A Model for Robotic Hand Based on Fibonacci Sequence
Anna Chiara Lai, Paola Loreti and Pierluigi Vellucci
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sezione di Matematica, Sapienza Universit
`
a di Roma, Italy
Keywords:
Robot Hand, Redundant Manipulator, Fibonacci Sequence, Discrete Control.
Abstract:
We study a robot hand model involving Fibonacci sequence. Fingers are modeled via hyper-redundant planar
manipulators. Binary controls rule the dynamics of the hand, in particular the extension and the rotation of each
phalanx. By estabilishing a relation with Iterated Function Systems, we investigate the reachable workspace
and its convex hull. Finally, we give an explicit characterization of the convex hull of the reachable workspace
in a particular case.
1 INTRODUCTION
The aim of this paper is to give a model of robot hand
whose links scale according to Fibonacci sequence
and to develop a theoretical background (related to
the theory of iterated function systems) in order to
study some geometrical features of a such a manip-
ulator. Fibonacci numbers attracted the interest of
researchers due to their fascinating algebraic prop-
erties (e.g. the relation with Golden Mean) and due
of their recurrence in natural phenomena. Examples
of relations with Fibonacci sequence can be found
in the branches of trees, in the arrangement of sun-
flowers seeds and, most interestingly for our model,
in some human anatomic proportions (see (Hamilton
and Dunsmuir, 2002)).
Self-similarity of configurations and an arbitrar-
ily large number of fingers (including the opposable
thumb) and phalanxes are the main features. Binary
controls rule the dynamics of the hand, in particular
the extension and the rotation of each phalanx. We
assume that each finger moves on a plane; every plane
is assumed to be parallel to the others, excepting the
thumb and the index finger, that belong to the same
plane. A discrete dynamical system models the posi-
tion of the extremal junction of every finger. A con-
figuration is a sequence of states of the system corre-
sponding to a particular choice for the controls, while
the union of all the possible states of the system is
named reachable workspace for the finger. The clo-
sure of the reachable workspace is named asymptotic
reachable workspace. Our model includes two binary
control parameters on every phalanx of every finger
of the robot hand. The first control parameter rules
the lenght of the k-th phalanx, that can be either 0 or
f
k
ρ
−k
, where f
k
is the k-th Fibonacci number and ρ is
a fixed scaling ratio, while the other control rules the
angle between the current phalanx and the previous
one. Such an angle can be either π, namely the pha-
lanx is consecutive to the previous, or a fixed angle
π −ω ∈ (0,π).
The structure of the finger ensures the set of possi-
ble configurations to be the projection of a particular
self-similar set. This is the key idea underlying our
model and our main tool of investigation.
We establish a connection between our model and
the theory of iterated function systems. This yields
several results describing the reachable workspace
and its convex hull.
1.1 Previous Work and Motivations
The fingers of our robot hand are planar manipu-
lators with rigid links and with a (arbitrarily) large
number of degrees of freedom, that is they belong
to the class of so-called macroscopically-serial hyper-
redundant manipulators (the term was first introduced
in (Chirikjian and Burdick, 1990)).
Hyper-redundant architecture was intensively
studied back to the late 60’s, when the first prototype
of hyper-redundant robot arm was built (Anderson
and Horn, 1967). The interest of researchers in de-
vices with redundant controls was motivated, among
others, by the ability to avoid obstacles and the ability
to perform new forms of robot locomotion and grasp-
ing (see for instance (Chirikjian and Burdick, 1995)).
A large number of papers were devoted in the lit-
erature to both continuously and discretely controlled
577
Lai A., Loreti P. and Vellucci P..
A Model for Robotic Hand Based on Fibonacci Sequence.
DOI: 10.5220/0005115205770584
In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 577-584
ISBN: 978-989-758-040-6
Copyright
c
 2014 SCITEPRESS (Science and Technology Publications, Lda.)

