Pitch Attitude Control of a Booster Rocket

Transcription

1 Research Inventy: International Journal Of Engineering An Science Issn: , Vol.2, Issue 7 (March 2013), Pp Pitch Attitue Control of a Booster Rocket Mr. Debabrata Roy 1, Prof. Raghupati Goswami 2, Dr. Sourish Sanyal 3, Prof. Amar Nath Sanyal 4 1 Assistant Professor, Seacom Engineering College, Jala Dhulagari, Howrah, West Bengal, Inia 2 Chairman of the Trust, Ieal Institute of Engineering,alyani Shilpanchal, alyani, Naia, West Bengal, Inia 3 Associate Professor, Acaemy of Technology, Aisaptagram, Hooghly WestBengal, Inia 4 Professor, Acaemy of Technology, Aisaptagram, Hooghly West Bengal, Inia Abstract-There are three important variables for a rocket moving through air an then through space. These are its yaw, pitch an roll. These variables are to be precisely controlle in orer to reach the target an prouce the esire effect. The paper eals with pitch attitue control of a booster rocket for launching a spaceship. Starting from its block iagram, the mathematical moel has been evelope an the analytical treatment has been mae by inigenously mae computer program an also by using MATLAB-tools for getting the esire results. Fulfillment of the specifications coul be mae suitably choosing the forwar path gain an the coefficient of rate feeback. eywors-feeback, Overshoot, Phase margin,pitch attitue, Rocket I. INTRODUCTION A rocket is essential for space travel. It is a self-propelle evice that carries its own fuel, as well as oxygen, or other chemical agent, neee to burn its fuel. Most rockets move by burning their fuel an expelling the hot exhaust gases that result. The force of these hot gases shooting out in one irection causes the rocket to move in the opposite irection. A rocket engine is the most powerful engine for its weight. Other forms of propulsion, such as jet-powere an propeller-riven engines, cannot match its power. Rockets can operate in space, because they carry their own oxygen for burning their fuel. Rockets are presently the only vehicles that can launch into an move aroun in space [17,18]. Space exploration is a challenging activity ue to its complexity. The complication arisesout of so many obstacles which we are to overcome [1,2,3,4,5] e.g. The vacuum of space Heat management problems The ifficulty of re-entry Orbital mechanics Micrometeorites an space ebris Cosmic an solar raiation The logistics of having restroom facilities in a weightless environment But the biggest problem of all is harnessing enough energy simply to get a spaceship off the groun. This is possible ue to high capacity of rocket engines. The rocket engines are easy to buil for flying inexpensively. But that is true only for the toy level. For propulsion through space it is so highly complicate [9,10] that space eneavours have been taken up only by someavance countries of the worl. Inia is one amongst the successful countries in space eneavours. Our ISRO is one of the pioneers in space activity. II. PRINCIPLE OF A ROCET ENGINE Rocket engines are reaction engines [17,18]. The basic principle is the Newton s 3 r. law i.e. action is equal to the reaction. A rocket engine throws mass in one irection an moves in the opposite irection by reactionary forces. The "strength" of a rocket engine is calle its thrust. It is generate by throwing burnt fuels backwar. The flight can be controlle by changing the rate of burning of the fuel an the irection of throw.motion along three principal axes is important while it moves through air an subsequently through space. The principal axes are the following: Vertical axis, or yaw axis an axis rawn from top to bottom, an perpenicular to the other two axes. 8

2 Pitch Attitue Control Of A Booster Rocket Lateral axis, transverse axis, or pitch axis an axis running from the pilot's left to right in pilote aircraft, an parallel to the wings of a winge aircraft. Longituinal axis, or roll axis an axis rawn through the boy of the vehicle from tail to nose in the normal irection of flight, or the irection the pilot faces. The motion along these three principal axes is to be closely controlle [1,2,3]. This is accomplishe by proportional-erivative (PD) feeback control. The erivative or rate feeback as well as the forwar path gain are to be ajustable [6,7,12]. III. THE MATHEMATICAL MODEL AND THE BLOC DIAGRAM The pitch attitue control of a booster rocket contains both attitue an rate gyros [6,7,12]. The block iagram [8] representation is given in fig. 1. We are aiming at fining out amplifier gain, an the feeback coefficients for best possible ynamic performance. The system shoul be almost critically ampe. The percentage overshoot has to be limite to 0.1% an the phase margin must be at least 30 o. Fig. 1. Pitch attitue control of booster rocket: block iagram The transfer function of the forwar an backwar path are given below: 6 G( S) ; H( s) 1 2 s s( s 16s 100) 1 The close loop transfer function is given as: 6 M( S) 3 2 s 16 s (6 100) s 6 2 IV. ANALYTICAL TECHNIQUES The general practice is to go in for root contour to fin out appropriate values of gain an feeback coefficients. But the metho is complex. So, we have treate the problem analytically [6,7]. The solution has been obtaine by a specially constructe program [14,15,16]. The program fins the characteristic equation choosing values of gain an feeback coefficients arbitrarily. Then a cut an try process has been followe to realize a amping factor of nearly unity for the ominant poles an at the same time a phase margin of atleast 30 o. The specifications coul not be fulfille in several attempts. The computer print-out change to worformat, is given below: The programme.fins out amplifier gain an feeback coefficients of a booster rocket The chosen parameters: Amplifier gain, = 87; Coefficient of attitue feeback, p = 1; coefficient of rate feeback, = 0.2 Roots of the characteristic equation are being foun out by Coates metho (3r orer convergence)[14,15] to reuce the no. of steps. The characteristic equation is given below: 3 2 s 16 s s st root: ; 2n & 3r roots: ± j The natural frequency of oscillation for the ominant roots = r/s The amping factor = Corresponing ampe frequency of oscillation = r/s The percent overshoot = The peak time = sec The settling time = sec The gain crossover occurs at an angular frequency of r/s Phase margin = o 9

