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In this paper we investigate factorizations of polynomials over the ring of dual quaternions into linear factors. While earlier results assume that the norm polynomial is real (“motion polynomials”), we only require the absence of real polynomial factors in the primal part and factorizability of the norm polynomial over the dual numbers into monic quadratic factors. This obviously necessary condition is also sufficient for existence of factorizations. We present an algorithm to compute factorizations of these polynomials and use it for new constructions of mechanisms which cannot be obtained by existing factorization algorithms for motion polynomials. While they produce mechanisms with rotational or translational joints, our approach yields mechanisms consisting of “vertical Darboux joints”. They exhibit mechanical deficiencies so that we explore ways to replace them by cylindrical joints while keeping the overall mechanism sufficiently constrained.

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

In this paper we present a method for representing surfaces using a set of dual quaternion control points, with the goal of fitting to point clouds. Each control point is defined by a position and radius, which specify the area of the surface it affects, and a dual quaternion defining the transformation it applies. A point is mapped using the surface by a weighted sum of the control points, in a similar method to dual quaternion skinning. A surface is then represented as the transformation of an original surface, such as a unit square plane, using the control points. We demonstrate how we may fit surfaces to point clouds using a modified iterative gradient descent algorithm, adding control points to regions of the surface that are most poorly modelled at the current step. These methods are applied to the problem of representing human breathing by fitting surfaces to a subject’s chest as recorded by an RGB-D (image plus depth) camera and parameterizing the breathing using each control point’s parameters. Variations in the breathing pattern are shown before and after exercise.

Assembles of scissor-like elements have plenty of links and joints leading into complex kinematic analysis. By using dual-quaternions in this paper is presented a novel approach to deal with. Forward, inverse kinematics analysis is developed on simple scissor-like element assembles. Aiming to improve the computational efficiency, dual-quaternion exponentiation is applied too. The various steps to compute the work-space, by using an algorithm, are discussed. Finallly, a numerical example is presented where is applied this novel approach.

Modern approaches for robot kinematics employ the product of exponentials formulation, represented using homogeneous transformation matrices. Quaternions over dual numbers are an established alternative representation; however, their use presents certain challenges: the dual quaternion exponential and logarithm contain a zero-angle singularity, and many common operations are less efficient using dual quaternions than with matrices. We present a new derivation of the dual quaternion exponential and logarithm that removes the singularity, and we show an implicit representation of dual quaternions offers analytical and empirical efficiency advantages compared to both matrices and explicit dual quaternions. Analytically, implicit dual quaternions are more compact and require fewer arithmetic instructions for common operations, including chaining and exponentials. Empirically, we demonstrate a 25%–40% speedup to compute the forward kinematics of multiple robots. This work offers a practical connection between dual quaternions and modern exponential coordinates, demonstrating that a quaternion-based approach provides a more efficient alternative to matrices for robot kinematics.

We propose a novel stereo laparoscopy video-based non-rigid SLAM method called EMDQ-SLAM, which can incrementally reconstruct thee-dimensional (3D) models of soft tissue surfaces in real-time and preserve high-resolution color textures. EMDQ-SLAM uses the expectation maximization and dual quaternion (EMDQ) algorithm combined with SURF features to track the camera motion and estimate tissue deformation between video frames. To overcome the problem of accumulative errors over time, we have integrated a g2o-based graph optimization method that combines the EMDQ mismatch removal and as-rigid-as-possible (ARAP) smoothing methods. Finally, the multi-band blending (MBB) algorithm has been used to obtain high resolution color textures with real-time performance. Experimental results demonstrate that our method outperforms two state-of-the-art non-rigid SLAM methods: MISSLAM and DefSLAM. Quantitative evaluation shows an average error in the range of 0.8-2.2 mm for different cases.

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

This paper presents a path planning strategy for robotic manipulators based on genetic algorithms, dual quaternions and artificial potential field, designing a multi-objective function that allow trajectories be planned avoiding collisions in the workspace and singularity-free kinematic restrictions for manipulators as an optimization problem, satisfying position and orientation conditions. Its analysis is based on the problem of generating a trajectory followed by a sequence of coordinated movements capable of moving the manipulator to perform tasks in the workspace, the problem is not only generated these movements, but also implement strategies that define the path with tools that are easy to implement and avoid obstacles autonomously. Robot kinematics solved by dual quaternion can be used to combine translation with orientation on robotic manipulators in a systematic way, simplifying calculation operations compatible with conventional methods. The artificial potential field approach has been extended to collision avoidance for all manipulator links. A genetic algorithm is used to solve the problem, which the fitness of the problem can be measured by a multi-objective function that involves the distance between the initial and desired position/orientation, minimum joint displacement, dual quaternion configuration, the use of attraction potential to the goal and a repulsion potential to the obstacles and its own links. This method has been implemented in MatLab© for an ABB© IRB1600 robot. Collision avoidance demonstrations have been performed by simulating equipment and static objects in the robot’s workspace.

