Wind field estimation for autonomous dynamic soaring

Jack W. LangelaanAerospace Engineering, Penn State University

email: jlangelaan@psu.edu

John Spletzer, Corey MontellaComputer Science and Engineering

Lehigh University

Joachim GrenestedtMechanical Engineering

Lehigh University

Abstract— A method for distributed parameter estimation ofa previously unknown wind field is described. The applicationis dynamic soaring for small unmanned air vehicles, whichseverely constrains available computing while simultaneouslyrequiring updates that are fast compared with a typical dynamicsoaring cycle. A polynomial parameterization of the wind fieldis used, allowing implementation of a linear Kalman filterfor parameter estimation. Results of Monte Carlo simulationsshow the effectiveness of the approach. In addition, in-flightmeasurements of wind speeds are compared with data obtainedfrom video tracking of balloon launches to assess the accuracyof wind field estimates obtained using commercial autopilotmodules.

I. INTRODUCTION

Dynamic soaring of birds was first described by LordRayleigh in 1883 [1] and discussed in more detail in 1889[2]. While it seems to be a much more recent idea, thepotential for dynamic soaring of aircraft has actually beenrecognized since at least 1909 [3], and dynamic soaringof full sized aircraft has been studied at several timesover the past century [4], [5], [6]. With the advent of lowcost electronics and the first demonstrations of radio-controldynamic soaring, the potential for long duration flight byunmanned aerial vehicles (UAVs) has recently become anarea of research [7]. The possibility of dynamic soaring inthe shear layer near the jet stream is discussed by Sachs andda Costa [8].

An overview of the instrumentation required for full scaledynamic soaring is discussed by Kiceniuk [9]. Detailedresults of dynamic soaring flight tests conducted at theDryden Flight Research Center are described by Gordon[10].

Typically dynamic soaring is cast as a trajectory optimiza-tion problem with an a priori known wind field [11]. Un-fortunately this will not be available during flight; moreover,to the authors’ knowledge there are currently no sensors thatcan be carried on a small UAV that can measure the 3Dwind field ahead of the vehicle. A means of mapping orpredicting the wind field using only on-board measurementsis therefore required. There are two components to windmapping: first, estimating wind velocity given measurementsaboard the vehicle; second, incorporating these estimates ina three-dimensional map of the mean wind field.

Since the purpose of the wind map is to enable real-timetrajectory planning it must satisfy several requirements, in-cluding: (a) it must be updated in real time; (b) the trajectoryplanner must be able to query the map very quickly; and (c)

uncertainty in the map must be computed to enable robusttrajectory planning.

In [12] Lawrance and Sukkarieh describe a method forwind field estimation based on Gaussian Process Regression.Their approach has the benefit of being applicable to arbitrarywind fields, but suffers from slow updates, making real-timeimplementation problematic. In contrast to that work, herewe seek to develop a method for wind field estimation thatuses known structure to simplify estimation.

During dynamic soaring, the cyclic trajectory implies thatthe volume of air “inhabited” by the vehicle is limited in size,with dimensions dependent on the trajectory parameters. Forsoaring in the near-ground shear layer the vertical dimensionis of order 10–100m with similar scales for the cross-wind and downwind dimensions. The preliminary resultsdescribed above for jet stream dynamic soaring show thata volume approximately 1 km × 1 km × 1 km is required.In addition, dynamic soaring (especially that employed byalbatrosses and human RC glider pilots) typically exploitswind fields with fairly simple structure. This wind fieldconsists of a mean spatially varying component with tem-porally varying terms (i.e. gusts) superimposed. Generallythe vertical gradient is significantly stronger than horizontalgradients, and the rate of change of the mean wind field isfairly slow compared with the time required to fly a singlecycle. Combined with the relatively limited volume occupiedduring dynamic soaring, this structure can be exploited tosimplify the mapping problem, making it tractable for real-time implementation. The estimation problem is thus cast asa parameter estimation problem, where a smooth functionis used to model the wind field and the parameters ofthis function are estimated using measurements obtained atdiscrete, distributed points.

