Model Order Reduction of an ISLANDED MICROGRID using Single Perturbation, Dir...
Time-Optimal Control of Wire Bonder Z-Axis
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Ruth and Joel Spira Laboratories of
Electromechanical System
Time-optimal Control of Z-Axis of Wire Bonder
Master Dissertation Defense
Deepak Agarwal
February, 2004
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Ruth and Joel Spira Laboratories of
Electromechanical System
Advisor
Dr. Meckl
Committee
Dr. Yao
Dr. Chiu
Lab Mates
Acknowledgements
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Ruth and Joel Spira Laboratories of
Electromechanical System
CONTENTS
Introduction
System Modeling and Analysis
Command Generation
Control Strategies
Results
Conclusion and future work
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INTRODUCTIONINTRODUCTION
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OBJECTIVES
Research Motivation
Improve bond-cycle time to improve productivity.
Develop methodology to control residual vibrations of bond head.
Develop time-optimal motion commands.
Develop controller design framework.
Introduction
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Introduction
Input Force Output Acceleration
Rigid body
displacement
TIME OPTIMAL COMMAND METHODLOGY
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Wire Bonder: Kulicke and Soffa Model 1418/19
Main Components
Computer
Work area
Horizontal Movement: X-Y Table Vertical Movement: Z-axis
SYSTEM SETUP
Introduction
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SYSTEM MODELING ANDSYSTEM MODELING AND
ANALYSISANALYSIS
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SYSTEM MODELING
System Modeling
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EQUATIONS OF MOTION
Equation of motion are formulated from Lagrange’s method.
Lagrange formulation
Equation of motion
i
i i i i
d T T U R
Q
dt q q q q
∂ ∂ ∂ ∂
− + + = ÷
∂ ∂ ∂ ∂ & &
System Modeling
1,2,...,i N=
( )( ) ( ) ( )1 1( ) ,
.
C C T AJ J J b L K G V Tθ θ θ θ θ θ θ
+ + + − = − ÷
&& &&
Nonlinear
inertial
effects
Nonlinear
damping
effects
Nonlinear
coriolis
effects
Net available
torque
Load torque
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EFFECTIVE INERTIA AND DAMPING
System Modeling
-150 -100 -50 0 50 100 150
0
1
2
3
4
5
x 10
-5
Crank angle (deg)
Nonlinearinertia(Kgm2
)
-150 -100 -50 0 50 100 150
1.45
1.5
1.55
x 10
-3
Crank angle (deg)
Nonlineardamping(Nmsec/rad)
( )J θ
CJ
Cb
( )
.
J θ
Operating Range
Operating Range
Operating range of crank rotation from -500
to 1000
.
Effective inertia is chosen as maximum value.
Effective damping is same as estimated damping from linear model.
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RIGID BODY DYNAMICS (PARAMETER ESTIMATION)
System Modeling
Z Motor Dynamics
c c m T AJ b K G Vθ θ τ+ = =&& &
Motor Frequency Response
Input: Voltage input to servo
amp
Output: Tachometer velocity
(volts)
Jc =3.32x10-5
kgm2
bc=1.494x10-3
Nm sec/rad
(estimated value)
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FLEXIBLE DYNAMICS (PARAMETER ESTIMATION)
System Modeling
Overall Frequency Response
Input: Servo amplifier
Voltage
Output: Tip Acceleration
(Volts)
ωn =4158 rad/sec
ς=0.032 (estimated value)
Overall linear transfer function
( )
( ) ( ) ( )
2
2 2
2.5
1 2
n
c c n n
K s
G s
J s b s s s
ω
τ ςω ω
=
+ + + +
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LINEARIZED AND NONLINEAR MODEL COMPARISON
L L
s
J s b+
Control Strategy
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MODEL VALIDATION (SERVO SYSTEM)
OPEN LOOP
Input: Voltage to amplifier
Output: Servo angular
displacement
Discrepancy
Uncertainties and
nonlinearities not modeled.
