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A Smith Predictor Based Generalized Predictive Controller for Mobile Robot Path-Tracking
Copyright © IFAC Intelligent Autonomous Vehicles, Madrid, Spain, 1998
A SMITH PREDICTOR BASED GENERALIZED PREDICTIVE CONTROLLER FOR MOBILE ROBOT PATH-TRACKING J.E. Normey-Rico*, J. G6mez-Ortega** and E.F.Camacho**
*Dpto. de Automa9tlO e Sistemas. Federal Univ. of Santa Catarina, Brazil. E-maiL:[email protected] *Dpto. Ingenieria de Sistemas y Automatica, Univ. de Sevilla. Escuela Superior de Ingenieros, Camino de Los Descubrimientos sin, 41092 Sevilla, Spain . Fax: +34-5-4556849, email: juango@cartuja. us. es
Abs~r<:tct:
This paper presents the application of a Smith predictor based generalized predlctlve controller (sPGPc) to the path-traking problem of a mobile robot. The equivalence between the generalized predictive controller (GPc) and a structure composed by an optimal predictor and a classical controller is given and it is shown that using a Smith predictor instead of an optimal predictor, increases the robustness of the system. Some results concerning the influence of the new predictor structure on the closed loop robustness are presented. Simulation and experimental tests carried out on a Labmate mobile robot validate the performance of the proposed strategy. Copyright © 1998lFAC
Keywords: Mobile robots, path tracking, predictive controllers, robustness.
Different approaches can be found for the mobile robot controller. In (Nelson and Cox, 1990) a linear proportional feedback of a set of different path tracking errors is implemented. In (Kanayama et al., 1990), a classical PID controller is used. Also geometric approaches are encountered in the literature. In this case, the control strategy consists of computing an analytical curve (for example a spline (Nelson, 1989) or a quintic polynomial (Shin and Singh, 1990)) that complies with several constraints (continuity in curvature, initial and final points, etc.). The structure of the curve is fixed and its parameters are calculated every sampling interval. In (Amidi, 1990), a strategy called pure pursuit is presented. The robot curvature is calculated in each sampling time as the curvature of a circumference that passes through the robot position and a lookahead point located on the reference path.
1. INTRODUCTION
One of the most important issues in the field of mobile robots is the path-tracking (PT) problem. The objective of PT is concerned with driving the vehicle as close as possible to a desired explicit path, with least control effort. When the environment is well known and no unexpected obstacles are considered, the reference path can be the result of an off-line path planning problem . But also, many avoiding obstacle strategies found in the literature use a decoupled scheme, where the reference path is replanned on-line when a unexpected obstacle is detected. PT methods are based on the estimated error between the current vehicle position and the reference path to be followed. In order to compute this error a reference point on the desired path must be chosen . There are several approaches for these selection . One approach chooses a reference point located on the desired path a predefined lookahead distance ahead the actual robot position (Amidi, 1990) . Another approach is used whith optimal control techniques where a set of consecutive reference points are chosen instead of an isolated point (Gomez Ortega, 1994; Gomez Ortega and Camacho, 1994) .
Also, optimal control techniques can be used for PT . In (Ollero and Amidi, 1991) the Generalized Predictive Control is chosen as control strategy. A linear model of the robot kinematics is considered for the prediction of the future robot positions. A quadratic cost function , where the tracking error and the control effort are penalized, is minimized. Also a predictive technique is encountered in (Gomez Ortega and Camacho, 1994) , (Gomez
153
where c5(i) and A(i) are respectively the error and control weighting factors, that are usually considered constant .
Ortega and Camacho, 1996) , but a non linear model of the robot is considered and a multilayer perceptron is used to implement the predictive controller in real time.
The future system outputs, Y(t + ilt) for i = NI , ... , N 2 , are predicted from a model of the process, from past inputs and outputs, and from the control actions foreseen for the future , U(k + ilt) for i=O, .. . ,Nu - 1, which are the unknown variables. In this way, J can be expressed as a function of only the future control actions.
The problem that we raise in this paper is that of driving a mobile robot to follow a previously calculated reference path , defined as a set of consecutive points. As the future references are known , it would seem that a predictive control technique is a suitable approach for this problem.
After this sequence is obtained, a receding horizon approach is considered. This consists of only applying the first calculated control action U(t) . This process is repeated at every sampling interval in such a way that the calculated open loop control law is applied in a closed loop manner. A block diagram of the system is shown in figure 1.
A common issue of these techniques is their high dependence on a predicition model. The uncertainties in this model can lead to a poor performance or make the system unstable. One of the more critical uncertainties is the dead time estimation. For the mobile robot , this uncertainty has several sources: the robot payload can changes over the time; the level of batteries charge decreases with time which implies a change in the dynamic of the locomotion system (for example the wheels acceleration, the dead time in the communications between the robot and a host computer in a teleoperation system) , etc. In this paper, a new Generalized Predictive Controller is proposed for mobile robot path tracking. This controller uses a Smith predictor instead of an optimal predictor, increasing the robustness of the controller.
Ptedi,-=Uve oontrollcr
Fulure predicted
The paper is organised as follows : In section 2 a brief description of the GPC is presented . Also the model considered for the mobile robot knimematics and the way in which the GPC is applied to the PT problem is shown in the same section . In section 3 the equivalence between the GPC and the predictor plus a clasical controller is shown. Section 4 is dedicated to present a new predictor structure and a robustness analysis of the proposed controller. A comparative simulation analysis between the proposed controller and the GPC is presented in section 5. The method has been tested on a Labmate mobile robot and some experimental results are presented in section 6.
Objetive
constraints
: .. _. -------- -----~~':~~~ ----- ----------- -- - -- -
-_.
