Angular Motion and Attitude Estimation Using Fixed and Rotating Accelerometers Configuration Ezzaldeen Edwan, Jieying Zhang and Otmar Loffeld
Center for Sensor Systems (ZESS) University of Siegen Siegen, Germany Email:
[email protected] AbstractfIXed
assuming constant angular velocity terms over one period and solving using synchronized demodulation algorithm as explained in [4]. Recently, a new approach for using a configuration of fixed and rotating accelerometers but with larger number of rotating accelerometers, has been proposed in [5]. Our contribution is extracting the angular velocity with a less number of rotating accelerometers and without assuming any approximations which might degrade the quality of the system. Moreover, we benefit from the developed algorithms for estimating angular motion from fixed accelerometer configurations [I].
In this paper, we present a novel configuration of
distributed
accelerometers
combined
with
rotating
accelerometers to infer the angular motion. Traditionally, fIXed distributed accelerometers are configured to form a gyro-free inertial measurement unit (GF-IMU). The main advantage of using rotating accelerometer over fixed one is having direct measurements of the angular velocity. This configuration can be used to fmd a complete attitude solution. For static case, the heading angle is computed from angular velocity due to Earth rotation sensed by the rotating accelerometer while the tilt angles are found from the projected gravity sensed by an accelerometer triad.
Keywords-Rotating accelerometers; system
GF-IMU;
II. MOTIVATION FOR Aro The fixed distributed accelerometer configuration gives an angular velocity vector with undetermined sign as shown by Costello [6]. This is mainly because the quadratic terms do not give a unique angular velocity vector solution. Instead, we get two solutions. For solving this problem, some researchers propose the fusion of the angular acceleration vector with quadratic terms of angular velocity to keep the solution in the right track of the sign [2] but in reality there is no guarantee for such a solution to work in all scenarios. For the determination of algebraic sign in a completely GF-IMU, there exist solutions using accelerometers that vibrate in a known matter to body as proposed initially by Merhav [4] and recently by Costello [5] where he gave a solution that does not require integration. The conventional low-cost gyros can be used to insure a correct sign convergence in the GF-IMU as shown in [7]. Other possible integration schemes include GPS [8, 9] and tri-axial magnetometers [10] to have a non-drifting orientation estimate; however, they are non-inertial solutions. The integration result will be determined by the quality of the used accelerometers. Theoretically, the knowledge of one component of the angular velocity is sufficient to give the complete correct sign solution if it has non-zero value.
North finding
I. INTRODUCTION There are intensive research efforts to deploy fixed accelerometers in inferring the angular motion due to their advantages over gyroscopes [I, 2]. Nevertheless, a little attention was given to the use of rotating accelerometers. The fixed accelerometers configurations have a simple setup; however, their use does not give an explicit expression for the angular velocity. The configuration of twelve separate mono axial accelerometers, described in [1], produces the angular information vector (AIV) that consists of a 3D angular acceleration vector and six quadratic terms of angular velocities. Unfortunately, the angular velocity vector is not expressed explicitly in the AIV. The appearance of the quadratic terms introduces a sign indeterminacy problem. Through the use of fixed and rotating accelerometers configuration, we get non vanishing Coriolis force terms. A rotating accelerometer is an accelerometer that rotates at a known angular velocity around its sensing axis. The whole rotating accelerometer assembly is mounted rigidly to the inertial measurement unit (IMU) frame. A.
Previous Research
III.
The principle of using a moving accelerometer is applied in commercial Micro Electromechanical systems (MEMS) gyros. The ADXRSI50 and ADXRS300 MEMS gyros from Analog Devices are based on a resonating mass that generates a Coriolis force [3]. One of the oldest approaches is to use only three rotating accelerometers but with many assumptions like
978-1-4673-0387-3/12/$31.00 ©2012 IEEE
CONFIGURATION OF FIXED AND ROTATING ACCELEROMETERS
We formulate our system based on acceleration equation of spatial motion of a point moving within a body which also moves with respect to the inertial frame. The measurable acceleration lr � a", ary an ] T of a moving point at =
8
distance l from the body center difference, is
rl
b b b b b .b b b b b Qir = Qio + Qor + 2Q!ib Qor + Q!ib xr + Q!ib x Q!ib xr . X
where Q�o is the acceleration vector at a fixed point inertial frame
The rotating accelerometers have the shown positions at time t=O with rB rotates around y axis with sensing axis along y axis, re rotates around z axis with sensing axis along z axis and rD rotates around x axis with sensing axis along x axis. For all rotating accelerometers, we assume they rotate at unique angular velocity wr with unique radius of rotation p and the center of rotation is placed at distance I from body frame center. The position vector of the rotating accelerometer rD on the traversed circular path is given as
, ignoring gravitational (I )
l with respect to
Q;o = �aox aoy aoz ) T
b Q!r is the acceleration vector of moving point r with respect to the body center _ abor = dQ�b = Il� abpx ab'P.'Y abpz ) T ' dt b Q!r is the velocity vector of point r.b with respect to the body db center � = � vbpx vbpy vbpz ) T , b 4 is turn rate of body frame with respect to inertial frame !Jlb is the angular acceleration vector,
I
pz
J�}
,
�x=
t�, -E �+
dt
Vbpz
rD
=
abpy abpz
wrpcosw t r
rD
d dt
-
rD
vb pz
-Pw; coswrt
=
Vbpy
2
.
