Conformal Geometric Algebra Framework Research Articles

Conformal Geometric Algebra–Based Geometric Modeling Method for Forward Displacement Analysis of 6-4 Stewart Platforms

Abstract This paper presents a novel geometric modeling method for direct displacement analysis of 6-4 Stewart platforms based on conformal geometric algebra (CGA). First, a geometric constraint relationship of four lines and a plane intersecting at a point is published. Second, a new coordinate-invariant geometric constraint equation of 6-4 Stewart platforms is deduced by CGA operation. Third, five polynomial equations are established by CGA theory. Fourth, based on the above six equations, a 5 × 5 Sylvester’s matrix is formulated by using Sylvester’s Dialytic elimination method and Gröbner bases method under the graded reverse lexicographical order. Finally, the coordinates of four points on the moving platform are revealed. Besides, a numerical example is used to prove the validity of the proposed method. The novelty of this study is that a whole geometric modeling method by geometric constraint relationship of four lines and a plane intersecting at a point is put forward under the CGA framework, which has good intuition and offers a novel idea for solving the other complex mechanisms. At the same time, Sylvester’s matrix constructed by this method is the smallest one in the known literature for forward displacement analysis of 6-4 Stewart platforms.

CGA-based novel modeling method for solving the forward displacement analysis of 3-RPR planar parallel mechanism

This paper put forward a new geometric modeling method for forward displacement analysis of 3-RPR planar parallel mechanisms (PPMs) under the conformal geometric algebra (CGA) framework. A geometric constraint relationship of four lines intersecting at a point is first given. Then, a new coordinate-invariant geometric constraint equation of 3-RPR PPMs is elaborated via CGA operation. Next, an univariate polynomial equation without algebraic elimination is published. At last, the coordinates of three points on the mobile platform are figured out. The best part of this research is that a complete geometric modeling method by geometric constraint relationship of four lines intersecting at a point is formulated under the CGA framework, which has good intuition and offers a novel idea for solving the other complex mechanisms. Three numerical examples are utilized to confirm the effectiveness of the proposed method.

"Fast GPU Interpolation for Medical Robotics Using the Conformal Geometric Algebra Framework"

Gerardo Martinez Teran, Oswaldo Urena-Ponce, Gerardo Soria Garcia, S Ortega-Cisneros and Eduardo Bayro Corrochano* Author Affiliations Department of Electrical Engineering and Computer Science, Mexico Received: September 01, 2021 | Published: September 15, 2021 Corresponding author: Eduardo Bayro Corrochano, Department of Electrical Engineering and Computer Science, Mexico DOI: 10.26717/BJSTR.2021.38.006201

Direct Kinematic Analysis of the Spatial Parallel Mechanism With 3-R(P)S Structure Based on the Point Pair Relationship

Abstract This paper demonstrates a novel geometric modeling and computational method of the family of spatial parallel mechanisms (PMs) with 3-R(P)S structure for direct kinematic analysis based on the point pair relationship. The point pair relationship, which is derived from the framework of conformal geometric algebra (CGA), consists of the relationship between the point and the point pair and two point pairs. The first research is on the distance relationship between the point and the point pair. Second, the derivation of the distance relationship between two point pairs is based on the aforementioned result, which shows the mathematical homogeneity. Third, two formulations for a point of the point pairs that satisfy the distance relationship between two point pairs are reduced. Fourth, the point pair relationship is applied to solve the direct kinematic analysis of the spatial parallel mechanism with 3-R(P)S structure. Finally, four numerical examples are provided to verify the validity of the proposed algorithm. Overall, the proposed method can be generalized for the direct kinematics of a series of spatial parallel mechanisms with 3-R(P)S structure.

