The Combined Use of Input Shaping and Nonlinear Vibration Absorbers

IFAC

Copyright @ IF AC Nonlinear Control Systems, SI. Petersburg, Russia, 200 I

c:

0

~

Publications www.elsevier.comllocate/ifac

THE COMBINED USE OF INPUT SHAPING AND NONLINEAR VIBRATION ABSORBERS William Singhose

Joel Fortgang

Department of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332 USA FAX 404-894-9342 [email protected] Abstract: Systems that exhibit flexible dynamics are susceptible to vibration caused by changes in the reference command, as well as, from external disturbances. Feedback control is an obvious choice to deal with these vibrations, but in many cases it is insufficient. Input shaping has proven very beneficial for reducing vibration that is caused by changes in the reference command. However, input shaping does not deal with vibration induced by external disturbances . On the other hand, vibration absorbers are an open-loop method that have long been used to decrease vibration from external sources, particularly sinusoidal disturbances. In this paper, nonlinear vibration absorbers are designed to reduce vibration from step disturbances. Furthermore, input shaping is combined with the absorbers in order to deal with the vibration induced by the reference command. The usefulness of this combined approach for dealing with both external and reference command disturbances is demonstrated with computer simulations. Copyright ~ 2001 IFAC Keywords: vibration, command signals, inputs, vibration absorbers.

C *u-Ac.

1. INTRODUCTION The vibration of flexible systems often limits speed and accuracy. The vibration of such systems is usually caused by changes in the reference command or from external disturbances. If the system dynamics are known, commands can be generated that will cancel the vibration from the system's flexible modes (Smith, 1957; Bhat and Miu, 1990; Singer and Seering, 1990). Input shaping is one such command generation scheme that is implemented by convolving a sequence of impulses with the command signal, as shown in Figure 1. The result of the convolution is used as the new reference command for the system. In general, the input shaper can contain any number of impulses and the reference command can be any function. Furtherm0re, input shaping can be used in conjunction with any type of feedback controller (Kenison and Singhose, 2000; Ooten and Singhose, 2000). A block diagram for the case with an input shaper outside the feedback loop and a ProportionalDerivative (PD) feedback controller is shown in Figure 2.

L

10

!J,.

10

!J,.

Figure 1. Input Shaping Process. controller was successfully implemented on a fivebar linkage manipulator in (Drapeau and Wang, 1993). When the input shaping is implemented outside of the feedback loop, its vibration-reducing effects do not work on external disturbances. For example, the vibration induced by the disturbances, D and W, shown in Figure 2, would not be reduced by the input shaper. The vibration must be dealt with by the feedback controller. Inserting the input shaping inside the feedback loop creates a time delay within the loop, thereby introducing a stability issue(Drapeau and Wang, 1993). Another possible solution is to add a vibration absorber to the plant dynamics, Gp (Hartog, 1985). A model for this augmented system under PD control is shown in Figure 3. Vibration absorbers, or mass dampers cancel unwanted dynamic responses of a

Input shaping has been implemented on a variety of systems. The performance of long-reach manipulators (Magee and Book, 1995), cranes (Noakes and Jansen, 1992; Fedderna, 1993; Singer et aI., 1997), and coordinate measuring machines (Singhose et aI., 1996a; Jones and Ulsoy, 1999) was improved with input shaping. In particular, input shaping has shown great promise when used in conjunction with feedback control on flexible robot arms . A combined input shaper and feedback

y

Figure 2. Input Shaper and Feedback Controller. 345

Section 2 gives a brief review of input shaping, while Section 3 describes the process used for designing the nonlinear vibration absorber proposed here. Section 4 uses computer simulations to demonstrate the effectiveness of combining input shaping with the Finally, conclusions are vibration absorber. presented in the last section.

Absorber

Primary System

2. REVIEW OF INPUT SHAPING The amplitudes and time locations of the impulses in an input shaper are determined by solving a set of constraint equations. Most types of constraints can be categorized as residual vibration constraints, robustness constraints, constraints on the impulse amplitudes, and the requirement of time optimality.

