Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering http://pii.sagepub.com/
Coordinated path-following control for a group of mobile robots with velocity recovery J Ghommam, H Mehrjerdi and M Saad Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 2010 224: 995 DOI: 10.1243/09596518JSCE914 The online version of this article can be found at: http://pii.sagepub.com/content/224/8/995
Published by: http://www.sagepublications.com
On behalf of:
Institution of Mechanical Engineers
Additional services and information for Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering can be found at: Email Alerts: http://pii.sagepub.com/cgi/alerts Subscriptions: http://pii.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.com/journalsPermissions.nav Citations: http://pii.sagepub.com/content/224/8/995.refs.html
>> Version of Record - Dec 1, 2010 What is This?
Downloaded from pii.sagepub.com by guest on April 14, 2012
995
Coordinated path-following control for a group of mobile robots with velocity recovery J Ghommam1*, H Mehrjerdi2, and M Saad2 1 Research Unit on Intelligent Control, Design, and Optimization of Complex Systems (ICOS), University of Sfax, Sfax, Tunisia 2 Department of Electrical Engineering, Ecole de technologie sup’erieure, Quebec Canada The manuscript was received on 17 October 2009 and was accepted after revision for publication on 26 May 2010. DOI: 10.1243/09596518JSCE914
Abstract: This paper addresses the problem of coordinated path following where multiple mobile robots are required to follow a prescribed path while keeping a desired inter-robot formation pattern. A combination of the Lyapunov techniques and graph theory is used to derive the formation architecture. Path following for each vehicle consists of converging the geometric error at the origin. Vehicles’ coordination is achieved by adjusting the speed of each vehicle along its path according to information on the positions and speeds of a subset of the other vehicles of the group. Unlike previous research that assume availability of the reference velocity to each mobile robot, the situation is considered where this information is only available to a leader of this formation. The control scheme relies on an adaptive design to estimate the reference velocity which the other mobile robots need to reconstruct to recover the desired formation. Simulations results are presented and discussed. Keywords: path-following, formation control, Lyapunov technique, graph theory, adaptive control, time-varying system, nested Matrosov theorem
1
INTRODUCTION
In the past few decades, decentralized coordination of autonomous vehicles has become an active area of research and has attracted the attention of multidisciplinary researchers because of its potential applications in both civilian and military sectors. Many efforts have been directed towards the deployment of groups of networked autonomous mobile robots that interact autonomously with one another and with the environment to improve significantly the efficiency, performance, reconfigurability, and robustness that individual vehicles currently cannot perform. The main challenge in coordination of mobile robots is to achieve a common group behaviour while they exchange information about their position. From a control engineering point of view, coordination can be roughly understood as controlling the position and *Corresponding author: Research Unit on Intelligent Control, Design, and Optimization of Complex Systems (ICOS), University of Sfax, Sfax Engineering School, BP W, 3038 Sfax, Tunisia. email:
[email protected] JSCE914
orientation of a group of mobile robots while they move forward to track a designated reference point that can be elected as a leader within the group, or just simply a virtual target that moves along the path with a prescribed dynamic. During the last few years, several motion coordination algorithms have been developed in the literature. Among the typical algorithms for motion coordination, two outstanding schemes that are currently arousing the researchers’ interest are the formation stability and the agreement problem. Fax and Murray [1] analyses the stability of vehicles’ behaviour modelled as a firstorder system relying on graph theory that captures the inter-vehicle communication topology. Ren [2], [3] contrived new consensus-type techniques to solve formation control problems for second-order systems or systems that can be converted to double-integrator dynamics via a feedback linearization. Formation control based on a consensus-type algorithm for vehicles with non-holonomic constraints were considered by Ren and Atkins [4]. However, the non-holonomic kinematics are transformed to double-integrator dynamics by controlling the hand position instead of the inertial Proc. IMechE Vol. 224 Part I: J. Systems and Control Engineering
Downloaded from pii.sagepub.com by guest on April 14, 2012
996
J Ghommam, H Mehrjerdi, and M Saad
position of the vehicles. Consequently, the vehicle heading is not controlled. Most of those works model the agents as point robots and consider only position control; in numerous applications the orientation of the agents plays an important role, which means that the agents must be modelled as rigid bodies. Taking into account the position and orientation of the rigid bodies in a formation of a team of mobile robots, Breivik et al. [5] proposed a guided formation control scheme based on a modular design procedure that makes the design completely decentralized in the sense that no variables need to be communicated between the formation members. In reference [6] the problem of coordinated path following of multiplewheeled robots was solved by resorting to linearization and gain scheduling techniques. Even though this solution is quite simple, it lacks global results, that is convergence of the vehicles to their paths and to the desired formation pattern is only guaranteed locally. In reference [7], the authors proposed a tracking consensus protocol for multi-agents with an active leader each modelled as a second-order dynamics and further explored the situation where the velocity of the active leader is unknown beforehand. An observer implemented on each follower to estimate the leader’s velocity is presented. Recently Peng and Yang [8] extended this consensus protocol to account for timevarying delay in channel communication. The results from the aforementioned two papers are basically obtained for linear agents or fully actuated systems, the dynamics of which can be