System Analysis
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MODEL VALIDATION (OVERALL SYSTEM)
System Analysis
0.88 0.9 0.92 0.94 0.96 0.98 1
-20
-10
0
10
20
Actual Response 15 deg
Simulated Response 15 deg
0.88 0.9 0.92 0.94 0.96 0.98 1
-10
0
10
TimeResponse(Volts)
Actual Response 0 deg
Simulated Response 0 deg
0.88 0.9 0.92 0.94 0.96 0.98 1
-10
-5
0
5
10
Time (sec)
Actual Response -65 deg
Simulated Response -65 deg
OPEN LOOP
Input: Square wave voltage
to amplifier
Output: Accelerometer
output (volts)
0.87 0.875 0.88 0.885 0.89 0.895 0.9
-20
-15
-10
-5
0
5
10
15
20
Actual Response 15 deg
Simulated Response 15 deg
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COMMAND GENERATIONCOMMAND GENERATION
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OBJECTIVES
Develop methods for generating exact zero-vibration commands.
Develop methods for generating commands that are robust to
modeling errors.
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WIRE BONDING CYCLE (Z WAVEFORMS)
Command Generation
Prime Focus: 5 High Acceleration Moves in the Cycle
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0 0.05 0.1 0.15 0.2 0.25
-40
-20
0
20
40Velcoity(rad/s)
0 0.05 0.1 0.15 0.2 0.25
-5000
0
5000
Time (sec)
Acceleration(rad/s2
)
Obtain Z-velocity waveforms experimentally
Differentiate to get acceleration profile
Command Generation
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DEVELOPMENT OF TIME OPTIMIZED WAVEFORMS
Utilize maximum available torque supplied by the actuator
THE APPROACH
Divide actuator saturation limit by peak acceleration in the
bonding cycle.
Use this scale factor to increase the current waveform acceleration.
Numerically integrate acceleration profile to generate
displacement.
Find the time-value where the current waveform’s displacement
equals the new displacement.
The reduced time is optimized-time for each move.
Generate a complete waveform.
Command Generation
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RESULTS
Current waveform
Residual vibration : 2 m/s2
Torque-optimized waveform
Residual vibration : about 40 m/s2
Time saving: 17.8%
Command Generation
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APPROACH FOR GENERATING TORQUE-OPTIMIZED
COMMANDS
1ST
STEP
Rigid body optimal move time estimation.
Tr – rigid body move time (sec)
b – viscous damping (Nm sec/rad)
J – rigid body inertia (kgm2
)
- rigid body move (rad)
Command Generation
2 2
1 2 0
fr
r
MAX
bb T b
TJ F J
e e
θ ×
+ ÷
− + =
fθ
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APPROACH
2nd
STEP
Non-dimension form of transfer function
is non-dimensional rigid body pole,
ς is the damping ratio of the flexible mode.
is the natural frequency of flexible mode
Command Generation
( ) ( )( )
*
2 * 2
4 1
( )
2 1n r
G s
s s s sT τ ςω
=
+ + +
*
τ
nω
( ) ( )( )( )
*
2 * 2 2
4 1
( )
2 1 2 1n r
G s
s s s s s sT τ ς ςω
=
+ + + + +
Non-robust case
Robust case
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APPROACH
3rd
step
Discretizing the continuous state space form of the system.
