Research Article  Open Access
Path Planning with Obstacle Avoidance Based on Normalized RFunctions
Abstract
Existing methods for path planning with obstacle avoidance need to check having the interference between a moving part and an obstacle at iteration and even to calculate their shortest distance in the case of given motion parameters. Besides, the tasks like collisionchecking and minimumdistance calculating themselves are complicated and timeconsuming. Rigorous mathematical analysis might be a practical way for dealing with the abovementioned problems. An Rfunction is a realvalued function whose properties are fully determined by corresponding attributes of their parameters, which is usually applied to express a geometrical object. Thus, a signed distance function based on Rfunctions is created to represent whether two objects intervene and their level of intervention or separation. As the signed function is continuous and differentiable, the gradient information of the objective function guides a moving part to avoid its obstacles and to approach its target position rapidly. Therefore, a path planning approach with obstacle avoidance based on normalized Rfunctions is proposed in this paper. A discrete convex hull approach is adopted to solve the problem that Rfunction is inappropriate to represent a geometric object with some curves or surfaces, and pendent points and edges are generated in Boolean operations. Besides, a normalized approach ensures accuracy calculation of signed distance function. Experimental results have shown that the presented approach is a feasible way for path planning with obstacle avoidance.
1. Introduction
In recent years, technologies of path planning with obstacle avoidance have received extensive attention in the fields of robotics, computer simulation, and game animation. In the product design process, path planning with obstacle avoidance is an essential element of the simulations of product assemblies and disassemblies. Path planning with obstacle avoidance is in essence an optimization problem, which usually introduces the evaluation criteria (such as the shortest path or the shortest planning time) to plan an optimal path with obstacle avoidance from a given initial position to the destination in highinterference environments [1].
To avoid obstacles, existing approaches for path planning are necessary to frequently detect whether two objects intervene or to estimate the shortest path between a moving part and an obstacle at the given location. Therefore, there is not a unified form for planning path with obstacle avoidance. Moreover, collisionchecking and minimumdistance estimating themselves are complex and timeconsuming tasks. To deal with the abovementioned problems, rigorous mathematical analysis might be a practical way for path planning with obstacle avoidance.
The attributes of an Rfunction are fully determined by corresponding attributes of their argument. Thus, the sign of a real function is fully determined by the sign of its argument. Rfunction is usually applied to represent a geometric object [2]. If g is an Rfunction that expresses a geometric object, we can use a signed distance function g(x)≤0 to describe interference constraints between this geometric object and a space point . Thus, g(x) <0, g(x) =0, or g(x)>0, respectively, represents that space point is the internal, the boundary, or the external of the geometric object.
Based on the abovementioned analysis, a path planning method with obstacle avoidance based on normalized Rfunctions is presented in this paper. Firstly, Boolean operations based on Rfunctions are introduced to generate an implicit function for expressing a geometric object. Secondly, a discrete convex hull method is adopted to solve problems that Rfunction is inappropriate to represent a geometric object with some curves or surfaces, and there are some pendent points and edges in Boolean operations. Then, a normalized method is introduced to transform the implicit function into the mth order normalization function so that the calculation of signed distance function is accurate. Finally, planning paths with obstacle avoidance for a singlepoint and a polygonal region are conducted to validate the presented approach. Experimental results have shown that the proposed method is a feasible way for path planning with obstacle avoidance.
The rest of this paper is organized as follows. First, related works are introduced in Section 2. Then, the theory of Rfunction is presented in detail in Section 3. In Section 4, a discreteconvex hull method for generating CSG models of geometric objects is introduced. Section 5 adopts the normalization method to transform g into the mth order normalization function. And path planning with obstacle avoidance based on the normalization function is given in Section 6. Following this section, experimental results applying the proposed method are presented in Section 7. Finally, the paper ends up with some conclusions in Section 8.
2. Related Works
Path planning with obstacle avoidance has been extensively applied for the robot navigation. For a robot, path planning carries out a series of translational motions and rotational motions from an initial position to the target position while obstacles are avoided in its working environment [3]. Most of path planning methods with obstacle avoidance can be classified into three categories: potential field method (PFM), samplingbased method, and heuristicbased algorithm.
