Maneuver Regulation for Accelerating Bodies in Atmospheric
Environments

arXiv:1708.01838v1 [] 6 Aug 2017

Juan-Pablo Afman1 , Eric Feron1 , John Hauser3
Abstract— In order to address the need for an affordable
reduced gravity test platform, this work focuses on the analysis
and implementation of atmospheric acceleration tracking with
an autonomous aerial vehicle. As proof of concept, the vehicle
is designed with the objective of flying accurate reduced-gravity
parabolas. Suggestions from both academia and industry were
taken into account, as well as requirements imposed by a
regulatory agency. The novelty of this work is the Proportional
Integral Ramp Quadratic PIRQ controller, which is employed
to counteract the aerodynamic forces impeding the vehicles
constant acceleration during the maneuver. The stability of the
free-fall maneuver under this controller is studied in detail via
the formation of the transverse dynamics and the application
of the circle criterion. The implementation of such a controller
is then outlined, and the PIRQ controller is validated through a
flight test, where the vehicle successfully tracks Martian gravity
0.378 G’s with a standard deviation of 0.0426.

Bremen tower [9], do so through centrifuges, which 1) are
still limited to the drop time of a microgravity experiment
and 2) induce a gravitational gradient across the experiment.
Conceptual efforts to overcome this limitation have been
proposed in recent work through the use of magnetism,
although they have not been implemented in practice to this
date [13]. Furthermore, a single 5.18 second drop in the
Zero-Gravity Research Facility at NASA Glenn, can cost
up to $9,000 U.S. dollars, and researchers are limited to
two drops per day [10]. Although there are benefits to the
drop tower infrastructure, the major drawback is the large
upfront cost for the infrastructure, its maintenance, operator
requirements, and capacity; if a large number of users must
be accommodated at the same time, a facility’s accessibility
diminishes considerably.

M OTIVATION
Enabling safe and reliable reduced-gravity environments
for short temporal intervals is the objective behind this work.
A wide range of scientific disciplines employ reduced-gravity
conditions as a tool to investigate phenomena of processes in
the absence of Earth’s gravitational field. These disciplines
include both life sciences as well as physical sciences. When
gravity is removed, other forces such as surface tension and
capillary forces, become predominant and drive a different
system dynamics. Gravity is a physical parameter, that together with pressure and temperature, define the state of
a system. Historically, major breakthrough and innovations
were achieved when systems were studied, for example,
at low temperatures. The duration of the reduced gravity
periods can range from a matter of seconds to months,
depending on the application and platform employed [1].
The scope of this work targets the development of an Earthbased vehicle with mission critical safety and autonomy, and
can deliver adequate gravitational fields to the hands of the
students and researchers. Although reduced-gravity research
has proven itself useful for public benefit through breakthroughs in the pharmaceutic, metallurgy, communications,
electronics [2]–[7], the means of providing an affordable and
reliable unmanned test facility remains an unsolved problem.
Two common Earth-based solutions employed for research experiments include drop towers [8]–[10] and the the
parabolic manned-aircraft [11], [26]. While drop towers can
offer anywhere from 1.4 and up to 9 seconds of microgravity,
the few towers capable of providing reduced-gravity, like the
1 J.P. Afman, E. Feron are with the Department of Aerospace Engineering
at the Georgia Institute of Technology, Atlanta, GA 30332, USA.
3 John Hauser is a private consultant.

Perhaps the most common way of performing Earthbased reduced gravity experiments is through parabolic
flight, which is is capable of providing an entire spectrum
of gravitational environments for approximately 17-23 sec.
intervals. Due to the special parabolic flight aircraft instrumentation, crew, and maintenance required, the operational
costs associated with this platform are inhibiting for most
researchers. The Zero Gravity Coorporation R is currently the
only commercial provider of microgravity flight services in
the United States. According to the Zero-G Weightless Lab R
pricing information, single flight costs range from $7,150
U.S. dollars + 5% tax per flight for a hand held object, to
$38,500 + 5% tax per flight for a larger experiment with
a footprint no larger than 10 f t 2 [14]. The other obvious
draw back expressed regular users is the lack of repetition,
as experiments often fail in flight and experiment data is not
aquired. Even at these high costs, NASA Inspector General
Paul K. Martin reported that the Zero Gravity Coorporation
has been incapable of delivering appropriate reduced-gravity
environments [15]. The Zero Gravity Corporation R later
pointed out that it had made significant investments in
configuring its plane to NASA specifications and training
pilots to conduct reduced gravity flights, which at minimum
highlights the importance of feedback control for performing
parabolic maneuver.
Although limited in scope, many studies can be performed
with only a few seconds of reduced-gravity. In order to
fulfill the gap faced by researchers from many fields, the
development of a multirotor-based platform for reducedgravity environments with increased availability, reliability,
and reduced costs, is formalized.

