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Flexible collaborative transportation by a team of rotorcraft
Hector Garcia de Marina1Ewoud Smeur2
Abstract— We propose a combined method for the collab-
orative transportation of a suspended payload by a team of
rotorcraft. A recent distance-based formation-motion control
algorithm based on assigning distance disagreements among
robots generates the acceleration signals to be tracked by the
vehicles. In particular, the proposed method does not need
global positions nor tracking prescribed trajectories for the
motion of the members of the team. The acceleration signals
are followed accurately by an Incremental Nonlinear Dynamic
Inversion controller designed for rotorcraft that measures and
resists the tensions from the payload. Our approach allows us
to analyze the involved accelerations and forces in the system
so that we can calculate the worst case conditions explicitly to
guarantee a nominal performance, provided that the payload
starts at rest in the 2D centroid of the formation, and it is not
under significant disturbances. For example, we can calculate
the maximum safe deformation of the team with respect to its
desired shape. We demonstrate our method with a team of four
rotorcraft carrying a suspended object two times heavier than
the maximum payload for an individual. Last but not least,
our proposed algorithm is available for the community in the
open-source autopilot Paparazzi.
I. INTRODUCTION
Robot swarms are envisioned to assist humans in lo-
gistics operations with better cost-effective approaches [1].
Rotorcraft are an essential part of this vision due to their
mechanical simplicity and relative expendable nature. In this
paper, we demonstrate a systematic method to deal with
the transport of objects that are too heavy for a single
rotorcraft, but not for a team of them. This added complexity
is supported by the fact that rotorcraft do not scale up well
aerodynamically speaking, and they become dangerous to
operate if their size is big enough. In particular, we focus on
carrying suspended loads. In this way, we give flexibility to
the team to change its shape and distribute the load among
the vehicles efficiently.
Formation control has been employed in the cooperative
transportation of objects on the ground [2]. In our approach,
we use rigidity theory [3] to describe the geometrical shape
of the team so that we can design the desired load distribution
among the rotorcraft in their steady-state, even while moving.
The coordinated motion of the team is induced by injecting
disagreements into the inter-robot distances of the formation.
The superposition of specific sets of disagreements creates
translational, rotational, and scaling motions of a distributed
rigid formation [4]. We combine the team motion controller
with Incremental Nonlinear Dynamic Inversion (INDI) to
1H. Garcia de Marina is with the Unmanned Aerial Systems Center,
Southern University of Denmark, Denmark. hgm@mmmi.sdu.dk.
2E. Smeur is with the MAVLab, Department of Aerospace Engineering,
Delft University of Technology, 2629HS Delft, The Netherlands.
track desired accelerations in the individual vehicles [5]. By
having prior knowledge about the created forces and mo-
ments of the actuators and their delays, the INDI tracks the
commanded accelerations coming from the formation con-
troller by using acceleration measurements, which includes
the tension force from the load. This measurement allows
the INDI controller to calculate the necessary increment in
the previous control action for the motors.
Our proposed method brings substantial positive differ-
ences with respect to the current approaches in the litera-
ture on collaborative transportation with aerial vehicles. For
example, we do not need centralized calculations or global
positioning, nor do we require previously calculated paths
to be tracked by the vehicles [6], [7], [8]. Therefore, the
usage of accurate motion capture systems is not a crucial
requirement for our approach. We will see that while we
employ them in our demonstration, we do not require global
positioning but we fake relative measurements that could
be provided by an onboard system [9]. Another practical
positive aspect of our approach is that we promote scalability.
The algorithm can be executed in a distributed way, i.e.,
robots can calculate their own desired accelerations based
only on local information, and the calculations can be done in
simple microcontrollers. Therefore we do not require heavy
computers for the practical task like in [10]. We do not
require the individual vehicles to track any trajectories either.
For example, such an approach, by a centralized motion
planning, seems pivotal for the motion of the rotorcraft
team in many works whose implementation is shown only
in simulations [11], [10], [12]. Nevertheless, they consider
active control over the payload.
The main contribution of this paper focuses on the practi-
cal demonstration of the combined approaches of the INDI
controller together with the motion of the formation by
disagreements for the collaborative transportation of objects.
In particular, we will show how these two techniques can
predict results on the deformation of the shape together with
the control of the velocity of the formation.
