AXIOMATIC Tri-Axial J1939 CAN, User Manual

Page: 12 / 67

UMAX0608XX‐1000. Tri‐Axial J1939 CAN Inclinometer. Version 9B
Page: 12-66
The Euler angles are calculated using the following formulas:
𝜃 𝑎𝑡𝑎𝑛2 𝑔 , 𝑔 𝑔
,
𝜃 ∈ 90
°
; 90
°
,
(8)
𝜙 𝑎𝑡𝑎𝑛2 𝑔 , 𝑔
,
𝜙 ∈ 180
°
; 180
°
,
where: 𝑔 𝑎 , 𝑔 𝑎 , 𝑔 𝑎 .
The roll angles for both: the unit rotation and Euler angles are the same: 𝜙 𝜙 .
The roll-over condition is observed when:
|𝜙 |
90
°
.
When the unit is parallel to the Earth surface, the Euler angles are zero:
𝜃 𝜙 0°
.
2.1.3.2.3 Gimbal Lock
The formulas for the roll angle
𝜙
and
𝜙
are numerically unstable when both:
𝑔 𝑔 0
.
This condition, called a gimbal lock, happens when the unit is placed in the vertical position
with the X axis parallel to the gravity vector, see Figure 6. When this happens, the unit
effectively loses one degree of freedom and the roll angles
𝜙
and
𝜙
become undefined and
can take any random value.
Z‐ axis
Y ‐ Axis
X‐ Axis
Plane (X
E
,Y
E
) is parallel to the Earth surface
Gravity vector
𝑔
is coincident
with the Z
E
axis
⃗
Z‐ axis
Y ‐ Axis
X‐ Axis
Figure 6. Gimbal Lock
The same condition occurs with the pitch angle
𝜃
when both:
𝑔 𝑔 0
.
The gimbal lock should be avoided in the inclinometer initial installation position. It should be
also avoided in the inclinometer working range when it leads to unstable angular
measurements.
The user can avoid the gimbal lock condition by changing orientation of the unit frame (a
coordinate system attached to the unit) using configuration parameters when necessary.