Feasible path planning for fixedwing UAVs using seventh order Bézier curves
 Armando Alves Neto^{1}Email author,
 Douglas G. Macharet^{1} and
 Mario F. M. Campos^{1}
https://doi.org/10.1007/s1317301200933
© The Brazilian Computer Society 2012
Received: 15 May 2012
Accepted: 16 October 2012
Published: 31 October 2012
Abstract
This study presents a novel methodology for generating smooth feasible paths for autonomous aerial vehicles in the threedimensional space based on a variation of the Spatial Quintic Pythagorean Hodographs curves. Generated paths must satisfy three main constraints: (i) maximum curvature, (ii) maximum torsion and (iii) maximum climb (or dive) angle. A given path is considered to be feasible if the main kinematic constraints of the vehicle are not violated, which is accomplished in our approach by connecting different waypoints with seventh order Bézier curves. This also indirectly insures the smoothness of the vehicle’s acceleration profile between two consecutive points of the curve and of the entire path by controlling the curvature values at the extreme points of each composing Bézier curve segment. The computation of the Pythagorean Hodograph is cast as an optimization problem, for which we provide an algorithm with fast convergence to the final result. The proposed methodology is applicable to vehicles in threedimensional environments, which can be modeled presuming the imposed constraints. Our methodology is validated in simulation with real parameters and simulated flight data of a small autonomous aerial vehicle.
Keywords
1 Introduction
Path planning is a fundamental task for any kind of autonomous mobile robot. Even though it might be possible for such robots to traverse their environments solely in a reactive way, the competence of planning and computing paths is an important feature for a large number of vehicles and a great variety of tasks.
In spite of its relevance, there are not that many works dealing with vehicles that move in threedimensional space. Furthermore, and less obvious, the increased freedom provided by a less restrictive environment poses many new challenges. Current problems, such as path planning for multiple Unmanned Aerial Vehicles (UAVs) and Autonomous Underwater Vehicles (AUVs), still call for better solutions.
The interest and research in UAVs have been increasingly growing, specially due to the decrease in cost, weight, size and performance of actuators, sensors and processors. As far as the capacity of covering a broad set of relevant applications is concerned, UAVs clearly have their niche of applications, which cannot be fulfilled by other types of mobile robots. One of their main advantages is in several types of monitoring and surveillance tasks, where they are able to navigate over large areas obviously faster than land vehicles, with a privileged view from above.
As their availability increases, so does the possibility of having multiple of such vehicles traversing a given volume of the air space. Therefore, there is a growing need to study and develop techniques for the generation of safe and feasible paths considering specific constraints of the different types of aerial robots. Robotics literature describes several such algorithms. However, one fundamental feature of a path planning algorithm is to insure that the vehicle will be able to execute the generated path, which means that limitations on vehicle movements must be obeyed (i.e., nonholonomic constraints). For example, curvature radius is one such restriction imposed on paths generated for typical Ackerman steering vehicles, since the sliding constraint of wheels dictates that the component of the wheel’s motion orthogonal to the wheel plane must be zero.
1.1 Unmanned Aerial Vehicles
UAVs can be divided into at least three categories: rotarywing aircrafts (e.g., helicopters and quadrotors), aerostatic aircrafts (such as airships and hot air balloons) and fixedwing aircrafts (airplanes). The technique described in this text will have its focus mostly in a fixedwing UAV, however, without the loss of generality, it can be applied to other types of vehicles (even ground and underwater robots).
Mobile robots frequently present some types of motion constraints that must be solved by path planning algorithms. Fixedwing aerial vehicles, for example, present dynamic behaviors where position and orientation variables are completely interdependent, which in turn imposes several nonholonomic constraints to the system. These constraints are embedded, for example, in the maximum values of lateral acceleration that can be imposed by those vehicles, which can be translated by the smallest curvature radius that the vehicle can perform in space. Fixedwing aircrafts present also other limitations on their mobility such as maximum climb or dive angles and minimum speed.
One of the fundamental motion constraints of a vehicle moving in threedimensional space is the climb (or dive) angle. This basically refers to the rate of change in altitude, which may be severely restricted for some types of aerial vehicles. There are aircrafts, for example, that have a very small angle of attack^{1} which is often limited by the control action of the guidance system. Other examples are underwater vehicles for which the climb angle is confined to short boundaries subtended by the configuration of their actuators and control surfaces.
In this study, a novel path planning methodology based on a variation of the Spatial Quintic Pythagorean Hodograph curves is proposed. The method takes into account three major motion constraints of a fixedwing UAV in the threedimensional space: maximum curvature, maximum torsion and maximum climb angle, but with special emphasis on the cases of limited climb (or dive) rates.
