Control Analysis on Robotic Boat System
List of Symbols and Abbreviation
RB Rescue Boat
EB Endangered Boat
RS Rescue Signal
MRP Mobile Robot Platform
WLAN Wireless Local Area Network
DOF Degree Of Freedom
Introduction
1.1 General Introduction
Artificial intelligence for system automation has been an active area of research for decades. In this respect dealing with an automation system will involve number of items across various subsystems and technical disciplines. The control and the flow of the robotic variables constitute major evaluation part and these variables are the lifeblood of the system. The distribution and the concurrency of these variables, activities must be followed in a systematic structure of a system.
The automated marine vehicles are gaining increasing attention due to the inherent difficulties in their manual navigation and control. Robotic marine vehicles that have been developed to reduce the risks of human life and to carry out tasks that would be impractical with a manned mission. The main obstacle to be overcome in the development of robotic boat is in the area of control strategy. Controlled system must meet certain criteria. For an automated marine vehicle the most important one is that the vehicle must be stable throughout its entire operational range. Without this there is the possibility that control and hence the vehicle itself may be lost. The focus of this paper is to present the transfer function model of both mechanical system and trackkeeping autopilot design of a robotic boat system and to investigate the stability of above transfer function models using various stability techniques.
1.2 Dissertation Aim
The aim of the dissertation is to present the control analysis of an automated rescue boat system.
1.3 Dissertation Question
How can an automated rescue boat be operated with consistent stability?
1.4 Dissertation Objective
To design mechanical and electromechanical models of an automated rescue boat.
To find out transfer functions – Mechanical and Electromechanical systems.
To develop algorithms for simulation.
1.5 Dissertation

Outline
This paper presents control system modeling and analysis of an automated rescue boat. Hence, the organization of this paper is as follows: Section 2 described about previous and related works, Section 3 discusses about the systems and Section 4 focuses on results based on experimental data. Last but not least, Chapter 5 discusses the conclusion of this project and the future development..
Motion Analysis
In mechanics, degrees of freedom (DOF) are the set of independent displacements and/or rotations that specify completely the displaced or deformed position and orientation of the body or system. This is a fundamental concept relating to systems of moving bodies.
A particle that moves in three dimensional spaces has three translational displacement components as DOFs, while a rigid body would have at most six DOFs including three rotations. Translation is the ability to move without rotating, while rotation is angular motion about some axis.
Figure 1: Definitions of 6DOF boat motion
2.1 Six Degree of Freedom (DOF)
In three dimensions, the six DOFs of a rigid body are sometimes described using these nautical names:
Moving up and down (heaving);
Moving left and right (swaying);
Moving forward and backward (surging);
Tilting forward and backward (pitching);
Turning left and right (yawing);
Tilting side to side (rolling).
2.2 Force and Moment
There are several ways to represent the coordinate system and associated nomenclature, we adopt the following notations:
Position vector in an Earthfixed frame ?_{1} =[x, y, z]^{T}
Vector of Euler angles ?_{2} = [?, ?, ?]^{T}
Surge, sway & heave velocity vectors ?_{1} = [u, v, w]^{T}
Roll, pitch and yaw velocity vectors ?_{2} = [p, q, r]^{T}
According to Newton’s 2nd Law, it follows
Where A , B and C are the forces acting along the x , y and z axis respectively and E ,F and G are the moments with respect to the x , y and z axis respectively. From (1), three force equations follow and from (2) three moment equations follow.
2.3 Coordinate System for Boat Navigation
For aquatic applications involving boats and ships, only three degrees of freedom are practically important. These lie in the plane parallel to the surface of the water, namely surge, sway and yaw. It is based upon the fact that the boat only moves in a plane parallel to the surface of water (will not go above or below water (zaxis)) and turn only along the z axis (without tilting or tipping over). For a relatively stable surface craft, this turns out to be a safe assumption and helps simplify the boat model.
Figure 2: Coordinate system used in boat navigation.
So, Position Vector used in the Earth fixed Frame ? = [x, y, ?]^{ T }
Corresponding surge, sway and yaw velocity vectors ? = [u, v, r]^{ T}
Corresponding Force and Moment Equation for 3 DOF boat motion:
Where m is the mass of the ship, I_{z} is the moment of inertia with respect to Z axis, u , v and r are the surge, sway and yaw speed and , , and are the surge, sway and yaw acceleration respectively.
Boat Steering Process
This section presents the mathematical model employed in the track keeping design of an automated robotic boat. For a constant speed straight line motion condition, linearization of (3) ~ (5) decouples the surge equation. Taking the Laplace transform of the coupled swayyaw system and cancellation of the sway term, the following 2nd order Nomoto model is obtained.
