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A method for determining the path of a landing air-craft from a cinecamera record
30 A method for determining the path of a landing air-craft from a cinecamera record by Ir. G. P R A S T . 1. Introduction. I n the Royal N e t h e r l a n d s N a v y , some d a t a w e r e needed on the t o u c h - d o w n values of v a r i o u s f l i g h t p a r a m e t e r s f r o m an a i r c r a f t l a n d i n g on t h e deck of a n a i r c r a f t c a r r i e r . As it w a s needed to g e t a l a r g e n u m b e r of d a t a and to get d a t a f r o m all a i r c r a f t of the ship (to allow statistical i n t e r p r e t a t i o n ) it w a s n o t possible to m a k e the m e a s u r e m e n t s w i t h i n s t r u m e n t s inside t h e a i r c r a f t , b u t the whole m e a s u r e m e n t h a d to be Llade w i t h i n s t r u m e n t s on b o a r d of t h e ship. The n o r m a l m e t h o d in a e r o n a u t i c a l r e s e a r c h f o r g e t t i n g t h i s s o r t of d a t a is u s i n g a cine-camera, w i t h w h i c h a record is m a d e f r o m the l a n d i n g of the a i r c r a f t . F r o m this r e c o r d the m o v e m e n t s of the a i r c r a f t a r e calculated. I n m o s t cases the c a m e r a is set u p on some distance of t h e l a n d i n g p a t h a n d t a k e s a side-view of t h e l a n d i n g a i r c r a f t . Special c a m e r a ' s (i.e. the " V i n t e n " t a k e - o f f c a m e r a ) h a s been built f o r t h i s p u r p o s e . In t h i s method t h e a s s u m p t i o n h a s to be m a d e t h a t the a i r c r a f t comes in over the centre-line of the r u n w a y , o r t h a t the a i r c r a f t comes in p a r a l l e l to this centreline. On t h e a i r c r a f t c a r r i e r h o w e v e r t h i s m e t h o d could not be adopted since: a.
The only possible position f o r the c i n e - c a m e r a is on b o a r d of the ship, t h e r e f o r e in f r o n t of the l a n d i n g a i r c r a f t ; b. I t can n o t be a s s u m e d t h a t the a i r c r a f t a p p r o a c h e s over the centre-line of the ship or parallel to t h i s centre-line; c. No m a r k e r s can be set u p on the flight-deck. These c i r c u m s t a n c e s m a d e it n e c e s s a r y to develop a n e w m e t h o d of analysis. I n this method also a c i n e - c a m e r a is used, which m a k e s a record of the l a n d i n g of t h e a i r c r a f t , b u t in this case the c a m e r a is set u p on the flight-deck, in f r o n t of the l a n d i n g a i r c r a f t , and no a s s u m p t i o n s h a v e been m a d e r e g a r d i n g t h e p a t h or a t t i t u d e of t h e a i r c r a f t . A s no m a r k e r s could be set u p on the flight-deck, some fixed points w h i c h a r e clearly to be seen on the flight-deck m u s t be used to d e t e r m i n e the a t t i t u d e of t h e c a m e r a w i t h respect to t h e flight-deck.
