A Computational Geometric Approach for Motion Generation of Spatial Linkages With Sphere and Plane Constraints

2018 ◽  
Vol 11 (1) ◽  
Author(s):  
Xiangyun Li ◽  
Q. J. Ge ◽  
Feng Gao
Keyword(s):  
Geometric Approach ◽  
Geometric Constraints ◽  
Dimensional Synthesis ◽  
Motion Generation ◽  
Geometric Framework ◽  
Kinematic Chains ◽  
Kinematic Mapping ◽  
Spherical Linkages ◽  
Spatial Displacements ◽  
The Given

This paper studies the problem of spatial linkage synthesis for motion generation from the perspective of extracting geometric constraints from a set of specified spatial displacements. In previous work, we have developed a computational geometric framework for integrated type and dimensional synthesis of planar and spherical linkages, the main feature of which is to extract the mechanically realizable geometric constraints from task positions, and thus reduce the motion synthesis problem to that of identifying kinematic dyads and triads associated with the resulting geometric constraints. The proposed approach herein extends this data-driven paradigm to spatial cases, with the focus on acquiring the point-on-a-sphere and point-on-a-plane geometric constraints which are associated with those spatial kinematic chains commonly encountered in spatial mechanism design. Using the theory of kinematic mapping and dual quaternions, we develop a unified version of design equations that represents both types of geometric constraints, and present a simple and efficient algorithm for uncovering them from the given motion.

A Task Driven Approach to the Synthesis of Spherical 4R Linkages Using Algebraic Fitting Method

2013 ◽  
Author(s):  
Ping Zhao ◽  
Xiangyun Li ◽  
Anurag Purwar ◽  
Kartik Thakkar ◽  
Q. J. Ge
Keyword(s):  
Linear Method ◽  
Geometric Constraints ◽  
Motion Generation ◽  
Constraint Manifold ◽  
Fitting Method ◽  
Kinematic Mapping ◽  
Motion Approximation ◽  
The Given ◽  
Best Fit ◽  
Additional Constraints

This paper studies the problem of spherical 4R motion approximation from the viewpoint of extraction of circular geometric constraints from a given set of spherical displacements. This paper extends our planar 4R linkage synthesis work to the spherical case. By utilizing kinematic mapping and quaternions, we map spherical displacements into points and the workspace constraints of the coupler into intersection of algebraic quadrics (called constraint manifold), respectively, in the image space of displacements. The problem of synthesizing a spherical 4R linkage is reduced to finding a pencil of quadrics that best fit the given image points in the least squares sense. Additional constraints on the pencil identify the quadrics that represent a spherical circular constraint. The geometric parameters of the quadrics encode information about the linkage parameters which are readily computed to obtain a spherical 4R linkage that best navigates through the given displacements. The result is an efficient and largely linear method for spherical four-bar motion generation problem.

Dimensional Synthesis and Constrained Motion Interpolation for Planar and Spherical 6R Closed Chains

2008 ◽  
Author(s):  
Anurag Purwar ◽  
Zhe Jin ◽  
Qiaode Jeffrey Ge
Keyword(s):  
Rational Curve ◽  
Dimensional Synthesis ◽  
Constrained Motion ◽  
Initial Attempt ◽  
Kinematic Constraints ◽  
Constraint Manifold ◽  
Cartesian Space ◽  
Closed Chain ◽  
Kinematic Mapping ◽  
The Given

In the recent past, we have studied the problem of synthesizing rational interpolating motions under the kinematic constraints of any given planar and spherical 6R closed chain. This work presents some preliminary results on our initial attempt to solve the inverse problem, that is to determine the link lengths of planar and spherical 6R closed chains that follow a given smooth piecewise rational motion under the kinematic constraints. The kinematic constraints under consideration are workspace related constraints that limit the position of the links of planar and spherical closed chains in the Cartesian space. By using kinematic mapping and a quaternions based approach to represent displacements of the coupler of the closed chains, the given smooth piecewise rational motion is mapped to a smooth piecewise rational curve in the space of quaternions. In this space, the aforementioned workspace constraints on the coupler of the closed chains define a constraint manifold representing all the positions available to the coupler. Thus the problem of dimensional synthesis may be solved by modifying the size, shape and location of the constraint manifolds such that the mapped rational curve is contained entirely inside the constraint manifolds. In this paper, two simple examples with preselected moving pivots on the coupler as well as fixed pivots are presented to illustrate the feasibility of this approach.