For the special case of fixedwing aircrafts, the main goal is to guarantee stallfree maneuvers. The general idea behind the method is to model the path as a seventh order spatial Bézier curve which is iteratively computed. This path satisfies all the required constraints and, at the same time, with a satisfactory length. It also guarantees smoothness of the acceleration profile of the entire path by controlling the curvature values of the curves that composes it. An algorithm with fast convergence to the final result is also described and evaluated, since the Pythagorean Hodograph computation is cast as an optimization problem.
Section 2 constitutes an overview of the related work in the literature, where the theoretical foundations of this work is discussed. Then, Sect. 3 presents the methodology, where the problem is formally posed along with the improvements, and the proposed optimization algorithm to calculate the paths. Results obtained for a virtual UAV and for a simulated model of an actual UAV called AqVS developed at Universidade Federal de Minas Gerais are presented are discussed in Sect. 4. Finally, Sect. 5 concludes the article and points to future research directions.
2 Related work
One of the most important factors for path planning is to produce paths that are feasible of being executed by the targeted vehicle. This means that during path generation, the movement restrictions of the vehicle must be considered (e.g., nonholonomic constraints). This problem has been thoroughly studied, and the literature available in the area abounds specially for manipulators and twodimensional mobile robots [18].
Some approaches of single vehicle path planning in less restrictive environments can be found in the literature [17, 23]. Voronoi diagrams [5, 7] and Vector Field [13, 19] are widely used techniques to generate paths for aerial robots with such constraints. However, very few works deal with the threedimensional case.
Rapidlyexploring Random Trees (RRTs) are also widely used, especially for solving the path problem for nonholonomic vehicles in broader scenarios. In [6], the authors present trajectory planning for both an automobile and a spacecraft. In the latter example, even though an obstaclefree environment is considered, the focus remains on the motion constraints that need to be satisfied for a safe entry of the spacecraft in Earth’s atmosphere. Other works like [14] use this technique to generate nominal paths for miniature aerial vehicles. The authors present an algorithm for terrain avoidance for the air platform BYU/MAV, which, among other things, enables the vehicle to fly through canyons and urban environments. And although these studies consider some restrictions inherent to aerial robots, none of them take into account important constraints such as climbing rate and spatial torsion on fixedwing vehicles.
There are also some works that only deal with the generation of safe paths for vehicles assuming an obstaclefree environment. Among those is the work of [22], which presents one of the first methods based on the Pythagorean Hodograph (a special type of Bézier curve) for the path planning for UAVs. The author discusses numerous advantages of using such a curve in the modeling of paths for vehicles with nonholonomic constraints.
In this study, we use a special technique to compute feasible paths, called PH Interpolation. The PH provides several properties that can be considered advantageous in the path planning problem such as uniformity in the parametric distribution, continuous parametric speed and capability to admit realtime interpolation. More details about these features will be detailed later.
The PH curves were presented for the first time for the twodimensional case by Farouki and Sakkalis [12]. Initially, only fifth order curves were considered since they are much simpler. A Hermite Interpolation algorithm was proposed in [11], where the authors demonstrated that there are four possible solutions for the curve in \(\mathbb{R }^2\) space. The chosen solution is the one that minimizes the cost function (bending energy function), based on the integral of the modulus of the curvature function (torsion in their case is assumed to be null).
The threedimensional case was initially presented by Farouki et al. [10]. In [9], the quaternion representation was used to deal with Hermite Interpolation issues, and the authors claim that the infinite cardinality of the set of solutions for the problem is due to an underdetermined system of equations that are needed to compute the final curve.
In order to significantly reduce the solution space, the authors suggest assigning a small set of values to the unknown variables, reducing the number of possible solutions. The best solution is obtained from the minimization of the cost function defined in [10], that is based on the integral of the sum of the curvature and torsion functions modulus of each curve.
Two of the first works involving the use of PH curve for robot path planning are Shanmugavel et al. [22] for the twodimensional case and in [21] for the threedimensional case. In order to guarantee that the PH did not violate the kinematic constraints of the vehicle, the authors proposed a modification where the PH curve was computed iteratively. For every step of the algorithm, gain values were increased until \({\vec{r}}(t)\) becomes realizable. However, only the curvature and torsion constraints were considered. As it will be shown later, for vehicles with bounded climb (or dive) angle values, it is not possible to minimize the problem for the spatial cost functions.
In [2], the authors present an improved cost function based on [10] that considers all three constraints: curvature, torsion and climb angle. This study represents an improvement with respect to [21] since it considers the limitation in the climb angle of fixedwing UAVs. A new cost function, computationally less expensive, was presented in [1], where only the climb angle constraint is used in the minimization function.
