Pteromys: Interactive Design and Optimization of
Free-formed Free-flight Model Airplanes
Nobuyuki Umetani1,2∗
Yuki Koyama1
Ryan Schmidt2
Takeo Igarashi1
1
2
The University of Tokyo / JST ERATO
Autodesk Research
automatic
adjustment
user’s edit

Figure 1: (Left) Our model airplane design tool analyzes the aerodynamic properties of a glider and optimizes while the user interactively
designs the plane. (Center) The user fabricates the airplane. (Right) The airplane actually flies.

Abstract1This paper introduces novel interactive techniques for designing
original hand-launched free-flight glider airplanes which can actually fly. The aerodynamic properties of a glider aircraft dependon their shape, imposing significant design constraints. We present
a compact and efficient representation of glider aerodynamics thatcan be fit to real-world conditions using a data-driven method. To
do so, we acquire a sample set of glider flight trajectories using
a video camera and the system learns a nonlinear relationship between forces on the wing and wing shape. Our acquisition system
is much simpler to construct than a wind tunnel, but using it wecan efficiently discover a wing model for simple gliding aircraft.
Our resulting model can handle general free-form wing shapes andyet agrees sufficiently well with the acquired airplane flight trajectories. Based on this compact aerodynamics model, we present a
design tool in which the wing configuration created by a user isinteractively optimized to maximize flight-ability. To demonstrate
the effectiveness of our tool for glider design by novice users, we
compare it with a traditional design workflow.

Humanity has always been fascinated with flight, and most of us
have experienced the joy of creating simple flying machines in
the form of paper airplanes. One only has to take model aircraftslightly more seriously to discover the wide range of forms that
simple hand-launched gliders can take on (Figure 2). The adventof personal fabrication devices, such as desktop 3D printers and
CNC cutters, has greatly expanded the ability of individuals to construct precise glider aircrafts. However, it is surprisingly complicated to design a simple glider that will actually fly (let alone flywell). There are complex constraints between the shapes, positions,and mass of the various plane components, which must obviously
be satisfied. Without domain knowledge of advanced aerodynamics, or extensive experimentation, novices (such as the authors) finddesigning even simple gliders to be quite frustrating. Designing
more complicated free-formed gliders is prohibitively difficult for
novices because the data and required knowledge for estimating
their aerodynamics have not yet been well established.

CR Categories: I.3.3 [Computer Graphics]: Computational Geometry and Object Modeling—Physically based modeling;
Keywords: fabrication, aerodynamics, optimization
Links:

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PDF

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∗ e-mail:n.umetani@gmail.com

V IDEO

Introduction

We introduce a novel computation technique that can model the
aerodynamics of free-flight gliders, and then use that model to automatically optimize a glider to maximize the flight-ability in order to
aid in the free-form glider design process. This optimization is efficient enough to run in real time, providing guidance when using ourinteractive glider design tool. Figure 1 illustrates an example of optimization during interactive shape editing. After the user sketches
the glider components (wings and fuselage), then our tool optimizes
the design so that it will actually fly. This design and optimization
cycle can be quickly iterated to reach a shape that is both aesthetically appealing and aerodynamically effective. Finally, the tool
outputs a blueprint of the glider, which can be sent to a CNC cutter
or 3D printer. The key technical challenge is efficiently estimating
the aerodynamics forces, which depend on the shape, orientation,

Figure 2: Example of existing hand launched gliders.

offline precomputation
actual flight
our physics model
sample airplanes
fitting
initialsimulation

wing force

wing shape

online design session
unflyable
real-time
optimization

user’s edit
flyableFigure 3: Our overall procedure consists of offline precomputationand online real-time design session. In the precomputation stage, acompact physics model that relates the wing’s shape to its aerodynamics force is represented using radial basis functions (RBFs), in
which the parameters are obtained by fitting the trajectories froma physics simulation to the acquired actual flights. This compact
physics representation is used in the interactive design session to
optimize the design of the glider in real-time.

and velocity of the aircraft. Although existing 3D computational
fluid dynamics simulations can provide this analysis, they are muchtoo expensive to compute in real-time.
We present a novel data-driven compact aerodynamics representation that can be acquired very inexpensively through simple experimentation to create a real-time simulation and optimization. Figure 3 shows an overview of our approach. First, we construct a setof actual gliders and take several flight videos for each glider. Then,
we optimize the parameters of our aerodynamics model to minimize
any deviation between the simulated trajectories and the set of acquired trajectories. Encoding the relationship between the planform(i.e., the wing shape from a top view) and the aerodynamic forces
is difficult because the dimensionality of the shape representation
is inherently high. We slice the wing into wing elements that areparallel to the streamlines to simplify this problem, and define thetotal glider forces as an aggregate of per-element forces. Our wing
elements have a low-dimensional parameterization, and as a result,a data-driven approach to the entire system becomes tractable.
Based on our aerodynamic force model, we can optimize the designof a glider to maximize flight characteristics such as long, stable
free flight. We must ensure that the optimized design will fly basedon as many initial conditions as possible, since a hand-launchedglider could have a wide range of initial velocities and orientations.
We propose a novel technique to evaluate the aerodynamic forces
without conducting an actual trajectory simulation, which allows
for the real-time optimization in our interactive design tool, in order
to efficiently consider many launching parameters. We performed auser study to compare the design experience using our tool with thatof a traditional design workflow, where the user iterates the designand evaluation several times. Our experiment shows that, using
our tool, even novice users can design original functional free-form
gliders that are difficult to design using a traditional workflow.
To summarize, our contributions include (1) a physics modelthat efficiently predicts the aerodynamics forces on arbitrary planforms, (2) a data-driven framework to estimate the parametersof our physics model from the acquired data, (3) an interactive
optimization-guided glider design tool.

