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New Tools for Computational Geometry and Rejuvenation of Screw Theory

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Conformal Geometric Algebraic (CGA) provides ideal mathematical tools for construction, analysis, and integration of classical Euclidean, Inversive & Projective Geometries, with practical applications to computer science, engineering, and physics. This paper is a comprehensive introduction to a CGA tool kit. Synthetic statements in classical geometry translate directly to coordinate-free algebraic forms. Invariant and covariant methods are coordinated by conformal splits, which are readily related to the literature using methods of matrix algebra, biquaternions, and screw theory. Designs for a complete system of powerful tools for the mechanics of linked rigid bodies are presented.
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... The conformal geometric algebra (CGA) introduced by Li and Hestenes [36] have been used to solve problems in vision, kinematics, and dynamics of robots due to its convenience and efficiency when representing points, lines, circles, spheres and planes. Ten years ago, Hestenes proposed the method to solve the kinematics and dynamics of open chain kinematics and give the definition of screws and wrenches represented with Clifford algebra and CGA [37]. At the same time, Selig also proposed a method using an eight-dimensional algebra to build rigid body dynamics, and he claimed that inertias, velocities, and momenta all can be represented as the elements of that algebra and all the relationships between physical quantities could be given by the standard operations [38]. ...
... A flag is a triple combination of a plane, a line on the plane, and a point on the line, and it can be used to frame a rigid body [58][59][60]. Hestenes wrote it in CGA form [37] as ...
... Spatial velocity (or screw) can be obtained by diff erentiating transform matrix with time [29]. Similarly, by diff erentiating motor with time, the spatial velocity in CGA can also be obtained, more details are in [37]. The motor in Eq. (13) ...
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Higher-order kinematics of mechanisms has been applied in servo motor control, human-robot interaction and machinery life design fields, etc. The representations of acceleration and jerk by screws have been fully developed by researchers with the methods of the differential of the matrix representation of SE (3) group. Clifford algebra, which is tighter and with higher computational efficiency than the matrix method, is another representation of the motions of rigid bodies. It has been used in position kinematics, grub task motion planning, and robot vision for its convenience of geometric representations and calculations. As far as we know, the work of higher-order kinematics of mechanisms based on Clifford algebra is rare. First, after recalling the based theory of motion representation in conformal geometric algebra (CGA), the mathematical relationships between flag and motor are built. Second, a method for the higher-order kinematics modeling of serial chain mechanisms is proposed. Finally, the higher-order kinematics of the 3-RRS parallel mechanism is built to prove the correctness of the algorithm. This work further enriches the application of CGA for the higher-order kinematics modeling of parallel mechanisms.
... Two kinds of GA are introduced, namely Cl (4,1,0) and Cl(6,0,0) [30]. Cl (4,1,0) is also named Conformal Geometric Algebra (CGA) [11], which comes from the fact that it handles the conformal transformations easily. In addition to this advantage, the visual representation and direct calculation for geometric entities such as point, line and plane are recognized widely as other two superior characteristics of CGA [11,19]. ...
... Cl (4,1,0) is also named Conformal Geometric Algebra (CGA) [11], which comes from the fact that it handles the conformal transformations easily. In addition to this advantage, the visual representation and direct calculation for geometric entities such as point, line and plane are recognized widely as other two superior characteristics of CGA [11,19]. These make CGA based approach attract more and more scholars and applied to the fields of mechanisms and robotics [9,12,18,23,27], such as inverse kinematic analysis of serial and parallel mechanisms [9,12,18]. ...