2. The camera's. In t h e f i r s t t r i a l s a n " E r n e m a n n " 36 m m c i n e - c a m e r a w a s used. This c a m e r a w a s provided w i t h a c h r o n o m e t e r (2;/6 rev./sec.) ; a p i c t u r e of the c h r o n o m e t e r and a p i c t u r e of the a i r c r a f t a r e obtained at the s a m e m o m e n t . The c a m e r a is o p e r a t e d by h a n d a t a b o u t 2 X 6 f r a m e s / s e e . F o r the l a t e r t r i a l s the c a m e r a in use w a s a " N e w m a n S i n c l a i r " 36 m m c i n e - c a m e r a provided w i t h a 2" lens. I t h a s a clockwork drive, capable of e x p o s i n g a b o u t 30 f r a m e s p e r sec. A p i c t u r e of a c h r o n o m e t e r (1./6 rev./sec.) m a y be t a k e n in a c o r n e r of each f r a m e . The time of e x p o s u r e is a b o u t 1/60 sec. which is s u f f i c i e n t l y s h o r t for r e c o r d i n g the f l i g h t p a t h of an a i r c r a f t , which is faced b y the c a m e r a . I n the e a r l i e r t r i a l s the c a m e r a h a s been set u p on t h e flight-deck itself, in o r d e r to m a k e a record of the p a t h of the a i r c r a f t , u p to the f i r s t point of teuch-down. In t h e l a t e r t r i a l s the c a m e r a w a s set u p in a rigid position on a p l a t f o r m a b o u t 6 m over t h e flight-deck, f o r g e t t i n g a good record of the final s t a g e of the l a n d i n g p a t h . I n t h a t position m o r e i n f o r m a t i o n could be derived on t h e a t t i t u d e d u r i n g and j u s t a f t e r touch-down.
3. The fixed points. F o r the fixed p o i n t s on the flight-deck, no special m a r k e r s could be set up, so some
31 points w h i c h w e r e c l e a r l y to be seen in m o s t p i c t u r e s h a d t o be u s e d . C o o r d i n a t e s of t h e s e points were m e a s u r e d w i t b r e s p e c t to t h e u s e d c o o r d i n a t e s y s t e m . 1. T h e c a l c u l a t i o n s .
4. 1. The coordinate systems. In t h e c a l c u l a t i o n s t h r e e image different coordinate systems (~ were used. A n X, Y, Zcoordinate s y s t e m w a s u s e d with its o r i g i n in t h e c e n t r e of t h e c a m e r a lens, i t s Zi axis n o r m a l to t h e f l i g h t deck, i t s X - a x i s p a r a l l e l to the c e n t r e - l i n e of t h e flight~ deck a n d i t s Y - a x i s n o r m a l to both X - a n d Z-axis. In % the i m a g e s a ~', ~ ' - s y s t e m a n d a $, ~]-system w e r e used. Fig. 1 These two systems are shown in fig. 1. T h e ~ ' , ~ ' - s y s t e m is u s e d in t h e r e a d i n g o f t h e i m a g e s u n d e r a r e a d i n g m i c r o s c o p e . This s y s t e m v a r i e s f o r e a c h i m a g e . T h e ~, M-system h a s i t s o r i g i n in t h e f o o t o f t h e perp e n d i c u l a r f r o m t h e c e n t r e o f t h e c a m e r a l e n s on t h e p l a n e of t h e i m a g e . T h e s e l a s t two s y s t e m s a r e r e l a t e d to e a c h o t h e r w i t h t h e h e l p o f t h e c o o r d i n a t e s of one of t h e f i x e d p o i n t s a.
~] :
~ ' - - ;7'a ~- ~]a"
(2)
The relation between X, Y, Z and ~, ~ is given by: - - X = f c o s a c o s f l + ~ ( c o s a s i n ~ sin 7 - - sin a cos 7) + ~ ( e o s a s i n f l c o s T ÷ s i n a s i n T ),
(3)
- - Y = f s i n a c o s f l A - ~ ( s i n a s i n fl sin7 A- cosa cos 7 ) + ~ ( s i n a s i n f l c o s T - c o s a s i n T ),
(4)
--Z=--f
(5)
sin fl + ~: cos fl sin 7 + ~] cos/~ cos 7
w h e r e a, fl a n d 7 a r e t h e a n g l e s w h i c h d e t e r m i n e t h e a t t i t u d e of t h e c a m e r a a n d f is t h e focal d i s t a n c e of t h e c a m e r a lens.
4.2. D e t e r m i n a t i o n of the a t t i t u d e of the camera. A s Xa, Ya, Za a r e t h e c o o r d i n a t e s o f t h e f i x e d p o i n t a a n d Xa, Ya a n d z a a r e t h e coordinates of its i m a g e , t h e s e two p o i n t s a r e l y i n g on a s t r a i g h t line t h r o u g h t h e c a m e r a lens so
X a = tara, } Ya = taYa' f Z a = taZa • .
