A Direct Method of Adaptive FIR Input Shaping for - Copy

316 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 18, NO. 1, FEBRUARY 2013

A Direct Method of Adaptive FIR Input Shaping forMotion Control With Zero Residual Vibration

Matthew O. T. Cole and Theeraphong Wongratanaphisan

Abstract—In this paper, we describe a method of adaptive feed-forward control that can achieve zero residual vibration in rest-to-rest motion of a vibratory system. When a finite impulse responsefilter is used to preshape a command input, zero residual vibrationis achieved for any input signal if the impulse response of the filtersatisfies a condition of orthogonality with respect to the impulseresponse of the system under control. An equivalent condition in-volving sets of measured I/O data is derived that forms the basis ofa direct method of adaptively tuning filter coefficients during mo-tion. The approach requires no prior model of the system and canbe applied to multimode and multiinput systems under arbitraryand nonrepetitive motions. Versions of the algorithm employingrecursive least-squares techniques are developed and analyzed. Asa special case of the general adaptation problem, tuning of impulse-based shapers with fixed impulse timings can also be achieved. Anexperimental implementation on a two-link rigid-flexible manipu-lator is presented. The method is thereby shown to be realizableand effective for real-world motion control problems.

Index Terms—Adaptive control, flexible structure, input shap-ing, motion control, vibration control.

I. INTRODUCTION

COMMAND preshaping or input shaping can be usefullyapplied to achieve zero residual vibration (ZRV) in rest-to-

rest motion of flexible structures, robots, mechanisms, and othervibration-prone systems. Rather than involving an offline con-struction of the command profile (e.g., as in [1]), the technique isused to modify a command input online to ensure that once thecommand reaches a terminal condition, then residual excitationof vibratory modes is canceled in finite time. This must be donein such a way that the effect on the overall motion of the systemis preserved. The established concept of input shaping in motioncontrol involves impulse-based shaping [2]–[6]. A more generaldiscrete-time finite impulse response (FIR) filtering can achievethese same goals, but the increased design freedom allows forfilter solutions with lower quadratic (H2) gain and improvedhigh-frequency filtering properties [7]. A key feature of boththese techniques is that the settling time of the system is fixedby the length of the filter/impulse sequence. Also, the only re-quired model parameters are natural frequencies and dampingratios for the system modes.

Manuscript received January 2, 2011; revised March 21, 2011, June 9, 2011,and August 26, 2011; accepted October 22, 2011. Date of publication December2, 2011; date of current version September 12, 2012. Recommended by Tech-nical Editor J. M. Berg.

The authors are with Chiang Mai University, Chiang Mai 50200, Thailand(e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMECH.2011.2174373

In the basic form, input shaping involves a model-based open-loop LTI operation on a command input. Consequently, the mainlimiting factor for performance is model accuracy. Various ap-proaches to improving robustness to model errors have beenconsidered, e.g., [2], [4], [9]. However, this either requires an in-crease in the shaper duration, and corresponding increase in thesettling time, or the introduction of negative impulses, which in-curs the risk of exciting unknown high-frequency modes [7], [8].Clearly, there is scope to improve performance by adaptingthe shaper according to measurements of actual vibration. Thedrawback is the need for additional sensors and increased con-troller complexity. However, this may be weighed against thepotential benefits in controlling systems with unknown or time-varying dynamics.

One approach to adaptive input shaping is to use system iden-tification methods, where the shaping filter is tuned to match arequired design that is already known as a function of identifi-able parameters in the system transfer function. A frequency-domain identification scheme was first proposed to adjust timeseparation of shaper impulses [10]. Time-domain identificationschemes have also been considered that are computationally lessdemanding [11]–[13]. How such schemes can be successfullyapplied to multimode systems, particularly, when measurementnoise is significant, is still an open question. Other time-domainschemes include learning algorithms that adapt according tomeasured residual vibration [14]–[16]. These schemes avoidexplicit identification of model parameters but are formulatedby assuming single-mode characteristics for the residual vibra-tion. The direct adaptive input-shaping (DAIS) method of Rhimand Book [13], [17] is also based on measurement of residualvibration but has the distinction of being directly applicable tomultimode systems. This is achieved by including a sufficientnumber of impulses in the shaper sequence and adjusting onlyamplitudes; ZRV can then be attained with arbitrary impulsetimings.

The adaptive control method in this paper is based on thegeneralized FIR input-shaping approach. First, the model-basedZRV orthogonality conditions given in [7] are transformedto an equivalent condition involving sets of system I/O data,as described in Section II. The DAIS technique presented inSection III is born out of an iterative scheme for recalculationof a filter solution as more I/O data become available. A maindistinction from previous direct methods [13]–[17] is that adap-tation can occur prior to measurement of actual residual vibra-tion and repetition of maneuvers is not required. Furthermore,through application of a modified recursive least-squares (RLS)algorithm, convergence to an optimal quadratic filter solutioncan be achieved. Simulation results focusing on noise effects

1083-4435/$26.00 © 2011 IEEE

COLE AND WONGRATANAPHISAN: DIRECT METHOD OF ADAPTIVE FIR INPUT SHAPING FOR MOTION CONTROL 317

Fig. 1. Adaptive filtering of a command signal to drive a vibratory system.

are presented in Section IV. Experimental results are given inSection V. In Section VI, conclusions are given.

