Visual-Inertial Navigation: A Tutorial
January 15, 2017
1 Notations
Table 1: Notations
R ∈ SO(3) robot orientation in the global frame
p ∈ R
3
robot position in the global frame
V ∈ R
3
robot velocity in the global frame
v ∈ R
3
robot velocity in the body frame
b
a
∈ R
3
accelerometer bias
f ∈ R
3
landmark position in the global frame
w
t
∈ R
3
gyroscope reading at the time t
a
t
∈ R
3
accelerometer reading at the time t
g ∈ R
3
The gravity in the global frame
2 From Original Model
Here we ignore the gyroscope bias. The motion model of IMU and landmark is
given in the following:
˙
R = RS(w
t
)
˙
p = Rv
˙
v = −S(w
t
)v + (a
t
− b
a
) + R
|
g
˙
b
a
= 0
˙
f = 0
(1)
Now we define two variables F and n:
F := R
|
(f − p)
n := R
|
g
(2)
According to Eq.1 and Eq.2, we get the dynamics of X (X = (F,v,n,b
a
))
˙
X = A
t
X + B
t
(3)
where A
t
=
−S(w
t
) −I
3
0
3,3
0
3,3
0
3,3
−S(w
t
) I
3
−I
3
0
3,3
0
3,3
−S(w
t
) 0
3,3
0
3,3
0
3,3
0
3,3
I
3
and B
t
=
0
3,1
a
t
0
3,1
0
3,1
. Please note that
the dynamics of X is linear time-varing system.
