UMAX0608XX‐1000. Tri‐Axial J1939 CAN Inclinometer. Version 9B
Page: 11-66
The unit rotation angles do not uniquely define the unit angular position in space. If this is
required, the Euler angles should be used.
2.1.3.2.2 Euler Angles
The Euler angles are coordinate system rotation angles performed in a specific order to rotate
the unit from its original position, parallel to the Earth surface, to its current position.
The Euler angles:
𝜃
and
𝜙
, together with the
𝜓
, are rotation angles about the Z
E
, Y
E
*
and X
axes performed in a standard (yaw, pitch, roll) rotation sequence used in aerospace and
defined in SAE J670 standard for automotive applications, see Figure 5.
X‐ Axis
Y ‐ Axis
Z‐ axis
Plane (X
E
,Y
E
) is parallel to the Earth surface
X
E
Y
E
Z
E
θ
E
‐ Pitch
φ
E
‐ Roll
X
E
*
Y
E
*
Z
E
*
ψ
E
‐ Yaw
1. ψ
E
< 0
2. θ
E
> 0
3. φ
E
< 0
Gravity vector
𝑔
is coincident
with the Z
E
axis
⃗
Figure 5. Euler Angles
The first rotation defines the
𝜓
– yaw angle. It is performed about the Z
E
axis of the Earth-
fixed coordinate system (X
E
,Y
E
,Z
E
) from the X
E
axis to the X
E
*
axis. An intermediate coordinate
system (X
E
*
,Y
E
*
,Z
E
*
) is a Z-down coordinate system whose X
E
*
and Y
E
*
axes are parallel to the
ground plane (X
E
, Y
E
), with the X
E
*
axis aligned with the vertical projection of the X axis onto
the ground plane. Since the yaw rotation
𝜓
on Figure 5 is opposite to the positive rotation
direction, shown by the red arrow about the Z
E
axis, the resulted angle is negative.
The second rotation defines the
𝜃
– pitch angle. It is performed about the Y
E
*
axis of the
intermediate coordinate system (X
E
*
,Y
E
*
,Z
E
*
) from the X
E
*
axis to the X axis. The pitch rotation
𝜃
on Figure 5 is in the positive rotation direction, defined by the yellow arrow about the Y
E
*
axis, and the resulted angle is therefore positive.
The final third rotation defines the
𝜙
– roll angle, as a rotation about the X axis from the Y
E
*
axis to the Y axis. The roll rotation
𝜙
on Figure 5 is negative. It is performed in the direction
opposite to the positive rotation direction shown by the green arrow about the X axis.
The set of the three: yaw, pitch, and roll Euler angles fully represents the angular position of
the inclinometer in space.
There is not enough information for the unit to calculate the yaw angle
𝜓
, based only on the
accelerometer data.
