Transcript Slide 1

MIDDLE EAST TECHNICAL UNIVERSITY
Aerospace Engineering Department
M.S. Thesis Presentation
on
Steering of Redundant Robotic Manipulators and Spacecraft
Integrated Power and Attitude Control-Control Moment Gyroscopes
Presentation By
: Alkan Altay
Thesis Supervisor
: Assoc. Prof. Dr. Ozan Tekinalp
M.S. Seminar – METU Aerospace Engineering Department January 2006
Presentation Outline




Redundant Actuator Systems

IPAC-CMG Systems

Robotic Manipulators

Mechanical Analogy
Steering of Redundant Actuators

Inverse Kinematics Problem & Solutions

Blended Inverse Steering Logic
Thesis Work and Results

Robotic Manipulator Simulations

IPAC-CMG Cluster & IPACS Simulations
Conclusion & Future Work
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Integrated Power and Attitude Control
System (IPACS)
IPAC – CMG Cluster
A Variable Speed CMG That
Stores Energy
IPACS
M.S. Seminar – METU Aerospace Engineering Department January 2006
3/34
Integrated Power and Attitude Control Control Moment Gyroscope (IPAC-CMG)
• A CMG variant, whose
flywheel spin rate is altered
by a motor/generator
Due to spin
acceleration


h  J .k

 dh   
τ
 h δh
dt




τ  J .k  J .δ. i
Due to gimbal
velocity
M.S. Seminar – METU Aerospace Engineering Department January 2006
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IPAC-CMG Cluster
-
Single IPAC-CMG, single direction
-
At least 3 IPAC-CMGs for 3-axis attitude control
n 

h cluster   h i
i 1
PYRAMID CONFIGURATION
-
1 redundancy
-
Nearly spherical momentum envelope with β= 54.73 deg,
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Robotic Manipulators
• An actuator system composed of
joints and series of segments
• Tasked to travel its end-effector
on a certain trajectory
• Redundancy Applied To Increase
Motion Capability
•
Mechanically analog to CMG cluster
M.S. Seminar – METU Aerospace Engineering Department January 2006
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The Mechanical Analogy
Total Ang. Mom.
IPAC-CMG Momentum
Torque
Steering Problem
h  h(δ, ω)
hi
 δ 
 
h  J (δ, ω). 
ω
 
x  x (θ)
li
x  J (θ).θ
Position
Link Length
End Effector Velocity
Steering Problem
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Inverse Kinematics Calculations
 Steering Laws
Steer the actuator through the desired path
Calculate the angular speed of each actuator
Invert a rectangular matrix ?
What if singular ?
Steering Laws For Redundant Systems
θ  J .x
1
?

Minimum 2-Norm Solution

Singularity Avodiance Steering Logic

Singularity Robust Inverses
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Moore Penrose Pseudo Inverse
(Minimum 2-Norm Solution)
θ MP  JT (JJT ) 1.x
(or δ MP  JT (JJT ) 1.τ)
•
Minimum normed vector; the solution that requires minimum energy
•
Singularity is a problem
•
Most steering laws are variants of this pseudo inverse
OTHER SOLUTIONS :
•
Singularity Avoidance Steering Logic
•
Singularity Robust Inverse, Damped Least Squares Method
•
Extended Jacobian Method, Normal Form Approach, Modified Jacobian Method
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Blended Inverse
Satisfy two objectives; realize the desired path in desired configuration
PROBLEM
SOLUTION
1 T

