- Email: [email protected]
Robust discrete time control of a nonholonomic mobile robot
ROBUST DISCRETE TIME CONTROL OF A NONHOLONOMIC MOB ...
14th World Congress ofTFAC
B-Id-04-6
Copyright CO 1999 IFAC 14th Triellllial World Congress, Beijing, P.R. China
ROBUST DISCRETE TIME CONTROL OF A
NONHOLONOMIC MOBILE ROBOT T. Leo * G. Orlando *
• Dipartimento di Elettronica ed A utomatica Universitd di Ancona, Via Brecce Bianche, 60191 Ancona, Italy Fax: ++39-071-~804334, E-Mail: orlando@ee.'Unia1l.it
Abstract: The tracking trajectory problem for a mobile robot violating the nonholonomic constraint is solved using discrete time sliding mode control. A simple discrete time model of the vehicle has been used, and the asymptotical boundness of the controlled variables has been theoretically proved. The proposed algorithm has been tested by simulation on the continuous time model of the robot, showing satisfactory performances. The simulation study reported in this note is preliminary to the experimental testing of the control algorithm, which is being carried out on a LABMATE vehicle at the Robotics Lab at the Dipartimento di Elettronica ed Automatica. Copyright© 1999lFAC Keywords: Nonholonomic Systems, Mobile Robots, Discrete Time Nonlinear Control, Sliding Mode Control, Robust ControL
1. INTRODUCTION
N onholonomic systems have received much attention by control researchers in recent literature (Kolmanovskyand McClamroch, 1995), (AstoHi, 1996), (Kolmanovsky et al., 1996), (D'Alessandro and Ferrante, 1997). In this context, the point stabilization and the tracking trajectory problems for a mobile robot have been addressed in different ways. (Canudas De Wit and Sordalen, 1992), (Guldner and Utkin, 1994), (Jiang and Nijmeijer, 1997L (Fierro and Lewis, 1997), (T.Leo and Orlando, 1998). As pointed out in (Kolmanovsky and McClamroch, 1995), (Fierro and Lewis, 1997), although an extensive theoretical framework has been developed in this field, relatively few results have been presented about the robustnes issue, Le. "the problem of control of nonholonomic systems when there are model uncertainties" (Kolmanovsky and McClamroch, 1995). In particular, "a very important situation requiring attention arises from model perturbations that destroy the nonholo-
nomic assumption .... There is much to be gained by studying the effects of these perturbations and by developing control designs that perform well in the presence of uncertainties" (Kolmanovsky and McClamroch, 1995). In (Canudas De Wit and Khennouf, 1995) a stabilizing control law for a mobile robot is derived, coping with a state
vanishing disturbance violating the nonholonomic constraint. In (Bennani and Rauchon, 1995) a design method for robust stabilization of flat and chained systems is proposed. Another important aspect to be considered, strictly related to the robustness issue, is the effectiveness of the theoretically developed control laws in a real situation. In fact, the implementability of such controllers is a key point for the development of robotic systems with a high degree of autonomy. In this context, an automatic navigation system is currently being developed on a LABMATE vehicle at the Robotics Lab at the Dipariimento di Elettronica ed Automatica (T.Leo et al., 1995), (Conte et al., 1995). The navigation system can be decomposed into a motion planner and a collision
623
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress oflFAC
ROBUST DISCRETE TIME CONTROL OF A NONHOLONOMIC MOB ...
avoidance module. Both blocks, however, rely on a suitable low-level control procedure, which has to drive the wheel actuators in order to ensure the tracking of the planned trajectory with a sufficient degree of precision, compatibly with the limited motor power and the limited response speed. In (T.Leo and Orlando, 1998) the point stabilization and the tracking trajectory problems for a mobile robot have been considered and correspondingly two different discrete time sliding mode control algorithms have been developed without considering robustness issues. The discrete time approach allows to avoid the well known problems due to the discretization of continuous time controllers (Gao et al., 1995), (Corradini and Orlando, 1997)
In the present paper the nonholonomic constraint of the vehicle is assumed to be violated by a bounded disturbance, which phisica1ly could represent a slipping situation, for example, or a situation in which the pure rolling condition is not verified. Consequently, the kinematic model is affected by disturbances not satisfying the matching conditions. The discrete time control law presented in (T.Leo and Orlando, 1998) for the tracking trajectory problem is here extended to cope with this kind of disturbances: a simple discrete time model of the vehicle has been used, and the asym. ptotical boundness of the controlled variables has been theoretically proved. Only the tracking trajectory problem has been considered in the present paper, in view of an experimental test on the mobile robot. However, the extension of the control law here proposed to the point stabilization problem jg straightforward. As a preliminary study, the proposed discrete time algorithms have been tested by simulation on the continuous time model of the robot. These results are going to be validated by an experimental test on the LABMATE vehicle. The paper is organized as follows. In Section 2 the kinematic model of the vehicle is reported, along with its discrete time approximated representation. The robust control law is presented in Section 3, while Section 4 contains some simulation results. Finally, conclusions are drawn in Section 5.
