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Tip Position Control of a Lightweight Flexible Manipulator Using a Fractional Order Controller C. A. Monje∗ , F. Ramos∗∗ , V. Feliu∗∗ and B. M. Vinagre∗ ∗

Industrial Engineering School,

University of Extremadura, Badajoz, Spain email: {cmonje,bvinagre}@unex.es ∗∗

Superior Technical School of Industrial Engineers,

University of Castilla-La Mancha, Ciudad Real, Spain email: {Francisco.Ramos,Vicente.Feliu}@uclm.es February 13, 2007

Abstract A new method to control single-link lightweight flexible manipulators in the presence of payload changes is proposed in this paper. Undoubtedly, the control of this kind of structures is nowadays one of the most challenging and attractive research areas, being remarkable its application to the aerospace industry, among others. One of the interesting features of the design method presented here is that the overshoot of the controlled system is independent of the tip mass. This allows a constant safety zone to be delimited for any given placement task of the arm, independently of the load being carried, thereby making it easier to plan collision avoidance. Other considerations about noise and motor saturation issues are also presented along the paper. To satisfy this performance, the overall control scheme proposed consists of three nested control loops. Once the friction and other nonlinear effects have been compensated, the inner loop is designed to give a fast motor response. The middle loop simplifies the dynamics of the system, and reduces its transfer function to a double integrator. A fractional derivative controller is used to shape the outer loop into the form

1

of a fractional order integrator. The result is a constant phase system with, in the time domain, step responses exhibiting constant overshoot, independently of variations in the load, and robust, in a stability sense, to spillover effects. Experimental results are shown when controlling the flexible manipulator with this fractional order derivator, that prove the good performance of the system.

1

Introduction

A considerable interest has been attracted to the control of lightweight flexible manipulators during the last two decades, becoming one of the most challenging research areas of robotic control. The necessity of this kind of robots arises from new robotic applications that require lighter robots that can be driven using smaller amounts of energy, such as the aerospace industry, where weight has to be minimized, or mobile robotics, where power limitations imposed by battery autonomy have to be taken into account. In addition, collisions of this type of robots present remarkably less destructive effects than those caused by traditional robots, since the kinetic energy of the movement is transformed into potential energy of deformation at the moment of impact. This fact allows us to perform some control strategy over the actuator before any damage takes place or, at least, to minimize it, which may lead, in a not very far future, to a robot-human cooperation without the actual dangers in case of malfunction, which is an emergent topic of research interest [1]. The challenge of controlling vibrations in flexible structures, such as robotic manipulators, has been approached with very different methods, from classical schemes [2], to nonlinear methods such as sliding control [3] or neural networks [4]. A recent published survey [5] gives a complete, detailed overview of all the work developed in this field since the late seventies. However, in spite of all the research devoted to modeling and controlling these kind of robots, there is no universal solution for the control, which is clearly demonstrated by the number of recent papers presenting new improved solutions for vibration control. The present work shows a new and simple control scheme for single-link flexible arms with a variable payload, based on the use of a fractional order derivative controller. This scheme uses measurement of the link deflection provided by a strain gauge placed at the base of the link.

2

This sensorial system lets us construct control schemes that are more robust than those based on accelerometer measurements (as was demonstrated in [6], too), being strain gauges simpler to instrument. The general control scheme proposed in this paper consists of three nested loops (see figure 1): 1. An inner loop that controls the position of the motor. This loop uses a classical PD controller to give a closed loop transfer function close to unity. 2. A simplifying loop using positive unity-gain feedback. The purpose of this loop is to reduce the dynamics of the system to that of a double integrator. 3. An outer loop that uses a fractional order derivative controller to shape the loop and to give an overshoot independent of payload changes. In figure 1, θm is the motor angle, θt the tip-position angle, Gb (s) the transfer function of the beam, and Ri (s), Re (s) the inner and outer loop controllers, respectively. The design of the first two loops follows [7]. The fractional order control strategy of the outer loop, which is based on the operators of fractional calculus (see [8]), is presented in this paper. A former use of this strategy was discussed in [9] considering a more ideal case and only simulation results. The state of the art in Fractional Order Control can be found in [10], [11], [12]. For a better understanding of this work, it is organized as follows. First, the physical model of the single-link flexible manipulator is presented in section 2, followed by a description of the general control scheme and the three nested loops in section 3. The design of the outer loop is explained in detail in section 4, and the effect of higher order dynamics is discussed in section 5. Next, section 6 presents the results obtained from the test of the control strategy in an experimental platform. Finally, some relevant concluding remarks are drawn in section 7.