Artificial PFM appeared earlier in [4], which is simple and highly efficient. However, disadvantages like local minima, nonsmooth movement, and oscillations also appear in artificial PFM. To escape from the location of the local minima, Kim [5] presented a new artificial potential function for path planning. Lee et al. [6] proposed new potential function with random force algorithm using PFM for solving the risks of a robotobstacle collision and local minima trap. To overcome artificial PFM with slow progress and system instability in the presence of obstacles and narrow passages, a modified Newton’s approach was applied to potential field [7]. Sfei et al. [8] proposed a new formula of repelling potential for reducing oscillations and avoiding collisions when the target approaches obstacles. To solve problems where a robot only follows the destination while obstacles are rejected in conventional artificial PFMs, Cosio et al. [9] proposed a path planning method with obstacle avoidance based on improved artificial PFM and genetic algorithm. Yin et al. [10] and Zhang et al. [11] adopted modified PFMs to solve path planning problem in dynamic environments.
The advantage of samplingbased path planning methods (SPPMs) has abilities to deal with planning problems in complex and/or time critical real world. At present, most influential SPPMs are probabilistic roadmaps (PRMs) and rapidly exploring random trees (RRTs) [12]. Although PRMs and RRTs both randomly sample connecting points from their state space, their graphs connecting points are generated in different ways [13]. More work related to SPPMs can be found in [14]. PRMs have better performance for path planning in highdimensional state space. As curves or straight lines are used for constructing PRMs, a robot may go anywhere in its free space. Visibility graph [15–17] and Voronoi diagram [18–20] are two typical roadmap approaches, which have achieved better results with dramatically various types of roads. RRT algorithms randomly build a spacefilling tree to efficiently handle problems with obstacles and differential constraints by searching nonconvex and highdimensional spaces [21]. Kothari et al. [22] proposed a RRT method to handle path planning problem of unmanned aerial vehicles with uncertainty. The probability of constraint violation and uncertain dynamic obstacles are, respectively, limited by a chance constraint and its extended framework. To further improve search efficiency, BiRRT algorithms with bidirectional search have appeared [23, 24]. Two RRTs, respectively, root at the initial and the target state, and algorithms seek states that are “common” for two trees. In [24], a BiRRT approach with the intelligence is adopted to handle path planning problem with complex cluttered environments.
A complete overview to heuristicbased algorithms for path planning, which can be found in [3].
3. Theory of RFunctions
3.1. Terminology Definition
The sign of an Rfunction is completely determined by the sign of its variable and has nothing to do with the value of its variable. For example, a function g_{1}=x_{1}x_{2}x_{3} is negative, if the number of its negative variables is odd or positive otherwise. In general, the domain of variable not just has positive or negative but has the form ≥a or <a (a is a real number). In a practical application, the real domain is usually divided into (∞, 0] and [+0, +∞). Then, the definition of an Rfunction can be expressed as follows.
Definition 1 (R function y). Give a function S(x) with real parameter x, which has the formsA real function is an Rfunction, if there is a logic function which makes the equation true. Here, the function is called g’s adjoint function. Some Rfunctions have the same adjoint function, such as functionsA group of Rfunctions with the same adjoint function is called an Rfunction branch. Key characteristics of an Rfunction are given in the following section (more characteristics can be found in [2]). (1)The set of Rfunctions has closure properties under a composition operator. Thus, a function can be obtained by compositing some Rfunctions, and it is also an Rfunction.(2)If and are Rfunctions from the same branch, then the function + is an Rfunction belonging to the same branch.
3.2. Sufficiency Completeness
The characteristics of an Rfunction show that a complex Rfunction may be generated from simple Rfunctions. If there is a group of Rfunctions, which can generate Rfunctions belonging to any branch by using composition operators, we define this group of Rfunctions as a function system with sufficiency completeness.
Rvachev [25] presented some useful Rfunctions with sufficiency completeness. Here, we give a function system for basic logic operation, such as “negation” (¬), “conjunction” (∧), or “disjunction” (∨).Here, a∈1, 1]. If a=0 or a=1, we call a system an system or an system. For function systems and , their logical operations can be defined as follows:And their “¬” operations have the same handling rule with function system.
