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One Degree of Freedom Approach for an Autonomous
Descent Vehicle Using a Variable Drag Parachute
Travis D. Fields,∗Jeffrey C. LaCombe†
and Eric L. Wang‡
University of Nevada, Reno, Reno, Nevada, 89557, USA
This paper describes a novel method for controlling the landing location of a descent
vehicle using a variable-drag parachute. Wind data is collected prior to descent and used
to predict the descent path from initial parachute deployment to landing. The landing
target location takes the form of a finite line segment on the ground surface. Guidance is
achieved by mo difying the effective surface area of the parachute. The control algorithm
uses a 1–DOF model to predict and control the landing location. Experimental results are
presented to demonstrate the feasibility of the approach. First, results from wind tunnel
testing quantify how throttling of a parachute through simple manipulation of parachute
shroud lines changes the cross sectional area and the resulting drag force. Second, the
accuracy of the descent path prediction algorithm was evaluated experimentally using a
stratospheric balloon flight. These data were used to validate the ability of the prop osed
control algorithm to accurately predict landing location. Together, these tests demonstrate
the potential for using inexpensive onboard flight hardware to autonomously sense and
control a descent vehicle to reach a linear target on the ground.
Nomenclature
ArReference area of parachute
Armax Maximum reference area of a given parachute
CdDrag coefficient of parachute
Cd,ave Average drag coefficient (typically calculated over full test)
drReference diameter of parachute
FdDrag force
FgForce due to gravity
gGravitational acceleration
mTotal mass of descent vehicle
pActual descent path projected onto ground, p∈(x, y)
ˆpPredicted descent path projected onto ground, ˆp∈(x, y)
PActual descent path, P∈(x, y, z)
ˆ
PPredicted descent path, ˆ
P∈(x, y, z)
rRange vector projected onto ground r∈(x, y) with magnitude r=|r|
RRange vector (vector between start and end points), R∈(x, y, z)
U∞Free stream air velocity measured in wind tunnel
w(z) Wind vector as a function of altitude, with magnitude (speed) w=|w|
z(t) Actual (measured) descent profile
ˆz(t) Predicted descent profile (algorithm not specified)
∗Graduate Research Assistant, Mechanical Engineering, 1664 N Virginia St. MS 312, Reno, NV, and AIAA Student Member.
†Associate Professor, Chemical and Materials Engineering, 1664 N Virginia St. MS 388, Reno, NV, and AIAA Member.
‡Associate Professor, Mechanical Engineering, 1664 N Virginia St. MS 312, Reno, NV.
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ˆzRK (t) Predicted descent profile determined using Runga-Kutta
ˆzT V (t) Predicted descent profile determined using terminal velocity assumption
zend Ending altitude (landing altitude)
znAltitude at nthtime step
zstart Starting altitude (release altitude)
˙zVertical speed (descent rate)
¨zVertical component of acceleration
∆ Overall error in landing location
∆θError in landing location bearing from release point
∆rError in landing location distance from release point
µDynamic viscosity of air
ρ(z) Air density as a function of altitude
θBearing from current location to release point
σCdStandard deviation in the drag coefficient
I. Introduction
The objective of this study is to investigate the potential for using an autonomous descent system to
reach a targeted landing location by controlling only one degree of freedom. This work contributes
to a growing area of interest over the last decade as various applications have emerged that require the
ability to drop cargo from high altitudes and guarantee touchdown at a desired location. Several approaches
have been researched including military–inspired autonomous circular parachutes controlled via pneumatic
muscle actuators1,2,3through the Affordable Guided Airdrop System (AGAS). This system actively steers
the parachute through the use of large pneumatic muscles which are capable of deforming the shape of the
parachute. This method requires large amounts of nitrogen, and also limits the number of actuations possible
(depending on amount of nitrogen gas). For this reason, in recent years, interest has transitioned towards
an autonomous parafoil descent vehicle.4,5,6These systems utilize a steerable ram air parachute (inflatable
wing), which can then be controlled via manipulation of control lines. A parafoil system is very complex,
requiring extensive modeling in order to yield accurate results. The approach developed here instead uses a
simpler one–dimensional control of the descent rate (detailed below) to control the landing location.
