Kinematic Nonholonomic Optimal Control: Skate Example

TP-01 3:00
TP07-1
3:50
Proc of the 36th IEEE CDC, San Dirgo, CA
Kinematic Nonholonomic Optimal Control: The Skate Example
S. Akileswar ~IICI.J. Baillisul*
of Aerospace and hleclm~id
Engineering,
Boston University:
Boston, MA! 02215.
loridl~~cngct.bu.edii and ,johnl>~~~(~ng;i.bu.c!clu
Department.
Abstract
bet\v~~cn cont.rol thwry urtcl ttonholottotnic
rnec*hattic*s ittclutlc! [5]. [O]? [lOI [8]. [13], [15]. and [1G].
Thr tnotiott of a skate ott ii plant! is OIIC of the sitnplc!s~.
T11e prc!Setlt. IGlIJel’ treats the problcnl
of ophItl;,tl killckiticxtatic rtonltolt~ttt~tttic systwts ant1 yet the problcttt of titntic (~d0t:it.y) Wtttrd of ill1 Wcdizd~
&to.
I3ccausc
rlcttwttittirtg
optitttal tttotit~tts erhi1,~it.s many intportattt.
the sgstcxrt is of low dittlensiorl
and txhihits
sotltt~ ratller
f(!iltWTS cotrttttott to il hr0;itl ClilSS of kintwtat~ic optinial
Spedid Structttrct, the ttt~ttlinear optimal control probletn
tY~rltr~Ji prol~lt!llls.
Altl1ough
thcsc prthlelns
for the skw!
cart be solwtl fairly explicitly.
Itttleed, tlic solutions we
lt;lw lwcn stttdieil in the literature.
tllc prfwtlt,
pilpC!r
filltl dlO\V IIS t0 Stlld~ t.Ile gC!OlIletr~
of Optinlitl triljcct,orit!s
t.akes the analysis to a rtew lcvcl of dctail. This article
in t:onsitlerable ddail. Ott the one hand, thr work tttitkes
tlcscritxs tltc attalytic struct.urc and ge0rrtetx-y of tltc op- (*ontitct. with rcwttt, rrsearch 011 tlte Sttb-R,icntanttii~n getirttal controls for point to point. kittcntatic cotttrol of the
0tncWy of R:’ (as described in [l] and [G]). -4s mtd
in
Skate. It. is sltonn t1titT.there are four types Of Opt~irllid txlsome of the work cit,ccl above. solutions to the optimal
jwt.orit?s for the skate. These are cltaractrrizcd
in terms of kinetnnt.ic wntrol problettt also provide an itppdittg
npil ~~ilrilIllct~‘ri~il~~i~~ll obtaitircl
by careful use of invariants
prodi
to ttiot,ioii-planttitig
for mobile robots whose tttoof tltc 0ptitrt;tl trtotiott. Xtitnerical cotttpitt.atiotts Wtihlish
t.ions art: subject to wlocity constraints.
For syst.ents itt
ltcy qunlitntiw
f(:iltureS
of 0ptiInal lIlotiOtIs such iLs the gc- which explicit solutions t,o the optimal corttrol problem
otnet~ry of gcotlwic ncighl~orltoods
illld cottjugtt~t? p0ittt.s.
are not easily obtaind,
we resort to carefully posed subWe briefly int1it.at.t~ how t,hosc rrsults may be extended
OptiIniil
control
problems.
(See.
[4] for work along these
and applictl to ot lwr proldcnts in thr Optirtlill ~oritrol of littcs.)
kittctttatic ttc-)ttltoloxtottlic nlechnrtical systctrts.
Cortsitlw it skate frep to mow with it,s knife edge 011 a.
flnt ~~1iUlC. With t.he asstttttptit~tt that the skittc cannot
Introduction
slitlc in any direction
trilllSV(!rSC
to it,s orieIltittioIl.
the
skatJe is an esatiiplr of a simple noriltolottottiit~ c-l~nxrtical
The fttttdittttf!rtt,;tl research issues in nortlirttw
cottt,rol
systtwt. This cxantplc has lawn st,ttdiccl by a ttttrttlwr of
tltoor!. itllC1 nonliolt~ttt~tnic ntcchattics xc Cl0Sdy r&ttcd
authors. indiitlittg [14].
tq. the uliifyitlg
t.lleIrtcl of no7li7lt~~~clbilitll.