Although the abovementioned function system and function system have the same logic function, their differential characteristics are various. For instance, function system is nondifferentiable at the origin of coordinate while other points are differentiable; but function system is nondifferentiable at the point = or =; and function system has morder differential operators at the origin of coordinate while other points have infinite differential operators. Differential characteristics of function system have great influence in practical applications. function system has better differential characteristics while the calculation of function system is simple.
3.3. RFunction Representation for a Geometric Object
Boolean operations (such as “intersection” and “union”) between basic primitives can be used to express a CSG model [26]. Then, a geometric object has the following form: Here, (i=1, 2, …, m) is a basic primitive; and is a regular Boolean function whose operations have forms , , and . If we use logical operators ∧, ∨, and ¬ to displace Boolean operations, G will become a logical function. Combined with the definition of Rfunction, a geometric object can be expressed as follows:
If we define the signed distance function (x) for a basic primitive as (6) can be rewritten as follows:Given a continuous function g(x) and its adjoint function G, a geometric object in (6) can be expressed as the following form [16]:In the above formula, (i=1, 2, …, m) is a real continuous function while is an implicit function of a geometric object. In the following, we give an example to illustrate the definition of implicit function . In Figure 1(a), basic regions and , respectively, have the following equations:It can be seen from Figure 1(a), rectangular region has logic function B=∧. Then, using function system to represent region has the following implicit function g:If in (10) and in (11) are plugged into (12), (2) has
(a) B=∧
(b) B=∨
4. DiscreteConvex Hull Method
In Section 3.3, using Rfunction to express a geometric object is based on its CSG model. However, a geometric object usually has various CSG representations. For example, rectangular region in Figure 1(a) has logic function B=∨ in Figure 1(b). CSG representations usually have various logic functions and implicit functions, but they have the same logic functionality. In fact, it is an intractable task to automatically generate a CSG representation for a complex object. Even so, existing algorithms can automatically generate CSG representation for polygon regions and polyhedrons. In [27], a polygon region is automatically divided into some Sidechains with the intersection of convex hull vertices. Then, the intersection of the halfspace defined by Sidechains can be adopted to express a polygon region while the halfspace is represented by the intersection or the union of the edges. If the intersection of two edges is a concave, the halfspace is the union; otherwise, the halfspace is the intersection. For instance, polygonal region in Figure 2 has logic function P=∧∧∧. Here, = ∧∨, = , = ∨, and =∧. Besides, convex hull method has been applied to polyhedrons in a threedimensional space [28, 29].
(a) A polygon region and its convex hull
(b) A Sidechain graph for polygon region P
However, the abovementioned method is unsuitable for a geometric object with some curves or surfaces. Then, the surface is approximately discretized into planes while the curve is approximately discretized into straight lines. Thus, plane region in Figure 3(a) has logic function P=∧∧∧ (see Figure 3(b)) and (see Figure 3(c)). In Figure 3(c), the curve is discretized into straight lines , , , and .
(a) Plane region P
(b) =
(c) P =
The abovementioned approach for dealing with a geometric object is called a discreteconvex hull method. If the geometric object is a polygon without a curve or a polyhedron without a surface, the implicit function is accurate; otherwise, the implicit function approximately expresses the geometric object.
Discreteconvex hull method not only automatically generates CSG representation for a geometric object but also solves the problem of Point Membership Classification (PMC) [30]. PMC problem might lead to the implicit function with 0 value for a nongeometric object on its boundary. There are the following two aspects. First, irregular Boolean intersection ∩ for a basic primitive might produce pendent points, pendent edges, or pendent planes (see Figure 4). As the operation ∧ for an Rfunction is irregular, the implicit function of a geometric object may has pendent points on its outside part. Second, rectangular region has logic function B= ∨ in geometry (see Figure 1(b)). But for mathematical analysis, the operation ∨ of an Rfunction cannot ensure that the implicit function has 0 value on B’s boundary. As PMC exists, the implicit function cannot define the interference constraint between a geometric object and a space point. Fortunately, a discreteconvex hull method avoids the abovementioned problems. The reasons are that two adjacent lines (planes) of a polygon (polyhedron) just intersect in one point (straight line), and this point (straight line) must belong to the boundary of this polygon (polyhedron). For the operation Boolean union, the discreteconvex hull method will not produce the PMC problem.