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This paper proposes a geometric algebra (GA) based approach to carry out inverse kinematics and design parameters of a 2-degree-of-freedom parallel mechanism with its topology structure 3-RSR&SS for the first time. Here, R and S denote respectively revolute and spherical joints. The inverse solutions are obtained easily by utilizing special geometric relations of 3-RSR&SS parallel positioning mechanism, which are proven by calculating relations among point, line and plane in virtue of operation rules. Three global indices of kinematic optimization are defined to evaluate kinematic performance of 3-RSR&SS parallel positioning mechanism in the light of shuffle and outer products. Finally, the kinematic optimal design of 3-RSR&SS parallel positioning mechanism is carried out by means of NSGA-II and then a set of optimal dimensional parameters is proposed. Comparing with traditional kinematic analysis and optimal design method, the approach employing GA has following merits, (1) kinematic analysis and optimal design would be carried out in concise and visual way by taking full advantage of the geometric conditions of the mechanism. (2) this approach is beneficial to kinematic analysis and optimal design of parallel mechanisms in automatic and visual manner using computer programming languages. This paper may lay a solid theoretical and technical foundation for prototype design and manufacture of 3-RSR&SS parallel positioning mechanism.
... When dealing with the nonlinear problems of dynamic and observation models, the UKF can achieve the same accuracy as the EKF [60], but when the peak and high-order moments of the state error distribution are prominent, the UKF can yield more accurate estimations. Conformal geometric algebra can provide an effective method of dealing with three-dimensional rigid body motion problems [61,62]. The rigid body motion equation described by CGA is consistent with the dual quaternion rigid body motion equation. ...
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On-orbit space technology is used for tasks such as the relative navigation of non-cooperative targets, rendezvous and docking, on-orbit assembly, and space debris removal. In particular, the pose estimation of space non-cooperative targets is a prerequisite for studying these applications. The capabilities of a single sensor are limited, making it difficult to achieve high accuracy in the measurement range. Against this backdrop, a non-cooperative target pose measurement system fused with multi-source sensors was designed in this study. First, a cross-source point cloud fusion algorithm was developed. This algorithm uses the unified and simplified expression of geometric elements in conformal geometry algebra, breaks the traditional point-to-point correspondence, and constructs matching relationships between points and spheres. Next, for the fused point cloud, we proposed a plane clustering-method-based CGA to eliminate point cloud diffusion and then reconstruct the 3D contour model. Finally, we used a twistor along with the Clohessy–Wiltshire equation to obtain the posture and other motion parameters of the non-cooperative target through the unscented Kalman filter. In both the numerical simulations and the semi-physical experiments, the proposed measurement system met the requirements for non-cooperative target measurement accuracy, and the estimation error of the angle of the rotating spindle was 30% lower than that of other, previously studied methods. The proposed cross-source point cloud fusion algorithm can achieve high registration accuracy for point clouds with different densities and small overlap rates.
... The latter issue asks for explicitly and concisely analysis of motions and constraints. For this purpose, Conformal Geometric Algebra (CGA) [10,25] is introduced in this paper. It has the merits of visual representation and direct calculation of geometric entities. ...
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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.
... Since this paper intends to analysis the mobility and singularity of parallel mechanisms in a visual and concise form, the key point of which is to obtain the unknown action lines of constraints (motion axis) by direct geometry entities calculations and linear dependency determination. According to the application, two common geometric algebras are used in this paper, namely, conformal geometric algebra (CGA) [31,32] and G 6 . Two superior characteristics of CGA are recognized widely as visual representation and direct calculation for geometric entities. ...
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The crucial procedure of mobility and singularity identification of parallel mechanisms is widely recognized as how to determine their motions (constraints) concisely and visually. In this paper, we propose a geometric algebra (GA) based approach to determine the motions/constraints, mobility and singularity of parallel mechanisms mainly utilizing the geometric and algebraic relations. Firstly, the motions, constraints and their relations are represented by conformal geometric algebra (CGA) formulas in a concise form by employing the characterized geometric elements with . Secondly, the mobility of parallel mechanism, including its number and property and the axes of motions, not only at origin configuration but also in the prescribed workspace, is obtained by the procedure proposed in this paper. Thirdly, the singularity of parallel mechanism is identified by the two indices proposed in this paper with shuffle and outer products. Finally, a typical example is given to illustrate the motions/constraints, mobility and singularity analysis. This approach is beneficial to kinematic analysis and optimal design of parallel mechanisms, especially for which would be carried out in automatic and visual manner using computer programming languages.
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