Since in-flight wind field estimation relies on in situmeasurements of wind speed, we also briefly discuss windvelocity measurements. Results of in-flight measurementsare compared with measurements of wind velocity collectedusing vision-based tracking of balloons. While time- andspace-synchronization of balloon and aircraft data is prob-lematic, we show that the aircraft can obtain reasonablemeasurements of wind velocity in-flight.

The remainder of this paper describes wind field estima-tion (Section II), presents results of Monte Carlo simulations(Section III), describes results of a hardware experiment(Section IV) and presents concluding remarks (Section V).

2012 IEEE International Conference on Robotics and AutomationRiverCentre, Saint Paul, Minnesota, USAMay 14-18, 2012

978-1-4673-1405-3/12/$31.00 ©2012 IEEE 16

II. WIND FIELD ESTIMATION

The problem at hand is to estimate a continuous, smoothfunction that captures the spatial variation of the mean windfield. The mean wind field is assumed to vary slowly withtime: here slowly means that several DS cycles can be flownbefore significant change in mean wind speed occurs.

Note that direct measurements of wind velocity are notavailable. Rather, these must be computed in-flight based ondata measured by an autopilot module onboard the aircraft.The wind field is then estimated using the computed windvelocities.

A. Wind velocity computation

A method for computing wind velocity and an associateduncertainty analysis using sensors typically available in anautopilot module (e.g. GPS position and velocity, air speed,rate gyros, accelerometers) is given in [13]. For convenienceresults are summarized here.

Components of the wind field can be computed fromground-relative velocity (available from GPS) and air speed:

wix = x− va cos γ cosβ cosψ

+va cos γ sinβ sinψ (1)wiy = y − va cos γ cosβ sinψ

−va cos γ sinβ cosψ (2)wiz = z − va sin γ (3)

Here wix, wiy, wiz are xyz components of the wind vec-tor expressed in the inertial frame, x, y, z are components ofaircraft velocity with respect to the inertial frame (obtainedfrom GPS), va is airspeed, γ is flight path angle with respectto the air, β is sideslip angle, and ψ is heading. It is assumedthat the autopilot module provides estimates of angles.

For zero-mean Gaussian measurement noise the uncer-tainty in computed wind field can be computed by linearizingthe above equations about the current flight condition so that

w ≈ g(z) + G(z− z) (4)

where w = [wix wiy wiz]T , z = [x y z va γ β ψ]T and

G = ∇g, the Jacobian of the wind computation equations(Equation 1, 2, 3) with respect to the measurements z. ThenΣw = GΣzG

T , where Σz is the noise covariance associatedwith ground speed, orientation and air data measurements.

As described in [13], a measure of the uncertainty inhorizontal magnitude is Rw = Σw(1, 1) + Σw(2, 2):

Rw = σ2x + σ2

y + σ2va cos2 γ

+σ2γv

2a sin2 γ + (σ2

β + σ2ψ)v2

a cos2 γ (5)

This immediately shows that a significant contributor touncertainty in computed wind field is the vehicle’s airspeed(through coupling with orientation uncertainty). The mostaccurate way to obtain in situ wind measurements is thususing a neutrally buoyant balloon!

Assuming that angles (flight path angle γ, heading ψ andsideslip β) can be obtained to an accuracy of 1◦, at typicaldynamic soaring speeds (ranging from 50 m/s to 150 m/s)

the uncertainty in computed wind magnitude ranges fromσ = 1.26 m/s to σ = 3.71 m/s.

B. Wind field estimation as distributed parameter estimation

1) Wind field parameterization: If DS is limited to onlythe boundary layer (so that only albatross-style DS is per-formed) then one can impose empirically-derived functionssuch as Prandtl’s power law relationship [14]:

u

U≈(h

δ

)1/7

(6)

where u is the wind speed, U is the free stream velocity,h is height above the surface and δ is the boundary layerthickness.

In this case a first order polynomial function in lnhcan perfectly model the wind field. However, such specificparameterizations can lead to inaccurate representations ifthey are inappropriately applied. A more generically ap-plicable parameterization is desirable: here a polynomialrepresentation is used.