2

Related Work

The study of aerodynamics rapidly advanced
after the Wright brothers’ first successful demonstration of heavierAirplane design.

than-air flight in the beginning of the 20th century. Modern airplane
design methodology is driven by wing theory, the study of wing performance under different shapes and conditions. Textbooks such as[Abbott 1959] and [Perkins and Hage 1949] provide good introductions to this topic, but focus on conventional planforms, such as
the rectangular, ellipsoid, and swept-back wings seen on virtually
all modern aircraft. These simple shapes are empirically known
to perform well, and aerodynamic optimization based on computational fluid dynamics has enabled these well-understood wings to
be fine-tuned to satisfy various performance goals [SobieszczanskiSobieski and Haftka 1997]. However, these elaborate, computationally intensive optimizations generally focus on small variations in
the conventional shapes. Our goal is not to optimize the performance, but rather to enable for the creative exploration of airplane
shapes while ensuring a basic flying capacity.
Since desktop fabrication devices suchas 3D printers or CNC cutters have become more widely available, accessible design tools for functional, easily fabricable objects are becoming more important [Schmidt and Ratto 2013]. Advanced computer graphics have assisted recent works in designing
more interesting functional objects such as mechanical contraptions[Zhu et al. 2012; Coros et al. 2013; Ceylan et al. 2013], beadwork [Igarashi et al. 2012], pop-up cards [Li et al. 2011]. We assemble our gliders out of the interlocking planar sections, a fabrication technique which has recently been studied in some detail [McCrae et al. 2011; Hildebrand et al. 2012; Schwartzburg and Pauly2013]. Other recent works have taken into consideration the optimization in the context of a particular fabrication technique, tosatisfy user-defined goals [Chen et al. 2013] or enhance the mechanical robustness of 3D-printed objects [Stava et al. 2012; Luoet al. 2012]. In the field of architecture, there are studies to design
fabricable structure such as [Panozzo et al. 2013; Song et al. 2013].
Functional fabrication.

Various prior
works have incorporated real-time physical simulation into their interactive design tools. Umetani et al.’s system [2011] assists theuser in designing physically viable objects using the real-time visualization of simulation results, while Umetani et al.’s system [2012]
additionally suggests alternative physically-valid designs when thecurrent design is invalid. In architectural design, Vouga et al. [2012]
and Whiting et al. [2012] used design optimization inside the modeling environment to create feasible masonry structures. Make itStand [Prévost et al. 2013] automatically optimizes the shape of a
3D-printed object so that it will be statically balanced when restingon flat ground. Our work applies a similar approach to a different aerodynamics problem domain and also presents a calibration
method for acquiring the parameters necessary for simulation.
Interactive design using physical simulation.

Directly simulatingcomplex physical phenomena such as the deformation of the microstructures of heterogeneous material is often prohibitively expensive for computer graphics applications. A standard way tohandle this problem, pioneered in computer graphics by Pai etal. [2001], is to propose a simpler physical model and fit its parameters to the empirical data. The frameworks to capture complex
physical phenomena using inexpensive experimental setups havebeen widely studied. Recent works have acquired images of deformed cloth and then used optimization techniques to fit the parameters of their simulation models to this acquired data to replicate
the physically-realistic cloth behavior [Wang et al. 2011; Otaduy
et al. 2012]. These sorts of approaches can be extended to quite
complex situations, such as the history-dependent plastic deformation of clothing [Miguel et al. 2013]. Similarly, Bickel et al. [2009;2010] have modeled the complex nonlinear deformation behavior
Measurement-based parameter fitting.

of isotropic, heterogeneous materials based on captured experimental data. In aerodynamics applications, controllable birds’ flapping
motions are synthesized by capturing their flights and modeling the
aerodynamics of their flapping motion [Ju et al. 2013]. In thisstudy, the true airflow around an airplane is too complex to simulate in real-time, we must use a simpler physical representationof the aerodynamic forces, and therefore, also use this approach of
measurement-based parameter fitting.

3

wing

User Interface

This section motivates the development of our simulation techniques described in the following sections with a brief descriptionof our interactive glider design tool. Please see the video materials for an interactive demonstration. As seen in the screen captureshown in Figure 4, the glider consists of wings and a fuselage constructed from rigid planar sections of a predefined thickness andmaterial. The user can edit the wings and fuselage by dragging
the polyline boundaries using as-rigid-as-possible curve manipulation [Igarashi et al. 2005]. Several simulated flight trajectories (§ 4)
are displayed below the shape view using a 2D height/distance plot
to show how a glider should fly. We show the possible trajectories
for six different angle/velocity launch conditions (velocity 7m/s,
9.0m/s and 11m/s horizontally and 10◦ vertically) to provide theuser with a sense of the flight characteristics of the current design.
As the wings and fuselage are manipulated, these trajectories are
updated in real-time.

glider’s design

simulatedtrajectories
Figure 4: Screen shot of our design tool.
The most difficult task in the design process is adjusting the angle and position of the wings (§ 6), because the aerodynamic effect of changing each parameter can be unintuitive. So, if thecurrent design would not fly, our tool automatically optimizes themounting angles and positions of the wings without changing theirshape. This optimization also happens at interactive rates, andtherefore, the design space is effectively constrained to the functional, physically-viable regions. However, from the users’ perspective, there is complete freedom to change the design and experiment with corrections that our tool suggests to ensure flyability.