Therefore, the implicit function of a geometric object based on discreteconvex hull method has 0 value on the object’s boundary. And the internal point and external point, respectively, have a value < 0 or > 0. Then, the sign of the function determines whether a space point and a geometric object have the interference. If g(x) represents an implicit function of a geometric object, g(x)≤0 shows that there is no interference between the space point and the object.
5. Normalized RFunction
The implicit function based on the abovementioned approach cannot accurately calculate the signed distance between a space point and an object’s boundary. For instance, the function has value of 0.84 at the point (1, 0) for the rectangular region given by the implicit function equation (13) in Section 3. But the minimum signed distance between the point (1, 0) and the boundary of the rectangular region is 1.0 in practices.
Let be the boundary of a geometric object and be a space point; the signed distance between the point and the boundary can be expressed as follows:where is the signed distance between the point and the point q; if , the signed distance is “+” (there is the interference between the point and the point q); otherwise, it is “”. If all space points satisfy the condition , the function is called G’s normal function. For example, let (a, b) and , respectively, be the centre and radius of a circle; the normal function of this circular region has the following form:
As most of geometric objects have some nonsmooth adjacent surfaces, their normal functions usually do not exist. Thus, we introduce the concept of normalized function for a geometric object. A similar normal function with essential differential attributes for a complex object is introduced, which satisfies the following conditions:where is a regular point on the boundary L and is the normal unit vector of the point . Then, is the morder normalized function of a geometric object. There are very small differences between normalized function near the boundary and signed distance function. And normalized function far from the boundary is gradually, continuously, and smoothly similar to signed distance function. In general, many Rfunctions have the abovementioned normalized properties. For an Rfunction without normalized properties, the following method can be adopted to normalize.
If the function , a point on the boundary satisfies the following conditions:Then, g can use the following formula to normalize into the firstorder form.Moreover, higherorder normalized method for the function can be found in [31]. Morder normalized functions and have the same logical operations ∨ and ∧ with function. Figure 5 lists the distribution of firstorder normalized function (it is also called similar signed distanced field; function (p=2) system is adopted) for the polygonal region in Figure 2.
The generation of normalized function of a geometric object can be summarized below.
Firstly, a discreteconvex hull method is adopted to generate the implicit function using the following three steps:(1)A geometric object is transformed into a polygon (polyhedral). Then, a halfspace implicit function that points to the interior of a geometric object is created based on its polygon (polyhedral) and edges (planes).(2)A convex hull method is introduced to generate CSG representation and logic function for this geometric object.(3)The logic function, the implicit function of an edge (plane), and appropriate Rfunction system are adopted to construct the implicit function for whole geometric object.
Secondly, the implicit function is normalized into .
6. Path Planning with Obstacle Avoidance
In obstacle environments, path planning with obstacle avoidance is to find an obstacle avoidance path for a moving part from a given initial posture to the final target posture. It is an optimization problem with noninterference between moving parts, noninterference between a moving part and a barrier, or kinematics constraints. For simplicity, we only discuss path planning with a moving part and noninterference between a moving part and a barrier in this paper. For the environments with obstacles, path planning with obstacle avoidance based on typical optimization can be expressed as follows:In the above formula, x is a motion variable and (x) ≤0 represents noninterference between a moving part and the ith obstacle and f(x) is an objective function.
For existing path planning approaches with obstacle avoidance based on numerical analysis, most of interference constraints cannot be explicitly defined using the abovementioned inequality. Thus, two procedures are adopted to solve these problems. First, the interference checking is used to qualitatively determine whether there is interference between a moving part and an obstacle. Second, if it is noninterference, the minimum distance between a moving part and an obstacle and gradient direction of motion parameters are quantitatively calculated, which are used to guide next iteration. Although existing algorithms can deal with them, they usually have heavy computing burden.
Fortunately, normalized Rfunction of a geometric object is not only applied to represent interference constraints between two objects, but also there are essential differential properties. Then, (19) can be rewritten as the following forms: Here, and DPt_{j}, respectively, express normalized Rfunction of the ith obstacle and a discrete point of moving part M’s surface.