Control tlieThis SyStettt hils tlvo lt~~11 C~C~WCSof frdottt.
The
ory is t~oncc~rtlcd with
prorlucitig
tri\~jectOriPS
using
collskiitc can 1ii0vc forwartl iii tlte plane a~otig it,s oricmbatiott
trolled t.lifLrtwt.iitl equatiotts in n-hich we tnodul;itt~ the
tlttd it, can &(j change it,s oriettt.i~tiott.
The cottfigutxniiqytit~tidc of wctor fields belongirtg to it nonintegrable
tiott
sput:t~
for
die
skate,
however.
is
a
three
c1itttt~nsiona.l
mecharlics
(= t~otilti~
. 0 1u t.’lvt: ) c1is.t. ri 1xl t.ion. Nt~rll~olOnoniic
xnanifdd, wit,11 natAural cottfigttratiott
variables being t.he
is cc)ttccrtttxl wit.h fintling tlte set. of possible tnotions for
oric~ntatiott of the skate mtl the position of its ccttter of
il s\-stctn sitl),jrct to ttottintegrablc
velocity t:otistraints.
rttass. Such a system catt also lx studied on SE(Z) as has
Sotiixttt!gral,ilit~
iii this contest means t.hat the velocIxen done in [il.
ity coIlstrilitlt.5
trrtttttot, bc itttegraterl to produce cqrtivitht.
cOttStr;titlt,s
011 the ctdigllrat~iolt
variables.
NotlinProblem Formulation
tegrablr wlwity
cottst~rairtts arise in ntechanicnl
syst,ems
Let, tdto ccttter of the 3kit.W t>c loc:~td at. {.r, J/} with
sitbjoc~t. t.0 rollittg cottstraittts (~~heclecl vehicles) or sliding
respect
to an ittertial
frame of reference fisetl to tltc plarte
c~onsttxitits,
Sut*h as the skate problcnt consiclcretl below.
Ott wlticlt t.hr skate rests Let, H lw tlw oriwttatiott
angla
Rw:ottt cotltril~utiorls
rwognizing
the wntlwtiorl
I.~c~twct~ll
0f the! skate wit,!1 respect.
to the positive
!/ axis.
Thus the:
“l’hc a~llllor woc~ld like to txpwss gwlitutlv
for suplwrt. from 1k
~:on~gllr~lt,ion
coortliIlates
of the sbltc Gltl be writtctt
as
Iltlitc!tl
Slates
<\ir I~OIW Office! of Scic~~l.ific
Rwt~;~rt:l~ under gr;trlt
I: l!lfiZO-W
(4
1 -OO.i!J
Y, 0).
1
0-7803-4187-2
1443-1448
0000
Dec 1997
TP-01 3:00
{xoJ”,H”} = {O,O,O}. Iu o t e 1lowever that there is a
configuration
Space compatibility
couditiou in ivllich lye
adopt, the convention that when the skate orientat,iou has
H = 0. it is alignecl \vith the positive y-axis. Also Observe
that. the resulting y axis is a possible trajectory
Of the
skate whereas the x axis is not.. Thus the shortest. pat.hs
to final destinations
on the y axis are qualit,at,ircly
very
different from those to fiual dest,iuations On the s x-5.5.
Using Pontryagin’s
maximum principle [12], the equatians goveruing the solutions of problem 1 can lx derived.
The costate vector is three dimensional
in this case. The
nwdificcl Ha~niltonian
H,, can be written as
x
H,,, = u’T)+ 16: + XI 111sin B + Xp~l cos 8 + X:~UZ.
Figure 1: A skate 011 a plane
The quabions
lw written
governing
as
Cilll
The! no-sliding
(cosH
const,rGnt
-siuB
The lcincnlatic
as
0)
t
0 ri
for the sliat,c~ Ci\Il be given as
XI
A.,
0 A,
(1)
= :icosH - !jsinO = 0.
=
and the optiuial
ecpat,iOns for the skate car1 now be writt,en
find
t7YZjeCtfJ7"' r
X1 sin 19+ X.Lcos 0
2
’
(4)
as
(5)
x3
2
TVe can use these equations to give us trajectories
which
are stationary
points in the space of admissible paths for
the cost funct.ioual.
In this paper: we analyze equat.ious
2! 4: 5 aud esplain t,he struct,ure of the resulting optimal
trajectories
of t,lie liinernatk
skate.
