Experimental Verification of Real-Time Control for Flexible ... - CiteSeerX

William Singhose Erika Biediger Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA

Hideto Okada NEC Toshiba Space Systems, Ltd., Yokohama, Japan

Saburo Matunaga Department of Mechanical and Aerospace Engineering, Tokyo Institute of Technology, Tokyo, Japan

1

Experimental Verification of Real-Time Control for Flexible Systems With On-Off Actuators A technique for driving a flexible system with on-off actuators is presented and experimentally verified. The control system is designed to move the rigid body of a structure a desired distance without causing residual vibration in the flexible modes. The on-off control actions are described by closed-form functions of the system’s natural frequency, damping ratio, actuator force-to-mass ratio, and the desired move distance. Given the closed-form equations, the control sequence can be determined in real time without the need for numerical optimization. Performance measures of the proposed controller such as speed of response, actuator effort, peak transient deflection, and robustness to modeling errors are examined. Experiments performed on a flexible satellite testbed verify the utility of the proposed method. 关DOI: 10.1115/1.2192837兴

Introduction

Moving flexible systems using on-off actuators is very challenging because each actuator action induces vibration. However, if the times at which the actuators are turned on and off are timed correctly, then the vibration can be canceled out 关1–11兴. In fact, the time-optimal zero-vibration control for many types of linear flexible systems is a multiswitch bang-bang command that is compatible with on-off actuators. The time locations of the control actions are obtained by performing a numerical optimization that minimizes the move time, while satisfying a set of constraint equations that ensure zero residual vibration. However, it would be very difficult to perform this optimization, obtain the control sequence, and verify the answer in real time 关12兴. Time-optimal control has the further drawback of being very sensitive to modeling errors 关13兴. This fact has motivated the development of robust time-optimal commands 关2–4,6兴. The robust approaches work by using additional constraint equations that ensure the vibration will be at a low level even when modeling errors exist. Although there is an extensive literature of promising techniques for generating low-vibration on-off commands, very little experimental evidence exists that demonstrate these techniques can be applied to real systems. A few experiments have been done for high-speed on-off control using a rotary table 关4兴 and a piezo-actuator 关8兴. More typically, the experimental work using on-off commands have verified low-speed, but fuel-efficient techniques 关14兴. One of the primary contributions of this paper is a thorough experimental investigation of a simple and practical method for generating high-speed on-off command sequences. To demonstrate the on-off control problem, consider the benchmark two-mass-spring-damper system 关15兴, where the actuator force can only have values of 1, 0, and −1. If the mass values and the spring constant are set equal to 1, then the system has an undamped natural frequency of 冑2 rad/ sec 共0.2251 Hz兲. Figure 1 shows the response of the second mass when the system has a damping ratio of 0.2 and three different types of on-off commands are used to move the system. The bang-bang command shown at the top of Fig. 2 produces a fast rise time. However, the system takes 15.29 sec to settle to within 1% of the move distance. The Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received November 25, 2004; final manuscript received July 19, 2005. Assoc. Editor: Santosh Devasia. Paper presented at the 2003 AIAA Guidance, Navigation, and Control Conference.

time-optimal zero-vibration 共TO-ZV兲 command 关1,2,9,11,13兴 shown in the middle of Fig. 2 was obtained via numerical optimization. It has switch times at 2.95, 4.57, and 4.99 sec and a total duration of 6.74 sec. The TO-ZV command reduces the settling time to 6.74 sec, while maintaining a very fast rise time. The third response in Fig. 1, and its corresponding command shown in Fig. 2, is labeled closed-form ZV. This command profile also results in zero vibration, but is does not require a numerical optimization. The closed-form description of this actuation function represents a huge advantage over optimization-based methods because it allows low-vibration on-off commands to be generated in real-time. When a flexible system needs to be maneuvered quickly without residual vibration, three main approaches exist for generating an on-off command sequence. First, an extensive series of calculations can be undertaken to obtain a time-optimal solution. This could include nonlinear optimization and verification of the resulting solution where guarantees of global optimality are challenging to obtain 关5,12兴. This approach requires significant computational abilities. Furthermore, it introduces a time delay into the system response while the optimization and solution checking are being performed. A second approach uses measurements of the system states to implement an on-off feedback control scheme 关10,16兴. This approach can continually adjust the actuator effort throughout the maneuver to better ensure that the desired motion is obtained. One method to accomplish this is to insert an intelligently designed pulse width modulator into a traditional control system that is designed to eliminate vibration 关16兴. Another technique creates a phase plane representation of the system subjected to time-optimal commands. The resulting switching surfaces can be used to implement an on-off feedback controller 关10兴. A drawback of these approaches is the need for accurate state information. Furthermore, the actuators may continually switch between positive and negative. This could lead to excessive actuator wear. In the case of on-off thruster control of spacecraft, this effect would lead to unacceptably large fuel usage. A third approach is to utilize on-off commands that are described in closed-form expressions 关17–19兴. When a motion is required, a command sequence is immediately known. However, the resulting command profiles will be slightly slower than timeoptimal solutions. The work presented here is based on this third approach—utilizing commands that are known in closed form. This paper builds on previous developments that were restricted to undamped systems. The command profiles developed here can be