2) Polynomial representation of wind fields: For conve-nience the wind field is represented by three components:horizontal magnitude, horizontal direction, and vertical com-ponent. Each component is assumed to vary independently.

w(h) =

N∑i=0

aihN−i (7)

φw(h) =

N∑i=0

bihN−i (8)

wz(h) =

N∑i=0

cihN−i (9)

where h = h−h0

∆h is altitude shifted to a reference datumand non-dimensionalized by a scale factor. Dynamic soaringtypically exploits spatial (especially vertical) gradient inmagnitude, hence for compactness equations here will bedeveloped only for vertical variation in magnitude.

The task of the estimator is thus to compute estimates ai ofthe polynomial parameters defining the wind field. Clearlywinds change with time as well as in space. The problemof wind parameter estimation is thus a dynamic estimationproblem, and as formulated it is a linear parameter estimationproblem, well suited to implementation as a Kalman filter.

C. Kalman Filter Design

Without an accurate model for the time rate of change ofwind field, wind velocity is assumed to vary as a randomwalk, so that

ak+1 = ak + n (10)

where a is the vector of parameters defining the windmagnitude field and n = N (0,Q).

Measurements of wind speed are taken as altitude varieswith time, so that

wk = Hkak + v (11)

17

where Hk =[hNk hN−1

k . . . 1]

(from Equation 7).Measurement noise v is assumed to be zero-mean and

Gaussian, so that v = N (0, Rw + Rgust). The two com-ponents reflect sensor dependent noise and the effect ofmeasured gusts. Recall that we seek to estimate the meanwind field, and gusts are treated as noise superimposed onthe mean wind field.

Note that Rw will vary with time: it is computed usingEquation 5 and thus it depends on airspeed and flight pathangle.

The Kalman filter equations for this distributed parameterestimation problem can now be summarized.

Predictionak|k−1 = ak−1|k−1 (12)

Pk|k−1 = Pk−1|k−1 + Q (13)

Correction

Pk|k =(P−1k|k−1 + HT

k (Rw +Rgust)−1Hk

)−1

(14)

Kk = Pk|kHTk (Rw +Rgust)

−1 (15)

ak|k = ak|k−1 + Kk

(wk −Hak|k−1

)(16)

Note that this is a 1D measurement update, requiring onlyone (low order) matrix inverse to compute the covarianceupdate. This can be performed very quickly even on low-power computation hardware.

D. Filter initialization

Under some conditions it may be possible to have ana priori estimate of the wind field (e.g. from satellitedata). However, this will not always be the case and ameans to initialize the Kalman filter must be provided. Heredelayed initialization is performed: wind is measured atseveral altitudes, the Vandermonde matrix is generated usingEquation 11 and a least squares solution of the wind fieldparameters is used to initialize the Kalman filter.

III. SIMULATION RESULTS

To assess the wind field parameter estimation Monte Carlosimulations of flight in a wind field typical of the shear layernear the jet stream were conducted. A DS cycle for a windfield defined by

w(h) = 0.01h− 100 (17)

with h in meters and w in m/s was computed using themethod described in [15] and the aircraft is assumed tofollow this trajectory exactly. At a rate of 10Hz the aircraftuses autopilot data to compute wind speed and uncertaintyin wind speed using Equation 1 through 3 and Equation 5.Sensor noise standard deviations are given in Table I. It isassumed that the wind field is gust-free.

The wind field estimator uses a first order polynomial todefine the wind field (matching truth).

1500

1000

500

0

500 1500

1000

500

0

50010000

10500

11000

11500

North (m)East (m)

Up

(m)

Fig. 1. Two loops of dynamic soaring cycle. Wind blows in the positiveEast direction, with magnitude increasing linearly with altitude.

