Rocket Launching Simulation in Vertical-Type PM LSM Rocket Launcher System including Rotational Motion Control

Transcription

1 Rocket Launching Simulation in Vertical-Type PM LSM Rocket Launcher System including Rotational Motion Control K. Yoshida, Y. Cao Graduate School o normation Science and Electrical Engineering Kyushu University chome Hakozaki, Higashi-ku, Fukuoka JAPAN Abstract: - To reduce the cost o launching rocket, various researches are perormed. A new vertical-type linear synchronous motor rocket launcher system is proposed by the irst author Yoshida. The linear launcher is accelerated to a peak speed with loading the rocket, and ater releasing the rocket, it is decelerated with no-load to be brought to rest at the upper end o underground guide tube near the earth surace. Ater the rocket is released rom the launcher and takes o with a high initial-speed, the rocket continues to ascend with no control subject to the Coriolis orce in the guide-direction under the orce o gravity in the rise-direction. This paper presents the rocket launching numerical simulation in vertical- type PM LSM rocket launcher system including rotational motion control. We have veriied the stability o rolling, pitching and yawing motion control by the simulation results in a small experimental model on the ground. Key-words: - PM LSM, Rocket launcher, Rotational motion, Simulation, Eulerian angles, Stability control 1 ntroduction Much practical easibility study on linear synchronous motor ( LSM rocket launcher system is presented on a basis o dynamics simulations. A vertical-type superconducting linear synchronous motor rocket launcher system is proposed by the irst author Yoshida [1][][3]. n previous papers [5][6], we proposed a new vertical-type permanent magnet (PM LSM rocket launcher system which is designed in our laboratory. t is composed o our acceleration guide-ways with armature-windings which are arranged symmetrically along a shat o about 3.8m. The PM LSM rocket launcher is accelerated to a peak speed with loading the rocket, and ater releasing the rocket, it is decelerated with no-load to be brought to rest at the upper end o guide tube. We analyzed the characteristics o three dimensional orces o PM LSM in the derived PM LSM rocket launcher system by using two-dimensional interpolation method [5]. Then, we derived the threedimensional motion equations o launcher and simulated three-dimensional motion control o rocket launcher [6]. However, it was assumed that there were no rotary motions o the rocket launcher. n act, the rocket launcher in which rotary motions are not restricted makes them with three degree o reedom. n this paper, we derived the PM LSM rocket launcher analysis model, based on the PM LSM three dimensional orces analysis results in such a case that the launcher produces complicated rotary motions. We also derived the three dimensional motion equations o PM LSM rocket launcher and proposed rotational motion control method o launcher. The rotational motion o rocket launcher can be controlled stably by using the Eulerian angles control method. The new control method has successully damped rolling, pitching and yawing motions by using eective armaturecurrent and mechanical load angle o each LSM. We simulated three dimensional rotational motion control o launcher with rocket on board. The stability o rocket launching has been veriied by the rotational motion control simulation results. PM LSM Rocket Launcher System Figure 1 shows the vertical-type PM LSM rocket launcher system, which has the acceleration guide tube o about 3.8m on the ground. Figure shows a square coniguration o the vertical-type PM LSM launcher system as a crosssection. n the igure, the linear launcher is delected by a in the y-direction and by b in the z-direction. The rocket launcher has 4 permanent magnets on each o our side-suraces o the launcher, on which there are total mounted 16 PM's. The rated air-gap δ between the ixed armature coil and the running launcher is 5mm in length.

2 3 Three Dimensional Rotational Motions 3.1 Three Dimensional Forces o PM LSM Table 1 shows speciications o vertical-type PM LSM rocket launcher model. Table 1 Parameters o PM LSM rocket launcher Height o launcher 6cm Wih o launcher 17cm Weight o launcher 5kg Weight o rocket kg Maximum mileage o launcher 3m Air-gap 5mm Pole pitch 15cm Moment o inertia around x-axis.16kg. m Moment o inertia around y- and z-axes kg. m Fig.3 Cross-section o one-side PM LSM Fig.1 Vertical-type PM LSM rocket launcher system Fig. PM LSM rocket launcher in horizontal section Figure 3 shows cross-section o one-side PM LSM with deviated position y and rotation angle ψ. n this PM LSM, the x-directed propulsion orce x and the z-directed levitation orce z are produced as x- and z-components o the tangential orce and the y- directed guidance orce y is also exerted as the normal orce in the y-direction. We analyzed the propulsion orce x, guidance orce y and levitation orce z o PM LSM using the analytical model o a PM LSM in such a case that the launcher produces complicated rotary motion within the range o the design. n the analytical results, three dimensional orces are inluenced by the higher harmonic o the magnetomotive orce spatial distribution and varied depending on launcher position x. The change o amplitude is dependent on the air-gap δ, deviated position in the y-direction y and the Eulerian angles (ψ, θ, φ. The phase is dependent on the mechanical load angle x and launcher position x.