(6)
A s Xa, Ya, Za, ~a, ~a a n d f a r e k n o w n , it is n o w possible to d e t e r m i n e a, fl a n d 7. T h i s d e t e r m i n a t i o n is done w i t h help of t h e t h r e e f i x e d p o i n t s f o r one i m a g e . T h e f o u n d v a l u e s for a, fl a n d 7 h a s been u s e d in all f u r t h e r c a l c u l a t i o n s . H e r e t h e a s s u m p t i o n is m a d e t h a t in each p i c t u r e t h e c a m e r a h a s t h e s a m e a t t i t u d e . T h i s a s s u m p t i o n m a y g i v e s o m e i n a c c u r a c y in t h e m e a s u r e d l a n d i n g p a t h , b u t it does n o t i n f l u e n c e t h e i m p o r t a n t d a t a , w h i c h a r e d e r i v a t i o n s o f t h e l a n d i n g p a t h , l a r g e l y .
32 4.3. The measured points o/ the aircraft. F o r each p i c t u r e f r o m t h e films, for t h r e e points of t h e a i r c r a f t (two w i n g t i p s a n d t h e top of t h e fin) a n d one fixed point, the coordinates ~:" and ~/ a r e m e a s u r e d u n d e r a m e a s u r i n g microscope. W i t h eq. (1) and (2), f r o m t h e s e coordinates t h e cooidinates ~ a n d ~/ a r e calculated f o r t h e t h r e e points of t h e a i r c r a f t . A f t e r t h a t w i t h help of the determ i n e d values of a, fl a n d 7 a n d t h e e q u a t i o n s (3), (4) a n d (5), the coordinates x, y and z f o r each of the i m a g e s of t h e fixed p o i n t s of t h e a i r c r a f t , m a y be calculated.
4.4. The equations for the aircraft position. The t h r e e points in t h e i m a g e w i t h coordinates x~, y¢ and z~ d e t e r m i n e t h r e e lines:
X i = t~x~, Yi tiY~ ,
(7)
=
Zt
=
tiZ~
•
and the a i r c r a f t h a s to f i t in t h e s e lines (see fig. 2). Z
...... -S: .........
Y Fig. 2. So t h e coordinates of t h e t h r e e points of t h e a i r c r a f t (X~, Y~, Z¢) m u s t be: t~x~, t~y¢, t¢z¢. W i t h t h e help o f t h e t h r e e d i s t a n c e s D~j b e t w e e n the fixed p o i n t s of t h e a i r c r a f t , t h e values of t¢ m a y be d e t e r m i n e d . L e t t h e d i s t a n c e f r o m p o i n t i o f t h e i m a g e to t h e focal p o i n t of t h e c a m e r a be r~ a n d t h e d i s t a n c e f r o m p o i n t i of t h e a i r c r a f t to t h e focal p o i n t Re. T h e n :
D~j2 = Re 2 + Rj 2 - - 2 R~R i cos ~ij, R~Rj cos q%j = X~Xj + Y~Yi + Z~Zj so
D~j2= (X~2 + y ? + Z ~ 2 ) + (Xj2-t-yj2 + Z j 2 ) - - 2 (X~Xj + YcYj + Z~Zj) and a s and Z~ = t~z~ also Dij2 : ti2 (xi2 ~- y 2 -{- zi2 ) -~- tj2 (xj2 -}- yj2 -~- zj2) - - 2 titj ( x~xj + y~yj + z~zj) : S u b s t i t u t i n g f o r i ] resp. 1 2, 2 3 a n d 3 1, t h r e e e q u a t i o n s w i t h t h e t h r e e u n k n o w n s t~. t 2 a n d t 3 a r e f o u n d as D~j a r e d i s t a n c e s m e a s u r e d on t h e a i r c r a f t a n d xe y~, z~ a r e
33 calculated f r o m the m e a s u r e d coordinates in the images. The solution of the e q u a t i o n s will give tl, t2 and t 3 so the c o o r d i n a t e s of the t h r e e points of the a i r c r a f t . F r o m these coordinates the position of the c e n t r e of g r a v i t y of the a i r c r a f t and t h e a t t i t u d e of t h e a i r c r a f t m a y be deduced.
