T ajec o y Planning o Spacec a
Rendez ous wi h On/O Th us e s ?
Ra ael Vazquez ∗F ancisco Ga ilan ∗Edua do F. Camacho ∗∗
∗Depa amen o de Ingenie ia Ae ospacial
∗∗ Depa amen o de Ingenie ia de Sis emas y Au oma ica
Uni e sidad de Se illa, Camino de los Descub imien os s/n, 41092,
Se illa, Spain (e-mails: [email p o ec ed], [email p o ec ed],
edua [email p o ec ed])
Abs ac : The objec i e o his wo k is o p esen a ajec o y planning algo i hm o spacec a
endez ous ha is able o inco po a e Pulse-Wid h Modula ed (PWM) con ol signals. The
algo i hm is based on linea iza ion a ound a p e iously compu ed solu ion. To ini ialize he
algo i hm, a i s solu ion needs o be ob ained. To do so, he ajec o y planning p oblem
is sol ed using Pulse-Ampli ude Modula ed (PAM) con ol signals; hese a e hen con e ed
o PWM signals, which a e used as an ini ial guess. I e a ing, he solu ion is e ined un il an
op imal alue is eached. Simula ions show ha his me hod con e ges a e a ew i e a ions.
The algo i hm is simple and as , hence i could be implemen ed online o used oge he wi h a
Model P edic i e Con olle .
Keywo ds: Spacec a au onomy, Space obo ics, Pulse-wid h modula ion, T ajec o y planning,
Op imal ajec o y, Linea iza ion.
1. INTRODUCTION
Technology enabling simple au onomous spacec a en-
dez ous and docking is becoming a g owing necessi y
as access o space con inues inc easing. A e decades
o de elopmen , many app oaches ha e been p oposed;
see Wo inden and Gelle (2008) o an his o ical accoun
o Fehse (2003) o he basics.
Classicaly (Wie (1998)), he p oblem is modeled using
impulsi e maneu e s, compu ing an ini ial and inal im-
pulse (∆V) o achie e endez ous. Impulsi e changes o
speed a e also used in ecen wo ks (Tong e al. (2007);
Gelle (2006)). La ely, app oaches based on ajec o y
planning and op imiza ion (B ege and How (2008)) and
p edic i e con ol (Richa ds and How (2003); Rossi and
Lo e a (2002); Asawa e al. (2006)) a e eme ging. Howe e ,
ypically hese me hods allow he con ol signal ( h us ) o
ake any alue in an allowable ange. This ype o con ol
signal is usually e e ed o as Pulse-Ampli ude Modula ed
(PAM).
A mo e ealis ic modeling o spacec a h us e s would
ake in o accoun ha ypically, h us e s a e ON-OFF
ac ua o s, i.e., he h us e s a e no able o p oduce any
alue o o ce, bu can be only swi ched on (p oducing he
maximum amoun o o ce) o o (p oducing no o ce).
Thus only he imes whe e he h us e s a e u ned ON o
OFF ( he swi ching imes) can be con olled. This ype
o con ol signal is usually e e ed o as Pulse-Wid h
Modula ed (PWM).
?The au ho s would like o acknowledge inan ial suppo o he
Spanish Minis y o Science and Inno a ion and o he Eu opean
Commission o unding pa o his wo k unde g an s DPI2008-
05818 and p ojec FPT ICT HD-MPC.
The objec i e o his wo k is o p esen a ajec o y
planning algo i hm o spacec a endez ous ha is able
o inco po a e PWM con ol signals. The algo i hm is
based on he ac ha , e en hough in he endez ous
model he PWM signals appea nonlinea ly, his depen-
dence on he swi ching imes is only weakly nonlinea .
Thus, linea iza ion a ound e e ence alues can ai h ully
cap u e he dependence o he sys em on he swi ching
imes, a leas close o he e e ence alues. To ob ain
