Collision-Free Path and Trajectory Planning Algorithm for Multiple

Collision-Free Path and Trajectory Planning Algorithm for Multiple-Vehicle Systems Anugrah K. Pamosoaji, and Keum-Shik Hong, Senior Member, IEEE

Abstract— An algorithm to generate collision-free paths and trajectories multiple-vehicle systems is presented. The algorithm is utilized to solve a problem of planning collision-free paths and trajectories of all the vehicles from their start points to goal points. Three-degree Bezier curves are utilized as the basic forms of the paths. Trajectories are generated subject to linear velocity, tangential velocity, and radial accelerations constraints of each vehicle. The presence of known static obstacles is considered. A Particle Swarm Optimization (PSO) technique is utilized to figure out the trajectories which minimize the accomplishing time of the slowest vehicle. Moreover, trajectories drive all vehicles to avoid collision to each other. Simulation results that show the trajectories of three vehicles are presented.

I. INTRODUCTION

P

roblems of path and trajectory planning in multiple-vehicle systems have been one of the common interests in robotics area. Various approaches were proposed to handle many issues. One direction of the approaches, which was based on path analysis theories were introduced by [1] and [2]. An algorithm to find the smoothest path connecting two given pairs of configurations (positions and orientations) was proposed. A solution of utilizing circular arcs and cubic spirals were introduced. Another solution with similar approaches was proposed by [3]-[7]. Lepetic, et. al. in [3] used circular arc and straight line as the planned path A solution which considered the composition of workspaces to build path was proposed in [6]. A case study of coordinated target assignment for Unmanned Air Vehicles (UAVs) was presented. A Voronoi-diagram-based path planner was used to build paths. The obstacles of this research were modeled as points. The research also considered interception problem, which was deal with the kinematic constraints of the vehicles. A decentralized approach to a problem of decoupling the assignment and navigation process was proposed by [8]-[11]. In the approach of [8], a hybrid automaton was used to distinguish available and assigned tasks. Widyotriatmo and Hong in [9] and Zavlanos and Pappas in [10] applied potential field to navigate the vehicles to their own goals. Issues of optimal

Manuscript received March 15, 2011. Anugrah K. Pamosoaji is with the School of Mechanical Engineering, Pusan National University, Busan, Korea (e-mail: nugie161@ pusan.ac.kr). Keum-Shik Hong is with the Department of Cogno-Mechatronics Engineering, Pusan National University, Busan, Korea (phone: +82-51-510-2454; fax: +82-51-514-0685; e-mail: [email protected]).

c 2011 IEEE 978-1-61284-250-9/11/$26.00

path planning for single vehicle systems were investigated in [11], and [12]. The work in [12] used a class of spline curves as the planned path and considered acceleration limits in generating tangential velocity along the path. A result in a robust approach for task assignment problem in a UAV case study was reported by [13]. The research studied a path planning to connect two configurations (position and orientation) by using at most two different-signed curvatures. In [11], a trapezoidal-velocity constraint was applied to find minimum-time paths and trajectories. The problem of minimum-time task assignment for a multiple-vehicle system in a complex workspace was introduced by [14]-[18]. In [14], the collision-free paths of the vehicles were generated by using Genetic Algorithm (GA). However, this solution did not cope with vehicle’s orientation. In this paper, we propose an algorithm to figure out the solution of a collision-free path and trajectory planning problem. We use a class of Bezier curves as a basic form of the paths, and the Particle Swarm Optimization (PSO) technique to figure out the path that can be traversed in minimum time. The contributions of this paper are described as follows. A procedure of planning minimum-time path and trajectory subject to linear velocity, angular velocity and angular acceleration is introduced. Procedure of finding collision-free trajectories for multiple-vehicle systems is proposed as well. The generation of velocity profiles only uses two linear velocity candidates, which is simpler than the solution of [19]-[24]. The rest of this paper is organized as follows. Section II describes the problem of time-optimal task assignment in a multiple-vehicle system. Section III discusses the proposed method, including the explanation of the 3-degree Bezier curves and Particle Swarm Optimization (PSO) technique to figure out the minimum-time path. Section IV shows the simulation results. Finally, Section V offers conclusions and directions for future works. II. PROBLEM DESCRIPTION In this paper, a multiple-vehicle system in a complex workspace is investigated. Suppose that we have three vehicles, i.e., vehicle 1, vehicle 2, and vehicle 3, whose positions are respectively in station 1, station 2, and station 3. We assumed that the destinations of all forklifts have been provided. Given a set of N depots and a group of M stations, where N and M are the number of depots and the number of stations, respectively. Each depot, by assumption,

