Receding Horizon UAV Path Planning Via Gradient-BasedOptimization of Ferguson SplinesKyle Ingersoll, Patrick DeFranco, Bryce Ingersoll

April 10, 2015

Abstract—Path planning is an integral task of manyunmanned air vehicle (UAV) applications. Minimizingtravel time while avoiding obstacle collision is an importantobjective of almost any UAV flight path. In this paper, weseek to minimize the path length from an initial positionto a final destination while avoiding collision with static,circular obstacles. We pose the path planning task as an op-timization problem inside a receding horizon framework.After each planning step, the first segment of the path isthen traversed and a new path is planned; this procedurecontinues until the final destination is achieved. Theobjective of the optimization is to minimize the distance tothe final destination; the objective function also includesa term that promotes smooth, navigable trajectories. Thepath optimization is constrained by the obstacles and bya minimum and maximum step length. Path segments aremodeled as Ferguson splines. The optimization of splinesresults in efficient, feasible paths even in complex obstaclefields. For particularly challenging scenarios, a multi-startapproach is used to increase robustness.

I. INTRODUCTION

Unmanned air vehicles (UAVs) have broad poten-tial applications, including infrastructure monitor-ing, police surveillance, retail delivery, etc. Muchof the current research surrounding UAVs focuseson achieving robust performance in a wide va-riety of environments. Examples of this type ofresearch include GPS-denied navigation and sense-and-avoid, the latter being imperative for the safeintegration of UAVs into the national airspace. Inboth of these specific areas, and in the field ofrobotics in general, path planning is an importanttask. Given a set of obstacles, path planning seeksto solve the problem of finding the best path toaccomplish some objective, often getting from onepoint to another in the shortest time. Advanced pathplanning algorithms take into account factors otherthan just physical obstacles. These additional factorsmight include threat to the agent incurred over the

path or the cumulative field of view of the agent asit traverses the path. In this paper, we consider thesimpler path planning problem: finding the shortestpath through an obstacle field.

A. Relevant literature

The UAV path planning problem has been ap-proached in a myriad of ways. In [1], Beard andMcLain present two widely referenced methods:Voronoi graphs and rapidly exploring random trees(RRT). The Voronoi graph method models eachobstacle as a point and then partitions the areainto a set of convex cells that each contain onlyone obstacle. The interior of each cell is closerto the obstacle contained in that cell than to anyother obstacle. When applied to path planning, theedges of the cells can be traced to obtain a paththrough the obstacle field. Two immediately appar-ent shortcomings of this method are 1) its inabilityto model obstacles with nonzero area and 2) that itproduces non-smooth paths. The first shortcomingcan be overcome by modeling real obstacles withseveral point obstacles configured in the shape ofthe real obstacles. The second shortcoming can becompensated for by over-estimating the size of theobstacles; thus, when the UAV overshoots the de-sired path, there is some factor of safety that it willnot collide with an obstacle. The second method,RRT, is an exploration algorithm that uniformly, butrandomly, searches an area. When using RRTs forpath planning, the algorithm checks for a feasiblepath from the branch ends to the final destinationat the end of each iteration. When a feasible pathis found, the algorithm then traverses the path andsearches for possible links between non-consecutivepath nodes, thus smoothing and shortening the path.The RRT method also produces non-smooth paths

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that are inherently difficult for a UAV to followprecisely.

Other path planning approaches include prob-abilistic road maps (PRMs) [2]–[4], graph-basedshortest path algorithms [5], and many other meth-ods [6], [7]. Each planning scheme presented in thecited works have advantages and disadvantages invarious situations. For a more thorough discussion,we refer the reader to the cited survey papers.

In this paper, we present a novel path planning al-gorithm that efficiently and robustly finds a feasiblepath through a complex obstacle field. Our approachintegrates and improves on aspects of [8] and [9]–[11]. These papers frame the path planning problemin terms of optimization.

The authors of [8] use receding horizon control,in which the controller plans several short segmentsin successive steps instead of planning the completepath all at once. At each time step, the controllerminimizes the time required to travel from the pathsegment’s end point to the final destination, wherethe path segment is constrained by some planninghorizon. The robot then precedes a given distancealong that path and then repeats the search.

This receding horizon approach has the advan-tage of reducing a very complex problem into sev-eral smaller, more tractable problems. Additionally,since the controller periodically recalculates thepath, it is more robust to errors, uncertainties, andunknowns in the system model or world map. It canalso better handle obstacles that change in shape orposition with time than a single-pass planner can.

