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Vibration Suppression of a Cart-Flexible Pole
System using a Hybrid Controller
Presented by
Dr. Ashish SinglaAssistant Professor
Mechanical Engineering Department
Thapar University, Patiala
INDIA
Oral Presentation @ iNaCoMM-2013 Dec 19, 2013
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OUTLINE
Fig. Single-link flexible manipulator.
Dynamic Modeling
Command Shaping
Robustness of the Shaper
Controller Design
Results and Discussions
Shaper
RoboticManipulator
+ n( , ) + K
+
+
Kopt
K1C
-1
Ko2L
G
F
H
Compensator
Inverse Dynamics
xds
-
-
xd Inverse
Kinematics
TrajectoryGenerator
inCartesian
Space
xds
xd
yd
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Compensatorto PlantError Control Output
(Regulator + ROO) (Flexible Model)
Feedback Loop(L)
Input
-+
Inverse Dynamics
Feedforward Path (NL)
++
Ufb
Uff
to Input
Shaper
Input
Shaper
Reference
Trajectory
OVERALL SCHEME
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DYNAMIC MODELING
Small angle approximation.
Euler-Bernoulli beam theory
Lagrange approach
Each link - finitenumber of elements.
Fig. Moving cart with a flexible pole
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COMMAND SHAPING
Convolution
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Basis of Command Shaping
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Basis of Command Shaping . . .
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Basis of Command Shaping . . .
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System
s Response to Multiple Impulses (Superposition Principle)
Impulse Response of a Second Order Under-damped System
=
where
Development of Constraint Equations
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Residual Vibration Amplitude (in %)
Amplitude of Unit Impulse
Residual Vibration Amplitude
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(1)
(2)
(3)
(4)
Two Impulse Sequence
Constraints: 2 Eqs, 4 Unknowns
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Robustness of Input Shaper
Three Impulse Sequence (ZVD)
Four Impulse Sequence
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Sensitivity Curve
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CONTROLLER DESIGN
Linear observer based optimal feedback term
Nonlinear Feedforward Control
The feedforward control action is given as
Best utilized for fast and repetitive moves.Efficientsolution offlinecomputation (along the nominal trajectory).
Control action
Nonlinear feedforward Control
Input shaper
The controller consists of three parts
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Linear Feedback Control
estimation of the unmeasurable vector.
observer gain matrix solution of the followingstate equation
measured/unmeasured variables.
Compensator based on reduced-order observer
Plant state-vector
Plants state equation (partitionedform)
Estimated state-vector
ROO Dynamics
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Optimal feedback gain matrix Kopt is obtained by
Control Law
min
Error Dynamics
where e(t) = xd(t)x(t) = tracking erroreo2(t) = x2(t)xo2(t) = estimation error.
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Complete Tracking System with ROO and Shaper
Shaper
Robotic
Manipulator
+ n( , ) + K
+
+Kopt
K1C-1
Ko2L
G
F
H
Compensator
Inverse Dynamics
xds
-
-
xd Inverse
Kinematics
TrajectoryGenerator
inCartesian
Space
xds
xd
yd
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CASE STUDY : Cart with a flexible pole
Control objective is to move the cartby one meter, while not letting thependulum to fall i.e. xd= {1 0 0 0 0 0}
T
OL= [0, 0,5.89i,639i].
Double pole at origin- unstable plant -cannot be controlled using uff only.
ufb is essential and calculated usingoptimal control theory
The only input u(t) is the horizontalforce applied to the cart and threeoutputs are x, 1 and 2 .