67

contains a load assigned to only one vehicle. Suppose that there exists a group of L vehicles assigned to execute load-picking task to the respective desired depots, as shown in Fig. 1. For further discussion, we express the vehicle’s configuration as ( xib , yib ,T ib ) . Moreover, the depots can be considered as static obstacles with configurations denoted as ( x dj , y dj ,T dj ) . In this paper, the orientations of the depots are considered as entrance orientations at which the associated assigned vehicles approach to. Therefore, if the i-th vehicle is assigned to serve the j-th depot, its configuration has to match with the desired configuration at the depot. The problem can be formulated as follows. Find collision-free paths and trajectories for all vehicles such that the following global cost function is minimized. f

min^ k tTf,i ` k nd min

^

`, i z j ,

(1)

Fig. 1. A multiple-vehicle system scenario. Each vehicle has to reach an assigned depot without any collision among them.

0 d | at | d at ,

(2)

0 d | ar | d ar , 0d|v|dv ,

(3) (4)

the values of u at the initial and the goal points, respectively, and Ci3 is a 3-degree Bernstein function [4].

Ob, i Ob, j

subject to

accomplishing time of the i-th vehicle; ar and ar are the

The main property of such class of curves is that the lowest and highest indexed control points are positioned at the start and goal points, respectively. Another property is that the vector from ( X 0 , Y0 ) to ( X 1 , Y1 ) and the vector

radial and maximum radial accelerations of each vehicle; at and at are the tangential and maximum tangential

from ( X 2 , Y2 ) to ( X 3 , Y3 ) make inclination angles of T 0 and T g with respect to the global x-axis, respectively. The

accelerations of each vehicle; and Ob,i represents the center

properties imply that ( X 1 , Y1 ) and ( X 2 , Y2 ) can be

of the i-th vehicle, respectively. It can be seen that the main problem is to search paths and trajectories of all the vehicles with minimum accomplishing time and minimum distance for any time t t 0 .

determined arbitrarily. However, since there are another given constraints, i.e., the fixed initial and goal orientations, i.e., T 0 and T f , ( X 1 , Y1 ) and ( X 2 , Y2 ) can be represented

III. THE PROPOSED METHOD

( X 1 , Y1 ) to the initial point and ( X 2 , Y2 ) to the goal point,

where k t and k nd are positive constants; Tf,i is the

A. Path Planning Using 3-Degree Bezier Curves In this work, 3-degree Bezier curves are utilized as path model. Bezier curves are curves which are characterized by a set of control points {( X 0 , Y0 ),..., ( X n , Yn )} , where n is the degree of the curves. In this paper, we use 3-degree Bezier curves to represent paths. The points ( X 0 , Y0 ) and ( X 3 , Y3 ) are dedicated to indicate initial and goal point, respectively. In this work, 3-degree Bezier curves are utilized as path model. Bezier curves are curves which are characterized The advantage of using the class of curves is that it copes with not only the positions of the initial and final points, but also the orientation of the vehicle at those points. A 3-degree Bezier curve is a set of points which satisfy the following formula. 3 ªXi º 3 i ªx º C u (1 u ) 3i , 0 d u d 1 , (5) « y» i 0 «Y » i ¬ ¼ ¬ i ¼

¦

by the d1 and d 2 , which are the distance between respectively. The control points can be expressed in terms of d1 and

d 2 and T 0 and T f as follows. ( X 1 , Y1 ) ( X 0 d1 cosT 0 , Y0 d1 sin T 0 ) , ( X 2 , Y2 )

( X 3 d 2 cos T f , Y3 d 2 sin T f ) .

