Random Matrix Approach: Toward Probabilistic Formulation of the Manipulator Jacobian

Abstract

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 appro-priately 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 Ja-cobian 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 kine-matic configuration and complexity in the probabilistic formulation; (ii) achieving the uncertainty model using limited available information; (iii) taking into account the work-ing 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. [DOI: 10.1115/1.4027871]

Matrix-Based Characterization of the Motion and Wrench Uncertainties in Robotic Manipulators

“…Here, we present the RMT-based formulation of the manipulator Jacobian matrix developed in our earlier studies [10], [12]. In a system with motion uncertainty, the inverse differential kinematic equation can be considered as a general stochastic differential equation given by dq dt = f (q, ω, t)…”

“…Let us now assume that the upper bound on the Frobenius norm of the J k is known and denoted as u, i.e., J F ≤ u. We showed that [12] this constraint along with the model described by Eq. ( 16) leads to the inequality tr nΣ ≤ u 2 − J k 2 F .…”

Wrench Uncertainty Quantification and Reconfiguration Analysis in Loosely Interconnected Cooperative Systems

“…The robustness of the second-order approximation as well as its superiority compared to the first-order method was verified in different numerical examples. Sovizi et al [14][15][16][17][18] proposed an uncertainty characterization method in complex robotic manipulators based on the random matrix theory (RMT). It was shown that RMT-based models can appropriately capture the uncertainty only using limited information about the system variation.…”