A Simplified Dynamics Block Diagram for a Four-Axis Stabilized Platform

Transcription

1 A Simplified Dynamic Block Diagram for a FourAxi Stabilized Platform Author: Hendrik Daniël Mouton a Univerity of Cape Town, Rondeboch, Cape Town, South Africa, 770 Abtract: t i relatively traightforward to derive comprehenive dynamic block diagram for two and threeaxi balanced tabilized platform. But trying to do it for four and higher order ytem i getting rapidly uch an amount of work, and the reulting block diagram o extenive, that enible implification from the tart if feaible could be very beneficial. Thoe term that are going to be inignificantly mall relative to the other term hould be left out right from the tart when deriving the equation. n thi paper a method i hown to implify the dynamic model of tabilized platform, with the fouraxi platform a the main example. The relevant equation are derived for the fouraxi example, the block diagram i compiled from the equation after adding imple control loop around the dynamic model to get to the final expanded block diagram, and then ome imulation reult are hown. The method i validated by howing that the imulation reult for a threeaxi ytem are exactly the ame for the comprehenive and implified block diagram, uing the ame method. t i propoed that it i therefore reaonable to expect imilar finding for the fouraxi cae. a Aociate Profeor H D Mouton, Mechanical Engineering, UCT, hennie.mouton@uct.ac.za

2 ntroduction: Variou ort of camera could typically be mounted on tabilized platform to give much improved quality image. t would provide tabilization againt rotational movement of the bae a for example naturally experienced during linear movement of the bae (land craft or aircraft or overhead porting camera mount). Reference [] and [] give very good overall view of major iue in tabilized tracking deign. Deigning uch platform require proper dynamic modelling of the mechanical configuration. t i not difficult to find the typical relevant equation to repreent the dynamic reference [3] and [4] are typical example, but converting them to enible block diagram to ait in imulation and to more eaily ee what i happening i not that readily available. So thi paper contain block diagram, being derived from baic dynamic equation uch a given in reference [5], but the main purpoe of the paper i to demontrate a method to develop implified but till valid block diagram. A threeaxi example: To demontrate the poible validity of the implification method to be given in thi paper, it i firt demontrated with a threeaxi tabilized platform. The derivation of the equation for the threeaxi cae i not going to be given here only the definition, the reaoning for ignoring certain term, a block diagram including control loop and then imulation reult. The block diagram will clearly indicate the comprehenive and implified model. The reult will how that for a uitable tet cenario the implified dynamic model give the ame reult a the comprehenive model. f the latter reult are indeed very imilar, it will be aumed to be valid to apply the ame method of implification to four and moreaxi tabilized platform. Of coure different configuration are poible for the threeaxi cae. But the demontration i going to be done on a pecific choen example. The aumption i that other configuration will give imilar reult. An outer to inner yawpitchroll configuration wa choen. t i aumed that the control loop in the end will provide good tabilization of the platform in roll, yaw and pitch a made poible by the three axe. Nomenclature for the threeaxi cae: From outer to inner, thee are the Euler angle and ubcript to be ued:

3 B will be ued a ubcript for the bae. ψ i the yaw angle between the bae and the outer yaw gimbal yaw will be ued a ubcript. θ i the pitch angle between the yaw gimbal and the pitch gimbal pit will be ued a ubcript. φ i the roll angle between the pitch gimbal and the platform to be tabilized rol will be ued a the ubcript. ω denote inertial angular rate. x, y and z correpond baically to the axe about which roll, pitch and yaw rotation happen repectively. The following picture hould clarify the configuration and the ymbol. X pit, X rol Y yaw, Y pit Z B, Z yaw Figure. Threeaxi configuration Small/large ignal ditinction for three axe: Becaue of the aumption above, the following ditinction between potentially large and mall ignal can be made. Table. Small/large ignal for threeaxi configuration ω xrol ω yrol ω zrol ω xpit ω ypit ω zpit ω xyaw ω yyaw ω zyaw Small Large

