A FRACTIONAL ADAPTATION SCHEME FOR LATERAL CONTROL OF AN AGV

Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and its Applications Porto, Portugal, July 19-21, 2006

A FRACTIONAL ADAPTATION SCHEME FOR LATERAL CONTROL OF AN AGV José Ignacio Suárez ;1 Blas M. Vinagre YangQuan Chen ;2

;1

GRASP, University of Extremadura, Badajoz, Spain CSOIS, ECE Dept. of Utah State Univ., Logan, USA

Abstract: The lateral control of an autonomous guided vehicle (AGV) is highly in‡uenced both for the longitudinal speed and the position input command (magnitude of the reference signal) of the vehicle. For that reason, a suitable strategy to govern the vehicle would be to use an adaptive and robust controller. In this paper an adaptive scheme is proposed which combines a model reference approach and a fractional order adjustment rule for a feedforward gain adjustment. Two parameters can be tuned to obtain robustness against speed and magnitude of the reference signal variations: adaptation gain, and derivative order of the adjustment rule. A model is developed for the vehicle, the design procedure is exposed, and simulation results are obtained to show the advantages of using the proposed fractional adaptation scheme. Keywords: autonomous vehicles, adaptive control, fractional calculus

1. INTRODUCTION The lateral control of an autonomous vehicle (AGV) is highly in‡uenced both for the longitudinal speed and the position input command (magnitude of the reference signal) of the vehicle. For that reason, a suitable strategy to govern the vehicle would be to use an adaptive and robust controller (Ackermann, 1997), (Netto et al., 2004). The model reference approach was developed around 1960 and MRAS (Model Reference Adaptive Systems) has become a standard part in textbooks on adaptive control (Åström and Wittenmark, 1989). The well known MIT rule for MRAS is to adjust or update the unknown parameter using gradient information.

1

Partially supported by Spanish Research Grant DPI2002-04064C05-03 (MCYT). 2 Supported in part by the TCO Bridging Fund of Utah State University (2005-2006).

Fractional calculus is a 300-years-old topic, being the theory of fractional-order derivative developed mainly in the 19-th century. Recent books (Podlubny, 1999) provide a good source of references on fractional calculus. Applying fractionalorder calculus to dynamic systems control is nowadays an increasing focus of interest (see (Vinagre and Chen, 2002)). In (Vinagre et al., 2002) two ideas were presented to extend the conventional MRAS method by using fractional order parameter adjustment rules and fractional order reference models. A step further, in this paper a fractional adaptation scheme is proposed for the lateral control of an AGV. This control scheme combines a model reference approach and a fractional order adjustment rule for a feedforward gain adjustment. The control scheme consists of two loops. The inner loop uses a Model-Following Controller (MFC) to obtain a stable system with gain depending on the vehicle speed, and the outer loop uses a fractional order adjustment rule for

2.2 Vehicle Dynamic Model

Fig. 1. Bicycle kinematic model feedforward gain adjustment. By doing so, two parameters can be tuned to obtain robustness against speed and magnitude of the reference signal variations: adaptation gain, and derivative order of the adjustment rule. This paper is organized as follows. In Sec. 2, the vehicle model is developed. Sec. 3 presents in detail the proposed fractional adaptation scheme for lateral control of the vehicle. In Sec. 4 illustrative simulations are given to demonstrate the e¤ectiveness of the proposed control scheme. Sec. 5 concludes this paper with some remarks on future research.

A simple four wheel car model is the known as the bicycle kinematic model shown in Fig. 1 ((Rajamani et al., 2003)). For simplicity we suppose that the two front and two rear wheels are lumped together. The vehicle is supposed to move in the X-Y plane and the so called guidance point is located in the middle of the rear wheel. The equations that determine the vehicle plane movement are the following:

(1)

where (x; y): are the coordinates of the guidance point in the plane X-Y, v: is the longitudinal velocity of the vehicle (or vehicle speed) at the guidance point, : is the yaw angle : is the instantaneous vehicle curvature.

