On Discrete-valued Modeling of Nonholonomic Mobile Robot Systems

Takuto KITA, Masato ISHIKAWA and Koichi OSUKA

Abstract— In this paper, we pursuit possibility of discrete-valued version of nonholonomic mobile robot systems. Insteadof the special Euclidean space SE(2), we suppose the hexagonalcellular space as the field of planar locomotion. We thenconsider discrete equivalent of wheeled mobile robot governedby nonholonomic kinematic constraints, which is defined bydiscrete-valued difference equations rather than continuous-valued differential equations. We also propose a model ofdiscrete-valued trailer system, which undergoes both nonholo-nomic constraints of wheels and nonholonomic constraints ofrigid linkage, followed by its extension to snake robot systemswith passive wheels and active joints. Finally, we examine thereachability of system states in the discrete settings, and discussthe minimal number of steps required to cover the entireneiborhood of the initial state.

I. INTRODUCTION

Nonholonomic mobile robot systems have been attractingmuch interest of nonlinear control theorists and roboticsresearchers since early 90s. Some notable properties ofsuch a system include: (1)the set of equilibria forms asubmanifold of the state space rather than an isolated point,(2)any equilibrium cannot be asymptotically stabilized bycontinuous state feedback[1], (3)nevertheless the equilibriumcan be reached from its neighborhood if it satisfies the Liealgebra rank condition[2].

In this paper, we propose to discuss discrete-valued ver-sion of nonholonomic planar mobile robot systems. Therobot’s configuration space SE(2) is supposed to be dis-cretized as a hexagonal cellular space[3], while the shapespace(or joint space, usually referred to Tn) is also dis-cretized as Zn

6 of modular arithmetic. As the conventionalnonholonomic system theory is based on nonintegrable na-ture of kinematic constraints, we start from consideringa discrete-valued version of nonholonomic constraints (aninteger-valued equation of integer variables), then discusshow the admissible motion that satisfies the constraint look,compared to the continuous ones. Here we emphasize thatthe issue addressed here is (relevant, though) different froma discretization of continuous nonlinear systems or nonlinearsampled-data systems ([4], [5], [6], [7]). Our standpoint isjust to observe what should happen, starting from the discreteconstraints as principal rules.

This paper is organized as follows. Section II preparesbasic properties hold on the hexagonal cellular space. InSection III, we introduce a discretized-version of singlecart system, followed by some elementary observations onholonomy, Lie bracketting and feedback control. Similarly,

The authors are with Dept. of Mechanical Enginering, Graduate Schoolof Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871,Japan. Email: ishikawa@mech.eng.osaka-u.ac.jp

trailer systems are illustrated in Section IV and snake robotis illustrated in Section V. In Section VI, we prepare somefundamental notions for stepwise reachability and discusshow they apply to the discrete nonholonomic mobile robotsystems. In Section VII, we conclude this paper by listingsome possible issues deserve to be discussed in the futureworks.

Notations: SE(2) denotes the two-dimensional specialEuclidean space, S denotes the unit circle, Tk =

∏ki=1 S

denotes the k-dimensional generalized torus. Z denotes theset of integers and Zn denotes the set of finite integersmodulo n. For any x ∈ Z, sat(x) ∈ Z implies the saturationfunction:

sat(x) =

1, (x > 0)0, (x = 0)

−1, (x < 0)(1)

Throughout the paper, t ∈ R denotes the time in continuouscase, while k ∈ Z denotes the time step in discrete case.Moreover, we often use the following short-form notations,Ci := cos θi, Si := sin θi, Cij = cos(θi − θj) and Sij =sin(θi − θj) to save the space.

II. BASIC PROPERTY ON HEXAGONAL CELLULAR SPACE

A. Coordinate Setting

Suppose a tessellation of the planar space R2 with unithexagons1, as shown in Fig. 1. Let O be a center of ahexagon. The x-axis is set as a line passing through O andis perpendicular to an edge, while y-axis passes through oneof its vertex (alternative definition can be possible). Thefollowing three constants will play important roles in thispaper (see Fig. 2):

α =12, β =

√3

2, γ =

π

3, (2)

satisfying fundamental relations

2α = 1, α2 + β2 = 1, α2 − β2 = −α. (3)

Each cell is addressed by the (x, y)-position of its center,such as (3, 4β) or (2 + α, 3β).