A Novel Geometric Modeling and Calculation Method for Forward Displacement Analysis of 6-3 Stewart Platforms

A novel geometric modeling and calculation method for forward displacement analysis of the 6-3 Stewart platforms is proposed by using the conformal geometric algebra (CGA) framework. Firstly, two formulas between 2-blade and 1-blade are formulated. Secondly, the expressions for two spherical joints of the moving platform are given via CGA operation. Thirdly, a coordinate-invariant geometric constraint equation is deduced. Fourthly, a 16-degree univariate polynomial equation without algebraic elimination by using the Euler angle substitution is presented. Fifthly, the coordinates of three spherical joints on the moving platform are calculated without judging the radical symbols. Finally, two numerical examples are used to verify the method. The highlight of this paper is that a new geometric modeling and calculation method without algebraic elimination is obtained by using the determinant form of the CGA inner product algorithm, which provides a new idea to solve a more complex spatial parallel mechanism in the future.

Projective Geometric Algebra as a Subalgebra of Conformal Geometric algebra

We show that if Projective Geometric Algebra (PGA), i.e. the geometric algebra with degenerate signature (n, 0, 1), is understood as a subalgebra of Conformal Geometric Algebra (CGA) in a mathematically correct sense, then flat primitives share the same representation in PGA and CGA. Particularly, we treat duality in PGA in the framework of CGA. This leads to unification of PGA and CGA primitives which is important especially for software implementation and symbolic calculations.

Geometric Intuitive Techniques for Human Machine Interaction in Medical Robotics

This work introduces a modern and intuitive geometric language to support and to enhance the handling of different tasks in medical robot vision. By reformulating screw theory (generalization of quaternions) in the conformal geometric algebra framework, we address the hand eye calibration, 3D model registration using Kinect, interpolation, haptics, virtual reality, graphics engineering, navigation and guided surgery. The contribution of this work is the use of conformal geometric algebra to solve some key computational issues in medical robot vision without the need to leave the mathematical framework. The experimental analysis shows promising possibilities for the use of this powerful geometric language to handle multiple tasks in minimal invasive medical robotics. For this goal, we use the geometric algebra language as a vehicle between the surgeon, haptics and the organ in the virtual and real world, this language relate the surgeon approach stimulating the surgeon’s intuition based on the utilization of geometric entities and geometric properties of the organ and the surgery itself. Readers can use this geometric language for different applications in graphic engineering and robotics as well.

A geometric algebra algorithm for the closed-form forward displacement analysis of 3-PPS parallel mechanisms

This paper presents an efficient graphical representation geometric algebra approach for the forward displacement analysis (FDA) of 3-PPS parallel mechanisms (PMs). Our approach based on conformal geometric algebra (CGA) reveals interesting intuitions and insight of the modelling and solution for FDA of this type PMs and reduces considerably computational burden and time. In this work, we first deduce the position of an arbitrary joint among three spherical joints on the moving platform through motors and represent it in terms of only one variable. Then, we formulate the position of either one joint of two remained spherical joints using the intersection of a sphere and two planes under the CGA framework. To obtain the geometric formulation of the third spherical joint position, we construct the outer product of two spatial balls with centers at two formulated spherical joints centers and two known planes. The positions of three spherical joints are all explicit geometric formulations of geometric entities of the parallel mechanisms. We achieve a coordinate-invariant equation expressed in terms of geometric entities via inner product of the last joint by itself. The symbolic 8th-degree univariate polynomial input-output equation is first proposed and all 8 sets of closed-form solutions can be obtained. The algorithm is the geometric algebra computation in a clear and coordinates-free way that avoids the use of rotation matrices, and complex algebraic modelling and nonlinear and multivariable elimination computations as most current approaches do. Finally, two numerical examples have been applied to demonstrate the efficiency of the algorithm comparing with the Dixon resultant method. The results show that this algorithm can simplify complexity of the problem and reduce computation time dramatically with strong geometrical intuition. The achieved symbolic univariate polynomial input-output equation can be applied directly to solve the DPA for all types of 3-PPS PMs instead of using any other methods. This work provides a novel geometric modeling and solution approach for the forward displacement analysis of spatial parallel mechanism and be of greatly useful for roboticists or engineers to design and develop 3-PPS PMs for kinds of applications.

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.

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.