a i s solu ion a ound which linea ize he sys ems, he
ajec o y planning p oblem is sol ed using PAM con ol
signals; hese a e hen con e ed o PWM signals, which
a e used as ini ializa ion o ou algo i hm. I e a ing, he
solu ion is e ined un il an op imal alue is eached. The
esul ing algo i hm is simple and as and could be imple-
men ed online, o wi h a Model P edic i e Con olle (such
as Ga ilan e al. (2009)), allowing he MPC o include
PWM con ol signals. While he idea o linea iza ion o
compu e op imal PWM con ol signals is, as a as we
know, o iginal, we no e ha local linea iza ion echniques
ha e been used o op imal ajec o y p oblems in o he
con ex s (see e.g. Kim e al. (2002)).
The s uc u e o his pape is as ollows. In Sec ion 2 we
in oduce he ma hema ical model o endez ous space-
c a used o ajec o y planning, bo h in he PAM and
PWM case. We ollow in Sec ion 3 whe e we o mula e
he planning p oblem, de ailing he equali y and inequali y
cons ain s, and he cos unc ion. Sec ion 4 desc ibes he
me hod we p opose o sol e he planning p oblem. In
Sec ion 5 we show simula ions o he p oposed me hod.
We close he pape wi h some ema ks in Sec ion 6.
2. MODEL OF SPACECRAFT RENDEZVOUS
In his wo k we use he he linea Hill-Clohessy-Wil shi e
(HCW) equa ions, as in oduced in Hill (1878) and Clo-
P oceedings o he 18 h Wo ld Cong ess
The In e na ional Fede a ion o Au oma ic Con ol
Milano (I aly) Augus 28 - Sep embe 2, 2011
978-3-902661-93-7/11/$20.00 © 2011 IFAC 8473 10.3182/20110828-6-IT-1002.02445
hessy and Wil shi e (1960). Those desc ibe he ela i e
posi ion o he spacec a i he a ge is o bi ing in a
ci cula keple ian o bi and app oaching ehicle is close
enough o he a ge .
The HCW model also assumes ha he a ge ehicle is
passi e and mo ing along a ci cula o bi o adius R.
Thus he angula speed o he a ge h ough i s o bi
is n=pµ
R3, whe e µis he g a i a ion pa ame e o he
Ea h, µ= 398600.4 km3/s2.
Deno e by k he ime ins an = 0+kT , whe e Tis
an adequa ely chosen sampling ime ( he sampling ime
chosen in his wo k is 60 s). We o mula e wo e sions o
he HCW equa ions in disc e e ime. In he i s e sion,
he con ol inpu s a e cons an du ing a whole sampling
ime. This is e e ed o as he PAM disc e e model. In
he second e sion, i is assumed ha h us e s a e no
able o p oduce any o ce alue, bu can be only swi ched
on (p oducing he maximum o ce) o o (p oducing no
o ce), and only once du ing each sampling ime. This is
e e ed o as he PWM disc e e model.
2.1 PAM disc e e model
Assuming he con ol signal cons an h ough he sampling
ime, i can be shown ha
x(k+ 1) = ATx(k) + BTu(k).(1)
In (1), x(k), u(k) deno e espec i ely he s a e a ime k,
and he inpu ampli ude ( om ime k o k+ 1). Also,
x= [x y z ˙x˙y˙z]T,u= [uxuyuz]T.(2)
In hese de ini ions, x,y, and zdeno e he posi ion o he
chase in a local– e ical/local–ho izon al (LVLH) ame
o e e ence ixed on he cen e o g a i y o he a ge
ehicle. In he LVLH ame, x e e s o he adial posi ion,
y o he in- ack posi ion, and z o he c oss- ack posi ion.
The eloci y o he chase in he LVLH ame is gi en by
˙x, ˙y, and ˙z. The a iables ux,uy, and uza e he inpu s
( h us ac ua ion) ac ing on he chase ehicle, e e ed o
he LVLH axes as indica ed by hei espec i e subsc ip s.
The ma ices ATand BTappea ing in (1) a e gi en by
AT=