converted through feedback linearization to double-integrator dynamics. This would limit the application of the derived consensus protocol to general non-holonomic systems. Motivated by these considerations, the problem of coordinated path-following control has recently come to the forum [9–11]. The strategy adopted to resolve such a problem unfolds into two main points. The first is the path following where the objective is to design a local control law that steers each mobile robot to its desired path and moves it along with a fixed velocity [12, 13]. The second is the coordination, where the objective is to adjust each vehicle’s velocity in order to synchronize it with the overall formation speed, which is known beforehand to all vehicles in the formation, and develop a coordinated controller which makes use of this information. A challenging problem will be to consider that this information is only available to one leader in the formation. In the current paper, a coordinated motion control strategy is derived for multiple mobile robots that builds on the authors’ previous work [14]. The techniques proposed by Ghabcheloo et al. [10, 15] we
exploited to handle the constraints imposed by the topology of the intervehicle communications network. The controller is derived in two stages: first, a path-following control law is used that drives each vehicle to its assigned path, regardless of the temporal speed profile adopted. This is done by making each vehicle approach a given virtual target that moves along the path. Second, the speed of the virtual target is adjusted to synchronize their position, thus achieving the coordination of the mobile robot along the paths. Unlike the work presented in references [10], [14], and [15] where by which authors assumed that the reference velocity is available to each vehicle within the group, in this paper, using the framework proposed by Bai et al. [16], this assumption is relaxed to the situation where only one leader possesses this information while the other members reconstruct the velocity profile to recover the desired formation pattern. The paper is organized as follows. Section 2 recalls the model for a unicycle-type mobile robot and formulates the path following and coordination problem. Section 3 gives a solution to path following for a single mobile robot followed by a strategy to coordinate multiple mobile robots along their paths by considering the topology of the intervehicle communication. Section 4 examines the situation where velocity speed of the formation is only known by a leader; an adaptive design is presented to show how the follower members estimate this velocity in order to recover the desired formation. Section 5 discusses robustness of the coordination controllers with respect to disturbances and time delay of the information being transmitted through the communication channel due to imperfections in low-cost hardware. Section 6 illustrates the performance of the pathfollowing controller together with the adaptive design through numerical simulation. Finally, in section 7, conclusions and new directions for research are given.
1.1
Preliminaries
The notation in this paper is as follows. A parameterized path is a geometric curve Nd ~ gdi [ R2 : Asi [ R such that g~gdi where gdi is continuously parameterized by the path variable si. A path particle is said to have a certain dynamic along its path gdi if the time derivative of its path variable converges to a given speed vL(t) as time goes to infinity. The following restrictions are imposed on the desired path: there exist positive
Proc. IMechE Vol. 224 Part I: J. Systems and Control Engineering
Downloaded from pii.sagepub.com by guest on April 14, 2012
JSCE914
Coordinated path-following control for a group of mobile robots
997
constants k1i and k2 such that C1
Lg 2 di Lsi ¢k1 , Vsi
C2
vL ðt Þ¢k2
Vt¢0
The first condition C1 means that the desired path to be followed is regular with respect to the path parameter si. In case the path presents some sharpness, one can divide the path into different regular paths and treat each path separately. Condition C2 implies that only the forward path following is considered.
2
Fig. 1
PROBLEM STATEMENT
The mathematical model for a unicycle-type mobile robot with two actuated wheels, shown in Fig. 1, is assumed to have the following form [17] g_ i ~Ji ðgi Þni Mi n_ i zCi ðg_ i Þni zDi ni ~Fi
ð1Þ
where gi 5 [xi, yi, yi]T denotes the position (xi, yi), and the heading yi of the ith robot in the earth fixed frame OXY, see Fig. 1, ni 5 [v1i, v2i]T, with v1i and v2i being the angular velocities of the wheels, F 5 [F1i, F2i]T with F1i and F2i being the control torques applied to the wheels of ith mobile robot. The mass matrix Mi, Coriolis matrix Ci(g˙i), damping matrix Di, and the rotation Ji(gi), in equation (1) are given by Mi ~
m11i
m12i
m12i
m11i
" Ci ðg_ i Þ~
Ji ðgi Þ~
0
, Di ~
ci y_ i
0
0
d22i
#
, {ci y_ i 0 " ri cosðyi Þ sinðyi Þ
b{1 i
2 cosðyi Þ sinðyi Þ
{b{1 i
#T
djji, j 5 1, 2 are the damping coefficients and ci ~ ð1=2bi Þri2 mci ai , where ai is the distance from the origin (xi, yi) to the centre of mass of the mobile robot, bi and ri are defined in Fig. 1. In the current paper, first the problem is considered of forcing the ith mobile robot to follow closely a virtual target moving with a desired speed profile on a given planar path V parameterized by (xd(s), yd(s)), with s being the path parameter. Assuming that individual path following controller has JSCE914
been implemented on each mobile robot, the aim is then to coordinate the entire group of robots in order to achieve the desired formation pattern. This will be done by adjusting the speed profile of each mobile robot within the group as a function of the parametrization states that capture the positions of the virtual targets on their corresponding paths gdi ðsi Þ. Having explained the two problems for path following and coordination, the formation control objective boils down to designing a control system F for each mobile robot such that limgi ðt Þ{gdi ðsi Þ~0
ð2Þ
lim si {sj ~0,
ð3Þ
t?‘
d11i
Interpretation of path-following errors for a single mobile robot [14]
t?‘
limjs_ i ðt Þ{vL ðt Þj~0 t?‘
The first objective (2) means that each robot must track its desired position and orientation, in the sense that each robot moves on the path and its linear velocity is tangential to its own path; the second objective (3) will guarantee that all the mobile robots are coordinated and propagate along the paths with a common desired velocity.