2

Modelling of the Single-Link Flexible Manipulator

The single-link lumped mass model that will be used in this paper is well known [6]. Particularly, a single tip mass that can rotate freely (no torque is produced at the tip) will be adopted for

3

the description of the link dynamics [6]. The effect of the gravity is assumed negligible since the arm moves in a horizontal plane. The motor has a reduction gear with a reduction relation n. The magnitudes seen from the motor side of the gear will be written with an upper hat, while the magnitudes seen from the link side will be denoted by standard letters. The dynamics of the link is described by c(θm − θt ) = ml2 ¨θt ,

(1)

where m is the mass at the end, l and c are the length and the stiffness of the bar, respectively, θm is the angle of the motor, and θt is the angle of the tip. The dynamics of the motor with a closed loop current control system (where the voltage V is proportional to the current output) is given by ··

·

ˆ coup + Γ ˆ Coul , KV = J ˆθm + ν ˆθm + Γ

(2)

where K is the motor constant, V is the voltage signal that controls the motor, J is the motor ˆ coup is the coupling torque between the motor and inertia, ν is the viscous friction coefficient, Γ ˆ Coul is the Coulomb friction. From now on we will suppose that the Coulomb the link, and Γ friction is negligible or is compensated by a term (see [13]) of the form

VCoul

à ! · ¯ Coul Γ = sign ˆθm , K

(3)

¯ Coul is an estimation of the Coulomb friction value. as shown in figure 2(a), where Γ On the other hand, the coupling torque equation between the motor and the link is

Γcoup = c(θm − θt ), ˆ= and finally, the conversion equations ˆθ = nθ and Γ

Γ n

(4)

complete the dynamic model.

Laplace transform is applied to (1) leading to the following transfer function

Gb (s) =

θt (s) ω2 = 2 0 2, θm (s) s + ω0 4

(5)

where ω 0 is the natural frequency of the link ω 20 =

c ml2 ,

which is mass dependant.

Combining all the previous equations, the transfer functions of the robot are θm (s) V (s) θt (s) V (s)

= =

¢ ¡ Kn s2 + ω 20 , s [Jn2 s3 + νn2 s2 + (Jn2 ω 20 + c) s + νn2 ω 20 ] Knω 20 . 2 3 2 2 s [Jn s + νn s + (Jn2 ω 20 + c)s + νn2 ω 20 ]

(6) (7)

It is evident that these robot equations are mass dependant (so is ω 0 ) and, therefore, changes in the mass will affect the system behavior.

3

General Control Scheme

The general control scheme is shown in figure 1, where it can be seen that it is composed of three nested loops: inner loop, simplifying loop and outer loop, commented previously. The features of the inner and outer loops have been previously detailed in [6], while the simplifying loop is now included to cope with the fractional order control strategy. Basically, this scheme allows us to design the loops separately, making the control problem simpler and minimizing the effects of the inaccuracies in the estimation of Coulomb and viscous frictions in control performance (as shown in [13]).

3.1

Inner Loop

The inner control loop, shown in figure 2(a), fastens the dynamic behavior of the motor. This purpose is achieved by means of a standard PD controller with proportional constant Kp and derivative constant Kv , which is tuned to make the motor dynamics critically damped. The second order critically damped expression obtained for the motor loop is

M (s) =

θm (s) 1 , = r θm (s) (1 + γs)2

(8)

where θrm (s) is the reference angle for the motor and γ is the motor dynamics constant, which q J is given by γ = Kp K . Theoretically, it is possible to make the motor dynamics as fast as desired by simply making γ → 0, but a very demanding speed would saturate the motor, with 5

γ 0.022

Kp 1

c(N ·m) 443.597

Table 1: Data of the motor-gear set Kv J(kg · m2 ) ν(N · m · s) K(N · m/V ) 0.025 24.24 · 10−4 51.66 · 10−4 3.399 Table 2: Data of the flexible link m(kg) l(m) r(m) E(GP a) I(m4 ) 1.9 0.866 0.008 68.9 1.86e-9

n 50

Ks 2.11

the subsequent malfunction of the controlled system. This fact implies that, although the motor dynamics can be made quite fast, we cannot consider it negligible in general. Effects of this dynamics can be studied by dividing the overall system into two time scale subsystems: the fast dynamics subsystem defined by the inner control loop and the slow dynamics subsystem defined by the flexible link. Then, interactions between both subsystems are studied. Singular perturbation techniques [14] define the general framework to carry on this study. Application of these techniques to flexible arms can be found in [15], where the arm dynamics is divided into a slow subsystem (which includes motor dynamics) and a fast subsystem (which includes high order vibration modes). A recent work based on this approach is presented in [16], where the dynamics of the highest vibration modes are neglected in the controller design but the maximum possible gain in the L2 sense of these removed dynamics is taken into account in such design. It is possible to modify these techniques in order to cope with the flexible arm control problem defined in this paper. In this sense, the effects of the servocontrolled motor fast dynamics on the slow dynamics - defined by the flexible link and the outer controller - can be quantified and used to adequately tune the outer controller. But the main purpose of this paper is to show the feasibility of a robust fractional controller designed by using a very simple technique that does not take into account this fast dynamics. We will show in section 4.5 that the effects of the inner loop fast dynamics can be reduced by adding a proportional term (tuned easily by simulations) to the fractional controller designed with our robust control technique. Then, a more elaborate analysis is not needed to achieve acceptable results with our arm. Tables 1 and 2 show the parameters of the motor-gear set and the flexible link used for the experimental tests, respectively. It is important to remark that the coupling torque is compensated within the inner loop by a term of the form 6

Vcoup =

1 Γcoup . Kn

(9)

Previous experimental works have proven the correctness of this assumption in direct driven motors, and motors with reduction gears as well [7]. It has been also demonstrated that, in the case of motors with gears, the effect of the coupling torque is very small compared to the motor inertia and friction, as its value is divided by n [6].

