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Trajectory planning for the tracking controlof systems with unstable zeros

Hyung-Soon Park, Pyung Hun Chang *, Doo Yong Lee

Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology,

Science Town, Taejon 305-701, Republic of Korea

Received 12 May 2000; accepted 31 May 2001

Abstract

In this paper, a trajectory control strategy for a nonminimum phase system is proposed. A

continuous-time version of the zero-phase error-tracking controller (ZPETC), a well-known

discrete-time feedforward controller for tracking control of a nonminimum phase system, is

derived. The performance of the continuous ZPETC is further enhanced by adopting a spe-

cially designed sinusoidal trajectory to compensate gain error. The sinusoidal trajectory has asynergic effect on tracking performance when combined with the continuous ZPETC. The

effectiveness of continuous ZPETC with sinusoidal trajectory is confirmed through simulation

and experiment of a nonminimum phase plant, a single link flexible arm.

2002 Elsevier Science Ltd. All rights reserved.

Keywords: Trajectory control; Trajectory generation; Unstable zero; Phase error; Gain error; Flexible arm

1. Introduction

It is well known that trajectory control may be categorized into two groups [11]:one is for a known desired trajectory and the other is for an unknown trajectory.

This paper concerns the former, especially when the trajectory can be expressed as an

explicit function of time.

Given a desired trajectory, the transfer function between the trajectory input and

the actual output should be unity for perfect tracking. This may be achieved by

introducing a feedforward controller (or a prefilter) that acts as an inverse of the

closed-loop system, cancelling the poles and zeros of the closed-loop system. In this

Mechatronics 13 (2003) 127139

* Corresponding author. Tel.: +82-42-869-3226; fax: +82-42-869-3210.

E-mail address: [email protected](P.H. Chang).

0957-4158/03/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.

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regard, the poles of the plant can be modified to desired locations by a feedback

controller so that they can be cancelled easily; even robustness against disturbances

can be assured [4] by a robust feedback control. But, the zeros of the plant are not

influenced by a feedback control; they can only be cancelled by a feedforwardcontroller.

In the nonminimum phase system that has unstable zeros, however, the unstable

zeros cannot be cancelled directly. These zeros are problematic because they cause

the phase and the gain errors in tracking.

The problem caused by the unstable zeros can be resolved, on the one hand, by

redefinition of the output variables [8] in a passive way. In an active way, on the

other hand, it can be solved by a specially designed control scheme. Zero-phase

error-tracking controller (ZPETC), originally developed by Tomizuka [12], elimi-

nates phase error caused by uncancelled zeros. It has attracted attention owing to its

effectiveness and simplicity. ZPETC itself has good tracking performance; further-more, the robustness of the overall control system can be assured when it is com-

bined with a robust feedback controller.

However, a continuous-time system must be converted to discrete-time to design

ZPETC. This transformation often requires quite an effort and may introduce ap-

proximation errors. As a remedy to this difficulty, we have developed a continuous-

time version of ZPETC in this paper.

The continuous-time version of ZPETC may suffer from noise problems associ-

ated with differentiating commands. Yet, if the command is pre-definedas in the case

of trajectory planning, the noise problems are avoidable. For most motion control

systems, the trajectory is designed and exerted as a command, i.e., the trajectory canbe designed so as to avoid such noise problems.

ZPETC cancels only the phase error but not the gain error, which becomes larger

for fast tracking. The gain error has undesirable effects on the tracking performance

such as residual vibration and velocity error at via points. There has been some

research work to address this problem. Adding zeros to ZPETC can enlarge the

frequency region of unity gain [7]. ZPETC with optimal gain guarantees the gain

error to be minimal for a given frequency range [5], and series approximation of

unstable zeros reduces the gain error [6].