Fig. 1. The predictive controller scheme. Prediction Model For an GPC formulation, a linear model of the mobile platform is needed to predict the future positions and headings of the robot. As a test bed a differential-drive mobile robot (Labmate (G6mez Ortega and Camacho, 1994)) has been used and its linearized model is obtained using a local reference frame . (see figure 2) :
2. GENERALIZED PREDICTIVE CONTROL PATH-TRACKING
B(t) ] [ y(t)
= (1-VTZ-1)
[
1
]
VT/2 ,(t -1- d)
where B(t) is the robot heading, y(t) is the local position, ,(t) is the curvature, d is the dead time, V is the linear velocity, T is the sample period and I is the identy matrix. In this work V is considered to be constant.
GPC strategy
The objective of Generalized Predictive Control (GPc) (D .W. Clarke and Tuffs, 1987) is to obtain a future control action sequence (U(t), U(t + lit), ... , U(t + Nu - lit)) in such a way that the future predicted system outputs + ilt) will be as close as possible to the desired references Yd(t + i) over the prediction horizon. This is accomplished by the minimization of a cost function J with respect to the control variables. Often, the cost function J is the sum of the squared future errors in reference tracking, e(t + ilt) = Y(t + ilt) - Yd(t + i), predicted over a prediction horizon N = N2 - N l , and also of the sum of the squared control actions foreseen for the future , U(t + ilt), over a control horizon Nu:
ht
Fig. 2. The local reference model. Nu
N2
J =
L
c5(i)e(t + ilt)
+ L A(i)[~U(t + i
- Il t
W
This linearized equations are valid only for small values of 6(). This fact implies that , when the
;=1
154
where w(t) represents the future references in the heading and position error y(t) and M, PI and P2 are constant matrices.
robot is not positioned on the desired path, approaching routes have to be considered. In this way, the mobile robot is always located on the reference path and the values of 68 are small. In this paper, the approaching path has been computed using the pure pursuit estrategy. That is, the next N reference points are predicted as if the robot should be controled with a pure pursuit scheme.
Thus the final control law is given by (see figure 3):
2N
The mobile robot curvature has been chosen as the control variable. Althought the system should has single input/single output (note that y(t) = (VT/2)8(t», it has been noticed that the system performance is much better when two outputs (8(t) and y(t» are considered. This is because the approaching path depends on the robot heading (Amidi, 1990). Thus, an error in 8(t) leads to not well fitted approaching trajectories. On the other hand , if the error in y(t) is not penalized, the robot could not approximate the desired global path.
122y(t + d - 1 It)
+ L j;w(t + d + i)
(5)
i=1
where the coefficients VT, 8(i) and ~(i).
lij
and j; are functions of
9 (1)
MOBILE ROBOT
The GPC Structure for the Mobile Robot In this work the GPC parameters are chosen in the following manner: Nu = N, NI = d + 1 and N2 = d + 1 + N . In order to compute the future control sequence U(t), U(t+ l), .. .it is necessary to compute Y(t+j I t). Using the prediction model of the mobile robot and an optimal predictor Y(t + d + j I t) is given by:
OPTIMAL PREDICTOR
Fig. 3. Control structure of the GPC controller for the mobile robot
+ j I t) = 20(t + j - 1 I t) 2 I t) + VT 6 ,(t + j - d - 1) (1)
O(t -O(t + j y(t + j
I t) = 2y(t + j
- 1 I t) - y(t
3. THE OPTIMAL PREDICTOR PLUS A CLASSICAL CONTROLLER
- 2 I t)+
+j
+(VT)2/2 6 ,(t + j - d - 1)
In this section it will be shown that a control structure based on an optimal predictor and a classical controller is equivalent to the GPC. This result will be used later to justify the use of the Smith predictor structure instead of the optimal one in the GPC.
(2)
or in a vectorial form :
1 [
O(t + d + 1 I t) O(t + d + 2 I t) [
y(t+d~Nlt) S
6,(t) 6,(t + 1)
=G
.
1
1
9<1+dl
1
[
6,(t +:N _ 1)
O(t + d I t) O(t+d-1It) [ y(t+dlt) y(t + d - 1 I t)
w(t+d+1) w(t+d+2)
+P2
.
[
,., t+d)
(3)
where G and S are constant matrices. Using this relation in the expresion of J, the minimization of the cost function gives:
M
ROBOT
+
y(l)
6,(t+:N-1)
O(t + d It) O(t + d - 1 I t) [ y(t+dlt) y(t + d - 1 I t)
6,(t) 6,(t+1)
9(1 )
MOBILE
Fig. 4. Equivalent control structure of the GPC controller for the mobile robot As stated in the previous section, the GPC can be represented by a linear control law computed using the predictions of the system outputs. The compelete control structure of the GPC, showed in the block diagram of figure 3, can easily be trans+ formed into the one shown in figure 4, concluding that the GPC is equivalent to an optimal predictor and a classical controller composed by a double PI and a pair of reference filters given by:
1
1
OPTIMAL PREDICTOR
(4)
w(t+d+2N) 155
Also note that the prediction of 0 and y is computed independently and so the final structure is not a multivariable scheme but the control action is composed by the addition of a pair of monovariable classical controllers.
9(1)
4. THE PROPOSED SPGPC In this section a robust analysis of the GPC is carried out and a new predictor structure is proposed in order to obtain better robust properties. The analysis is made for each controlled variable because of the decoupled structured plant model.
