International audienceIn birds, the neck exhibits remarkable performances and serves as a dextrous arm for performing various tasks. Accordingly, it is an interesting bioinspiration for designing new manipulators with enhanced performances. This paper proposes a preliminary bird neck model using several stacked tensegrity crossed bar mechanisms. It addresses several issues regarding kinetostatic and dynamic modelling, design and control

Abourachid, A.

Böhmer, C

Chablat, Damien

Chevallereau, Christine

Fasquelle, Benjamin

Furet, Matthieu

Wenger, Philippe

English

Archive Ouverte en Sciences de l'Information et de la Communication

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Modelling, design and control of a bird neck using
tensegrity mechanisms
A. Abourachid, C Böhmer, Philippe Wenger, Damien Chablat, Christine
Chevallereau, Benjamin Fasquelle, Matthieu Furet
To cite this version:
A. Abourachid, C Böhmer, Philippe Wenger, Damien Chablat, Christine Chevallereau, et al.. Mod-
elling, design and control of a bird neck using tensegrity mechanisms. ICRA’2019 Worskhop on
Tensegrity, May 2019, Montréal, Canada. ￿hal-02355070￿
Modelling, design and control of a bird neck using tensegrity
mechanisms
B. Fasquelle1, M. Furet1, A. Abourachid2, C. Böhmer2, D. Chablat1, C. Chevallereau1, P. Wenger1,
Abstract—In birds, the neck exhibits remarkable perfor-
mances and serves as a dextrous arm for performing various
tasks. Accordingly, it is an interesting bioinspiration for de-
signing new manipulators with enhanced performances. This
paper proposes a preliminary bird neck model using several
stacked tensegrity crossed bar mechanisms. It addresses several
issues regarding kinetostatic and dynamic modelling, design and
control.
I. INTRODUCTION
The bird neck features outstanding dexterity and dynamic
performances. Birds are capable of using their neck for dex-
trous tasks interacting with the environment (e.g. a vulture
tearing food from a carcass), as well as for tasks demanding
high force transmissions and accelerations (e.g. the wood-
pecker hitting a tree trunk). Contrary to hydrostats such as
the elephant trunk or the cephalopod tentacle, bird necks have
a spine that is composed of several elements (the vertebrae)
like the snake. This work is carried out in the frame of the
AVINECK project, a collaborative, multidisciplinary project
with biologists and aims at proposing a bird neck robotic
model. The concept of tensegrity has been chosen in this
project as a general paradigm able to link the interests of
biologists and roboticists. A tensegrity structure is made
of compressive and tensile components held together in
equilibrium [1], [2]. Tensegrity structures were first used in
art [3] and have then been applied in civil engineering [4]
and robotics [5], [6], [7], [8], [9], [10], [11], [12]. There are
suitable to model muskuloskelet structures where the bones
are the compressive components and the muscles and tendons
are the tensile elements [13]. A preliminary, planar bird neck
robotic model is considered in this paper. This model is
built upon stacking a series of X-mechanisms i.e. crossed
four-bar mechanisms with springs along their lateral sides.
These mechanisms are inspired from the Snelson’s X-shape
mechanisms [2]. Although simplified due to its planarity, this
model goes beyond the only available bird neck model in the
literature that uses a simple planar articulated linkage [14],
as it can be more easily actuated with cables that play the
role of tendons and muscles. Snelson’s X-shape mechanisms
This work was conducted with the support of AVINECK Project ANR-
16-CE33-0025
1 Benjamin Fasquelle, Matthieu Furet, Damien Chablat,
Christine Chevallereau and Philippe Wenger are with Labo-
ratoire des Sciences du Numerique de Nantes, CNRS, Ecole
Centrale de Nantes, France {Benjamin.Fasquelle,
Matthieu.Furet, Christine.Chevallereau,
Philippe.Wenger}@ls2n.fr
2 Anick Abourachid and Christine Böhmer are with Museum Na-
tional d’Histoire Naturelle, Paris, France abourach@mnhn.fr,
boehmer.christine@gmail.com
lli, kli
lri, kri
αi
(i− 1)
(i+ 1)
Fig. 1: Parametrization of a stack of mechanisms.
have been studied by a number of researchers, either as a
single mechanism [5], [7], [15], [16] or assembled in series
[17], [18], [9], [19], [20], [21]. In this paper, each X-shape
mechanism is actuated with two lateral tendons threaded
through the spring attachment points like in [9], [19], [20],
and [21]. The resulting manipulator is supposed to operate in
a vertical plane and is thus subject to gravity, unlike in [9],
where the mechanism was used in a snake-like manipulator
