Continuity Smooth Path Planning Using Cubic Polynomial Interpolation with Membership Function

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1 J Electr Eng Technol Vol., No.?: 74-?, 5 ISSN(Print) 975- ISSN(Online) Continity Smooth Path Planning Using Cbic Polomial Interpolation with Membership Fnction Seong-Ryong Chang* and Uk-Yol Hh Abstract Path planning algorithms are sed to allow mobile robots to avoid obstacles and find ways from a start point to a target point. The general path planning algorithm focsed on constrcting of collision free path. However, a high continos path can make smooth and efficiently movements. To improve the continity of the path, the searched waypoints are connected by the proposed polomial interpolation. The existing polomial interpolation methods connect two points. In this paper, point grops are created with three points. The point grops have each polomial. Polomials are made by matching the differential vales and simple matrix calclation. Membership fnctions are sed to distribte the weight of each polomial at overlapped sections. As a reslt, the path has continity. In addition, the proposed method can analyze path nmerically to obtain crvatre and heading angle. Moreover, it does not reqire complex calclation and databases to save the created path. Keywords: Continos-crvatre path, Geometric continity, Interpolation, Path planning, Cbic polomial, Mobile robot, Robot motion, Smooth path, Spline interpolation, Vehicles navigation. Introdction The aim of path planning for mobile robots and vehicles is finding a path from the start to the target point with avoiding obstacles. The potential field method [-3], A* and D* algorithm [4-6] and RRT algorithm [7, 8] are proposed for searching path. The prposes of global path planning are the avoidance of stationary obstacles and solving mazes. On the other hand, considering the continity, the path can be improved to se real movement, stable motion, smooth movement and optimal path. In addition, it can be sed to analyze a planned path before driving. The continity of the path is defined as,, in geometry [9]. continity is jst contacted at each point by a straight line. The continity path is eqal to the primary differential vale at each point. The path will crve from continos. The continity path has an identical second-order differential vale at each point. This means the same crvatre at each point. In particlar, the continity means continity of the th order differential vale at each point []. In order to se smooth path in the real environment, the continity shold be moreover continity. It means the continos path has a continos velocity fnctions and acceleration fnctions. So this path has continoscrvatre. As a reslt, if the mobile robot and vehicles Corresponding Athor: Dept. of Electrical Engineering, Inha University, Korea. (yhh@inha.ac.kr) * Dept. of Electrical Engineering, Inha University, Korea. (themir@ naver.com) Received: Jly 4, 3; Accepted: October 7, 4 follow the smooth path, the velocity is not changed abrptly in the path. Ths, they can be moved with the continos velocity and acceleration withot stop motion. This movement is efficient with time, energy and damics. Kanayama and Hartman docmented a definition of path crvatre and the derivative path crvatre []. Fraidchard and Ahactzin sggested the continoscrvatre path planning for cars []. Lamirax and Lammond proposed steering method for a car-like vehicle providing smooth paths [3]. Kito et al. described the path planning with maximm crvatre, crvatre derivative and crvatre continity [4]. Yang and Skkarieh reported the continos-crvatre path-smoothing algorithm with efficient and analytical [5]. Villagra et al. indicated the efficiency and comfort of the smooth path and speed planning for the vehicle transportation system [6]. The obtained path from RRT connect algorithm [7, 8] consists of (the initial configration), waypoints, (the goal configration) and a line connecting the points in consective order [7, 7, 8]. A line connecting path is continity. RRT connect algorithm [7, 8] can only solve mazes and stationary obstacle avoidance problems. The paths reqired to improve the continity to apply at the real environment. Generally, in order to improve the continity of path, the path shold be reconstrcted sing an interpolation method considering the continity. The interpolation method estimates at with only given and [9, ]. Polomial interpolation and spline interpolation are sed in general. Polomial interpolation aims to find a polomial inclding whole points [9]. The order of the polomial is increased by the nmber of points. A high 74