the control inputs into properly generated motor torques, by means of inner control loops. Since the angular and the displacement velocities are the only variables directly available to the designer, a kinematic model of the mobil robot will be considered, in order to synthesize the control law. The vehicle position is described by the coordinates (x,y) of the midpoint between the two driving wheels, and by the orientation angle (J with respect to a fixed frame, as shown in Fig.1. y
y
o
The LABMATE vehicle is equipped with two driving wheels mounted on the same axis, and additional free wheels. The motion and orientation are achieved by independent actuators, i.e. DC motors providing the necessary torques to the wheels. The vehicle can be controlled by setting its angular and its displacement velocities (control variables): a dedicated DSP board then provides for mapping
x
Fig.l - Schematic representation of the vehicle.
The kinematic model has the following form:
vet) cos(9} vet) sin(O) o = wet)
:i; = { 'if =
(2.1)
where wet) and vet) are the angular and the displacement velocities, respectively. Model (2.1) corresponds to the hypothesis of"pure rolling and non slipping condition" for the vehicle. Assume that a bounded disturbance 1JCt), 11J(t)1 ::; p, with p > 0 known constant, violates the nonholonomic constraint associated to (2.1) (Canudas De Wit and Khennouf, 1995), Le.: x(t)sin(OCt» - yet) cos(6(t» = 1J(t)
(2.2)
Equation (2.2) could represent a situation in which the non slipping condition is not verified, and corresponds to the following kinematic model, in which the disturbance doesn't satisfy the matching conditions: {
2. MODEL OF THE MOBILE ROBOT
Y'
d; = vet) cos(9) + l1(t) sin(8) iI = vet) sin(B) -1J(t) cos(t/)
e = wCt)
(2.3)
System (2.3) represents the plant to be controlled. As far as the discretizazion of model (2.3) is concerned, it is assumed that cos6(t), sinO(t) and the disturbance l1(t) are slowly varying with respect to the sampling time Tc. With this assumption, considering a zero-order-hold for the control variables wet), vet) and integrating equations (2.1) from kTe to (k + 1)Te, the following apprOximated discrete time model is obtained:
624
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST DISCRETE TIME CONTROL OF A NONHOLONOMIC MOB ...
Xk+1 {
Yk+l 01.+1
=
Xk
+ TcVk cos(lh) + Tc1Jk sin (Ok)
= Yk + TcVk sin(ek ) = 81. + TcWk
-
14th World Congress ofTFAC
In order to achieve a discrete time sliding mode on (3.5), the variable 810+1 needs to be calculated:
T cl1,. COS(8k)
810+1
(2.4) The subscript k indicates the variable evaluated in t = kTc •
= (tk -
0,. - TcWk
(3.6)
( 6Yk~1 - e~~l) (;r:}
(3.6)
where: ale
= arctan
(d)
.6.XIe+l - e"+l
3. DISCRETE TIME SLIDING MODE CONTROL
Considering that I1k is bounded and defining the following quantity:
In this section, a control law for the robust tracking of an assigned trajectory will be derived. It will be shown that the tracking errors remain bounded in a region whose amplitude depends on the uncertainties. The control algorithm and the proof of its convergence will be derived using the discrete time model (2.4). Its effectiveness, however, will be tested by simulation considering the continuous time model as the plant to be controlled, in order to have results closer to the real situation. If the mobile robot has to track the assigned trajectory Zd(t) = [x(d) (t)y(d) (tWCd) (t)Y, this latter needs to satisfy the nonholonomic constraint
Le.: . (d) (t) tan(r9(d) (t») = -y-x(d) (t)
(et)
8X.Hl
+
( )
- TcVk sin(8 k )
Pkz -
)
T"Vk COS(Ok) (3.8)
it can be shown by simple geometric considerations that: (3.9) being 61: an unmeasurable bounded quantity, 161. I ~ ../2Tcp. The following auxiliary control input duced: _ W,. =
{ -Bk
-Br.