 Terr .Rx err }
min
{
θ
err .Q.θ err  x
θ 2
where,
x err  J.θ  x
θ BI  (Q  J T .R.J) 1 (Q.θ des  J T .R.x )
θ err  θ  θ desired
and Q and R are symmetric
positive definite weighting
matrices
The proper desired quantity is
injected through this term
Pre-planned
Steering
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Robotic Manipulator Simulations
3-link planar robot manipulator
dynamics :
Direct Kinematical
Relationship
0.01i  0.1i  ui for i  1,2,3
with   0.5 and   10 rad/sec
Steering Logic
ui 
 i
if  i  umax
umax. sgn( i ) if  i  umax
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Robotic Manipulator Simulations
Case I)
(Test
0
-0.5
for 0  t  105 sec
x1  x10  0.5 sin(
x
2
and 135  t  210 sec
[m]

t)
15

x2  x20  1  cos( t )
15
x1  x10  sin(

t)
15
x2  x20  2.5  0.5 cos(

t)
15
x realized
for 105  t  135 sec
x0
x end
-1
-1.5
-2
-2.5
-3
-1
0
x1
1
2
[m]
AIMS :
• Repeatability performance of B-inverse on a
routinely followed closed path
• Tracking performance of B-inverse, when supplied
with false θ des
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Robotic Manipulator Simulations
Case I –MP-inverse Results)
(Test
80
100
200
180
60
160
50
manipulability measure
joint velocities [ deg/sec ]
joint angles [ deg ]
220
140
40
120
0
20
10
100
10
15
0
5
10
-10
0
5
-20
-5
0
-30
0
100
200
-10
0
100
200
time [ sec ]
-5
0
100
5
4
3
2
1
0
0
100
200
time [ sec ]
200
M.S. Seminar – METU Aerospace Engineering Department January 2006
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80
140
60
120
40
100
20
80
0
joint angles
node
120
100
80
60
-20
60
40
5
0
10
10
0
0
-5
5
4
3
2
1
0
0
100
200
time [ sec ]
-10
-10
-10
(Test
manipulability measure
joint velocities [ deg/sec ]
joint angles [ deg ]
Robotic Manipulator Simulations
Case I –B-inverse Results)
-20
-15
0
100
200
-20
0
100
200
time [ sec ]
0
100
200
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Robotic Manipulator Simulations
Case II)
(Test
0
x0
for 0  t  40sec
x end
-1
2
[m]
-0.5
x

x1  x10  1  cos( t )
30

x2  x 20  sin( t )
30
x realized
-1.5
-2
-3
-2.5
x1
-2
-1.5
-1
[m]
AIM :
• The singularity avoidance performance of
B-inverse
• MP-inverse drives the system close to an
escapable singularity at [ x1 , x2 ] = [-2 , 0 ]
Escapable Singularity
M.S. Seminar – METU Aerospace Engineering Department January 2006
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220
100
200
50
100
50
0
180
0
-50
160
-50
-100
10
40
30
0
20
20
-10
0
10
-20
-20
0
-30
0
20
40
-40
(Test
manipulability measure
joint velocities [ deg/sec ]
joint angles [ deg ]
Robotic Manipulator Simulations
Case II –MP-inverse Results)
0
20
time [ sec ]
40
-10
0
20
2.5
2
1.5
1
0.5
0
0
20
time [ sec ]
40
40
M.S. Seminar – METU Aerospace Engineering Department January 2006
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280
140
260
120
-50
joint angles
node
-100
-150
240
100
-200
220
80
200
60
-300
6
5
5
4
0
2
-5
(Test
manipulability measure
joint velocities [ deg/sec ]
joint angles [ deg ]
Robotic Manipulator Simulations
Case II –B-inverse Results)
-250
0
-5
5
4
3
2
1
0
0
20
time [ sec ]
40
-10
0
-2
-10
0
20
40
-15
-15
0
20
time [ sec ]
40
-20
0
20
40
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Robotic Manipulator Simulations
Case II – Results)
(Test
Escapable Singularity Simulations
Steering with MP-inverse
Steering with B-inverse
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Robotic Manipulator Simulations
Case III)
1
(Test
x realized
x end
0
2
for 0  t  30sec
x0
x

