Motion Planning for Omnidirectional Wheeled Mobile Robot by Potential Field Method

Li, Weihao; Yang, Chenguang; Jiang, Yiming; Liu, Xiaofeng; Su, Chun-Yi

doi:https://doi.org/10.1155/2017/4961383

Journal of Advanced Transportation

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Intelligent Autonomous Transport Systems Design and Simulation

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Research Article | Open Access

Volume 2017 | Article ID 4961383 | https://doi.org/10.1155/2017/4961383

Motion Planning for Omnidirectional Wheeled Mobile Robot by Potential Field Method

Weihao Li,¹Chenguang Yang,^1,2Yiming Jiang,¹Xiaofeng Liu,^3,4,5and Chun-Yi Su¹

Academic Editor: Cheng S. Chin

Received30 Dec 2016

Accepted07 Feb 2017

Published12 Mar 2017

Abstract

In this paper, potential field method has been used to navigate a three omnidirectional wheels’ mobile robot and to avoid obstacles. The potential field method is used to overcome the local minima problem and the goals nonreachable with obstacles nearby (GNRON) problem. For further consideration, model predictive control (MPC) has been used to incorporate motion constraints and make the velocity more realistic and flexible. The proposed method is employed based on the kinematic model and dynamics model of the mobile robot in this paper. To show the performance of proposed control scheme, simulation studies have been carried to perform the motion process of mobile robot in specific workplace.

1. Introduction

In recent decades, omnidirectional mobile robot (OMR) has attracted increasing attention and investigation from the research communities [1–3]. One of advantages of OMR using omnidirectional wheels is that it does not have nonholonomic constraint which exists in differentially driven mobile robot [4–6]. 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 [18–21]. 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 [22–24]. 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 [25–27]. 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 [28–31]. 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 [32–35]. 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.

2. The Kinematics Equation of Omnidirectional Wheel

From Figure 2, the following equation can be obtained:where , , and are generalized velocity of point in Cartesian coordinate system and , , and are generalized velocity of point in coordinate system. is the th roller’s central velocity vector.

When the th omnidirectional wheel’s central speed is mapped to Cartesian coordinate system, then

The system moves in two-dimensional space, so, according to geometric relationship and [36], the wheels’ speed can be represented bywhere 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 aswhere is not zero, so as the . Define , , and . is the angle between and -axis. Then the inverse kinematics equation of th omnidirectional wheel isdefine ; thenDefineThe inverse kinematics solution of wheel speed to system center isThe 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 mm and mm, the actual parameter of OMR can be substituted into (8); then

From (10), , which means that this robot can achieve omnidirectional movement.

3. 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 is a force vector on the central point of mobile robot in the Cartesian coordinate. is a symmetric positive definite matrix, and is the mass of mobile robot.

The transfer matrix which transfers coordinate into coordinate isConsequently, and , respectively, mean the central point O’s displacement and force vector in mobile coordinate . Therefore, (12) becomesFinally, the dynamics of the omnidirection mobile robot can be described aswhere and are the moment of inertia of mobile robot around its central axis and the corresponding torque, respectively. Among them, ,, and can be drawn bywhere is the angle between wheel and -axis, and , , and .According to [37], the dynamics model of drive system of each wheel is assumed aswhere 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. is angular acceleration of each wheel. is radius of the wheel. is drive factor. is the input torque of each wheel.

The speed of each omnidirectional mobile robot’s wheel can be described as . According to [38], the dynamics model of OMR can be described as the following equation:where

Let us define , , and .

4. 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 two-dimensional 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 is a positive scaling factor, is the distance between the OMB’s mass point and the goal , and or . For , the attractive force isThe repulsive potential function iswhere is a positive scaling factor, is the minimal distance between the OMB’s mass point and the hindrance, 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 and are two unit vectors pointing from the obstacle to the OMR and from the OMR to the goal. According to (23) and (25), 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, , , and , which satisfies (8), and and have the following relationship with 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 bywhere is the optimal velocity of the robot.

As the total force can be calculated, its angle is known. The difference angle between and the orientation of the robot isThus the angular velocities can be ensured bydefine as a positive gain.

5. 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.

5.1. Kinematic Controller

With (8) and , then the kinematics model of mobile robot can be transformed intoDefine as the state of mobile robot in and . Therefore

With the help of zero-order hold (ZOH), a continuous-time system can be described as a discrete-time formwith a sampling period . According to (31), (32), and (33) can be rewritten as

The cost function for the MPC can be defined aswhere is the stage cost.where is prediction horizon where and is control horizon where . and are appropriate weighting matrices. means the predicted state of the OMR and 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

According to [42], (38) can be transformed into

The kinematic equation (34) can be described into the following form:where and are the continuous nonlinear function, , is the state vector, is the input vector, andDefine the following vectors:The predicated output can be determined by the following form:wheredefine .

Hence, the original optimization problem (35) can be transformed intosubject toProblem (45) can be rewritten as a QP problemsubject towhere the coefficients are

5.2. Dynamics Controller

According the dynamics model of OMR, then

Applying (33) into (50) and the discrete-time dynamics model of the OMR can be described asAccording to and , thenApplying (52) into (40), then, we can drawsubject towhere, respectively, and means the lower bounds and upper bounds. is the motor inputs. is the velocity. is the acceleration.

According to (53), the predictive model can be formulated aswhere .

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 bysubject to

6. Experimental Part

In this section, we design a 100 m × 100 m workplace within 6 obstacles . We define the velocity of the OMR as 1 m/s and the start point of the OMR as while the destination is . We define the prediction horizon and the control horizon . 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 () is modified by MPC as on the left of simulation.

Connecting Figures 6, 8, and 9, we can see that the moment the change of 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 .

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.

6.2. 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.

7. 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.

Disclosure

Chun-Yi Su is on leave from Concordia University, Canada.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was partially supported by National Nature Science Foundation (NSFC) under Grant 61473120, Guangdong Provincial Natural Science Foundation 2014A030313266 and International Science and Technology Collaboration Grant 2015A050502017, Science and Technology Planning Project of Guangzhou 201607010006, and State Key Laboratory of Robotics and System (HIT) Grant SKLRS-2017-KF-13.

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Copyright

Copyright © 2017 Weihao Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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