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Collision Avoidance Systems:Computing Controllers which Prevent Collisions
By Adam CataldoAdvisor: Edward Lee
Committee: Shankar Sastry, Pravin Varaiya, Karl Hedrick
PhD Qualifying Exam
UC Berkeley
December 6, 2004
Cataldo 2
Talk Outline
• Motivation and Problem Statement
• Collision Avoidance Background– Potential Field Methods– Reachability-Based Methods
• Research Thrusts– Continuous-Time Methods– Discrete-Time Methods
Cataldo 3
Motivation—Soft Walls
• Enforce no-fly zones using on-board avionics
• A collision occurs if the aircraft enters a no-fly zone
Cataldo 4
The Research Question
• For what systems can I compute a collision avoidance controller?– Correct by construction– Analytic
System Model,Collision Set
Control Law,Safe Initial States
Cataldo 5
Collision Avoidance Problem(Continuous Time)
000 )(
)(),(),(,)(
Xptx
tdtutxtftx
RUu U
RDd
RRnx
nT RTtx
d
Xu
)(
,
,t
thatso and find 0
R
Cataldo 6
Collision Avoidance Problem(Discrete Time)
00 )(
)(),(),(,)1(
ptx
tdtutxtftx
ZUu
ZDd
ZRnx
nT RTtx
d
Xu
)(
,
,t
thatso and find 0
R
Cataldo 7
Potential Field Methods(Rimon & Koditschek, Khatib)
• Provide analytic solutions, derived from a virtual potential field
• No disturbance is allowed
• Dynamics must be holonomic
Oussama Khatib: Real-time Obstacle Avoidance for Manipulators and Mobile Robots
Reachability-Based Avoidance(Mitchell, Tomlin)
0)0(
)(),(),()(
px
tdtutxftx
],0[ sUu
],0[ sDd
],0[ snx R
compact
0)( pgpT nRTtx
stumBp
B
s
ns
)(
],,0[,,,,
such that find
0
R
)()(],0[tumtd
t
Cataldo 9
Hamilton Jacobi Equation(Mitchell, Tomlin)
T
0),( psVpB ns R
n
DeUv
pstpgpV
evpfptVp
ptVt
R
],0,[ ),(),0(
,0),,(),(minmax,0min),(
Cataldo 10
etutxftxtVtx
tu
De),(),()(,ˆ)(/min
maximize )(let
Computing Safe Control laws(Mitchell, Tomlin)
gf ,
0ˆ)/(
until ),(in approx.
Vt
ptV
VpV ˆ)/(,ˆ
offline
online
Cataldo 11
Applied to Soft Walls(Master’s Report)
• Works for a many systems
• Storage requirements may be prohibitive– 40 Mb for the Soft Walls example
• Cannot analyze qualitative system behavior under numerical control law– switching surfaces, equilibrium points, etc.
Cataldo 12
Analytic Computation:Soft Walls Example
)()(
)(sin
)(cos
)(
)(
)(
tdtu
t
t
t
ty
tx
1)()()(),(),( 22 tytxttytx
)(tx
)(ty)(t
2
5.1
],[)(
],[)(
e
ev
eetd
vvtu
Cataldo 13
Change of Variables
)(
)(sin)()()(
)(cos)(
tr
ttdtut
ttr
1)()(),( trttr
)(t
)(tr
0)(,0
0))(sin(,
)(or 0))(sin(,
)(),(,
t
tv
ttv
ttrtk
Cataldo 14
Lyapunov Function
)(),( ttrV
),2/()()(sin)(2)(
)2/,[)()(sin)(2)(
2/|)(|)(
)(),(22
22
tkkttkrtr
tkkttkrtr
ttr
ttrV
ek /2
safety implies
1)(),( ttrV
Cataldo 15
A Sufficient Condition(Leitmann)
00 )(
)(),(),(,)(
ptx
tdtutxtftx
RUu
RDd RRnx
)(,)( txtktu
)(,)( txtmtd
nT R
Ttx
Dm
Ukn
n
)(
,:,t
thatprove :given
RRR
RR
Cataldo 16
A Sufficient Condition(Leitmann)
• Find a Lyapunov function over an open set encircling the collision set which ensures against collisions
),( ,
,,
in increasing )(,
,:
,tx such that,
: find
0
qsVptV
AqSpst
ttxtV
Dm
S
SV
n
RR
RRnS R
nT R
nA R
One Possible Extension
1
121, pppT nRR
)(),(),(),()(
)()(
2122
211
tdtutxtxftx
txftx
R compact
)(
)(
Dtd
Utu
)(maxarg
continuous ,
21*
21
12
pfp
ff
np
R
*2212
*2*2212
21
,,,maxminarg
,,,minmaxarg
),()(let
ppevppf
ppppevppf
ppktu
DeUv
DeUv
Cataldo 18
One Possible Extension
)(infinf)(),( 121 xtxtxV
td
)(),(),(),()(
)()(
2122
211
tdtutxtxftx
txftx
)(),( if safe 0201 txtxV
Cataldo 19
Open Questions
• When can we find our control law analytically?