4−3C0 0 S
n
2(1 −C)
n0
6(S−nT) 1 0 −2(1 −C)
n
4S−3nT
n0
0 0 C0 0 S
n
3nS 0 0 C2S0
−6n(1 −C) 0 0 −2S4C−3 0
0 0 −nS 0 0 C












,(3)
BT=
















1−C
n2
2nT −2S
n20
2(S−nT)
n2−3T2
2+ 41−C
n20
0 0 1−C
n2
S
n21−C
n0
2(C−1)
n−3T+ 4S
n0
0 0 S
n
















,(4)
whe e S= sin nT and C= cos nT.
Compac o mula ion
Nex we de elop a compac o mula ion ha simpli ies he
no a ion o he p oblem. The s a e a ime k+ 1, gi en
he ini ial s a e a ime 0 (which is deno ed as x0) and
he inpu signals om ime 0 o ime k, is compu ed by
applying ecu si ely Equa ion (1):
x(k) = Akx0+
k−1
X
j=0
Ak−1−jBu(j).(5)
De ine now xSand uSas a s ack o Nps a es and inpu
signals, espec i ely, spanning om ime 1 o ime Np,
whe e Npis he planning ho izon:
xS=



x(1)
x(2)
.
.
.
x(Np)




,uS=



u(0)
u(1)
.
.
.
u(Np−1)




.
Then,
xS=












ATx0+BTu(0)
A2
Tx0+
1
X
j=0
A1−j
TBTu(j)
.
.
.
ANp
Tx0+
Np−1
X
j=0
ANp−1−j
TBTu(j)












,(6)
which can be w i en as
xS=Fx0+GuS,(7)
whe e Gis a block lowe iangula ma ix wi h i s non-
null elemen s de ined by (G)ij =Ai−j
TBTand he ma ix
Fis de ined as:
F=




AT
A2
T
.
.
.
ANp
T





.(8)
2.2 PWM disc e e o mula ion
Conside ing now he case o ON-OFF h us e s, i is
assumed ha he h us e s a e no able o p oduce any
alue o o ce, bu can be only swi ched on o o . Also,
i is assumed ha he e is an aligned pai o h us e s
o each di ec ion i= 1,2,3 wi h opposing o ien a ion.
To dis inguish be ween he posi i e and nega i e hey a e
deno ed as u+
iand u−
i, whe eas he espec i e maximum
h us is e e ed o as u+
imax and u−
imax , espec i ely.
Finally, du ing each sample ime each h us e is allowed
o i e only once.
Thus, he PWM ou pu o each ime in e al kis com-
ple ely desc ibed by wo new con ol a iables o each
pai o h us e s: he pulse wid h κ+
i(k) and he pulse
s a ime τ+
i(k) ( o he posi i ely o ien ed h us e in
he di ec ion i) and simila ly κ−
i(k) and τ−
i(k) o he
nega i ely o ien ed h us e in he di ec ion i. Then, o
∈[kT, (k+ 1)T], we ha e:
u+
i( ) = 


0, ∈kT, kT +τ+
i(k),
u+
imax , ∈kT +τ+
i(k), kT +τ+
i(k) + κ+
i(k),
0, ∈kT +τ+
i(k) + κ+
i(k),(k+ 1)T,
(9)
18 h IFAC Wo ld Cong ess (IFAC'11)
Milano (I aly) Augus 28 - Sep embe 2, 2011
8474
and simila ly o he nega i ely o ien ed h us e s. The
new a iables con ol a iables e i y κ+
i(k)>0, τ+
i(k)>
0 and τ+
i(k) + κ+
i(k)< T, and simila ly o he nega i ely
o ien ed h us e s. The las cons ain p e en he PWM
signal o spill o e o he nex ime in e al.
Call he PWM con ol a iables as uP(k):
uP(k) =



















τ+
1(k)
κ+
1(k)
τ−
1(k)
κ−
1(k)
τ+
2(k)
κ+
2(k)
τ−
2(k)
κ−
2(k)
τ+
3(k)
κ+
3(k)
τ−
3(k)
κ−
3(k)



















.(10)
To ind he HCW equa ions o PWM inpu s, de ine
b1
=













1−C
n2
2(S−n )
n2
0
S
n
2(C−1)
n
0













, b2
=













2n −2S
n2
−3 2
2+ 41−C
n2
0
21−C
n
−3 + 4S
n
0













,
b3
=










0
0
1−C
n2
0
0
S
n










,(11)
which a e he h ee columns o he ma ix B in (4), bu
de ined o di e en .
Then he sys em e olu ion equa ion (1) is eplaced in he
PWM case by
x(k+ 1) = Ax(k) + BPWM(uP(k))umax,(12)
whe e
BPWM(uP(k)) =











AT−τ+
1(k)−κ+
1(k)b1
κ+
1(k)
AT−τ−
1(k)−κ−
1(k)b1
κ−
1(k)
AT−τ+
2(k)−κ+
2(k)b2
κ+
2(k)
AT−τ−
2(k)−κ−
2(k)b2
κ−
2(k)
AT−τ+
3(k)−κ+
3(k)b3
κ+
3(k)
AT−τ−
3(k)−κ−
3(k)b3
κ−
3(k)











T
,(13)
umax =








u+
1max
−u−
1max
u+
2max
−u−
2max
u+
3max
−u−
3max








.(14)
No ice ha sys em (12) is nonlinea in he PWM a iables
uP(k).
Compac o mula ion
The compac o mula ion de eloped be o e can be eadily
adap ed o PWM inpu s. Equa ion (7) is now w i en as
xS=Fx0+GPWM(uPS)umaxS,(15)
whe e uPSis a s ack ec o wi h all he PWM sig-
nals, GPWM is a block lowe iangula ma ix wi h
i s non-null elemen s de ined by (GPWM(uPS))ij =
Ai−jBPWM(uP(j−1)), and he cons an ec o umaxS
is de ined as:
umaxS=



umax
umax
.
.
.
umax











N imes.(16)
3. FORMULATION OF THE PLANNING PROBLEM
Nex we o mula e ou planning p oblem, in oducing
he cons ain s (bo h equali y- and inequali y- ype con-
s ain s) and he objec i e unc ion ha has o be mini-
mized. The o mula ion is done o bo h PAM and PWM
con ol signals.
3.1 Cons ain s on he p oblem
Inequali y cons ain s on he s a e
Fo sensing pu poses (see B ege and How (2008)), du ing
endez ous i is equi ed ha he chase ehicle emains
inside a line o sigh (LOS) a ea. To simpli y he cons ain ,
we conside a 2-D LOS a ea as shown in Figu e 1. This
LOS egion is he in e sec ion o a cone, gi en by he
equa ions y≥cLOS(x−x0) and y≥ −cLOS(x+x0), and
he egion y≥0.
Fig. 1. Line o Sigh egion.
The LOS cons ain is ALOS x(k)≤bLOS, whe e
ALOS ="0−10000
cLOS −10000
−cLOS −10000#, bLOS ="0
cLOSx0
cLOSx0#.
(17)
Using he compac o mula ion ha was de eloped in
Sec ion 2.1, he cons ain s equa ions o he s a e can
be ew i en as:
AcxS≤bc,(18)
18 h IFAC Wo ld Cong ess (IFAC'11)
Milano (I aly) Augus 28 - Sep embe 2, 2011
8475
whe e Acand bca e gi en by:
Ac=



ALOS
ALOS
...
ALOS




,bc=



bLOS
bLOS
.
.
.
bLOS




.(19)
Then, o he case o PAM con ol and using equa ion (7),
one can e o mula e he LOS cons ain s as cons ain s o
he con ol signals in he ollowing way:
AcGuS≤bc−AcFx0,(20)
and simila ly o he case o PWM con ol.
Equali y cons ain s on he s a e
Equali y cons ain s a e o mula ed o ensu e ha he
chase spacec a a i es a he o igin wi h ze o eloci y a
he end o he planning ho izon. Thus, hese cons ain s
can be w i en as x(Np) = 0. De ining Aeq as
Aeq =



0
...
0
Id6×6




,(21)
he equali y cons ain can be w i en as:
AeqxS=0.(22)
Then, o he case o PAM con ol and using equa ion
(7), one can e o mula e he a i al condi ions as equali y
cons ain s o he con ol signals in he ollowing way:
AeqGuS=−AeqFx0,(23)
and simila ly o he case o PWM con ol.
Inpu cons ain s
Fo he case o PAM con ol, one would ha e limi a ions
on he magni ude o he con ol, gi en by
umin ≤uS≤umax.(24)
Fo he case o PWM con ol, he cons ain s a e gi en by
κ±
i(k)>0, τ±
i(k)>0 and τ±
i(k) + κ±
i(k)< T.
3.2 Objec i e unc ion
The objec i e unc ion o be minimized in he planning
p oblem is he 1-no m o he con ol signal, which is
p opo ional o uel consump ion.
PAM con ol inpu s
Fo he case o PAM con ol inpu s, he selec ed con ol
unc ion is gi en by:
JP AM =
Np−1
X
k=0
kuT(k)k1=kuSk1.(25)
PWM con ol inpu s
Fo he case o PAM con ol inpu s, using (9) i can be
seen ha he selec ed objec i e unc ion is gi en by:
JPWM =
Np−1
X
k=0
3
X
i=1 u+
imax κ+
i(k) + u−
imax κ−
i(k).(26)
4. COMPUTATION OF THE OPTIMAL CONTROL
INPUT
As shown in Sec ion 2.2, he disc e e HWC equa ions in
he PWM case a e nonlinea in he swi ching imes. Thus,
i is nei he easy no as o sol e an op imiza ion p oblem
using he swi ching imes as con ol a iables.
To ind he op imal con ol inpu wi hou needing o sol e
a nonlinea planning p oblem, he ollowing scheme is
p oposed:
S ep 1. A PAM linea op imiza ion p oblem is sol ed.
S ep 2. A PAM/PWM il e is used o con e he PAM
inpu signals o PWM signals (i.e., swi ching imes) ha
p oduce a e y simila sys em ou pu .
S ep 3. The plan equa ions, in he PWM o mula ion,
a e linea ized a ound he p e ious s ep solu ion, hus
ob aining a linea plan wi h espec o he swi ching
ime. A linea op imiza ion p oblem is hen posed and
sol ed. The esul ing solu ion is aken as a be e
app oxima ion owa ds he eal solu ion.
S ep 4. Repea he linea iza ion p ocess o S ep 3
a ound he new solu ion. Op imize again o ind a be e
e inemen . The p ocess is i e a ed un il he solu ion
con e ges.
Nex , we desc ibe all he s ep in ou scheme.
4.1 Compu a ion o PAM con ol inpu
To compu e he op imal con ol plan (wi h PAM con ol
signals), one seeks he con ol signal ha minimizes he
cos unc ion o e he planning ho izon, sa is ying a he
same ime he PAM cons ain s:
min
uS
JP AM (uS) (27)
subjec o: AcGuS≤bc−AcFx0
AeqGuS=−AeqFx0
umin ≤uS≤umax.
Since he cos unc ion and he cons ain s a e linea , hen
(27) can be eadily sol ed.
4.2 Ini ial PWM solu ion: A PAM/PWM il e
The PAM solu ion ound when sol ing (27) is ans o med
o an equi alen PWM solu ion using a PAM/PWM il e .
This il e is o mula ed in he li e a u e ( o ins ance
in Shieh e al. (1996); Ieko e al. (1999)) whe e se e al
me hods a e p oposed. These me hods allow o, gi en he
PAM inpu s o a sys em, compu e equi alen PWM inpu s
ha p oduce a sys em ou pu op imally app oxima ing he
ou pu o he sys em when d i en by he PAM signals.
Following hese e e ences, a PAM con ol signal can be
op imally app oxima ed by a PWM con ol signal by using
he ollowing ules o each ime ins an kand di ec ion i:
(1) Use he posi i e o nega i e h us e acco ding o he
sign o he PAM signal ui(k).
(2) The pulse wid h mus be compu ed using he P in-
ciple o Equi alen A eas: κ±
i(k) = T|ui(k)|
u±
imax
, whe e
u±
imax is he maximum le el o he (posi i e o nega-
i e) h us e i.
18 h IFAC Wo ld Cong ess (IFAC'11)
Milano (I aly) Augus 28 - Sep embe 2, 2011
8476
(3) Since only one impulse pe sample ime is conside ed,
i mus be alloca ed a he cen e o he sampling
in e al, ha is τ±
i(k) = 1
2(T−κ±
i(k)).
The PWM signals uP(k) cons uc ed by his me hod
p oduce an almos iden ical ou pu o he sys em d i en
by PAM signals. Howe e , he PWM a e no necessa ily
op imal since hei cons ain s a e qui e di e en ; in ac
hey migh e en no e i y he cons ain s. Thus, his so-
lu ion is only used as an ini ializa ion o he op imiza ion
algo i hm p oposed nex .
4.3 Re ined PWM solu ion: An op imiza ion algo i hm
No ing ha mos nonlinea i ies appea ing in he sys em
equa ions (12) a e o he o m cos n o sin n , whe e n
is he o bi al angula eloci y, and no ing n 1, we
app oxima e (12) by linea izing BPWM(uP(k)) a ound
uP(k). Then, he sys em equa ions a e:
x(k+1) = Ax(k)+BPWM(uP(k))umax+B∆(uP(k))∆(k),
(28)
whe e
B∆=





