Figure 3. System Under PD Control with Nonlinear Vibration Absorber. system by introducing a new vibration mode. One of the earliest applications of this principle occurred in 1883 on the British warship the HMS Inflexible which used a volume of water in its hull as the secondary oscillating mass (Hunt, 1979). The deck motion resulting from the near sinusoidal forcing of the ocean swells was reduced by the counteracting motion of the water volume. Since that time, the theory, as well as the application of vibration absorbers has advanced. An early vibration-absorber patent granted to Frahm in 1911 also dealt with controlling oscillation in ships(Fraham, 1911). The early theory only considered the ideal undamped case(Hartog, 1985). With the advent of computational techniques it has been possible to design an absorber for a variety of situations and criteria, including damping and non-linearity in both the absorber and primary system (Pennestri, 1998). Implementation of multiple absorbers and active absorbers whose parameters are controlled by an external source have also been developed(Chao et aI., 1997).

To constrain the residual vibration , we need an expression for the residual vibration amplitude as a function of an impulse sequence. If we assume the system can be modeled as a second-order harmonic oscillator (this analysis can be extended to a superposition of second-order systems) then the system response from a single impulse is(Bolz and Tuve, 1973): yo(t) ==

[~~~~2 e-'W(t-tO)}in(w~(t - to)),(1)

where Ao is the amplitude of the impulse, to is the time the impulse is applied, w is the natural frequency, and S is the damping ratio. The response from a sequence of impulses is just a superposition of the response given in (1). Using the simplification:

wd ==

W~I- S2 ,

(2)

the response to a sequence of impulses after the time of the last impulse is:

Yl;(t)=

The use of vibration absorbers has crossed a broad range of applications, including architecture: chimneys, bridges, windmills, communication towers ; rotational machinery: pumps, generators, engines; and consumer goods: dishwashers, refrigerators, passenger seating (Hunt, 1979; Sun et aI., 1995). However the application to robotics and pre-planned trajectories has been limited. Traditionally absorbers were designed to cope with periodic, generally sinusoidal excitations, with few excursions into other areas. Absorbers have been designed for random excitations (Nigam and Narayanan, 1994), as well as for the "time optimal case" in (Bartel and Krauter, 1971) defined by energy dissipation for an impulse response , but the step trajectory application addressed here has not been fully explored.

i[~AiliJ 2 e-sCtJ(t-t;l]Sin(liJd(t-t& i=1

(3)

1-'

Given (3), an expression for the amplitude of residual vibration can be formed by using the trigonometric identity: n

L Bi sin(

wt + !Pi) ==

Ar sin( Wt + lI'),

(4)

i=l

where,

A~ (~BiOO,(~;)r +(~Bi"n(~;)r . =

(5)

Given the expression in (3) , the coefficients in the summation of (4) are:

Ai w -'liJ(t-ti) I-Re .

B. -

This paper will deal with the application of a classic vibration absorber scheme, the non-linear vibration absorber. Many non-linear absorber approaches have been considered, including bumpers (Gonsalves et aI., 1993), non-linear dampers , and cubic springs (Shaw et aI., 1989; Queini, 1999). In this study a cubic spring is employed in order to deal with a common occurrence in manufacturing and robotic applications , a step trajectory or disturbance . The absorber approach is augmented with Input Shaping to provide good vibration reduction for both external and command-induced vibration.

(6)

To calculate the residual vibration amplitude, we evaluate (5) at the time of the last impulse, t = tn . Substituting (6) into (5) gives:

Ar == ~ e-SliJtn ~r-[c-(w-,-s)--:C-+-[s-(w-,-S)-c-]2

f

-y 1- S2

(7)

where , n

c( w, 0 == L Aie'(j)[i cos( Wdti) and

(8)

i=l n

S(w,S)== LAieSliJti sin(wdtd . i=l

346

(9)

One way to avoid the problem of large amplitude impulses is to require all of the impulses to have positive values . If the amplitudes are all positive, then each individual impulse must be less than one so that (l2) can be satisfied. In this paper only positive amplitudes are used in the input shaper.

25

c .:2 ....~ 20 ,,

""

,Q

>

..,, ,

15

CQ

S 10

,,

c~ ~

"" ~

=-

,, ,,

..,

~

5

,, ,

,

,,

. , . : . . . ....