A downside of receding horizon path planningis “entrapment,” or its tendency to become stuck,particularly when facing obstacles with concaveshapes [8]. As far as the planner can see, the bestpath is into the obstacle. After taking a step, itfinds the concave interior of the obstacle and cannotproceed further.

Our work also builds on the work done for groundrobots in [9]–[11], in which the authors optimizepath segments described by Ferguson splines forground robot navigation. We make a few key im-provements in our work. First, the works describedin the list papers optimize the trajectory from startto finish in a single pass. Our path planning probleminvolves a more complicated obstacle field thatwould be more difficult to plan in a single pass,so we implement a receding horizon approach.In addition, those papers use a penalty that is a

function of the distance from the obstacles whichconsequently penalizes paths that may be a safedistance from obstacles. In our work, we implementthe distance from obstacles as a constraint, thusallowing the path to go right to the border of theobstacles. This allows for shorter paths that are stillfeasible.

We choose to use a multi-start gradient-basedmethod for our optimization rather than particleswarm optimization (PSO) or artificial bee colony(ABC) optimization. Gradient-based optimizationquickly finds valids paths and we show the multi-start approach still avoids being trapped in localminima of the obstacle field.

B. Our Implementation

Our algorithm uses the receding horizon frame-work of [8] in combination with the Ferguson splineapproach of [9]. We design our objective functionto favor paths that are navigable by a typical fixed-wing UAV. Section II defines the problem statementand simulation environment. Section III describesour proposed method. Section IV presents severalpaths produced by our method. Finally, Section Vpresents concluding remarks and potential futureresearch directions.

II. PROBLEM STATEMENT

We seek to find a feasible and short path throughan arbitrary obstacle field. The beginning locationof the path is at (0, 0) and the final destinationis at (100, 100). In the space (x, y) ∈ [5, 95], weplace nobs obstacles; this ensures that the beginningand ending locations lie in the feasible region. Theobstacle locations are selected using Latin hyper-cube sampling (LHS). Two sets of nobs/2 obstaclesare overlayed with successive calls of LHS. Thedesign of the obstacle field is meant to ensure theobstacles are relatively evenly distributed but to alsoallow for the challenges of overlapping obstaclesand concavities in the obstacle field. This type offield is meant to be representative of an inner-city orforest environment. The obstacles are circular withradii uniformly ranging from rmin = 3 to rmax = 7.Non-circular obstacles can either be conservativelymodeled with a single circular obstacle or moreclosely modeled with several smaller circular ob-stacles. A given randomly generated obstacle field

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may contain concavities; a good path planner shouldbe reasonably robust to these types of challenges.

We also seek to plan a path that could be followedby a typical fixed-wing UAV. In our UAV model, weassume the simple case of level flight with no wind.Thus, only turning dynamics are considered. Wefurther ignore side slip and assume that all turns arecoordinated turns. A coordinated turn is describedby

χ̇ =g

Vgtanφ cos(χ− ψ),

where χ is the aircraft course angle, ψ is the aircraftheading, φ is the roll angle, g is the gravitationalconstant, and Vg is the aircraft ground speed. Theturn radius of the aircraft is described by

R =Vg cos γ

χ̇,

where γ is the flight path angle (vertical climb). Inlevel flight γ = 0, and with no wind or side slip,χ = ψ. Combining the equations and making thementioned assumptions, the turn radius R is definedby

R =V 2g

g tanφ. (1)

III. OUR METHOD

In this section, we describe the overall frameworkof our approach (Section III-A) and provide a de-tailed formulation of the optimization problem weseek to solve (Section III-C).

A. Overall Framework

Our path planner uses a receding horizon frame-work. We start by planning the first three path seg-ments. The UAV traverses the first segment and thenplans three more segments. This process is repeateduntil the end of the third segment is within somesmall gate distance of destination. When this occurs,the final three path segments are re-optimized tominimize their total length, and the path planningis complete.

Each path segment is modeled as a Fergusonspline. Ferguson splines are described by

X(t) = P0F1(t) + P1F2(t) + P ′0F3(t) + P ′1F4(t)

where the basis functions are given by

F1(t) = 2t3 − 3t2 + 1

F2(t) = −2t3 + 3t2

F3(t) = t3 − 2t2 + t

F4(t) = t3 − t2,where P0 and P1 are the beginning and end pointsof the spline and P ′0 and P ′1 are the derivatives of thespline at these points. In our application, t ∈ [0, 1].

As defined in [12], Ferguson splines can have oneof any three boundary conditions. In this work, wechoose the constraint of having defined derivativesat the boundaries. Having the derivatives of theend points be explicit spline parameters is a majoradvantage of using Ferguson splines as it allows thewhole path to be smooth path by implicitly requiringderivatives to match at the spline joints.

