![Fig. 2: Normal force at the end-effector when the robot is in contact. With FT sensors feedback (top plot), the measured force remains close to the desired value. With no feedback (bottom plot), the force drifts of ±1 [N].](/figures/fig-2-normal-force-at-the-end-effector-when-the-robot-is-in-10od8c71.webp)
Preprint version, final version at http://ieeexplore.ieee.org/ IEEE Robotics and Automation Letters 2020
Direct Force Feedback Control and Online Multi-task Optimization for
Aerial Manipulators
Gabriele Nava
1, 2
, Quentin Sabl
´
e
3
, Marco Tognon
3
, Daniele Pucci
1
, Antonio Franchi
3,4
Abstract— In this paper we present an optimization-based
method for controlling aerial manipulators in physical contact
with the environment. The multi-task control problem, which
includes hybrid force-motion tasks, energetic tasks, and po-
sition/postural tasks, is recast as a quadratic programming
problem with equality and inequality constraints, which is
solved online. Thanks to this method, the aerial platform
can be exploited at its best to perform the multi-objective
tasks, with tunable priorities, while hard constraints such as
contact maintenance, friction cones, joint limits, maximum
and minimum propeller speeds are all respected. An on-board
force/torque sensor mounted at the end effector is used in
the feedback loop in order to cope with model inaccuracies
and reject external disturbances. Real experiments with a
multi-rotor platform and a multi-DoF lightweight manipulator
demonstrate the applicability and effectiveness of the proposed
approach in the real world.
I. INTRODUCTION
Nowadays, Unmanned Aerial Vehicles (UAVs) are
widespread and employed in several application scenarios
such as surveillance, remote monitoring and aerial photogra-
phy. The recent development of aerial manipulators physi-
cally interacting with the environment, significantly enlarged
the number of feasible tasks spanning from manipulation
and grasping of objects to the contact-based inspection [1].
Such interaction tasks often require the aerial manipulator to
precisely regulate both position of the contact point (possibly
moving) and the amount of force that is exerted on it.
There is a rich literature concerning the modeling and
control of aerial manipulators [2]. Regarding the motion
control, different methods can be applied, e.g., full dy-
namic inversion [3], flatness-based [4], and adaptive sliding
mode [5]. On top of those, if the system is over actuated
with respect to the desired task, a nullspace-based behavioral
control [6] or a task priority controller [7] can be applied
at the kinematic level exploiting the redundancy to achieve
secondary tasks (e.g., obstacle avoidance, minimum energy
consumption, etc.). In particular, the latter acts as a local
motion planner providing the reference trajectory of each
degree of freedom of the robot to the low level motion
controller. The main limitation of this control architecture
1
Dynamic Interaction Control, Istituto Italiano di Tecnologia, Genova, Italy
gabriele.nava@iit.it,
2
DIBRIS, University of Genova, Genova, Italy,
3
LAAS-CNRS, Universit
´
e de Toulouse, CNRS, Toulouse, France,
marco.tognon@laas.fr, quentin.sable@laas.fr
4
Robotics and Mechatronics lab, Faculty of Electrical Engineering,
Mathematics & Computer Science, University of Twente, Enschede, The
Netherlands a.franchi@utwente.nl
This research was partially supported by the ANR, Project ANR-18-
CE33-0001 ‘The Flying Coworker’.
Fig. 1: CAD rendering of the aerial manipulator designed
and developed at LAAS-CNRS.
is the inability to consider additional hard constraints to be
respected, e.g., input/state bounds, dynamics of the system
which is not taken into account at the kinematic level, friction
cone in case of interaction, etc. The mentioned constraints
are very important because, if not respected, they might bring
the whole system to instability in real scenarios.
From the interaction control side, one of the most common
strategy is to use an admittance filter. In [8] and [9], such
technique has been employed to control the interaction force
in the case of a fully-actuated platform equipped with a
rigid tool, and of an under-actuated platform equipped with a
robotic arm, respectively. In [10], a passivity-based controller
has been proposed as well. However, in those works, the
force-control is only indirect. In fact, the force is not directly
measured but rather estimated using the robot dynamics.