Ka f

(3)

with, : the time constant of the steering actuator, Ka : the gain (supposed Ka = 1) and f : the front wheels input command signal.

cos sin tan

1 (4)

and by making use of (2), we can obtain x_ = v y_ = v _ = v L

(5)

By combining (3) and (5) and by applying the Laplace transformation we obtain the simpli…ed linear dynamic model of the lateral movement of the vehicle: Gv (s) =

Y (s) v 2 =L = 2 s ( s + 1) f (s)

(6)

Note in (6) that the DC gain depends on the vehicle velocity and also note the two poles at the origin that make the vehicle unstable.

in

tan (2) L where L is the distance from the front to the rear axle. =

+

To obtain a lateral linear model of the vehicle we can use the small angle approximations

2.1 Vehicle Kinematic Model

The relationship between the steering angle Fig. 1 and the curvature is given by:

_=

2.3 The Whole Vehicle Model

2. VEHICLE MODEL

x_ = v cos y_ = v sin _ =v

The kinematic model is not enough for simulation purposes. We also must take into account the dynamic behavior of the vehicle. Some authors presents only a lateral dynamic model ((Chaib et al., 2004), (Tsugawa et al., 1999)), but they do not consider the dynamic of the steering system. The latter has signi…cant e¤ects on the vehicle performance that cannot be ignored during the controller design ((Brennan et al., 1998)). Some authors use the steering system actuator dynamic ((Heredia et al., 1998), (Rodriguez-Castaño et al., 2003)) as a simpli…ed model with good results. The following equation gives a …rst order actuator dynamic:

3. CONTROLLER SCHEME 3.1 Adaptation of a Feedforward Gain The lateral position of the vehicle, and more precisely the gain, depends on the vehicle velocity

adjusted based on the error between the reference model (ym ) and the vehicle (y) outputs: e=y

Fig. 2. MRAS scheme for a feedfoward gain adaptation

Fig. 3. MFC structure that stabilizes the vehicle as shown in (6), so we use the adaptive scheme depicted in Fig. 2 and proposed, with = 1, by (Åström and Wittenmark, 1989). This is a simple MRAS in which the problem is to adjust the vehicle feedforward gain to the value 0 . From the Fig. 2, with = 1, we can observe that the error signal is given by e=y

0

ym = G(s)(

)uc

(7)

and hence ym @e = uc G(s) = 0 @

(8)

By applying the MIT rule we obtain the following formula for adjusting the feedforward gain: d = dt

0

e

@e = @

0

e

ym

=

eym

(9)

ym

(10)

The classical objective of the MRAS is to minimize the error signal by means of an adjustment mechanism: the MIT rule, Lyapunov functions, etc. Instead of using one these minimization method, we propose to design the MFC for a given vehicle velocity, and then to use the adaptation of the feedforward gain for improving the vehicle performance at other speeds. The design of a MFC can be looked at as a simultaneous zero and pole placement approach. The controller has two terms (Fig. 3): a feedforward term that anticipates the vehicle response, intended to cancel its dynamics, and a feedback term for compensating for any error in the feedforward design. We can represent the lateral position of the vehicle by: B(s) u (11) A(s) where u is the command input, and A(s) and B(s) are two polynomial in the domain of the complex variable s. It is assumed that: y=

A and B are relatively prime, i.e. they do not have any common factors, A is monic, i.e. the coe¢ cient of the highest power in A is unity, and deg(A) deg(B). We want the vehicle to behave like:

where = 0 = 0 is the adaptation gain that determines the adaptation rate.

Bm (s) uc (12) Am (s) where Am (s) and Bm (s) are the desired denominator and numerator of close-loop system, respectively, and uc is the lateral position command.

