In this chapter, we will introduce some tracking systems with IMUs. Related works are about tracking or motion by accelerometers, gyroscopes and magnetometer. Finally the six linear accelerometer system will be introduced. It describes a tracking system which composes of six linear accelerometers without any gyroscope or magnetometer. It designs one accelerometer at the center of face of a cube and the sensing axis of each accelerometer is along the respective cube face diagonal.
The research for tracking, motion and navigation system is currently carried out in many laboratories with using of accelerometers, gyroscopes and magnetometers. This is the basic idea for tracking and navigation by using accelerometers, gyroscopes and mag-netometers. They usually use accelerometers to get acceleration of objects and integrate the acceleration to calculate the moving distance. Gyroscopes and magnetometers are usually used to calculate the rotation or change of orientation.
3.1 Accelerometer + Gyroscope
In [14], it represented a simplified strapped down system for navigation. A strapped down navigation system comprised of gyroscopes and accelerometers without magnetometers.
They could be mounted on body or vehicle. Gyroscopes provided attitude angle and an-gular rate signals along three axes. Accelerometers meant provide signals representative of the acceleration along three independent axes. First transformation matrix connected to the attitude angle output of the gyroscopes to accelerometers transforms the gyroscope and the accelerometer signals from body coordinates to gyroscope coordinates. Second
transformation matrix connected to the output of the gyros, transforms the gyro coor-dinates into navigation coorcoor-dinates. The algorithm flow is shown as Figure 3.1. It is a simplified and basic system for tracking or motion. Because of gyroscopes had expensive price and error increased with double integration by the time. Attitude determination systems that used inexpensive sensors had been discussed.
Gyroscopes (Captured)
Accelerometers
Transformation
to gyro coor. Transformation to inertial coor.
Torquing Rates(ω)
Gravitational
acceleration Inertial velocity Inertial position
Figure 3.1: A simplified strapped down system [From ”Simplified strapped down inertial navigation utilizing bang-bang gyro torquing”].
3.2 Accelerometer + Magnetometer
In [15], it replaced expensive gyroscopes with magnetometers. This IMU composed of accelerometers and magnetometer. It used the earth’s magnetic field and gravity as the two measured quantities, a low-cost attitude determination system is proposed. It used the magnetic field vector and the acceleration, a unique plane containing the two vectors can be defined, and if the components of those two vectors can be measured in two non-aligned coordinate frames, then the rotation needed to align the two coordinate frames can be determined. In [16], the author also used a accelerometer and a tri-magnetometer to determine the orientation of a static or slow-moving rigid body. It presented a geometrically intuitive 3-degree-of-freedom (3-DOF) orientation estimation algorithm with physical meaning which is called the factored quaternion algorithm (FQA).
It is defined orientation by rotating it about its z-axis by an angle ψ (azimuth or yaw rotation), then about its y-axis by angle θ (elevation or pitch rotation), and finally about
its x-axis by angle ϕ (bank or roll rotation). The elevation and roll rotation are determined from accelerometers. The azimuth rotation is determined from magnetometers. The value of ax, ay, az is from a tri-accelerometer. And the value of bmx, bmy, bmz is from a tri-magnetometer in body coordinate system. The emx, emy, emz are magnetic vector in Earth coordinate system. The initial magnetic vector is
h
Having obtained all three rotation quaternion, the quaternion estimates representing the orientation of rigid body is finally given by
b
q = qa qe qr
Although they are cheaper than the gyroscope, the magnetometer is easily interfered by other electronic device. When a ferrous object is close to those systems, they will get bigger noise and lost their accuracy.
