In this paper, we formulate the manipulator Jacobian matrix in a probabilistic framework based on the random matrix theory (RMT). Due to the limited available information on the system fluctuations, the parametric approaches often prove to be inadequate to appropriately characterize the uncertainty. To overcome this difficulty, we develop two RMT-based probabilistic models for the Jacobian matrix to provide systematic frameworks that facilitate the uncertainty quantification in a variety of complex robotic systems. One of the models is built upon direct implementation of the maximum entropy principle that results in a Wishart random perturbation matrix. In the other probabilistic model, the Jacobian matrix is assumed to have a matrix-variate Gaussian distribution with known mean. The covariance matrix of the Gaussian distribution is obtained at every time point by maximizing a Shannon entropy measure (subject to Jacobian norm and covariance positive semidefiniteness constraints). In contrast to random variable/vector based schemes, the benefits of the proposed approach now include: (i) incorporating the kinematic configuration and complexity in the probabilistic formulation; (ii) achieving the uncertainty model using limited available information; (iii) taking into account the working configuration of the robotic systems in characterization of the uncertainty; and (iv) realizing a faster simulation process. A case study of a 2R serial manipulator is presented to highlight the critical aspects of the process.

References

References
1.
Smith
,
R.
, and
Cheeseman
,
P.
,
1986
, “
On the Representation of and Estimation of Spatial Uncertainty
,”
Int. J. Rob. Res.
,
5
(
4
), pp.
56
68
.10.1177/027836498600500404
2.
Dissanayake
,
M. W. M. G.
,
Newman
,
P.
,
Clark
,
S.
,
Durrant-Whyte
,
H. F.
, and
Csorba
,
M.
,
2001
, “
A Solution to the Simultaneous Localization and Map Building (SLAM) Problem
,”
IEEE Trans. Rob. Autom.
,
17
(
3
), pp.
229
241
.10.1109/70.938381
3.
Montemerlo
,
M.
,
Thrun
,
S.
,
Koller
,
D.
, and
Wegbreit
,
B.
,
2002
, “
FastSLAM: A Factored Solution to the Simultaneous Localization and Mapping Problem
,” AAAI National Conference on Artificial Intelligence, Edmonton, Canada, pp.
593
598
.
4.
Simmons
,
R.
, and
Koenig
,
S.
,
1995
, “
Probabilistic Robot Navigation in Partially Observable Environments
,” International Joint Conference on Artificial Intelligence, San Mateo, CA, pp.
1080
1087
.
5.
Durrant-Whyte
,
H.
, and
Bailey
,
T.
,
2006
, “
Simultaneous Localization and Mapping: Part I
,”
IEEE Rob. Autom. Mag.
,
13
(
2
), pp.
99
110
.10.1109/MRA.2006.1638022
6.
Wang
,
Y.
, and
Chirikjian
,
G. S.
,
2006
, “
Error Propagation on the Euclidean Group With Applications to Manipulator Kinematics
,”
IEEE Trans. Rob.
,
22
(
4
), pp.
591
602
.10.1109/TRO.2006.878978
7.
Wang
,
Y.
, and
Chirikjian
,
G. S.
,
2006
, “
Error Propagation in Hybrid Serial-Parallel Manipulators
,” IEEE International Conference on Robotics & Automation, Orlando, FL, pp.
1848
1853
.
8.
Telatar
,
I. E.
,
1999
, “
Capacity of Multi-Antenna Gaussian Channels
,”
Eur. Trans. Telecommun.
,
10
, pp.
585
595
.10.1002/ett.4460100604
9.
Tulino
,
A. M.
, and
Verdu
,
S.
,
2004
,
Random Matrix Theory and Wireless Communications
,
Now Publishers, Inc.
, Delft, Netherlands.
10.
Mallik
,
R. K.
,
2003
, “
The Pseudo-Wishart Distribution and Its Application to MIMO Systems
,”
IEEE Trans. Inf. Theory
,
49
(
10
), pp.
2761
2769
.10.1109/TIT.2003.817465
11.
Kang
,
M.
, and
Alouini
,
M. S.
,
2003
, “
Largest Eigenvalue of Complex Wishart Matrices and Performance Analysis of MIMO MRC Systems
,”
IEEE J. Sel. Areas Commun.
,
21
(
3
), pp.
418
426
.10.1109/JSAC.2003.809720
12.
Zanella
,
A.
,
Chiani
,
M.
, and
Win
,
M. Z.
,
2009
, “
On the Marginal Distribution of the Eigenvalues of Wishart Matrices
,”
IEEE Trans. Commun.
,
57
(
4
), pp.
1050
1060
.10.1109/TCOMM.2009.04.070143
13.
Soize
,
C.
,
2000
, “
A Nonparametric Model of Random Uncertainties for Reduced Matrix Models in Structural Dynamics
,”
Probab. Eng. Mech.
,
15
(
3
), pp.
277
294
.10.1016/S0266-8920(99)00028-4
14.
Soize
,
C.
,
2001
, “
Maximum Entropy Approach for Modeling Random Uncertainties in Transient Elastodynamics
,”
J. Acoust. Soc. Am.
,
109
(
5
), pp.