Fig. 5. Equivalent structure for robustness analysis and using that P = Pn + DP , (RoPn + QI}) = G n and (RyPn + Qy) = (VT /2)G n the characteristic equation is:
The plant will be represented by the matrix transfer function given in equation 2 and unstructured uncertainties will be considered. lt will be assumed from now on that the behaviour of the process is described by a family of linear models. Note that in this case the two elements of the matrix have the same poles and zeros and only a different gain. From now on P will represent the monovariable transfer function from 'Y to O. Thus the real plant P will be in a vicinity of the nominal plant Pn , that is: P = Pn + 6P. If d is the dead time of the plant it is possible to write, in a discret representation: P = Gz- d and for the nominal case Pn = Gnz-dn . Thus G represents the plant without the dead time and G n is its nominal value.
1 + (Cl
+ 11 t) = (1- A(z-I»
zO(t)
+ B(Z-l)r(t -
CGn I I DP Imax = 11I+CRI} I where C = Cl
B(t
It)
= «1 -
d)
i
A(Z-l ))z)iO(t)+
+ 2:(1- A(z-l)i-l B(Z-l)r(t -
d + j - 1) (6)
i=l
Using the same procedure for y, it is posible to obtain the final block diagram representation of the optimal predictor and the controller that can be seen in figure 5. In this case, the polynomials Qo , Ro , Qy and Ry have the following expresions:
Qy(Z-I) = ~TQo(Z-I)
=
+ (VT/2)C2 .
In order to improve robustness a new generalized predictive controller, the Smith predictor based generalized predictive controller SPGPC is proposed. This new control strategy is based on the Smith predictor structure (Smith, 1958) and its block diagram is shown in figure 6. The main ideas of this strategy are: (i) the prediction at time t is computed by a model of the plant without the dead time, and (ii) in order to correct mismatch between the plant and model , the error between the output of the process and the model (including the dead time) is feedback, as indicated in figure 6. lt has been shown by some of the authors (Normey Rico and E.F.Camacho, 1996) that this control structure has better robustness properties than stuctures based on optimal predictors. In this case the blocks Cl and C2 represent the linear control law given by the GPC algorithm. Comparing this structure with the block diagram of figure 5 it is easy to note that RI} = Ry = 1 and so the uncertainty norm-bound is defined by the following expresion:
= 2 .. .. , B(t + j I t) is computed as: +j
(10)
Then using the expresion of RI} (7), it easy to show that I RI} (jw) 12 1 for all frequencies. In fact the modulus of the vector 2 - z-l vary between 1 and 2. Thus, only small uncertainties will be aceptable. Note that I RI}(jw) I will increase with the dead time and this implies that the robustness of the optimal predictor based strucure will be worst for plants with high dead time.
where A(z-l) = .0.A(Z-I) and B(z-l) = 6B(z-l) Then, for j
(9)
and so the uncertainty norm-bound is defined by the following expresion:
In order to analyse the robustnes of the system the block diagram of figure 4 will be used . The relation between the prediction and the input and output of the plant can be computed directly from the polinomials A and B and the dead time of the plant (Pn = Bz~d - l). For the robot heading it follows : B(t
+ (VT /2)C2 )(G n + RI}DP) = 0
CGn I I DP ISM = 11 +ICI
(11)
The last expresion shows that the robust stability bound is the same as if the system had no dead time in the transfer function (M.Morari and E .Zafiriou. , 1989). Then , if the blocks Cl and C2 are the same in the GPC and in the SPGPC (this condition is obtained using the same tuning parameters) , the one computed with the Smith predictor structure has better robust properties. No-
VT,/n (zd n - Ro) .
From the block diagram of figure 5 it is easy to see that this equation is :
156
O(t) = RWr(t - 1) - w/(t - 1)
9(t)
2W Mobile Robot
y(t)
V
= R Wr (t -
1)
+ W/ (t -
1)
2 Model without
if O(t)
= 0, then:
the Delay
+ 1) = O(t) Xg(t + 1) = Xg(t) + VT cos O(t) Yg(t + 1) = Yg(t) + VT sin O(t) O(t
+
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ()~+--------~
Fig. 6. Smith Predictor where Wr and w/ are respectively the rigth and left wheel velocities .
tice that in the absence of model uncertainties or perturbations, both the GPC and the SPGPC give the same nominal performance, independly of the selection of the controller parameters (look ahead, N, A and 8) that can be computed using a classical procedure like in (Ollero and Amidi, 1991) or a fuzzy methodology as the one presented in (Ollero et al., 1993) .
The control is computed using a normal GPC and also a SPGPC with the same controller parameters: A = 0.5, 80 = 0.5, 8y = 1 and N = 15. In both cases the sampling time is T = 0.5 seconds, the look ahead parameter is fixed to 0.6 meters and the linear velocity V = O.lmj s. In figure 7 the nominal performance of both controllers is compared (d n = 2). As can be seen the SPGPC (dotted line) and the normal GPC (solid line) have similar responses. In figure 8 the performance of both controllers is compared when the real dead time (dr ) is 4. Note that the normal GPC (solid line) can not follow the reference and that the SPGPC (dotted line) has the same performance as in the nominal case. Note the unstability in the GPC case.
For the computation of the controller parameters the following procedure must be executed: • compute the prediction of the output using the local open loop model of the mobile robot without considering disturbances, that is, using only polynomials A and B and the dead time d; • add the missmatch between the output and the prediction to each prediction. For this case i = 0,1:
nominal case
O(t+d-i It) = O(t+d-i I t)+O(t-i) -O(t-i) y(t+d-i I t)
= y(t+d-i I t)+y(t-i)-y(t-i)
• compute the coeficients lij and fi using the same procedure as in the normal GPC , as explained in section 2; • compute the control law (6,) using the same control equation as in the GPC (equation 5) .
r
~:
Note that the predictive approach proposed in this paper is very simple and does not need any training stage like the one proposed in (G6mez Ortega and Camacho, 1994).