2 Seong-Ryong Chang and Uk-Yol Hh order polomial occrs Rnge s phenomenon []. This type of polomial is nsitable for applying a large nmber of points. Spline interpolation prodces a polomial between points by the differential vale at the points [9]. Cbic spline interpolation is widely sed and satisfies the continos []. Polomial interpolation and spline interpolation involve nmerical analysis to determine the fnction of connecting characteristic points that satisfy the condition. In other words, they can be sed nder the condition of an always increasing [9]. Therefore, normally interpolation methods are nsitable for mobile robots and vehicles with complex movement. Therefore modified interpolation method has applied a mobile robots and vehicles. Piazzi et al. attempted to smooth path planning sing the qantic -splines [], the -splines [3, 4] and the -splines [5]. However, methods of Piazzi have complex calclations sch as high order polomial and trigonometric fnctions which reqire the high performance hardware. Yang et al. proposed a continos-crvatre path-smoothing algorithm sing cbic Bézier crves [6]. Bézier crve is a poplar method to create crves for compter graphics [9]. Many smooth path stdies have sed this method, bt the planned path is not visited entire waypoints becase this method reqires control points to decide the crvatre. So some waypoints are sed to the control points or reqire an algorithm to decide the control points. In the path planning method, the planned path mst visit every waypoint. Therefore, a method sing Bézier crves is not sitable to apply the smooth path planning. On the movement of D plane, is not changed in horizontal movement and shows no variations in vertical movement on the D map. The parametric method is sed in the present paper. The parametric method expresses and by. The parameter is the accmlation vale of the distance between waypoints. This parameter is always a positive, increasing fnction and does not reqire the condition. The polomial of and can be obtained sing parameter in any movement. Spline interpolation makes a polomial containing two points sing point vales and differential vales. In addition, it makes another polomial of the next two points in the same method. In this paper, the point grop is made by three points. The qadratic polomial and cbic polomial inclde three points obtained by point vales and differential vales. The next grop is made by two previos points and one new point. The qadratic polomial and cbic polomial abot the new grop is derived and extended to the next grops. The neighboring polomial occrs in the overlapped section. Membership fnctions distribte the weight of each polomial to make an improved continity path in an overlapped section by parameter. In addition, the planned smooth path cold visit every waypoint nlike any other smooth path planning methods. The proposed method provides the difference vales. The crvatre and heading angle can be calclated sing the differential vales of and. These data can be applied to an analysis of the planned path and real movement. In addition, a controller and control algorithm se these data. The crvatre is concerned with the damics. In other words, the robots and vehicles have the maximm velocity abot crvatre. If the controller and control algorithm se this data, the ser can check a driving time with stable movement and the maximm velocity before real driving. This paper is organized as follows: section describes a method for obtaining a qadratic polomial and cbic polomial with three points. The difference between the proposed method and the general spline interpolation method is explained. Section 3 proposes a method for making a rotated polomial sing the parametric method. In section 4, the distribtion of weight is explained in a polomial overlapped section. In section 5, the continity, crvatre and heading angle fnctions are described. Section 6, the proposed method is simlated to find the alternation of the primary differential vales, second-order difference vales. These vales prove that the proposed method can make a continity path. In addition, the crvatre and heading angle are obtained by the proposed method. These provide a data for real driving and path analysis. Spline Interpolation and Polomial Interpolation. Spline interpolation Spline interpolation connects each point by a polomial. A cbic polomial interpolation is sed in general. The method for obtaining the polomial involves matching the primary differential vale and second-order differential vale at each point [9, ]. The planned path is the matched crvatre at each point that satisfies the continity. The continity reqires the matching of primary and second-order differential vales at each point. Matching of the differential vale is difficlt in a qadratic polomial becase the second-order differential vale is constant in a qadratic polomial. Therefore, a qadratic polomial can only se the continity graph. Ths, the minimm order for satisfying over continity is three, which means a cbic polomial. This is why a cbic polomial sed generally. The widely sed spline interpolation reqires a change in. shold be increased. () 743

