m.s. seminar – metu aerospace engineering department january 2006 steering of redundant robotic...
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M.S. Seminar – METU Aerospace Engineering Department January 2006
Steering of Redundant Robotic Manipulators and Spacecraft Integrated Power and Attitude Control-Control Moment
Gyroscopes
M.S. Thesis Presentation
on
MIDDLE EAST TECHNICAL UNIVERSITY Aerospace Engineering Department
Presentation By : Alkan Altay
Thesis Supervisor : Assoc. Prof. Dr. Ozan Tekinalp
M.S. Seminar – METU Aerospace Engineering Department January 2006
Presentation Outline
Robotic Manipulator Simulations
IPAC-CMG Cluster & IPACS Simulations
IPAC-CMG Systems
Robotic Manipulators
Mechanical Analogy
Inverse Kinematics Problem & Solutions
Blended Inverse Steering Logic
Redundant Actuator Systems
Steering of Redundant Actuators
Thesis Work and Results
Conclusion & Future Work
2/34
M.S. Seminar – METU Aerospace Engineering Department January 2006
Integrated Power and Attitude Control System (IPACS)
IPACS
A Variable Speed CMG That Stores Energy
IPAC – CMG Cluster
3/34
M.S. Seminar – METU Aerospace Engineering Department January 2006
Integrated Power and Attitude Control - Control Moment Gyroscope (IPAC-CMG)• A CMG variant, whose flywheel spin rate is altered by a motor/generator
ikτ
hδhh
τ
kh
...
.
δJJdtd
J
Due to spin acceleration
Due to gimbal velocity
4/34
M.S. Seminar – METU Aerospace Engineering Department January 2006
IPAC-CMG Cluster- Single IPAC-CMG, single direction
- At least 3 IPAC-CMGs for 3-axis attitude control
- 1 redundancy
- Nearly spherical momentum envelope with β= 54.73 deg,
n
iicluster
1
hh
PYRAMID CONFIGURATION
5/34
M.S. Seminar – METU Aerospace Engineering Department January 2006
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
6/34
M.S. Seminar – METU Aerospace Engineering Department January 2006
The Mechanical Analogy
Total Ang. Mom. Position
IPAC-CMG Momentum Link Length
Torque End Effector Velocity
Steering Problem Steering Problem
7/34
θθJxω
δωδ,Jh
h
θxxωδ,hh
i
).( ).(
l
)( )(
i
M.S. Seminar – METU Aerospace Engineering Department January 2006
Inverse Kinematics Calculations Steering Laws
Steering Laws For Redundant Systems
Minimum 2-Norm Solution
Singularity Avodiance Steering Logic
Singularity Robust Inverses
xJθ .1?
Steer the actuator through the desired path
Calculate the angular speed of each actuator
Invert a rectangular matrix ?
What if singular ?
