The Application of Magnetic Sensors in Self-Contained Local Positioning

Nowadays, GPS (Global Positioning System) has been widely used to position airplanes, ships, automobiles, migrating animals, and human beings, etc. However, GPS has its own limitations. Other than the user groups for whom GPS is operated, there is no service level agreement with anyone else. As more and more people become dependant on GPS in more and more countries, the political control of GPS raised doubt about its value. Moreover, in the event of system failure, equipment depending on GPS, such as airplanes or ocean-going vessels, needs alternative positioning systems. Furthermore, GPS relies on complicated infrastructure and cannot provide sufficient accuracy, stability and coverage for environments such as indoors and highly urbanized areas (known as “urban canyon”) due to the lack of lineof-sight (LOS) to the satellites of GPS. As a result, most commercial and social activities taking place in such environment can not benefit from positioning services offered by GPS.

A typical application of LPS in transportation is so-called ITS. This new technology is still being developed to control traffic jam and improve road safety. ITS also reduces vehicle wear, travel time, and fuel consumption through managing competing factors such as vehicle type, load, and routes. Other than conventional services such as electronic toll collection and emergency vehicle notification, sophisticated services are also provided to control urban traffic and to guide drivers with automatic routes for optimal performance. LPS, integrated with GPS, makes these services possible.
In the long run, both the technology and society environments will be changed greatly. Integrated circuits (IC) will be faster, smaller and cheaper according to Moore's law. The positioning will be more accurate and robust with advanced algorithms and new sensors. Building automation and indoor three-dimensional (3D) dynamic mapping will illustrate and control the whole building environment in much greater details such that both the quality of indoor environment and the operation cost-effectiveness will be significantly improved. Virtual world and real world will be merged to create new sports, games, entertainment and arts. People will have more wealth, enjoying more social networking, and receive more cares. As a result, LPS will surely have more and more applications. Personal positioning devices will be popular to every working persons and school students. Value added services will be provided through LBS (Location Based Services) and ITS. GIS (geographic information system) and LPS will be combined to provide quick emergency response and automatic rescuing. Much smarter supply chain management, traffic management, and work force management will be available to businesses. LPS may induce a commercial exploitation in the future.

Strap-down inertial positioning
Some LPS systems reply on certain infrastructures and are difficult to setup and maintain. They are not suitable for applications that require quick setup, such as fire fighting operations, and emergency rescue services. We recommend self-contained LPS to locate objects residing or moving within a covered area. These objects are equipped with intelligent sensor modules, normally in form of microelectromechanical system (MEMS) component that can sense both the direction and speed of movement and compute the position of the objects within a given coordinate system. In this section we introduce a selfcontained pedestrian tracking technique for the establishment of a complete infrastructurefree positioning system. The technique is the synergism of two existing technologies of dead www.intechopen.com The Application of Magnetic Sensors in Self-Contained Local Positioning 69 reckoning and inertial positioning. In the implementation of such a technology, the magnetic sensors take an important role in achieving acceptable performance.

Dead reckoning
Examples of existing self-contained LPS systems include dead reckoning systems for marine navigation (McBrewster, 2009), wheel sensing systems for automobiles (Thiessen & Dales, 1983), and pedometers for human beings (Chiang, 2007). Dead reckoning is a method of estimating an object's present position by projecting its courses steered and speeds over ground from the last known position. The dead reckoning position is only a rough approximation because it does not consider the effect of leeway, currents, helmsman error, or gyro error.
All dead reckoning systems compute the position of an object based on the last known position and the relative movement from that position. The relative movement is calculated according to the direction and speed of motion. This principle is employed in the aforementioned three existing self-contained LPS. Despite the advantage of independence on complicated infrastructure, these systems normally suffer from low accuracy.