Results
as
skote~
vt!ctor field
can be cqxwsecl
(2)
141(t) is the forward velocity of the skate in its hCildillg
clircct~ion and 712(t) is the angular vclocit,y Of t.he skate
about.
its cent~w of mass. These cau bc considered as controls mtl we cm f0rIriulatc the optimal control plolJlerr1
Problem 1 For the kinematic
II,
(t)?
‘11.J (t)
S~U~!l~.
that
th! 7wu.1ti71g
t11c cnst ~UlM2i071
functions
112= --,
6) =,Ll(~~~)+l‘2(H)= (::;z).
costate
0
0
i ul (X2 sin B - X Lcos B) )*
control
IL1 = -
the optinial
(3)
co7Ltrols
rrri7Limizes
Lemma 1 For un optid
ll&)’
trajectory
of the skate,
+ 1L2(g2 = w7rstn7~t.
(6)
(f)'
+ug(t)'Ldt
J1Ul
und
at
j0.in.s
POhtS
time
t
(2":
y//o,a,>
ut ti7ne t =
0
11nd
{x1
! y1
( 81
Proof A str;ligIit,fo~~~~~~rtlverification
can be achieved by
differentiating
t.lle left. hand side of equation G iIllt1 using
equations 1 and 5.0
If we call t.hc corresponding
s&e trajectory
lY, then
we cau d&w CT: such that. 7~: + 7~: = C:. As the costate
space is throc dimensional,
we riced oue more conserved
quantity in t,hc: control space to be ahle to integrat.e this
optimal control pr0Llem.
}
= 1.
Note 1 It shouldbe noted that the trajectories that solve
the cLbone problt~rn are ide7l.k:al to the “v&L7Lo7r~,ic”trcljectorks ([14/. [3]) of the skate. These var~ntio~d trrLjecto7Yc!sdo 7Lot rnotlel the dyntlmics of the skate. The r11/7m7r~its of the .shte are ~rrtwdeledco7,rectly by the d’cllembf~rt
Lngrwge eqwltions which take into account the nonholo7107niC C07l~.St7~Ui7l.t
a7Ld its f$eCt 011 the dy7lMl7l~~c~. The
(1‘A41e7nbe7,tLagrunge equatims are esse7~tinlly fowe bnlrace equations but not ~tm7*iatio~rrcLl
i71 ~natrrr~f~.
Lemma 2 For ax optimal trajectory
uJt)'
ill(t)"
=
+ -
of the skate,
co71Stur1t.
u.z(t)’
Proof Notice that both X1 a.nd X2 are constant
as per
equation 4. He~ice A? + AZ = constant. Now using equatious 4 aud 5, we can verify that
ill(t)’
xf + /q = ILL(t)’ + -.
IId
As we are free to choose the origin of our coordiuate q3tern:
without loss Of gruera1it.y we can assuule
2
TP-01 3:00
Thus rho lolnn~a is proved.
Let, r<; I-,(! defined
0
by q(f)”
+
s
=
Thew conservation
laws show that in fact, (I~(t), I(?( t)
arc .Jacohi elliptic functions
a.nd satisfy (>scillat,or equations for quartic potfwtial functions.
Using equations
6
and 7; ma can prow the following relations
(CF + ZC~)*l.l(t)’ - 1Ll(t)” + ill(f)” = Ii$C;,
ti.Jf)2 + u.‘(t)” + (A-F - 2C’;t)2/,2(t)2 = c;(I(-;
..
i: 4 &se
Type 1 I<r = 0,
l
Type 2 I<‘p= & ,
l
Type 3 I<r > Cr.,
l
Type4Kr<Cr.
In trms of the phase portrait in figure 2. t,he four classes
corrwpontl
to the non-liypwholic:
cquililwimn
(Type 1).
the l~yp~~rt~olic equilibrium
with its stable and unstable
manifolds (Tylx? 2): c10sed orbits (Type 1) and running
mode orb
(Type 3). Qualitat.ive differences in 4 orbits
lead to qualitat.ive
differences in the skat.e t,rajectories.
Type 1 paths art! pure rotational
skate trajectories.
Type
2 pat.hs are asymptotic
skate brajectories
(See figure 3).