3
CONTROL DESIGN
In previous work [14], the present authors proposed an algorithm for the coordination of multiple mobile robots using exclusively the backstepping technique combined with the principle of the virtual structure that yields uniform boundedness and asymptotic convergence of the position error, as well as the coordination states. However, with this strategy, the formation control law designed for each robot required Proc. IMechE Vol. 224 Part I: J. Systems and Control Engineering
Downloaded from pii.sagepub.com by guest on April 14, 2012
998
J Ghommam, H Mehrjerdi, and M Saad
knowledge of the path parameter state, the measurements of position and heading errors of itself and the other robots in the group – which amounts to significant intervehicle data communication – and this is not suitable for operations in environments with restrictive communication options. In the following, a slight modification of the method developed in reference [14] is suggested to account for a small amount of intervehicle communication.
3.1.1
The feedback law for Fi for each mobile robot i obtained in reference [14] together with equation (7) solves the path-following problem. In particular the path-following error gei and the speed error ji converges asymptotically to zero as time goes to infinity.
3.1.2 3.1
Path-following design
In this subsection, the result obtained in reference [14] is briefly recalled to solve the path-following problem. Define the position error gei in a frame attached to the reference path V(si) as the difference between the positions of the robot and of the virtual target, and ji the virtual target speed error gei ~RTi ðgi Þ½gi ðt Þ{gdi ðsi ðt ÞÞ ji ~s_ i {vL ðt Þ
ð4Þ
where R(g) is an orthonormal transformation matrix. Notice that in reference [14], the speed error state ji was defined as the error between the along-path distance of vehicle i and the centre of the virtual structure. Exploiting the technique design in reference [18], a local controller for Fi has been designed that makes the time derivative of the following Lyapunov function 1 1 1 2 VPFi ~ kgei k2 z nTi Mi ni z j 2 2 2kcci i
ð5Þ
takes the form
i 1 h_ ji ji zdi ðgei Þ V_ PFi ~{gTei Pgei {nTi Knei ni z kcci
ð6Þ
3
1 0 0 6 7 P~4 0 0 0 5, Knei 0 0 1 is a positive matrix. To solve the path-following problem, one needs to assign a feedback law for j˙i in order to make V˙1i negative; this is done by choosing j_ i ~{kji ji {di ðgei Þ
Remark 3.1
Notice that by construction, the term di(gei) goes to zero since the path-following error states converge asymptotically to zero. For the sequel, in order to simplify the development, set di(gei) 5 0.
3.2
Coordinated controller
So far, a strategy has been designed regarding how to steer a single mobile robot to move along a given path with a desired speed profile. Consider now a group of n mobile robots I : ~f1, . . . , ng, each with its own path-following controller implemented on board. To achieve coordination among the members of the group, it is necessary to assign a common speed profile to each path so that the mobile robots propagate along them while holding a specified geometric formation pattern. Inspired by the work of Ghabcheloo et al. [10, 15], a control variable is introduced in the form of a correction term vcci ðt Þ that is added to the feedback law (7); this signal will serve as an additional controller to be designed in order to achieve coordination of the vehicles. The dynamics of the coordination subsystem with di(gei) 5 0 can therefore be written as follows s_ i ~ji zvL ðt Þ j_ i ~{kji ji zvcci ðt Þ
where kcci is a positive gain, di(gei) captures the terms associated to the speed error state ji, 2
Proposition 3.1
ð7Þ
ð8Þ
To take into account the communication constraints among the vehicles, the neighbourhood set N j is defined that represents the set of vehicles that communicate with vehicle i, that is all vehicles that exchange information with vehicle i. The restriction is that a vehicle in the formation only exchanges its coordination state si with its neighbour in the topology communication. To solve the coordination problem, the following decentralized coordination law for vcc(t) is proposed as follows vcci ðt Þ~{kcci
X
si {sj
ð9Þ
j[Nj
Proc. IMechE Vol. 224 Part I: J. Systems and Control Engineering
Downloaded from pii.sagepub.com by guest on April 14, 2012
JSCE914
Coordinated path-following control for a group of mobile robots
where kcci is the positive gain introduced in equation (5). The coordination subsystem together with the coordination control law can be rewritten in compact form as s_ ~jzvL ðt Þ1 j_ ~{Kj j{Kcc Ls
ð10Þ
where s~½si i [ I , j~½ji i [ I , 1~½1i [ I , and the n6n matrix L is the Laplacian of an undirected graph G that models the underlying communication topology of the robots in formation. It is well known (e.g. see references [10] and [12] for more details) that LT 5 L 5 MMT > 0 and L1 5 M1 5 0 where M is the incidence matrix of the graph that captures the topology communication among the mobile robots. 3.2.1
3.2.3
Remark 3.2
With the above coordination control law (9), it was shown that the coordination problem is simply reduced to aligning the coordination states si asymptotically, that is making si 2 sj R 0 as t R ‘. The coordination control can, however, be extended to achieve different convergence results such as the differences in the information states converge to desired values, i.e. si 2 sj R Dij(t), where Dij(t) 5 di 2 dj denotes the desired (time-varying) deviation between si and sj. The coordination control law (9) can be modified as vcci ðt Þ~d_ i ðt Þzkji di {kcci
X j[Nj
si{di { sj {dj ð13Þ
The following corollary for relative state deviations can be given.