If θ is an ’angle’ on this space, it should be confined to0,±γ,±2γ,±3γ, · · · , where 3γ and −3γ are identified. Asa slight abuse of notation, we identify γi (i ∈ Z) with iitself, in case it will cause no confusion. Namely, the spaceof angles is identified with the set of integers modulo 6:

Z6 = 0, 1, 2, 3, 4, 5 ≡ 0,±1,±2, 3, (4)

1Hexagon is not the only choice of tessellation; we chose it for someof its preferable properties, e.g., distance between neighboring centers isalways 1, in contrast to rectangular tessellation.

978-1-4673-2126-6/12/$31.00 © 2012 IEEE

O

Fig. 1. Coordinate Settings in the Hexagonal Cellular Space

Fig. 2. Basic constants α, β, γ

The integer 3 ∈ Z6 will be treated as discrete counterpartof π ∈ S. Similarly, cos θ actually implies cos γθ for anydiscrete angle θ ∈ Z6. Cosine and sine of discrete angles aresummarized in Fig. 3. Fundamental trigonometric identities,such as angle addition formulae, naturally hold as in thecontinuous case.

Fig. 3. Cosine and Sine on the Hexagonal Cell

In summary, we define the whole configuration space ofplanar rigid body, say SEd(2), as follows:

SEd(2) =

x

yθ

=

nx + odd(y)αnyβnθ

∣∣∣∣∣∣ nx, ny ∈ Z, nθ ∈ Z6

' Z × Z × Z6. (5)

where odd(y) is 1 if y is an odd integer, 0 otherwise.

B. Basic Difference Calculus

For a function f(θ), we define

∆θf := f(θ + ∆θ) − f(θ) (6)

where |∆θ| ≤ 1. ∆θf is simply denoted by ∆f if theargument is obvious. Note that ∆f depends on both θ and∆θ. By definition, ∆f = 0 if ∆θ = 0.

Differentiation of trigonometric functions are derived asfollows. First, note that cosine and sine of small angles areformulated by (see Fig. 3)

cos ∆θ = 1 − α∆θ2, sin∆θ = β∆θ (if |∆θ| ≤ 1) (7)

Therefore

cos(θ + ∆θ) − cos θ = cos θ cos ∆θ − sin θ sin∆θ − cos θ

= − sin∆θ sin θ + (cos ∆θ − 1) cos θ

= −β∆θ sin θ − α∆θ2 cos θ

sin(θ + ∆θ) − sin θ = sin θ cos ∆θ + cos θ sin∆θ − sin θ

= sin ∆θ cos θ + (cos ∆θ − 1) sin θ

= β∆θ cos θ − α∆θ2 sin θ

Thus we have the basic difference formulae

∆ cos θ = −β∆θ sin θ − α∆θ2 cos θ (8)∆ sin θ = β∆θ cos θ − α∆θ2 sin θ (9)

In contrast to continuous differentiation, we should note thatthe differences are neither linear nor symmetric with respectto ∆θ, due to the presence of ∆θ2. This asymmetry will yieldthe discrepancy between the continuous and discrete casesin the following discussion. Moreover, differential algebraicrelations such as (sin θ)′ = cos θ and (cos θ)′ = − sin θ donot hold in discrete case, while the following phase shiftrelations are satisfied:

∆ cos θ = cos(θ + 2∆θ)∆ sin θ = sin(θ + 2∆θ).

TABLE IDISCRETE CALCULUS OF TRIGONOMETRIC FUNCTIONS

∆ cos θ

θ = −2 θ = −1 θ = 0 θ = 1 θ = 2 θ = 3∆θ = 1 1 α −α −1 −α α∆θ = −1 −α −1 −α α 1 α

∆ sin θ

θ = −2 θ = −1 θ = 0 θ = 1 θ = 2 θ = 3∆θ = 1 0 β β 0 −β −β∆θ = −1 β 0 −β −β 0 β

III. SINGLE CART

A. Review of Continuous version

Let us begin with recalling an elementary example ofplanar single cart system, shown in Fig. 4(left). The statevector of this system is ξ = (x0, y0, θ0) ∈ X , X := SE(2)where (x0, y0) indicates its position and θ0 indicates itsorientation angle relative to the x -axis. We assume thatthe cart is not allowed to slide sideways, thus the followingnonholonomic constraint

y0 cos θ0 − x0 sin θ0 = 0. (10)

must be satisfied. The kinematic state equation is given bythe following differential equation

