A Random Matrix Approach to Manipulator Jacobian

Abstract

Traditional kinematic analysis of manipulators, built upon a deterministic articulated kinematic modeling often proves inadequate to capture uncertainties affecting the performance of the real robotic systems. While a probabilistic framework is necessary to characterize the system response variability, the random variable/vector based approaches are unable to effectively and efficiently characterize the system response uncertainties. Hence in this paper, we propose a random matrix formulation for the Jacobian matrix of a robotic system. It facilitates characterization of the uncertainty model using limited system information in addition to taking into account the structural inter-dependencies and kinematic complexity of the manipulator. The random Jacobian matrix is modeled such that it adopts a symmetric positive definite random perturbation matrix. The maximum entropy principle permits characterization of this perturbation matrix in the form of a Wishart distribution with specific parameters. Comparing to the random variable/vector based schemes, the benefits now include: incorporating the kinematic configuration and complexity in the probabilistic formulation, achieving the uncertainty model using limited system information (mean and dispersion parameter), and realizing a faster simulation process. A case study of a 6R serial manipulator (PUMA 560) is presented to highlight the critical aspects of the process. A Monte Carlo analysis is performed to capture the deviations of distal path from the desired trajectory and the statistical analysis on the realizations of the end effector position and orientation shows how the uncertainty propagates throughout the system.

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“…The reduced Jacobian matrix has right and left Eigen-vectors as (4) which is resulting in participation factors that can be defined as [21], [22].…”

“…Generally, the modal analysis relies on the matrix of diagonal eigen-values of matrix J R (). Thus, the reduced Jacobian matrix determines the condition of the given electrical system with respect to voltage stability [22].…”

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“…In this section, our random matrix formulation of the manipulator Jacobian matrix developed in [4], [5] is generalized to dynamic EOM of the robotic systems. The EOM of the deterministic mean dynamic system can be expressed as:…”

“…• Computational efficiency. In this paper, we generalize our RM-based uncertainty model for manipulator Jacobian matrix, developed in [4], [5], that is based on the non-parametric probabilistic model proposed by Soize [6], [7], to dynamic equations of motion (EOM) of the robotic systems. This provides a systematic approach that adequately satisfies the requirements noted above and can be beneficial in the design procedure of the complex dynamic systems (from the uncertainty perspective).…”

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.…”