4. 5. Solution of the equations for the aircraft position. F o r the n u m e r i c a l solution of the t h r e e e q u a t i o n s an i t e r a t i v e m e t h o d h a s been used which w a s adopted to the electronic c a l c u l a t i n g m a c h i n e of the M a t h e m a t i c a l C e n t r e in A m s t e r d a m , which m a c h i n e h a s been used f o r the calculations. With t h i s i t e r a t i v e method, several solutions f o r the t~ m a y be found, as the t h r e e second o r d e r e q u a t i o n s h a s eight solutions. H o w e v e r in n e a r l y all cases only one of t h e s e solutions will give a possible a t t i t u d e of the a i r c r a f t . T h i s good solution is f o u n d w h e n good a s s u m p t i o n s are m a d e f o r tl, t 2 and t 3. T h e s e a s s u m p t i o n s a r e : t~, t 2 and t 3 < 0 (the a i r c r a f t and the image are lying on e i t h e r side of the lens), and
lt2t > It~l, its] as point 2 is the tip of the fin, which h a s a l w a y s a g r e a t e r distance to t h e c a m e r a t h a n the wingtips. The f i r s t a s s u m p t i o n for t~, t 2 and t 3 is the solution of the t h r e e e q u a t i o n s f o r the attitude of the a i r c r a f t f r o m the f o r e g o i n g picture. As this is a l r e a d y a v e r y good a s s u m p t i o n , t h e iterative method needs only one step. Only in the f i r s t p i c t u r e of a film m o r e s t e p s m a y be needed.
5. Accuracy of the results. F r o m an estimation of the possible e r r o r s in the d i f f e r e n t f a c t o r s which influence the r e s u l t s m a y be s h o w n t h a t the following total possible e r r o r s m a y be expected: in in in in in in
the the the the the the
X-coordinate: Y-coordinate: Z-coordinate: angle of roll: angle of incidence and the angle of b a n k : deduced velocities:
1.00 m 0.20 m 0.10 m 1o 3o 1%
These v a l u e s h a v e been checked e x p e r i m e n t a l l y by t a k i n g a c i n e - c a m e r a record of an a i r c r a f t at r e s t on t h e flight-deck a n d c o m p u t i n g the position of this a i r c r a f t f r o m a n u m b e r of f r a m e s of this record. C o m p a r i s o n of the positions f o u n d in this w a y indicated t h a t the given v a l u e s f o r the possible e r r o r s are correct. Also the v a r i a n c e of the several p o i n t s f r o m one l a n d i n g a r o u n d the m e a n curve are in a g r e e m e n t w i t h these e s t i m a t e d e r r o r s . The total possible e r r o r is influenced by e r r o r s w h i c h are caused b y : e r r o r in a t t i u d e and position of the c a m e r a ; e r r o r in d e t e r m i n a t i o n of the focal distance of the c a m e r a ; difference in s h r i n k i n g of the film d u r i n g development; s n a k i n g of the film while r u n n i n g ; s n a k i n g of the film u n d e r the r e a d i n g microscope; e r r o r s m a d e in film reading. Some of these e r r o r s m a y be diminished by m o r e c a r e f u l u s i n g of this method. F o r example : e r r o r in focal distance and position and a t t i t u d e of the c a m e r a m a y be diminished by u s i n g m o r e fixed points, and calculate an a p p a r e n t focal distance a f t e r s h r i n k a g e , the attitude of the c a m e r a and the position of the f i h n in the c a m e r a f o r each inmg~'.