The use of a 1–DOF control system contrasts with the other (more complex) approaches discussed above.
The principal constraint is that the vehicle travels along the wind direction throughout the entire descent.
Typically, the wind direction is a function of altitude, and thus, while always traveling “downwind”, the
heading and horizontal speed will vary with altitude. Guidance of the vehicle is achieved by trimming the
parachute to control how long the vehicle spends at each altitude. For example, if a westward course is
desired, the parachute will fully open to slow the vehicle while it is passing through an altitude with a
westward wind direction. Similarly, it will speed the descent when passing through a layer where the wind
blows in the “wrong” direction. The end result is a system in which the horizontal distance traveled during
descent can be beneficially controlled, for purposes such as delivering payloads to a linear (extended) target
such as a road or valley, independent of the release point. The specific point along the linear target cannot,
however, be controlled using this 1–DOF control approach.
While remaining cognizant of the limitations of the 1–DOF control approach, it has the advantage of
simplicity both in the control algorithm, as well as the hardware design. These are anticipated to lead to
reduced system costs and increased operational reliability. Control of the descent rate is envisioned to occur
by throttling of a parachute by venting or by cinching the shroud lines together, effectively decreasing the
cross-sectional area, affecting the drag force generated by the parachute. This variable–drag parachute is used
in conjunction with location and wind data, and a control algorithm to determine the optimal configuration
of the parachute needed to reach the desired landing location.
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II. Methods
As introduced above, the focus of this research is concentrated on developing a simple, low–cost approach
to autonomously control the landing location of a parachute descent vehicle. This is accomplished using
operating principles similar to those used to navigate a hot air balloon, i.e., control of altitude to take
advantage of differing wind directions at different altitudes. To accomplish this with a parachute system,
an adjustable parachute is used with wind data (speed and direction) and an algorithm that controls the
descent rate as a function of altitude; thus governing the horizontal distance traveled during the descent.
The predictive capabilities of the model will permit the control system to land the descent vehicle close to a
desired target “finish line” (Figure 1). As mentioned above, this control scheme is inherently 1–DOF, in that
the vehicle is assumed to always travel along the heading of the wind streamline (at all altitudes). Control
is achieved simply by deciding how long the vehicle should spend in each segment of the air column, each
with its own wind speed and direction. Foreknowledge of the wind speed and direction at each altitude is
therefore critical for the performance of the control system. For this investigative study, the control algorithm
assumes only a single actuation for the remainder of the descent. This algorithm is then periodically updated
to correct for errors in wind prediction, at which time the algorithm again assumes only a single actuation
for the remainder of the descent.
(x1,y1)
(x2,y2)
(xf, yf)
(x0,y0)
r
Projection
of descent
path, p
Finish line
segment
x
y
q
Figure 1: Projection of descent path, p, onto ground surface (z=zf) with target finish line segment shown.
Adjustments to the descent rate will scale pand r, but will not affect θ. The solution technique adjusts P
so that xf, yf, is coincident with the finish line segment.
The first necessary aspect of this work is a predictive model that calculates where the parachute will land
after release from a particular altitude and location. The second component of this work feeds into the first,
and concerns characterizing the range of descent rates that are achievable by throttling a parachute through
simple manipulation of the shroud lines (effectively decreasing the parachute area). Figure 2demonstrates
the approach used to throttle the parachute. The third component of this work builds on these results to
formulate a control algorithm to enable an autonomous descent vehicle to reach a linear target on the ground.
A. Development of the Descent Control System Algorithm
1. Overview of System
The complexities of controlling a descent vehicle are well–documented.7However, even with the simplified
1–DOF control approach employed here, there are still challenges resulting from imperfect knowledge of the
wind speeds and directions throughout the air column. However, if some knowledge of the wind is availabe,
such as from weather forecast data, then a descent path control algorithm may be used to navigate the vehicle
to a specified landing target. Typically, for many geographical areas, wind data is periodically measured at
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dr
dr
Shortened
shroud lines
(b) Choke(a) No choke
Figure 2: Parachute with and without throttling via shroud line choking. The effect of choking the shroud
lines (right) is to reduce the cross-sectional area of the parachute and reduce the drag force, Fd.