Instead of reducing the gain error through modifying system characteristics, we

attempt to achieve the same goal through trajectory design. By actively using the

condition that the trajectory be pre-defined, we design it in such a fashion that it can

explicitly compensate for the gain error. To this end, we adopt a trajectory planning

method that generates trajectories consisting of sinusoidal components with pre-

scribed frequencies. In our approach, hence, a concurrent design is attempted for

both the trajectory and the control system to maximize the control performance.

The intended utility of the trajectory, therefore, is twofold: to provide a smooth

trajectory to avoid noise problems; and to reduce the aforementioned gain error.

Note that the trajectory is also meant to satisfy kinematic constraints at a series of

pre-determined via points.In Section 2, the continuous-time version of ZPETC is developed. Section 3

presents the method to design a smooth sinusoidal trajectory optimizing the tracking

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performance. Verifications are shown in Section 4, through simulation and experi-

ments with a flexible arm which is known as a typical nonminimum phase system.

Finally, conclusions are drawn in Section 5.

2. Continuous zero-phase error-tracking controller

Zeros are usually observed in mechanical systems that have noncolocated sensors

and actuators, and the stability of zeros is related to the energy propagation char-

acteristics. Unstable zeros are found in dispersive flexible systems that have non-

propagating waves [10]. In discrete-time domain, fast sampling rate introduces

additional unstable zeros for systems which have a relative degree greater than 2 [1].

The unstable zeros, both in continuous-time domain and discrete-time domain, may

not be cancelled directly by a feedforward controller. These uncancelled zeros lead totwo types of tracking error: phase error and gain error.

Continuous ZPETC (CZPETC) can be designed for the reduction of the phase

error in a similar way with ZPETC [12]. Let us consider a transfer function expressed

in continuous-time domain

ys

rs

Ns

Ds; 1

where

Ds a0 a1sa2s2 ansn;

Ns b0 b1sb2s2 bms

m nPm:

For a nonminimum phase system, Ns can be divided into two parts

Ns NasNus; 2

where

Nas ba0 ba1s b

amks

mk stable zeros; and

Nus bu0bu1s b

uks

k unstable zeros;

k is a natural number less than m:

Let the CZPETC be defined by the following equation:

rs

yds

Ds

Nas

Nus

Nu02: 3

Then the overall transfer function between the input, yds, and the output, ys, is

ysyds

NusNusNu0

2 : 4

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Substituting jx for s,

Nujx bu0 bu1jx b

u2 jx

2 bukjx

k

ReNu j ImNu;

Nujx bu0 bu1jx b

u2 jx

2 1kbukjxk

ReNu j ImNu

and therefore

yjx

ydjx

Re2Nu Im2Nu

Nu02

jNujxj2

Nu02 : 5

Eq. (5) shows that the overall transfer function between the desired trajectory and

the output does not have an imaginary part; thus the phase shift is always 0 for all

the frequencies.In Eq. (3), the numerator has an order greater than the denominator, which

implies CZPETC acts as a high pass filter. Thus CZPETC may seem to be sus-

ceptible to noise amplification. But this does not cause a problem when the input is

a known smooth function as in most cases of planned trajectories. In other words,

the differentiation is free from noise when the derivatives of ydt are given ana-lytically.

In Eq. (5), the gain of the overall transfer function is approximately equal to unity

for lower frequencies, but it becomes larger than unity for higher frequencies. This

causes the gain error for fast tracking applications. In point-to-point motions, this

also causes velocity error at the final point resulting in residual motion after the finalpoint. This gain error is also problematic for trajectories which go through pre-

defined via points.

However, as mentioned by Tomizuka [12], ifydtis a single frequency sinusoidalsignal with a known frequency component, x, the overall transfer function can be

rescaled by replacing Nu02 with Nux2. Furthermore, if the trajectory input canbe designed to have a finite number of frequency components, the gain error can be

compensated by the modified design of CZPETC, i.e.,

ris

ydis

DsNus

NusjNujxij2; 6

where ydi is a component of the trajectory input that has the ith frequency, xi. The

method to design such a trajectory with finite frequency components, and the cor-

responding feedforward controller is discussed in the following section.