34 However t h i s procedure a s k s for a lot of e x t r a calculation. Also t h e error m a y be diminished by u s i n g a camer~ with l a r g e r film size and longer focal distance, as to get better pictures a n d h i g h e r r e a d i n g conditions. In the described e x p e r i m e n t s f r o m which the d a t a only h a v e been used for statistical i n t e r p r e t a t i o n , no h i g h e r a c c u r a c y w a s needed.
6. Other possible method. In the described e x p e r i m e n t s a stereoscopic method p e r h a p s could h a v e been used better. In t h a t case a l a r g e distance between the c a m e r a ' s should have been used, to g e t enough accuracy. (Not less t h a n some meters.) Therefore p e r h a p s a method with two electrical driven a n d synchronised c a m e r a ' s , one on each side of the flight-deck m i g h t be a v e r y successfull method, which should not h a v e required so m u c h calculation. Indeed the stereoscopic method h a s been u n d e r consideration between the f i r s t and later experiments, b u t as no i n s t r u m e n t s for t h a t method were available, the described method, which h a d shown to be succesfull in the f i r s t t r i a l s h a s been chosen a g a i n for the later experiments.
The only possible position f o r the c i n e - c a m e r a is on b o a r d of the ship, t h e r e f o r e in f r o n t of the l a n d i n g a i r c r a f t ; b. I t can n o t be a s s u m e d t h a t the a i r c r a f t a p p r o a c h e s over the centre-line of the ship or parallel to t h i s centre-line; c. No m a r k e r s can be set u p on the flight-deck. These c i r c u m s t a n c e s m a d e it n e c e s s a r y to develop a n e w m e t h o d of analysis. I n this method also a c i n e - c a m e r a is used, which m a k e s a record of the l a n d i n g of t h e a i r c r a f t , b u t in this case the c a m e r a is set u p on the flight-deck, in f r o n t of the l a n d i n g a i r c r a f t , and no a s s u m p t i o n s h a v e been m a d e r e g a r d i n g t h e p a t h or a t t i t u d e of t h e a i r c r a f t . A s no m a r k e r s could be set u p on the flight-deck, some fixed points w h i c h a r e clearly to be seen on the flight-deck m u s t be used to d e t e r m i n e the a t t i t u d e of t h e c a m e r a w i t h respect to t h e flight-deck.
2. The camera's. In t h e f i r s t t r i a l s a n " E r n e m a n n " 36 m m c i n e - c a m e r a w a s used. This c a m e r a w a s provided w i t h a c h r o n o m e t e r (2;/6 rev./sec.) ; a p i c t u r e of the c h r o n o m e t e r and a p i c t u r e of the a i r c r a f t a r e obtained at the s a m e m o m e n t . The c a m e r a is o p e r a t e d by h a n d a t a b o u t 2 X 6 f r a m e s / s e e . F o r the l a t e r t r i a l s the c a m e r a in use w a s a " N e w m a n S i n c l a i r " 36 m m c i n e - c a m e r a provided w i t h a 2" lens. I t h a s a clockwork drive, capable of e x p o s i n g a b o u t 30 f r a m e s p e r sec. A p i c t u r e of a c h r o n o m e t e r (1./6 rev./sec.) m a y be t a k e n in a c o r n e r of each f r a m e . The time of e x p o s u r e is a b o u t 1/60 sec. which is s u f f i c i e n t l y s h o r t for r e c o r d i n g the f l i g h t p a t h of an a i r c r a f t , which is faced b y the c a m e r a . I n the e a r l i e r t r i a l s the c a m e r a h a s been set u p on t h e flight-deck itself, in o r d e r to m a k e a record of the p a t h of the a i r c r a f t , u p to the f i r s t point of teuch-down. In t h e l a t e r t r i a l s the c a m e r a w a s set u p in a rigid position on a p l a t f o r m a b o u t 6 m over t h e flight-deck, f o r g e t t i n g a good record of the final s t a g e of the l a n d i n g p a t h . I n t h a t position m o r e i n f o r m a t i o n could be derived on t h e a t t i t u d e d u r i n g and j u s t a f t e r touch-down.