where is discrete system matrix
discrete input matrix
4th
step
Least square optimization such that
subject to constraint
Command Generation
( ) k
ϕ = +0UΦ x CU
T
e= A
Φ
1
[ k−
=CΦ Γ 2k−
Φ Γ 3k−
Φ Γ ......... ]Γ
0
T
e dξ
ξ= ∫
A
Γ b
0[u=U 1u 2u 1......... ]ku −
1
2
min( )T
η ϕ ϕ=
min 1 maxku u u−< <
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SIMULATION RESULTS
0 0.05 0.1 0.15 0.2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Time (sec)
Shapedtorque(Nm)
Command Generation
0 0.05 0.1 0.15 0.2
-1000
0
1000
Time (sec)
Tipacceleration(m/sec2
)
0.092 0.094 0.096 0.098 0.1 0.102 0.104 0.106 0.108
-2
0
2
Time (sec)
Tipacceleration(m/sec2
)
0 0.05 0.1 0.15 0.2
-0.01
-0.005
0
Time (sec)
Tipdisplacement(m)
Open loop linear
system
INPUT
OUTPUT
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COMMAND GENERATION FOR NONLINEAR SYSTEM
Command Generation
Rigid body moves Inertia value Damping value
Descent to first bond. 1.08Jc bc
Ascent to loop
height/Descent to second
bond
1.15Jc bc
Ascent to clamp height. 1.6Jc bc
Rise to reset height 1.5Jc bc
Effective values of inertia and damping used for different moves
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CONTROL STRATEGYCONTROL STRATEGY
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CONTROL DESIGN FRAMEWORK - 2 DOF CONTROLLER
DESIGN
Control Strategy
The most effective control strategy is one which combines:
•Shaped command
•Feedback controller
•Feedforward controller
Command
Generator
Feedforward
Controller
Feedback
Controller Plant
Output
Desired
Action
Reference
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TASK DELEGATION
Shaped commands
Serve as reference for the system.
Incorporates robustness into shaped commands to variations in natural
frequency and damping ratio.
Feedforward controller
Makes transient response faster.
Cancels plant nonlinearities.
Feedback controller
Reduces the effect of model uncertainties.
Good regulation in presence of disturbances.
Control Strategy
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SIMULATION RESULTSSIMULATION RESULTS
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PERFORMANCE SPECIFICATIONS
Performance specifications
Residual vibration level less than 10 m/s2
Cycle time less than 285.6 msec
Inputs used for performance comparison
Bang-bang
Ramped sinusoids
Results
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OPEN LOOP SIMULATION RESULTS
Input: Shaped torque
Problem: Drift in displacement
Solution: Feedback control
Results
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CLOSED LOOP SIMULATION RESULTS (WITH
FEEDBACK)
Results
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-2000
-1000
0
1000
Time (sec)
Accleration(m/sec2
)
0.118 0.12 0.122 0.124 0.126 0.128 0.13 0.132 0.134 0.136 0.138
-100
0
100
200
Time (sec)
Accleration(m/sec2
)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-10
-5
0
5
x 10
-3
Time (sec)
Displacement(m)
Input: Shaped torque
Problem: Residual vibration
Solution: Feedforward control
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CLOSED LOOP SIMULATION RESULTS (WITH
FEEDBACK AND FEEDFORWARD)
P
L L
s
G
J s b
=
+
Results
( )( ) ( ) ( , ) ( )m loadJ J b Lτ θ θ θ θ θ θ τ θ= + + − +&& & &&
Feedforward control strategies
Linear feedforward
Rigid body inversion
Linear plant inversion
Nonlinear feedforward
Rigid body inversion
Nonlinear plant inversion
(computed torque)
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CLOSED LOOP SIMULATION RESULTS (WITH
FEEDBACK AND FEEDFORWARD)
Results
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1000
-500
0
500
1000
Time (sec)
TipAccleration(m/sec2
)
Closed Loop Acceleration of Nonlinear Model with Nonlinear Feedforward
Closed Loop Acceleration of Nonlinear Model with Linear Feedforward
0.17 0.172 0.174 0.176 0.178 0.18 0.182 0.184 0.186 0.188 0.19
-15
-10
-5
0
5
10
15
Time (sec)
TipAccleration(m/sec2
)
Closed Loop Acceleration of Nonlinear Model with Nonlinear Feedforward
Closed Loop Acceleration of Linear Model with Linear Feedforward
Reduction in residual vibration.