hyper-redundant manipulators. Our approach, based
on discrete actuators, is motivated by their precision
with low cost compared to actuators with continu-
ous range-of-motion. Moreover the resulting dis-
crete space of configurations reduces the cost of po-
sition sensors and feedbacks. In (EbertUphoff and
Chirikjian, 1996) the inverse kinematics of discrete
hyper-redundant manipulators is investigated.
In general the number of points of the reachable
workspace increases exponentially, the computational
cost on the optimization of the density distribution of
the workspace is investigated in (Lichter et al., 2002).
Note that the concept of a binary tree describ-
ing all the possible configurations underlies above
mentioned approaches. In our method the self-
similar structure of such a tree gives access to well-
established results on fractal geometry and iterated
function systems theory. Robotic devices with a sim-
ilar fractal structure are described in (Moravec et al.,
1996).
The relation between iterated function systems,
expansions in non-integer bases and planar manipu-
lator was investigated in (Lai and Loreti, 2011) and in
(Lai and Loreti, 2013), by assuming the ratio between
the lenghts of two consecutive links to be equal to a
constant ρ > 1, so that the length l
k
of the k-th link is
equal to 1/ρ
k
. In the present paper we extend this in-
vestigation to the case l
k
= f
k
/ρ
k
where, as mentioned
above, f
k
is the k-th Fibonacci number.
This assumption yields a non-trivial generaliza-
tion of the purely self-similar case l
k
= 1/ρ
k
and,
on the other hand, aims to mimic the recurrence of
Fibonacci sequence in proportions of human limbs
(Hamilton and Dunsmuir, 2002). In our model ev-
ery link (phalanx) is controlled by a couple of bi-
nary controls. The control of the rotation at every
joint is a common feature of all above mentioned ma-
nipulators. The study of a control ruling the exten-
sion of every link has twofold applications. In one
hand it can be physically implemented by means of
telescopic links, that are particularly efficient in con-
strained workspaces (see (Aghili and Parsa, 2006)).
On the other hand, our model can be considered a dis-
crete approximation of continuous snake-like manip-
ulators - see for instance (Andersson, 2008).
1.2 Organization of Present Paper
The paper is organized as follows. In Section 2 we in-
troduce the model. A characterization of the asymp-
totic reachable workspace via Iterated Function Sys-
tems is given Section 3. In Section 4 we describe the
convex hull of the asymptotic reachable set and we
explicitely characterize it in a particular case.
2 THE MODEL
In our model the robot hand is composed by H fin-
gers, every finger has an arbitrary number of pha-
lanxes. We assume junctions and phalanxes of each
finger to be thin, so to be respectively approximated
with their middle axes and barycentres and we also
assume the junctions of every finger to be coplanar.
Inspired by the human hand, we set the fingers of our
robot as follows: the first two fingers are coplanar and
they have in common their first junction (they are our
robotic version of the thumb and the index finger of
the human hand) while the remaining H −1 fingers
belong to parallel planes. By choosing an appropriate
coordinate system oxyz we may assume that the first
two fingers belong to the plane p
(1)
: z = 0 while, for
h ≥ 2, h-th finger belongs to the plane p
(h)
: z = z
(h)
0
for some z
(2)
0
,...,z
(H)
0
∈ R.
We now describe in more detail the model of a
robot finger. A configuration of a finger is the se-
quence (x
k
)
K
k=0
⊂ R
3
of its junctions. The configu-
rations of every finger are ruled by two phalanx-at-
phalanx motions: extension and rotation. In particu-
lar, the length of k-th phalanx of the finger is either 0
or
f
k
ρ
k
, where f
k
is the k-th fibonacci number, namely
(
f
0
= f
1
= 1;
f
k+2
= f
k+1
+ f
k
k ≥0.
(1)
while ρ > 1 is a fixed ratio: this choice is ruled by a
binary control we denote by using the symbol u
k
, so
that the lenght l
k
of the k-th phalanx is
l
k
:=k x
k
−x
k−1
k=
u
k
f
k
ρ
k
.
As all phalanxes of a finger belong to the same
plane, say p, in order to describe the angle be-
tween two consecutive phalanxes, say the k −1-th
and the k-th phalanx, we just need to consider a one-
dimensional parameter, ω
k
. Each phalanx can lay on
the same line as the former or it can form with it
a fixed planar angle ω ∈ (0, π), whose vertex is the
k −1-th junction. In other words, two consecutive
phalanxes form either the angle π or π −ω. By in-
troducing the binary control v
k
we have that the angle
between the k−1-th and k-th phalanx is π−ω
k
, where
ω
k
= v
k
ω.
To describe the kinematics of the finger we adopt
the Denavit-Hartenberg (DH) convention. To this end,
first of all recall that our base coordinate frame oxyz
is such that oxy is parallel to p (hence to every plane
p
(h)
) and we consider the finger coordinate frame
ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics
578

