# Innovations in Motion Control: Part II—Vibration Control

People and animals can move quickly and accurately without the rigid or precise (i.e., expensive) structural characteristics of typical numerically controlled motion systems (e.g., robots). In this, the second in a three-part series explaining how modern computer control emulates these properties, we’ll consider using a favorable reference signal to minimize a mechanical system’s vibrations.

A classic example of this reference motion selection technique is the natural control that crane operators use (now embedded in some crane control systems) to minimize the load’s vibrations at the cables’ ends. The motion required at the crane’s top (where the motors for horizontal motion are located) differs significantly from that desired for the load at the cable’s end. However, if the operator performs a move "just right," the motions at both ends of the cable come to rest simultaneously.

In fact, minimizing whole-system vibration, particularly at the end of arm (EOA), by shaping this motor motion profile is a fairly well-recognized technology.1 It’s sometimes called command, or input, shaping.2

## How It’s Done

Consider the simple case of controlling a two-mass system’s position (Figure 1). As per standard practice, we’ve chosen a smooth, S-shaped curve for the reference signal XR. This profile, used for the case of starting and stopping at rest, is often one of limited acceleration and velocity (and perhaps jerk, rate of change of acceleration.) This shape accomplishes two things relative to a simple step change in commanded position. First, it causes less machine terminal vibration. Second, it represents an achievable motion. A step isn’t achievable in any real system, so why call for it as the reference position?

Now, suppose we know the natural frequencies and damping ratios (both often very small) for the mechanical structure’s elastic modes. We could put what’s called a command shaping filter after the reference signal generator (Figure 1), such that XR’s components contain little if any of those frequencies. (Communications people think of this as a notched filter.) We’d then expect the system’s motion to have little vibration, reducing or eliminating settling times accordingly.

But a very important problem would remain. Classic notched filters have a significant time-spreading effect: The curve, although still S-shaped, would lengthen over time. Further, a servo system combined with the electromechanical structure has oscillation frequencies that depend on the feedback system’s own gains. That is, the system’s frequencies aren’t just the mechanics’ natural vibration modes. If the feedback system works well, however, these oscillations are also significantly damped.

Among several known notched-filter improvements is the optimal arbitrary time-delay (OAT) filter.

## The OAT Filter

Essentially, a command shaping filter molds the desired motion to an elastic system without exciting the system modes’ resonance. For real systems, the fundamental—and most significant—frequencies are usually less than 100 Hz. In Figure 1, the usual S-shaped commanded motion XR is reshaped to XRF. The command shaper takes the form of a finite impulse response filter; its parameters are determined by the resonant frequencies and damping ratios of the flexible system’s undesired elastic modes. For a linear time-invariant system, the command shaper forms its output by adding together a scaled input function and delayed multiples of that function. This sum cancels the residual vibrations caused by all the terms taken together.

XRF(now)=A0XR(now)+A1XR(now – T)+A2XR(now – 2T)

This particular technique—the OAT filter—requires three terms for single elastic mode cancellation: the scaled input and two of its delays. Calculating the scaling factors A0, A1, and A2 requires not only knowledge of the frequencies and damping ratios that occur for the structure/feedback control system combination but also an arbitrary specification of the time delays T and 2T. If the command shaper coefficients are properly chosen, the filter can use any T to cancel the given resonance poles.3

Because all modern control systems are in the discrete time domain, T must be chosen as an integer multiple of the controller’s sampling time. The OAT filter’s freedom in choosing the time delay eases its implementation in a digital control system.

However, there’s a slight delay, 2T, in achieving the S-shaped motion. Theoretically, this delay can be two control system sampling intervals. But as a practical matter, a delay of approximately one cycle of the damped natural frequency is often best. Thus, if we were trying to kill a 50-Hz oscillation that would otherwise exist, there’d be a 20-msec increase in XRF’s effective length.

Figure 1. Two-mass control system with OAT filter.

## Combined OAT Filter and Learning

It’s possible to greatly reduce the tracking-error amplitude based on learned feed forward ("Innovations in Motion Control: Learned Effort," Motion Control, March/April 2002). What if we use both learning and command shaping? The two methods are complementary; both decrease the time to make a motion. By learning, we can not only track a commanded motion much better but also cancel some of its vibrations. Using the OAT filter reduces all vibrations’ magnitude.

