Design of Mini ROV
ME 6105: Modeling and Simulation in Design
HW4: Uncertainty Analysis
Dazhong Wu
Qing Chen
Nov. 4, 2009
1
Contents
Task 1: Become Familiar with ModelCenter ............................................................. 3
Task 2: Identify and Model the Uncertainty in your Design Analysis Model.......... 9
Task 3: Elicit a Detailed CDF for the Most Significant Uncertain Variables ........ 12
Task 4: Determine the Distribution of the Output of your Model ........................... 15
Task 5: Lessons learned ............................................................................................. 24
References .................................................................................................................... 33
2
Task 1: Become Familiar with ModelCenter

Customizing the fileWrapper for my Modelica model, and then install a Driver for
Latin Hypercube Sampling as shown below in ModelCenter:
Fig. 1 ModelCenter

Revise my Modelica model
At this point, we need to go back and revise your Modelica model to make it more
robust. In addition, the Modelica model is revised greatly based on the comment of
HW2. Some equations in HW2 were improved based on the references. The
following are updated models.
3
DC model:
The DC model is the same as what we did in class. The parameters are from
references.
damper
pin_p
resistor
inductor
R=0.49
L=1.08e-4
d=0.01
k=2.78...
pin_n
emf
inertia
fixed
flange_b
J=0.0001
Fig. 2 DC Motor
Battery:
pin_p
+
-
pin_n
constantVoltage=12
ground
Fig. 3 Battery
Propeller model:
This model is to calculate two important outputs of propeller. One is thrust. The other
is torque. The torque will function as a load to the shaft of the motor. Therefore, there
is feedback loop in this model. The flange_a is connected to the motor. In addition,
two formulas [1-4] for calculating thruster and torque are from references and online
4
materials. The formulas are generally used in literature but the accuracy depends on
the specific conditions. That is to say the coefficients in the formula need to be
determined by experiments data. Also, this formula does not account this relative
speed between ROV and water, which will be considered in the complete ROV model.
Thruster
Thrust
Torque
Fig. 4 Propeller
DragResist:
The resistance force to ROV should be considered. So we come up with the
DragResis model. The formula in this model is broadly used in references. The
associated parameters are water density, front area, water resistance coefficient and
speed.
5
1
DragResist=  Av 2CD
2
Fig. 5 Drag resistance from water
BatteryPower:
This model is to test the output power of the battery.
pin_p
voltageSensor
V
pin_n1
InstantaneousPow er
abs1
product
Pow er
abs
pin_p1
currentSensor
k=2
A
integrator
pin_n2
abs2
TotalPow er
I
k=1
abs
TwoPowerCost
InstantaneousPower
TotalPower
Fig. 6 Battery Power
ROV:
6
Fig. 7 ROV

Review of fundamental objectives:
Before analyzing the effect of uncertainty on our system, we will review fundamental
objectives (Fig. a), means objective (Fig. b) and corresponding attributes (Table a) so
that we will be more clear that what we are achieving. We identified two measures of
effectiveness for our system, which are battery power cost and translational speed.
These attributes help us to assess how our objectives are accomplished, which are
minimizing cost and maximizing performance.
7
Maximize Client's
Satisfaction
Maximize
reliability
Material
reliability
Telemetry
Maximize
Performance
Minimize Cost
Minimize
Initial
Cost
Minimize
Operating
Cost
Minimize
Maintenance
Cost
Maximize
Accelerate
Maximize
Top Speed
Fig. a fundamental objectives hierarchy
Fig. b means objectives network
Table a Objectives and attributes
Objective
Attributes (unit)
Can be modeled in Dymola?
Minimize battery power
cost
Power (w)
YES
Maximize translational
speed
Speed (m/s)
YES
8
Task 2: Identify and Model the Uncertainty in your Design
Analysis Model

Identify uncertain variables
It should also be noted that the design variables were not supposed to be included in
the uncertainty analysis. If they are included, this would have very likely yielded
results indicating these design variables dominated the impact on the effectiveness
measures. It is difficult to come up with 10 uncertain variables. As we are not sure
whether a specific variable plays an important role or not, then we include it in our
analysis. Ideally, we could do two sets of uncertainty analysis. One is including all
uncertain variables, which incorporates some design variables. The other includes
only uncertain variables. However, due to the time constrain, we will just do one
uncertainty analysis which includes design variables. All the uncertain variables and
initial triangular distribution are shown in Table 1.
Table 1 Uncertain variables and initial triangular distributions
Uncertainty Parameter
Lower value
Mean value
Upper value
Battery Voltage
12
24
48
ROV Front area A (m2)
0.1
0.5
1
5.0e-3
1
2
0.1
0.49
0.8
Motor EMF K (V/rad/s)
3.0e-4
2.78e-3
3.0e-1
Motor Inertia J(Nm/rad/s2)
5.0e-5
1.0e-4
1.0e-3
Motor inductor L(H)
1.0e-5
1.08e-4
5.0e-3
Motor damper (Nm/rad/s)
1.0e-4
1.0e-2
1.0e-1
Thrust coeff KT
8.0e-4
5.0e-3
2e-2
Drag coeff CD
Motor resistor R(Ω)
9
Propeller Diameter (m)
Thrust coeff KO