This paper is divided as follows. In Section II we introduce
the guidance system for the formation and motion of the
rotorcraft. The controller responsible for tracking the guid-
ance signals is described in III. We continue by presenting in
Section IV a qualitative description of the forces involved in
the system and how to quantitatively calculate tolerances in
the formation and control gains from a worst case condition.
We show the performance of our system in Section V with
a team of four rotorcraft transporting a heavy object for an
individual vehicle. We finally end the paper in Section VI
with some conclusions.
II. GUI DANC E SYS TEM F OR TH E FORMATIO N-MOT ION
CONTROL OF SECOND-ORD ER ROBOT S
A. Undirected rigid formations
Consider a team of n≥2robots and denote by pi∈
R2, i ∈ {1, . . . , n}their 2D positions in the horizontal plane
parallel to the ground with respect to a fixed navigation
frame of coordinates. The guidance system for the formation-
motion of the team generates an acceleration signal to be
tracked by the robots. In particular, the guidance will assume
that the vehicles can generate such acceleration sufficiently
fast in the fixed navigation frame as it is a common
assumption in the literature [12]. Although our rotorcraft
(quadcopter) can only generate a force along one axis in
its body frame, we will see that the INDI controller adapts
the attitude and the thrust of the vehicle fast enough so that
the vehicle can track a sufficiently slow varying acceleration
signal effectively, while it rejects disturbances such as the
non-constant tension from the ropes. In particular, as we
will see, the exponential nature of the proposed guidance
and the INDI control will allow us to set the time constants
for the exponential decay of their signals by appropriately
selecting their gains. The guidance system considers the
following dynamical model for the formation-motion task
of the vehicles (˙p=v
˙v=u,(1)
where p, v ∈R2nare the stacked vector of positions and
velocities in the plane parallel to the ground respectively,
and u∈R2nis the stacked vector of accelerations generated
by the guidance system.
A robot does not need to measure its relative position
with respect to all the robots in the team, but only with
respect to its neighbors. The neighbors’ relationships are
described by an undirected graph G= (V,E)with the
vertex set V={1, . . . , n}and the ordered edge set E ⊆
V × V. The set Niof the neighbors of robot iis defined
by Ni
∆
={j∈ V : (i, j)∈ E}. We define the elements of
the incidence matrix B∈R|V|×|E | for Gby bik ={1,−1}
if {Etail
k,Ehead
k}or 0otherwise, where Etail
kand Ehead
kdenote
the tail and head nodes, respectively, of the edge Ek, i.e.,
Ek= (Etail
k,Ehead
k). For undirected graphs, how one sets
the direction of the edges is not relevant for the stability
results or the practical implementation of the algorithm [13].
The proposed formation control algorithm is based on the
distance-based approach, i.e., we are defining shapes by
only controlling distances between neighboring robots. These
shapes are based on the rigidity graph theory [3]. One of
the advantages of controlling distances instead of relative
positions is the freedom of the shape to be rotated and
translated without modifying the controller. This fact will
give us enough freedom to design both the motion and
formation controller at once.
The stacked vector of the sensed relative positions by the
robots can be calculated as
z= (BT⊗I2)p, (2)
where I2is the 2×2identity matrix, and the operator
⊗denotes the Kronecker product. Note that each vector
zk=pi−pjstacked in zcorresponds to the relative
position associated with the edge Ek= (i, j). The introduced
concepts and notations are illustrated in Figure 1. As an
example, in this paper we will focus on a regular square
with the neighbors defined by the incidence matrix
B=1 0 0 1 1 0
−1 1 0 0 0 1
0 0 1 0 −1−1
0−1−1−1 0 0 .(3)
Let d:= d1, . . . , dkT, k ∈ {1,...,|E|} be the stacked
column vector of fixed distances, associated to their corre-
sponding edges, which defines the desired regular square.
Then, the error signals to be minimized are given by
ek(t) := ||zk(t)|| − dk.(4)
The control action for each robot in order to stabilize the
regular square can be derived from the gradient descent of
the potential function involving all the error distances to be
minimized and the kinetic energy of the agents
V=c1
2
|V|
X
i=1 ||vi||2+c2
2
|E|
X
k=1
(||zk(t)|| − dk)2,(5)
where c1, c2>0, which leads to the following control action
[14] for each robot i
iui=−c1ivi−c2X
j∈Ni
i(pi−pj)
||pi−pj||(||pi−pj|| −d(i,j)),(6)
where each desired distance d(i,j)=d(j,i)is associated with
its corresponding dk, and the superscript iover the vectorial
quantities is used for the representation of a vector with
respect to the local frame of coordinates of robot i.
