Dynamic systems were considered subject to control by two agents, one of whom desires that no trajectory of the system emanating from outside a given set, intersects that set no matter what the admissible actions of the other agent. Constructive conditions sufficient to yield a feedback control for the agent seeking avoidance were employed to deduce an evader control for the planar pursuit-evasion problem with bounded normal accelerations

Leitmann, G.

Liu, H. S.

English

NASA Technical Reports Server

General Disclaimer 
One or more of the Following Statements may affect this Document 
 
 This document has been reproduced from the best copy furnished by the 
organizational source. It is being released in the interest of making available as 
much information as possible. 
 
 This document may contain data, which exceeds the sheet parameters. It was 
furnished in this condition by the organizational source and is the best copy 
available. 
 
 This document may contain tone-on-tone or color graphs, charts and/or pictures, 
which have been reproduced in black and white. 
 
 This document is paginated as submitted by the original source. 
 
 Portions of this document are not fully legible due to the historical nature of some 
of the material. However, it is the best reproduction available from the original 
submission. 
 
 
 
 
 
 
 
Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19780002890 2020-03-22T07:12:56+00:00Z
,	
_	
n
r
i
NASA CR 152043
jj
Evasion in the Plane
G. Leitmann i. and H. S. Liu'h.
I'
(NASA-CR-152043) EVASION IN THE PLANE 	 N78-10833
(California Univ.) 13 p HC A02/MF A01
CSCL 12A
Unclas
"	 G3/67 52032
t} _
July 1977
NSG - 2242
"
I
i
,a
h University of l California, Berkeley, California, USA
ItGraduate student, Department of Electrical Engineering and
Computer Science, University of California 	 ^-
sr
c	 ^.
'j
i
Abstract
:.	 I
We consider dynamical systems subject to control by two agents
one of whom desires that no trajectory of the system emanating from
outside a given set, intersects that set no matter what the admissible
actions of the other agent. Constructive conditions sufficient to
yield a feedback control for the agent seeking avoidance were given
earlier. These are employed here to deduce an evader control for
>a
the planar pursuit-evasion problem with bounded normal accelerations.
9
qq
_i
9
a
1
I	
r
a
c
a	 `^	 3
e
t^
z	 !^
t
r.
1. Introduction
A problem 'of collision avoidance arises whenever two, or more,
objects move in space. Here we consider the case of two objects
moving in the same plane, e.g., two ships. One object (evader) is
capable of determining the relative position and velocity of the
other object (pursuer). The pursuer may be active or passive; that is,
he may desire collision, or he may be unable to measure the evader's
relative position and velocity and thereby cause collision through
inadvertence. Each controls his motion by means of his normal
acceleration whose values are constrained. The evader desires to
maneuver so as to avoid collision no matter what the actions of the
pursuer.
The problem outlined above belongs to the following class of
problems. There is given a dynamical system subject to control by
two agents, one of whom desires that no trajectory of the system,
emanating from outside a prescribed set, intersects that set no matter
what the admissible actions of the other agent. 	 Such problems have
been discussed in Refs. 1-9, among others. There the treatment is
within the framework of differential games, Refs. 6 and 7, either as
games of kind (qualitative games) or games of degree (quantitative
games). In the former approach, the players seek a saddlepoint for
time of collision or for miss-distance, Refs. 1-3 and S, 9,and in the
latter,barriei^s are sought which separate regions in which collision
can be brought about from regions in which avoidance can be assured,
Refs. A and 5. These techniques usually require numerical integration.
Furthermore, only necessary_ conditions are employed so that avoidance
-2-
UN
.	 4
cannot be assured. In Ref. 10 we propose an alternative approach,
namely, the constructive utilization of conditions sufficient to
guarantee avoidance. Before discussing the planar avoidance problem
in Sec. 3, we state the general avoidance problem and the results of
Ref, 10. The more general case in which each player has his own
target on which fie desires termination is treated in Ref. 11.
2. General problem Statement and Results
Let
d.
P 1 ( )	 Rn x R -} the nonempty subsets of R 
1	 i = 1, 2
be feedback controls (strategies) belonging to given classes of possibly set-
valued functions, U i , with control values ui ranging in prescribed sets, Ui
(which may depend on state and time); that is, given (x, t) E Rn x R
d.
uI E p '
 