C ONCEPT R EVIEW AND R EQUIREMENT D EFINITION
Review of prior art
There are major benefits, both financial and logistics, for
enabling reduced gravity environments by means of removing the pilot from the vehicle, and decreasing infrastructure
and operational costs. Some of these benefits include mission
reliability, quick turn-around time, and minimum maintenance. These advantages have been the major incentive
behind the work of several pioners, who have dedicated
their efforts towards enabling autonomous reduced gravity
environments over the past 22 years [16]–[21].
To the best knowledge of the authors, every unmanned
attempt to this date has been fixed a fixed winged aircraft,
and sadly each attempt has failed to produce reliable and
repeatable reduced-gravity environments. It is noted here
that these failures are not due to the vehicle configuration,
but rather that they have fallen victims of attempting to
perform gravitational tracking in atmospheric environments
through the means of a single Proportional Integral control
law. From classical control theory, it is well understood that
a single integrator is incapable of dynamically inverting the
quadratic growing disturbance; a disturbance that rises due to
the non linearity associated with aerodynamic forces while
performing the maneuver.
In 1995, Meshland [16] published his experimental work
on the BAR-1 aircraft, and the conceptual design of the
BAR next. Cancellation of lift was obtained by controlling
the elevator. The BAR-1 employed a feedback control law
designed to minimize the error between a desired G set-point,
and the actual measurement. Furthermore, this control input
was augmented by a feed forward controller that too into
account the G set-point, and the vehicle’s forward airspeed.
Unfortunately, the experimental results for both microgravity and reduced-gravity illustrate the controller’s inability
to reliably compensate to produced acceptable experimental
conditions. No further work was found regarding the BAR
next concept. In 2006, Kraeger [17] proposed augmenting the
BAR next concept previously conceptualized by Meshland,
through the conceptual design of a free-wing unmanned
aircraft. In his work, he shows that for this configuration,
a relatively simply PI control architecture is sufficient to
fly highly accurate reduced-gravity trajectories by applying
torque to the wing. Unfortunately, this work never left the
conceptual design phase, and hence validation of his claims
were never confirmed experimentally. In 2009, Hofmeister
and Blum [18] proposed a jet-powered reduced-gravity test
bed employing a modified unmanned aircraft from URModellbau. The G-regulating feedback control structure was
a simple PID, where the integral component was chosen to
regulate the continuous set point deviation. Unfortunately,
the experimental results for a micro-gravity maneuver show
a 0.4 G RMS, which they claim is due to the aircraft being
incapable of adjusting its absolute airspeed due to the vehicle’s inertia. In 2010, Higashino and Kozai [19] attempted
the single integrator feedback control law that had failed
in previous work, only to find similar experimental results.

In 2011, a project led by Jeffery Fox from NASA Johnson
Space Center [21] effort proposed a reduced-gravity platform employing a commercially-available Troybuilt DV8R
remote-controlled jet aircraft. To the best of the author’s
knowledge, feedback control was never implemented and the
work was never made publicly availabe. In 2016, Hathaway
also proposed the use of a PID feedback control architecture,
but his work never left the simulation phase. [20]
Requirements and Configuration
A product discovery effort led by the research team, composed of interviews performed at over a dozen universities,
research institutions and private organizations, revealed the
requirements and constraints involved in the the development
of an autonomous reduced-gravity platform. These are now
itemized and described below.
• Affordability: Vehicle must cost less than $25,000 USD.
• Reliability: Vehicle must be capable of producing repeatable and reliable results.
• Safety and Convenience: Vehicle must address inconveniences faced by other platforms while providing the
same level of safety.
• Time Exposure: Vehicle must be able to provide at
minimum 2 sec. of reduced-gravity conditions.
• Payload: Vehicle must be able to carry a 0.5 kg payload
and a volume of at least 10 cm3
• Regulatory Constraints: Vehicle must not violate regulations relative to the operation of civilian drones.
In order to further define the reliability factor, flight
specifications employed by NASA and the Zero Gravity
Corporation R were employed. Table 1 shows the flight specifications and requirements employed by NASA for judging
the adequacy of reduced gravity time offered by the Zero
Gravity Corporation R .
TABLE I
F LIGHT S PECIFICATION R EQUIREMENTS [15]
Reduced-Gravity (g’s)
0.00 g +/- 0.02
0.00 g +/- 0.05
0.10 g +/- 0.05
0.16 g +/- 0.05
0.20 g +/- 0.05
0.30 g +/- 0.05
0.38 g +/- 0.05
0.40 g +/- 0.05
0.50 g +/- 0.05

Continuous Time (sec)
10
17
20
20
20
20
20
20
20

The decision to use an autonomous UAV was made
since it fits perfectly within the requirements and constraints
discovered during the product discovery process. From a
financial standpoint, most UAVs are mechanically simple
and massively available on the commercial market free of
restrictions, which makes them available and affordable to
any academic research institution for open research, which
makes them possible to meet the $25,000 USD price tag.
Second, autonomy and feedback control can be implemented
in order to produce repeatable and reliable results on every
flight. With fault detection algorithms executing at 333

with approximately 5 times control bandwidth, which is
capable of fighting drag disturbances independent of their
direction through positive and negative thrust deflections, all
while maintaining attitude control authority independent of
the thrust required during a reduced-gravity tracking flight.
Figure 2 illustrates one of the four variable-pitch mechanisms employed by the prototype, where pitch actuation is
performed by a high-speed, high-torque digital servo.

