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In this paper, we extend the Chen and Moore determinants of quaternion Hermitian matrices to dual quaternion Hermitian matrices. We show the Chen determinant of dual quaternion Hermitian matrices is invariant under addition, switching, multiplication, and unitary operations at the both hand sides. We then show the Chen and Moore determinants of dual quaternion Hermitian matrices are equal to each other, and they are also equal to the products of eigenvalues. The characteristic polynomial of a dual quaternion Hermitian matrix is also studied. [ABSTRACT FROM AUTHOR]

Eigenvalue decomposition of quaternion Hermitian matrices is a crucial mathematical tool for color image reconstruction and recognition. Quaternion Jacobi method is one of the classical methods to compute the eigenvalues of a quaternion Hermitian matrix. Using quaternion Jacobi rotations, this paper brings forward an innovative method for the eigenvalue decomposition of dual quaternion Hermitian matrices. The effectiveness of the proposed method is confirmed through numerical experiments. Furthermore, a dual complex matrix representation for the color image is developed, and the dual quaternion Jacobi method is applied to the eigenvalue problems of dual complex Hermitian matrices. This approach achieves successful results in the color images reconstruction and recognition. Compared to the quaternion matrix representation of the color image, this approach makes computations more convenient when dealing with problems related to color image processing. [ABSTRACT FROM AUTHOR]

Geometric error is a crucial factor influencing the spatial accuracy of CNC machine tools. A novel methodology for modeling, measuring, and identifying geometric errors in multi-axis machine tools is proposed in this paper. Firstly, a synthetic volumetric error model is established by utilizing dual quaternions for multi-axis CNC machine tools. To characterize the relative position relationship between the tool and the workpiece, the kinematics model automatically incorporates the complex coupling between position and orientation motion in an implicit way, enabling a concise and compact representation of the kinematics. Secondly, measure path planning includes candidate measurement positions which are screened to obtain the optimal position group by using observation indices and the modified Detmax method. Thirdly, a parametric modeling method based on exponential cosine fitting is proposed for representing both angular and linear errors, and an improved sparrow search algorithm and nested parameter uncertainty optimization are established to process curve fitting–based optimization of the geometric error term. The fitting of the exponential cosine model is quantified with model uncertainty, and the nested uncertainty optimization method is employed to improve geometric error terms with a poor fitting effect. Finally, the effectiveness is demonstrated through experimental comparisons. The geometric error of a single axis has an average decreased of 69.7%, and the compensation rate of roundness driven by two-axis synchronization is an average of 68.7%. This method offers the advantage of quantifying the minimum optimal number of measurements and positions, improving the efficiency of parametric modeling. [ABSTRACT FROM AUTHOR]

We provide a complete classification of paradoxical closed-loop n-linkages, where n ≥ 6 , of mobility n - 4 or higher, containing revolute, prismatic or helical joints. We also explicitly write down strong necessary conditions for nR-linkages of mobility n - 5 . Our main new tool is a geometric relation between a linkage L and another linkage L ′ resulting from adding equations to the configuration space of L. We then lift known classification results for L ′ to L using this relation. [ABSTRACT FROM AUTHOR]

Considering that a unit dual quaternion can describe elegantly the rigid transformation including rotation and translation, the point-wise weighted 3D coordinate transformation using a unit dual quaternion is formulated. The constructed transformation model by a unit dual quaternion does not need differential process to eliminate the three translation parameters, while traditional models do. Based on the Lagrangian extremum law, the analytical dual quaternion algorithm (ADQA) of the point-wise weighted 3D coordinate transformation is proved existed and derived in detail. Four numerical cases, including geodetic datum transformation, the registration of LIDAR point clouds, and two simulated cases, are studied. This study shows that ADQA is valid as well as the modified procrustes algorithm (MPA) and the orthonormal matrix algorithm (OMA). ADQA is suitable for the 3D coordinate transformation with point-wise weight and no matter rotation angles are small or big. In addition, the results also indicate that if the distribution of common points degrades from 3D or 2D space to 1D space, the solvable correct transformation parameters decrease. In other words, all common points should not be located on a line. From the perspective of improving the transformation accuracy, high accurate control points (with small errors in the coordinates) should be chosen, and it is preferred to decrease the rotation angles as much as possible. [ABSTRACT FROM AUTHOR]