Proposition 3.2
Consider the coordination subsystem (8) with di(gei) 5 0 and assume that the graph that models the communication topology G is connected. Let L be the Laplacian of G and Kj ~diag kji and Kcc ~ diagðkcci Þ be two positive definite diagonal matrices. The decentralized coordination control (9), solves the coordination problem defined in equation (3).
3.2.4
Proof
Define the graph-induced coordination error X ~ MT s, the closed loop (10) rewrites X_ ~MT j ð11Þ
where the property has been used that M vL(t)1 5 0. Let the Lyapunov function T
ð12Þ
The time derivative of Vcc along the solutions of equation (10) is V_ cc ~{jT K{1 cc K j j which is negative semi-definite. Note that V˙cc implies that j 5 0 and in turn implies that MTs 5 0, and in consequence Ls 5 0. Since Vcc is radially unbounded and the system is time invariant, then a direct application of the LaSalle’s invariance principle [19] shows that the origin is globally asymptotically stable, which completes the proof. JSCE914
Proof
Defines˜i 5 si 2 di,j˜ 5 j 2 d˙(t),whered(t) 5 [d1, d2, …, dn]T and the coordination error as X~~MT~s Equations (11) can be written as
j_ ~{Kj j{Kcc MX
1 1 Vcc ~ jT Kcc{1 jz X T X 2 2
Corollary 3.1
The coordination control law (13) ensures that si 2 sj R Dij(t) as t R ‘ if and only if directed graph G is connected.
3.2.5 3.2.2
999
_ X~~MT ~j _ j~~{Kj ~j{Kcc MX~
ð14Þ
The methodology used in the proof of Proposition 2 can now be exploited to show that s˜i(t) R s˜j(t) as t R ‘ if and onlyifG isconnected.Therestoftheproofthenfollowsthe fact that s˜i(t) R ˜sj(t) as t R ‘ is equivalent to si 2 sj R Dij(t) as t R ‘. The architecture of the general coordinated pathfollowing control (CPFC) system can be summarized in Fig. 2. In this figure, the complete control structure for vehicle i and interconnection between communicating vehicles in the formation are given along with relevant equation number. Using wireless communication, vehicles i and j communicate their path parameter to each other. All other required information for vehicle i is obtained locally using local sensory information as shown in this figure. Proc. IMechE Vol. 224 Part I: J. Systems and Control Engineering
Downloaded from pii.sagepub.com by guest on April 14, 2012
1000
J Ghommam, H Mehrjerdi, and M Saad
Fig. 2 Coordinated path-following control system architecture
4
ADAPTIVE DESIGN FOR FORMATION VELOCITY RECOVERY
In section 3 it was assumed that the reference velocity vL(t) assigned to each mobile robot in the group is known. Now the situation is considered where an elected leader within the group only knows this information a priori, while the remaining members estimate this reference velocity to recover the desired formation pattern. Following reference [16], it is assumed that the desired speed velocity vL(t) is parameterized as vL ðt Þ~wi ðt Þh
ð15Þ
where h is available only to the leader and wi(t) is a scalar function available to each mobile robot and satisfies the persistent excitation (PE). which means that for all t0 > 0 ð t0 zz t0
wi ðt Þ2 dt¢c
ð16Þ
with some constants f . 0 and c . 0. Since wi(t) is time-varying it is required, that all the mobile robots have synchronized clocks. To ensure coordination, the remaining mobile robots must estimate the unknown parameter h by hˆi and reconstruct vˆL(t) from hi v^Li ðt Þ~wi ðt Þ^
ð17Þ
For the sequel it is assumed that the leader is vehicle i 5 1, thus the dynamic of s1 is kept the same as in equation (4) while the path parameters for the remaining vehicles i.e. i 5 2, 3, …, n and the coordination control as well are replaced with s_ ~^jzv L ðt Þ ^j_ i ~{Kj ^j{Kcc Ls
ð18Þ
where v L ðt Þ~½vL ðt Þ, v^Li ðt ÞTi [ I {f1g , the estimate velocity vˆLi(t) is now obtained from equation (17). It is necessary to design an updated law for hˆi so that the vehicles in the formation estimate the desired velocity profile and synchronize with the leader, that is si 5 sj 5 s1. The following proposition solves this new coordination problem.