−A0
T−τ+
1−κ+
1
b1
κ+
1
u+
1max
−A0
T−τ+
1−κ+
1
b1
κ+
1
+AT−τ+
1−κ+
1b10
κ+
1u+
1max
A0
T−τ−
1−κ−
1
b1
κ−
1
u−
1max
A0
T−τ−
1−κ−
1
b1
κ−
1
−AT−τ−
1−κ−
1b10
κ−
1u−
1max
−A0
T−τ+
2−κ+
2
b2
κ+
2
u+
2max
−A0
T−τ+
2−κ+
2
b2
κ+
2
+AT−τ+
2−κ+
2b20
κ+
2u+
2max
A0
T−τ−
2−κ−
2
b2
κ−
2
u−
2max
A0
T−τ−
2−κ−
2
b2
κ−
2
−AT−τ−
2−κ−
2b20
κ−
2u−
2max
−A0
T−τ+
3−κ+
3
b3
κ+
3
u+
3max
−A0
T−τ+
3−κ+
3
b3
κ+
3
+AT−τ+
3−κ+
3b30
κ+
3u+
3max
A0
T−τ−
3−κ−
3
b3
κ−
3
u−
3max
A0
T−τ−
3−κ−
3
b3
κ−
3
−AT−τ−
3−κ−
3b30
κ−
3u−
3max





























T
,(29)
and whe e he alue o κand τ a iables a ime kis
used. The a iable ∆(k) ep esen s he inc emen s o
dec emen s wi h espec o uP(k):
∆(k)=



















∆τ+
1(k)
∆κ+
1(k)
∆τ−
1(k)
∆κ−
1(k)
∆τ+
2(k)
∆κ+
2(k)
∆τ−
2(k)
∆κ−
2(k)
∆τ+
3(k)
∆κ+
3(k)
∆τ−
3(k)
∆κ−
3(k)



















.(30)
The ma ices used in (29) a e de ined as:
A0
=






3nS 0 0 C2S0
6n(C−1) 0 0 −2S4C−3 0
0 0 −nS 0 0 C
3n2C0 0 −nS 2nC 0
−6n2S0 0 −2nC −4nS 0
0 0 −n2C0 0 −nS