Wh
is intro-
+ ,8(lskl- Tcpy2) if Is.:! > Tcp.;2 if IS1o1 :::; TcpV2
being {3 a design parameter, -1 < {3 < 1. Taking into account (3.7), (3.8), (3.9), and choosing:
(3.2)
As a consequence, the sampled reference trajectory Z~d) = [x~d)y~d)r9~d)JT fulfil Is the following relationship: 1.
(6Y~~1 + pr,:)
(3.10) (3.1)
~
arctan
ii
xCd) (t) sin(O(d) (t») -
tan(O(d)
_=
(tk
Wk
1 (a" - Ok Te
= -
8,. - Wk)
(3.11)
equation (3.6) becomes: 8k+l
= 810 + Oh + Wh
(3.12)
Le.:
Cd)
=~ Llx(d)
(3.3)
(3.13)
k
being dyed) - y(d) _ y(
By imposing the condition for the existence of a discrete time sliding mode on (3.5):
Introducing the tracking errors e~"') = x" - x~d), = Yk - y~d), = eh - O~d), and defining the quantities p~"') = .6.xkd) - ei"') ,pill) = 6yid) - 11 ), 60(d) O(d) _ B(d) p(/I) 6e(d) _ is) the k k+l 1.1' k k ,. , tracking error dynamics is described by:
it can be easily shown that control law (3.10)(3.11) guarantees 810 to remain in a neighbourhood of zero (sector), whose amplitude is Tcp.../2.
eiv)
4/1)
-p~"') + {er~l = -pi") +
4
Te v}, COS(Ok) + Tc'f/" sinCe,,) Tcv,. sin(e k ) - T,,1Jk cos(8k )
4~1 = e}.~l = _p~ll) + TcWk
8"Llsk+l <
When
BIo
tan(8 k
-~(6sk+dl
(3.14)
is inside the sector, it holds:
+ Ok) =
(:t:~)
(3.15)
Le.
(3.4) The following sliding surface is considered: BI<
= arctan
( pC")) p~"')
- Ok
=0
(3.16) (3.5)
625
Copyright 1999 IF AC
ISBN: 008 0432484
ROBUST DISCRETE TIME CONTROL OF A NONHOLONOMIC MOB ...
14th World Congress ofTFAC
4. RESULTS
The solution of (3.16) with respect to cos(O,.) and sin(O,.) gives: cosefh,) sin(lh,)
=
=
cos§,.p~!Il)
+ sino,.p~JI)
J(piz»),a + (p~1I»2
cos OkP~") - sin OkP~"')
J(Pk
Z
)2
+
(3.17)
(PkY ))2
The first two equations of system (3.4) with cos(B,.) and sin(8,.) replaced by (3.17) become:
As discussed in the Introduction, a preliminary simulation study on the kinematic model of the vehicle has been performed before the experimental testing can be achieved. In all the simulations, the sampling time chosen is the same used by the navigation system, Le. To = 0.58. The reference trajectory considered is an experimental planned trajectory (Fig.2). The time evolution of the disturbance l1(t) is represented in Fig.3: it simulates a sudden slipping of the vehicle wheels.
(3.18)
From (3.18) the square norm of the error for variables x, y at time instant k + 1 can be calculated:
(e~11)2 + (e~~1)2 '(COSOkVk -
=
I/~ +
T c 1}k sin 0,.)
2J (p~:Il))2 + (p~1I»)2.
Fig.2 - Reference trajectory in the :c - y plane.
+ T;f]~ + (pi:ll»2 + (pip)? (3.19)
being 911
I/k
= TcVk.
=
(e~;c))'~
Define the quantity:
+ (ekP)2+ _T;p2 + (p~z»)2 + (p~JI»)2)
(1 2v'2J
Fig.3 - Disturbance 1}(t}.
being 7 a design parameter, -1 < "f < 1, the square norm of the error for variables x, 'SI decreases until it reaches the region defined by: (3.21)
g,. ::; 0
When e~:r:), e~lI) are inside the region defined by (3.21), and is inside the sector, it follows from
8,.