4.1
Proposition 4.1
Consider the coordination subsystem described by equation (18), where vL(t) is uniformly bounded and piecewise continuous, and its components are parameterized as equation (17) with the parameter hˆi updated as _ h^~{CWT Ls
ð19Þ
where hˆ 5 [hˆi]16n, W 5 diag(w1, …, wn), and C 5 CT is the adaptive gain matrix. Then the equilibrium point (jˆ, Ls, hˆ) is globally uniformly asymptotically stable.
Proc. IMechE Vol. 224 Part I: J. Systems and Control Engineering
Downloaded from pii.sagepub.com by guest on April 14, 2012
JSCE914
Coordinated path-following control for a group of mobile robots
4.1.1
To see this, note that Y1 5 0 implies jˆ 5 0 and it follows that the last two terms of equation (26) cancel out, Y2 becomes
Proof
The error variable h˜i is denoted by ~ hi ~ ^ hi {h,
i~2, . . . ,n
ð20Þ
Since the speed profile for the formation is known for vehicle 1, then set h˜1 5 0, w1 5 1, and construct a column vector h˜ 5 [h˜1, …, h˜n]T. Clearly, from equation (19) it can noted that _ h~~{CWT Ls
ð21Þ
the coordination dynamic (18) with the update law (21) can be written as follows s_ ~^ jzvL 1zWh~
ð28Þ
Next, a third auxiliary function is introduced as follows V3 ~{ðLv~L ÞT s
ð29Þ
where v˜L 5 Wh˜. The time derivative of V3 is given by _T V_ 3 ~h^ WLszh^T W_ Ls{v~LT L^j{v~LT LWh~
ð30Þ
Y1 ~0
_ h~~{CWT Ls
ð22Þ
To prove that the equilibrium point (jˆ, Ls, h˜) is globally uniformly asymptotically stable, the nested Matrosov theorem [20] is used. The first auxiliary function is as follows 1 T {1 ^ 1 T 1 V1 ~ ^ j Kcc jz s Lsz h~T C{1 h~ 2 2 2 which time derivative along the solutions of the second and the last equation of (22) yields the following negative semi-definite derivative ^ jT K{1 V_ 1 ~{^ cc Kj j : ~Y1 ¡0
ð24Þ
where the fact that LvL1 5 0 has been used, thus guaranteeing uniform global stability. The second auxiliary function is jT Ls V 2 ~^
ð25Þ
The derivative of V2 yields
jT R ^ jz ^ jT LWh~ : ~Y2 ~{xT Qxz^
[V_ 3 ~{v~LT Lv~L : ~Y3 ¡0
Y2 ~0,
ð31Þ
To show equation (31), notice that Y2 5 0 implies jˆ 5 Ls 5 0, therefore the first and second terms of equation (30) vanish. Equation (31) follows from the definition of v˜L. At this final step another auxiliary function is defined V4 ~{h~T F ðt Þh~ where F ðt Þ~ F_ ðt Þ~e t
Б
ð‘
t
ð32Þ
e ðt{tÞ WðtÞWðtÞT dt. Note that
e {t WðtÞWðtÞT dt
t
d ze dt t
ð‘ e
{t
T
WðtÞWðtÞ dt
t
~F ðt Þ{Wðt ÞWðt ÞT
ð33Þ
Consequently, the time derivative of the Lyapunov function V4 is given as
jT L ^ jz ^ jT LWh~{^ jT Kj Ls{sT LKcc Ls V_ 2 ~^ ð26Þ
0
1 1 K P j B 2 C where x 5 (jˆT, sTL)T, Q~@ 1 A, and R 5 P + L; Kj Kcc 2
. P is chosen such that Q > 0 that is P¢ kj2 4kcc I{L where I is the identity matrix with appropriate dimensions. Now it is claimed that
JSCE914
V_ 2 ~{xT Qx¡0
where it is claimed that
_ ^ j~{Kj ^ j{Kcc Ls
Y1 ~0[Y2 ¡0
1001
ð27Þ
_ V_ 4 ~{h~T F ðt Þh~zv~LT v~L {2h~T F ðt Þh~ : ~Y4
ð34Þ
and claim that 2 Y2 ~0, Y3 ~0[Y4 ~{ce {f h~ ¡0
ð35Þ
This can be shown easily since Y3 5 0 implies that v˜L 5 0 and Y2 5 0 implies that jˆ 5 0 and Ls 5 0, it is straightforward to conclude from equation (19) that the last two terms of equation (34) vanish. Since Yi 5 0, i 5 1, 2, 3, 4, this implies from the nested Matrosov theorem [20] that the equilibrium Proc. IMechE Vol. 224 Part I: J. Systems and Control Engineering
Downloaded from pii.sagepub.com by guest on April 14, 2012
1002
J Ghommam, H Mehrjerdi, and M Saad
(gˆ, Ls, h˜) 5 (0, 0, 0) is globally uniformly asymptotically stable. Since h˜ R 0 then for the ith, i 5 2, …, n mobile robot, it can be seen that vˆLi R vL(t). Moreover, since Ls R 0, coordination of mobile robots is achieved, that is si 5 sj 5 s1 for all i, j. This completes the proof.
5
ROBUSTNESS DISCUSSION
The control laws for path following have been designed under the assumption that there are no environmental disturbances and the vehicle parameters are certain. However, in practice, there is uncertainty in the kinematics and the dynamics because some geometric parameters may not be known exactly; there may also be uncertainty in the map. In order to make the controller robust, some robustifying terms can be added or it is possible to resort to an adaptive version, as shown in reference [14]. Under additive environmental disturbances, namely the friction of the vehicles’ wheels on the ground, it can still be shown that the path-following error IgeiI is ultimately uniformly bounded (UUB). Boundedness can be reduced by changing the control gains if the frictions are not too large. Yet one should also consider robustness from the sensory point of view; one should consider: (a) that sensors deliver uncertain responses; (b) that different physical sensors, even if apparently identical, may perform differently because of slight differences in the electronics and mechanics or because of their different positions on the robot. Later, in the simulation results section, the case will be considered where output variables of some sensors may be noisy or some perturbations are added to the model. The case is also considered of limited and unreliable information exchange among the robots, which can cause some communication delay.
6
NUMERICAL SIMULATIONS
In this section simulations are carried out for coordinated control of a group of three identical mobile robots to illustrate the effectiveness of the proposed controllers developed in this paper. First simulations are run for the non-adaptive design in section 3.2. Two different scenarios are proposed: first, in the absence of disturbances the coordination control is simulated to execute an in-line formation, that is the three mobile robots are requested to follow a sinusoidal
path by having them aligned along a common vertical line. In the second step of the simulations, the three vehicles are considered in the coordination process while the formation shapes are reconfigured on the fly, along three concentric circles (i.e. for the first 50 s the coordinated vehicles are fixed to maintain an inline formation pattern, and then afterwards a triangular shape for the rest of testing time). In a third step, a formation of three vehicles along smoothly varying reference trajectories is explored in non-ideal conditions such that measurement noise is added to sensor measurements, including uncertain terms like friction. Coordination control with delayed information is illustrated for three robots executing an in-line formation in ideal conditions (i.e. without disturbances). Finally, simulations are run for the adaptive design of section 4, in which the reference velocity vL(t) 5 4 sin(t) m/s 5 w(t)h, where w(t) 5 sin(t) available for all vehicles and h 5 4 is only available for vehicle 1 and has to be estimated by the two remaining mobile robots using the parametrization (17) and the update law (19). The physical parameters of the robot are taken from [17]. The robots parameters are given in Table 1. 2 2 m11i ~0:25b{2 i ri mi bi zIi zIwi , 2 2 m12i ~0:25b{2 i ri mi bi {Ii mi ~mci z2mwi ,
6.1
Ii ~mci a2i z2mwi b2i zIci z2Imi
Coordination in free environment disturbances
In the first simulation, the reference trajectories are taken as xdi(s) 5 si and ydi(si) 5 10i sin(0.1si) for all i 5 1, …, 3. Figure 3(a) shows the case where three robots are required to move along a sinusoidal paths as an in-line formation with a common speed vL ~2 m=s. It can be seen that all robots track their reference trajectories well. Notice in Fig. 3(b) how Table 1 Parameters of the mobile robot Parameters
Values
Unit
ri bi ai mci mwi Ici Iwi Imi
0.15 0.75 0.3 30 1 15.625 0.005 0.0025
m m m kg kg kg m2 kg m2 kg m2
Proc. IMechE Vol. 224 Part I: J. Systems and Control Engineering
Downloaded from pii.sagepub.com by guest on April 14, 2012
JSCE914
Coordinated path-following control for a group of mobile robots
1003
Fig. 3 Coordination of three wheeled robots along a sinusoidal paths
vehicles adjust their speed along the path so as to achieve coordinations. This could be explained as follows: during the first few seconds, the vehicles quickly move to reach their trajectories. As soon as they are moving on their paths, the vehicles slow down to wait for their neighbours in order to synchronize as long as they propagate. In Fig. 3(c) and Fig. 3(d) it can be observed that the position and orientation tracking errors converge to zero and remain there as Proposition 3.2 suggests. 6.2
Coordination with reconfiguration man euvre