The Nomoto models that were derived under the assumption of constant speed can be used to describe the steering behavior for small rudder angles, when the loss of speed is negligible, and to describe the behavior during the stationary part of the zigzag maneuvering, where the speed remains constant as well. However, the parameters of the models are different for different rudder angles.
According to pondinlab trial databased identification, the values of the parameters T2 and T3 in (6) are almost same. This suggests further simplification of (6) is possible and the 1^{st} order Nomoto model follows
Where k is the radar gain and is the yaw mode time constant. Equation (7) can be represented by a differential equation in the time domain as
Equation (7) can be written as the following 2nd order model with the definition of r = where ? is the heading angle,
…………….. (9)
Figure 3: Definition of motions in the horizontal plane
The kinematics of the ship can be described according to the above Figure 3 as bellows:
…….. (10)
…….. (11)
………………………………. (12)
As the system defined by (10) ~ (12) is nonlinear in the variables u, v, ? defies direct application of the linear control design method. Besides, linearization can be done by rotating the earthfixed coordinate system to make the desired heading zero and by moving the coordinate origin to coincide with the starting point.
However as the heading angle ? is small, we have sin ? ? ? and cos ? ? 1. Assuming that the surge speed u is nearly constant, and the surge speed is much larger than the sway speed i.e. u >> v
We have the following linear equations:
Moreover by assuming that the sway speed is nearly constant and taking the Laplace transform of above equations,
Substitute the relationship defined by equation ( 9)
The transfer functions from the rudder to the x, and y positions are obtained as
Finally, it is possible to write the above in a unified expression as below
Mechanical System Analysis
4.1 Mechanical System Model
We will design a mechanical system model for an automated rescue boat which will include the interactions among various components within the rescue boat and with it’s surroundings. The mechanical system of a robotic boat is shown in the following figure:
Figure 4: Mechanical system model
Where M is the Mass of the physical Structure of the boat including stator of the motor, J_{1} is Inertia of the rotor of the motor, J_{2} is inertia of the propeller, D_{2} is the Damping between rotor and stator , D_{L} is the Damping between propeller and water surface.
After simplification we get the following equivalent inertia() , torque() and damping() ,
Figure 5: Modified Mechanical system model
4.2 Transfer Function of Mechanical System
The input –output relationship of any mechanical system can be charaterized by using transfer function. The transfer function of a system is a mathematical model in that it is an operational method of expressing the differential equation that relates the output variable to the input variable. It is a property of a system itself,independent of the magnitude and nature of the input or driving function.if the transfer function of a mechanical system is known , the output or response can be studied for various forms of inputs with a view toward understanding the nature of the system.
The laplace transform of the equations of motions of above mechanical system (fig 5) can be written as
If is the linear distance travelled by the rotating propeller and f_{a} is the axial force which drives the boat in the forward direction, we can consider the following relations
Replacing the values of and f(s) in equation (13) and (14) , we get
[k=]
Now, _{}
After mathematical manipulationwe get the final transfer function
This transfer function relates the output displacement x(s) to input force f_{a}(s) .
Electromechanical System Analysis
A system which is a hybrid of electrical and mechanical variables is called electromechanical system. An RB is an electromechanical component that yields a displacement output for voltage input (an electrical input results in a mechanical output).
Figure 6: An Electromechanical System (DC Motor)
From the above figure we can write the Loop gain equation, relating armature current (), back emf () and input voltage ().
From equation (12)
From equation (13)
Therefore,
This equation relates the output angular displacement of the propellar with input voltage .
Simulation Work
The stability of the track keeping control system, mechanical system and Electromechanical analysis of robotic boat whose mathematical model was developed earlier are simulated and examined in this section. We use Matlab for simulation and examine the response of the system to allow more ?exibility in interaction with other components of the overall navigator model.
We will analyze the system for stability using step response, impulse response, pole zero map, root locus and bode plot.
Step response is the time behavior of the outputs of a general system when its inputs change from zero to one in a very short time. Knowing the step response of a dynamical system gives information on the stability of such a system, and on its ability to reach one stationary state when starting from another.
The response of a system to an impulse which differs from zero for an infinitesimal time, but whose integral over time is unity; this impulse may be represented mathematically by a Dirac delta function.
The use of poles and zeroes and their relationship to the time response of a system is a useful technique to qualitatively describe the stability of the system. The poles of a transfer function are the values of the Laplace transform variable, s that causes the transfer function to become infinite. On the other hand, the zeros of a transfer function are the values of Laplace transform variable, s that causes the transfer function to become zero.