4 Becaue it i aumed that the tabilization i going to be good, diturbance torque can be ignored that contain multiplication with any of the ignal in thi table that are claified a Small. Block diagram and imulation reult for three axe: Thi lead to comprehenive and implified block diagram. The implified block diagram i the ame a the comprehenive one but with all block coloured with grey omitted ee the firt block diagram to follow. By looking at the increae in diturbance torque when modelling the dynamic of the three axe from the inner axi to the outer axi a repreented in the comprehenive block diagram, an idea can be formed how expanded comprehenive four and fiveaxi platform model would be. Rather imple tabilization loop are added in the block diagram to follow, with ω g ζ g ω g ω g repreenting feedback from gyro mounted on the platform, meauring ω xrol, ω yrol and ω zrol. A proportional controller and one lead compenator cloe each loop. The command input to the three loop are zero becaue the intention of the loop i to control ω xrol, ω yrol and ω zrol at zero depite the bae motion ω xb, ω yb and ω zb applied to the ytem. They are applied one after the other but overlapping a hown in the firt graph below. Their amplitude are.0 rad/ and their frequencie linearly increae from.0 Hz to 50.0 Hz. n the later graph they will be hown on top of one another imply becaue many other ignal mut be hown on the ame graph.

5 Figure. Applied bae motion The amplitude of the meaurement ω xrol, ω yrol and ω zrol are indication of the quality of tabilization the maller thee ignal the better the tabilization. ω zrol i ignificantly larger than ω xrol and ω yrol a can be expected from uch a threeaxi ytem. t will later be een that the fouraxi ytem provide much better tabilization with baically the ame control loop, but that apect i not the focu of thi paper. The initial yaw, pitch and roll angle, ψ, θ and φ, were et at 45 to prevent ome diturbance torque to be very mall becaue of mall angle. f ome diturbance torque were mall becaue of mall angle it would have defied the validation of the implification propoed here. Remember: n the block diagram below the implified block diagram i the ame a the comprehenive one but with all block coloured with grey omitted.

6 gg g g xrol K T rol rol [ m] 00[ rad/ ] g 0.7 g 0.0 K rol T rol 0.T rol T xrol xrol zrol yrol xrol zrol in co in co yrol ypit xpit in co in co K T K T pit pit 3.0 [ m] yaw yaw [ m] zyaw in co xyaw xb zpit xpit in co yb yyaw xyaw zyaw Typitrol Tzyawpitrol zb T pit 0.T pit yzpitrol zpit zpit xyaw xyaw zxyawpitrol T yaw 0.T yaw K pit xpit zpit ( zrol )co xrol xpit zrol yyaw yyaw yzyawpitrol yyaw xyaw Kyaw 0.0 xrol ( yrol xrol )in ( xpit )co 0. 0 xpit ypit yrol ypit yrol co zrol ( zpit ypit)in zpit in zrol ypitrol ypit yrol yrol xrol zrol yrol zrol ( xrol )coco yrol ( zrol )inco xrol ( yrol )in zrol gg g g yrol yyaw xpit zpit co zpitrol in xpitrol xrol yrol zrol xpit ypit zpit xyaw yyaw zyaw 3.0e 4 [ kg. 5.0e 4 [ kg. 5.0e 4 [ kg..0e 4 [ kg..0e 4 [ kg. 3.0e 4 [ kg. 4.0e 4 [ kg. 3.3e 4 [ kg. 3.0e 4 [ kg. xpitrol ypitrol zpitrol yzpitrol zyawpitrol zxyawpitrol yzyawpitro l xpit ypit zpit yrol zyaw xrol zrol zpitrol yrol yzpitrol co co zrol in zpitrol co co xpitrol zrol yrol in in in xpitrol in yrol zrol co co zrol in co yrol zyaw yzyawpitrol xrol in zyawpitrol zb zrol gg g g Figure 3. Comprehenive and implified block diagram of threeaxi yawpitchroll