Gy (m/s2)

Gx (m/s2)

Hz, it is possible to address the adequate level of safety
requested. Furthermore, placing these vehicles on the hands
of students and researchers covers the inconveniences face
with all other Earth-based platforms. Lastly, time exposures,
payloads, and regulations can all be met through proper
vehicle
sizing. Federal regulations posed by the US Federal
100
Aviation
Administration
permits its use as long as it remains
80
Acceleration
0g Reference
within
eyesight,
under 400 f t (approximately 120 m) and
60
with
Crash Reading: 25 kg). In
40 a mass lower than 55 lbs (approximately
~8 g's
order
to
increase
the
convenience
factor,
the choice for an
20
unmanned
vehicle
capable
of
vertical
takeoff-and-landing
0
VTOL
implies minimum real-state requirements.
-20
Several
modifications to the conventional multi-rotor ar-40
chitecture
are necessary to obtain a vehicle capable of
-60
295
296
297
298
299
300
301
302 a
delivering
proper
reduced-gravity
environments.
Tracking
120
parabolic trajectory requires the use of both positive and
100
negative
thrust forces in order to compensate aerodynamic
80
drag
in ascent and descent portions of the parabolic flight;
60
Crash
Reading:
something
that a conventional fixed-pitch
multi-rotor
simply
~10 g's
40
cannot perform. Furthermore, since the thrust profile on a
20
fixed-pitch
quadrotor is a quadratic function of the propeller
0
rotational
velocity, control authority tends to zero as the
-20
thrust
required goes to zero and hence, vehicle instability
-40
occurs.
Fig.2961 shows
experimental
data 300for a 301
fixed-pitch
295
297
298
299
302
multi-rotor attempting to track a reduced-gravity profile.
30
20

Uncontrolled recovery leads to crash

Fig. 2.

Variable pitch mechanism

Hence, the remainder of the work focuses on the development of an autonomous vairable-pitch multi-rotor, capable of
performing a temporal parabola resulting in reduced gravitational environments. This platform is not intended to replace
any existing platforms, but rather act as a complimentary tool
that researchers can use to speed up their development prior
to the large investment associated with other Earth-based, or
even space-based platforms.

Gz (m/s 2)

10
Drag force effects

0

M ATHEMATICAL M ODELING & S YSTEM I DENTIFICATION

-10
-20
Approaching
Zero-G's

-30
-40
295

296

297

298

299

300

301

302

300

301

302

Throttle Input

100
Initiate Drop

80

Recovery
(second
attempt)

Initiate
Autorecovery

60
40

Throttle (%)
Current (A)

20
0
295

296

297

298

299

Time (sec)

Fig. 1.

Zero G tracking attempt with fixed pitch quadrotor

It is noted that both the slow responsiveness of fixed-pitch
rotors, due to rotational inertia, as well as aerodynamic drag
forces, prevent the vehicle from properly tracking a constant
acceleration field. Furthermore, with thrust forces nearly
zero, the the fixed-pitch vehicle is incapable of regulating
its attitude during the maneuver and hence the flight test
results end up with the loss of a vehicle.
To overcome the issues observed during fixed-pitch multirotor experiments, the research team looked towards the
benefits of variable-pitch multi-rotors. This modification in
the approach, which employs a constant rotor speed and n
independent pitch actuators, results in a responsive system

The development of a model inside of a simulation environment enables the evaluation of modeling assumptions
and control algorithms. Fig. 3 is a high level illustration of
the model developed inside the Simulink environment.
In order to visualize the model properly, a direct link is
established between Simulink and FlightGear. A link to a
video of the Simulink-FlightGear simulation is provided on
the caption of Fig. ??
Equations of Motion
A full 6 DOF model is derived. We begin by introducing
the linear kinematic equations of motion for a rigid body,
which is an underlying assumption for the concept vehicle,
and given by the Newton-Euler equations (1).
u̇ = vr − wq − gsinθ + Fx /m
v̇ = wp − ur + gsinφ cosθ + Fy /m

(1)

ẇ = uq − vp + gcosφ cosθ + Fz /m
Note that the previous equations employ notation from the
~ = [u, v, w]T . Similarly, the angular
linear velocity vector U
kinematic equations of motion are given by the Newton-Euler
equations (2); the cross products of inertia are neglected.
ṗ = qr(Iyy − Izz )/Ixx + Mx /Ixx
q̇ = pr(Izz − Ixx )/Iyy + My /Iyy
ṙ = pq(Ixx − Iyy )/Izz + Mz /Izz

(2)

Note that the previous notation is composed of components
from the angular velocity vector ~Ω = [p, q, r]T and the inertia
tensor. The set of forces and moments acting on the system,
~ = [Mx , My , Mz ] respectively,
denoted by ~F = [Fx , Fy , Fz ] and M
are organized as
~F = Fgravity + Fpropulsion + Faerodynamic
(3)
~
M = Mdrag−torque + M propulsion
Fig. 4 illustrates the commonly used variables necessary to
describe a variable pitch quadrotor.