~ •,...... ,,, I
'"
1~~--~--~~--~--~-L--~~~-
0 .75
In order to illustrate the results given in the previous sections a simulation example is presented. The reference path is a straight line trajectory (dashed line in figure 7) and the kninematic behaviour of the mobile robot is computed using the complete non linear model given by the following equations (where x g, Yg are the global coordinates of the mobile robot) :
¥ 0,
0 .9
0 .95
1 1.05 x(metars)
1.1
1.15
1.2
1.25
6. EXPERIMENTAL RESULTS The proposed SPGPC was implemented in a LABMATE mobile robot shown in figure 9. The mobile robot configuration (position and heading) has been computed at each sampling time using an odometry algorithm, based on internal sensors (incremental encoders) of the LABMATE mobile robot. This is not a error-free approach , and the robot's configuration estimation errors are accumulative. Thus, for long paths, the odometry system should be updated with data from an external sensor system (video camera, range laser, etc) ,
then:
O(t
0 .85
Fig. 7. Mobile robot path-tracking for the SPGPC (dotted line) and GPC (solid line). Nominal case.
5. SIMULATION RESULTS
if O(t)
0 .8
+ 1) = O(t) + O(t)T
+ 1) = Xg (t) + O~) (sin(O(t + 1)) - sin(O(t))) Yg(t + 1) = Yg(t) - O~) (cos(O(t + 1) - cos(O(t))) Xg (t
157
7. CONCLUSIONS
error delay case
A Smith Predictor based Generalized Predictive Controller has been proposed for a mobile robot Path Tracking problem. A robustness analysis has been carried out showing the better robust performance of this new controller. Some simulation and experimental results have shown that classical GPC can not be used for this problem in the presence of dead time uncertainties, because the mobile robot became unstable. The new control strategy gives similar nominal performance than the normal GPC and also robust performance for this case.
.'~"
,
6.==75----:0:':: .8----:0::-:8:;-5---::0':.9 ---::07. .9S:---:-,-71.0=-S----:,:L .,-,:-': . ':;-5--=".::-2-=-'1.25
8. REFERENCES Amidi, O. (1990). Integrated Mobile Robot Control. Carnegie Mellon University Robotic Institute (Pittsburgh) . Technical Report CMURI-TR-90-17. D.W. Clarke, C. Mothadi and P.S. Tuffs (1987). Generalized Predictive Control. Part I The Basic Algorithm and Part 11 Extensions and Interpretations. Automatica 23(2), 137- 160. G6mez Ortega, J . (1994). Navegaci6n en robots m6viles basada en tecnicas de control predicitvo neuronal. PhD thesis. Escuela Sup. de Ingenieros, Univ. de Sevilla. G6mez Ortega, J . and E. F. Camacho (1994) . Neural Network MBPC for Mobile Robots Path Tracking. Robotics and Computer Integrated Manufacturing Journal 11(4), 271- 278. G6mez Ortega, J. and E. F. Camacho (1996) . Neural Predictive Control for Mobile Robot Navigation in a Partially Structured Static Environment. In: To appear in the Proc. of the 13th IFAC World Congress. San Francisco. Kanayama, Y., Y. Kimura, T . Noguchi and F. Miyakazi (1990) . A Stable Tracking Control Method for an Autonomous Mobile Robot. In: Proc. IEEE Int. ConJ. on Robotics and Automation. pp. 384- 389. M.Morari and E.Zafiriou. (1989). Robust Process Control. Prentice Hall. Nelson, W. and I. Cox (1990). Autonomous Robot Vehicles. Chap. Local Path Control for an Autonomous Vehicle, pp. 38- 44. SpringerVerlag. Nelson, W. L. (1989). Continous SteeringFunction control of robot carts. IEEE Transactions on Industrial Electronics 36(3), 330337. Normey Rico, J.E. and E.F.Camacho (1996) . On closed loop performance of predictor based structures. IEEE T. on Control System Technology (in revision). Ollero, A. , A. Garcfa-Cerezo and J. L. Martfnez (1993). Fuzzy Supervisory Path Tracking of Mobile Robots. In: Proc. First IFAC Int. Workshop. Hampshire. Ollero, A. and O. Amidi (1991). Predictive Path Tracking of Mobile Robots. Aplications to the CMU Navlab . In: Proc. IEEE Fifth Int. ConJ. on Advanced Robotics. Pisa. pp. 1081- 1086. Shin, D. H. and S. Singh (1990) . Vision and Navigation. The Carnegie Mellon Navlab. Chap. Vehicle and Path Models for Autonomous Navigation, pp. 283- 307. Kluwer Academic Publishers. Smith, O.J.M (1958) . Feedback Control Systems. Mc Graw Hill.
x (met.rs)
Fig. 8. Mobile robot path-tracking for the SPGPC (dotted line), GPC (solid line). Dead time uncertainties case. with a predefined frequency. This issue is separated from the control problem and has not been adressed in the paper.
Fig. 9. Labmate Mobile robot Figure 10 shows an experimental test, where the reference path is drawn in dashed line. The controller parameters have been chosen as follows: A = 0.5, 80 = 0.5, 8y = 1, N = 10, T = 0.2, look ahead = 0.6 meters and V = O.lm/s. In this case, it is assumed that the error in the dead time is 0.4 seconds (two sampling periods). Figure 10 shows the quite good performance of the proposed controller in spite of the error in the dead time estimation. Note also that the reference path chosen has small radious of curvature, which makes more dificult the path reference following experimental r.suns
,
, ,
O~-L-~-~~-~-~~~~-~~
o
0.5
' .5
2.5
35
45
x(meters)
Fig. 10. Mobile robot path-tracking for the SPGPC . Path reference in dashed line.