The simulation results are reported in FigA, (reference trajectory and actual trajectory in the x-V plane), in Figs.5-7 (tracking errors), in Figs.8-9 (control variables). The controller performances are satisfactory: the system reacts to the disturbance recovering the planned trajectory with acceptable control efforts.
(3.3) and (3.5):
5. CONCLUSIONS
(3.22) When (3.21) and (3.22) are satisfied, the tracking errors remain bounded.
In this note, the tracking trajectory problem for a wheeled mobile base has been addressed, considering the presence of distmbances violating the nonholonomic constraint. A discrete time robust
626
Copyright 1999 IF AC
ISBN: 008 0432484
ROBUST DISCRETE TIME CONTROL OF A NONHOLONOMIC MOB ...
14th World Congress ofTFAC
sliding mode control has been proposed, whose convergence has been theoretically proved. As a preliminary result, the effectiveness of the control law has been tested by simulation on the continuous time model of the vehicle. However, as a future development, the experimental testing of the control algorithm is going to be carried out.
6. REFERENCES
Fig.4 - Reference (thin line) and vehicle (thick line) trajectories in the a; - 'Y plane.
liS)
Fig.5 - Tracking error e~"')
Fig.6 - Thacking error e~II).
Astolfi, A. (1996). Discontinuous control of nonholonomic systems. Systems and Control Letters 27,37-45, Bennani, M.K. and P. Roucholl (1995), Robust stabili'Lation of flat and chained systems. Proc. of 3rd European Control Conference pp. 2630-2635. Canudas De Wit, C. and H. Khennouf (1995). Quasi continuous stabilizing controllers for nonholonomic systems: design and robustness considerations. Proc. of 3rd European Control Conference pp. 2630-2635. Canudas De Wit, C. and O.J. Sordalen (1992). Exponential stabilization of mobile robots with nonholonomic constraints. IEEE 7rans. on Automatic Control 37, 1791-1797. Conte, G., S. Longhi and R. Zulli (1995), Nonholonomic motion planning using distance field. Proc. of IFAC Conference on intelligent Autonom()1J,S vehicles pp. 93-98. Corradini, M.L. and G. Orlando (1997). A discrete adaptive variable structure controller for MIMO systems, and its application to an underwater ROV. IEEE 1rans. Contr. Syst. Technology 5,349-359. D' Alessandro, D. and A. Ferrante (1997). Optimal steering for an extended class of nonholonomic systems using lagrange functionalso Automatica 33, 1635-1646. Fierro, R. and F. L. Lewis (1997). Robust practical point stabilization of a nonholonomic robot using neural networks. Journal of intelligent and robotic systems 20, 295-317. Gao, W., Y. Wang and A. Homaifa (1995). Discrete-time variable structure control systems. IEEE 1rans. Industrial Electronics 42, 117-122. Guldner, J, and V.I. Utkin (1994). Stabilizationof nonholonomic mobile robots using lyapunov functions for navigation and sliding mode control. Proc. of the 33rd conference on decision and control pp. 2967~2972.
627
Copyright 1999 IF AC
ISBN: 008 0432484
ROBUST DISCRETE TIME CONTROL OF A NONHOLONOMIC MOB ...
Jiang, Z.P. and R. Nijmeijer (1997). Tracking control of mobile robots: a case study in backstepping. Automatica 33, 1393-1399. Kolmanovsky, 1. and N.R. McClamroch (1995). Developments in nonholonomic control problems. IEEE Control Systems 15,20-36. Kolmanovsky, L, M. Reyhanoglu and N.R. McClamroch (1996). Switched mode feedback control laws for nonholonomic systems in extended power form. Systems and Control Letters 27, 29-36. T.Leo and G. Orlando (1998). Discrete time sliding mode control of a nonholonomic mobile robot. to be presented at Nonlinear Oontrol
14th World Congress ofTFAC
IlsI
Fig.S - Control variable
VI<.
Systems Design Symposium, NOLCOS98. T.Leo, S. Longhi and R. Zulli (1995). On-line collision-avoidance for a robotic assistance system. Proc. of [FAC Workshop on Human-
Oriented Design of Advanced Robotic Systems DARS 95 pp. 89-95.