(x, t ) ^ Ui c R 1	 ,	 i = 1, 2
Let
f(-) : R n x R x R 
d	 d
1 x R 2+ R n
be a prescribed function, and for given p i (•) E U i	  = 1, 2,
define a set-valued function F( • ) by
F(x, t) 6 fz E R n	 z = f(x, t, u l , u 2 ), u  E p i (x, t))
= f(x, t , p I (x , t), p2 (x, t)),
Then a dynamical system, e.g. Refs. 12 and 13, is defined by the relation
x E F(x, t) ,	 (1)
Given (xC , t0 ) e.6 , where A is an open set (or the closure of
an open set) in Rn x R , solutions of (1) are absolutely continuous
functions on intervals of - R
X(-) : [to , tl ] -r R n	 x(to) = x o	(2)
such that
91AIGIN "
^' pOOFt
	j
I
-4-	 1
X(t) e f(x(t), t, pl (x( t ), t), p 2 (x(t), t))	 (2)	
(i .
a. e.	 [to , t 1 1 .
Now let there be given an anti-target, T , in A , that is a given set
into which no solution of (1) must enter for some p l (-) c U 1 and all
p 2 (•) c 11 2 . Consider a closed subset, A , of A such that A D T
and consider also the closure, A E , of an open subset of A such that
AE DA and DA, nDAnintA=^
We call A the avoidance set and
AA6Ac\A
the safety zone. If a solution avoids A then it cannot enter T	 and if
a strategy P
I N is used in AA that guarantees avoidance of A for all
p2 (•) , then a solution originating outside of A cannot reach A
For given p l (•) c- 11 1 , let K denote the set of all trajectories of
(1) for all (x 
o' 
to ) e AA x R and all p 2 ( • ) e U2 . Then, given system
(1) and sets U 1 and U2 	a prescribed set A is avoidable if there is a
p 
1 (-) e L1 1 and AA x R #	 such that
K n A = 0	 (3)
Note that (3) implies global avoidance, that is, avoidance for all
( xo , te ) c- (A \ A) x R. Avoidance set A may be any set containing anti-
target T ; often it is different from T .
The following theorem and corollary are proved in Ref. 10.
Theorem A given set A is avoidable if there exist a nonempty set A A and two
functions, a strategy p l (•) c L1 1 and a C1 function V( • ) : S f R ,
S(open) D AA
 x R , such that for all (x, t) E AA x R
E
i
(i) V(x, 0 > V(x', t')
	
V x' G DA ,	 V t' ;^- t,
and V u l
 e 01 (x, t)(ii)N , t + vx V(x, t) f(x, t, u l , u2 ) > o
V u 2 a U 2 , where p ( ) is the restriction of p l ( • ) to
A x R .
Let
H(x, t, u1, u 2 ) Q av(x, t + Ox V(x t) f(x t u l 	u 2 ) .
Then the Theorem has a
Corollary_ Given (x, t) a AA x R , if there is a
("u l , u2 ) e U1 x U2 such that
(i)	 H(x, t, ul; u 2 ) =	 max	 min	 H(x, t, u 1 , u2)
u I G U 
1 
u2N2
and
0i)	 li(x, t, u l , u 2 ) > 0
then condition (ii) of the Theorem is met. Furthermore, "ul e "p l (x, t)
provided the resulting pi 	 c- ul .
Note that the Corollary is constructive in that it may permit construction
of 01  .3
3, Avoidance in the Plane
Consider two agents, called pursuer P and evader E , moving in a
plane. Let V
P 
and vE be the velocities (relative to an inertial
reference frame) of P and E , respectively. We suppose that their
speeds, vp = 1 Vp j and v E =.IvE I , are constants and that vE > v  .
Referring to Figure 1, the kinematic equations of motion are 
3Usually, u`l = p l (x, t) a.e „ that is, except on discontinuity manifolds.
Note that here, unlike in Ref. 5, position is relative to the pursuer.
..	 i
_6-il
r=vE Cos R-VP' sill 0
e = r (vE sin R - vP cos 0) - uP	 (Q)
R= r (v E sinR - vP cos 0) +E
where
uE^OE	 ,	 uP^OP
are the controls of E and P	 respectively; that isy E and P control
their motions by means of their normal acceleration components. These are
constrained; namely,
uE	
:'E ( g iven)	 up I < up (given)	 (5)
Evader, E	 wishes to avoid having pursuer, P , approach more
closely than a given distance r 	 that is, the anti-target
T = {(r, 0., R) ® A I r < r}	 (6)
wi th`l
A = {(r, 0, R) 1 r e; R+ , 0 E R	 ,	 I R 1 < ir} ,
There arises now the question of selecting an avoidance set, A .
To allow E maneuverability, one wants r "sufficiently" large when
"min	 but when r = rmax one can allow r = r , where5
"min --vE - vp (02i 2mr, R = ^ T)
Max"
	