8/34
M.S. Seminar – METU Aerospace Engineering Department January 2006
Moore Penrose Pseudo Inverse (Minimum 2-Norm Solution)
).)((or .)( 11 τJJJδxJJJθ TTMP
TTMP
• Minimum normed vector; the solution that requires minimum energy
• Singularity is a problem
• Most steering laws are variants of this pseudo inverse
9/34
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
Blended InverseSatisfy two objectives; realize the desired path in desired configuration
}...{2
1min err
Terrerrerr
T xRxθQθθ
xθJx .err desirederr θθθ
and Q and R are symmetric positive definite weighting matrices
where,
PROBLEM SOLUTION
)...()..( 1 xRJθQJRJQθ Tdes
TBI
The proper desired quantity is injected through this term
10/34
Pre-plannedSteering
M.S. Seminar – METU Aerospace Engineering Department January 2006
rad/sec 10 and 5.0 with
3,2,1for 1.001.0
iuiii
3-link planar robot manipulator dynamics :
Robotic Manipulator Simulations
max
max
max if
if
)sgn(. u
u
u i
i
i
i
i
uSteering Logic
Direct Kinematical Relationship
11/34
M.S. Seminar – METU Aerospace Engineering Department January 2006
Robotic Manipulator Simulations (Test Case I)
AIMS :
• Repeatability performance of B-inverse on a routinely followed closed path
• Tracking performance of B-inverse, when supplied with false desθ
)15
cos(1
)15
sin(
0
22
0
11
txx
txx
)15
cos(5.05.2
)15
sin(5.0
0
22
0
11
txx
txx
sec 210135 and
sec 1050for
t
t
sec 135105for t
12/34
-1 0 1 2
-3
-2.5
-2
-1.5
-1
-0.5
0
x1 [ m ]
x 2
[ m
]
xrealized
x0
xend
M.S. Seminar – METU Aerospace Engineering Department January 2006
Robotic Manipulator Simulations (Test Case I –MP-inverse Results)
13/34
0
50
100
join
t an
gles
[
deg
]
20
40
60
80
100
120
140
160
180
200
220
0 100 200-30
-20
-10
0
10
join
t ve
loci
ties
[ d
eg/s
ec ]
0 100 200-10
-5
0
5
10
time [ sec ]0 100 200
-5
0
5
10
15
0 100 2000
1
2
3
4
5
time [ sec ]
man
ipul
abili
ty m
easu
re
M.S. Seminar – METU Aerospace Engineering Department January 2006
Robotic Manipulator Simulations (Test Case I –B-inverse Results)
14/34
-20
0
20
40
60
80
join
t an
gles
[
deg
]
40
60
80
100
120
140
60
80
100
120
joint anglesnode
0 100 200-15
-10
-5
0
5
join
t ve
loci
ties
[ d
eg/s
ec ]
0 100 200-20
-10
0
10
time [ sec ]0 100 200
-20
-10
0
10 0 100 2000
1
2
3
4
5
time [ sec ]
man
ipul
abili
ty m
easu
re
M.S. Seminar – METU Aerospace Engineering Department January 2006
Robotic Manipulator Simulations (Test Case II)
AIM :
• The singularity avoidance performance of B-inverse
• MP-inverse drives the system close to an escapable singularity at [ x1 , x2 ] = [-2 , 0 ]
)30
sin(
)30
cos(1
0
22
0
11
txx
txx
sec400for t
Escapable Singularity
15/34
-3 -2.5 -2 -1.5 -1-2
-1.5
-1
-0.5
0
x1 [ m ]
x 2
[ m
]
xrealized
x0
xend
M.S. Seminar – METU Aerospace Engineering Department January 2006
Robotic Manipulator Simulations (Test Case II –MP-inverse Results)
16/34
160
180
200
220
join