various locations and altitudes using sounding balloons, and is then used in forecast models. However, use
of such forecast data can be detrimentally inaccurate if the descent vehicle is deployed at a different time
and/or location than the data used in the forecast. Inaccurate wind data will result in improper parachute
trim commands being issued by the control algorithm. An alternative is to make direct measurements of
the wind field immediately prior to deployment of the descent vehicle using a sounding balloon or a suitable
dropsonde deployed from an aircraft.8
Figure 1depicts the projection of the descent path onto the ground, p, in relation to the targeted “finish
line”, which takes the mathematical form of a line between two points, (x1, y1) & (x2, y2). The algorithm
will attempt to control the descent rate so that it lands somewhere along this finish line. The descent path
projection poriginates at the projected release point (x0, y0), and terminates at (xf, yf), where the projected
path’s start and end are connected by range vector r(of length r), at a bearing angle θfrom the release
point.
When only actuating the parachute once (constant Arfor remainder of descent), θremains constant.
Thus the landing location (or finish line) can be reached if the range vector, r, intersects the finish line
assuming the landing location is within the performance characteristics of the parachute. Then, by choosing
the appropriate parachute area (thereby controlling descent rate), the range can be scaled so that the descent
vehicle will land on the finish line.
The descent path system’s control algorithm includes the following components:
1. Obtain wind field data (direction and speed data as a function of altitude), w(z). This can come from
forecast data, or can be directly measured immediately prior to descent.
2. Identify the current location and target landing coordinates (assumed to be known at the start).
3. Using an initial estimate of the desired descent profile and the wind field data, w(z), calculate the
reference predicted descent path ˆpprojected onto the ground. The shape of this path will be invariant
with parachute throttling, and can thus be directly scaled for different values of the parachute area,
Ar.
4. Determine the necessary parachute area, Ar, that scales the reference descent path so that its end falls
along the targeted finish line.
5. Adjust throttling of parachute to achieve the necessary parachute area, Ar.
6. In practice, this calculation can be periodically updated (repeat steps 2-5), and adjustments to the
parachute trim can be made to adapt to changing conditions during the descent.
2. Descent Profile Prediction
Implementation of the above control algorithm requires knowledge of the wind velocity field, w(z), as hor-
izontal motion is strictly dictated by only this. For simplification, it is assumed that all drag forces are in
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the vertical direction, with no lift forces due to lateral relative wind speeds. By expressing the equation
of motion in the vertical direction, and applying a numerical solver, an altitude versus time expression is
generated, denoted as the descent profile prediction, ˆz(t). Figure 3shows a graphical representation of the
descent profile prediction, ˆz(t), compared with the actual descent profile, z(t).
0 500 1000 1500 2000
0
5
10
15
20
25
30
Time [s]
Altitude [km]
ˆzRK(t)
z(t)
Figure 3: Predicted descent profile, ˆz(t), as calculated using the Runge-Kutta numerical solver. The predicted
descent profile shows a strong correlation with the actual descent profile, z(t); however, there is a propagation
of errors resulting in a progressively–growing deviation between predicted and actual.
The equation of motion in the vertical direction is expressed in terms of gravitational and drag forces
acting on the parachute. This is written as
m¨z=Fd−Fg.(1)
In Equation (1), the gravitational force is the conventional Fg=mg and the drag force is assumed to be
in the vertical direction, and of the form
Fd=1
2ρ(z)ArCd˙z2.(2)
Equations (1) and (2) combine to yield
¨z=ρ(z)Cd˙z2
2m−g. (3)
It is important to note that the density cannot be assumed constant and therefore a closed–form solution
to z(t) using Equation (3) does not exist. However, using standard atmospheric data,9Equation (3) can be
solved numerically using the Runge–Kutta method to produce the altitude as a function of time, ˆz(t).