3.2

Simplifying Loop

As commented previously, the response of the inner loop (position control of the motor) is significantly faster than the response of the outer loop (position control of the tip). The motor position is first supposed to track the reference position with negligible error and the motor dynamics will be considered later. That is, the dynamics of the inner loop can be approximated by "1" when designing the outer loop controller. Taking this into account, a strategy for simplifying the dynamics of the arm, shown in figure 2(b), is proposed. For the case of a beam with only one vibrational mode, a simplifying loop can be implemented that reduces the dynamics of the system to a double integrator by simply closing a positive unitygain feedback loop around the tip position (β = 1). Then, the equation relating the output and input of the loop is θt (s) =

ω 20 1 u(s) + 2 P (s), s2 s

(10)

where P (s) represents disturbances with the form of a first order polynomial in s, which models initial deviations in tip position and tip velocity [6]. In equation (10), the dynamics of the arm has been reduced to a double integrator dynamics, simplifying the control strategy proposed in this work, as will be seen later. The stability study by using Nyquist diagrams shows that the condition β = 1 is not critical to get stable control systems, being sufficient to implement a feedback gain close to 1.

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3.3

Outer Loop

The block diagram for the outer loop used in this work is shown in figure 3. As it is observed in the scheme, an estimation of the tip position, θet , is used to close the loop. We actually feed back the deformation measurements of two strain gages, placed at the base of the link in a half-bridge, 2-active-gages configuration, to control the arm. These sensors provide the value of the coupling torque Γcoup between the arm and the motor by means of expression

Γcoup =

1 2.00 EI , 2 Ks r

(11)

where Ks is the gage factor provided by the manufacturer, r is the outer beam radius, E the Young’s modulus of the beam material and I the cross section inertia of the beam. These values can be found at table 2. On the other hand, represents the deformation measured by the strain gauges, which is proportional to a voltage signal and calibrated with a dynamic strain amplifier. The

1 2

gain is due to the chosen configuration, which doubles the sensed value while remains

insensitive to temperature variations and cancels the compressive/tensile strain. Coupling torque is used to decouple motor and link dynamics and estimate tip position [6]. Combining equations (1), (4) and the fundamental frequency definition, we can obtain the relation between Γcoup and the output θm , which yields

GΓ (s) =

cs2 . s2 + ω 20

(12)

Given the values of θm and Γcoup , experimentally measured, the value of θt can be estimated according to equation (4) by θet = θm − Γcoup /c. This estimated tip angle is used to close the control loop, as shown in figure 3, where H(s) = − 1c . The main purpose of this work is to design the outer controller Re (s) (see figure 3) so that the time response of the controlled system has an overshoot independent of the tip mass and the effects of disturbances are removed. This will lead to the use of a fractional derivative controller, as will be detailed in the next section. Besides, in this particular case the outer controller will be designed in the frequency domain for the specifications of phase margin (damping of the response), and crossover frequency (speed

8

of the response). In order to guarantee a critically damped response (overshoot Mp = 0), a phase margin ϕm = 76.5◦ is selected. Besides, the response is desired to have a rise time around 0.3sec, so the crossover frequency is fixed to ω cg = 6 rad/sec. The crossover frequency defines the speed of response of the closed loop system. The practical constraint that limits the speed of response of the arm, and hence the value of ω cg , is the maximum torque provided by the motor. This maximum torque limits the speed of response of the inner motor loop, and hence its bandwidth. The control scheme proposed here works ideally if the dynamics of the inner loop is negligible. Consequently, its bandwidth must be much larger than the desired crossover frequency ω cg . However, it must be taken into account that the experimental results to be presented in this work show the behavior of the arm assuming nonnegligible inner loop dynamics, since the value of the torque provided by the motor limits the speed of response of this loop. This fact may change slightly the final frequency specifications found for the system, as will be shown later.

4

Design of the Outer Loop Controller Re (s)

With the inner and simplifying loops closed, the reduced diagram of figure 4 is obtained, which is based on equation (10). From this diagram, the expression for the tip position is

θt (s) =

1 1+

s2 Re (s)ω 20

θrt (s) +

1 1+

s2 Re (s)ω 20

P (s) . Re (s)ω 20

(13)

The controller Re (s) has a twofold purpose. One objective is to obtain a constant phase margin in the frequency response, in other words, a constant overshoot in time response to a step reference for varying payloads. The other is to remove the effects of the disturbance, represented by the initial conditions polynomial, on the steady state. To attain these objectives, most authors propose the use of some kind of adaptive control scheme (see [7]). We propose here a fractional order derivative controller with enhanced robustness properties to achieve the above two objectives, without needing any kind of adaptive algorithm. Some methods have been developed in the last years to tune fractional PID controllers with robustness properties to changes in one parameter (typically the plant gain) [17], [18]. These methods are based on optimization procedures. Our approach is much simpler as it is specifically tailored to our

9

particular arm dynamics, leading to very straightforward tuning rules.