B. Motion control
The control action (6) only achieves the task of converging
(exponentially fast) to the desired shape where all the agents
will be stopped [14]. In order to create motion we ask the
agents to disagree on the distances d(i,j), i.e., the second
term in (6) is written as
c2Pj∈Ni
i(pi−pj)
||pi−pj|| (||pi−pj||−(d(i,j)+µ(i,j)))=
=c2Pj∈Ni
i(pi−pj)
||pi−pj|| (||pi−pj||−d(i,j))−c2Pj∈Ni
i(pi−pj)
||pi−pj|| µ(i,j),
(7)
where µij ∈Ris a designed disagreement and it is equal
to zero in (6). Note that the second term in (7) will be the
responsible of the desired motion of the agent iduring its
steady state [14], e.g., once the regular square is achieved. In
particular, these acceleration signals are linear combinations
of the unit relative positions izk
||zk|| where the coefficients are
the disagreements µ(i,j).
We design three rigid motions for the team: two orthogo-
nal translations and one rotational following the systematic
method explained in [14]. For each of these motions, each
rotorcraft must follow a specific velocity with respect to the
body frame Obas illustrated in Figure 1. We first write the
3
4
2
1
||z1||
||z2||
||z3||
||z4||
||z5||
||z6||
Ob
Fig. 1: We ask the rotorcraft to control their distances in between to
form a regular square, where the payload is placed at the centroid
but in a different altitude. The red, blue and green color vectors
are velocities that create translational and rotational motions of
the formation with respect to Ob. They are constructed as linear
combinations of zkwith the disagreements µij .
desired velocities for each motion as linear combinations
of izk
||zk|| so we can construct the following matrix with
disagreements based on the incidence matrix (3)
Av="µ(1,2) 0 0 µ(1,4) µ(1,3) 0
µ(2,1) µ(2,4) 0 0 0 µ(2,3)
0 0 µ(3,4) 0µ(3,1) µ(3,2)
0µ(4,2) µ(4,3) µ(4,1) 0 0 #.(8)
Let us write in compact form the stacked vector of all
control actions (6) with disagreements as in (7) in the
navigation frame
u=−c1v−c2BDzD˜ze+AD˜zz, (9)
where ˜z∈R|E| is the stacked vectors of all 1
||zk|| , we define
the operator X:= X⊗I2, and Dxis the operator that takes
each stacked element (vector or scalar) of xand places them
in a diagonal block matrix. The matrix Awill be function of
the previously calculated Av. In particular, we have dropped
c2in the third element of (9) since, as we will see, it can
be cancelled out by scaling up or down the disagreements in
(8).
We need to introduce different calculations with respect
to [14] in order to figure out the matrix Asince in this work
we deal with a different potential function (5) that leads to
work with the unit vectors zk
||zk|| in the control action (9). We
first need to define the velocity error ev:= v−AvD˜zz.
Proposition 2.1: The matrix Ais calculated from Avand
is given by A=c1Av+Aa, where Aa=AvrDdBTAvr,
and Avrdefines the desired steady-state rotational motion.
Proof: Let us define the stacked vector v∗(t)∈R2|V|
of desired velocities for the agents. In particular, we have
designed it from linear combinations of the unit vectors of
the desired relative positions by employing (8). Therefore we
can write
v∗(t) = AvDdz∗(t),(10)
where we recall that dis the stacked vector of desired
distances and z∗∈R2|E| is the stacked vector of desired
relative positions with respect to Obin Figure 1. Now we
calculate the stacked vector with all the desired accelerations
d
dtv∗(t) = AvDd
d
dtz∗(t)
=AvDdBTv∗(t)
=AvDdBTAvDdz∗(t).(11)
Let A=c1Av+Aaso that (9) can be written as
u=−c1ev−c2BDzD˜ze+AaD˜zz, (12)
therefore once e, ev= 0, i.e., the formation is at the desired
shape and velocity, we have that u∗(t) = AaDdz∗(t), and
noting that for pure translational motion the accelerations
are identically zero because d
dt z∗(t)=0in such a case,
then from (11) and (12) we have that
AaDdz∗(t) = AvrDdBTAvrDdz∗(t)
Aa=AvrDdBTAvr.(13)
The key for showing the exponential stability of eand evlies
on the fact that under the control (9) for A= 0 these signals
are exponentiall stable [14]. Then, the disagreements, i.e.