Since the method is strongly model based, it is thus prone
to error in case of parameter uncertainties. Furthermore, if
the system is affected by external disturbances, it might
not be possible to discriminate them from the interaction
forces. A first attempt in using a direct force feedback can
be found in [11] where a force sensor has been attached to
the end-effector. Nevertheless, this feedback is not used to
precisely control the interaction force but is rather used in
an impedance control framework to make the end-effector
compliant.
In this work we propose a whole-body force control
strategy for aerial manipulators that allows, in a unique
formulation, to achieve hybrid position/force control, con-
sidering multi-task optimization under hard-constraints. We
shall show that with the proposed control framework, the
aerial manipulator can perform different contact-based tasks
using the same strategy, only changing the tasks priority and
contact-constraints. In our framework, we consider several
control objectives organized in a weighted prioritization. The
controller is implemented in the form of a Quadratic Pro-
gramming (QP) optimization, which is a rather known tech-
nique in the field of humanoid robotics [12], [13], but whose
applicability and effectiveness was never demonstrated until
now in the challenging field of aerial manipulators – where
the physical interaction tasks are made very complex by,
e.g., the absence of stabilizing contacts, limited propeller
forces, inaccurate and time varying aerodynamics models,
and mechanical vibrations.
Furthermore, to provide an accurate interaction control,
our controller implements a direct force-feedback employing
a 6-axis force-torque sensor (FT) mounted on the manipu-
lator end-effector. We shall show through real experiments
that the FT sensor feedback ensures the robustness of the pro-
posed control law with respect to disturbances and modeling
errors. The effectiveness of the overall control framework
has been shown through real experiments for three different
tasks: 1) pushing under external disturbances, 2) sliding
while pushing and 3) high-acceleration motions exploiting
contacts, but of course these are just a few of the many real
world tasks can be achieved with the proposed method.
The remainder of the paper is organized as follows:
Sec. II recalls the notation, the dynamics model of the aerial
manipulator and the rigid contact model. Sec. III details the
task-based control strategy and the QP optimization problem.
Simulations and experimental results with a real hexarotor
equipped with a three degrees of freedom manipulator are
presented in Sec. IV. Conclusions and perspectives end the
paper.
II. MODELING
We consider an aerial manipulator composed of an aerial
platform equipped with a robotic manipulator as in Fig. 1.
Given the flying nature of the system, we model it as a
floating base system [14]. We assume that the robotic arm
is composed of n + 1 links, connected by n actuated joints
with a single degree of freedom each.
To describe the state of the aerial manipulator we define
a world frame F
W
= {O
W
, x
W
, y
W
, z
W
}, with arbitrarily
placed origin O
W
, and unit axes (x
W
, y
W
, z
W
) such that z
W
points in the opposite direction of the gravity vector. An
additional frame F
R
= {O
R
, x
R
, y
R
, z
R
} is rigidly attached
to the aerial platform base. Finally, we rigidly attach a frame
F
E
= {O
E
, x
E
, y
E
, z
E
} to the end-effector of the robotic arm
such that z
E
is parallel to the last link of the manipulator.
The robot configuration is then given by q =
(p
R
, R
R
, q
A
) ∈ C ⊆ R
3
× SO(3) × R
n
where p
R
∈ R
3
and R
R
∈ SO(3) represent the position and orientation of
F
R
with respect to (w.r.t.) F
W
, respectively, while q
A
∈ R
n
are the manipulator joint angles which characterize the
configuration of the robotic arm. The velocity relative to q
is given by v = [v
>
R
ω
>
R
v
>
A
]
>
∈ R
6+n
, where v
R
∈ R
3
and
ω
R
∈ R
3
are the aerial platform linear and angular velocities,
and v
A
∈ R
n
are the joints velocities. More precisely, ω
R
is the angular velocity of F
R
w.r.t. F
W
expressed in F
R
,
which satisfies
1
˙
R
R
= S(ω
R
)R
R
.