158
A SMITH PREDICTOR BASED GENERALIZED PREDICTIVE CONTROLLER FOR MOBILE ROBOT PATH-TRACKING J.E. Normey-Rico*, J. G6mez-Ortega** and E.F.Camacho**
*Dpto. de Automa9tlO e Sistemas. Federal Univ. of Santa Catarina, Brazil. E-maiL:[email protected] *Dpto. Ingenieria de Sistemas y Automatica, Univ. de Sevilla. Escuela Superior de Ingenieros, Camino de Los Descubrimientos sin, 41092 Sevilla, Spain . Fax: +34-5-4556849, email: juango@cartuja. us. es
Abs~r<:tct:
This paper presents the application of a Smith predictor based generalized predlctlve controller (sPGPc) to the path-traking problem of a mobile robot. The equivalence between the generalized predictive controller (GPc) and a structure composed by an optimal predictor and a classical controller is given and it is shown that using a Smith predictor instead of an optimal predictor, increases the robustness of the system. Some results concerning the influence of the new predictor structure on the closed loop robustness are presented. Simulation and experimental tests carried out on a Labmate mobile robot validate the performance of the proposed strategy. Copyright © 1998lFAC
Keywords: Mobile robots, path tracking, predictive controllers, robustness.
Different approaches can be found for the mobile robot controller. In (Nelson and Cox, 1990) a linear proportional feedback of a set of different path tracking errors is implemented. In (Kanayama et al., 1990), a classical PID controller is used. Also geometric approaches are encountered in the literature. In this case, the control strategy consists of computing an analytical curve (for example a spline (Nelson, 1989) or a quintic polynomial (Shin and Singh, 1990)) that complies with several constraints (continuity in curvature, initial and final points, etc.). The structure of the curve is fixed and its parameters are calculated every sampling interval. In (Amidi, 1990), a strategy called pure pursuit is presented. The robot curvature is calculated in each sampling time as the curvature of a circumference that passes through the robot position and a lookahead point located on the reference path.
1. INTRODUCTION
One of the most important issues in the field of mobile robots is the path-tracking (PT) problem. The objective of PT is concerned with driving the vehicle as close as possible to a desired explicit path, with least control effort. When the environment is well known and no unexpected obstacles are considered, the reference path can be the result of an off-line path planning problem . But also, many avoiding obstacle strategies found in the literature use a decoupled scheme, where the reference path is replanned on-line when a unexpected obstacle is detected. PT methods are based on the estimated error between the current vehicle position and the reference path to be followed. In order to compute this error a reference point on the desired path must be chosen . There are several approaches for these selection . One approach chooses a reference point located on the desired path a predefined lookahead distance ahead the actual robot position (Amidi, 1990) . Another approach is used whith optimal control techniques where a set of consecutive reference points are chosen instead of an isolated point (Gomez Ortega, 1994; Gomez Ortega and Camacho, 1994) .
Also, optimal control techniques can be used for PT . In (Ollero and Amidi, 1991) the Generalized Predictive Control is chosen as control strategy. A linear model of the robot kinematics is considered for the prediction of the future robot positions. A quadratic cost function , where the tracking error and the control effort are penalized, is minimized. Also a predictive technique is encountered in (Gomez Ortega and Camacho, 1994) , (Gomez
153
where c5(i) and A(i) are respectively the error and control weighting factors, that are usually considered constant .
Ortega and Camacho, 1996) , but a non linear model of the robot is considered and a multilayer perceptron is used to implement the predictive controller in real time.
The future system outputs, Y(t + ilt) for i = NI , ... , N 2 , are predicted from a model of the process, from past inputs and outputs, and from the control actions foreseen for the future , U(k + ilt) for i=O, .. . ,Nu - 1, which are the unknown variables. In this way, J can be expressed as a function of only the future control actions.
The problem that we raise in this paper is that of driving a mobile robot to follow a previously calculated reference path , defined as a set of consecutive points. As the future references are known , it would seem that a predictive control technique is a suitable approach for this problem.
After this sequence is obtained, a receding horizon approach is considered. This consists of only applying the first calculated control action U(t) . This process is repeated at every sampling interval in such a way that the calculated open loop control law is applied in a closed loop manner. A block diagram of the system is shown in figure 1.
A common issue of these techniques is their high dependence on a predicition model. The uncertainties in this model can lead to a poor performance or make the system unstable. One of the more critical uncertainties is the dead time estimation. For the mobile robot , this uncertainty has several sources: the robot payload can changes over the time; the level of batteries charge decreases with time which implies a change in the dynamic of the locomotion system (for example the wheels acceleration, the dead time in the communications between the robot and a host computer in a teleoperation system) , etc. In this paper, a new Generalized Predictive Controller is proposed for mobile robot path tracking. This controller uses a Smith predictor instead of an optimal predictor, increasing the robustness of the controller.
Ptedi,-=Uve oontrollcr
Fulure predicted
The paper is organised as follows : In section 2 a brief description of the GPC is presented . Also the model considered for the mobile robot knimematics and the way in which the GPC is applied to the PT problem is shown in the same section . In section 3 the equivalence between the GPC and the predictor plus a clasical controller is shown. Section 4 is dedicated to present a new predictor structure and a robustness analysis of the proposed controller. A comparative simulation analysis between the proposed controller and the GPC is presented in section 5. The method has been tested on a Labmate mobile robot and some experimental results are presented in section 6.
Objetive
constraints
: .. _. -------- -----~~':~~~ ----- ----------- -- - -- -
-_.