TABLE ISENSOR NOISE STANDARD DEVIATION FOR MONTE CARLO

SIMULATIONS

parameter variable 1σ noise/uncertaintyorientation φ, θ, ψ σφ = σψ = 0, σθ = 1◦

air data va, α, β σv = 0.2 m/s, σα = σβ = 1◦

ground speed x, y, z 0.1 m/s

Results of a representative run illustrate wind field es-timation. Two loops of cyclic dynamic soaring are shownin Figure 1. A time history of the estimated wind fieldparameters is shown in Figure 2, with a0 denoting the verticalgradient of the horizontal wind and a1 denoting the windspeed at the datum (in this case h0 = 0). The estimatedparameters are well within estimated error bounds, with goodconvergence in parameters seen by about 0.4 loops. Fordynamic soaring the key parameter is wind gradient, andthis is well estimated.

Figure 3 shows the evolution of the estimated wind fieldduring DS cycles, with plot times coinciding with the opensymbols in Figure 1. The dotted red lines show the 2σenvelope of the uncertainty in estimated wind field. At timezero there is no estimate of the wind field. It is interestingto note the change in wind field uncertainty: at integer times(when the aircraft is at the bottom of a cycle) the wind fielduncertainty is lowest at the bottom. At half-integer times(when the aircraft is at the top of the cycle) wind fielduncertainty is reversed.

Results of Monte Carlo simulations show that the parame-ter estimation is consistent (i.e. the estimated error accuratelypredicts the actual error).

A measure of the accuracy of estimating the wind field isa mean error residual:

R =1

N

N∑i=1

|w(hi)− w(hi)| (18)

18

5 0 5 10 1510000

10200

10400

10600

10800

11000

w (m/s)

altit

ude

(m)

0.00 loops

5 0 5 10 15w (m/s)

0.50 loops

5 0 5 10 15w (m/s)

1.00 loops

5 0 5 10 15w (m/s)

1.50 loops

5 0 5 10 15w (m/s)

2.00 loops

Fig. 3. Time history of wind field estimation for a representative run. Blue lines show the true wind velocity, solid red shows the estimated field anddotted red shows 2σ uncertainty bounds on the estimated wind field.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.05

0.025

0

0.025

0.05

a 0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2200150100

500

50100150200

Loop fraction

a 1

Fig. 2. Wind parameter estimation for a representative run. Blue linesshow true parameter values, solid red shows the estimated parameter anddotted red shows 2σ bounds.

where hi ∈ {h1 h2 . . . hN}, a set of altitudes for evaluatingactual and predicted wind. Figure 4(bottom) shows the meanresidual (solid red) and the envelope of maximum/minimumresiduals (dotted red) as DS cycles are flown. The meanerror in wind magnitude for the estimated wind field isapproximately 2 m/s, consistent with the error in the windcomputation.

IV. EXPERIMENTAL RESULTS

A. Aircraft Platform

The aircraft development platform used in this work wasdesigned and fabricated in-house at Lehigh University, and

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

100

200

300

400

500

estim

ate

erro

r

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

Loop fraction

win

d re

sidua

l

Fig. 4. Top: 2-norm of estimate error (mean: red; minimum/maximum:grey) and mean estimated error (blue). Bottom: Time history of residualerror in estimated wind field.

is shown at Figure 5. The planform is based upon an inverseZimmerman design. It is made of two half ellipses, bothhaving the same minor axis but the forward ellipse having amajor axis that is three times that of the rear ellipse. A fixedfin was used in conjunction with two elevons. The designhas very benign stall characteristics, is capable of operatingsafely in turbulent air, and can glide down steeply to landwhen required. Full stall landings at almost zero airspeed areeasy with this aircraft.

This is clearly not a dynamic soaring aircraft, but it issuitable for collecting wind field data.

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Fig. 5. Uglo 6 development platform used in this work. During testing,the aircraft was flown manually while telemetry data were logged via theon-board Piccolo SL autopilot system.

B. Ground Truth Wind Velocity Estimation

As stated previously, one of the primary objectives of thiswork was to demonstrate real-time wind mapping with anactual aircraft. In order to characterize the performance ofthe proposed approach, “ground truth” data were needed.While these were readily available for simulated wind fields,estimating wind velocities in proximity to an actual aircraftduring flight was significantly more challenging. To addressthis requirement, we employed a vision-based approach forground truth wind field estimation.

