A Random Matrix Approach to Manipulator Jacobian

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. ∗Address all correspondence to this author. INTRODUCTION Motion planning of robotic manipulators entailing either direct or inverse kinematics has traditionally been examined in deterministic settings. However, there are several sources of the uncertainties that can substantially affect the overall performance of the robot. These variations arise from inherently uncertain environment, limited range and resolutions of sensors that are also subjected to disturbing noises (sensing uncertainties), failure in accurate actuation (control uncertainties), modeling inaccuracy (simplifications) and computational limitations [1–3]. Hence, a systematic approach to characterize the variations of the system response is required. Many researchers have incorporated the uncertainty effects in the motion planning of the mobile robotic systems [4–9] that generally deals with global localization of the robot subjected to several workspace and uncertainty constraints. Jung and Ravani [10] considered the uncertainty in Jacobian matrix due to ignoring the actuator Jacobian (mapping from actuator space to joint space). They used neural network method for online compensation of the uncertainty in the Jacobian matrix. Cheah et al. [11] investigated the kinematic as well as dynamic uncertainties in tracking control of the robot end effector (EE). The uncertain kinematic and dynamic parameters of the system are updated based on an online updating law that results in convergence to the desired EE trajectory. Other works as [12–14] have also used the estimation of Jacobian matrix due to kinematic uncertainties. Wang and Chirikjian [15] developed a nonparametric second order approximation of error propagation whose one of its main applications is in the robot kinematic analysis and motion planning in presence of uncertainties. The main part of the paper discusses efficient approximation of mean and covariance of two uncertain frame density functions convolution, while the mean and covariance of each random frame are known. The propagation of spatial uncertainties covariance using first order approximation method has also been studied in an earlier work by Su and Lee [16]. The main objective of this paper is to develop a systematic way to characterize the propagation of uncertainties in a manipulator system. In this work, we focus on manipulator jointactuation uncertainties, which can be induced by joint flexibilities, joint clearances, imperfections, complex motor assemblies, etc. However, we assume that: (i) at each time step the joint space variables can be accurately observed by the sensors; and (ii), the structure of the manipulator is known and the exact EE position can be obtained corresponding to a given realization of random joint space configuration through forward kinematic relations. So, at each time step (tk) the joint space variables are known (accurately observed by the sensors) and the configuration of the system at subsequent time step (tk+1) is uncertain (due to actuation uncertainties). The novelty of our work arises from formulating the Jacobian matrix of the manipulator as a random matrix. The process noise acting on the uncertain system is modeled by randomness of the kinematic structure Jacobian matrix. This offers several significant advantages when compared to the random variable/vector based approaches. First, the proposed method incorporates the kinematic configuration (and complexity of the manipulator structure) within the probabilistic formulation that facilitates a better characterization of real system uncertainties. In contrast, in the random variable/vector based methods, probabilistic formulation of the system uncertainties does not depend on the structural configuration/complexity of the robot. Second, the random matrix formulation is based on maximum entropy method (developed in next sections) and uses the available system information. In the subsequent sections, we will show that the only information used to construct the probability model of the Jacobian matrix are its mean and a scalar value dispersion parameter that represents the level of uncertainty in the system. The mean can be determined by substituting the joint variables values into closed-form formulation of the Jacobian; while the scalar dispersion can be estimated from system knowledge or from experimental measurements. In addition, this method provides a faster simulation by eliminating the substitution of the realizations of the random joint variables into nonlinear functions. To the best of our knowledge, except the work we developed in [17], this is the first time that random matrix theory is employed to model the kinematic Jacobian matrix of the uncertain manipulator systems. The rest of this paper is organized as follows: In the next section, a basic background in manipulators forward and inverse kinematics is reviewed and numerical method to solve the inverse kinematic problem is presented. The probabilistic formulation of Jacobian matrix using random matrix theory is developed immediately afterwards. Next, PUMA 560 serial manipulator is specifically considered to implement the developed method. A three-dimensional path following task is assumed for the robot EE and the Monte Carlo numerical simulation is performed to obtain sufficient number of system response realizations that are statistically analyzed. Finally, the discussion is presented in the last section. FORWARD AND INVERSE KINEMATICS A routine problem in manipulators kinematic analysis is to express the end effector position and orientation as a function of joint space variables (forward kinematics) and vice versa (inverse kinematics). The forward kinematic relation may be stated as: