lntemational Conference on Instrumenlation, Control & AutomationtcA2009October 20-22, 2009, Bandung, Indonesia

Abstract

Most of unmanned aerial vehicles (UAVs) and au-tonomous underwater vehicles (AUVs) can be fallen intothe field of nonholonomic systems, especially subjectedto an acceleration constraint. In this paper, underactuatedcontrol systems are described for such vehicles with sixstates and four inputs. First, a discontinuous approach andan invariant manifold approach are introduced to constructkinematics-based controllers, in which a canonical modelfor such vehicles, called chained form, is assumed to beapplied. Next, combining such a result and dynamicalmodel information, a dynamics-based controller is shownto be derived for generating practical force andlor torqueinputs, where a backstepping approach is utilized to incor-porate the previous input derived by the kinematics-basedcontroller into the dynamics-based controller, togetherwith a partial linearization using a degenerated dynamicalmodel.

1 Introduction

Recently, wheeled mobile robots (with two independentdriving wheels) are general as a powered-wheel chairin our daily-life or an automated guided vehicle (AGV)in factory automation. The mechanical feature of this

' type of robots is to use two independent motors forcontrolling three generalized coordinates, consisting of twotranslational motions within a flat-plane and one rotationalmotion around the axis orthogonal to the plane, from theviewpoint of kinematics. It can be also regarded that twooperational variables, which are composed of the forwardvelocity of the robot and the rotational speed due to asteering (or handling), are available to control such threegeneralized coordinates. From this point of view, such acontrol system is called an underactuated control system[1]. Note however that this type of robots has a constraintthat it cannot move sideways directly, which is callednonholonomic constraint tzl,t3l, so that it cannot achieve

02009 rcA, tsBN 978-979-8861-05-5

Unmanned Vehicles Control Systems: The Development ofUnderactuated Control Systems for Vehicles with

Six States and Four Inputs

Keigo Watanabe* and Kiyotaka Izumil* Department of Intelligent Mechanical Systems,

Division of Industrial Innovation Sciences,Graduate School of Natural Science and Technology, Okayama University,

3-1-1 Tsushima-naka, Kita-ku, Okayama 700-8530, JapanTel : +8 I -86-25 1 - 8064, Fax : +8 1 -86- 25 | -8064, E-mail: watanabe @ sys.okayama-u. ac jp

t Department of Advanced Systems Control Engineering,Graduate School of Science and Engineering, Saga University,

1 Honjomachi, Saga 840-8502, JapanTel : +8 1 -952- 28-8696, Fax: +8 1 -952-28- 85 87, E-mail: izumi @ me.saga-u.ac jp

a movement to any place unless any cut operation ofsteering (or handling) is applied.

On the other hand, most of aerial robots and underwatervehicles or robots t4l, t5l, [6] have a constraint that thereare no any thrusts (or skids) in the body-side directionin three-dimension, which is similar to skids (or sidewaymotions) due to terrestrial wheels. Therefore, any skilfulcontrol is required for unmanned aerial vehicles (UAV)or autonomous underwater vehicles (AUV) to achieve theeffective motion control of such vehicles.

In this paper, some underactuated control methods areintroduced to control VTOL aerial and underwater robotswith six-states and four inputs, where the former ones aresometimes called X4-flyer, Quadrotor, etc. [7], t8l, I9l,ll0l, illl, whereas the latter ones are the extension ofX4-flyer to the underwater robots [12]. In what follows,kinematics level controllers are first introduced for suchsystems, in which a discontinuous control originally de-veloped by Astolfi t131, t14l and an invariant manifoldcontrol proposed by Khennouf and Canudas de Wit [15],[16] are applied for developing such kinematics-basedcontrollers. Next, in order to apply the result of kinematics-based controllers to the case when generating practicalforce or torque inputs, i.e., a case of using a dynamics-based controller, a backstepping approach is introduced[17], together with use of panial linearization using a de-generated model for the original dynamical model. Finally,some simulation examples are presented to demonstratethe effectiveness of the introduced underactuated controlsystems for an X4-flyer and a lateral type AUV.

2 Kinematics-based Control Systems

Ler us consider the coordinate system and constructionof X4-flyer shown in Fig. l.

Let E : {E, Eo E"} denote a right-hand inertial framesuch that E, denotes the vertical direction downwards intothe earth. Let the vector { : [r A z)T denote the position

International Conference on Instrumentation, Control & AutomationrcA2009Oclober 20-22, 2009, Bandung, lndonesia

fzh

Fig. 1. Coordinate system ofX4-flyer

of the centre of mass of the airframe in the frame -Erelative to a fixed ongin O e E.

Let c be a (righrhand) body fixed frame for the air-frame. When defining the rotational angles 4 : [$ 0 llraround X-,Y-, and Z-axis in the frame c, the orientationof the rigid body is given by a rotation R'. c -'+ -8, wherel? € $t3x3 is an orthogonal rotation matrix given by

| "O"rb sSsgc$ - cdstb c$s?alt + s$stll

p: I c?s$ s$s?st! * c$ct! c$s9st! - s(;ct! |

(1)

L -t0 s$c0 c(;cO j

where co denotes cos c and sa implies sin o.From the rotational matrix, the kinematic equation for

X4-flyer, q : S(q)a, can be reduced to

(2)because the X4-flyer,nur.