Figure 4(a) displays a formation of three vehicles changing shapes on the fly, where the robots primarily achieve a straight line formation, followed by a wedge formation (see Fig. 5). This example captures the situation where the robot in the apex of the triangular shape enters a narrow tunnel and plays the role of the formation leader, whereas the others are free to move by coordinating their positions with the leader. Figure 4(b) depicts the coordination errors convergence. In the inline formation the coordination errors s1 2 s2 5 s1 2 s3 5 s2 2 s3 5 0 converge to zero. In the triangle formation however s1 2 s2 and s1 2 s3 converge to 230 and s2 2 s3 converges to zero as desired. JSCE914
6.3
Coordination in an uncertain environment
A problem of growing importance in any environments is the location of mobile robots. It is considered that the fleet of vehicles is located through global positioning system (GPS) sensor. Uncorrected GPS signals may come in different forms and arise from a variety of different sources, namely, the highsignal-frequency noise and the long-term drift [21]. In order to perform realistic simulations, a highfrequency noise introduced into the range sensing is considered, which has a normal distribution of 0.2 m2 variance. Friction is also introduced, in the dynamic model of equation (1) and represented as the vector F¯i 5 [a1i sign(vi) + b1ivi, a2i sign(wi) + b2iwi]T, where a1i 5 0.05, a2i 5 0.25, b1i 5 0.01, and b2i 5 0.3 for all vehicles in the formation. For the second scenario, the reference paths are chosen as follows: for the first 30 s, xdi(si) 5 si, ydi 5 60 tanh(0.1si) + 10(i 2 1) and xd 5 (50 + 10i) 10 sin(0.1si), yd 5 10i cos(0.1si) + 50 (for all i 5 1, …, 3) for the rest of the simulation time. This choice implies that the reference path is a hyperbolic path for the first 30 s and is a circle for the remaining time. Figure 6(a) illustrates the transient behaviour of the formation vehicles as they assemble and maintain a triangular form with constant pattern D 5 Proc. IMechE Vol. 224 Part I: J. Systems and Control Engineering
Downloaded from pii.sagepub.com by guest on April 14, 2012
1004
J Ghommam, H Mehrjerdi, and M Saad
Fig. 4
Coordination of three wheeled robots with reconfiguration manouevre
[Dij]361 5 [22.5, 1.5, 22.5]T. Figure 6(c) depicts the error position yei, i 5 1, …,3 and shows their convergence to zero even in the presence of highfrequency noises and frictions. Notice, however the high initial error of robots 1 and 3, which is attributable to the fact that the reference trajectories start at some points that are far from the initial robots’ positions. It is also of interest to notice from Fig. 6(d) that the robots initially move with a fast oscillating heading until they reach their desired path, then exhibit a steady-state error with mean value of 0.1 rad, which varies a little at the change in the radius of the reference trajectories, then recovers to previous mean steady-state values. Figure 6(b) shows how vehicles adjust their speed along the path so as to achieve coordinations. It can be observed that when vehicles keep tracking a straight line path their linear velocities are almost the same; however, at the break point when the radius of the reference trajectories changes, linear velocities are no longer the same to keep the formation pattern intact.
6.4
Effect of time delay on coordination
In most applications, the information being transmitted in a network of mobile robots usually exhibits a
time delay t. To illustrate the effect of time delay on the coordination process, a simple formation problem is considered of three wheeled mobile robots along parallel straight lines. If there is a small communication delay in the range of [0 s, 0.5 s], according to Figs 7 (a) and (b) the coordinated controllers still work and this shows that the controllers are robust to small communication delay. However, if this delay exceeds this range (e.g. see Figs 7(b) and (c)), the coordinated controllers fail and the error coordinating states will no longer converge to zero.
6.5
Finally, simulations are shown for three wheeled mobile robots moving in formation along three rectilinear paths using the proposed adaptive design in which the velocity vL(t) 5 h1 sin(t) is time varying and is only available for mobile robot 1; that is for robot 1, the parameter h1 5 4, whereas the other two vehicles have to estimate this velocity according to equation (15) and the update law (19). It is clear from Fig. 8(a) that with the adaptive design from section 4, the coordination states converge altogether to zero as t R ‘. Figure 8(b) illustrates how well the parameter convergence is guaranteed. Figure 8(c) shows that the speed dynamic of each path parameter converges to the reference velocity (objective (3)), and Fig. 8(d) illustrates that the estimated velocities vˆL2(t) and vˆL3(t) converge to the corresponding reference velocities vˆL1(t) 5 4 sin(t).