For path planning with obstacle avoidance, we must ensure that planning paths have noninterference between a moving part and a barrier. Then, it requires that each iteration point will not default for solving (20). The reason is that interference constraint conditions cannot be strictly ensured for some artificial potential field methods [11]. Therefore, a feasible SQP method in [32] is adopted to solve (20). SQP is a local optimum algorithm based on the gradient with the superliner convergence.
Based on the above analysis in Sections 3.3, 4, 5, and 6, the flow diagram of the proposed method is given in Figure 6.
7. Experimental Results
To validate the proposed approach in this paper, some experiments have been performed for path planning with obstacle avoidance on the following three conditions:(1)The generation of the implicit function is similar in a twodimensional space and a threedimensional space, and the representation of the implicit function is relatively complex in a threedimensional space. For simplicity, examples are conducted in a twodimensional space.(2)Compromise differential properties and the computational complexity for a normalized function; an function system (p=2) is adopted to create the implicit function for an obstacle.(3)The tests are conducted on a computer with DELL OPTIPLEX 760 and Windows XP and Matlab (R2001a).
First, a singlepoint moving part is moved to the target position from an initial position through the translation movement in direction and direction, and the obstacles , , …, have been avoided (see Figure 7). Besides, the obstacle has an arc . Then, it is easy to get normalized function of an obstacle and to create interference constraints for (20).
(a) Obstacle avoidance planning environments
(b) A planning path for O5’s arc c with 4 discrete straightline segments
(c) A planning path for O5’s arc c with 6 discrete straightline segments
(d) Normalized functions of obstacles affect the iteration direction
In Figures 7(b) and 7(c), O_{5}’s arc is, respectively, discretized into 4 and 6 straightline segments, and their corresponding planning paths are listed. As there is a higher discrete approximation for O_{5}’s arc in Figure 7(c), its planning path is more reasonable. In general, the curve of an obstacle has more discrete straightline segments; normalized function is more complex, which will lead to higher computational complexity. The time is spent on planning paths for O_{5}’s arc with various discrete approximations is given in Table 1.

Figure 7(d) is partial enlargement of dashed boxes in Figure 7(c), which is listed to further illustrate the role of normalized function in path planning. Red and blue plots in Figure 7(d), respectively, express outer distance field of obstacles and . At an iteration point I, and , respectively, express negative gradient direction of normalized function of obstacles and , and represents negative gradient direction of the objective function. To avoid obstacles and , and make final search direction be revised to at iteration point I.
Second, a moving part is a polygonal region, which has three degrees of freedom (the translation movement in and directions, and the rotation motion in direction), and , , , and are obstacles (see Figure 8). To generate interference constraints between a moving part and an obstacle, M’s boundary is uniformly discretized into 142 points. And M’ is the target posture of a moving part. The translation and rotation motions of a moving part make bold edges of and the obstacle are aligned, and their midpoints and are coincident in Figure 8(a). Therefore, the objective function for (20) can be expressed as follows:where (x) is a function of external normal vector of M’s bold edge for motion parameter x and ≡(1 0 0) expresses external normal vector of O_{1}’s bold edge and (x) represents a function of the midpoint of M’s bold edge for the parameter x and and describe weight coefficients. As the cosine value of the vectorial angle is smaller (≥1 and ≤1), we, respectively, set =100 and =1.
(a) Obstacle avoidance planning environments
(b) Path planning with obstacle avoidance
(c) The partial enlargement of dashed boxes in Figure 8(b)
Figure 8(b) has listed obstacle avoidance planning paths for a polygonal region M, and dashed lines represent M’s posture. To observe clearly, solid lines are adopted to express M’s typical postures. And Figure 8(c) is partial enlargement of dashed boxes in Figure 8(b). It can be seen from Figure 8, moving part successfully avoids obstacles and on the basis of posture adjustments. Experimental results have shown that normalized functions of the obstacles can well approximate their signed distanced fields, and their gradient information may guide obstacle avoidance planning paths.
The proposed approach for path planning with obstacle avoidance is based on semianalytic geometry with Rfunctions, which has rigorous mathematical analysis. In general, the characteristics of the algorithm are that the results usually have higher accuracy while the processing speed of the algorithm is relatively slow. In particular, the processing time for various instances has greater diversity.
