_
r ¼ v
_
v ¼ a
c
þ g 2x
l
m
v
_
q ¼
1
2
XðxÞq
ð1Þ
where r and v are vehicle position and velocity vectors, q is
the vehicle attitude quaternion vector with respect to the
target-fixed reference system, x is the vehicle angular speed
with respect to the target-fixed reference system expressed
in the body-fixed reference system, x
l
m
is Mars rotation
angular speed expressed in the target-fixed reference sys-
tem, a
c
is the control acceleration generated by the descent
engines, g is the local gravity vector which is assumed to be
a constant vector, and X(x) is the quaternion matrix given
by
XðxÞ¼
0 x
x
x
y
x
z
x
x
0 x
z
x
y
x
y
x
z
0 x
x
x
z
x
y
x
x
0
2
6
6
6
4
3
7
7
7
5
ð2Þ
Since the Coriolis acceleration caused by the Mars rota-
tion is much smaller than the control and the gravitational
acceleration, it can be omitted in the dynamics. Therefore,
Eq. (1) can be simplified as
_
r ¼ v
_
v ¼ a
c
þ g
_
q ¼
1
2
XðxÞq
ð3Þ
2.2. Sensor models
Three types of navigation sensors are included in the
proposed navigation scheme. IMU is the basic sensor used
for dead reckoning. IDR measures vehicle beam-direction
distance to the ground which contains vehicle altitude
information and the line-of-sight velocity which contains
vehicle velocity information. Thus, the aim of involving
IDR is to correct the recursive altitude and velocity. The
radio senso r measures the distance between the vehicle
and the MO as well as their line-of-sight relative velocity
to correct the horizontal position.
2.2.1. Inertial Measurement Unit
The IMU is composed of an accelerometer and a gyro-
scope. Even the most advanced IMU suffers from misalign-
ment, bias, and integration drift (Busnardo et al., 2011).
Here, the accelerometer and gyroscope are both considered
to be corrupted by a drift-rate bias and drift-rate noise.
Thus, the output of IMU is given by
a
o
¼ a
r
þ b
a
þ g
a
x
o
¼ x
r
þ b
g
þ g
g
ð4Þ
where a
r
and w
r
are the true acceleration and angular rate,
b
a
and b
g
are the drift-rate biases of the accelerometer and
gyroscope, g
a
and g
g
are the drift-r ate noises assumed to be
Gaussian white noise. The drif t-rate bias is not a constant
but an integration of a Gaussian white noise, shown in Eq.
(5)
db
a
dt
¼ n
a
db
g
dt
¼ n
g
ð5Þ
The IMU data is not treated as observation but used for
dead reckoning. Hence the IMU noise is not considered as
observation noise but state noise.
2.2.2. Integrated Doppler Rada r
To determine the vehicle altitude and Car tesian velocity
in the targe t-fixed reference system, the IDR should carry
at least three beams in different directions (Amzajerdian
et al., 2012). In this paper, a six-beam IDR, similar to the
MSL Terminal Descent Sensor, is considered. This sensor
contains six antennas each with a unique beam direction.
Distribution of six beams is shown in Fig. 2, with one beam
pointed along the vehicle vertical axis (nadir), three beams
inclined a degrees from the vertical axis and evenly distrib-
uted in azimuth, and two beams inclined b degrees from
the nadir axis, each rotated c de grees from the vehicle for-
ward axis in azimu th (Pollard and Chen, 2009).
IDR beams
MO
descent stage
radio signal
Mars atmospheric interface
Fig. 1. Schematic diagram of TDS/MO navigation.
1890 T. Qin et al. / Advances in Space Research 54 (2014) 1888–1900