• When can we find the corresponding Lyapunov function analytically?
• Can we build up complex models from simple ones?
Cataldo 20
Bisimilarity and Collision Avoidance
21
00
,)(obs
)(
)(),(),()1(
tx
Xtx
tdtutxftx
11
12
unsafe statedisable this transition
• When is the system bisimilar to an finite-state transition system (FTS)?
• If the system is bisimilar to an FTS, can I compute a control law from a controller on the FTS?
Example: Controllable Linear Systems (Tabuada, Pappas)
00 )(
)()()1(
Xtx
tButAxtx
W
ZRmu ZRnx
WT
Ttx
k
W )( always
such that find
)()( txktu
semilinear sets on W
),...,(
),...,,(
),...,,(min
1
1
mAAspanW
bAAbbspan
bAAbbspank
ik
ii
in
iii
LTL formula
Cataldo 22
The Result(Tabuada, Pappas)
• There exists a bisimilar FTS for observations given as semilinear subsets of W
• A feedback strategy k which enforces the LTL constraint exists iff a controller for the FTS which enforces the constraint exists
Ttx
TX
W )( always
,,0
mnk RR :
Cataldo 23
Bounded Control Inputs
• If we want to extend this for disturbances, we will need to be able to bound the control inputs
• Adding states won’t work; we may lose controllability
)()(
)(
00
0
)1(
)1(tu
I
B
ty
txA
ty
tx
1
0,
00
10 :example BA
Cataldo 24
Research Questions
• When we have bounds on the control input, when can we find a bisimilar FTS?
• For systems with disturbances, when can we find a bisimilar FTS?
• For nonlinear systems with disturbances, when can we find a bisimilar FTS?
Cataldo 25
Where is this Going?
• Build a toolkit of collision avoidance methods
• These methods must give correct by construct control strategies
• We should be able to analyze the control strategies
Cataldo 26
Conclusions
• I plan to develop new collision avoidance methods
• Many approaches to collision avoidance have been developed, but methods which produce analytic control laws have limited scope
• In the end, we would like to automate controller design for problems such as Soft Walls
Cataldo 27
Acknowledgements
• Aaron Ames
• Alex Kurzhanski
• Xiaojun Liu
• Eleftherios Matsikoudis
• Jonathan Sprinkle
• Haiyang Zheng
• Janie Zhou
Cataldo 28
Additional Slides
Cataldo 29
Global Existence and Uniqueness(Sontag)
• Given the initial value problem
• There exists a unique global solution if– f is measurable in t for fixed x(t)– f is Lipschitz continuous in x(t) for fixed t– |f| bounded by a locally integrable function in t for
fixed x
nnf
ptx
txtftx
RRR
:
)(
)(,)(
00
Cataldo 30
Potential Functions(Rimon & Koditschek)
T
T
goalq
000 )(),(
)()()(),()()(
Xtqtq
tutqgtqtqftqtqM
)(),()()( tqtqdtqVtu
goalt
qtq
Ttqtt
dVX
)(lim
and )(,
such that and ,, find
0
0
Holonomic Constraints(Murray, Li, Sastry)
• Given k particles, a holonomic constraint is an equation
• For m constraints, dynamics depend on n=3k-m parameters
• Obtain dynamics through Lagrange's equation
0),...,( ,: 13 k
krrgg RR
)()(),()(
)(),()(
,:
tutqtqtq
Ltqtq
tq
L
dt
d
q n
RR
Cataldo 32
Information Patterns(Mitchell, Tomlin)
• In computing the unsafe set, we assume the disturbance player knows all past and current control values (and the initial state)