The root locus is the locus of the poles of a transfer function as the closed loop gain of a system is varied. The root locus is a useful tool for analyzing single input single output dynamic systems. A system is stable if all of its poles are in the lefthand side of the splane (for continuous systems).
A Bode plot is a graph of the logarithm of the transfer function of a linear, timeinvariant system versus frequency, plotted with a logfrequency axis, to show the system’s frequency response. It is usually a combination of a Bode magnitude plot (usually expressed as dB of gain) and a Bode phase plot (the phase is the imaginary part of the complex logarithm of the complex transfer function).
6.1 Track keeping control:
Figure 7 plot of tack keeping model

6.2 Mechanical system responses:





6.3 Electromechanical system responses:









Related Works
A number of researchers [1, 2, 4, 5] had referred to a water based robotic system. There are several research projects in the field of autonomous robotic boat.
Sarker and Hussain [1] present a simple prototype of sensing and control of a water based robotic system where initial works on RB relationship, velocity, computer environment and so forth are discussed.
Leghari et al. [3] proposes a model of a multimobile robots system where preliminary principles and methods towards an automated system are outlined.
Dhariwal and Sukhatme [4] present an algorithm for estimating robotic boat location by integrating various sensor inputs. Multisensors are adopted in this system.
The previous works fall short of attention to the control analysis of automated rescue boat system.
Conclusion and Future Works
In this paper of a simple control analysis of robotic boat system is presented. The transfer function model of track keeping system and mechanical system of a robotic boat are developed. The stability of both the track keeping system and mechanical system of a robotic boat has been analyzed. Several techniques have been applied for this purpose.
This stability analysis implies an attractive possibility for future application of robotic boats. There are a number of major research challenges that still need to be overcome for a full potential system, such as, fuzzy logic based automated control, fault analysis, wind effects (lateral). The wave model is not considered, there are some difference in the results of the simulation and experiment. In the next step, we will take the wave model into account. This is an ongoing research and the system needs continuous improvement.
Appendix A
MATLAB Code
Track keeping control
clear all;
close all;
clear memory;
k=.8;
t=.63;
num=[k];
den=[t 1 0 0];
system=tf(num,den) %Transfer Function
figure(1);
step(num,den) %Step Response of the Transfer Function
figure(2);
pzmap(num,den) %PoleZero Map of the Transfer Function
figure(3);
rlocus(num,den) %Root locus of the Transfer Function
figure(4);
bode(num,den) %Bode Plot of the Transfer Function
Appendix B
MATLAB Code
Mechanical system responses
clear all;
close all;
clear memory;
m=1;
j=1;
k=.8;
ke=1;
r=1;
d2=1;
de=1;
kr=k*r;
a2=(j/(kr));
a1=((d2+de)/kr)+d2/r;
b4=(m*j)/kr;
b3=(j*d2+m*(d2+de))/kr;
b2=(j*ke+d2*de)/kr;
b1=(ke*(d2+de))/kr;
num=[a2 a1 0];
den=[b4 b3 b2 b1 0];
syst=tf(num,den) %Transfer Function
figure(1);
step(num,den) %Step Response of the Transfer Function
figure(2);
pzmap(num,den) %PoleZero Map of the Transfer Function
Figure(3);
rlocus(num,den) %Root locus of the Transfer Function
Figure(4);
bode(num,den) %Bode Plot of the Transfer Function
Appendix C
MATLAB Code
Electromechanical system responses
clear all;
close all;
clear memory;
m=50;
je=50;
k=.8;
ke=5;
kt=1;
kb=1;
Ra=1;
d2=1;
de=1;
a1=d2*(1+1/k);
b4=(Ra/kt)*(je*mje*(d2)^2);
b3=(Ra/kt)*(je*d2+m*(d2+de)(d2+de)*(d2)^2)+kb*m;
b2=(Ra/kt)*(je*ke(d2+de)*d2)+kb*d2;
b1=(Ra/kt)*(ke*(d2+de)kb*ke);
num=[m a1 ke];
den=[b4 b3 b2 b1 0];
syst=tf(num,den) %Transfer Function
figure(1);
step(num,den) %Step Response of the Transfer Function
figure(2);
pzmap(num,den) %PoleZero Map of the Transfer Function
figure(3);
rlocus(num,den) %Root locus of the Transfer Function
figure(4);
bode(num,den) %Bode Plot of the Transfer Function
References
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