4.1

Condition for Constant Phase Margin

The condition for a constant phase margin can be expressed as ∙ ¸ ω2 arg Re (jω) 0 2 = constant, ∀ω, (jω)

(14)

and the resulting phase margin ϕm is

ϕm = arg [Re (jω)] .

(15)

For a constant phase margin 0 < ϕm < π/2 the controller that achieves this must be of the form Re (s) = Kc sα , α =

2 ϕ , π m

(16)

so that 0 < α < 1. This Re (s) is a fractional derivative controller of order α. The two definitions used for the general fractional integro-differential operation are the Grünwald-Letnikov (GL) definition and the Riemann-Liouville (RL) definition [8]. The GL definition is [ t−a h ] α a Dt f (t)

−α

= lim h h→0

X

(−1)j

j=0

µ ¶ α f (t − jh), j

(17)

where [·] means the integer part, while the RL definition is

α a Dt f (t) =

dn 1 Γ(n − α) dtn

Z

a

t

f (τ ) dτ , (t − τ )α−n+1

(18)

for (n − 1 < α < n) and where Γ(·) is the Euler’s gamma function. For convenience, Laplace domain notion is usually used to describe the fractional integrodifferential operation. The Laplace transform of the RL fractional derivative/integral (18) under zero initial conditions for order α (0 < α < 1) is given by

±α £{a D±α F (s). t f (t)} = s

10

(19)

Taking this notation into account, Re (s) corresponds to a fractional derivative controller of order α.

4.2

Condition for Removing the Effects of Disturbance

From the final value theorem, the condition to remove the effects of the disturbance is 1

lim

s→0 1

+

s2 Re (s)ω 20

P (s) = 0. Re (s)ω 20

(20)

Substituting Re (s) = Kc sα and P (s) = as +b (initial tip position and velocity errors different from zero), this condition becomes

lim

1

s→0 1

lim

s→0 1

+ 1

+

b s−α Kc ω 20

= 0,

(21)

a 1−α s Kc ω 20

= 0,

(22)

s2−α Kc ω 20

s2−α Kc ω 20

which implies that α < 1.

4.3

Ideal Response to a Step Command

Assuming that the dynamics of the inner loop can be approximated by unity and that disturbances are absent, the closed loop transfer function with controller (16) is

Fcl (s) =

θt 1 Kc ω 20 = , 2 r = s θt s2−α + Kc ω 20 1 + Re (s)ω 2

(23)

0

which exhibits the form of Bode’s ideal loop transfer function [19]. The corresponding step response is

θt (t) = £−1

½

Kc ω 20 2−α s(s + Kc ω 20 )

¾

= Kc ω 20 t2−α E2−α,3−α (−Kc ω 20 t2−α ),

(24)

where Eδ,δ+1 (−Atδ ) is the two-parameter Mittag-Leffler function [8]. The overshoot is fixed by 2 − α, which is independent of the payload, and the speed by Kc ω 20 , that is, by the payload and the controller gain. In fact, notice that this expression can be normalized with respect to time 11

by θt (tn ) = tn2−α E2−α,3−α (−t2−α ), n

(25)

1

where tn = t(Kc ω 20 ) 2−α . This equation shows that the effect of a change in the payload implies a change in ω 0 that only means a time scaling of the response θt (t). To obtain a required step response, it is then necessary to select the values of two parameters. The first one is the order α to adjust the overshoot between 0 (α = 1) and 1 (α = 0), or, equivalently, a phase margin between 90◦ and 0◦ . The second one is the gain Kc to adjust the crossover frequency, or, equivalently, the speed of the response for a nominal payload. Note that increasing α decreases the overshoot but increases the time required to correct the disturbance effects (see [20]).

4.4

Controller Design

As commented above, the design of the controller thus involves the selection of two parameters: • α, the order of the derivative, which determines: (a) the overshoot of the step response, (b) the phase margin, or (c) the damping. • Kc , the controller gain, which determines for a given α: (a) the speed of the step response, or (b) the crossover frequency. These parameters can be selected by working in the complex plane, the frequency domain or the time domain. In the frequency domain, the selection of the parameters of the fractional order derivative controller can be regarded as choosing a fixed phase margin by selecting α, and choosing a crossover frequency ω cg , by selecting Kc for a given α. That is

α=

2 ω cg ϕ , Kc = 2 . π m ω0

(26)

According to table 2, where the parameters of the flexible manipulator are presented, the fundamental frequency of the system is ω 0 = 17.7 rad/sec. The frequency specifications required for the controlled system, commented previously, are: phase margin ϕm = 76.5◦ , and crossover frequency around ω cg = 6 rad/sec. Therefore, the parameters of the fractional derivative con-

12

troller are α = 0.85 and Kc = 0.02. With this controller, and under the assumption of negligible inner loop dynamics, the Bode plots obtained for the open loop system are shown in figure 5(a), where it can be observed that at the crossover frequency ω cg = 6 rad/sec the phase margin is ϕm = 76.5◦ , fulfilling the design specifications. The simulated step responses of the controlled system for m = 0.6 kg, m = 1.9 kg, m = 3.2 kg and m = 6 kg are shown in figure 5(b). It is observed that the overshoot of the response remains constant to payload changes, being Mp = 0, fulfilling the robustness purpose. For the nominal mass (m = 1.9 kg), a rise time tr = 0.3 sec is obtained. This controller has been implemented as described in section 4.6, except for the constant k = 0.25, which has been introduced later to compensate the effects of the nonnegligible inner loop dynamics, as explained next.