Does not satisfy performance
specification.
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BANG-BANG vs. RAMPED SINUSOID INPUT
INPUT COMPARISON
Results
Bang-bang input profile
Better available torque
utilization.
Better time-optimality.
Ramped sinusoid input profile
Better vibration suppression
capability.
More robust to plant
uncertainties.
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BANG-BANG vs. RAMPED SINUSOID INPUT
CLOSED LOOP ACCELEARTION COMPARISON
Results
Bang-bang input is time-optimal
Ramped sinusoid input has lesser
vibration level.
Tradeoff
Time-optimality vs. Vibration
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1000
-500
0
500
1000
Tim e (sec)
TipAccleration(m/sec2
)
0.045 0.05 0.055 0.06
-15
-10
-5
0
5
10
15
Tim e (sec)
TipAccleration(m/sec2
)
nonlinear model with bang-bang input
nonlinear model with ramped sinusoid
nonlinear model with bang-bang input
nonlinear model with ramped sinusoid
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ROBUSTNESS TO PARAMETER VARIATIONS
Investigate the system performance when the following
parameters are varied:
Rigid body viscous damping
Rigid body inertia
Natural frequency
Damping
Results
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± 10% VARIATION IN RIGID BODY DAMPING
Results
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± 10% VARIATION IN RIGID BODY INERTIA
Results
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± 5% VARIATION IN NATURAL FREQUENCY
Results
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± 5% VARIATION IN MODAL DAMPING
Results
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BONDING CYCLE
ATTRIBUTES
CURRENT
WAVEFORM
OPTIMIZED-
TIME
WAVEFORM
BANG-BANG
PROFILE
RAMPED
SINUSOID
PROFILE
Total time for
5 moves (msec)
122.80 55.72 29.10 40.87
Saving in total
move time (msec)
--- 54.63% 76.30% 66.72%
Total non-move
time (msec)
160.8 160.88 162.8 163.03
Total bonding cycle
time (msec)
285.6 216.6 191.9 203.9
Time saving per
Cycle
--- 24.15% 32.8% 28.6%
Highest Residual
Vibration Amplitude
(m/s2
)
2 40 5 5
QUANTITATIVE SUMMARY
Results
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CONCLUSIONS
Developed approach is likely to work for nonlinear systems in which:
Nonlinearities are tied in with the rigid-body mode.
Resonant mode is well defined.
Robustness to modeling errors in natural frequency and damping ratio
can be incorporated into the design of shaped commands.
The effective control strategies combine shaped command, feedback,
and feedforward controller to achieve fast motion with minimum
residual vibration.
Wire bonder performance can be significantly improved as
A better level of residual vibration has been achieved.
Bonding cycle time has been reduced.
Conclusion
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FUTURE WORK
Generate synchronized controlled motions using X, Y and Z servos
which complete the bonding process, as z-axis model and control
strategies have been developed.
Implement developed control strategies on the actual system.
Modify current approach to accommodate negative velocity regions in
the profile where bonding force is applied.
Use nonlinear model for the optimization technique to generate shaped
motion profiles.
Future Work
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QUESTIONS ?
Notas del editor
Welcome to my thesis defense. Topic of my today’s defense is time-optimal of Z axis of wire bonder.
I will like to thank my advisor Dr. Meckl for his support and guidance. Also I will like to thank Dr. Yao and Dr. Chiu for serving on my graduate committee.
I will briefly touch upon these topics in my presentation. I start with the introduction followed by system modeling and analysis, command generation, control strategies, results and then I will conclude with conclusion and future work.
Current research is motivated by the following points. Improve the bonding-cycle time. By reducing the cycle time of the current waveform the productivity can be improved. …..Develop controller design framework that takes care of plant uncertainties and achieve good regulation.
