Motion Planning for Omnidirectional Wheeled Mobile Robot by Potential Field Method

1Key Lab of Autonomous Systems and Networked Control, Ministry of Education, South China University of Technology, Guangzhou 510640, China 2Zienkiewicz Centre for Computational Engineering, Swansea University, Swansea SA1 8EN, UK 3College of IoT Engineering, Hohai University, Nanjing, China 4Jiangsu Key Laboratory of Special Robots, Hohai University, Nanjing, China 5Changzhou Key Laboratory of Robotics and Intelligent Technology, Changzhou, China


Introduction
In recent decades, omnidirectional mobile robot (OMR) has attracted increasing attention and investigation from the research communities [1][2][3].One of advantages of OMR using omnidirectional wheels is that it does not have nonholonomic constraint which exists in differentially driven mobile robot [4][5][6].With the input of the rotating speed of each omnidirectional wheel, the mobile robot can easily move wherever the user wants.This simplifies the control law which can be achieved easily.As it is shown in Figure 1, omnidirectional wheel consists of wheel and rollers, which means that the speed of the whole omnidirectional wheel is the combination of wheel speed and roller speed.Robot's control is very complicated, and sometimes it is necessary to consider state constraint of the robot to complete the control design [7,8].
Since the path planning problem has been put forward, it has been studied by numbers of researchers [9].A large number of research results have been proposed.The path planning algorithm develops from the earliest grid method, artificial potential field method [10], visibility graph [11] to C-space method [12],  * algorithm, and  * algorithm [13].Now, it is also studied to combine fuzzy logic algorithm [14,15], adaptive algorithm [16,17], and neural network algorithm [18][19][20][21].In recent years, potential field method is more and more mature and widely used in omnidirectional mobile robots, because of its logical simplicity and obstacle avoidance capability.Many researches have proved the excellent capability of navigation and obstacle avoidance [22][23][24].Hence, potential field method is utilized in this paper for the motion planning of omnidirectional wheeled mobile robot.To overcome the local minima problem and the goals nonreachable with obstacles nearby (GNRON) problem, the repulsive potential functions for motion planning contain the distance between robot and obstacle.
A popular way to control a mobile robot is to design the kinematic control based only on the kinematics equation [25][26][27].Since 1995, people have put forward an integral dynamics model of a mobile robot [4].Using dynamics model to control robot's motion is a common way [28][29][30][31].This paper combines the kinematics as well as dynamics equation of the omnidirectional wheel and potential field method, to control and navigate the mobile robot.In addition to these contributions in this paper, model predictive control (MPC) is utilized in motion planning for robust controller performance [32][33][34][35].This paper is organized as follows: in Section 2, the kinematics equation of omnidirectional wheel and how the mobile robot built with 3 omnidirectional wheels can achieve the omnidirectional motion are discussed.In Section 3, the dynamics model of 3-omnidirectional wheel mobile robot is explained.In Section 4, a novel potential field method, which can overcome the GNRON problem, is introduced.In Section 5, MPC has been introduced.Both kinematics model and dynamics model have been applied in MPC.In Section 6, the simulations illustrating the effectiveness of the proposed method are presented.Finally, conclusion is given in Section 7.

The Kinematics Equation of Omnidirectional Wheel
From Figure 2, the following equation can be obtained: where V  , V  , and   are generalized velocity of point    in Cartesian coordinate system and V   , V   , and    are generalized velocity of point    in       coordinate system.V  is the th roller's central velocity vector.
When the th omnidirectional wheel's central speed is mapped to Cartesian coordinate system, then x 㰀

훼 i ir
Figure 2: The motion relationship between system center and the  omnidirectional wheel.
The system moves in two-dimensional space, so, according to geometric relationship and [36], the wheels' speed can be represented by where   and   are the position of the th wheel's mass point in  coordinate.According to (2)-(3), the system inverse kinematics equations can be defined as where det( 1 ) is not zero, so as the det( 2 ).
define   =   −   ; then Define The inverse kinematics solution of wheel speed to system center is The Jacobian matrix systems inverse kinematics equation is According to Figures 2-4, since this paper discusses a mobile robot built by three omnidirectional wheels,   ,   , and   , are fixed.The actual values are shown in Table 1.From Table 1 and specification that  = 50.67mm and  = 118.18mm, the actual parameter of OMR can be substituted into (8); then From (10), rank() = 3, which means that this robot can achieve omnidirectional movement.