We study an isomorphism between the group of rigid body displacements and the group of dual quaternions modulo the dual number multiplicative group from the viewpoint of differential geometry in a projective space over the dual numbers. Some seemingly weird phenomena in this space have lucid kinematic interpretations. An example is the existence of non-straight curves with a continuum of osculating tangents which correspond to motions in a cylinder group with osculating vertical Darboux motions. We also look at the set of osculating conics of a curve in projective space, suggest geometrically meaningful examples and briefly discuss and illustrate their corresponding motions.

This paper seeks to bring mathematical clarity to the modelling of rigid motion origami involving quaternions and dual quaternions. We illustrate with an analysis of a single vertex Muria-ori pattern utilizing quaternions and an analysis of a multi-vertex Muria-ori pattern utilizing dual quaternions. We construct a complete lists of formulas that determine the edge vectors, normal vectors and fold angles throughout the folding transformations. In addition we provide explicit functional representations of the resulting folding transformations .

This paper describes various representations of the space of planes. The main focus is on the plane space representation in the symmetry plane detection in \(E^3\) where many candidate planes for many pairs of points of the given object are created and then the most often candidate is found as a mode in the candidate space, so-called Mode-based approach. The result depends on the representation used in the mode-seeking process. The most important aspect is how well distances in the space correspond to similarities of the actual planes with respect to the input object. So, we describe various usable distance functions and compare them both theoretically and practically. The results suggest that, when using the Mode-based approach, representing planes by reflection transformations is the best way but other simpler representations are applicable as well. On the other hand, representations using 3D dual spaces are not very appropriate. Furthermore, we introduce a novel way of representing the reflection transformations using dual quaternions.

Various mathematical formulations are used to describe mechanism and robot kinematics. This mathematical formulation is the basis for kinematic analysis and synthesis, i.e., determining displacements, velocities and accelerations, on the one hand, and obtaining design parameters on the other hand. Vector/matrix formulation containing trigonometric functions is arguably the most favored approach used in the engineering research community. A less well-known but nevertheless very successful approach relies on an algebraic formulation. This involves describing mechanism constraints with algebraic (polynomial) equations and solving these equation sets that pertain to some given mechanism or robot, with the powerful tools of algebraic and numerical algebraic geometries. In the first section of this chapter, the algebraic formulation of Euclidean displacements using Study coordinates (or dual quaternions) will be recalled. Then it will be shown how constraint equations of different kinematic chains can be derived using either geometric insight, elimination methods or the linear implicitization algorithm (LIA). LIA will be described in detail because it can be used even without deep kinematic and geometric insight into the properties of the kinematic chain. In case the kinematic chain consists of only elementary joints, the constraint equations consist of a set of polynomial equations. In the language of algebraic geometry, the corresponding polynomials form an ideal which can be treated with almost classical methods that might not be well known in the engineering community. Therefore in the third section, these methods will be recalled and it will be shown how they can be used to derive the properties of the kinematic entities. In the last section, we apply the introduced algebraic methods to the complete analysis of the 3-UPU-TSAI parallel manipulator.

This article proposes a new dual quaternion based approach for motion interpolation . The highlight is that dual quaternions act in the usual way on points, even if the Study condition is not fulfilled. This induces a fibration of kinematic image space into straight lines that describe the same rigid body displacement. This allows to use standard interpolation schemes for (piecewise) rational curves in a linear space rather than on the curved Study quadric.

This paper investigates constraint singularity induced multifurcation of a novel 7R (R for revolute joint) linkage. Kinematic structure of the 7R linkage is described first for the purpose of geometric analysis. According to geometric properties of the kinematic structure, D-H parameters and kinematics equations in dual quaternion are derived subsequently. The study further explores analytical form of mechanism constraint-screw systems corresponding to distinct motion branches reconfigured from the 7R linkage based on reciprocity in screw theory. The constraint analysis reveals inherent properties of motion branch changes induced by constraints variation and geometric restriction of joints in these motion branches. This leads to identification of multifurcation of the reconfigurable 7R linkage, meaning motion-branch transitions between the non-overconstrained 7R linkage and overconstrained 6R and 4R mechanisms.

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

We demonstrate that millimeter-level manipulation accuracy can be achieved without the static camera registration typically required for visual servoing. We register the camera online, converging in seconds, by visually tracking features on the robot and filtering the result. This online registration handles cases such as perturbed camera positions, wear and tear on camera mounts, and even a camera held by a human. We implement the approach on a Schunk LWA4 manipulator and Logitech C920 camera, servoing to target and pre-grasp configurations. Our filtering software is available under a permissive license (Software available at http://github.com/golems/reflex).

In the preceding chapters we discussed mathematical methods for analyzing robot motion. We will now see how a robot can interact with its environment in a medical application.

We consider Bezier-like formulas with weights in quaternion and geometric (Clifford) algebra for parametrizing rational curves and surfaces. The simplest non-trivial quaternionic case of bilinear formulas for surface patches is studied in detail. Such formulas reproduce well known biquadratic parametrizations of principal Dupin cyclide patches, and are characterized in general as special Darboux cyclide patches. We also survey the most general case of geometric algebra generated by pseudo-Euclidean space. This extends the previous constructions to any dimensions and any signatures of ambient space. Applications include Bezier curves and surfaces in the conformal model of Euclidean space, bilinear Clifford-Bezier patches on isotropic cyclides, and rational offset surface modeling.