:n]Irrne thrust in the Z-

direction, where o : LiJb 4 0 *) , where 2o denotes the

Z-directional translational velocity ^d lO

etd]t l, ti,"

rotational angular velocity vector in the airframe.

2,1 Discontinuous Control Approach 1: No Use ofCanonical Form

A method of discontinuous coordinate tiansformationproposed by Astolfi [13] is introduced at the origin to makethe controlled object discontinuous in advance, so that theorigin is stabilized by a continuous feedback law.

In Eq. (2), exchange the order of i and 2 and defineu1 : 26,u2 : d,us : 0, and u4 : $. In addition, new

@2009 lcA. rsBN 978-979-8861-05-5

control inputs are defined as follows:

u1 : (cos Scos?)q, 'tL2: r,2(3)

U4:1)4

Yz:

(6)

(7)

and, as a o-process to make the present system discontin-uous, the following coordinate transformation is applied:

A+: Q,

afraz: =, A3: -

zz

Us:g, ya:qt G)

f " ' ll " t ll " " lLU4 J

(5)

When defining the coordinates before a transformationas 21 : z and Zz [y yI 0 ,/,]r, the abovetransformation is equivalent to the situation that o : zand V : lU s d, 0z tl,,zlT are chosen in Y1 : za:l.d Y2 : V(Zt, z2)lo(21), which are the coordinatesafter the transformation. Note here that o(0) : 6 un6V(0,22): ly , 0 0 0l ' l0 arc sat isf ied.

Differentiating these with respect to time and arrangingit gives the following transformed system:

Then, the transformed coordinates Y1 : y1 and

lA" y, yn ys Aalr are reduced to

Yr: g11(Yy,Y2)u1

l"'1i ' r : grr(Y1.Y2)u1+ gzz(Y2) lusl

l"nJ

nlcos/sin0cosi t*s in/s inry ' 0 0 0l

- . _cos/sindsin' / - s inry 'cosy'r 0 0 O | | " . , Icosry 'cose 0 0 0l lq I

0 10 ol ld l0 0 1 0l L, , l ,J0 0 0 1J

where

9t (Y1,Yz) :

gzr (Yt,Y z) :

fo o oll0 o 0l

s22(Y2):11 0 0 lt0 1 0l

lo o 1JWhenthat

using the input u1 - -kAt, it is easy to find [18]

6 cosf s in0sind-sinScosry ' ^ . T| --------6;7;GF- - v2 |I cos </ sin 0 cos d*sin d sin'iy' ^. II cos o5 cos d .'D I ns,,u, : -k l

B l=Jvr)Loi, r ,

000

000100010001

cos 4, cos d

cos d, sin S cos ilr+siD d siD ri,

/ cos d s in d s in d-sin d cos d - . \ f

\-------GEo;7-- - c2l yl

/ cos qb sin 0 cos dr*sin c5 sin r,/r ^. \ 1

\ -------;E}-c"" o- - ca ) G

00o

lnternational Conference on lnstrumentation, Control & AutomationtcA2009October 20-22, 2009, Bandung, Indonesia

Partially differentiating /(Y2) with respect to Yz and

evaluating the Jacobian matrix at the origin gives the

following approximation:A+11/^\ l

f (Yr) - " r \ - z) | Y^ (9)t - oYz ly, :o- "

Therefore, the linearization about the origin('Uz : 'gt : 'y4 = 'As : 'A6: 0) becomes

Yz:f(Yr)*gn

which is apparently controllable, so that it is easy to finda linear state feedback (as a continuous function) that

stabilizes the system asymptotically. Note however thatthis is not globally stable, i,e., it is stable locally aroundthe origin.

2.2 Discontinuous Control Approach 2: Use of ChainedForm

Whendef in ingq: [ " y * O 0 r l t ) r andu:

l"u Q 0 ,lt]' , it can be rewritten in the symmetric affineform of six states-four inputs:

Q : frur * fzuz -t feut * f a'us, g € mo 02)

I ) A Chained Form Transformation: In order to satisfya sufficient condition for implementing a chained formtransformation of the input vector fields [19], [20], theinput vector fields are changed to

where Gs and Gr are involutive, and G1 has rank 5.

Since nz +ns ln+*4: 6, i f n2 : n3: 1 and n+ : 0,then the conditions for determining hr . . . ha are given by

d,hrGo, dLon,hzLGo, dLSrlhLGo

dhrGt

where note that the scalar ha can be selected arbitrarilybecause of na: g.

Consequently, when choosing

hr : z, hz: U, h3: t r , hn: lb (14)

it is found that

o(q) :

[k ohl0 k 0

=lo o olo 0 0L0 0 0

| " '1l ra | (10)

L"njo ol fsr l [o o ol-k ol lur l lo o ol fuzl0 0l lvnl+; t o ol lual0 0l lebl l0 t 0 l [u+Jo oJ fuol Lo o 1J

(u)

[01l0 l

, gz: l? ll0 lLol

1tandsinTy' - tan/$

tanl costf,t * tan/qL*000

ts l is llo l lo l" : l? l " : l8 lLol Lt l

The corresponding distributions are given by

I n'' I

l:T::^'"1:IT:J

z1tan d sinTy' - tr" Oi# |

Y | / r5\ban d cos ry' + tan /414 |

t'"

n l4, I

f .Qt: --?