4

expensive for real-time use, so we leverage wing theory, which isan empirical method to estimate the aerodynamic forces on a wing.
We briefly review wing theory in § 4.1. The original wing theory
is limited to simple planforms (the wing configuration as viewed
from above), and cannot be directly applied to free-form planforms.
We extend wing theory to more complex shapes by introducing thewing element in § 4.2. In § 4.3, we explain how to simulate glider
flight based on these wing forces.

Aerodynamic ModelIn this section, we explain how free-flights of gliders are simulated
given their design. During flight, the airflow interacts with the outershape of the glider and generates a stress distribution on the glider
surface. The aggregate of these stresses produces lift force, making
flight possible. In addition to the lift forces, there is drag force that
resists the airflow, and torque force, which changes the orientationof the glider. Stable flight is possible only when these aerodynamic
forces are in balance. Therefore, glider design involves manipulating the shape to balance the relationship between these forces.
Obtaining aerodynamic forces through 3D fluid simulation is too(a) local coordinate

flightdirection(b) side view of a wing

Figure 5: (a) Glider configuration. The center of gravity is placedat the origin, and the XY-plane is the plane of symmetry. (b) Aerodynamic forces on mounting position P: fd , fl and torque τz .

Most gliders are mirror-symmetricaround the fuselage, and we follow this convention. We orient theglider so that the XY-plane is the plane of symmetry, and the glider
flies in the direction of the X axis, and Y is the vertical axis (Figure 5-(a)). The origin is placed at the center of gravity of the glider.
The rotation of the glider about the Z-axis is called the pitch. We
assume that the glider is a combination of rigid planar plates, eachof which is either a wing or a fuselage. The fuselage is perpendicular to the Z-axis, while the wings rotate around the Z-axis, and cantranslate within the XY-plane. The angle between the wing and
the X-axis is called the mounting angle Θ (see Figure 5-(b)).
Glider local coordinates.

4.1

Wing Theory

Our approach is based on wing theory, which estimates the aerodynamics forces on a wing using a scaling law and experimental
data. This theory is easy to apply for many airfoils (the sectional
shape of a wing) and planforms, so is commonly used in the initialdesign stage of flying objects. We can only summarize this theory,see [Abbott 1959] for complete details.
Basic wing theory assumes that there is no interference between the wings and fuselage, thus each wing’s force canbe independently computed. Of course, the wings generate airflow
and pressure distributions that have a global structure, but if the
distance between the wings is sufficiently large, this interference isusually not significant.
Assumptions.

Similarly, wing theory assumes that the forces on the glider are determined only by the current velocity and orientation (this is thequasi-static assumption). During stable flight, the velocity and orientation generally change little and can be ignored. We emphasizethat our simulation framework is only suitable for predicting stableflight, where the temporal change in angle and velocity is small.
During trajectory simulation, large changes in these values indicate
unstable flight, and the simulation results are no longer valid.
Since we assume that the glider is perfectly
rigid, flight is not affected by the shape of the wing/fuselage connection. Hence we assume that the wing and fuselage are mechanically coupled at a single mounting point P, and forces transfer atthis point. The mounting points are purely virtual and can be placed
Forces on a wing.

at arbitrary locations. We place P inside both the plane of symmetry and the plane of the wing.

4.2

We assume the wing is moving in still air at a constant velocity
V that has a zero Z-directional component. Combined with oursymmetric plane configuration, the airflow around the wing andits stress distribution are also mirror-symmetric. At the mounting
point, the representative force is the integral of the stress distribution on the wing surface. In the symmetric configuration the translational force stays inside the plane of symmetry. Torque at themounting point is the integral of the cross product of the surface
stress and displacement of the surface from the mounting point. The
torque has only a Z-directional component since the other components cancel out.

We introduce a new discretization of a wing that is based on thewing element (see Figure 6) to efficiently handle a wide range ofplanforms. In aerodynamics terminology, the upwind edge of awing is called the leading edge, the downwind edge is called thetrailing edge and the length in between is called the chord length.
A wing is discretized into wing elements by slicing it with a plane
perpendicular to the Z-axis at a constant interval ∆z. The wing
element then takes on a trapezoidal shape to approximate the slice.
The average horizontal length l of the trapezoid corresponds to thechord length of the wing. The angles at the leading and trailingedges of the trapezoid are denoted as ξl and ξt .

Therefore, the forces on the mounting point can be described using
three components f = (fd , fl , τz ), as shown in Figure 5-(b). The
first component fd is the drag force, and is in the opposite directionof the velocity. The second component fl is the lift force, whichis the perpendicular component of the aerodynamic force. The lastcomponent τz is the torque around the Z direction, which is called
the pitching moment, measured at the mounting point.

Our assumption behind the wing element discretization is that the
air velocity relative to wings is much faster in the (X, Y) direction,compared to that of Z. Therefore, we can use a simplified model forthe shape difference in the Z-direction. If the shape is sufficiently
smooth, the planform of the wing element can be captured using
these two angle parameters, ξl and ξt .