Inertial positioning
An inertial positioning system (IPS) indirectly obtains the speed and direction, or displacement, through integrating the measured accelerations and the angular velocity of an object over time. Accelerometers and gyroscopes are used to continuously measure and record translational and rotational motions. Thus inertial positioning and dead reckoning are different in the data acquired (acceleration v. s. speed, angular velocity v. s. course direction) and the devices used to measure these data (e. g. accelerometer v. s. speedometer). For a dead reckoning system, the distance can also be acquired through direct estimation. For example, in a pedometer, the step length is simply estimated, not measured or calculated. The distance is estimated by multiplying the estimated step length and the number of steps counted. Generally speaking, inertial positioning, or inertial navigation, is a modern technology superseding the dead reckoning, a relatively older technology.
A traditional inertial positioning system is gimbaled (Sarton and George, 1959). A mechanical device called gimbal-stabilized platform is used to establish a reference system in vehicles such as submarines, surface vehicles, aircrafts and space crafts. The sensors, the gyroscopes and accelerometers, or pick-ups, are mounted up on the stabilized platform to sense specific forces. The advantage of gimbaled systems is that the computation of position, velocity, orientation and angular velocity is less complex.
As shown in Fig. 1, with the initial local velocity v l (0) and initial local displacement s l (0) of an object known respectively, we are able to track the position of the object by subtracting the acceleration due to gravity, g, and then integrating the remaining acceleration a l (t)-g, once over a time period t to obtain velocity, and twice to obtain displacement in the local geographic frame (l-frame):  Using the rectangular rule, the above integration can be implemented in discrete-time form , denoting δt as the integration interval: Unfortunately, there are several problems with the gimbaled systems. Other than frictions from bearings and the dead zones of motors, extra power is needed to align the platform with the navigational frame. Moreover, gimbaled systems need high quality electromechanical parts including motors, slip rings and bearing, recalibration which is difficult, and regular maintenance by certified personnel in a clean room through a lengthy recertification process. Consequently, the traditional position systems are mostly used for airplanes, vessels, and intercontinental ballistic missiles.

Strap-down inertial positioning
The strap-down IPS replaces the traditional gimbaled system for low-cost applications. The accelerometers and rate gyroscopes are rigidly mounted in the body of a tagged object thus there is no relative movement between them. This is a major hardware simplification compared with the stabilized platform in tradition system. The most significant advantage of the strap-down IPS in comparison with the gimbaled IPS is the considerably reduced size and weight, which normally result in lower cost, power consumption, and hardware complexity.
However, the processing of inertial navigation is more sophisticated. Fundamentally, for IPS, a number of Cartesian coordinate reference frames, such as the body frame (b-frame), positioning frame and local geographic frame (l-frame) have to be rigorously defined and precisely related. We define the b-frame as an orthogonal axis set, which is aligned with the roll, pitch and yaw axes of the body to be positioned. We also define the positioning frame as an l-frame which has its origin at the original position for positioning and axes aligned with the directions of north, east and the local vertical (down). In a strap-down IPS, the computational complexity is increased because output data are measured in the b-frame rather than the l-frame. An algorithm has to be applied to keep track of the orientation of the sensor module (and the object) and rotate the measurements from the b-frame to the l-frame. Fig. 2 shows this procedure.

Reference frames and rotations
Denoting axes x l , y l , and z l respectively represent north, east and down in the l-frame fixed to the surface of the earth, the orientation of the device will be estimated in this coordinate frame. Axes x b , y b and z b represent the orthogonal axes in the b-frame.
In order to describe the components of a vector of arbitrary orientation in l-frame with respect to the original b-frame, or, in a new frame with respect to an old frame, after a certain rotation, a few rotation representations must be applied. We use the direct cosines representation to develop algorithms for tracking the orientations of sensors. To do this, a direct cosine matrix (DCM) has to be established by transformation of three sequential rotations from the original axes in the l-frame to the new body axes. These rotations continue as a Ψ rotation about z; a θ rotation about y, resulting from the first rotation; and, finally, a  rotation about x, resulting from the second rotation. The resulting DCM is as follows:

Tracking the orientation of a moving object
The orientation of a moving object is tracked by integrating the angular velocity signal obtained from rate gyroscope in the swing phase of walking.
At a time instant t, let =[ x y z ] T be the corresponding angular velocity sample, denote the norm of angular velocity asξ,i.e.,ξ=∥ x , y , z ∥,and a matrix With a short sampling interval t, the DCM at time t, R(t), can be calculated from the DCM at time t-t, R(t-t), according to below equation With the DCM updated, it becomes possible to project the acceleration signal a b (t) from the accelerometers in the b-frame into acceleration a l (t) in the l-frame: After the local acceleration is obtained, the velocity and the displacement of the object can be determined according to (1) and (2) respectively to locate the object.