o
I
x
I
y
I
z
I
associated to the 4 ×4 homogeneous trans-
form
A
I
=



cosω
I
−sin ω
I
0 x
I
sinω
I
cosω
I
0 y
I
0 0 1 z
0
0 0 0 1



for some ω
I
∈ [0,2π). In particular if x and x
0
are re-
spectively coordinates of a point with respect to oxyz
and o
I
x
I
y
I
z
I
then

x
1

= A
I

x
0
1

.
Remark 1. When only one finger is considered one
may assume the base coordinate frame to coincide
with the finger coordinate frame: this reduces A
I
to
the identity and it could be omitted in the model. The
need of a coordinate frame for the finger rises when
more than one finger, especially in the case of co-
planar, opposable fingers, is considered.
Now, the (DH) method consists in attaching
to every phalanx, say the k-th phalanx, a coordi-
nate frame o
k
x
k
y
k
z
k
, so that x
k
coincides with o
k
and x
k
− x
k−1
is parallel to o
k
x
k
. Note that the
coordinates of x
k+1
with respect to o
k
x
k
y
k
z
k
are
(
u
k+1
f
k+1
ρ
k+1
cosω
k+1
,
u
k+1
f
k+1
ρ
k+1
sinω
k+1
,0).
Since we are considering a planar manipulator, for
every k > 1 the geometric relation between the coor-
dinate systems the k −1-th and the k-th phalanx is
expressed by the matrix
A
k
:=





cosω
k
−sin ω
k
0
u
k
f
k
ρ
k
cosω
k
sinω
k
cosω
k
0 −
u
k
f
k
ρ
k
sinω
k
0 0 1 0
0 0 0 1





where the rotation matrix


cosω
k
−sin ω
k
0
sinω
k
cosω
k
0
0 0 1


represents the rotation of the coordinate frame
o
k
x
k
y
k
z
k
with respect to o
k−1
x
k−1
y
k−1
z
k−1
and the
vector

u
k
f
k
ρ
k
cosω
k
,−
u
k
f
k
ρ
k
sinω
k
,0

represent the po-
sition of o
k
with respect to o
k−1
x
k−1
y
k−1
z
k−1
.
Set
T
k
:= A
I
k
∏
j=0
A
j
.
By definition T
k
is the composition the transforms
A
I
,A
0
,...,A
k
and, consequently, it represents the re-
lation between the base coordinate frame oxyz and
o
k
x
k
y
k
z
k
. In particular
T
k
=

R
k
P
k
0 1

where R
k
is a 3 ×3 rotation matrix and the entries of
the vector P
k
are the coordinates of o
k
(= x
k
) in the
reference system oxyz. Expliciting T
k
one has
R
k
=




cos

ω
I
+
∑
k
j=0
ω
j

−sin

ω
I
+
∑
k
j=0
ω
j

0
sin

ω
I
+
∑
k
j=0
ω
j

cos

ω
I
+
∑
k
j=0
ω
j

0
0 0 1




and
P
k
= P
I
+
k
∑
j=0
R
j


u
j
f
j
ρ
j
0
0


=




x
I
+
∑
k
j=0
u
j
f
j
ρ
j
cos

∑
j
n=0
ω
n

y
I
−
∑
k
j=0
u
j
f
j
ρ
j
sin

∑
j
n=0
ω
n

z
I




Then, for every k ≥0


x
k
y
k
z
k


=




x
I
+
∑
k
j=0
u
j
f
j
ρ
j
cos

∑
j
n=1
ω
n

y
I
−
∑
k
j=0
u
j
f
j
ρ
j
sin

∑
j
n=1
ω
n

z
I




(2)
3 CHARACTERIZATION OF THE
REACHABLE WORKSPACE
VIA ITERATED FUNCTION
SYSTEMS
We fix as initial state (x
I
,y
I
,z
I
) = (0, 0,0) and assume
ω
I
= 0. By employing the isometry between R
2
and C
and by considering that our manipulator is essentially
planar, we may rewrite (2) as
(
x
k
=
∑
k
j=0
u
j
f
j
ρ
j
e
−iω
∑
j
n=0
v
n
x
I
= 0.
(3)
We aim to study the asymptotic reachable workspace
R
∞
˙=
(
∞
∑
k=0
u
k
f
k
ρ
k
e
−iω
∑
k
j=0
v
j
|(v
j
),(u
j
) ∈ {0,1}
∞
)
In order to have a more compact notation, infinite
binary (control) sequences (u
j
) and (v
j
) are equiva-
lently denoted by u and v, respectively. We set
x(u,v) ˙=
∞
∑
k=0
u
k
f
k
ρ
k
e
−iω
∑
k
j=0
v
j
AModelforRoboticHandBasedonFibonacciSequence
579