These forces on a wing change due to many different parameters,such as the density of air ρ, the wing’s speed V = |V|, the wing’s
shape S, and the angle of attack α, which is the relative pitch angle
between the tangential direction of the wing and the wing’s velocity
V. We can reduce the dimensionality of these parameters by introducing Reynolds’s scaling law [Shevell 1988]. Let us assume thatthe two-dimensional planform S is uniformly scaled with a scalingcoefficient L ∈ R of S(L). This law states that the stress distribution on a wing’s surface is proportional to ρV 2 /2 when the
Reynolds’s number Re = ρV L/µ is constant, where the µ is theviscosity coefficient. Aerodynamic forces fd , fl are the stress integrated over the wing’s surface and thus are proportional to L2 , andthus to the wing’s area A. Torque τz is the integral of the crossproduct between the surface stress and moment arm, thus it is proportional to L3 , or L · A. Therefore, the drag and lift forces can becompactly written as(fd , fl , τz )T =1
(Cd , Cl , Cm L)T ρV 2 A,
2(1)

where Cd is the drag coefficient, Cl is the lift coefficient, and Cm
is the pitching moment coefficient. These coefficients are dimensionless and are parameterized with the angle of attack α, Reynolds
number Re, and normalized planform S̄ with an area of A = 1.
The field of aerodynamics engineering has extensively studied how
these coefficients change under different conditions (α, Re, S̄).
However, the variation in wing shape S̄ in these analyses is limited
to simple planforms such as rectangular, taper, back-sweep, and ellipsoid wings. Typical approaches for obtaining these coefficientsfor complex planforms are using wind-tunnel or large-scale fluid
simulations. In § 5, we describe a simpler approach.

: chord length

leadingedge

trailingedge

flight direction
fuselage

Figure 6: Our wing element discretization of wings.

Wing Element Discretization

Let wing S have wing elements si (i = 1, . . . , N ). For each si ,we compute the amount of torque at chosen point pi be 0.25l downwind from the leading edge. For a rectangular wing, this point iscalled the aerodynamic center and it is empirically known that thepitching moment here is insensitive to the angle of attack [Abbott
1959]. Each wing element contributes drag, lift, and pitching moment to mounting point P as



fdi
cd
1
i
 fl  =  cl
 ρV 2 ai ,
2
i
i
i
i
cm l + cd d sin α + cl d cos α
τz


(2)

where ai is the area of the wing element and di is how far pi isfrom the mounting point in the upwind direction. These drag, lift,
and pitching moment coefficients c = (cd , cl , cm ) ∈ R3 for a wing
element are parameterized with a four-dimensional parameter vector  = (α, Re, ξl , ξt ) ∈ R4 , where Re = ρV l/µ is the Reynolds’
number for the element. We assume the use of planar wings, andtherefore, the sectional shape is rectangular and thus symmetric,so cd becomes an even function and cl , cm become odd functions,with respect to the angle of attack. These forces are aggregated tocompute the total forces at the mounting point.
Our discretization is motivated by the blade element [Perkins andHage 1949] used in propeller design. Propeller blades are decomposed into blade elements, and per-element forces integrated to estimate forces on the entire blade. This technique is common forpropeller blades because the sectional shape and angle of attack significantly vary along the blades’ span, whereas wings usually havea constant sectional shape and angle of attack. Our wing also hasconstant sectional shapes and angle of attack, but the element shapealong the wing may significantly change. The main difference between the wing and blade elements is that we include parameters
ξl , ξt to capture these planform differences along the span.
Coefficient representation by scattered data interpolation.

We take a scattered data interpolation approach obtain coefficient cfor our new wing element model. Let the relationship between input parameter  and output coefficient c be represented as function
c() : R4 → R3 . We discretely represent this function by smoothlyinterpolating the output coefficient value c(k ) = (ckd , ckl , ckm ) at
several sampling points k = (αk , Rek , ξlk , ξtk ), where k ∈ K and
K is a set of sampling points. We explain how to construct these
discrete sampling sets in § 5.

Following [Bickel et al. 2009], we
use Radial Basis Functions (RBFs)
for the interpolation between sampling points (i.e., RBF centers). We
carefully construct the RBF interpolant so that it automatically enforces that cd is an even function
and cl , cm are odd functions, withrespect to the angle of attack α.
We enforce this parity by using a ghost sampling point ¯k =(−αk , Rek , ξlk , ξtk ) for each sampling point k that is located symmetric to the hyper plane α = 0 in the input parameter space.
The output coefficient value of the ghost sampling point is set atc(¯k ) = (ckd , −ckl , −ckm ). The interpolation becomes




X k
c() =
w · ψ || − k || + w̄k · ψ || − ¯
k || ,
(3)k∈K

where ψ : R → R is a scalar kernel function and wk =k(wdk , wlk , wm
) ∈ R3 is the weight for the k-th sampling point. We
solve the weights for the drag, lift, and pitching moment coefficients, respectively, using standard RBF techniques. The weights
for the ghost sampling points become w̄dk = wdk for the drag cok
kefficient and w̄lk = −wlk , w̄m= −wm
for the lift and moment
coefficients, enforcing the parity of these coefficients.
We use the Gaussian kernel ψ(r) = exp(−r2 ), where the distancer is the square norm of the difference of parameter . Smootherharmonic kernels ψ(r) = r and ψ(r) = r3 resulted in unstableflight simulations because they can output unrealistically large coefficients when the input parameter is far from the sampling points.

4.3

Free-flight Simulation of Glider(px , py ) is the location of the center of gravity. The plane flies at the
speed V = ||ṗ(t)|| in the direction θ(t) = − tan−1 {ṗy (t)/ṗx (t)},where the sign of θ takes on positive values when the glider is descending. Since the glider is flying against still air at a relative angle
γ = θ + β, this is the angle of attack. Note that we set the signs ofangles such as θ, β and Θ such that larger angles increase the angleof attack by following tradition of aerodynamics (Figure 7).
Figure 7 shows the aerodynamic forces acting on a glider. Let Sj (j = 1, . . . , #wing) be a wing in the glider.
The angle of attack for each wing is αj = γ + Θj , where Θj is themounting angle for wing j. We compute the (fdj , flj , τzj ) forces forP
fdj
wing j using the method described in § 4.1. Let Fd = #wing
j=1
be the total drag force generated in the flying direction and Fl =P#wing j
fl be the total upward force generated perpendicular to
j=1
the flying direction. We measure the pitching moment at the glider’scenter of gravity (i.e., the origin). Since the mounting point and thecenter of gravity are different, we take the linear forces into accountPτzj + Pjx fl + Pjy fd ,for the total pitching moment Tz = #wing
j=1
TT
where (Px , Py ) = R(γ)(PX , PY ) . Finally, the equation of
motion can be written as
Equation of motion.