x1  x10  sin( t )
30

x2  x20  1  cos( t )
30
[m]
0.5
-0.5
-1
0
0.5
1
1.5
x1
[m]
2
2.5
AIM :
• Singularity transition performance of
B-inverse
• The path passes an inescapable
singularity at [ x1 , x2 ] = [ 0 , 0 ]
Inescapable Singularity
M.S. Seminar – METU Aerospace Engineering Department January 2006
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joint velocities [ deg/sec ]
joint angles [ deg ]
60
360
280
340
260
320
240
300
220
280
200
0
260
180
5
80
50
60
0
40
20
4
(Test
manipulability measure
Robotic Manipulator Simulations
Case III –MP-inverse Results)
3
40
-50
20
-100
2
1
0
0
20
40
0
0
20
time [ sec ]
40
-150
0
20
2.5
2
1.5
1
0.5
0
0
20
time [ sec ]
40
40
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Robotic Manipulator Simulations
Case III –B-inverse Results)
450
10
250
400
5
200
350
0
150
300
-5
-10
joint velocities [ deg/sec ]
joint angles
node
300
manipulability measure
joint angles [ deg ]
15
100
250
4
10
2
8
0
6
-2
4
-4
2
-6
(Test
0
20
40
0
50
0
-5
2.5
2
1.5
1
0.5
0
0
20
time [ sec ]
40
-10
0
20
time [ sec ]
40
-15
0
20
40
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Robotic Manipulator Simulations
Case III – Results)
(Test
Inescapable Singularity Simulations
Steering with B-inverse
M.S. Seminar – METU Aerospace Engineering Department January 2006
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IPAC-CMG Cluster Simulations
Torque and
Power
Commands
Rate
Command
to each
IPAC-CMG
Realized
Torque
and Power
STEERING
ALGORITHMS
IPAC-CMG Cluster
AIMS :
•
Investigate the performance of IPAC-CMG cluster
•
Investigate the performance of B-inverse
M.S. Seminar – METU Aerospace Engineering Department January 2006
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IPAC-CMG Cluster Simulations
Two different simulation models are employed to steer IPAC-CMG cluster
Generic simulation model
B-inverse simulation model
( used in MP-inverse simulations )
M.S. Seminar – METU Aerospace Engineering Department January 2006
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IPAC-CMG Cluster Simulations
1
1
0.4
0.5
0.5
0.3
0
-0.5
-0.5
0.1
-1
-1
60
1
1
50
[ N.m.s ]
-150
-200
30
0
h
-0.5
-0.5
-300
IPAC-CMG Flywh. Spin Interval [kRPM]
10
-350
0
0
0
50
50
100
150
100
150
200
t [ sec ]
t [ sec ]
200 -1
0
Min Ang.Mom.of each IPAC-CMG [Nms]
y
-250
20
0.5
z
40
[ N.m.s ]
Power
0.5Command
h
P h [ N.m.s
[ Watt
]
]
command
x
200
0
0.2
-100
150
z [ N.m ]
0.5
y [ N.m ]
x [ N.m ]
Torque Command
0
50
Initial Flywheel Spin Rates -1(kRPM)
100
150
200
Initial
Angles (deg)
t [ secGimbal
]
0
50
7.7
15 – 60
[40, 40, 40, 40]
100
150
200
t [ sec ] [0, 0, 0, 0]
M.S. Seminar – METU Aerospace Engineering Department January 2006
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IPAC-CMG Cluster Simulations –
inverse Results
MP-
Flywheel
and Momentum
Power
Spin
Profiles
Rates
TorqueEnergy
& Angular
Realized
Gimbal
Angle
History
Singularity
Measure
50
z [ N.m ]
[ deg ]
[ kRPM ]
[ Watt ]
realized
2
-200036
2
y [ N.m ]
P
-0.5
-0.534
-250
-132
-300
0 0
0
-0.5
5050
100
100
150
150
200
200
-1
-350
-1
100 0 150
200
0
50
100
150
0
50
100 140150
200
t [ sec ]
t [ sec
1 ]
1
t [ sec ]
0
200
50 50 100100-0.5
150150 200
t
[
sec
]
t [ sec ]
200
-1
0
50
0.538
0.5
036
[ N.m.s ]
[ N.m.s ]
40
200
-0.534
-132
0 0
100
150
t [ sec ]
0
z
0.5
100
150
t [ sec ]
0.5
h
x
h
0
0
50 50 100100 150150 200
200
36
0 32
0 0
5
0.5
60
20
10
1
[ kRPM ]
34
0.5
0.538