,(31)
and
b10
=










S
n
2(C−1)
n
0
C
−2S
0










, b20
=










2−2C
n
−3 + 4S
n
0
2S
−3+4C
0










, b30
=








0
0
S
n
0
0
C








.(32)
Equa ion (33) is now compac ly w i en as
xS=Fx0+G∆(uPS)∆S+GPWM(uPS)umaxS,(33)
whe e G∆(uPS) is a block lowe iangula ma ix
wi h i s non-null elemen s de ined by (G∆(uPS))ij =
Ai−jB∆(uP(Np−j−1)) and ∆Sis a s ack ec o o he
inc emen in he PWM a iables ∆(k). The LOS inequal-
i y cons ain s (20) can be e o mula ed as cons ain s o
he PWM a iables in he ollowing way:
AcG∆∆S≤bc−AcFx0−AcGPWMumaxS,(34)
whe e he dependence o GPWM(uPS) and G∆(uPS) on
uPShas been omi ed o simplici y. Simila ly, he equali y
cons ain s become:
AeqG∆∆S=−AeqFx0−AeqGPWMumaxS.(35)
The cons ain s on he ∆(k) a e as ollows:
−∆κ±
i(k)≤κ±
i(k),−∆τ±
i(k)≤τ±
i(k) (36)
∆τ±
i(k)+∆κ±
i(k)≤T−τ±
i(k)−κ±
i(k),(37)
|∆(k)| ≤ ∆MAX ,(38)
whe e (38) is used o a oid oo la ge a ia ions in each
i e a ion s ep. These cons ain s can be summa ized as
A∆(k)∆S(k)≤b∆(k).(39)
Finally, he objec i e unc ion can be ew i en as a
unc ion o he PWM a iables and hei inc emen s as
J(uPS,∆S) = JPWM(uPS) + J∆(∆S), whe e
J∆(∆S) =
Np−1
X
k=0
3
X
i=1 u+
imax ∆κ+
i(k) + u−
imax ∆κ−
i(k).(40)
Thus, a linea p og amming p oblem wi h PWM ou pu s
can be posed as ollows:
min
∆S
J∆(∆S) (41)
s. .: AcG∆∆S≤bc−AcFx0−AcGPWMumaxS,
AeqG∆∆S=−AeqFx0−AeqGPWMumaxS,
A∆∆S≤b∆.
The solu ion o (41), ∆S, can be used o ecompu e new
PWM a iables:
uNEW
PS=uPS+∆S.(42)
Then, he alues o uNEW
PSis used o ecompu e he
a ious ma ices appea ing in (41), and he op imiza ion
p oblem is sol ed again. I e a ing, i is expec ed ha he
solu ion keeps imp o ing un il a solu ion close o he global
op imum is eached. Nex , his is shown in simula ions.
5. SIMULATION RESULTS
We nex show simula ions o he ajec o y planning
p oblem o spacec a endez ous using ou algo i hm.
18 h IFAC Wo ld Cong ess (IFAC'11)
Milano (I aly) Augus 28 - Sep embe 2, 2011
8477