Type 3 paths art’ zigzag paths as seen in figure 4 and type
4 IJilthS arc t~lunl~ling paths as swn in figure 5. Sote that.
i1S tllc! I)llaSc~plot esist.s On tllC txlgfmt
bundle of S’ , l>oth
t,yIw 3 ilIlt t!pc
4 skate trajectories
xix from periodic (b
0rl)it.s. Iii ot.lwr wxtls? the mining
inodc ci orbits illX! also
periodic. ml closrd in TS’ . The plots illustrate tlw qualitati\vct diftcrcrciiws Iwtweeii the mrious skate trajectories.
- C;t).
The solutions to these equstions arc indeed t.he Jacobi
elliptic fiinc*t,ions [ll]. We shall espress the optimal coiitrol fuiictims
as functions
on a circle which is a more
compact rc:[)reseiit~at.ic,ii, and iiivolves noting t,he geornetric content. of the first hnma.
We can use bho coortlinnt,c~s
oii this circle to describe t,he optimal input functions.
Y’
.I
Figure
l
A-;.
I
portrait
for a. skate ’
Lernrna 3 The optirml irqmtfmctio~~~.s
07’f!fyk71
by
1~1(f) = Cr sin Q(t).
WI(t) = Cr coscb(f):
Figure
ullLc?rid(k) .wLtisfir~s
$(t)” + Ct sin’ d(t) = KF.
Cr = Icr
Not all of tlwse t~rajcctorks
that satisfy the first order
Trajectories
of some
nccwsary conditions
are OptilIlill.
typos RI‘C always optimal whereas those of otlicr types
liavc (wijugatc~ points. Iii this case, the nuincrical search
for StiltiOIl;lY~ trajc~t.ories
coniiec*tiiig two p0iut.s will be
coutluctod for each of the types scparatcly.
In part,ic:nlar,
if a soliition is foulid iii a class of triLjcct0lk
1~110u~n t.0
tw always
opt~illlnl
m-t! iicwl not starch
the other classes.
Iii gmcral cusps are observed in the skate traject,ories
when projcwtod to the sy plant?. Trajectories
of first type
nwcr
havr auy cusps.
Trajectories
of the second type
show a Inasilllunl
of one cusp. Trajectories
of the fourth
and third type show a repeating
seywnce of cusps. but
optimal trajectories
of these types have no more than one
or two c~lsps rcqx~ctiwly
(3
P7mf A straiglitforwird
vcrificatimi
cm he done ilsiilg
lemmas 1 arid 2.0
Figure 2 shorn the resulting phase portrait. for ri, for
Cr = 1. .Thc cliffmwt, kiutls of orbits in this phase plot
Ski1t.f trajectories.
corrrspontl
to qui1lit,;ltiv~!ly diffwwt
The following proposit.iom
enumcratc
and &scribe
the
vilriorls
0ptiIli;ll
skate trajcct~ories.
C%issificwtion
3: A Type 2 trnjrctory.
of Paths
Tlmc arv follr tliffcwnt classes of pa.tlls that sat.isfy the
first, o&r
~icwss;wy c:oiitlitioiis.
They can classified as
fObVS.
3
TP-01 3:00
Proposition 2 Givm the positions of the sknte ot initial
the t = 0 wd jknl time t = 1: if it can he con7l,ectect
1);~
11 tryjfaYry
only
that
has
motion
linear
on&
a7rgula~
motio~l
(j:
=
ri
((3 = 0). then this trajectory
=
0)
07
is alruays
optitnol.
Proof The proof is c.1rar.U
Proposition 3 Whm Cr = K r : the h2jectovy
asymptotdcctlly tds
towu~ds II strui~~ht line trajectory.
Tlw angle mule 6y the qppfote
to the positive y nz%s is poporI
tiord to cp(0) - @(Cc). VJLere C++(m) = )$A qqb) = fir/2.
Y
Figure!
I
4: A
Type
3 trajectory,
Cl-
<
PlYJOfSee[z] for the proof.0
#I.
It. is clear that t,ype 2 trajject.orics never meet any trajcctories of the first type. Although it was riot proved
analytically,
such trajectories
are cheaper than any type>
3 or 4 tlajcctories
~onnrctiug
the same two points. This
was chcckctl uumorically
iind is clear from the geonietare curvcyl
ric OiXX~lTYlt~ioll that types 3 and 4 trajectories
more thi\u type 2 trajcmories.