Synthesis of Planar Mechanisms for Pick and Place Tasks With Guiding Locations

2013 ◽  
Author(s):  
Pierre Larochelle
Keyword(s):  
Algebraic Geometry ◽  
Constraint Equation ◽  
Dimensional Synthesis ◽  
Motion Generation ◽  
Planar Mechanisms ◽  
Open Chain ◽  
Synthesis Algorithm ◽  
Kinematic Chains ◽  
Synthesis Technique ◽  
Kinematic Inversion

A novel dimensional synthesis technique for solving the mixed exact and approximate motion synthesis problem for planar RR kinematic chains is presented. The methodology uses an analytic representation of the planar RR dyads rigid body constraint equation in combination with an algebraic geometry formulation of the exact synthesis for three prescribed locations to yield designs that exactly reach the prescribed pick & place locations while approximating an arbitrary number of guiding locations. The result is a dimensional synthesis technique for mixed exact and approximate motion generation for planar RR dyads. A solution dyad may be directly implemented as a 2R open chain or two solution dyads may be combined to form a planar 4R closed chain; also known as a planar four-bar mechanism. The synthesis algorithm utilizes only algebraic geometry and does not require the use of a numerical optimization algorithm or a metric on planar displacements. Two implementations of the synthesis algorithm are presented; computational and graphical construction. Moreover, the kinematic inversion of the algorithm is also included. An example that demonstrates the synthesis technique is included.

A Task Driven Approach to Unified Synthesis of Planar Four-Bar Linkages Using Algebraic Fitting of a Pencil of G-Manifolds

2013 ◽  
Author(s):  
Q. J. Ge ◽  
Ping Zhao ◽  
Anurag Purwar
Keyword(s):  
Least Squares ◽  
Singular Values ◽  
Geometric Constraints ◽  
Motion Generation ◽  
Motion Approximation ◽  
Value Decomposition ◽  
Four Bar Linkage ◽  
The Given ◽  
Best Fit ◽  
Additional Constraints

This paper studies the problem of planar four-bar motion approximation from the viewpoint of extraction of geometric constraints from a given set of planar displacements. Using the Image Space of planar displacements, we obtain a class of quadrics, called Generalized- or G-manifolds, with eight linear and homogeneous coefficients as a unified representation for constraint manifolds of all four types of planar dyads, RR, PR, and PR, and PP. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of G-manifolds that best fit the image points in the least squares sense. This least squares problem is solved using Singular Value Decomposition. The linear coefficients associated with the smallest singular values are used to define a pencil of quadrics. Additional constraints on the linear coefficients are then imposed to obtain a planar four-bar linkage that best guides the coupler through the given displacements. The result is an efficient and linear algorithm that naturally extracts the geometric constraints of a motion and leads directly to the type and dimensions of a mechanism for motion generation.

A Fourier Descriptor Approach to Integrated Type and Dimensional Synthesis of Coupled Serial Mechanism for Motion Generation

2019 ◽  
Author(s):  
Hao Lv ◽  
Yuanfei Han ◽  
Xiangyun Li ◽  
Liuxian Zhu
Keyword(s):  
Fourier Descriptor ◽  
Dimensional Synthesis ◽  
Type Synthesis ◽  
Motion Generation ◽  
End Effector ◽  
Novel Approach ◽  
Descriptor Approach ◽  
Harmonic Components ◽  
The Given ◽  
Serial Mechanism

Abstract Coupled serial mechanism is a class of mechanisms that couple the relative rotation of successive links utilizing gears or cable-pulley systems. They can be used to generate complex end-effector trajectories or motions with a single actuator. With the employment of Fourier descriptors, a novel approach to integrate type synthesis and dimensional synthesis of such mechanisms is proposed in this paper. Through the Fourier analysis of two arbitrary trajectories from the given motion, the simplest trajectory that contains the least number of harmonic components is identified. Then, characteristic information of those harmonics such as their numbers, amplitudes and initial phases are used to determine the topology and dimensions of the corresponding coupled serial mechanism, thus effectively solving the motion synthesis problem of this type of mechanisms. Finally, three examples are given to demonstrate the validity of the proposed method.

