Random Matrix Based Input Shaping Control of Uncertain Parallel Manipulators

Input shaping control techniques have proven to be very effective in improving performance and lowering actuation requirements for Linear Time-Invariant (LTI) systems. However, deployments tend to be very sensitive to variation of the system natural frequencies and damping ratios. This limits the application of input shaping schemes in the nonlinear systems (e.g., robotic manipulators) or systems with uncertain parameters. The main focus of this paper is to address the parametric uncertainty in the system and the design of an input shaping control based upon the estimated parameters. The variations induced by system non-linearity are tackled by the method of the linearization. The uncertainty is characterized by treating the linearized system mass matrix as a random matrix. This provides the probability density function of the eigenvalues of the underdamped system that are then used to design the input shaping control for the uncertain system. We believe that such a characterization (and desensitization to production variability) is especially important for successful deployment of newer generations of 3D printed robotic systems. This is verified via a Monte Carlo simulation — the statistics of the percent residual vibration verifies the superiority of the input shaping control scheme based on the estimated parameter values (compared to the one based on the nominal parameters). ∗Address all correspondence to this author. INTRODUCTION Over the past few decades, input shaping (IS) technique has shown some success in the control of the high-speed dynamical systems with high precision applications. A main challenge in the control of these systems is the controversy of the speed and system residual vibration. In a closed-loop control system, to achieve a fast response, it is desired to tune the controller gains such that the system remains underdamped while its stability is ensured. The payoff of the short rise time is the excess overshoot and long settling time of the residual vibration that must be eliminated or reduced due to the high required precision. The IS method dates back to the work by Singer and Seering [1] in 1990. As the name suggests, it shapes the input command using a series of the impulses such that the superposition of the responses corresponding to each impulse yields zero (or near zero) vibration. Refer to [2, 3] for a detailed discussion of the IS technique. Despite of its effectiveness in linear time invariant (LTI) systems, the standard IS method is highly sensitive to the variation of the system natural frequencies (ωi’s) and damping ratios (ζi’s), the parameters based on which the input shaper is designed. This limits the application of the standard IS method in nonlinear or uncertain systems where ωi and ζi are time varying or are not exactly known. Robust IS technique is one of the most common strategies to overcome the variation of the system natural frequencies. In addition to the zero residual vibration amplitude, Copyright c © 2014 by ASME it introduces sensitivity constraints that requires the derivatives of the (residual) vibration amplitude respect to the modeling natural frequency to be equal to zero (ZVD) [1, 4]. A measure of the robustness called specified insensitivity (SI) was also proposed by Singhose et al. [5]. In their work, a frequency sampling method [6] was used for (approximately) satisfying the derivative constraints. It was shown that the rise time increases with the degree of the robustness of the input shaper. In addition to the robust IS method, there exist different other treatments to the parameter variation problem including adaptive [7,8] and nonlinear [9, 10] input shapers that are often computationally demanding. Linearization method is a common way to tackle the system nonlinearity in IS control designs. For example, Singh et al. [11] implemented the IS method to control flexible/rigid link robots in which the natural frequencies and damping ratios were obtained from the linearized system. In a more recent study, Kozak et al. [12] examined different input shapers designed based on the linearized dynamics of the multi-degrees-of-freedom (multiDOF) parallel manipulators. Different input shaping strategies were also examined by Narayanan [13] for reduction of the residual vibration in the end effector (EE) of the 2-RRR and 3-RRR manipulators. However, methods to address the variability of the system parameters for the uncertain dynamical systems (especially in the context of IS control design) appears to not have been explored. One may solely use the robust IS schemes, however, as discussed above, they suffer from the increased rise time and the optimized design can only be achieved using the information obtained from the distribution of the uncertain parameters. The appropriate choice of the modeling frequency and the upper and lower bounds of the variations (for robust IS control designs) are some of the useful information that can be obtained from the random parameter probability density functions (pdf’s). Hence, in this paper we will examine the extension of IS control techniques to include uncertainty characterization. In particular, an IS control scheme will be designed for a parallel manipulator (PM) whose mass parameters are uncertain. Such situations can easily occur in real-world settings as: (i) timevarying loads are being transported; or (ii) low-cost techniques (such as desktop 3D printing) are employed for production. To characterize the uncertainty, the linearized system mass matrix is considered to be random and a probabilistic model is then constructed using random matrix theory. This approach provides a closed form pdf of the (undamped) eigenvalues of the random system that are used to design an IS control scheme for the uncertain PM. The rest of this paper is organized as follows. In the next section, we review the dynamics of a kinematically redundant planar PM (3-(P)RRR) [14] and derive the linearized closed-loop system equations of motion (EoM). Next, we construct the random matrix based probability model for linearized system with random mass matrix that, in turn, facilitates the characterization of the pdf of the random system eigenvalues. Afterward, the design of IS control scheme for the uncertain 3-(P)RRR PM, based on the mathematical tools provided in the preceding section, is discuessed. Moreover, using a Monte Carlo analysis, the performance of the designed IS control scheme is evaluated and compared to the results obtained from an IS control method based on the nominal parameter values. Finally a brief discussion and directions for future work are presented in the last section. KINEMATICALLY REDUNDANT 3-(P)RRR planar PM Parallel platforms are well suited for the high speed and precise manipulations and machining due to the higher stiffness and lower inertia compared to their serial counterparts. In this work, we consider a 4-DOF kinematically redundant planar PM (3-(P)RRR) to implement the IS control method for fast and accurate manipulations. The kinematics and dynamics of the 3(P)RRR PM [14], shown in Fig. 1, is reviewed in this section and its corresponding linearized closed-loop EoM are derived to be used in the implementation of the IS control strategies in the subsequent sections. The loop-closure constraints for ith serial limb of the PM can be written as fi = [ xGi + li1 cos(θi)+ li2 cos(θi +ψi)− xE − ri cos(φE +βi) yGi + li1 sin(θi)+ li2 sin(θi +ψi)− yE − ri sin(φE +βi) ]