7 The reult for the comprehenive and implified model are hown below, demontrating that the propoed implification are valid ince the reult are identical. Figure 4. Comprehenive threeaxi graph reult

8 Figure 5. Simplified threeaxi graph reult

9 The choen fouraxi example: Like for all axe, different configuration are alo poible for the fouraxi cae. The principle to be dicued here i applicable to all configuration. An outer to inner yawpitchyawroll configuration wa choen. t i aumed that the control loop in the end will provide good tabilization of the platform in roll, yaw and pitch a made poible by the three inner axe. The outer yaw axi i only for rough alignment typically not controlled by an inertial rate loop like the inner three axe, but by a relative rate loop enuring that the inner yaw gimbal angle remain relatively mall. Nomenclature for the fouraxi cae: From outer to inner, thee are the Euler angle and ubcript to be ued: B will be ued a ubcript for the bae. β i the yaw angle between the bae and the outer yaw gimbal Yaw will be ued a ubcript. θ i the pitch angle between the outer yaw gimbal and the pitch gimbal pit will be ued a ubcript. ψ i the yaw angle between the pitch gimbal and the inner yaw gimbal yaw will be ued a ubcript. φ i the roll angle between the inner yaw gimbal and the platform to be tabilized rol will be ued a the ubcript. ω denote inertial angular rate. x, y and z correpond baically to the axe about which roll, pitch and yaw happen repectively. The following picture hould clarify the configuration and the ymbol.

10 Y Yaw, Y pit Z pit, Z yaw X yaw, X rol Z B, Z Yaw Figure 6. Fouraxi configuration Small/large ignal ditinction for four axe: Becaue of the aumption above, the following ditinction between potentially large and mall ignal can be made. Table. Small/large ignal for fouraxi configuration ω xrol ω yrol ω zrol ω xyaw ω yyaw ω zyaw ω xpit ω ypit ω zpit ω xyaw ω yyaw ω zyaw Small Large Becaue it i aumed that the tabilization i going to be good, diturbance torque can be ignored that contain the term that were claified a Small in the table above. Derivation of equation for the four axe: Apply T = d dt a (W) ω a W with T the net torque and W = ω to the platform (rol), the inner yaw gimbal (yaw), the pitch gimbal (pit) and the outer yaw gimbal (Yaw). d dt a refer to the derivative w.r.t. a noninertial axe ytem. ω a i the inertial angular rate vector of the noninertial axe ytem.

11 ω i the inertial angular rate vector of the body under conideration. i the Moment of nertia matrix of the body under conideration. To the platform ( rol ): xrol 0 0 W = [ 0 yrol 0 ] [ 0 0 zrol ω xrol ω yrol ω xrol ω yrol ] ω a = [ ] ω zrol ω zrol t i aumed that the ymmetry i uch that the inertia matrix can be repreented in the diagonal form. T xrol T rol = [ T yrol ] = [ T zrol xrol ω xrol yrol ω yrol zrol ω zrol ω yrol ω zrol ( zrol yrol ) ] [ ω zrol ω xrol ( xrol zrol ) ω xrol ω yrol ( yrol xrol ) ] T xrol xrol ω xrol () T yrol yrol ω yrol T zrol zrol ω zrol To the inner yaw gimbal ( yaw ): xyaw 0 0 ω xyaw ω xyaw W = [ 0 yyaw 0 ] [ ω yyaw ] ω a = [ ω yyaw ] 0 0 zyaw ω zyaw ω zyaw t i aumed that the ymmetry i uch that the inertia matrix can be repreented in the diagonal form. T xyaw xyaw ω xyaw ω yyaw ω zyaw ( zyaw yyaw ) T yaw = [ T yyaw ] = [ yyaw ω yyaw] [ ω zyaw ω xyaw ( xyaw zyaw )] T zyaw zyaw ω zyaw ω xyaw ω yyaw ( yyaw xyaw ) T xyaw xyaw ω xyaw T yyaw yyaw ω yyaw T zyaw zyaw ω zyaw T yaw i the net torque applied to the inner yaw gimbal. Call the torque applied to thi gimbal from the outide T yawrol