System Identification
In order to obtain a high fidelity simulation of the vehicle
and its environment, system identification was performed
on real hardware. This task enabled further understanding
of actuator dynamics for a fixed motor speed, while also
revealing the relationship between thrust and blade angle for
a fixed motor speed. The command ranges for a given blade
pitch are symmetric around the neutral point. The maximum
commanded deflections were set in radians at approximately
δr = ±0.09. An average static-thrust curve was obtained
by sweeping through the blade deflection envelope several
times, revealing the mean of the expected non-linear static
thrust curve for a given blade pitch as illustrated in Fig. 6.
Note that blade pitch is commanded through Pulse Width
Modulation PWM demand to each servo.

Fig. 4.
3D Render of a concept vehicle illustrating commonly used
aerospace notation

The rotational kinematic equations were mechanized using
quaternions [31]. The inertial velocities are derived from
the body-axis velocities by a coordinate transformation (flatEarth equations are used) and integrated to obtain inertial position. A fourth-order Bogacki-Shampine integration method
[32] is used, with an integration step of 0.003 seconds.
Figure 5 illustrates the vehicle performing the maneuver in
the simulation environment.

Fig. 6.
Average static thrust curve as function of the Pulse Width
Modulation PWM envelope

The Assault Reaper 500 R servos employed by the variable pitch mechanisms were subjected to small-amplitude
frequency sweeps, and pulse signals while the vehicle was
operational and fixed to a load cell. Since the system was
fully operational, proper loading was established on the servos. A normalized linear second-order system was acquired
through a least squares optimization algorithm, given in
transfer function form by
G p (s) =

Fig. 5.

Illustration of the simulation environment

Fig. 3.

1
0.0008s2 + 0.045s + 1

.

(4)

Its important to highlight that G p (s) combines the dynamics
from 1) the governor of the Electric Speed Controller ESC,
2) the brush-less DC motor 3) the digital servos and 4) the

Model Based Design in the Simulink Environment

propellers. Hence, this transfer function provides dynamics
for a combined thrust-producing actuator.
As described through the angular kinematic equations 2,
mass moment of inertia is also an important parameter for the
accurate dynamic modeling of aerospace vehicles. The bifilar
device, a two support wire vertical-axis torsional pendulum,
was employed to estimate the inertia of the physical test
bed, where a nonlinear equation was simplified by assuming
the angular motion to be small and by omitting damping.
This simplification produces a linear equation that yields the
expression that can be used to estimate mass moment of
inertia:
mgd 2 T 2
(5)
Inn =
16Lπ 2

flight sequence are reminiscent of prior work [35]. Figure 7
illustrates the automated logic embedded in the experimental
vehicle.

2

where the measured period T = ω2πn , d is the distance between
the two strings, L is the length of the strings, m the mass of
the body, and g represents gravity (9.807m/s2 ). The mean of
several consecutive tests are summarized as estimated values
in Table II. Products of inertia were neglected, and a mass
estimate was obtained using a high-accuracy laboratory scale.
TABLE II
E STIMATED PARAMETERS
Parameter
Ixx
Iyy
Izz
m

Estimate
0.0068 (kg · m2 )
0.0171 (kg · m2 )
0.0207 (kg · m2 )
1.265 (kg)

Other parameters such as drag coefficients Cd , planform
areas S, and the effects of vertical dynamics on thrust T (w)
were estimated and validated through correlation from flight
experiments.
Attitude Control
Since the scope of the work lies on the reduced-gravity
maneuver regulation, this work places little emphasis on the
attitude of the vehicle. However, once the characteristics of
the model were accounted for, the model was linearized
along its three primary axes and closed loop stability was
aquired while ensuring adequate phase and gain margins. The
attitude control law employed in the vehicle is commonly
known as a cascade feedback control law. After a thorough
review of control algorithms for autonomous quadrotors [33],
the cascade control strategy was chosen due to its simplicity,
precision, tracking ability, fast convergence and robustness.
Closed-loop stability for vehicle rates along each axis was
achieved first, through individual Proportional-Integral PI
controllers. This was followed by the closed-loop stability
of the vehicle’s attitude, where a proportional controller was
employed for the pitch and roll axis. Yaw angle is not a
controlled state in this framework.
M ANEUVER D ESCRIPTION AND A NALYSIS
The maneuver can be discretized by five autonomous state
transitions, which include hover, ascend, track, recover and
land. These automatic transitions during an overall aggressive

Fig. 7.