The system equations of motions can be computed by
applying the Euler-Poincar
´
e formalism [15, Ch. 13.5] :
M (q) ˙v + c(q, v) + g(q) =
w
>
R
τ
>
A
>
+ J
E
(q)
>
w
E
, (1)
where M (q) ∈ R
(n+6)×(n+6)
is the mass matrix, c(q, v) ∈
R
n+6
accounts for the Coriolis and centrifugal effects, g(q) ∈
R
n+6
is the gravity vector, w
R
∈ R
6
is the control wrench
(forces and moments) applied to the aerial vehicle by the
propellers, and τ
A
∈ R
n
are the joint torques of the manip-
ulator. We assume that the interaction with the environment
occurs at the manipulator end-effector only. When the robot
end-effector is in contact with the environment, the external
wrench w
E
∈ R
6
has to be included in the equations. The
Jacobian J
E
(q) ∈ R
6×(n+6)
is the map between the system
velocity v and the linear and angular velocities of the
end-effector at the contact location. It is assumed that for
the applications proposed in this paper the aerodynamics
phenomena such as, e.g., wind effect, blade flapping, cross
interference between propellers, and ground/wall effect do
not significantly affect the robot dynamics, and therefore can
be neglected. Considering the aerial vehicle actuated by a set
of p ∈ N rigidly attached propellers, the control wrench w
R
can be conveniently rewritten as a function of the propellers
angular rates. In particular we consider, as usual, a quadratic
relation between propeller angular rates and corresponding
generated thrust:
w
R
= G
w
(q)ω
2
P
:
= G
w
(q)(ω
P
ω
P
), (2)
where ω
P
∈ R
p
is the vector of propellers angular rates, is
the component-wise product between two vectors, and G
w
∈
R
6×p
is the mapping between the propellers square angular
rates and the control wrench. In particular we can partition
G
w
in two blocks, G
w1
∈ R
3×p
and G
w2
∈ R
3×p
such that
G
w
= [G
>
w1
G
>
w2
]
>
. G
>
w1
and G
>
w2
map the propellers square
angular rates into control force and moment, respectively.
In this work, we consider both cases of 1) under-actuated,
and 2) fully-actuated vehicles. For under-actuated vehicles,
0 < rank(G
w1
) < 3 which means that the total thrust can-
not change in all directions without reorienting the whole
platform. This is the case of standard quadrotors where
the propellers are all collinear and the thrust direction is
fixed w.r.t. F
R
. On the contrary, for fully-actuated vehicles,
rank(G
w1
) = 3 (it has to be that p ≥ 6) which means that
the total thrust can change in all directions. In both cases,
G
w2
= 3, i.e., there is full control on the moment applied by
the aerial platform.
A. Contact Modeling
We assume that the robot interacts with a rigid and planar
environment.
2
In an industrial environment, such assump-
tions may occur while executing tasks such as polishing,
1
The skew operator S(?) : R
3
→ R
3×3.
is defined such that for two generic
vectors x, y ∈ R
3
, S(x)y = x × y
2
The planarity assumption is adopted here for simplicity and could be
replaced by a less stringent assumption of surface smoothness with small
changes in the model and control law.
Preprint version, final version at http://ieeexplore.ieee.org/ 2 IEEE Robotics and Automation Letters 2020
inspection or welding. Possible types of interaction from the
end-effector constraints point of view are:
• Fully constrained: the end-effector position and orientation
remain always constant w.r.t. the inertial frame;
• Only the position is constrained: the end-effector can
freely rotate, but it cannot change its position. This is the
case of a single contact point;
• Only the normal translation is constrained: the translation
in the direction normal to the contact plane is constrained,
while the end-effector is free to move in the perpendicular
directions. The end-effector can also rotate;
• Rotations and the normal translation are constrained: the
end-effector position is constrained as in the previous case.