3 Continity Smooth Path Planning Using Cbic Polomial Interpolation with Membership Fnction On the other hand, the paths of the robots or vehicles are not satisfied (). They have a variety of movements. or is not variable in vertical or horizontal movement. Therefore, general spline interpolation is nsitable for expressing the path trajectory.. Qadratic polomial interpolation The qadratic polomial interpolation is a method to connect three points continosly. There are three points like as, and. The polomial connecting these three points can be expressed as eqation (). ( ) P : x, y R n n n y= ax + bx+ c () The primary and second-order differential vales of qadratic polomial are (3) and (4). dy ax b dx = + (3) d y = a (4) dx The crvatre k is obtained sing eqation (5) by (3) and (4). k = a 3 ( + ( ax+ b) ) Fig.. Qadratic polomial with 3 points ( and ) (5) The graph of the qadratic polomial has a concave or convex shape (Fig. ). To se the qadratic polomial interpolation, () shold be satisfied. If () is not satisfied, the graph can be expressed, as shown in Fig.. In Fig., the line is a rotation polomial graph of connecting P 5, P 4 and P 3 in serial order, and the dot line is the rotation polomial graph with P, P and P 6 in consective order..3 Cbic polomial interpolation The cbic polomial is the third-order fnction. The basic form can be expressed as (6). ( ) P : x, y R n n n 3 y= ax + bx + cx+ d (6) The second-order differential fnction is needed to check the G continity. The primary and second-order differential fnctions are (7) and (8). dy 3ax bx c dx = + + (7) d y = 6ax+ b (8) dx The crvatre k (9) can be calclated by (7) and (8). k = 6ax+ b ( ax bx c) Fig. 3 shows the basic graph of cbic polomial. More than for conditions are needed to se a cbic polomial interpolation by (6). On the other hand, a normal polomial graph is nconformable to apply the path of a mobile robots or vehicles. The path shold be shortest and realizable. In Fig. 3, there are convex and detor paths when connecting and. In particlar, the path shold be as short and realizable as possible with continity. Therefore, if done correctly, the general cbic (9) Fig.. Graph of the rotated qadratic polomial ( and ) Fig. 3. Graph of rotated cbic polomial 744

4 Seong-Ryong Chang and Uk-Yol Hh polomial interpolation is nsitable for application to mobile robots or vehicles. The proposed method is the rotated form of the cbic polomial. Fig. 4 is the rotated graph of cbic polomial. The graph can be expressed nsatisfied ()..4 Rotated polomial graph The center point is set to obtain a polomial of three points connected in serial order. Three points exist, and. is the center point. If the three points connecting the graph are rotated centered on, two cases exist. The first case is a straight line on and. In this case, the crvatre is nlimited and is the simplest way of movement. The second case is the crve on and. In this case, the graph will be convex or concave. The optimal smooth path is a convex or a concave graph by maintaining continity. A qadratic polomial is a polomial inclding three points with continity. If it is rotated, the center point will approach the vertex. () are the general rotation fnctions. is the rotation angle. () () are impossible to obtain a graph that connects each point in consective order. In addition, it is difficlt to express with a qadratic polomial or cbic polomial, sch as () and (). () and () shold be sed to find,,, and. On the other hand, trigonometric fnctions hinder soltions from being fond. () () Fig. 5 shows three cases, where, and are connected. The seqential path cannot be derived by () or (). Therefore, another method is needed to connect the points seqentially and rotated graph. This method shold be able to apply a qadratic polomial and cbic polomial. 