IEEE - Aerospace and Electronic Systems - January 2020 - 14

Calibration Method of Laser Detection System With Position Sensitive Detector

Figure 6.
Sketch of the measurement mode with the installation angle error.

oG x3 yG zG and o1 x3 y3 z3 lies on the origin position,
the origin position of the frame oG x3 yG zG is
ðÀf; 0; 0Þ in the frame o1 x3 y3 z3 .

Figure 5.
Sketch of the transformation relation between the frame o4 x4 y4 z4
and o1 x3 y3 z3 .

where RIg

2
1
0
¼ 4 0 cos Ig
0 Àsin Ig

2
cos Iu
R Iu ¼ 4 Àsin Iu
0

sin Iu
cos Iu
0

3
0
0 5,
1

3
2
0
cos Ic
sin Ig 5, RIc ¼ 4
0
Àsin Ic
cos Ig

3
0 sin Ic
1
0 5,
0 cos Ic

Iu , Ic , Ig are installation angle

errors.
The transformation matrix R31 between the frame
o1 x3 y3 z3 and o1 x1 y1 z1 can be expressed as
&

½ x3 y3 z3 T ¼ R31 ½ x1
R 31 ¼ Rg R 21
2

1
0
where Rg ¼ 4 0 cos g
0 sin g
of the projectile.

y1

z1 T

(2)

3
0
Àsin g 5. g is the roll angle
cos g

The transformation relation between the frame o4 x4 y4 z4
and o1 x3 y3 z3 can be expressed as
½ x4

y4

z 4  T ¼ ½ x3

y3

z 3 T þ ½ 0

Dy3

Dz3 T
(3)

where Dy3 , Dz3 represent the longitudinal and lateral
installation displacement errors, respectively.

MEASUREMENT MODEL
Definitions of coordinates, which will appear in this subsection, are introduced as follows.
 The PSD frame (oG x3 yG zG ): the PSD frame is
located at the PSD. Its origin located at the PSD
center, and the PSD frame oG x3 yG zG is in the same
direction with the rolling measurement frame
o1 x3 y3 z3 . The only difference between the frame
14

A sketch of the measurement mode with the installation
angle error is shown in Figure 6 when the projectile roll
angle is equal to g. The positive direction of the laser unit
vector L is the direction in which the projectile points to
the target. The laser unit vector L is determined by the laser
angles u and c . The laser angle c is the angle between
the laser unit vector L and the plane o1 x1 y1 , and the laser
angle u is the angle between the projection of the unit vector L in the plane o1 x1 y1 and the x1 -axis. Then the laser
unit vector L can be expressed as
L ¼ ½ cosc cosu

cosc sinu

sinc T :

(4)

When the LDS receives the laser unit vector L , an
imaging point p will be detected. The coordinates of the
imaging point p are ð0; yG ; zG Þ in the frame oG x3 yG zG
and ðÀf; y3 ; z3 Þ in the frame o1 x3 y3 z3 . The projections of
the laser unit vector L in the frame o1 x3 y3 z3 are V x3 , V y3
and V z3 , respectively. The transformation matrix R31 in
(2) can be expressed as R31 ¼ ½ V x3 ; V y3 ; V z3 . Noting that the angles between the laser unit vector L and the
unit vectors V x3 , V y3 , and V z3 are ax , ay , and az , respectively. According to the principle of vector point multiplication, the following expression can be obtained:
Vx ÁL
cosax ¼  3 
¼ V x3 Á L :
V x 3 jL j

(5)

In the same way, the following mathematical relationship can be obtained:
&

cosay ¼ V y3 Á L
:
cosaz ¼ V z3 Á L

(6)

The cosine value in the three directions can be denoted
as M ¼ ½ cosax cosay cosaz T , in the frame o1 x3 y3 z3 ,
according to the imaging point p, M can be expressed as

IEEE A&E SYSTEMS MAGAZINE

JANUARY 2020
IEEE - Aerospace and Electronic Systems - January 2020

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