8. Conclusions
A path planning method with obstacle avoidance based on normalized Rfunctions is presented in this paper. First, an Rfunction is introduced to represent a geometrical object, which ensures that normalized Rfunction near the boundary of a geometrical object has little difference with signed distance functions, and other parts are continuously and smoothly similar to signed distance functions. Thus, normalized Rfunction can be applied to define interference constraints between a space point and a geometrical object. As for the generation of interference constraint between two geometrical objects, this needs to plug discrete points of a geometrical object’s boundary into normalized Rfunction of another object. Finally, path planning with obstacle avoidance for a singlepoint and polygonal region is conducted to validate the presented approach. Experimental results have shown that the proposed method is a feasible way for path planning with obstacle avoidance.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by research team of Wuhan Technical College of Communication (no. CX2018B03).
References
 J. C. Latombe, Robot Motion Planning, Springer Science & Business Media, 2012.
 V. Shapiro, Theory of RFunctions and Applications: A Primer, Cornell University, 1991.
 T. T. Mac, C. Copot, D. T. Tran, and R. De Keyser, “Heuristic approaches in robot path planning: A survey,” Robotics and Autonomous Systems, vol. 86, pp. 13–28, 2016. View at: Publisher Site  Google Scholar
 O. Khatib, “Realtime obstacle avoida nce for manipulators and mobile robots,” International Journal of Robotics Research, vol. 5, no. 1, pp. 90–98, 1986. View at: Google Scholar
 D. H. Kim and S. Shin, “Local path planning using a new artificial potential function composition and its analytical design guidelines,” Advanced Robotics, vol. 20, no. 1, pp. 115–135, 2006. View at: Publisher Site  Google Scholar
 J. Lee, Y. Nam, and S. Hong, “New potential functions with random force algorithms using potential field method,” Journal of Intelligent & Robotic Systems, vol. 66, no. 3, pp. 303–319, 2012. View at: Publisher Site  Google Scholar
 J. Ren, K. A. McIsaac, and R. V. Patel, “Modified Newton's method applied to potential fieldbased navigation for mobile robots,” IEEE Transactions on Robotics, vol. 22, no. 2, pp. 384–391, 2006. View at: Publisher Site  Google Scholar
 J. Sfeir, M. Saad, and H. SaliahHassane, “An improved Artificial Potential Field approach to realtime mobile robot path planning in an unknown environment,” in Proceedings of the 2011 9th IEEE International Symposium on Robotic and Sensors Environments, ROSE 2011, pp. 208–213, Canada, September 2011. View at: Google Scholar
 F. Arambula Cosío and M. A. Padilla Castañeda, “Autonomous robot navigation using adaptive potential fields,” Mathematical and Computer Modelling, vol. 40, no. 910, pp. 1141–1156, 2004. View at: Publisher Site  Google Scholar
 L. Yin, Y. Yin, and C.J. Lin, “A new potential field method for mobile robot path planning in the dynamic environments,” Asian Journal of Control, vol. 11, no. 2, pp. 214–225, 2009. View at: Publisher Site  Google Scholar  MathSciNet
 Q. Zhang, S. Yue, Q. Yin, and Y. Zha, “Dynamic ObstacleAvoiding Path Planning for Robots Based on Modified Potential Field Method,” in Intelligent Computing Theories and Technology, vol. 7996 of Lecture Notes in Computer Science, pp. 332–342, Springer Berlin Heidelberg, Berlin, Heidelberg, 2013. View at: Publisher Site  Google Scholar
 J. Lee, O. Kwon, L. Zhang, and S.E. Yoon, “A selective retractionbased RRT planner for various environments,” IEEE Transactions on Robotics, vol. 30, no. 4, pp. 1002–1011, 2014. View at: Publisher Site  Google Scholar
 S. Karaman and E. Frazzoli, “Samplingbased algorithms for optimal motion planning,” International Journal of Robotics Research, vol. 30, no. 7, pp. 846–894, 2011. View at: Publisher Site  Google Scholar
