# Extension of Leader-Follower Behaviours for Wheeled Mobile Robots in Multirobot Coordination

Authors: P. Paniagua-Contro; E. G. Hernandez-Martinez; O. González-Medina; J. González-Sierra; J. J. Flores-Godoy; E. D. Ferreira-Vazquez; G. Fernandez-Anaya
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper presents the extension of leader-follower behaviours, for the case of a combined set of kinematic models of omnidirectional and differential-drive wheeled mobile robots. The control strategies are based on the decentralized measurements of distance and heading angles. Combining the kinematic models, the control strategies produce the standard and new mechanical behaviours related to rigid body orn-trailer approaches. The analysis is given in pairs of robots and extended to the case of multiple robots with a directed tree-shaped communication topology. Combining these behaviours, it is possible to make platoons of robots, as obtained from cluster space or virtual structure approaches, but now defined by local measurements and communication of robots. Numerical simulations and real-time experiments show the performance of the approach and the possibilities to be applied in multirobot tasks.


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## Body

## 1. Introduction

The coordination of multiple mobile robots has found a wide field of applications in the industry, surveillance, home services, logistics, among others [1]. It extends the classical problems of point convergence and trajectory tracking of a unique mobile robot to the case of collective behaviours like the convergence to formation patterns, formation tracking, dispersion, containment, interrobot collision avoidance, etc. The design of formation patterns has been studied in structured and behavioural approaches and the approaches inspired on multicellular mechanism, according to [2]. The possible interactions between robots are represented by a communication topology which is predefined by the designer or by the proximity of the robots. The control laws require the feedback of the global position of the robots or their relative displacements, distances, and angles according to the sensor installed in the workspace or locally in the robots [3–5].The problem of formation tracking requires the simultaneous convergence of the robots to a formation pattern and a target or trajectory. The most basic scheme of multirobot formation tracking is the case of two robots, where a leader follows the trajectory, and the follower agent must satisfy a relative posture with respect to the leader. In a decentralized manner, the control strategy depends on the local measurements of distance and heading or absolute angles. A pioneer work can be found in [6], where a follower robot is formed with respect to one or two leaders. In [7] the desired position of the follower is given by a virtual unicycle robot. The estimation of the posture of a single leader is given in [8]. The leader-follower behaviour designed for two robots is extended to multiple robots in [9], to achieve directed tree-shaped configurations. All the previous works analyse the local convergence only and infer the global convergence in the settling times, but they avoid formal proofs about the stability of the whole system, with an arbitrary number of robots.On the other hand, the study of the global convergence of distance-based formation control has been studied in the context of formation graphs [10]. Such is the case of [11] using distances and PI controllers, time-varying formation in [12], the addition of area constraints in [13, 14] for triangle formations, or the use of Model Predictive Control based on neural-dynamic optimization in [15], among others. In the context of formation graphs, the extension of the leader-follower formations is falling in the case of directed graphs with a unique leader node, as presented in [16]. A unified, distributed formation control scheme that accommodates an arbitrary number of group leaders and an arbitrary information flow among vehicles is presented in [17]. The architecture requires the local information exchange among neighbours only. An extended consensus algorithm is applied on the group level to estimate the time-varying group trajectory information. Using the estimated group trajectory information, a consensus-based distributed formation control strategy is then applied for the vehicle level control. In [18], a merging strategy for cooperative vehicles is addressed. Each vehicle possesses a tracking controller considering both tracking and coupling error signals, where all vehicles in the group know their positions. The trajectories of the cooperative vehicles are planned, to guarantee that each vehicle keeps a certain distance from the rest of the vehicles, and are designed by using the virtual structure approach. The merging strategy is validated in an experimental platform consisting of a group of nonholonomic mobile vehicles representing a basic bicycle model for cars with a zero vehicle length. Using the Lyapunov theory, it is showed that the strategy is global asymptotically stable for the group. In [19], the authors propose a hierarchical control scheme based on the definition of multirobot task functions. The scheme faces the following tasks for controlling the behaviour of multirobot systems: (1) the time parametrization of tasks and (2) smooth task transitions. It is demonstrated that both aspects can be solved by means of terminal attractors. Also the robustness and versatility of the proposed scheme with global tasks in simulation and one experiment considering local tasks with a set of wheeled robots are verified.For the case of wheeled mobile robots moving in the plane, the kinematic and dynamic models commonly used in industrial and service applications are the differential-drive mobile robots and the omnidirectional (with mecanum and omnidirectional wheels) robots [20]. Example of implementation of these wheeled mobile robots is found in [21], for the case of real object transportation. Also, the work in [22] addresses a tractor convoy used in agriculture to move loads and following waypoints. Finally, in [23] the object transportation in an industrial area collaborating with unmanned aerial robots is presented.Experimental work for the previous wheeled mobile robots, mostly for the case of differential-drive robots, has been presented in [24] considering the dynamics of their on-board leader trackers. The on-board cameras perspective and calibration are studied in [25], and the bounded single-view distance estimation of cameras is presented in [26]. Control laws to maintain the visibility of the leader in the presence of obstacles are given in [27]. Nonlinear estimators for aircrafts using a seeker sensor are presented in [9]. Finally, the possible reassignation of the leader based on an affection fuzzy measurement is proposed by [28].For the extension of the leader-follower strategies, to the case of collaborative object transportation or target tracking, some global interrobot structures are imposed for the robots. The first case is the generation of virtual trajectories or navigation functions for the followers, moving with respect to the leader, as in [29]. The second strategy is the creation of virtual robots or targets placed on the center of circles to generate polygons-shaped formations for the followers [30]. This idea is extended for column subgroups of robots with center of rotation in a hierarchical setup in [31]. The third concept is the definition of virtual structures, where the desired position of the robots is defined in a structure which is moved along a trajectory. Therefore, the control for each robot is designed to converge to its global position in the structure. Some examples are given in [32], with a human operator moving the virtual structure, [33] for the case of intelligent vehicles, [34] using the dynamical model of the virtual structure and [35] for some infinitesimally and minimally rigid frameworks. A special virtual structure is the so-called V-shape formation, where a leader is placed in a root node, and lines of followers are placed with respect to a global angle related to the root node [36]. Finally, the concept of imposing a moving global structure is also referred to as the cluster space specifications in [37, 38], which imposes the dynamics of an object to be transported and specifies the positions of the robots under these object. Note that all the previous strategies impose for the robots the knowledge of global coordinates, velocities, and other dynamical references to be measured.Inspired by [6] and our previous work of distance-based formation control given in [4, 39], this paper contributes to the context of the leader-follower formation of wheeled mobile robots. The originality and contributions are summarized in the next points:(i)
This paper addresses the leader-follower behaviour applied to the kinematic model of the omnidirectional robot. A differential-drive model can be obtained by cancelling the lateral linear velocity. Therefore, the approach becomes heterogeneous combining the most common wheeled vehicles used in the practice.(ii)
Four basic leader-follower control laws are defined due the combination of the omnidirectional and differential-drive robots. The local convergence is proved. It extends the results of [6] and the case presented in [39], limited only to two omnidirectional robots. The control laws are designed to be decentralized using the measurement of distance and the heading angle with respect to the leader, avoiding the use of global references as mentioned before. Furthermore, they require the velocities from the leader which can be obtained from a wireless communication.(iii)
It is proved that the robots converge to the classical rigid motion behaviour and the standardn-trailer. Also, by the heterogeneous concept of the system, we introduce a new mechanical device called omnitrailer. As far as we know, the demonstration of these local behaviours in the settling time has not been reported in the literature.(iv)
It is shown that the extension of the four control laws can generate formation tracking, useful for object transportation or collaborative tasks if the robots satisfy a directed tree-shaped topology. By numerical simulation, convoy-like formation of rigid platoons of robots can be achieved. Thus, it is possible to generate complex behaviours like virtual structures or cluster space specifications, avoiding the information of a global reference framework.(v)
Finally, the control approach is evaluated by a numerical simulation and experimental work using, in a first step, a motion capture system.The paper is organized as follows. The problem definition is given in Section2. Some preliminaries are given in Section 3, while the main results about the control strategies are defined and analysed in Section 4. The extension to groups of mobile robots in directed tree-shaped formation tracking is given by numerical simulations in Section 5. Real-time experiments are presented in Section 6. Finally, some conclusion remarks and future work are presented in Section 7.


