PDC Control for Mobile Robot Formations with Virtual Reference Based on Separation-Bearing

– This paper presents a development of leader-follower formation control using separation-bearing control (SBC) and Parallel Distribution Compensation (PDC) control. The formation control involves tracking of each desired trajectory by leader and follower robots. The follower trajectory is generated using SBC approach with respect to predefined trajectory of the leader. This design is used to improve formation control when initial error is given to leader. In order to maintain the formation and avoid internal collision, the error tracking of each robot must be kept near zero. Each robot is controlled by kinematic and dynamics controller which is designed using PDC and PID. The velocity reference for dynamic robots is limited. The simulation result shows the tracking errors for position and orientation with initial lateral error set at 0.5 m are less than 0.5 m and 1.2 rad which then converges to the desired value. Thus, the good trajectory formation tracking is achieved.


I. Introduction
The coordination of multi-robot which performs collective tasks have been widely developed for past decades. These types of implementation can be seen in rescue or military robots [1]. Since it involves multiple robots, the formation control require strategy to implement [2]. The control is used to form or maintain formation in point to points motion or tracking reference trajectory [3], [4].
There are several approaches for formation strategy such as behavior-based [5], [6] virtualstructure [7], and leader-follower [8]. Formation using leader-follower is begin with leader moving along predefined trajectory and the follower maintain its position and orientation with respect to the leader [9]. One of the approaches which often be used is separation-bearing controller (SBC) [10]. Based on leader position and desired separation distance, the follower will maintain its position and orientation. The problem with this approach is the value of linear and angular velocity which generated from the kinematic controller of follower. When leader tracks a circle trajectory, follower linear and angular velocity have the same value when follow the leader at inner and outer circle reference. Thus, the velocity generated from the SBC can only be used if the variables are distance and bearing angle. However, posture result is accurate. As it stated in [11], SBC can be used when the initial error position is not given to the leader. Especially, if the system involves robot dynamics model. In [12], the problem of formation is transformed into trajectory tracking. This paper examines how to coincide the center of mass and wheel axis despite the motion and reference problems are not stated. For individual robot, parallel distributed compensation (PDC) control can be used to minimize error tracking [13]. This method is also suitable to solve tracking problem when dynamics model is used [14]. However, if the number of rules is quite big, the feasibility of the linear matrix inequality (LMI) solver is difficult to be obtained. In this study, SBC controller is used as posture reference for each follower. The kinematic controller is designed using PDC and auxiliary velocity based on model error. Then, dynamics controller is designed using PID controller and forward gain. Using kinematic model of nonholonomic robot, the linear and angular velocity

II. Research Method
The diagram block of the system is shown in Fig.  1. The references for each robot are position ( , xy), orientation ( ), linear velocity ( v ), and angular velocity (  ). The position and orientation references generated from SBC with predefined distance and bearing angle. For linear and angular velocity reference, the values are generated using kinematic model based on SBC output.

II.1. Formation trajectory generator
The motion of multi robot in formation need to be coordinated in order to prevent internal collisions between robot. The coordination is arranged in trajectory generator. First, the desired trajectory for leader is defined as follows: where lr v and lr  are linear and angular velocities reference for leader. lr P is vector posture of leader virtual reference.
There are two followers, 1 F and 2 F , that will be used in this study as it shown in Fig. 2 The position and orientation virtual references of each follower can be derived from (3) as follows: The different reference value between two followers in (4) is the angle bearing as it shown in Fig. 2.

II. 2. Kinematic and Dynamics Model of the robot
The robots are assumed to have nonholonomic constraints with pure rolling constraint and no lateral slip motion [15]. Let 1,2,3 i  denote leader, follower 1 (F1), and follower 2 (F2). The forward kinematic model of the robot can be expressed as follows:  (5) where p is the robot posture ( , , ) xy , R  and L  are the angular velocities of right and left wheel, R is the wheel radius, and 2L is the distance between right and left wheel.
The coordinates configuration of the robot can be expressed as: where R  and L  are the angle of right and left wheel. Let define wheels angular velocities as: Using Lagrange dynamic approach, the dynamics model of the robot can be written as follows: (8) with:    (11) where di v and di  are the auxiliary velocities control.
As in [15], the following form is chosen to be the auxiliary velocities control:

III. 3. Kinematic and Dynamics Controller
The kinematic controller of each robot is designed and stabilized using PDC control law based on Takagi (15) Using (15), the matrix m A and m B can be expressed as: The premises are constructed using maximum and minimum value as well as the membership function in order to obtained the PDC control law as follows: where m F is feedback gain. In order to obtained the stability of the controller, the LMI equations (18)-(21) is used to calculate the feedback gain.
The result of input velocities control in (12) will be used as reference for the dynamics robot system. Because the dynamics model (8) uses the right and left wheel angular velocities instead of linear and angular velocities, the input velocities (12) will be converted using the following form: where r K is feedforward gain, p K is proportional gain, d K is derivative gain, and i K is integral gain.

III. Result and Discussion
The simulation of the designed controller is implemented using MATLAB and Simulink. In order to verify the proposed method, the simulation sets are conducted using single and formation robot with and without initial error. Trajectory reference for formation is a circle. The linear velocity limitation is 15 rad/s while the angular velocity is 15  rad/s. The predefined formation is arranged with 1   For error tracking with initial error (0, 0.5,0)  is shown in Fig. 7.  The kinematic error is quite big around 6  rad/s but the steady state is achieved within 7 2 10   rad/s. For formation trajectory tracking, the result of single robot above is used as the leader reference. The formation trajectory tracking with and without initial error is shown in Fig. 11. The steady state of each error is achieved for both F1 and F2. The orientation errors for both robots are bit higher but still within target around 1 10   rad. The posture error for tracking with initial error is show in Fig. 13 and Fig. 14.

IV. Conclusion
The development of formation control by utilizing the separation-bearing control as trajectory reference generator for follower robot and parallel distribution compensation as control law for kinematic controller of each robot is presented. The separation-bearing control used the desired trajectory which is given to leader robot as reference in order to virtually form and maintain the predefined formation throughout the tracking process. From the simulation results, the good formation trajectory tracking can be achieved with small errors both without and with initial error. Particularly, the value is not exceeding the given initial error which is 0.5 m (lateral error). In spite of the rise and settling time are slower than the result of tracking without initial error, the steady state of the posture and velocity errors is obtained. Thus, it can be stated that the proposed method can be used to solve initial error problem in formation control when the robot dynamics model is involved.