moving on the ground. This paper reports on some recent
results obtained by our team on the AVINECK project.
II. KINEMATIC AND DYNAMIC MODELLING
An assembly of N X-shape tensegrity mechanisms stacked
in series is considered (see figure 1). The mechanisms i
are numbered from i = 1 (base) to i = N (head). Each
mechanism i is a class-2 tensegrity mechanism [4] consisting
of 3 bars and two pretensioned springs. The first mechanism
is fixed to the ground and all the rigid links and springs are
connected to each other with perfect revolute joints. Each
X-mechanism i has a base bar and a top bar of length b and
two crossed bars of length L (with the assembly condition
b ≤ L ), thus defining an antiparallelogram. In parallel to the
left spring (resp. to the right spring), a tendon (not shown in
figure 1) applies a force fli (resp. fri).
If the position (x,y) of the end effector is defined in the
middle of the top bar of the last mechanism, the direct
kinematic model can be expressed as follows [20] [21]:
{
x =
∑n
i=1 li(αi) cos(
pi−αi
2 +
∑i
j=1 2αj)
y =
∑n
i=1 li(αi) sin(
pi−αi
2 +
∑i
j=1 2αj)
(1)
with:
li(αi) = lli + lri =
√
L2 − b2 cos2(αi2 ) (2)
where αi is the relative angle of the top bar of the mechanism
i with respect to the top bar of mechanism i− 1.
The dynamic model of the stacked mechanism is computed
thanks to Lagrange’s equations for a multi-dof (degree of
freedom) system:
d
dt
(
∂T
∂α˙i
)
− ∂T
∂αi
+
dV
dαi
+ = Qi, i = 1, 2, ..., ndof (3)
where T is the kinetic energy, V the potential energy, Q the
generalized forces and ndof is the number of dof (i.e the
number of stacked X-mechanisms). The equation of motion
can be written in the following form:
M(α)α¨+C(α, α˙)α˙+G(α) = Q(α) (4)
where M is the inertia matrix, such that T = 12 α˙
>Mα˙,
C is the matrix of coriolis effects that can be deduced from
M [22], G is the vector of potential effects, G = dVdα ,and Q
is the vector of generalized forces. In [23], the expression
of M(α), G(α) are explicitly derived using an iterative
procedure.
In this study, each mechanism is assumed fully actuated
in an antagonist way, namely, a pair of tendons applies two
positive forces fli and fri in parallel to the left and right
spring, respectively. The generalized forces vector Q defined
in (3) is the torque associated to the generalized coordinate
α :
Q = Zlfl + Zrfr (5)
with fl = [fl1, ..., flN ]>, fr = [fr1, ..., frN ]>, Zl and Zr are
two diagonal matrices whose entries are respectively − dllidαi
and −dlridαi , i = 1, ..N .
III. CONTROL
Each mechanism i has a desired trajectory αdi . Let α
d
be the vector of all those trajectories, and α the vector of
the measured mechanism orientations. The control law is
built from equation (4). Low velocities are assumed and the
Coriolis effects are thus neglected: C = 0N×N . Thus, the
desired torque Qd is:
Qd =M(α)
(
α¨d+kdIN (α˙
d−α˙)+kpIN (αd−α)
)
+G(α)
(6)
where kd and kp are the gains of the control law. The required
forces fl and fr must satisfy:
Qd = Zl(α)fl + Zr(α)fr (7)
The solution that minimizes the norm of the sum of the forces
||fl+ fr|| is chosen i.e. since forces must remain positive, for
each mechanism one force is zero and the other one produces
the motion.
100 200 300 400 500 600 700 800 900 1000
100
200
300
400
500
600
700
Fig. 2: First frame of the video, where the middle line of the
neck is added in white, and the placed mechanisms are in
green. The bird is a green woodpecker (Picus viridis).
IV. EXTRACTION OF A NECK MOVE FROM A VIDEO
Our goal is to reproduce movements made by a wood-
pecker with the proposed tensegrity manipulator. To get bird
neck movements, recorded videos of bird movements would
be used. However, on alive birds, the feathers around the
neck make it almost impossible to distinguish the neck. The
extraction of a move was made from a video of a bird corpse
from which the feathers were removed, moved by hand.
The video was made with a green woodpecker.
The motion capture was performed frame by frame. For
each frame, the line at the middle of the neck is detected
and the mechanisms are then placed along this line.
A. Positioning the mechanisms along the line
To reproduce the neck trajectory with the tensegrity ma-
nipulator, the position of each mechanism at each frame is
required. The number of mechanims N , the lengths of the
top bars bi and the diagonal bars Li of each mechanism i
are fixed. The orientation of the base of the neck γ0 is given
by the orientation of the string at this place. The middle of
the base bar x0 is given by the bottom of the middle line.
The mechanisms are placed one by one from the bottom.