7
Fig. 5 Desired formation configuration
Coordination with velocity recovery
CONCLUSIONS
The current paper has addressed the problem of manouering a group of mobile robots along a given
Proc. IMechE Vol. 224 Part I: J. Systems and Control Engineering
Downloaded from pii.sagepub.com by guest on April 14, 2012
JSCE914
Coordinated path-following control for a group of mobile robots
Fig. 6
Coordination of three wheeled robots along piecewise paths
path, while holding a desired intervehicle formation pattern. The coordination scheme is first executed while taking into account that the reference velocity is known for all members of the group. Second, the proposed solution is then extended to consider coordination where the reference velocity is only available to an elected leader in the formation, while the other mobile robots estimate this information
Fig. 7 JSCE914
1005
with an adaptive design. The solution proposed is built on an adaptive Lyapunov-based technique and graph theory. Stability analysis of the coordination control with velocity recovery is carried out by using Matrosov’s generalized theorem. Current work is underway to implement the proposed coordination controller to real prototypes. Future work will extend the coordination controller in the current paper to
Time history of the coordination error states with different time delays Proc. IMechE Vol. 224 Part I: J. Systems and Control Engineering
Downloaded from pii.sagepub.com by guest on April 14, 2012
1006
J Ghommam, H Mehrjerdi, and M Saad
7 8
9
10
11 Fig. 8
Time evolution of the states a coordinations with the adaptive design 12
robust adaptive control in order to compensate for non-model dynamics, sensors, and actuators perturbations and unknown environments.
13
ACKNOWLEDGEMENT
14
The authors are very grateful to the anonymous reviewers for insightful remarks and for useful comments on an earlier draft.
15
F Authors 2010 REFERENCES 1 Fax, A. and Murray, R. M. Information flow and cooperative control of vehicle formations. IEEE Trans., Automatic Control, 2004, 50, 1465–1476. 2 Ren, W. On consensus algorithms for doubleintegrator dynamics. IEEE Trans., Automation Control, 2008, 53, 1503–1509. 3 Ren, W. On consensus algorithms for doubleintegrator dynamics. In Proceedings of the 46th Conference on Decision and Control, New Orleans, USA, pp. 2295–2300. 4 Ren, W. and Atkins, E. Coordination of multiple micro-air vehicles using consensus schemes. In Proceedings of AIAA Infotech@Aerospace Conference, Arlington, USA, pp. 2005–7067. 5 Breivik, M. and Fossen, T. I. Guided formation control for wheeled mobile robots. In Proceedings of the 9th IEEE Conference on Control Automation Robotics and Vision, Singapore, 5–8 July 2006, pp. 1–7. 6 Ghabcheloo, R., Pascoal, A., Silvestre, C., and Kaminer, I. Coordinated path following control of
16
17
18
19 20
21
multiple wheeled robots using linearization techniques. Int. J. Systems Sci., 2006, 37, 399–414. Guo, J. L. W. and Chen, S. Peng, K. and Yang, Y. Leader-following consensus problem with a varying-velocity leader and timevarying delays. Physica A, 2009, 388, 193–209. Aguiar, A. P. and Pascoal, A. M. Coordinated pathfollowing control for nonlinear systems with logicbased communication. In Proceedings of the 46th Conference on Decision and Control, New Orleans, Louisiana, USA, 2007, pp. 1473–1479. Ghabcheloo, R., Pascoal, A., Silvestre, C., and Kaminer, I. Coordinated path following control of multiple wheeled robots with bidirectional communication constraints. Int. J. Adaption Control Signal, 2007, 21, 133–157. Skjetne, R., Moi, S., and Fossen, T. Nonlinear formation control of marine craft. In Proceedings of the 41st Conference on Decision and Control, Las Vegas, NV, 2002, vol. 2, pp. 1699–1704. Pedro Aguiar, A. and Hespanha, J. P. Trajectorytracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty. J. Automatica, 2007, 8, 1362–1379. Skjetne, R., Fossen, T., and Kokotovic, P. V. Robust output maneuvering for a class of nonlinear systems. J. Automatica, 2004, 40, 373–383. Ghommam, J., Saad, M., and Mnif, F. Formation path following control of unicycle-type mobile robots. In Proceedings of IEEE Conference on Robotics and Automation, Pasadena, California, USA, May 2008, pp. 1966–1972. Ghabcheloo, R., Pedro Aguiar, A., Pascoal, A. M., and Silvestre, C. Synchronization in multi-agent systems with switching topologies and non-homogeneous communication delays. In Proceedings of the 46th Conference on Decision and Control, New Orleans, Louisiana, USA, pp. 2327–2332. Bai, H., Arcak, M., and Wen, J. T. Adaptive design for reference velocity recovery in motion coordination. 2008, 57(8), 602–610. Fukao, T., Nakagawa, H., and Adachi, H. Adaptive tracking control of nonholonomic mobile robot. IEEE Trans., Robotics Automatic, 2000, 16(5), 609– 615. Jiang, Z. P. and Nijmeijer, H. Tracking control of mobile robotic case study in backstepping. J. Automatica, 1997, 33, 1393–1399. Khalil, H. K. Nonlinear systems, 1992 (Macmillan, New York). Loria, A., Panteley, E., Popovic, D., and Teel, A. R. A nested matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous system. IEEE Trans., Automatics Control, 2005, 50(2), 183–198. Lund, H. H., Miglino, O., and Nolfi, S. Evolving mobile robots in simulated and real environments. Artificial Life, 1995, 4(2), 417–434.
Proc. IMechE Vol. 224 Part I: J. Systems and Control Engineering
Downloaded from pii.sagepub.com by guest on April 14, 2012
JSCE914