Fig. 1. The predictive controller scheme. Prediction Model For an GPC formulation, a linear model of the mobile platform is needed to predict the future positions and headings of the robot. As a test bed a differential-drive mobile robot (Labmate (G6mez Ortega and Camacho, 1994)) has been used and its linearized model is obtained using a local reference frame . (see figure 2) :
2. GENERALIZED PREDICTIVE CONTROL PATH-TRACKING
B(t) ] [ y(t)
= (1-VTZ-1)
[
1
]
VT/2 ,(t -1- d)
where B(t) is the robot heading, y(t) is the local position, ,(t) is the curvature, d is the dead time, V is the linear velocity, T is the sample period and I is the identy matrix. In this work V is considered to be constant.
GPC strategy
The objective of Generalized Predictive Control (GPc) (D .W. Clarke and Tuffs, 1987) is to obtain a future control action sequence (U(t), U(t + lit), ... , U(t + Nu - lit)) in such a way that the future predicted system outputs + ilt) will be as close as possible to the desired references Yd(t + i) over the prediction horizon. This is accomplished by the minimization of a cost function J with respect to the control variables. Often, the cost function J is the sum of the squared future errors in reference tracking, e(t + ilt) = Y(t + ilt) - Yd(t + i), predicted over a prediction horizon N = N2 - N l , and also of the sum of the squared control actions foreseen for the future , U(t + ilt), over a control horizon Nu:
ht
Fig. 2. The local reference model. Nu
N2
J =
L
c5(i)e(t + ilt)
+ L A(i)[~U(t + i
- Il t
W
This linearized equations are valid only for small values of 6(). This fact implies that , when the
;=1
154
where w(t) represents the future references in the heading and position error y(t) and M, PI and P2 are constant matrices.
robot is not positioned on the desired path, approaching routes have to be considered. In this way, the mobile robot is always located on the reference path and the values of 68 are small. In this paper, the approaching path has been computed using the pure pursuit estrategy. That is, the next N reference points are predicted as if the robot should be controled with a pure pursuit scheme.
Thus the final control law is given by (see figure 3):
2N
The mobile robot curvature has been chosen as the control variable. Althought the system should has single input/single output (note that y(t) = (VT/2)8(t», it has been noticed that the system performance is much better when two outputs (8(t) and y(t» are considered. This is because the approaching path depends on the robot heading (Amidi, 1990). Thus, an error in 8(t) leads to not well fitted approaching trajectories. On the other hand , if the error in y(t) is not penalized, the robot could not approximate the desired global path.
122y(t + d - 1 It)
+ L j;w(t + d + i)
(5)
i=1
where the coefficients VT, 8(i) and ~(i).
lij
and j; are functions of
9 (1)
MOBILE ROBOT
The GPC Structure for the Mobile Robot In this work the GPC parameters are chosen in the following manner: Nu = N, NI = d + 1 and N2 = d + 1 + N . In order to compute the future control sequence U(t), U(t+ l), .. .it is necessary to compute Y(t+j I t). Using the prediction model of the mobile robot and an optimal predictor Y(t + d + j I t) is given by:
OPTIMAL PREDICTOR
Fig. 3. Control structure of the GPC controller for the mobile robot
+ j I t) = 20(t + j - 1 I t) 2 I t) + VT 6 ,(t + j - d - 1) (1)
O(t -O(t + j y(t + j
I t) = 2y(t + j
- 1 I t) - y(t
3. THE OPTIMAL PREDICTOR PLUS A CLASSICAL CONTROLLER
- 2 I t)+
+j
+(VT)2/2 6 ,(t + j - d - 1)
In this section it will be shown that a control structure based on an optimal predictor and a classical controller is equivalent to the GPC. This result will be used later to justify the use of the Smith predictor structure instead of the optimal one in the GPC.
(2)
or in a vectorial form :
1 [
O(t + d + 1 I t) O(t + d + 2 I t) [
y(t+d~Nlt) S
6,(t) 6,(t + 1)
=G
.
1
1
9<1+dl
1
[
6,(t +:N _ 1)
O(t + d I t) O(t+d-1It) [ y(t+dlt) y(t + d - 1 I t)
w(t+d+1) w(t+d+2)
+P2
.
[
,., t+d)
(3)
where G and S are constant matrices. Using this relation in the expresion of J, the minimization of the cost function gives:
M
ROBOT
+
y(l)
6,(t+:N-1)
O(t + d It) O(t + d - 1 I t) [ y(t+dlt) y(t + d - 1 I t)
6,(t) 6,(t+1)
9(1 )
MOBILE
Fig. 4. Equivalent control structure of the GPC controller for the mobile robot As stated in the previous section, the GPC can be represented by a linear control law computed using the predictions of the system outputs. The compelete control structure of the GPC, showed in the block diagram of figure 3, can easily be trans+ formed into the one shown in figure 4, concluding that the GPC is equivalent to an optimal predictor and a classical controller composed by a double PI and a pair of reference filters given by:
1
1
OPTIMAL PREDICTOR
(4)
w(t+d+2N) 155
Also note that the prediction of 0 and y is computed independently and so the final structure is not a multivariable scheme but the control action is composed by the addition of a pair of monovariable classical controllers.
9(1)
4. THE PROPOSED SPGPC In this section a robust analysis of the GPC is carried out and a new predictor structure is proposed in order to obtain better robust properties. The analysis is made for each controlled variable because of the decoupled structured plant model.