12 T yaw = T yawrol T rol T yawrol = T yaw T rol Now take the roll angle φ between the yaw and rol axe into account: Y yaw Y rol Z rol Z yaw Figure 7. Roll angle tranformation T xyawrol T xyaw T xrol T yawrol = [ T yyawrol ] = [ T yyaw T yrol co φ T zrol in φ] T zyawrol T zyaw T zrol co φ T yrol in φ T xyawrol xyaw ω xyaw xrol ω xrol T yyawrol yyaw ω yyaw yrol ω yrol co φ zrol ω zrolin φ T zyawrol zyaw ω zyaw zrol ω zrol co φ yrol ω yrolin φ The following are the equation for the angular rate and acceleration: ω xrol = ω xyaw φ ω yrol = ω yyaw co φ ω zyaw in φ () ω zrol = ω zyaw co φ ω yyaw in φ ω xrol = ω xyaw φ ω yrol = ω yyaw co φ ω zyaw in φ φ ω zrol ω zrol = ω zyaw co φ ω yyaw in φ φ ω yrol Subtitute ω xrol, ω yrol and ω zrol in T xyawrol, T yyawrol and T zyawrol. T xyawrol ω xyaw( xyaw xrol ) φ xrol T yyawrol ω yyaw( yyaw yrol co Φ zrol in φ) ω zyaw( yrol zrol ) in φ co φ φ ω zrol yrol co φ φ ω yrol zrol in φ

13 T zyawrol ω zyaw( zyaw zrol co Φ yrol in φ) ω yyaw( yrol zrol ) in φ co φ φ ω yrol zrol co φ φ ω zrol yrol in φ Define: xyawrol = xyaw xrol (3) yyawrol = yyaw yrol co φ zrol in φ (4) zyawrol = zyaw zrol co φ yrol in φ (5) yzyawrol = yrol zrol in φ (6) T xyawrol ω xyaw xyawrol φ xrol T yyawrol ω yyaw yyawrol ω zyaw yzyawrol φ ω zrol yrol co φ φ ω yrol zrol in φ T zyawrol ω zyaw zyawrol ω yyaw yzyawrol φ ω yrol zrol co φ φ ω zrol yrol in φ (7) To the pitch gimbal ( pit ): xpit 0 0 ω xpit ω xpit W = [ 0 ypit 0 ] [ ω ypit ] ω a = [ ω ypit ] 0 0 zpit ω zpit ω zpit t i aumed that the ymmetry i uch that the inertia matrix can be repreented in the diagonal form. T xpit xpit ω xpit ω ypit ω zpit ( zpit ypit ) T pit = [ T ypit ] = [ ypit ω ypit] [ ω zpit ω xpit ( xpit zpit )] T zpit zpit ω zpit ω xpit ω ypit ( ypit xpit ) T xpit xpit ω xpit T ypit = ypit ω ypit ω zpit ω xpit ( xpit zpit ) T zpit zpit ω zpit T pit i the net torque applied to the pitch gimbal. Call the torque applied to thi gimbal from the outide T pityawrol. T pit = T pityawrol T yawrol T pityawrol = T pit T yawrol