ξ = g1(ξ)u1 + g2(ξ)u2 (11)

g1(ξ) :=

cos θ0

sin θ0

0

, g2(ξ) :=

001

where u1 ∈ R is the forwarding velocity and u2 ∈ R isthe heading angular velocity. Each point ξ ∈ X can be anequilibrium by setting u = 0. As a well-known consequenceof Brockett’s theorem[1], this system is not asymptoticallystabilizable by any smooth time-invariant state feedbacklaw. This system is controllable in the sense that any twoequilibria can be reached from each other[2], because itscontrollability Lie algebra

C(ξ) := spang1(ξ), g2(ξ), [g1, g2](ξ) (12)

where [g1, g2](ξ) =∂g2

∂ξg1 −

∂g1

∂ξg2 =

sin θ0

− cos θ0

0

has dimension 3 at ∀ξ ∈ X .

continuous discrete

admissible

instanteneous direction

admissible

1-step direction

Fig. 4. Single cart: nonholonomic constraint of rolling wheel

B. Discrete version

Suppose a single cart placed on the hexagonal cellularspace (Fig. 4, right). The state vector of this system is ξ =(x0, y0, θ0) as same as in the continuous case, but it must bean element of X = SEd(2).

Next, let us think of a condition which prevents the cartfrom sliding sideways. Let ∆x0 denote the progress of thevariable x0 from the current step k to the next step k+1,i.e.,

∆x0 = x0[k+1] − x0[k]. (13)

∆y0 and ∆θ0 are defined in the same manner. Then thediscrete version of the nonholonomic constraint is given by

∆y0 cos θ0 − ∆x0 sin θ0 = 0 (14)

Suppose u1 ∈ −1, 0, 1 is the forwarding velocity and u2 ∈−1, 0, 1 is the heading angular velocity. Then the stateequation of the cart is immediately obtained as∆x0

∆y0

∆θ0

=

cos θ0

sin θ0

0

u1 +

001

u2, (15)

or equivalently,

∆ξ = g1(ξ)u1 + g2(ξ)u2 (16)

g1(ξ) :=

cos θ0

sin θ0

0

, g2(ξ) :=

001

C. Holonomy and Lie bracket motionBased on the model obtained above, let us investigate

discrete version of holonomy, i.e., the net effect of periodicinputs. Fig. 5 shows primitive 8 patterns of four step periodicinput signals with unit amplitude. The input (a’) is the time-reversal signal of (a’) and vice versa, and so for other pairs.

(a)

(b)

(c)

(d) (d')

(c')

(b')

(a')

Fig. 5. Primitive periodic input patterns

Fig. 6 shows the effect of these inputs starting from theorigin. The effect of (a’) is just the opposite to that of (a)and so for the other pairs. In essence, the holonomy is splitinto two types, the effect of (a)(b) and that of (c)(d).

input (a)(b)input (c)(d)

input (c')(d')input (a')(b')

Fig. 6. Effect of Lie bracket motions of the single cart

As an analogy from the continuous case, we expect itpossible to analyze this effect by some discrete counterpartof Lie bracket. For this purpose, let us first define discreteversion of Jacobian matrix.

∆g1 = g1(ξ + ∆ξ) − g1(ξ) =

−β∆θ0S0 − α∆θ20C0

β∆θ0C0 − α∆θ20S0

0

where S0 = sin θ0, C0 = cos θ0. The problem here is that∆g1 is not linear with respect to ∆ξ due to the presence of∆θ2

0 .Now, recall that ∆θ2

0 = ∆θ0 in case of ∆θ0 ≥ 1, while∆θ2

0 = −∆θ0 in case of ∆θ0 ≤ −1. This motivates usto define two branches of Jacobians J+(g1) and J−(g1), asfollows:

J+(g1) :=

0 0 −βS0 − αC0

0 0 βC0 − αS0

0 0 0

(17)

J−(g1) :=

0 0 −βS0 + αC0

0 0 βC0 + αS0

0 0 0

(18)

which satisfy

∆g1 =

J+(g1)∆ξ, if ∆θ0 ≥ 0J−(g1)∆ξ, if ∆θ0 ≤ 0 (19)

Using J+ and J−, we can define the following two branchesof Lie brackets:

[g1, g2]+(ξ) := J+(g2)g1 − J+(g1)g2 (20)

=

βS0 + αC0

−βC0 + αS0

0

[g1, g2]−(ξ) := J−(g2)g1 − J−(g1)g2 (21)

=

βS0 − αC0

−βC0 − αS0

0

.