We can observe an interesting phenomenon in Figure 1’s classical form of servo control with the PD control on the motor. Our commanded motion is the motor position, but it’s the EOA’s motion we actually want to control—thus, the use of very rigid structures. If we use the standard S-shaped curve without the OAT filter, adding learning actually worsens the EOA’s oscillations, although the motor’s motion is faster and more accurate. The vibratory increase arises from our very nearly achieving the motor’s commanded motion. As indicated earlier, that motion isn’t designed to minimize vibrations; a "sloppier" motor position control actually reduces both residual vibrations and the EOA’s settling time. Most real systems detune the motor feedback loop to account for this.

Figure 2 plots results of an experimental system with a linear motor driving a single axis.4 An arm, fastened to the armature, has an accelerometer at its end (Figure 2a). Note how adding an OAT filter (Figure 2b) greatly reduces the settling time. Further, learned feed forward (Figure 2c) worsens accelerations vis-à-vis proportional-integral-derivative control only. However, the armature’s tracking error (not shown) has been reduced by a factor of greater than 10. When both an OAT filter and learning are used, we get low vibration, negligible settling time (Figure 2d), and nearly perfect tracking (not shown).

Based on achieving a programmed motion of the motor, combining an OAT filter with learning produces results superior to using either the PD system or the OAT filter alone. In this case, the frequency and damping ratio used in the OAT filter is that of the second mass and spring, as a system, because the motor mass (armature) is effectively commanded to follow the reference signal the OAT filter provides.

We could have partially achieved this result by making the motor PD control much tighter than normal and using the OAT filter alone—that is, the OAT filter minimizes the need to detune the feedback loop. However, practical systems can’t have infinite PD gains that are theoretically possible in the ideal analog control of a lumped mass.

Friction is a subtle consideration in Figure 1; it makes the system nonlinear and thus noncompliant with the command shaping methods’ requirements. However, we can consider the system to be linear if the motor position can exactly track its desired motion. Hence, we have an additional reason to combine the learned open-loop effort with the OAT filter. MC

Figure 2. Plots of experimental results: (a) original system; (b) OAT filter added; (c) learned feed forward added; (d) OAT filter and learned feed forward both added.

 Part III will provide modern motion control system techniques related to relative position estimation.

 Acknowledgements The authors have been fortunate to be associated with a very talented pool of Georgia Tech graduate students and colleagues who, over the years, have shown us how to build motion machines much more intelligently. Many companies and government agencies have played a part, but we most recently recognize the support of the National Center for Manufacturing Science, Visteon, and CAMotion, Inc., a Georgia Tech spin-off company. Dr. Nader Sadegh has caused us to recognize the importance of learned feed-forward effort.

#### Make Contact!

Steve L. Dickerson, Sc.D., is chairman of CAMotion, Inc. Contact him at 813 Ferst Drive, Atlanta, GA 30332-0405; tel: (404) 894-3255; fax: (404) 894-9342; www.camotion.com. Wayne J. Book, Ph.D., is HUSCO/Ramerez Professor of Fluid Power and Motion Control. Contact him at the Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405; tel: (404) 894-3247; fax: (404) 894-9342; www.gatech.edu.

1Magee, D. P. and W. J. Book, "Eliminating Multiple Modes of Vibration in a Flexible Manipulator," Proceedings of the 1993 IEEE International Robotics and Automation Conference, Vol. 2, pp. 474–479, Atlanta, 2–7 May 1993.

2Singer, N.C. and W. P. Seering, "Preshaping Command Inputs to Reduce System Vibration," ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 112, No. 1, March 1990, pp. 76–82.

3McGee, D. P. and W. J. Book, "Optimal Arbitrary Time-Delay Filter and Method to Minimize Unwanted System Dynamics," Patent No. 6,078,844, 20 June 2000.

4Rhim, S., A. Hu, N. Sadegh, and W. J. Book, "Combining A Multirate Repetitive Learning Controller With Command Shaping For Improved Flexible Manipulator Control," ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 123, No. 3, September 2001, pp. 385–390.