Knowledge graph aims to describe the concepts, the entities and the complex relations of them in the real world. Recently, a series of quaternion rotation models that usually considering the relation vector as the rotation between head and tail entities, have been extensively studied. The results showed these models had the advantages of simplicity and efficiency. However, they are quite weak in capturing the representation and the feature interaction between entities and relations, resulting in insufficient expressiveness of the underlying models, because these models only focus on the linear relations between entities and relations. In order to solve this problem, this paper proposes a novel knowledge graph embedding model called DualDE, which dynamically maps the dual quaternions to the knowledge graph. Specifically, DualDE uses a dynamic mapping mechanism to construct the entity transition vector and the relation transition vector, and continuously adjusts the embedding position of the entity vector in the dual quaternion space according to the dual quaternion multiplication rules. In addition, DualDE can dynamically construct a variety of complex relation types, such as one-to-many, many-to-one and many-to-many. The experimental results based on three standard data sets show that the DualDE model is superior to the existing knowledge graph embedding models on many metrics. [ABSTRACT FROM AUTHOR]

Robotic-assisted surgical system has introduced a powerful platform through dexterous instrument and hand-eye coordination intuitive control. The knowledge of laparoscopic vision is a crucial piece of information for robot-assisted minimally invasive surgery focusing on improved surgical outcomes. Obtaining the transformation with respect to the laparoscope and robot slave arm frames using hand-eye calibration is essential, which is a key component for developing intuitive control algorithm. We proposed a novel two-step modified dual quaternion for hand-eye calibration in this study. The dual quaternion was exploited to solve the hand-eye calibration simultaneously and powered by an iteratively separate solution. The obtained hand-eye calibration result was applied to the intuitive control by using the hand-eye coordination criterion. Promising simulations and experimental studies were conducted to evaluate the proposed method on our surgical robot system. We extensively compared the proposed method with state-of-the-art methods. Results demonstrate this method can improve the calibration accuracy. The effectiveness of the intuitive control algorithm was quantitatively evaluated, and an improved hand-eye calibration method was developed. The relationship between laparoscope and robot kinematics can be established for intuitive control. [ABSTRACT FROM AUTHOR]

3D coordinate transformation is frequently encountered in geodesy applications. In addition, it is frequently encountered in computer vision, robotics, engineering surveying, and GiS applications. Seven parameter Helmert transformation is one of the widely used 3D transformation algorithms. Nowadays, the quaternion algorithms are finding increasingly common usage due to some disadvantages of Helmert transformation based on Euler angles; in this article, a new 3D Cartesian coordinate transformation with the dual quaternion method is explained in detail, and its advantages over the classical transformation problem algorithm are emphasized. Quaternions are not preferred in 3D transformation applications due to the complexity of the subject. As a result, readers generally stay away from the subject of quaternion. In this article, a new dual quaternion algorithm with eight parameters (DQA) is presented. In addition, a MATLAB function DQA_3d_transformation will be introduced in the article so that readers can easily perform 3D transformations with dual quaternions. We hope that after reading this article, the reader’s perspective on dual quaternions may change slightly. [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 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]

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]

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]

This paper presents some improvements to the robot kinematic control strategy based on linear programming, as well as its application to a nonholonomic mobile manipulator. In addition to being computationally efficient, this approach enables the inclusion of inequality and equality constraints in the system control inputs and has formal guarantee of stability. We first propose a new positive definite function of the error variation to avoid joint movements when the robot end-effector stabilizes at a point different from the desired one. In addition, the nonholonomic constraint of the mobile base is imposed as an equality constraint, and inequality constraints are defined to avoid both violation of joint limits and collisions between the mobile base and obstacles in the plane. Last, a performance comparison between the linear programming strategy and an approach based on the pseudoinverse of the whole-body Jacobian matrix is presented. Experimental results show that the controller based on linear programming has low computational cost, and the robot is able to control its end-effector without colliding with obstacles in the plane and without violating its joints limits. However, it tends to generate more abrupt control signals than the continuous controller based on the pseudoinverse of the whole-body Jacobian matrix. [ABSTRACT FROM AUTHOR]

This paper presents the whole-body control of a nonholonomic mobile manipulator using feedback linearization and dual quaternion algebra. The controller, whose reference is a unit dual quaternion representing the desired end-effector pose, acts as a dynamic trajectory generator for the end-effector, and input signals for both nonholonomic mobile base and manipulator arm are generated by using the pseudoinverse of the whole-body Jacobian matrix. In order to deal with the nonholonomic constraints, the input signal to the mobile base generated by the whole-body motion control is properly remapped to ensure feasibility. The Lyapunov stability for the proposed controller is presented and experimental results on a real platform are performed in order to compare the proposed scheme to a traditional classic whole-body linear kinematic controller. The results show that, for similar convergence rate, the nonlinear controller is capable of generating smoother movements while having lower control effort than the linear controller. [ABSTRACT FROM AUTHOR]