B. Vehicle DynamicsA disadvantage of Ferguson splines is that it is

difficult to analytically find the maximum curvaturealong a given spline, which means it is difficultto understand feasibility of the path for a givenset of vehicle dynamics. We experimented withcubic and quadratic Bezier curves to model the pathsegments and considered using other approaches,such as fixed radius circles or many discrete linesegments. We decided that it was more valuableto continue using Ferguson splines instead of otherpath definitions; although those methods could havemade defining a minimum turn radius more straight-forward, they would have been more difficult tocontrol or would have increased the complexity ofthe optimization problem.

Due to the nature of our objective function (seeSection III-C2), paths generated will generally wraptightly around an obstacle, move in a straight linebetween obstacles, or turn slightly in the direction ofa better path that becomes apparent as the planninghorizon progresses. With careful definition of theobstacles to meet a minimum radius and safetymargin, the minimum turn radius constraint can stillbe met with reasonable assurance. For example,in our tests we have a minimum obstacle radiusof 3 meters. Using Equation (1), we can calculatethat with a roll angle of 60 degrees, the maximumairspeed in the tightest turn would be just over 7meters per second, which is feasible for many small,foam-type UAVs. Such calculations could be run for

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a given obstacle field to set feasible safety marginsfor obstacle size and to determine what type ofvehicle can fly through a given field.

In considering vehicle navigation, we ignore thesecond order dynamics of the aircraft. Constraintson higher order dynamics would ensure that thechange in turn radius does not exceed the rollcapabilities of the aircraft. If the turns are smoothand the obstacles reasonably spaced, it is safe toassume that this constraint could be ignored, as wasdone in [13].

The dynamic constraints focus on fixed-wingUAVs. When using helicopter or multi-rotor UAVs,the turn constraints no longer apply. Like manyground robots, rotorcraft are able to move at arbi-trarily low forward speeds and even rotate in place,thus allowing them to make a turn of any curvature.

C. Formulation of the Optimization Problem

1) Design Variables: Our optimization problemhas 12 design variables: (P1x , P1y) and (P ′1x , P

′1y),

the location and derivative of the first segment’s endpoint, which is also the second segment’s begin-ning point; (P2x , P2y) and (P ′2x , P

′2y), the location

and derivative of the second segment’s end pointand the third segment’s end beginning point; and(P3x , P3y) and (P ′3x , P

′3y), the location and derivative

of the third segment’s end point. The location andderivative of the first segment’s beginning point isimplicitly constrained to be the current location andderivative.

2) Objective Function: At each planning step,we minimize the objective function

f(x) = d

(1 + α

`

`min

)where

d =√

(xfinal − P3x)2 + (yfinal − P3y)2,

the distance between the end point of the third splineand the final destination. This objective functionrewards 1) minimizing the distance to the final des-tination and 2) taking a direct path. The importanceof these tasks is balanced by the scalar α. In oursimulations, we set α = 0.2.

The curvature reduction term

α`

`min(2)

was used in [9]–[11]. This metric penalizes longpaths that traverse only a short distance—that is,paths with high curvature. By tuning α, we canreach a good balance between straight paths andoptimal ending points. A similar effect could beachieved by performing two optimization problems:first minimizing the distance to the final destina-tion and then minimizing the path length given afixed (P3x , P3y) computed with the first optimiza-tion problem. We believe that the single objectivefunction is a more elegant approach. We note thatwith the objective function, there is no incentive tofind a minimum curvature path when the third splinearrives at the final destination because d = 0 forcesf = 0, regardless of curvature. In this case, we usetwo optimization steps to straighten the path.

3) Constraints: We impose constraints associ-ated with the length of the path segments and withthe obstacles. Each path segment must be longerthan a minimum step size P`min and shorter thana maximum step size P`max , set to 3 and 15, re-spectively, in our simulations. Enforcing a minimumstep size helps the path escape from concavitiesin the obstacle field. Limiting the maximum stepsize enables faster solutions by reducing the designspace. The length of a parametric curve is given by

` =

∫ β

α

√(dx

dt

)2

+

(dy

dt

)2

dt. (3)

Unfortunately, when Equation (3) is applied to theFerguson spline, the closed-form solution to theintegral becomes difficult to compute analytically.Consequently, we estimate the length of each pathsegment by approximating the spline as a sum ofsmall, straight elements,

`est =n∑i=1

√(∆x)2 + (∆y)2

where n equals 50. The step size constraints are thengiven as

P`max ≥ `est

P`min ≤ `est.