14 Now take the yaw angle ψ between the pit and yaw axe into account: X pit X yaw Y yaw Y pit Figure 8. Yaw angle tranformation T xpityawrol T xpit T xyawrol co ψ T yyawrol in ψ T pityawrol = [ T ypityawrol ] = [ T yrol T yyawrol co ψ T xyawrol in ψ] T zpityawrol T zpit T zyawrol T xpityawrol xpit ω xpit (ω xyaw xyawrol φ xrol ) co ψ (ω yyaw yyawrol ω zyaw yzyawrol φ ω zrol yrol co φ φ ω yrol zrol in φ) in ψ xpit ω xpit (ω xyaw xyawrol φ xrol ) co ψ (ω yyaw yyawrol ω zyaw yzyawrol ) in ψ with φ ω yrol and φ ω zrol ignored. T ypityawrol ypit ω ypit ω zpit ω xpit ( xpit zpit ) (ω yyaw yyawrol ω zyaw yzyawrol φ ω zrol yrol co φ φ ω yrol zrol in φ) co ψ (ω xyaw xyawrol φ xrol ) in ψ ypit ω ypit ω zpit ω xpit ( xpit zpit ) (ω yyaw yyawrol ω zyaw yzyawrol ) co ψ (ω xyaw xyawrol φ xrol ) in ψ with φ ω yrol and φ ω zrol ignored. T zpityawrol zpit ω zpit ω zyaw zyawrol ω yyaw yzyawrol φ ω yrol zrol co φ φ ω zrol yrol in φ zpit ω zpit ω zyaw zyawrol ω yyaw yzyawrol with φ ω yrol and φ ω zrol ignored. The following are the equation for the angular rate and acceleration: ω xyaw = ω xpit co ψ ω ypit in ψ

15 ω yyaw = ω ypit co ψ ω xpit in ψ (8) ω zyaw = ω zpit ψ ω xyaw = ω xpit co ψ ω ypit in ψ ψ ω yyaw ω yyaw = ω ypit co ψ ω xpit in ψ ψ ω xyaw ω zyaw = ω zpit ψ Subtitute ω xyaw, ω yyaw and ω zyaw in T xpityawrol, T ypityawrol and T zpityawrol. T xpityawrol xpit ω xpit (ω xpit co ψ ω ypit in ψ ψ ω yyaw ) xyawrol co ψ (ω ypit co ψ ω xpit in ψ ψ ω xyaw ) yyawrol in ψ (ω zpit ψ ) yzyawrol in ψ φ xrol co ψ = ω xpit( xpit xyawrol co ψ yyawrol in ψ) ω ypit( xyawrol yyawrol ) in ψ co ψ ω zpit yzyawrol in ψ ψ ω yyaw xyawrol co ψ ψ ω xyaw yyawrol in ψ ψ yzyawrol in ψ φ xrol co ψ T ypityawrol ypit ω ypit ω zpit ω xpit ( xpit zpit ) (ω ypit co ψ ω xpit in ψ ψ ω xyaw ) yyawrol co ψ (ω xpit co ψ ω ypit in ψ ψ ω yyaw ) xyawrol in ψ (ω zpit ψ ) yzyawrol co ψ φ xrol in ψ = ω ypit( ypit yyawrol co ψ xyawrol in ψ) ω zpit yzyawrol co ψ ω xpit( xyawrol yyawrol ) in ψ co ψ ω zpit ω xpit ( xpit zpit ) ψ ω xyaw yyawrol co ψ ψ ω yyaw xyawrol in ψ ψ yzyawrol co ψ φ xrol in ψ T zpityawrol zpit ω zpit (ω zpit ψ ) zyawrol (ω ypit co ψ ω xpit in ψ ψ ω xyaw ) yzyawrol = ω zpit( zpit zyawrol ) ω xpit yzyawrol in ψ ω ypit yzyawrol co ψ ψ ω xyaw yzyawrol ψ zyawrol Define: xpityawrol = xpit xyawrol co ψ yyawrol in ψ (9) ypityawrol = ypit yyawrol co ψ xyawrol in ψ (0) zpityawrol = zpit zyawrol () xypityawrol = xyawrol yyawrol in ψ () yzpityawrol = yzyawrol co ψ (3) zxpityawrol = yzyawrol in ψ (4)