. l '1 -

"orqaotd'9t: fs,

which can be reduced to

10oolL'nrn, Lo"Llnrhz Lo"L[rh2 Ls^L1srh2 |L?rn" Lo,Llnrfu L%Lrsl fu Ls^Lls, fu |Ltn,nn Ln,Lf;,ha Ln"Ll,ha Ls^L\,h J100o -*JT*" #i% - tan/jg$cos,/o ;##;? !ffi + tan/ffi sin,/000

oltanr/ff i *tandcos'y' ' ltan </Er4 - tune rinrl, I

trsl

, l

E (q)

ItI_L

9z: f z(13)

9+: f a

9t:

Go : span{gz, gz, ga}

Gt : span{92t 93, 94t adsrgz, adsrgs}

@2009 rcA, rsBN 978-979-8861-05-5

Therefore, the new control input vector n :r1T'

lrt u, u3 ua]' can be reduced to

u:E(q)u (17)

and the new state variable vector z: [€o (o 0 rlo ry n)Tcan be reduced to

z: o(q) (18)

Thus, the resultant system with a chained form transfor-mation is described by

fq'l | ,'l lo l l : ,

' : l t : l : l ' [ '| 4 ' I l \oatLiol | 'n

(ie)

Intemational Conference on Instrumentation, Control & AutomationtcA2009October 20-22, 2009, Bandung, Indonesia

2) Discontinuous Control: In order to make the abovesystem discontinuous, applying a coordinate transforma-tion as a o process yields

Ut: €o, Uz: eo,

When defining^ 21 : €o and Zz

[(o (r 'to 'ry 'yo)' as the coordinates with notransformation, this is equivalent to select 1641 6r - {02 and

v : [(01o (r{o r7o€o lr to€o]r in the transformed so that

coordinates Y1 : 'Eo, Yz V (Zt,22) l"Qr).Here, it is satisfied that a(0) : 0 and V(0,22) :

[o 0 o 41 o1'+0.Differentiating the above new state variables, defining

u3 : {e03, ua: {n'tta and rearranging it gives

U

@z-ut) f r_u

At(ot ,

- 9r t - \ J-\y+ -

yo,t ,111

_ ss.Y1

Furthermore, defiryng Y1 At and Yz

lu" 'au an 9s 9o]' Yields

9tt (Y1,Yz) : I

9zt(Yr,Yz):

gzz(Yz):

Setting ur : -kyr gives

!21 X Lt1 :

i - . ' ) ' ' ' '

il'tl'

l "u| ..1( i

0246Time t lsl

Fig,Z. Controlled state variables in a chained form

Yz: f (Yz) -r gzz

which is shown to be controllable. Therefore, it is easy tofind a continuous function as a linear state feedback thatcan globally asymptotically stabilize this system.

3) Simulation Example: Setting k : 1 and assigningtheclosed- looppolesas[-1 -2 -3 -4 -5] forthe o transformed system, the resultant feedback gainmatrix is obtained as

Therefore, the input [r, 0" 6a]T can be described by

(2s)

The simulation results are shown inFig.2, Fig. 3, Fig. 4,Fig. 5, and Fig. 6, where the initial state vector was setto so : [1.5 1.5 -2.0 n/IO nl:r0 rl70]r and thedesired value was q, : [0 0 0 0 0 0]' .

Fig. 2 shows the time-responses of the state variables forthe system with the chained form transformation, whereasFig. 3 denotes the control inputs for such a chained formsystem.

It is found from Figs.4 and 5 that the latitude, horizontalposition, and all attitude angles converged to the desired

-q

'Eo

oEa(t

a3: i -Eo eo)?0qo

tlo rlrAn:

eo, Yr:

* .

Ir' Il l i : l :dt I 'us I

l r ' lLY6 )

1,,,1l t ' lLtn l

o o'l fvz Io ol lvel0 0l lu+l2k 0l ls ' r lo t, l Lsol

l rrll t ' | e3)Lon l

Io o o

: l - r t l Il0 0 - /cL0 0 0

[ r o o' ll0 0 0l. l3 ; Sllo o 1l

f " ILfi]

QD100000010000001

[o lI ar-ut I-u I o^:;o,lL-uul

|- ,r, -0r,,* I

l-^,_ki*)

lil il(24)

fe -18 1 -6 olr :11 -3 10 -82 ol

L0 0 0 0 2)

I ur1

f;;l : -,lt:)l t4 l I y ' I

Lou j

@2009 lcA, tsBN 978-979-8861-05-5

2fvr)

V2

!.I

I

I+-

lntemational Conference on Instrumentation, Control & AutomationtcA2009October 20-22, 2009, Bandung, Indonesia

0246Time 1[s]

Fig. 3. Discontinuous control inputs

0246Time t [s]

Fig. 4. Position results

values in a shot time by applying the proposed discontin-

uous control method, where the actual control inputs for

the aerial robot is shown in Fig. 6.