(6)
-w psmw t r r
rD
Similarly for accelerometer rB ' the position vector of the rotating accelerometer on the traversed circle is given as Px Py pz
(7) pcoswrt
ro
The velocity vector at point
2
We consider the configuration of rotating and fixed accelerometers shown in Figure I where the fixed accelerometer triads are symbolized as A,B,C,D and the monoaxial rotating accelerometers are symbolized as
=
d Px dt Py
is derived as
rB
(8)
pz
ro
The acceleration vector at point rB is the time derivative of the velocity vector and it is given as
z
abpx
D
-Wrpsmw t r
0
-
vbpz
ro
.
2
Vb x
d p = Vbpy dt
abpy abpz
-Pw; coswrt ' the position
ro
(9)
Similarly for accelerometer re vector of the rotating accelerometer on the traversed circle is given as
y
pcosw t r
Px
psinw t+l r
Py pz Figure I.
pz
The acceleration is the time derivative of the velocity and hence the acceleration vector is derived as 0 abpx vbpx
(2)
-Wy -Wz WxWy -Wz WxWz +Wy Px + WxWy +Wz -W; - W; WyWz -Wx Py WxWz -Wy WyWz +Wx -Wx -Wy pz
r B,re,rD·
rD
(4)
+l
-
In (l), the superscript b denotes that the quantity is expressed in the body frame. The gravitational difference vector can be ignored for relatively small distribution distance. We rewrite (I) in matrix form as 0 -wz Wy vbpx arx aox abpx ary aoy + abpy +2 Wz 0 -wx vbpy a", aoz abpz -wy Wx 0 vbpz (3) 2 2 2
p sin w t r
The velocity is the time derivative of the position and hence the velocity vector is derived as Vbpx 0 Px d = Py -pw sinw t Vbpy (5) r r
is cross product operation defined as skew symmetric form
�=
pcoswrt
Py
l
x
o
Px
The velocity vector at point
Fixed and rotating accelerometers configuration
9
o
rc
re
is derived as
(10)
Vbpx Vbpy
x d P = dt Py
-PWr sin wrt WrP cos wrt
-
Vbpz
rc
pz
of the angular velocity appear in any two combinations of the rotating accelerometers equations as shown in (13).
(11)
IV. STATIC SCENARIO CASE ( NORTH-FINDING SYSTEM) For a complete static attitude finding system that finds heading by sensing Earth rotation, the required number of accelerometers can be reduced to an accelerometer triad and a rotating accelerometer as shown in Figure 2. Reduction is possible because the angular acceleration terms become zeros and the quadratic terms are extremely small that they can be ignored. One common approach is to use a rotating accelerometer that has sensing axis along gravity to find the north for the leveled case [12-14]. Though the use of one accelerometer might sound reasonable, the system is sensitive to errors in misalignment from horizontal position besides the need for leveling equipment.
0
rc
The acceleration vector at point rc is the time derivative of the velocity vector and it is given as -w; p cos wrt abpx vbpx abpy abpz
=
rc
d dt
-
-Pw; sin wrt
Vbpy vbpz
(12)
0
rc
Since we are interested in the values of acceleration where the rotating accelerometers are mounted as shown in Figure 1, we substitute the position, velocity and acceleration values given in (4)-(12) into (3) and form the following vector equation
z
a} a: wz wr Psn i w/ +wywr pcosw/ a;B = a; +2 wxwr psinwr t +wz wrPCOswr t + azr c (13) (WxWy -w')pcoswr t +(wxwz +wy)(psinw/ +/) (wywz -wx)pcosw/ +(wxwy +w.)(psn i w/ +/) (wxwz -Wy)pcoswr t +(wywz +wx)(psn i wr t +I) Equation (13) can be simplified by sampling at the moment
y
where the sine terms are equal to zero. In such moments the cosine terms are equal to 1. The simplified equations becomes
axr D aAx WyWr P r A B ay = ay +2 Wz Wr P wxwr p azr c aAz (WxWy -w,)p +(wxwz +wy) 1 + (wyWZ -wx)P+(WXwy +w.)1 (wxwz -wy)p +(wyWZ +wx) 1
x
Figure 2.
A.
(14)
a;c = a1 +2PWr(wy sinwrt +wx cos wrt) .
(17) The index pulse determines the initial position of the rotating accelerometer. For a local navigation frame NED, the gravity and Earth rotation vectors are given as wie cos