Journal of Dynamic Systems, Measurement, and Control Copyright © 2006 by ASME

JUNE 2006, Vol. 128 / 287

Fig. 3 Sketch of proposed command profiles

Fig. 1 Response of benchmark flexible system

described by simple functions of the system natural frequency ␻, damping ratio ␨, the desired slew distance xd, and the actuator force-to-system mass ratio ␣. The command profiles presented here are not the time-optimal profiles; those cannot be described in closed form. However, the profiles are very nearly time optimal. A technique for making very small motions is presented in Sec. 3. Important qualities of the command profiles, such as move duration, actuator effort, maximum transient deflection, and robustness to modeling errors are discussed in Sec. 4. Experiments using a satellite testbed that floats on air bearings are discussed in Sec. 5. The primary contributions of this paper are the extension of the closed-form on-off command profiles to damped systems, the technique for making very small motions, and the experimental results.

2

Closed-Form On-Off Commands

The method used here to derive commands in real-time is based on three ideas. First, a bang-bang command is a desirable template function because it produces a fast response. 共If actuator effort or fuel usage is of concern, then the ideas presented here can be extended to bang-coast-bang command profiles 关18兴.兲 Second, the three transitions in actuator effort that compose a bang-bang command 共zero to positive, positive to negative, and negative to zero兲 can each be accomplished without causing residual vibration. Third, once vibration-reducing actuator transitions are derived, then the rigid-body motion can be set by properly choosing the time duration between the three transitions. If each of the three transitions does not cause vibration, then the entire command will not cause vibration. This process is shown schematically in Fig. 3 and is summarized by the following four steps: 共1兲 Generate a low-vibration transition from zero-to-

positive actuator effort by switching the actuator on and off between time zero and time tn. 共Transition 1 in Fig. 3.兲 共2兲 Generate a low-vibration transition from positive actuator effort to negative effort between times tn+1 and tn+m. 共Transition 2 in Fig. 3.兲 共3兲 Generate a low-vibration transition from negative actuator effort back to zero between times tn+m+1 and the end of the command at tn+m+p. 共Transition 3 in Fig. 3.兲 共4兲 Specify the rigid-body displacement by choosing the time durations between transitions 1 and 2 共tn+1 − tn兲 and between transitions 2 and 3 共tn+m+1 − tn+m兲. Using this process, the flexible dynamics are canceled by each of the three actuator transitions. The rigid-body motion requirements are satisfied by specifying the time duration between the transitions. By breaking this complicated problem down into two simpler problems, a closed-form solution can be obtained. That is, the on-off command sequence can be obtained by simply plugging the system parameters and desired move distance into a set of equations that immediately give the actuator switch times. 2.1 Zero-to-Positive Command Transitions. There are several ways to transition the actuator state from zero-to-positive effort without causing residual vibration. These methods require turning the actuator on and off at specific times that depend on the vibration frequency and damping ratio. The simplest way to accomplish such a transition for an undamped system with vibration period T is to turn the actuator on for T / 6 seconds, turn the actuator off for T / 6 seconds, and then turn the actuator back on 关8兴. The vibration caused by the three switches in this process adds up to zero. This can be better understood by interpreting the actuator transition as a step input convolved with an impulse sequence of the form

冋册 Ai ti

=

冤

1 −1 1 0

T 6

T 3

冥

共1兲

where Ai and ti are the impulse amplitudes and time locations. This interpretation is shown in Fig. 4. Note that the time locations of the impulses in Eq. 共1兲 correspond to the switch times in the resulting on-off command. The process of convolving a function with a sequence of impulses to generate a command signal is called input shaping and the impulse sequence is called an input shaper 关20兴. The motivation for performing this deconvolution is that the vibration properties of the command profile can be largely