Henw they are optimal t,ill
they Iuec% another type 2 trajectory.
skate i‘ti n P:nr.r
Proposition 4 Periodic
tl:O*ll.
uvN~
& 0r~bit.slead to skate
cOnfi,y~~m-
4 in UJhkh
ti
i’i
~lf!l’l.Odif:
the same funflf~me1rtcr.1
pwiorl (sag y). We note that
tl’fljUtOl?.f~.~
Of
&]‘tX
.“3 llld
~(7i.p + s) - :c(np - p + s) = Cr sin d(0)
y( HI' + S) Figure 5: A TJ~V 4 traject,ory,Cr > Kr
-I-’
+
S)
for all such periodic 6, tmjectories.
P7wf See [2] for the pro0f.O
Having classifitxl tllc different trajectories
that satisfy
t.lie first order nec*essary conditions for optirnality, equivid~ric~ rdations
for pi, orbits corresponding
to tllr smie
IXlthSof tllo kiricnlatic Sk&c! Can be tlc95vcd.
Proposition
T/(np
Definition 1 Th.a slope constant for, cm optirnol
tory of the kin.crrzutic
.skute is defined ILLS 9.
trujcc-
1 Two r/ orl~its, rl, II’?, we mch thnt
I
c;, = (Jqq.
I\-;‘,= aqy--;,,
Cr,(0)= dT.*(OL
(8)
5 .
3
.
if and only if the correspondingpaths J?,, r2 of the &herrutic skntc we ~~-2Ja7nnretel-lzntrons
of each other. 171,
other ~OI&
kmemutic
the physical paths arc the sumc alth.ough the
sktctc mnygtuwerse the puth at difjt7en.trntes.
Proof SW [2] for the proof.0
Corollary
the
1 hJov%n,y slower along u tmjrdo7y
cost incw7ed in reaching IIpoint.
Pwof A fairly
straightforward
vcrifkation
decwwm
i
‘,
provrs
Figure
the
statc111c11t.0
4
!5
‘
G: Different
Paths
with same slope (~otlstiult
TP-01 3:00
Definition 2 The f~rdnr~~~~&~l trajectory
trq~cct0ric.s of t:ijl~,e.s
.3 und 4 is defined us
sq:!/7Jw~t
Proposition
for
of
thd trajectoq com3ponding to the first period of the cop
173~J07111i11.g 4
the
yiecc
lJrbl2.
In the cast of the’ Reed-Shepp ca.r, shortest paths we
cornposed of sequences of clualitatirely
different. segments.
In the cast of ollr Sk&!, however, optimal trajcct.orics are
within a given type.
always fOlmd
iics
that
5 There are
accomplish
t~Jo
flistinct
optimal
trajuto-
of ttt.eskctc. Om
is dei~ivcd from a C/I
orb%trulidchis outr1.ct la t cm1 rfio tion
of the tmjwtorics
side the sepwatrix (Type 3). The other trujectarg
rived ff~~,~n a (j orbit mhkh is inside the sepcantrix
is de( TTQJC
4).
Proof Using proposition 4. we know that if (p(O) = 0 then
after wi:h lwriod the trajcct.ory niccts the 1 axis. 111othilr
Corollary 2 Periodic $ oi~bits lead to skilte ti~~jcct0rie.s words, when S(O) = 0: H f1lIldilIIl~!Ilt~ll trajcxxory scglricmt
i:orreslmnds
to effectively sliding the skate in the conof types 3 and 4 uJlr.if:h are corfqmscd of rqmtetl tu~nslatd
straiiid
directioii.
fil?i,da?ri.errtill
triljectiytt segments.
Fix t,hc: total timt of trawl to lw p. Give it fixed Cr.
Proof Consider a 4 orbit. with poriutl 1~. From the last. RT Cilll fiIlt1 tW0 VdllW Of K1- mllich IlilVl! the SilIll(? f1111&1propositioli,
InCl1till pc,riocl p for tlwir (/, orbits.
Consider both these
{H(p), f+qlJ), f&l)} = {O(O), ((0). i(O)}. If we ClEmgc~ coIt’ll orbits with b(O) = 0. Both these pa.ths will cost t.he
ordilliltes such t.llat the sknto position at, time p is tlw SillllC for equal tiiiies of trawl. AAlSO iifT.C!r the first. period,
origin. tlicu w arc in a sit.uat.ion irlaitic7d to t1i;l.t ilt. t.lley will ~nc?et~at t,he 3: axis. Tlwcforc
t,herc! arc t.wo funtime f = 0. (i.0. The equat.ions of motion arc\ idcntidaniental trajectory
scgirx3it.s to cnch point, in tlio .r axis.