Their values at ξ = 0 are:

g1(0) =

100

, g2(0) =

001

[g1, g2]+(0) =

α−β0

, [g1, g2]−(0) =

−α−β0

,

which are consistent with the actual displacements shown inFig. 6.

IV. TRAILER SYSTEMS

A. Review of Continuous version

Suppose a cart towing ` trailers as shown in Fig. 7(left).Each of the carts 0, · · · , `−1 has a free joint on the centerof its wheel axis, which connects the following cart to itself.Length of each connecting link is supposed to be 1. The statevector is

ξ = (x0, y0, θ0, · · · , θ`)T ∈ XX := SE(2) × T`−1

where (x0, y0) denotes position of the truck (cart 0) and θi

denotes orientation of the cart i for i = 0, · · · , `. This systemundergoes ` + 1 nonholonomic constraints

yi cos θi − xi sin θi = 0, i = 0, · · · , `−1 (22)

and ` holonomic constraints of rigid linkage as well:xi = xi+1 + cos θi+1

yi = yi+1 + sin θi+1, i = 0, · · · , `−1 (23)

We also have to pay attention to the joint limitation

|θi+1 − θi| < π, i = 0, · · · , ` − 1

By taking all the constraints into account, the state equa-tion is obtained as

ξ = g1(ξ)u1 + g2(ξ)u2 (24)

g1(ξ) :=

cos θ0

sin θ0

0− sin(θ1 − θ0)

− sin(θ2 − θ1) cos(θ1 − θ0)...

, g2(ξ) :=

00100...

where u1 is the forwarding velocity and u2 is the headingangular velocity of the truck (cart 0). It is easy to show thatthis system is also controllable by analyzing its controllabil-ity Lie algebra.

continuous discrete

Fig. 7. Trailer: holonomic constraints of rigid linkage

B. Discrete version

Now let us turn to consider the discrete version (Fig. 7,right). Each cart is placed on the hexagonal cells SEd(2),thus each joint angle is the difference between adjoining cartorientation, e.g., θi+1 − θi. We also assume that the jointangles are limited to

|θi+1 − θi| < 3, i = 0, · · · , ` − 1

The state vector is

ξ = (x0, y0, θ0, · · · , θ`)T ∈ XX := SEd(2) × Z`−1

6 .

Control inputs are assigned to the velocity of the trucks,i.e., u1 is the forwarding velocity and u2 is the headingangular velocity of the front cart, respectively.

∆x0C0 + ∆y0S0 = u1

∆θ0 = u2 (25)

Nonholonomic constraint of each wheel

∆yi Ci − ∆xi Si = 0, i = 0, · · · , ` (26)

Holonomic constraints of rigid linkage:xi−1 = xi + Ci

yi−1 = yi + Si, i = 1, · · · , ` (27)

The holonomic constraints should be kept in every step;hence the constraints in the next step

(xi−1 + ∆xi−1) = (xi + ∆xi) + cos(θi + ∆θi)(yi−1 + ∆yi−1) = (yi + ∆yi) + sin(θi + ∆θi)

(28)should also hold for i = 1, · · · , `.

The state vector of this system is

ξ = (x0, y0, θ0, · · · , θ`) ∈ SEd(2) × Z`−16

In order to obtain difference equation for this system, wehave to eliminate ∆x1, · · · ,∆x`, ∆y1, · · · ,∆y`, x1, · · · , x`,y1, · · · , y` from (25)–(28) and derive explicit expressionof ∆ξ. (We eliminate 4` variables from 5` + 3 equations,resulting in `+3 solutions). First, substituting (27) and (8)(9)into (28), we have

∆xi−1 = ∆xi − β∆θiSi − α∆θ2i Ci

∆yi−1 = ∆yi + β∆θiCi − α∆θ2i Si

(29)

or equivalently,∆xi = ∆x0 +

∑ij=1

(β∆θiSi + α∆θ2

i Ci

)∆yi = ∆y0 +

∑ij=1

(−β∆θiCi + α∆θ2

i Si

) (30)

