In this paper we apply the Fourier transform on the Euclidean motion group to solve problems in kinematic design of binary manipulators. In recent papers it has been shown that the workspace of a binary manipulator can be viewed as a function on the motion group, and it can be generated as a generalized convolution product. The new contribution of this paper is the numerical solution of mathematical inverse problems associated with the design of binary manipulators. We suggest an anzatz function which approximates the manipulator’s density in analytical form and has few free fitting parameters. Using the anzatz functions and Fourier methods on the motion group, linear and non-linear inverse problems (i.e., problems of finding the manipulator’s parameters which produce the total desired workspace density) are solved.

1.
Chirikjian
G. S.
,
Ebert-Uphoff
I.
,
1998
, “
Numerical Convolution on the Euclidean Group with Applications to Workspace Generation
,”
IEEE Transactions on Robotics and Automation
, Vol.
14
, No.
1
, Feb. 1998, pp.
123
136
.
2.
Chirikjian, G. S., Kyatkin, A. B., 2000, Engineering Applications of Noncommutative Harmonic Analysis, CRC Press, to appear.
3.
Driscoll
J. R.
,
Healy
D. M.
,
1994
, “
Computing Fourier Transforms and Convolutions on the 2-Sphere
,”
Advances in Appl. Mathematics
, Vol.
15
, pp.
202
250
.
4.
Ebert-Uphoff, I., Chirikjian, G. S., 1996, “Inverse Kinematics of Discretely Actuated Hyper-Redundant Manipulators Using Workspace Density,” Proceedings 1996 IEEE International Conference on Robotics and Automation, Minneapoils MN April 1996, pp. 139–145.
5.
Ebert-Uphoff
I.
,
Chirikjian
G. S.
,
1998
, “
Discretely Actuated Manipulator Workspace Generation by Closed Form Convolution
,”
ASME JOURNAL OF MECHANICAL DESIGN
, Vol.
120
, No.
2
, June 1998, pp.
245
251
.
6.
Elliott, D. F., Rao, K. R., 1982, Fast Transforms: Algorithms, Analyses, Applications, Academic Press, New York, London, 1982.
7.
Kyatkin
A. B.
,
Chirikjian
G. S.
,
1998
, “
Regularized Solutions of a Nonlinear Convolution Equation on the Euclidean Group
,”
Acta App. Mathematicae
, Vol.
53
, pp.
89
123
, 1998.
8.
Kyatkin, A. B., Chirikjian, G. S., 1999, “Computation of Robot Configuration and Workspaces Via the Fourier Transform on the Discrete Motion Group,” to appear, International Journal of Robotics Research.
9.
Matoba, H., Ishikawa, T., Kim, C., Muller, R. S., 1994, “A Bistable Snapping Mechanism,” IEEE Micro Electro Mechanical Systems 1994, pp. 45–50.
10.
Murray, R. M., Li, Z., Sastry, S. S., A Mathematical Introduction to Robotic Manipulation, CRC Press, Ann Arbor MI, 1994.
11.
Orihara
A.
,
1961
, “
Bessel Functions and the Euclidean Motion Group
,”
Tohoku Math. J.
Vol.
13
, 1961, pp.
66
71
.
12.
Opdahl, P. G., Jensen, B. D., Howell, L. L., 1998, “An Investigation into Compliant Bistable Mechanisms, Proc. 1998 ASME Mechanisms Conference, Sept. 13–16, 1998, Atlanta, GA, Paper DETC98/MECH-5914.
13.
Rockmore
D. N.
,
1994
, “
Efficient Computation of Fourier Inversion for Finite Groups
,”
Journal of the ACM
, Vol.
41
, 1994, pp.
31
66
.
14.
Sugiura, M., 1990, Unitary Representations and Harmonic Analysis, 2nd edition, Elsevier Science Publisher, The Netherlands, 1990.
15.
Talman, J., 1968, Special Functions, W. A. Benjamin, Inc., Amsterdam, 1968.
16.
Vilenkin, N. J., Klimyk, A. U., 1991, Representation of Lie Group and Special Functions, Vol. 1–3, Kluwer Academic Publishers, The Netherlands, 1991.
17.
Vilenkin
N. J.
,
1956
, “
Bessel Functions and Representations of the Group of Euclidean Motions
,”
Uspehi Mat. Nauk.
Vol.
11
, 1956, pp.
69
112
. (in Russian).
This content is only available via PDF.
You do not currently have access to this content.