II. SYNTHESIS OF FIR INPUT-SHAPING FILTERS GIVING ZRV

A. ZRV Orthogonality Condition

Consider a discrete time FIR filter H of order K connectedin series with a stable linear system G with infinite-durationimpulse response, i.e., g = {g0 , g1 , g2 , . . .}. The input shaper Hhas the impulse response h = {h0 , h1 , . . . , hK }, which is thetarget of optimization/adaptation, while G and I form the systemunder control, as shown in Fig. 1. The system I represents theoverall dynamics of the target object, for which the output p is anoverall motion state and the actual target for control. The outputof G is a vibratory state that depends on u but is unrelated to theoverall motion of the target object. The input-shaping approachrequires the determination of a filter H that ensures that whenthe command/reference input r reaches and remains zero, orsome steady-state value, the output y reaches and remains zeroin finite time. This will be referred to as the ZRV condition. Theassumption here is that by driving the state y to zero, unwantedvibration components in the position state p are also eliminated.

The output y occurring in response to a command signal ris given by the convolution y = f ∗ r, where f = g ∗ h is theimpulse response of the overall system. Following an arbitrarycommand input r of a finite duration Lr , the ZRV condition,i.e., yn = 0, n > Lr + Lf , is achieved if h is chosen such thatf has a finite duration Lf :

fn = 0, n ≥ Lf ⇔K∑

k=0

gn−k hk = 0, n ≥ Lf . (1)

This requires that the impulse response series hk and gn−k

are orthogonal for all shifts n ≥ Lf . If G has no delayed directfeedthrough, then the orthogonality condition can be consideredfor n ≥ Lf = K + 1. For a finite number of shifts K + 1 ≤n ≤ N , the resulting ZRV condition can be written in the matrix

form as

ΓN h = 0

ΓN =

⎡

⎢⎢⎣

gK +1 gK . . . g1gK +2 gK +1 . . . g2

......

...gN gN −1 . . . gN −K

⎤

⎥⎥⎦, h =

⎡

⎢⎢⎣

h0h1...

hK

⎤

⎥⎥⎦. (2)

The matrix ΓN , whether formed from direct measurement, sys-tem modeling, or identification procedures, can be used for syn-thesizing an FIR input shaper h. However, in the next section,it is shown that a direct synthesis of h satisfying (2) can also beachieved using a general set of I/O data.

B. Direct Synthesis From I/O Data

Consider the following matrix description of I/O mappingsfor G, derived from the convolution sum formula:⎡

⎢⎢⎢⎢⎣

yN yN −1 . . . yN −K

yN −1 yN −2 . . . yN −K−1...

......

yK +1 yK . . . y1yK yK−1 . . . y0

⎤

⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎣

uN uN −1 . . . uN −K

uN −1 uN −2 . . . uN −K−1...

......

uK +1 uK . . . u1uK uK−1 . . . u0

⎤

⎥⎥⎥⎥⎦

⎡

⎢⎢⎣

g0 0 . . . 0g1 g0 . . . 0...

.... . .

...gK gK−1 . . . g0

⎤

⎥⎥⎦

+

⎡

⎢⎢⎢⎢⎣

uN −K−1 uN −K−2 . . . u0uN −K−2 uN −K−3 . . . 0

......

...u0 0 . . . 00 0 . . . 0

⎤

⎥⎥⎥⎥⎦

×

⎡

⎢⎢⎣

gK +1 gK . . . g1gK +2 gK +1 . . . g2

......

...gN gN −1 . . . gN −K

⎤

⎥⎥⎦

which can be written more concisely as

YN = UN Φ + VN ΓN . (3)

For a dataset of some length N ≥ K + 1 + 2M , where M isthe number of vibratory modes involved in the I/O mapping(implying rankΓN = 2M ), it follows from (3) that if h satisfiesthe ZRV orthogonality condition, i.e., ΓN h = 0, then

YN h − UN Φh = VN ΓN h = [0][N +1−K ]×1 . (4)

Noting that Φh = f = [ f0 f1 . . . fK ]T , it follows that

YN h − UN f = 0. (5)

For a given set of I/O data, f = Φh will satisfy (5), wheneverΓN h = 0. However, under certain rank conditions relating tothe choice of input signal, the existence of a vector f satisfying(5) for a given h is a necessary and sufficient condition for


ΓN h = 0. This ZRV condition, formulated in terms of I/O data,is a direct consequence of the following theorem.

Theorem 1: Given two matrices

YN =

⎡

⎢⎢⎣

yN yN −1 . . . yN −K

yN −1 yN −2 . . . yN −K−1...

......

yK yK−1 . . . y0

⎤

⎥⎥⎦

UN =

⎡

⎢⎢⎣

uN uN −1 . . . uN −K

uN −1 uN −2 . . . uN −K−1...

......

uK uK−1 . . . u0

⎤

⎥⎥⎦

where

rank[YN UN ] = K + 1 + 2M

and y ∈ RN +1 , u ∈ R

N +1 satisfy y = g ∗ u, where g is the im-pulse response of a causal delay-free system having 2M modalcomponents, then h ∈ R

K +1 gives a finite duration f = g ∗ hif and only if there exists x ∈ R

K +1 such that

YN h + UN x = 0. (6)

In this case,

fn ={−xn , 0 ≤ n ≤ K0, n > K

}.