Fig.9 - Control variable Wk.
628
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress ofTFAC
B-Id-04-6
Copyright CO 1999 IFAC 14th Triellllial World Congress, Beijing, P.R. China
ROBUST DISCRETE TIME CONTROL OF A
NONHOLONOMIC MOBILE ROBOT T. Leo * G. Orlando *
• Dipartimento di Elettronica ed A utomatica Universitd di Ancona, Via Brecce Bianche, 60191 Ancona, Italy Fax: ++39-071-~804334, E-Mail: orlando@ee.'Unia1l.it
Abstract: The tracking trajectory problem for a mobile robot violating the nonholonomic constraint is solved using discrete time sliding mode control. A simple discrete time model of the vehicle has been used, and the asymptotical boundness of the controlled variables has been theoretically proved. The proposed algorithm has been tested by simulation on the continuous time model of the robot, showing satisfactory performances. The simulation study reported in this note is preliminary to the experimental testing of the control algorithm, which is being carried out on a LABMATE vehicle at the Robotics Lab at the Dipartimento di Elettronica ed Automatica. Copyright© 1999lFAC Keywords: Nonholonomic Systems, Mobile Robots, Discrete Time Nonlinear Control, Sliding Mode Control, Robust ControL
1. INTRODUCTION
N onholonomic systems have received much attention by control researchers in recent literature (Kolmanovskyand McClamroch, 1995), (AstoHi, 1996), (Kolmanovsky et al., 1996), (D'Alessandro and Ferrante, 1997). In this context, the point stabilization and the tracking trajectory problems for a mobile robot have been addressed in different ways. (Canudas De Wit and Sordalen, 1992), (Guldner and Utkin, 1994), (Jiang and Nijmeijer, 1997L (Fierro and Lewis, 1997), (T.Leo and Orlando, 1998). As pointed out in (Kolmanovsky and McClamroch, 1995), (Fierro and Lewis, 1997), although an extensive theoretical framework has been developed in this field, relatively few results have been presented about the robustnes issue, Le. "the problem of control of nonholonomic systems when there are model uncertainties" (Kolmanovsky and McClamroch, 1995). In particular, "a very important situation requiring attention arises from model perturbations that destroy the nonholo-
nomic assumption .... There is much to be gained by studying the effects of these perturbations and by developing control designs that perform well in the presence of uncertainties" (Kolmanovsky and McClamroch, 1995). In (Canudas De Wit and Khennouf, 1995) a stabilizing control law for a mobile robot is derived, coping with a state
vanishing disturbance violating the nonholonomic constraint. In (Bennani and Rauchon, 1995) a design method for robust stabilization of flat and chained systems is proposed. Another important aspect to be considered, strictly related to the robustness issue, is the effectiveness of the theoretically developed control laws in a real situation. In fact, the implementability of such controllers is a key point for the development of robotic systems with a high degree of autonomy. In this context, an automatic navigation system is currently being developed on a LABMATE vehicle at the Robotics Lab at the Dipariimento di Elettronica ed Automatica (T.Leo et al., 1995), (Conte et al., 1995). The navigation system can be decomposed into a motion planner and a collision
623
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress oflFAC
ROBUST DISCRETE TIME CONTROL OF A NONHOLONOMIC MOB ...
avoidance module. Both blocks, however, rely on a suitable low-level control procedure, which has to drive the wheel actuators in order to ensure the tracking of the planned trajectory with a sufficient degree of precision, compatibly with the limited motor power and the limited response speed. In (T.Leo and Orlando, 1998) the point stabilization and the tracking trajectory problems for a mobile robot have been considered and correspondingly two different discrete time sliding mode control algorithms have been developed without considering robustness issues. The discrete time approach allows to avoid the well known problems due to the discretization of continuous time controllers (Gao et al., 1995), (Corradini and Orlando, 1997)
In the present paper the nonholonomic constraint of the vehicle is assumed to be violated by a bounded disturbance, which phisica1ly could represent a slipping situation, for example, or a situation in which the pure rolling condition is not verified. Consequently, the kinematic model is affected by disturbances not satisfying the matching conditions. The discrete time control law presented in (T.Leo and Orlando, 1998) for the tracking trajectory problem is here extended to cope with this kind of disturbances: a simple discrete time model of the vehicle has been used, and the asym. ptotical boundness of the controlled variables has been theoretically proved. Only the tracking trajectory problem has been considered in the present paper, in view of an experimental test on the mobile robot. However, the extension of the control law here proposed to the point stabilization problem jg straightforward. As a preliminary study, the proposed discrete time algorithms have been tested by simulation on the continuous time model of the robot. These results are going to be validated by an experimental test on the LABMATE vehicle. The paper is organized as follows. In Section 2 the kinematic model of the vehicle is reported, along with its discrete time approximated representation. The robust control law is presented in Section 3, while Section 4 contains some simulation results. Finally, conclusions are drawn in Section 5.