0.01
0.2
0.5
5.0e-7
6.8e-5
5.0e-3
Main effects analysis based on the central composite experiment
Perform main effects analysis based on the central composite experiment as shown
in Fig. 8. Main Effects for power cost and speed are showen in Fig. 9 and Fig.10
respectively.
As we can see in Fig. 9, the main effects on power cost are
dCMotoremf, thrust coefficient (propellerKT), torque coefficient (propllerKO), battery
voltage, propeller diameter, dCMotorresistorR and water resistance coefficient
(CdragCoeff). In Fig. 10, only battery voltage has main effects on speed. Based on
the equations included implicitly in the model as discussed in the first section, the
power cost should be related to battery voltage, thrust coefficient, torque coefficient,
propeller diameter, water resistance coefficient etc. therefore, this result is what we
expected. However, we are surprised to see that only battery voltage has main effect
on speed. The reason why this happened might be our battery model is not perfect
model, which can not reflect the real power output. In addition, It might be because
we include many design variables, which will very likely yield results indicating those
design variables will dominate the impact on the effectiveness. In this case, the
battery voltage (design variable) is a design variable of the battery, which will
dominate the impact on main effects to battery power output, as stated in previous
section.
10
Fig. 8 Screenshot of Model Center setup for Central Composite method
Fig. 9 Main Effects on power cost
11
Fig. 10 Main Effects on speed
Task 3: Elicit a Detailed CDF for the Most Significant
Uncertain Variables
Based on previous main effects analysis, we determined two important sources of
uncertainty:
1. battery voltage
2. thrust coefficient (propellerKT)
For the most important sources of uncertainty, we will elicit our beliefs through a
series of elicitation questions as discussed in class. Take thrust coefficient for
example, when we elicit a detailed CDF for thrust coefficient, we will ask ourselves a
series of “lottery” questions as what we did in class. These lottery questions were
the following:
12
"At what value of thrust coefficient are we indifferent between a lottery where we get
$10 if the actual thrust coefficient is greater than this value and $0 if it is less than
this value and a lottery where we have a 50% (p) chance of winning $10 and 50% (1p) chance of winning $0?" By doing this, the value was elicited at a probability of 0.5
(p). This question was repeated for each value of p in order to elicit the entire curve.
Figure 11 shows the elicitation of the cumulative probability distribution for battery
voltage. Figure 12 shows the elicitation of the cumulative probability distribution for
thrust coefficient.
13
Fig. 11 Elicited CDF and PDF for battery voltage
As we can see in Fig. 11, we believe battery voltage should follow uniform
distribution. It is because battery voltage is actually a design variable. We are not
sure what value it should be. So we believe it should follow uniform distribution within
the upper and lower bound.
Fig. 12 Elicited CDF and PDF for thrust coefficient
As we can see in Fig. 12, three key points are displayed on PDF graph. The
14
probability that thrust coefficient happens in between 2e-4 and 2.5e-4 is very high.
The thrust coefficient which is 2.25e-4 has the largest probability to happen. It could
conclude that I believe thrust coefficient will happen more likely within this range,
which is between 2e-4 and 2.5e-4.
Task 4: Determine the Distribution of the Output of your
Model
For the sake of exploring the nature of the uncertainty in your problem, perform a
Monte-Carlo analysis and investigate the histogram of the model output. All the
uncertain variables I identified in task 2 are included. 1000 runs were used for this
simulation. Figure 13 shows a screenshot of the Probabilistic Analysis GUI in which I
have entered all the distribution data.
15
Figure 13: Monte Carlo simulation design variable range values
The Monte Carlo results are shown in Fig. 14-24. Take three histograms of these
results which are Fig. 14, Fig. 16 and Fig. 22 respectively for example to see if they
match our expectations. These three histograms show the distribution of Front Area,
battery voltage and thrust coefficient. It can be seen that the histograms are identical
with triangular distribution shown in the following table. Most histograms match the