Fig. 2 Command profiles used to produce the responses in Fig. 1

288 / Vol. 128, JUNE 2006

Fig. 4 Input shaping to generate a zero-to-positive transition

Transactions of the ASME

Table 1 Switch times for zero-to-positive transitions

Fig. 5 Vibration cancelation with a sequence of positive and negative impulses

determined by examining only the impulse sequence 关21,22兴. Superposition of the responses from each of the three impulses given in Eq. 共1兲 results in zero residual vibration. This effect is demonstrated in Fig. 5. Given that the input shaper results in zero residual vibration, then the command formed by convolving the shaper with a step input will also cause no residual vibration 关20兴. The impulse time locations given in Eq. 共1兲 were determined by forcing the residual vibration to equal zero at the end of the impulse sequence. For a sequence containing n impulses this zero vibration 共ZV兲 constraint equation is given by 关20兴 0 = e−␨␻tn冑关C共␻, ␨兲兴2 + 关S共␻, ␨兲兴2

共2兲

where n

C共␻, ␨兲 =

兺Ae i

␨␻ti

cos共␻冑1 − ␨2ti兲

共3兲

␨␻ti

sin共␻冑1 − ␨2ti兲

共4兲

i=1 n

S共␻, ␨兲 =

兺Ae i

i=1

Note that Eq. 共2兲 produces two constraint equations because the cosine summation Eq. 共3兲, and the sine summation Eq. 共4兲, are squared under the radical and must equal zero independently. The time locations of the impulses given in Eq. 共1兲 are obtained by setting ␨ equal to zero, using the appropriate values of Ai from Eq. 共1兲, and then setting Eqs. 共3兲 and 共4兲 equal to zero. The resulting equations can be solved to obtain the impulse time locations given in Eq. 共1兲 关8兴. Although the switch times given by Eq. 共1兲 transition the command from zero-to-positive effort without residual vibration, it is not the fastest possible transition. It is of interest because the resulting command does not contain negative pulses. This leads to a fuel-efficient 共FE兲 transition 关23兴. The transition given in Eq. 共1兲 will be referred to as the zero-vibration fuel-efficient 共ZV-FE兲 zero-to-positive transition. On the other hand, the zero-vibration time-optimal 共ZV-TO兲 transition requires a negative force pulse. For undamped systems this transition can be obtained by using an input shaper described by 关17兴

冋册 Ai ti

=

冤

1

−2

2

0

cos−1共1/4兲 T 2␲

cos−1共− 1/4兲 T 2␲

冥

共5兲

2.2 Damped Systems. The transitions described by Eqs. 共1兲 and 共5兲 work perfectly only on undamped systems. When the system has damping, the impulse time locations must be slightly shifted in time to account for the effects of damping. Note that the impulse amplitudes are limited to integer values so that on-off commands are generated. The impulse time locations for damped Journal of Dynamic Systems, Measurement, and Control

systems are determined by satisfying Eq. 共2兲 with a nonzero value of ␨. Given that the most problematic vibrations occur with modes having low damping ratios, this problem was numerically solved for 0 艋 ␨ 艋 0.2. Third-order polynomial curves were then fit to the data so that the maximum error in any time location was less than 1%. The curve fits that describe the impulse time locations for the zero-to-positive transitions are given in Table 1. There are four transitions listed: the ZV-FE and ZV-TO, mentioned above, and the ZVD-FE and ZVD-TO that are more robust to modeling errors. These robust commands will be explained in the next section. The second column in Table 1 lists the impulse amplitudes corresponding to each transition. Given that the command profiles must switch between only three states 共positive, zero, negative兲, these impulse values are restricted to the set of integers 关−2 , −1 , 1 , 2兴. The remainder of the table can be used to calculate the switch times as a function of ␨. The table lists the coefficients, M0-M3 for each of the switch times described by a third-order polynomial ti = 共M 0 + M 1␨ + M 2␨2 + M 3␨3兲T

共6兲

Note that the first switch time in each transition is set to zero. 2.3 Robustness to Uncertainty. When the modeling parameters ␻ and ␨ are not exact, the commands will produce some amount of residual vibration. To explore this effect, the amplitude of residual vibration can be plotted versus the modeling error. This information is often shown as the percentage residual vibration 共residual vibration amplitude divided by the amplitude from a step input兲 versus the error in frequency or damping ratio. The frequency axis is often plotted as a normalized frequency 共actual frequency ␻a divided by the modeling frequency ␻m兲. The vibration amplitude used to form the normalized amplitude value is calculated at the instant the command is completed. The solid curve in Fig. 6 shows such a sensitivity curve for the input shaper 共ZV-FE兲 given in Eq. 共1兲. The figure shows that as the actual frequency deviates from the modeling frequency, the amplitude of residual vibration increases rapidly. Therefore, small errors in frequency can lead to noticeable vibration 关24兴. The nonrobust nature of the ZV shapers shown in Fig. 6 has motivated the development of more robust input shapers. One type of robust shaper requires that the derivative of the vibration amplitude, with respect to the frequency, equal zero at the modeling parameters 关20兴. That is JUNE 2006, Vol. 128 / 289