3 Pitch Attitue Control Of A Booster Rocket The Nyquist plot oes not intersect the real axis. So the gain margin is infinity. The phase margin is aequate. But the time-omain performance is not acceptable. The percentage overshoot is very high, much above the specifie value. Therefore, we make recourse to MATLAB-tools to get an acceptably goo esign. V. MATLAB-BASED ANALYSIS Now we make recourse to MATLAB-tools for refinement of the esign [11,12,13]. We set the amping coefficient arbitrarily at = 0.1 an check the t-omain an f-omain performance. We fin that the overshoot is as high as 10.3%. Then we graually increase the value of to 0.14 in steps of 0.02 an check the same. The t-omain an f-omain (Boe plot) responses are given in fig. 2 to fig. 7. It is observe that the best results are obtaine for a amping coefficient of = Fig 2 Step response with = 0.1 Fig. 3 Boe plot for = 0.1 For = 0.1, we get a rather high overshoot of 10.3% at a peak time of s. The rise time is s an the settling time is s. The phase margin is 31.1 o at 14.1 r/s. This esign is not acceptable as the overshoot is high. We want almost critical amping. Fig 4 Step response with =

4 Phase (eg) Magnitue (B) Pitch Attitue Control Of A Booster Rocket Fig 5 Boe plot with = 0.12 For = 0.12, we get a much reuce overshoot of 1.53% at a peak time of 0.68 s. The rise time is s an the settling time is s. The phase margin is 33.2 o at 15.1 r/s. This esign is much better. But we shall try to further reuce the overshoot an achieve critical amping. Fig 6 Time omain response for = Boe Diagram Gm = Inf B (at Inf ra/sec), Pm = 34.1 eg (at 16.2 ra/sec) Frequency (ra/sec) Fig. 7 Boe plot for =

5 Pitch Attitue Control Of A Booster Rocket For = 0.14, we get an overshoot of % at a peak time of 1.02 s. The rise time is s an the settling time is s. The phase margin is 34.1 o at 16.2 r/s. The overshoot is now extremely small. We have almost reache critical amping. We accept the thir esign as final. The experimental results for ifferent values of are given in table 1 TABLE 1 COMPARISON OF RESULTS (PA CONTROL) Damping coeff. Rise time Peak time Peak overshoot Settling time Phase margin = s s 10.3% s 31.1 o = s 0.68 s 1.53% s 33.2 o = s o The gain margin is infinity for all the cases. VI. CONCLUSION A booster rocket is use to launch a space-ship from launch-pa to its spatial flight orasatellite to its esire orbit. The rocket has a powerful engine which operates on the principle Newton s 3 r. law. It is propelle controlle combustion of fuel as well as the burning material within the rocket. At first, it moves through air an then through space. The spee has to be continuously controlle as the rocket leaves the launch-pa moves to higher an higher altitue[17,18]. The motion control has to be exercise along three principle axes viz. yaw, pitch an roll. The paper eals with the esign an performance evaluation of automatic control system intene forpitch attitue control of the rocket[1]. The control is of proportional-erivative (PD) type, with controllable rate feeback so as to keep the system only very slightly uner ampe retaining the phase margin above a specifie value. It was trie at first to reach the target by analytical treatment an self-mae program. But the attempt faile. Then recourse was mae to MATLAB-tools [11,12,13]. The specifications coul be reache by simultaneous change of the forwar path gain an the coefficient of rate feeback. REFERENCES Journal Papers: [1] M.D. Shuster, A survey of attitue representations, J. Astronautical Science., 41,pp , [2] Y. Umetani an. Yoshia. Resolve motion rate control of space manipulators with generalize jacobian matrix, in IEEE Transactions on Robotics an Automation,5(3): pp , [3] E. Papaopoulos an S. Dubowsky, On the nature of control algorithms for free-floating space manipulators, in IEEE Trans. on Robotics an Automation,7(6): pp , December, [4] D. Nenchev, Y. Umetani an. Yoshia, Analysis of a reunant free-flying spacecraft/ manipulator system, in IEEE Trans. on Robotics an Automation, 8(1):1-6, [5] O. Egelan an J.R. Sagli, Coorination of motion in a spacecraft/manipulator system, in Int. Journal of Robotics Research, 12(4): pp. 366{379, August, Books: [6]. Ogata, Moern control engineering, Pearson Eucation [7] S.M. Shinners, Moern control system theory an esign, John Wiley an Sons. [8] A.M. Law an W.D. elton, Simulation, moeling an analysis, Mcgraw-Hill, New York, 2 n. Eition, [9] J.R. Wertz (E.): Spacecraft attitue etermination an control,.luwer Acaemic Publishers, Dorrecht, [10] R.H. Battin, An introuction to the mathematics an methos of astroynamics, AIAA Eucation Series, Reston, VA (1999). [11] R. Pratap, Getting strate with MATLAB7,Oxfor, Inian E, [12] J.J. D Azzo, C.H. Houpis an S.N. Shelon, Linear control system analysis an esign with MATLAB, 5e, Marcel Dekker Inc. New York, BASEL [13] A.J. Grace, N. Laub, J.N. Little an C. Thomson, Control system tool box for use with MATLAB, User Guie, Mathworks, [14] E.reyszig, Avance engineering mathematics, Wiley, New York, [15] V. Rajaraman, Computer-oriente numerical methos, PHI, 1995 [16]. Deb, Optimization for engineering esign, PHI, 2010 E- References: [17] Free Encyclopeia of Wikipeia, 12