Discrete Asymptotic Reachability via Expansions in Non-integer Bases

Anna Chiara Lai and Paola Loreti

Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sezione di Matematica, Sapienza Universit

a di Roma,

Via A. Scarpa 16, Roma, Italy

Keywords:

Discrete Control Theory, Expansion in Non-integer Bases, Robot Hand.

Abstract:

Aim of this paper is to show the connection between the theory of expansions in non-integer bases and discrete

control systems. This idea is supported by an example of application, in the framework of robotics. We show

how a model of multi-phalanx self-similar robot hand may be studied by means of results and techniques

coming from non-standard numeration systems and related tools, like Iterated Function Systems (IFS) and,

more generally, fractal geometry.

1 INTRODUCTION

Discrete control systems are discrete in time control

systems with a ﬁnite control set. They are employed

for local approximations of continuous systems, but

also to model intrinsically discrete phenomena. The

study of the dynamics of a polyhedron rolling on a

plane or on a spheric surface, whose interest is mo-

tivated by robotics, or systems with a bounded trans-

mission channel for controls are examples of appli-

cations of discrete control systems to practical prob-

lems.

On the other hand the theory of expansions in

non-integer bases knew an increasing interest from

researchers in mathematics and theoretical computer

science, due its applications to, among others, com-

puter arithmetics, data compression and cryptogra-

phy. Expansions in non-integer bases were introduced

in (R

enyi, 1957) and they generalize the classical po-

sitional number systems, e.g., binary and decimal nu-

meration system, to the choice of a non-integer base

and of an arbitrary digit set.

Aim of this paper is to show the connection be-

tween the theory of expansions in non-integer bases

and discrete control systems. This idea is supported

by an example of application, in the framework of

robotics. We show how a model of multi-phalanx

self-similar robot hand may be studied by means of

results and techniques coming from non-standard nu-

meration systems and related tools, like Iterated Func-

tion Systems (IFS) and, more generally, fractal geom-

etry.

The paper is organized as follows. Section 2 con-

tains an overview on expansions in non-integer bases.

To the best of our knowledge, the ﬁrst application of

theoretical results in non-integer base expansions in

control theory are due to Chitour and Piccoli (Chitour

and Piccoli, 2001): in Section 3 we recall some of

their results on this topic. Section 4 and Section 5 are

devoted to a model of robot hand, whose properties

are investigated by means of arguments and results

coming from the theory of expansions in non-integer

bases.

2 EXPANSIONS IN

NON-INTEGER BASES

A positional number system is a couple (λ,A), s.t. the

base λ is greater than 1 in modulus and the alphabet

A is a subset of C. A value x ∈ C is representable

if there exists an inﬁnite sequence with digits in A,

named expansion of x in base λ,

x =

∞

∑

j=1

The set R(λ,A) =

∑

∞

j=1

| c

∈ A

is called repre-

sentable set or, sometimes, fundamental domain.

Example 1. In the case of binary numeration system

we have R(2,{0, 1}) = [0,1]. More generally if λ < 2

then R(λ,{0, 1}) = [0, 1/(λ − 1)] (R

enyi, 1957).

Example 2. If λ = 3 and A = {0, 2} then R(λ, A) is

the Middle Third Cantor set.

360

Lai A. and Loreti P..

Discrete Asymptotic Reachability via Expansions in Non-integer Bases.

DOI: 10.5220/0004092403600365

In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 360-365

ISBN: 978-989-8565-22-8

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

Example 2 points out the intimate relation be-

tween non-integer base expansions and self-similar

fractal sets. Self-similar sets, namely sets that are

similar to a part of themselves, are classically gen-

erated by iterated function systems (Falconer, 1990).

An iterated function system (IFS) is a set of contrac-

tive functions f

: C → C. We recall that a function

in a metric space (X ,d) is a contraction, if for every

x,y ∈ X

d( f (x), f (y)) < c · d(x, y)

for some c < 1. In (Hutchinson, 1981) Hutchin-

son showed that every ﬁnite IFS, namely every IFS

with ﬁnitely many contractions, admits a unique non-

empty compact ﬁxed point R w.r.t the Hutchinson op-

erator

F : S 7→

[

j=1

(S)

Moreover for every non-empty compact set S ⊆ C

lim

k→∞

(S) = R

The attractor R is a self-similar set. This result was

lately generalized to the case of inﬁnite IFS (Mihail

and Miculescu, 2009).

Remark 1. If conv(R) is the convex hull of R, then

conv(R) ↓ R for k → ∞.