4.5

Effect of the Nonnegligible Inner Loop Dynamics

In the practical case presented in this work, the dynamics of the inner loop is not negligible, being given by the transfer function

M (s) =

θm (s) 1 , = r θm (s) (1 + γs)2

(27)

with γ = 0.022. Notice that M (s) is independent of the value of the tip payload as its effects on the motor dynamics are removed by the compensation term Vcoup in (9) based on the measurement of the motor-beam coupling torque. The introduction of M (s) implies that the response of the controlled system will be affected by this dynamics, since the simplifying loop does not result in a double integrator anymore. Besides, it is important to remark that step inputs are not very appropriate for robotic systems, being more suitable the use of smoother references to avoid surpassing the physical limitations of the robot, such as the maximum torque allowed to the links before reaching the elastic limit or the maximum feasible control signal value, (V for a DC motor-amplifier set). For that reason, a 4th order polynomial reference θrt has been used in our case. It has been observed that with the introduction of M (s) the settling time of the response gets longer. To reduce it, a proportional part k is introduced in the controller to make the output 13

converge faster to its reference. However, it must be taken into account that the introduction of this constant affects the frequency response of the system, changing the specifications. Therefore, there must be a trade off between the fulfilment of the frequency specifications and the settling time required, resulting k = 0.25 in our case. Then, the final controller is

Re (s) = 0.25 + 0.02s0.85 .

(28)

Only a slight modification of the frequency specifications is obtained with this controller, resulting ω cg = 6.6 rad/sec and ϕm = 70◦ . The time responses of the system for payload changes are shown in figure 6. Bigger masses than 3.2kg have not been considered since they could cause the beam to reach its elastic limit and, hence, they will be neither simulated nor experimented. For the nominal mass an overshoot Mp = 0% is obtained. As far as the robustness is concerned, a slight change in the overshoot of the response appears when the payload changes, due to the effect of the nonnegligible inner loop dynamics. However, only a 0.59% variation in the overshoot is obtained for the different masses.

4.6

Fractional Order Controller Implementation

No physical devices are available to perform the fractional derivatives, then approximations are needed to implement fractional controllers. These approximate implementations of FOC can be classified into either continuous [21], [22], [23] or discrete methods [22], [24]. In this particular case, an indirect discretization method is used. That is, firstly a finite dimensional continuous approximation is obtained, and secondly the resulting s-transfer function is discretized. It must be taken into account that the fractional derivative s0.85 has been implemented as 1 s0.85 = s · s−0.15 = s s0.15 , that is, an integer derivative plus a fractional integrator. This way, ω2

the resulting open loop system in the ideal case would be Re (s) s20 =

ω 20 −0.15 , s s

guaranteeing the

cancellation of the steady-state position error due to the effect of the pure (integer) integral part. Therefore, only the fractional part Rd (s) = s−0.15 has been approximated. To obtain a finite-dimensional continuous approximation of the fractional integrator, a frequency domain identification technique is used, provided by the Matlab function ”invfreqs”. An integer order transfer function that fits the frequency response of the fractional order integrator 14

Rd in the range ω ∈ (10−2 , 102 ) is obtained. Later, the discretization of this continuous approximation is made by using the Tustin rule with prewarp frequency ω cg and sample period Ts = 0.002sec, obtaining a 5th-order digital IIR filter

Rd (z) =

−0.1124z −5 + 0.7740z −4 − 2.0182z −3 + 2.5363z −2 − 1.5523z −1 + 0.3725 . −0.4332z −5 + 2.6488z −4 − 6.3441z −3 + 7.4747z −2 − 4.3462z −1 + 1

(29)

Therefore, the resulting total fractional order controller is a 6th -order digital IIR filter given by Re (z) = 0.25 + 0.02

5

µ

1 − z −1 Ts

¶

Rd (z).

(30)

Robustness to higher vibration modes

This section studies the robustness of the developed fractional controller to nonmodeled higher vibration modes. These modes can influence the closed loop system in two ways: 1) they are fed back to the controller and, if they were not taken into account in the controller design, the global system can become unstable, 2) the estimator of the tip position based on expression (4) no longer remains correct, exhibiting high frequency estimation errors that are fed back to the closed loop system. In order to avoid these destabilizing effects we propose a lemma based on the next dynamic property. Let us consider the transfer function GΓ (s) between the motor angle and the motor-beam coupling torque (the other measured variable). Then we say that this transfer function exhibits the interlacing property of the poles and zeros on the imaginary axis if it verifies that: ¡ ¢ ¡ ¢ ¡ ¢ s2 s2 + 21 · · · s2 + 2i · · · s2 + 2n GΓ (s) = 2 , (s + ω 20 ) (s2 + ω 21 ) · · · (s2 + ω 2i ) · · · (s2 + ω 2n ) where ω i−1 <

i

< ωi ,

(31)