3. Parametric Qadratic Polomial 3. Qadratic polomial with parametric method In this section, the qadratic polomial and cbic polomial are expressed by parameter. If the three points are existed, the center point shold be approached the vertex of a graph to rotate the polomial and the points shold be connected in serial order. To rotate the polomial, and are separated by parameter. The parameter is the accmlation vale of the straight line distance between points. As a reslt, parameter is always a positive and increasing vale and the graph has rotated shape. can express to (3) with three points. = ( ) ( ) = + + x x y y ( ) ( ) = + x x + y y (3) () can change to (4) and (5) sing parameter. (4) (5),,,, and can be calclated by (6) and (7). a x x( ) bx = x( ) c x ( 3) 3 3 x a y y( ) by = y( ) c y ( 3) 3 3 y (6) (7) Fig. 4. Cases of three points connected qadratic polomial graph Fig. 5. Graph of the cbic polomial with three points 745

5 Continity Smooth Path Planning Using Cbic Polomial Interpolation with Membership Fnction Primary and second-order differential vale of (6) and (7) are expressed as (8) and (9). dx a x b x d = +, d x a x d = (8) dy a y b y d = +, d y a y d = (9) (6) and (7) make a polomial with three points. The three points are then applied to (6) and (7) in more than three points path., and are sed to obtain th polomial. () th polomial is made by, and. (3) is changed to the general fnction as (). (6) and (7) can express to () and () with () th points. m = ( ) ( ), ( n) = x x + y y n n n n n n= a x n n n x n bx = ( ) n n n x n cx n n x + + n+ n+ ( ) () () ( ) y n n y y = n n n n y n+ ( n ) ( ) ( ) a n b y () c + n+ n+ ( n m ) th points are needed in a () th qadratic polomial. Every polomial contacts the whole points and each polomial had continity inclding three points, and. For example, two polomials are the existence with for points. Each polomial has the overlapped section (Fig. 6). 3. Cbic polomial with parametric method Parameter consists of qadratic polomial obtained Fig. 6. Two qadratic polomial graph inclding for points by (6) and (7). The qadratic polomial can be expressed sing parameter. The polomial is in contact with three points in serial order and has a rotated graph form. Each polomial has continity. On the other hand, it is nsre in more than three points. To make continity, a second-order differential vale is matched at each point. Therefore a higher order polomial is needed. (4) and (5) can be expanded to a cbic polomial. (3) and (4) are the basic form of cbic polomial sing parameter. (3) (4) (3) and (4) will be (5) and (6) in more than three points case. is -axis polomial inclding, and. is -axis polomial inclding, and. (5) (6) ( (5) and (6) are cbic polomials inclding,, in consective order.,, and of (5) and (6) cannot be obtained by three points becase another condition is needed to match the second-order differential vale at each point. (7) and (8) are added to the conditions. ( ) ( ) d P d P = (7) d d ( ) ( ) d P d P = (8) d d (7) and (8) apply to (5) and (6). (9) and (3) are then obtained. 6 6 (9) 6 6 (3) (5), (6), (7) and (8) can be combined into matrix (3) ( (3) 746

6 Seong-Ryong Chang and Uk-Yol Hh (3) can change to (3) via an inverse matrix ( (3),,, and can be obtained by. On the other hand, they do not exist if ). Therefore, polomial of mst se a qadratic polomial by () and (). The range of vales is changed, and (3) can be expressed as (33).,, 6 6 ( (33) As a reslt, the and polomials are connecting, and by a qadratic polomial. In addition, is expanded to,,, to obtain a cbic polomial with matching second-order differential vale at each point. The parametric method provides a variation of and. They are connected with parameter. This method is sefl to express complex movements. 4. Polomial Interpolation with Membership Fnction Polomials of th points are consist of qadratic polomials and cbic polomials with a matching second-order differential vale. On the other hand, there are overlapped sections between th polomial and () th polomial. Membership fnctions distribte the weight of the polomial in overlapped sections. They are fnctions of the parameter. is a grop of and is a grop of. The smooth path is denoted as (34),, (34) polomial are adjsted by the distance between the points in overlapped sections. In particlar, the th polomial has the largest weight at the th point and decreases as it approaches the () th point. The weight of the () th polomial is zero at th point, bt it increases by approaching the ( ) th point. (35) and (36) are the membership fnctions expressing parameter., (35), (36) In Fig. 7, is zero at and at. At the same time, is at and zero at. (37) is (34) combined (35) and (36). Fig. 7. Graph of the membership fnctions (37) The general form can be expressed as (38).,:, (38) The weight of the th polomial and ( ) th and effect to the first section ( to ) 747

