Extended IMU Wind Estimation Theory

Extended IMU Wind Estimation
Theory
A. Chavarr´ıa-Krauser
May 1, 2014
An extension of the IMU Wind Estimation theory by William Premerlani
is presented. That theory is based on the assumption that a change in over
ground velocity originates only from a rotation of the aircraft. A method
to overcome this strong assumption is presented here.
The method proposed by William Premerlani is based on the assumptions of a
constant wind vector w and a constant airspeed modulus kvk, [1]. Premerlani suggests
to estimate the airspeed modulus by
kvk =
kv 02 − v 01 k
,
kf 2 − f 1 k
(1)
where v 0i is the inertial velocity (from e.g. GPS) and f i the “fuselage vector” (from
IMU) at two different time points i = 1, 2.
The main idea of the derivation is to express v 0 by the airspeed v and the wind
component w
v0 = v + w .
(2)
Using w = const, the time derivatives of the two speeds are found to be equal
dv 0
dv
=
.
dt
dt
(3)
The airspeed v is expressed by its modulus and a direction vector t of modulus one
v = kvk t .
(4)
The direction vector t can be expressed by the “fuselage vector” f , which is provided
by an IMU, and residual pitch and yaw errors given by the angle of attack α and the
sideslip angle β
t = RT (α, β)f ,
(5)
1
where R(α, β) is a rotation matrix. Hence
v = kvk RT (α, β)f ,
(6)
and
dv 0
d
=
kvk RT (α, β)f .
(7)
dt
dt
At this point is where the approach presented here deviates from the original idea
of Premerlani. He assumes that kvk and either that RT (α, β) is constant or that it
deviates not much from the identity matrix I. Eq. (1) is obtained then by producing
the modulus of the simplified equation.
If that assumptions are dropped, Eq. (7) renders
dkvk T
dRT (α, β)
df
dv 0
=
R (α, β)f + kvk
f + kvk RT (α, β)
.
dt
dt
dt
dt
(8)
The last term corresponds to the one used in the original approach, while the other
two terms are dropped. Because of the additive terms, it is not possible to take
the modulus and calculate kvk. This equation is vectorial, has three unknowns
p := (kvk, α, β)T ∈ R3 and should suffice. For the sake of a simpler noation, we
will drop from here on α and β in RT (α, β). The reader should keep that dependency
in mind.
Provided that dt/dt = dRT /dt f + RT df /dt 6= 0, a scalar multiplication of Eq. (8)
with dRT /dt f + RT df /dt gives
kvk =
dkvk T
dv 0
−
R f
dt
dt
2
dRT
df
dRT
T df ·
f + RT
f
+
R
.
dt
dt
dt
dt (9)
If the assumption of Premerlani are used (dkvk/dt = 0 and dRT /dt = 0), the equation
reduces into
2
df dv 0
T df
,
·R
kvk =
dt
dt dt where the property kRT xk = kxk for all x of rotation matrices was used. By means
of Eq. (8) an equation corresponding to Eq. (1) is obtained
0 . dv df kvk = dt dt The difference lies when the assumption dkvk/dt = 0 is dropped. Note that because
kf k = 1 it follows that f · df /dt = 0 and because RT is a rotation matrix also
RT f · RT df /dt = 0 is valid. Thus we find again
0
2
dv
dkvk T
df df .
kvk =
−
R f · RT
dt
dt
dt
dt (10)
df 2
dv 0
T df
·R
=
.
dt
dt dt 2
At the first glance it looks like kvk is independent of dkvk
dt and Eq. (1) is still valid.
This is, however, not the case and instead of Eq. (1) we would obtain
! 0 2
df 2
dv dkvk dv 0
T
2
,
−
(11)
·
R
f
kvk = dt dt dt dt
which reduces into Eq. (1) for dkvk/dt = 0. In contrast to the original approach, there
is no advange here in making the last step. Both expressions still contain the rotation
matrix and the first one needs less computational steps.
In a first approximation the rotation matrix can be assumed to be the identity
matrix, RT ≈ I, which corresponds to zero angle of attack α and zero sideslip β. An
estimate of the airspeed modulus accounting for acceleration is then
2
dv 0 df df ,
(12)
kvk =
·
dt dt
dt where dv 0 /dt and df /dt are readily available and can be approximated by differences,
such as done in Eq. (1)
kvk =
(v 02 − v 01 ) · (f 2 − f 1 )
.
kf 2 − f 1 k2
(13)
In the next order of accuracy, the system depends on α and β and the calculation of
kvk cannot take place independently of these. Though possible, a non-linear systems
of equations would have to be solved, for example using a Newton-Scheme.
References
[1] William Premerlani, IMU Wind Estimation Theory, December 2009,
URL: http://gentlenav.googlecode.com/files/WindEstimation.pdf
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