Development Program

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To: ME 5554 / AOE 5754 / ECE 5754 Students

Subject: Phase I Development Program (i.e. your Midterm Project)

Background

Kraken Robotics, located in Newfoundland Canada, has developed a family of Hovering

Autonomous Underwater Vehicles (H-AUV) known as ThunderFish

(https://krakenrobotics.com/products/thunderfish/). These H-AUVs are equipped with

revolutionary Synthetic Aperture Sonar (SAS) systems that enable high-resolution

seabed imaging. Your Objective for this project is to design a state-feedback control

system that can control the H-AUV depth and longitudinal position. To do this, we must

first develop LTI equations of motion.

Equations of Motion

Figure 1 indicates the conventional coordinate system nomenclature defined by:

DTNSRDC Revised Standard Submarine Equations of Motion , J. Feldman, 1979.

Figure 1. Standard coordinate system nomenclature for AUVs.

where: O is the origin of the Earth-fixed (absolute) coordinate system

 is the roll angle about the -axis (not using for this project)

 is the pitch angle about the -axis (rad)

 is the yaw angle about the -axis (not using for this project)

 is the position in the -direction (m)

 is the position in the -direction (not using for this project)

 is the position in the -direction (m)

VIRGINIA TECH

and: o is the origin of the AUV body-fixed (local) coordinate system, which is

located at the center of mass

 is the roll angle rate about the

axis (not using for this project)

 is the pitch angle rate about the

axis (rad/s)

 is the yaw angle rate about the

axis (not using for this project)

is called the Surge direction (m)

is called the Sway direction (not using for this project)

is called the Heave direction (m)

 is the velocity in the

direction (m/s)

 is the velocity in the

direction (not using for this project)

 is the velocity in the

direction (m/s)

For this project, we are only interested in pitch (q), heave ( z ), and surge ( x ) motions.

This model is commonly referred to as a pitch-axis model. The linearized mapping

between the AUV-fixed coordinates and the world coordinates are given by:

(

)

=

(

)

(

)

=

(

)

(

)

=

(

)

The following assumptions have been used to simplify the model:

The AUV is neutrally buoyant, i.e. its weight is equal to its buoyancy
The AUV-fixed coordinate system is located at the AUV center of mass
The center of mass is at the same point as the center of buoyancy
The pitch angle is relatively small

The linearized dynamic equations of motion are obtained from Newtons second law:

3 (

)=()=

@AB

()

3 (

)=()=

@AB

()

3 (

)

=

(

)

=

@AB

(

)

where: is the mass of the AUV (kg)

is the mass moment of inertia about the

axis (kgm

)

The linearized external forces and moments are given by:

@AB

(

)

=

(

)

(

)

+

(

)

@AB

(

)

=

(

)

(

)

(

)

(

)

+

(

)

+

(

)

@AB

(

)

=

(

)

(

)

(

)

(

)

(

)

(

)

VIRGINIA TECH

where:

is the

direction added mass from accelerating water (kg)

is the

direction hydrodynamic drag coefficient (Ns/m)

is the

direction added mass from accelerating water (kg)

is the

direction hydrodynamic drag coefficient (Ns/m)

is the

direction added mass caused by rotation (kgm)

is the

direction hydrodynamic drag caused by rotation (kgm/s)

is the added rotational inertia about the

axis (kgm

)

is the moment drag coefficient about the

axis (Nms)

is the added rotational inertia about the

axis (kgm)

is the moment drag coefficient about the

axis (Ns)

is the position of the stern (rear) control surface in the

direction (m)

is the position of the bow (front) control surface in the

direction (m)

and the external control inputs are:

is the thrust in the

direction (N)

is the stern thrust in the

direction (N)

is the bow thrust in the

direction (N)

The pitch angle , the depth z , and the position x are directly measured with sensors to

represent the outputs of the model.

VIRGINIA TECH

Problem 1. [ 10 pts.] Derive a complete state-space model for the AUV, including a

state equation and an output equation. Hint: The MathWorks LiveScript feature has a

built-in equation-editor which is very easy to use (simpler than LaTeX), and allows you

to embed formatted equations directly in your mat lab script.

Problem 2. [ 2 pts.] Using the numerical values and formulas listed below, compute

the following parameters at the top of your Matlab script.