Fig. 6 Sensitivity curves for the ZV-FE and ZVD-FE transitions

0=

d −␨␻t 关e n冑关C共␻, ␨兲兴2 + 关S共␻, ␨兲兴2兴 d␻

共7兲

By enforcing this zero derivative constraint, the vibration tends to stays near zero as the actual frequency shifts away from the modeling frequency. The price of the improved robustness of the zero vibration and derivative 共ZVD兲 shaper is an increase in the shaper duration. Therefore, the system rise time is slightly longer with ZVD shapers than with ZV shapers. However, Fig. 6 shows that the ZVD-FE transition is considerably more robust to errors in the modeling frequency than the ZV-FE transition. Figure 7 shows the sensitivity curves for errors in damping ratio. Figure 7共a兲 shows the curves for the ZV-FE transition when the damping ratio used to design the shaper ␨m varies between 0 and 0.2. Figure 7共b兲 shows the corresponding information for the ZVD-FE transitions. Note that enforcing the zero-derivative-withrespect-to-frequency constraint given in Eq. 共7兲 also causes the derivative with respect to damping ratio to be zero. The shapers

Fig. 7 Sensitivity curves as a function of damping ratio. „a… ZV Transitions and „b… ZVD transitions.

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Fig. 8 On-off command transitions

are less sensitive to damping ratio errors than errors in frequency, but the shapes of the sensitivity curves are similar to the “V” and “U” shapes of the frequency sensitivity curves shown in Fig. 6. Note that when investigating robustness to errors in damping ratio, a normalized damping ratio is not used. The damping ratio itself is a nondimensional quantity. Furthermore, to formulate a normalized damping ratio such as ␨a / ␨m can lead to very misleading and nonsensical results. The most obvious case is when the system is modeled as having no damping 共␨m = 0兲. For this relatively common model, the normalized damping ratio is undefined. Also consider a case of a small, nonzero damping ratio, say ␨m = 0.01. For a very small error in damping ratio, say ␨a = 0.02, the normalized damping ratio would indicate a 100% error in the model. While a 100% shift would be a substantial error if it occurred in the frequency parameter, this is a nearly inconsequential change in damping ratio. The actual dynamic response of the system would be nearly identical to the modeled response. Given the sensitivity curves for the ZVD shaper shown in Figs. 6 and 7, it is obvious that the actual parameters can deviate considerably from the modeling frequency before the residual vibration becomes significant. However, there may be other types of uncertainty in a given system. For example, the frequency at which the control loop is running may be somewhat inconsistent, or it may be slow compared to the vibration frequencies that the controller is attempting to suppress. Luckily, input shaping is also fairly robust to these types of uncertainties 关25兴. Input shapers providing more robustness than the ZVD approach are easily calculated 关4,26–28兴, but we restrict our attention to ZVD shapers in this paper because robustness is not our central theme. The development of closed-form commands and their experimental evaluation is our primary interest. Four possible transitions from zero-to-positive actuator effort are shown in the top part of Fig. 8. The ZV transitions are designed by enforcing the zero residual constraint given in Eq. 共2兲, while the ZVD transitions also satisfy Eq. 共7兲. The transitions are further labeled time optimal or fuel efficient. The switch times for each of the transitions were given in Table 1. 2.4 Positive-to-Negative Transitions. The transition from positive-to-negative actuator effort can be shaped differently than Transactions of the ASME

Table 2 Switch times for positive-to-negative transitions

the first transition because a two-unit change in command value is required. Four possible transitions are shown in the middle section of Fig. 8 and their switch times are given in Table 2. A square root function provides a better fit for the ZV-FE transition than a polynomial function. The other transitions are fit with a third-order polynomial as was done in Table 1. The switch times of these transitions were determined using the same constraint Eqs. 共2兲 and 共7兲 that were used for the zero-to-positive transitions. The differences arise because the impulse amplitudes must be different in order to accomplish the two-unit change from positive to negative actuator effort. 2.5 Negative-to-Zero Transitions. The third transition in the command takes the actuator state from negative back to zero. This transition requires a one-unit change, just like the first transition. However, due to damping, the negative-to-zero transitions are not antisymmetric to the first transitions. Four possible negative-tozero transitions are shown at the bottom of Fig. 8 and their switch times are given in Table 3. Once again, the same constraint equations were used, but the impulse amplitudes have changed. For example, the ZV-TO zero-to-positive transition has impulse amplitudes of 关1 , −2 , 2兴 while the ZV-TO negative-to-zero transition has impulse amplitudes of 关2 , −2 , 1兴. 2.6 Complete Command Profiles. The above transitions can now be used to generate command profiles that perform rest-torest motion without residual vibration. To do this, select a shaper for each of the three command transitions and then determine the necessary time between transitions to accomplish the desired rigid-body motion. The time between transitions can be determined from simple rigid-body mechanics. If the system has a total mass of M and the position of the center of mass is x, then x¨共t兲 =