the control inputs into properly generated motor torques, by means of inner control loops. Since the angular and the displacement velocities are the only variables directly available to the designer, a kinematic model of the mobil robot will be considered, in order to synthesize the control law. The vehicle position is described by the coordinates (x,y) of the midpoint between the two driving wheels, and by the orientation angle (J with respect to a fixed frame, as shown in Fig.1. y
y
o
The LABMATE vehicle is equipped with two driving wheels mounted on the same axis, and additional free wheels. The motion and orientation are achieved by independent actuators, i.e. DC motors providing the necessary torques to the wheels. The vehicle can be controlled by setting its angular and its displacement velocities (control variables): a dedicated DSP board then provides for mapping
x
Fig.l - Schematic representation of the vehicle.
The kinematic model has the following form:
vet) cos(9} vet) sin(O) o = wet)
:i; = { 'if =
(2.1)
where wet) and vet) are the angular and the displacement velocities, respectively. Model (2.1) corresponds to the hypothesis of"pure rolling and non slipping condition" for the vehicle. Assume that a bounded disturbance 1JCt), 11J(t)1 ::; p, with p > 0 known constant, violates the nonholonomic constraint associated to (2.1) (Canudas De Wit and Khennouf, 1995), Le.: x(t)sin(OCt» - yet) cos(6(t» = 1J(t)
(2.2)
Equation (2.2) could represent a situation in which the non slipping condition is not verified, and corresponds to the following kinematic model, in which the disturbance doesn't satisfy the matching conditions: {
2. MODEL OF THE MOBILE ROBOT
Y'
d; = vet) cos(9) + l1(t) sin(8) iI = vet) sin(B) -1J(t) cos(t/)
e = wCt)
(2.3)
System (2.3) represents the plant to be controlled. As far as the discretizazion of model (2.3) is concerned, it is assumed that cos6(t), sinO(t) and the disturbance l1(t) are slowly varying with respect to the sampling time Tc. With this assumption, considering a zero-order-hold for the control variables wet), vet) and integrating equations (2.1) from kTe to (k + 1)Te, the following apprOximated discrete time model is obtained:
624
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST DISCRETE TIME CONTROL OF A NONHOLONOMIC MOB ...
Xk+1 {
Yk+l 01.+1
=
Xk
+ TcVk cos(lh) + Tc1Jk sin (Ok)
= Yk + TcVk sin(ek ) = 81. + TcWk
-
14th World Congress ofTFAC
In order to achieve a discrete time sliding mode on (3.5), the variable 810+1 needs to be calculated:
T cl1,. COS(8k)
810+1
(2.4) The subscript k indicates the variable evaluated in t = kTc •
= (tk -
0,. - TcWk
(3.6)
( 6Yk~1 - e~~l) (;r:}
(3.6)
where: ale
= arctan
(d)
.6.XIe+l - e"+l
3. DISCRETE TIME SLIDING MODE CONTROL
Considering that I1k is bounded and defining the following quantity:
In this section, a control law for the robust tracking of an assigned trajectory will be derived. It will be shown that the tracking errors remain bounded in a region whose amplitude depends on the uncertainties. The control algorithm and the proof of its convergence will be derived using the discrete time model (2.4). Its effectiveness, however, will be tested by simulation considering the continuous time model as the plant to be controlled, in order to have results closer to the real situation. If the mobile robot has to track the assigned trajectory Zd(t) = [x(d) (t)y(d) (tWCd) (t)Y, this latter needs to satisfy the nonholonomic constraint
Le.: . (d) (t) tan(r9(d) (t») = -y-x(d) (t)
(et)
8X.Hl
+
( )
- TcVk sin(8 k )
Pkz -
)
T"Vk COS(Ok) (3.8)
it can be shown by simple geometric considerations that: (3.9) being 61: an unmeasurable bounded quantity, 161. I ~ ../2Tcp. The following auxiliary control input duced: _ W,. =
{ -Bk
-Br.