Computing ∆yi−1Ci − ∆xi−1Si leads us

∆yi−1Ci − ∆xi−1Ci = ∆yiCi − ∆xiSi + β∆θi

= β∆θi

where the nonholonomic constraints (26) are used. Thus ∆θi

can be obtained by recursive computation

β∆θi = ∆yi−1Ci − ∆xi−1Si

= ∆y0Ci − ∆x0Si

−i−1∑j=1

(β∆θj(CiCj + SiSj) + α∆θ2

j (SiCj − CiSj))

= −Si0u1 −i−1∑j=1

(β∆θjCij − α∆θ2

j Sij

)(31)

where Cij = cos(θi − θj), Sij = sin(θi − θj).1) Single trailer: The simplest case is a single trailer

system (` = 1), whose state vector is ξ = (x0, y0, θ0, θ1)T .The state equation is given by

∆ξ = g1(ξ)u1 + g2(ξ)u2,

g1(ξ) =

C0

S0

0−S10/β

, g2(ξ) =

0010

(32)

Forwarding motion this trailer system is not difficult toimagine from the single cart case. Backward motion is alsopossible, e.g., by a skillful steering shown in Fig. 8.

initial

final

reference

path

Fig. 8. Backward parking of the single trailer for linear reference path

Lie bracket can be obtained as follows: since ∆θ0S10 =−β∆θ0C10 − α∆θ2

0S10, ∆θ1S10 = β∆θ0C10 − α∆θ20S10,

we have

J+(g1) :=

0 0 −βS0 − αC0 00 0 βC0 − αS0 00 0 0 00 0 C10 + αS10/β −C10 + αS10/β

[g1, g2]+ =

βS0 + αC0

−βC0 + αS0

0−C10 − αS10/β

, [g1, g2]+(0) =

α−β0−1

However, it is still difficult to connect the effect of periodic

inputs and the Lie brackets for the time being.2) Double trailer: In the case of ` = 2, the state vector

is ξ = (x0, y0, θ0, θ1, θ2). Behavior of the first four statevariables is as exactly same as in the previous case (32),while ∆θ2 can be derived using (31) as follows:

β∆θ2 = −S20u1 − β∆θ1C21 − α∆θ21S21

= −S20u1 + βS21C10u1β − α

β2S21S

210u

21

= −S21C10u1 −α

β2S21S

210u

21.

Therefore the state equation is not linear in u1 any longer.

C. Trailer with off-axle hitching

In the case of off-axle hitching where the hinge joint isnot precisely at the center of the rear axis (see Fig. 9), itsbehavior is slightly different from the previous case. In thiscase, the holonomic constraint (27) is replaced by

xi−1 = xi + Ci + Ci−1

yi−1 = yi + Si + Si−1, i = 1, · · · , ` (33)

The state equation is obtained by solving (33) and (26) for(∆x0,∆y0, ∆θ0, · · · ,∆θ`). The single trailer case (` = 1)is given as follows:

∆ξ =

C0

S0

0−S10/β

u1 +

00u2

−C10u2 − αS10u22/β

(34)

Unlike the on-axle case, the right-hand side is not linearin u2 any longer, and the steering input u2 affects both∆θ0 and ∆θ1. Indeed, this system is much alike to thesnake robots(Sec. V) where the only difference the inputassignment, rather than to the on-axle trailer systems.

Fig. 9. Trailer with off-axle hitching

V. SNAKE ROBOTS

Once we regard the joints active, we can discuss thediscrete version of snake robots and how it resembles thecontinuous version. Suppose a chain of `+1 carts at (xi, yi)with orientation θi, where a active joint is inserted betweeneach pair of adjoining carts, as shown in Fig. 10. The statevector ξ is (x0, y0, θ0, · · · , θ`)T as same as in the trailersystems.

active joint

active joint

Fig. 10. Snake robot (` = 3)

The constraints to be taken into account are the sameas the nonholonomic constraints (26) and the holonomicconstraints (33) in the case of off-axle hitching. However,the resulting state equation in closed-form will look quitedifferent. Suppose the simplest case where ` = 3. Aftercomplicated but straightforward computation, the constraintsreduces to the following form:∆x1