Autonomous logic represented as StateFlow diagram [36]

The maneuver is initiated through a manual switch, and
fault detection logic executes at each time step ensuring the
safe operation of the maneuver. Once triggered, the vehicle
initiates at ”hover” mode, where the vehicle is commanded
to hover at 2 meters from the ground. An altitude control law
regulates the vehicles vertical speed and position, regulating
such that the vehicle behaves as a first-order system with
zero overshoot. Once the altitude error reaches a predefined
threshold for some period of time, the ascent phase is initiated with a slight delay and warning sound from the vehicle,
after which maximum thrust is commanded. This state results
in an aggressive rise, providing the ideal conditions upon
which to initiate the reduced-gravity maneuver. In order to
maximize the time of the reduced-gravity maneuver, this
phase is operational until the aircraft reaches the point at
which the desired maximum height will be reached with the
impending reduced-gravity tracking. At this point, the tracking state is activated and the vertical acceleration controller
takes over. During this phase, a controller compensates the
error of the vertical acceleration measurement signal with
thrust output commands. Input shaping is used to provide a
smooth first order transition, where the desired acceleration
is initiated based on the value of the accelerometer at the first
time step, and is gently driven towards the desired value. This
phase will continue until the aircraft can no longer recover
from the dive, or if the maximum thrust is reached, where
an actuator saturation fault is signaled. At this point, the
state transition logic enters the recovery stage. Similar to
the ascend stage, the aircraft once again commands maximal
thrust in order to stop the vehicle descent. The descent stage

ends once the vertical velocity of the maneuver is close to
zero. Lastly, the vehicle is commanded to gently descend
down to ground level, where the end of the autonomous
maneuver is reached.

ad − g. The state and input trajectories corresponding to this
(quadratic in t) output are also quadratic

Dynamics of Vertical Flight

with coefficients determined by enforcing c p x̃ p (t) = ba2d t 2 +
ad − g and x̃˙ p (t) = A p x̃ p (t) + b p ũ p (t). Indeed, examining the
quadratic terms, we see that

The vertical velocity of a flying vehicle, with v > 0 when
descending, evolves according to

x̃ p (t) = x0 + x1 t + x2 t 2 /2
ũ p (t) = u0 + u1 t + u2 t 2 /2

0 = A p x2 + b p u2

v̇ = g − bv|v| + a p
where −bv|v| models the acceleration due to aerodynamic
drag (acting to retard the motion) and the thrusting acceleration a p = y p is produced by a motor-propeller-servo system
which we will model as a linear time-invariant system of the
form G p (s) = cTp (sI − A p )−1 b p with unit DC gain, G p (0) =
−cTp A−1
p b p = 1. The accelerometer senses the applied (nongravity) acceleration, a p − bv|v|, filtered by the vehicle,
accelerometer, and mounting structures (and possibly other
filters), modeled as an LTI system Ga (s) = cTa (sI − Aa )−1 ba
with Ga (0) = 1. The actuator and sensing systems are taken
to be stable and minimum phase with minimal state space
realizations. The vertical velocity dynamics, with sensing and
actuation, in block diagram form is shown in figure 8.

2ba2d = cTp x2
so that x2 = −A−1
p b p u2 (G p (s) stable implies that A p invertible) which leads to 2ba2d = −cTp A−1
p b p u2 = u2 and finally
x2 = −2ba2d A−1
p b p . Proceeding in a similar manner with the
linear and constant terms, we find that
u2 = 2ba2d
x2 = −2ba2d A−1
p bp
u1 = 2ba2d cTp A−2
p bp
T −2
−1
x1 = −2ba2d (A−2
p bp + cp Ap bp Ap bp)
T −2
2
u0 = 2ba2d (cTp A−3
p bb + (c p A p b p ) ) + ad − g
T −2
−2
x0 = −2ba2d [A−3
p bp + cp Ap bp Ap bp
T −2
2
−1
+ (cTp A−3
p bb + (c p A p b p ) ) A p b p ]

− A−1
p b p (ad − g)
providing a dynamic inversion of the quadratic output.

Fig. 8.

Vertical velocity dynamics

Feasibility Study
Fig. 10.

We are interested in enabling a free-fall type maneuver
in which the vehicle accelerates downward with a constant
desired acceleration ad > 0, resulting in a linearly increasing
velocity v = ad t ≥ 0, automatically compensating for the
naturally occurring drag force/acceleration. When ad = 1 g,
the goal is indeed free fall. In some cases, we merely choose
to accelerate downward at less than one g. For the time being,
we will consider the falling scenario, leaving the toss and fall
case for a future discussion.

Fig. 9.

A desired falling trajectory/maneuver.

To understand what is needed to perform such a maneuver,
consider a trajectory with ṽ(t) = ad t so that the output
of the propeller/servo system needs to be ỹ p (t) = ba2d t 2 +

Realization of a PIRQ control system

Now, the required quadratic input ũ p (t) for the drag
compensated maneuver should be provided by the output
ỹc (t) of our (to be designed) controller C(s) with zero input
ũc (t) ≡ 0. This can be accomplished using a chain of three
integrators as shown in figure 10, providing constant (or
step), linear (or ramp), and quadratic components in its
output under zero input conditions. The transfer function of
this PIRQ (Proportional-Integral-Ramp-Quadratic) controller
is
kP s3 + kI s2 + kR s + kQ
C(s) =
s3
with state space realization


0
1
0
0


 0
0
1
0 
Ac bc

=
 0
0
0
1 
cTc dc
kQ kR kI
kP
and where the PIRQ coefficients are all positive (and subject
to some further [stability] conditions below).