During flight testing, brightly colored balloons containingan air-helium mixture were released serially from the groundso their trajectories would carry them in the vicinity of theaircraft flight path. The balloons were then tracked over timeusing what amounted to a wide baseline (e.g., 50-70 meter)stereo vision system using a pair of Point Grey Chameleon1280x960 video camera systems that logged images at arate of 2 Hz. Point correspondences between the two setsof camera images were then recovered manually for eachballoon track during a post-processing phase. Using thesecorrespondences in conjunction with a three-dimensional re-construction approach based upon Hartley’s method [16], therelative position and orientation of both camera systems, aswell as the positions of the tracked balloons, were recoveredto a scale factor. The scale factor was obtained by measuringthe camera baseline. The balloon position estimates werethen transformed to an earth-centered coordinate frame usingGPS position estimates for the cameras, as well as measure-ments of camera azimuth and elevation from a compass andinclinometer, respectively. With the balloon positions known,their velocities – and as a consequence the Northing/Eastingcomponents of the wind field – were estimated using a finitedifference approach vs. time with temporal smoothing to helpmitigate high frequency noise.

To validate the the vision-based approach, initial experi-ments involved tracking a tethered balloon rig carrying anEagleTree eLogger V4 with a 10 Hz WAAS enabled GPSmodule as payload [17]. The motivation was to use thelogged GPS velocities for benchmarking the vision system’s

tracking performance. A total of 6 launches were conductedfrom different initial positions at standoff distances of ≈110 − 180 meters from the camera systems. The balloonrig was released at ground level, and allowed to rise withminimal resistance while attached to a 125 meter long, 0.15mm diameter tether. Each trial was considered completedonce the end of the tether was reached. Results from theseexperiments showed that the mean absolute deviation be-tween the two velocity estimates vs. time for all trials was0.35 m/s (minimum 0.22 m/s, maximum 0.55 m/s). Resultsfrom a single launch are shown at Figure 6 (right). Notethat these error levels represent the compounding of both thevision tracker and GPS velocity errors. These results indicatethat the vision-based tracking system provides an effectivemeans for estimating the wind velocity at standoff distancesin excess of 200 meters.

0 10 20 30 40 50 60 700

1

2

3

4

time (seconds)

velo

city

(m/s)

Fig. 6. Balloon horizontal velocity magnitude estimates vs. time for thevision system (solid black line) and GPS (dashed blue line). In this trial,the mean absolute deviation vs. time between the two approaches was 0.24m/s.

C. Wind field estimation using flight data

To assess the performance of wind field estimation, a flighttest was conducted using the Uglo 6 aircraft. The aircraftwas flown manually (radio controlled) while telemetry datawere logged via the on-board Piccolo SL autopilot system.During the approximately 6.5 minute flight, 9 balloons werelaunched and tracked with the vision system to estimate thewind field. Figure 7 shows the aircraft flight path, as well asthe balloon tracks.

When post-processing the aircraft telemetry data, we ob-served GPS velocity drops during significant portions ofthe flight. These drops compromised the aircraft velocityestimates, and as a result the algorithm’s ability to estimatewind velocity. As a result, we constrained our analysis to a 80second window where GPS velocity estimates were mostlyavailable.

Figure 8 shows a composite of the wind velocity estimatesvs. altitude from portions of 3 balloon tracks that overlappedin time with the flight window. The discrete velocity esti-mates are shown as red “x” markers. The solid blue lineshows a quadratic fit of the data which was used as groundtruth for comparison with wind field estimation conducted

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Fig. 7. Aircraft flight path (black solid line) and balloon tracks (dotted colorlines) from flight testing. The latter were recovered by the vision trackingsystem

Fig. 8. Wind velocity magnitude vs. altitude as estimated by the visiontracking system. Discrete measurements are shown, as well as a quadraticfit to the data.

using post-processed data from the Piccolo SL autopilotmodule.