3 Invariant Manifold APProach

3.1 TWo-stage Switching Method

Assume that the controlled object is to be a system

transformed into a chained form with four-inputs and six-

stdtes:zL:ut

2z: az

i3: zzul

iq: as

2g: Z4l)1

28: t '4

246Time 1[sl

Fig. 5. Attitude results

- tlt

-*-- U2

- '- ' - Ll3-----. . u,

o 2 rime tlsl

6

Fig.6. Control inPuts

In fact, under the feedback control using Eq. (27), it yields

that

o

-{

=eoO

E

oO

E

o

o

1.inQ): i3- ) l ,1zz* ztrz l

z1.

: - lz2t t1 - ztr t2 l :0z

Lr l: z2't t ! - - lu1z2 -f zyt2lz

(30)

' where it is assumed that a feedback control law is derived

for a stabilizing problem that any initial state vector

z(0) : [rr(0) ..ro(0)]r should be converged to the

origin.1) Invariant Manifolds: When the following inputs in

a state feedback control with a gain k > 0 are applied to

Eq. (26),

u1: -k27, Uz: -kzZ' Q7)

'tB : -ls+^, 1t4: -fo4u (28)

from the constant term of zs(t), the following can be

selected as one candidate for manifolds:

s1(z) : zs(t) - laft)22ft) Qs)

02009 tcA. rsBN 978-979-8861-05-5

Therefore, using the above feedback control gives

s1(z) : ssn51. (31)

so that s1(z) becomes an invariant manifold. Therefore,

if s(z): 0 holds at any time t : T, then s1 (z) : 0 for

t > T, so it is found from Eqs. (26) and (27) that 21 :

-kzt and 2z : -kzz, that is, a(t) and z2(t) wdetgo

asymptoticallY zt + 0 and z2 -

0' In addition, it follows

that zs -

0 because s1(z) :0 is satisfied.

Similarly, when applying the inputs in feedback control

laws (21) and (28) to the system in (26), from the constant

term of z5(t), the following can be selected as one

candidate of the second invariant manifold,

,z(z): z1(t) - !a(t)za(t)

Actually, under the feedback control of using Bqs' (27)

and (28), it yields that

1 1,s2(z) : i5 -

|[taza * z1t4] : z4't t -'r[t'tz+ * 4q]

1.:

, lz+t t t - z1' I tg l : g (33)

(26)

(32)

0I\l/

IIIII

-x- - - - v- - - . - z

so this can be also confirmed to be an invariant manifold.

lntemational Conference on Instrumentation, Control & AutomationtcA2009October 20-22, 2009, Bandung, Indonesia

2) Anractive Control to Manifuld: In this control usingan invariant manifold proposed by Khennouf and Canudasde Wit [15], an attractive control to an invariant manifoldfor a control duration t <7, i.e., the first step control to

obtain s1 (z) : 0 and sr(r) : 0 is performed, whereasfor t ) T each state is converged to the origin along the

invariant manifold by using a feedback control on such an

invariant manifold, as the second step control.Now, define s(z) : ls1(z) tr(.)l ' as an invariant

rnanifold vector. As the Lyapunov function of s(z), se-lecting the following

v(r) : r""'1t1"1r1 G4)

and as the control inputs, selecting

u1: -Js1(z)221 1'2: Js1(z)2t ,13: fs2(z)zr, l ) 0(35)

it follows that

where

w(z):-21*2Ll -T a2

22z4

which constitutes other invariant manifold. From this fact,it holds that w(z) : w(z(O)) : const. Therefore, Eq.(39) becomes negative as long as tu(z(O)) I 0, so thats*0as1---+oo.

l) A Quasi-continuous Exponential Stabilizer: Usingthe above result, let us consider a quasi-continuous ex-ponential stabilizer that replaces two types of controllersat two-step approach with one controller.

Combining Bql (27), (28), and (38), il follows that

For the closed-loop system of Eq. (26) with this controller,it is found that

i l t :2(zpt + z2u2) :2(-kz? - kr i l : -2kw (43)

v(z\:" ' l : ' ! , ) l : 1" ' lzzut - ztuzlLsz\z)) 2 lzat:1 -zt\)

: - is 'W(z)s<0 (36)

lfi] :,,h1,? +] ]:i:t 4i;!

- zps):i?r,\t"? + 4)) : -{^:;

'|s1 :

t \2241

. t ,s2 :

t \24u1

- zpz) : i?r,'it,? + A) : -r*:^;

U

-2. l> o (37)

Solving these responses gives

Note here that Eq. (36) becomes negative as long asW (z(0)) I 0, so that s --+ 0 when I ---+ oo. In fact,t : T if ll"(r)ll < E is satisfied and thereafter a controllaw on the invariant manifold is assumed to be applied,where e denotes a practical allowance value for achievingor approximating the invariant manifold.

The attractive control to the manifold discussed aboveis theoretically continuous with infinite time. A discon-tinuous control method that uses a sign function can alsobe introduced to assure an attractive control to a manifoldwlth finite time (see Watanabe et al. [21]).