M p̈x = Fx , M p̈y = Fy − M g, I β̈ = Tz ,(4)

where (Fx , Fy )T = R(θ) (Fd , Fl )T , M is the mass of the glider,and I is the rotational inertia of the pitching rotation measured at
the center of gravity.
The trajectory of a glider is obtained by numerically integrating (4).
We use Verlet integration for this time integration. Weißmann etal. [2012] described a more accurate time integration scheme basedon the Kirchhoff tensor, but we found a simple Verlet to be sufficientfor our application.

We have thus far illustrated the forces on an individual wing given
the discrete sampling set of the relationship between a wing element
and the coefficients of aerodynamic force. In this section, we simulate the flights of gliders using these parameterized wing forces.
This simulation is used for the trajectory estimation during interactive shape design, as well as for real-time design optimization (§ 6),
and our parameter acquisition procedure (§ 5.1).

current
flightdirection
Figure 8: Twenty gliders used to train our physics model.

5
Figure 7: Configuration of forces on glider during flight.

As a glider flies, its position and orientation
continuously change. We use 3D world coordinates (x, y, z) withthe y axis being the upright, vertical axis to describe this flight trajectory. The glider is launched at time t = t0 , and we assume
a perfectly symmetric launching condition so that the gliders fly
inside the xy-plane. The transformation from the glider local coordinates to the world coordinates can be written as (x(t), y(t))T =R(β(t))(X, Y )T + (px (t), py (t)) and z(t) = Z, where (X, Y, Z)
and (x, y, z) are an identical point represented in local and world
coordinates, respectively. R is a two-dimensional rotation matrix,β ∈ R is the glider’s pitching angle in world coordinates, andWorld coordinates.

Fitting Parameters

In the previous section, we explained how the trajectory of a glider
is simulated given the glider’s geometry and aerodynamic coefficients c() of the wing elements. This section describes how to
obtain these coefficients by using a data-driven method. These coefficients are computed by constructing a set of training gliders, andthen fitting simulated trajectories to those we acquired.

5.1

Data Acquisition

We manually designed a set of 20 gliders, shown in Figure 8, to fit
with the physically-valid parameters for our wing element model.
The gliders have varying planforms (rectangular, tapered, swept,etc.), which we hoped would result in good coverage of the parameter space ξl , ξt of the wing element. Flight data cannot be acquired

(a)

(b)

Figure 9: (a) Our launching device and (b) Our flight trajectory
tracking interface.
for invalid gliders, so none were included. Note that eliminating
invalid gliders does not bias the sampling because we do not selectively exclude specific wing element shapes from sampling.

where L is the set of launches across all the training gliders. The
number of tracked point Nl for each launch generally ranged from15 to 20. However, the exact launch conditions were difficult tomeasure using our simple launcher. Therefore, we also obtained(i) the launch conditions (velocity, orientation, and timing) fromthe optimization together with (ii) the aerodynamic coefficients of
the wing elements. These two optimizations are coupled; if wechange the aerodynamic coefficients of the wing elements the optimized launch condition estimates will change, and vice versa. So,
we interleave these two optimizations to simultaneously optimize
both sets of parameters.
For each launch l ∈ L, the launch parameters σ l = (V0 , θ0 , β0 , t0 )are estimated by fitting the initial trajectory points qi :arg minσl

We constructed our gliders using 1mm
thick foam core board, which is a polystyrene sheet cladded withpaper. This material is light and easily cut using a laser cutter. We
added slits to the wings and fuselage so that the planar plates caninterlock with each other, and glued small L-shaped plastic braces
at the intersections to reinforce the joint. We attached a paperclip
of a known weight at the head of the glider to add mass. The density of the board is measured beforehand to compute the moment ofinertia from the glider’s geometry and the paperclip’s position.

M
X||pl (t0 + m∆t) − qlm ||2 ,

(5)

m=1

Glider construction.

We constructed the simple launcher shown in
Figure 9-(a) to produce consistent glider launches. The launch force
is provided by a rubber band. During a launch, the glider slides ona rail that is placed on a flat plate attached to a tripod, allowing thelaunch pitch angle to be adjusted. Note that our optimization takes
care of determining the exact launch velocity V0 , pitch angle β0 ,and time t0 , so we do not need to precisely measure these values.

where pl (t0 + m∆t) denotes the simulated glider position after
time t = t0 + m∆t and we chose M = 10. The optimization of(5) starts from a guess where V0 and β0 are computed from two
points right after the launch, t0 is zero, and θ0 is equal to β0 .
Given the initial launch parameters, the coefficients at the sampling
points ck are optimized to fit the trajectory for all the acquired
launches, across all the training glider shapes.

Launching device.

We used video captures from the side and back of thelaunching device to capture the sample glider trajectories. We used
off-the-shelf SONY α77 DSLR cameras to capture 1920 × 1180
video at 30 frames per second. As the gliders should fly within theplane of symmetry, we only needed the side camera to capture thetrajectories; the rear camera feed was used to verify this assumption. Several of our gliders did initially veer off-course, but manual
symmetrization of the gliders corrected these artifacts.

Capturing.

