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Requirements and Evaluation of a Smartphone Based Dead … 7
2.2 Error-State Model
The model has the error-state vector
T;
x ¼ Dp; Dv; Dr; Db x ; Db f ; DcT
ð3Þ
where Db x and Db f are the errors of the IMU biases and
DcT ¼ cðdt rec ; d_t rec Þ
is the GNSS receiver’s clock error dt rec and its drift d_t rec , which are essential to
consider the raw measurements of the GNSS receiver. The error propagation model
is in-depth derived in [11] and given in its continuous form by
0
1
0 I 0 0 0 0
0 0 ðC^n b f i ib Þ C^n b 0 0
0 0 0 0
_x C^n ¼
b 0
0 0 0 0 0 0
x; ð4Þ
B
C
@ 0 0 0 0 0 0 A
0 0 0 0 0 F DcT
where I is the 3 3 identity matrix, C^n b is the direct cosine matrix to rotate the IMU
coordinates to the estimated navigation frame, f i ib is the skew-symmetric matrix
representation of the vector cross product of f i ib and F DcT is given by
F DcT ¼ 0 1
: ð5Þ
0 0
The absolute states are always corrected with the estimated errors after every
observation step of the KF. Afterwards, the error-states are reset to zero, so that a
nonlinear error propagation is not necessary.
2.3 Observation Models
For correcting the determined position, velocity and orientation of the strapdown
algorithm, the measurements of a GNSS receiver and a barometer are used.
Internally, a GNSS receiver measures the geometric distances to the satellites
which are summed up with other error terms. In particular the clock error cT of the
GNSS receiver produces a significant error. The measurement of the geometric
distance, added by the clock error term, is referred to as pseudorange ~q i of satellite
i. From a set of pseudoranges of different satellites, the GNSS receiver’s position
solution ~p n GNSS can be deduced.