Proof: According to the rank condition on [ YN UN ] ∈R

[N +1]×2(K +1) , there exists a set of L = K + 1 − 2M linearly

independent solutions

[hx

]

i

to the equation

[ YN UN ][

hx

]= 0. (7)

Substituting YN from (3) gives

VN ΓN h + UN (Φh + x) = 0. (8)

Therefore, the equation

[ VN ΓN UN ][

hz

]= 0 (9)

has a corresponding set of L solutions[

hz

]

i

=[

I 0Φ I

][hx

]

i

.

As g has 2M modal components and no delays, then rank ΓN =2M , and so, the equation ΓN h = 0 also has L linearly indepen-dent solutions for h. These solutions for h, together with z = 0,are the complete set of solutions to (9) and thus give the completeset of solutions to (7). This direct correspondence of solutionsimplies that there is x satisfying (6) only if ΓN h = 0, and inwhich case, it is given by x = −Φh = −g ∗ h = −f. �

Theorem 1 provides a means to calculate a solution for hdirectly from I/O data simply by finding a solution pair (h, x)that satisfies (6). The condition that [ YN UN ] has the max-imum possible rank of K + 2M + 1 requires that the inputsignal contains components that excite all the system modes.

Note that this requirement can still be satisfied for input sig-nals that already achieve ZRV as modes can be excited duringtransients. Although knowledge of the dynamics embedded ing is not necessary for synthesis, approximate values for naturalfrequencies may be useful for selecting a suitable duration forthe input shaper.

C. Extension to MIMO Systems

Multi-input multi-output (MIMO) versions of the direct syn-thesis follow in a straightforward manner. Consider, for illus-tration, a system having two input signals that excite vibratorymodes observable in measurement signal channel z(k) , where kis the channel index. The relevant two-input versions of (3) is

Y(k)N = U

(1)N Φ(k1) + V

(1)N Γ(k1)

N + U(2)N Φ(k2) + V

(2)N Γ(k2)

N .(10)

Quantities here relate to each input channel as indicated in thesuperscript. If both inputs can excite the same modes, thenorthogonality to Γ(k1)

N and Γ(k2)N are equivalent, and so, the

same filter can be applied on each input channel:

Γ(k1)N h = 0, Γ(k2)

N h = 0. (11)

Equations (10) and (11) lead to

[V

(1)N V

(2)N

][ Γ(k1)N

Γ(k2)N

]h

= Y(k)N h − U

(1)N Φ(k1)h − U

(2)N Φ(k2)h = 0. (12)

The corresponding condition for ZRV, in terms of I/O data, isthe existence of (x(k1) , x(k2)) satisfying

Y(k)N h + U

(1)N x(k1) + U

(2)N x(k2) = 0. (13)

This follows from the same line of argument as given forTheorem 1, i.e., the complete set of L = K + 1 − 2M solu-tions (h, x(k1) , x(k2)) satisfying (13) are those for which h sat-isfies Γ(k1)

N h = 0, Γ(k2)N h = 0. The rank condition that must

be satisfied in this case is that rank[Y

(k)N U

(1)N U

(2)N

]=

2(K + 1) + 2M . With multiple output measurement signals, aset of equations in (13), one for each measurement channel, canbe considered for which a common solution for h must be found.

D. Exact Solutions

For sufficiently large filter order K, there will be no uniquesolution to (6), and so, additional specifications may be consid-ered. These may relate to additional control requirements, suchas bounds on control signals or on filter gains, required settlingtime, etc. Model-based designs, where combinations of these re-quirements were considered, with solutions based on (1), werepresented in [7]. Usually, design requirements for h will includethe normalization condition

K∑

k=0

hk = 1. (14)

This ensures that steady-state values of the filter input and outputsignals are equal and that the total area is conserved (

∑u =


∑r). Condition (14) thereby helps to ensure that the net effect of

a command input, in terms of the overall motion of the controlledobject, is preserved. When hk are all nonnegative, condition(14) also ensures that peak absolute values of the shaped inputsignal u never exceed the peak absolute values of the unshapedcommand r, which is a useful characteristic when there arelimits for control input signals that should not be exceeded.

To ensure condition (14) is satisfied, the filter impulse re-sponse can be formed as

h = Tw + b. (15)

Here, b is a predetermined vector given by

b =1

K + 1

⎡

⎣1...1

⎤

⎦

[K +1]×1

(16)

and T ∈ R[K +1]×K is an orthogonal complement of b so that

K∑

k=0

hk = (K + 1) bT h = (K + 1) bT b = 1. (17)

