [ABSTRACT FROM AUTHOR]

We give a constructive proof for the existence of a unique rational motion of minimal degree in the dual quaternion model of Euclidean displacements with a given rational parametric curve as trajectory. The minimal motion degree equals the trajectory's degree minus its circularity. Hence, it is lower than the degree of a trivial curvilinear translation for circular curves. [ABSTRACT FROM AUTHOR]

This paper presents the implementation of the high performance adaptive control algorithm for 6-DOF systems described by dual quaternions. The fast adaptation is achieved from the introduced stable adaptation dynamics in the closed-loop system which cannot be expected in the conventional adaptive controller designed upon the certainty equivalence framework. The development of the proposed adaptive control algorithm relies on the two newly introduced algebraic operations. The one is a dual filter for a dual relative error signal. The dual filter essentially has a linear low-pass filter structure and enables us to recast the system dynamics into a linear parameter-affine system dynamics in dual quaternions combined with dual control inputs. The other is an isomorphism between a dual quaternion and a vector for the error measurement. Using this isomorphism, the stability proof of the closed-loop dual quaternion system based on the proposed Lyapunov-like function is rewritten in the form of familiar matrix/vector algebra. In order to demonstrate the performance improvement, numerical simulation results are given, which compare the proposed adaptive control method with the conventional one from the certainty-equivalence principle. [ABSTRACT FROM AUTHOR]

In this paper, we study closed linkages with six rotational joints that allow a one-dimensional set of motions. We prove that the genus of the configuration curve of such a linkage is at most five, and give a complete classification of the linkages with a configuration curve of genus four or five. The classification contains new families. [ABSTRACT FROM AUTHOR]

In this paper, a new forward kinematic algorithm based on dual quaternion for a six-degree-of-freedom (DOF) general Stewart Platform is proposed. The algorithm is established after taking into account the pose parameters of joints and the D–H parameters of each branch chain, which yields a precise mathematic model with a total of 174 geometric parameters. The validity of the algorithm is tested using an inverse kinematic algorithm of a general serial manipulator. The forward kinematic algorithms for the 6-6R and 6-2RP3R mechanisms can be expressed by a unified mathematic model. The proposed algorithm can solve the difficult problem of forward kinematics for the unsolved 6-6R parallel manipulator and can also be applied to error analysis, error synthesis, kinematic calibration, stiffness analysis and the accurate kinematic simulation by analysing the effect of geometric parameters on the platform. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to introduce the application of dual quaternions in one interesting problem in structural biology, i.e., the description of protein structure. The secondary protein structure is a specific geometric shape and the description uses Chasles theorem which states that any rigid body displacement can be described by a screw motion. We will briefly introduce the theory of dual quaternions in connection with the screw motion. Consequently, it is shown that modeling based on dual quaternions is an elegant mathematical method and a compact formula for the description of secondary protein structure is derived using the dual quaternion calculus. [ABSTRACT FROM AUTHOR]

The inverse kinematics solution of a dexterous robotic finger has a significant impact on the real-time control of the robotic hand. Therefore a rapid method for solving is needed. The classical homogeneous matrix transformation is the most popular method used in robot kinematics. However, for the multi degree-of-freedom (DOF) robotic finger, the matrix parameters cost much storage and the inverse matrix calculation requires a large amount of computational cost. So it is not conducive to the real-time control of the robotic hand. Therefore, a method based on dual quaternions is presented for analysing the kinematics of a multi-DOF (4-DOF) robotic thumb. Firstly, the kinematics equation is expressed by dual quaternions. Then the multivariate kinematic equations are converted to binary quadratic equations with methods of separating variables and variable substitution, which is relatively easy to obtain the closed-form solution of the inverse kinematics. Finally, it proves that the dual quaternions method has advantages over the homogeneous matrix transformation in storage and computational cost by the specific numbers for the robotic thumb, which is conducive to the real-time control of robotic hand. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a special kind of overconstrained 6R closed linkage which we call angle-symmetric 6R linkage. These are linkages with the property that the rotation angles are equal for each of the three pairs of opposite joints. We give a classification of these linkages. It turns out that there are three types. First, we have the linkages with line symmetry. The second type is new. The third type is related to cubic motion polynomials. [Copyright &y& Elsevier]

After a review of some properties of dual quaternions, De Moivre's and Euler's formulas for the matrices associated with these quaternions are studied. In special case, De Moivre's formula implies that there are uncountably many matrices of unit dual quaternions satisfying An = I4 for n≥3. Also; we give the relation between the powers of matrices of dual 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. [ABSTRACT FROM AUTHOR]