## 2. Problem Definition

LetN={R1,…,Rn} be a group of n omnidirectional robots (see Figure 1). The kinematic model of the omnidirectional robots is given by(1)ξ˙i=Rθiui,i=1,…,n,where R(θi) is the rotation matrix defined by (2)Rθi=cosθi-sin⁡θi0sinθicos⁡θi0001,ξi=xiyiθi⊤∈R3 is the state vector with xi, yi∈R as the position in the plane of the i-th agent, θi∈R is the orientation with respect to the horizontal axis, ui=vxivyiwi⊤ is the control input vector with vxi∈R as the longitudinal velocity, vyi∈R is the lateral velocity, and wi∈R is the angular velocity.Figure 1
Schematic diagram for the leader-follower approach.Remark 1.
It is worth mentioning that if the lateral velocityvyi=0, for all time in (1), one can obtain the kinematic model of a differential-drive robot given by(3)ξ˙i=cosθi0sinθi001vxiwii=1,…,n.It is clear that a physical differential-drive robot and any kind of physical omnidirectional robot, suppressing its lateral velocity vyi, can perform the motion given in (3). In the latter case, the same robot can be switched between (1) and (3), as shown below.The focus of this article is to extend the results presented in [39] to a group of n agents. In this, a dynamic model, of a multiagent heterogeneous system composed of omnidirectional and differential-drive robots, based on the distance and angle between a pair of robots will be developed,i.e.,(4)η˙ji=d˙jiα˙jie˙θji⊤=fdji,αji,eθji,uj,ui,where dji∈R+ is the distance measured from the geometrical center of agent Rj to the geometrical center of the agent Ri, with R+ as the set of all positive real numbers, djix and djiy∈R+ are the components of the distance vector dji with respect to a global frame, αji∈R is the formation angle measured from the distance vector dji to a local frame attached to the agent Ri, and uj and ui are the vector control inputs, for the leader and follower, respectively, while eθji is the orientation error angle between the leader and the follower. Note that (4) is a general representation of the model and, as it will be shown, particular cases will be defined later on. Once the model is obtained, a control strategy is designed, such that (i)
limt→∞⁡(dji-dji∗)=0, where dji∗ is the desired distance;(ii)
limt→∞⁡(αji-αji∗)=0, where αji∗ is the desired angle.This means thatRi will keep a distance and heading orientation with respect to Rj. Depending on the kinematics model of both robots (omnidirectional or differential-drive), some control laws, developed in Section 4, will generate, in the steady state, different mechanical structures.


## 3. Preliminaries

A reminder of the rigid motion equation, the kinematic model of the standard 1-trailer, and a new rigid structure will be given in this section. This will help us to validate the main results of this article.


### 3.1. Rigid Body Motion Equation

The velocity relationship between any two points of a rigid body (see Figure2) can be expressed by the rigid body motion equation [20] given as(5)vQ=vP+wB×rPQ,where vP and vQ are the velocities of the points P and Q, respectively, wB is the angular velocity of the rigid body, rPQ is the position vector from point P to point Q, θQ and θP are the orientations of the velocities of points Q and P, respectively, and L is the distance between P and Q. Considering a two-dimensional space, therefore, the rigid body motion equation (5) has to satisfy the following expressions:(6a)vQcos⁡θQ=vPcos⁡θP+Lθ˙Bsin⁡θB,(6b)vQsin⁡θQ=vPsin⁡θQ-Lθ˙Bcos⁡θB.Figure 2
Motion equation of two points of a rigid body.


### 3.2. Kinematic Model of a Standard 1-Trailer

From Figure1, suppose that vyi=vyj=0 and αji=0. This means that both agents are differential-drive robots. Consider that the agents are linked through mechanical joints, obtaining the scheme for the standard 1–trailer with kinematic model given by (Rouchon et al. [40]; Orosco-Guerrero et al. [41]; Bushnell et al. [42]) (7a)x˙j=vxjcos⁡θj,(7b)y˙j=vxjsin⁡θj,(7c)θ˙j=wj,(7d)θ˙i=vxjsin⁡θj-θidji,where xjyj⊤∈R2 are the coordinates of the midpoint of the wheels axis of the tractor and θj and θi are the orientation of the tractor and trailer, respectively, with respect to the horizontal axis, while vxj and wj are the control inputs corresponding to the linear and angular velocities of the tractor, respectively. Therefore, we have the following evident result.Lemma 2.
The linear velocity of the trailervxi depends on the linear velocity of the tractor vxj and the orientations θj and θi and is given by(8)vxi=vxjcos⁡θj-θi.


### 3.3. Omnitrailer

Due to the heterogeneous structure of the trailer system, it is possible to have an omnidirectional robot as a tractor (leader robot) and, therefore, Lemma2 can be extended. From Figure 1 suppose that vyi=0; hence, Ri is a differential-drive robot, which is pulled by an omnidirectional robot and they are linked by a rigid bar of length dji. In consequence, we have the following evident Lemma.Lemma 3.
The linear velocityvxi and the angular velocity wi of the differential-drive robot depend on the velocities vxj and vyj and the angles θj and θi and is given by(9a)vxi=vxjcos⁡θj-θi-vyjsin⁡θj-θi,(9b)θ˙i=1djivxjsin⁡θj-θi+vyjcos⁡θj-θi.


## 3.1. Rigid Body Motion Equation

The velocity relationship between any two points of a rigid body (see Figure2) can be expressed by the rigid body motion equation [20] given as(5)vQ=vP+wB×rPQ,where vP and vQ are the velocities of the points P and Q, respectively, wB is the angular velocity of the rigid body, rPQ is the position vector from point P to point Q, θQ and θP are the orientations of the velocities of points Q and P, respectively, and L is the distance between P and Q. Considering a two-dimensional space, therefore, the rigid body motion equation (5) has to satisfy the following expressions:(6a)vQcos⁡θQ=vPcos⁡θP+Lθ˙Bsin⁡θB,(6b)vQsin⁡θQ=vPsin⁡θQ-Lθ˙Bcos⁡θB.Figure 2
Motion equation of two points of a rigid body.


## 3.2. Kinematic Model of a Standard 1-Trailer

From Figure1, suppose that vyi=vyj=0 and αji=0. This means that both agents are differential-drive robots. Consider that the agents are linked through mechanical joints, obtaining the scheme for the standard 1–trailer with kinematic model given by (Rouchon et al. [40]; Orosco-Guerrero et al. [41]; Bushnell et al. [42]) (7a)x˙j=vxjcos⁡θj,(7b)y˙j=vxjsin⁡θj,(7c)θ˙j=wj,(7d)θ˙i=vxjsin⁡θj-θidji,where xjyj⊤∈R2 are the coordinates of the midpoint of the wheels axis of the tractor and θj and θi are the orientation of the tractor and trailer, respectively, with respect to the horizontal axis, while vxj and wj are the control inputs corresponding to the linear and angular velocities of the tractor, respectively. Therefore, we have the following evident result.Lemma 2.
The linear velocity of the trailervxi depends on the linear velocity of the tractor vxj and the orientations θj and θi and is given by(8)vxi=vxjcos⁡θj-θi.