The vector w ∈ RK parameterizes the remaining design free-

dom. If T is chosen such that TT T = I, then

hT h = (Tw + b)T (Tw + b) = wT w + bT b (18)

and so, the norm of h and w have the simple correspondence‖h‖2 = ‖w‖2 + 1

K +1 .Considering (15) in (6) and defining the error in (6) as the

ZRV error prediction e = {e0 , e1 , . . . , eN } give

e = [YN T UN ][

wx

]+ YN b (19)

which implies that an ideal filter solution would achieve e = 0.If the matrix YN is constructed from measured signal data, itmay be subject to noise, disturbances, measurement error, andthe possibility that plant behavior is not perfectly linear. Insuch cases, we cannot expect [YN UN ] to be rank deficient,and so, a solution giving e = 0 is implausible. This issue is themain consideration when constructing a solution from I/O data,whether offline from a batch of data or within a real-time adapta-tion algorithm. For the offline calculation, it should be possibleto separate the components of the matrix [YN T UN ] associ-ated with the system dynamics from those due to noise/error.For example, the solution may be constructed from the first(K + 1 + 2M ) dominant components obtained by a singular-value decomposition P = [YN T UN ] = UΣVT . In the idealcase, the diagonal matrix, i.e., Σ = diag{σ1 , σ2 , . . . , σ2K +1},contains only K + 1 + 2M nonzero singular values, and thus,

a reduced order construction has the form P = U ΣVT, where

only K + 1 + 2M rows/columns have been retained. The so-lution that minimizes e and for which ‖w‖2 + ‖y‖2 takes theminimum value follows as

[wx

]= −V Σ

−1UTYN b. (20)

This solution is similar to the minimum quadratic (H2) gainshaper presented in [7] except that it is optimal in the sense

that the combined quadratic cost ‖h‖2 + ‖f‖2 is minimized,rather than only ‖h‖2 . The quadratic gain, which is equal tothe Euclidean norm of the impulse response vector, provides abound on the I/O mapping according to

|un | ≤ ‖h‖ ‖r‖, |yn | ≤ ‖f‖ ‖r‖, r = [rn−K−1 , . . . , rn ]T .(21)

Therefore, solution (20) will help minimize peak values ofshaped input and transient vibration response. Also, unlikeimpulse-based shapers, the solution h contains no isolated im-pulses. Equation (20) corresponds to the case, where there isequal penalty weighting of ‖f‖ and ‖h‖. However, a scaling ofUN or YN can be used when a cost of the form ‖h‖2 + β‖f‖2

is more usefully minimized.For an adaptive algorithm, the cost of calculation (20) may

be too high for real-time operation if a large batch of I/O datais to be considered in each update. Therefore, algorithms basedon the recursive solution are proposed that can adaptively tuneinput-shaping filters to achieve ZRV. The approach describedin the following section is based on minimizing the predictionerror e through continuous updates of the filter coefficients in h.

III. ADAPTIVE FIR INPUT SHAPERS

A. RLS Algorithm

A least-squares solution for (19), which minimizes the costJ = eT e, can be calculated (nonrecursively) from[

wx

]

N

= −(γ−1I +

[TT Y T

N

UTN

][YN T UN ]

)−1 [TT Y T

N

UTN

]YN b.

(22)The term γ−1I prevents the inverted matrix being singular andensures that the filter converges to the optimal quadratic solutionfor h. These points will be explained further once the RLSalgorithm has been described. Equation (22) has the form

[wx

]

N

= −S−1N qN (23)

where the updates to these quantities at each time step are givenby

qN +1 = qN + sN ψTN b, SN +1 = SN + sN sT

N

sN =[

TT ψN

υN

],

ψN = [ yN yN −1 . . . yN −K ]T

υN = [uN uN −1 . . . uN −K ]T .

(24)

To avoid calculating the inverse of the correlation matrix SN

every time step, which may be impossible for high-order filters(large K) due to the computation time required, the inversematrix RN = S−1

N can be updated directly according to thematrix inversion formula

RN +1 = RN −(

11 + sT

N RN sN

)RN sN sT

N RN . (25)

The resulting algorithm is similar to a standard RLS adaptivefilter except here the error eN relates to the ZRV predictionrather than an actual output error. The N th filter update is de-fined by [h]N = T [w]N + b and the adaptive states are updated


according to

eN = sTN

[wx

]

N

+ ψTN b (26)

dN = −(

11 + sT

N RN sN

)RN sN (27)

[wx

]

N +1=

[wx

]

N

+ dN eN (28)

RN +1 = (I + dN sTN )RN . (29)

With this algorithm, the matrix R and vectors w and x should,in theory, remain bounded. However, the potential for numericalinstability should be recognized.

The algorithm is initialized with R0 = γI, and so, the con-vergence rate is dependent on the choice of γ, with larger valuesgiving faster convergence. Nevertheless, providing the system ispersistently excited, the algorithm will converge on the optimalquadratic solution given by (20). To prove this, consider againthe singular-value decomposition PN = [YN T UN ] = UΣVT .Then, from (22),[

wx

]

N

= −(γI + PTN PN )−1PT

N YN b (30)

= −V (Σ2 + γI)−1ΣUT YN b (31)

= −V diag{

σ1

γ + σ21, . . . ,

σK +1+2M

γ + σ2K +2M +1

}UT

YN b.

(32)

In the limit as N → ∞ and γ/σi → 0, then[

wx

]

N

→ −V diag{

1σ1

, . . . ,1

σK +1+2M

}UT

YN b (33)

which is solution (20).