and we define the shift operator on R
∞
σ : x(u,v) 7→ x(σ(u), σ(v))
so that if x = x(u, v) then
σ(x) =
∞
∑
k=0
u
k+1
f
k
ρ
k
e
−iω
∑
k
j=0
v
j+1
.
Finally we define the auxiliary set
Q
∞
=
{
(x,σ(x)) | x = x(u,v); u, v ∈ {0,1}
∞
}
.
Note that Q
∞
∈ R
∞
× R
∞
and π(Q
∞
) = R
∞
where
π : C
2
→ C denotes the projection of a bidimensional
complex vector on its first component.
We characterize Q
∞
and, consequently, R
∞
via the
linear maps F
00
,F
10
,F
01
,F
11
: C
2
→C
2
defined as fol-
lows
F
uv
(z) = e
−iωv

A
ρ
z +

u
0

for u,v ∈{0,1}
where z ∈ C
2
and
A
ρ
˙=

1
ρ
1
ρ
2
1 0

.
In order to describe the action of F
uv
’s on Q
∞
, for any
u,v ∈ {0, 1}
∞
set
¯
u(u) ˙=uu and
¯
v(v) ˙=vv. In other
words
¯u
k
(u) =
(
u if k = 0
u
k−1
otherwise .
¯v
k
(v) =
(
v if k = 0
v
k−1
otherwise .
Lemma 2. Let u, v ∈ {0, 1}
∞
, u, v ∈ {0, 1}. Set x =
x(u,v) and ¯x = x(
¯
u(u),
¯
v(v)). One has
F
uv
(x,σ(x)) = ( ¯x,σ( ¯x)) = ( ¯x, x). (4)
Remark 3. F
uv
acts on x(u,v) by prepending to
the control sequences u and v the controls u and v.
Lemma 2 also implies that F
uv
(Q
∞
) ⊂ Q
∞
for every
u,v ∈{0,1}.
Proof of Lemma 2. By definition of F
uv
and of σ, and
recalling σ(
¯
u(u)) = u and σ(
¯
v(v)) = v, one has
F
uv
((x,σ(x)) =

e
−iωv

1
ρ
x +
1
ρ
2
σ(x) + u

,σ( ¯x)

.
Then it is left to prove
e
−iωv

1
ρ
x +
1
ρ
2
σ(x) + u

= ¯x.
Recalling f
0
= f
1
, one has
e
−iωv

1
ρ
x +
1
ρ
2
σ(x) + u

= ue
−iωv
+
∞
∑
k=0
u
k
f
k
ρ
k+1
e
−iω(
∑
k
j=0
v
j
+v)
+
∞
∑
k=0
u
k+1
f
k
ρ
k+2
e
−iω(
∑
k
j=0
v
j
+v)
= ue
−iωv
+
u
0
f
0
ρ
e
−iω(v
0
+v)
+
∞
∑
k=1
u
k
f
k+1
ρ
k+1
e
−iω(
∑
k
j=0
v
j
+v)
= u f
0
e
−iωv
+
u
0
f
1
ρ
e
−iω(v
0
+v)
+
∞
∑
k=1
u
k
f
k+1
ρ
k+1
e
−iω(
∑
k
j=0
v
j
+v)
=
¯u
0
f
0
e
−iω ¯v
0
+
¯u
1
f
1
ρ
e
−iω ¯v
1
+
∞
∑
k=2
¯u
k
f
k
ρ
k
e
−iω
∑
k
j=0
¯v
j
=
∞
∑
k=0
¯u
k
f
k
ρ
k
e
−iω
∑
k
j=0
¯v
j
= ¯x.
Before stating next result, we recall that
throughtout this paper the scaling ratio ρ is assumed
greater than the Golden Mean.
Proposition 4. Q
∞
is the unique compact subset of
C
2
satisfying
[
u,v∈{0,1}
F
uv
(Q
∞
) = Q
∞
. (5)
Proof. First of all we show (5) by double inclusion.
The inclusion ⊆ directly follows by Lemma 2 – see
also Remark 3. Thus it suffices to show that for every
¯x ∈ R
∞
there exist x ∈R
∞
and u,v ∈{0,1} such that
F
uv
(x,σ(x)) = ( ¯x,σ( ¯x)).
Let
¯
u,
¯
v ∈{0, 1}
∞
be a couple of the control sequences
satisfying x = x(u,v). Then, again by Lemma 2,
F
¯u
1
¯v
1
(σ( ¯x)),σ(σ( ¯x)))) = ( ¯x,σ( ¯x)).
Since R
∞
is closed with respect to σ, then x ˙=σ(¯x) ∈
R
∞
and this completes the proof of (5).
Now, let us prove the uniqueness of Q
∞
. First of
all we note that for every u,v ∈ {0, 1}, F
uv
is a linear
map and consider its spectral radius R(ρ). R(ρ) is
hence the greatest modulus of the eigenvalues of A
ρ
.
If ρ > ϕ, where ϕ is the Golden Mean, then R(ρ) =
√
5ρ+ρ
2ρ
2
< 1. Consequently the induced norm of A
k
ρ
||A
k
ρ
|| ˙= max
z∈C
2
,z6=(0,0)−
||A
k
ρ
z|| → 0 as k →∞.
ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics
580

