Conformal Geometric Algebra Research Articles

Rail wear inspection based on computer-aided design model and point cloud data

Railway transportation demands more efficient and accurate rail wear inspection systems to ensure the train operation safety. To obtain continuous three-dimensional data, a structured light rail wear inspection system is developed in this article, and the data processing method for aligning the point cloud from structured light scanning to the nominal computer-aided design model of the rail is investigated. For further data registration, the point cloud of the computer-aided design model is generated and the normal vectors of these points are calculated. In the coarse registration, conformal geometric algebra is applied to align the segmented point clouds by intuitive geometric calculations to those from the computer-aided design model. In the fine registration, the accurate alignment of the point clouds is implemented by the iterative closest point algorithm. The vertical and lateral wear amounts are obtained on the cross section by slicing the aligned three-dimensional point clouds data. Finally, the proposed system and method are validated by comparing the vertical wear amounts with the two-dimensional laser scanning and contact measurement results.

A conformal geometric algebra method for virtual hand modeling and interaction

Virtual hand usually simulates human hands by mapping the actual shapes of hands to virtual environment and modeling the hand-object manipulations in human machine interactions. For describing virtual hand and its manipulation in a unified framework, a method based on conformal geometric algebra (CGA) is proposed to solve the problems of virtual hand modeling and interaction in this paper. With the vertex blending based on CGA, the artifacts on the finger joints are improved, which enhances the realism of the virtual hand model. With the same tool, the collision detection between the virtual hand and the manipulated objects is implemented. The gesture of grasp, pinch, and hold are recognized, and the corresponding manipulation rules are established by CGA calculation. To test these three typical manipulations, the manipulated objects are imported and the manipulation effectiveness is evaluated.

A Curvature-Based Descriptor for Point Cloud Alignment Using Conformal Geometric Algebra

This paper presents a descriptor for course alignment of point clouds using conformal geometric algebra. The method is based on selecting keypoints depending on shape factors to identify distinct features of the object represented by the point cloud, and a descriptor is then calculated for each keypoint by fitting two spheres that describe the local curvature. The method for estimating the point correspondences is to a larger extent based on geometric arguments than the method of Kleppe et al. (IEEE Trans Autom Sci Eng, 2017), which results in improved performance. The accuracy of the curvature-based descriptor is validated in experiments, and is shown to compare favorably to state-of-the-art methods in an experiment on course alignment of industrial parts to be assembled with robots.

Quadric Conformal Geometric Algebra of $${\mathbb {R}}^{9,6}$$ R 9 , 6

Geometric Algebra can be understood as a set of tools to represent, construct and transform geometric objects. Some Geometric Algebras like the well-studied Conformal Geometric Algebra constructs lines, circles, planes, and spheres from control points just by using the outer product. There exist some Geometric Algebras to handle more complex objects such as quadric surfaces; however in this case, none of them is known to build quadric surfaces from control points. This paper presents a novel Geometric Algebra framework, the Geometric Algebra of $${\mathbb {R}}^{9,6}$$ , to deal with quadric surfaces where an arbitrary quadric surface is constructed by the mere wedge of nine points. The proposed framework enables us not only to intuitively represent quadric surfaces but also to construct objects using Conformal Geometric Algebra. Our proposed framework also provides the computation of the intersection of quadric surfaces, the normal vector, and the tangent plane of a quadric surface.

Motor Parameterization

In this paper, we consider several parameterizations of rigid transformations using motors in 3-D conformal geometric algebra. In particular, we present parameterizations based on the exponential, outer exponential, and Cayley maps of bivectors, as well as a map based on a first-order approximation of the exponential followed by orthogonal projection onto the group manifold. We relate these parameterizations to the matrix representations of rigid transformations in the 3-D special Euclidean group. Moreover, we present how these maps can be used to form retraction maps for use in manifold optimization; retractions being approximations of the exponential map that preserve the convergence properties of the optimization method while being less computationally expensive, and, for the presented maps, also easier to implement.

Geometric techniques for robotics and HMI: Interpolation and haptics in conformal geometric algebra and control using quaternion spike neural networks

In this work, by reformulating screw theory (generalization of quaternions) in the conformal geometric algebra framework, we address the interpolation, virtual reality, graphics engineering, haptics. We derive intuitive geometric equations to handle surface operations like in kidney surgery. The interpolation can handle the interpolation and dilation in 3D of points, lines, planes, circles and spheres. With this procedure, we interpolate trajectories of surgical instrument. Using quaternions, we formulate the quaternion spike neural network for control. This new neural network structure is based on Spike Neural Networks and developed using the quaternion algebra. The real valued training algorithm was extended so that it could make adjustments of the weights according to the properties and product of the quaternion algebra. In this spike neural network, we are taking into account two relevant ideas the use of Spike neural network which is the best model for oculo-motor control and the role of geometric computing. As illustration. the quaternion spike neural network is applied for control of robot manipulator. The experimental analysis shows promising possibilities for the use of this powerful geometric language to handle multiple tasks in human–machine interaction and robotics.

