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k1

BA

(d)(c)

(b)(a)

k1k1k1

k3

k2

k2

k3

k1k1

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k3

BA

k1k2

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in real time

system

in advance

parameter

control state

optimality criteria

runtime parameters

generation

trajectory

controller

on-line

information

region and parameter

recipe

control objectives

system model (ODE)

tolerance

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[origin destination] pair

(synthesis)(analysis)

stack of state-space portraits

tolerance

first passonly

dynamicsclassifier

tolerancechecker

MAPPER PATH FINDER

model

region information (R, m)

parameter information (range, step)

control objectives (A, B)

iteration depthparameter infooptimality procedures

model

FIRST

PASS

SETUP

A

B C

trajectory length and step

region corners and grid spacing

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o

second pass

first pass

d

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orig

dest

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A B

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u

theta

--

motor

velocitysense

positionsense

g1 g2

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(c)

Re

Im

0a2

a2 = 0

2

(a)

a2 = (a1/2)

(a1/2)2

a2

(b)

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different from

range?

k out of 

previous portrait?

]

section 4.4

section 4.3

section 4.2

k by

increment k 

hl

[A, B]; corners, m; ODE model;t, length;

generate

trajectories

dynamicsclassify

trajectories

discretize

stack of 

portraits

average k 

and old k 

Y

N

N

Y

Y

N

Di

D

k; [k k  

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(0 0) (3 0)

(3 3)

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3

parameter scale

d

d

d

bifurcation tochaotic attractor

chaotic attractorapproaches objectives

limit cycle

pass

number

1

1

1

1

1

2

2

3

4

4

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[A B]

Y

section 5.3

section 5.4

section 5.2

find best

segment

splice intopath and check 

tolerance

compute new

region info

invoke

mapper

pass?

 junction info

compute

recipe

tolerance

set of mapsoptim. cost

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a

b

(a)

[h, b][d, g][f, c][a, e]

[d, b][a, c]

[a, b]

(b)

= path finder invocation

a

b

c

d

e

f 

g

h

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dest

orig

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(3 3)

(0, 3)

(3 0)

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xx x

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box of [A, B] by

overrange factor

t; trajectory length;

expand bounding

search region corners

B

A

]hl k [k k 

;00

mRODE model;

trajectory length

timestep and

determine

[A, B]; tolerance;

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(b)(a)

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x(0)

+ Ix(0)

D

x(0)

x(t)

(b)(a)

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(c)(a) (b)

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T

L

T’

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(b) (c)(a)

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r3 = 10

(8.82, -4.37, 57.44)

(8.8230, -4.3716, 57.2139)

S3(8.10, 29.3, 63.7)

r1 = 40

r0 = 50

S0

S1

(8, 29, 64)

A(-24.5, -20, 68)

B

r2 = 10S2

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frequency

1)(per

A xxxx

x

x

(per = 1)periodic

x

20

3

2

15105

1

drive

drive

periodic chaotic

amplitude

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amplitude magnitude

1

1

T-T time

fre

w 2woo

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range

VCO input voltage

phase

difference

lock range

capture

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