In practice, ZV shapers can be very sensitive to modeling errors (Tallman and Smith, 1958). To demonstrate this effect, the amplitude of residual vibration can be plotted as a function of a modeling error. The solid line in Figure 4 shows such a sensitivity curve for the ZV shaper. The vertical axis is the percentage vibration given by (l1), while the horizontal axis is the normalized frequency formed by dividing the actual frequency of the system, ffia , by the modeling frequency , wm' Notice that the residual vibration increases rapidly as the actual frequency deviates from the modeling frequency. The robustness can be measured quantitatively by measuring the width of the curve at some low level of vibration . This non-dimensional robustness measure is called the shaper's frequency insensitivity, loo. The 5% insensitivity has been labeled in Figure 4.

' 5%

0 0.7

0.8

0.9

1.0

l.l

Normalized Frequency

1.2

1.3

(roJ
Figure 4 . Sensitivity Curves for ZV and ZVD Shapers.

To express the amplitude in a non-dimensional manner, (7) can be divided by the amplitude of residual vibration from a single impulse of unity magnitude. The resulting percentage residual vibration expression gives us the ratio of vibration with input shaping to that without input shaping. By expressing the constraint in this way, we can set the residual vibration to a desired percentage of the amount that occurs without input shaping . The amplitude of residual vibration from a single unitymagnitude impulse applied at time zero is simply: w (10)

In order to increase the robustness of the input shaping process, the shaper must satisfy additional constraints . One such constraint sets the derivative of (11), with respect to frequency, equal to zero (Singer and Seering, 1990) . That is:

o= ~ V(w,S)

(14) dw When (l1) - (l4) are satisfied with V = 0, the result is a Zero Vibration and Derivative (ZVD) shaper. By comparing the 5% insensitivities shown in Figure 4, it can be concluded that the ZVD shaper is significantly more robust to modeling errors than the ZV shaper.

Dividing (7) by (10) yields the percentage vibration equation:

V(w ,S)=Ar =e-(wtn~[c(w, s)f+[s(w,s)f Ai

(11)

If V(w,S) is set equal to zero at the mode ling parameters, (Wm, Sm), then a shaper that satisfies the equation is called a Zero Vibration (ZV) shaper.

Other types of robustness constraints have also been proposed (Singer and Seering, 1992; Singhose et aI., 1994; Singhose et aI., 1996b). One of these robust methods, called Specified-Insensitivity (SI) shaping, suppresses a specified range of frequencies . The most straightforward method for generating a shaper with specified insensitivity to frequency errors is the technique of frequency sampling (Singer and Seering, 1992; Singhose et aI. , 1996b). This method requires repeated use of the vibration amplitude equation, (11). In each case, V(w,S) is set less than or equal to a tolerable level of vibration, Vtor

A constraint must be applied to ensure that the shaped command produces the same rigid-body motion as the unshaped command. To satisfy this requirement the impulse amplitudes must sum to one: (12) i=l

Due to the transcendental nature of (11), there will be multiple possible solutions . To make the solution time optimal, subject to the residual vibration constraints , the shaper duration must be made as short as possible. Therefore , the time optimality requirement is : (13) min(t n ) ,

• V(c(ws. I V;o! ~ e-~O)sIn t;)t" + (S(w s. t;)t . s = I..m'

(15)

where Ws represents the m unique frequencies at which the vibration is limited.

where tn is the time of the final impulse. For example, if the shaper needs to suppress vibration for frequency errors of ±20% , then the constraint equations limit the vibration to below V tol at specific frequencies between 0.8wm and 1.2wm. In effect, the amplitude of residual vibration is constrained at a regular sampling period over the frequency interval. This procedure is illustrated in Figure 5 for V tol = 5% . Given that the SI shapers can be designed to suppress any desired range of frequencies , they prove advantageous over the ZVD shapers. This flexibility in the robustness properties of the SI shapers is

The input shaper that satisfies the constraints discussed above with minimum time duration is undefined because the impulse amplitudes are driven to positive and negative infinity by (13). Large amplitude impulses produce a shaped input with correspondingly large values, and at some point actuator saturation will occur. Therefore. it is necessary to place a limit on the amplitude of the impulses.