However, the end-effector can rotate about the normal
direction only;
We model those interactions by a set of holonomic con-
straints which describe the limitations of the end-effector
motion [16], [17]. Often it is easier to express those con-
straints w.r.t. local reference frame F
E
. Differentiating such
constraints w.r.t. time one obtains:
S
c
E
v
>
E
E
ω
>
E
>
= 0,
where
E
v
E
and
E
ω
E
are the end-effector linear and angular
velocities w.r.t. F
E
, while S
c
∈ R
n
c
×6
is a selector of the
constrained directions of motion. n
c
∈ N
>0
represents the
number of motion constraints applied to the end-effector.
Note that in the local reference frame the selector matrix
S
c
usually remains constant during each interaction task.
Using the kinematic relation, the contact constraints can be
expressed as a function of the robot velocities v:
S
c
¯
R
v
>
E
ω
>
E
>
= J
c
E
v = 0, (3)
where
3
J
c
E
:
= S
c
¯
RJ
E
∈ R
n
c
×6+n
,
¯
R = blkdiag(R
E
, R
E
)
with R
E
∈ SO(3) being the rotation matrix describing the
orientation of F
E
w.r.t. F
W
. The equations complementary
to (3) represent instead the directions of motion of the end-
effector that remain free to move, and can be written as:
S
f
¯
R
v
>
E
ω
>
E
>
= J
f
E
v =: v
f
, (4)
where J
f
E
:
= S
f
¯
RJ
E
∈ R
6−n
c
×6+n
with S
f
∈ R
6−n
c
×6
the
selector matrix complementary to S
c
.
Rewriting the system dynamics (1) including (2) and
taking into account the contact model (3)-(4) gives:
M ˙v + c + g = Gu + J
c>
E
f
c
+ J
f >
E
f
f
(5a)
J
c
E
˙v +
˙
J
c
E
v = 0, (5b)
where G = blkdiag(G
w
, I
3
), u = [ω
2
P
>
τ
>
A
]
>
, f
c
∈ R
n
c
are
the contact forces and/or moments, while f
f
∈ R
6−n
c
repre-
sent forces and moments that may arise in the unconstrained
directions of motion, e.g., viscous friction during motion.
Equation (5b) is the time differentiation of (3) and highlights
the constraints on the accelerations ˙v.
3
From now on, with the aim of compactness, we avoid to report the state
dependence of some quantities as the Jacobian and mass matrix, and so on.
III. CONTROL DESIGN
Let us denote the vector of outputs of interest (called tasks)
with y = [y
1
. . . y
m
]
>
∈ R
m
, where m ∈ N
>0
. The control
method is composed of an outer loop and an inner loop.
The outer-loop assumes that a certain time-derivative for each
task, defined by the symbol
4
a = [y
(r
1
)
1
. . . y
(r
m
)
m
]
>
, is directly
controllable by a virtual input a
?
, i.e.:
a = a
?
. (6)
Based on this assumption any kind of stabilizing controller
(PID, sliding mode, robust control, etc) can be applied by
the outer loop designing a
?
such that to steer y along a
sufficiently smooth desired trajectory y
d
(t). In what follows,
we apply to the outer loop a PID controller, whose effec-
tiveness in stabilizing the desired output has been verified
experimentally. The role of the inner-loop is instead to
compute the real system inputs in order to verify as much as
possible (6) via the resolution of a constrained optimization
problem [12]:
minimize
u
(a − a
?
)
>
W
a
(a − a
?
), (7)
subjected to several constrains, e.g.: 1) input and state bound-
aries, 2) contact stability constraints, 3) system dynamics,
etc. The (semi) positive and diagonal weight matrix W
a
∈
R
m×m
allows to defines soft priorities among all the tasks.
As demonstrated in other robotic fields,
5
this technique
can handle complex system dynamics and a large number
of tasks. An additional advantage is the flexibility in adding
and removing both tasks and constraints. As an example, if
strict tasks priorities are required, the optimization problem
can be easily modified by transforming some of the elements
of the cost function into equality constraints.
The input-output asymptotic stability is obtained if it exists
u
?
such that a = a
?