## 3.3. Omnitrailer

Due to the heterogeneous structure of the trailer system, it is possible to have an omnidirectional robot as a tractor (leader robot) and, therefore, Lemma2 can be extended. From Figure 1 suppose that vyi=0; hence, Ri is a differential-drive robot, which is pulled by an omnidirectional robot and they are linked by a rigid bar of length dji. In consequence, we have the following evident Lemma.Lemma 3.
The linear velocityvxi and the angular velocity wi of the differential-drive robot depend on the velocities vxj and vyj and the angles θj and θi and is given by(9a)vxi=vxjcos⁡θj-θi-vyjsin⁡θj-θi,(9b)θ˙i=1djivxjsin⁡θj-θi+vyjcos⁡θj-θi.


## 4. Control Strategy

In this section, the dynamic model, based on distance and orientation, referred in (4), is developed considering, in a first step, a group of n omnidirectional robots. Then, by cancelling some velocities it is possible to obtain a dynamic model for a heterogeneous system, composed by omnidirectional and differential-drive robots. In this context, four different dynamic models are developed.O - O Leader-Follower Scheme. Both agents Ri and Rj are omnidirectional robots.U - O Leader-Follower Scheme. Agent Rj is a differential-drive robot and Ri is an omnidirectional robot.U - U Leader-Follower Scheme. Both agents Ri and Rj are differential-drive robots.O - U Leader-Follower Scheme. Agent Rj is an omnidirectional robot and agent Ri is a differential-drive robot.In a second step, four control strategies are designed, by means of linearizing techniques, to achieve rigid platoons of robots. Finally, a global control strategy is developed taking into account a group ofn mobile robots.


### 4.1.O-O Leader-Follower Scheme

Based on Figure1, the distance dji and the angle αji, with i=1,…,n-1 and j=1,…,n, j≠i, are given by(10a)dji=xj-xi2+yj-yi2=djix2+djiy2,(10b)αji=θi-tan-1⁡yj-yixj-xi,where djix=xj-xi and djiy=yj-yi. The time-derivative of (10a) and (10b) is given by(11a)d˙ji=djixd˙jix+djiyd˙jiydji,(11b)α˙ji=θ˙i-djixd˙jiy-djiyd˙jixdji2,with(12a)d˙jix=vxjcos⁡θj-vyjsin⁡θj-vxicos⁡θi+vyisin⁡θi,(12b)d˙jiy=vxjsin⁡θj+vyjcos⁡θj-vxisin⁡θi-vyicos⁡θi.Substituting (12a) and (12b) into (11a) and (11b) and considering that djix=djicos⁡(θi-αji) and djiy=djisin⁡(θi-αji), therefore (11a) and (11b) can be expressed as(13)η˙ji=fηjiηjiuj+gηjiηjiui,i≠j,with(14a)fηji=cos⁡eθji+αji-sin⁡eθji+αji0-sin⁡eθji+αjidji-cos⁡eθji+αjidji0001,(14b)gηji=-cos⁡αjisin⁡αji0sin⁡αjidjicos⁡αjidji100-1,where ηji=djiαjieθji⊤ is the state vector and eθji=θj-θi. Since both agents are omnidirectional mobile robots, therefore, uj=vxjvyjwj⊤ and ui=vxivyiwi⊤. Note that (13) is a special case of the general model given in (4).The static state feedback control, for system (13), is given as(15)ui=gηji-1-fηjiuj+pji,where pji is defined by(16)pji=d˙ji∗-kdedjiα˙ji∗-kαeαji-kθeθji,with kd, kα, and kθ being positive design gains, and edji=dji-dji∗ and eαji=αji-αji∗ are the distance error and the formation orientation error, respectively.Theorem 4.
Consider system (13) in closed-loop with (15) and (16); therefore, the error coordinates edji, eαji, and eθji tend to zero,i.e.,limt→∞⁡edji=limt→∞⁡eθji=limt→∞⁡eαji=0.Proof.
The dynamics of the error coordinates are given by(17)e˙ji=η˙ji-η˙ji∗,where eji=edjieαjieθji⊤, ηji∗=dji∗αji∗0⊤. Substituting (13), (15), and (16) into (17), then, one obtains(18)e˙ji=-Kjieji,where Kji=diag⁡{kd,kα,kθ} is a diagonal matrix. Note that -Kji is a Hurwitz matrix if kd, kα, and kθ>0; hence, limt→∞⁡edji=limt→∞⁡eθji=limt→∞⁡eαji=0 and the agent Ri keeps a distance and orientation with respect to the agent Rj.Remark 5.
Note that theO-O Scheme was studied in [39] considering only two omnidirectional robots. However, part of this work extends the results for a group of n omnidirectional robots.


### 4.2.U-O Leader-Follower Scheme

Proposition 6.
LetRj be a differential-drive robot and Ri be an omnidirectional robot. In this sense, vyj=0 and the system (13) is rewritten as(19)η¯˙ji=f¯ηjiηjiu¯j+g¯ηjiηjiu¯i,i≠j,where g¯ηji=gηji, as in (14b), u¯j=vxjwj⊤, u¯i=vxivyiwi⊤, and (20)f¯ηji=cos⁡eθji+αji0-sin⁡eθji+αjidji001.The static state linearizing feedback control law is given by(21)u¯i=g¯ηji-1-f¯ηjiu¯j+pji,with the same auxiliary control given in (16).Remark 7.
Note that in theO-O and U-O leader-follower scheme the determinant of the matrices gηji and g¯ηji is the same,i.e.,det⁡(gηji)=det⁡(g¯ηji)=1/dji. Since the parameter dji is the distance from agent Rj to Ri, therefore dji>0 and matrices gηji and g¯ηji will be always invertible. Furthermore, if the parameter dji is close to zero, it implies that the agents are very close to each other, which is not feasible, due to the dimensional restrictions of the robots. Hence, in this scheme, the control performance is not constrained.


### 4.3.U-U Leader-Follower Scheme

Proposition 8.
Let agentsRi and Rj be differential-drive robots, hence, vyi=vyj=0, and functions fηji and gηji, of (14a) and (14b), are rewritten as (22)f^ηji=cos⁡eθji+αji0-sin⁡eθji+αjidji001,g^ηji=-cos⁡αji0sin⁡αjidji10-1.Taking into account the output function h^ji=djiαji⊤, the following system is obtained:(23)h^˙ji=f^hjih^ji,eθjiu^j+g^hjih^jiu^i,with u^j=vxjwj⊤, u^i=vxiwi⊤ and (24)f^hji=cos⁡eθji+αji0-sin⁡eθji+αjidji0,g^hji=-cos⁡αji0sin⁡αjidji1.Since det⁡(g^hji)=-cos⁡αji, with αji≠±π/2, then, it is possible to define a linearizing feedback control law given by (25a)u^i=g^hji-1-f^hjiu^j+p^ji,(25b)p^ji=d˙ji∗-kdedjiα˙ji∗-kαeαji,,where edji=dji-dji∗, eαji=αji-αji∗ are the distance error and the formation orientation error, respectively, while kd and kα are positive design gains. Due to the simplified model obtained for the differential-drive robots, the control law (25a) and (25b) is related only for two control variables.