Abstract: This paper addresses the synchronized control problem of relative position and attitude for spacecraft with input constraint. First, using dual quaternion, the kinematic and dynamic models of the six-degree-of-freedom relative motion of spacecraft are introduced. Second, a new adaptive sliding mode control scheme is proposed to guarantee the globally asymptotic convergence of relative motion despite the presence of control input constraint, parametric uncertainties and external disturbances. A detailed stability analysis of the resulting closed-loop system is included. Finally, simulation results are presented to illustrate the validity and effectiveness of the proposed controller, which has the following properties: (1) explicit accounting for the problem of input constraint, (2) fast convergent rate and accurate results can be obtained, (3) no chattering phenomenon is present in the control torque and control force, (4) self-adaptive regulation law is dynamically adjusted to ensure the tracking errors tend to zero asymptotically, (5) the upper bounds of unknown variables are estimated dynamically. [Copyright &y& Elsevier]

Abstract: Coordinated control problem of translation and rotation between two spacecraft is investigated in this paper. Using dual quaternion, we establish the relative coupled dynamic model in which the coupling effect between translational and rotational motion is indicated. Based on the logarithm of dual quaternion, a model-independent PD-like controller is proposed for spacecraft tracking control problem. The convergence of the closed loop system in the presence of external disturbances is proven theoretically. The validity of the proposed approach is demonstrated by numerical simulations. The advantages of the proposed controller are displayed by comparing with others. [Copyright &y& Elsevier]

Both traditional and dual quaternion-based velocity update algorithms for strapdown inertial navigation system (SINS) are comparatively studied. The leading cause of difference in accuracy between the two algorithms is analyzed and determined in principle. By defining both truncation and approximate integral errors, the difference in accuracy can be not only described qualitatively but also computed quantitatively. The validity of theoretical analysis and derivation is verified through simulation testing. The significant superiority of the dual quaternion algorithm over the traditional one in accuracy is demonstrated through simulation results under highly-dynamic and large-maneuver conditions. Comparison results under both ideal and real dynamic conditions show that the dual quaternion algorithm is superior to the traditional one in overall performance. [ABSTRACT FROM AUTHOR]

We present a new formulation method to solve the kinematic problem of multiarm robot systems. Our major aims were to formulize the kinematic problem in a compact closed form and avoid singularity problems in the inverse kinematic solution. The new formulation method is based on screw theory and quaternion algebra. Screw theory is an effective way to establish a global description of a rigid body and avoids singularities due to the use of the local coordinates. The dual quaternion, the most compact and efficient dual operator to express screw displacement, was used as a screw motion operator to obtain the formulation in a compact closed form. Inverse kinematic solutions were obtained using Paden-Kahan subproblems. This new formulation method was implemented into the cooperative working of 2 Stäubli RX160 industrial robot-arm manipulators. Simulation and experimental results were derived. [ABSTRACT FROM AUTHOR]

In this paper, Euler's and De Moivre's formulas for complex numbers and quaternions are generalized for the dual quaternions. Also, the matrix representation of dual quaternions is expressed. [ABSTRACT FROM AUTHOR]

In this paper, it is investigated eigenvalues and eigenvectors of the dual Hamilton operators. Moreover, it is examined a special type dual quaternion equation using these eigenvalues and eigenvectors. Finally, it is given the nth power of a dual quaternion. [ABSTRACT FROM AUTHOR]

Abstract: A line symmetric motion is the motion obtained by reflecting a rigid body in the successive generator lines of a ruled surface. In this work we review the dual quaternion approach to rigid body displacements, in particular the representation of the group SE(3) by the Study quadric. Then some classical work on reflections in lines or half-turns is reviewed. Next two new characterisations of line symmetric motions are presented. These are used to study a number of examples one of which is a novel line symmetric motion given by a rational degree five curve in the Study quadric. The rest of the paper investigates the connection between sets of half-turns and linear subspaces of the Study quadric. Line symmetric motions produced by some degenerate ruled surfaces are shown to be restricted to certain 2-planes in the Study quadric. Reflections in the lines of a linear line complex lie in the intersection of the Study quadric with a 4-plane. [ABSTRACT FROM AUTHOR]

The goal pursued by this article, is to evaluate the potential of sliding-mode control and estimation techniques, to address fault tolerance against a large class of actuator faults, including loss of controllability of the faulty actuator, for autonomous rendezvous between a chaser spacecraft and a passive spacecraft on a circular orbit. The proposed solution is based on the dual quaternion formalism, to describe in a single equation, rotational and translational spacecraft dynamics, solar array flexible modes, propellant sloshing, the most dimensioning space disturbances, and their coupling. Such a modelling formalism enables to propose a six degree-of-freedom fault tolerant control architecture, which relies on the generalized super-twisting control algorithm nested with a nonlinear fault estimator. An anti-windup strategy based on polytope algebra is applied to the control algorithm, to prevent instability due to actuator saturation when faults occur. Asymptotic stability of the proposed fault-tolerant control scheme is formally proved with respect to a wide variety of faults, providing that the first derivatives of the fault estimation error versus time and the sliding surface, are bounded. Intensive simulations from a functional engineering simulator that accurately simulates the rendezvous mission, are presented in the paper, as well as capture-oriented criteria. The presented results demonstrate that the proposed fault-tolerant solution is able to cover any kind of thruster faults, including total loss of controllability of the faulty thruster, as well as solar array flexible modes, propellant sloshing, gravity gradient, the second zonal harmonic, atmospheric drag and magnetic disturbances. [ABSTRACT FROM AUTHOR]