t an
gles
[
deg
]
-50
0
50
100
-100
-50
0
50
100
0 20 40-30
-20
-10
0
10
join
t ve
loci
ties
[ d
eg/s
ec ]
0 20 40-40
-20
0
20
40
time [ sec ]0 20 40
-10
0
10
20
30
0 20 400
0.5
1
1.5
2
2.5
time [ sec ]
man
ipul
abili
ty m
easu
re
M.S. Seminar – METU Aerospace Engineering Department January 2006
Robotic Manipulator Simulations (Test Case II –B-inverse Results)
17/34
200
220
240
260
280
join
t an
gles
[
deg
]
60
80
100
120
140
-300
-250
-200
-150
-100
-50joint anglesnode
0 20 40-2
0
2
4
6
join
t ve
loci
ties
[ d
eg/s
ec ]
0 20 40-15
-10
-5
0
5
time [ sec ]0 20 40
-20
-15
-10
-5
0
5
0 20 400
1
2
3
4
5
time [ sec ]
man
ipul
abili
ty m
easu
re
M.S. Seminar – METU Aerospace Engineering Department January 2006
18/34
Robotic Manipulator Simulations (Test Case II – Results)
Steering with MP-inverse Steering with B-inverse
Escapable Singularity Simulations
M.S. Seminar – METU Aerospace Engineering Department January 2006
Robotic Manipulator Simulations (Test Case III)
AIM :
• Singularity transition performance of B-inverse
• The path passes an inescapable singularity at [ x1 , x2 ] = [ 0 , 0 ]
)30
cos(1
)30
sin(
0
22
0
11
txx
txx
sec300for t
Inescapable Singularity
19/34
0 0.5 1 1.5 2 2.5
-1
-0.5
0
0.5
1
x1 [ m ]
x 2
[ m
]
xrealized
x0
xend
M.S. Seminar – METU Aerospace Engineering Department January 2006
Robotic Manipulator Simulations (Test Case III –MP-inverse Results)
20/34
0
20
40
60
join
t an
gles
[
deg
]
260
280
300
320
340
360
180
200
220
240
260
280
0 20 400
1
2
3
4
5
join
t ve
loci
ties
[ d
eg/s
ec ]
0 20 400
20
40
60
80
time [ sec ]0 20 40
-150
-100
-50
0
50
0 20 400
0.5
1
1.5
2
2.5
time [ sec ]
man
ipul
abili
ty m
easu
re
M.S. Seminar – METU Aerospace Engineering Department January 2006
Robotic Manipulator Simulations (Test Case III –B-inverse Results)
21/34
-10
-5
0
5
10
15
join
t an
gles
[
deg
]
250
300
350
400
450
50
100
150
200
250
300joint anglesnode
0 20 40-6
-4
-2
0
2
4
join
t ve
loci
ties
[ d
eg/s
ec ]
0 20 400
2
4
6
8
10
time [ sec ]0 20 40
-15
-10
-5
0
0 20 400
0.5
1
1.5
2
2.5
time [ sec ]
man
ipul
abili
ty m
easu
re
M.S. Seminar – METU Aerospace Engineering Department January 2006
22/34
Robotic Manipulator Simulations (Test Case III – Results)
Steering with B-inverse
Inescapable Singularity Simulations
M.S. Seminar – METU Aerospace Engineering Department January 2006
IPAC-CMG Cluster Simulations
23/34
IPAC-CMG Cluster
Torque and Power Commands
STEERING ALGORITHMS
Realized Torque and Power
Rate Command to each IPAC-CMG
AIMS :
• Investigate the performance of IPAC-CMG cluster
• Investigate the performance of B-inverse
M.S. Seminar – METU Aerospace Engineering Department January 2006
IPAC-CMG Cluster Simulations
24/34
Generic simulation model
( used in MP-inverse simulations )
B-inverse simulation model
Two different simulation models are employed to steer IPAC-CMG cluster