A6= 0, are treated as parametric disturbances as suggested
in [15]. Therefore, if they are small enough (or c1and c2
big enough), then we do not modify the exponential nature
of the convergence of eand ev.
III. ROTOR CRA FT CO NTRO L
A. Incremental Nonlinear Dynamic Inversion controller
We have taken the concept of Incremental Nonlinear
Dynamic Inversion (INDI) [16], and applied it to track
the linear accelerations generated by the guidance system
for the formation-motion of the rotorcraft. The idea behind
INDI is that forces and moments acting on the vehicle
are, according to classical mechanics, proportional to the
acceleration and angular acceleration of the vehicle. The
acceleration can be measured with the accelerometer, and
the angular acceleration can be derived from the gyroscope.
From a desired increment in linear and angular acceleration,
the required increment in inputs can be easily calculated
using a control effectiveness matrix. Since the 3D tensions
from the rope will be measured by the accelerometer and
gyroscope as well, they will be naturally counteracted by
the controller, without having to model the load. Though the
tension of the ropes may introduce small moments on the
rotorcraft, the ropes are attached close to the center of mass
and the expected effect on the attitude dynamics is negligible.
To obtain a measurement of the linear acceleration a0
of a rotorcraft, we take the specific force measurement of
the accelerometer, and add the gravity vector to that. The
acceleration a small time step ahead can now be predicted
with
a−a0=1
mG(η0, T0)(u−u0),(14)
where a∈R3is the acceleration in the navigation frame
(the subscript 0denotes for current value), mis the mass
of the Bebop, η=φ θ ψare the three attitude angles
roll,pitch and yaw, we group the pitch, roll and thrust in
u=φ θ T T, and we finally define the matrix of partial
derivatives of the thrust vector
G(η, T ) = T(cφsψ −sφcψsθ)T cφcψ cθ sφsψ+cφcψsθ
−T(sφsψsθ+cφcψ)T cφsψ cθ cφsψsθ−sφcψ
−T cθsφ −T sθcφ cφcθ .(15)
The measured acceleration a0incorporates disturbances, and
the force from the rope tension. Because the acceleration
measurement is noisy due to vibrations, we employ a low
pass Butterworth filter, denoted with a subscript f. In order
to synchronize the signals, a0and u0will both be filtered
such that they will have the same delay. The filtered signals
are incorporated in (14) and the equation is inverted to obtain
the following incremented controller
∆u=mG−1(ηf, Tf)(νa−af),(16)
where νais the desired 3D linear acceleration for the
rotorcraft in the navigation frame. In particular, the first two
components of νafor each rotorcraft are given by (9) and the
third or vertical component is generated by the PD controller
νaz=kp(pz−pzd)−kvvz,(17)
where the gains kpand kvare chosen according to the
stability analysis given in [16].
Remark 3.1: To calculate Gin (16) we need to know Tf,
which is experimentally estimated by a quadratic function
fT(ω2
f)in a static airflow regime. Errors due to the sim-
plifications in the modelling are expected to have a low
impact on the performance. This is explained because of the
incremental nature of the controller, i.e., if an increment of
thrust does not give the desired acceleration, then another
increment is applied in a similar way as an integral controller
does. However, note that the INDI measures the disturbances
and the tensions from the rope, and have a feedforward
knowledge of the vehicle and its actuators.
B. Effect of the payload on the control
As we have discussed, no explicit knowledge about the
payload is incorporated in the control of the rotorcraft since
the INDI controller is naturally designed for dealing with
non-modelled external forces by measuring them directly.
For example, due to the incremental nature of the controller,
feedforward control increments will be applied on the actua-
tors as long as the desired acceleration is not reached, which
means that steady state offsets will not occur. Consequently,
external forces are incorporated in the control either by
compensating them, or by taking advantage from them if for
example the tension of the load and the desired acceleration
have the same direction.