∆y1

∆θ1

=1

D(φ)Rθ1Γ(φ)

(β∆φ1

β∆φ2

)(35)

where

D(φ) = β(Sφ1 − Sφ2 + sin(φ1 − φ2)) − δSφ1Sφ2 ,

Rθ1 =

C1 −S1 0S1 C1 00 0 1

,

Γ(φ) =

−β(1 + Cφ2) − δαSφ2 −β(1 + Cφ1) + δαSφ1

0 0Sφ2 Sφ1

δ = sgn(∆θ1).

and φ1 := θ1−θ0, φ2 := θ2−θ1 are the joint angles, Cφi :=cos φi, Sφi := sin φi. For simplicity of expression, we chose(x1, y1, θ1) instead of (x0, y0, θ0) to represent configurationof the entire robot.

We assign the control inputs u = (u1, u2)T as u1 = ∆φ1,u2 = ∆φ2, respectively. The control inputs should satisfy|ui| ≤ 2 so that |∆θi| ≤ 1.

The system falls into singular posture when D(φ) = 0, orφ1 = φ2, which implies that the snake is in arched shape;this condition is exactly as same as in the continuous case[8].

Now let us see how it locomotes under periodic controlinputs. Suppose the initial state is ξ = (2, 2, 1, 1,−1)T asshown in Fig. 12(a), and consider a pair of periodic inputsthat resembles (sin t, cos t) as shown in Fig. 11 Stepwisechanges of the state variables (x1, y1, θ1, φ1, φ2)T are shownin Table II and Fig. 12.

TABLE IISNAKE ROBOT: FORWARD LOCOMOTION

(x1, y1, θ1) (φ1, φ2) (u1, u2)Initial State (2, 2, 1) (1,−1) (1,1)

STEP1 (2, 3, 1) (2, 0) (0,-2)STEP2 (2, 3, 2) (2,−2) (-1,1)STEP3 (2, 3, 1) (1,−1)

Fig. 11. Input pattern for the snake robot

Thus the robot moves in the upper-right direction, whichis parallel to the middle link without changing its shape. Thisresult indicates this model partialy reproduce the behavior ofthe continuous case.

VI. REACHABILITY ANALYSIS

It is an important problem to observe the region thatmobile robots can reach from given initial state. In thissection, we define stepwise reachability set as the collectionof reachable states within given number of steps.

A. Definitions

Suppose the state equation of a discrete-valued nonholo-nomic mobile robot systems is expressed as the followingdifference equation of integral values:

ξ[k + 1] = ξ[k] + ∆ξ = G(ξ[k], u[k]) (36)

Let u[k]|k ∈ Z+ be a series of inputs to be applied.

(a) Initial State (b) STEP1

(c) STEP2 (d) STEP3

Fig. 12. Snake robot: forward locomotion

Then the stepwise evolution of the system state is given by

ξ[1] = G(ξ[0], u[0]) = G1(ξ[0], u[0])ξ[2] = G(ξ[1], u[1]) = G2(ξ[0], u[0], u[1])...

......

ξ[k] = G(ξ[k − 1], u[k − 1]) = Gk(ξ[0], u[0], ..., u[k − 1])(37)

where Gk is recursively defined by

Gk+1(ξ[0], u[0], · · · , u[k−1]):= G(Gk−1(ξ[0], u[0], · · · , u[k − 2]), u[k − 1])

G1(ξ[0], u[0]) := G(ξ[0], u[0]).

Definition 1: (Stepwise Reachability Set) For the systems(36), k-stepwise reachability set from the state ξ[0], denotedby Λ(ξ[0], k), is defined as

Λ(ξ[0], k) :=Gk(ξ[0], u[0], · · · , u[k−1]), u[j] ∈ Ω, j = 0, · · · , k−1

where Ω is the set of all admissible inputs.Definition 2 (Neighborhood): For an integer-valued state

ξ ∈ ZN , its neighborhood is defined as

N(ξ) := ξ+(δ1, · · · , δN )T , δi ∈ −1, 0, 1, i = 1, · · · , N.