3. The fixed points. F o r the fixed p o i n t s on the flight-deck, no special m a r k e r s could be set up, so some
31 points w h i c h w e r e c l e a r l y to be seen in m o s t p i c t u r e s h a d t o be u s e d . C o o r d i n a t e s of t h e s e points were m e a s u r e d w i t b r e s p e c t to t h e u s e d c o o r d i n a t e s y s t e m . 1. T h e c a l c u l a t i o n s .
4. 1. The coordinate systems. In t h e c a l c u l a t i o n s t h r e e image different coordinate systems (~ were used. A n X, Y, Zcoordinate s y s t e m w a s u s e d with its o r i g i n in t h e c e n t r e of t h e c a m e r a lens, i t s Zi axis n o r m a l to t h e f l i g h t deck, i t s X - a x i s p a r a l l e l to the c e n t r e - l i n e of t h e flight~ deck a n d i t s Y - a x i s n o r m a l to both X - a n d Z-axis. In % the i m a g e s a ~', ~ ' - s y s t e m a n d a $, ~]-system w e r e used. Fig. 1 These two systems are shown in fig. 1. T h e ~ ' , ~ ' - s y s t e m is u s e d in t h e r e a d i n g o f t h e i m a g e s u n d e r a r e a d i n g m i c r o s c o p e . This s y s t e m v a r i e s f o r e a c h i m a g e . T h e ~, M-system h a s i t s o r i g i n in t h e f o o t o f t h e perp e n d i c u l a r f r o m t h e c e n t r e o f t h e c a m e r a l e n s on t h e p l a n e of t h e i m a g e . T h e s e l a s t two s y s t e m s a r e r e l a t e d to e a c h o t h e r w i t h t h e h e l p o f t h e c o o r d i n a t e s of one of t h e f i x e d p o i n t s a.
~] :
~ ' - - ;7'a ~- ~]a"
(2)
The relation between X, Y, Z and ~, ~ is given by: - - X = f c o s a c o s f l + ~ ( c o s a s i n ~ sin 7 - - sin a cos 7) + ~ ( e o s a s i n f l c o s T ÷ s i n a s i n T ),
(3)
- - Y = f s i n a c o s f l A - ~ ( s i n a s i n fl sin7 A- cosa cos 7 ) + ~ ( s i n a s i n f l c o s T - c o s a s i n T ),
(4)
--Z=--f
(5)
sin fl + ~: cos fl sin 7 + ~] cos/~ cos 7
w h e r e a, fl a n d 7 a r e t h e a n g l e s w h i c h d e t e r m i n e t h e a t t i t u d e of t h e c a m e r a a n d f is t h e focal d i s t a n c e of t h e c a m e r a lens.
4.2. D e t e r m i n a t i o n of the a t t i t u d e of the camera. A s Xa, Ya, Za a r e t h e c o o r d i n a t e s o f t h e f i x e d p o i n t a a n d Xa, Ya a n d z a a r e t h e coordinates of its i m a g e , t h e s e two p o i n t s a r e l y i n g on a s t r a i g h t line t h r o u g h t h e c a m e r a lens so
X a = tara, } Ya = taYa' f Z a = taZa • .
(6)
A s Xa, Ya, Za, ~a, ~a a n d f a r e k n o w n , it is n o w possible to d e t e r m i n e a, fl a n d 7. T h i s d e t e r m i n a t i o n is done w i t h help of t h e t h r e e f i x e d p o i n t s f o r one i m a g e . T h e f o u n d v a l u e s for a, fl a n d 7 h a s been u s e d in all f u r t h e r c a l c u l a t i o n s . H e r e t h e a s s u m p t i o n is m a d e t h a t in each p i c t u r e t h e c a m e r a h a s t h e s a m e a t t i t u d e . T h i s a s s u m p t i o n m a y g i v e s o m e i n a c c u r a c y in t h e m e a s u r e d l a n d i n g p a t h , b u t it does n o t i n f l u e n c e t h e i m p o r t a n t d a t a , w h i c h a r e d e r i v a t i o n s o f t h e l a n d i n g p a t h , l a r g e l y .