IV. FOR C E AN D WOR ST CA SE ANA LYS IS
In this section, as an illustrative example, we conduct
an analysis to guarantee the safety of the system while it
is close to a steady state, e.g., close to the desired shape
with the desired velocity. We analyze the forces involved,
in particular the ones in a worst case, and connect them
to the physical limits of the vehicles together with the
Motors
Rigid body
KF attitude
PD alt. Eq. (17)
Guidance Eq. (9)
Attitude
Control
(INDI)
INDI a
Eq. (16)
p, v
pneighbors
Ω
Ω
a
νa
ηωc
∆Td, ηdBEB OP
Fig. 2: Block diagram of the elaborated signals in the closed
loop system. We actuate over the Bebops commanding desired
rpm ωcfor the motors. We measure the angular velocities (Ω) and
linear accelerations (a) for the INDI controllers and the attitude
estimation. We measure the positions pto fake direct measurements
of zwith a vision localization system.
demands from the guidance system in Section II and the
controllers in Section III. For the analysis, our vehicles are
four rotorcraft Bebop from the company Parrot that have
been characterized in our laboratory [16]. Each rotorcraft
weighs around 400 grams, an individual motor/propeller can
generate 1.6Newtons of thrust once it is rotating at 160 Hz
in hover flight, and a motor/propeller generates a moment
around 0.0006 m/s2/rpm.
We describe the forces involved in the system in a steady
state configuration as starting point. We set the desired shape
as the regular square in Figure 1, where all the ropes have the
same length so that the payload can be placed at the center of
the square while hanging. This configuration allows an equal
distribution of the payload’s weight across all the vehicles.
The force diagram on the left of Figure 3 focuses on the
vertical plane connecting vehicles 4and 2. It shows how the
vehicles need to tilt to compensate the horizontal components
of the ropes’ tensions. In particular, we define tilt as the angle
formed by the thrust force and the horizontal plane parallel
to the ground. If the rotorcraft are in equilibrium, Mis the
mass of the payload, and lis the fixed length of the rope,
then the tension satisfies
||TM2|| =Mg
4
l
ql2−d2
2
4
,(18)
which increases clearly with d2. Therefore, we will set d2the
shortest as possible depending on the sizes of the rotorcraft
and the expected disturbances that will vary ||z2|| (equal to
d2in the steady state).
The diagram on the right side of Figure 3 is in the plane
described by the vehicles 1and 3. We describe the situation
where we demand the same horizontal acceleration to the
vehicles while keeping the altitude constant. In such a case,
we see that the force Rin vehicle 1will create more tension
on its rope since the vehicle is more tilted, and consequently
it needs a higher F1to compensate for gravity. As a result,
the vehicle 1will need to lift more weight from the payload
than its neighbor. Because the altitude is constant, and both
Mg
TM2
TM4
Tr4Tr2m2gm4g
F4F2
||z2||
Mg
m1gm3g
F3
F1
||z3||
R
R R
Fig. 3: Diagram forces. On the left, the plane defined by the
payload and the vehicles 2and 4in a configuration of equilibrium.
On the right, the plane defined by the payload and the vehicles 1
and 3where the guidance system demands the same acceleration
from both vehicles.
vehicles equally accelerate, then the payload experiences the
same force Ras well.
We take the above described cases to calculate conserva-
tive bounds on the maximum deformation of the formation,
i.e., on the norm of the error signal e(t), on the gains c1
and c2and the disagreements µrin (9) for a demanded
motion once a rotorcraft is close to the equilibrium. We
have measured that one actuator of the Bebops can generate
around 1.6Newtons of force at the 85% of its capacity, so
we consider 6.4Newtons as a reference thrust force for
the following worst case condition. We first estimate the
maximum safe tilt for a rotorcraft. During the experiments,
the initial positions of the rotorcraft are close to the desired
one. Therefore, the payload will always stay close to the
centroid as we will see according to the calculations in this
section. We consider that a vehicle will share a third of
the weight of the payload as a conservative worst case. In
particular, we will deal with a payload equal to the mass
of a single vehicle. Therefore the maximum vertical force
to compensate will be (0.4 + 0.4
3)g= 5.22 Newtons. A
simple trigonometric calculation reveals that the maximum
tilt for a rotorcraft is arccos 5.22
6.4= 0.62 radians. Therefore,
we impose a maximum angle of 20 degrees (0.35 rads)
for both pitch and roll so that the tilt is below 0.62 rads.