### 4.4.O-U Leader-Follower Scheme

Proposition 9.
LetRj be an omnidirectional robot and Ri be a differential-drive robot. In this context, vyi=0 and the functions f~ηji=fηji, given in (14a), and g~ηji=g^ηji, given in (22). Taking into account the output function h~ji=djiαji⊤, the following system is obtained:(26)h~˙ji=f~hjih~ji,eθjiu~j+g~hjih~jiu~i,with u~j=vxjvyjwj⊤, u~i=vxiwi⊤ and (27)f~hji=cos⁡eθji+αji-sin⁡eθji+αji0-sin⁡eθji+αjidji-cos⁡eθji+αjidji0,g~hji=-cos⁡αji0sin⁡αjidji1.Since det⁡(g~hji)=-cos⁡αji, with αji≠±π/2, then, it is possible to define a linearizing feedback control law given by(28)u~i=g~hji-1-f~hjiu~j+p~ji,with p~ji=p^ji.Remark 10.
Recall that the angleαji is the angle measured from the distance vector dji to a local frame attached to the agent Ri. In this sense, an angle αji=±π/2 means that the component of the velocity perpendicular to the wheels of the differential-drive mobile robots is aligned to the distance vector dji, and, due to the nonholonomic restriction of this kind of vehicles, given by x˙isin⁡θi-y˙icos⁡θi=0, then the motion, perpendicular to the linear velocity, is zero. Therefore, the restriction of αji≠±π/2 implies the nonholonomic restriction of the differential-drive mobile robots.Remark 11.
By a similar procedure to Theorem4, it can be shown that from system (19) with control (21) the error coordinates will converge to zero,i.e.,limt→∞⁡edji=limt→∞⁡eθji=limt→∞⁡eαji=0. Furthermore, from system (23) with control (25a) and (25b) and system (26) with control (28), the errors edji and eαji will converge to zero,i.e.,limt→∞⁡edjilimt→∞⁡eαji=0.


### 4.5. Global Control Strategy

This subsection states a relevant contribution of the article. Specifically, a global control strategy is developed to ensure that a group of agents, composed by omnidirectional and differential-drive mobile robots, converge to a desired distance and formation angle.Recall thatN is a group of n agents where Rn is the leader, represented by an omnidirectional mobile robot, while the rest are the followers that could be omnidirectional or differential-drive mobile robots, and suppose that j>i, then, the dynamic model of the group can be represented by(29)ζ˙=LU,where ζ is the state vector containing the distances, formation angles, and orientations with respect a pair of agents, U is the control input, and L is an upper triangular block matrix.Remark 12.
Without loss of generality, consider that the agents are under the well-known directed open-chain formation, and let us assume that the agents are ordered as follows:N={O,U,U,O,…,O,U,U,O,O}, then, the state vector ζ, the control input U, and the matrix L are given as follows: (30)ζ=η¯21h^32h~43η54⋮η¯n-3n-4h^n-2n-3h~n-1n-2ηnn-1ξn,U=u¯1u^2u~3u4⋮u¯n-4u^n-3u~n-2un-1un,(31)L=g¯η21f¯η2103×203×303×3⋯03×303×203×203×303×302×3g^h32f^h3202×302×3⋯02×302×202×202×302×302×302×2g~h43f~h4302×3⋯02×302×202×202×302×303×303×203×2gη54fη54⋯03×303×203×203×303×3⋮⋱⋮03×303×203×203×303×3⋯g¯ηn-3n-4f¯ηn-3n-403×203×303×302×302×202×202×302×3⋯02×3g^hn-2n-3f^hn-2n-302×302×302×302×202×202×302×3⋯02×302×2g~hn-1n-2f~hn-1n-202×303×303×203×203×303×3⋯03×303×203×2gηnn-1fηnn-103×303×203×203×303×3⋯03×303×203×203×3Rθn,with 0a×b as an a×b zero matrix. Note that matrix L is still an upper triangular block matrix.The linearizing feedback control law is given by(32)U=L-1Pd,where(33)Pd=p21p^32p~43p54⋮pn-3n-4p^n-2n-3p~n-1n-2pnn-1pn are the auxiliary controls defined previously.Lemma 13.
MatrixL is always invertible for all αji≠±π/2 and j≠i.Proof.
Since matrixL is an upper triangular block matrix, hence, the determinant is given by the product of the determinants of the diagonal matrices. Specifically, for n=2, the determinant of matrix L is defined as (34)det⁡L=1d21, while for n≥3, the determinant of matrix L is obtained as (35)det⁡L=r∏p,qncos⁡αqq-1dpp-1, where p=2,5,6,9,10…,n, q=3,4,7,8,11…,n, and (36)r=-1,ifn=3,7,11,…1,otherwise.Proposition 14.
Consider system (29) in closed-loop with the control (32), then, the errors edji and eαji will converge to zero,i.e.,limt→∞⁡edji=limt→∞⁡eαji=0.Proof.
Let us define the error coordinates as follows:(37)e=ζ-ζ∗, where ζ∗ is the desired vector of distances and orientations and is given as follows: (38)ζ∗=η¯21∗h^32∗h~43∗η54∗⋮η¯n-3n-4∗h^n-2n-3∗h~n-1n-2∗ηnn-1∗ξn∗, with η¯ji∗=dji∗αji∗0⊤, h^ji∗=dji∗αji∗⊤, h~ji∗=dji∗αji∗⊤, ηji∗=dji∗αji∗0⊤, and ξn∗=mxmyθd⊤, for j≠i, where mx, my, and θd are the desired trajectory and orientation for the leader. The dynamics of the error coordinates are given by(39)e˙=ζ˙-ζ˙∗,and substituting (29) with control (32) into (39), then one has (40)e˙=-Ke, where(41)K=diag⁡K21,K32,K43,K54,K65,…,Knn-1,Kn, with K21=K54=K65=Kn(n-1)=diag⁡kd,kα,kθ, K32=K43=diag⁡kd,kα, and Kn=diag⁡kxn,kyn,kθn. It becomes evident that if kd, kα, kθ, kxn, kyn, and kθn>0, hence, matrix K is Hurwitz, and the errors will converge to zero,i.e.,limt→∞⁡e=0.Remark 15.
The following results are interesting due to the fact that it is possible to show that the group of agents can emulate the kinematic behaviour of rigid bodies. As far as we know, this problem has not been addressed in the literature, considering the distance and the formation angle between a pair of robots.