~1.) Tlwrcfolo the triljectory gelleri\t.d
tlurirq the nest
point on tlic
WI can c~l~oosc~ Cl- to rcw~li t,lie appropriate
lmiorl n-ill lw iilcntical t.0 the first. period (ignoring the
x Gs. Thaw arc t,lle OII~,Ytraject.ories that rcdl
the reIlrt trallSliltioll).O
quid
tlrStillilt.iOll
at the rrqnired
time thus proving
the
prop0sition.O
Corollary 3 Pn7iodi.c d, orbit ile~~%vctl trr~jijcctories rrot
having thr SI~IW slope constant never med.
Note 3 Ghen
iGtia1
w&d find
corl~glrrntio?rs of the
Proof Sw [‘L]for tlic proof.0
Skate,
UJl! CfI’l1 ~l~~l~~lI~el~~~l~l~~~
SWfCh
the jkt
t~llJ0 ~l#ES
Of
aptinrol trujectories.
These are eaq to scorch in the case
Corollary 4 l$pcs 3 end 4 tmjectoks
have conjugate
of the skate. If no t~ajwtorie.s
cali bc fowd,
th.en UK
or
at their fi9*sf irrpoints at the end of their first period
secmh for (1 typr! 4 krujectorg by restricting our search. to
tcmection ,with a i?jpe 2 trajectory
I<,- -c Cl-. If no type 4 trajectoiy iwf), be found, th
we
Proof Consitlcr a type 4 trajectory with Ir’r < Cr. For tlw cm searchfor ii type 3 trajectory b:y res trict%ry our sca~‘~‘h
to I<,. > Cl-. The sewch
z'sthus done ,inorder of inewmsa111c slopct c:011sta11t. ;~rlclKLllE of C,. WC can find a value of
irrg
level
of
nrrmcricul
diffic
ft fg.
I<r > Cr. swh that, t,he corresponding
type 3 tmjectory
llas the sillll(! period as the original.
This is clear from
-'?.l
-0.05
0.05
0.1
0
the 11;1tim of t,lio optiriinl control phase space. At, die twl
of the first period, Ijot these trajectories
mm. as they
llil\rtl the WIIW slope c~onSt;lnt. HCIIW the point, where thl>y
0.002mwt. is 3 coii,jlgate point for both the tr:~jrc:torics. On tlw
othm llmd~ if such a trajectory rnect,s i>n type 2 traject+ory
during tlic: first, period then this point of intersection
is
ills<) a coll,jugate poillt. for the type 3 or 4 trajectory.
thus
prol-iug the corollary.0
::
0
Note 2 l)pes 3 and 4 trajectories
are not optimnl after they intersect a tgpe 2 puth. This was ohserved in
nvmei~icnl computntions but is also clear from thr! geometry of tlrc p&s.
At the point of intersection, the type
2 path is r:hrzqwr than. the type 3 or 4 path. Henm the
type 3 07’ 4 path is riot optimal after it intersects it type
2
pi&h.
Multi
pw1.011
trujectori13
we
cdninys
n)ore
5
-c.o02.
Figure?
exprrr-
.S~“IJL’
than a siqle period trnjectoql conrrect%n,q two poifits.
This ~~1s ohse~wd
571 nurncricnl comput(Ltion.s
but it is nlso
CREW!jeo7,letl.icnllll. It ~uta.sirlso 0b.seri.d nwuerici~ll~~ that
t7~~j~~ct1~ric.s
of type 3 and 4 a.7.enot optirrinlafter thei
Wr
7: CoIlajrli?;at,e
can
conjugate
filrbda.lrrelltol
trr~jectoq stywent.
5
IIow
locii
cornput,e
set,s of the
o~~tinial
trajectories
ti1.tiVdg
wc! caIl
on
tleicribe
surface:
t.h(.1 gcodf:sic
origin
and
isocost,
using
their
COST = 0.01
Iiciglibortior~ds
011~ l~n~mleclgct
con.jug;ltr
t.ht! c:OllSt>iIlt
and
of all t.he
points.
QualiCOSt surfarcs
tO be
TP-01 3:00
0.02.
I
0.
-0.02.