Inertial sensors
As mentioned above, inertial positioning relies on the measurement of accelerations and angular velocities. The estimate of changes in rotated angles can be obtained through integrating the angular velocity from a gyroscope over time. The estimate of changes in velocities and positions in the l-frame can be obtained through integrating the local acceleration over time once and twice respectively. The local acceleration is the projection of body acceleration, measured from accelerometers, onto the l-frame.

Gyroscopes and accelerometers
A wide range of sensors can detect or measure angular movement. These devices range from the conventional mechanical gyroscopes and optical gyroscopes, to the ones based on atomic spin.
An accelerometer measures the physical acceleration it experiences relative to freefall, not to the coordinate systems. There are many types of accelerometers. Generally, all these devices are amenable for strap-down applications. However, the accuracy ranges widely from micro-g to fractions of g due to the variety of designs. Acceleration can be internally measured without using any external references, which are needed for the measurement of velocity. This is why it is preferable to develop algorithms based on acceleration for LPS.
MEMS technology, used to form structures with dimensions in the micrometer scale, is now being employed in manufacturing state-of-the-art MEMS-based inertial sensors.

Performance evaluation
Practically, the measuring errors of accelerometer and rate gyroscope result in errors during positioning. Assuming the average output from an accelerometer when it is not undergoing any movement is b a , the accumulated positioning error after double integration over time period t is e s (t)= b a t 2 /2. This means that the positioning error caused by a constant bias of accelerometer increases quadratically through double integration over time. Such an error is called integral drift.
Similarly, assuming the offset of the output angular velocity from true value is b g , the error of a gyroscope over time period t, we have e g (t)= b g t. This linearly increasing angular error, i.e. orientation error, can cause the error of the rotation matrix R(t). The resulting incorrect projection of acceleration signals onto the global axes causes two problems. Firstly, the acceleration is integrated in the incorrect direction, causing interaction of velocity and position between (not along) axes. Secondly, acceleration due to gravity can not be removed completely. The residue acceleration originated from gravity will become a "bias" to the true acceleration due to movement of the object.
In the strap-down navigation algorithm, the acceleration due to gravity, g, is deducted from the globally vertical acceleration signal before integration. When there are angular errors, there are tilt errors. Here a tilt error refers to the angle between the estimated vertical direction and true vertical direction. The tilt error e  in radian will cause the projection of gravity onto the horizontal axes, resulting in a component of the acceleration due to gravity with magnitude: e a h =g×sine  . This component can be treated as a residual bias due to gravity, remaining in the globally horizontal acceleration signals. In the mean time, in globally vertical axis, there is a residual bias of magnitude: e a v =g×(1-cose  ). Fortunately this problem is much less severe because for small e  , we have e a v ≈0. Therefore, a small tilt error will mainly cause positioning error in the globally horizontal plane.
In some cases, Such as human walk, the mean absolute acceleration measured is much smaller than the magnitude of gravity. As a contrast, a tilt error of 0.05° can cause a component of the acceleration due to gravity with magnitude near g/1000. This residue bias can cause a positioning error of 15.4 meters after only a minute of integration, or error of 0.49 meter after only 10 seconds (as demonstrated by the red dashed line in Fig. 3). Therefore, gyroscope errors, which propagate in the positioning algorithm, are critical errors affecting the accuracy of pedestrian tracking. Before the development of the algorithm presented in this chapter, it was believed that positioning with data from inertial sensors was not possible due to the quadratic growth of errors caused by sensor drift during double integration.