347

35 1.6

= = 30

- - ·Linear Absorber

~

...all

~

25

III

;> 20 III

-...= QQ

all

15

III 101

10

III

=-

1.2

'"= 8. 0.8 '"

III

Cl::

0.4

5 0

0.8 1.2 Normalized Frequency (roa/C%t) Figure 5. Specified-Insensitivity Shaper. 35

-ZVD

- S I (I,., =0.5) --·SI (I,., =0.7)

= 30 =

~

...

all

25

~

0.7

;> 20 III

QQ

S

j4-_ _

15

/

I

1.4

I

~0.:::.5_ _--,,>\;./

III 101 III

=-

5

10

15

20

25

30

Time Figure 7: Effect of Linear Vibration Absorber. performance index that is minimized is the peak overshoot, Mp, and the settling time, ls, defined as the time when the system settles to within five percent of the original system's peak overshoot. The original system's Mp is used instead of the absorber system peak overshoot in order to de-couple the peak overshoot and settling time characteristics of the response.

= 10 ...

0

Previous work has shown that absorber effectiveness increases with its mass (Sun et aI., 1995). However this increase in mass leads to an increase in actuator effort. Therefore, the mass can be used as a given parameter defined by the performance specifications of the driving hardware. What remains is to look through the absorber variables and compute the response characteristics. Once these are known for a variety of possible absorbers, a weighting function is formed using the normalized improvement of the peak overshoot and settling time: J=ls+a.Mp (16) The weighting in this cost function can be chosen to meet desired performance requirements. Here a is set equal to 0.1, meaning more value is placed on the settling time than the peak overshoot. Once the absorber that best fits these requirements is found, the size of the search mesh is reduced while at the same time making the mesh finer. This produces a rapid convergence to an optimum solution.

5 0 0.8 1.2 1.4 Normalized Frequency (CIlaIronJ Figure 6. ZVD and SI Sensitivity Curves. 0.6

demonstrated in Figure 6, where the sensitivity curves for 10) = 0.5 and 0.7 are compared to that of the ZVD shaper. Any shaped command will have its rise time increased by the duration of the shaper as was shown in Figure 1. Because the duration of the ZVD shaper is twice that of the ZV shaper, the ZVD shaper increases the rise time more than the ZV shaper. This increased rise time is the price that is paid for the increased robustness to modeling errors. With the SI shapers, increasing robustness increases rise time in a non linear manner. This leads to certain operating points that are advantageous (Singhose et al ., 1996b).

Once the optimal linear absorber is found , the nonlinear component of the spring is added. In this case, a cubic spring constant is used, resulting in a spring force of: 3

3. NONLINEAR VffiRATION ABSORTBER DESIGN The parameters for a nonlinear vibration absorber proposed here are computed through the implementation of a brute-force optimization technique. First, the optimum linear absorber is calculated for the desired point-to-point motion, then this optimum linear absorber is augmented with a non linear component in the spring to improve the response even further. The inclusion of this absorber with an input-shaping scheme will enable the system to cope with both reference command and external disturbances.

f's = kIll + k3 tl

(17)

where 6 is the spring deflection. Another search is then performed with this non-linear coefficient being the variable. Once again, the characteristics of the step response are calculated and the non-linear absorber parameters are selected. As an example, consider the case where the baseline system is characterized by ml=l00, kl=l00, and a damping ratio, 1;1 = 0.1. Then the optimal linear absorber with a 7% mass would be described by r2=0.855 (the ratio between the natural frequencies of the absorber and primary system) and damping ratio, 1;2=0.281. The improved settling time with this linear absorber is shown in Figure 7.

To determine the optirnallinear absorber, the primary system step response is first calculated. This is used as a standard to compare other responses. Once the parameters of the response are known, particularly peak overshoot and settling time, an absorber is added to the primary mass as shown in Figure 3. A systematic scan of possible absorber parameters is then performed in order to find the best one. The variables of this linear absorber are its spring constant k, damping b, and mass, m2. The

When the nonlinear term is added to the absorber, the response is highly dependent on the amplitude of nonlinearity. As the nonlinear term is increased, the

348

1.04 1.03

-No Absorber - - • Linear Absorber

1.6

1.02 ~

1.01

8 c. '" ~

0.99

~

1.2

8-

~ 0.8

0.98

0.4

0.97

o

0.96 10

20

15

25

30

35

40

Time Figure 8: Effect of Spring Nonlinearity.

o

15 20 25 Time Figure 10: Effect of Adding SI Shaping. 5

10

30

1.2

1.15

- - . Linear Absorber -Nonlinear Absorber

1.1 ~ 1.05

Q;

8.'"