Modeling and Pose Control of Robotic Manipulators and Legs using Conformal Geometric Algebra

Controlling the pose of a manipulator involvesfinding the correct configuration of the robot’s elementsto move the end effector to a desired position and orientation.In order to find the geometric relationshipsbetween the elements of a robot manipulator, it is necessaryto define the kinematics of the robot. We presenta synthesis of the kinematical model of the pose forthis type of robot using the conformal geometric algebraframework. In addition, two controllers are developed,one for the position tracking problem and another for theorientation tracking problem, both using an error feedbackcontroller. The stability analysis is carried out forboth controllers, and their application to a 6-DOF serialmanipulator and the legs of a biped robot are presented.By proposing the error feedback and Lyapunov functionsin terms of geometric algebra, we are opening a newvenue of research in control of manipulators and robotlegs that involves the use of geometric primitives, suchas lines, circles, planes, spheres.

Geometric Entities Voting Schemes in the Conformal Geometric Algebra Framework

Traditional methods for geometric entities resort to the Hough transform and tensor voting schemes for detect lines and circles. In this work, the authors extend these approaches using representations in terms of k-vectors of the Conformal Geometric Algebra. Of interest is the detection of lines and circles in images, and planes, circles, and spheres in the 3-D visual space; for that, we use the randomized Hough transform, and by means of k-blades we code such geometric entities. Motivated by tensor voting, we have generalized this approach for any kind of geometric entities or geometric flags formulating the perceptual saliency function involving k-vectors. The experiments using real images show the performance of the algorithms.

Robust tracking of bio-inspired references for a biped robot using geometric algebra and sliding mode control

SUMMARYControlling a walking biped robot is a challenging problem due to the robot's complex and uncertain dynamics. In order to tackle this problem, we propose a sliding mode controller based on a dynamic model that we obtained using the conformal geometric algebra (CGA) approach. An important contribution of this paper is the development of algorithms using the CGA framework. The CGA framework permits us to use lines, points, and other geometric entities to obtain the Lagrange equations of the system. The references for the joints of the robot were obtained in a bio-inspired way following the kinematics of a walking human body. The first and second derivatives of the reference signal were obtained via an exact robust differentiator based on a high-order sliding mode. We analyzed the performance of the proposed control schemes by using bio-inspired walking patterns and simulations.

Total Least Squares Fitting of k-Spheres in n-D Euclidean Space Using an (n+2)-D Isometric Representation

We fit k-spheres optimally to n-D point data, in a geometrically total least squares sense. A specific practical instance is the optimal fitting of 2D-circles to a 3D point set. Among the optimal fitting methods for 2D-circles based on 2D (!) point data compared in Al-Sharadqah and Chernov (Electron. J. Stat. 3:886---911, 2009), there is one with an algebraic form that permits its extension to optimally fitting k-spheres in n-D. We embed this `Pratt 2D circle fit' into the framework of conformal geometric algebra (CGA), and doing so naturally enables the generalization. The procedure involves a representation of the points in n-D as vectors in an (n+2)-D space with attractive metric properties. The hypersphere fit then becomes an eigenproblem of a specific symmetric linear operator determined by the data. The eigenvectors of this operator form an orthonormal basis representing perpendicular hyperspheres. The intersection of these are the optimal k-spheres; in CGA the intersection is a straightforward outer product of vectors. The resulting optimal fitting procedure can easily be implemented using a standard linear algebra package; we show this for the 3D case of fitting spheres, circles and point pairs. The fits are optimal (in the sense of achieving the KCR lower bound on the variance). We use the framework to show how the hyperaccurate fit hypersphere of Al-Sharadqah and Chernov (Electron. J. Stat. 3:886---911, 2009) is a minor rescaling of the Pratt fit hypersphere.

