Inertial Navigation System (INS) has been developed for a wide range of vehicles. In general, an INS is mainly built up by a set of inertial measurement units (IMU), which consists of accelerometers and gyros, and the IMU is mounted on a platform. Besides, a processor is required to transform the measured data of acceleration and angular rate into useful navigation information: position, velocity, and attitude. [3] However, INS inevitably has its disadvantage in the errors of position, velocity, and attitude, which are usually caused by alignment, measurement, calculation, and initial states and often accumulate and grow divergently with time in the integral process. Therefore, INS needs external sensor to compensate, and the most popular method is using GPS to correct the divergent error in INS.
INS can provide continuously position interpolation during the period of receiving GPS signal. The other purpose of INS would be used in automatic car control and that is to provide the velocity of vehicle, then the velocity would be feedback to control the vehicle.
3.1 System Overview
There are two general types of INS, gimballed and strapdown, which will be described in this section.
3.1.1 Gimballed System
Figure.3.1 shows the gimballed system, where the accelerometers and gyros are mounted on a stabilized platform system; hence, it is also called the stabilized platform system [18]. To be a stabilized reference coordinates, the gimbals should be well controlled and then isolated from the vehicle’s motion. Furthermore, the
accelerometers can be horizontally aligned to get rid of the gravity effect by rotating the platform. Although the gimballed system has high accuracy, some disadvantages, such as complex hardware design, high cost, and large power consuming, make the design of this system difficult.
3.1.2 Strapdown System
Contrary to gimballed INS system, the accelerometers and gyros (IMU) are mounted on a platform fixed on the vehicle in strapdown INS (SDINS). The IMU can measure the acceleration and angular velocity of vehicle, which are used to calculate the variation of position, velocity, and attitude. In the computing process, numerical errors caused by integrating the accelerations and angular rates would be produced accumulatively and divergently. Besides, the calculation of DCM (Direct cosine matrix) and Euler angle is quite complicated such that a high-performance processor is often needed. Fortunately, the technology of processor has been increasingly improved and good enough to do this work.
X-GYRO
Y-GYRO Z-GYRO
TORQUE MOTOR
TORQUE MOTOR OUTER
GIMBAL
MODEL GIMBAL X-ACCEL
Z-ACCEL
Y-ACCEL
TORQUE MOTOR
RESOLVER
RESOLVER RESOLVER
STABLE PLATFORM
Figure 3.1 The gimballed INS system
It is known that the gimballed INS system, better than the SDINS in accuracy, is commonly adopted for long-term navigation. However, the SDINS is generally employed for short-term navigation due to its small size, low cost, power saving, and easy design. For car-navigation, Hence the SDINS is available for car-navigation, which requires short-term data, with external sensors, like GPS, which can compensate the disadvantages in long-term work.
3.2 Coordinate Frames And Transformation
In general, there are five coordinate frames commonly used in an inertial navigation system. This section will first briefly describe these five coordinate systems and then show two transformations, between body and navigation frames and between geodetic and Earth-centered earth-fixed frames.
3.2.1 Coordinate Frames
A. Body Frame
The body frame, also called the vehicle coordinate frame, is symboled as b-frame.
The measurements acquired by various inertial sensors can easily apply to the body frame. Usually, the body frame is rigidly attached to the vehicle’s center of gravity and its three axes are conventionally defined along the forward, right, and down directions as shown in Figure. 3.2, especially when adopted for car navigation.
xb
yb
zb
Figure 3.2 Body (Vehicle) coordinate frame
B. Navigation Frame [3]
The navigation frame, denoted as n-frame, is also called local geodetic frame with origin fixed on the vehicle and three axes pointing to the true north, east, and down as shown in Figure. 3.3. Clearly, the navigation frame intrinsically moves with the vehicle and then it is not suitable for specifying the vehicle’s position on the earth.
In fact, this frame plays a main role to provide local north, east, down directions and velocities, which is useful for navigation systems with sensors generally aligned with the local horizontal and vertical planes.
C. Earth-Centered Earth-Fixed (ECEF) Frame
The earth-centered earth-fixed frame, which is symboled as e-frame, is constructed with origin at the earth’s mass center and the three axes rotate with earth.
Furthermore, WGS-84 (World Geodetic System, 1984) is one kind of ECEF frame.
Besides, the x-axis and the y-axis respectively point to the prime meridian of 0o longitude and the meridian of 90o longitude through the tangent plane of the equator, and the z-axis points to North Pole as shown in Figure. 3.4.
y = East
z = down x = north(true)
Equator
Prime Meridian
Figure 3.3 Navigation frame or tangent plane reference coordinate frame
D. Geodetic Frame
The geodetic frame, which is symboled as g-frame, describes the position of a moving body by the longitude, latitude, and altitude, denoted as l, L, and h, where the 0o longitude is defined at Greenwich meridian and the 0o latitude is defined at the equator. The meridian longitudes start from 0olongitude at Greenwich meridian to the west and the east each up to 180oand the parallel latitudes start from 0olatitude at the equator to the north and the south each up to 90o. The altitude is defined as a distance from the local sea level to the point where the vehicle locates as shown in Figure 3.5.
E. Inertial frame
This frame is the most fundamental coordinate system that is symboled as i-frame and defined in which Newton’s laws of motion apply. The frame is static or in uniform linear motion system without accelerating. The definitions of axes are
y z
x Equator plane
Prime meridian
Figure 3.4 ECEF rectangular coordinate system
P a
b N
h P
x
y z
rotation about z
l
Figure 3.5 Geodetic reference coordinate system
generally built base on the star in universe. The x and z axes point toward the vernal equinox and along the earth’s spin axis. The y-axis is defined to complete the right-handed coordinate system. In this analysis, the frame is attached to the center of earth, and is not rotating.
3.2.2 Transformation Between ECEF and Geodetic Frames
This section will discuss the transformation between ECEF and geodetic (Longitude, Latitude, and Altitude)[17]. Besides, the WGS-84 ellipsoid parameters are used throughout this discussion. The position in the ECEF frame is calculated as follows.
N h cos L coslx (3.2.1)