(7)

In this paper, we simplify the expression of path in (5) as x(u ) a 0 a1u a 2 u 2 a3u 3 , (8)

y (u ) where

b0 b1u b2 u 2 b3u 3 ,

(a0 , b0 ) ( X 0 , Y0 ) ,

(9)

(10)

(a1 , b1 )

(3 X 0 3 X 1 , 3Y0 3Y1 ) ,

(11)

(a 2 , b2 )

(3 X 0 6 X 1 3 X 2 , 3Y0 6Y1 3Y2 ) ,

(12)

(a 3 , b3 ) ( X 0 3 X 1 3 X 2 X 3 ,

Y0 3Y1 3Y2 Y3 ) .

where u is a curve parameter whose values 0 and 1 indicate 68

(6)

2011 IEEE 5th International Conference on Robotics, Automation and Mechatronics (RAM)

(13)

points. It implies that if there is no obstacle intercepting the area, the vehicle’s path is guaranteed to be collision-free. As shown in Fig. 2, two via-points P and Q are placed as

C P

P1

Mb Ob,i

P2 r' O j rj

Mg

A2

connectors for four Bezier-paths which perform two path alternatives. Let r ' be the distance of O j and one of the via

rb

points , i.e., P or Q . P1 is defined as the intersection point

E Q

of POb,i and the boundary of the j-th obstacle, and P2 is the

D

intersection between PG and the obstacle. Let LO G {( x, y ) | ( x, y ) OOb,i (1 O )G, 0 d O d 1} , (16)

A1 Fig. 2. Alternative paths in the presence of an obstacle.

b,i

LA A

{( x, y ) | ( x, y )

1 2

OA1 (1 O ) A2 , 0 d O d 1} ,

(17)

and CO j be the closed set of the j-th obstacle. The decision to make new via-points is established if the following condition is satisfied. LO G CO j {} , (18) b, i

L A A CO j {} .

(19)

1 2

In the case of new via-points generation phase, the via-point candidates P and Q (see Fig. 2) are generated by Fig. 3. An intersection point of two paths. The planned trajectory has to guarantee both vehicles not to collide to each other at the intersection point.

solving the solution of an intersection point of the extension of lines Ob,i P1 and OG P2 . P1 and P2 are the tangents to

We also use the derivative of (8) and (9) with respect to u as x' (u ) a1 2a 2 u 3a3u 2 , (14)

the boundary of the obstacle passing through the center of the vehicle and the goal point G , respectively. We assume that the environment is not extremely narrow. Therefore, the selected via-point is the one with closest distance to the goal point G . The obstacle-polygons intersection checking can be used to figure out the maximum distance of the free control points A0 and A3 to the initial and goal points, respectively. The

y ' (u )

b1 2b2 u 3b3u 2 ,

(15)

where u [0,1] is a path parameter .

B. Collision-Free Path Analysis In the presence of an a priori known obstacle, a procedure of placing a via-point is proposed. A property of Bezier path is used, that is, if the enveloping polygon of the path does not have intersection with any obstacle, the path is collision-free. As shown in Fig. 2, the obstacle is modeled as its environment, i.e., a circle with radius r j rb , where r j and rb represents the radius of the obstacle and the radius

of the vehicle, respectively. According to the property, a path from the center of the j -th obstacle, i.e., O j , to the goal point, i.e., G , generated by only used 1 Bezier path cannot guarantee the path to be collision-free. At least a combination of two Bezier paths is needed to solve the problem. We need to construct paths between any two vehicles such that their distance is as far as possible. By such the way, any collision of any two vehicles, especially at the intersection point of their paths, can be avoided, as shown in Fig. 3. According to a property of three-degree Bezier curves, that is, the entire parts of the path are interiors of the area of polygonal area bounded by lines connecting its control

combination of (d1,i , d 2,i ) can be obtained by the following steps. The boundary of the obstacle is expressed as ( x x j ) 2 ( y y j ) 2 (r j rb ) 2 .

(20)

By substituting the following convex equation, O A1 (1 O ) A2 ( x, y ) T ,

(21)

to (20), we obtain

O2 Ob,i G

2

2O (( x A2 x j )( x A1 x A2 )

( y A2 y j )( y A1 y A2 )) x 2j y 2j x A22 y A22

2( x j x A2 y j y A2 ) (r j rb ) 2

0.

