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Likely one gets
m32 * 7(94n254*395n55*03n53) (12-2)
Mag = -(r4n54*nron55*n3n53) (12—3)
where eq.(12-3) is identical to eq.(10).
3.4 Evaluation of ky
Writing the first six expressions of
eqs. (4) in the form of
ü
m» 1^31 p4*m34n^241:
m341^32. 7? PBotma4n55,
m21"33 ?. P3*m3 423.
m32n341 = d4*m355n» 1:
Maas = d5*m32n22*
m32^n33 * d3t*n3»5n»3.
multiplying the first with the forth, the
second with the fifth and the third with
the sixth of each side of the above ex-
pressions and summing up them, one can
calculate the right side of it. And the
left side becomes
m534m55(n342*n352*n332) = Mo4Moo
--$ink4cosk4- -1/2sin2k,.
This procedure produces four candidates
for k4-
Then n34.n32 and n are evaluated for
each candidate for Kq - They are evaluated
from following different equations for
better precision.
a) for -3/4% <k4<-M/4 or T/4<k,4<3/4T
VB 4)
n32* (potmg4n22)/C-=in ky),
n33 *(pgtma34no3)/cos k4 (13—1)
b) fon -T/Ask4«T/4 or 3/AT«k4«5/4tr
nee msanaq37coskq:
n327(05*m35022) 76093.
n337(83*ma45n53)/cosk4 (13-2)
3.5 Evaluation of £5, wo
From eqs.(8-2)
sin W4COS d = Nog,
COS w5,COS gy =nag. (14)
Since n33 >0, which means cos @, # 0,
cos Bo = ros? + nas * (15)
There are four candidates for £5. And for
each candidate for £5, angle wy, is evalu-
ated by
sin w4- no5g3/cos gs,
203
COS W5 * n33/COS PB». (16)
3.8 Evaluation of ka
From eqs.(8-2);
(-cos w5)sin ks
+ (sin wo sin £g5)cos ko * n1
( cos W5)cos ko
+ (sin Wo sin $5)sin Ko = noo
( sin wa)sin ko
+ (cos Wo sin $5)cos Ka, 5 Naz (17)
(-sin W5)coS ko
+ (cos wasin $5)sin Ko * ngo,
one solves the first two equations to get
gin ks and cos kp. They are always solva-
ble, even df sin g is zero. And this Ko
is tested by substituting it into the
third and forth equations. Any sets of
candidates for do and Wo that do" not
satisfy both are abandoned.
3.1 Strict relative orientation and deter-
mination of the sign of a base length
Since the precision of approximations
evaluated above is usually not suffi-
cient, one should execute relative orien-
tation again using those approximations.
An independent model is thus obtained,
which ‘igs either Fig.1-1 or 1-2.
Next the sign of a base length is deter-
mined the way that if Zp coordinates of
objects in the independent model coordi-
nate system are lesser than O0, it is set
plus, and if Zp coordinates are greater
than 0, it is set minus.
4. EVALUATION OF ORIENTATION PARAMETERS
IN THE OBJECT SPACE COORDINATE SYSTEM
4.1 Model connection in the global model
coordinate system
Independent models thus produced are
linked to make a global model by usual
successive orientation. Scales of succes-
sive models are adjusted by scaling base
lengths. As a result exposing positions
and rotation matrices associated with the
global coordinate system XMYmZM are deter-
mined.
e global model!
object space
4.2 Transformation from t
coordinate system to th
coordinate system
When an object space coordinate system XYZ
is given, global model coordinates XyYwZw
are further transformed to the object
coordinates. Here let us consider the case
the object space coordinate system is
implicitly given in the form of a few of
3-D control points. In most industrial
measurements this is common. And in this
case one can calculates orientation param-
eters automatically in the following way.
Similar transformation