= 500 (kg)

= 9. 5  2 (kgm/s)

= 25 (m)

=V

XY

Z

(kgm

)

=V

[YY

Z

(kgm

)

= 1. 1  3 (Nms)

=V

[Y

Z (kg)

=V

_Y

Z

(kgm)

= 94 (Ns/m)

= 320 (Ns)

=V

WY

Z (kg)

=aV

Z (m)

= 4. 7  2 (Ns/m)

=V

Z (m)

=V

XY

Z (kgm)

deA

= 3 6 (N)

where:

|

deA

|

deA

|

deA

Problem 3. [ 10 pts.] Use the variables defined in Problem 2 to construct the A , B , C ,

and D matrices for your linear state-space model, then use these matrices to generate

an LTI object called ol_sys. You are NOT allowed to directly substitute numerical values

for the parameters above into your matrices!

You must also define short descriptive names for each of the states, inputs, and outputs

and embed these names in the LTI state-space model object using the SET command.

Problem 4. [ 2 pts.] Apply the MINREAL function in Matlab to your LTI object from

Problem 3 to determine whether your state-space model is a minimum realization.

Explain your results.

Problem 5. [ 10 pts.] Show that the open-l oop system is Completely Controllable using

all of the control inputs. Determine (and clearly document) whether any subsets of

control inputs results in a Completely Controllable system. Justify your results.

Problem 6. [ 6 pts.] Compute the open loop poles, the natural frequencies with units of

Hz, and the damping rat ios for each eigenvalue. Tabulate and display your results

using the Matlab TABLE function.

VIRGINIA TECH

Problem 7. [ 10 pts.] Define an initial condition vector using the following initial values:

(

0 )

= 50

(

0 )

= 100

(

0 )

= 350

(

0 )

= 300

(

0 )

= 45

(

0 )

= 0

Use the INITIAL function in Matlab to simulate the initial value response of the open-

loop system. Using the output data from this simulation with the LSIMINFO function,

estimate and display the 5% settling times for each of the open-loop response outputs.

Plot the open-loop response outputs on a single figure. You may combine signals with

the same units on the same axes, but if signals have different units, they must be in a

separate axes. Properly annotate your figure with axis labels, grid lines, and a legend.

Place a marker on each output curve at the time corresponding to the 5% settling time.

Problem 8. [ 15 pts.] Design a full-state feedback controller that meets the following

performance requirements when the closed-loop system is simulated using the same

initial condition vector that was used in Problem 7:

Each control signal must be bounded by the limit specified in Problem 2
The AUV must completely enter the tunnel without contacting the barrier
Minimize the time it takes to enter the tunnel

VIRGINIA TECH

The Matlab function animate_auv.p is provided for you to generate the plot above

using output data from your simulation. To use this function, you must provide the x(t),

z(t), and theta(t) data, where x and z are in meters, and theta is in radians. You can

also input an optional Boolean value (savemovie) to save the animation to a .mpg file.

animate_auv(t,x,z,theta,savemovie)

Document a minimum of five design iterations that demonstrate that you attempted to

achieve the requirements. Your documented design iterations must show sufficient

improvement and must clearly indicate that you made an earnest attempt to meet the

requirements.

Your documentation for each iteration should at least include:

The desired closed-loop poles you chose
An indication of whether you meet the requirement for entering the tunnel without

contact

An indication of whether you meet the requirement for control signal saturation
5% settling times for each of the outputs

Final Design Plot. [ 5 pts.] For your FINAL closed-loop design ONLY, plot a single

figure window with three axes (3 rows and 1 column). The topmost axes should include

the two position outputs, the middle axes should be the pitch angle, and the bottom

axes should include both control signals. Properly annotate each axes with clearly

defined axis labels, grid lines, and legends.

Place a marker on each output curve at the time corresponding to the 5% settling time.

Submission Rules:

You MUST use the Matlab LiveScript feature to generate your Midterm Project

results. This will serve as a formatted  report that includes your Matlab code, any

outputs to the command window, and any plots you generate.

You MUST submit both the .mlx (LiveScript) file as well as a PDF version of your

LiveScript output.

You MUST use the Output Inline option in LiveScript. You CANNOT use the

Output on Right option which places plots on the right-hand side of the page

and the code on the left-hand side.

You MUST insure that the plots you generate are sufficiently large that the PDF

displays them at a publication-quality size (i.e. preferably with a width that is

close to the margin-to-margin edges).

You MUST NOT display long columns of output data in your LiveScript!!