The relevant portion of the flight path is shown in Figure 9.Note that this is not a typical DS trajectory: rather it consistsof orbits with slowly changing altitude.

Flight data from the autopilot was post-processed to com-pute the 3D wind vector and compute an estimate of thewind field using a second order polynomial fit. At the endof the flight window the estimated wind profile was

wf (h) = −2.2631× 10−4h2 + 0.0272h+ 1.262 (19)

with h height above ground in meters and wf is wind speedin m/s. The standard deviations of the terms above are 4.14×10−4 for the quadratic term, 0.037 for the linear term and0.769 for the constant term. The balloon-derived velocityprofile is

wb(h) = −3.259× 10−4h2 + 0.04940h+ 1.261 (20)

This is well within the range predicted by the post-processed in-flight measurements.

Figure 10 show convergence of the estimated parametersover time. Because the flight path changed altitude slowly

050

100150

200 500

50100

1500

10

20

30

40

50

60

70

80

North (m)East (m)

Up

(m)

Fig. 9. Portion of flight path used for wind field estimation.

0 10 20 30 40 50 60 70 800.05

0.025

0

0.025

0.05

a 0

0 10 20 30 40 50 60 70 8010

5

0

5

10

a 1

0 10 20 30 40 50 60 70 8020

10

0

10

20

time (s)

a 2

Fig. 10. Wind parameter estimation from flight. Blue lines show parametervalues obtained from balloon tracking data, solid red shows the estimatedparameter and dotted red shows 2σ bounds.

over time, the temporal convergence of the parameter esti-mates is quite slow. More dramatic altitude variation wouldallow faster convergence.

A time history of the estimated wind field is shown inFigure 11. The balloon-obtained second order polynomialcurve fit is shown in blue, and shows higher wind speedthan obtained from in-flight data. As before, at time zerothere is no estimate of wind field. As time progresses onecan see the estimated wind field approach ground truth (theballoon data). With a longer flight window, we would expectto see further convergence with the ground truth estimate.

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0 1 2 3 40

10

20

30

40

50

60

70

80

w (m/s)

altit

ude

(m)

time: 0.00

0 1 2 3 4w (m/s)

time: 25.20

0 1 2 3 4w (m/s)

time: 44.64

0 1 2 3 4w (m/s)

time: 60.64

0 1 2 3 4w (m/s)

time: 79.12

Fig. 11. Time history of wind field estimation from flight data. Blue lines show curve fit of wind velocity obtained from balloon data, solid red showsthe estimated field and dotted red shows 2σ uncertainty bounds on the estimated wind field. Times coincide with open blue circles in the flight path ofFigure 9.

V. DISCUSSION AND CONCLUSIONS

With an eye to enabling autonomous dynamic soaring on asmall unmanned aircraft, this paper has: (a) presented resultscomparing wind measurements using a vision-based balloontracking system with wind measurements obtained using anaircraft in flight; (b) described a method based on distributedparameter estimation to compute a map of a wind field; (c)presented results of a Monte Carlo simulation to assess theeffectiveness of the parameter estimation.

The wind field is represented using a polynomial approx-imation. This is well suited to shear layers in the vicinity ofthe jet stream and the atmospheric boundary layer, howeverthis method may be subject to errors related to both under-approximation and over-approximation. While performanceof the wind estimator is good in the linear wind shearcase examined here (both the gradient and the wind speedat a reference altitude were well-estimated), some tuningof parameters in the Kalman filter should further improveperformance. Future work will focus on using splines toparameterize the wind field. This will obviate potentialissues with scaling and will allow parameterization of morecomplex wind fields than those found near the jet stream orthe atmospheric boundary layer (for example, shear layerscaused by temperature inversions or flow separation in thelee of a ridge).

With the availability of a wind field estimator that includesestimates of parameter uncertainty, robust dynamic soaringtrajectories can be generated without a priori knowledge ofthe wind field (although clearly exploration will be requiredto generate the wind field estimate).

VI. ACKNOWLEDGMENTS

This research was funded by the National Science Foun-dation under grants IIS-1065024 and IIS-1065202.

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