To ensure the obstacle constraints, we need to findthe minimum distance between all obstacles and apath segment. Once again, the closed-form solutionto this problem is difficult to compute analytically.Consequently, we sample each segment at ns uni-form intervals and require that each sampled point

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along the spline meet the constraint requirement.In our simulations, ns = 30, i.e. each segment issampled at t = 0.0345, 0.0690, . . . , 0.9655, 1. Theseconstraints then take the form

ri ≤√

(Xx(t)− xobsi)2 + (Xy(t)− yobsi)

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where ri and (xobsi , yobsi) are the radius and positionof the ith obstacle, respectively. We note that t = 0must already be a feasible point. At each planningstep, the optimizer only considers obstacles thatmight pose an active constraint in the current look-ahead window, i.e. obstacles that meet the require-ment√

(Xx(0)− xobsi)2 + (Xy(0)− yobsi)

2 ≥ 3P`max +ri.

Considering only the nearby obstacles improves theefficiency of the method.

4) Final Optimization Problem: As mentionedin Section III-C2, once the final destination hasbeen achieved, we re-optimize the final three pathsegments. In this problem, we seek to minimize `est.We retain the constraints described in Section III-C3except we consider all obstacles in the field ratherthan just those in the current look-ahead window toreduce code complexity.

5) Gradients: Gradients of both the objectivefunction and constraint functions are computed us-ing the complex step approximation [14] given by

∂f

∂xi=

Imf(xi + j)

h

where h = 10−30, and the complex variable jis only added to the ith element of x. Though amore efficient gradient calculation method couldbe used to improve performance, the complex stepmethod gives accurate gradients and was simple toimplement. Exact gradients supplied by the complexstep method result in quick and stable convergence.

6) Initial Guesses and Multi-Start: At the be-ginning of each planning step, the locations andderivatives of P1, P2, and P3 are set to that of P2, P3,and P3 from the previous planning step. Althoughthis means that P3 starts in an unfeasible region, itallows that point to freely expand into space guar-anteed to be devoid of obstacles. It also generallyimproves time to convergence because the algorithmdoes not have to “redo the work” it performed in theprevious planning step. Beginning with a random,feasible guess has been found to make the single-start approach more robust. However, it invariably

increases time to convergence and designs longerpaths.

When using a multi-start approach, the first guessis that described in the preceding paragraph. Wethen generate a random initial guess within the min-imum and maximum step size radii. We then checkthat the random initial guess meets the step size andobstacle constraints; if the constraints are satisfied,the optimization problem is solved using that initialguess. If the constraints are violated, we continueto randomly generate initial guesses until a feasibleguess is found or we reach the maximum number ofiterations, usually set to 10. If the optimization rou-tine converges to a solution, satisfies all constraints,and results in a lower objective function value thanthe previous minimum, that solution is retained. Weperform the multi-start approach described above 10times. Experimental results shows that multi-startadds considerable robustness to our method.

7) Solver: We use the Matlab (Mathworks, Inc.)function fmincon to solve the constrained gradient-based optimization problem. The (x, y) locations ofthe spline control points are confined by the bounds(−10, 110); the derivatives of the control points areconfined by the bounds (−1000, 1000).

IV. RESULTS

We present planned paths through four randomlygenerated obstacles fields. For all obstacle fields,nobs = 50. To repeatedly simulate specific obstaclefields, the random number generator in Matlab wasseeded with 1, 2, 3, and 4. Consequently, the obsta-cle fields will be referred to as rng(1), rng(2), rng(3),and rng(4) respectively. Table I reports the numberof function evaluations required by each path (asreported by fmincon), the number of required pathsegments, and the total path length.

Figure 1 displays the evolution of the path in therng(2) obstacle field for the multi-start approach;Figure 2 displays the resulting path for the single-start approach in the same obstacle field. Videos

TABLE I: Path Planning Results

Planning Scenario Func. Eval. Segments Path LengthSingle-Start, rng(2) N/A N/A N/ASingle-Start, rng(4) N/A N/A N/AMulti-Start, rng(1) 10205 11 157.7259Multi-Start, rng(2) 11275 11 153.6096Multi-Start, rng(3) 11256 11 150.8103Multi-Start, rng(4) 6849 11 145.0263

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Fig. 1: The path produced using the multi-start approach in the rng(2) obstacle field. A viable path was found with littledifficulty, even with groups of obstacles forming concavities.

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showing the evolution of these paths are located athttps://www.youtube.com/watch?v=XJn0Dau1Q84(single-start) and https://www.youtube.com/watch?v=wiOkw5A-8Gw (multi-start). In this obstaclefield, only the multi-start approach found aviable path to the final destination. We notethat in the fourth panel of Figure 1, the path istemporarily trapped in an obstacle concavity—alocal minimum of our objective function. Themulti-start approach enables the path to escapethis concavity and continue towards the finaldestination. Our objective function does possesssome inherent robustness to concavities that oftentrap the single-start method.