A Task Driven Approach to Simultaneous Type Synthesis and Dimensional Optimization of Planar Parallel Manipulator Using Algebraic Fitting of a Family of Quadrics

2013 ◽  
Author(s):  
Xiangyun Li ◽  
Ping Zhao ◽  
Q. J. Ge ◽  
Anurag Purwar
Keyword(s):  
Least Squares ◽  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Surface Fitting ◽  
Image Space ◽  
Synthesis Process ◽  
Dimensional Synthesis ◽  
Planar Parallel Manipulator ◽  
Kinematic Mapping ◽  
The Given

This paper studies the rigid body guidance problem for 3-DOF planar parallel manipulators (PPM) with three-triad assembly. We present a novel, unified, and simultaneous type and dimensional synthesis approach to planar parallel manipulator synthesis by using kinematic mapping, surface fitting, and least squares techniques. Novelty of our approach lies in linearization of a highly non-linear problem and the fact that the nature of the given motion or displacement drives the synthesis process without assuming triad topology or their geometry. It has been well established that by using planar quaternions and kinematic mapping, workspace related constraints of planar dyads or triads can be represented as algebraic constraint manifolds in the image space of planar displacements. The constraints associated with planar RR-, PR- and RP-dyads correspond to a single quadric in the image space, while that of each of the six planar triads (RRR, RPR, PRR, PPR, RRP and RPP) map to a pair of quadrics and the space between them. Moreover, the quadrics associated with RRR- and RPR-triads are of the same type as that of RR dyads, of PRR- and PPR-triads as that of PR-, and RRP- and RPP-triads as that of RP-dyad. This simplification nicely extends a dyad synthesis problem to a triad synthesis one. The problem is formulated as the least-squares error minimization problem to find a trinity of quadrics that best fit the image points of task displacements. The fitting error corresponding to each single quadric of the trinity is regarded as variation (thickness) of that quadric, which turns that quadric into a pair of quadrics. Hence, three dyads with minimal surface fitting errors can be converted to three triads in the Cartesian space.

scholarly journals A Task-Driven Approach to Unified Synthesis of Planar Four-Bar Linkages Using Algebraic Fitting of a Pencil of G-Manifolds

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Q. J. Ge ◽  
Anurag Purwar ◽  
Ping Zhao ◽  
Shrinath Deshpande
Keyword(s):  
Least Squares ◽  
Singular Values ◽  
Singular Value ◽  
Geometric Constraints ◽  
Motion Generation ◽  
Value Decomposition ◽  
Four Bar Linkage ◽  
The Given ◽  
Best Fit ◽  
Additional Constraints

This paper studies the problem of planar four-bar motion generation from the viewpoint of extraction of geometric constraints from a given set of planar displacements. Using the image space of planar displacements, we obtain a class of quadrics, called generalized- or G-manifolds, with eight linear and homogeneous coefficients as a unified representation for constraint manifolds of all four types of planar dyads, RR, PR, and PR, and PP. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of G-manifolds that best fit the image points in the least squares sense. This least squares problem is solved using singular value decomposition (SVD). The linear coefficients associated with the smallest singular values are used to define a pencil of quadrics. Additional constraints on the linear coefficients are then imposed to obtain a planar four-bar linkage that best guides the coupler through the given displacements. The result is an efficient and linear algorithm that naturally extracts the geometric constraints of a motion and leads directly to the type and dimensions of a mechanism for motion generation.

