1. A method of determining a leader’s strategy for a security domain by solving a Stackelberg game, the method comprising:
fixing the policy of a leader to a k-uniform policy according to a Stackelberg game for a security domain;
formulating using a processor an optimization problem the follower solves to find its optimal response to the leader’s policy;
maximizing the payoff of the leader, given that the follower uses an optimal response that is function of the leader’s policy, and formulating the problem as a mixed integer quadratic program (MIQP);
performing a decomposition of the MIQP to produce a decomposed MIQP, wherein the decomposition addresses the size of the MIQP;
reformulating the leader payoff problem by linearizing the MIQP to form a decomposed mixed-integer linear program (MILP);
solving using the processor the decomposed MILP; and
directing a patrolling strategy of an agent within the security domain based on the solved decomposed MILP.
2. The method of claim 1, wherein the optimization problem the follower solves is of the form,
\u2211
j
\u2208
Q
\u2062
\u2211
i
\u2208
\u2135
\u2062
\u2062
1
k
\u2062
C
ij
\u2062
x
i
\u2062
q
j
,
wherein the leader is the row player and the follower the column player, the vector of strategies of the leader are denoted by x, q denotes the vector of strategies of the follower, X and Q denote the index sets of the leader and follower’s pure strategies, respectively, k is the size of the multiset, and the payoff matrix C correspond to Cij that is the reward of the follower when the leader takes pure strategy i and the follower takes pure strategy j.
3. The method of claim 2, wherein the optimization problem is subject to constraints,
\u2211
j
\u2208
Q
\u2062
q
j
=
1
and q\u22670.
4. The method of claim 2, wherein the MIQP is of the form,
max
\u2062
\u2062
\u2211
i
\u2208
\u2135
\u2062
\u2062
\u2211
j
\u2208
Q
\u2062
\u2062
1
k
\u2062
R
ij
\u2062
q
\u2061
(
x
)
j
\u2062
x
i
,
and wherein the payoff matrix R corresponds to Rij, the reward of the leader.
5. The method of claim 4, wherein the MIQP is subject to constraints,
\u2211
i
\u2208
\u2135
\u2062
x
i
=
k
and xi\u2208{0, 1, . . . , k}.
6. The method of claim 2, wherein the MIQP is of the form,
max
\u2062
\u2062
\u2211
i
\u2208
\u2135
\u2062
\u2062
\u2211
j
\u2208
Q
\u2062
\u2062
1
k
\u2062
R
ij
\u2062
x
i
\u2062
q
j
,
and wherein the payoff matrix R corresponds to Rij, the reward of the leader.
7. The method of claim 6, wherein the MIQP is subject to constraints,
\u2211
i
\u2062
x
i
\u2062
=
k
,
\u2062
\u2211
i
\u2208
Q
\u2062
q
j
=
1
,
\u2062
0
\u2264
(
a
–
\u2211
i
\u2208
\u2135
\u2062
1
k
\u2062
C
ij
\u2062
x
i
)
\u2264
(
1
–
q
j
)
\u2062
M
,
\u2062
x
i
\u2208
{
0
,
1
,
\u2026
\u2062
,
k
}
,
\u2062
and
q
i
\u2208
{
0
,
1
}
.
wherein a is the follower’s maximum reward value, and M is a large constant.
8. The method of claim 1, wherein for the k-uniform policy x, the value xi is the number of times pure strategy i is used in the k-uniform policy, and wherein the k-uniform policy is selected with probability xik.
9. The method of claim 1, further comprising performing a decomposition for multiple adversaries.
10. The method of claim 9, wherein performing a decomposition comprises constructing a decomposed MIQP of the form,
\u2211
i
\u2208
\u2135
\u2062
\u2211
l
\u2208
L
\u2062
\u2211
j
\u2208
Q
\u2062
p
l
k
\u2062
R
ij
l
\u2062
x
i
\u2062
q
j
l
.
11. The method of claim 10, wherein the MIQP is subject to constraints,
\u2211
i
\u2062
x
i
\u2062
=
k
,
\u2062
\u2211
j
\u2208
Q
\u2062
q
j
l
=
1
,
\u2062
0
\u2264
(
a
l
–
\u2211
i
\u2208
\u2135
\u2062
1
k
\u2062
C
ij
l
\u2062
x
i
)
\u2264
(
1
–
q
j
l
)
\u2062
M
,
\u2062
x
i
\u2208
{
0
,
1
,
\u2026
\u2062
,
k
}
,
\u2062
and
q
j
l
\u2208
{
0
,
1
}
.
wherein a is the follower’s maximum reward value, and M is a large constant.