Example 3. The Middle Third Cantor set C :=

R(3,{0, 2}) is the attractor of F = { f

, f

}, where

: x 7→

(2 + x).

that is

C = f

Remark that [0,1] is the convex hull of C. We have

([0,1]) ↓ C.

In general any representable set R(λ, A) is the at-

tractor of the IFS of the IFS F

λ,A

:= { f

: x 7→

x− a

∈ A} and this yields an operative way to approxi-

mate R(λ,A) from above given its convex hull – see

Remark 1.

Example 4. If λ ∈ R and if A is a ﬁnite subset of R

then conv(R(λ, A)) = [min A/(λ− 1),max A/(λ− 1)].

Beyond the representability issue, the theory of

expansion in non-integer bases includes several unex-

pected results laying on, among others, ergodic the-

ory, automata theory, algebraic number theory. For

instance it is well known that representations in dec-

imal numeration system are univoque, i.e., there ex-

ists only one decimal representation for (almost) ev-

ery number. The exceptional cases are only repre-

sented by the ambiguity 0.999·· · = 1. When we con-

sider non-integer bases and real digit set the scenario

is deeply different: indeed if the base is sufﬁciently

small, then there exists for almost every number, a

continuum of different expansions (Sidorov, 2003).

Moreover the Golden Ratio G plays a special role in

the case A = {0, 1}: indeed if λ ≤ G then every num-

ber in R(λ,A) can be represented in at least two differ-

ent ways. The redundancy of these numeration sys-

tems leads to the study of particular expansions and

related symbolic dynamical systems. For instance the

greedy expansions privilege the choice of great digits

enyi, 1957) and they minimize the truncation error

in the case of positive real bases. Particular expan-

sions in negative and complex base are also discussed

in (Komornik and Loreti, 2007) and (Komornik and

Loreti, 2010).

We conclude this section with a result on repre-

sentability in complex base, that we will apply in Sec-

tion 5.

Theorem 1. (Lai, 2011) The set of representable

numbers in base λ = ρe

i2π/n

and arbitrary ﬁnite al-

phabet A = {a

< ·· · < a

} ⊂ R is a convex polygon

(containing the origin and with 2n edges if n is even

and with n edges if otherwise) if and only if

max

i=1,...,m−1

i+1

− a

≤

maxA − minA

− 1

. (1)

Remark 2. If A = {0, 1} then (1) is equivalent to ρ ≤

1/n

3 DISCRETE CONTROL

SYSTEMS AND EXPANSIONS

IN NON-INTEGER BASES

In (Chitour and Piccoli, 2001), the controllability of

linear discrete control systems is investigated. Among

other results, the paper contains a deep investigation

of the unidimensional case, i.e., the study of the sys-

tem

k+1

= λx

+ a

∈ A

with | λ |> 1 and A ⊂ R. In this case the reachable

set is R(λ,A) = {

∑

j=0

| a

∈ A, n ∈ N}. To ex-

plain the relation between R(λ,A) and the expansions

in non-integer base, we introduce the notion of integer

part in base λ. Let x ∈ λ

R(λ,A) for some N, then

x = c

−N

N−1

+ ··· + c

−1

λ + c

+ c

−1

+ ···

for some (c

) with digits in A. The numerical value

−N

N−1

+ · ·· + c

−1

λ + c

is called integer part of x

in base λ.

Remark 3. Due of the redundancy of expansions in

non-integer bases (Sidorov, 2003), a real number may

have distinct (or none) integer parts in base λ.

DiscreteAsymptoticReachabilityviaExpansionsinNon-integerBases

361

In other words, R(λ,A) is the set of all the integer

parts of any real number in base λ and with alphabet

Also by means of this relation with the theory of

expansions in non-integer bases, Chitour and Piccoli

pointed out a connection between the algebraic prop-

erties of λ and the topology of R(λ,A). Before stating

some of their results, we recall the following deﬁni-

tions.

Deﬁnition 1. An algebraic integer λ is the real solu-

tion of a monic polynomial with integer coefﬁcients,

namely

+ a

n−1

+ ··· + a

n−1

λ + a

= 0

for some a

,. .. ,a

∈ Z . If the polynomial P(X) :=

n−1

+···+a

n−1

X +a

is irreducible, namely

it cannot be factorized as product of polynomials with

integer coefﬁcients, and if P(λ) = 0, then P(X ) called

the minimal polynomial of λ.

A real or complex number

λ is a algebraic conju-

gate of λ if it is a root of the minimal polynomial of λ,

namely

λ) = 0.

A Pisot number is an algebraic integer λ > 1

whose algebraic conjugates are less than 1 in mod-

ulus.