In this study, the dual complex numbers defined as the dual quaternions have been considered as a generalization of complex numbers. In addition, the dual unitary matrices that are more general form than unitary matrices were obtained. Finally, the group of the dual symplectic matrices was attained by using symplectic structure upon dual quaternions. In particular, the group of symplectic matrices that are isomorphic to dual unitary matrices was studied. [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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• An algebraic method is proposed to synthesize multi-mode Bennett-based mechanisms. • Several mechanisms with different number of joints are constructed. • Explicit poses of joint axes of the generated mechanisms are obtained. • Multimode characteristics of the generated mechanisms are demonstrated. • 7R Bennett-based mechanisms could have four motion modes comparing to two modes in general. This paper systematically deals with the synthesis of multi-mode single-loop 6R, 7R and 8R Bennett-based mechanisms from an algebraic viewpoint. Based on the factorization of motion polynomials over dual quaternions, an algebraic method is proposed to synthesize multi-mode single-loop 6R, 7R and 8R Bennett-based mechanisms. Using this method, several multi-mode single-loop Bennett-based mechanisms with different number of joints are constructed depending on explicit poses of joint axes. Then motion mode analysis of the 7R mechanism is carried out by formulating and solving a set of kinematic loop equations using tools from algebraic geometry. The analysis demonstrates that this multi-mode 7R mechanism has four motion modes, including a two degree-of-freedom (DOF) double Bennett mode, a 2-DOF hybrid mode, a 1-DOF rotation mode and a 1-DOF spatial 7R mode. Meanwhile, multimode characteristics of the single-loop 6R and 8R mechanisms also are concisely demonstrated in light of reconfiguration analysis. This work provides an algebraic representation framework for further investigation on multi-mode mechanisms that composed of two or more single-loop overconstraint mechanisms. [ABSTRACT FROM AUTHOR]

This paper aims to provide an efficient way for analyzing operation modes and revealing corresponding motion characteristics of lower-mobility parallel mechanisms (LMPMs) using unit dual quaternions. Unit dual quaternions are first classified into 132 cases corresponding to 132 elementary operation modes based on the number of constant zero components. Meanwhile, the kinematic interpretation of these cases is presented in detail. Then a general method for analyzing operation modes and revealing motion characteristics of LMPMs is proposed according to unit dual quaternions and geometrical constraints. By this means, operation modes of LMPMs with complicated constraint conditions can also be analyzed, where a prime decomposition of the ideal corresponding to constraint equations in this condition is infeasible. Taken a 3-RSR LMPM and a 3-RPS LMPM as examples, the operation modes and motion characteristics can be obtained by the proposed approach. It is shown that the former LMPM has seven operation modes including two elementary operation modes and five extra operation modes. Under certain operation modes, the zero-torsion motion of the 3-RSR LMPM can not even be achieved. On the other hand, the latter has two operation modes in which the parasitic motion exists. To gain a deeper insight into the physical meaning of the operation modes, axodes are analyzed and drawn by mean of the velocity operator. It is demonstrated that the 3-RSR LMPM can realize an equal-diameter spherical pure rolling movement with variable diameters and the 3-RPS LMPM can achieve a rolling movement accompanied by sliding. [ABSTRACT FROM AUTHOR]

In traditional pose measurement algorithms, there are many visual measurement solutions for feature points and lines but few for feature circles. In this paper, the feature circle is described by the rotation characteristic of a point around a rotating axis in dual quaternion algebra. Dual vectors are used to describe the feature points, and Plücker lines are used to describe the rotation axis; that is, the dynamic characteristic of the feature point rotating around the Plücker line is used to describe the feature circle, and then the relative position and attitude information between the leader and follower rigid bodies are calculated by the principle of binocular vision measurement. The simulation results show that the algorithm can meet the requirements of the relative pose measurement accuracy of rigid bodies. [ABSTRACT FROM AUTHOR]

Quaternions, which are found in many fields, have been studied for a long time. The interest in dual quaternions has also increased after real quaternions. Nagaraj and Bharathi developed the basic theories of these studies. The Serret–Frenet Formulae for dual quaternion-valued functions of one real variable are derived. In this paper, by making use of the results of some previous studies, helixes and harmonic curvature concepts in Q D 3 and Q D 4 are considered and a characterization for a dual harmonic curve to be a helix is given. [ABSTRACT FROM AUTHOR]

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. 10.1186/s40623-015-0263-6). [ABSTRACT FROM AUTHOR]