12. The method of claim 9, wherein performing a decomposition comprises constructing a decomposed MILP of the form,
\u2211
i
\u2208
\u2135
\u2062
\u2211
l
\u2208
L
\u2062
\u2211
j
\u2208
Q
\u2062
p
l
k
\u2062
R
ij
l
\u2062
z
ij
j
.
13. The method of claim 12, wherein the MILP is subject to constraints,
\u2211
i
\u2208
\u2135
\u2062
\u2211
j
\u2208
Q
\u2062
z
ij
l
=
k
,
\u2062
\u2211
j
\u2208
Q
\u2062
z
ij
l
\u2264
k
,
\u2062
\u2211
j
\u2208
Q
\u2062
q
j
l
=
1
,
\u2062
0
\u2264
(
a
l
–
\u2211
i
\u2208
\u2135
\u2062
1
k
\u2062
C
ij
l
\u2062
x
i
\u2061
(
\u2211
h
\u2208
Q
\u2062
z
ih
l
)
)
\u2264
(
1
–
q
j
l
)
\u2062
M
,
\u2062
\u2211
i
\u2208
Q
\u2062
z
ij
l
=
\u2211
j
\u2208
Q
\u2062
z
ij
l
,
\u2062
z
ij
l
\u2208
{
0
,
1
,
\u2026
\u2062
,
k
}
,
\u2062
and
\u2062
\u2062
q
j
l
\u2208
{
0
,
1
}
.
wherein a is the follower’s maximum reward value, and M is a large constant.
14. The method of claim 1, wherein the agent comprises a robot or an unmanned aerial vehicle (UAV).
15. The method of claim 14, wherein the security domain comprises infrastructure to be protected.
16. The method of claim 14, wherein the security domain comprises one or more airport terminals to be protected.
17. A computer-executable program product comprising a non-transitory computer-readable medium with resident computer-readable instructions, the computer readable instructions comprising instructions for:
fixing the policy of a leader to a k-uniform policy according to a Stackelberg game for a security domain;
formulating an optimization problem the follower solves to find its optimal response to the leader’s policy;
maximizing the payoff of the leader, given that the follower uses an optimal response that is function of the leader’s policy, and formulating the problem as a mixed integer quadratic program (MIQP);
performing a decomposition of the MIQP to produce a decomposed MIQP, wherein the decomposition addresses the size of the MIQP;
reformulating the leader payoff problem by linearizing the decomposed MIQP to form a decomposed mixed-integer linear program (MILP);
solving the decomposed MILP, wherein the leader’s strategy for a security domain can be determined; and
directing a patrolling strategy of an agent within the security domain based on the solved decomposed MILP.
18. The computer-executable program product of claim 17, wherein the computer-readable instructions comprise an optimization problem the follower solves of the form,
\u2211
j
\u2208
Q
\u2062
\u2211
i
\u2208
\u2135
\u2062
1
k
\u2062
C
ij
\u2062
x
i
\u2062
q
j
,
wherein the leader is the row player and the follower the column player, the vector of strategies of the leader are denoted by x, q denotes the vector of strategies of the follower, X and Q denote the index sets of the leader and follower’s pure strategies, respectively, k is the size of the multiset, and the payoff matrix C correspond to Cij that is the reward of the follower when the leader takes pure strategy i and the follower takes pure strategy j.
19. The computer-executable program product of claim 18, wherein the computer-readable instructions including the optimization problem include constraints,
\u2211
j
\u2208
Q
\u2062
q
j
=
1
and q\u22670.
20. The computer-executable program product of claim 18, wherein for the k-uniform policy x, the value xi is the number of times pure strategy i is used in the k-uniform policy, and wherein the k-uniform policy is selected with probability xik.
21. The computer-executable program product of claim 18, wherein the MIQP is of the form,
max
\u2062
\u2062
\u2211
i
\u2208
\u2135
\u2062
\u2211
j
\u2208
Q
\u2062
1
k
\u2062
R
ij
\u2062
q
\u2061
(
x
)
j
\u2062
x
i
,
and wherein the payoff matrix R corresponds to Rij, the reward of the leader.
22. The computer-executable program product of claim 21, wherein the MIQP is subject to constraints,
\u2211
i
\u2208
\u2135
\u2062
x
i
=
k
and xi\u2208{0, 1, . . . , k}.