Then there exists k
ρ
such that if k ≥ k
ρ
then F
k
uv
is a
contraction. Since the quantity k
ρ
is independent on u
and v, one has that any concatenation of length k ≥ k
ρ
of F
uv
’s, say
G
u
k
,v
k
˙=F
u
k
1
v
k
1
◦···◦F
u
k
k
v
k
k
,
is a contraction. Consequently one may consider the
Hutchinson operator
G(·) ˙=
[
u
k
,v
k
∈{0,1}
k
G
u
k
,v
k
(·)
and deduce by (5)
G(Q
∞
) = Q
∞
. (6)
Since G is generated a finite set of contractive maps,
namely by an Iterated Function System, then
Q
∞
is the only compact subset
of C
2
enjoying (6)
(U)
In order to seek a contradiction, assume now that there
exists a compact set X ⊂ C
2
different from Q
∞
satis-
fying (5). Then X also satisfies (6): the uniqueness
condition (U) provides the required contradiction and
concludes the proof.
Remark 5. For an estimate of k
ρ
, see (Lai et al.,
2014).
4 CHARACTERIZATION OF THE
CONVEX HULL OF THE
REACHABLE WORKSPACE
Throughtout this section we employ Proposition 4 in
order to characterize the co(R
∞
), co denoting the con-
vex hull of a set. We begin by the following general
fact.
Lemma 6. Let {F
1
,. ..,F
H
} be a finite set of linear
maps on a metric space X and assume that there exists
and it is unique a compact set Q satisfying
F (Q) ˙=
H
[
h=0
F
h
(Q) = Q.
If
F (Y ) ⊆Y (7)
for some Y ⊂ X then
Q ⊆Y (8)
Proof. By iterating (7) for one has for every k
Y ⊇ F (Y ) ⊇ F
2
(Y ) ⊇ ··· ⊇ F
k
(Y )
then as k →∞, the set sequence F
k
(Y ) converges to a
set
¯
Y satisfying
F (
¯
Y ) =
¯
Y ⊆Y.
By the uniqueness of Q one has
¯
Y = Q and this com-
pletes the proof.
Theorem 7. With the notations of previous Section,
let V ⊂ Q
∞
be such that
F (V ) ˙=
[
u,v∈{0,1}
F
uv
(V ) ⊆ co(V ). (9)
Then
co(R
∞
) = co(π(V )).
Proof. The linearity of F
uv
’s and (9) imply
F (co(V )) ⊆ co(V ). (10)
This together with Proposition 4, implies that we may
apply Lemma 6 to Q
∞
and Y = co(V ) and deduce
Q
∞
⊆ co(V ). By assumption we also have V ⊂ Q
∞
,
then co(Q
∞
) = co(V ). The claim hence follows by
the fact that R
∞
= π(Q
∞
) and that projection is a con-
vex map.
Next result gives a more operative description of
co(R
∞
).
Theorem 8. Let W be a compact subset of Q
∞
. If
π(F (W )) ⊆ π(co(W )) (11)
then
co(R
∞
) = co(π(W )).
Proof. We show the claim by double inclusion. The
inclusion ⊇ is trivial, since we assumed W ⊆ Q
∞
and,
consequently π(W ) ⊆R
∞
. Now we show by induction
that if (11) holds then for every k
π(F
k
(W )) ⊆ π(co(W )). (12)
The case k = 1 is given by (11) itself. We then assume
as inductive hypothesis
π(F
k−1
(W )) ⊆ π(co(W ))
so that we get for every ( ˆw,σ( ˆw)) ∈W
F
k
( ˆw) = F (w, σ(w)) with w ∈π(co(W ))
In particular, w =
∑
k
λ
k
w
k
for some w
k
∈ π(W ) and
some convex combinators λ
k
. Since W ⊆ Q
∞
, if w
k
∈
π(W ) then (w
k
,σ(w
k
)) ∈W . Then, by (11)
π(F
k
( ˆw)) =
∑
k
λ
k
π(F (w
k
,σ(w
k
))) ⊆ co(π(W )).
(13)
AModelforRoboticHandBasedonFibonacciSequence
581