M.S. Seminar – METU Aerospace Engineering Department January 2006
0.1
0.2
0.3
0.4
0.5
x [
N.m
]
-1
-0.5
0
0.5
1
y [
N.m
]-1
-0.5
0
0.5
1
z [
N.m
]
0 50 100 150 2000
10
20
30
40
50
60
t [ sec ]
h x [
N.m
.s ]
0 50 100 150 200-1
-0.5
0
0.5
1
t [ sec ]
h y [
N.m
.s ]
0 50 100 150 200-1
-0.5
0
0.5
1
t [ sec ]
h z [
N.m
.s ]
IPAC-CMG Cluster Simulations
25/34
0 50 100 150 20030
35
40
45
50
t [ sec ]
Eki
netic
[
Wat
t-h
]
0 50 100 150 200-350
-300
-250
-200
-150
-100
t [ sec ]
Pco
mm
and
[ W
att
]
Torque Command
Power Command
Min Ang.Mom.of each IPAC-CMG [Nms]
7.7
IPAC-CMG Flywh. Spin Interval [kRPM]
15 – 60
Initial Flywheel Spin Rates (kRPM) [40, 40, 40, 40]
Initial Gimbal Angles (deg) [0, 0, 0, 0]
M.S. Seminar – METU Aerospace Engineering Department January 2006
IPAC-CMG Cluster Simulations – MP-inverse Results
26/34
-0.1
0
0.1
0.2
0.3
0.4
x [
N.m
]
-1
-0.5
0
0.5
1
y [
N.m
]-1
-0.5
0
0.5
1
z [
N.m
]0 50 100 150 200
0
5
10
15
20
25
t [ sec ]
h x [
N.m
.s ]
0 50 100 150 200-1
-0.5
0
0.5
1
t [ sec ]
h y [
N.m
.s ]
0 50 100 150 200-1
-0.5
0
0.5
1
t [ sec ]h z
[ N
.m.s
]
real
comm
Torque & Angular Momentum RealizedSingularity Measure
0 50 100 150 2000
0.5
1
1.5
t [ sec ]
Sin
gula
rity
Mea
sure
0 50 100 150 200-100
-80
-60
-40
-20
0
1 [
deg
]
0 50 100 150 200-1
-0.5
0
0.5
1
2 [
deg
]
0 50 100 150 2000
20
40
60
80
100
t [ sec ]
3 [
deg
]
0 50 100 150 200-1
-0.5
0
0.5
1
t [ sec ]
4 [
deg
]
Gimbal Angle History
0 50 100 150 20032
34
36
38
40
1
[
kRP
M ]
0 50 100 150 20032
34
36
38
40
2
[
kRP
M ]
0 50 100 150 20032
34
36
38
40
t [ sec ]
3
[
kRP
M ]
0 50 100 150 20032
34
36
38
40
t [ sec ]
4
[
kRP
M ]
Flywheel Spin Rates
0 50 100 150 20030
35
40
45
50
t [ sec ]
Eki
netic
[
Wat
t-h
]
0 50 100 150 200-350
-300
-250
-200
-150
-100
t [ sec ]
Pre
aliz
ed
[ W
att
]
Energy and Power Profiles
M.S. Seminar – METU Aerospace Engineering Department January 2006
IPAC-CMG Cluster Simulations – B-inverse Results
27/34
Torque Error & Ang. Mom. Profile
-4
-3
-2
-1
0
1x 10
-6
x err
or
[ N
.m ]
-1
0
1
2
3x 10
-8
y err
or
[ N
.m ]
-6
-4
-2
0
2x 10
-8
z err
or
[ N
.m ]
0 50 100 150 2000
10
20
30
40
50
60
t [ sec ]
h x [ N
.m.s
]
0 50 100 150 200-5
0
5
10
15x 10
-4
t [ sec ]
h y [ N
.m.s
]
0 50 100 150 200-5
0
5
10
15x 10
-4
t [ sec ]h z
[ N.m
.s ]
Singularity Measure
0 50 100 150 2000
0.5
1
1.5
2
t [ sec ]
Sin
gula
rity
Mea
sure
Gimbal Angle History
0 50 100 150 200-150
-100
-50
0
1 [
deg
]
0 50 100 150 200-50
0
50
100
150
200
2 [
deg
]
nodes
0 50 100 150 2000
50
100
150
200
t [ sec ]
3 [
deg
]
0 50 100 150 200-60
-40
-20
0
t [ sec ]
4 [
deg
]
0 50 100 150 20032
34
36
38
40
1
[
kRP
M ]
0 50 100 150 20032
34
36
38
40
2
[
kRP
M ]
0 50 100 150 20032
34
36
38
40
t [ sec ]
3
[
kRP
M ]
0 50 100 150 20032
34
36
38
40
t [ sec ]
4
[
kRP
M ]
Flywheel Spin Rates
0 50 100 150 20030
35
40
45
50
t [ sec ]
Eki
netic
[
Wat
t-h
]
0 50 100 150 200-350
-300
-250
-200
