unknown transformation parameters. The volume of each
Type I, Type II, and Type III tetrahedron are shown in Equa-
tions 6, 7, and 8, respectively. These are the observation
equations to determine the calibration parameters in terms of
3D rigid body transformation. The volume of Type I, II, and III
tetrahedrons are denoted by V
1
, V
2
and V
3
, respectively.
V
X Y Z
x y z
x y z
x y z
1
1 1 1
1 1 1
2 2 2
3 3 3
1
6
1
1
1
1
=
(6)
V
X Y Z
X Y Z
x y z
x y z
2
1 1 1
2 2 2
1 1 1
2 2 2
1
6
1
1
1
1
=
(7)
V
X Y Z
X Y Z
X Y Z
x y z
3
1 1 1
2 2 2
3 3 3
1 1 1
1
6
1
1
1
1
=
(8)
In the equations (6, 7, and 8),
X
i
,
Y
i
,
Z
i
are coordinates of
points in control plane and
x
i
,
y
i
,
z
i
are the coordinates of li-
dar plane points in control coordinate frame. However, trans-
formation between lidar sensor frame and control coordinate
frame are unknowns. Hence,
x
i
,
y
i
,
z
i
need to be expressed
in terms of a 3D rigid body transformation that also refers to
boresight parameters as shown in Equation 9:
(
)
x
y
z
R
x
y
z
t
t
t
i
i
i
i
i
i
x
y
z
=
′
′
′
+
α β γ
, ,
.
(9)
In the above equation
x
i
,
y
i
,
z
i
are the coordinates of lidar
point cloud in the control coordinate system. The coordinates
of lidar points in
SBF
are given by
x
′
i
,
y
′
i
, and
z
′
i
. The angles
α
,
β
, and
γ
denote boresight rotation angles with respect to
three perpendicular axes and
t
x
,
t
y
and
t
z
. refers to boresight
translation parameters. Each observation equation in the form
of Equations (6, 7, and 8) must be expanded by plugging in
Equation 9. Then, the equations need to be linearized and
partial derivatives have to be taken with respect to unknown
calibration parameters. Considering the length of these deri-
vations, only the partial derivatives with respect to one of the
boresight rotation angle (
α
) for a Type I tetrahedron is given in
Equation 10 as an example.
∂
∂
=
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
V V
x
x V
x
x V
x
x V
y
y V
1
1
1
1
1
2
2
1
3
3
1
1
1
1
α
α
α
α
α
y
y
V
y
y V
z
z V
z
z V
z
z
2
2
1
3
3
1
1
1
1
2
2
1
3
3
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
α
α
α
α
α
. (10)
The partial derivatives (Equation 11) with respect to lidar
point coordinates
x
1
,
y
1
,
z
1
,
x
2
,
y
2
,
z
2
,
x
3
,
y
3
,
z
3
, are taken by
expanding the volume Equation 6 for Type I:
∂
∂
V
x
1
1
=
z
2
y
3
–
z
3
y
2
–
Z
1
y
3
+
z
3
Y
1
+
Z
1
y
2
–
z
2
Y
1
∂
∂
V
x
1
2
= –
z
1
y
3
+
z
3
y
1
+
Z
1
y
3
–
z
3
Y
1
–
Z
1
y
2
+
z
2
Y
1
∂
∂
V
x
1
3
=
z
1
y
2
–
z
2
y
1
–
Z
1
y
2
+
z
2
Y
1
–
Z
1
y
1
+
z
1
Y
1
∂
∂
V
y
1
1
= –
z
1
x
3
+
z
3
x
2
+
Z
1
x
3
–
z
3
X
1
–
Z
1
x
2
+
z
2
X
1
(11)
∂
∂
V
y
1
2
=
z
1
x
3
–
z
3
x
1
+
z
3
X
1
–
Z
1
x
3
+
Z
1
x
1
–
z
1
X
1
∂
∂
V
y
1
3
= –
z
1
x
2
+
z
2
x
1
+
Z
1
x
2
–
z
2
X
1
–
Z
1
x
1
+
z
1
X
1
∂
∂
V
z
1
1
= –
x
2
y
3
+
y
2
x
3
+
X
1
y
3
–
Y
1
x
3
–
X
1
x
2
+
Y
1
x
2
∂
∂
V
z
1
2
= –
y
1
x
3
+
x
1
y
3
+
Y
1
x
3
–
X
1
y
3
+
X
1
y
1
–
Y
1
x
1
∂
∂
V
z
1
3
= –
x
1
y
2
+
y
1
x
2
+
X
1
y
2
–
Y
1
x
2
–
X
1
y
1
+
Y
1
x
1
.
Before taking derivative of
x
i
,
y
i
,
z
i
with respect to the un-
knowns, rotation matrix
R
(
α
,
β
,
γ
) can be simplified to
dR
for
all small angles of
α
,
β
,
γ
as:
dR
d d
d
d
d d
≈
−
−
−
−
1
1
1
γ
β
γ
α
β α
.
(12)
Hence, the partial derivatives with respect to
α
, for in-
stance, are written as:
∂
∂
=
−
R
α
0 0 0
0 0 1
0 1 0
.
(13)
Therefore, by using Equation 9:
∂
∂
= − ′
α
x
y
z
z
y
1
1
1
1
1
0
.
(14)
As stated above, a similar derivation is needed for other
tetrahedron types and remaining boresight parameters. Then,
by using the linearized observation equations of all tetrahe-
drons formed between the corresponding planes, the calibra-
tion parameters that minimize the volume of each tetrahe-
dron is determined. It should be noted that each tetrahedron
volume equation is considered as an observation equation for
the
LSQ
adjustment. As there are six unknown parameters,
at least six tetrahedrons that are formed between conjugate
planes in lidar and control surface are needed. However, to
avoid degeneracy, tetrahedrons that represent three mutually
perpendicular conjugate planes are required. In the absence of
such vertical and horizontal conjugate planes, multiple slop-
ing planes will be required (Schenk, 1999).
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
October 2018
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