We prove that every bounded rational space curve of degree d and circularity c can be drawn by a linkage with 92d-6c+1 revolute joints. Our proof is based on two ingredients. The first one is the factorization theory of motion polynomials. The second one is the construction of a motion polynomial of minimum degree with given orbit. Our proof also gives the explicit construction of the linkage. [ABSTRACT FROM AUTHOR]

A translational surface is a rational tensor product surface generated from two rational space curves by translating one curve along the other curve. Translational surfaces can also be generated from two rational space curves by dual quaternion multiplication. Using the mathematics of dual quaternions, we provide a necessary and sufficient condition for a rational tensor product surface to be a translational surface. Examples are provided to illustrate our theorems and flesh out our algorithms. [ABSTRACT FROM AUTHOR]

A six-axis vibration isolation system is essential to high-precision space systems for attenuating vibrations on precise instruments. The kinematic optimal design is researched for the space six-axis vibration isolator via Stewart mechanism. Jacobian matrix is the basis of the kinematic performance index. However, the conventional Jacobian matrix is not usually dimensionally homogeneous due to the inhomogeneous physical units, caused by the different mathematical representations of the rotation and translation. In this paper, we propose a dual quaternion approach to derive the dimensionally homogeneous Jacobian matrix of a general six-axis parallel mechanism. Two quaternions are used to parameterize the rotation and translation of the platform. The dimensionally scaling factor for the generalized Jacobian matrix is defined as the ratio of the norms of the two quaternions. The dimensionally homogenous Jacobian matrix is then obtained and applied to the optimal design of the six-axis vibration isolator. The performance index of isotropy is considered to make the isolator minimum kinematic coupling in its working configuration. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Nowadays, G01 blocks are still the most widely utilized representation form for the tool paths in the five-axis machine tools. Tangency and curvature discontinuities of the linear tool paths may lead to frequent feedrate fluctuation and acceleration variation, which inevitably deteriorate the machining efficiency and quality. To solve this problem, this paper proposed a dual quaternion B-spline approximation method of dominant points to generate smooth tool paths for five-axis CNC machine tools. First, the dominant points, which can characterize the shape of the linear path, are selected for the tool tip location and the tool orientation simultaneously. Such scheme can significantly compress the number of the control points and release the computational load. Then, the linear tool paths are converted to dual quaternions and approximated to dual quaternion B-spline, which can realize the parameter synchronization for the tool tip and tool orientation, and avoid the fitting oscillatory as well. To verify the proposed method, a trajectory generation method for the dual quaternion B-spline is developed. Simulations and experiments on a five-axis machine tool are conducted, and the results demonstrate the feasibility and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we present an algorithm for human motion capture of the real-time motion trajectory of human arms based on wireless inertial 3D motion trackers. It aims to improve the accuracy of inertial motion captures and quickly reconstruct some human movements. To evaluate the performance of the proposed dual quaternion algorithm, we present the prototype design. The wireless inertial measurement system and Kinect device are introduced simultaneously in capturing human motion. The dual quaternion algorithm incorporates features of the quaternion rotation and translation. So the singular points of Euler angles can be avoided. Dual quaternion algorithm and DCM(direction cosine matrix) are used to reconstruct human arm movements respectively. Compared with the computing speed in Matlab, the speed of the dual quaternion is faster than it of DCM. To the end, we propose a 3D ADAMS human robotic model for simulating the motion trajectory using dual quaternion algorithm. The results show that the dual quaternion can achieve capabilities of a positive DCM solving, which completed between body segments rotating and translating the coordinate system transformation. Also it can effectively drive in real-time a human model to animate movement, and provide a good algorithm. [ABSTRACT FROM AUTHOR]

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]