F共t兲 M

共8兲

The velocity and position at the end of the command profile are then Journal of Dynamic Systems, Measurement, and Control

Table 3 Switch times for negative-to-zero transitions

␯d =

冕

tf

F共t兲 dt M

0

冕冕 tf

xd =

0

tf

0

F共t兲 2 dt M

共9兲

共10兲

For rest-to-rest motion, Eq. 共9兲 must equal zero and Eq. 共10兲 must equal the desired move distance. Because the forcing function F共t兲 is on-off, the integrals in Eqs. 共9兲 and 共10兲 can be evaluated in closed form. Profiles for undamped systems are particularly easy to obtain because they are symmetric about their midpoints and, therefore, the integrals in Eqs. 共9兲 and 共10兲 only need to be integrated up to the midpoint. For example, suppose a command is formed by choosing a fuel-efficient ZV shaper for each of the three command transitions. The resulting command would be called a ZV FE-FE-FE command and nearly all of its switch times would be known immediately from Tables 1–3. The only unknown command switch times would be the start of the second and third transitions. For the undamped case, the second ZV-FE transition reduces from the general three-switch function given in Table 2 to a two-switch function. This simplification occurs because the time of the second switch goes to zero when ␨ = 0 and it aligns with the first switch in the transition. Therefore, the spacing between the start of the middle transition t4 and its end t5 is known to be T / 2. The time of the maneuver midpoint tm is therefore

tm = t4 +

T 4

共11兲

The value of t4 is now determined from rigid-body mechanics. At midmaneuver the position of the center of mass xtm must be at one half of the desired move distance. By simply integrating the rigid-body equation of motion with respect to time, an expression for the mass center position as a function of the switch times is obtained JUNE 2006, Vol. 128 / 291

xtm =

xd ␣ = 关− t22 + t23 − t24兴 + ␣tm关t2 − t3 + t4兴 2 2

共12兲

The force-to-mass ratio in this case is ␣ = umax / M. Using Eqs. 共11兲, 共12兲, and the known values of t2 and t3 from Table 1, the beginning of the second transition, t4, is found to be

冋册 Ai ti

=

冤

1

−1

1

0

T 6

T 3

t4 =

冉 冑冉 冊 冊 T 12

2

冑冉 冊 T 12

2

+

xd ␣

共13兲

Therefore, for an undamped system the command profile is described by eight switches

−1

−T + 12

−T + 12

xd + ␣

−1

1

−1

1

T t4 + 2

t5 + t4 − t3

t5 + t4 − t2

t5 + t4

冥

共14兲

Note that the entire command is given in a closed-form expression that can easily be evaluated in real time. It is a simple function of the frequency, damping, move distance, and force-to-mass ratio. The complete profiles for damped systems are a little more difficult to obtain because they are not antisymmetrical about the midpoint and the above integrals must be evaluated up until the final switch time. The resulting equations are difficult to solve by hand, but relatively easy to solve using MATHEMATICA. For example, the ZV FE-FE-FE command for damped systems is described by the following nine switches:



Ai ti

=

1

−1

1

−2

1

−1

1

−1

1

0

t1b

t1c

t4

t4 + t2b

t4 + t2c

t7

t7 + t3b

t7 + t3c

where t1a, t1b, and t1c refer to the three switches in the first transition, t2a, t2b, and t2c refer to those in the second transition, etc. This command was graphically represented in the bottom of Fig. 2. Recall that most of the switch times come from Tables 1–3. The times at which the second and third transitions start t4 and t7 in Eq. 共15兲, are given by t4 = − t1b + t1c + 0.5t2b − 0.5t2c

+ 0.5冑4t21b − 4t1bt1c − 3t22b + 2t2bt2c + t22c + 4t23b − 4t3bt3c + 4xd/␣ 共16兲 t7 = t1b − t1c − t2b + t2c + t3b − t3c + 2t4

共17兲

To summarize, the equations in Tables 1–3 and Eqs. 共15兲–共17兲 can be used in real time to generate an on-off command to rapidly move a lightly damped flexible system without residual vibration. Note that very small values of damping do not significantly affect the command and/or the response. However, moderate damping necessitates the use of the damped solutions given in Tables 1–3, rather than the undamped profiles given previously in the literature 关17兴. Consider the case when the benchmark system has a damping ratio of 0.2. If the damping is ignored and a command profile is designed assuming zero damping, then the response will have noticeable residual vibration as shown in Fig. 9.