B. Single-cart system

Let us turn to consider how k-stepwise reachability setgrows as k increases, focusing on the case of single cart.Fig. 13 shows a visualization of k-stepwise reachability set ofthe single cart from ξ[0] = (0, 0, 0)T . We varied k from 1 to4. In Fig. 13, thick-lined hexagons imply the reachable cellsby k steps. These cells contain some colored triangles, whichimply the reachable “orientation” by k steps. For instance,the 1-step reachability set cosists of

(0, 0, 0)T , (0, 0, 1)T , (0, 0,−1)T , (1, 0, 0)T , (−1, 0, 0)T .

At k = 1, the cart can move only in the initial orientationdue to the nonholonomic constraint (14). It cannot stepsideways. Next, in 2-stepwise reachability set, the cart canmove to cells around the initial cell. However, the orientationof the cart is different from initial orientation. Therefore, thecart can not take any state. Finaly, in 4-stepwise reachabilityset, the cart can move to all the neiboring cells aroundthe initial one, and can take any orientations there. Thisobservation can be summarized as

argmink

Λ(ξ[0, k]) ⊇ N(ξ[0]) = 4.

This clearly gives us a sufficient condition for controllability.By repeating this primitive process of (at most) 4 steps, thestate of the single cart can be transferred to any desired state.In addition, Fig. 13 shows the same property as continuousmodel that it is easy for the cart to move in the same directionas initial orientation.

Initial State

STEP2

STEP3

STEP4

Fig. 13. Stepwise reachability set of the single cart system

C. Trailer system

In the same manner, we can consider k-stepwise reacha-bility set for the single trailer system. Suppose initial state ofthe single trailer is ξ[0] = (0, 0, 0, 0)T . Fig. 14 shows resultsfrom 1-stepwise to 7-stepwise reachability set of the 1st cart(the rear cart). In this case,

argmink

Λ(ξ[0, k]) ⊇ N(ξ[0]) = 7

Therefore it is clear that the state of the trailer can be trans-ferred to any state by repeating the primitive 7 steps. Also,Fig. 14 indicates us the difficuty of “backward locomotion”of the trailer compared to the single cart, as analogous tocontinuous cases.

VII. CONCLUDING REMARKS

In this paper, we tried to discuss possibility of discrete-valued version of nonholonomic mobile robot systems. Weshowed that, many intrinsically consistent properties can bederived starting from simply defined discrete constraints. Wealso examined k-stepwise reachability set Λ(ξ[0], k) for thesesystems, to confirm possibility to maneuver the system stateto any states. In addtion, for discrete-value nonholonomicmobile robot systems the same property as continuous modelof the single cart and the single trailer.

There remain numbers of interesting issues to be discussedin future works, for example:

1) Controllability: Any form of general test for control-lability, or discrete-version of Chow’s theorem, shouldbe established and related with Lie brackets. It is quiteinteresting how controllability of this system is relatedwith stepwise reachability set.

Initial State

STEP2

STEP4

STEP6

STEP1

STEP3

STEP5

STEP7

Fig. 14. Stepwise reachability set of the single trailer system

2) Stability, stabilization and stabilizability: Characteri-zation of stability must be crucial in the first place.Lyapunov approach may have a difficulty, in the sensethat the converse theorem is not likely to hold indiscrete-valued (i.e., discontinuous) cases. Moreover,discrete version of Brockett’s theorem[1] must be aquite interesting issue.

3) Controller design: Design of discontinuous, time-varying or hybrid controllers have been central is-sues of continuous nonholonomic systems. Someidea of existing design approaches, e.g., time-varyingapproaches[9], may remain effective in discrete cases.

4) Underlying mechanics: It is important to ask ifthere exist discrete equivalents of energy, Lagrangian,Hamiltonian or variational principle that are consistentwith the current results. It would also be interestingto relate it with discrete mechnics proposed by Mars-den et al[10]. The current work can be considered aLebesgue-type approach to discrete mechanics, in con-trast that the aforementioned one[10] can be regardedas a Riemann-type approach.

The authors expect the current work to be a first step to-ward establishment of discrete-valued nonholonomic systemtheory.

VIII. ACKNOWLEDGMENTS

This work has been partially supported by the Aihara In-novative Mathematical Modelling Project, the Japan Societyfor the Promotion of Science (JSPS) through the ”FundingProgram for World-Leading Innovative R&D on Science andTechnology (FIRST Program),” initiated by the Council forScience and Technology Policy (CSTP).

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