With zero input, the controller state for the falling maneuver will have the form

q0 + r0t + s0t 2 /2

r0 + s0 t
x̃c (t) = 
s0


so that, equating the controller output with the maneuver
actuator input, we find that the required controller initial
condition is


kR2 − kI kQ
1
kR
u2
u0 − u1 +
2

 
kQ
kQ
 kQ


q0

1
kR

x̃c (0) =  r0  = 
u1 − u2

k
k
Q
Q
s0


1
u2
kQ












This shows that the controller and actuator are capable of
providing the necessary signals to compensate the drag in
an ideal maneuver.

Fig. 12.

Maneuver Dynamics Considered

The dynamics of the system in figure 12 is given by
  
 
A p b p cTc −b p dc cTa
ẋ p
xp
 ẋc  =  0
Ac
−bc cTa   xc 
ba cTp
0
Aa
ẋa
xa




0
b p dc
+  bc  (ad − g) +  0  ,
0
−ba bv2

v̇ = cTp x p − bv2 + g .

(6)

(7)

While the linear feedback loop has been stabilized by
the chosen PIRQ controller, the injection of the nonlinear
feedback −bv2 into the loop may cause trouble.
Fig. 11. PIRQ control system for rejecting an idealized drag acceleration
disturbance

In fact, this control system can be used to determine,
in a feedback manner, the internal trajectory leading to
asymptotic rejection of the disturbance −ba2d t 2 without
knowledge of b (or even ad ), shown in figure 11. Here 1/s3
provides an internal model for the (idealized maneuver drag)
disturbance t 2 . Asymptotic disturbance rejection is obtained
for the linear feedback system in figure 11 provided that C(s)
stabilizes the feedback loop. This is possible when G p (s) and
Ga (s) are (exponentially) stable and minimum phase. This is
accomplished by choosing the location of the zeros of
s3 +

kQ
kI 2 kR
s + s+
kP
kP
kP

and the overall gain kP to bring the three compensator
poles (at 0) into the open left half plane. With stability, the
actuator and controller states will converge x p (t) → x̃ p (t) and
xc (t) → x̃c (t) as t → ∞ and the sensor state will converge to
its constant value, xa (t) → x̃a = −A−1
a ba (ad − g).
As noted, the feedback system in figure 11 is idealized in
the sense that the drag disturbance is modeled as a function of
time when, in reality, this disturbance depends on the velocity
state v as shown in figure 12.

In fact, what kind of stability might we expect for the
system depicted in figure 12? Clearly, we are hoping that the
velocity increases according to v̇ = ad so it doesn’t seem like
we are looking for stability for v, though perhaps we would
like to somehow regulate v̇ to the desired acceleration ad .
Also, note that we cannot accelerate for too long before we
exceed the operating envelope, which in reality is composed
of saturation on actuators, as well as a critical recovery height
which prevents the vehicle from hitting the ground.
What we would like is to make the desired maneuver
exponentially attractive. By maneuver, we mean an invariant
curve in the combined state space with state (v, x p , xc , xa ).
When discussing maneuvers, it is useful to keep in mind the
case of a periodic orbit (and the notion of orbital stability),
a particularly nice example of a maneuver.
Above, for a given desired velocity trajectory ṽ(t) = ad t
with fixed ad > 0, we were able to construct the corresponding state trajectory (x̃ p (t), x̃c (t), x̃a ) for t ≥ 0. This
trajectory traces out a curve in (v, x p , xc , xa ) space and since
t 7→ ṽ(t) = ad t is monotonically increasing (for ad > 0), we
may use its inverse t¯(v) = v/ad to provide a parametrization
of the maneuver by v. Indeed, defining x̄ p (v) = x̃ p (t¯(v)),
x̄c (v) = x̃c (t¯(v)), and x̄a = x̃a , we obtain the desired maneuver
(v, x̄ p (v), x̄c (v), x̄a ) = (v, x̄(v)), v ≥ 0. We will use γad to refer
to the maneuver corresponding to the desired acceleration
ad , although we will not explicitly label x̄ p (v), etc., with ad .

Next,
ż p = ẋ p − x̄0p (v) v̇

x̄ p (v) = x0 + x1 v/ad + x2 (v/ad )2 /2

= A p (x̄ p (v) + z p ) + b p cTc (x̄c (v) + zc ) − b p dc cTa (x̄a + za )

x̄0p (v) = x1 /ad + x2 /ad (v/ad )

+ b p dc (ad − g) − x̄0p (v) (ad + cTp z p )

T −2
−1
−1
= −2bad ((A−2
p b p + c p A p b p A p b p ) + A p b p (v/ad ))

= A p z p + b p cTc zc − b p dc cTa za − x̄0p (v)cTp z p

T −2
−1
−1
x̄0p (ad θ ) = −2bad ((A−2
p bp + cp Ap bp Ap bp) + Ap bp θ )

where we have made use of (9) and (11). Using (10), we see
that
żc = Ac zc − bc cTa za − x̄c0 (v)cTp z p


q0 + r0 (v/ad ) + s0 (v/ad )2 /2

r0 + s0 (v/ad )
x̄c (v) = x̃c (v/ad ) = 
s0


and finally, using (8) and (12),
ża = Aa za + ba cTp z p .