3.2 Quasi-continrious Exponential Stabilizing Method

Here, as slightly different control inputs from the aboveone, selecting the following inputs [22] given by

u1-- -Js1Q)22

. u2: fs1(z)4 (38)/ - - \ / -2 t -2\

1ry : - J s1(z) ( 3+ \ + J 's2Q) ( = |\zr / \ zr /

it results in

i t : - {sr@)w6(z)s(z) !o (3e)z

where WaQ) is a diagonal matrix, whose elements are

lr l l : ' t t22: z?+'3> 0 (40)

Here, sefting u(z) : u)rr : w22 under the condition ofEq. (38) gives

dt:21"" '*22u2):g (41)

02009 tcA. rsBN 978-979-8861-05-5

which yield that as f --+ @, 1r) * Q, sr -

0, sz + 0, sothat all z\, ..., z6 asymptotically converge to zeros. Notehowever that the first term of Eq. (42) may diverge as

4 ---+ 0, but observing that

s{z(0))e-f t (4e)

(s0)

u (z (0)) e- zkt z L(0) e- kt

these numerators converge Io zero faster than the denom-inators, i f !>3k.

2) Simulation Example: Let us consider a lateral X4-AUV model depicted in Fig. 7. Since the lateral typeX4-AUV has only the total thrust in the X-direction, thekinematics is given by

(5 1)

w : *(r(o))e-2k'

s l : s1 (z(oDs-f t

sz: sz(z(o))e-I t

(46)

(47)

(48)

cosdcosry' 0 0 0lcosdsinry ' 0 0 Ol[ i . r l

-s ind 0 0 0l l4 l

3 ; ?3lL; lo oo1, l

t- r'tt l

l l t ll i l -lq l -l0 lLr/l

where r.l : l*u O d ,h1', in which i6 denotes the X-L' 'J

r - . , .17directional translational velocity and l<t' 0 ltl is the

rotational angular velocity vector in thet body fr'ame.

lntemational Conference on Instrumentation, Control & AutomationtcA2009October 20-22, 2009, Bandung, Indonesia

Z(b) Coordinate frames

Fig. 7. A lateral X4-AUV

The kinematic mode of the present AUV can be rewrit-ten by the following symmetric affrne system with sixstates and four inputs:

Q : ftut * fzuz -t fsuz * f+uq, g € m6 62)

By using the coordinate and input transformation with

z:Q(q), a:E(q)u (53)

- - - - - zt - - - - - - 2t- - - - L2 ' ' - - .5

Time t lsl

Fig. 8. Controlled state variables in a chained form

____ v2

-.- . . v3

Time t [s]

Fig. 9. Quasi-continuous conlrol inputs

4 Dynamics-based Control Systems

4.1 Backstepping Approach

The dynamical model of X4-flyer or lateral-type X4-AUV is described in the following matrix form:

NI(q)q +v^(q,ds + G(q) : B(q)r (54)

wherc M(q) € J?6xo is the symmetric, positive definiteinertia matrix, Vrn(q,q) € nu*u is the centrifugal andCoriolis matrix, G(d e ft6 is the gravitational vector,B(q) e $?6x4 is the input transformation matrix, and r €Ra is a generalized force vector consisting of force ortorque components.

I) Degenerate State-Space Model: To derive a de-generate state-space model, the following kinematic anddynamical models are combined:

q: S(q)u (55)

tut(i ls *V*(e,s)A + G@) : B(q)r (s6)

Differentiating Eq. (55) with respect to time, substituting itand its derivative into Eq. (56), and premultiplying the bothsides by S"(q) yields the degenerate state-space modeldescribed by

[ , t (q)b +v^(q,q)a + G(q) : B(q)t (s i )

N

.9o'c6

o

at)

=Ico()

12

this system can be transformed into Eq. (19) or Eq. (26),where

| - , - lI

tan? |

| I t Io(o) : l - r"""4 |

| : " ' lLol[1 0 0 0 Ilo o o #'= |

E(s) : l ;6 - I - tano{un!, 1

" cos2 0 cos r/, cos 1,i, I

Lo1 o o I

Assume that the feedback gains are set to &0.6 and / : 2, the initial state vector is g0 :

[3 3 3 rll0 rl10 lllo)r, and the reference is to beq, : [0 0 0 0 0 0]' .The results are shown in Fig. 8,Fig. 9, Fig. 10, Fig. 11, and Fig.12.

@2009 tcA, lsBN 978-979-8861-05-5

(a) Schema of a lateral X4-AUV

lntemational Conference on Instrumentation, Control & AutomationlcA2009October 20-22, 2009, Bandung, Indonesia

-x- - - - v-"--- z

o4812Time t [s]

Fig. 10. Position responses

It'\ - $

' \Ai \ . - .--- \ ,

\_

l r 't,

,/

Time t lsl

Fig. 11. Attitude responses

where

u(d: sr(q)u1fls(q) (58)/

V^(q,q): t , l *@)S(q) + W"k,ds(q)) rselG(q) : s'(q)c(q) (60)B(q) : sr (q)B(q) (61)

2) Partial Linearization of Degenerate State-SpaceMbdel: To partially linearize the dynamical model, thefollowing nonlinea.r feedback

, : B-'(q) {u(d" +V*(q, q)o + G(q)} (62)

is applied to Eq. (57), where a is an acceleration vector.From this nonlinear feedback, Eqs. (55) and (57) can bereduced to

(63)(64)

4ATime t lsl

Fig. 12. Control inputs

sections. Here, assume that ,r.r is a virtual input, its desiredvalue o4"u is a stabilizing function that may be replacedby the previous kinematics-based controller, and 16 :

u -'ud,et is an error variable [i8].Using the error variable in the kinematic model gives

q : S(q) (aa"" * ru) (66)

When selecting the Lyapunov energy function such as

ac

;o

oE0

I -v :5r f i ra (67)

it is easy to find that the control input u given by

u: -cr t * t )a l ' " ^ ' ' 'tffs(sl@des * 16) (68)

gives

q : s(q)a

' i : : a

V:-rTCra<0 (6e)

so that the system can be asymptotically stable, whereC € $axa is a positive definite matrix.