共15兲

Therefore, the damped solutions presented here extend the effective performance of this method to a much wider array of systems.

3

Commands for Small Motions

One limitation of the closed-form commands presented in the previous section should be pointed out. The commands cannot be used for very small move distances. A problem arises because the time duration of the three command transitions is fixed by the flexible-body dynamics. The minimum move distance occurs when the transitions occur sequentially without delay. The robust 共ZVD兲 transitions require more time to complete than the nonrobust 共ZV兲, so using these commands increase the smallest possible move distance. However, the minimum move distances are very small—often corresponding to move durations that last only one or two periods of the vibration. If there is the need to make very small motions, then a simple method can be used. First, if robust transitions are being used and the minimum move distance is too large, then the control algorithm switches to nonrobust transitions. This decreases the minimum possible move to a very small value. For example, using a robust undamped ZVD FE-FE-FE command, the minimum move distance is: 关xmin兴ZVD

FE-FE-FE =

0.599075␣T2

共18兲

On the other hand, the minimum move distance for the undamped ZV FE-FE-FE command is 关xmin兴ZV

FE-FE-FE =

0.16667␣T2

共19兲

Second, if the minimum move distance is still too large, then the control algorithm switches to the simplest zero-vibration command. This command contains only two actuator pulses—one positive pulse to start the system moving and a second negative pulse to stop the system. The second pulse is applied after one period of the damped vibration Td. The time durations of the pulses determine the move distance. The minimum move distance is then determined by the shortest burst that the actuator can perform. The switch times in this two-pulse command can be described by

冋册 Ai

Fig. 9 Response of damped system to undamped commands

292 / Vol. 128, JUNE 2006

ti

=

冤

1

−1

−1

1

0

xd ␣

Td

Td +

xd ␣

冥

共20兲

Transactions of the ASME

Fig. 12 Transient deflection from various commands Fig. 10 Move duration for closed-form fuel efficient and time optimal commands

Note that when the move distances are on these small scales, the amount of energy created by the actuators will be corresponding small. Therefore, there is no need to force the command to be very robust. It will be robust because even if the model is very inaccurate, the low amount of actuator effort cannot excite large amounts of vibration. This method will give exactly zero vibration when the system has zero damping, but as damping increases, so will the amount of residual vibration. However, recall that the actuator bursts will be very small, so any residual vibration will also be very small. The complete algorithm for real-time generation of on-off commands for any move distance, and any parameter values of a lightly damped system can be stated as follows: 共1兲 Use Tables 1–3 to determine three command transitions. 共2兲 Determine the time of the second and third transitions by using velocity and position boundary conditions. See for example, Eqs. 共14兲 and 共15兲. 共3兲 If the rigid-body boundary conditions cannot be satisfied, then the desired move distance is too short and the control algorithm switches to nonrobust commands or to the simple two-pulse command. This algorithm can immediately calculate the command for any move distance of any lightly damped system.

4

Evaluation and Comparison of Command Profiles

When designing command profiles for flexible systems, there are several design tradeoffs that must be considered. Rapid motion is almost always desired, but actuator effort, maximum transient deflection, and robustness to modeling errors must also be within acceptable bounds. This section will evaluate these quantities for the closed-form commands as a function of the move distance and