, i.e., if the optimal control input is
a feedback linearizing control. This implies that a careful
choice of a is required. However, the definition of a may not
be easy in case the robot has to perform a complex operation
in a real environment. Furthermore, the presence of several
constraints may prevent to find a solution that guarantees (6)
for all the components of a. In this case, the solution will
privilege the high priority tasks while keeping bounded the
the error on lower priority tasks. The priority among tasks,
and consequently the behavior of the robot, can be modified
by properly choosing W
a
.
In the following, considering a contact-based application
requiring a hybrid position/force control of the end-effector,
we define the appropriate set of tasks and the implementation
of the inner- and outer-loop controllers.
1) Control Task Definition: For our case study, we define
the output as y = (p
R
, R
R
, q
A
, p
f
, f
c
, r
a
) and the correspond-
ing time-derivatives to be assigned by the outer loop as
a =
˙v
>
R
˙ω
>
R
˙v
>
A
˙v
>
f
f
>
c
r
>
a
>
. The tasks p
R
, R
R
and q
A
represent the full aerial manipulator configuration, and can
4
For a given variable x ∈ R, x
(r)
indicates the derivative of order r of x.
5
As, e.g., in humanoid control.
Preprint version, final version at http://ieeexplore.ieee.org/ 3 IEEE Robotics and Automation Letters 2020
be used to cover several real world objectives, from simple
changes of the overall position, to more complex coordinated
maneuvers in cluttered environment. The task p
f
∈ R
6−n
c
and
its time-derivative v
f
represent the end-effector positions and
velocities in the unconstrained directions of motion. The role
of v
f
and f
c
is to specify the hybrid force-motion behavior
of the aerial manipulator perform, when in contact. Finally,
the term r
a
is a propeller regularization task which is defined
as a function of the propeller forces and such that r
a
= 0
when all the forces are the same. The regularization term is
designed as follows:
r
a
= D
r
a
ω
2
P
,
where D
r
a
∈ R
p−1×p
is a matrix whose elements are 1 along
the main diagonal and −1 right above the main diagonal, and
all other elements are equal to zero. The goal of this task is
to balance the propeller forces in order to avoid energetically
unfavorable solutions where some propellers deliver almost
zero thrust and others are close to saturation.
6
Notice that
the virtual input associated to the tasks f
c
and r
a
are the
variable themselves (zero-order time-derivative) being such
tasks algebraic functions of the input.
2) Inner-Loop (QP-based Optimization): Let us consider
the extended input u
0
= [u
>
f
>
c
]
>
∈ R
n+p+n
c
in which the
contact wrench is added to the real control inputs u. In
this way, inverting the dynamics (5a) and the time derivative
of (4) one can express a as a linear function of u
0
:
a = H(q)u
0
+ h(q, v), (8)
where H(q) ∈ R
m×(n+p+n
c
)
and h(q, v) ∈ R
m
contain all the
terms that do not depend on u
0
. More specifically, the matrix
H(q) and the bias vector h(q, v) are given by:
H(q) =
M
−1
G M
−1
J
c>
E
J
f
E
M
−1
G J
f
E
M
−1
J
c>
E
0
(n
c
×p)
1
(n
c
×n
c
)
D
r
a
0
(p−1×n
c
)
,
h(q, v) =
M
−1
(J
f >
E
f
f
− c − g)
J
f
E
M
−1
(J
f >
E
f
f
− c − g) +
˙
J
f
E
v
0
0
.
To ensure the satisfaction of the contact constraint the
equation (5b) is added to the optimization problem (7) as a
constraint. Such input extension avoids the explicit inversion
of J
c
E
and the related singularity issues that may arise in
some configuration.
For the considered interaction-based scenario, the inner-
loop control problem (7) may be formulated as
minimize
u
0
(a − a
?
)
>
W
a
(a − a
?