B. Adaptive Impulse-Based Shapers

Impulse-based shapers are essentially time delay filters and socan also be realized as discrete-time FIR filters. The describedalgorithm can thus be applied directly for adaptive tuning if thetiming of impulses is fixed in advance. A shaper with Q impulsesand the total duration K + 1 is realized by an FIR filter with theimpulse response

h = A h (34)

where h ∈ RQ is the vector of impulse amplitudes and the matrix

A ∈ R[K +1]×Q contains ones and zeros to assign the elements

of h to the appropriate nonzero coefficients of h. The structureof the filter synthesis problem is unchanged if the (reduced)filter coefficient vector h is expressed as

h = Tw + b, b =1Q

⎡

⎣1...1

⎤

⎦

Q×1

, T = b⊥. (35)

In this case, the parameterization is over w ∈ RQ−1 and the

ZRV prediction error is given by

eN = sTN

[wx

]

N

+ ψTN Ab, sN =

[TT AT ψN

υN

]. (36)

Otherwise, the algorithm will remain unchanged.It should be recognized that it is the full-order FIR filter that

arises as a general solution to the discrete-time equation forZRV (6). Filters with isolated impulses are a restricted subsetof solutions obtained only when additional constraints are im-posed. In one sense, the full-order filter can be considered as thecase where the maximum number of impulses is used. However,the properties of the filter are better understood by treating theimpulse response of the filter as a continuous function ratherthan a series of impulses [7]. This is in fact the reality whenthe smoothing action of the digital-to-analog conversion is alsotaken into account. Note that the term “order” is being used hereto refer to the number of nonzero coefficients and not the actualfilter order K.

C. Noise Effects

The ZRV error prediction (19) may be expressed in terms ofthe FIR filter operations for h and f as

e = Hy + Fu (37)

where y = y + d is a measurement contaminated by randomwhite noise d with zero mean value and variance σ2 = E(d2).The corresponding noise-perturbed quadratic cost for a dataseries of length N is

J = eT e = hT Y TN YN h + 2hT Y T

N UN f + fT UTN UN f (38)

where YN = YN + DN . The expectation value of the noise-perturbed cost is given by

E(J) = J + hT E(DTN DN )h (39)

= J + σ2N‖h‖2 . (40)

This simple analysis shows that the presence of noise will penal-ize the quadratic norm/gain of h, and thus, it should be expectedthat adaptation will produce a suboptimal solution, where ‖h‖ isreduced at the expense of allowing some residual vibration. Theeffect of noise will be lessened if the desired shaper h alreadyhas the low quadratic norm. The minimum possible norm (sub-ject to

∑h = 1) occurs when h has all equal coefficients. These

results suggest that better noise performance may be expectedfor filters with more impulses and longer durations as both canhelp to reduce ‖h‖.

IV. SIMULATION RESULTS

A simulation study has been carried out principally to ex-amine the influence of measurement noise, which is the mainlimiting factor for achievable performance. The study is basedon a benchmark system previously considered in [13], which isa single-degree-of-freedom spring–mass–damper system with


Fig. 2. Reference trajectory used for simulation and corresponding responsewithout input filtering.

the transfer function

G(s) =Y2(s)Y1(s)

=2ζωns + ω2

n

s2 + 2ζωns + ω2n

. (41)

The output y2 is the displacement of the mass, while the inputy1 is the displacement at the other end of the spring/damper.The model parameters are ωn = 40π rad/s and ζ = 0.1. Thesampling period for control/adaptation is Ts = 0.001 s.

The unmodified trajectory for the displacement y1 and re-sulting acceleration of the mass are shown in Fig. 2. The mo-tion pattern is chosen to be smooth but with no intervals ofuniform acceleration so that actual residual vibration can onlybe measured or evaluated when motion stops (after 1 s). Themotion pattern thus presents a challenging case to demonstrategood adaptation and residual vibration suppression. Results withadaptive input shaping are shown in Fig. 3. A 3-impulse filter(Q = 3) has been chosen to allow easy visualization of thechanging coefficient values. The filter impulse response has thetotal duration of K + 1 = 41 samples but the filter has onlythree equally spaced nonzero coefficients. These results showclearly that adaptation occurs in response to measurement of theinitial vibration so that a near-optimal filtering is achieved by thetime motion stops, and residual vibration is effectively canceled.In Fig. 3(a), the adaptation is based on noisy measurements ofmass acceleration with noise variance σ2 = 0.1, which resultsin a signal-to-noise ratio (SNR) of 21 dB. Results with a highernoise level (σ2 = 1) are shown in Fig. 3(b). It can be seen thatat the start of motion the filter coefficients adapt at a similarrate, and to similar values, as in the previous case. However, af-ter approximately 0.4 s, the measured vibration becomes lower,and so, the effect of noise, which tends to drive the coefficientstoward equal values, can be seen more clearly. This behavior canbe understood in terms of the algorithm trying to minimize theperturbed cost (40), which penalizes ‖h‖2 in proportion to σ2 .Nonetheless, residual vibration is still significantly reduced andit can be concluded that the adaptation algorithm shows goodrobustness to measurement noise.

To allow comparison with full-order FIR filters, simulationswith filters of various total durations were undertaken. Conver-gence of the filter solution was achieved under repetition of themotion pattern in Fig. 2. Levels of residual vibration for both

Fig. 3. Simulation results for the adaptive 3-impulse shaper with measurementnoise (a) SNR = 21 dB and (b) SNR = 11 dB.