CGA-Based approach to direct kinematics of parallel mechanisms with the 3-RS structure

This paper presents a unified geometric modeling and solution procedure for direct kinematic analysis of a class of parallel mechanisms based on conformal geometric algebra (CGA). After locking the actuated joints, such parallel mechanisms will be turned into a 3-RS structure, which is composed of two triangular platforms connected by three RS serial chains in parallel. Using the proposed approach, the univariate polynomial equation for the direct kinematic analysis of these parallel mechanisms can be derived in three steps. Firstly, the positions of two of the three spherical joints on the moving platform are formulated by the intersection, dissection and dual of the basic geometric entities under the frame of CGA. Secondly, a coordinate-invariant equation expressed in terms of geometric entities is derived via CGA operation. Thirdly, a univariate polynomial equation is obtained directly from the aforementioned coordinate-invariant equation by using tangent-half-angle substitution. Several case studies are then presented to verify the solution procedure. The novelties of this approach lie in that: (1) The formulation is concise and coordinate-invariant and has intrinsic geometric intuition due to the use of CGA; (2) No algebraic elimination procedure is required to derive the univariate polynomial equation; and (3) The proposed approach is applicable to the direct kinematics of this family of parallel mechanisms with any link parameters.

Geometric Error Modeling of Parallel Manipulators Based on Conformal Geometric Algebra

An approach for geometric error modeling of parallel manipulators (PMs) based on the visual representation and direct calculation of conformal geometric algebra is introduced in this paper. In this method, the finite motion of an open-loop chain is firstly formulated. Through linearization of the finite motion, error propagation of the open-loop chain is analyzed. Then the error sources are separated in terms of joint perturbations and geometric errors. Next, motions and constraints of PMs are analyzed visually by their reciprocal property. Finally geometric error model of PMs are formulated considering the actuations and constraints. The merits of this new approach are twofold: (1) complete and continuous geometric error modeling can be achieved since finite motions are considered, (2) visual and analytical computation of motions and constraints are applied for transferring geometric errors from the open-loop chain to the PM. A 2-DoF rotational PM is applied to demonstrate the geometric error modeling process. Comparisons between simulation and analytical models show that this approach is highly effective.

Finite Motion Analysis of Parallel Mechanisms with Parasitic Motions Based on Conformal Geometric Algebra

Finite motion analysis of parallel mechanisms (PMs) denotes formulating the map between finite motion of end-effector and those of its component limbs. By employing conformal geometric algebra (CGA), this paper presents an analytical and accurate method to analyze the finite motions of PMs with parasitic motions. Herein, parasitic motions are defined as the dependent motions in the constraint Degrees-of-Freedom (DoFs) of PMs. Firstly, description of rigid body transformations based on CGA is reviewed. Then, the intersection algorithm of finite motions is introduced by exploiting the algebraic properties of CGA. Based on this, a method to formulate the finite motions of PMs with parasitic motions is proposed. Finally, Z3 mechanism is sketched as example by utilizing the approach. This method facilitates the invention of new mechanisms and can also be applied in the finite motion analysis of other kinds of PMs.

A Geometric Modeling and Computing Method for Direct Kinematic Analysis of 6-4 Stewart Platforms

A geometric modeling and solution procedure for direct kinematic analysis of 6-4 Stewart platforms with any link parameters is proposed based on conformal geometric algebra (CGA). Firstly, the positions of the two single spherical joints on the moving platform are formulated by the intersection, dissection, and dual of the basic entities under the frame of CGA. Secondly, a coordinate-invariant equation is derived via CGA operation in the positions of the other two pairwise spherical joints. Thirdly, the other five equations are formulated in terms of geometric constraints. Fourthly, a 32-degree univariate polynomial equation is reduced from a constructed 7 by 7 matrix which is relatively small in size by using a Gröbner-Sylvester hybrid method. Finally, a numerical example is employed to verify the solution procedure. The novelty of the paper lies in that (1) the formulation is concise and coordinate-invariant and has intrinsic geometric intuition due to the use of CGA and (2) the size of the resultant matrix is smaller than those existed.

Mobile Robot Path Planning based on Conformal Geometric Algebra and Teaching-Learning Based Optimization

A crucial task for mobile robots is to move safely through its environment. Therefore, the determination of a safe path planning algorithm is a key problem. In this paper, the authors present a path planning approach based on Conformal Geometric Algebra and Teaching-Learning Based Optimization. The proposed approach solves the path planning for 2D and 3D environments. The simulation results proved the effectiveness of the proposed method.