vE+vP (0 = -2 2n7r, R=0)
This is accomplished, for instance, by
A = {(r, 0, R) e o I r - r < b(1 -1• sin 0) + cR 2 }	 (7)
for given constants b > 0 , c > 0 .
'I 
As will be seen subsequently, it suffices to consider { R	 < IT
5 Here, n is any integer.
ORIGINAL PAGU 16
OF POOR QUALr y
To satisfy condition (i) of'the Theorem we choose
	 V( • ) such that
V(r, 0,
	
R, t) = r - r - b(1
	 + sin 0)	 - 0 (8)
To apply condition (ii) of the Theorem we form
H(r,	 0,	 R,	 t,	 u E ,	 up )	 =	 ( v E cos R - vp sin 0)
- b cos 0 [
	 (v E sin R - v p cos 0) - up]r
(9)
- 20
r
 (vp cos 0 - vE sin R) + uE7
First we check to see what is required to assure	 H a 0 for all
U P	 satisfying constraints	 (5),	 and for all	 (r,	 0 .,	 R) c AA . S At	 R = 0
b	 pH = (v E - v p sin 0) +	 cos t 0 + b up cos 0r
N v E - vp sin 0 + b up cos 0 > v E -	 vp 2 + b 2 up2
Thus, to satisfy condition (ii) of the Theorem we make the conservative
choice
/vE	 vpb < (10)
Lip
Next we impose the conditions of the Corollary. 	 Since QE	 and	 up
are separated in
	 li	 ,	 it follows readily that
a  = - u 	 for	 R > 0
u"E =	 u E	 for	 R < 0 (11)
u 
	
e [ - 5 E , u E a	 for	 R = 0
and
AA has not been defined yet; it depends on AE . For instance, one might let
Ac = {(r, 0, R) c- A I r 6 r + b(l + sin 0) + cR2 i• e , e = constant > 0} .
I•
u
p
 = u
p
	for cos 0 < 0
up = - u P 	for cos 0 > 0	 (12)
up e [ - up, up] for cos 0 = 0
Now we investigate conditions onu E
 which assure satisfaction of
(ii) of the Corollary; namely, in view of (11) and (12), for all
(r, 0, R) a AA and all	 ( u E , u P ) satisfying (5)	 i
min max N = v E cos R - vp sin 0 + b (vp cos 0 - v E sin R) cos 0
uP
	uE
+ C (v E
 sin R - v P
 cos 0)
(13)
+ 2 c 131 u E - b Icos 01 u p Z^' 0 0
To obtain a conservative estimate for the required value of O
E
 we
rewrite (13) as
5 	 max	 2 cl R [ VP sin 0 - vE cos R
(r, 0, R) e 0A
+ r( v E sin R- v P cos 0) cos 0
+ 2 r R (v p cos 0- v E sin R) a • b 1cos 01 6p
whence
uE a	 max [ vP +	 1 ( 3 v	 ' + b^ ur2
R c[ 0, Tr]
	 r	 2 c R
- vE cos
by
R +	 E sin R)]
r
Letting
•Q
g(R)	 _ ,vP
	
b 2 5- vE cos
b vE	
RR + - VE sin
r
h(R) 
=
2 c p
ORIGINAL PAGE IS
OF POOR Qp"ITI
(lh)
i
-g-
t
we rewrite (14) as
VP,
uE a	 +	 g (R)	 h(R)	 (15)R c
max	 ff]
An even more conservative bound is .hen found by replacing
^f g (R)	 h(0)
4 e [0, Tr]
by
max g (R)
	