-150
-100
t [ sec ]
Pre
aliz
ed
[ W
att
]
Energy and Power Profiles
M.S. Seminar – METU Aerospace Engineering Department January 2006
IPACS Simulations
28/34
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] [39, 40, 41, 42]
Initial Gimbal Angles [deg] [-75, 0, 75, 0]
M.S. Seminar – METU Aerospace Engineering Department January 2006
IPACS Simulations
29/34
Spacecraft IPACS Simulation Model
M.S. Seminar – METU Aerospace Engineering Department January 2006
IPACS Simulations
30/34
0
20
40
60
[ de
g ]
-1
0
1
[
deg
]
0 50 100 150 200 250 300-1
0
1
t [ sec ]
[ d
eg ]
0 50 100 150 200 250 300-300
-200
-100
0
100
200
t [ sec ]
Pco
mm
and
[ W
att
]
Attitude Command
Power Command
M.S. Seminar – METU Aerospace Engineering Department January 2006
IPACS Simulations – MP-inverse Results
31/34
Attitude Profile
0
200
400
600
[ de
g ]
-4
-2
0
2
[ de
g ]
0 50 100 150 200 250 300-4
-2
0
2
t [ sec ]
[ d
eg ]
RPYreal
RPYcomm
0 100 200 3000
0.1
0.2
t [ sec ]
Sin
gula
rity
Mea
sure
Singularity Measure
0 50 100 150 200 250 30035
36
37
38
39
40
41
42
t [ sec ]
[ kR
PM
]
1
2
3
4
IPAC-CMG Flywheel Spin RatesGimbal Angles
-95
-90
-85
-80
-75
-70
1
[ de
g ]
-3
-2
-1
0
1
2
[ de
g ]
0 100 200 30070
75
80
85
90
t [ sec ]
3
[ de
g ]
0 100 200 300-1
0
1
2
3
t [ sec ]
4
[ de
g ]
-0.4
-0.2
0
0.2
0.4
0.6
x [
N.m
]
-0.04
-0.02
0
0.02
0.04
y [
N.m
]
real
comm
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
z [
N.m
]0 100 200 300
22
22.5
23
23.5
24
24.5
t [ sec ]
h x [
N.m
.s ]
0 100 200 300-0.4
-0.2
0
0.2
0.4
0.6
t [ sec ]
h y [
N.m
.s ]
0 100 200 3000.4
0.5
0.6
0.7
0.8
0.9
t [ sec ]h z
[ N
.m.s
]
Torque and Angular Momentum History
0 100 200 30040
42
44
46
48
50
t [ sec ]
Eki
netic
[
Wat
t-h
]
0 100 200 300-400
-200
0
200
400
t [ sec ]
Pre
aliz
ed
[ W
att
]
Energy and Power Profile
M.S. Seminar – METU Aerospace Engineering Department January 2006
IPACS Simulations – B-inverse Results
32/34
Attitude Profile
-50
0
50
100
roll,
[ d
eg ]
-2
0
2
4x 10
-6
pitc
h,
[ de
g ]
0 50 100 150 200 250 300-4
-2
0
2x 10
-6
t [ sec ]
yaw
,
[ de
g ]
RPYreal
RPYcom
0 100 200 3000
0.5
1
1.5
t [ sec ]
Sin
gula
rity
Mea
sure
Singularity Measure
-80
-70
-60
-50 1
[ de
g ]
0
50
100
150
2 [
deg
]
nodes
0 100 200 30060
80
100
120
140
t [ sec ]
3 [
deg
]
0 100 200 300-100
-80
-60
-40
-20
0
t [ sec ]
4 [
deg
]
Gimbal Angles
-4
-2
0
2
4x 10
-8
x err
or
[
N.m
]
-6
-4
-2
0
2
4x 10
-9
y err
or
[
N.m
]-2
-1
0
1
2
3x 10
-8
z err
or
[
N.m
]0 100 200 300
23.2
23.4
23.6
23.8
24
24.2
t [ sec ]
h x [
N.m
.s ]
0 100 200 300-0.4
-0.2
0
0.2
0.4
0.6
t [ sec ]
h y [
N.m
.s ]
0 100 200 3000.6
0.7
0.8
0.9
1
t [ sec ]h z
[ N
.m.s
]
Torque Error and Ang.Mom. Profile
0 50 100 150 200 250 30035
36
37
38
39
40
41
42
t [ sec ]
[ kR
PM
]
1
2
3
4
IPAC-CMG Flywheel Spin Rates
0 100 200 30040
42
44
46
48
50
t [ sec ]
Eki
netic
[
Wat
t-h
]
0 100 200 300-400
-200
0
200
400
t [ sec ]