Figurr
8: Conjugate
locii ou isocost surface:
Cost, = 0.1
Figurr
twistwl, tl~~fo~~ncd cllipsoitls in {x? !/! H} spnw. They look
like saddles. Due t,o the structure of t,he t,ri\jcctories satisfying the first, order necessary condit.ions, the set of points
nccessiblr via Type 2 trajectories
are t.he uorltrivial conjugate sets for the Type 3 and Type! 4 trajwtories.
The
trivial c’onjugatc~ sets for these trajectories
a.re t,lie 1: and
y axes. Thr ac*companying figures (7! 8, 9. 10) show the
nontrivial wujugate sets on constant cost surfaces for tliffercnt cost,s. Observe how the nontrivial
conjugate locii
scpcrate and move away from each other with increasing
cost. On a isocost surface, these locii are topologically
equimlrnt
to the seperatrix of the 4 phnw potrait.
Thcl
trivial COIl,jl.IgiltC! st?bs being p0iut.s caullot, bo we11 in the
R
yl0t.s.
-1
-0.5
0
0.5
1
0.2.
:\
o-
-0.2.
tl
Figure
9: Coujugatc> locii on isocost surfam:
Cost = 1
Conclusion
There are t\vo essent.ial features of t,lw optimal steering
problem for the nonholouomic
skate that. have pemitted us to obtiiiu t,hchdet,ailcd result,s prcscntctl iu t,liis papc’r. First.. t,hc optimal wntrol problem is s(~l>a~able in
10: Conjugate
locii on isocost surface:
Cost. = 9
t,hc smsc that the necessary conditions
involve a differential equation for the control ii, which involve no ternis
The other
tlepcmtling esplicitly on the stat,c or cost&e.
feat,urc we have esploitecl is the existence of two independent, conserved quant,ities which have permitted
us t,o
represent the optimal controls in terms of *Jacobi elliptic functions.
We have further described the solution in
term of a phase port.rait representation.
The uncoupling
of the optimal control equations and the system coufigurat.ion qmtious
in the scpa.rable case allows us to analyze
t.wo lower tlirncnsional problcnis separately and study the
structiire of the optimal system t,raject,ories.
In general,
we will hwe a. phase port.rait. associat,ed
with the optimal controls.
The different kinds of orbits
in t.he phase portrait give rise to various kinds of optimal cont,rols. The structure of the phase portrait can be
uwtl t,o study the optimal controls and their conjugate
points. For clsamplr in the case of a rolling penny, we get.
a family of optimal control phase plots that cm be obMined as perturbations
of the optimal control phase plot
of the skate. In this case me get. six kinds of orbits in the
associated optima.1 phase portraits and thus six kinds of
opt,irnal trajectories
for the rolling penny.
III the case of t,he skate. we see that t.hcre are two surfaces that arise from the ‘stable and unstable rnanifolcls
of the Q, phase portrait.
These are the sets of conjugate
points associated with the opt,imal controls. -Also we see
that generic optimal paths tend t,o have oue Or no cusps
when projected onto t.lie sy plane of the skate’s configuration space. Note that, the closer a point. on the 4 phase
port,rait is to the stable-unstable
manifolds,
the longer
the time of optima&
of the a.ssociated path.
Although the inverse problem of finding a path given
teruiinal configuration
st.ill weds a numerical
solution,
this numerical search can be adapt,ed to the st.ruct.ure of
the 4 phase portraits ~vhich yields a quicker and more sta-
TP-01 3:00
hlC SWrdl routirm
\Ye can start by first s(,yir(:hing qllilibriunl rdat~d optiulnl paths (Type 1) ant1 tllc?n progrpss
to tk: Stil~)Ic-II~~StnLlt, IIliAfold relatcxl OIWS (Type 2). Fitlidl~ WC*Ciltl wi-idcn the search to gelicy+ points (T~yes 3
Uld 4) 011 tAlCaph;lSc! portrait. st,arting CloW t.0 t,lio St,ilbl('unstirblr niaiiifoltls mtl Inovirig ;way frown it,.
[l] -4. Agrachev.
~ontwd
[l::] R.3.f. Mmay
and S. Sastry. Xonholononni(: motion
planning:
Stwring
using sinusoids.
IEEE Tm~s~ct%ons OIL.4utomatic Contlal. 38(5):i00-i1G.
1993.
Nl ,J. I. X:eirnark a.nd F. -il. Fufaev.
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