The dynamics-based control for a case when applyinga backstepping method is shown in Fig. 13.

4.2 Underactuated Control for X4-Flver UnmannedAerial Vehicle (UAV)

Here, the simulation of X4-flyer is demonstrated byusing the backstepping method, but with a discontinuouscontroller as a kinematics-based controller [23]. The phys-ical parameters used in the simulation are shown in Table I.

TABLE I

MoDEL PARAMETERS FoR AN X4-FLYERWhen defining the state vector as * : lq o]' and theinput vector as ?r: a, from Eqs. (63) and (64), the state-space model in closed-loop system can be described by

et:J@)+s(n)u: f t (X)" ] * f 9 l" (6s)L u J Lr: l

where 1 is a unit matrix.3) Application of Backstepping Method: In Eq. (63),

if u car' be regarded as an input, then the system canbe controlled by the discontinuous control approach orthe invariant manifold approach presented in the previous

@2009 tcA, tsBN 978-979-8861-05-5

n1,II,faT

r

bd

GravityMassDistanceRoll InertiaPitch InertiaYaw InertiaRotor InertiaThrust FactorDrag Factor

9.806650.468 kg0.225 m4.9 x 10-3 kg .*2

4.9 x 10-3 kg.-28.8 x 10-3 kg.*23.4 x 10-5 kg.-22.9 x 1-0-b1.1 x 10-6

f=::l

I

-u1___- u2- ---- u3' - . - - - .u,

Fig. 13. Block diagram of dynamical control using a backstepping method, where ,: : ST (q)q because ST (g),5 (d = I

Intemational Conference on lnstrumentation, Control & AulomationlcA2009October 20-22, 2009, Bandung, I ndonesia

Note also that each matrix in the dvnamical model canbe reduced to

M(q):

c

(I

ln 'L 0 0 0 0 0ll0 n 'L 0 0 0 0llo o m o o oll0 0 o r ,0 0ll0 0 0 0 Io 0lLo o o o o I .J

f0 0 0 0 0 0 Il0 0 0 0 o o ll0 0 0 0 0 0 |v*(s 'd=lo o o o J,e+I. l ) - I " .Ollo o o -J,e-I" l ) o. I ,d lLoooh0-I ,ool

Time t lsl

Position control via dynamical controller

048Time t lsl

Fig. 15. Attitude control via dynamical controller

The simulation results are shown in Fig. 14 and Fig. 15.It is seen from this result that all the generalized co-ordinates are converged to the desired values smoothly.The corresponding inputs due to the kinematics and thebackstepping are shown in Fig. 16 and Fig. 17 respectively,and the torque inputs due to the dynamics are depicted inFig. 18.

4.3 Underactuated Control for Lateral-type X4-Autonomous Underwater Vehicle (AUV)

Here, the simulation of lateral-type X4-ALIV is dernon-strated by using the backstepping method, but with a quasi-continuous exponential stabilizer as a kinematics-basedcontroller. The physical parameters used in the simulationare shown in Table tr [24].

Note also that each matrix in the dvnamical model can

Eo

Io lI^ I

lu lG(q): l -Tt l

lo lL0ll -(crbs?cn1'+s$sl,t) 0 0 0ll -kdses$-sdc4,) 0 0 0l

B(q)=l - 'd"o o o ol. I o r o ol

I o o , o lL 0 0 0 1j

When k : 1 and the closed-loop poles are determined as(-L, -2, -3, -4, -5) for the kinematics-based controlthe feedback coefficient matrix is obtained by

[s-80 o olr ' :10 0 I _24 0l (70)

Looo o 6.1so that [u2 fu ta]r is represented by

I v"1[,,r.l I vr Ilor l : -Flanl tz l lLa.J | ,' I

lYs )The positive definite matrix in Lyapunov function is set

to C : diag(2O, 20,20,20). In addition, assume that theinitial value of generalized coordinates is set to qo :

[0 0 0 r/70 nl10 nl^70]r and the desired value isq, : [ -B 3 3 0 0 0] ' .

lnverseInput

Transformation

-X*--- v- - - - - z

-a- - - - 0----- 1/

II

@2009 rcA. tsBN 978-979-8861-05-5

lntemational Conference on Instrumentation, Control & AutomationrcAzo09Oclober 20-22, 2009, Bandung, Indonesia

o

TABLE II

MoDEL PARAMETERS FOR A LATERAL X4-AUV

Distance 0.1 mRoll Inertia 7.774 x IO-2 kg ' tn2Pitch lnertia 7.714 x 70-2 kg ' *2Yaw Inertia 1.714 x 10-2 kg'*2Fluid Density 1023.0 kg/-3

n@):

where

diag(m,m,m) : m6I * My Q2)

diag (1,, I* 1") : J1' -t Jy Q3)

Here, 1 denotes the unite matrix, My is an addedmass matrix, and J 7 is an added moment of inertiamatrix. Assuming that the fluid density is p and thepresent AUV form is a sphere, it is found that A[y :

diag(lpzrls,lptrl3,firrls) and Jy : 0. Furthermore,assume that the X4-AUV is in the state of neutral buoyancyto neglect the potential energy, so that G(q) : O.