Reduced-order observer is designed
by taking the carts position as theonly output variable. Fig. Moving cart with a flexible pole
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Mass of the cart in Kgs M 1
Mass of the first
(second) link in Kgs
m1 (m2) 0.05
Length of the first
(second) link in meters
l1 (l2) 1
Spring stiffness at first
joint in Nm/rad
k1 5
Spring stiffness at
second joint in Nm/rad
k2 500
State weighting
matrix
Q diag([5000 500 0 20 0 0])
Control cost
matrix
R [50]
Vector of desired
ROO poles
v [-20 -2260i -6069i]T
Optimal gain
matrix
Kopt [10 -1.29 -0.43 4.80 0.02 -0.01]
Impulse
First Mode Second Mode Third Mode
Mag. Time Mag. Time Mag. Time
1 0.9183 0.0000 0. 2792 0.0000 0.250 0.0000
2 0. 0799 1.4694 0. 4984 0. 5350 0.500 0.0049
3 0.0017 2.9387 0. 2224 1.0701 0.250 0.0098
System Parameters Controller Parameters
Wn= [ 3.02 5.87 638.92]T (Hz), = [0.706 0.0362 1e-5]TShaper Parameters
Parameters
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Plot 1(a) : One Mode Shaping Desired Position
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Plot 1(b) : One Mode Shaping Position Response
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Plot 1(c) : One Mode Shaping Velocity Response
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Plot 1(d) : One Mode Shaping Estimation Error
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Plot 2(a) : Two Mode Shaping Position Response
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Plot 2(b) : Two Mode Shaping Velocity Response
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Plot 3(a) : Three Mode Shaping Position Response
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Plot 3(b) : Three Mode Shaping Velocity Response
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ANIMATION
Unshaped Response 2-Mode Shaped Response
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Table : Level of Vibration/Force Reduction
Variable Bang-Bang
One Mode Shaping Two Mode Shaping Three Mode Shaping
Value % Reduction Value % Reduction Value % Reduction
|x(t)max| 1.0614 1.0112 4.73 % 1.0000 5.78 % 1.0000 5.78 %
|1(t)max| 13.0915 12.030 8.11 % 2.488 80.99 % 2. 488 81.00 %
|2(t)max| 0.0429 0.0393 8.32 % 0.0086 80.01 % 0.0078 81.83 %
|dx(t)max| 1.3069 1.2002 8.17 % 0.7903 39.53 % 0.7901 39.54 %
|d1(t)max| 1.1555 1.0613 8.16 % 0.2349 79.67 % 0.2272 80.33 %
|d2(t)max| 0.0616 0.0585 10.05 % 0.0148 75.97 % 0.0089 86.36 %
|u(t)max| 10.00 8.9206 10.79 % 4.3165 56.83 % 4.1920 58.08 %
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SUMMARY
The control scheme (LQR-ROO-CS) is sufficient to track large movements of flexiblerobotic manipulators.
The nonlinear feedforward control is derived using inverse dynamics, which provides
the major contribution in the control effort during tracking problems. The feedbackloop is designed with linear observer based optimal regulator which ensuresstabilization and performance objectives. Finally, the command shaping isincorporated to obtain the desired non-oscillatory response.
Large reductions in vibration levels as well as in input torque magnitudes are observed
when compared to controllers implemented without command shaping.
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FUTURE DIRECTIONS
The work has been implemented successfully on planar manipulators which can beextended to spatial manipulators.
The work can also be extended to redundant manipulators for Obstacle avoidance,
Increase in manipulability, Singularity avoidance,
Fault-tolerant design.
The feedback channel can be made more robust using Kalman filters.
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Thank You
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REFRENCES
M. O. Tokhi, Z. Mohamed, and M. H. Shaheed. Dynamic characterization of a flexiblemanipulator system. Robotica, 19(5):571580, 2001.
Z. Mohamed and M. O. Tokhi. Vibration control of a single-link flexiblemanipulator usingcommand shaping techniques. Proceedings of the Institution of Mechanical Engineers.
Part I: Journal of Systems and Control Engineering, 216(2):191
210, 2002.
NC Singer and WP Seering. Preshaping command inputs to reduce system vibration.Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME,112(1):7682, 1990.
WE Singhose, WP Seering, and NC Singer. Shaping inputs to reduce vibration: a vectordiagram approach. IEEE International Conference on Robotics and Automation, pages922927, 1990.W. Singhose, W. Seering, and N. Singer. Residual vibration reduction using vectordiagrams to generate shaped inputs. Journal of Mechanical Design, Transactions Of theASME, 116(2):654659, 1994.
J. Vaughan, A. Yano, and W. Singhose. Comparison of robust input shapers. Journal ofSound and Vibration 1 : 81 2008.
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Arun Banerjee and William Singhose. Command shaping in tracking control of a two-linkflexiblerobot. Journal of guidance, control, and dynamics, 21(6):10121015, 1998.
Z. Mohamed and M. O. Tokhi. Command shaping techniques for vibration control of aflexiblerobot manipulator. Mechatronics, 14(1):6990, 2004.
M. Z. M. Zain, M. O. Tokhi, and Z. Mohamed. Hybrid learning control schemes with inputshaping of a flexiblemanipulator system. Mechatronics, 16(3-4):209219, 2006.
WE Singhose and NC Singer. Effects of input shaping on two-dimensional tra jectoryfollowing. Robotics and Automation, IEEE Transactions on, 12(6):881887, 1996.
M. Romano, B.N. Agrawal, and F. Bernelli-Zazzera. Experiments on command shapingcontrol of a manipulator with flexiblelinks. Journal of Guidance, Control, and Dynamics,25(2):232239, 2002.
A. Tewari. Modern control design with MATLAB and SIMULINK. Chichester; Wiley.
J J Craig Introduction to Robotics: Mechanics and Control Addison Wesley LongmanP bli hi C I B MA USA 8
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