### 4.6. Rigid Body Behaviour

It has been shown [39] that, when system (13) with control (15) and (16), reaches the steady state, then limt→∞⁡edji=limt→∞⁡eαji=limt→∞⁡eθji=0 and the leader-follower scheme, composed by two omnidirectional mobile robots, will emulate the kinematic behaviour of a rigid body.Based on the aforementioned, the following theorem states another contribution of this article.Theorem 16.
Consider that, in system (13), the agents Rj and Ri can be either an omnidirectional or a differential-drive robot. Suppose that system (13) has reached the steady state and dji∗=d¯ji∈R+ and αji∗=0; therefore, the leader-follower scheme will emulate the kinematic behaviour of a rigid body, the kinematic behaviour of a standard 1-trailer or the kinematic behaviour of the omnitrailer.The proof will be done constructively, taking into account the cases given in Sections4.2, 4.3, and 4.4.Proof (U-O leader-follower scheme).
Consider that system (19) with control (21) has reached the steady state,i.e.,limt→∞⁡edji=limt→∞⁡eαji=limt→∞⁡eθji=0, then, substituting the control law (21) into the kinematic model (1), in steady state, one has (42)x˙i=vxjcos⁡θi-d¯jiwjsin⁡αji-θi,y˙i=vxjsin⁡θi-d¯jiwjcos⁡αji-θi,θ˙i=wj.Due to limt→∞⁡eθji=0, this means that limt→∞⁡θi=θj, limt→∞⁡αji=αji∗=0, and the above expression is rewritten as (43)x˙i=vxjcos⁡θj+d¯jiwjsin⁡θj,y˙i=vxjsin⁡θj-d¯jiwjcos⁡θj,θ˙i=wj.Note that, from (3), x˙j=vxjcos⁡θj and y˙j=vxjsin⁡θj, then, one obtains(44a)x˙i=x˙j+d-jiwjsin⁡θj,(44b)y˙i=y˙j-d-jiwjcos⁡θj,(44c)θ˙i=wj.Comparing these last expressions with respect to the rigid body motion equations (6a) and (6b), one can conclude that the follower will move such that the whole system will behave as a rigid body.Proof (U-U leader-follower scheme).
Consider that system (23) with control (25a) and (25b) has reached the steady state,i.e.,limt→∞⁡edji=limt→∞⁡eαji=0; therefore, substituting the control law (25a) and (25b) into the kinematic model (3) and considering dji∗=d¯ji, in steady state, one has (45)x˙i=vxjcos⁡αji∗cos⁡θicos⁡αji∗+eθji,y˙i=vxjcos⁡αji∗sin⁡θicos⁡αji∗+eθji,θ˙i=vxjd¯jicos⁡αji∗sin⁡eθji.Then, letting αji∗=0 and θi=θj-eθji, one obtains(46a)x˙i=vxjcos⁡θj-eθjicos⁡eθji,(46b)y˙i=vxjsin⁡θj-eθjicos⁡eθji,(46c)θ˙i=vxjd¯jisin⁡eθji.Note that the linear velocity of agent Ri can be obtained from (46a) and (46b) as follows:(47)vxi=x˙i2+y˙i2=vxjcos⁡eθji.From Lemma 2 and comparing (47) with (8) and (46c) with (7d) one can note that they have the same structure. This means that the agent Ri will behave as a trailer and agent Rj as a tractor; hence, the whole system will behave as a standard 1-trailer.Proof (O-U leader-follower scheme).
Consider that system (23) with control (28) has reached the steady state,i.e.,limt→∞⁡eji=limt→∞⁡eαji=0; therefore, substituting the control law (28) into the kinematic model of the follower (3) and considering that limt→∞⁡edji=limt→∞⁡eαji=0, dji∗=d¯ji, in steady state, one has (48)x˙i=cos⁡θicos⁡αji∗vxjcos⁡αji∗+eθji-vyjsin⁡αji∗+eθji,y˙i=sin⁡θicos⁡αji∗vxjcos⁡αji∗+eθji-vyjsin⁡αji∗+eθji,θ˙i=1d¯jicos⁡αji∗vxjsin⁡eθji+vyjcos⁡eθji.Then, letting αji∗=0 and θi=θj-eθji, one obtains(49a)x˙i=cos⁡θj-eθjivxjcos⁡eθji-vyjsin⁡eθji,(49b)y˙i=sin⁡θj-eθjivxjcos⁡eθji-vyjsin⁡eθji,(49c)θ˙i=1d-jivxjsin⁡eθji+vyjcos⁡eθji.Note that the linear velocity of agent Ri can be obtained from (49a) and (49b) as follows:(50)vxi=x˙i2+y˙i2=vxjcos⁡eθji-vyjsin⁡eθji.From Lemma 3 and comparing (50) with (9a) and (49c) with (9b), one can conclude that the leader-follower scheme will achieve the same kinematic behaviour of the omnitrailer given in Section 3.3.


## 4.1.O-O Leader-Follower Scheme

Based on Figure1, the distance dji and the angle αji, with i=1,…,n-1 and j=1,…,n, j≠i, are given by(10a)dji=xj-xi2+yj-yi2=djix2+djiy2,(10b)αji=θi-tan-1⁡yj-yixj-xi,where djix=xj-xi and djiy=yj-yi. The time-derivative of (10a) and (10b) is given by(11a)d˙ji=djixd˙jix+djiyd˙jiydji,(11b)α˙ji=θ˙i-djixd˙jiy-djiyd˙jixdji2,with(12a)d˙jix=vxjcos⁡θj-vyjsin⁡θj-vxicos⁡θi+vyisin⁡θi,(12b)d˙jiy=vxjsin⁡θj+vyjcos⁡θj-vxisin⁡θi-vyicos⁡θi.Substituting (12a) and (12b) into (11a) and (11b) and considering that djix=djicos⁡(θi-αji) and djiy=djisin⁡(θi-αji), therefore (11a) and (11b) can be expressed as(13)η˙ji=fηjiηjiuj+gηjiηjiui,i≠j,with(14a)fηji=cos⁡eθji+αji-sin⁡eθji+αji0-sin⁡eθji+αjidji-cos⁡eθji+αjidji0001,(14b)gηji=-cos⁡αjisin⁡αji0sin⁡αjidjicos⁡αjidji100-1,where ηji=djiαjieθji⊤ is the state vector and eθji=θj-θi. Since both agents are omnidirectional mobile robots, therefore, uj=vxjvyjwj⊤ and ui=vxivyiwi⊤. Note that (13) is a special case of the general model given in (4).The static state feedback control, for system (13), is given as(15)ui=gηji-1-fηjiuj+pji,where pji is defined by(16)pji=d˙ji∗-kdedjiα˙ji∗-kαeαji-kθeθji,with kd, kα, and kθ being positive design gains, and edji=dji-dji∗ and eαji=αji-αji∗ are the distance error and the formation orientation error, respectively.Theorem 4.
Consider system (13) in closed-loop with (15) and (16); therefore, the error coordinates edji, eαji, and eθji tend to zero,i.e.,limt→∞⁡edji=limt→∞⁡eθji=limt→∞⁡eαji=0.Proof.
The dynamics of the error coordinates are given by(17)e˙ji=η˙ji-η˙ji∗,where eji=edjieαjieθji⊤, ηji∗=dji∗αji∗0⊤. Substituting (13), (15), and (16) into (17), then, one obtains(18)e˙ji=-Kjieji,where Kji=diag⁡{kd,kα,kθ} is a diagonal matrix. Note that -Kji is a Hurwitz matrix if kd, kα, and kθ>0; hence, limt→∞⁡edji=limt→∞⁡eθji=limt→∞⁡eαji=0 and the agent Ri keeps a distance and orientation with respect to the agent Rj.Remark 5.
Note that theO-O Scheme was studied in [39] considering only two omnidirectional robots. However, part of this work extends the results for a group of n omnidirectional robots.


## 4.2.U-O Leader-Follower Scheme

Proposition 6.
LetRj be a differential-drive robot and Ri be an omnidirectional robot. In this sense, vyj=0 and the system (13) is rewritten as(19)η¯˙ji=f¯ηjiηjiu¯j+g¯ηjiηjiu¯i,i≠j,where g¯ηji=gηji, as in (14b), u¯j=vxjwj⊤, u¯i=vxivyiwi⊤, and (20)f¯ηji=cos⁡eθji+αji0-sin⁡eθji+αjidji001.The static state linearizing feedback control law is given by(21)u¯i=g¯ηji-1-f¯ηjiu¯j+pji,with the same auxiliary control given in (16).Remark 7.
Note that in theO-O and U-O leader-follower scheme the determinant of the matrices gηji and g¯ηji is the same,i.e.,det⁡(gηji)=det⁡(g¯ηji)=1/dji. Since the parameter dji is the distance from agent Rj to Ri, therefore dji>0 and matrices gηji and g¯ηji will be always invertible. Furthermore, if the parameter dji is close to zero, it implies that the agents are very close to each other, which is not feasible, due to the dimensional restrictions of the robots. Hence, in this scheme, the control performance is not constrained.


## 4.3.U-U Leader-Follower Scheme

Proposition 8.
Let agentsRi and Rj be differential-drive robots, hence, vyi=vyj=0, and functions fηji and gηji, of (14a) and (14b), are rewritten as (22)f^ηji=cos⁡eθji+αji0-sin⁡eθji+αjidji001,g^ηji=-cos⁡αji0sin⁡αjidji10-1.Taking into account the output function h^ji=djiαji⊤, the following system is obtained:(23)h^˙ji=f^hjih^ji,eθjiu^j+g^hjih^jiu^i,with u^j=vxjwj⊤, u^i=vxiwi⊤ and (24)f^hji=cos⁡eθji+αji0-sin⁡eθji+αjidji0,g^hji=-cos⁡αji0sin⁡αjidji1.Since det⁡(g^hji)=-cos⁡αji, with αji≠±π/2, then, it is possible to define a linearizing feedback control law given by (25a)u^i=g^hji-1-f^hjiu^j+p^ji,(25b)p^ji=d˙ji∗-kdedjiα˙ji∗-kαeαji,,where edji=dji-dji∗, eαji=αji-αji∗ are the distance error and the formation orientation error, respectively, while kd and kα are positive design gains. Due to the simplified model obtained for the differential-drive robots, the control law (25a) and (25b) is related only for two control variables.