1 ≤ i ≤ n, and c > 0. This property is verified by uniform single-link

flexible manipulators with distributed mass and a payload at the tip as illustrated next. The governing equation of a flexible link (Euler-Bernouilli equation) can be normalized by p defining T = ρl4 /EI - where ρ is the mass per unit length - and introducing the dimensionless

time tn = t/T . Consequently, the tip payload is also normalized with respect to the beam mass: mn = ρl/m. Transfer functions GΓ (s) are obtained for the normalized beam for different mn 15

ratios, and the poles and zeros associated to the first six modes are calculated. We assume that modeling six modes is enough to study spillover effects in most flexible manipulators. Figure 7 shows the values of these poles and zeros for mass ratios ranging from negligible link mass (mn = 0.01) to the case of a link mass 10 times larger than the tip payload (mn = 10). This figure shows that the aforementioned interlacing property is verified by any uniform beam at least in the specified range of variation of the link mass. The next lemma proves that if this property is verified, the controller proposed in the previous section is robust to nonmodeled higher vibration modes (spillover). Lemma. Assume that our flexible arm verifies the interlacing property (31), and that the inner loop dynamics is negligible (M (s) ≈ 1). Then any outer loop controller of the form Re (s) = k + Kc sα ,

Kc > 0 ,

0≤α≤1,

(32)

combined with a tip position estimator of the form given by θet = θm − Γcoup /ˆ c,

(33)

cˆ ≥ c ,

(34)

where parameter cˆ verifies

keeps stable the closed loop system. Proof . Figure 3 shows the block diagram of the outer control loop. Operating this block diagram we obtain the equivalent transfer function Γcoup (s) GΓ (s) ¡ ¢ = r 1 Θt (s) 1 + cˆ GΓ (s) −1 + Re−1 (s)

(35)

If GΓ (s) verifies the above interlacing property, the alternation between poles and zeros of expression (31) produces a Nyquist diagram of GΓ (jω) of the form shown in figure 8(a). It exhibits as many half-turns in the infinity as vibration modes has the transfer function (as many as terms ¡ 2 ¢ s + ω 2i are in the denominator of this transfer function). This plot shows that the closed loop ¢ ¡ system associated to GΓ (s) is marginally stable. The product 1cˆ −1 + Re−1 (s) subtracts phase 16

to the system from zero, when frequency is very small, to 180o , when frequency tends to infinity, hence progressively rotating the Nyquist of GΓ (jω), but never crossing the negative x-axis, as figure 8(b) shows for a three vibrational modes example. In order to guarantee that the Nyquist plot does not embrace the point (-1,0), it must be verified that ¡ ¢ 1 GΓ (s) −1 + Re−1 (s) ≥ 1 . ω→∞ c ˆ lim

(36)

Assuming that Re (s) is of the form (32) and taking into account (31), it easily follows that condition (36) becomes cˆ ≥ c, and expression (34) is proven. In addition, if Re (s) is of the form (32), after some operations we have that

ξ(jω) =

¢ k (1 − k) + Kc ω α (1 − 2k) cos π2 α − Kc2 ω 2 − jKc ω α sin π2 α 1¡ ¡ ¢ . (37) −1 + Re−1 (jω) = cˆ cˆ k2 + 2Kc kω α cos π2 α + Kc2 ω 2α

The imaginary component of this expression is negative ∀ω ≥ 0 provided that 0 ≤ α ≤ 1 and K ≥ 0. Then ∠ξ(jω) ≤ 0, ∀ω ≥ 0, and it subtracts phase from GΓ (jω) at all frequencies, as the Nyquist stability condition requires. ¤ Remark . Conditions (32)-(34) make the closed loop system stable for any single-link flexible arm that fulfils the interlacing property (31), independently of the number of high frequency modes considered.

6

Experimental Results

The control strategy proposed here, with the use of the outer loop controller in (30), has been tested in the experimental platform of the picture in figure 9, whose dynamics corresponds to the one described previously for the single-link flexible manipulator. In this section, the experimental results obtained are presented. The robustness of the system has been tested by changing the payload at the tip. Motor and tip position records are shown. Simulated control signals are plotted together with the experimental motor control signals for comparison purposes. Note that the simulations neglect Coulomb friction, whilst in the experimental platform a compensation term, +0.3 V for positive motor velocities and −0.25 V for negative ones, has been added to the control signal. These compensation values have been found by a trial-error process. 17

Figure 10(a) shows the measurements of the tip angle θt and the motor angle θm , while 10(b) shows the motor voltage V , both figures corresponding to a mass m = 3.2kg. A relay type control appears in the transient and steady states due to Coulomb friction compensation. For this reason the experimental voltage signal obtained presents quick oscillations and is not zero in the steady state. Figure 11 shows the measurements of the tip and motor angles and motor voltage obtained for the nominal mass m = 1.9kg. And finally, figure 12 shows the results when m = 0.6kg. Through figures 10(b), 11(b) and 12(b), it can be observed that the peak of the control signal keeps lower than the saturation limit and remains almost constant in the presence of payload changes, with a value around 0.65V , making this control strategy very suitable for motor saturation problems. Another important aspect to remark is that the fractional derivator part of the controller, 1 s0.85 , is implemented by s0.85 = s s0.15 . That is, the fractional integrator part acts like a low pass

filter of the signal that enters the derivative operator and reduces the noise introduced through the control loop. Therefore, with the fractional controller the system is not only more robust to payload changes, but also to noise presence.