Although Runge-Kutta provides an accurate numerical solution, ˆz(t), a more computationally–efficient
method results from the assumption that the vehicle is at terminal velocity at all times. The velocity at
each time step can then simply be calculated by using Equation (3) and setting ¨z= 0 as it is assumed the
descent vehicle does not accelerate during each time step (requires sufficiently–small time step). Note that
the terminal velocity is treated as changing between each time step due to increasing air density. For the
nth time step, the descent rate takes the form:
˙zn=s2mg
ρ(zn)Cd
(4)
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In order to calculate the entire predicted descent profile,ˆzT V (t), the terminal velocity must be calculated
at each time step (using the corresponding air density). Once the velocity has been calculated for a particular
time step, the change in altitude can be determined by using the zero–acceleration kinematics equation:
ˆzT V
n= ˆzT V
n−1+ ˙zn−1∆t(5)
The terminal velocity assumption was tested by comparing simulations run using both the Runga–Kutta
and terminal velocity approaches. Figure 4shows the difference (residual error) between Runge–Kutta
and terminal velocity simulations. Integration time steps were chosen through convergence testing, which
showed highest accuracy in the terminal velocity simulations is obtained by integrating at the same interval
as measured data is sampled (1Hz). With the maximum error near deployment, the cumulative error over an
entire descent is ≈135mwhich is well–within predictive limits set by changing wind speeds. An important
detail is that whereas the terminal velocity technique approximates the Runge-Kutta simulation very well,
both predictive techniques be differ from the actual descent profile. From these results, it is concluded that
the terminal velocity predictive technique is sufficiently accurate, and because of its simplicity, will thus be
used for all descent prediction profile calculations (ˆz(t) = ˆzT V (t)).
510152025
−40
−20
0
20
40
60
ˆzRK - ˆzT V [m]
Altitude [km]
Figure 4: Residual error between Runge-Kutta simulation and terminal velocity simulation for typical descent
vehicle (starting altitude is 28.7km). The error decreases during the descent; however, the difference between
the Runge-Kutta and terminal velocity simulations is small (less than 70m).
Incorporating the descent profile prediction with predicted wind data, a descent path prediction, ˆ
P, can
be generated. As discussed previously, the wind data is incorporated into the simulation by assuming the
descent vehicle travels at the wind velocity at the respective altitude. By varying the parachute surface area,
Ar, the various possible landing locations can be determined. This process is depicted visually in Figure 5,
where five different values of Arwere considered to produce the desired range value of r= 10.3km.
It is important to note, for this study, if the descent vehicle is incapable of reaching the finish line
(outside of the performance envelope), it will attempt to get as close as possible. Future work will investigate
the use of multiple landing locations as well as continuous actuation (resulting in 2D control). This two-
dimensional control results from throttling at different times, presenting the opportunity to “ride” various
winds (depending on desired landing location). This will allow for basic two dimensional control (landing
location is a point rather than line segment). Additionally, in this study, if the wind data is not available
for some segment of the descent phase, the algorithm will extrapolate available data to predict the wind
direction and velocity.
3. Testing the Descent Control System Algorithm
To test the predictive portion of the algorithm outlined above (as well as coefficient of drag measurements
discussed later), a set of simulations were conducted using experimental data obtained by dropping GPS log-
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−1 0 1 2 3 4 5
−2
0
2
4
6
8
10
12
X−Distance (km)
Y−distance (km)
CdA = 0.91m2
CdA = 1.18m2
CdA = 1.42m2
CdA = 1.60m2
(xo,
yo)
r
Finish
Line
Alternate
Landing
Locations
Figure 5: Typical simulated descent profile (projection) for various parachute throttling values and the
parachute reference area, Ar. Note that regardless of the apparent area, Ar, the descent path will always
terminate along the range vector r.
ging payloads via weather balloon from altitudes ranging from 10–30 km. The release altitudes were reached
using conventional latex weather balloons, with GPS location and altitude data recorded and telemetered
to the ground tracking station. A figure illustrating a typical experimental setup is shown in Figure 6.
Parachute
Communications
Payloads
Balloon
Parachute
Deployed
Figure 6: High altitude bal-
loon payload system configura-
tion. When balloon bursts or
is released, the parachute inflates
and the system descends (inset).
Ideally, errors resulting from inaccurate wind data must be minimized.
That is, a well-characterized air column, w(z), is needed from the outset.
This information was obtained during the initial ascent phase of the
balloon (up to release altitude), gathering horizontal speed data, which
was assumed to be equal to the wind speed, giving a local and current
w(z) data set.