16 T xpityawrol ω xpit xpityawrol ω ypit xypityawrol ω zpit zxpityawrol ψ ω yyaw xyawrol co ψ ψ ω xyaw yyawrol in ψ ψ yzyawrol in ψ φ xrol co ψ T ypityawrol ω ypit ypityawrol ω zpit yzpityawrol ω xpit xypityawrol ω zpit ω xpit ( xpit zpit ) ψ ω xyaw yyawrol co ψ ψ ω yyaw xyawrol in ψ ψ yzyawrol co ψ φ xrol in ψ (5) T zpityawrol ω zpit zpityawrol ω xpit zxpityawrol ω ypit yzpityawrol ψ ω xyaw yzyawrol ψ zyawrol To the outer yaw gimbal (Yaw): xyaw 0 0 W = [ 0 yyaw 0 ] [ 0 0 zyaw ω xyaw ω yyaw ω xyaw ω yyaw ] ω a = [ ] ω zyaw ω zyaw t i aumed that the ymmetry i uch that the inertia matrix can be repreented in the diagonal form. T xyaw T Yaw = [ T yyaw ] = [ T zyaw xyaw ω xyaw yyaw ω yyaw zyaw ω zyaw ω yyaw ω zyaw ( zyaw yyaw ) ] [ ω zyaw ω xyaw ( xyaw zyaw ) ω xyaw ω yyaw ( yyaw xyaw ) ] T Yaw i the net torque applied to the outer yaw gimbal. Call the torque applied to thi gimbal from the outide T Yawpityawrol. T Yaw = T Yawpityawrol T pityawrol T Yawpityawrol = T Yaw T pityawrol Now take the pitch angle θ between the Yaw and pit axe into account: X pit X Yaw Z pit Z Yaw Figure 9. Pitch angle tranformation

17 T xyawpityawrol T xyaw T xpityawrol co θ T zpityawrol in θ T Yawpityawrol = [ T yyawpityawrol ] = [ T yyaw T ypityawrol ] T zyawpityawrol T zyaw T zpityawrol co θ T xpityawrol in θ T zyawpityawrol zyaw ω zyaw ω xyaw ω yyaw ( yyaw xyaw ) (ω zpit zpityawrol ω xpit zxpityawrol ω ypit yzpityawrol ψ ω xyaw yzyawrol ψ zyawrol ) co θ (ω xpit xpityawrol ω ypit xypityawrol ω zpit zxpityawrol ψ ω yyaw xyawrol co ψ ψ ω xyaw yyawrol in ψ ψ yzyawrol in ψ φ xrol co ψ) in θ zyaw ω zyaw ω xyaw ω yyaw ( yyaw xyaw ) (ω zpit zpityawrol ω xpit zxpityawrol ω ypit yzpityawrol ψ ω xyaw yzyawrol ψ zyawrol ) co θ (ω xpit xpityawrol ω ypit xypityawrol ω zpit zxpityawrol ψ ω xyaw yyawrol in ψ ψ yzyawrol in ψ φ xrol co ψ) in θ with ψ ω yyaw ignored. The following are the equation for the angular rate and acceleration: ω xpit = ω xyaw co θ ω zyaw in θ ω ypit = ω yyaw θ (6) ω zpit = ω zyaw co θ ω xyaw in θ ω xpit = ω xyaw co θ ω zyaw in θ θ ω zpit ω ypit = ω zyaw θ ω zpit = ω zyaw co θ ω xyaw in θ θ ω xpit Subtitute ω xpit, ω ypit and ω zpit in T zyawpityawrol. T zyawpityawrol zyaw ω zyaw ω xyaw ω yyaw ( yyaw xyaw ) (ω zyaw co θ ω xyaw in θ θ ω xpit ) zpityawrol co θ (ω zyaw co θ ω xyaw in θ θ ω xpit ) zxpityawrol in θ (ω xyaw co θ ω zyaw in θ θ ω zpit ) zxpityawrol co θ (ω xyaw co θ ω zyaw in θ θ ω zpit ) xpityawrol in θ (ω yyaw θ ) yzpityawrol co θ (ω yyaw θ ) xypityawrol in θ ψ ω xyaw ( yzyawrol co θ yyawrol in ψ in θ) ψ ( zyawrol co θ yzyawrol in ψ in θ) φ xrol co ψ in θ = ω zyaw( zyaw zpityawrol co θ zxpityawrol in θ xpityawrol in θ) ω xyaw[ zxpityawrol (co θ in θ) ( xpityawrol zpityawrol ) in θ co θ] ω yyaw( yzpityawrol co θ xypityawrol in θ)