Other works, such as [4, 22], present the application of PH curves in the path planning and trajectory planning problem for multiple vehicles. Both, however, deal only with the twodimensional case, without taking into account torsion and climb angle constraints.
The adaptation of the fifth order PH curve was firstly proposed by Alves Neto et al. [3]. The main objective was to create Bézier curves with continuous curvature profiles among several waypoints in \(\mathbb{R }^2\). That work presents a methodology based on a variation of the RRTs that generate feasible trajectories for autonomous vehicles with nonholonomic constraints in environments with obstacles.
In this study, we expand to \(\mathbb{R }^3\) the methodology described in [3], keeping the continuity in the curvature profile and taking into account all three constraints, namely curvature, torsion and climb (or dive) angle. Our method also allows the connection of two different curves without violating the constraints of the whole path. We propose a new cost function that will include just the climb angle constraint, indirectly ensuring the others constraints, that effectively reduce the computational cost of the method. We also present an optimization function based on the bending energy function in order to minimize the total path length.
3 Methodology
In this section, we initially present the theoretical formalization of our problem. We describe mathematically the robot’s constraints and the Pythagorean Hodograph formulation. We then provide the foundations for calculating the PH curves for which the union of several paths keeps the continuous curvature profile in the final plan. Finally, we show an optimization problem formulation that will allow us to generate short paths that are feasible by the robot.
3.1 Problem formalization
Each waypoint \({p_{}}\) is described by threeposition (x, y, z) and twoorientation (\(\psi ,\theta \)) variables. The variable \(\psi \) is an angle that describes the waypoint orientation parallel to the XY plane in relation to the \(\mathbf X \) axis. We define \(\theta \) as the waypoint orientation measured in relation to the \(\mathbf X \) axis and parallel to XZ plane. Here, we must say that the roll angle (\(\phi \) in conventional aeronautical notation) is not considered, since it has no physical sense regarding the curve in the 3D space. In other words, the curve represents a trajectory to be followed, not the actual trajectory being executed by the robot.
3.1.1 Constraints
In order to be considered a feasible path for a given robot, the curve \({\vec{r}}(t)\) must respect the kinematic constraints of the vehicle. The three motion constraints mentioned before are the maximum curvature (\({\kappa _{\mathrm{max}}}\)), the maximum torsion (\({\tau _{\mathrm{max}}}\)) and the maximum climb (or dive) angle (\({\theta _{\mathrm{max}}}\)) that are realizable by the robot in the threedimensional space. It is possible to completely define a curve in \(\mathbb{R }^3\) as a function of curvature and torsion only [16].
As far as the underlying physics of the system is concerned, the curvature may be defined as a quantity that is directly proportional to the lateral acceleration of the robot in the space. The value of \({\kappa _{\mathrm{max}}}\) is inversely proportional to the minimum curvature radius (\({\rho _{\mathrm{min}}}\)) of the curve that the vehicle is capable of executing, which is also proportional to the maximum lateral acceleration of the vehicle.
In the next section, we detailed the necessary steps to realize the union in Eq. (2) in order to guarantee curvature and climb angle continuity for \({\mathcal{R }}\). As expressed by Eq. 3, the curve produced by the path planning algorithm must be continuously derivable, and it must also be second order differentiable (\(C^2\)).
3.1.2 Spatial Pythagorean Hodograph curves
The path planning problem is then reduced to find a solution to an Hermite Interpolation problem. One important advantage of using this model is that the resulting curve is infinitely continuous, so that the curvature, torsion and inclination functions are always smooth.
To guarantee that the PH does not violate the kinematic constraints of the vehicle, Shanmugavel et al. [21] propose a modification to iteratively compute the PH. It uses gain factors to modify the position of some of the control points in every step of an iterative algorithm, until \({\vec{r}}(t)\) becomes realizable. We will discuss these characteristics later in this article. However, Shanmugavel’s method only consider the curvature and torsion constraints. As it will be shown later, for vehicles with bounded values of climb (or dive) angle, it is not possible to minimize the problem for that spatial cost function. Therefore, we propose a new cost function that incorporates all three variables along with limitations on their maximum values. The \({\theta _{\mathrm{max}}}\) constraint will be included in the PH computation, which will then generate feasible paths for robots in threedimensional space, under all the aforementioned constraints.