-0.5
0.5
1.0
-1.0
-0.8
-0.6
-0.4
-0.2
0.2
Figure 1: Convex hull of R
∞
with ρ = ϕ + 1.
Now, note that F
k
(W ) is a non-decreasing sequence
of compact sets, consequently as k → ∞ it tends to
some compact set
¯
W satisfying F (
¯
W ) =
¯
W . By
Proposition 4 we get
¯
W = Q
∞
. Consequently
R
∞
= π(Q
∞
) = lim
k→∞
π(F
k
(W )) ⊆ π(co(W )) (14)
The claim follows by noting that above inclusion im-
plies co(R
∞
) ⊆ π(co(W )).
4.1 Explicit Description of co(R
∞
) in a
Particular Case
We now consider the case ω = π/3 and we show that
co(R
∞
) is a polygon whose vertices are
v
1
˙=
∑
∞
k=0
f
k
ρ
k
; v
2
˙=e
−iω
∑
∞
k=0
f
k
ρ
k
;
v
3
˙=e
−iω
+ e
−i2ω
∑
∞
k=1
f
k
ρ
k
; v
4
˙=e
−i2ω
∑
∞
k=1
f
k
ρ
k
;
v
5
˙=1 + e
−i2ω
∑
∞
k=2
f
k
ρ
k
.
(see Figure 1).
In order to apply Theorem 8, it is useful to intro-
duce the symbols 0 and 1 to denote infinite sequences
of 0’s and 1’s, respectively, and to note that
v
1
= x(1,0); v
2
= x(1,10);
v
3
= x(1,110); v
4
= x(01,110);
v
5
= x(101,0110);
so that, recalling the definition σ(x(u, v)) =
x(σ(u),σ(v)) where σ(u) denotes the unit shift of u,
one gets
σ(v
1
) = v
1
σ(v
2
) = v
1
;
σ(v
3
) = σ(v
4
) = v
2
σ(v
5
) = v4.
Let W = {(v
h
,σ(v
h
) | h = 1, .. ., 5}. By Theorem 8
one has
co(R
∞
) = co({v
h
| h = 1,. .. ,5})
if for every h = 1,...,5 and for every u,v ∈ {0,1}
π(F
uv
(v
h
,σ(v
h
))) ∈ π(co(W )). (15)
We will show above inclusion by distinguishing
the cases h = 1,. .. ,5, but first we remark that
(0,0) ∈ co(V ). Consequently if z ∈ V then for every
c ∈ [0,1], cz ∈co(V ).
Claim 1.
π(F
uv
(v
1
,σ(v
1
)) ∈ co(V ). (16)
Proof. First notice
π(F
uv
(v
1
,σ(v
1
)) = x(u1,v0)
Thus
x(u1,v0) = ue
−iωv
+
∞
∑
k=1
f
k
ρ
k
Case 1.1; u = v = 0: one has
x(01,00) =
∞
∑
k=1
f
k
ρ
k
= v
1
−1
Inclusion (16) hence follows by v
1
> 1.
Case 1.2; u = 0,v = 1: one has
x(01,10) = e
−iω
∞
∑
k=1
f
k
ρ
k
= v
2
−e
−iω
= cv
2
where c = (1 −1/v
1
) (indeed v
2
= e
−iω
v
1
).
Case 1.3; u = 1,v = 0: imediate, indeed
x(11,00) = 1 +
∞
∑
k=1
f
k
ρ
k
= v
1
.
Case 1.4; u = 1,v = 1: imediate, indeed
x(11,10) = e
−iω
+ e
−iω
∞
∑
k=1
f
k
ρ
k
= v
2
.
Claim 2.
π(F
uv
(v
2
,σ(v
2
)) ∈ co(V ). (17)
Proof. First notice
π(F
uv
(v
2
,σ(v
2
)) = x(u1,v10)
Thus
x(u1,v10) = ue
−iωv
+ e
−iω(v+1)
∞
∑
k=1
f
k
ρ
k
Case 2.1; u = v = 0:
x(01,010) = e
−iω
∞
∑
k=1
f
k
ρ
k
= x(01,10)
Then (17) follows by Claim 1, see Case 1.2.
Case 2.2; u = 0,v = 1: imediate, indeed
x(01,110) = e
−i2ω
∞
∑
k=1
f
k
ρ
k
= v
4
.
ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics
582