(22)

By solving (22) and set A1 and A2 such that (22) has only one distinct root, the optimal combination of (d1,i , d 2,i ) can

be obtained. C. Path and Trajectory Planning For path and trajectory planning, the accomplishing time


69

shown in Fig. 4. D. Particle Swarm Optimization (PSO) Since the searching of minimum-length path is a time consuming process, and in 3-degree Bezier curve, the free parameters that can be tuned are d1 and d 2 , we use the Particle Swarm Optimization (PSO) technique to search the minimum-length path. A particle is modeled as a pair of the distances, i.e., (d1 , d 2 ) , and the dynamics is modeled as

Fig. 4. Linear velocity profile of a vehicle.

ª'd1,i (k 1) º « » «¬'d 2,i (k 1)»¼

Tf,i is set to be free. Moreover, the function which maps the

domain of the parameter u to the domain of time t is unknown as well. The transformation from u to t is established by the following steps. First, given a set of control points of a path, the length of the paths is calculated using linear approximation technique. The length of the i-th path, denoted by Li is expressed as follows. Li

Ns

¦ ³

t k 't

k 0 tk

v(W ) dW ,

(20)

where N s is the number of segments of the path. To deal with the maximum tangential velocity, normal and tangential acceleration constraints, we have to analyze the curvature of the path as x' (u ) y ' ' (u ) y ' (u ) x' ' (u ) K (u ) . (21) [( x' (u )) 2 ( y ' (u )) 2 ]3 / 2 The curvature (21) is used to determine the linear velocity v such that the radial acceleration a r , which is formulated by

the following formula, a r v 2 (t ) / | K (u ) | ,

(22)

and the tangential acceleration at , which is given by at

dv(t ) / dt ,

(23)

does not violate a r and at , respectively. To solve this problem, we use the relationship model of proposed by [6] as follows. at2 at2

a r2 ar2

1.

a r and at

(24)

Two kind of linear velocities are considered. The first is linear velocity considering radial acceleration constraint, and the second is the one considering tangential acceleration constraint. As shown by (22), the velocity which considers radial acceleration constraint only depends on the curvature K. In the other side, the velocity which considers tangential acceleration constraint is independent of the curvature K. The resultant linear velocity, as the work in [3], is the minimum value between both the velocities at any time t, as 70

ªd1,i (k ) º ª'd1,i (k 1) º », « »« ¬«d 2,i (k )¼» ¬«'d 2,i (k 1)¼»

ªd1,i (k 1) º « » ¬«d 2,i (k 1)¼»

(25)

ªd1best º ª'd1,i (k ) º ,i ( k ) k c0 « » c1r1,i « best » «d 2,i (k )» «¬'d 2,i (k )»¼ ¬ ¼

swarm ªd1best (k )º ªd1,i (k ) º ,i »« c 2 r2k,i « », swarm d 2,i (k )¼» «d 2best » « k ( ) ¬ ¬ ,i ¼

(26)

best T where [d1best ,i ( k ), d 2,i ( k )] is the previously best position

of

the

i-th

swarm [d1,best (k ), i

particle

until

swarm d 2,best (k )]T i

the

k-th

iteration;

is the best position in the

entire swarm at the k-th iteration ; ('d1,i (k ), 'd 2,i (k )) is the rate of the i-th particle position change at the k-th iteration; c 0 , c1 and c2 , are the inertia scaling factor, the constant cognitive scaling factor and social scaling factor; and r1k,i and r2k,i are the random numbers uniformly distributed in [0, 1] , respectively. In our PSO technique, each particle will evaluate the cost function described in (1) at its current position. A limitation of (d1,i , d 2,i ) is applied to prevent long paths. In addition, the domain of (1) is segmented into 2 classes. The first class is the domain whose generated paths contain at least two points from them with smaller distance than the minimum distance required. This class of area is called “restricted area”. Otherwise, it is called “safe area”. A switch procedure is applied to prevent the searching process being trapped to a restricted area on the domain, i.e., res safe and k nd as the value (d1,i , d 2,i ) space. Let us define k nd of k nd for a particle which searches in the restricted area and the safe area on the (d1,i , d 2,i ) space, respectively. res safe ! k nd . By switching the Moreover, we determine that k nd

value of k nd , the directions of the particles’ motion are changed once a particle finds a safe area. IV. SIMULATION RESULTS The simulation results are presented in this section. We simulated a scenario of task assignment described in Fig. 1


Fig. 5. The generated velocity profiles of the three vehicles. Fig. 7. Distances of any two vehicles.