The ascent phase (wind characterization) of a typical test lasts 60–90
minutes, after which deployment of the parachute descent system occurs,
usually when the balloon bursts, or the payload is remotely released from
the balloon from the ground via radio link. Descent typically takes 45–60
minutes, depending on parachute configuration and release altitude.
After landing, the payload was recovered, and the data describing the
position as a function of time was recovered from the data logger (radio
downlink of these data served as a backup). The ascent phase data was
used to construct a w(z) wind data set, and the descent phase data served
as a comparison test for the (unchanging) parachute used in the experi-
ment. This provided actual descent path data, Pas a function of time,
for comparison with simulation results, ˆ
P, generated by the algorithm
(conducted post–mission). Unlike the algorithm described previously, in
these experiments the parachute size was fixed at the start of the descent.
The simulation algorithm was therefore run with a similar constraint and
the simulated landing location was compared with the experimentally–
measured location. Thus far, 19 such experimental data sets have been
obtained for validation of simulations, with system masses ranging from
1.67–6.14kg.
B. Characterizing Parachute Performance
The second critical component of this 1–DOF descent control system
concerns accurately controlling the drag force generated by the parachute,
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as described by Equation (2). As discussed above, the parachute drag (Fd) will be controlled through
throttling of the parachute (Figure 2). To characterize this, a set of scaled wind tunnel tests were conducted
using varying parachute sizes and wind speeds to simulate the range of Reynolds numbers expected during
a controlled descent experiment.
Figure 7illustrates the setup used for wind tunnel testing of a dimensionally–scaled parachute. The
available facilities limited the parachute diameter to ∼0.6m, when placed in the diverging flow section at
the wind tunnel exit. It is recognized that this flow is not as steady and uniform as the conventional test
section of the tunnel, but nonetheless provided the best results with the smallest available parachute that
was geometrically similar to the larger parachutes used in the balloon–deployed tests (described above). The
drag force of the parachute, Fdwas measured using a digital force gauge (Omega DFG51-100, ±0.9N). The
wind speed was measured using a portable flow meter (Kestral 2000, ±3%). The wind speed was measured at
the location of the force gauge at the end of the shroud lines for repeatability and proximity to the parachute
location. To quantify any effects due to performing the tests in the divergent section of the wind tunnel,
data were collected that showed only minor reduction in velocity (less than 8%) within the divergent section
of the tunnel. Wind speed measurements were also made with the parachute both inflated and deflated to
confirm that flow blockage in the tunnel affected the wind speed by less than 5%.
Figure 7: Wind tunnel setup used to measure force and velocity.
Initial drag force measurements showed excessive oscillations of ∼50% of the mean, due to oscillatory
movements of the parachute in the flow. This was addressed by constraining the lateral motion (x/y plane) at
the top of the parachute at point A in Figure 7. This stabilization significantly reduced drag force oscillations
in the force measurements, with oscillations of ∼15–20% of the mean. Each experiment was run for ∼60
seconds at approximately 30 samples/sec, yielding records of ∼1800 drag force measurements which were
used to obtain the mean drag force.
Four different wind speeds were used with five different throttling positions (Ar) for each speed, yielding
twenty total experiments. The reference area of the parachute, Ar, was calculated by first measuring the
reference diameter, dr, of the throttled (choked) parachute at its widest point, and calculating the area
normal to the wind flow, Ar= (π/4)d2
r. Uncertainties of all measured parameters were combined to produce
reported values (and error bars in figures) using the standard propagation of error methodology.
The measured data were then used to calculate the Reynolds number and coefficient of drag using
Re =ρU∞dr
µ, and (6)
Cd=2Fd
ρArU2
∞
.(7)
III. Results and Discussion
A. Parachute Performance
The wind tunnel data was used to estimate the Reynolds number and coefficient of drag using Equations (6)
and (7). The resulting calculated values of Cdand Re are plotted in Figure 8, which shows how the coefficient
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of drag would vary over the Reynolds numbers experienced during a high altitude drop test. Cd,ave would
be the resulting average drag coefficient over an entire flight test (as shown below).