Fig. 4. Comparison of residual vibration due to noise-related errors for a rangeof filter durations and types (SNR = 58 dB).

impulse-based filters and FIR filters are shown in Fig. 4. Thenoise levels are the same in all cases (SNR = 58 dB). Keyresults concerning noise effects are that the 3-impulse filter per-forms comparatively poorly when the duration is close to twicethe natural period (as the required solution has large negativeimpulses) and that lower residual vibration is achieved with afull-order filter for all durations considered.

In general, a system with M vibratory modes requires a seriesof Q ≥ 2M + 1 impulses to ensure vibration cancellation canbe achieved with arbitrary time separations (as noted in [17]).


Fig. 5. Comparison of filter solutions obtained from the simulation for thecase K = 40. (a) Filter coefficients. (b) Filter frequency response magnitude.

This is because, in addition to the constraint∑

h = 1, there aretwo linear constraint equations (orthogonality conditions) for hassociated with each mode [7]. However, using the minimumvalue for Q may be a poor choice for adaptive input shaping withfixed timings as, depending on the system natural frequency, therequired solution may have large quadratic norm ‖h‖ and thusbe prone to noise errors.

A. Characteristics of Optimized Filter Solutions

Final solutions for the case K = 40, obtained under repetitionof the motion pattern in Fig. 2, but with no measurement noise,are shown in Fig. 5. The frequency response of the full-orderfilter is distinct in character when compared with the impulse-based filters as it is aperiodic and has low-pass properties. Notethat all three filters give exact cancellation of residual vibra-tion as they all have a sufficient number of impulses and wereobtained without measurement noise.

For a fully adapted filter, the settling time of the system isequal to the shaper duration (K + 1)Ts . Therefore, decreasingK will reduce the time taken to extinguish residual vibration

Fig. 6. Filter solutions with different total durations. A: Minimum durationpositive filter solution (K = 26). B: Full-order filter solution (K = 26). C:Minimum duration positive full-order filter solution (K = 35).

following any given command input. A key issue in relation tothis is whether to allow negative coefficients/impulses. In situ-ations where the shaped input is used directly as an actuationsignal, it may be useful to impose the nonnegativity constraintshk ≥ 0 as this will ensure that peak values of the shaped signaldo not exceed those of the original command input. In this case,the shortest duration solution is the 2-impulse ZV shaper de-sign (see [2]). Some indirect methods of adaptation have beenproposed that involve varying impulse timings, as well as am-plitudes, in order to achieve such a solution [10]–[15].

For the algorithm in this paper, a nonnegativity constraint isnot imposed explicitly. Nonetheless, a positive shaper solutioncan always be achieved by adjusting the value of K until the non-negativity condition is satisfied. In doing this, a 3-impulse shaperwill achieve the minimum-duration positive solution when themiddle impulse approaches zero. A full-order filter solution ofthe same duration will have some negative coefficients. In Fig. 6,both these solutions are shown, as well as the minimum-durationpositive full-order solution, which has a duration 23% longer.

In situations where the filter is used to shape a positionreference signal for a servo control loop, the inherent input-smoothing property of the full-order filters, derived from the factthat the impulse response of the filter has no isolated impulses,may be advantageous. For a given input signal, the smoothinghelps to reduce errors within the feedback loop and can thusallow faster overall motions without causing saturation of ac-tuators [7]. In such situations, the positivity constraint may notbe useful, and so, how short the shaper can be made must bedecided based on an alternative measure, such as the value ofthe quadratic gain.

V. EXPERIMENTAL RESULTS

A. Experimental Flexible Arm Manipulator

The test system used in the experimental study is a lab-scaletwo-link planar manipulator (see Fig. 7). The first link is a stiffaluminum beam with length 210 mm. The end link consists ofa thin acrylic beam of length 220 mm with a brass tip-mass thatcan be changed in order to select the natural frequency of the firstflexible mode within the range 1–5 Hz. The joints are actuatedby dc motors using harmonic drives to eliminate backlash. Thejoint angles θ1 and θ2 are measured by incremental encoders.Strain gauges fitted close to the base of the end link measure


Fig. 7. Two-link manipulator with lightweight flexible end link.

Fig. 8. Schematic of control structure for shaping joint-angle referencetrajectories.

TABLE ISERVO CONTROL LOOP PARAMETERS

bending and this provides a vibratory signal, which can be usedto evaluate link deflection and residual vibration. This signal isalso the vibration variable y used in the adaptation algorithm.The system is operated in the horizontal plane to minimizegravitational effects.

To achieve high positioning accuracy, feedback control ofjoint angles is used with input shaping/filtering applied to thereference signals, as shown schematically in Fig. 8. The joint-angle control loops operate with high gain PD feedback of themeasured joint angles. The system configuration is similar tothat considered in a comparative study of standard input-shapingand alternative control techniques [18]. Further information onthe servo control loops is given in Table I.

The servo control loop and adaptive input shaper algorithmwere implemented on a PC-based control system (AMD DualCore 2.8 GHz processor). Clearly, the digital processor speedwill limit the achievable sampling rate and filter order. For thecurrent system, a sampling frequency of 1000 Hz was usedfor the servo control loops. However, the input-shaping filterand adaptation iterations were operated with a lower samplingfrequency of 500 Hz. An input-shaping filter with duration equalto half the natural period of vibration has the filter order K closeto 100.