-150
comm
4
[ kRPM ]
3
3 [ deg ]
15
38
real
4 [ deg ]
0 100 40
50
80
20
[ N.m.s ]
[ kRPM ]
-100 32
0 0
25
0
36
-10040

y
35
30
-0.1
-60
-80 34
E
0
40
-40
1
1
1
h
kinetic
0.1
45
38
1
0.2
1.5
-20
1 [ deg ]
[ Watt-h ]
0.3
x [ N.m ]
0 40
50
Singularity Measure
0.4
-0.5 150
100
150
5050 100
t
[
sec
]
t [ sec ]
200
-1
0
200
200
50
100
150
t [ sec ]
200
M.S. Seminar – METU Aerospace Engineering Department January 2006
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IPAC-CMG Cluster Simulations –
B-inverse Results
Torque
Singularity
Energy
Error &
and
Ang.
Measure
Power
Mom.
Profiles
Profile
Gimbal
Flywheel
Angle
Spin
History
Rates
30
50
100
100
0
36
100
30
20
34
50
32
0 0
0
10
0
50
100
t [ sec ]
50
50
150
150
150
50
10
34
0
-300
-350
200
50
50
0
38
100 -20
150
t [ sec ]36
200
200
50
100
150
0
-2
-4
100
100
50
150
150
200
200
-6
100
150
-4
15t [ sec ]
200
20010
-40
34
32
-60
00
x 10
x 10
40
0
0
0
-250
32
-50
00
200
200
5
100
150
150
100
t
[
sec
]
t [ sec ] -5
200
z error [ N.m ]
[ Watt ]
0
realized
1
100 -1 150
-4
t [ sec15] x 10
hy [ N.m.s ]
3 [ deg ]
3 [ kRPM ]
[ N.m.s ]
40
50
50
0.5
0
38
150
50
x
0
40
200
60
0
32
-150
00
100
-200
36
50
hz [ N.m.s ]
-3
1
2
P
35
1.5
2
150 -150
38
nodes
 [ deg ]
2 2 [ kRPM ]
1
-100
34
-2
-4
h
36
40
3 
 [ deg ]
4 4 [ kRPM ]
-1
-50
-8
200
40 -100
x 10
2
Singularity Measure
[ Watt-h ]
kinetic1 [ deg ]
[ kRPM ]

4538
0
E
x error [ N.m ]
1
-8
5040
0
y error [ N.m ]
-6
x 10
50
50
200
5
0
100
150
150
100
t
[
sec
]
t [ sec-5]
0
t [ sec ]
50
200
200
100
150
200
t [ sec ]
M.S. Seminar – METU Aerospace Engineering Department January 2006
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IPACS Simulations
Spacecraft Inertias [ kgm2 ]
[15, 15, 10]
Initial Orientation of S/C [deg]
[0, 0, 0]
IPAC-CMG Flywh. Spin Interval [kRPM]
15 - 60
Initial Flywheel Spin Rates [kRPM]
Initial Gimbal Angles [deg]
[39, 40, 41, 42]
[-75, 0, 75, 0]
M.S. Seminar – METU Aerospace Engineering Department January 2006
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IPACS Simulations
Spacecraft IPACS Simulation Model
M.S. Seminar – METU Aerospace Engineering Department January 2006
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IPACS Simulations
 [ deg ]
60
40
20
1
0
-1
 [ deg ]
Attitude
Command
 [ deg ]
0
1
0
-1
0
50
100
150
t [ sec ]
200
250
300
200
250
300
P
Power
Command
command
[ Watt ]
200
100
0
-100
-200
-300
0
50
100
150
t [ sec ]
M.S. Seminar – METU Aerospace Engineering Department January 2006
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IPACS Simulations –
MP-inverse Results
[ N.m.s ]
7536
100
200
t [ sec ]
 2 [ deg ]
[ Watt ]
-400
z [ N.m ]
realized
P
-200
-1
-2
0.01
0
-0.01
-3
-0.02
3
-0.03
02
100
200
0.9
t [300
sec ]
200

