This paper presents the design of a kinematic controller for coordinating a team of mobile robots into specified formation configurations using the leader-follower framework. First, the leader-follower strategy is reformulated as a trajectory tracking problem. Then, a discrete model predictive control (DMPC) is integrated with a discrete sliding mode (DSM) control to guide the follower robots in tracking the leader's trajectory while preserving the required formation geometric configuration. The suggested control scheme ensures precise trajectory tracking and a robust formation maintenance with a constrained and chattering free control inputs. Simulation results demonstrate the efectiveness, eficiency, and practicality of the proposed control strategy for real-world scenarios.
research to address these issue. For example, authors in
[26] addressed the formation control of nonholonomic
Formation control is a fundamental aspect of multi-robot mobile robots. a globally finite-time stable sliding
systems, it allows for the robots to operate cohesively mode controller has been designed. Then, a continuous
by regulating states like position and orientation to reaching law has been derived to mitigate the chattering
achieve specific geometric configurations. Its broad caused by control limitations and computation time
applicability spans various domains such as exploration, delays. In [27] a second order sliding mode controller
rescue missions, surveillance, and transportation, which has been developed , based on the relative motion
make it a critical focus in robotics research. Recent states and without the leader velocity measurement, to
developments in the field have led to the exploration of stabilize the robots towards the required time-varying
various control methodologies, including behavior-based formation and to avoid the the chattering phenomena.
[
On the other hand, employing model predictive
control MPC for the formation control of nonholonomic
mobile robots can efectively account for physical
limits of the robots, making it capable for yielding
an optimal formation tracking and maintenance. The
authors in [
The key contribution of this paper lies in the
development of a controller that combines discrete model pre- Figure 1: Diferential-drive wheeled mobile robot.
dictive control MPC and discrete sliding mode control to
achieve a robust and accurate formation control of
nonholonomic wheeled mobile robots. The integration of Where is the robot angular velocity and is the
MPC allows for optimal formation producing and track- robot linear velocity.
ing with constrained states and inputs, while the sliding
mode control ensures robustness against kinematic per- In practice, the robot model is subjected to kinematic
turbations subjected to the robots model in practice. By uncertainty and input disturbances. Hence, a more
realleveraging the strengths of both control techniques, the istic model of the robot can be expressed as follow [
2.1. Nonholonomic mobile robot Figure 2 show the basic architecture of the leaderkinematic model follower formation approach. Where the posture of the leader robot is = [ ] and the posture of Consider the diferential-drive wheeled mobile robot the follower robot is given by = [ ] and shown in Figure 1. Let = [ ] be the robot center the desired posture for the follower robot is given by of mass posture, where (, ) denotes the position of the = [ ] . robot in the global Cartesian frame ( ) and is the The leader-follower approach can be seen as a trajectory orientation angle. tracking problem where the follower robot must track
This robot satisfy the following pure rolling and non- the trajectories generated by the leader robot in-order to slipping nonholonomic constraints given by: preserve the required separation distance and heading angle Φ , and to form the predefined formation shape.
̇ cos − ̇ sin = 0 (1) Hence the desired posture can be given as [
By using the nonholonomic constraints in (1), the kinematic model of the robot can be described as follow:
⎡ cos ̇ = ⎣ sin 0 0 ⎦ 1 0 ⎤ ︂[ ]︂ = () (2)
⎡ ⎤ ⎡ + cos (Φ + ) ⎤ = ⎣ ⎦ = ⎣ + sin (Φ + ) ⎦ atan2 (̇, ̇ + )
(4)
Where = 0, 1 is the driving direction ( 0 for the forward motion and 1 for reverse) and 2 is the fourquadrant inverse tangent function. To accomplish the formation objective, the follower robot need to follow the reference trajectory consist of the set of the desired postures , which implies that the following must satisfy: lim ( − ) = 0 →∞
(5)
Since the leader-follower formation is converted to a trajectory tracking problem, the tracking error model of the formation can be written as:
The linearized model of the tracking error dynamics (12) can be written in discrete form as : ( + 1) = ()() + () (13) ⎡ ⎤
cos = ⎣ ⎦ = ⎣ − sin 0 ⎡ sin cos 0 0 ⎤ ⎡ − 0 ⎦ ⎣ − ⎦ 1 − ⎤
(6)
The tracking error dynamics of the formation can be
obtained by taking the time derivative of (6) and by using
equations (3) and (1) as follow [
Where and are the linear and angular feedfor- lowing sliding mode function is defined: ward control input defined as: And ∈ R× , : number of states variable, ∈ R× m : number of input variable and : is the sampling time.