r0 /ad + s0 /ad (v/ad )

s0 /ad
x̄c0 (v) = 
0

Collecting these results, we obtain
v̇

x̄0p (v) ad
x̄c0 (v) ad
cTa x̄a

= A p x̄ p (v) + b p cTc x̄c (v)

θ̇ = 1 + (1/ad ) cTp z p
  
A p − x̄0p (ad θ )cTp b p cTc
ż p
 żc  =  −x̄c0 (ad θ )cTp
Ac
ża
ba cTp
0






(8)
(9)
(10)

= ad − g

(11)
(12)

In order to characterize the nature of the closed loop
dynamics about the maneuver, it is helpful to write the dynamics in a maneuver adapted, transverse coordinate system,
x p = x̄ p (v) + z p
xc = x̄c (v) + zc
xa = x̄a + za
with transverse coordinates (z p , zc , za ) about the maneuver
(v, x̄ p (v), x̄c (v), x̄a ),v ≥ 0.
First, using (8), note that the evolution of v simplifies from
its nonlinear form (7) to
v̇ = ad + cTp z p .

 
−b p dc cTa
zp
−bc cTa   zc 
za
Aa

 
−b p dc cTa
zp
−bc cTa   zc 
za
Aa

By a set of straightforward calculations, one may show
that

= Ac x̄c (v)

0 = Aa x̄a + ba (ad − g)

b p cTc
Ac
0

Changing coordinates by parametrizing the velocity by θ
with v = ad θ results in θ̇ = 1 on the maneuver giving

Working from defining properties above, we see that the
maneuver curves satisfy
cTp x̄ p (v) = bv2 + ad − g

ad + cTp z p

  
A p − x̄0p (v)cTp
ż p
 żc  =  −x̄c0 (v)cTp
ża
ba cTp



kR
1
u1 /ad − u2 /ad
r0 /ad =
kQ
kQ


1
kR
T −2
= 2bad
c A bp −
kQ p p
kQ
1
s0 /ad = u2 /ad
kQ
1
= 2bad
kQ

 
kR
T
−2
+θ
cp Ap bp −
2bad 
kQ

x̄c0 (ad θ ) =
1
kQ 
0

=

T −2
−1
−1
x̄0p (ad θ ) = −2bad [(A−2
p bp + cp Ap bp Ap bp) + Ap bp θ ]
 T −2

(c p A p b p − kR /kQ )/kQ + θ /kQ

x̄c0 (ad θ ) = 2bad 
1/kQ
0

so that

 

T −2
−1
−(A−2
−A−1
p bp + cp Ap bp Ap bp)
p bp
  

  T −2

1/kQ 

x̄0 (ad θ ) 
 (c p A p b p − kR /kQ )/kQ  
= 
 +
 0 
1/k

θ
Q

 
2bad


0 
0
0
0
=: x̄1 + x̄2 θ
defining the vectors x̄1 and x̄2 which depend only on the
(realization) parameters of the propeller/servo system and
on the PIRQ control system gains and not on the desired
acceleration ad . As expected, x̄0 (ad θ ) is affine in θ since
x̄(v) was quadratic (in v). Using ρ T = (zTp , zTc , zTa ) and c̄T =
(cTp , 0, 0), the maneuver dynamics is thus given by
θ̇ = 1 + (1/ad ) c̄T ρ
T

(13)
T

ρ̇ = (Ā − 2bad x̄1 c̄ − 2bad x̄2 θ c̄ ) ρ
=: A(θ ) ρ

(14)

where Ā is the system matrix from (6) that characterizes the
dynamics of the linear feedback system in, e.g., figure 11.
If we can find (constant) P, Q > 0 such that
A(θ )T P + P A(θ ) + Q ≤ 0
for all θ ∈ [0, θ1 ], then the maneuver will be exponentially
attractive on [0, θ1 ]. Indeed, the transverse Lyapunov function
V (θ , ρ) = ρ T P ρ
Fig. 13.

satisfies

A transverse Lyapunov function for maneuver stability

V̇ (θ , ρ) = ρ T (A(θ )T P + P A(θ )) ρ
≤ −ρ T Q ρ < 0 .
Maneuver Regulation
For each fixed positively invariant ad > 0, the nonlinear
system possesses a one dimensional manifold of the form
γad = {(v, x̄c (v), x̄a ), v ≥ 0} ,

(15)

which we call maneuver and formalize as follows.
Definition Consider a dynamical system D over the statespace X , defined by the differential equation


d
D = x(t)
x = f (x), t = [0 T ] .
dt
A closed, one-dimensional set M defined as
M = {x ∈ X | m(x) = 0 }
is called a maneuver of D if there exists x0 ∈ X and T > 0
such that x(0) = x0 and
x(t) ∈ D and x(t) ∈ M , 0 ≤ t ≤ T.
Hence, the maneuver M is an invariant one-dimensional
set for the system D.
We now consider the transverse stability of the maneuver
(v, x̄c (v), x̄a ), v ≥ 0. Maneuver stability, and regulation, is a
natural generalization of the notion of orbital stability [39]–
[42]. To this end, we express the controller and accelerometer
states as xc = x̄c (v) + zc and xa = x̄a + za where the planar
sections zc , za , provide coordinates on the moving transverse
to the maneuver. We write the closed loop dynamics in
(v, zc , za ) coordinates.
Maneuver stability analysis
We now ask the question of ”maneuver stability analysis”.
Intuitively, we would like the system’s behavior in the vicinity of the maneuver to exhibit the characteristics sketched
in Fig. 13, whereby any departure from the the maneuver
results in the vehicle ”converging back” to it.
Asking about ”maneuver stability” is a slight abuse of
language, because stability is an asymptotic behavior, while
the maneuver discussed in this paper has finite duration.
We therefore reformulate the question in terms of invariant
sets and the transverse dynamics of the controlled system as
follows: Consider the closed-loop system. We consider the