Denote bλc the integer part of λ. A regular num-

ber is a real number λ such that for every P(X) with

coefﬁcients in A

= {0,±1, .. ., ±bλc}

P(λ) 6= 0.

Example 5. The Golden Mean, i.e., the greatest root

of the polynomial P(X ) = X

− X − 1 is a Pisot num-

ber.

All transcendental numbers are regular.

Theorem 2. (Chitour and Piccoli, 2001) For every

λ > 1 it is possible to design an appropriate control

set A such that R(λ,A

) is dense in R. In particular if

λ is regular and setting A

:= {0,±1,...,±bλc} we

have that R(λ,A

) is dense in R.

Algebraic properties of λ are relevant also in the

following result.

Theorem 3. (Chitour and Piccoli, 2001) Let λ be a

Pisot number. If A ⊂ Z then R(λ, A) is a discrete.

4 A MODEL FOR A ROBOT

HAND BASED ON EXPANSIONS

IN NON-INTEGER BASES

We introduce a model for a robot hand, which main

features are the following: an arbitrary number of ﬁn-

gers, moving on parallel planes excepting the index

Figure 1: ρ = 2

1/3

and ω = 2π/3. We have a full-extension

conﬁguration, i.e., e

= e

= 1, and the rotation controls

= 0 and r

= 1. In particular r

= 1 implies that the

angle between the ﬁrst phalanx and the second phalanx is

π − ω = π/3.

ﬁnger and the opposable thumb. Every ﬁnger is char-

acterized by an arbitrary number of phalanxes and a

constant ratio between phalanxes. The motion of ev-

ery phalanx is ruled by a couple of controls belonging

to the unit interval. In particular every phalanx can

extend and/or rotate or simply do nothing, the latter

case corresponding to a null couple of controls. The

physical parameters describing a particular ﬁnger are

• ω ∈ (0, 2π)\{π} representing the greatest rotation

(so that π − ω is the greatest angle between pha-

lanxes);

• ρ > 1 representing the scaling factor of phalanxes;

while the control features are the following:

• x

is the position of the k-th junction on the com-

plex plane;

• u

∈ E = {e

= 0 < e

< ·· · < e

= 1} is the ex-

tension control for the k-th phalanx;

• v

∈ R = {r

= 0 < r

< · ·· < r

= 1} is the rota-

tion control for the k-th phalanx;

We set the dynamics of a ﬁnger on the complex

plane, so that rotations are modeled with complex ex-

ponentiation, and we assume with no loss of general-

ity x

, the initial junction of the ﬁnger, to be set in the

origin. The resulting discrete control system is







= x

k−1

−i

∑

n=1

= 0.

(2)

We brieﬂy remark some features of the ﬁnger de-

rived from (2). The complex value x

− x

k−1

−i

∑

n=1

represents the k-phalanx. In particular

its lenght depends on the extension control u

| x

− x

k−1

On the other hand, this phalanx is oriented on

the complex plane according to the argument of

− x

k−1

, that is

∑

n=1

ω. This implies that

the system keeps memory of all rotations and, in

ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics

362

k−1

≡ x

k+1

k+2

(a) v

k+1

= 0;

k−1

≡ x

k+1

k+2

(b) v

k+1

= 1.

Figure 2: In both cases u

k+1

= 0, u

k+2

= 1 and v

k+2

= 1.

particular, unextended rotations affect subsequent

motions (see Figure 2). We approach the study of the

reachable set of System (2) by means of the theory

of expansions in non-integer bases. We have the

following correspondence

base ↔ physical properties of the ﬁnger

alphabet ↔ control set

representability ↔ reachability

Indeed, by setting

λ := ρe

i ω

(3)

we have that R

(ω,ρ), the reachable set in time k,

satisﬁes

(ω,ρ) =

(

∑

j=1

| c

= u

iω( j−

∑

n=1

)

∈ E; v

∈ R

Equation above implies that reachable points are rep-

resentable numbers in base λ and with alphabet

A := {u

iω( j−

∑

n=1

)

; u

∈ E, v

∈ R}. (4)

Consider now the asymptotic reachable set

∞

(ω,ρ) :=

∞

[

k=1

(ω,ρ)

(

∞

∑

j=1

| c

= u

iω( j−

∑

n=1

)

∈ E; v

∈ R

From the theory of expansions in non-integer

bases we have the following result.

Proposition 1. Let ρ ∈ (1,∞), ω ∈ [0,2π], E,R ⊂

[0,1] and let λ and A be respectively like in (3) and

(4). Then R

∞

(ω,ρ) is the attractor of the IFS

ρ,ω,E,R

:= { f

: 1/λ(x + a

) | a

∈ A}.