Indeed, this relatively small angle helps the INDI controller
since for its design we have assumed small increments in
the commanded attitude signals. This estimation will leave
around 6.4 sin(0.35) = 2.18 Newtons of maximum available
force for a rotorcraft in the horizontal plane.
We then estimate what would be a conservative bound for
a horizontal force in opposition to the one created by the
thrust of the vehicle. Although present, we will not consider
any aerodynamic drag since we will set the maximum
vehicle’s speed in 1m/s, and the tension from the rope is
substantially bigger than any drag at that speed. Focusing on
vehicle 2, from equation (18) and basic trigonometry we can
derive the expression for the bound on the horizontal tension
of the rope
Th=Mg
3||z2||
2ql2−||z2||2
4
,(19)
where we assumed that the horizontal distance between the
vehicle and the payload is approximately ||z2||
2since the
initial positions for the rotorcraft are close to the desired
square, and the controllers will keep such a situation if
they can cope with the predicted worst cases in this current
analysis. We set the length of the rope to approximately
l=√1.25 meters, and for equation (19) we consider a very
conservative ||z2|| = 2 meters since in the experiments we
will set d2close to 1meter. As a result, an upper bound to
This Mg
3, and because the maximum available horizontal
force in the worse case for the tilt is 2.18 Newtons, we can
conclude that the maximum norm of the acceleration to be
asked by the guidance system is
max(||¨pi||) = 2.18 −M g
3
mi
= 2.18 m/s2,(20)
where mi=M= 0.4Kg are the mass of the vehicle iand
the payload. Consequently, we impose on the vehicles that
the maximum for each of the acceleration coordinates in the
plane horizontal to ground is 2.18 ×sin(0.79) = 1.54 m/s2,
i.e., a vector of magnitude 2.18 with 45 degrees with respect
to the X and Y axis. For example, starting from the equation
(9) for the vehicle 4(and the same argument can be extended
to the rest of vehicles) we can calculate conservative values
for c1, c2and µrfrom the following expression
(¨p4x=−c1evx−c2(e4+ 0.7e2) + µr||z4||
¨p4y=−c1evy−c2(e3+ 0.7e2)−µr||z3||.(21)
The first term in the equations in (21) is related to the desired
speed in one of the horizontal directions. For example, we
recall that for a desired translational motion, the desired
velocity is designed by the disagreements µtin Section II. By
design, we will not demand a higher speed than 1m/s, so we
can safely assume max{||ev||} = 1. The second term refers
to the control of inter-vehicle distances. Recall that we start
close to the equilibrium with a desired square of side 1meter,
therefore we can assume a very conservative worst case of
max{||ek||} = 1 meter since we consider as worst case for
the horizontal tension a distance of 2meters between vehicles
on the side of the squared formation, e.g., max{||z4||} = 2
meters. Note that looking at Figure 1, e2almost equally
contributes to both components, while for example e3can be
safely omitted in the xcomponent. Finally, the disagreement
µrconstructs the desired centripetal acceleration towards the
centroid, so the formation spins around it. Therefore, in order
to satisfy the following worst case condition
1.54m/s2≥c1max{||ev||}+c2(max{e4}+0.7 max{e2})+µrmax{||z4||},
(22)
we assign c1= 0.17,c2= 0.55 and max{µr}= 0.2, where
µrsets the limit to the maximum angular velocity of the
spinning motion of the formation. With these chosen values,
the system remains stable and within its physical limits.
Firstly, the forces from the payload in this analysis have
been found substantially smaller than the disturbances that
can be handled by the Bebops [16] with the INDI controller.
Secondly, the incremental accelerations demanded by the
guidance system are bounded and within the physical limits
to be tracked effectively. Thirdly, the stability analysis of the
guidance system [14] guarantees the maximum magnitude of
the error signals evand e. In particular, if the accelerations
are tracked correctly and the distances between the vehicles
have an initial error of less than 0.5meters, then following
[14] it can be derived that the maximum for a single error
distance cannot be more than a meter if the requested velocity
has a speed smaller than a meter per second. All in all, the
correct performance of the team carrying a load is guaranteed
under the described nominal conditions in this section.