The Dynamics Equation of OMR
Three coordinates       , , and          are constructed as in Figure 4.   is a specific point in the workspace, and  is the central point of the mobile robot while    is the central point of each wheel. is the angle between the front of OMR and   .The vector   = [  ,   ]  indicates the position of .According to Newton's second law, equivalent to Consequently, = [, ]  and  = [  ,   ], respectively, mean the central point O's displacement and force vector in mobile coordinate .Therefore, (12) becomes Finally, the dynamics of the omnidirection mobile robot can be described as where   and   are the moment of inertia of mobile robot around its central axis and the corresponding torque, respectively.Among them,   ,   , and   can be drawn by where  is the angle between wheel and -axis, and  1 = 30 ∘ ,  2 = 30 ∘ , and  3 = 90 ∘ .
According to [37], the dynamics model of drive system of each wheel is assumed as where    is the drive power of each wheel.  is the moment of inertia of wheel around its central axis. is viscous friction constant between wheel and ground.ω  is angular acceleration of each wheel. is radius of the wheel.ℎ is drive factor.  is the input torque of each wheel.
The speed of each omnidirectional mobile robot's wheel V  can be described as   .According to [38], the dynamics model of OMR can be described as the following equation: where Let us define  1 = √ 3 sin −cos ,  2 = − √ 3 cos −sin , and  3 = √ 3 cos  − sin .

Potential Field for OMR's Motion Planning
Using the potential field algorithm for OMR's path planning will be modified to produce a virtual force to navigate mobile robot and obstacle avoidance.For simple theoretical analysis, mobile robot is considered as a mass point and moves in twodimensional space whose position can be denoted by  = [, ]  .The distance as well as the angle between robot and goal, robot and obstacles can be detected by ultrasonic sensors.Inspired by [23], then the attractive potential function caused by goal can be calculated by the following equation: where  att is a positive scaling factor,   (,  goal ) = ‖( goal − )‖ is the distance between the OMB's mass point and the goal  goal , and  = 1 or 2. For  = 2, the attractive force is The repulsive potential function is where  rep is a positive scaling factor, (,  obs ) is the minimal distance between the OMB's mass point  and the hindrance,  0 denotes the level of the influence of the hindrance to the robot and it is defined as a positive constant, and  is a positive constant.The repulsive force is OR = ∇(,  obs ) and  RG = −∇(,  goal ) are two unit vectors pointing from the obstacle to the OMR and from the OMR to the goal.According to ( 23) and ( 25),  total is drawn by the following equation: According to [23], for the robot with 3 omnidirectional wheels, the real input is the 3 angular velocities of the omnidirectional wheels,  1 ,  2 , and  3 , which satisfies (8), and V   and V   have the following relationship with V and : The mobile robot needs to decelerate as soon as it nears the obstacle, while its velocity will be higher when it is far from the obstacle, so the robot's velocity is chosen by the distance between OMB and obstacle.Thus the velocity of the OMR in       can be determined by where V  is the optimal velocity of the robot.
As the total force  total can be calculated, its angle   is known.The difference angle between   and the orientation of the robot   is Thus the angular velocities  can be ensured by define  as a positive gain.

Model Predictive Control for Omnidirection Mobile Robot
In recent years, MPC has been widely used in motion control Internet of things applications [35,39].MPC has low requirements for model's accuracy and it is suitable for step response model and linear and nonlinear model.The control problem is described as a cost function's optimization problem.The input which is constrained by some specific conditions and minimizes the cost function is the optimal input.One of MPC's advantage is its rolling optimization [40] that means, according to its reference, it can optimize a cost function to get an optimal input vector at every sample time.According to the MPC method introduced by [41], because the reference is produced by potential field, what we need is a discrete-time model with constraints.In the following section, according to [35] two discrete-time controllers, kinematic controller, and dynamics controller are proposed.

Kinematic Controller.
With (8) and  = , then the kinematics model of mobile robot can be transformed into Define  = [  ,   , ]  as the state of mobile robot in       and () = () −1 .Therefore With the help of zero-order hold (ZOH), a continuoustime system can be described as a discrete-time form  ( + 1) =  () + q () . ( with a sampling period .According to (31), (32), and ( 33) can be rewritten as ( + 1) The cost function for the MPC can be defined as where   (, ) is the stage cost.
where  is prediction horizon where  ≥ 1 and   is control horizon where 1 ≤   ≤ .  and   are appropriate weighting matrices.( +  | ) means the predicted state of the OMR and Δ( +  | ) means the input increment of the controller.  (, ) is the most used standard quadratic form in practice.By way of solving the following finite-horizon optimal control problem (FHOCQ) online: The current control () = [(), ()]  can be ensured at the instant time .Because the torques generated by motors are limited by the performance of the motors, () has upper bound and lower bound and the change of () is also constrained.Thus 6 Journal of Advanced Transportation According to [42], (38) can be transformed into The kinematic equation ( 34) can be described into the following form: where  1 and  2 are the continuous nonlinear function, ]  is the input vector, and Define the following vectors: The predicated output can be determined by the following form: where Hence, the original optimization problem (35) where the coefficients are (49)

Dynamics Controller.
According the dynamics model of OMR, then Applying ( 33) into (50) and [ ẍ , ÿ , η ] = [ V   , V   , ω ] the discrete-time dynamics model of the OMR can be described as According to  = [V   , V   , ]  and  = [  ,   , ]  , then Applying ( 52) into (40), then, we can draw subject to where, respectively, min and max means the lower bounds and upper bounds. is the motor inputs. is the velocity.q is the acceleration.
The quadratic objective function (QBF) of the robot's state and the motor input under a predictive horizon  and a control horizon   can be determined by the following equation: where   and   are appropriate weighting matrices.
Hence, the dynamics predictive motor torque can be obtained by subject to (58)

