On the Direct Static Problem of a Planar Rigid Body Suspended From Two Cables

2014 ◽  
Author(s):  
Pascal Dion-Gauvin ◽  
Clément Gosselin
Keyword(s):  
General Solution ◽  
Rigid Body ◽  
Mathematical Formulation ◽  
Static Problem ◽  
Static Equilibrium ◽  
A Posteriori ◽  
Real Solutions ◽  
Validation Procedure ◽  
Univariate Polynomial ◽  
Planar Body

This paper presents a general solution to the direct static problem of a planar body suspended from two cables. First, the conditions for static equilibrium are stated and a mathematical formulation of the problem is derived. A twelfth degree univariate polynomial is then obtained using the resultant of two intermediate polynomials. It is shown that up to twelve real solutions can be obtained, thereby confirming that the polynomial is of minimal degree. Since the condition used in the derivation is necessary but not sufficient, the roots must be tested a posteriori for validity. Simple mathematical conditions are provided that allow such a verification. Finally, two examples are provided to illustrate the results and highlight the importance of the proposed root validation procedure.

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.

Polynomial Solution to the Five-Position Synthesis of Spatial CC Dyads via Dialytic Elimination

2000 ◽  
Author(s):  
Chintien Huang ◽  
Yu-Jui Chang
Keyword(s):  
Polynomial Solution ◽  
Polynomial Equation ◽  
Elimination Method ◽  
Real Solutions ◽  
Numerical Example ◽  
Univariate Polynomial ◽  
Position Synthesis

Abstract This paper presents a polynomial solution to the five-position synthesis of spatial cylindrical-cylindrical dyads. The solution procedures start with the simplification of the synthesis equations derived by Tsai and Roth. The simplified equations are solved by Sylvester’s dialytic elimination method to obtain a univariate polynomial equation of degree six, which gives at most 6 CC dyads for the five-position synthesis. A numerical example with six real solutions is provided.

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.

scholarly journals Additional Rigidly Rotating Solutions in the String Model of Hadrons

1984 ◽  
Vol 37 (1) ◽  
pp. 1 ◽  
Author(s):  
CJ Burden ◽  
LJ Tassie
Keyword(s):  
General Solution ◽  
Rigid Body ◽  
String Model ◽  
Azimuthal Velocity ◽  
Relativistic String ◽  
String Equation ◽  
Body Rotation ◽  
Alternative Derivation ◽  
Rigid Body Rotation ◽  
Velocity Of Light

Solutions to the relativistic string equation are found which correspond to rigid body rotation about the z-axis with azimuthal velocity greater than the velocity of light. If the solutions lie entirely in the x-y plane they are rotating epicycloids, complimentary to the hypocycloid solutions found previously. The use of a general solution to the string equation in terms of two arbitrary world-lines with null tangents provides an alternative derivation of the rigidly rotating solutions.

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