A Visual Glove Based on Conformal Geometric Algebra

In this paper, a visual glove based on Conformal Geometric Algebra is proposed. To estimate the motion of hand in an intuitive way, Conformal Geometric Algebra is introduced in that it can describe relationship of objects in 3D Euclidean space geometrically. For measuring human hand joints angles with glove like device, the physiological structure of human hand is analyzed and an inverse kinematic model is constructed for obtaining the joint angles of fingers. In the Conformal Geometric Algebra framework, a visual glove without sensors and signal process parts on itself is designed. The features on glove surface are extracted in a binocular computer vision system, and the self-calibration algorithm for the glove is implemented. With the calibrated glove, all the geometric objects for inverse kinematic computing by Conformal Geometric Algebra method are reconstructed in 3D space. With the support of this visual glove, the motion of hand model is demonstrated in the virtual environment.

Visual Servoing on the Sphere Using Conformal Geometric Algebra

This paper deals with a visual servoing scheme, which uses a paracatadioptric sensor as visual input. The paracatadioptric sensor provides a wide field of view while maintaining the single center of projection, which is a desirable property of these sensors. The projection induced by this sensor is nonlinear. In this paper a linear model of this projection is presented, the model is developed using the conformal geometric algebra framework which allows to represent nonlinear conformal transformations using a special type of multivectors called versors. With this model we relate the feature time variation with the camera velocity to design a velocity controller used in a visual servoing task.

Integration of Hough Transform of lines and planes in the framework of conformal geometric algebra for 2D and 3D robot vision

This paper presents the application of 2D and 3D Hough Transforms together with conformal geometric algebra to build 3D geometric maps using the geometric entities of lines and planes. Among several existing techniques for robot self-localization, a new approach is proposed for map matching in the Hough domain. The geometric Hough representation is formulated in such a way that one can easily relate it to the conformal geometric algebra framework; thus, the detected lines and planes can be used for algebra-of-incidence computations to find geometric constraints, useful when perceiving special configurations in 3D visual space for exploration, navigation, relocation and obstacle avoidance. We believe that this work is very useful for 2D and 3D geometric pattern recognition in robot vision tasks.

A 3D GIS spatial data model based on conformal geometric algebra

We propose a new Geographic Information System (GIS) three-dimensional (3D) data model based on conformal geometric algebra (CGA). In this approach, geographic objects of different dimensions are mapped to the corresponding basic elements (blades) in Clifford algebra, and the expressions of multi-dimensional objects are unified without losing their geometric meaning. Geometric and topologic computations are also processed in a clear and coordinates-free way. Under the CGA framework, basic geometrics are constructed and expressed by the inner and outer operators. This expression combined geometrics of different dimensions and metric relations. We present the structure of the framework, data structure design, and the data storage, editing and updating mechanisms of the proposed 3D GIS data model. 3D GIS geometric and topological analyses are performed by CGA metric, geometric and topological operators using an object-oriented approach. Demonstrations with 3D residence district data suggest that our 3D data model expresses effectively geometric objects in different dimensions, which supports computation of both geometric and topological relations. The clear and effective expression and computation structure has the potential to support complex 3D GIS analysis, and spatio-temporal analysis.

Body sensor calibration and construction of 3D maps for robot navigation using the framework of conformal geometric algebra

SUMMARYThe research, described in this paper, concerns the robot indoor navigation, emphasizing the aspects of sensor model and calibration, environment representation, and self-localization. The main point is that combining all of these aspects, an effective navigation system is obtained. We present a model of the catadioptric image formation process. Our model simplifies the operations needed in the catadioptric image process. Once we know the model of the catadioptric sensor, we have to calibrate it with respect to the other sensors of the robot, in order to be able to fuse their information. When the sensors are mounted on a robot arm, we can use the hand-eye calibration algorithm to calibrate them. In our case the sensors are mounted on a mobile robot that moves over a flat floor, thus the sensors have less degrees of freedom. For this reason we develop a calibration algorithm for sensors mounted on a mobile robot. Finally, combining all the previous results and a scan matching algorithm that we develop, we build 3D maps of the environment. These maps are used for the self-localization of the robot and to carry out path following tasks. In this work we present experiments which show the effectiveness of the proposed algorithms.

Body sensor calibration and construction of 3D maps for robot navigation using the framework of conformal geometric algebra

Body sensor calibration and construction of 3D maps for robot navigation using the framework of conformal geometric algebra