0
0.2
0.4
0.6
0.8
1
1.0E+05 2.0E+05 3.0E+05 4.0E+05 5.0E+05
Drag Coefficient, C
d
Reynolds Number, Re
0.144
0.128
0.107
0.091
0.082
Cd,ave= 0.73
m2
Figure 8: Coefficient of drag versus Reynolds number for 51 cm (20 in) parachute. Cd,ave is an average of
0.73 over the range of 2×10−5≤Re ≤4×10−5experienced during a controlled–descent test deployed from
a high–altitude balloon. Symbols represent differing parachute areas, Ar.
Wind tunnel results show a reasonably constant Cd(Cd,ave ≈0.73), which demonstrates that the shape
of the parachute is approximately constant regardless of the throttle position. As the wind tunnel testing was
performed on scaled parachutes, drop testing was necessary to validate the results obtained from the wind
tunnel. 19 drop tests were performed with varying starting altitudes (10-30km), as well as payload-parachute
configurations as described in Table 2. The amount of throttling applied to a particular test is evident when
comparing the parachute surface area, Ar, with the maximum (unthrottled) parachute area, Armax.
The drop test results show large variations in the mean Cd(range = 0.7−1.5), with large standard
deviations, σCd, as well. To investigate trends in the average Cd, it was plotted as a function of parachute
loading (defined as payload mass divided by Ar) in Figure 9. Additionally, the average Cdwas plotted as
a function of deployment altitude (starting altitude) as shown in Figure 10. Clearly from both figures, the
average drag coefficient is not constant between tests, and there are no visible trends induced by the setup
tested (i.e. release altitude). The typical Cdfrom a Ar= 1.59m2parachute is approximately 1.0. This value
is used as the constant coefficient of drag for all simulations as discussed in Section B.
No statistically-significant trends are present in the highly-variable drop test data (Figures 9&10). It
is important to note that although the Cddata shows high variability, the error can in fact come from
incorrect parachute area or an error in Cditself as the measured parameter is the descent velocity, ˙z. Since
the measured parameter is the descent velocity, all accrued errors are simply placed in the coefficient of drag
for convenience. As the error in the descent profile is directly related to errors in descent velocity (used
in the coefficient of drag), an adaptive algorithm must be used to correct the coefficient of drag parameter
based on actual data measured during the descent phase. The adaptive algorithm development and results
are presented later in Section C.
B. Descent Control System Algorithm
Figure 5, presented earlier, illustrates the descent path prediction algorithm’s capability of iteratively iden-
tifying the necessary parachute area needed to land on the targeted “finish line”. This demonstration of the
algorithm is complemented here by the simulation tests of the descent control algorithm, conducted using
experimental data as wind inputs, and experimentally observed landing positions as validation tests for the
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Table 2: Parachute data gathered from weather balloon drop testing.
Test Mass Armax Arzstar t Cd,ave σCd
(kg) (m2) (m2) (m)
1 6.14 2.14 2.14 27922 1.2 0.1
2 5.70 1.59 1.59 24438 1.0 0.2
3 5.70 1.59 1.59 23934 1.0 0.3
4 4.84 1.59 1.59 14848 1.0 0.1
5 2.44 4.56 4.56 10581 1.2 0.3
6 2.44 1.59 1.59 12881 1.1 0.1
7 4.84 1.59 1.59 15860 1.0 0.1
8 4.84 4.56 4.56 11898 1.0 0.1
9 2.44 4.56 1.59 7504 1.1 0.2
10 2.49 1.59 0.84 8281 1.3 0.2
11 1.91 1.59 0.64 11462 1.2 0.1
12 4.51 4.56 3.05 34394 0.7 0.2
13 5.24 4.56 3.87 32767 1.1 0.1
14 3.04 1.59 1.59 28739 0.8 0.1
15 1.98 1.59 1.00 11878 1.3 0.2
16 3.21 1.59 1.38 13653 1.2 0.1
17 1.75 2.14 1.59 27909 1.2 0.3
18 1.67 1.59 0.56 24132 1.5 0.2
19 2.40 1.59 1.59 29515 1.0 0.3
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
0 5 10 15 20 25 30 35 40
Average Drag Coefficent, Cd,ave
Parachute Loading [N/m²]
Figure 9: Average coefficient of drag versus parachute loading (defined as vehicle mass divided by parachute
area, Ar). Each data point comes from an average of the coefficient of drag calculated at each time step.