3.3 Accelerometer + Gyroscope + Magnetometer
In order to improve the accuracy or reduce the error of orientation, much research had focused on using IMUs system that composed of accelerometers, gyroscopes and magne-tometer. They not only used gyroscopes to calculate orientation but also magnetometers had been used to enhance accuracy and reduce error. In [7], it described a self-contained
method for relative position tracking of a human engaged in various types of motion in-volving discrete steps. This method is based on the use of the inertial and magnetometer sensor module attached to the foot. This modules contained three orthogonally mounted angular rate sensors, three orthogonal linear accelerometers and three orthogonal magne-tometers. In generally, the output of an accelerometer will be integrated twice to obtain displacement information. However it is susceptible to drift errors. A drift correction can be applied to the accelerometer with double integration method so that the final estimated velocity should be zero when a foot instant contacts with the ground. These inertial and magnetic sensor modules are primarily designed for tracking orientation. Although this system is high accuracy, it is not cheap and also is easily interfered by electronic devices.
3.4 IMU + Kalman filter
Not only using multi kinds of sensors but also applying Kalman filter to improve accuracy or reduce drift error on IMUs system. The Kalman filter is the most widely used state estimator for tracking applications. The filter is the general solution to the recursive linear minimum mean square estimation problem. In [17], it used two different adaptive Kalman filter. Each individual process noise covariance value will provide optimum estimates of the target’s states only when underlying model, which is represented by that specific process noise covariance, is correct. For example a low level process noise covariance will match the target dynamics during an almost straight line motion period but fail to track when the target moves into a turn manoeuvre. The first filter is adjusted at each time step according to the estimated turn rate. The turning rate is estimated from acceleration divided by the estimated speed of target. Second uses a scale filter which is estimated from the available data of sensors reading. The first algorithm exploits the idea ”the choice of the process covariance level must be made according to the expected turn rate”
and utilizes an empirical turn rate-process noise covariance level curve. And the second algorithm introduces a scale factor which represents the current magnitude of process noise, i.e., target unpredictability, at time t as estimated from the available data. But a proper Kalman filter is complex and hard to design.
2ρ
Figure 3.2: Six linear accelerometer system [From ”Gyroscope free Strapdown inertial measurement unit by six linear accelerometers”].
3.5 Six Linear Accelerometers
Gyroscopes are not inexpensive and magnetometers are easily interfered by other elec-tronic devices. In 1994 Jeng-Heng [18] presented a method to determine the kinematics of a rigid body by using only linear accelerometers. In their work, six linear accelerometers are used for a complete description of a rigid body motion. They design one accelerometer at the center of each face of a cube and the sensing axis of each accelerometer is along the diagonal of respective cube face, shows as Figure 3.2 (a). The line of cube side is 2ρ and xb, yb, zb are body frame of cube. Ob is the center of this cube. Figure 3.2 (b) shows relation of the inertial frame(OI) and a rotating moving frame (Ob). The −→
R = −→r + −→ρ then the accelerometer of point P can be calculated by double integration −→
R . S and T are 3 × 6 matrices After calculating, the formula will be produced :
,where A is the acceleration from accelerometers, ∂−→ω is the angular acceleration, the ω is the angular velocity by integrating ∂−→ω , ∂−→v is inertial acceleration.
S = 1
√2
1 −1 0 0 1 −1
−1 0 1 −1 0 −1
0 1 −1 −1 1 0
, T = 1
√2
1 1 0 0 −1 −1
1 0 1 −1 0 1
0 1 1 1 1 0
Although this system only uses accelerometer, it needs six accelerometer, fixed graph and fixed direction when installs the accelerometer. Because of the high accurate at the installing position is difficult, it needs to lathe to reach this request If the direction of accelerometer does not aligned, it will cause error. The major request is high accuracy for those accelerometers.
Some related works usually use gyroscopes or magnetometers to calculate change of orientation and accelerometer to get distance information. But they have the problem of expensive price and easily are influenced by electric/ferrous devices. Although the system of six linear accelerometers can also use pure accelerometers to tracking objects, it is hard to design and lathe is needed. In this work the g-sensor constellation system not only considers the price and is influenced by other device but also considers the sensor error itself. The pure accelerometers can reduce the cost and does not be influence by electric devices. And the LSM is a method to correct the error itself in this rigid graph system.