18 ω xyaw ω yyaw ( yyaw xyaw ) θ ω xpit ( zpityawrol co θ zxpityawrol in θ) θ ω zpit ( zxpityawrol co θ xpityawrol in θ) ψ ω xyaw ( yzyawrol co θ yyawrol in ψ in θ) θ ( yzpityawrol co θ xypityawrol in θ) ψ ( zyawrol co θ yzyawrol in ψ in θ) φ xrol co ψ in θ Define: zyawpityawrol = zyaw zpityawrol co θ zxpityawrol in θ xpityawrol in θ (7) zxyawpityawrol = zxpityawrol co θ xpityawrol zpityawrol in θ (8) yzyawpityawrol = yzpityawrol co θ xypityawrol in θ (9) T zyawpityawrol ω zyaw zyawpityawrol ω xyaw zxyawpityawrol ω yyaw yzyawpityawrol ω xyaw ω yyaw ( yyaw xyaw ) θ ω xpit ( zpityawrol co θ zxpityawrol in θ) θ ω zpit ( zxpityawrol co θ xpityawrol in θ) ψ ω xyaw ( yzyawrol co θ yyawrol in ψ in θ) θ ( yzpityawrol co θ xypityawrol in θ) ψ ( zyawrol co θ yzyawrol in ψ in θ) φ xrol co ψ in θ (0) Now take the yaw angle β between the B and Yaw axe into account: X B X Yaw Y Yaw Y B Figure 0. Outer yaw angle tranformation ω xyaw = ω xb co β ω yb in β ω yyaw = ω yb co β ω xb in β () ω zyaw = ω zb β

19 Block diagram and imulation reult for four axe: From equation () to () and with the addition of friction, the following block diagram can be compiled. Rather imple tabilization loop are added in the block diagram, with ω g ζ g ω g ω g repreenting feedback from gyro mounted on the platform, meauring ω xrol, ω zrol and ω yrol. A proportional controller and one lead compenator cloe each loop. The outer yaw gimbal ha a tachometer, repreented by ω t ζ t ω t ω t, feeding back the relative angular rate β and alo a proportional controller and one lead compenator. t i driven by the inner yaw gimbal angle ψ through a gain K ctrl to effectively keep ψ at a mall angle therefore aiding good tabilization. The ize of the meaurement ω xrol, ω yrol and ω zrol i an indication of the quality of tabilization the maller thee ignal the better the tabilization. They can be een in the graph below to be much maller than the previou threeaxi cae a expected. The initial Yaw, pitch and roll angle β, θ and φ were et at 45 to prevent ome diturbance torque to be very mall becaue of mall angle.