) (10a)
subject to: u
l
≤ u ≤ u
u
(10b)
6
This task is particularly important in the case that the number of
propellers and their arrangement is redundant for the execution of the
remaining tasks. as, e.g., in the case of an underactuated floating base with
six or more propellers all pointing in the same direction.
Cf
c
≤ b (10c)
M ˙v + c + g = Gu + J
c>
E
f
c
+ J
f >
E
f
f
(10d)
J
c
E
˙v +
˙
J
c
E
v = 0 (10e)
The constraint (10b) takes into account the bounds on the
control inputs, such as saturations in the joint torques and
propeller velocities, while (10c) ensures the respect of con-
tact stability conditions such as friction cone (approximated
with linear inequalities) and positivity of the normal force.
Constraints (10e)-(10d) correspond to the dynamics of the
aerial manipulator when in contact with the environment.
Observing (10), it is easy to verify that is an instance of
a Quadratic Programming (QP) problem. In fact, the cost
function and constraints are a quadratic and linear functions
of the optimization variable u
0
, respectively. The QP problem
allows for a fast and efficient solution that provides online
the control inputs u
?
, such that u
0?
= [u
?
>
f
?
c
>
]
>
is solution
of (10).
3) Outer-Loop (Desired Dynamics): Given a desired out-
put trajectory y
d
(t) the task virtual inputs a
?
are chosen as
follows:
a
?
=
˙v
d
R
− K
DR
(v
R
− v
d
R
) − K
PR
(p
R
− p
d
R
)
˙ω
d
R
− K
DRω
(ω
R
− ω
d
R
) − K
PRω
e
Rω
˙v
d
A
− K
DA
(v
A
− v
d
A
) − K
PA
(q
A
− q
d
A
)
˙v
d
f
− K
DF
(v
f
− v
d
f
) − K
PF
(p
f
− p
d
f
)
f
d
c
− K
PC
(f
m
c
− f
d
c
) − K
IC
R
t
0
(f
m
c
− f
d
c
)dt
0
(11)
where the K
∗
are symmetric and positive definite matrices
and the rotation error e
Rω
∈ R
3
is given by e
Rω
=
1
2
[R
>
R
R
d
R
−
R
d
R
>
R
R
]
∨
. Direct feedback from FT sensors measurements
is used in the force control task for computing the force error
f
m
c
− f
d
c
, with f
m
c
being the measured contact force.
IV. RESULTS
A. Experimental Setup
The control framework presented in Sec. III is tested with
the Open Tilted Hexarotor (OTHex) [18]. The OTHex is a
custom-made aerial vehicle composed of six coplanar-center
propellers. The tilted arrangement of the propellers allows
the multi-directional thrust property and therefore guarantees
the local full actuation of its dynamics. However, despite
the local full actuation, the propeller saturation still limits
substantially its ability to exert a force sideways, as it will
be clearly shown in some phases of the experiments, where
tilting of the whole OTHex is also requested by the controller
in order to get the needed pushing force.
The OTHex has been equipped with a three degrees of
freedom serial manipulator as in Fig. 1. The manipulator’s
end-effector is composed of a pointed tool rigidly attached
to a 6-axis force/torque sensor. The manipulator is controlled
with a velocity control loop that commands desired mo-
tors velocities to three Dynamixel motors. The reference
velocities for the velocity control loop are computed by
numerically integrating the commanded joints acceleration
Preprint version, final version at http://ieeexplore.ieee.org/ 4 IEEE Robotics and Automation Letters 2020
Weights for the selected tasks during experiments
p
R
R
R
q
A
p
f
f
c
r
a
Push with dist. 1 1 0.1 0 2 1e-5
Push and slide 0.1 1.5 0.025 2.5 1 1e-5
Push away 2 2 0.05 0 0 1e-5
Simulation 1 0.01 0.01 0 1 1e-5
TABLE I: Weights of the selected tasks for the QP cost
function, during the three experiments and in simulation.
˙v
?
A
, that are obtained inverting the system’s dynamics (5)
when u
0
= u
0?
.