The camera parameters (angle of view, relative position to launcher, etc.) were determined based on the reference points of an already known configuration. We developed asimple tool in which the user manually annotates the location of
the center of gravity in each frame of the video to recover the flighttrajectories, as shown in Figure 9-(b). We estimated the acquisition precision to be ±5cm. See the supplementary material for theactual captured video and acquired trajectories.

Trajectory tracking.

5.2

Parameter Estimation Algorithm

We fit our simulated trajectories to the acquired trajectory to obtain
the aerodynamic coefficients of the wing elements. As explained in§ 4.2, we model the aerodynamic coefficients of a wing element as
an interpolated function c() that is sampled at discrete parametersk . Therefore, we can obtain the function by optimizing the valuesof the coefficients at the sampling points ck = c(k ).
In the previous section, the trajectories of actual gliders weretracked for each launch l ∈ L as discrete trajectory points qlm ∈R2 {m = 1, . . . , Nl } with the camera’s frame rate ∆t = 1/30,

arg min E, E =ck , k∈K

NlXX||pl (t0 + m∆t) − qlm ||2 .

(6)l∈L m=1

The two optimization problems in (5) and (6) are solved using thegradient descent method. The gradients were computed using theforward-differncing, where E is computed at two different parameters with an empirically chosen difference. Even with this simple
approach, we observed a rapid convergence to the optimal parameters for both the launch conditions and wing element parameters.
One remaining question is how to obtain the sample points k . We
used a greedy method created by [Bickel et al. 2009]. We startwith a single point 0 that has the averaged parameters of the wing
elements obtained for all the training gliders. We find the launch
conditions and wing element coefficients for this single sampling
point using the above optimization. After convergence, we add thenext sample point k at the location where the largest difference between the simulated and acquired trajectories occurs. We thus reduce the approximation error until it converges by iteratively adding
the sampling points in this fashion.

6

Design Optimization

In the previous sections, we explained how to model the physics
of a glider to simulate its flying trajectories. We now illustrate ourdesign optimization based on the physics model. The optimization
changes the mounting angle Θj and the mounting positions PjX
and PjY for all the wings. Therefore, the optimization searches thedesign in a 3 × (#wing)-dimensional design space. The launchingconditions such as the glider’s velocity and pitch angle for a handlaunch are different for each launch. So, we optimize the glider’sperformance for a certain range of velocities and orientations, instead of optimizing performance for a specific launching condition.
In § 4.3, the glider’s dynamics were simulated from the forces of the wing. We denoted function F(V, γ) : R2 → R3 , which maps the glider’s speed V andangle of attack γ to (Fd , Fl , Tz ). With this function, from (4), a
Characteristic function of glider.

glider’s flight path can be computed given the launching configuration. Therefore, we assume that this function characterizes the aerodynamics property of the entire glider. Since we use hand launching to fly the glider, the initial velocity and initial angle varies alot. So, we take into account the region of interest in the input
parameters Ω = [Vmin , Vmax ] × [γmin , γmax ], which is the rectangular region of the range of speed and angle of attack, and thentry to optimize the overall glider’s performance in this region instead of taking into consideration the specific launching configurations. We chose Vmin = 6m/s, Vmax = 9m/s, γmin = −10◦ , andγmax = 10◦ , which are typical in hand launching.
What property in this characteristic function F in region Ω makes
a better glider? The most important observation is that when aglider flies stably the pitching moment approaches zero: Tz = 0.
This is because the aerodynamic forces Fd , Fl are very sensitive tothe glider’s angle of attack, and if the pitching moment constantly
has a large value, it quickly changes the angle of attack and thenthe forces are difficult to balance. We constructed a cost function
to evaluate a glider’s performance based on this observation. The
cost function consists of three different performance evaluations: (i)pitch stability, (ii) vertical force balance, and (iii) less drag. Eachof these three properties are represented as a cost function and theoverall cost function is the sum of them. We respectively explainthese properties in the following paragraphs.
Pitch stability means the pitching angle is stable
under a small perturbation. This is the most important property fora long and stable flight. Without pitch stability, the glider rapidlychanges its pitching angle and angle of attack, resulting in a sharp
plummet or a quick flip after launch. The pitch stability is achievedby producing restoring torque to make the deviation small when theangle of attack deviates from the angle that makes Tz = 0. Thatis, a small angle of attack should produce a positive pitching moment and large angle of attack should produce a negative pitching
moment. In the Ω region, this property is enforced if the angle of attack is too small γ = γmin , then there is a positive pitching moment
Tz (V, γmin ) > 0, or if the angle of attack is too large γ = γmax ,then there is a negative pitching moment (see Figure 10). This canbe represented with the cost function:
Z VmaxTz− (V, γmin ) + Tz+ (V, γmax )dV,
(7)
C1 =

ascends and the possibility of a stall increases. Therefore, the lift
force should be equal to the gravitational force in the given range.
The cost function is described as
Z VmaxC2 =kFl (V, γ) − M gk2 dV, where Tz (V, γ) = 0. (8)Vmin

Note that the actual vertical aerodynamic force is not Fl but Fy =Fl cos θ+Fd sin θ, where θ is the defending angle (§ 4.3). However,if the glider flies horizontally the defending angle becomes small
and the difference between the lift and vertical forces also becomes
small.
As the glider flies, the drag force pulls the glider
backward and the speed attenuates. If the drag force is small theglider flies longer. Therefore, we designed the cost function as
Z VmaxC3 =kFd (V, γ)k2 dV, where Tz (V, γ) = 0(9)Less drag.

