#  The Computation of All 4R Serial Spherical Wrists With an Isotropic Architecture 

Damien Chablat

Jorge Angeles

_J. Mech. Des_. Jun 2003, 125(2): 275-280 (6 pages)

https://doi.org/10.1115/1.1561041

**Published Online:** June 11, 2003 

A spherical wrist of the serial type with _n_ revolute (R) joints is said to be isotropic if it can attain a posture whereby the singular values of its Jacobian matrix are all equal to n/3. What isotropy brings about is robustness to manufacturing, assembly, and measurement errors, thereby guaranteeing a maximum orientation accuracy. In this paper we investigate the existence of redundant isotropic architectures, which should add to the dexterity of the wrist under design by virtue of its extra degree of freedom. The problem formulation, for n=4, leads to a system of eight quadratic equations with eight unknowns. The Bezout number of this system is thus 28=256, its BKK bound being 192. However, the actual number of solutions is shown to be 32. We list all solutions of the foregoing algebraic problem. All these solutions are real, but distinct solutions do not necessarily lead to distinct manipulators. Upon discarding those algebraic solutions that yield no new wrists, we end up with exactly eight distinct architectures, the eight corresponding manipulators being displayed at their isotropic postures. 

Issue Section:

Technical Papers

Keywords:

manipulator kinematics, Jacobian matrices, robust control

Topics:

Algebra, Architecture, Computation, Design, Isotropy, Jacobian matrices, Manipulators, Reflection, Errors, Manufacturing, Degrees of freedom, Manipulator kinematics, Robust control, Robustness

1\. 

Salisbury

, 

J. K.

, and 

Craig

, 

J. J.

, 

1982

, “

Articulated Hands: Force Control and Kinematic Issues

,” 

Int. J. Robot. Res.

, 

1

(

1

), pp. 

4

–

17

.

2\. 

Klein

, 

C. A.

, and 

Miklos

, 

T. A.

, 1991, “Spatial Robotic Isotropy,” Int. J. Robot. Res., 10(4) August.

3\. 

Golub, G. H., and Van Loan, C. F., 1989, _Matrix Computations_ , The Johns Hopkins University Press, Baltimore.

4\. 

Tchon´

, 

K.

, 

2000

, “

Singularities of Euler Wrist

,” 

Mech. Mach. Theory

, 

35

, pp. 

505

–

515

.

5\. 

Long, G. L., Paul, R. P., and Fischer, W. D., 1989, “The Hamilton Wrist: A Four-Revolute Spherical Wrist Whiteout Singularities,” _Proc. IEEE Int. Conf. Robotics and Automation_ , pp. 902–907.

6\. 

Farhang

, 

K.

, and 

Zargar

, 

Y. S.

, 

1999

, “

Design of Spherical 4R Mechanisms: Function Generation for the Entire Motion Cycle

,” 

ASME J. Mech. Des.

, 

121

, pp. 

521

–

528

.

7\. 

Angeles, J., 2002, _Fundamentals of Robotic Mechanical Systems_ , Second Edition, Springer-Verlag, New York.

8\. 

Kroto

, 

H. W.

, 

Heath

, 

J. R.

, 

O’Brien

, 

S. C.

, 

Curl

, 

R. F.

, and 

Smalley

, 

R. E.

, 

1985

, “

C60: Buckminsterfullerene

,” 

Nature (London)

, 

318

, pp. 

162

–

163

.

9\. 

Angeles, J., and Chablat, D., 2000, “On Isotropic Sets of Points in the Plane. Application to the Design of Robot Architectures,” Lenarcic, J., and Stanisic, M. M., eds., _Advances in Robot Kinematic_ , Kluwer Academic Publishers, pp. 73–82.

10\. 

Emiris, I., 1984, “Sparse Elimination and Applications in Kinematics,” Ph.D. Thesis, UC Berkeley.

11\. 

Bernstein

, 

D. N.

, 

1975

, “

The Number of Roots of a System of Equations

,” 

Funct. Anal. Appl.

, 

9

(

2

), pp. 

183

–

185

.

#### Purchase this Content

$25.00