23. The computer-executable program product of claim 18, wherein the MIQP is of the form,
max
\u2062
\u2062
\u2211
i
\u2208
\u2135
\u2062
\u2211
j
\u2208
Q
\u2062
1
k
\u2062
R
ij
\u2062
x
i
\u2062
q
j
,
and wherein the payoff matrix R corresponds to Rij, the reward of the leader.
24. The computer-executable program product of claim 23, wherein the MIQP is subject to constraints,
\u2211
i
\u2062
x
i
=
k
,
\u2062
\u2211
i
\u2208
Q
\u2062
q
j
=
1
,
\u2062
0
\u2264
(
a
–
\u2211
i
\u2208
\u2135
\u2062
1
k
\u2062
C
ij
\u2062
x
i
)
\u2264
(
1
–
q
j
)
\u2062
M
,
\u2062
x
i
\u2208
{
0
,
1
,
\u2026
\u2062
,
k
}
,
\u2062
and
q
i
\u2208
{
0
,
1
}
.
wherein a is the follower’s maximum reward value, and M is a large constant.
25. The computer-executable program product of claim 17, further comprising instructions for performing a decomposition for multiple adversaries.
26. The computer-executable program product of claim 25, wherein performing a decomposition comprises constructing a decomposed MIQP of the form,
\u2211
i
\u2208
\u2135
\u2062
\u2211
l
\u2208
L
\u2062
\u2211
j
\u2208
Q
\u2062
p
l
k
\u2062
R
ij
l
\u2062
x
i
\u2062
q
j
l
.
27. The computer-executable program product of claim 26, wherein the MIQP is subject to constraints,
\u2211
i
\u2062
x
i
=
k
,
\u2062
\u2211
j
\u2208
Q
\u2062
q
j
l
=
1
,
\u2062
0
\u2264
(
a
l
–
\u2211
i
\u2208
\u2135
\u2062
1
k
\u2062
C
ij
l
\u2062
x
i
)
\u2264
(
1
–
q
j
l
)
\u2062
M
,
\u2062
x
i
\u2208
{
0
,
1
,
\u2026
\u2062
,
k
}
,
\u2062
and
q
j
l
\u2208
{
0
,
1
}
.
wherein a is the follower’s maximum reward value, and M is a large constant.
28. The computer-executable program product of claim 25, wherein performing a decomposition comprises constructing a decomposed MILP of the form,
\u2211
i
\u2208
\u2135
\u2062
\u2062
\u2211
l
\u2208
L
\u2062
\u2211
j
\u2208
Q
\u2062
p
l
k
\u2062
R
ij
l
\u2062
z
ij
j
.
29. The computer-executable program product of claim 28, wherein the MILP is subject to constraints,
\u2211
i
\u2208
\u2135
\u2062
\u2211
j
\u2208
Q
\u2062
z
ij
l
=
k
,
\u2062
\u2211
j
\u2208
Q
\u2062
z
ij
l
\u2264
k
,
\u2062
\u2211
j
\u2208
Q
\u2062
q
j
l
=
1
,
\u2062
0
\u2264
(
a
l
–
\u2211
i
\u2208
\u2135
\u2062
1
k
\u2062
C
ij
l
\u2062
x
i
\u2061
(
\u2211
h
\u2208
Q
\u2062
z
ih
l
)
)
\u2264
(
1
–
q
j
l
)
\u2062
M
,
\u2062
\u2211
i
\u2208
Q
\u2062
z
ij
l
=
\u2211
j
\u2208
Q
\u2062
z
ij
l
,
\u2062
z
ij
l
\u2208
{
0
,
1
,
\u2026
\u2062
,
k
}
,
\u2062
and
q
j
l
\u2208
{
0
,
1
}
.
wherein a is the follower’s maximum reward value, and M is a large constant.
30. The computer-executable program product of claim 17, further comprising instructions for directing the patrolling of a robot or an unmanned aerial vehicle (UAV) based on the solved MILP.
31. The computer-executable program product of claim 30, wherein the patrolling domain comprises infrastructure to be protected.
32. The computer-executable program product of claim 30, wherein the patrolling domain comprises one or more airport terminals to be protected.