Consider the discrete-time system (13), then the fol = + (), = {︃ = ± √︀̇2 + ̇2 = ˙¨− ˙¨
˙2+˙2
Where (+) for the forward motion and (− ) for backward motion . By neglecting the input disturbance ∆ = ︀[ ︀] , then defining the control input vector of the follower robot as the sum of the feedforward and feedback control action: () = () (14) (8)
Where ∈ R× Is the gain matrix. For a discrete-time
system (13), a quasi-sliding mode reaching law is given
as in [
Then, we introduce the following cost function: =1 =1 = ∑︁ (( + 1) − ( + ))2
+ ∑︁ (() − ())2 Where () is the discrete sliding mode equivalent control given by:
() = ()− 1[()()] The cost function in (21) can be described in quadratic form as: ((), , ) = ‖(( + 1) − ( + 1))‖2 + ‖(() − ())‖2
︁) − 1 [︁ ( () − ( + 1)) − ] Where and are weighting matrices given as: ( + 1) − () = ( + 1) + ()
= ()() + () − () Then, solving for () gives the following control input: () = − ()− 1[()() − ()
+ () + sign(())]
The main idea of predictive sliding mode control is to find a control law
() that drive the predictive sliding function vector ( + 1) to a reference sliding function vector ( + 1), by minimizing a quadratic cost function ((), , ).
Consider the discrete sliding mode problem for the system (13), taking the reaching law (15) as a reference sliding surface results the following : ⎩ () = () ⎨ ⎧ ( + p) = (1 − qs) ( + − 1) − sign (( + − 1)) − 1 =1 The value of the sliding function vector (14) at time instant can be obtained as: ( + ) = ∏︁ ( + )() + ∑︁(︁ ∏︁ ( + ) ︁) =1 − 1 =1
as fol× ( + − 1) + ( + − 1) ( + 1) = [( + 1), ( + 2), . . . , ( + )] Where is the prediction horizon, then ( + 1) can be described as: low: ( + 1) = ()() + ()() (20) 4. Simulation results () = [ (), ( + 1)(), . . . , ˜(, 0)] () = ⎢⎢⎢ ( + 1)
. ⎥robot is assigned as a leader and the second robot is
The control parameters for the DSM and PDSM
methods were determined by trial and error as follow : =
3, = 10− 3
diag[
, = − , = 1.8/ While the kinematic input disturbances defined in (3) is selected as follow: ∆ = ︂[ ︂] = ︂[ .01() ]︂
.01() control
= [0.25 1.5 − / 4] .
In this simulation example, the discrete predictive sliding mode controller (24) is used to control the formation, the leader robot is assumed to be moving in a 8-shape trajectory produced by (25). The required separation distance is chosen as = 0.15 m and the bearing angle is selected as = 3/ 2, while the initial robots posture are given as: = [0 0 / 4] and
The real-time trajectories for both robots are shown in Figure 3 . The follower robot tracking errors and its velocities commands are shown in Figure 4 and Figure 5, respectively. (25) (26) (27)
This section presents a comparison between the discrete sliding mode DSM control in (17) and the discrete predictive sliding mode DPSM control in (24). In this example, the leader robot is following a circular trajectory given by equation (28) with a constant angular = ∫︀0 ()2, integral time absolute error
Figure 6, shows the formation trajectories based on both DSM and DPSM control schemes. While Figure 7 depict a comparison between the formation tracking errors, and the control inputs of the follower robot using the suggested control techniques are illustrated in Figure 8 error
To compare the formation tracking performances, ifve error indexes are employed. Namely, mean square ror = ∫︀ ()2, integral time square error 1 ∑︀1 ()2, integral square er = 0 = ∫︀0 | ()|, and integral absolute error = ∫︀
0 | ()|. Where () is given in (29) and is the time of simulation.
() = ()2 + ()2 + ()2
(29) The obtained results of the comparison between the formation tracking performances are listed in Table 1.
In the first example, the simulation results of the leader-follower formation in an 8-shape trajectory is successfully performed. As shown in Figure 3, the follower robot efectively follows the leader, maintaining the required distance and keeping the desired heading angle. The tracking errors of the follower robot steadily decreases until it reaches zero in the presence of the input disturbances, as seen in Figure 4. Additionally, Figure 5 illustrate that the robot velocities adhere to the imposed constraints without any chattering.
The comparison between the formation trajectories in Figure 6 shows that the formation problem is successfully solved based on both proposed control strategies, where the tracking errors gradually reach the origin over time as depicted in Figure 7. However, from the control laws of the follower robots illustrated in Figure 8, it can be noted that the DPSM control scheme can generate a chattering free control signals that respect the physical input limits of the robot. Moreover, the analysis of the tracking performances in Table 1 show that DPSM control technique has a better formation tracking accuracy when compared to the DSM control strategy.
To summarize, the above simulation outcomes indicate that the proposed predictive sliding mode DPSM controller can perform an accurate formation tracking with a practical and chattering free control inputs.
The leader-follower-based formation control for wheeled nonholonomic mobile robots has been addressed in this paper. Initially, the trajectory following problem was expanded into a formation control problem. Then, by the utilization of linear formation tracking error dynamics, we have designed a discrete sliding mode controller to guide the follower robots in maintaining their formation relative to the leader and achieving the desired spatial geometric configuration. Furthermore, to optimize control eforts and mitigate the challenging chattering phenomenon, we integrated a discrete model predictive control DMPC with the DSM approach. The suggested method eficacy was demonstrated through simulation results and comparative studies.
During the preparation of this work, the authors used ChatGPT, Grammarly in order to: Grammar and spelling check, Paraphrase and reword. After using this tool/service, the authors reviewed and edited the content as needed and take full responsibility for the publication’s content.