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Research Papers

Random Matrix Approach: Toward Probabilistic Formulation of the Manipulator Jacobian

[+] Author and Article Information
Javad Sovizi

Mechanical and Aerospace Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: javadsov@buffalo.edu

Sonjoy Das

Assistant Professor
Mechanical and Aerospace Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: sonjoy@buffalo.edu

Venkat Krovi

Associate Professor
Mechanical and Aerospace Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: vkrovi@buffalo.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received December 19, 2013; final manuscript received June 13, 2014; published online October 21, 2014. Assoc. Editor: Jongeun Choi.

J. Dyn. Sys., Meas., Control 137(3), 031012 (Oct 21, 2014) (10 pages) Paper No: DS-13-1521; doi: 10.1115/1.4027871 History: Received December 19, 2013; Revised June 13, 2014

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.

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Figures

Grahic Jump Location
Fig. 1

2R serial chain: (a) kinematic configuration, (b) desired trajectory and Yoshikawa manipulability (hot colors represent higher manipulability)

Grahic Jump Location
Fig. 2

Standard deviation of ωθ1 and ωθ2 (Wishart perturbation matrix approach) with ordinates on the left axis. The Jacobian determinant, EE, and joint velocities of the nominal (deterministic) manipulator over the time of the simulation with ordinates on the right axis.

Grahic Jump Location
Fig. 3

KS joint density estimate of ωθ1 and ωθ2 at t = 1, 5, 7.5, and 9 s—Wishart perturbation matrix approach

Grahic Jump Location
Fig. 4

Ellipsoids of covariance matrix of ω at different time points (Wishart perturbation matrix approach). Note that the center of ellipsoids are located at the EE position in the corresponding time point. The contour plot represents Yoshikawa measure of the manipulability.

Grahic Jump Location
Fig. 5

States uncertainty propagations (Wishart perturbation matrix approach). Hot colors represent higher densities.

Grahic Jump Location
Fig. 6

Standard deviation of the system states versus the time (Wishart perturbation matrix approach)

Grahic Jump Location
Fig. 7

Standard deviation of ωθ1 and ωθ2 (Gaussian Jacobian matrix approach) with ordinates on the left axis. The Jacobian determinant, EE, and joints velocities of the nominal (deterministic) manipulator over the time of the simulation with ordinates on the right axis.

Grahic Jump Location
Fig. 8

Ellipsoids of calculated Σ at different time points (Gaussian Jacobian matrix approach). Note that the center of ellipsoids are located at the EE position in the corresponding time point. The contour plot represents Jacobian Frobenius norm measure of the manipulability.

Grahic Jump Location
Fig. 9

KS joint density estimate of ωθ1 and ωθ2 at t = 1, 5, 7.5, and 9 s—Gaussian Jacobian matrix approach

Grahic Jump Location
Fig. 10

States uncertainty propagations (Gaussian Jacobian matrix approach). Hot colors represent higher densities.

Grahic Jump Location
Fig. 11

Standard deviation of the system states versus the time (Gaussian Jacobian matrix approach)

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