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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

Chapter 1 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 track-keeping 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 electro-mechanical models of an automated rescue boat. To find out transfer functions â€“ 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. Heave

Yaw

Roll

Surge Pitch Sway

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 Earth-fixed frame η1 =[x, y, z]T Vector of Euler angles η2 = [φ, θ, ψ] T Surge, sway & heave velocity vectors ν1 = [u, v, w] T Roll, pitch and yaw velocity vectors ν2 = [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

Îˇ = [x, y, Ďˆ] T

Corresponding surge, sway and yaw velocity vectors

Î˝ = [u, v, r] T

Corresponding Force and Moment Equation for 3 DOF boat motion:

Where m is the mass of the ship, Iz 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 sway-yaw 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 pond-in-lab trial data-based 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 = heading angle,

where ψ is the

…………….. (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, ψ defies direct application of the linear control design method. Besides, linearization can be done by rotating the earth-fixed 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 ψ is small, we have sin ψ ≈ ψ and cos ψ ≈ 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 itâ€™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, J1 is Inertia of the rotor of the motor, J2 is inertia of the propeller, D2 is the Damping between rotor and stator , DL 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 â€“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 manipulation we get the final transfer function

This transfer function relates the output displacement x(s) to input force f a(s) .

Chapter 5 Electro-mechanical 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 Electro-mechanical System (DC Motor)

From the above figure we can write the Loop gain equation, relating armature current ( input voltage ( ).

From equation (12)

From equation (13)

Therefore,

), back emf (

) and

This equation relates the output angular displacement

of the propellar with input voltage

.

Chapter 6 Simulation Work The stability of the track keeping control system, mechanical system and Electro-mechanical 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 ďŹ‚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 left-hand side of the s-plane (for continuous systems). A Bode plot is a graph of the logarithm of the transfer function of a linear, time-invariant system versus frequency, plotted with a log-frequency 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:

Pole-Zero Map

Step Response

1

0.8

0.8

0.7 0.4

0.5

0.2

Imaginary Axis

Amplitude

0.6

0.6

0.4

0 -0.2

0.3

-0.4

0.2

-0.6 -0.8

0.1 -1 -1.6

0

0

1

2

3

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

Real Axis

4

Time (sec) Bode Diagram 50 Root Locus 6

Figure 7 plot of tack keeping model

Magnitude (dB)

Mechanical system responses:

-2

-4

-6

-100 -150 -180

0

Phase (deg)

Imaginary Axis

2

-50

Pole-Zero Map

-225

1 0.8 0.6

-6

-4

-2

0

2

Real Axis

-270 0.4 -1 10 Imaginary Axis

4

6.2

0

10

0.2

0

10

1

10

2

Frequency (rad/sec)

0 -0.2 -0.4 -0.6 -0.8

Figure 7: Impulse response, Pole-Zero map, Root Locus, Bode plot -1 of track keeping model -3

-2.5

-2

-1.5

-1

-0.5

0

Real Axis

Bode Diagram 20

Magnitude (dB)

0

Root Locus 1

-40 -60 -80 180

0.5

135 Phase (deg)

Imaginary Axis

-20

0

90 45

-0.5

0 -1

10 -1 -5

-4

-3

-2

-1

0

1

2

10

0

10

1

Frequency (rad/sec)

Figure 8: Step Pole-Zero map, Root Locus , Bode plot of Mechanical system model Realresponse, Axis

2

10

6.3

Electro-mechanical system responses:

Pole-Zero Map

Imaginary Axis

0.4

0.2

0

-0.2

-0.4

-0.03

-0.02

-0.01

0

Real Axis

Bode Diagram

Roo t L oc us Magnitude (dB)

1.5 1

0 -0.5

Phase (deg)

Im a gina ry A x is

0.5

-1 -1.5 -0 .1 5

-0 .1

-0 .05 Rea l A x is

0

0.05

0 .1

80 60 40 20 0 -20 -40 -90 -135 -180 -225 -3 10

10

-2

10

-1

Frequency (rad/sec)

Figure 9: Step response, Pole-Zero map, Root Locus, Bode plot of Electro-mechanical system model

10

0

Chapter 7 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 multi-mobile 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. Multi-sensors are adopted in this system. The previous works fall short of attention to the control analysis of automated rescue boat system.

Chapter 8 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)

%Pole-Zero 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)

%Pole-Zero 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 Electro-mechanical 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*m-je*(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)

%Pole-Zero 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 [1] Sarker, M.M.H. and Hussain, Z.M. “Sensing and control of a water based robotic system”, International Journal of Electrical Engineering in Transportation, Vol.4, No.2, pp. 41– 45, 2008. [2] Leghari, F., Jakaria, A., Sarker, M.M.H., Hoque, M.A. and Islam,K.K. “Towards a model of a multimobile robot system”, Proc. on Int’l Multi-topic Conf., pp.239-242, IEEE, 2008. [3] Sarker, M.M.H. and Tokhi, M.O. “A Multi-agent System Model of a Diesel Engine using RuleInduction”, Expert Systems With Applications, Elsevier, Vol.36, Issue. 3, Part 2, pp. 6044–6049, 2009.

[4] Dhariwal,A. and Sukhatme, S.G. “Experiments in robotic boat localization”, Proc. Int’l Conf. on Intelligent Robots and Systems, IEEE, pp.1702-1708, 2007. [5] Wang, H., Wang, D., Tian, L., Tang, C. and Zhao, Y. “An automatic tracking system for marine navigation”, IEEE, pp. 180-183, 2001. [6] Klafter,R.D., Chmielewski,T.A. and Negin, M. “ Robotics Engineering: An integrated approach”, Printice-Hall, India (PHI), 2005. [7] Norman S. Nise, “Control Systems Engineering”, 4th Edition, 2008. [8] Mata et al. “Object learning and detection using evolutionary deformable models for mobile robot navigation”, ROBOTICA, Cambridge University Press, vol. 26, pp. 99–107, 2008. [9] Donoso-Aguirre, F. et al. “Mobile robot localization using the Hausdorff distance”, ROBOTICA, Cambridge University Press, vol. 26, pp. 129–141, 2008. [10] Nomenclature for Treating the Motion of a Submerged Body Through a Fluid. SNAME. The Society of Naval Architects and Marine Engineers, 1950, no. 1-5. [11] Amerongen, J. van, "Ship Steering", Theme: Control Systems, Robotics and Automation, edited by Unbehauen, H.D. , in Encyclopedia of Life Support Systems, (EOLSS), Developed under the Auspices of the UNESCO, EOLSS Publishers, Oxford, UK , 2003