Fig. 5. Equivalent structure for robustness analysis and using that P = Pn + DP , (RoPn + QI}) = G n and (RyPn + Qy) = (VT /2)G n the characteristic equation is:
The plant will be represented by the matrix transfer function given in equation 2 and unstructured uncertainties will be considered. lt will be assumed from now on that the behaviour of the process is described by a family of linear models. Note that in this case the two elements of the matrix have the same poles and zeros and only a different gain. From now on P will represent the monovariable transfer function from 'Y to O. Thus the real plant P will be in a vicinity of the nominal plant Pn , that is: P = Pn + 6P. If d is the dead time of the plant it is possible to write, in a discret representation: P = Gz- d and for the nominal case Pn = Gnz-dn . Thus G represents the plant without the dead time and G n is its nominal value.
1 + (Cl
+ 11 t) = (1- A(z-I»
zO(t)
+ B(Z-l)r(t -
CGn I I DP Imax = 11I+CRI} I where C = Cl
B(t
It)
= «1 -
d)
i
A(Z-l ))z)iO(t)+
+ 2:(1- A(z-l)i-l B(Z-l)r(t -
d + j - 1) (6)
i=l
Using the same procedure for y, it is posible to obtain the final block diagram representation of the optimal predictor and the controller that can be seen in figure 5. In this case, the polynomials Qo , Ro , Qy and Ry have the following expresions:
Qy(Z-I) = ~TQo(Z-I)
=
+ (VT/2)C2 .
In order to improve robustness a new generalized predictive controller, the Smith predictor based generalized predictive controller SPGPC is proposed. This new control strategy is based on the Smith predictor structure (Smith, 1958) and its block diagram is shown in figure 6. The main ideas of this strategy are: (i) the prediction at time t is computed by a model of the plant without the dead time, and (ii) in order to correct mismatch between the plant and model , the error between the output of the process and the model (including the dead time) is feedback, as indicated in figure 6. lt has been shown by some of the authors (Normey Rico and E.F.Camacho, 1996) that this control structure has better robustness properties than stuctures based on optimal predictors. In this case the blocks Cl and C2 represent the linear control law given by the GPC algorithm. Comparing this structure with the block diagram of figure 5 it is easy to note that RI} = Ry = 1 and so the uncertainty norm-bound is defined by the following expresion:
= 2 .. .. , B(t + j I t) is computed as: +j
(10)
Then using the expresion of RI} (7), it easy to show that I RI} (jw) 12 1 for all frequencies. In fact the modulus of the vector 2 - z-l vary between 1 and 2. Thus, only small uncertainties will be aceptable. Note that I RI}(jw) I will increase with the dead time and this implies that the robustness of the optimal predictor based strucure will be worst for plants with high dead time.
where A(z-l) = .0.A(Z-I) and B(z-l) = 6B(z-l) Then, for j
(9)
and so the uncertainty norm-bound is defined by the following expresion:
In order to analyse the robustnes of the system the block diagram of figure 4 will be used . The relation between the prediction and the input and output of the plant can be computed directly from the polinomials A and B and the dead time of the plant (Pn = Bz~d - l). For the robot heading it follows : B(t
+ (VT /2)C2 )(G n + RI}DP) = 0
CGn I I DP ISM = 11 +ICI
(11)
The last expresion shows that the robust stability bound is the same as if the system had no dead time in the transfer function (M.Morari and E .Zafiriou. , 1989). Then , if the blocks Cl and C2 are the same in the GPC and in the SPGPC (this condition is obtained using the same tuning parameters) , the one computed with the Smith predictor structure has better robust properties. No-
VT,/n (zd n - Ro) .
From the block diagram of figure 5 it is easy to see that this equation is :
156
O(t) = RWr(t - 1) - w/(t - 1)
9(t)
2W Mobile Robot
y(t)
V
= R Wr (t -
1)
+ W/ (t -
1)
2 Model without
if O(t)
= 0, then:
the Delay
+ 1) = O(t) Xg(t + 1) = Xg(t) + VT cos O(t) Yg(t + 1) = Yg(t) + VT sin O(t) O(t
+
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ()~+--------~
Fig. 6. Smith Predictor where Wr and w/ are respectively the rigth and left wheel velocities .
tice that in the absence of model uncertainties or perturbations, both the GPC and the SPGPC give the same nominal performance, independly of the selection of the controller parameters (look ahead, N, A and 8) that can be computed using a classical procedure like in (Ollero and Amidi, 1991) or a fuzzy methodology as the one presented in (Ollero et al., 1993) .
The control is computed using a normal GPC and also a SPGPC with the same controller parameters: A = 0.5, 80 = 0.5, 8y = 1 and N = 15. In both cases the sampling time is T = 0.5 seconds, the look ahead parameter is fixed to 0.6 meters and the linear velocity V = O.lmj s. In figure 7 the nominal performance of both controllers is compared (d n = 2). As can be seen the SPGPC (dotted line) and the normal GPC (solid line) have similar responses. In figure 8 the performance of both controllers is compared when the real dead time (dr ) is 4. Note that the normal GPC (solid line) can not follow the reference and that the SPGPC (dotted line) has the same performance as in the nominal case. Note the unstability in the GPC case.
For the computation of the controller parameters the following procedure must be executed: • compute the prediction of the output using the local open loop model of the mobile robot without considering disturbances, that is, using only polynomials A and B and the dead time d; • add the missmatch between the output and the prediction to each prediction. For this case i = 0,1:
nominal case
O(t+d-i It) = O(t+d-i I t)+O(t-i) -O(t-i) y(t+d-i I t)
= y(t+d-i I t)+y(t-i)-y(t-i)
• compute the coeficients lij and fi using the same procedure as in the normal GPC , as explained in section 2; • compute the control law (6,) using the same control equation as in the GPC (equation 5) .
r
~:
Note that the predictive approach proposed in this paper is very simple and does not need any training stage like the one proposed in (G6mez Ortega and Camacho, 1994).