Figure 3 displays the evolution of the path in therng(4) obstacle field for the single-start approach;Figure 4 displays the resulting path for the multi-start approach in the same obstacle field. Videosshowing the evolution of these paths can be found athttps://www.youtube.com/watch?v=1h9zIFSA0xk(single-start) and https://www.youtube.com/watch?v=9LRA9OPhM5g (multi-start). This obstaclefield is particularly difficult because of thebarrier of contiguous obstacles centered at aboutx = 50, y = 40. The multi-start approach istrapped behind this barrier for one planning period,but is then able to proceed towards the finaldestination. The multi-start approach requiresrelatively few planning periods and plans a paththat is only slightly longer than the obstacle-freedistance to the final destination. In the end, thesingle-start approach is unable to find a viable path;it finds a way to circumvent the aforementionedbarrier—though in a way not likely compatiblewith fixed-wing UAV dynamics—but then is stilltrapped later by other obstacles.

Figure 5 shows the path planned by the multi-start approach through the rng(1) obstacle field.Notice in this figure that the third and fourth pathsegments (olive green and purple) begin travelingupwards before proceeding to the right and thenupwards. At these planning steps, the future pathsegments actually extended upwards into the openarea directly above the fourth path segment. Themulti-start approach allowed the path to re-routeaway from this dangerous area (containing a deepconcavity). It does not necessarily do so intelli-gently, the optimizer always seeks only to reduce thefinal objective function at each step. The multi-startapproach simply has the advantage of having more

opportunities to find the globally optimum path ateach step.

Increasing the P`max parameter effectively permitsthe optimizer to see farther at each planning stepand can help reduce path length by minimizingthe number of these types of directional changes.However, increasing P`max also expands the designspace, thus increasing computational time.

Figure 6 shows the path planned by the multi-start approach through the rng(3) obstacle field. Inthis randomly-generated obstacle field there happensto be a relatively obstacle-free channel leading inthe general direction of the final destination, whichgreatly facilitated the path planning.

V. CONCLUDING REMARKS

There are a number of straightforward extensionsto this path planning method. These include con-sidering non-circular obstacles, modeling vehicledynamics, and extending to 3D space. Implementingobstacles of different shapes would not be difficult,it would simply require redefinition of the con-straints, though it would increase the complexity ofthe problem. It may also just be just as efficient tomodel complex shapes with overlapping circles.

Another extension of the path planning algorithmis a more thorough investigation of how constraintson vehicle dynamics can be applied to constraintson spline parameters. Though we did a cursoryexploration of this area, we did not have successin applying motion models to the spline shapes.Velocity dynamics could be included in this model:since splines use parametric representations, theyalready implicitly represent vehicle velocity. In thiscase, the optimization could be run for fastest path,not just the shortest path. If our algorithm wereimplemented in hardware, the curvature of the pathwould be sampled at each point in time and thesecurvature values could be used to inform groundspeed commands to the UAV’s flight controller.

In addition, the method here only considerssearches in a 2D space. The search could be ex-tended to 3D space, though some extension of 2DFerguson splines into 3D space would need to beconsidered.

There are also more extensive changes that couldbe made to develop this path planner into a func-tional method. The first is determining if there isa reformulation of the problem so that it has a

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deterministic computation time. For example, themethods listed in [6], [7] often have a proven big-O computational complexity. With the constrainedoptimization approach used here, it is difficult toguarantee when the algorithm will converge. Itwould be worth reconsidering the problem to see ifour receding horizon approach could be presentedas, for example, a convex optimization problem,which has very predictable behavior. At the veryleast, analytic or sensitivity-based gradients could beused rather than complex step to reduce computationtime.

A second extensive change would be to determinehow the receding horizon approach could be morerobustly implemented. If the optimizer were to gettruly stuck, perhaps in a wide, deep concavity, somemethod of backtracking and picking a new routewould be necessary. One possibility is to keep amap of the space and update regions as visited as thepath goes through them. If the planner gets stuck,it could then pick a favorable not-previously-visitedregion of the map, backtrack to a branch point, andproceed into the new region.

In pursuing these research directions, it will beimportant to do a more thorough comparison toother UAV path planners. There are many otherapproaches to path planning that are being studiedand published. Though we performed a cursoryliterary search on the topic, the wide variety ofmethods available warrants a broader search throughthe discipline. A complete investigation would in-clude performance benchmark comparisons to otherefficient path planners.

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