An iterative method is introduced successfully to solve the inverse kinematics of a 6-DOF manipulator of a tunnel drilling rig based on dual quaternion, which is difficult to get the solution by Denavit-Hartenberg (D-H) based methods. By the intuitive expression of dual quaternion to the orientation of rigid body, the coordinate frames assigned to each joint are established all in the same orientation, which does not need to use the D-H procedure. The compact and simple form of kinematic equations, consisting of position equations and orientation equations, is also the consequence of dual quaternion calculations. The iterative process is basically of two steps which are related to solving the position equations and orientation equations correspondingly. First, assume an initial value of the iterative variable; then, the position equations can be solved because of the reduced number of unknown variables in the position equations and the orientation equations can be solved by applying the solution from the position equations, which obtains an updated value for the iterative variable; finally, repeat the procedure by using the updated iterative variable to the position equations till the prescribed accuracy is obtained. The method proposed has a clear geometric meaning, and the algorithm is simple and direct. Simulation for 100 poses of the end frame shows that the average running time of inverse kinematics calculation for each demanded pose of end-effector is 7.2 ms on an ordinary laptop, which is good enough for practical use. The iteration counts 2-4 cycles generally, which is a quick convergence. The method proposed here has been successfully used in the project of automating a hydraulic rig. [ABSTRACT FROM AUTHOR]

Virtual hands play key roles in three-dimensional interaction in various virtual reality applications. However, their visual realism has seldom been seriously discussed in the community of human computer interaction, not to mention the challenging balance between their visual/motion realism and real-time performance. In this paper, a novel approach for virtual hand interaction modeling based on dual quaternion is proposed. Specifically, dual quaternion blending (DQB) is introduced as pseudo muscular layer of our virtual hand model instead of traditional linear blending (LB) and quaternion blending (QB) due to its advantages in muscular deformation. A framework for virtual hand interaction is proposed which supports both gesture interactions and 3D manipulations. Experimental results show that our proposed virtual hand interaction model is suitable for most virtual reality applications and achieves good visual/motion realism while maintains real-time performance. [ABSTRACT FROM AUTHOR]

This paper proposes a new type of control laws for free rigid bodies. The start point is the dual quaternion and its characteristics. The logarithm of a dual quaternion is defined, based on which kinematic control laws can be developed. Global exponential convergence is achieved using logarithmic feedback via a generalized proportional control law, and an appropriate Lyapunov function is constructed to prove the stability. Both the regulation and tracking problems are tackled. Omnidirectional control is discussed as a case study. As the control laws can handle the interconnection between the rotation and translation of a rigid body, they are shown to be more applicable than the conventional method. [ABSTRACT FROM AUTHOR]

In this paper, a new representational model based on dual quaternionic matrices is proposed for classical electromagnetism. After demonstrating the isomorphic matrix representations of dual quaternions, Maxwell’s equations and the constitutive relations for electromagnetism are expressed in terms of dual quaternionic matrices. For this purpose, new 8 × 8 matrices connected with quaternion basis elements have been introduced. [ABSTRACT FROM AUTHOR]

Determination of relative three-dimensional (3D) position, orientation, and relative motion between two reference frames is an important problem in robotic guidance, manipulation, and assembly as well as in other fields such as photogrammetry. A solution to pose and motion estimation problem that uses two-dimensional (2D) intensity images from a single camera is desirable for real-time applications. The difficulty in performing this measurement is that the process of projecting 3D object features to 2D images is a nonlinear transformation. In this paper, the 3D transformation is modeled as a nonlinear stochastic system with the state estimation providing six degrees-of-freedom motion and position values, using line features in image plane as measuring inputs and dual quaternion to represent both rotation and translation in a unified notation. A filtering method called the Gaussian particle filter (GPF) based on the particle filtering concept is presented for 3D pose and motion estimation of a moving target from monocular image sequences. The method has been implemented with simulated data, and simulation results are provided along with comparisons to the extended Kalman filter (EKF) and the unscented Kalman filter (UKF) to show the relative advantages of the GPF. Simulation results showed that GPF is a superior alternative to EKF and UKF. [ABSTRACT FROM AUTHOR]

Topology and performance are the two main topics dealt in the development of robotic mechanisms. However, it is still a challenge to connect them by integrating the modeling and design process of both parts in a unified frame. As the properties associated with topology and performance, finite motion and instantaneous motion of the robot play key roles in the procedure. On the purpose of providing a fundamental preparation for integrated modeling and design, this paper carries out a review on the existing unified mathematic frameworks for motion description and computation, involving matrix Lie group and Lie algebra, dual quaternion and pure dual quaternion, finite screw and instantaneous screw. Besides the application in robotics, the review of the work from these mathematicians concentrates on the description, composition and intersection operations of the finite and instantaneous motions, especially on the exponential-differential maps which connect the two sides. Furthermore, an in-depth discussion is worked out by investigating the algebraical relationship among these methods and their further progress in integrated robotic development. The presented review offers insightful investigation to the motion description and computation, and therefore would help designers to choose appropriate mathematical tool in the integrated design and modeling and design of mechanisms and robots. [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]