7 Continity Smooth Path Planning Using Cbic Polomial Interpolation with Membership Fnction (38) can change to (4) and (4) to find differential vales. Fig. 8. Membership fnctions and polomials and and make the path in the last section ( to ). On the other hand, memberships fnction and polomials are combined in the other sections. Fig. 8 is the weight variation of the polomial by parameter. The membership fnctions change the weight of each polomial by a parameter in the overlapped section. As a reslt, separated smooth paths are combined to one smooth path that maintain the continity. 5. Continity, Crvatre and Heading Angle The proposed method can calclate the crvatre and heading angle. This method can be sed to analyze the planned path and backgrond data of real movements. The primary differential vale and second-order differential vale are needed to obtain the crvatre. They can be expressed as (8) and (9) in the qadratic polomial. (39) are differential vales of, sing (8) and (9) in case of the qadratic polomial. dp dp d d ( ) ( ) d P d d P d = a + b = a + b ( ) ( ) = a (39) = a (4) are the primary and second-order differential vales of the cbic polomial by (9) and (3). dp dp d d ( ) = 3a + b + c ( ) d P d P = 3a + b + c d d ( ) ( ) = 6a + b (4) = 6a + b,:,: (4),:,: (4) and are expressed as the qadratic polomial connecting, and. The cbic polomials are shown in other sections. This can maintain the continity by (3) and (3). The crvatre can be obtained sing (4) and (4). k = X Y Y X ( ) 3 X + Y The heading angle can be calclated sing (44). ( ) ( ) dy θ = tan = tan x x dx n n (43) (44) The differential vales are sed to analysis of the 748

8 Seong-Ryong Chang and Uk-Yol Hh planned path for the velocity and acceleration control. The crvatre concerns the maximm velocity. In addition, the heading angle indicates the direction of robots and vehicles, and they se to obtain an anglar speed. 4 vs. x 8 6. Simlation In this section, two simlations are shown. The first simlation is a comparison with a Qintic -splines []. This spline method connects each waypoint sing the fifthorder fnctions and trigonometric fnctions. The proposed method created a smooth path sing given waypoints. The second simlation shows the smooth path from an RRT algorithm searched path. This path has continity. The proposed method improved the continity to continity. 6. Case of [] Qintic -splines for the iterative steering of vision-based atonomos vehicles [] reported motion planning sing the qintic - spline. The continity path is shown with the five points sing the proposed method. Table is the given waypoints of []. Table. Points of [] x y Fig.. vs. vs. y 5 5 Fig.. vs. vs. dx The proposed method can be applied to []. Fig. 9 shows the reslts by (38). In [], paths between each waypoint consist of qntic fnction and trigonometric fnctions. However, the proposed method can create the path by sing the cbic polomial. Figs. and show the variation of and by. These are continos shape. To check continity, the differential vales shold be observed. Figs. and 3 show the primary differential vales of and by (4). The graph has continos shape. dy dx Fig.. vs. vs. dy Fig. 3. vs..4 vs. ddx. ddx Fig. 4. vs. Fig. 9. [] (black line) and the cbic polomial interpolation with a membership fnction (red line) This means that the path has continity at least. Therefore, the planned path is continos on the velocity. To confirm continity, the second-order differential vales are checked. Figs. 4 and 5 show the second-order 749

9 x Continity Smooth Path Planning Using Cbic Polomial Interpolation with Membership Fnction.4 vs. ddy. ddy Fig. 5. vs. vs. Crvatre k Crvatre.. Fig. 8. The path from RRT algorithm [8], an example of how to solve a qery with the roadmap 5 5 Fig. 6. vs. crvatre vs. Heading angle theta 5 theta differential vales by (4). Figs. 4 and 5 have not open sections with any points. These are show that path is continity. These graphs indicate that the planned path has acceleration continity. Fig. 6 shows the variation of crvatre by (43). This graph is continos and closed in whole sections. These are featres of the continity path. This data can be applied to control of the velocity and acceleration for real driving. Fig. 7 shows the variation of heading angle sing (4) and (44). can be important data of the driving simlation, real driving and analyze of planned path. Figs. -5 show that the path is the continity. Fig. 9 compares path of [] with the proposed path. The figre shows some difference between the path of [] and the proposed path in to section. This is becase path of [] has an internal heading angle at, bt the proposed path has no internal heading angle. The internal heading angle is the constraint of simlation. On the other hands, it is not needed in real driving. The crvatre graph (Fig. 6) is continos, and Fig. 4 and Fig. 5 are not open in whole sections. In addition, the variation of heading angle can be obtained. As a reslt, proposed path has continity with checking variation of differential vales and crvatre. 