Steven D. Jones Suspa Inc., Grand Rapids, Ml 49548 Mem. ASME
[email protected]
A. Galip Ulsoy Department of Mectianical Engineering and Applied Mectianics, Ttie University of Mictiigan, Ann Arbor, Ml 48109-2125 Fellow ASME ulsoy@umicfi,edu
An Approach to Control Input Shaping With Application to Coordinate Measuring Machines Dimensional measurements obtained with Coordinate Measuring Machines (CMMs) are negatively affected by self-induced structural vibrations. In this paper, a control strategy that reduces the structural vibrations in a CMM is outlined and experimentally demonstrated. The control strategy, designated the Feedforward Filter, is developed by establishing the relationship between contemporary controller input shaping techniques and traditional notch filtering methods. Issues on both robustness and multiple mode vibrations are addressed. Controller input development takes place in the discrete time domain. This method provides results identical to those for optimal command input preshaping obtained through non-linear programming methods and requires considerably less computational effort. Experimental results show a 50 percent reduction in the peak-to-peak magnitude of structural vibrations as compared to unshaped bang-bang trajectories.
Introduction Coordinate Measuring Machines (CMMs) are being integrated into mass production operations to supply dimensional feedback for near real-time process control purposes. CMMs have traditionally been viewed as low productivity, laboratory machines since their massive mechanical structures must be moved slowly to reduce the structural vibrations that result in poor measurement quality. In this paper, a motion control strategy, which reduces vibrations in CMMs that occur at higher operating speeds, is developed and experimentally evaluated. During the past decade, numerous researchers have directed their efforts to modeling flexible structures and developing control strategies to manage the complexities of positioning nonrigid structures such as flexible beams, robot arms, disk drive heads, etc. CMMs are a special class of robots, and nearly all of the research on flexible robots has been directed toward those with non-Cartesian designs (containing at least one revolute joint). High accuracy CMMs are exclusively prismatic (three linear axes) in design so that equal spatial resolution can be maintained throughout the work zone. The basic difference in structural design has a number of significant effects on controller design. First, changes in configuration of non-Cartesian robots result in large changes in inertia, while prismatic structures like CMMs typically utilize counter balances to eliminate any effects of changing inertia with respect to the controller. Second, the vibrations that are detrimental to CMM accuracy and repeatability are usually of an order or two less in magnitude than those commonly investigated with non-Cartesian robots. In addition, the lack of accuracy or repeatability in a CMM is the result of compliance between the linear transducer and the probing mechanism, not a difference in command position and actual position, which is the focus of much of the robotics research. Finally, varying loads, which have significant effects on controller performance in machine tools and robots, are not an issue with CMMs. Because of these differences, a CMM control strategy, called a Feedforward Filter (FF Filter), that preshapes the command input to existing CMM controllers has been developed here. The preshaping strategy does not require modification of the Contributed by tlie Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript
received by the Dynamic Systems and Control Division July 1994. Associate Technical Editor: Tsu-Chin Tsao.
control loop, but only a transformation of the reference or command inputs to the controller. This simple method does not require the introduction of additional sensors or actuators, and avoids the need for existing actuators to deliver forces in a frequency range coincident with the CMMs structural resonant frequency. It provides results, with considerably less computation effort, identical to those obtained using Input Command Preshaping (ICP); a technique for reducing structural vibrations proposed by Meckl and Seering, Singer and Seering and Hyde and Seering [7, 11, 4 ] . ICP is based on hnear systems theory and is used to shape the inputs to a system to eliminate unwanted vibration. Issues related to shaping inputs were first addressed by Smith [12] through a technique called posicast. Smith's work was later extended to systems with multiple modes by Cook [2], Aspinwall [1] investigated command shaping acceleration profiles to minimize the residual responses of flexible structures. Experimental results demonstrating the control of flexible structures utilizing an adaptive feedback control were shown by Tzes et al. [13]. These ideas were then extended by Hillsley and Yurkovich [3] by developing techniques to combine endpoint acceleration and joint information feedback. Analysis in the discrete-time domain and the development of a digital filter with an arbitrary sample rate was performed by Murphy and Watanabe [8]. Applications to CMM vibration control using the ICP strategy have recently been demonstrated by Seth et al. [7] and Singhose et al. [10]. A comprehensive list of research publications on utilizing shaping techniques can be found on the Internet at http://web.mit.edu/shaping/www/. CMMs are ideal candidates for application of vibration reduction strategies like ICP or FF Filtering. The main reason is that configuration changes for well designed prismatic positioning devices, like CMMs, may not result in significant variations in resonant frequencies. Another reason is that time delays, usually attributed to plant inversion strategies, do not create a problem since inputs to the system are known in advance (i.e., the path program is stored in computer's memory prior to execution). Review of Input Command Preshaping The basic idea behind Input Command Preshaping as presented by Hyde and Seering [3] is appropriately constructing inputs to a physical system with some inherent compliance to eliminate unwanted vibrations. The input commands are preshaped by constructing a series of impulses to achieve the desired response. A train of impulses can suppress residual vibra-
242 / Vol. 121, JUNE 1999
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JL ^
1+K
T=0
.
1+K
T=
TIME
avlK"^ Fig. 1
Preshaping Robustness Issues. The two impulse sequence will cancel unwanted structural vibrations when both the system damping ratio ( Q and natural frequency (w) are known exactly. In actual applications these parameters are rarely known precisely and often vary due to changes in structural configurations. The issue of robustness is addressed by differentiating the governing equations (3) and (4) with respect to LO (which may be shown to be equivalent to differentiating with respect to Q . The differentiated equations are then introduced as constraints on residual vibration in the governing equations which result in an additional impulse and a sequence that is less sensitive to changes in the parameters.
A
Graphical illustration of two impulse sequence
tion by forcing the resulting amplitude of vibration to zero. In order to construct the preshaped command for a single mode of vibration, one begins by analyzing the response of a second order system to an impulse: yi{t) = A,e-«-