Structure from Motion Using Bio-Inspired Intelligence Algorithm and Conformal Geometric Algebra

Structure from Motion algorithms offer good advantages, such as extract 3D information in monocular systems and structures estimation as shown in Hartley & Zisserman for numerous applications, for instance; augmented reality, autonomous navigation, motion capture, remote sensing and object recognition among others. Nevertheless, this algorithm suffers some weaknesses in precision. In the present work, we extent the proposal in Arana-Daniel, Villasenor, Lopez-Franco, & Alanis that presents a new strategy using bio-inspired intelligence algorithm and Conformal Geometric Algebra, based in the object mapping paradigm, to overcome the accuracy problem in two-view Structure form motion algorithms. For this instance, we include two new experiments and the inclusion of the circle entity; the circle carries stronger information about its motion than other geometric entities, as we will show.

Multimodal Medical Image Registration Based on Feature Spheres in Geometric Algebra

Multi-modal image registration in medical image analysis is very challenging as the appearance of body structures in images from different imaging devices can be very different. In this paper, we propose a new method [GA-speeded up robust features (SURF)], which incorporates the geometric algebra (GA) into SURF framework, to detect features from images. We model the volumetric data and register the multi-modal medical images using feature spheres formulated in conformal geometric algebra (CGA). Specifically, we first extract features from medical images using GA-SURF. Second, we construct the feature spheres using the feature points and find the correspondence of feature spheres in the two images using CGA. With that, we can register the images based on the correspondence of the feature spheres. The experimental results evaluated by RIRE have shown that our method can register the multi-modal images with high accuracy. The maximum registration error is less than $4mm$ .

Design and motion tracking of a strip glove based on machine vision

Abstract A type of machine vision based data glove with certain encoded features on its surface, strip glove is proposed in this paper. The machine vision device replaces the sensors attached on the glove and captures the codes on the glove to obtain the hand motion data. To handle the complex geometrical transformations in the strip glove motion, conformal geometric algebra is introduced as the mathematical tool. In the framework of conformal geometric algebra, the degrees of freedom in hand motion are assign to different planes geometrically to form palm and finger planes. The mathematical model of strip glove is established for the pose solution of the strip glove. The motion tracking of the strip glove is implemented in a binocular system. The particle filter algorithm is applied for tracking the colored blocks on a plane. A continuous motion model of finger segments is constructed in the binocular machine vision system by conformal geometric algebra according to hand motion characteristics, and the hand motion data is obtained during the tracking. Based on particle filter algorithm, the conformal geometric algebra method is applied to set particle parameters for the solution of the local occlusion problems caused by hand pose variation, and the tracking stability of the strip glove is improved.

Type synthesis of 1T2R and 2R1T parallel mechanisms employing conformal geometric algebra

Abstracts This paper proposes an analytical method based on conformal geometrical algebra to implement and unify type synthesis of one translational and two rotational (1T2R) parallel mechanisms and two rotational and one translational (2R1T) parallel mechanisms. Firstly, conformal geometrical algebra is introduced in this paper to describe finite motions and carry out type synthesis of parallel mechanisms in an algebraic way. Secondly, the finite motions of 1T2R and 2R1T parallel mechanisms are described and their relations are explored, and this results in unifying the type synthesis of these two types of parallel mechanisms in the same procedure. Then, a parametric and algebraic generation method is proposed to obtain available limbs and axes layout of joints with these limbs. Finally, the assembly conditions among available limbs are given to synthesize systematically 2R1T parallel mechanisms with non-overconstrained or overconstrained property, as well as 1T2R parallel mechanisms with the same properties. This method proposed in this paper is valid in carrying out finite motion representation and type synthesis of parallel mechanisms in an algebraic and analytical way, and then is beneficial to an automatic manner using computer programming languages. Furthermore, the method may solve type synthesis of special parallel mechanisms, such as parallel mechanisms with varied output rotational axes.

Conformal Geometric Algebra Voting Scheme Implemented in Reconfigurable Devices for Geometric Entities Extraction

This paper presents an implementation of the conformal voting scheme using a reconfigurable hardware approach in the frame of conformal geometric algebra $\mathcal {G}_{3,1}$ . This algorithm is able to extract geometric entities, such as circles and lines from edged images. The conformal voting scheme is divided into two main stages: a local stage computed using neighborhoods in the image, and a global stage using the results of the local voting stage. In this implementation, we focused on the most computationally demanding stage: the local stage, which is computed in the field-programmable gate array, while the global voting stage continues to be executed on the PC. The results show an average speedup of 3x depending on the features of the input image and a coefficient of variation near to zero in the times of repeated experiments. Finally, our method can be used for multiple applications in real-time image processing.