## 4.4.O-U Leader-Follower Scheme

Proposition 9.
LetRj be an omnidirectional robot and Ri be a differential-drive robot. In this context, vyi=0 and the functions f~ηji=fηji, given in (14a), and g~ηji=g^ηji, given in (22). Taking into account the output function h~ji=djiαji⊤, the following system is obtained:(26)h~˙ji=f~hjih~ji,eθjiu~j+g~hjih~jiu~i,with u~j=vxjvyjwj⊤, u~i=vxiwi⊤ and (27)f~hji=cos⁡eθji+αji-sin⁡eθji+αji0-sin⁡eθji+αjidji-cos⁡eθji+αjidji0,g~hji=-cos⁡αji0sin⁡αjidji1.Since det⁡(g~hji)=-cos⁡αji, with αji≠±π/2, then, it is possible to define a linearizing feedback control law given by(28)u~i=g~hji-1-f~hjiu~j+p~ji,with p~ji=p^ji.Remark 10.
Recall that the angleαji is the angle measured from the distance vector dji to a local frame attached to the agent Ri. In this sense, an angle αji=±π/2 means that the component of the velocity perpendicular to the wheels of the differential-drive mobile robots is aligned to the distance vector dji, and, due to the nonholonomic restriction of this kind of vehicles, given by x˙isin⁡θi-y˙icos⁡θi=0, then the motion, perpendicular to the linear velocity, is zero. Therefore, the restriction of αji≠±π/2 implies the nonholonomic restriction of the differential-drive mobile robots.Remark 11.
By a similar procedure to Theorem4, it can be shown that from system (19) with control (21) the error coordinates will converge to zero,i.e.,limt→∞⁡edji=limt→∞⁡eθji=limt→∞⁡eαji=0. Furthermore, from system (23) with control (25a) and (25b) and system (26) with control (28), the errors edji and eαji will converge to zero,i.e.,limt→∞⁡edjilimt→∞⁡eαji=0.


## 4.5. Global Control Strategy

This subsection states a relevant contribution of the article. Specifically, a global control strategy is developed to ensure that a group of agents, composed by omnidirectional and differential-drive mobile robots, converge to a desired distance and formation angle.Recall thatN is a group of n agents where Rn is the leader, represented by an omnidirectional mobile robot, while the rest are the followers that could be omnidirectional or differential-drive mobile robots, and suppose that j>i, then, the dynamic model of the group can be represented by(29)ζ˙=LU,where ζ is the state vector containing the distances, formation angles, and orientations with respect a pair of agents, U is the control input, and L is an upper triangular block matrix.Remark 12.
Without loss of generality, consider that the agents are under the well-known directed open-chain formation, and let us assume that the agents are ordered as follows:N={O,U,U,O,…,O,U,U,O,O}, then, the state vector ζ, the control input U, and the matrix L are given as follows: (30)ζ=η¯21h^32h~43η54⋮η¯n-3n-4h^n-2n-3h~n-1n-2ηnn-1ξn,U=u¯1u^2u~3u4⋮u¯n-4u^n-3u~n-2un-1un,(31)L=g¯η21f¯η2103×203×303×3⋯03×303×203×203×303×302×3g^h32f^h3202×302×3⋯02×302×202×202×302×302×302×2g~h43f~h4302×3⋯02×302×202×202×302×303×303×203×2gη54fη54⋯03×303×203×203×303×3⋮⋱⋮03×303×203×203×303×3⋯g¯ηn-3n-4f¯ηn-3n-403×203×303×302×302×202×202×302×3⋯02×3g^hn-2n-3f^hn-2n-302×302×302×302×202×202×302×3⋯02×302×2g~hn-1n-2f~hn-1n-202×303×303×203×203×303×3⋯03×303×203×2gηnn-1fηnn-103×303×203×203×303×3⋯03×303×203×203×3Rθn,with 0a×b as an a×b zero matrix. Note that matrix L is still an upper triangular block matrix.The linearizing feedback control law is given by(32)U=L-1Pd,where(33)Pd=p21p^32p~43p54⋮pn-3n-4p^n-2n-3p~n-1n-2pnn-1pn are the auxiliary controls defined previously.Lemma 13.
MatrixL is always invertible for all αji≠±π/2 and j≠i.Proof.
Since matrixL is an upper triangular block matrix, hence, the determinant is given by the product of the determinants of the diagonal matrices. Specifically, for n=2, the determinant of matrix L is defined as (34)det⁡L=1d21, while for n≥3, the determinant of matrix L is obtained as (35)det⁡L=r∏p,qncos⁡αqq-1dpp-1, where p=2,5,6,9,10…,n, q=3,4,7,8,11…,n, and (36)r=-1,ifn=3,7,11,…1,otherwise.Proposition 14.
Consider system (29) in closed-loop with the control (32), then, the errors edji and eαji will converge to zero,i.e.,limt→∞⁡edji=limt→∞⁡eαji=0.Proof.
Let us define the error coordinates as follows:(37)e=ζ-ζ∗, where ζ∗ is the desired vector of distances and orientations and is given as follows: (38)ζ∗=η¯21∗h^32∗h~43∗η54∗⋮η¯n-3n-4∗h^n-2n-3∗h~n-1n-2∗ηnn-1∗ξn∗, with η¯ji∗=dji∗αji∗0⊤, h^ji∗=dji∗αji∗⊤, h~ji∗=dji∗αji∗⊤, ηji∗=dji∗αji∗0⊤, and ξn∗=mxmyθd⊤, for j≠i, where mx, my, and θd are the desired trajectory and orientation for the leader. The dynamics of the error coordinates are given by(39)e˙=ζ˙-ζ˙∗,and substituting (29) with control (32) into (39), then one has (40)e˙=-Ke, where(41)K=diag⁡K21,K32,K43,K54,K65,…,Knn-1,Kn, with K21=K54=K65=Kn(n-1)=diag⁡kd,kα,kθ, K32=K43=diag⁡kd,kα, and Kn=diag⁡kxn,kyn,kθn. It becomes evident that if kd, kα, kθ, kxn, kyn, and kθn>0, hence, matrix K is Hurwitz, and the errors will converge to zero,i.e.,limt→∞⁡e=0.Remark 15.
The following results are interesting due to the fact that it is possible to show that the group of agents can emulate the kinematic behaviour of rigid bodies. As far as we know, this problem has not been addressed in the literature, considering the distance and the formation angle between a pair of robots.