Case 2.3; u = 1,v = 0:
x(11,010) = 1 + e
−iω
∞
∑
k=1
f
k
ρ
k
= v
2
+ 1 −e
−iω
= cv
1
+ (1 −c)v
2
with c = 1/v
1
.
Case 2.4; u = 1,v = 1: imediate, indeed
x(11,110) = e
−iω
+ e
−i2ω
∞
∑
k=1
f
k
ρ
k
= v
3
.
Claim 3.
π(F
uv
(v
3
,σ(v
3
)) ∈ co(V ). (18)
Proof. First notice
π(F
uv
(v
3
,σ(v
3
)) = x(u1,v110)
hence
x(u1,v110) = ue
−iωv
+
e
−iω(v+1)
ρ
+ e
−iω(v+2)
∞
∑
k=2
f
k
ρ
k
Case 3.1; u = v = 0:
x(01,0110) =
e
−iω
ρ
+ e
−i2ω
∞
∑
k=2
f
k
ρ
k
= c
1
v
2
+ c
2
v
4
with c
1
= 1/(ρv
1
) and c
2
= 1 −1/(ρ|v
4
|). The claim
hence follows by the fact that c
1
+ c
2
∈ [0,1] and 0 ∈
co(V ).
Case 3.2; u = 0,v = 1: recalling ω = 2π/3
x(01,1110) =
e
−i2ω
ρ
+
∞
∑
k=2
f
k
ρ
k
=
= c
1
v
1
+ c
2
v
4
where c
1
= 1 − 1/v
1
− 1/(ρv
1
) and c
2
= 1 −
1/(ρ|v
4
|). Now, notice that v
1
= ρ
2
/(ρ
2
−ρ −1) and
|v
4
|= v
1
−1, consequently c
1
+c
2
∈[0,1]. The claim
hence follows by 0 ∈ co(V ).
Case 3.3; u = 1,v = 0:
x(11,0110) = 1 +
e
−iω
ρ
+ e
−i2ω
∞
∑
k=2
f
k
ρ
k
= c
1
v
1
+ c
2
v
2
+ (1 −c
1
−c
2
)v
4
with c
1
=
ρ
3
−2ρ−1
ρ
(
ρ
2
+ρ+1
)
and c
2
=
ρ
2
+1
ρ
(
ρ
2
+ρ+1
)
. The claim
hence follows by c
1
,c
2
∈ [0,1].
Case 3.4; u = 1,v = 1:
x(11,1110) = e
−iω
+
e
−i2ω
ρ
+ e
−i3ω
∞
∑
k=2
f
k
ρ
k
= c
1
v
1
+ c
2
v
2
+ (1 −c
1
−c
2
)v
3
with c
1
=
2ρ+1
ρ
3
and c
2
=
(2ρ+1)
(
ρ
2
−ρ−1
)
ρ
3
(ρ+1)
. The claim
hence follows by c
1
,c
2
∈ [0,1].
Claim 4.
π(F
uv
(v
4
,σ(v
4
)) ∈ co(V ). (19)
Proof. First notice
π(F
uv
(v
4
,σ(v
4
)) = x(u01,v110)
Thus
x(u01,v110) = ue
−iωv
+ e
−iω(v+2)
∞
∑
k=2
f
k
ρ
k
Case 4.1; u = v = 0: the claim follows by 0 ∈co(V ),
indeed
x(001,1110) = e
−i2ω
∞
∑
k=2
f
k
ρ
k
= v
4