Feature preserving multiresolution subdivision and simplification of point clouds: A conformal geometric algebra approach

Due to the huge volume and complex structure, simplification of point clouds is an important technique in practical applications. However, the traditional algorithms often lose geometric information and have no dynamic expanding structure. In this paper, a new simplification algorithm is proposed based on conformal geometric algebra. First of all, a multiresolution subdivision is constructed by the sphere tree, which computes the minimal bounding spheres with the help of k‐means clustering, and then 2 kinds of simplification methods with full advantages of distance computing convenience are applied to carry out self‐adapting simplification. Finally, several comparisons with original data or other algorithms are implemented from visualization to parameter contrast. The results show that the proposed algorithm has good effect not only on the local details but also on the overall error rate.

Triple Conformal Geometric Algebra for Cubic Plane Curves

The triple conformal geometric algebra (TCGA) for the Euclidean ‐plane extends CGA as the product of 3 orthogonal CGAs and thereby the representation of geometric entities to general cubic plane curves and certain cyclidic (or roulette) quartic, quintic, and sextic plane curves. The plane curve entities are 3‐vectors that linearize the representation of nonlinear curves, and the entities are inner product null spaces with respect to all points on the represented curves. Each inner product null space entity also has a dual geometric outer product null space form. Orthogonal or conformal (angle preserving) operations (as versors) are valid on all TCGA entities for inversions in circles, reflections in lines, and by compositions thereof, isotropic dilations from a given center point, translations, and rotations around arbitrary points in the plane. A further dimensional extension of TCGA also provides a method for anisotropic dilations. Intersections of any TCGA entity with a point, point pair, line, or circle are possible. The TCGA defines commutator‐based differential operators in the coordinate directions that can be combined to yield a general n‐directional derivative.

Real‐time rendering under distant illumination with conformal geometric algebra

Precomputed radiance transfer (PRT) methods established for handling global illumination (GI) of objects from area lights in real time and many techniques proposed for rotating the light using linear algebra rotation matrices. Rotating area lights efficiently are crucial part for computer graphics since it is one of the main components of real‐time rendering. Matrices commonly used for handling such rotations are not quite efficient and require high memory consumption; as a result, the need for proposing new more efficient rotation algorithms has been established. In this work, we use the conformal geometric algebra (CGA) as the mathematical background for “GI in real‐time” under distant Image‐based lighting (IBL) illumination, for diffuse surfaces with self‐shadowing by efficiently rotating the environment light using CGA entities. Our work is based on spherical harmonics (SH), which are used for approximating natural, area‐light illumination as irradiance maps. Our main novelty is that we extend the PRT algorithm by representing SH for the first time with CGA.The main intuition is that SH of band index 1 are represented using CGA entities and SH with band index larger than 1 are represented in terms of CGA‐SH of band 1. Specifically, we propose a new method for representing SH with CGA entities and rotating SH by rotating CGA entities. In this way, we can visualize the SH rotations, rotate them faster than rotation matrices, and we provide a unique visual representation and intuition regarding their rotation, in stark contrast to usual rotation matrices, and we achieve consistently better visual results from Ivanic rotation matrices during light rotation. Via our CGA expressed SH, we provide a significant boost on the PRT algorithm since we represent SH rotations by CGA rotors (4 numbers) as opposed to 9 × 9 sparse matrices that are usually used. With our algorithm, we pave the way for including scaling (dilation) and translation of light coefficients using CGA motors.

Type Synthesis of Parallel Tracking Mechanism With Varied Axes by Modeling Its Finite Motions Algebraically

Parallel tracking mechanism with varied axes has great potential in actuating antenna to track moving targets. Due to varied rotational axes, its finite motions have not been modeled algebraically. This makes its type synthesis remain a great challenge. Considering these issues, this paper proposes a conformal geometric algebra (CGA) based approach to model its finite motions in an algebraic manner and parametrically generate topological structures of available open-loop limbs. Finite motions of rigid body, articulated joints, and open-loop limbs are formulated by outer product of CGA. Then, finite motions of parallel tracking mechanism with varied axes are modeled algebraically by two independent rotations and four dependent motions with the assistance of kinematic analysis. Afterward, available four degrees-of-freedom (4-DoF) open-loop limbs are generated by using revolute joints to realize dependent motions, and available five degrees-of-freedom (5-DoF) open-loop limbs are obtained by adding one finite rotation to the generated open-loop limbs. Finally, assembly principles in terms of minimal number and combinations of available open-loop limbs are defined. Typical topological structures are synthesized and illustrated.