## 4.6. Rigid Body Behaviour

It has been shown [39] that, when system (13) with control (15) and (16), reaches the steady state, then limt→∞⁡edji=limt→∞⁡eαji=limt→∞⁡eθji=0 and the leader-follower scheme, composed by two omnidirectional mobile robots, will emulate the kinematic behaviour of a rigid body.Based on the aforementioned, the following theorem states another contribution of this article.Theorem 16.
Consider that, in system (13), the agents Rj and Ri can be either an omnidirectional or a differential-drive robot. Suppose that system (13) has reached the steady state and dji∗=d¯ji∈R+ and αji∗=0; therefore, the leader-follower scheme will emulate the kinematic behaviour of a rigid body, the kinematic behaviour of a standard 1-trailer or the kinematic behaviour of the omnitrailer.The proof will be done constructively, taking into account the cases given in Sections4.2, 4.3, and 4.4.Proof (U-O leader-follower scheme).
Consider that system (19) with control (21) has reached the steady state,i.e.,limt→∞⁡edji=limt→∞⁡eαji=limt→∞⁡eθji=0, then, substituting the control law (21) into the kinematic model (1), in steady state, one has (42)x˙i=vxjcos⁡θi-d¯jiwjsin⁡αji-θi,y˙i=vxjsin⁡θi-d¯jiwjcos⁡αji-θi,θ˙i=wj.Due to limt→∞⁡eθji=0, this means that limt→∞⁡θi=θj, limt→∞⁡αji=αji∗=0, and the above expression is rewritten as (43)x˙i=vxjcos⁡θj+d¯jiwjsin⁡θj,y˙i=vxjsin⁡θj-d¯jiwjcos⁡θj,θ˙i=wj.Note that, from (3), x˙j=vxjcos⁡θj and y˙j=vxjsin⁡θj, then, one obtains(44a)x˙i=x˙j+d-jiwjsin⁡θj,(44b)y˙i=y˙j-d-jiwjcos⁡θj,(44c)θ˙i=wj.Comparing these last expressions with respect to the rigid body motion equations (6a) and (6b), one can conclude that the follower will move such that the whole system will behave as a rigid body.Proof (U-U leader-follower scheme).
Consider that system (23) with control (25a) and (25b) has reached the steady state,i.e.,limt→∞⁡edji=limt→∞⁡eαji=0; therefore, substituting the control law (25a) and (25b) into the kinematic model (3) and considering dji∗=d¯ji, in steady state, one has (45)x˙i=vxjcos⁡αji∗cos⁡θicos⁡αji∗+eθji,y˙i=vxjcos⁡αji∗sin⁡θicos⁡αji∗+eθji,θ˙i=vxjd¯jicos⁡αji∗sin⁡eθji.Then, letting αji∗=0 and θi=θj-eθji, one obtains(46a)x˙i=vxjcos⁡θj-eθjicos⁡eθji,(46b)y˙i=vxjsin⁡θj-eθjicos⁡eθji,(46c)θ˙i=vxjd¯jisin⁡eθji.Note that the linear velocity of agent Ri can be obtained from (46a) and (46b) as follows:(47)vxi=x˙i2+y˙i2=vxjcos⁡eθji.From Lemma 2 and comparing (47) with (8) and (46c) with (7d) one can note that they have the same structure. This means that the agent Ri will behave as a trailer and agent Rj as a tractor; hence, the whole system will behave as a standard 1-trailer.Proof (O-U leader-follower scheme).
Consider that system (23) with control (28) has reached the steady state,i.e.,limt→∞⁡eji=limt→∞⁡eαji=0; therefore, substituting the control law (28) into the kinematic model of the follower (3) and considering that limt→∞⁡edji=limt→∞⁡eαji=0, dji∗=d¯ji, in steady state, one has (48)x˙i=cos⁡θicos⁡αji∗vxjcos⁡αji∗+eθji-vyjsin⁡αji∗+eθji,y˙i=sin⁡θicos⁡αji∗vxjcos⁡αji∗+eθji-vyjsin⁡αji∗+eθji,θ˙i=1d¯jicos⁡αji∗vxjsin⁡eθji+vyjcos⁡eθji.Then, letting αji∗=0 and θi=θj-eθji, one obtains(49a)x˙i=cos⁡θj-eθjivxjcos⁡eθji-vyjsin⁡eθji,(49b)y˙i=sin⁡θj-eθjivxjcos⁡eθji-vyjsin⁡eθji,(49c)θ˙i=1d-jivxjsin⁡eθji+vyjcos⁡eθji.Note that the linear velocity of agent Ri can be obtained from (49a) and (49b) as follows:(50)vxi=x˙i2+y˙i2=vxjcos⁡eθji-vyjsin⁡eθji.From Lemma 3 and comparing (50) with (9a) and (49c) with (9b), one can conclude that the leader-follower scheme will achieve the same kinematic behaviour of the omnitrailer given in Section 3.3.


## 5. Numerical Simulations

It is worth mentioning that the previous control strategies are defined between a pair of robots, where the follower robot is controlled with respect to a unique leader. If these behaviours are extended for the case of more than two robots, then it is possible to consider a graph topology with a directed tree-shaped configuration with a leaderRn following a desired trajectory. This topology can be constructed defining that, in each pair of robots Ri (follower) and Rj (leader), it is satisfied that j>i,∀i=1,…,n-1, and j=1,…,n with j≠i.For the first numerical simulation, Figure3 displays a comparison between the kinematic behaviour of a standard 1-trailer, given by (7a), (7b), (7c), and (7d), and the U-U leader-follower scheme, given by (23). The trailer receives commands of linear and angular velocities such that is following a Lemniscate of Gerono with parametrization ξd:R+→R2 given by (51)ξd=2cos⁡π10t1.5sin⁡π5t⊤. The initial conditions for the standard 1-trailer are xj(0)yj(0)θj(0)θi(0)⊤=1000⊤ while for the follower are ξi(0)=0.500⊤, with αji∗=0 rad, dji∗=0.3 m and using the Matlab Simulink ode4 function solver [43]. Specifically, Figure 3(a) shows the trajectory in the plane for both, the standard 1-trailer, and the U-U leader-follower scheme. It becomes evident that, in spite of the follower has different initial conditions with respect to the trailer, the follower has the same behaviour as the trailer; while Figure 3(b) shows the orientation for the trailer and the follower. Note that both orientation angles converge to the same value, allowing us to validate that the U-U leader-follower scheme will behave as a standard 1-trailer.Figure 3
Comparison behaviour between a standard 1-trailer and theU-U leader-follower scheme.
(a)
Trajectory in the plane for a standard 1-trailer and theU-U leader-follower scheme (b)
Orientation angle of the trailer and the followerIn order to show the performance of the control law (32) and the different combinations of the leader-follower scheme, Figure 4 depicts a numerical simulation with n=13. The leader robot O13 is controlled to follow a Lemmiscate of Gerono defined as (52)ξ13∗=mx,my,arctan⁡m˙ym˙x⊤, with mx=4cos⁡(π/10t), my=3sin⁡(π/5t), and K13=diag⁡{1,1,3}. Since the agents are under the leader-follower topology, then, the desired distance and the formation angle are the same for all the pair of robots,i.e.,dji∗=0.2 m and αji∗=0 rad, while the control parameters are K21=K54=K65=K98=K109=K1312=diag⁡{2,2,3} and K32=K43=K76=K87=K98=K1110=K1211=diag⁡{2,2}. The initial conditions for the robots are (53)ξ10=0.890.0850⊤,ξ20=1.14-0.09050⊤,ξ30=1.33-0.2880⊤,ξ40=1.52-0.1040⊤,ξ50=1.74-0.1860⊤,ξ60=1.89-0.1170⊤,ξ70=2.02-0.1530⊤,ξ80=2.139-0.1120⊤,ξ90=2.35-0.0240⊤,ξ100=2.3520.1540⊤,ξ110=2.59-0.0980⊤,ξ120=2.900⊤,ξ130=300⊤.Particularly, Figure 4(a) illustrates the trajectory in the plane of all the robots. In this sense, the pairs of robots (O5,O4), (O9,O8), and (O13,O12) are under the O-O scheme while the pairs of robots (U2,O1), (U6,O5), and (U10,O9) are under the U-O scheme. In both cases, the agents achieve the rigid body behaviour. For the U-U scheme, integrated by the pairs (U3,U2), (U7,U6), and (U11,U10), it achieves the behaviour of the standard 1-trailer. The omnitrailer behaviour appears in the pairs (O4,U3), (O8,U7), and (O12,U11) with the O-U scheme. Note that, in all the cases, the pairs of robots satisfy j>i. On the other hand, Figure 4(b) depicts the distance among the robots, where it becomes evident that each pair of agents converges to the desired distance dji∗=0.2 m. Finally, Figure 4(c) displays the formation angles of each pair of agents, which converge to the desired formation angle αji∗=0 rad.Figure 4
Simulation withn=13 combining the four different leader-follower schemes.
(a)
Trajectory in the plane (b)
Distance among the agents (c)
Formation angles


