Dynamic modeling and control of a 6-DOF micro-vibration simulator

Highlights

• A 6 degree-of-freedom (DOF) micro-vibration simulator (6-MVS) is proposed.

• The dynamics equations, which consider the effect of the flexure joints, are derived.

• A robust proportional-integral control scheme is carried out.

Abstract

A micro-vibration simulator with multiple degrees of freedom is required for performance testing of sensitive instruments in a micro-vibration environment on-board spacecraft before launch. In this study, a novel 6-DOF micro-vibration simulator (6-MVS) is proposed, which can reproduce a micro-vibration environment with a wide bandwidth of disturbance frequencies. The complete inverse dynamic equations of the proposed 6-MVS are derived using the Kane method, which is very suitable for processing by computer. The validity of the derived dynamic equations is then verified by co-simulation. The structural performance of the 6-MVS is investigated using the finite element method. Based on this dynamic model, a robust proportional-integral (PI) control scheme is then performed. The control performance of the proposed controller is evaluated by co-simulation. The analysis and simulation results show that the proposed robust PI controller has excellent robustness and stability and the 6-MVS can exactly produce the required micro-vibration spectrum.

Introduction

There is an increasing requirement for precision pointing and extreme stability for current and forthcoming optical remote sensors, which have a larger aperture and a higher resolution imaging. The James Webb space telescope [1], terrestrial planet finder [2] and Space Interferometry Mission [3] are such examples where microarcsecond pointing and nanometer levels of motion stability are required. However, micro-vibrations generated by on-board motion equipment in spacecraft (for example, reaction/momentum wheel assemblies (R/MWA), cryo-coolers, thrusters, solar array drive mechanisms, etc.) can greatly degrade the performance of optical payloads with high pointing accuracy and stability [4].

Since micro-vibrations have characteristics of low amplitude, a wide frequency range, and multiple directions, current shake tables are unable to reproduce the required micro-vibration environment. Hostens et al. [5, 6] have proposed a six degree-of-freedom (DOF) vibration simulator, which can be used to generate high-amplitude and narrow-band vibrations. Park et al. [7] have developed a multiple-degree-of-freedom micro-vibration emulator to test jitter in spacecrafts, which can generate the disturbance spectrum of flight RWAs. However, this device has some coupling effects causing differences between the target input and the measured data responses along different axes. For space payloads, ground experiments are essential before launch, including image quality testing of the optical system in the micro-vibration environment. However, no suitable micro-vibration simulator exists that is qualified for this work. The usual solution is to adopt the actual disturbance resources or use dummy resources in the ground experiments. The R/MWA is generally regarded as one of the largest disturbance sources onboard a spacecraft [8]. Therefore, a real R/MWA is usually used for the micro-vibration test. However, it is uncommon to use all the flight R/MWAs to conduct the ground validating experiments because of scheduling issues or product assurance activities. Therefore, development of a micro-vibration shaking platform, which can replace the real flight R/MWAs, is considered to be an important adjunct to the development processes for space missions.

Parallel manipulators such as the Gough-Stewart platform (GSP) have been recently employed in various applications, because they have the advantages of high maneuverability, precision, high stiffness, and a large payload driven capability compared with serial manipulators. Li and Xu [9] have presented a three-prismatic-revolute-cylindrical parallel kinematic machine, and have investigated its dynamic modeling and robust control. A six degree-of-freedom parallel kinematic machine has been developed by Dong et al. [10], which is used for the motion simulation of hazardous chemical transportation. The Gough-Stewart platform, also known as the hexapod, is one of the most widely used parallel manipulators [11], [12], [13], [14], [15]] and its kinematics, dynamics and control problems have been studied by many researchers. Oftadeh et al. [16] have presented explicit dynamics formulation for the GSP and utilized the Lagrange method to verify the resulting dynamics equations. Dasgupta and Mruthyunjaya [17] have derived an inverse dynamic formulation using the Newton-Euler approach for the GSP, with frictional forces occurring in the joints; the mass of inertia of the pods was also taken into consideration in their study. Staicu [18, 19] has developed a recursive matrix approach in kinematics and dynamics modeling of parallel robots, which can reduce the number of equations and computation operations significantly. Jiang et al. [20], [21], [22], [23], [24]] have investigated an optimal design of the GSP with dynamic isotropy, as well as the influence of passive joint damping. Behrouz et al. [25] have developed a full parameter model of the GSP damped vibrations, which includes parametric expressions of the damped eigenfrequencies and the corresponding eigenvectors.

The control strategies for the parallel manipulator can be divided into two categories: control in the joint space and control in the task space [26]. The former control scheme can be readily employed in industry, but does not always guarantee high performance for parallel manipulators [27]. Kim et al. [28] have proposed a robust nonlinear control scheme in the joint space for an electro-hydraulic parallel manipulator based on the Lyapunov redesign method, but coupling is not taken into account, which should not be ignored for high performance tracking controllers. Wu et al. [29] presented an improved robust nonlinear controller, which is composed of the linear control part, nonlinear part and excitation compliment part. This proposed controller has the advantages of fine adjustability, low power consumption and a wide frequency range of isolation in all directions. But its pivotal objective is to attenuate the micro-vibrations. Superior control performance can potentially be provided using the control scheme in the task space. Han et al. have published a series of reports on robust controls for 6-DOF parallel manipulators, which include a computed force and velocity control, proportional plus derivative control and decoupling control schemes, etc. [30], [31], [32], [33], [34]]. Kim et al. [35] have proposed a robust nonlinear task space control with a friction estimator for a dynamoelectric GSP. However, most studies have focused mainly on displacement or velocity trajectory tracking control, while acceleration trajectory tracking control of a parallel manipulator with multiple degrees of freedom is still rare. Although some acceleration trajectory control strategies have by been reported in Refs. [36, 37], they are only suitable for shake tables with a single degree of freedom.

In this study, a 6-DOF micro-vibration simulator (6-MVS) has been developed, which can reproduce micro-vibrations with different amplitudes and frequencies. While a conventional GSP has stretched rods, the structural configuration of the proposed 6-MVS has been improved and has rods of fixed length. Since the mass of the rods in a conventional GSP is relatively heavy, a GSP may have low and closely-spaced local natural frequencies [38]. As a result, the natural frequencies of the overall system are reduced. The improved configuration simulator that is presented in this study can solve this problem, and its structural characteristics have been analyzed using the finite element method (FEM). Moreover, flexure joints are adopted to avoid nonlinear effects due to friction, backlash, and micro-impacts which are produced by traditional joints with bearings. The inverse dynamics models, which consider the effect of the flexure joints, were established using the Kane method. A co-simulation was then adopted to verify the validity of the dynamics models, which combined ADAMS with MATLAB/Simulink. Finally, a robust proportional-integral (PI) controller based on the inverse dynamics model was designed. This control strategy was designed for the acceleration control of the parallel manipulator, which considered the effects of uncertainties such as modeling errors, unknown loads, and parameter measurements. Its performance was analyzed in theory and simulation, including stability, precision and robustness of the proposed controller.