3 Because the air-gap δ, delected position y and the Eulerian angles (ψ, θ, φ are a unction o the three dimensional orces, the propulsion, guidance and levitation orces can be transormed according to the theoretical analysis as ollows: (a Propulsion orce x = KFx 1( δ, y, ψ, θ, φ 1sin x 6π + KFx( δ, y, ψ, θ, φ 1sin x + x (b Guidance orce y = KFy1( δ, y, ψ, θ, φ 1cos x 6π + KFy( δ, y, ψ, θ, φ 1cos x + x (c Levitation orce z = KFz1( δ, y, ψ, θ, φ 1cos x + KFz ( δ, 6 π y, ψθφ,, 1 cos x + x where is the pole pitch, 1 is eective armaturecurrent, δ is air-gap length between PM and armature coil, x is mechanical load angle, K Fx1 and K Fx are coeicients o propulsion orce, K Fy1 and K Fy are coeicients o guidance orce, K Fz1 and K Fz are coeicients o levitation orce. 3. Equations o Motion n the PM LSM rocket launcher system, neglecting airresistance, the equations o three dimensional motions are simply described as ollows: (a beore the rocket separates rom launcher ( ML + MR ax = x1 + x + x3 + x4 ( ML + MR g ( ML + MR ay = z z4 + y1 y3 ( M + M a = + L R z z1 z3 y4 y (b ater the rocket separates rom launcher MLax = x1+ x + x3+ x4 MLg MRax = M R Rg M a = + L y z z4 y1 y3 MRayR = M a = + M a = L z z1 z3 y4 y R zr (1 ( (3 (4 (5 (6 (7 (8 (9 (1 (11 (1 where a x, a y and a z are x-, y- and z-directed accelerations o the launcher, respectively. g is the acceleration o gravity. M L and M R is the mass o the launcher and rocket, respectively. n the PM LSM rocket launcher system, the Eulerian angles (ψ, θ, φ are necessary to describe the rotary motion o the rocket launcher. The rolling angle ψ is a rotational angle in the x-axis circumerence. The pitching angle θ is a rotational angle in the y-axis circumerence. The yawing angle φ is a rotational angle in the z-axis circumerence. By the analytical result, the equations o three dimensional rotational motions including rolling, pitching and yawing motions are simply described, neglecting air-resistance, as ollows: ( ψ = yθ =.775 ( x 1 + x φ =.775 x y z y z ( z1s1 zs1 z3s1 z4s1 ( z1n1 zn1 z3n1 z4n1 ( z1s zs z3s z4s ( z1n zn z3n z4n ( z x x4 3 ( y1s1 + ys1 + y3s1 + y4s1 y + y + y + y + y + y + y + y ( 1N1 N1 3N1 4N1 ( 1S S 3S 4S ( y1n yn y3n y4n (13 (14 (15 where x, y, z are moments o inertia around x-, y- and z-axes. 4 Control Method 4.1 Control Method or Three Dimensional Motions n the vertical-type PM LSM rocket launcher system, each LSM is controlled independently. The demand patterns o mechanical load angle x and eective armature-current 1 are obtained or the demand threedimensional motion pattern rom the motion equation. The three dimensional motions o the launcher are controlled by eective armature-current 1 and mechanical load angle x o each LSM.

4 n order to ollow the demand patterns in the x-, y- and z-directions, the control law or F x, F y and F z based on PD regulator becomes as ollows: (a beore the rocket separates rom launcher Fx = ( ML + MR ax + ( ML + MR g + KPx( x x + K ( x x + K ( x x (16 x Dx F = ( M + M a + K ( y y y L R y Py + K ( y y + K ( y y y Dy F = ( M + M a + K ( z z z L R z Pz + K ( z z + K ( z z z Dz (b ater the rocket separates rom launcher F = M a + M g + K ( x x x L x L Px + K ( x x + K ( x x x Dx F = M a + K ( y y y L y Py + K ( y y + K ( y y y Dy F = M a + K ( z z z L z Pz + K ( z z + K ( z z z Dz (17 (18 (19 ( (1 where F x, F y and F z are command three dimensional orces o launcher, respectively. K Pi, K i and K Di ( i = x, y, z are eedback gains. x and x are x- directed position and its demand pattern o the launcher, respectively. 4. Control Method or Rolling Motion By the Eulerian angles control method, the rolling motion is controlled by using the controlled-value o levitation orce sc that is generated by each LSM. To control rolling motion, the eedback control law becomes as ollows: = K ( ψ ψ + K sc Pψ Dψ d( ψ ψ ( where ψ is rolling angle o the launcher and K Pψ, K Dψ are eedback gains. 4.3 Control Method or Pitching and Yawing Motions The control method o pitching motion and yawing motion is similar because there is a symmetry o the PM LSM rocket launcher system. The pitching motion is controlled by using the controlled-value o propulsion orces that is generated by LSM1 and LSM3. Similarly, the yawing motion is controlled by using the controlledvalue o propulsion orces that is generated by LSM and LSM4. To control pitching and yawing motions, the controlled-value o propulsion orces,, xc1 xc and xc3 xc4 are shown below ( θ xc 1 = x 1 = K ( θ θ xc3 x3 Pθ K ( θ θ K ( θ < = + K ( θ θ xc1 x1 Pθ xc3 x3 ( φ = = xc x θ xc4 x4 Pφ + K ( θ θ + K θ = K ( φ φ K φ xc x Pφ xc4 x4 ( φ φ K ( φ < = + K ( φ φ = + K ( φ φ + K φ d( θ θ d( θ θ (3 (4 (5 (6 (7 (8 (9 (3 where θ is pitching angle o the launcher, φ is yawing angle o the launcher and K Pθ, K θ, K, K Dθ Pφ, K φ, K Dφ are eedback gains. 4.4 nstantaneous Armature-current From command mechanical load angle x n and command eective armature-current 1n, each LSM's command values o instantaneous armature-current can be derived. The instantaneous three phase armature-current o each LSM are shown below Dθ Dθ Dφ Dφ d( φ φ d( φ φ π iun = 1 n cos ωt+ xn π ivn = 1 n cos ωt + xn π 3 π 4 iwn = 1 n cos ωt+ xn π 3 π ωt = x ( t where i un, i vn and i wn ( n = 1,, 3, 4 are each LSM s command instantaneous armature-current. (31 (3