32 4.3. The measured points o/ the aircraft. F o r each p i c t u r e f r o m t h e films, for t h r e e points of t h e a i r c r a f t (two w i n g t i p s a n d t h e top of t h e fin) a n d one fixed point, the coordinates ~:" and ~/ a r e m e a s u r e d u n d e r a m e a s u r i n g microscope. W i t h eq. (1) and (2), f r o m t h e s e coordinates t h e cooidinates ~ a n d ~/ a r e calculated f o r t h e t h r e e points of t h e a i r c r a f t . A f t e r t h a t w i t h help of the determ i n e d values of a, fl a n d 7 a n d t h e e q u a t i o n s (3), (4) a n d (5), the coordinates x, y and z f o r each of the i m a g e s of t h e fixed p o i n t s of t h e a i r c r a f t , m a y be calculated.
4.4. The equations for the aircraft position. The t h r e e points in t h e i m a g e w i t h coordinates x~, y¢ and z~ d e t e r m i n e t h r e e lines:
X i = t~x~, Yi tiY~ ,
(7)
=
Zt
=
tiZ~
•
and the a i r c r a f t h a s to f i t in t h e s e lines (see fig. 2). Z
...... -S: .........
Y Fig. 2. So t h e coordinates of t h e t h r e e points of t h e a i r c r a f t (X~, Y~, Z¢) m u s t be: t~x~, t~y¢, t¢z¢. W i t h t h e help o f t h e t h r e e d i s t a n c e s D~j b e t w e e n the fixed p o i n t s of t h e a i r c r a f t , t h e values of t¢ m a y be d e t e r m i n e d . L e t t h e d i s t a n c e f r o m p o i n t i o f t h e i m a g e to t h e focal p o i n t of t h e c a m e r a be r~ a n d t h e d i s t a n c e f r o m p o i n t i of t h e a i r c r a f t to t h e focal p o i n t Re. T h e n :
D~j2 = Re 2 + Rj 2 - - 2 R~R i cos ~ij, R~Rj cos q%j = X~Xj + Y~Yi + Z~Zj so
D~j2= (X~2 + y ? + Z ~ 2 ) + (Xj2-t-yj2 + Z j 2 ) - - 2 (X~Xj + YcYj + Z~Zj) and a s and Z~ = t~z~ also Dij2 : ti2 (xi2 ~- y 2 -{- zi2 ) -~- tj2 (xj2 -}- yj2 -~- zj2) - - 2 titj ( x~xj + y~yj + z~zj) : S u b s t i t u t i n g f o r i ] resp. 1 2, 2 3 a n d 3 1, t h r e e e q u a t i o n s w i t h t h e t h r e e u n k n o w n s t~. t 2 a n d t 3 a r e f o u n d as D~j a r e d i s t a n c e s m e a s u r e d on t h e a i r c r a f t a n d xe y~, z~ a r e
33 calculated f r o m the m e a s u r e d coordinates in the images. The solution of the e q u a t i o n s will give tl, t2 and t 3 so the c o o r d i n a t e s of the t h r e e points of the a i r c r a f t . F r o m these coordinates the position of the c e n t r e of g r a v i t y of the a i r c r a f t and t h e a t t i t u d e of t h e a i r c r a f t m a y be deduced.
