Inverse Geometrico-Static Problem of Underconstrained Cable-Driven Parallel Robots With Three Cables1

2013 ◽  
Vol 5 (3) ◽  
Author(s):  
Marco Carricato
Keyword(s):  
Complete Solution ◽  
Static Problem ◽  
Parallel Robots ◽  
Solution Sets ◽  
Real Solutions

This paper studies underconstrained cable-driven parallel robots (CDPRs) with three cables. A major challenge in the study of these robots is the intrinsic coupling between kinematics and statics, which must be tackled simultaneously. Effective elimination procedures are presented which provide the complete solution sets of the inverse geometrico-static problems (IGPs) with assigned orientation or position. In the former case, the platform orientation is given, whereas the platform position and the cable lengths and tensions must be computed. In the latter case, the platform position is known, whereas the platform orientation and the cable lengths and tensions are to be calculated. The described problems are proven to admit at the most 1 and 24 real solutions, respectively.

Direct Geometrico-Static Problem of Underconstrained Cable-Driven Parallel Robots With Three Cables1

2013 ◽  
Vol 5 (3) ◽  
Author(s):  
Marco Carricato
Keyword(s):  
Complete Solution ◽  
Static Problem ◽  
Parallel Robots ◽  
Homotopy Continuation ◽  
Complex Field ◽  
Efficient Computation ◽  
End Effector ◽  
Tensile Forces ◽  
Univariate Polynomial ◽  
The Ideal

This paper studies the direct geometrico-static problem (DGP) of underconstrained cable-driven parallel robots (CDPRs) with three cables. The task consists in determining the end-effector pose and the cable tensile forces when the cable lengths are assigned. The problem is challenging, because kinematics and statics are coupled, and they must be tackled simultaneously. An effective elimination procedure is proposed and a least-degree univariate polynomial free of spurious factors is obtained in the ideal governing the problem. This is proven to admit 156 solutions in the complex field. Several approaches for the efficient computation of the complete solution set are presented, including an eigenproblem formulation and homotopy continuation.

Synthesis of Defect Free Six Bar Linkages for Body Guidance Through up to Six Finitely Separated Positions

1994 ◽  
Author(s):  
Timothy Tylaska ◽  
Kazem Kazerounian
Keyword(s):  
Design Problem ◽  
Design Methodology ◽  
Complete Solution ◽  
Higher Order ◽  
Synthesis Problem ◽  
Design Constraint ◽  
Solution Sets ◽  
Constraint Equations ◽  
Design Considerations ◽  
Highly Nonlinear

Abstract In the synthesis of watt I six bar linkage, for finitely separated design positions, or in higher order design, constraint equations become highly nonlinear and transcendental. This paper presents a method to decouple the synthesis problem to the synthesis of two path generator 4-bar linkages. Based on this decoupled system, an explicit design methodology is developed, enabling a three, four, five or six body guidance position Watt I linkage to be designed while the designer has choice of some body pivots and ground pivots. Numerical procedures for higher number of positions are also discussed. The methodology allows the designer to obtain an entire set of solutions to a particular design problem. As a spin off from this work, a methodology is also presented to obtain complete solution sets of four bar path generators capable of passing through up to seven precision points, with a procedure that can be eventually extended to eight and nine path points. Design considerations such as branching and transmission angles are also considered.

Synthesis of Six-Bar Timed Curve Generators of Stephenson-Type Using Random Monodromy Loops

2020 ◽  
Vol 13 (1) ◽  
Author(s):  
Aravind Baskar ◽  
Mark Plecnik
Keyword(s):  
Rigid Body ◽  
Complete Solution ◽  
Numerical Algebraic Geometry ◽  
Algebraic Equations ◽  
Solution Sets ◽  
Root Finding ◽  
Rigid Body Motions ◽  
Joint Variable ◽  
Four Bar Linkage ◽  
Parameter Continuation

Abstract Synthesis of rigid-body mechanisms has traditionally been motivated by the design for kinematic requirements such as rigid-body motions, paths, or functions. A blend of the latter two leads to timed curve synthesis, the goal of which is to produce a path coordinated to the input of a joint variable. This approach has utility for altering the transmission of forces and velocities from an input joint onto an output point path. The design of timed curve generators can be accomplished by setting up a square system of algebraic equations and obtaining all isolated solutions. For a four-bar linkage, obtaining these solutions is routine. The situation becomes much more complicated for the six-bar linkages, but the range of possible output motions is more diverse. The computation of nearly complete solution sets for these six-bar design equations has been facilitated by recent root finding techniques belonging to the field of numerical algebraic geometry. In particular, we implement a method that uses random monodromy loops. In this work, we report these solution sets to all relevant six-bars of the Stephenson topology. The computed solution sets to these generic problems represent a design library, which can be used in a parameter continuation step to design linkages for different subsequent requirements.

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