B. Rest-To-Rest Operation of End Link Only

The first set of results presented involves operation of the endlink only. This ensures linear system dynamics and allows highersampling rates and filter orders than would be possible with op-eration of both links (two-input case). These results provide areference set with useful indications of achievable performancefor a single-degree-of-freedom system. A piecewise linear po-sition command with constant speed intervals is considered thatproduces rest-to-rest motion of the end link through rotationsof ±60◦. The command profile is repeated in order to assessadaptation.

1) Full-Order FIR Filter: System behavior with adaptivefiltering of the joint-angle reference command is shown inFig. 9(a). The angular position of the tip mass was calculated bysumming joint angle measurements with the angular displace-ment error estimated from measurements of link strains. Forthe first 8 s of operation the command signal was unmodified.The maximum angular error in the tip position was about 16◦,while the settling time for vibration was approximately 4 s (for2% settling criteria). The natural frequency of vibration witha 5-g tip mass was 2.85 Hz. After the first motion interval, theadaptive filtering was activated. The filter, operating with a sam-pling frequency of 500 Hz, has an order of 81 and, therefore,has an impulse response duration of 0.16 s, or 0.456 vibrationperiods. The RLS algorithm (26)–(29) involves a matrix R oforder 2K + 1 = 161. The adaptive algorithm was initializedwith γ = 0.01, which gave moderate rates of initial adaptationbut still resulted in a significant vibration reduction after onlyone motion cycle. The actuation signal shown is the outputsignal of the PD controller within the servo control loop andcorresponds to the motor current. Saturation limits were set at±0.8 A in accordance with the peak current rating of the motor.Note that, because the full-order FIR filtering removes first-order discontinuities from the command input, the large spikesin the actuation signal that occurred prior to filtering are nolonger present. After five to six motion cycles further improve-ments in residual vibration were unnoticeable [see Fig. 9(b)].Note that the effect of the filtering is to modify the command in-put over the time interval of 0.16 s that follows each (first-order)discontinuity in the unshaped command. Residual vibration ofthe link is extinguished by the end of this interval.

The effect of using different values for γ is shown in Fig. 10.Larger values lead to faster convergence. However, a value thatis too large (in this case, around γ = 1) can cause larger oscil-lations in the initial vibration of the system.


Fig. 9. End-link motion under adaptive input shaping with full-order filter. Command filtering and adaptation start after 8 s. (a) Initial response. (b) Responseafter convergence.

Fig. 10. End-link vibration under adaptive input shaping with different initializations R0 = γI .

2) Three-Impulse Filter: Results for the same motion patternbut using a 3-impulse adaptive filter are shown in Fig. 11. Thefilter impulse response has only three equally spaced nonzerocoefficients (Q = 3). The total duration is the same as the previ-ous case, and so, the matrix R has the order K + Q + 1 = 84.The filter is initialized as h = { 1

3 , 13 , 1

3 }, in accordance with(35) and w = 0. Coefficients adapt toward the optimal valuesfrom the start of motion. Note that, for these results, the filterwas active and adapting from 0 s. Also, the filter adaptation ratewas set quite high (γ = 0.1), such that after the initial phase ofmotion any further vibration excitation is effectively canceledby the shaping filter.

According to the theoretical arguments previously outlined,the 3-impulse shaper that achieves ZRV on a single-mode sys-

tem is a unique solution. For a shaper with duration of 0.5vibration periods, the solution is the 2-impulse ZV shaper, forwhich only the two end coefficients are nonzero. Here, the filteris slightly shorter in duration and so has a negative impulse:h = {0.62,−0.15, 0.53} (see [8]). Note that, as the filter con-tains discrete impulses, spikes in the motor current caused byfirst-order discontinuities in the reference input are still evidentunder input shaping. Peak values of the current can be comparedwith those shown in Fig. 9 for the full-order filter. The effectthat input smoothing has on the peak values is clearly evident.Nonetheless, for these experimental conditions, saturation wasavoided, and so, residual vibration was not noticeably differentfor each case.


Fig. 11. End-link motions with an adaptive three-impulse shaper.

C. General Planar Motions

In this section, attention is turned to operations with simulta-neous rotation of both links. This involves a significant increasein the computational load. Separate (identical) shaping filtersoperated on each command coordinate. To use the full-orderfilter, the sampling frequency was reduced to 250 Hz and filterlength set to K + 1 = 41, giving the total duration 0.16 s.

The motion pattern shown in Fig. 12 involves a single rest-to-rest maneuver with smooth command signals that producesignificant vibration excitation only at the start and end of mo-tion. This presents a fairly challenging task for successful adap-tation and residual vibration cancellation. The initial configura-tion (θ1 = 0, θ2 = 0) involves alignment of both links. Withoutinput filtering, large deflection (>20◦) and residual vibrationoccurred in the end link [see Fig. 12(b)]. With adaptive filtering,the algorithm reacts to the vibration measured at the start of mo-tion to achieve immediate and effective vibration cancellationat the end of the maneuver [see Fig. 12(c)]. The final result is a93% reduction in residual vibration. Note that this result cannotbe obtained with algorithms that utilize measurement of residualvibration only, as the input signal is constantly fluctuating andthere can be no direct measurement of residual vibration untilafter the motion cycle has ended.