3.2 Design of the Model Following Controller

The desired close-loop behavior is obtained by means of the following controller (Fig. 3):

0

y=

The transfer function of the plant should be stable, but as shown in (6), it is unstable due to the double integrator. This would make the MRAS not to work properly. For solving this di¢ culty we propose to implement a MFC that makes the system stable. Then we will have two control loops: the outer loop for the adaptation of the feedforward gain and the inner loop for making the system stable by means of a MFC (Fig. 3).

where T =R is the feedforward term and S=R is the feedback term. By substituting (11) in (13) we obtain:

In the MFC design, we must …nd a controller that ful…lls two conditions: …rst, the performance of the vehicle is speci…ed by a reference model so that the closed loop system behaves like a reference model. And secondly, the parameters of the controller are

By comparing (12) with (14) we can obtain the parameters of the controller T , S and R. A detailed method for obtaining those parameters with the lowest degree controller is described more thoroughly in (Åström and Wittenmark, 1989).

u=

T (s) uc R(s)

y=

S(s) y R(s)

BT uc AR + BS

(13)

(14)

In our case, we have supposed the transfer function of the vehicle given in (6). Let us de…ne, T (s) R(s) S(s) s0 (s) = R(s) t0 (s) =

R(s)=1

(15)

R(s)=1

Then, taking into account (13), the closed-loop transfer function is:

G(s) =

Y (s) v 2 =L t0 (s) = 2 Uc (s) s3 + 1 s2 + vL s0 (s)

(16)

For obtaining a causal controller, see (Åström and Wittenmark, 1989), we must ful…ll the condition:

deg(Am )

deg(Bm )

3). The traditional scheme has an adaptation law given by (9) in which the rate of change of the parameter depends solely on the adaptation gain . As proposed by (Vinagre et al., 2002), it is possible to make the rate of change depending on both the adaptation gain and the derivative order , by using the adaptation law:

deg(A)

deg(B)

(17)

what implies, in our case, that the pole excess, deg(Am ) deg(Bm ), of the reference model must be at least 3. Then we choose the reference model given by the following transfer function: p3 Ym (s) = 3 = Uc (s) (s + p) p3 = 3 s + (3p)s2 + (3p2 )s + p3

Gm (s) =

(18)

This has a triple pole at s = p. This system has no overshoot on its step response and its dc gain is unity. By comparing (16) with (18) we get t0 with order 0 and s0 with order 1. So s0 must be like this:

d = eym (22) dt where is a real number denoting the fractional order derivative. The Grünwald-Letnikov de…nition of the fractional di¤erentiation is [t a=h] a Dt

f (t) = lim h h!0

X

( 1)j

j=0

j

f (t

jh) (23)

where [ ] means the integer part.

4. SIMULATIONS Next we show the simulation results carried out with the adaptive scheme depicted in Fig. 2 for = 1 and = 0:85 (not an optimum value). In all the simulations, all the parameters remain unchanged except for . The simulations consists of introducing a square input command, hence the vehicle must do lane changes alternatively, so that we can see the performance and the evolution of both the integer and non-integer adaptation mechanisms. We have chosen 0 = 1, = 0:5 and hence p = 0:667.

4.1 First Test s0 (s) = as + b

(19) This test shows the in‡uence of the speed in the vehicle performance. Firstly, a set of test at di¤erent speeds, from 20 to 80 Km=h (in steps of 20), were carried out with:

Then we obtain:

b = t0 (s) =

Lp3 v2

3 Lp2 a= v2 with the necessary condition p = 1=(3 ).

(20) (21)

Note that from (20) and (21) the controller parameters depend on the vehicle speed, so we must set those parameters up at a desired speed. In our simulations, unless otherwise indicated, we have calculated the controller parameters for 20 Km=h.

= 0:01, uc , a square wave from 3 to +3 metres, v = 20, 40, 60 and 80 Km=h, MFC parameters calculated for 20 Km=h. Fig. 4 and 5 show good performance for both the fractional (FC) and the integer controller (IC). In the …rst cycle, the IC is faster than the FC, but in the next cycles the IC has a longer overshoot than the FC. With the increasing speed, the FC is less a¤ected.

3.3 Fractional Adaptation Scheme 4.2 Second Test As shown in Fig. 2 the reference model is given by (18) with 0 = 1, and the plant (16) is now composed of the vehicle and the MFC (see Fig.