Fig. 11 Fuel usage

Journal of Dynamic Systems, Measurement, and Control

comparisons will be made to time-optimal on-off commands. All results in this section are based on simulations of the benchmark system. 4.1 Move Duration. Figure 10 compares the move duration of four different types of commands as a function of the desired move distance for an undamped system. Two of the profiles are the ZV and ZVD versions of the closed-form FE-FE-FE commands. The other two commands are the time-optimal commands that satisfy the ZV and ZVD constraints. Note that these are multiswitch bang-bang commands that would be very difficult to generate in real time 关1–3,12兴. However, they do provide the theoretical lower bound on move duration. The closed-form ZV commands average 9.4% longer than the time-optimal ZV commands over the range 10艋 xd 艋 40. The closed-form ZVD commands average 15.5% longer than the timeoptimal ZVD commands over the same range. Note that at small move distances 共less than six units兲, the ZVD FE-FE-FE command becomes equivalent to the ZV FE-FE-FE command, as discussed in Sec. 3. Furthermore, for very small moves 共less than 2.0 units兲, both the ZV and ZVD FE-FE-FE commands reduced down to the simple two-pulse command. The results shown in Fig. 10 indicate that a moderate penalty of 10–15% in move time is the cost of using commands that are known in closed form and can be generated in real time. The following sections will show that the closed-form commands are actually superior to time-optimal commands in the other performance measures. 4.2 Fuel Usage. Actuator effort is often an important concern. In the case of flexible spacecraft the amount of thruster fuel burned is always an overriding concern. This consideration has motivated several studies in fuel-optimal and fuel-efficient slewing 关14,23,29–31兴. Figure 11 compares the fuel usage 共defined as the amount of time that the actuators are turned on兲 for the four types of command profiles. Once again, notice that for very small move distances, the ZV and ZVD FE-FE-FE commands coincide. The closed-form ZV profile uses an average of 16.8% less fuel than the time-optimal ZV profile, while the closed-form ZVD uses an average of 17.7% less fuel than the time-optimal ZVD profile over the range 10艋 xd 艋 40. 4.3 Maximum Transient Deflection. Another important consideration is the maximum deflection that occurs during the motion. The shaped profiles control the residual vibration, but no provision is made for limiting the transient deflection. For the benchmark two-mass system, the deflection is the compression or extension of the connecting spring. Figure 12 shows the transient deflection corresponding to the three responses shown in Fig. 1. Both the time-optimal and closed-form ZV commands produce zero residual vibration, but the ZV FE-FE-FE command greatly reduces the transient deflection as well. Note that the time-optimal ZV command produces essentially the same deflection as the bang-bang command. JUNE 2006, Vol. 128 / 293

Fig. 13 Maximum transient deflection

Fig. 15 10% Insensitivity versus move distance

The significant reduction in deflection is not limited to the single case shown in Fig. 12. The maximum transient deflection for a wide range of move distances is shown in Fig. 13. Both the ZV and ZVD closed-form commands limit the transient deflection to 0.5, while the time optimal commands induce considerably more deflection for virtually all move distances. If transient deflection is of primary concern, then techniques presented previously should be employed to limit the deflection to an acceptably low level 关32兴.

profiles over a wide range of move distances. A similar analysis can be used to determine robustness to errors in damping ratio and the results are similar. As a further note, Fig. 14 clearly shows that the commands are very effective on structures with closely spaced modes. For example, the ZV FE-FE-FE command would completely suppress the frequency it was designed for 共␻a / ␻m = 1兲, but it will also totally eliminate modes that are located at 0.92 and 1.08 times the primary design frequency. The ZVD FE-FE-FE command would eliminate any closely spaced modes that were within ±20% of the primary frequency.

4.4 Robustness to Modeling Errors. As was shown throughout this paper, many types of on-off profiles produce motion with zero residual vibration if the system model is exact. When the model is not exact, the amplitude of residual vibration is highly dependent on the command. As was demonstrated in Figs. 6 and 7, the ZVD transitions are much more robust than ZV transitions. When the transitions are assembled together to form complete on-off command sequences, the same relative robustness qualities persist. The level of robustness, however, is dependent on the move distance and the type of profile. Figure 14 shows the sensitivity curves of closed-form and timeoptimal commands when xd = 10. The ZVD profiles are more robust than the ZV profiles, as expected. To measure the robustness in a quantitative manner, the width of the frequency suppression region can be measured 关27兴. The figure shows the suppression range for each command assuming 10% residual vibration is acceptable. These nondimensional robustness measures are called the 10% insensitivities. For the case shown in Fig. 14, the ZV FE-FE-FE profile is 68.9% more insensitive than the time-optimal ZV profile, while the ZVD FE-FE-FE profile is approximately 263% more insensitive than the time-optimal ZVD profile. Using this measure, the robustness of the commands can be predicted over a range of move distances, as shown in Fig. 15. The closedform profiles are considerably more robust than the time-optimal

Fig. 14 Sensitivity curves for commands producing a ten unit move

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5

Experimental Investigation

To verify the utility of the proposed on-off command sequences, experiments were conducted using the two-dimensional spacecraft dynamics simulator at the Tokyo Institute of Technology. The setup, shown in Fig. 16, was constructed to verify the Robot Satellite Cluster Systems 关33兴. The facility has a 3 m ⫻ 5 m flat glass floor, three satellite simulators that float on air bearings, and an image processing system measuring twodimensional position. A schematic diagram of a dynamics and intelligent control simulator for satellite clusters 共DISC兲 unit is shown in Fig. 17. Position and attitude of the DISC units are controlled by air thrusters whose commands are sent via a wireless local area network 共LAN兲. The system mimics many of the important dynamic effects in on-off thruster control of satellites. For example, on-off thrusters never produce a perfect pulse in force. There is a transient rise and fall in the thrust force. Furthermore, the thrust force is never perfectly constant when it reaches its peak value. Both of