0.8
0.2
0
0.7
y
z
0.6
-1
0
100
200
300
0
100
200
100
150
200
250
50t 50
100
150
200
250
[ sec
]
t [ sec0.5
]
-0.2
t t[ sec
]
[ sec
]
h
x
h
0
0
0.02
t [ sec1 ]
0.4
70
0
-4 35
0 0
22.5
200
100
200
300
0 0.6
t [ sec ]
0
100
-2
23
22
-0.04
-4 85
0
23.5
0
0.03
real
comm
[ N.m.s ]
90
38
-0.02
1
h
0
-2
0.1
2 8037
24
[ N.m.s ]
40
39
0-95
 [ deg
 3 ] [ deg ]
24.5
42
0
400
 4 [ deg ]
-0.4
2-90
 [ deg ]
E
-0.2
0-8540
y [ N.m ]
44
46
RPYcomm
0.02
Singularity Measure
0
0.2
400-75
41
200-80
[ kRPM ]
[ Watt-h ]
48
0.04
RPY0.2
real

0.4
600 42
-70
 [ deg ]
50
 1 [ deg ]
0.6
kinetic
x [ N.m ]
Torque
Energy
IPAC-CMG
andSingularity
Angular
Attitude
and
Gimbal
Power
Flywheel
Momentum
Angles
Profile
Profile
Spin Rates
History
Measure
300
-0.4
0.4
0
100
200
t [ sec ]
300
0
300
1
2
3
4
300
300 300
100
200
t [ sec ]
300
M.S. Seminar – METU Aerospace Engineering Department January 2006
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IPACS Simulations –
inverse Results
B-
Torque
IPAC-CMG
Energy
Error
Gimbal
and
Attitude
Flywheel
Ang.Mom.
Power
Angles
Profile
Spin
Profiles
Profile
Rates
Singularity
Measure
[ N.m.s ]
x 10
360
y
35
60 0
0-4
0
0
100
200
t [ sec ]
50
[ N.m ]
z error
Watt] ]
[ [deg
0.6
00.4
0
1
-60
-80
-0.2 100
0
150 -100
sec
150]
sec ]
100 t [ 200
t [ sec ]
RPYreal
100
200 RPYcom
300
-2
t [ sec ]
100
200
-40 ]
t [ sec
0.2
0
-1
-20
100
200
300
50
100 t [
t [ sec -0.4
]
300
-200
0
h
yaw, 
23.4
-2
50
1
[ N.m.s ]
-6
80
23.6
23.2
100
[ deg ]
3 [ deg ]
[ N.m.s ]
x
h
23.8
2
[ N.m ]
0
0
-4000
0
300
2
z
140380
37
-4
100
200
t [ sec ]-6
120 -2
24
-2
200
100
h
0
0
1
0.5
x 10
nodes
P
-6
x 10
3
400
2
y error
40
-50
-70 44
Singularity Measure
roll,  [ deg ]
[ Watt-h ]
46
-8
150
x 10
4 [ deg ]
24.2
0
4
42
39
-80 40
2
-2
-4
48
-6041
[ kRPM
[ deg ]E ]
kinetic
0
4
1.5
50
50
 
pitch,
x error
2
1 [ deg ]
[ N.m ]
4
-9
42
-50100
2
realized
-8
x 10
200
0
300
300  1
0.9
2
3
0.8
4
0.7
250
100
200
200
250
t [0.6
sec ]
0
300
300
300
100
200
t [ sec ]
300
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Conclusion
•
B-inverse is employed in robotic manipulators :