From the figure it is evident there are no significant trends when compared with the loading on the parachute.
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0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
0 5 10 15 20 25 30 35 40
Average Drag Coefficient, Cd,ave
Deployment Altitude [km]
Figure 10: Average coefficient of drag versus deployment altitude. Each data point comes from an average
of the coefficient of drag calculated at each time step. There are no significant trends evident in the data
when compared with the deployment altitude.
algorithm’s descent path predictions. An example comparison between a predicted and experimentally–
observed descent path is shown in Figure 11, shown in latitude and longitude coordinates. In the shown
case, the predicted landing location is 6195m from the actual landing location, after release from 24 km.
39.2
39.3
39.4
39.5
−119.5 −119.4 −119.3 −119.2 −119.1 −119
0
5
10
15
20
25
Latitude
Longitude
Altitude (km)
Prediction
Actual
Figure 11: Predicted descent path, ˆ
P, versus experimentally–observed path, P, after release from 24 km.
Landing locations differ by 6195m.
Summarizing data from the 19 simulation tests are shown in Table 3. Key tabulated parameters in-
clude the payload mass, the release (start) and landing (end) altitudes, and the errors in the landing
location in both the distance, ∆r=rpred −rexpt, and bearing, ∆θ=θpr ed −θexpt. The overall error,
∆ = p(xpred −xexpt)2+ (ypr ed −yexpt)2is normalized using the linear 3–D distance traveled, |R|, which
incorporates contributions from both the height of release, and the strength of the wind. All tests were
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compared with predictions using a Cd= 1.0.
Table 3: Descent simulation test data using data from 19 drop tests.
Test Mass Arzstart zend ∆r∆θ|∆|
|R|
(kg) (m2) (m) (m) (m) (deg.)
1 6.14 2.14 27922 1837 1158 -1.86 4.2%
2 5.70 1.59 24438 1621 456 1.29 2.2%
3 5.70 1.59 23934 1501 1290 -0.94 5.5%
4 4.84 1.59 14848 1261 931 2.02 4.0%
5 2.44 4.56 10581 1596 5550 -5.67 12.8%
6 2.44 1.59 12881 1348 3220 0.54 6.4%
7 4.84 1.59 15860 1690 237 1.18 2.0%
8 4.84 4.56 11898 1563 59 2.89 4.9%
9 2.44 1.59 7504 1254 557 -4.01 7.2%
10 2.49 0.84 8281 1494 1343 4.16 11.1%
11 1.91 0.64 11462 1521 1543 3.20 7.6%
12 4.51 3.05 34394 1450 3347 5.43 10.1%
13 5.24 3.87 32767 1302 6593 3.12 14.2%
14 3.04 1.59 28739 1325 1976 4.01 7.2%
15 1.98 1.00 11878 1326 918 0.33 6.6%
16 3.21 1.38 13653 1337 214 0.02 1.5%
17 1.75 1.59 27909 1294 12569 3.29 17.4%
18 1.67 0.56 24132 1543 6195 2.35 13.2%
19 2.40 1.59 29515 1422 2994 0.58 5.0%
C. Adaptive CdAlgorithm
A proof-of-concept algorithm was used on test data in which Cd,ave = 1.5 (Test #18) to improve the descent
profile prediction by modifying the initial drag coefficient guess, Cd= 1.0. This adaptive algorithm was
developed to use the previous five minutes of real- time descent data. By inputting the best guess drag
coefficient, Cd= 1.0, measured Ar, and mass, the simulation was performed as per the descent prediction
algorithm discussed previously. The average Cdwas then calculated using the previous real-time data via the
terminal velocity equations. Figure 12 shows the improved predictive capabilities that result from estimating
the actual Cdmid-descent using actual descent data. The rapid decrease in root mean squared error (RMSE)
indicates the predicted descent profile, ˆz(t), quickly converges to the actual profile, z(t). It is important to
note the drag coefficient is not constant over an entire drop test, and therefore the converging solution will
not equate to the full test average drag coefficient. Using this refinement to the predicted descent profile,
ˆz(t), the full predicted descent path, ˆ
P, can be determined more accurately.
