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Research Papers

Analytical Determination of the Proximity of Two Right-Circular Cylinders in Space

[+] Author and Article Information
Saurav Agarwal

Robotics Laboratory,
Department of Engineering Design,
Indian Institute of Technology Madras,
Chennai 600 036, India
e-mail: agr.saurav1@gmail.com

Rangaprasad Arun Srivatsan

Robotics Institute,
Carnegie Mellon University,
5000 Forbes Avenue,
Pittsburgh, PA 15213
e-mail: rarunsrivatsan@cmu.edu

Sandipan Bandyopadhyay

Robotics Laboratory,
Department of Engineering Design,
Indian Institute of Technology Madras,
Chennai 600 036, India
e-mail: sandipan@iitm.ac.in

1The author contributed to this work as a Project Officer working in the Robotics Laboratory, Department of Engineering Design, Indian Institute of Technology Madras.

2Corresponding author.

Manuscript received August 11, 2015; final manuscript received November 23, 2015; published online March 8, 2016. Assoc. Editor: David Dooner.

J. Mechanisms Robotics 8(4), 041010 (Mar 08, 2016) (10 pages) Paper No: JMR-15-1218; doi: 10.1115/1.4032211 History: Received August 11, 2015; Revised November 23, 2015

This paper presents a novel analytical formulation for identifying the closest pair of points lying on two arbitrary cylinders in space, and subsequently the distance between them. Each cylinder is decomposed into four geometric primitives. It is shown that the original problem reduces to the computation of the shortest distance between five distinct combinations of these primitives. Four of these subproblems are solved in closed form, while the remaining one requires the solution of an eight-degree polynomial equation. The analytical nature of the formulation and solution allows the identification of all the special cases, e.g., positive-dimensional solutions, and the curve of intersection when the cylinders interfere. The symbolic precomputation of the results leads to a fast numerical implementation, capable of solving the problem in about 50 μs (averaged over 1 × 106 random instances of the most general case) on a standard PC. The numerical results are verified by repeating all the calculations in a general-purpose commercial cad software. The algorithm has significant potential for applications in the various aspects of robotics and mechanisms, as their links can be modeled easily and compactly as cylinders. This makes tasks such as path planning, determination of the collision-free workspace, etc., computationally easier.

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References

Ketchel, J. , and Larochelle, P. , 2006, “ Collision Detection of Cylindrical Rigid Bodies for Motion Planning,” IEEE International Conference on Robotics and Automation, Orlando, FL, May 15–19, pp. 1530–1535.
ElMaraghy, W. , Valluri, S. , Skubnik, B. , and Surry, P. , 1994, “ Intersection Volumes and Surface Areas of Cylinders for Geometrical Modeling and Tolerancing,” Comput.-Aided Des., 26(1), pp. 29–45. [CrossRef]
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Biermann, D. , Joliet, R. , and Michelitsch, T. , 2008, “ Fast Distance Computation Between Cylinders for the Design of Mold Temperature Control Systems,” Advances in Computational Intelligence—Theory and Practice, Reihe CI-259/08, SFB 531, Technical University of Dortmund, Dortmund, Germany.
Hudgens, J. , and Arai, T. , 1993, “ Planning Link-Interference-Free Trajectories for a Parallel Link Manipulator,” IECON’93, Maui, HI, Nov. 15–19, Vol. 3, pp. 1506–1511.
Merlet, J.-P. , 1995, “ Determination of the Orientation Workspace of Parallel Manipulators,” J. Intell. Rob. Syst., 13(2), pp. 143–160. [CrossRef]
Patel, R. V. , Shadpey, F. , Ranjbaran, F. , and Angeles, J. , 2005, “ A Collision-Avoidance Scheme for Redundant Manipulators: Theory and Experiments,” J. Field Rob., 22(12), pp. 737–757.
Chittawadigi, R. , and Saha, S. , 2013, “ An Analytical Method to Detect Collision Between Cylinders Using Dual Number Algebra,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Tokyo, Nov. 3–7, pp. 5353–5358.
Srivatsan, R. A. , and Bandyopadhyay, S. , 2013, “ An Analytical Formulation for Finding the Proximity of Two Arbitrary Cylinders in Space,” Conference on Advances in Robotics, AIR’13, ACM, New York, p. 48.
Vranek, D. , 2002, “ Fast and Accurate Circle-Circle and Circle-Line 3D Distance Computation,” J. Graphics Tools, 7(1), pp. 23–31. [CrossRef]
Karnam, M. K. , Srivatsan, R. A. , and Bandyopadhyay, S. , “ Computation of the Safe Working Zone of Parallel Manipulators,” Mech. Mach. Theory (in press).
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Chirikjian, G. , and Burdick, J. , 1994, “ A Modal Approach to Hyper-Redundant Manipulator Kinematics,” IEEE Trans. Rob. Autom., 10(3), pp. 343–354. [CrossRef]
Selig, J. , 2005, Geometric Fundamentals of Robotics (Monographs in Computer Science), Springer-Verlag, New York.
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Figures

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Fig. 1

(a) The cylinder Ei and its constituent geometric primitives—the cylindrical surface Si; bottom end cap Dib, top end cap Dit, and their bounding circles, Cib,Cit, respectively; the axis, Li. (b) The relative position and orientation of the two cylinders. The local coordinate system of the first cylinder serves as the global frame of reference.

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Fig. 2

One cylinder inside the other cylinder

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Fig. 3

Flowchart of the algorithm at the first level depicting the branching of the original problem into three subproblems. The final results are obtained via the equations given in each terminal block.

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Fig. 4

Flowchart of the algorithm for finding proximity of a disk and a cylindrical surface. The final results are given by the equations mentioned in each terminal block.

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Fig. 5

Flowchart of the algorithm for finding proximity of two disks

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Fig. 9

Examples of special cases. These include the cases when there is interference and the solution of proximal points being one-/two-dimensional. (a) One point interference: the solution is given by P(S, S). (b) One common face: the two cylinders are parallel to each other and the faces overlap when seen along the axes. It has a two-dimensional solution. (c) Interfering cylinders: the curve of intersection is given by the formulation for finding P(S, S) described in Sec. 3.1. (d) Solution given by P(S, S). The axes of the cylinders are parallel to each other and the proximal points form line segments Ls1 on the cylinder E1 and Ls2 on the cylinder E2. (e) Solution given by P(D, L). The axes of the cylinders are orthogonal to each other and the proximal points form line segments LD on the disk and LS on cylindrical surface.

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Fig. 10

Summary of numerical experiments depicting actual occurrences of each case and interference at each level: (a) near field and (b) far field. The values, e.g., “51.45(34.12)” in Case 1 for near field, denote that the observed occurrences of the particular case are 51.45% and actual interference detected is 34.12%.

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