Pre
aliz
ed
[ W
att
]
Energy and Power Profiles
M.S. Seminar – METU Aerospace Engineering Department January 2006
Conclusion
33/34
• B-inverse is employed in robotic manipulators :
Singularity Avoidance
Singularity Transition
Repeatability
• IPACS is discussed :
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
Future Work
B-inverse in highly redundant robotic mechanisms
34/34
Detail Design of IPAC-CMG Capabilities of B-inverse
)...()..( 1 xRJθQJRJQθ Tdes
TBI
M.S. Seminar – METU Aerospace Engineering Department January 2006
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
#/30
M.S. Seminar – METU Aerospace Engineering Department January 2006
Singularity Avoidance Steering Logic
nxJJJθ ..).( 1 TT
Particular Solution
Homogeneous Solution
θθJx ).(
0. nJ
Addition of null motion, n, in the proper amount (determined by γ)
12/40
M.S. Seminar – METU Aerospace Engineering Department January 2006
Singularity Robust Solutions
• 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
Singularity Robust Inverse : xJJIJθ .).( 1 TT
SR k
k = 0 for m > mcr
k0(1-m/m0)2 for m < mcr
13/40
M.S. Seminar – METU Aerospace Engineering Department January 2006
Singularity Robust SolutionsNew generation of solutions, offering accurate and smooth singularity transitions, not mature yet
• Extended Jacobian Method
• Normal Form Approach
• Modified Jacobian Method
Extends the jacobian matrix with additional functions, creating a well –conditioned one, belonging to a “virtual” system
square matrix
)(
)(
θ
θJJ
fvir
singularity
Proposes to transform the kinematics to its quadratic normal form, employing equivalence transformation, around singularities
Proposes to replace the linearly dependent row of Jacobian Matrix, to remove the singularity, with a derivative of a configuration dependent function
J
)(θf
14/40
M.S. Seminar – METU Aerospace Engineering Department January 2006
Thesis Objectives
Blended Inverse on IPAC-CMG clusters
Blended Inverse on Redundant Robotic Manipulators
Spacecraft Energy Storage & Attitude Control
IPAC-CMG based IPACS
0 100 200-50
0
50
100
join
t an
gles
[
deg
]
0 100 20040
60
80
100
120
140
time [ sec ]0 100 200
40
60
80
100
120
0 100 200-15
-10
-5
0
5
join
t ve
loci
ties
[ d
eg/s
ec ]
0 100 200-20
-10
0
10
20
time [ sec ]0 100 200
-20
-10
0
10
20
joint anglesknot
3/40
M.S. Seminar – METU Aerospace Engineering Department January 2006
Spacecraft Energy Storage and Attitude Control
• Spacecraft store & drain energy periodically.
• Rotating flywheels for smooth attitude control
• Integrate energy storage & attitude control
Electrochemical Batteries
vs.Flywheel Energy Storage Systems (FES)
4/40
M.S. Seminar – METU Aerospace Engineering Department January 2006
Blended Inverse
How to select ?desθ
)...()..( 1 xRJθQJRJQθ Tdes
TBI
11/40
Pre-planned Steering
cur
curknotdes tk
t
θθθ
pk
tkttk cur
,...,1 where
)1(