3. Trajectory planning method

3.1. Review of trajectory planning methods

There are two common approaches to planning trajectories [2]. In the first ap-

proach, a desired path is described explicitly by an analytic function, and the goal is


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to determine the trajectory whose resulting path closely approximates the desired

path. In the second approach, the trajectory is determined by specifying the position,

velocity, and acceleration at a number of points along the desired path.

One of the widely used methods of the second approach, polynomial trajectoryplanning, needs a relatively small amount of computation, and can satisfy the ki-

nematic constraints [3]. A polynomial trajectory with CZPETC (or ZPETC) pro-

duces no phase shift between the trajectory input and the output. However, in

general there will still be a gain error because we do not know the frequencies of the

polynomial trajectory, a priori. If the trajectory has a known finite number of fre-

quencies, the gain error can be reduced to 0. In the following, we introduce a tra-

jectory planning method which can make the generated trajectory have finite

frequency components.

3.2. Sinusoidal trajectory planning

Consider the following trajectory that is composed ofn sine functions:

ydt A0Xni1

Aisinxitt0 t06 t6 tf; 7

wheret0, tf and Ai i 1; 2;. . .;n are the initial time, the final time and the gain ofeach sinusoidal function, respectively. The unknown variables of the above equation

are xis and Ais. We can assign arbitrary values to xis, which are less than

p=tft0and not duplicated. Then, the n unknownAis can be determined from theuser-defined constraints at t0 and tf.

A trajectory which has the position, velocity and acceleration constraints att0or tfcan be determined through the following procedure. Substituting t0 into Eq. (7) gives

ydt0 A0. Hence,A0is uniquely determined by the initial position constraint. Otherconstraints can be expressed in the following way:

A0A1 sinx1Dt A2 sinx2Dt A3 sinx3Dt A4 sinx4Dt ydtf;

A1x1A2x2A3x3A4x4 _ydydt0;

A1x1 cosx1Dt A2x2 cosx2Dt A3x3 cosx3Dt A4x4 cosx4Dt _ydydtf;

A1x21 sinx1Dt A2x

22 sinx2Dt A3x

23 sinx3Dt A4x

24 sinx4Dt yydtf;

where Dt tft0.In the above equations, xis i 1; 2; 3; 4 have different values which are arbi-

trarily chosen to satisfy xi < p=tft0 so that ydt may not have a fluctuatingshape. Then the unknown variables,Ai i 1; 2; 3; 4, are determined from the aboveequations. This procedure can be straightforwardly extended to generate trajectories

with more constraints.

Note that differentiating Eq. (7) twice and substitutingt0 give trivial constraint on

acceleration at t0, i.e., yydt0 0, which implies that the acceleration constraints atvia points must be 0 for continuity of acceleration. This causes no problem with

stationary to stationary motion because the velocity and the acceleration at both the


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initial and the final time are always 0. For the trajectories with nonstationary via

points, the sinusoidal trajectory may have the disadvantage that the trajectory

cannot have arbitrary acceleration at the via points. However, this approach still

guarantees the continuity of acceleration and the smooth trajectory. If it is necessary

to have nonzero acceleration at the via points, a simple combination of sine and

cosine functions could be applied.

If the xis are chosen to be unduplicated and less than p=tft0, the sinusoidaltrajectory has a similar shape to the polynomial-type trajectory. Both the sinusoidal

trajectory and the polynomial trajectory satisfy the constraints and have similar

shapes. But the sinusoidal trajectory has a finite number of frequencies whereas thepolynomial trajectory does not.

Frequency components of the sinusoidal trajectory can be arbitrarily set to avoid

undesirable hazardous frequencies. Furthermore, the sinusoidal trajectory has a

synergic effect when combined with CZPETC (or ZPETC), because one can com-

pensate for the gain error for each included frequency component, resulting in better

tracking performance. As shown in Fig. 1, CZPETCs can be designed to compensate

for the gain errors for each component of the sinusoidal trajectory, so that the

output has a unity gain over the input trajectory.