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x (km)
y (km)
PWM
PAM
Fig. 2. Solu ion ajec o y, showing he ini ial ajec o y
wi h PAM inpu s (dashed) and he inal ajec o y
wi h PWM inpu s (solid).
! " # $ % &
%'!(
%'"
%'""
%'"$
%'"&
%'"(
%'#
%'#"
%'#$)*!+!#
,-./01*23*45016542,7
!"#$%&'(
Fig. 3. Value o objec i e unc ion depending on i e a ion
numbe .
! "!! #!!! #"!! $!!! $"!! %!!!
!&
!$
!
$'(#!!"
)*+,
-'*./+$,
! "!! #!!! #"!! $!!! $"!! %!!!
!"
!
"'(#!!"
)*+,
-0*./+$,
Fig. 4. Ini ial PAM con ol signal.
The planning ho izon Npwas chosen as 50. The alue o
u±
imax was chosen as 10−4km/s2. Ini ials condi ions we e
0= [0.2 0.6 0.2]Tkm, 0= [0.0015 0.002 −0.001]Tkm/s.
The solu ion ajec o y is shown in Fig. 2 (solid). The
PAM solu ion ha was used o ini ialize he algo i hm
is shown in he same igu e (dashed). In his and o he
igu es only he xand ycoo dina es a e shown o he
sake o b e i y. As i can be seen in Fig. 3, he algo i hm
con e ged a e 6 i e a ions o a solu ion wi h a cos
unc ion app oxima ely 2.5% smalle . Each i e a ion ook
less han a second on a con en ional compu e , using
MATLAB’s linp og o calcula e he solu ion. Compa ing
he ini ial PAM (Fig. 4) and PWM (Fig. 5) con ol
signals, i can be seen ha o bo h he con ol e o is
concen a ed a he beginning, wi h some mid-cou se and
inal maneu e s ha a e di e en o bo h cases.
! "!! #!!! #"!! $!!! $"!! %!!!
!$
!#
!
#
$&'#!!(
)*+,
-&*./+$,
! "!! #!!! #"!! $!!! $"!! %!!!
!$
!
$&'#!!(
)*+,
-0*./+$,
Fig. 5. Final PWM con ol signals.
6. CONCLUDING REMARKS
We ha e p esen ed an algo i hm o compu e he op imal
PWM con ol signals applied o endez ous o spacec a .
The algo i hm uses he HCW model wi h LOS and equal-
i y cons ain s, bu can be gene alized o mo e complica ed
models and cons ain s. Since he algo i hm is e y as ,
i could be used online o implemen an MPC scheme
(e.g. Ga ilan e al. (2009)) including PWM con ol signals.
REFERENCES
Asawa, S., Nagashio, T., and Kida, T. (2006). Fo ma ion
ligh o spacec a in ea h o bi ia MPC. In SICE-
ICASE In e na ional Join Con e ence.
B ege , L. and How, J.P. (2008). Sa e ajec o ies o
au onomous endez ous o spacec a . J Guid Con
Dynam., 31(5), 1478–1489.
Clohessy, W.H. and Wil shi e, R.S. (1960). Te minal
guidance sys ems o sa elli e endez ous. Jou nal o
he Ae ospace Sciences, 27(9), 653–658.
Fehse, W. (2003). Au oma ed Rendez ous and Docking o
Spacec a . Camb idge Uni e si y P ess.
Ga ilan, F., Vazquez, R., and Camacho, E.R. (2009). Ro-
bus model p edic i e con ol o spacec a endez ous
wi h online p edic ion o dis u bance bound. In P o-
ceedings o AGNFCS’09, Sama a, Russia,.
Gelle , D. (2006). Linea co a iance echniques o o bi al
endez ous analysis and au onomous onboa d mission
planning. J Guid Con Dynam., 29(6), 1404–1414.
Hill, G. (1878). Resea ches in luna heo y. Ame ican
Jou nal o Ma hema ics, 1(3), 5–26, 129–147, 245–260.
Ieko, T., Ochi, Y., and Kanai, K. (1999). New design
me hod o pulse-wid h modula ion con ol sys ems ia
digi al edesign. J Guid Con Dynam., 22(1), 123–128.
Kim, H.J., Shim, D.H., and Sas y, S. (2002). Nonlinea
model p edic i e acking con ol o o o c a -based
unmaned ae ial ehicles. In P oceedings o ACC 2002.
Richa ds, A.G. and How, J. (2003). Pe o mance e alua-
ion o endez ous using model p edic i e con ol. AIAA
Pape 2003-5507.
Rossi, M. and Lo e a, M. (2002). A mul i a e p edic i e
app oach o o bi con ol o small spacec a . In P o-
ceedings o ACC 2002.
Shieh, L.S., Wang, W.M., and Sunkel, J. (1996). Design
o PAM and PWM con olle s o sampled-da a in e al
sys ems. J Dyn Sys Meas Con ., 118(4), 673–681.
Tong, C., Shijie, X., and Songxia, W. (2007). Rela i e
mo ion con ol o au onomous endez ous based on
classical o bi elemen di e ences. J Guid Con Dy-
nam., 30(4), 1003–1014.
Wie, B. (1998). Space ehicle dynamics and con ol. AIAA.
Wo inden, D.C. and Gelle , D.K. (2008). Na iga ing he
oad o au onomous o bi al endez ous. Jou nal o
Spacec a and Rocke s, 44(4), 898–909.
18 h IFAC Wo ld Cong ess (IFAC'11)
Milano (I aly) Augus 28 - Sep embe 2, 2011
8478