Fig. 16 Two-dimensional spacecraft dynamics simulator

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Fig. 19 Vibration amplitude versus appendage frequency Fig. 17 Schematic diagram of a DISC unit

these effects are readily apparent in the air thruster system used to move the DISC units. Another realistic inconvenience is the time delays in the command sequence due to the use of the LAN to issue thruster commands. In order to test the real-time on-off commands presented here, a flexible appendage with an endpoint tracking target was attached to the DISC units, as shown in Fig. 16. Both the length and endpoint mass of the flexible beam were adjustable. By varying these parameters, the robustness of the commands to errors in both frequency and damping could be tested. Note that the beam is fairly wide and the endpoint tracking target is large. These surfaces produce a nonlinear damping force that is a function of the beam length and endpoint mass. The details of this dynamic phenomenon were not rigorously investigated. Instead, the damping was modeled as linear viscous friction. Given that the damping forces were fairly small and the on-off commands have some robustness to damping errors, this modeling inaccuracy did not greatly degrade the effectiveness of the commands. Several hundred experiments were conducted to evaluate various on-off thruster commands for a variety of move distances and appendage dynamics. In each case, a DISC unit was initially allowed to float freely on the air bearings. Then, a sequence of thruster firings was performed. The overhead camera recorded the location of the main unit, as well as the endpoint of the flexible appendage. The nominal length of the adjustable appendage was set to give a lightly damped natural frequency of 1.05 Hz. The control system calculated on-off thruster firings based on these baseline dynamics. For example, the endpoint deflection of the appendage resulting from a 35 deg rotational motion is shown in Fig. 18. As seen in this figure, a bang-bang command induces both large transient deflection and residual vibration. Although the response is dominated by the lightly damped mode at 1.05 Hz, higher-order

modes are readily apparent, especially during the initial phase of the bang-bang response. This multimode nature of the system is a further complication that mimics actual spacecraft. However, given the dominance of the low mode, the commands were only designed to suppress the low frequency. If the higher modes were a significant problem, then the commands could be designed to suppress the higher modes 关19,34,35兴. Figure 18 demonstrates that both the ZV and ZVD FE-FE-FE commands significantly reduce the excitation of the flexible dynamics. In this particular case, the ZV FE-FE-FE command reduced the residual vibration to 17% of the amount induced by the bang-bang command, while the ZVD FE-FE-FE command reduced it to 12%. Note that the response to the ZV command still contains a noticeable component of the 1.05 Hz mode. This is due to the poor robustness of the ZV commands combined with the various inaccuracies in the model and the nonlinearities in the experimental setup. The ZVD response contains a much smaller 1.05 Hz component, as well as the higher frequencies that were not targeted for suppression. To thoroughly test the robustness of the on-off commands to uncertainties and changes in the flexible dynamics, a series of maneuvers were performed with various appendage lengths. The control system was not modified for each change, so it continued to create on-off command sequences based on the incorrect system parameters. Figure 19 shows the residual vibration amplitude as a function of the appendage frequency. Note that the damping ratio changes slightly as the appendage length is changed, but the effect is small, so it was neglected. Both the ZV and ZVD FEFE-FE commands produce a low vibration response over a wide range of flexible appendage frequencies. As expected, the ZVD commands are more robust to modeling errors than the ZV commands. The above experimental data was collected from 35 deg rotational slews. To verify the control system’s ability to provide lowvibration motion over a range of maneuver distances, another series of experiments was performed. Figure 20 shows the amplitude of residual vibration over a range of slew distances and

Fig. 18 Appendage deflection for a rotational motion

Fig. 20 Residual vibration amplitude versus slew distance

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Fig. 21 Maximum transient deflection versus slew distance

Fig. 21 shows the maximum transient deflection. The data clearly demonstrate that the commands give consistent low-vibration motion over a large range of move distances.

6

Conclusions

On-off command profiles for rest-to-rest motion of flexible systems were presented. The commands are described by closed-form functions of the system parameters and desired displacement. Although the closed-form profiles are not the time optimal profiles, they are nearly time optimal in most cases. The closed-form profiles are significantly more robust to modeling errors and they cause less transient deflection. Furthermore, the closed-form commands use considerably less fuel with only minor increases in move duration time. Experiments conducted on a spacecraft testbed verified vibration-reducing properties of the proposed control system. The experiments demonstrated robustness to modeling errors and consistency over a range of move distances.

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