4. Simulation and experiments

4.1. Modeling of the selected plant

In the previous sections, CZPETC, a feedfoward control algorithm for nonmin-

imum phase system, and the synergic strategy with sinusoidal trajectory are pre-

sented.

To verify the proposed method, a nonminimum phase system with unstable zeros

in continuous-time domain is considered. Dispersive flexible systems with non-propagating energy are known to have unstable zero dynamics [10]. They are gov-

erned by a fourth-order differential equation, and a flexible arm is a good example.

Fig. 1. CZPETC with sinusoidal trajectory: strategy for gain error compensation.


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To derive the dynamic equations of the flexible arm having planar motion, we

consider the position of a point on the beam with virtual rigid-body motion and

deflection with respect to the rigid-body coordinates by using a BernoulliEuler

beam model. We use the rigid-body coordinates that is attached at the base with theclamp-free boundary condition as shown in Fig. 2(a) [9].

Considering the rigid body motion and the first mode, the linearized dynamic

equation of the flexible arm can be written as follows:

M11 M12M21 M22

hh1

hh2

F1 0

0 F2

_hh1

_hh2

0 0

0 K

h1

h2

s

0

: 8

The state-space representation of the above equation is as follows:

_xxt Axt Bst; 9

yCxt; 10

x

h1

_hh1

h2

_hh2

2666664

3777775; A

0 1 0 0

0 M22F1

D

M12K

D

M12F2

D

0 0 0 1

0 M21F1

D

M11K

D

M11F2

D

266666664

377777775;

B

0

M22D

0

M21

D

266666664

377777775; C

1

0

1

0

2666664

3777775

T

;

where D M11M22M12M21.

Fig. 2. The model of the single link flexible arm.


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The output to be controlled is the rotating angle of the tip. It can easily be shown

that the above system is controllable and observable by constructing controllability

and observability matrices.

4.2. Application of CZPETC with time delay contol

We combine CZPETC with a robust feedback controller, time delay controller

[14]. The state-space model of the flexible arm, represented by Eq. (8), does not

satisfy the matching condition, so that time delay control (TDC) cannot be applied

directly. However, there always exists a transformation which converts the model to

a controllable companion form which satisfies the matching condition, provided that

the system is controllable [13].

Through the coordinate transformation, the closed-loop dynamics can be repre-

sented as the following transfer function:

ys

rs

am1M22 M12s2 F2sK

Ds4 am4s3 am3s2 am2sam1

am1M22M12sbasbu

Ds4 am4s3 am3s2 am2sam1; 11

where D M11M22 M212 and amis i 1; 2; 3; 4 are controller gains determined in

TDC and ba and bu represent the stable and the unstable zero of the plant, respec-

tively. According to Eq. (11), CZPETC is designed as follows:

rs

yds

Ds4 am4s3 am3s

2 am2sam1

am1M22 M12sba

sbu

b2u: 12

If a sinusoidal trajectory is designed for the synergic effect, i.e., compensation of the

gain error, CZPETC for theith trajectory input,yditwith frequency component, xi,is designed as follows:

rs

ydis

Ds4 am4s3 am3s

2 am2sam1

am1M22 M12sba

sbu

b2u x2i

: 13

4.3. The results of the simulation

In the simulation, time delay controller (TDC), a robust feedback controller in

continuous-time domain, is designed so that the closed-loop characteristic polyno-

mial becomess354.The trajectories, ydt, are required to satisfy the following constraints:

ydt0 0 deg; ydtf 60 deg;

_ydydt0 _ydydtf 0 deg=s;

yydt0 yydtf 0 deg=s2

;

where t0 0 s and tf0:5 s.