20 gg g g xrol K T rol rol [ m] K ctrl.0[( rad/ )/ rad] 00[ rad/ ] g 0.7 g K rol T rol 0.T rol K yaw Tyaw 0.T yaw T xrol T ypityawrol gg g T zyawrol g xrol xrol zrol in co in co yrol zyaw yyaw zyaw zyawrol zpit yyaw yyaw yzyawrol xyaw yyaw in co in co xyaw ypit xyaw yyaw K T K T yaw yaw pit pit xpit [ m] [ m] xpit zpit xyaw yyaw K T in co in co xyaw Yaw Yaw [ m] zyaw zyaw zxyawpityawrol yzyawpityawrol yyaw xyaw K ctrl yyaw in co in co T zyawpityawrol 00[ rad/ ] t 0.7 t T yaw 0.T yaw K yaw xb yb zb T pit 0.T pit K pit yzpityawrol zpit xpit xypityawro l xpit zpit xpit zpit co in zpityawrol zxpityawro l zxpityawrol co in xpityawrol 0.0 xpit zpit co yyawrol yzpityawro l in xrol xyaw xyaw yzyawrol yzpityawrol zyawrol co yyawrol in in co in xypityawrol co yzyawrolin in co in xrol ypityawrol ypit zyaw zyawpityawrol zb gg g g yrol yyaw t tt t xrol yrol zrol xyaw yyaw zyaw 3.0e 4 [ kg. 5.0e 4 [ kg. 5.0e 4 [ kg..0e 4 [ kg. 3.0e 4 [ kg..0e 4 [ kg. xpit ypit zpit xyaw yyaw zyaw 4.0e 4 [ kgm. ] 3.0e 4 [ kgm. ] 3.3e 4 [ kgm. ] 5.0e 4 [ kgm. ] 4.4e 4 [ kgm. ] 4.0e 4 [ kgm. ] xyawrol yyawrol zyawrol yzyawrol xyaw yyaw zyaw yrol xrol yrol zrol zrol co co in zrol yrol in in xpityawrol ypityawrol zpityawrol xypityawrol yzpityawrol zxpityawrol xpit ypit zpit xyawrol yzyawrol yzyawrol xyawrol yyawrol zyawrol co in co co yyawrol in yyawrol xyawrol in in zyawpityawrol zxyawpitya wrol yzyawpitya wrol zyaw zxpityawro l yzpityawro l zpityawrol co co co xpityawrol xypityawro l zxpityawro l in in zpityawrol in xpityawrol in Figure. Simplified block diagram of fouraxi yawpitchyawroll

21 The imulation reult i the following. Figure. Simplified fouraxi graph reult A could be expected, ω xrol, ω yrol and ω zrol, which are indication of the quality of tabilization, are much maller than for the threeaxi cae. Concluion: Baed on the demontration with the threeaxi example where the comprehenive and implified model were compared and found to give identical reult, and the reaoning for ignoring certain term when deriving the implified equation and model, it i concluded that the propoed implification method i valid alo for moreaxi, like the fouraxi configuration. The aving in the amount of work to be done to derive model for four and higher number of axe platform by uing thi method i ubtantial a can be appreciated a bit by comparing the comprehenive and implified threeaxi block diagram. Actually the amount of work to be aved will increae dramatically a the number of axe i increaed.

22 Summary: n thi paper a method wa hown to implify the dynamic model of tabilized platform, with the fouraxi platform a the example vehicle. The relevant implified equation for a pecific fouraxi configuration were derived, the reulting block diagram wa given, and then ome imulation reult were hown after adding imple control loop around the dynamic model. The validity of the implification method wa demontrated by howing that identical imulation reult were achieved for a comprehenive and implified threeaxi model example. Reference: [] Michael K. Maten nertially Stabilized Platform for Optical maging Sytem Tracking Dynamic Target with Mobile Senor EEE CONTROL SYSTEMS MAGAZNE, February 008, Digital Object dentifier 0.09/MCS [] J. M. Hilkert nertially Stabilized Platform Technology Concept and Principle EEE CONTROL SYSTEMS MAGAZNE, February 008, Digital Object dentifier 0.09/MCS [3] Edmund Gorczycki Dynamic Conideration Relating to the Behavior of nertial SpaceStabilized Platform Journal of the Aeronautical Science (ntitute of the Aeronautical Science), 957, Vol.4: 3038, 0.54/ [4] Melvin Feinberg Rigid Body Dynamic Stable Platform olation Sytem Guidance and Control, 964: 4437, 0.54/ [5] Murray R Spiegel

23 Schaum Outline of Theory and Problem of Theoretical Mechanic, pp. 538 Schaum Publihing Company, 967