4. 5. Solution of the equations for the aircraft position. F o r the n u m e r i c a l solution of the t h r e e e q u a t i o n s an i t e r a t i v e m e t h o d h a s been used which w a s adopted to the electronic c a l c u l a t i n g m a c h i n e of the M a t h e m a t i c a l C e n t r e in A m s t e r d a m , which m a c h i n e h a s been used f o r the calculations. With t h i s i t e r a t i v e method, several solutions f o r the t~ m a y be found, as the t h r e e second o r d e r e q u a t i o n s h a s eight solutions. H o w e v e r in n e a r l y all cases only one of t h e s e solutions will give a possible a t t i t u d e of the a i r c r a f t . T h i s good solution is f o u n d w h e n good a s s u m p t i o n s are m a d e f o r tl, t 2 and t 3. T h e s e a s s u m p t i o n s a r e : t~, t 2 and t 3 < 0 (the a i r c r a f t and the image are lying on e i t h e r side of the lens), and
lt2t > It~l, its] as point 2 is the tip of the fin, which h a s a l w a y s a g r e a t e r distance to t h e c a m e r a t h a n the wingtips. The f i r s t a s s u m p t i o n for t~, t 2 and t 3 is the solution of the t h r e e e q u a t i o n s f o r the attitude of the a i r c r a f t f r o m the f o r e g o i n g picture. As this is a l r e a d y a v e r y good a s s u m p t i o n , t h e iterative method needs only one step. Only in the f i r s t p i c t u r e of a film m o r e s t e p s m a y be needed.
5. Accuracy of the results. F r o m an estimation of the possible e r r o r s in the d i f f e r e n t f a c t o r s which influence the r e s u l t s m a y be s h o w n t h a t the following total possible e r r o r s m a y be expected: in in in in in in
the the the the the the
X-coordinate: Y-coordinate: Z-coordinate: angle of roll: angle of incidence and the angle of b a n k : deduced velocities:
1.00 m 0.20 m 0.10 m 1o 3o 1%
These v a l u e s h a v e been checked e x p e r i m e n t a l l y by t a k i n g a c i n e - c a m e r a record of an a i r c r a f t at r e s t on t h e flight-deck a n d c o m p u t i n g the position of this a i r c r a f t f r o m a n u m b e r of f r a m e s of this record. C o m p a r i s o n of the positions f o u n d in this w a y indicated t h a t the given v a l u e s f o r the possible e r r o r s are correct. Also the v a r i a n c e of the several p o i n t s f r o m one l a n d i n g a r o u n d the m e a n curve are in a g r e e m e n t w i t h these e s t i m a t e d e r r o r s . The total possible e r r o r is influenced by e r r o r s w h i c h are caused b y : e r r o r in a t t i u d e and position of the c a m e r a ; e r r o r in d e t e r m i n a t i o n of the focal distance of the c a m e r a ; difference in s h r i n k i n g of the film d u r i n g development; s n a k i n g of the film while r u n n i n g ; s n a k i n g of the film u n d e r the r e a d i n g microscope; e r r o r s m a d e in film reading. Some of these e r r o r s m a y be diminished by m o r e c a r e f u l u s i n g of this method. F o r example : e r r o r in focal distance and position and a t t i t u d e of the c a m e r a m a y be diminished by u s i n g m o r e fixed points, and calculate an a p p a r e n t focal distance a f t e r s h r i n k a g e , the attitude of the c a m e r a and the position of the f i h n in the c a m e r a f o r each inmg~'.
34 However t h i s procedure a s k s for a lot of e x t r a calculation. Also t h e error m a y be diminished by u s i n g a camer~ with l a r g e r film size and longer focal distance, as to get better pictures a n d h i g h e r r e a d i n g conditions. In the described e x p e r i m e n t s f r o m which the d a t a only h a v e been used for statistical i n t e r p r e t a t i o n , no h i g h e r a c c u r a c y w a s needed.
6. Other possible method. In the described e x p e r i m e n t s a stereoscopic method p e r h a p s could h a v e been used better. In t h a t case a l a r g e distance between the c a m e r a ' s should have been used, to g e t enough accuracy. (Not less t h a n some meters.) Therefore p e r h a p s a method with two electrical driven a n d synchronised c a m e r a ' s , one on each side of the flight-deck m i g h t be a v e r y successfull method, which should not h a v e required so m u c h calculation. Indeed the stereoscopic method h a s been u n d e r consideration between the f i r s t and later experiments, b u t as no i n s t r u m e n t s for t h a t method were available, the described method, which h a d shown to be succesfull in the f i r s t t r i a l s h a s been chosen a g a i n for the later experiments.