All experiments are performed in an indoor arena using
a Motion Capture (MoCap) system. The Control algorithm
is implemented in Matlab-Simulink and runs on an external
PC at a frequency of 250 [Hz]. The inner-loop optimization
problem in (10) is resolved run-time by means of qpOASES
solver [19]. As it often occurs in real world applications, the
hard QP equality constraints are softened by moving (10e)
into the set of virtual inputs (11). This modification helps to
reduce the discontinuities in the control input when the con-
tact constraints are activated/deactivated. Contact constraints
are then heavily weighted by properly designing matrix W
a
,
to enforce the achievement of the corresponding task in (11).
We performed three different experiments and a simulation,
that will be all detailed in the next paragraphs. The QP
weights used for the different tasks during each experiment
are listed in Table I. The videos of the experiments are
available in the attached multimedia material.
B. Pushing with Disturbances
During this experiment, the robot’s end-effector gets in
contact with a rigid surface. The robot is then required
to push against the surface with a normal force of 5 [N].
After few seconds, a virtual force disturbance of 3.5 [N]
is applied at the OTHex base link. The disturbance persists
for 5 seconds, and it is removed after. The purpose of the
experiment is to understand the benefit of adding direct
force feedback from the force/torque sensor when tracking a
reference force in presence of external disturbances. To this
purpose, we compared two different scenario: in the ’no FT’
case, the reference output force from (11) is computed as
f
c
= f
d
c
, thus not adding any feedback from the measured
contact forces. In the ’with FT’ case, the reference output
force is computed including feedback terms as in (11).
Figure 2 describes the behavior of the normal force during
the push with disturbance experiment for the two different
cases. In particular, the plot shows the measured ( f
m
c
z
) and
commanded ( f
?
c
z
) normal forces during the interaction task.
The dashed black line is the desired normal force. The red
region denotes the period of time during which the end-
effector is in contact with the surface, and the blue region is
the time period when the disturbance is applied.
If feedback from force/torque sensors is used in the control
law (top plot of Fig. 2), the commanded vertical force
(blue line) increases to 6N when the disturbance is applied.
This due to the presence of force feedback, that attempts
at compensating for the exernal disturbance. In fact, as a
0
2
4
6
8
0 5 10 15 20
-2
0
2
4
6
8
Fig. 2: Normal force at the end-effector when the robot is in
contact. With FT sensors feedback (top plot), the measured
force remains close to the desired value. With no feedback
(bottom plot), the force drifts of ±1 [N].
consequence of the new commanded force, the measured
force f
m
c
z
(red line) remains close to 5N after a short transient
phase. When no force feedback is present instead, as in the
bottom plot of Fig. 2, the commanded force does not change
magnitude and the error between measured and desired force
increases up to 1 [N]. We recall that in this second case the
force/torque sensor information is only used as ground-truth,
but is not actively employed in the control algorithm.
The top plot of Fig. 3 shows the the propeller commanded
angular rates ω
?
P
during the experiment with FT sensor
feedback. When the external force is applied, three propellers
reach saturation, which is represented in the figure by the
dotted horizontal lines. Therefore, the solution a = a
?
cannot
be achieved anymore, and the QP penalizes tasks with lower
priority, such as the OTHex orientation, in the attempt of
maintaining a small error on the higher priority task. This
effect is visible in the bottom plot of Fig. 3, where the error
along the pitch angle of the OTHex increases up to 7 [deg]
during the saturation phase.
C. Push and Slide
In this experiment, the robot pushes against the surface
with a normal force of 2.5 [N], while sliding along the surface
following a reference end-effector trajectory. We made use
of FT sensors feedback for improving the tracking of the
normal force, and the FT sensor information is also used
for compensating the viscous friction forces acting on the
surface while sliding.
The top part of Figure Fig. 4 compares the measured and
the desired end-effector trajectory along the contact surface.
Despite some noise due to vibrations and compliance of
the arm, the tracking error always remains small (< 1 [cm]
Preprint version, final version at http://ieeexplore.ieee.org/ 5 IEEE Robotics and Automation Letters 2020