~ •,...... ,,, I
'"
1~~--~--~~--~--~-L--~~~-
0 .75
In order to illustrate the results given in the previous sections a simulation example is presented. The reference path is a straight line trajectory (dashed line in figure 7) and the kninematic behaviour of the mobile robot is computed using the complete non linear model given by the following equations (where x g, Yg are the global coordinates of the mobile robot) :
¥ 0,
0 .9
0 .95
1 1.05 x(metars)
1.1
1.15
1.2
1.25
6. EXPERIMENTAL RESULTS The proposed SPGPC was implemented in a LABMATE mobile robot shown in figure 9. The mobile robot configuration (position and heading) has been computed at each sampling time using an odometry algorithm, based on internal sensors (incremental encoders) of the LABMATE mobile robot. This is not a error-free approach , and the robot's configuration estimation errors are accumulative. Thus, for long paths, the odometry system should be updated with data from an external sensor system (video camera, range laser, etc) ,
then:
O(t
0 .85
Fig. 7. Mobile robot path-tracking for the SPGPC (dotted line) and GPC (solid line). Nominal case.
5. SIMULATION RESULTS
if O(t)
0 .8
+ 1) = O(t) + O(t)T
+ 1) = Xg (t) + O~) (sin(O(t + 1)) - sin(O(t))) Yg(t + 1) = Yg(t) - O~) (cos(O(t + 1) - cos(O(t))) Xg (t
157
7. CONCLUSIONS
error delay case
A Smith Predictor based Generalized Predictive Controller has been proposed for a mobile robot Path Tracking problem. A robustness analysis has been carried out showing the better robust performance of this new controller. Some simulation and experimental results have shown that classical GPC can not be used for this problem in the presence of dead time uncertainties, because the mobile robot became unstable. The new control strategy gives similar nominal performance than the normal GPC and also robust performance for this case.
.'~"
,
6.==75----:0:':: .8----:0::-:8:;-5---::0':.9 ---::07. .9S:---:-,-71.0=-S----:,:L .,-,:-': . ':;-5--=".::-2-=-'1.25
8. REFERENCES Amidi, O. (1990). Integrated Mobile Robot Control. Carnegie Mellon University Robotic Institute (Pittsburgh) . Technical Report CMURI-TR-90-17. D.W. Clarke, C. Mothadi and P.S. Tuffs (1987). Generalized Predictive Control. Part I The Basic Algorithm and Part 11 Extensions and Interpretations. Automatica 23(2), 137- 160. G6mez Ortega, J . (1994). Navegaci6n en robots m6viles basada en tecnicas de control predicitvo neuronal. PhD thesis. Escuela Sup. de Ingenieros, Univ. de Sevilla. G6mez Ortega, J . and E. F. Camacho (1994) . Neural Network MBPC for Mobile Robots Path Tracking. Robotics and Computer Integrated Manufacturing Journal 11(4), 271- 278. G6mez Ortega, J. and E. F. Camacho (1996) . Neural Predictive Control for Mobile Robot Navigation in a Partially Structured Static Environment. In: To appear in the Proc. of the 13th IFAC World Congress. San Francisco. Kanayama, Y., Y. Kimura, T . Noguchi and F. Miyakazi (1990) . A Stable Tracking Control Method for an Autonomous Mobile Robot. In: Proc. IEEE Int. ConJ. on Robotics and Automation. pp. 384- 389. M.Morari and E.Zafiriou. (1989). Robust Process Control. Prentice Hall. Nelson, W. and I. Cox (1990). Autonomous Robot Vehicles. Chap. Local Path Control for an Autonomous Vehicle, pp. 38- 44. SpringerVerlag. Nelson, W. L. (1989). Continous SteeringFunction control of robot carts. IEEE Transactions on Industrial Electronics 36(3), 330337. Normey Rico, J.E. and E.F.Camacho (1996) . On closed loop performance of predictor based structures. IEEE T. on Control System Technology (in revision). Ollero, A. , A. Garcfa-Cerezo and J. L. Martfnez (1993). Fuzzy Supervisory Path Tracking of Mobile Robots. In: Proc. First IFAC Int. Workshop. Hampshire. Ollero, A. and O. Amidi (1991). Predictive Path Tracking of Mobile Robots. Aplications to the CMU Navlab . In: Proc. IEEE Fifth Int. ConJ. on Advanced Robotics. Pisa. pp. 1081- 1086. Shin, D. H. and S. Singh (1990) . Vision and Navigation. The Carnegie Mellon Navlab. Chap. Vehicle and Path Models for Autonomous Navigation, pp. 283- 307. Kluwer Academic Publishers. Smith, O.J.M (1958) . Feedback Control Systems. Mc Graw Hill.
x (met.rs)
Fig. 8. Mobile robot path-tracking for the SPGPC (dotted line), GPC (solid line). Dead time uncertainties case. with a predefined frequency. This issue is separated from the control problem and has not been adressed in the paper.
Fig. 9. Labmate Mobile robot Figure 10 shows an experimental test, where the reference path is drawn in dashed line. The controller parameters have been chosen as follows: A = 0.5, 80 = 0.5, 8y = 1, N = 10, T = 0.2, look ahead = 0.6 meters and V = O.lm/s. In this case, it is assumed that the error in the dead time is 0.4 seconds (two sampling periods). Figure 10 shows the quite good performance of the proposed controller in spite of the error in the dead time estimation. Note also that the reference path chosen has small radious of curvature, which makes more dificult the path reference following experimental r.suns
,
, ,
O~-L-~-~~-~-~~~~-~~
o
0.5
' .5
2.5
35
45
x(meters)
Fig. 10. Mobile robot path-tracking for the SPGPC . Path reference in dashed line.
158