Now we present several tests at the same speed of 40 Km=h, but with di¤erent input signals. The

Fig. 4. Test 1 at v = 20 Km=h with MFC calculated for 20 Km=h

= 0:01 and

Fig. 7. Test 2 with uc = 3, = 0:1, v = 40 Km=h and MFC calculated for 20 Km=h sponse gets worse. So it is interesting to calculate the larger value of the input command signal for which the vehicle becomes unstable. From Fig. 2 we can see that,

Fig. 5. Test 1 at v = 80 Km=h with MFC calculated for 20 Km=h

= 0:01 and

ym = 0 Gm (s)uc (24) y = G(s)uc where, in the case of perfect model following we have G(s) = Gm (s). Then substituting (18) in (24) it follows, in the time domain, that: ... a0 y m + a1 y•m + a2 y_ m + a3 ym = p3 0 uc ... a0 y + a1 y• + a2 y_ + a3 y = p3 uc

(25)

where a0 = 1, a1 = 3p, a2 = 3p2 and a3 = p3 Fig. 6. Test 2 with uc = 1, = 0:1, v = 40 Km=h and MFC calculated for 20 Km=h aim is to show the in‡uence of the magnitude of the reference signal in the vehicle behavior. The conditions of the test are: = 0:1, uc , three di¤erent square waves: 1 (from 1 to +1), 2 and 3 metres, v = 40 Km=h, MFC parameters calculated for 20 Km=h. Fig. 6 shows a similar performance to that of the …rst test, with faster response in the IC in the …rst cycle and poorer behavior in the next cycles. As depicted in Fig. 7, the IC tends to show a greater oscillatory response, with the increasing in the input command, than the FC. Note that is not chosen for an optimum performance, but only for showing that an improvement in the vehicle behavior is possible.

4.3 Limits of Stability From the results in the second test we can observe that by increasing the input signal the vehicle re-

And, for the case of the IC, the adaptation law is _=

(y

ym )ym

(26)

By di¤erentiating the second equation of (25) and substituting in (26) we get, .... ... a0 y + a1 y + a2 y• + a3 y_ + p3 uc ym y = 2 = p3 u c ym + p3 u_ c

(27)

which is a time-varying linear di¤erential equation. Now, let us do an experiment following (Åström and Wittenmark, 1989). Firstly, assume that the adaptation law is disconnected and the input command signal uc is a constant u0c . Then, the reference model output will go towards an equilibrium 0 value ym . Now consider that the adaptation law is connected when the equilibrium is reached. In this special situation, (27) becomes a di¤erential equation with constant coe¢ cients and has the equilibrium solution 0 y = ym (28) which, by applying the Routh’s test, is stable if

for optimal tuning of the couple fractional orderadaptation gain. REFERENCES

Fig. 8. Response of the vehicle with IC and FC when u0c = 2:43 s 8p 0 (29) uc < 9 0 Then, for

= 0:1, p = 0:667 and u0c < 2:43

0

= 1 we obtain (30)

To check this results we have simulated the system with the following conditions: = 0:1, uc = 2:43 metres, v = 40 Km=h, parameters of the MFC calculated for 40 Km=h.(instead of 20 because the simulation is at 40Km=h). In Fig. 8 we can check that the vehicle with the IC is in its limit of stability, oscillating around the limit value u0c = 2:43. But what it is more important is that with the FC using the same parameters the vehicle performance is still stable and good. This demonstrates that we can choose an adequate value of that guarantees a proper performance for a given range of input amplitudes (uc ).

5. CONCLUSIONS AND FUTURE WORKS A new fractional adaptation scheme is proposed in this paper for the lateral control of an AGV which combines MFC and a fractional order adjustment rule for a feedforward gain adjustment. We can see from the simulations results that the fractional adaptation scheme has a good in‡uence in the vehicle response. The transient performance of IC is dependent on the model uncertainties and disturances, while FC is less dependent on that. By using a fractional order adjustment rule, we can vary the rate of change of the adaptation mechanism without changing the adaptation gain . The simulations results demonstrate that we can spread the variation margins of both, the vehicle speed and the input command, improving the relative stability of the system, or even preserving the absolute stability. The work in progress includes the analysis of stability bounds with fractional order adjustment rules and the design of strategies

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