 M. Elbanhawi and M. Simic, “Samplingbased robot motion planning: a review,” IEEE Access, vol. 2, pp. 56–77, 2014. View at: Publisher Site  Google Scholar
 T. Siméon, J.P. Laumond, and C. Nissoux, “Visibilitybased probabilistic roadmaps for motion planning,” Advanced Robotics, vol. 14, no. 6, pp. 477–493, 2000. View at: Publisher Site  Google Scholar
 J. Kim, M. Kim, and D. Kim, “Variants of the quantized visibility graph for efficient path planning,” Advanced Robotics, vol. 25, no. 18, pp. 2341–2360, 2011. View at: Publisher Site  Google Scholar
 M. Zimmermann and C. König, “Integration of a visibility graph based path planning method in the ACT/FHS rotorcraft,” CEAS Aeronautical Journal, vol. 7, no. 3, pp. 391–403, 2016. View at: Publisher Site  Google Scholar
 E. Masehian and M. R. AminNaseri, “A voronoi diagramvisibility graphpotencial field compound algorith for robot path planning,” Journal of Robotic Systems, vol. 21, no. 6, pp. 275–300, 2004. View at: Publisher Site  Google Scholar
 P. Bhattacharya and M. L. Gavrilova, “Roadmapbased path planning  Using the voronoi diagram for a clearancebased shortest path,” IEEE Robotics and Automation Magazine, vol. 15, no. 2, pp. 58–66, 2008. View at: Google Scholar
 Y. V. Pehlivanoglu, “A new vibrational genetic algorithm enhanced with a Voronoi diagram for path planning of autonomous UAV,” Aerospace Science and Technology, vol. 16, no. 1, pp. 47–55, 2012. View at: Publisher Site  Google Scholar
 S. M. LaValle, “Rapidlyexploring random trees: A new tool for path planning,” Tech. Rep., Computer Science Department, Iowa State University, 1998. View at: Google Scholar
 M. Kothari and I. Postlethwaite, “A probabilistically robust path planning algorithm for UAVs using rapidlyexploring random trees,” Journal of Intelligent & Robotic Systems, vol. 71, no. 2, pp. 231–253, 2013. View at: Publisher Site  Google Scholar
 S. M. LaValle and J. J. Kuffner Jr., “Randomized kinodynamic planning,” International Journal of Robotics Research, vol. 20, no. 5, pp. 378–400, 2001. View at: Publisher Site  Google Scholar
 A. H. Qureshi and Y. Ayaz, “Intelligent bidirectional rapidlyexploring random trees for optimal motion planning in complex cluttered environments,” Robotics and Autonomous Systems, vol. 68, pp. 1–11, 2015. View at: Publisher Site  Google Scholar
 V. L. Rvachev, “On the analytical description of some geometric objects,” Reports of Ukrainian Academy of Sciences, vol. 153, no. 4, pp. 765–767, 1963. View at: Google Scholar  MathSciNet
 B. Mirtich, “VClip: Fast and robust polyhedral collision detection,” pp. 177–208. View at: Google Scholar
 D. Dobkin, L. Guibas, J. Hershberger, and J. Snoeyink, “An efficient algorithm for finding the CSG representation of a simple polygon,” Algorithmica. An International Journal in Computer Science, vol. 10, no. 1, pp. 1–23, 1993. View at: Publisher Site  Google Scholar  MathSciNet
 Y. S. Kim and D. J. Wilde, “A convex decomposition using convex hulls and local cause of its nonconvergence,” Journal of Mechanical Design, vol. 114, no. 3, pp. 459–467, 1992. View at: Publisher Site  Google Scholar
 Y. S. Kim and D. J. Wilde, “A convergent convex decomposition of polyhedral objects,” Journal of Mechanical Design, vol. 114, no. 3, pp. 468–476, 1992. View at: Publisher Site  Google Scholar
 V. Shapiro, “Wellformed set representations of solids,” International Journal of Computational Geometry & Applications, vol. 9, no. 2, pp. 125–150, 1999. View at: Publisher Site  Google Scholar  MathSciNet
 A. Biswas and V. Shapiro, “Approximate distance fields with nonvanishing gradients,” Graphical Models, vol. 66, no. 3, pp. 133–159, 2004. View at: Publisher Site  Google Scholar
 Z. Zhu and K. Zhang, “A new SQP method of feasible directions for nonlinear programming,” Applied Mathematics and Computation, vol. 148, no. 1, pp. 121–134, 2004. View at: Publisher Site  Google Scholar  MathSciNet
Copyright
Copyright © 2018 Songqiao Tao and Juan Tan. 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.