6. Case of the roadmap Fig. 7. vs. heading angle This simlation is an improving continity. The proposed Fig. 9. The path from RRT algorithm [8] vs. continity path sing proposed method method ses only waypoints. If searching algorithm obtains a collision-free linear path and waypoints, the proposed method can improve to the continity sing searched waypoints. Fig. 8 shows a path from RRT algorithm [7, 8]. This is a simply connected path at each point with continity. The proposed method was applied to Figs. 6 and 8 can be express as Fig. 9. This figre shows the continity smooth path inclding every waypoint. In order to check the continity of the path, the differential vales were obtained. Figs. and show the variation in and by (38) vs. x Fig.. vs. 75

10 y Seong-Ryong Chang and Uk-Yol Hh vs. y Crvatre.. -. vs. Crvatre k Fig.. vs Fig. 6. vs. crvatre.5.5 vs. dx Fig. 7 shows the variation of the heading angle by (44). vs. Heading angle theta dx Fig.. vs. theta vs. dy Fig. 7. vs. heading angle dy Fig. 3. vs. vs. ddx The continity path sing the RRT algorithm [7, 8] change to the continity path by the proposed method. The proposed method cold obtain the change in crvatre and heading angle data. This simlation indicates the combination with the searching algorithm. The searching algorithm cannot create the smooth path. However, it can provide waypoints. The proposed method ses these waypoints and the process of finding a smooth path is simple. ddx ddy Fig. 4. vs. vs. ddy Fig. 5. vs. Figs. -5 show the variation in the primary differential vales and second-order differential vales by (4) and (4). They have not open sections in every point. As a reslt, the proposed path is continity that jst checking these graphs. Fig. 6 shows the change in crvatre by (43). 7. Conclsion Most searching algorithms do not consider the continity of a path. The proposed method aimed to improve the continity. Polomials derived sing point grops consisted of three points. They were extended to other grops and sed to create polomials. The point grops considered the differential vales at the points. Differential vale matching improved the continity, and a parametric method was sed to obtain a polomial with complex movement. The parameter fnctions consisted of a linear distance variable. Membership fnctions were sed to distribte the weight of each polomial s inflence. As a reslt, the continity path was obtained in the whole sections. The proposed method cold analyze the path nmerically. This shows that the crvatre and heading angle can be fond sing differential vales. The crvatre and heading angle are sefl data to design a control algorithm considering damics. The proposed method was verified by two simlations. The first simlation compared with the qintic -spline. A similar reslt was obtained and cold analyze variation 75

11 Continity Smooth Path Planning Using Cbic Polomial Interpolation with Membership Fnction of,, the crvatre and the heading angle. The second simlation prodced a continity path from RRT algorithm path. The RRT algorithm cold obtain only a continity path. On the other hand, with the RRT algorithm, a path cold change to the continity path sing the proposed method. When there were waypoints by the searching algorithms, the continos path between waypoints cold be obtained calclations sing the proposed method. An increase in the volme of data is not necessary and the path can be pdated pon a change in environment. This is becase the proposed method ses a simple matrix form and does not se trigonometric fnctions or high order fnctions which reqire high performance hardware and complex calclations compare with other smooth path planning algorithms. As a reslt, the proposed method can apply to simple system. In addition, the control algorithm cold be stable and simple. The proposed method can be applied to mobile robots, vehicles, game algorithms and compter graphics. The method can be sed in flying robots and aircraft if it can be expanded to 3D space. The method of the expanding points grop can be sed to improve the continity of the path in local path planning. Acknowledgements This stdy was spported by the National Research Fondation of Korea (NRF) grant fnded by the Korea government (MEST) (No.