## 6. Experimental Work

The approach is tested using the four mobile robots with four mecanum wheels like the ones shown in Figure5(a). The robots are actuated by servomotors Dynamixel® AX-12W and controlled by a microcontroller NXP® model LPC1768 with Bluetooth communication to a PC computer using the module HC-05. In a first step, the position and orientation of the robots are measured by a Vicon® motion capture system composed by 6 cameras model Bonita® (Figure 5(b)). The motion capture measures within an available workspace area of 3 m by 0.7 m. Note in Figure 5(a) that the reflective markers were placed on the top of the robots in order to be identified by the Vicon® system. The control algorithm runs at 117 ms rate with a resolution of ±0.5 mm.Figure 5
Experimental setup composed by omnidirectional and differential-drive mobile robots and six Vicon cameras.
(a)
Omnidirectional wheeled mobile robots (b)
Experimental setupAccording to Figure6, the velocities of each wheel can be calculated by(54)ωi1ωi2ωi3ωi4=1r1-1-L+l11L+l11L+l1-1-L+lvxivyiωi,where L=0.1 m, l=0.05 m, and r=0.0275 m. Note that (54) can be adequate according to the number of wheels of the robot, for example, for the three omnidirectional wheeled mobile robot.Figure 6
Omnidirectional wheeled mobile robot.Figures7, 8, and 9 show an experiment of the combined four control laws presented in the Section 4. The leader robot is O4 (omnidirectional robot in black color). The pair (O4,O3) implements the control O-O scheme, achieving a rigid body behaviour. The pair (O4,U2) enables the control law O-U scheme, obtaining an omnitrailer behaviour. Finally, the pair (U2,O1) implements the control law U-O scheme performing another rigid body behaviour. The desired distance and heading angles are given by d43∗=d42∗=d21∗=0.3 m and α43∗=-π/2 rad, α42∗=0 rad, and α21∗=π/2. The control parameters are Kji=diag⁡{6,6,2} for the pairs (O4,O3) and (U2,O1) and K42=diag{5,1} for the pair (O4,U2). The initial conditions of the robots are given by (55)ξ10=0.6163-0.10531-12.41⊤ξ20=0.9072-0.17935-12.38⊤,ξ30=1.2902-0.16269-1.274⊤,ξ40=0.9943-0.11272-12.71⊤,in meters and radians, respectively. The leader robot is controlled to follow a circled-shape trajectory with radius equal to 1 m centered on the origin and oriented to the velocity vector of the trajectory;i.e., the desired values for the leader are given by(56)ξ4∗=mx,my,arctan⁡m˙ym˙x-π2⊤, where mx=sin⁡2π/25t and my=cos⁡2π/25t. Thus, the control law for the leader robot is established as u4=R-1(θ4)ξ˙4∗-k4(ξ4-ξ4∗), with k4=diag⁡6,6,2.5.Figure 7
Trajectories in the plane withn=4 agents. The agent U2 is an omnidirectional robot with lateral velocity vy2=0 allowing him to act as a differential-drive robot.Figure 8
Distance error, formation angle error, and orientation error.Figure 9
Control inputs for the mobile robots.The trajectories of the robots are shown in Figure7. The colors of the links between robots in Figure 7 identify the type of control law that it is being used,i.e., gray color for O-O scheme, magenta color for O-U scheme, and cyan color for U-O scheme. Note that, in all cases, it is satisfied that j>i. Furthermore, in this experiment, the agent U2 is an omnidirectional mobile robot with vy2=0 obtaining the behaviour of a differential-drive robot. It is important to point out that, from a global point of view, a leader-follower setup with subsets or platoons of robots is achieved. This kind of system can be applied in collaborative transportation tasks where the platoons of robots load together a big work-piece and all the platoons are formed in a convoy-like configuration.From Figure8, it becomes clear that each follower converges to the desired distance and the desired formation angle. Finally, Figure 9 depicts the control inputs for each robot. Noise effects in the trajectories appear due to the nonmodelled friction of the wheels on the floor, sensor measurements, and actuator errors, among others.A second experiment withn=5 agents is displayed in Figures 10, 11, and 12. In this case, the group of agents is composed by a differential-drive mobile robot and four omnidirectional robots, where one of them acts as a differential-drive robot. The initial conditions of the robots are given by (57)ξ10=1.62-0.112.68⊤,ξ20=1.69-1.142.21⊤,ξ30=1.02-1.61.16⊤,ξ40=0.33-0.99-0.2⊤,ξ50=0.34-0.20.93⊤,in meters and radians, respectively. The leader robot is controlled to follow the same circled-shape trajectory of the previous experiment. The trajectories in the plane are depicted in Figure 10. The leader robot is O5 (omnidirectional robot in magenta color). The pair (O5,U4) implements the control O-U scheme, achieving the omnitrailer behaviour, illustrated by an orange line. The pair (O5,O1) enables the control law O-O scheme while the pair (U4,O2) enables the control law U-O scheme, obtaining, in both cases, a rigid body behaviour, depicted by a gray and cyan line, respectively. Finally, the pair (U4,U3) implements the control law U-U scheme performing the standard 1- trailer behaviour, represented by the yellow line.Figure 10
Trajectories in the plane withn=5 agents. The agent U4 is an omnidirectional robot with lateral velocity vy4=0 allowing him to act as a differential-drive robot while U3 is a differential-drive robot.Figure 11
Distance error, formation angle error, and orientation error.Figure 12
Control inputs for the mobile robots.The desired distance and heading angles are given byd51∗=d42∗=0.4 m, d54∗=d43∗=0.6 m, α51∗=α42∗=π/2 rad, and α54∗=α43∗=0 rad. The control parameters are the same as in the previous experimenti.e.,Kji=diag⁡{6,6,2} for the pairs (O5,O1) and (U4,O2) and K54=K43=diag⁡{5,1} for the pairs (O5,U4) and (O4,U3).The distance error, the formation angle error, and the orientation error are displayed in Figure11. Note that each follower converges to the desired distance and the desired formation angle. On the other hand, the control inputs for each mobile robot are shown in Figure 12.Finally, in order to show that agentU4 behaves as a differential-drive robot, Figure 13 illustrates the nonholonomic restriction of this agent. This signal was obtained by applying a numerical method to calculate the derivative of x4 and y4. Note that, in steady state, the nonholonomic restriction converges to a band with magnitude given in terms of the measurement error. This means that the nonholonomic restriction is satisfied by the agent U4.Figure 13
Nonholonomic restriction for the agentU4 which is an omnidirectional robot acting as a differential-drive robot.


## 7. Conclusions

This work extends the leader-follower behaviours of a combined set of several kinematic models of omnidirectional and differential-drive robots. This kind of robots becomes actually in the most common wheeled vehicles used in the industry and service tasks. By the combination of these models, four basic leader-follower control laws are designed using the measurements of distance and the heading angle with respect to a leader, avoiding the use of a global reference framework. It is proved that these control laws generate, in the settling time, some mechanical structures related to rigid body, the standard 1-trailer, and a new structure called omnitrailer. Extending this local behaviour to the case of multiple groups of agents, a directed tree-shaped topology can be constructed, where the leader robot is the unique root node.It is shown by numerical simulations and real-time experiments that the combination of the leader-follower setups can generate formation tracking, useful for object transportation or other collaborative tasks, liken-trailer and convoy behaviours of rigid platoons of robots. This complex behaviour becomes a more decentralized approach than the global referenced strategies like virtual structures or cluster space specifications. As future work, the formation topologies will be extended for other directed and undirected graphs and the local sensors will be implemented in the robots to measure the distances and the heading angles.



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*Source: 4957259-2019-04-02.xml*