33. A computer system comprising:
a processor configured to:
determine a leader’s strategy for a security domain by solving a Stackelberg game;
formulate a leader’s a k-uniform policy according to the Stackelberg game;
formulate an optimization problem the follower solves to find its optimal response to the leader’s policy;
maximize the payoff of the leader, given that the follower uses an optimal response that is function of the leader’s policy, and formulating the problem as mixed integer quadratic program (MIQP);
perform a decomposition of the MIQP to produce a decomposed MIQP, wherein the decomposition addresses the size of the MIQP;
reformulate the leader payoff problem by linearizing the decomposed MIQP to form a decomposed mixed-integer linear program (MILP);
solve the decomposed MILP; and
direct a patrolling strategy of an agent within the security domain based on the solved decomposed MILP.
34. The system of claim 33, wherein the processor is configured and arranged to direct a patrolling route of a robot or an unmanned aerial vehicle (UAV).
The claims below are in addition to those above.
All refrences to claim(s) which appear below refer to the numbering after this setence.
1. A patient support apparatus comprising
an air supply,
an air fluidization therapy bed including a fluidization space and a fluidizable medium positioned in the fluidization space, the air fluidization therapy bed receiving a flow of pressurized air from the air supply, the flow of pressurized air operable to fluidize the fluidizable medium, and
a pulsator coupled to the air fluidization therapy bed and positioned to transmit air pulses through the fluidized fluidizable medium to impart a percussive force to a body supported on air fluidization therapy bed.
2. The patient support apparatus of claim 1, wherein the pulsator is in fluid communication with the air supply and operable to form air pulses from at least a portion of the flow from the air supply.
3. The patient support apparatus of claim 2, wherein the frequency and magnitude of the air pulses are variable.
4. The patient support apparatus of claim 3, wherein the patient support apparatus further comprises a percussionvibration controller operable to control the operation of the pulsator.
5. The patient support apparatus of claim 4, wherein the patient support apparatus further comprises a controller and a user interface, the user interface operable to receive inputs from a user indicative of at least one desired operating parameter of the percussionvibration controller, the controller operable to modify the operation of the percussionvibration controller to achieve the desired operating parameter.
6. The patient support apparatus of claim 1, wherein the patient support apparatus comprises a first conduit, a tap coupled to the first conduit, a second conduit coupled to the tap and the pulsator, a third conduit coupled to the tap and the air fluidization therapy bed, the third conduit smaller than the first conduit and the second conduit.
7. The patient support apparatus of claim 6, wherein the pulsator comprises a valve that is openable to allow air to flow through the second conduit and the valve of the pulsator.
8. The patient support apparatus of claim 7, wherein the valve of the pulsator is in communication with the space of the air fluidization therapy bed such that when the valve opens, air flows through the valve into the space.
9. The patient support apparatus of claim 8, wherein the valve opens and closes to create air pulses in the space of the air fluidization therapy bed.
10. The patient support apparatus of claim 7, wherein the air supply has a variable output, the output being modified to compensate for the flow of air used to provide the percussive forces.
11. The patient support apparatus of claim 1, wherein the air supply has a variable output, the output being modified to compensate for the flow of air used to provide the percussive forces.
12. A patient support apparatus comprising
an air supply,
a first patient support zone comprising an air fluidization therapy bed including a fluidization space and a fluidizable medium positioned in the fluidization space, the air fluidization therapy bed receiving a flow of pressurized air from the air supply, the flow of pressurized air operable to fluidize the fluidizable medium,
a second patient support zone comprising a plurality of inflatable support bladders, and
a sonic emitter coupled to the second portion and positioned to transmit sonic waves through the bladers to impart a percussive force to a body supported on the second portion.
13. The patient support apparatus of claim 12, wherein the sonic emitter has a variable frequency and amplitude.
14. The patient support apparatus of claim 13, wherein the patient support apparatus further comprises a sonic controller operable to control the frequency and amplitude of the sonic emitter.
15. The patient support apparatus of claim 14, wherein the patient support apparatus further includes a controller operable to control the fluid supply and the sonic controller.
16. The patient support apparatus of claim 15, wherein the patient support apparatus further includes a user interface, the user interface operable to accept a user input indicative of a desired operating parameter of the sonic emitter.
17. The patient support apparatus of claim 16, wherein the controller modifies operation of the sonic controller to achieve the desired operating parameter of the sonic emitter.
18. The patient support apparatus of claim 12, wherein the second patient support zone is movable relative to the first patient support zone.
19. The patient support apparatus of claim 18, wherein the patient support apparatus further comprises a sonic controller operable to control the frequency and amplitude of the sonic emitter.
20. The patient support apparatus of claim 19, wherein the sonic emitter is coupled to the second patient support zone to move therewith.