Error in landing location is predominantly due to errors in either the descent profile, ˆz(t), or differences
between predicted and actual descent wind data, w(z). Using the adaptive algorithm to improve the descent
profile reduces only one contribution to the error in landing location. Figure 13 shows the residual error in
landing location normalized by the total distance remaining to be traveled (from current location to landing
location). The initial error remains at ≈17% until five minutes has elapsed at which time the adaptive
algorithm begins using previous data to improve the descent profile. Immediately after the adaptive algorithm
begins, the error quickly decreases down to the minimum (≈5%); however, the error quickly increases to
20+%. This increase in error is due to the other main cause in error, changing wind speeds between prediction
and descent. This error is quantified on the second y-axis in Figure 13, in which the largest wind residual
(difference between predicted and actual horizontal wind speed) occurs at low altitude. This causes the
prediction to deteriorate as the simulation progresses to lower altitudes as only poor wind data remains.
The larger the percentage of poor wind data, the larger the error will be. It is important to note the residual
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errors are an artifact of the wind predicting technique (using a high-altitude balloon); however, similar errors
can occur whenever the predicted wind data is reported from either a different time and/or location (i.e.
reported data from NOAA). Future work will investigate a robust control algorithm capable of predicting
and adjusting to these wind errors through the use of a dropsonde.
0 500 1000 1500 2000 2500
0
5
10
15
20
25
Time [s]
Altitude [km]
z(t)
ˆz(t)
0 500 1000 1500 2000 2500
0
5
10
15
20
25
Time [s]
Altitude [km]
0 500 1000 1500 2000 2500
0
5
10
15
20
25
Time [s]
Altitude [km]
0 500 1000 1500 2000 2500
0
5
10
15
20
25
Time [s]
Altitude [km]
Figure 12: Sequence of four descent profile simulations at 0, 10, 20, and 30 minutes respectively. Each
simulation is calculated using the average drag coefficient from the previous five minutes of descent data.
By estimating the drag coefficient in flight, the predicted descent profile is drastically improved.
0510152025
0
10
20
30
Percentage Error in Landing Location
Altitude [km]
0510152025 -20
-10
0
10
Actual - Predicted Wind [m/s]
Figure 13: Percentage error in landing location normalized by the total distance traveled (calculated for
each iteration). Initial error drop signals the start of the adaptive algorithm (after five minutes). The error
slowly increases as the vehicle descends due to differing predicted (gathered during ascent) and actual wind
velocities.
IV. Conclusions
An approach was developed to autonomously navigate towards a finish line using a variable drag parachute
(via throttling). First, the parachute performance was quantified using wind tunnel experiments. These
experiments showed constant coefficient of drag regardless of throttle position. Next, the parachute perfor-
mance was verified using high-altitude drop tests. These tests clearly show large variations in both mean
coefficient of drag and scatter in the coefficient of drag (standard deviation). In order for the descent ve-
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hicle to accurately navigate towards the desired line segment, the descent profile (z(t)) must be adequately
predicted. As the drop tests showed difficulty in making precise profile predictions using a priori knowledge
(measurements), an adaptive algorithm was developed to utilize descent data during the early stages of
descent to improve estimates of the coefficient of drag (thereby improving descent profile prediction). Using
this adaptive algorithm, the errors in z(t) were significantly reduced, leaving only errors due to changing
wind velocities between prediction and descent.
Once the descent profile has been characterized, the full descent path prediction can be performed. This
led to development of the descent control system algorithm. This algorithm uses the descent profile, and
determines the horizontal distance traveled at each altitude using predicted wind velocity data. This provides
a full three-dimensional profile of the descent path. Using only a single actuation, the algorithm iterates
to find the required parachute area needed to land on the desired line segment (or finish line). Simulations
show promise for the algorithm, although improvements may be necessary in control robustness to overcome
wind prediction errors at low altitudes.
V. Acknowledgement
Funding provided by the Nevada NASA Space Grant Consortium through NASA grant number NNX10AJ82H.
References
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