8.

~

0.8 - _. Linear Absorber + Shapiog

0.6

~

Q;

-Nonlinear Absorber + Shaping

Cl: 0.4

Cl: 0.95

0.2

0.9

o

0.85 10

15

20 Time

25

o

15 20 25 30 Time Figure 11: Comparison of Linear and Nonlinear Shaped Responses.

30

Figure 9: Improved Settling with Nonlinear Absorber.

5

10

linear modeling, it is possible that using It ID conjunction with the non linear absorber might not work as well. However, Figure 11 shows that the linear and non linear shaped responses are virtually identical. The non linearity does not greatly effect the shaped response because the shaper was designed to suppress a range of frequenc ies. This robustness allows the shaping to work well with the nonlinear absorber. The resulting solution provides the effectiveness of a nonlinear vibration absorber for damping out external disturbances and the effectiveness of input shaping for reducing vibrations induced by the reference command.

initial effect is positive in that the system settles faster. However, if the non linear term is made too large, then the response takes longer to settle. This dependence on the non linear term is shown in Figure 8, where the value of the non linear term is increased from 0 to 0.5 in steps of 0.05. Based on the results in Figure 8, a good choice for the nonlinear term is 0.4. Figure 9 shows the improved response when this nonlinear absorber is used. 4. COMBINED USE OF INPUT SHPAING AND A NON-LINEAR VffiRATION ABSORBER Although the addition of a vibration absorber can improve settling time after step disturbances, input shaping provides a much better way to decrease vibration that results from step changes (or any other types of changes) in the reference command. To demonstrate this effect, an input shaper was designed for the system above with the linear absorber. This system has two frequencies described by 0)1 =0.8292, ~I=O. 1711 , 0>2=1.0312, and ~2=O.2178. To suppress the vibration of these two closely-spaced modes, an SI shaper was designed to suppress the residual vibration of frequencies in the range 0.79 < 0) < 1.046. A frequency range was chosen slightly larger than the known frequency range to accommodate for modeling errors and the nonlinearities that will be introduced by the non linear absorber. The damping ratio used in the design was 0.2. The resulting shaper is given by: Ai] = [0.4273 0.4461 0.1267] (18) [ tj 0 3.3867 6.7511 Figure 10 shows how the step response is improved when the shaping is combined with the linear absorber. Because the shaping is designed based on

5. CONCLUSIONS The application of input shaping and the inclusion of a vibration absorber each offer an improvement in the characteristic performance of a system. The input shaping eliminates vibration induced by the reference command, while the vibration absorber tackles the problem of external disturbances. The inclusion of non-linearity in the vibration absorber improves its ability to deal with external disturbances. The robust input reduces vibration induced by the reference command, even with the inclusion of the nonlinearity of the vibration absorber. The combined approach described here offers vibration reduction from both external disturbances and from changes in the reference command. REFERENCES Barte!, D. L. and Krauter, A. 1. (1971) . Time Domain Optimization of a Vibration Absorber, Transactions of the ASME, Journal of Engineering for Industry, Vol. 93(3), pp. 799-804. Bhat, S. P. and Miu, D. K. (1990). Precise Pointto-Point Positioning Control of Flexible Structures,

349

Singer. N .• Singhose. W. and Kriikku. E. (1997). An Input Shaping Controller Enabling Cranes to Move Without Sway. ANS 7th Topical Meeting on Robotics and Remote Systems. Augusta. GA. Singer. N. C. and Seering. W . P . (1990) . Preshaping Command Inputs to Reduce System Vibration. J. of Dynamic Sys.. Measurement. and Control. Vo!. 112(March). pp. 76-82. Singer. N. C. and Seering. W. P. (1992). An Extension of Command Shaping Methods for Controlling Residual Vibration Using Frequency Sampling. IEEE International Conference on Robotics and Automation. Nice. France. Vo!. 1. pp. 800-805. Singhose, W .• Seering. W. and Singer. N. (1994) . Residual Vibration Reduction Using Vector Diagrams to Generate Shaped Inputs. J. of Mechanical Design. Vo!. 116(June), pp. 654-659. Singhose. W .• Singer. N. and Seering. W .• (l996a). Improving Repeatability of Coordinate Measuring Machines with Shaped Command Signals. Precision Engineering. VO!. 18(April). pp. 138-146. Singhose. W. E .• Seering. W. P. and Singer. N. C. (l996b). Input Shaping for Vibration Reduction with Specified Insensitivity to Modeling Errors. JapanUSA Sym. on Flexible Automation. Boston. MA. Smith. o. J . M . (1957). Posicast Control of Damped Oscillatory Systems. Proceedings of the IRE. Vo!. 45(September). pp. 1249-1255. Sun. J. Q .• Jolly. M. R. and Norris. M. A. (1995). Passive. Adaptive. and Active Tuned Vibration Absorbers- A Survey. Transactions of the ASME. Vo!. 117(. pp. 234-242. Tallman. G. H. and Smith. O. J. M . (1958) . Analog Study of Dead-Beat Posicast Control. IRE Transactions on Automatic Control. March). pp. 1421.