1979
1996
. 10.1121/1.1360716
15.
Soize
,
C.
,
2005
, “
Random Matrix Theory for Modeling Uncertainties in Computational Mechanics
,”
Computer Meth. Appl. Mech. Eng.
,
194
(
12–16
), pp.
1333
1366
.10.1016/j.cma.2004.06.038
16.
Soize
,
C.
,
2001
, “
Transient Responses of Dynamical Systems With Random Uncertainties
,”
Probab. Eng. Mech.
,
16
(
4
), pp.
363
372
.10.1016/S0266-8920(01)00026-1
17.
Soize
,
C.
,
2003
, “
Random Matrix Theory and Non-Parametric Model of Random Uncertainties in Vibration Analysis
,”
J. Sound Vib.
,
263
(
4
), pp.
893
916
.10.1016/S0022-460X(02)01170-7
18.
Soize
,
C.
, and
Chebli
,
H.
,
2003
, “
Random Uncertainties Model in Dynamic Substructuring Using a Nonparametric Probabilistic Model
,”
J. Eng. Mech.
,
129
(
4
), pp.
449
457
.10.1061/(ASCE)0733-9399(2003)129:4(449)
19.
Adhikari
,
S.
,
2007
, “
Matrix Variate Distributions for Probabilistic Structural Dynamics
,”
AIAA J.
,
45
(
7
), pp.
1748
1762
.10.2514/1.25512
20.
Adhikari
,
S.
,
2008
, “
Wishart Random Matrices in Probabilistic Structural Mechanics
,”
ASCE J. Eng. Mech.
,
134
(
12
), pp.
1029
1044
.10.1061/(ASCE)0733-9399(2008)134:12(1029)
21.
Adhikari
,
S.
, and
Friswell
,
M. I.
,
2007
, “
Random Matrix Eigenvalue Problems in Structural Dynamics
,”
Int. J. Numer. Methods Eng.
,
69
(
3
), pp.
562
591
.10.1002/nme.1781
22.
Adhikari
,
S.
, and
Sarkar
,
A.
,
2009
, “
Uncertainty in Structural Dynamics: Experimental Validation of Wishart Random Matrix Model
,”
J. Sound Vib.
,
323
(
3–5
), pp.
802
825
.10.1016/j.jsv.2009.01.030
23.
Adhikari
,
S.
, and
Chowdhury
,
R.
,
2010
, “
A Reduced-Order Non-Intrusive Approach for Stochastic Structural Dynamics
,”
Comput. Struct.,
88
(
21–22
), pp.
1230
1238
.10.1016/j.compstruc.2010.07.001
24.
Adhikari
,
S.
,
Pastur
,
L.
,
Lytova
,
A.
, and
Du-Bois
,
J. L.
,
2012
, “
Eigenvalue-Density of Linear Stochastic Dynamical Systems: A Random Matrix Approach
,”
J. Sound Vib.
,
331
(
5
), pp.
1042
1058
.10.1016/j.jsv.2011.10.027
25.
Das
,
S.
, and
Ghanem
,
R.
,
2009
, “
A Bounded Random Matrix Approach for Stochastic Upscaling
,”
Multiscale Model. Simul.
,
8
(
1
), pp.
296
325
.10.1137/090747713
26.
Das
,
S.
,
2008
, “
Model, Identification, & Analysis of Complex Stochastic Systems: Applications in Stochastic Partial Differential Equations and Multiscale Mechanics
,” Ph.D. thesis, University of Southern California, Los Angeles, CA. See also URL http://digarc.usc.edu/assetserver/controller/view/search/etd-Das-20080513
27.
Gupta
,
A. K.
, and
Nagar
,
D. K.
,
2000
,
Matrix Variate Distributions
,
Chapman & Hall/CRC
,
Boca Raton, FL
.
28.
Jaynes
,
E. T.
,
1957
, “
Information Theory and Statistical Mechanics
,”
Phys. Rev.
,
106
(
4
), pp.
620
630
.10.1103/PhysRev.106.620
29.
Shannon
,
C.
,
1948
, “
A Mathematical Theory of Communication
,”
Bell Syst. Tech. J.
,
27
(
3
), pp.
379
423
.10.1002/j.1538-7305.1948.tb01338.x
30.
Kapur
,
J. N.
, and
Kesavan
,
H. K.
,
1992
,
Entropy Optimization Principles With Applications
,
Academic
,
San Diego, CA
.
31.
Ghanem
,
R.
, and
Das
,
S.
,
2009
, “
Hybrid Representations of Coupled Nonparametric and Parametric Models for Dynamic Systems
,”
AIAA J.
,
47
(
4
), pp.
1035
1044
.10.2514/1.39591
32.
Bishop
,
C. M.
,
2006
,
Pattern Recognition and Machine Learning
,
Springer
, New York.
33.
Kim
,
J. O.
, and
Khosla
,
P. K.
,
1991
, “
Dexterity Measures for Design and Control of Manipulators
,” IEEE RSJ. International Workshop on Intelligent Robots and Systems, Osaka, Japan, pp.
758
763
.
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