1 −
1
ρ|v
4
|

.
Case 4.2; u = 0,v = 1: the claim follows by 0 ∈co(V )
and ω = 2π/3, indeed
x(001,1110) =
∞
∑
k=2
f
k
ρ
k
< v
1
.
Case 4.3; u = 1,v = 0:
x(101,0110) = 1 + e
−i2ω
∞
∑
k=2
f
k
ρ
k
= v
5
Case 4.4; u = 1,v = 1: inclusion (19) follows by
x(101,1110) = e
−iω
+
∞
∑
k=2
f
k
ρ
k
=
1
v
1
v
2
+

1 −
1
v
1
−
1
ρv
1

v
1
.
Claim 5.
π(F
uv
(v
5
,σ(v
5
)) ∈ co(V ). (20)
Proof. First notice
π(F
uv
(v
4
,σ(v
5
)) = x(u101,v0110)
Thus
x(u101,v0110) =ue
−iωv
+
1
ρ
e
−iω(v+1)
+ e
−iω(v+2)
∞
∑
k=3
f
k
ρ
k
.
Case 5.1; u = v = 0:
x(0101,00110) =
1
ρ
+ e
−i2ω
∞
∑
k=3
f
k
ρ
k
= c
1
v
1
+ c
2
v
4
AModelforRoboticHandBasedonFibonacciSequence
583

with c
1
= 1/(ρv
1
) and c
2
=
3ρ+2
ρ
2
(ρ+1)
. The claim hence
follows by c
1
,c
2
,1 −c
1
−c
2
∈ [0,1] and 0 ∈co(V ).
Case 5.2; u = 0,v = 1:
x(0101,10110) =
1
ρ
e
−iω
+ e
−i2ω
∞
∑
k=3
f
k
ρ
k
= c
1
v
1
+ c
2
v
4
with c
1
= 2/ρ
4
+ 3/ρ
3
and c
2
= 1/(ρv
1
). The claim
hence follows by c
1
,c
2
,1 −c
1
−c
2
∈ [0,1] and 0 ∈
co(V ). Case 5.3; u = 1,v = 0:
x(1101,00110) = 1 +
1
ρ
+ e
−i2ω
∞
∑
k=3
f
k
ρ
k
c
1
v
1
+ c
2
v
4
+ (1 −c
1
−c
2
)v
5
with c
1
= 1/v
1
and c
2
= 1/ρ
2
. The claim hence fol-
lows by c
1
,c
2
∈ [0,1].
Case 5.4; u = 1,v = 1: recalling ω = 2π/3
x(1101,10110) =

1 +
1
ρ

e
−iω
+
∞
∑
k=3
f
k
ρ
k
= c
1
v
1
+ c
2
v
2
with c
1
= 2/ρ
4
+ 3/ρ
3
and c
2
=
p
3
−2p−1
p
3
. The claim
hence follows by c
1
,c
2
,1 −c
1
−c
2
∈ [0,1] and 0 ∈
co(V ).
5 CONCLUSIONS
We introduced a robot hand model composed by an
arbitrarily large number of hyper-redundant binary
planar manipulators. The lenght of each link scales
according to the Fibonacci sequence. Our assump-
tions (e.g. binary controls, kinematic redundancy,
planar motion...) have twofold motivations. In one
hand they facilitate the development of a theory relat-
ing fractal geometry and automatic control. On the
other hand they appear validated by practical motiva-
tions in a wide literature.
We described the kinematics of each finger by giv-
ing an explicit formula for the position of the end-
effectors (Section 2). We then addressed the investi-
gation of the reachable workspace, by characterizing
it as a projection of the attractor of a suitable IFS (Sec-
tion 3). The relation with iteration function systems
also allows to describe the convex hull of the reach-
able workspace: this technique is finally applied to
the explicit characterization in a particular case.
The results in the present paper extend techiques
previously developed for the case of links with a con-
stant ratio. The several explicit results obtained also
in this more complicated case suggest that the relation
with IFSs is a deep connection and a powerful theo-
retical tool for the investigation of automatic control.
In this paper we studied the purely discrete case
in order to give closed formulae and to enlight the re-
lation with IFSs. However we plan to investigate the
continuous case in a future work. The issues concern-
ing the practical implementation of our model are be-
yond the purposes of the present paper; but of course
it would be interesting to establish the link between
the theoretical approach and its application. Other
open problems include a tuning of parameters in or-
der to avoid self-intersecting configurations, grasping
algorithms and optimal control strategies.
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ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics
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