5 a, a L R (m/s x, x L R (m 4. x R 3. x L (a Positions o launcher and rocket a L a R (c Accelerations o launcher and rocket (mm v, v (m/s L R y v R v L (b Speeds o launcher and rocket (d Delection o launcher in the y-direction.1 (mm (rad.5 z - ψ (e Delection o launcher in the z-direction ( Rolling angle o the launcher.5.5 (rad (rad θ -.5 φ -.5 (A 1, LSM1 (A 1, LSM (g Pitching angle o the launcher (i Eective current o LSM (k Eective current o LSM x /, LSM1, LSM x / (h Yawing angle o the launcher (j Mechanical load angle o LSM (l Mechanical load angle o LSM Fig.4 Simulation results

6 (A 1, LSM3 (A 1, LSM (m Eective current o LSM (o Eective current o LSM4, LSM3 x /, LSM4 x / (n Mechanical load angle o LSM (p Mechanical load angle o LSM4 Fig.4 Simulation results 5 Rocket Launching Simulation Figure 4 shows rocket launching simulation results. Figure 4 (a ~ (c show the positions, speeds and accelerations o the launcher and rocket, respectively. From these igures, we can see that the launcher ollows very well the demand patterns o position, speed and acceleration. Figure 4 (d ~ (h show delections in the y- and z- directions, rolling angle, pitching angle and yawing angle o the launcher, respectively. To veriy the stability o 3-D rotational motion control, we simulated rocket launcher starting rom a lot o dierent positions. From the results, the launcher can be controlled stably in the short time when it starts rom horizontal and rotational displacements. Figure 4 (i ~ (p show results o the mechanical load angle x and the eective armature-current 1 o each LSM. t is ound rom Fig. 4 (i that the maximum value o eective armature-current oversteps 38 A when launching rocket starts rom horizontal displacements with (, 3mm, mm and rotational displacements with (.rad,.6rad,.6rad. 6 Conclusion n the vertical-type PM LSM rocket launcher system proposed here, propulsion, levitation and guidance motions have been controlled simultaneously by our LSM's. The Eulerian angles control method proposed by us has successully applied in the rotational motion control simulation, including rolling, pitching and yawing motions. We have veriied the stability o three dimensional rotational motions control including rolling, pitching and yawing motions in the PM LSM rocket launcher system with launching rocket. Reerences: [1] K.Yoshida, T.Ohashi, K.Shiraishi, H.Takami : Dynamics Simulations o Superconducting LSM Rocket Launcher System, Proc. J. o Space Tech. and Science, Vol.9 No., 1993, pp [] K.Yoshida, T.Ohashi, K.Shiraishi, H.Takami : Feasibility Study o Superconducting LSM Rocket Launcher System, Proc. o nd nt. Symp. on Mag. Susp. Tech. NASA Con. Pub. No. 347 Part, 1994, pp [3] K.Yoshida, K.Hayashi, H.Takami : Electromechanical Dynamics Simulations o Superconducting LSM Rocket Launcher System in Attractive- Mode, Proc. o 3rd nt. Symp. on Mag. Susp. Tech. NASA Con. Pub. No.3336 Part, 1996, pp [4] L.S.Tung, R.F.Post, E.Cook and J.Martinez-Frias: The NASA nductrack Model Rocket Launcher at the Lawrence Livermore National Laboratory, Proc. o 5th SMST,, pp [5] K. Yoshida and Y. Cao : - D nterpolation o Propulsion, Levitation and Guidance Forces in Vertical-type PM LSM Rocket Launcher System, Proc. o 7th SMST, 3, pp [6] K. Yoshida and Y. Cao : Three Dimensional Motions Control Simulation or Radial Displacements in Vertical-type PM LSM Rocket Launcher, Proc. o 4th PEMC, 4.