Wh
is intro-
+ ,8(lskl- Tcpy2) if Is.:! > Tcp.;2 if IS1o1 :::; TcpV2
being {3 a design parameter, -1 < {3 < 1. Taking into account (3.7), (3.8), (3.9), and choosing:
(3.2)
As a consequence, the sampled reference trajectory Z~d) = [x~d)y~d)r9~d)JT fulfil Is the following relationship: 1.
(6Y~~1 + pr,:)
(3.10) (3.1)
~
arctan
ii
xCd) (t) sin(O(d) (t») -
tan(O(d)
_=
(tk
Wk
1 (a" - Ok Te
= -
8,. - Wk)
(3.11)
equation (3.6) becomes: 8k+l
= 810 + Oh + Wh
(3.12)
Le.:
Cd)
=~ Llx(d)
(3.3)
(3.13)
k
being dyed) - y(d) _ y(
By imposing the condition for the existence of a discrete time sliding mode on (3.5):
Introducing the tracking errors e~"') = x" - x~d), = Yk - y~d), = eh - O~d), and defining the quantities p~"') = .6.xkd) - ei"') ,pill) = 6yid) - 11 ), 60(d) O(d) _ B(d) p(/I) 6e(d) _ is) the k k+l 1.1' k k ,. , tracking error dynamics is described by:
it can be easily shown that control law (3.10)(3.11) guarantees 810 to remain in a neighbourhood of zero (sector), whose amplitude is Tcp.../2.
eiv)
4/1)
-p~"') + {er~l = -pi") +
4
Te v}, COS(Ok) + Tc'f/" sinCe,,) Tcv,. sin(e k ) - T,,1Jk cos(8k )
4~1 = e}.~l = _p~ll) + TcWk
8"Llsk+l <
When
BIo
tan(8 k
-~(6sk+dl
(3.14)
is inside the sector, it holds:
+ Ok) =
(:t:~)
(3.15)
Le.
(3.4) The following sliding surface is considered: BI<
= arctan
( pC")) p~"')
- Ok
=0
(3.16) (3.5)
625
Copyright 1999 IF AC
ISBN: 008 0432484
ROBUST DISCRETE TIME CONTROL OF A NONHOLONOMIC MOB ...
14th World Congress ofTFAC
4. RESULTS
The solution of (3.16) with respect to cos(O,.) and sin(O,.) gives: cosefh,) sin(lh,)
=
=
cos§,.p~!Il)
+ sino,.p~JI)
J(piz»),a + (p~1I»2
cos OkP~") - sin OkP~"')
J(Pk
Z
)2
+
(3.17)
(PkY ))2
The first two equations of system (3.4) with cos(B,.) and sin(8,.) replaced by (3.17) become:
As discussed in the Introduction, a preliminary simulation study on the kinematic model of the vehicle has been performed before the experimental testing can be achieved. In all the simulations, the sampling time chosen is the same used by the navigation system, Le. To = 0.58. The reference trajectory considered is an experimental planned trajectory (Fig.2). The time evolution of the disturbance l1(t) is represented in Fig.3: it simulates a sudden slipping of the vehicle wheels.
(3.18)
From (3.18) the square norm of the error for variables x, y at time instant k + 1 can be calculated:
(e~11)2 + (e~~1)2 '(COSOkVk -
=
I/~ +
T c 1}k sin 0,.)
2J (p~:Il))2 + (p~1I»)2.
Fig.2 - Reference trajectory in the :c - y plane.
+ T;f]~ + (pi:ll»2 + (pip)? (3.19)
being 911
I/k
= TcVk.
=
(e~;c))'~
Define the quantity:
+ (ekP)2+ _T;p2 + (p~z»)2 + (p~JI»)2)
(1 2v'2J
Fig.3 - Disturbance 1}(t}.
being 7 a design parameter, -1 < "f < 1, the square norm of the error for variables x, 'SI decreases until it reaches the region defined by: (3.21)
g,. ::; 0
When e~:r:), e~lI) are inside the region defined by (3.21), and is inside the sector, it follows from
8,.