free-fall maneuver to be stable if the transverse dynamics
can be shown to have an exponentially decreasing quadratic
abstraction. In other terms, there exists a positive definite
matrix P and a positive number ε such that for any initial
transverse condition ρ0 at the maneuver onset, the function
V (ρ) defined as
V (ρ) = ρ T Pρ
(16)
satisfies the inequality
dV (ρ(t))
≤ −2εV (ρ(t)).
(17)
dt
The inequality (17) indicates that the transverse dynamics
converges exponentially to zero during the maneuver execution, with decay rate ε. Following established knowledge,
finding an appropriate P amounts to finding a positivedefinite P that satisfies
A(θ )T P + P(A(θ )) ≤ −2εP

(18)

for all admissible values of θ . The existence of such a P can
be obtained via a straightforward application of the circle
criterion: A(θ ) = A0 + θ A1 and A1 is a rank one matrix,
that is A1 = bc, where b (resp. c) are columns (resp. row)
vectors of appropriate dimension. Therefore, given a range of
values [θmin θmax ], the exponential stability of the systemis
guaranteed if there exists η > 0 such that
c( jωI − Ã − εI)−1 b <

1
,
θr2

(19)

where Ã = A0 + (θmax − θmin )A1 and θr = (θmax − θmin )/2.
Thus the circel criterion can be checked easily by computing
the appropriate Nyquist plot and looking for its maximum
modulus. Since Ã is stable, it is always possible to find ε > 0
such that (19) is satisfied. Computing P satisfying (18) can
then be done by solving the Riccati equation
(Ã + εI)T P + P(Ã + εI) + PbbT Pθr2 + cT c = 0.
The closed-loop system model below satisfies the circle
criterion and therefore displays ”maneuver stability”.

25

Accz/g 0 (Gs)

1
17

17.5

18

18.5

19

17

17.5

18

18.5

19

17

17.5

18

18.5

19

17

17.5

18
Time (sec)

18.5

19

Fig. 15.

Martian 0.378 G maneuver

30

(m)
Vz (m/s)

Altitude LIDAR (m)

15

20
10
0

0
-20

10
5

16

17

18

19

20

Recover
Track
Climb
Hover

Thrust (%)

100

Time (sec)

15

2

Desired Gs
Raw Acc z/g 0

20

20

0
15

3

0

LIDAR

On July 14th 2017, after much anticipation and tremendous efforts, the worlds first autonomous Martian gravity
parabola (0.378Gs) was performed, meeting tolerances of
±0.1G0 s for a period of approximately 1.5 seconds, with
a mean G of 0.3804 and a standard deviation of 0.0426. The
Martian parabola performed by the vehicle is illustrated in
Figure 14, where the LIDAR Lite data has been processes
by a double sided filter. The different colors of the parabola
represent the active autonomous state during the maneuver,
also described at the bottom of Figure 14

Altitude

I MPLEMENTATION AND VALIDATION

0

-100
16

17

18

19

20

Time (sec)

Fig. 14. Filtered Altitude and Automata. Different colors represent the
active autonomous state during the maneuver

CONCLUSION
Most importantly, Figure 15 validates the martian parabola
by illustrating the transient of the raw accelerometer data,
normalized by g0 = −9.807m/s2 , during the tracking maneuver shown in red. The vehicle’s altitude, velocity and percent
throttle during the maneuver are also illustrated.
Although the desired acceleration was set to 0.378 G’s,
the reader should note that input shaping was performed by
employing a first order transfer function with DC gain of one.
Its purpose is to begins the maneuver by taking the current
accelerometer value and gently driving it towards the desired
G value based on the transfer function’s time constant. The
input shaping is can be is highlighted by the desired G blue
line in figure 15.
While this data shows a successful implementation of the
PIRQ controller, the authors acknowledge that higher fidelity
acceleration tracking can be achieved by properly initializing
each of the three integrators. It is also important to note that
during the flight tests, strong winds were observed. Lastly,
there are unmodeled dynamics from the RPM governor
controller that could be introducing instabilities along the
z translation, which will have to be revised in the future.

The objective of this paper was to present the controller
design and implementation for regulating reduced-gravity
maneuvers on a custom variable-pitch multirotor. This work
has validated the proposed PIRQ controller both theoretically, using the circle criterion, and experimentally on a flight
test vehicle. Future work will involve further improving the
hardware and controller design to increase the accuracy of
the maneuver.
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