Remark 4.1: For chosen the gains c1and c2, we checked
in simulation that the desired acceleration signals have a time
constant of around 1second for its exponential decay. The
chosen gains for the INDI controller indicates a time constant
of around 0.01 seconds. Therefore, we can guarantee that
the guidance system is at least 100 times slower in order
to ensure the stability of the two slow-fast interconnected
systems.
Remark 4.2: The payload is assumed at rest at the initial
conditions, e.g., it is not swinging. Since the payload starts
close to the centroid of the formation, and the guidance
system guarantees that the centroid is under control, then the
forces on the payload are very similar to the ones required to
achieve the desired velocities of the formation as a whole.
Therefore, as we will see in the experiments, we predict
to do not have significant swing motions on the payload.
If there are small disturbances on it, the INDI would detect
them since changes on the tensions of the ropes are measured
by the accelerometers. So, both the INDI and the guidance
system will compensate such disturbances.
V. EXPERIMENTAL RESULTS
The experimental results1are conducted in a controlled
area, e.g., no wind or obstacles around. The Bebops are
equipped with the Paparazzi autopilot2and their positions
are obtained with an Optitrack camera system. An operator
with a gamepad commands the movements in Figure 1 where
the sticks set the magnitude of the disagreements of the
matrix Ain Proposition 2.1, i.e., the speed of the motions.
The operator with the gamepad also controls the scale of
the formation and its altitude at once, e.g., to control when
the payload is lifted. The guidance system runs in a ground
control station. It processes the relative positions of the
robots and the input from the gamepad at the fixed frequency
of 4Hz. We show in Figure 4 a caption of the experiments
and the trajectories described by the team while it is steered
by the operator. We would like to remind that the operator
commands the formation as a single super-vehicle, i.e., the
operator maneuvers the team as a single solid rigid body. In
this experiment, the payload flies at around an altitude of 1
m over the ground. The Figure 5 shows the actual distances
between vehicles and their velocities in the navigation frame
of coordinates. In particular, we show the distances of the
1A high-definition video of this section can be found at
https://www.youtube.com/watch?v=HUZH46Oxc5c.
2http://wiki.paparazziuav.org/wiki/
−4−2024
−4
−3
−2
−1
0
1
2
3
4
Fig. 4: On the left, a caption of the rotorcraft formation transporting
the payload. On the right, the trajectories (in meters) described by
the team of rotorcraft during the collaborative transportation.
240 250 260 270 280 290 300 310 320 330
Time [s]
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
Vel. [m/s] / Distances [m]
||z2|| ||z3|| ||z4|| ||z5||
Fig. 5: Distances (on top) and X/Y velocity components of the ve-
hicles (on bottom) evolution during the collaborative transportation.
Black-dashed lines represent the desired inter-vehicle distances. For
the translational motion, all vehicles experience the same velocities
for the xand ycoordinates respectively. In the seconds 253 and
278 the vehicles are asked to change abruptly their velocities. The
velocities describe a sinusoid for the spinning motion in the second
305 (circle on the left in Figure 4).
two diagonals and two of the sides of the square as defined
in Figure 1. The operator changes the scale of the formation
(black-dashed line) over time at the same time he maneuvers
the formation. In fact, in order to show the robustness of the
system, during some parts of the experiment the operator
changes quickly the direction of the desired velocity, e.g.,
check seconds 253 and 278. One can check that our worst-
case analysis holds since the worst error distance we noticed
is less than 30 cm around second 296 during a spinning
motion of the formation. In the video of the experiments,
one can also check the predicted non-swing motion of the
payload during its transportation.
VI. CONCLUSIONS
This paper has experimentally validated the predicted
stability properties of a combined method (motion by dis-
agreements + INDI) for the collaborative transportation of
a payload by a team of rotorcraft. In particular, the whole
setup allows us to perform an accurate analysis of forces
and accelerations in the system. Consequently, this approach
enable the possibility of performing a worst-case analysis
such that a nominal operation can be guaranteed. Experi-
ments are conducted with a team of four rotorcraft carrying
a heavy payload, impossible to be lifted by a single vehicle.
The experimental results match with the predicted nominal
behaviors.
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