Before proceeding further, I will like to briefly touch upon the basic idea of my research. If suppose we want to move from one point to another in minimum time we have to give this input which switches between max positive and max negative force instantly. The problem with input is that it excite the resonant vibration inherent in the structure. The output being the acceleration of mass 2.
But if we use another input which has some additional switches the output acceleration is this. The acceleration in second case has lot lesser residual vibration. The objective is to design such inputs that minimizes those residual vibration.
This is the wire bonder setup. It consists of 2 main components Computer and the work area. This is the z axis whose motion we are modeling.
The setup shown in the previous slide can be represented by the following schematic. The bond head is driven by the cam, which is in turn driven by the Z servo. The cam is attached to the bond head through a kinematic linkage. The linkage mechanism can be approximated by the vertical slider crank mechanism that is also shown by this figure. In the following discussion, a slider crank mechanism is considered that helps in carrying out the kinematic and dynamic analysis of the motion of z-axis of the system.
Lagrange method is used to come up with the equation of motion. This equation show the EOM. It consists of 4 parts which captures the nonlinear inertial effects damping effects and the coriolis effects. The 4th term is the net available torque. T1 is the load torque. Each of these are a nonlinear function of crank angle.
This slide shows the nonlinear variations of inertia and damping on the crank angle. The operating range of crank rotation is from -50 to 100. Red line shows the linear inertia and damping. Blue line shows the total inertia and damping. The effective value of inertia for the linearized system is chosen as the max value. Effective damping is the same as the estimated damping.
After coming up with the nonlinear motor model, the system parameters are estimated by considering the linear motor model. For estimating the motor viscous damping the frequency response if the motor is considered. This figure shows the frequency response of the actual motor and the linear model. As shown in the figure the model frequency response quite closely follows the actual frequency response. From this the viscous damping is estimated to be this.
If the overall frequency response of the system is observed, we will notice a peak. This peak can be modeled by the 2nd order transfer function. From this we can get the estimate of the natural frequency and damping ratio. Once we estimated all the parameters, the final transfer function can be represented in this way. (Js+b) is the rigid body pole the pole at 2200 is added to match the frequency response better.
This slide shows a comparison of the linear and the nonlinear models. In the linearized model the nonlinear motor model is simply replaced by linear servo model and the kinematics block is replaced by kinematics gain. Jl and bl are the linearized value of inertia and damping.
The motor model is validated in time domain. The first figure shows the actual input which is used for the model. The red one shows the simulated output and the blue one shows the actual output. There is some mismatch between the actual and the simulated response of the system. The above mismatch can be attributed to the plant uncertainties and nonlinearties, which are not captured in the model.
The overall model is also validated in time domain. As shown in figure the actual data is collected for various crank angles and the model is simulated for those angles. The simulated response quite closely captures the nonlinear trends of the actual system.
The objective of command generation is to generate zero-vibration motion commands that are robust to modeling errors.
The idea behind developing optimized waveforms is that current waveform does not utilize peak actuator effort. So there is further scope of reducing the cycle time of the current waveform if the peak actuator effort is applied. Here is the approach we followed.
The time optimized waveform is applied to the model. Peak residual vibration amplitude of around 40m/sec2 is obtained which is 20 times more than the current waveform peak residual vibration amplitude of 2m/sec2. Thus time optimality is achieved at the cost of increased level of residual vibration.
The first step in command generation is solving this equation for Tr.
The linear system we have is ill-conditioned for the optimization routine which we will using later to generate time-optimal command. So we decided to non-dimensionalized the transfer function.
For generating robust time optimal command one more complex set of poles are added.
The bang-bang force profile has the effect of placing a zero over each of the flexible poles of the system. The additional robustness can be achieved by placing multiple zeros at the nominal location of the poles of the system.
The 3rd step is to discretize the continuous state space form of the transfer function.