For each mechanism i, the orientation of the base bar γi and
its center position xi are given by
γi = γ0 +
i−1∑
j=1
αj (8)
xi = x0 +
i−1∑
j=1
√
L2j − b2jcos(
αj
2
)2cos(γj +
αj
2
) (9)
With a scan on the position αi, a set of potential positions
are tested. A mechanism is considered well positioned if the
middle of its top bar is near the line and if the orientation
of its top bar is near the perpendicular from the tangent to
the line where the mechanism is located. Hence, the choosen
position αi is the one that minimizes the following function:
v(αi) = d(xi) + λ|(γi + αi)− αline| (10)
where d(xi) is the distance between the line and xi, αline
is the angle of the perpendicular from the tangent to the
line where the top bar of the mecanism cuts it, and λ is a
parameter that can be used to modify the influence of the
two parts.
Fig. 3: The manipulator in Simscape. Only the solid bodies
are visible.
B. Simulated movement
As the angle of the base of the neck does not change much
during the video, it is fixed to γ0 = −pi4 . The manipulator
is composed of N = 11 mechanisms, because the green
woodpecker neck has 11 vertebrae. The Lb = 1.52 ratio is
choosen by hand to have a manipulator with a size similar
to the neck.
To define the trajectories of the mechanisms, the posi-
tions obtained previously were smoothed separately for each
mechanism.
V. SIMULATION RESULTS
A. Neck parameters
The manipulator studied is made of N = 11 identical
mechanisms with L = 0.152m and b = 0.1m. The springs
free length is defined as l0 = L − b (which is the smallest
length of the springs reached in the flat configurations αi =
±pi). All bars are cylinders of diameter d = 0.01m and made
of ABS with a volumic mass 1050 kg/m3. All the springs
have the same stiffness k = 100N.m−1.
An head with a mass of 270g is put on the last mechanism.
It is modelized like a sphere in the dynamic model. The
Figure 3 shows the manipulator in Simscape. The cables and
the springs are not observable.
The minimal and maximal bounds of the actuation forces
are defined as fmin = 0N (the tendons can only pull) and
fmax = 100N , respectively.
B. Simulation results
Simulations are performed with Matlab Simulink and
Simscape Multibody. Figure 4 shows the positions of the
mechanisms during the simulation and the desired configu-
ration based on the data extract for the video of the neck bird
motion. The tracking is good and it has been shown in [23]
that the use of the dynamic model allow to obtain better
time [s]
po
sit
io
n 
[ra
d]
Mechanism1
time [s]
po
sit
io
n 
[ra
d]
Mechanism2
time [s]
po
sit
io
n 
[ra
d]
Mechanism3
time [s]
po
sit
io
n 
[ra
d]
Mechanism4
time [s]
po
sit
io
n 
[ra
d]
Mechanism5
time [s]
po
sit
io
n 
[ra
d]
Mechanism6
time [s]
po
sit
io
n 
[ra
d]
Mechanism7
time [s]
po
sit
io
n 
[ra
d]
Mechanism8
time [s]
po
sit
io
n 
[ra
d]
Mechanism9
time [s]
po
sit
io
n 
[ra
d]
Mechanism10
time [s]
po
sit
io
n 
[ra
d]
Mechanism11
Fig. 4: Position during the simulations. The Reference curve
(in black) corresponds to the desired trajectory, the red one
is the result of the simulation.
result that a PD control law. The forces applied with the
dynamic control law on each mechanism are shown in Figure
5. The present control law limits the forces applied by using
only one antagonist actuator while the other one produces
no force. A control of the stiffness of the mechanism can be
also included to react to the environment.
VI. CONCLUSION AND PERSPECTIVES
This paper analyzed the modeling, design and control of
a tensegrity robot inspired from a bird neck and using a
stack of X-shape tensegrity mechanism actuated by cables.
The kinematic model was used to define a reference motion
of each mechanism based on a video of the motion of
a bird neck. The dynamic model was used to build a
convenient control law to track the reference motion with
an actuation composed of two independant forces acting on
each mechanism.
A prototype is currently under development in order to
carry out experimental work as illustrated in figure 6.
The current work concerns:
• The choice of the actuation,
• The interaction with the environment,
time [s]
f l 
[N
]
time [s]
f r
 
[N
]
Fig. 5: Forces applied during the movement.
Fig. 6: The prototype under construction.
• The extension to 3D motion of each mechanism.
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Modelling, design and control of a bird neck using tensegrity mechanisms

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Modelling, design and control of a bird neck using tensegrity mechanisms

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