max h(R)	 for	 R e {R E! [0r IT]	 g(R) a 0) l
i
In this connection we utilize condition (10) whence
g(0) _ _ vE d ,	 v 	 + b z 5 P2 < 0
so that	 R c- (0, n]	 for	 g(R) = 0	 We arrive at the very conservative
bound
E	
vim,	
2c	 v 	 , 1 + ( b /r)2
r
v 2 ^• b2 u
2	
1	 P	 P
`+ 3 v	 + U2	 Lcos_l	 (16)
p	 P	 vE	 1	 + (b/i^)21
- 
cos-
To reiterate, given pursuer and evader speeds	 v p	 and	 vE
i
respectively, pursuer control bound	 u P	 ,	 missdistance	 r	 and
constants	 b > 0 , c > 0 , with	 b	 subject to (10), the use of evader
g control	 (11) with
	
u E	 satisfying	 (16)	 guarantees collision avoidance;
n
of course, the evader needs to implement such a control only on	 AA , t
As assumed in the definition of	 :A',	 R e [-Tr, Tr]	 since 1	 j
= yPcos e T :u{
=.t
	
E
7g R•IT
	 r {.	 d
and by (16)
Pu E	 r<r ^)
Pr^,•of course,	 U l	 admits piecewise continuous functions.
-10-
I
^L
r
LI	 Or
whence it follows that
Ala =n<0 ale= -n>0 .
To illustrate the aforegoing results consider
vE = 300 m/s
v 
	
= 225 m/s
u 
	 = 1 rad/s "r	 = 3000 m
Then	 (10) becomes
b c [0,	 198.4)
For example, with c = 6 x 104	and
(i ) b = 100 uE ;;, 0.083
(ii) b = 10 uE > 0.081
(iii) b =	 1 6  > 0.081
Finally, we can draw these conclusions:
(i) The bound on uE	given by (16) is quite insensitive to changes
in the value of	 b , and it can be decreased by increasing the
value of	 c (that is,	 by increasing the size of the avoidance
set	 A ).
(ii) For given	 a ,	 the contour of A	 is	 "nearly" circular (more
so at	 a = it than at	 a = 0 ).
References
1. HAGEDORN, P. and DRE=tiiMELL, J, V., A Differential Game with Two
Pursuers and One Evader, Journal of Optimization Theory and
Applications, Vol, 18, No, 1, 1976,
2, DREAKWELL, J. V. and ME•RZ, A., Toward a Complete Solution of the
Homicidal Chauffeur Game, Proceedings of First International
Conference on the Theory and Applications of Differential Games,
University of Massachusetts, Amherst, 1969,
3. FOLEY, M. A, and SCHMITENDORF, W, E., A Class of Differential
Games with Two Pursuers versus One Evader, IEEE Tr;anslctions
on Automatic Control, Vol. AC-19, No, 3, 1974,
4. MERZ, A., Optimal Evasive Maneuvers in Maritime Collision
Avoidance, Navigation, Vol. 20, No. 2, 1973,
5. VINCENT, T. L., Avoidance of Guided Projectiles,The Theor y and
Appliration of Differential fames (edited by J. D. Grote ,
RoliFelru@ fishing Company, Doutrecht, 1975, P. 267.
61 1SIMCS, R., Differential Games, John Wiley and Sons, New York,
Nevi 1'orl<, 1965.
7. DLAQUIERE. A.,.GERARD, F. and LEITMANN, G., Quantitative and
Qualitative Games, Academic Press, New York, New York, IM07
a. KRASOVSKIY, N, H., A Differential Game of Approach and
Evasion.
	
1., Engineering Cybernetics, No. 2, 1973.
9. KRASOVSKIY, H. N., A Differential Game of Approach and
Evasion. 2., Engineering Cybernetics, No. 3, 1973.
10. LEITMANN, G. and SKOWROiiSKI, J., Avoidance Control, Journal
of Optimization Theory and Applications, to appear.
11. GET7_, 14. and LEITMANN, G., Qualitative Differential Games with
Two Targets, to appear,
12. FILIPPOV, A, F., Classical Solutions of Differential Equations
with Multi-Valued Right-Nand Side, SIAM Journal of Control,
Vol. 5, No, 4, 1967.
13. ROMP, E, ,, On Generalized Dynamical Systems Defined by a
Contingent Equation, Journal of Differential Equation's,
Vol, 1, pp. 186 f., 191;5.
r	 i
j
6 r
v	
P
.we^oa....ae.w's..xvnnt.nworawmw.nnrn'.n::•^:.a.r^.Y+.xn+.rtrxva^aYma.'4rmaoa^n.amvn.^rr L  c.3 V't`ol
Figure 1, Coordinate System
ORIGINAL UAZrPY
or
,
 POOR Q,
f
l


Evasion in the plane

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server