Vmin

The lift-drag ratio and is frequently used for evaluating an airplane’s quasi-static performance, where all the forces are in balance and the lift and drag forces do not change. On the other hand,during actual glider flight the lift and drag forces constantly change,
and it is difficult to directly apply this lift-drag-ratio. However, witha small C2 , the lift and gravity forces are similar. So, minimizing
C3 and C2 together means the overall lift-drag ratio is improved.

Pitch stability.

Vmin

Minimizing cost function. The second and third cost functions
are difficult to compute because they are the integrations of nonlinear function F over the line, where the pair of V and γ makeTz = 0. Therefore, we discretely sample the values of the forces
inside Ω and interpolate them to make it easier to handle. We use
one-dimensional interpolation; we sample several fixed velocities,
and then with that given velocity, we linearly interpolate the valueof the forces that are sampled with different γ.

We set up a total cost function as C(F) = w1 C1 + w2 C2 + C3 ,where we choose w1 = 100 and w2 = 1. Note that the first
cost function is the most important one for the plane’s flight, so wechoose a large number. We minimize this cost function online during editing using the steepest descent method. We use the numericaldifferentiation to compute gradients of the total cost function withrespect to design parameters: the wing positions, mounting angles.

where Tz− (V, γ) takes −Tz (V, γ) if the value Tz is negative andEtakes on a zero if the value is positive. In the same manner,
Tz+ (V, γ) takes on Tz (V, γ) if the value Tz is negative and zero
if positive.

wind tunnel
wing element

30

1.0

25
20

0.5

15
10
5
0

1

5

10

(a) convergence

15

|K|

0

5

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15

[deg]

(b) plot of coefficients

Figure 11: (a) Convergence of fitting error E with increasing number of sampling points and (b) change of drag and lift coefficientswith respect to angle of attack α for wind tunnel data and the wing
element model.

2

(a)

1

(b)

7

Results

7.1

Validation of Aerodynamics Model

Figure 10: (a) Stable and (b) unstable glider design.
Pitch stability ensures that Tz is nearzero and the glider flies stably. However, with insufficient lift forces
the glider quickly descends, but with too much lift force the glider
Vertical force balance.

We introduced wing elements to represent the aerodynamic properties of free-form wings. The parameters of our wing element

acquired trajectory

fitted trajectoryacquired trajectory
empirical wing element

empirical wing element2

our simulation
ballistic trajectory

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Figure 12: Simulated trajectories (red) produced after fitting ourwing element model to the acquired trajectories (blue). Green trajectories shows simulation using empirical wing elements.

model are learned from empirical trajectory data, which consistsof 113 flights of the 20 gliders. Figure 11-(a) plots the differences
between the measured trajectories and the simulated trajectories resulting from the parameter-fitting procedure illustrated in § 5.2. We
observe that as the number of sampling points increases, the error
steadily decreases, until approximate convergence. In all further
experiments we use 10 RBF centers.
Figure 11-(b) compares the aerodynamic lift and drag coefficientsof a rectangular wing (modeled by our wing element with ξl =0, ξt = 0) and experimental wind-tunnel data for a planar wingwith 30mm chord length and 0.3mm thickness, taken from [Abbott
1959]. We choose this experimental data because the Reynolds’
number is similar to our problem (Re ' 1.0 × 104 ). The coefficient data obtained from wind-tunnel tests with this Reynolds’
number is frequently used for construction of gliders with typicalwing shapes. Since the airfoils are different, these plots should not
be identical, but the similarities show that our obtained model captures basic traits of the aerodynamics coefficients, such as that thelift coefficient stops increasing when the angle of attack is around10◦ (This phenomena is called stall). The drag coefficient is larger
than in the wind tunnel data because our wing is thicker.
Figure 12 shows a comparison of the recorded flight and simulation
results using the parameters obtained by using (5) and (6). In the
figure, we also show a trajectory simulated with a method similar
to the blade element, discretizing it with rectangular wing elements,
and using the empirical relationship between the aerodynamic coefficients and attacking angle from Figure 11-(b). We refer to this-2

Figure 13: Trajectory plots produced from our acquired data, oursimulation, simulation with the empirical wing element, and ballistic motion, for complex glider designs that are not included in the
training data.
naı̈ve approach as the empirical wing element. When the wing
shape is close to rectangular (e.g., Fig. 12-first row), the trajectories
simulated with our wing element and the empirical wing element
produce similar trajectories, however if the wing element angles
are large (e.g., the forward-swept wing glider in Fig. 12-third row),
our approach produces a trajectory that better fits the acquired trajectory. These results, and those for our set of training gliders foundin the supplemental materials, show that our simple wing element
model is capable of reproducing many real glider flights.
We validated our wing element by simulating the trajectories of four
complex gliders that were not included in the training data. Fig. 13
compares the acquired trajectories with those generated by conducting a simulation using our wing element and the empirical wing element. We also show the ballistic trajectories that were computed
when there were no aerodynamics forces. The trajectories fromour more advanced wing element show a closer approximation to
the experimental data than those based on the empirical wing element. Fig. 14 shows more complex gliders that are created with oursystem. They each fly more than 10m with casual hand launches.
Please see the accompanying video for their flight examples.