J. of Dynamic Sys., Meas., and Control. Vo!. 112(4). pp. 667-674. Bolz. R. E. and Tuve. G. L. (1973). CRC Handbook of Tables for Applied Engineering Science. Boca Raton. FL: CRC Press. Inc .• pp. 1071. Chao. c.-P .• Shaw, S. W. and Lee. C.-T. (1997) . Stability of the Unison Response for a Rotating System With Multiple Tautochronic Pendulum Vibration Absorbers. Journal of Applied Mechanics. Vo!. 64(March), pp. 149-156. Drapeau. V. and Wang. D. (1993) . Verification of a Closed-loop Shaped-input Controller for a Fivebar-linkage Manipulator. IEEE Int. Con! on Robotics and Automation. Atlanta. GA. Vo!. 3. pp. 216-221. Feddema. J. T. (1993). Digital Filter Control of Remotely Operated Flexible Robotic Structures. American Control Con! . San Francisco. CA. Vo!. 3. pp. 2710-2715. Fraharn. H. (1911) . Device for Damping Vibration in Bodies. United States. Gonsalves. D. H.. Neilson. R. D. and Barr. A. D. S. (1993). The Dynamics of a Non-Linear Vibration Absorber. Journal of Mechanical Engineering Science. Vo!. 207(C6). pp. 363-374. Hartog. J. P. D. (1985). Mechanical Vibrations . New York: Dover Publications. Inc. Hunt. J. B .• (1979). Dynamic Vibration Absorbers. London: Mech. Eng. Publications LTD. Jones. S. and Ulsoy. A. G. (1999). An Approach to Control Input Shaping with Application to Coordinate Measuring Machines. J. of Dynamics, Measurement, and Control, Vo!. 121(June). pp. 242247. Kenison. M. and Singhose. W . (2000) . Concurrent Design of Input Shaping and Feedback Control for Insensitivity to Parameter Variations. 6th International Workshop on Advanced Motion Control. Nagoya. Japan. Magee. D. P. and Book. W. J. (1995). Filtering Micro-Manipulator Wrist Commands to Prevent Flexible Base Motion . American Control Con! . Seattle. W A. pp. 924-928. Nigam. N. C. and Narayanan. S. (1994) . Applications of Random Vibrations. New York: Springer-Verlag and Narosa Publishing House. pp. 557. Noakes . M . W. and Jansen. J. F . (1992) . Generalized Inputs for Damped-Vibration Control of Suspended Pay loads. Robotics and Autonomous Systems. Vo!. 10(2). pp. 199-205. Ooten. E. and Singhose. W. (2000). Command Generation with Sliding Mode Control for Flexible Systems. 6 th International Workshop on Advanced Motion Control. Nagoya. Japan. Pennestri. E. (1998). An Application of Chebyshev's Min-Max Criterion to the Optimal Design of a Damped Dynamic Vibration Absorber. Journal of Sound and Vibration . Vo!. 217(4). pp. 757-765. Queini. S. S. C .• CharMing; Nayfeh. Ali H. (1999) . Dynamics of a Cubic Nonlinear Vibration Absorber. Nonlinear Dynamics. Vo!. 20. pp. 283295. Shaw. J .• Shaw. S. W. and Haddow. A. G. (1989). On the Response of the Non-Linear Vibration Absorber. Internatioanl Journal of Non-Linear Mechanics. Vo!. 24(4). pp. 281-293.

350