-5564). References [] Y. K. Hwang and N. Ahja, A potential field approach to path planning, Robotics and Atomation, IEEE Transactions on, vol. 8, pp. 3-3, 99. [] S. S. Ge and Y. J. Ci, New potential fnctions for mobile robot path planning, Robotics and Atomation, IEEE Transactions on, vol. 6, pp. 65-6,. [3] Y. Wang and G. S. Chirikjian, A new potential field method for robot path planning, in Robotics and Atomation,. Proceedings. ICRA'. IEEE International Conference on,, pp [4] P. E. Hart, N. J. Nilsson, and B. Raphael, A formal basis for the heristic determination of minimm cost paths, Systems Science and Cybernetics, IEEE Transactions on, vol. 4, pp. -7, 968. [5] A. Stentz, The focssed D* algorithm for real-time replanning, in International Joint Conference on Artificial Intelligence, 995, pp [6] S. Koenig and M. Likhachev, D* Lite, in Proceedings of the national conference on artificial intelligence,, pp [7] J. J. Kffner Jr and S. M. LaValle, RRT-connect: An efficient approach to single-qery path planning, in Robotics and Atomation,. Proceedings. ICRA'. IEEE International Conference on,, pp [8] S. M. LaValle, Planning algorithms: Cambridge niversity press, 6. [9] G. E. Farin, Crves and srfaces for CAGD [electronic resorce]: a practical gide: Morgan Kafmann,. [] B. A. Barsky and A. D. DeRose, Geometric continity of parametric crves, 984. [] Y. Kanayama and B. I. Hartman, Smooth local path planning for atonomos vehicles, in Robotics and Atomation, 989. Proceedings., 989 IEEE International Conference on, 989, pp [] T. Fraichard and J.-M. Ahactzin, Smooth path planning for cars, in Robotics and Atomation,. Proceedings ICRA. IEEE International Conference on,, pp [3] F. Lamirax and J.-P. Lammond, Smooth motion planning for car-like vehicles, Robotics and Atomation, IEEE Transactions on, vol. 7, pp ,. [4] T. Kito, J. Ota, R. Katski, T. Mizta, T. Arai, T. Ueyama, et al., Smooth path planning by sing visibility graph-like method, in Robotics and Atomation, 3. Proceedings. ICRA'3. IEEE International Conference on, 3, pp [5] K. Yang and S. Skkarieh, An analytical continoscrvatre path-smoothing algorithm, Robotics, IEEE Transactions on, vol. 6, pp ,. [6] J. Villagra, V. Milanés, J. Pérez, and J. Godoy, Smooth path and speed planning for an atomated pblic transport vehicle, Robotics and Atonomos Systems, vol. 6, pp. 5-65,. [7] N. A. Melchior and R. Simmons, Particle RRT for path planning with ncertainty, in Robotics and Atomation, 7 IEEE International Conference on, 7, pp [8] H. Choset, K. M. Lch, S. Htchinson, G. Kantor, W. Brgard, L. E. Kavraki, et al., Principles of robot motion: theory, algorithms, and implementations: MIT press, 5. [9] J. Stoer, R. Blirsch, R. Bartels, W. Gatschi, and C. Witzgall, Introdction to nmerical analysis vol. : Springer New York, 993. [] A. Gilat and V. Sbramaniam, Nmerical methods for engineers and scientists: Wiley, 7. [] J. F. Epperson, On the Rnge example, Amer. Math. Monthly, vol. 94, pp , 987. [] A. Piazzi, C. Lo Bianco, M. Bertozzi, A. Fascioli, and A. Broggi, Qintic G -splines for the iterative steering of vision-based atonomos vehicles, Intelligent Transportation Systems, IEEE Transactions on, vol. 3, pp. 7-36,. 75

12 Seong-Ryong Chang and Uk-Yol Hh [3] A. Piazzi, C. Garino Lo Bianco, and M. Romano, η 3 Splines for the Smooth Path Generation of Wheeled Mobile Robots, IEEE TRANSACTIONS ON ROBOTICS, vol. 3, pp , 7. [4] A. Piazzi, C. G. L. Bianco, and M. Romano, Smooth path generation for wheeled mobile robots sing η3- splines, Motion Control, pp ,. [5] A. Piazzi, M. Romano, and C. G. L. Bianco, G3- splines for the path planning of wheeled mobile robots, in Eropean Control Conference, 3. [6] K. Yang, D. Jng, and S. Skkarieh, Continos crvatre path-smoothing algorithm sing cbic B zier spiral crves for non-holonomic robots, Advanced Robotics, vol. 7, pp , 3. Seong-Ryong Chang received his B.S and M.S degree in Electronics from Dankook University in 999 and, and a Ph.D. degree in Electrical Engineering from Inha University of Incheon, Korea in 3. His research interests are path planning, robot motion, mobile robot and embedded system. Uk-Yol Hh received his B.S, M.S and Ph.D degrees in Electrical Engineering from Seol University in 974, 978 and 98. He is crrently a professor with the Department of Electrical Engineering, Inha University of Incheon, Korea. His research interests are in the areas of intelligent control, design of fzzy controller design and mobile robots. 753