Fig. 6. The acceleration profiles of the three vehicles.

to show the performance of the proposed methods. We apply the following constants: ar 2 m / s 2 , at 2 m / s 2 , and v

2 m / s . The initial conditions for all vehicles were

described as follows. For the 1st vehicle, the initial position was at (0.5 m, 0 m) and the initial orientation was 0 rad. The 2nd vehicle was initially positioned at (1.2 m, 1.5 m) and was oriented at S / 2 rad. The 3rd vehicle was initially positioned at (1.5 m, 2 m) and the orientation was at S / 2 rad. The goals to be achieved by all vehicles were described as follows. The goal position for the 1st vehicle was at (2.0 m, 2.0 m) and the entry orientation was at 0 rad. The goal position for the 2nd vehicle was at (2.0 m, 0.5 m) and the entry orientation was at S rad. The goal position for the 3rd vehicle was at (0.7 m, 1.5 m) and the entry orientation was at S rad. In order to search the best path and trajectories, we apply a Particle Swarm Optimization (PSO) algorithm with some parameters described as follows. The constant inertial scaling factor, cognitive scaling factor and social scaling factor were set as c 0 1 , c1 2 and c 2 2 , and r1k,i and r2k,i were set as random in the range of [0, 1]. The PSO optimizer then searches the minimum-time path to reach the waypoint. If there is no significant obstacle to

Fig. 8. Generated trajectories of the three vehicles.

avoid, the planner generates path to the goal. The number of particles used is 30 and the number of maximum allowed res safe and k nd are set to search iteration is 50. The values of k nd be 1000 and 10, respectively, to make a big differences between restricted and safe areas. The resulted velocity profiles of the three vehicles are shown in Fig. 5. Vehicle 1, 2, and 3 started at zero initial velocities and reached their goals in 8.2 s, 7.7 s, and 5.1 s, respectively. The simulation successfully kept the velocities of the three vehicles less than or equal to the maximum allowable velocities. Moreover, the velocities were reduced to the required velocities at the goal points without violate the velocity and acceleration constraints (2)-(4). The acceleration profiles for the three vehicles are shown in Fig. 6. The planner successfully kept the absolute accelerations less than 2 m/s2. In addition, the planner successfully satisfied the required goal velocities of the three vehicles, which were set as zeros. The simulation result in Fig. 7 shows the distance profiles of any two vehicles. It can be shown that the distances of any two vehicles were more than 0.38 m to any remaining ones. Fig. 8 shows the trajectories of the three vehicles. It can be shown that the


71

path of the 3rd vehicle is intersected to the path of the 1st vehicle. Even though the paths are intersected, the associated vehicles did not collide to each other.

[7]

[8]

V. CONCLUSION An algorithm to generate collision-free paths and trajectories in a multiple-vehicle system is presented. The algorithm used a class of 3-degree Bezier curve as the path to connect the initial configuration to the goal configuration. The control point parameters of the curve are used in the Particle Swarm Optimization (PSO) algorithm to figure out the parameters that cause the path to traverse in minimum time by each vehicle without any collision among them. The proposed method has an advantage, that is, the effort needed to calculate the resultant linear velocity is reduced since we used only two classes of linear velocities. The first class is the linear velocity which considers radial acceleration constraint, and the second one is the linear velocity which considers tangential acceleration constraint. From the presented simulations, it is shown that the proposed method successfully kept the velocities and accelerations of the vehicles obeying the velocity and acceleration constraints. In the future, the work will be continued to handle some issues related to re-planning and obstacle avoidance in multiple-vehicle systems.

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

ACKNOWLEDGEMENT

This work was supported by the World Class University program through the National Research Foundation of Korea funded by The Ministry of Education, Science and Technology, Korea (grant no. R31-2008-000-20004-0) and by the Regional Research Universities Program (Research Center for Logistics Information Technology, LIT), granted by the National Research Foundation of Korea under the Ministry of Education, Science and Technology, Korea.

[18]

[19]

[20]

[21]

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