The feedback gains &; : 0.6 and I :2 are set for thequasi-continuous exponential stabilizer as a kinematics-based controller. The positive definite matrix in Lyapunovfunction is set to C : diag(S, 5,5,5). In addition, assumethat the initial value of generalzed coordinates is set to

So: [0 0 0 nl4 rl4 nl4)' and the desired value is

s, : [5 5 5 0 0 0] ' .The simulation results are shown in Fig. 19 and Fig. 20.

It is seen from this result that all the generalized co-ordinates are converged to the desired values smoothly.The corresponding inputs due to the kinematics and thebackstepping are shown in Fig. 21 and Fig. 22 respectively,and the torque inputs due to the dynamics are depicted inFig.23.

5 Conclusion

This paper has introduced a brief discussion on un-manned vehicle control, especially for an X4-flyer orX4-autonomous underwater vehicle, which has six-statesand four-inputs. Since these vehicles are typically non-holonomic systems, we have proceeded to design anunderactuated control system by using a kinematics:basedcontroller and also extending it to the case when derivinga dynamics-based controller that provides the practical

force and/or torque inputs. One of the key approachesthat this paper did not discuss is to use a second-ordercanonical form, such as in chained form, power form,double integrator model, etc., to provide the force or

rnbII

Iafz

p

4Time t [s]

Fig. 16. Control inputs due to kinematic controller

ll'4\__I

I

| -Uri ---- ui| - ' --- us| . , , - "" . ud

04Time t [s]

Fig. 17. Control inputs due to backstepping method

- 'lz

___- T{

----- t re

Time t [s]

Fig. 18. Torque inputs due to total dynamical model

be reduced to

c

Eco()

cosdcosTy' 0 0 0lcos9sinry ' 0 0 0l

-s ind 0 0 0lo 1 o ol0 0 I 0 lo o o l l

EzoYo

lm 0 0 0 0 0lt0 n7 o

3 3 3lM(q) :13 3 T r , o ollo o o o Io olL0 o 0 0 0 I . )

;0 0 0 0 0 0 Ilo o o o o o Il0 0 0 0 0 0 |

v^(a,d: lo o o o. r .1 j ) - r re. llo o o -1"4) o J,a+1,01L0 0 0 Io0 -J,Q-I ,d o J

@2009 tcA. lsBN 978-979-8861-05-5

Intemational Conference on lnstrumentation, Control & AutomationtcA2009Odober 20-22, 2009, Bandung, lndonesia

E

E.

=o(!

o

of

Y

-x- -* ' v' ' - - - z

-u1___- u2- ' - ' - U3-.- . - - -u,

- jL,0'0 4 8 12Time t [s]

Fig. 20. Attitude control via dynamical controller

04812Time t [s]

Fig, 19. Position control via dynamical controller

o0v

| ' ' .

Fig.22.

Time t [s]

Control inputs due to backstepping method

Time t [s]

Fig.23. Torque inputs due to total dynamical model

[6] K. Watanabe and K. Izumi, "Skilful control for underactuated robotsystems: From the ground to the air and underwater," Proc, of the2nd International Conference on Underwater System Technology:Theory and Applications 2008 (USYS '08), Bali, Indonesia, PaperrD 39.2009.

[7] P. Pounds, R. Mahony, P Hynes, and J. Roberts, "Design of a four-rotor aerial robotl' Proc. of tlu 2002 Australasian Conf, on Roboticsand Automation, Auckland, New Zealand, pp. 145-150, 2002.

l8l S. D. Hanford, L. N. Long, and J. F. Horn, 'A small semi-autonomous rotary-wing unmanned air vehicle (UAY)]' Proc. ofInfotech@ Ae ro space Confe rence, Arlington, Virginia, Paper No.2005-7077.200s.

[9] S. Bouabdallah and R. Siegwart, "Backstepping and sliding-modetechniques applied to an indoor micro quadrotor," Proc, of IEEEInt. Conf. on Robotics and Automation, pp.2259-2264,7005.

[10] A. Tayebi and S. McGilvray, 'Attitude stabilization of a VTOLquadrotor aircraft," IEEE Trans. on Control Systems Technology,1a(3), pp. 562-57 1, 2006.

[1] P. Castillo, A. Dzul, and R. Lonazo, "Real-time stabilization andtracking of a four-rotor mini rotorcraft," IEEE Trans. on ControlSystem Theology, l2(4), pp.5i0-516, 2004.

[12] K. Watanabe, K. Izumi, K. Okamura, and R. Syam, "Discontinuousunderactuated control for lateral X4 autonomous underwater vehi-clesi' Proc. of the 2nd International Conference on UnderwaterSystem Teclmology: Theory and Applications 2008 (USLS '08),Bali, Indonesia, Paper ID 14,2008.

[13j A. Astolfi, 'Asymptotic stabilization of nonholonomic systems withdiscontinuous control," Ph.D. Thesis, Swiss Federal Institute ofTechnology (ETH) in Zurich, Switzerland, 1996.