•
Singularity Transition
Repeatability
IPACS is discussed :



•
Singularity Avoidance
Comparison to Current Technologies
Algorithm Construction
Theoretical Performance
B-inverse is employed in IPACS :


In IPAC-CMG Clusters & S/C IPACS
Singularity Avoidance & Multi Steering
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Future Work
B-inverse in highly redundant
robotic mechanisms
θ BI  (Q  J T .R.J) 1 (Q.θ des  J T .R.x )
Detail Design of IPAC-CMG
Capabilities of B-inverse
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Singularity in Robotic Manipulators and
CMG Systems
•
Physically, no end effector
velocity (torque) can be
produced in a certain direction
•
Controllability in that
direction is lost.
•
Mathematically, Jacobian
Matrix loses its rank.Thus;
1.
det(J)= 0 ( or det(JJT)=0 )
2.
Singularity Measure
m=det(JJT)
3.
J-1 ( or (JJT)-1 ) becomes
undefined
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Singularity Avoidance Steering Logic
x  J(θ).θ
θ  J T .(JJT ) 1.x   .n
Particular
Solution
Homogeneous
Solution
J.n  0
Addition of null motion,
n, in the proper amount
(determined by γ)
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Singularity Robust Solutions
Singularity
Robust Inverse :
θ SR  JT (k.I  JJT ) 1.x
0
k=
for m > mcr
k0(1-m/m0)2 for m < mcr
• Disturbs the pseudo solution near singularities to artificially
generate a well –conditioned matrix
• Increases the tracking error, causes sharp velocity changes
around singularities
•
Another example may be the Damped Least Squares Method
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Singularity Robust Solutions
New generation of solutions, offering accurate and smooth singularity
transitions, not mature yet
•
Extended Jacobian Method
Extends the jacobian matrix with
additional functions, creating a well –
conditioned one, belonging to a
“virtual” system
•
J vir
 J ( θ) 


 f ( θ) 
square matrix
singularity
Normal Form Approach
Proposes to transform the kinematics to its
quadratic normal form, employing equivalence
transformation, around singularities
•
Modified Jacobian Method
Proposes to replace the linearly dependent
row of Jacobian Matrix, to remove the
singularity, with a derivative of a configuration
dependent function



J









f (θ)
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Thesis Objectives
joint angles [ deg ]
100
50

Spacecraft Energy Storage
& Attitude Control

IPAC-CMG based IPACS

Blended Inverse on IPACCMG clusters
joint velocities [ deg/sec ]
Blended Inverse on Redundant
Robotic Manipulators
joint angles
knot
120
120
100
100
80
80
0
60
60
-50

140
0
100
200
40
0
100
200
time [ sec ]
40
5
20
20
10
0
10
-5
0
0
-10
-10
-10
-15
0
100
200
-20
0
100
200
time [ sec ]
-20
0
100
200
0
100
200
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Spacecraft Energy Storage and Attitude
Control
• Rotating flywheels for smooth
attitude control
• Spacecraft store & drain
energy periodically.
Electrochemical Batteries
vs.
Flywheel Energy Storage Systems (FES)
•
Integrate energy storage &
attitude control
M.S. Seminar – METU Aerospace Engineering Department January 2006
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Blended Inverse
θ BI  (Q  J T .R.J) 1 (Q.θ des  J T .R.x )
How to select θ des?
Pre-planned Steering
θ  θ cur
θ des  knot
kt  tcur
(k  1)t    tcur  kt  
where k  1,..., p
M.S. Seminar – METU Aerospace Engineering Department January 2006
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