7

Conclusions

A new method to control single-link lightweight flexible arms in the presence of payload changes has been presented in this work. The overall controller consists of three nested control loops. Once the Coulomb friction and the motor-beam coupling torque have been compensated, the inner loop is designed to give a fast motor response. The simplifying loop reduces the system transfer function to a double integrator. The fractional order derivative controller is used to shape the outer loop into the form of a fractional order integrator. The result is an open loop constant phase system whose closed loop responses to a step command exhibit constant overshoot, independently of variations in the load. A study of the effect of spillover has been carried out, where system stability to any nonmodeled higher order dynamics has been proven. The fractional order controller has been tested in an experimental platform by using discrete implementations. From the results obtained it can be concluded that an interesting feature of

18

the fractional control scheme is that the overshoot is independent of the tip mass. This allows a constant safety zone to be delimited for any given placement task of the arm, independently of the load being carried, thereby making it easier to plan collision avoidance. It must be remarked that with the fractional order controller the control signal is less noisy than with a standard P D controller, since the fractional integrator acts like a low pass filter and reduces the effects of the noise introduced in the control loop. Besides, with this control strategy, changes in the payload imply only slight variations in the maximum value of the control signal, avoiding possible saturation issues.

Acknowledgments This work has been financially supported by the Spanish Government Research Program via Project DPI-2003-03326 (MEC) and by the Spanish Research Grant 2PR02A024 of the Junta de Extremadura.

References [1] Zinn M, Khatib O, Roth B, Salisbury JK. Playing it safe [Human-Friendly Robots]. IEEE Robotics and Automation Magazine. 2004;11(2):12—21. [2] Cannon RH, Schmitz E. Initial Experiments on the Endpoint Control of a Flexible One-Link Robot. International Journal of Robotics Research. 1984;3(3):62—75. [3] Chen YP. Regulation and Vibration Control of a Fem-Based Single-Link Flexible Arm Using Sliding-Mode Theory. Journal of Vibration and Control. 2001;7(5):741—752. [4] Su Z, Khorasani K. A Neural-Network-Based Controller for a Single-Link Flexible Manipulator Using the Inverse Dynamics Approach. IEEE Transactions on Industrial Electronics. 2001;48(6):1074—1086. [5] Benosman A, Le Vey G.

Control of Flexible Manipulators:

2004;22(5):533—545.

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A Survey.

Robotica.

[6] Feliu V, Ramos F. Strain Gauge Based Control of Single-Link Very Lightweight Flexible Robots to Payload Changes. Mechatronics. 2005;15(5):547—571. [7] Feliu JJ, Feliu V, Cerrada C. Load Adaptive Control of Single-Link Flexible Arms Based on a New Modeling Technique. IEEE Transactions on Robotics and Automation. 1999;15(5):793— 804. [8] Podlubny I. Fractional Differential Equations. Academic Press, San Diego. 1999;. [9] Feliu V, Vinagre BM, Monje CA. Fractional Control of a Single-Link Flexible Manipulator. In: Proceedings of the ASME IDETCT/CIE. Long Beach, California; 2005. . [10] Oustaloup A. The CRONE Approach: Theoretical Developments and Major Applications. In: Proceedings of the Second IFAC Workshop on Fractional Differentiation and its Applications. Porto, Portugal; 2006. p. 39—69. [11] Chen YQ. Ubiquitous Fractional Order Controls? In: Proceedings of The Second IFAC Workshop on Fractional Derivatives and Its Applications (FDA’06). Porto, Portugal; 2006. . [12] Le Mehauté A, Tenreiro JA, Trigeassou JC, Sabatier J. Fractional Differentiation and Its Applications. Ubooks; 2005. [13] Feliu V, Rattan KS, Brown HB. Control of Flexible Arms with Friction in the Joints. IEEE Transactions on Robotics and Automation. 1993;9(4):467—475. [14] Kokotovic PV. Applications of Singular Perturbation Techniques to Control-Problems. SIAM Review. 1984;26(4):501—550. [15] Siciliano B, Book W. A Singular Perturbation Approach to Control of Lightweight Flexible Manipulators. International Journal of Robotics Research. 1988;7(4):79—90. [16] Karimi HR, Yazdanpanah MK, Patel RV, Khorasani K. Modeling and Control of Linear Two-time Scale Systems: Applied to Single-Link Flexible Manipulator. Journal of Intelligent and Robotics Systems. 2006;45:235—265.