[4] A. Astolfi, "Discontinuous control of nonholonomic systems,"Systent and Control Lefters,27(I), pp.3745, 1996.

[i5] H. Khennouf and C. Canudas de Wit: "On the construction ofstabilizing discontinuous controllers for nonholonomic systems,"Proc. of IFAC Nonlinear Control Systems Design Symp. (NOLCOS'95), Tahoe City, CA, pp.147-752, 7995.

[16] H- Khennoufand C. Canudas de Wit: "Quasicontinuous exponentialstabilizers for nonholonomic systems," Proc. of IMC 13th WorldCongress, San Francisco, USA, pp. 49-54, 1996.

[17] R. Fieno and F. L. Lewis, "Control of nonholonomic mobile robot

-- -- V2

----- v3

o 4 Time t tsls

12

Fig. 21. Control inputs due to kinematic controller

torque inputs. As a result, several results obtained for thekinematics-based approach would be inherited to design adynamics-based controller [25], 1261, 1271.

References

[1] T. Mita, Introduction to Nonlinear Control Theory - Skill Controlof Underactuated Robots -, Shoko-do, Tokyo, 2000.

I2l L. G. Bushnell, D. M. Tilbury, and S. S. Sastry "Steering three-input nonholonomic systems: The fire truck example," InternationalJournal of Robotics Research, l4(4), pp. 36G381, 1995.

[3] S. Iannitti and K. M. Lynch, "Minimum control-switch motionsfor the snakeboard: A case study in kinematically controllableunderactuated systems," IEEE Trans. on Robotics,20(6), pp.994-1006, 2004.

[4] S. Bouabdallah, P. Munieri, and R. Siegwart, "Towards autonomousindoor micro YTOLI' Autonomous Robots,18, pp. 171-183, 2005.

[5] J. Yuh, "Design and control of autonomous underwater robots: Asvrvey :' Autonomous Robots, 8( I ), pp. 7 -24, 2O0O.

@2009 tcA. tsBN 978-979-8861-05-5

lntemational Conference on Instrumentation, Control & AutomaliontcA2009October 20-22, 2009, Bandung, Indonesia

using neural networks," IEEE Trans. on Neural Networks,9(4), pp.589-600, 1998.

[18] K. Watanabe, K. Tanaka, K. Izumi, K. Okamura, and R. Syam,"Discontinuous control and backstepping method for the under-actuated control of VTOL aerial robots wilh four rctorsl' Procof International Conference on Intelligent Unrnanned Systems(ICIUS2007), Ba[, Indonesia, pp. 224-231, 2007.

[19] K. Tanaka, K. Watanabe, K. Izumi, and K. Okamura, 'An underac-tuated control for VTOL aerial robots with four rotors via a chainedform transformation," Proc. of the 13th Inl. Svmposium on AnificialLife and Robotics (AROB 13th'08J, Beppu, Oita, Japan, pp' 775-778,2008.

[20] K. Watanabe, K. Okamura, K. Tanaka, and K. Izumi, 'A discon-tinuous control of VTOL aerial robots with four rotors through achained form transformation," Proc. of2008 Int. Conf. on Control,Autom.ation and Systems (ICCAS 2008), Seoul, Korea, pp. 804-809,2008.

[21] K. Watanabe, K. Izumi, and K. Okamura, "Kinematics-based con-trol of underactuated vehicles with four-inputs and six-states byapplying invariant manifolds," Proc. of 2009 IEEE InternationalConference on Networking, Sensing and Connol (IEEE ICNSC'19),Okayama, Japan, pp. 84G851, 2009.

l22l K.lzum| K. Okamura, Y. Saito, and K. Watanabe, "Underactuatedcontrol based on a manifold for an autonomous underwater vehicle,"Proc. of the 2009 JSME Conf on Robotics and Mechatronics(Robomec'09). Fukuoka, Japan, 1P1-B08, 2009.

[23] K. Okamura, K. Watanabe, K. Izumi, and K. Tanaka, 'An un-deractuated control for VTOL aerial robots with four rotors by abackstepping method," Proc. of the 2008 JSME Conf. on Roboticsand Mechatronics (Robomec'08), Naganno, Japan, 1P1-F20, 2008.

[24] K. Okamura, K. Watanabe, and K. Izumi'An underactuated controlfor an autonomous underwater vehicle," Proc. of the 26th SICEKyushu Branch Annual Conference, Okinawa, Japan, 10445, 2008.

t25l W. L. Xu and W. Huo, "Variable structure exponential stabilizationof chained systems based on the extended nonholonomic integra-tor," Systems Control Lett., 4t, pp.225-235,2000.

t26l W. L. Xu and B. L. Ma, "Statrilization of second-order non-holonomic systems in canonical chained form," Robotics andAutonomous Systetns, 34, pp. 223-233, 2001.

[27] T. Yoshikawa, K. Kobayashi, and T. Watanabe, "Design of adesirable Trajectory and convergent control for 3-d.o.f manipulatorwith a nonholonomic constraint," Proc. of the 2000 IEEE Int. Conf.on Robotics and Autornation, San Francisco, CA, pp. 1805-1810,2000.

@2009 lcA. lsBN 978-979-8861-05-5