7.2

User Study

We performed an informal user study to demonstrate the effectiveness of our design tool. Our goal was to examine how significant an
impact our automatic flyability correction had on the glider design
process compared to traditional design/test iterations. Four com-8

Figure 14: Airplanes designed with our system.
puter science students (A,B,C,D) participated in the study. They
were familiar with 3D modeling but had no prior experience or
knowledge with airplane design. We tested two different glider
design environments. In the first environment, the participants designed gliders in a conventional way, without continuous feedback
from the simulation and optimization. Since actually making theglider and observing its flight pattern would be very time consuming, instead the participants could preview the simulated flight results, but only three times per task, mimicking design/test iterations. In the second environment, the user designed gliders with
our real-time optimization enabled, which automatically corrected
their designs to ensure flyability. Each participant designed a different original glider in each design environment. The participants
were divided into two groups; participants A and B tried the conventional design environment first, and then designed with our flyability correction enabled. Participants C and D completed the sametasks in the reverse order.
Each subject was given 15 minutes to design an original glider that
could fly more than 5m horizontally when launched from a 1.5m
height. The eight resulting gliders were constructed and tested using the launching device described in § 5.1, and we used a radar gun(D&M Pocket Radar TM ) to verify the actual launching velocity.
We collected 10 flight distances for each plane, with the launching
speeds ranging from 7 to 9m/s.
Figure 15 shows the designed gliders and their flight distances. All
the participants used our optimization functionality frequently, totry to improve flight distances both during exploration and also
while finalizing the design. With the traditional design/test iterative process, the plane designs not only stayed closer to the conventional glider shapes, but many of the planes also failed to fly long
distances. With our optimization-guidance enabled, the subjects
explored more design variations, and also achieved longer flights.

Discussion

We presented an interactive glider design tool that uses a real-time
aerodynamics-driven optimization to constrain the design space toshape configurations that are likely to actually fly. To do so, we introduced a compact wing element model that allowed us to simulate
the aerodynamics of free-form wings, and recovered physicallyvalid simulation parameters from the captured trajectories of a setof sample gliders. Our analysis results show that this simplified
model is effective, and our user study shows that novice users caneasily design novel flyable gliders.
We obtained a reasonable physics model by conducting inexpensive
experiments, and this required only a video camera and many different sample gliders. This approach was much simpler and moreaccessible than wind tunnel experiments or expensive CFD simulations, and could be widely deployed in various design tasks. Our
methodology is applicable under more complex conditions such as
variable airfoil shapes and different flying conditions, and may evenbe useful for other functional design tasks such as for kites, windmills, submarines, or sailing boats.
Our techniques are suitable for designing hobby-grade handlaunched gliders with creative shapes which can be constructed easily and inexpensively. However, our techniques would need extensive validation to be applied to the design of human-scale manned
gliders, where aerodynamic performance is more critical. We believe our prototype system could still be useful in those contexts asa tool for brainstorming or education.
Construction using other material.

Aerodynamic forces depend on theouter shape of the glider, and therefore,
we can use the same optimized simulation parameters for different materials
if the thickness remains consistent andthe density is known. The right figure
shows a glider constructed from lasercut acrylic resin board. The acrylic resin plane flies roughly 9.5m
at a horizontal launch speed of 8m/s. See the accompanying video
for more detail.
In the glider design tool, we only
compute aerodynamic forces on a wing, ignoring contributions
from the fuselage. This is because if a glider flies in the plane ofsymmetry, the airflow around the thin fuselage results in very small
aerodynamic forces. However, fuselage shape does affect the rotational stability about the Y-axis (i.e., yaw stability). If the front
part of the fuselage is significantly larger than the rear portion, theglider flips about the Y-axis and falls down. For example, in theuser study, the glider created by user B in Figure 15 has a large
head compared to its tail, thus it did not fly stably. Although it is
not observed in our study, the same rotation instability can happen
for the X-axis (i.e., roll instability) when the upper part of a glider
is significantly larger than the lower part. Furthermore, guaranteeing stability in the presence of imperfect symmetry in construction
and launching condition is an important future work.
Limitations and future work.

A

B

C

D

3.3
2.2

3.3
2.0

4.5
2.6

3.2
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8.5
6.3

(4.2)
(3.3)

9.8
8.6

11.0
8.5

Trad.

max [m]med [m]Sim.
+
Opt.
max [m]med [m]Figure 15: Result of the user study. The top row (Trad.) shows
the user’s glider designs with traditional design/test iteration. The
bottom row (Sim.+Opt.) shows designs with our tool. med and max
stands for the maximum and median flight distances of 10 flights.
Participant B’s design was design was unstable due to Y-axis rotation (discussed in the next section).

Our aerodynamic model cannot handle the interference
from multiple wing elements
in a streamline (see the rightfigure), in these cases the system gives unreliable results. Our toolis limited to symmetric glider designs assembled from thin rigid
plates. Symmetric wings simplify the problem because there are
only three independent component forces acting on the wing, andthe planar components simplify both the fabrication and aerodynamic analysis. Using a thin plate for the fuselage results in lower
drag forces, and at the scale and velocity of hand-launched gliders,the performance of the planar wings is comparable to more complex airfoils. However, we would like to use sculpted 3D shapes toincrease the users’ design freedom. This will involve modeling andlearning the aerodynamic coefficients of a 3D asymmetric object,which is a challenging problem to address in our future work.
Another potential direction is increasing the controllability. We
currently optimize for gliders that fly straight and flat within theplane of symmetry. A more complex goal behavior, such as for a
glider that flies along a designed trajectory or stays airborne for a
maximum amount of time, is an interesting possibility. Moreover,thanks to the recent advances in battery technology, micro-motors,
and wireless radios, it is possible to provide thrust and control evenfor small, lightweight gliders. These are interesting venues to explore in the future.
We thank the anonymous reviewers and
Kenichi Rinoie for their comments. We appreciate the assistanceof Daisuke Sakamoto, Yutaro Hiraoka and Junghyun Kim in our
experiments. Yuki Koyama is funded by JSPS research fellowship.
Acknowledgements.

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