The simulation results are reported in FigA, (reference trajectory and actual trajectory in the x-V plane), in Figs.5-7 (tracking errors), in Figs.8-9 (control variables). The controller performances are satisfactory: the system reacts to the disturbance recovering the planned trajectory with acceptable control efforts.
(3.3) and (3.5):
5. CONCLUSIONS
(3.22) When (3.21) and (3.22) are satisfied, the tracking errors remain bounded.
In this note, the tracking trajectory problem for a wheeled mobile base has been addressed, considering the presence of distmbances violating the nonholonomic constraint. A discrete time robust
626
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ROBUST DISCRETE TIME CONTROL OF A NONHOLONOMIC MOB ...
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sliding mode control has been proposed, whose convergence has been theoretically proved. As a preliminary result, the effectiveness of the control law has been tested by simulation on the continuous time model of the vehicle. However, as a future development, the experimental testing of the control algorithm is going to be carried out.
6. REFERENCES
Fig.4 - Reference (thin line) and vehicle (thick line) trajectories in the a; - 'Y plane.
liS)
Fig.5 - Tracking error e~"')
Fig.6 - Thacking error e~II).
Astolfi, A. (1996). Discontinuous control of nonholonomic systems. Systems and Control Letters 27,37-45, Bennani, M.K. and P. Roucholl (1995), Robust stabili'Lation of flat and chained systems. Proc. of 3rd European Control Conference pp. 2630-2635. Canudas De Wit, C. and H. Khennouf (1995). Quasi continuous stabilizing controllers for nonholonomic systems: design and robustness considerations. Proc. of 3rd European Control Conference pp. 2630-2635. Canudas De Wit, C. and O.J. Sordalen (1992). Exponential stabilization of mobile robots with nonholonomic constraints. IEEE 7rans. on Automatic Control 37, 1791-1797. Conte, G., S. Longhi and R. Zulli (1995), Nonholonomic motion planning using distance field. Proc. of IFAC Conference on intelligent Autonom()1J,S vehicles pp. 93-98. Corradini, M.L. and G. Orlando (1997). A discrete adaptive variable structure controller for MIMO systems, and its application to an underwater ROV. IEEE 1rans. Contr. Syst. Technology 5,349-359. D' Alessandro, D. and A. Ferrante (1997). Optimal steering for an extended class of nonholonomic systems using lagrange functionalso Automatica 33, 1635-1646. Fierro, R. and F. L. Lewis (1997). Robust practical point stabilization of a nonholonomic robot using neural networks. Journal of intelligent and robotic systems 20, 295-317. Gao, W., Y. Wang and A. Homaifa (1995). Discrete-time variable structure control systems. IEEE 1rans. Industrial Electronics 42, 117-122. Guldner, J, and V.I. Utkin (1994). Stabilizationof nonholonomic mobile robots using lyapunov functions for navigation and sliding mode control. Proc. of the 33rd conference on decision and control pp. 2967~2972.
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ROBUST DISCRETE TIME CONTROL OF A NONHOLONOMIC MOB ...
Jiang, Z.P. and R. Nijmeijer (1997). Tracking control of mobile robots: a case study in backstepping. Automatica 33, 1393-1399. Kolmanovsky, 1. and N.R. McClamroch (1995). Developments in nonholonomic control problems. IEEE Control Systems 15,20-36. Kolmanovsky, L, M. Reyhanoglu and N.R. McClamroch (1996). Switched mode feedback control laws for nonholonomic systems in extended power form. Systems and Control Letters 27, 29-36. T.Leo and G. Orlando (1998). Discrete time sliding mode control of a nonholonomic mobile robot. to be presented at Nonlinear Oontrol
14th World Congress ofTFAC
IlsI
Fig.S - Control variable
VI<.
Systems Design Symposium, NOLCOS98. T.Leo, S. Longhi and R. Zulli (1995). On-line collision-avoidance for a robotic assistance system. Proc. of [FAC Workshop on Human-
Oriented Design of Advanced Robotic Systems DARS 95 pp. 89-95.
Fig.9 - Control variable Wk.
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ISBN: 008 0432484