• The control player knows nothing (except the initial state)
• This is conservative• In computing a control law, we assume the
control player will at least know the current state
Cataldo 33
Relation to Isaacs Equation• Isaacs Equation:
• W(t,p) gives the optimal cost at time t
(terminal value only)
n
DeUv
pstpgpW
evpfptWp
ptWt
R
],0,[ ),(),0(
,0),,(),(minmax),(
0)(,,,,0minmax),(
umuptgptWmu
causal} |:{ mm
Cataldo 34
Relation to Isaacs Equation• Isaacs Equation:
• The min with 0 term gives the minimum cost over [t,0]
n
DeUv
pstpgpV
evpfptVp
ptVt
R
],0,[ ),(),0(
,0),,(),(minmax,0min),(
0)(,,,,minminmax),(]0,[
umuptgptVtmu
causal} |:{ mm
Viscosity Solutions(Crandall, Evans, Lions)
0),(H ),(
minimum local a is ),)(()2
0),(H ),(
maximum local a is ),)(()1
allfor if
0),(H ),(
forsolution viscositya is 0
ptt
hp,pt
t
h
pthV
ptt
hp,pt
t
h
pthV
Ch
ptt
Vp,pt
t
V
CV
n
n
RRR
RRR
Cataldo 36
Convergence of V
• At each p, V can only decrease as t decreases
• If g bounded below, then V converges as
• It may be the case that all values are negative, that is, no safe states
n
DeUv
pstpgpV
evpfptVp
ptVt
R
],0,[ ),(),0(
,0),,(),(minmax,0min),(
t
Cataldo 37
Applying Optimal Control:Soft Walls Example
)(),( ttrV
)(),( ttrb
safeunsafe
1
)(),()(),()(),(ˆapply ttrkttrbttrk
1
Cataldo 38
Lyapunov-Like Condition(Leitmann)
• Given a C1 Lyapunov function V:S, A is avoidable under control law k if
• Note that this can be generalized when V is piecewise C1
AA
AA
ttApSpt
ptVptV
,,),(
when),,(),( )1
R
D,),( when
,0)),,(,,(),(),(
)2
eSpt
etpkptfptVt
ptVp
Cataldo 39
Lyapunov-Like Condition(Leitmann)
• Let {Yi} be a countable partition of S, and let {Wi} be a collection of open supersets of {Yi}, that is, WiYi
1Y
3Y
2Y
SR
Cataldo 40
Lyapunov-Like Condition(Leitmann)
• Given a continuous Lyapunov function V:S, A is avoidable under control k if
AA
AA
ttApSpt
ptVptV
,,),(
when),,(),( )1
R
D,),( when
,0)),,(,,(),(),(
and , with ),( )2 1
eYtp
etpktpftpVt
tpV
VVWCV
i
ipi
YYiiiii
R
Cataldo 41
Transition System
mapn observatio :obs)5
states initial ofset )4
nsobservatio ofset )3
relationn transitio)2
states ofset )1
obs,,,,
0
0
Q
Q
QQT
00**
10 ,,..., :run dinitialize QqQqqr
)(obs,run dinitialize L(T) :language ** rwrOw
Cataldo 42
Bisimulation
RpppqQp
pqRqq
qqRqq
QqRqqQq
TTQQR
QQTQQT
),(,,
,),( 3)
)(obs)(obs),( 2)
),(, 1)
if to from simulation a is
obs,,,, obs,,,,
2122222
11121
221121
2,02211,01
1221
22,022211,0111
)()(
to and to from ssimulation:
2121
122121
TLTLTT
TTTTTT
Cataldo 43
Linear Temporal Logic (LTL)
• Given a set P of predicates, the following are LTL formula:
211211
21
U,,,
are so then formula, LTL are , if
,,,
Ppfalsetrue
Cataldo 44
Semilinear Sets
• The complement, finite intersection, finite union, or of semilinear sets is a semilinear set
• The following are semilinear sets
},{ ~ ,,
where0~T
R
ba
bxax
n
n
Cataldo 45
Computing Safe Control Laws(Tabuada, Pappas)
LTL Formula Buchi Automaton
Finite Transition System
Discrete-Time System
Finite-StateSupervisor
Hybrid,Discrete-Time
State-FeedbackControl Law