Experimental Part
In this section, we design a 100 m × 100 m workplace within 6 obstacles (  ,  = 1, 2, 3, 4, 5, 6).We define the velocity of the OMR as 1 m/s and the start point of the OMR as (3, 90) while the destination is (70, 28).We define the prediction horizon  = 3 and the control horizon   = 2.As the inputs of kinematic controller and dynamics controller are different, we set up a different constraint.The save distance between the OMR and obstacle is adjustable and the accurate value of the OMR's model is based on the OMR we built physically.The simulation is carried out in MATLAB.
6.1.Simulation of Kinematics Controller.In this section, simulation is carried out based on kinematics model.The OMR is navigated by potential field method combining MPC.In this simulation, we assume the goal position is already known, and the start position and start velocity are defined in advance.The process of this simulation is shown by the following flow chart in Figure 5, where   means the th obstacle.First, we define the originate position of the mobile robot and the originate velocity.Then, we sent the goal position to the mobile robot.With the help of potential field method, mobile robot can draw the next status, which is used as the reference for the MPC.After the MPC process, the predictive status is sent to mobile robot for navigation and to the previous process for the next reference.
The area of workplace is within 100 m × 100 m and there are some obstacles randomly distributed in it.With the help of potential field and MPC, the OMR can adjust its velocity and finally reach the goal which is shown by Figures 6 and 7.
According to Figures 6 and 7, we can see that the OMR successfully reaches the goal and can smoothly self-avoid the obstacle.The velocity of the OMR is shown by Figures 8, 9, and 10.
At the beginning of simulation, we define the originate velocity of the robot as 1.When the mobile robot starts moving, the velocity (V  ) is modified by MPC as V  on the left of simulation.
Connecting Figures 6, 8, and 9, we can see that the moment the change of V  is rapid is the moment that mobile robot encounters an obstacle and it needs to change its direction.Figure 10 is the velocities of three omnidirectional wheels; the composition of all three velocities is V  .
According to potential field method, we can get the total force angle, and then we can determine .The change of  is shown by Figure 11.
The result of potential field method in navigation is outstanding.The OMR can smoothly avoid obstacles and successfully guide itself to the goal.With supplement of MPC, the motion of mobile robot is constrained and the velocity becomes more realistic and flexible.The robustness of system is enhanced.

Simulation of Dynamics Controller.
In this section, similar to the last section, we change kinematics model into dynamics model and simulate a similar process which OMR is navigated by potential field and MPC.In this simulation, we assume the goal position is already known, and the start position and start velocity are defined in advance too.
According Figures 12 and 13, it shows that, with the control of dynamics controller and navigation of potential field, the OMR achieves its goal and successfully avoids the  obstacles.When the OMR approaches an obstacle, potential field algorithm generates a repulsive force which makes the OMR turn around and prevent itself from hitting the obstacle.
With the help of MPC, the velocity is constrained, which can  prevent some value that is beyond the real produced by the potential field method from navigating the OMR.
Comparing Figures 14 and 15 to Figure 12, it is easy to see that when the velocity changes sharply is when the OMR is too close to the obstacle and it needs to slow down to avoid possible collision.According to Figure 16, when the OMR needs to change its direction, it changes three torques, respectively, to generate a combined effort to steer its motion.

Conclusion
In this paper, a novel potential field method has been used to navigate a class of OMR.With the consideration of distance between the robot and the goal, robot, and obstacle, the GNRON problem is solved.In addition, it discusses the kinematic as well as dynamics model of mobile robot.For improving system's robustness, it combines potential field and MPC, so the motion planning is more complete.Finally, simulation results show that the proposed control scheme is more appropriate for omnidirectional mobile robot's navigation.

Figure 1 :
Figure 1: The structure of  omnidirectional wheel.

Figure 3 :
Figure 3: The structure of th omnidirectional wheel.

Figure 4 :
Figure 4: All three coordinates in OMR system.

Figure 5 :Figure 6 :
Figure 5: The velocity of the OMR during the simulation.

Figure 7 :
Figure 7: The trajectory of the OMR controlled by kinematic controller in       coordinate.

Figure 8 :
Figure 8: The velocity of the OMR controlled by kinematic controller during the simulation.

VtFigure 9 :
Figure 9: The change of velocity of the OMR controlled by kinematic controller.

Figure 13 :Figure 14 :VtFigure 15 :
Figure 13: The trajectory of the OMR controlled by dynamics controller in       coordinate.

Figure 16 :
Figure 16: The change of torque of the OMR.
, and   =   sin   .  is the angle between    and -axis.Then the inverse kinematics equation of th omnidirectional wheel is