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[17] Monje CA, Calderón AJ, Vinagre BM, Chen YQ, Feliu V. On Fractional P I λ Controllers: Some Tuning Rules for Robustness to Plant Uncertainties. Nonlinear Dynamics. 2004;38(14):369—381. [18] Valério D, Sá Da Costa J. Ziegler-Nichols Type Tuning Rules for Fractional PID Controllers. In: ASME International Design Engineering Technical Conference and Computer and Information in Engineering Conference. Long Beach, California, USA; 2005. . [19] Bode HW. Relations between Attenuation and Phase in Feedback Amplifier Design. Bell System Technical Journal. 1940;19:421—454. [20] Oustaloup A. La Commade CRONE: Commande Robuste d’Ordre Non Entier. Paris: Hermes; 1991. [21] Oustaloup A, Levron F, Nanot F, Mathieu B. Frequency-Band Complex Noninteger Differentiator: Characterization and Synthesis. IEEE Transaction on Circuits and Systems I: Fundamental Theory and Applications. 2000 January;47(1):25—40. [22] Valério D. Fractional Robust System Control. Instituto Superior Técnico, Universidade Técnica de Lisboa; 2005. [23] Podlubny I, Petráš I, Vinagre BM, O’Leary P, Dorˇcák L.

Analogue Realizations of

Fractional-Order Controllers. Nonlinear Dynamics. 2002;29(1-4):281—296. [24] Chen YQ, Moore KL. Discretization Schemes for Fractional-Order Differentiators and Integrators. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 2002;49(3):363—367.

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List of Figures 1

Proposed general control scheme . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Figure (a) shows the block diagram for the inner loop. Figure (b) shows the block

23

diagram for the simplifying loop . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3

Basic scheme of the outer control loop . . . . . . . . . . . . . . . . . . . . . . . .

25

4

Reduced diagram for the outer loop . . . . . . . . . . . . . . . . . . . . . . . . .

26

5

Figure (a) shows the Bode plots of the open loop system with the fractional order derivative, considering negligible inner loop dynamics. Figure (b) shows the simulated time responses of the system with the fractional order derivator to a step input for different payloads, considering negligible inner loop dynamics . . . . . .

6

Time responses of the system with controller Re(s), considering nonnegligible inner loop dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

27

28

The interlacing property between poles (dashed lines) and zeroes (solid lines) of transfer function GΓ (s) is numerically demonstrated for a wide range of beam masses in this picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

29

Nyquist plots used in the study of spillover: (a) GΓ (s) proves to be marginally stable when interlacing property is fulfilled; (b) the addition of the controller Re (s) and the simplifying loop subtracts phase, achieving a stable behavior for controlled system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

9

Photo of the experimental platform

31

10

Experimental results obtained using controller Re (s) for m = 3.2kg. The experi-

. . . . . . . . . . . . . . . . . . . . . . . . .

mental tip angle θt and motor angle θm obtained are shown in figure (a). Figure (b) shows a comparison between the experimental and simulated motor voltage V. 32 11

Experimental results obtained using controller Re (s) for m = 1.9kg. The experimental tip angle θt and motor angle θm obtained are shown in figure (a). Figure (b) shows a comparison between the experimental and simulated motor voltage V. 33

12

Experimental results obtained using controller Re (s) for m = 0.6kg. The experimental tip angle θt and motor angle θm obtained are shown in figure (a). Figure (b) shows a comparison between the experimental and simulated motor voltage V. 34

22

Figure 1: Proposed general control scheme

23

Figure 2: Figure (a) shows the block diagram for the inner loop. Figure (b) shows the block diagram for the simplifying loop

24

Figure 3: Basic scheme of the outer control loop

25

Figure 4: Reduced diagram for the outer loop

26

Figure 5: Figure (a) shows the Bode plots of the open loop system with the fractional order derivative, considering negligible inner loop dynamics. Figure (b) shows the simulated time responses of the system with the fractional order derivator to a step input for different payloads, considering negligible inner loop dynamics

27

Figure 6: Time responses of the system with controller Re(s), considering nonnegligible inner loop dynamics

28

Figure 7: The interlacing property between poles (dashed lines) and zeroes (solid lines) of transfer function GΓ (s) is numerically demonstrated for a wide range of beam masses in this picture

29

Figure 8: Nyquist plots used in the study of spillover: (a) GΓ (s) proves to be marginally stable when interlacing property is fulfilled; (b) the addition of the controller Re (s) and the simplifying loop subtracts phase, achieving a stable behavior for controlled system.

30

Figure 9: Photo of the experimental platform

31

Figure 10: Experimental results obtained using controller Re (s) for m = 3.2kg. The experimental tip angle θt and motor angle θm obtained are shown in figure (a). Figure (b) shows a comparison between the experimental and simulated motor voltage V.

32

Figure 11: Experimental results obtained using controller Re (s) for m = 1.9kg. The experimental tip angle θt and motor angle θm obtained are shown in figure (a). Figure (b) shows a comparison between the experimental and simulated motor voltage V.

33

Figure 12: Experimental results obtained using controller Re (s) for m = 0.6kg. The experimental tip angle θt and motor angle θm obtained are shown in figure (a). Figure (b) shows a comparison between the experimental and simulated motor voltage V.

34