3.2 Smooth path calculation
In this section, we present a method to calculate Bézier curves that fulfills the three constraint conditions mentioned before. We begin by mentioning an interesting characteristic of Bézier curves. They are infinitely differentiable functions, which mean that one curve has continuous curvature profiles. However, if we connect two or more curves to compose a path \({\mathcal{R }}\) among several waypoints, it is possible that discontinuity in accelerations will show up at the connecting points, violating the \(C^2\) condition. Then, in order to keep the continuity, we must calculate Bézier curves whose curvature values at extreme points are identical.
Then, we will always have zero curvature value at the extreme points of each Bézier, making its composition with others curves (\({\mathcal{R }}\)) a \(C^2\) function, as required. The main reason which led us to choose a seventh order function is now clear. We must keep the three initial and three final control points of the Bézier aligned. So we need at least other two points to create a function with at least one point of inflexion, in order to make a more flexible path.
The PH provides several properties that can be considered advantageous in the path planning problem, of which, the most relevant for this study are: (i) the uniform distribution of points along the curve, which contributes to the smoothness of the path; (ii) the parametric speed (first order derivative) that provides a continuous velocity profile; and finally (iii) the capability to admit realtime interpolator algorithms for computer numerical control.
Once the Bézier points are computed, we must guarantee that the generated curve does note violate the kinematic constraints of the vehicle, according to the constraints conditions of Eq. (6). Hence, we must determine the values of \({c_0}\) and \({c_7}\) in Eq. (12). As there is no closed form solution, these values will be estimated, in the next section, as the solution to an optimization problem .
The problem is initially approached by assuming that the extreme points of the curve (\({\mathbf{p }_{0}}\) and \({\mathbf{p }_{7}}\)) are directly determined by the initial and final poses \({p_{i}}\) and \({p_{f}}\), respectively. All the remaining points will depend on these pose vectors and on the \({c_0}\) and \({c_7}\) gains.
3.3 Iterative algorithm
The combinatorial arrangement of these five values for each \(\phi _k\) leads to a total of 125 solutions for the PH curve. The authors also argue that for most cases, \(\phi _1\) may be equal to \(\tfrac{\pi }{2}\), without the loss of generality, reducing the number of solutions to 25. This value, generally, makes the final result to minimize the variation under the \(\mathbf Z \) axis, what also reduces the climb angle function.
This solution, however, is not satisfactory for the problem considered here, since it does not take into account the climb angle function in the energy computation. As it will be shown later, the PH curves that minimize the cost function, may present climb (or dive) angles unattainable for a given robot.
We verified experimentally that, in most cases, the smallest values of \(\mathcal{E }\) were obtained when \(\phi _1=\tfrac{\pi }{2}\) (as seen in the previous case), and when the difference between \(\phi _0\) and \(\phi _2\) was the largest possible. Therefore, taking \(\phi _0\) in the \(\phi _k\) interval and \(\phi _2=\phi _0\), makes it possible to reduce the number of solutions to only five.
Finally, the values of \({c_0}\) and \({c_7}\) (the gains for which the spatial PH fulfills the requirements described in Eq. (6) still remains to be determined. This naturally leads to an optimization approach, since there is no closed solution for this problem. Increasing these gains tends to minimize the function \(\omega (t)\) as a whole, which is indirectly linked to the inverse of the parametric “speed” of the curve. However, that produces, as a consequence, an increase in path length. Thus, the ideal values of \({c_0}\) and \({c_7}\) are those which produce a feasible curve for a given robot, but also minimizes the length of \(s\).
This method promotes a convergence to the result with a very small number of iterations, producing a curve of reasonable length.
4 Experiments and results
In this section, we present and discuss the results of our experimental validation. Firstly, we describe the experiments for a simulated airplane, modeled only by the three kinematic constraints, namely curvature, torsion and climb angle. Then, we apply the proposed method to a simulated model based on a real UAV developed at the Universidade Federal de Minas Gerais/Brazil. All experiments were performed using the® software (Version R2008b) running on a computer with Ubuntu/Linux 10.04 operating system.
4.1 Virtual airplane
Firstly, we use a single simulated UAV. This step was designed to show the applicability of seventh order Bézier curves to paths of aircraft with relatively small values of maximum climb angle and with torsion radius much smaller than curvature radius. This is a key feature of our approach, since it would be otherwise very complex to be handled by other methods.

\({\rho _{\mathrm{min}}}=10\) m,

\({\sigma _{\mathrm{min}}}=100\) m,

\({\theta _{\mathrm{max}}}=\tfrac{\pi }{6}\) rad.