Although the input-shaping methodology has been developedprimarily for point-to-point motions, for some applications, itmay be necessary to consider the effect on the spatial path(locus) of motion. Previous studies have shown that significantimprovements in path following can be achieved with appli-cation of fixed model-based designs [19], [20]. To explore theeffect of adaptation on path following, a five-segment constant

Fig. 12. Test case involving single rest-to-rest maneuver with two-link mo-tion. (a) Command signals. (b) Response without adaptive input shaping.(c) Response with adaptive input shaping.

speed (150 mm/s) trajectory for the tip mass was considered.The corresponding command signals for the joint angles werecalculated from rigid-body inverse kinematics. When these sig-nals were used directly for control the results in Fig. 13(a) wereobtained. Tracking of the joint command signals caused signif-icant vibration of the end-link during motion, as evident in thetrajectory of the tip mass. The arc of motion about the final po-sition gives a clear indication of residual vibration. As previous,tip position is estimated from strain measurement at the base ofthe end link.

Adaptive filtering was applied to the joint angle commandsignals using a five-impulse shaper (Q = 5) with the samplefrequency 500 Hz and the total duration K + 1 = 81 (0.16 s).The filter was initialized with equal impulses. Tracking alongthe path was improved significantly under adaptation and levelsof residual vibration at the end of motion greatly reduced [seeFig. 13(b)]. Modification of the path of motion at the cornerpoints is due to the temporal overlap of changing x and y com-mands. This is caused by shaper delay and can be overcomeby including a dwell period between motion segments equalto the shaper duration [20]. These results demonstrate that theadaptation algorithm can work successfully on a two-input sys-tem driven simultaneously by two differing command inputs.For nonlinear MIMO systems, there is motivation to considerapplying the input shaping in a general transformed coordinatespace. This might be with a view to improving path following


Fig. 13. Rest-to-rest motion with the piecewise rectilinear command trajectoryfor tip mass. Adaptive input shaping operates on joint coordinates. (a) Unshapedtrajectory. (b) Trajectory shaping with an adaptive filter.

or providing compensation for varying modal dynamics, whichin general will be configuration dependent. Tests with adaptivefiltering applied to the reference trajectory in the Cartesian co-ordinates (prior to transforming to joint coordinates) have alsobeen undertaken and show similar levels of residual vibrationreduction. These results indicate good robustness to such co-ordinate transformations and, though beyond the scope of thecurrent paper, provide motivation for further development inthis area.

VI. CONCLUSION

In this paper, we have introduced an adaptive control methodbased on the FIR input-shaping methodology for achieving ZRVin rest-to-rest motion of flexible structures and other vibratory

systems. Mathematical conditions for ZRV have been derivedthat allow shaping filter solutions to be calculated directly fromarbitrary sets of system I/O data. This provides a main contribu-tion of the current study over previous time-domain approachesto adaptive input shaping. Recursive algorithms for real-timeupdating of filter coefficients have also been proposed and eval-uated.

The method can be applied with very little knowledge ofthe system dynamics and is directly applicable to multimodesystems, which would introduce considerable complications foridentification-based schemes. Results indicate that, with appro-priate initialization of the algorithm, good robustness and fastconvergence rates can be achieved. Finally, it should be empha-sized that the technique is unlike repetitive control or learning-based algorithms in that the convergence to a ZRV filter solutioncan be achieved for arbitrary nonrepetitive maneuvers.

REFERENCES

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[7] M. O. T. Cole and T. Wongratanaphisan, “Optimal FIR input shaper de-signs for motion control with zero residual vibration,” Trans. ASME, J.Dyn. Syst., Meas., Control, vol. 133, pp. 021008-1–021008-9, 2011.

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[18] Z. Mohamed, J. M. Martin, M. O. Tokhi, J. Sa da Costa, and M. A. Botto,“Vibration control of a very flexible manipulator system,” Control Eng.Pract., vol. 13, pp. 267–277, 2013.

[19] V. Drapeau and D. Wang, “Verification of a closed-loop shaped-inputcontroller for a five-bar-linkage manipulator,” in Proc. IEEE Int. Conf.Robot. Autom., 1993, vol. 3, pp. 216–221.

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Matthew O. T. Cole was born in Leamington Spa,U.K., in 1971. He received the M.A. degree in natu-ral sciences from the University of Cambridge, Cam-bridge, U.K., in 1995. He received the M.Sc. andPh.D. degrees in mechanical engineering from theUniversity of Bath, Bath, U.K., in 1995 and 1999 re-spectively.

Since 2003, he has been a Faculty Member at Chi-ang Mai University, Chiang Mai, Thailand, where heis currently an Associate Professor in the Departmentof Mechanical Engineering. His research interests in-

clude the application of signal processing and control technology in machinesystems.

Theeraphong Wongratanaphisan was born inPetchaburi, Thailand, in 1972. He received the B.Eng.degree in mechanical engineering from Chiang MaiUniversity, Chiang Mai, Thailand, in 1993. He wasawarded a scholarship by the Thai Government tostudy at Lehigh University, Bethlehem, PA, fromwhere he received the M.S. and Ph.D. degrees in me-chanical engineering in 1996 and 2001, respectively.

Since 1994, he has been with the Departmentof Mechanical Engineering, Chiang Mai University,where he is currently an Associate Professor. His re-

search interests include system design, control, and robotics.

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