triangular distribution. Basically, they match our expectations compared with the
expected triangular distribution.
Uncertainty Parameter
Lower value
Mean value
Upper value
Battery Voltage
12
24
48
ROV Front area A (m2)
0.1
0.5
1
16
Drag coeff CD
5.0e-3
1
2
0.1
0.49
0.8
Motor EMF K (V/rad/s)
3.0e-4
2.78e-3
3.0e-1
Motor Inertia J(Nm/rad/s2)
5.0e-5
1.0e-4
1.0e-3
Motor inductor L(H)
1.0e-5
1.08e-4
5.0e-3
Motor damper (Nm/rad/s)
1.0e-4
1.0e-2
1.0e-1
Thrust coeff KT
8.0e-4
5.0e-3
2e-2
0.01
0.2
0.5
5.0e-7
6.8e-5
5.0e-3
Motor resistor R(Ω)
Propeller Diameter (m)
Thrust coeff KO
Figure 14: Monte Carlo Histogram of the uncertainty of Front Area
17
Figure 15: Monte Carlo Histogram of the uncertainty of water resistance coefficient
Figure 16: Monte Carlo Histogram of the uncertainty of battery voltage
18
Figure 17: Monte Carlo Histogram of the uncertainty of motor resistance
Figure 18: Monte Carlo Histogram of the uncertainty of motor emf
19
Figure 19: Monte Carlo Histogram of the uncertainty of motor inertia
Figure 20: Monte Carlo Histogram of the uncertainty of motor inductance
20
Figure 21: Monte Carlo Histogram of the uncertainty of motor damper
Figure 22: Monte Carlo Histogram of the uncertainty of thrust coefficient
21
Figure 23: Monte Carlo Histogram of the uncertainty of propeller diameter
Figure 24: Monte Carlo Histogram of the uncertainty of torque coefficient
22
23
Figure 25: Monte Carlo Histogram of ROV speed
Fig. 25 shows the Monte Carlo distribution of ROV speed. The mean speed: 27.3471
m/s; Std Dev: 7.37; Skewness: 0.4425; According to the PDF and CDF shown in Fig.
25, the speed expected distribution approximately follows Gaussian distribution.
Task 5: Lessons learned
The difficulties we encountered are the following:
1. Making Modelica model robust and accurate;
2. Customizing the fileWrapper for my Modelica model;
3. Identifying what uncertain variables should be considered when model the
uncertainty and what lower bound and upper bound should be;
4. Analyzing the main effects graph;
24
5. Elicitation process and provide an intuitive justification for the most important
characteristics of the CDF
Based on what we learned, we will not include the design variables as uncertainties
but only the uncertain variables to model uncertainty in my design analysis model.
Except for the new model which employed multi-body modules, we also modified the
old model in some extents. One significant modification is the incorporation of the
actual propeller thrust force and torque equation in relation with the propeller
diameter and the rotating speed.
Another modification of the old model is concerning the water resistance. In the old
model, the water resistance is a linear relationship to the speed, but after examining
the related literature, we conclude that water resistance should better be a linear
function of velocity square. So we modified the hull object regarding this change.
25
Revised Thruster Model in which the formulas of thrust force and torque on thrust
shaft are incorporated into.
F=kt* w^2 * D^4
T=kq*w^2 * D^5
In which:
F: thruster force;
T: torque on propeller shaft;
kt: thruster force coefficient;
w: angular velocity;
D:propeller diameter.
26
The torque exerted by the water resistance is fed back to the thruster shaft via
flange_a.
We also studied the modified old model in ModelCenter
Design of Experiment
Obviously propeller diameter have big influence on the vehicle speed.
27
Monte Carlo Simulation
28
29
Study the influence of thruster diameter on the energy output
As the diameter increases to 0.33, something unusual happened. The energy
30
suddenly became very large. To find the reason we run the model in Dymola, and
change the diameter from 0.12 to 0.35 to see what happened.
D=0.12m
D=0.3m
31
D=0.35m, the system began oscillating and unstable
As a result the energy consumed is keeping increasing since the system cannot
reach a stable state.
32
References
[1] A mechatronic positioning system actuated using a micro DC-motor-driven
propeller–thruster, Shung-Fei Yang and Jyh-Horng Chou.
[2] Conceptual Design of an AUV Equipped with a Three Degrees of Freedom
Vectored Thruster, Emanuele Cavallo, Rinaldo C. Michelini and Vladimir F. Filaretov
[3] The concept of anti-spin thruster control, yvind N. Smogeli, Asgeir J. Sørensena,
and Knut J. Minsaas
[4] http://www.grc.nasa.gov/WWW/K-12/airplane/thrsteq.html
33