MotionGen: Interactive Design and Editing of Planar Four-Bar Motions for Generating Pose and Geometric Constraints

2017 ◽  
Vol 9 (2) ◽  
Author(s):  
Anurag Purwar ◽  
Shrinath Deshpande ◽  
Q. J. Ge
Keyword(s):  
Geometric Constraints ◽  
Dimensional Synthesis ◽  
Motion Generation ◽  
Unified Framework ◽  
Synthesis Algorithm ◽  
Approximate Synthesis ◽  
High Utility ◽  
High Level ◽  
Generation Problem ◽  
Practical Constraints

In this paper, we have presented a unified framework for generating planar four-bar motions for a combination of poses and practical geometric constraints and its implementation in MotionGen app for Apple's iOS and Google's Android platforms. The framework is based on a unified type- and dimensional-synthesis algorithm for planar four-bar linkages for the motion-generation problem. Simplicity, high-utility, and wide-spread adoption of planar four-bar linkages have made them one of the most studied topics in kinematics leading to development of algorithms and theories that deal with path, function, and motion generation problems. Yet to date, there have been no attempts to develop efficient computational algorithms amenable to real-time computation of both type and dimensions of planar four-bar mechanisms for a given motion. MotionGen solves this problem in an intuitive fashion while providing high-level, rich options to enforce practical constraints. It is done effectively by extracting the geometric constraints of a given motion to provide the best dyad types as well as dimensions of a total of up to six four-bar linkages. The unified framework also admits a plurality of practical geometric constraints, such as imposition of fixed and moving pivot and line locations along with mixed exact and approximate synthesis scenarios.

Synthesis of Planar Mechanisms for Pick and Place Tasks With Guiding Positions

2015 ◽  
Vol 7 (3) ◽  
Author(s):  
Pierre Larochelle
Keyword(s):  
Algebraic Geometry ◽  
Constraint Equation ◽  
Dimensional Synthesis ◽  
Motion Generation ◽  
Open Chain ◽  
Synthesis Algorithm ◽  
Kinematic Chains ◽  
Synthesis Technique ◽  
Pick And Place ◽  
Kinematic Inversion

A novel dimensional synthesis technique for solving the mixed exact and approximate motion synthesis problem for planar RR kinematic chains is presented. The methodology uses an analytic representation of the planar RR dyad's rigid body constraint equation in combination with an algebraic geometry formulation of the exact synthesis for three prescribed positions to yield designs that exactly reach the prescribed pick and place positions while approximating an arbitrary number of guiding positions. The result is a dimensional synthesis technique for mixed exact and approximate motion generation for planar RR dyads. A solution dyad may be directly implemented as a 2R open chain or two solution dyads may be combined to form a planar 4R closed chain, also known as a planar four-bar mechanism. The synthesis algorithm utilizes only algebraic geometry and does not require the use of a numerical optimization algorithm or a metric on elements of SE(2); the group of planar displacements. Two implementations of the synthesis algorithm are presented; computational and graphical construction. Moreover, the kinematic inversion of the algorithm is also included. Two examples that demonstrate the synthesis technique are included.

Kinematic Geometry of Wheeled Vehicle Systems

1996 ◽  
Author(s):  
S. V. Sreenivasan ◽  
P. Nanua
Keyword(s):  
Geometric Approach ◽  
Companion Paper ◽  
Kinematic Chains ◽  
Uneven Terrain ◽  
Geometric Conditions ◽  
Kinematic Study ◽  
Vehicle Systems ◽  
Motion Characteristics ◽  
The One ◽  
Wheeled Vehicles

Abstract This paper addresses instantaneous motion characteristics of wheeled vehicles systems on even and uneven terrain. A thorough kinematic geometric approach which utilizes screw system theory is used to investigate vehicle-terrain combinations as spatial mechanisms that possess multiple closed kinematic chains. It is shown that if the vehicle-terrain combination satisfies certain geometric conditions, for instance when the vehicle operates on even terrain, the system becomes singular or non-Kutzbachian — it possesses finite range mobility that is different from the one obtained using Kutzbach criterion. An application of this geometric approach to the study of rate kinematics of various classes of wheeled vehicles is also included. This approach provides an integrated framework to study the kinematic effects of varying the vehicle and/or terrain geometric parameters from their nominal values. In addition, design enhancements of existing vehicles are suggested using this approach. This kinematic study is closely related to the force distribution characteristics of wheeled vehicles which is the subject of the companion paper [SN96].

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