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In addition to the above constraints, the continuity of the CZPETCs output is

considered. Therefore, a sixth-order polynomial trajectory and a sinusoidal trajec-

tory of five frequency components are designed. In the sinusoidal trajectory plan-

ning, the frequencies are arbitrarily selected as 1:0p, 1:1p, 1:2p, 1:3p, 1:4p.As shown in Fig. 3, CZPETC with TDC can follow the desired trajectory, both

the polynomial and the sinusoidal, with no phase shift. But the gain error exists for

the polynomial trajectory, whereas CZPETC with the sinusoidal trajectory com-pensates the gain error. Even if there exists the gain error as in the polynomial

trajectory, the position constraints can be satisfied. But the gain error causes the

velocity constraint to be violated as shown in Fig. 4. This brings undesirable effects

on the response after tf, which are significantly reduced when CZPETC with sinu-

soidal trajectory is applied.

4.4. Experiments

The experimental setup of the flexible arm is shown in Fig. 5. The flexible beam ismade of tempered spring steel approximately 400 mm long. To minimize the tor-

sional effect, the width of beam is set to 30 mm, which is relatively longer than the

Fig. 3. Simulation result: position response of CZPETC with TDC (sinusoidal trajectory vs. polynomial

trajectory).


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thickness of the beam, 0.5 mm. To measure the rotation angle of the tip, the follower

arm is attached to the flexible beam. The effect of the follower arm can be treated as

the equivalent mass at the tip. Bearings are used at the connection between the beam

and the follower arm to minimize the frictional effect.

The parameters of the system are estimated by the least-square method, from

which the zeros are then calculated. The trajectories, both polynomial and sinusoi-

dal, are planned with the same constraints used in the simulation, except that

tf0:7 s. The sampling time of the controllers is set to 0.4 ms.CZPETC is derived in continuous-time domain, but implemented using a digital

computer with a fast sampling rate. Although this implementation may seem

mathematically incorrect, the control scheme works well when the sampling fre-

quency is sufficiently large compared to the bandwidth of the plant.

TDC is designed so that the closed-loop has fourfold poles at )22 rad/s, i.e., the

closed-loop characteristic polynomial is s224. Fig. 6 shows the experimental

results of CZPETC with TDC and the polynomial trajectory. Fig. 7 shows the results

of CZPETC with TDC and the sinusoidal trajectory. As shown in Fig. 6, the gainerror causes the undesirable effect on the response after tf. Meanwhile, Fig. 7 shows

that CZPETC with the sinusoidal trajectory reduces the gain error. All the experi-

Fig. 4. Simulation result: velocity response of CZPETC with TDC (sinusoidal trajectory vs. polynomial

trajectory).


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ments are carried out with the spring disturbance caused by the line of the encoder 2,

and the robustness of TDC is confirmed.

Fig. 5. Experimental setup: flexible arm.

Fig. 6. Experimental result: CZPETC with TDC (polynomial trajectory).


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5. Conclusions

In this paper, a trajectory planning method has been proposed to be used withCZEPTC for trajectory control of nonminimum phase systems. A sinusoidal tra-

jectory is generated by the proposed method, once a CZEPTC is designed for a

closed-loop system. The design of CZEPTC, which can be used with various types of

feedback controllers including TDC, is made in continuous-time domain to avoid the

effort of converting a continuous-time system to a discrete-time system. CZPETC

cancels the phase error caused by the uncancelled unstable zeros. The gain error,

which is problematic in tracking controls, can be compensated by the simultaneous

design of CZPETC and the sinusoidal trajectory.

The performance of the proposed method is confirmed through the simulation

and experiments on the flexible arm, a typical nonminimum phase system. It isshown that CZPETC with a feedback controller can follow the desired trajectory

with no phase shift. It is also confirmed that the simultaneous use of the sinusoidal

trajectory and CZPETC reduces the gain error to a substantial degree, thereby

demonstrating its effectiveness.

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Fig. 7. Experimental result: CZPETC with TDC (sinusoidal trajectory).


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