In particular, for k → ∞

ρ,ω,E,R

(conv(R

∞

(ρ,ω, E,R)) ↓ R

∞

(ρ,ω, E,R).

5 THE BINARY CASE

Through this Section we focus on the case E = R =

{0,1}, namely the case of binary controls. Since E

and R are ﬁxed, we shall omit them in the notations.

5.1 Convex Hull of the Reachable Set:

A Particular Case

Through a combinatorial approach it is possible to de-

termine the convex hull of the reachable set R

(ρ,ω),

by characterizing the extremal conﬁgurations (see

Figure 3).

-1

- 2.0

-1.5

-1.0

- 0.5

0.5

1.0

1.5

(a) Full extension and no rotation

-1

- 2.0

-1.5

-1.0

- 0.5

0.5

1.0

1.5

(b) Full extension and initial rota-

tion (v

= 1).

-1

- 2.0

-1.5

-1.0

- 0.5

0.5

1.0

1.5

= 0,v

= v

= 1 and full ex-

tension and no-rotation of latter pha-

lanxes.

-1

- 2.0

-1.5

-1.0

- 0.5

0.5

1.0

1.5

(d) full extension, v

= v

= 1 and

no-rotation of latter phalanxes.

Figure 3: Extremal conﬁgurations for R

1/3

,2π/3).

Theorem 4. For every k ∈ N and for every ρ > 1

the convex hull of R

(ρ,2π/3) is a polygon whose ex-

tremal points are reachable points obtained by the fol-

lowing control sequences

• full extension, no rotation:

u = (1, 1,. .. ,1) v = (0,0, 0,. .. ,0)

DiscreteAsymptoticReachabilityviaExpansionsinNon-integerBases

363

• full extension, one initial rotation:

u = (1, 1,. .. ,1) v = (1,0, 0,. .. ,0)

• full extension, two initial rotations:

u = (1, 1,. .. ,1) v = (1,1, 0. .. ,0)

• no initial extension, two initial rotations:

u = (0, 1,. .. ,1) v = (1,1, 0. .. ,0)

5.2 Reachability around the Origin

We are interested on the topological properties of

∞

(ρ,ω) and we get started by asking whether it con-

tains an open ball centered in the origin, that is to say,

if there exists a region of the plane that can be reached

by the ﬁnger with arbitrary precision in ﬁnite time.

To this end we consider a particular class of con-

ﬁgurations called full-extension conﬁgurations. They

represent all the conﬁgurations where the rotation

control is constantly equal to one, i.e., v

= 1 for ev-

ery j ∈ N. The corresponding alphabet A

f e

is {0,1}

and the corresponding reachable set is

f e

∞

(ρ,λ) := {

∞

∑

j=1

| a

∈ {0,1}}

By applying Theorem 1, we have the following result.

Theorem 5. The representable set in base λ = ρe

iω

and with alphabet A is contained in R

∞

(ρ,ω). More-

over if ω = 2π/n, with n ≥ 3 and if ρ ≤ 2

1/n

then

f e

∞

(ρ,ω) is a polygon with 2n edges if n is odd and

with n edges if n is even.

5.3 Asymptotic Reachable Set and IFS

In view of Proposition 1 the asymptotic reachable set

∞

(ρ,ω) =

(

∞

∑

j=1

−iω

∑

n=1

| u

∈ {0,1}

)

is the ﬁxed point of the IFS

: x 7→

−iω

: x 7→

(1 + x) f

: x 7→

−iω

(1 + x).

We now explain the role of the above IFS and its rela-

tion with the controls. Let x ∈ R

∞

(ρ,ω). Then

x =

∞

∑

j=1

−iω

∑

n=1

and for every h = 1,...,4

(x) =

−iωv

+ x) =

∞

∑

j=0

−iω

∑

n=1

for some u

∈ {0, 1} In particular to every function

in the IFS corresponds a couple of control sequences

↔ u = 0 and v = 0 f

↔ u = 0 and v = 1

↔ u = 1 and v = 0 f

↔ u = 1 and v = 1.

To apply f

to a point x ∈ R

∞

(ρ,ω) is equivalent to

preﬁx the corresponding control couple (u

) to the

control sequence of x.

Example 6. Let x ∈ R

∞

1/3

,π/6) with control se-

quences

u = (1, 1,1, 1,0, 0,. ..) v = (0, 1,0, 1,0, 0,. ..)

then f

(x) ∈ R

∞

1/3

,π/6) and its control sequences

are

u = (1,1, 1,1, 1,0, 0,. .. ) v = (1, 0,1, 0,1, 0,0, .. .)