Using a discrete-time model, the objective is to find an input, bounded by the actuator limits, that reduces the norm of the difference between the desired and the actual state vector to zero at the end of the control time. Assuming the system is linear time invariant, the optimal transition from -xk to 0 will be the same as the optimal transition from 0 to xk. Thus we can arbitrarily assign the final state to be 0 and the initial state to be the negative of the desired transition. By doing this, we can form the problem as finding the smallest time such that the 2-norm of equation is zero.
The time-optimal command generated by previous method is tested on the linear open loop system. This plot shows the output of the model which is tip acceleration. As shown in figure the output residual vibration level is 2m/sec2. We require this level to be less than 10m/sec2.
The command for the nonlinear system is generated by considering different value of inertia for different moves. Different value of inertia values are given in this table.
The 2-DOF controller design is choosen for implementing the control strategy. The most effective control strategy combines the shaped commands, feedback and feedforward controller.
The task delegated to each of the component is as follows. The shaped commands serve as the reference to the system. Certain amount of robustness to variations in system parameters like natural frequency and damping ratio can be incorporated into the design of shaped commands.
Feedforward controller makes the transient response faster and cancels the plant nonlinearities.
Feedback controller reduces the effect of model uncertainties and provide good regulation in presence of disturbances.
Performance specifications are as follows: the residual vibration level should not exceed 10m/sec2 and the cycle time should be less than the 285.6 msec which is the current cycle time.
Two inputs are used for performance comparison; bang-bang and ramped sinusoids.
This slide present the open loop simulation results. The input being the shaped torque and the output being the tip acceleration. The problem with this implementation is that there is drift in displacement and the residual vibrations are not within the bounds of 10m/sec2. The obvious solution would be to use the feedback controller.
This slide present the simulation results after feedback controller is implemented. The drift in displacement has been eliminated by controller implementation. The problem with this implementation is that the residual vibrations are still not within the bounds of 10m/sec2. The obvious solution would be to use the feedforward controller.
Two techniques of feedforward control strategy are explored. The first one is the inversion of linear motor model given by this transfer function. The second one is the inversion of nonlinear model which is given by this nonlinear equation.
Both the feedback strategies are tested on the model. A shown in figure the blue one shows linear feedforward and the red one with nonlinear feedforward. The residual vibration with nonlinear feedforward are lot smaller as compared with linear feedforward. Also linear feedforward does not satisfy the residual vibration constraints.
Before moving to comparison of results of bang-bang and ramped sinusoid. Here is input comparison. Bang-bang profile utilizes the available torque in a much better way as compared to ramped sinusoid. As yu can observed from the figure the cycle time of the bang-bang profile is less than that of ramped sinusoid.
Ramped sinusoid has better vibration suppression capability and is more robust to modeling uncertainties.
This is the output of the closed loop implementation using bang-bang and ramped sinusoid input profile. As can be seen from the figure the level of residual vibration for the ramped sinusoid input is lesser as compared with the ban-bang input. So there is trade-off between time-optimality and the vibration level.
The robustness to variations in model is also considered. The parameters that are varied are rigid body inertia, rigid body damping, natural frequency and damping ratio.
This figure shows the output acceleration comparison for two inputs when the rigid body damping is varied ± 10%. Both the inputs are quite robust to this variations.
This figure shows the output acceleration comparison for two inputs when the rigid body inertia is varied ± 10%. Both the inputs are quite robust to this variations and remain with the bounds.
This figure shows the output acceleration comparison for two inputs when the natural frequency is varied ± 5%. The output of the bang-bang profile does not settle within the bounds thus more sensitive to variations in natural frequency.
This figure shows the output acceleration comparison for two inputs when the damping ratio is varied ± 5%. The output of the bang-bang profile does not settle within the bounds thus more sensitive to variations in damping ratio. The output of the ramped sinusoid settle within the vibration bounds.
This table compares the total move and non-move times for different moves and for different moves. Bang-bang profile achieves highest time saving as compared to ramped sinusoid. But the ramped sinusoid input is more robust to parameters variations.