Initial and final waypoints for the virtual UAV
Waypoint  \(x\) (m)  \(y\) (m)  \(z\) (m)  \(\psi \) (rad)  \(\theta \) (rad) 

\({p_{i}}\)  0  0  0  \(\tfrac{\pi }{2}\)  \(\frac{\pi }{6}\) 
\({p_{f}}\)  50  20  50  \(\tfrac{\pi }{2}\)  0 
Besides the generated path, it is also possible to see the configuration of the control points computed for the curve. Those points were obtained using the values \(\phi _0=\frac{\pi }{2}\), \(\phi _1=\frac{\pi }{2}\) and \(\phi _2=\frac{\pi }{2}\) for our seventh order Bézier curve.
Also, it is important to notice that the extreme points in the curvature profile of Fig. 2a are both null, as we explained before. This means that, when we use this method to generate a path among several waypoints, all curvature values between single paths will be zero, and the total curvature profile of \({\mathcal{R }}\) will be continuous.
Waypoint set \({\langle {p_{a}}, {p_{h}} \rangle }\) for the virtual UAV
Waypoint  \(x\) (m)  \(y\) (m)  \(z\) (m)  \(\psi \) (rad)  \(\theta \) (rad) 

\({p_{a}}\)  0  0  0  0  0 
\({p_{b}}\)  200  0  100  \(\tfrac{\pi }{2}\)  \(\tfrac{\pi }{6}\) 
\({p_{c}}\)  500  500  400  \(\tfrac{\pi }{2}\)  0 
\({p_{d}}\)  500  1,000  200  \(\tfrac{\pi }{2}\)  0 
\({p_{e}}\)  500  500  500  \(\pi \)  \(\tfrac{\pi }{6}\) 
\({p_{f}}\)  300  200  300  \(\tfrac{\pi }{4}\)  0 
\({p_{g}}\)  0  300  200  \(\pi \)  0 
\({p_{h}}\)  500  1,000  100  \(\pi \)  0 
4.2 AqVS UAV
AqVS main characteristics [15]
Variable  Value 

Total weight  25 N 
Wingspan  2 m 
Length  1.6 m 
Wing loading  27–29 N/m\(^2\) 
Wing area  0.42 m\(^2\) 
Cruise speed  13.9 m/s (50 km/h) 
Operational ceiling  150 m (above ground level) 
Operational radius  10 km 
Curvature, radius \(({\rho _{\mathrm{min}}})\)  150 m 
Torsion, radius \(({\sigma _{\mathrm{min}}})\)  300 m 
Maximum, climb, angle \(({\theta _{\mathrm{max}}})\)  \(\tfrac{\pi }{30}\) rad 
Waypoint set \({\langle {p_{a}},{p_{e}} \rangle }\) for the AqVS/UAV
Waypoint  \(x\) (m)  \(y\) (m)  \(z\) (m)  \(\psi \) (rad)  \(\theta \) (rad) 

\({p_{a}}\)  0  0  1,013  0  0 
\({p_{b}}\)  2,000  0  1,023  0  \(\tfrac{\pi }{40}\) 
\({p_{c}}\)  2,000  2,000  1,033  0  0 
\({p_{d}}\)  2,000  0  1,023  0  0 
\({p_{e}}\)  0  200  1,013  \(\pi \)  0 
5 Conclusion and future work
We presented a methodology for path planning in threedimensional environments for autonomous vehicles that consider at least three motion constraints: maximum curvature, maximum torsion and maximum climb angles. The methodology is an extension of the Spatial Pythagorean Hodographs where such constraints are explicitly taken into account.
The use of analytical curves, such as Bézier curves, allowed for greater flexibility of this model with a low computational cost. The design of these curves takes into account very simple kinematics and dynamics constraints, which imply the simplification of the model of the vehicle at few points of operation.
The proposed methodology uses an elastic bending energy function for the resolution of the Spatial Pythagorean Hodograph that minimizes the climb angle function of the curve \({\vec{r}}(t)\), generating a solution that is adequate for vehicles with limited climb (or dive) angle capability. The methodology was also used to generate paths with satisfactory results for a simulated model of a real SUAV. As part of our ongoing research, we plan on incorporating other constraints to the cost function, such as maximum translation speed.
As further investigation, we will also consider environments with static and dynamic objects, which may demand path replanning in real time, which is of major relevance to the case of multiple and cooperating vehicles moving in threedimensional space.
Defined as the angle that the chord of the wing, viewed laterally, makes with respect to the wind. This is also another way to define the climb angle.
Declarations
Acknowledgments
The authors gratefully thank Prof. Paulo Iscold from the Center for Aeronautics Studies (CEA) of UFMG for the flight data of the AqVS/UAV. This work was developed with the support of CNPq, CAPES and FAPEMIG.
Authors’ Affiliations
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