Integration of PDR with inertial positioning
Precise tracking of people, especially of first responders, in a harsh environment, remains an open research area. For a practical person tracking system, the measuring device of the system should be portable or wearable for the person being tracked. Inertial positioning can be applied in people tracking theoretically. However, the integral drift of accelerometers and gyroscopes, as well as the tilt errors of gyroscopes, make the inertial positioning not practical. As a result, as mentioned earlier, the characteristic of the movement of the person should be incorporated to improve positioning accuracy. If pedestrian dead reckoning (PDR) is integrated with IPS, the performance will be significantly improved, as shown is Fig. 3.

Pedestrian dead reckoning
PDR is the application of dead reckoning in pedestrian tracking. Due to the difficulty in measuring the speed of a walking person directly, PDR estimate a pedestrian's present position based on estimated step length and heading from the last known position. Using probability models, Mezentsev et al (2005) confirm that the main source of errors in a PDR system is related to the estimation of the step length and the heading. Different random noise models can be applied to heading estimation and step length estimation (Mezentsev et al, 2005). Suppose the modeled mean step length s is a constant, and the ith length error i is normally distributed with a standard deviation σ, then the true ith step length satisfy s i =s+ i . Assuming the step errors are uncorrelated during a walk, the distance error variance after N steps is Hence, the error variation equals the product of the step count and the variation of the driving noise. Accordingly, the position error maximizes when the pedestrian walk continuously along a straight line with a constant step length error. Therefore, if not assisted by other methods, PDR is not able to provide precise positioning. For further improvement of accuracy, the kinetics of human walk is investigated.

Human gait and phase detection
According to Rose and Gamble (1994), the human motion during a walk is cyclical and repeatable. Such a basic pattern is significantly constant among individuals. The gait cycle during human walk comprises two phases: a stance phase and a swing phase. In the stance phase, a foot keeps in touch with the ground from the time when the heel strikes the ground to the time when the toes leave the ground. The swing phase starts when the foot is taken away and carried forward. When the foot touches the ground again, the swing phase ends and the next stance phase starts. Each gait cycle is completed and restart in such a way. The other foot performs the same process, but lags 180° behind. Fig. 4 shows how the stance phase is started and ended. For a certain point at the front part of sole, its ground speed is almost zero from T 3 to T 5 (the period ΔT). If this time periodΔT can be dependably detected every stride, referential data may be acquired and used to reset computations and get rid of accumulating errors, the critical problem in pedestrian tracking. Huang et al (2010) advocate the integration of inertial navigation and pedestrian dead reckoning. With the stance phase detected, the velocity and angles can be updated in the stance phase and swing phase respectively. Thus both the integral drift of velocity and positions, as well as the tilt error caused by integral errors of rotated angles, can be greatly reduced. The positioning performance of the integration is illustrated in Fig. 3. The details of these updates are explained in the sub-sections that follows.

Fig. 4. Stance phase in a stride
In order to detect stance phase we can use sensor data, including all three components of angular velocity, acceleration, and even magnetic field, provided by a typical inertial/magnetic sensor module. Because the y-axis angular rate is the most significant indicator of stride events, Huang et al (2010) use the y-axis angular rate to detect the stance phases instead.

Zero-Velocity update and swing phase integral error reduction
Theoretically, the velocity in the l-frame is zero in the stance phase. In practice, zero-velocity updates (ZUPT) are performed when the acceleration and rate gyro measurements drop below empirically determined thresholds for a certain period of time. The main purpose of these updates is to mitigate both movement measured in the stance phase of a gait and increasing integral drifts. ZUPT is used by Stiring and Fyfy (2005), Mezentsev and Lachapelle (2005). Stirling and Fyfy stopped integrating and reset the velocity before each swing phase. The positioning accuracy of their system is of 10 to 20 percent of distance traveled.
In order to obtain a higher tracking accuracy, Huang et al (2010) further use additional methods to remove integral drifts. Some methods should be applied in the swing phase of each stride.

Orientation update using magnetic sensors
Huang et al (2010) update orientation using magnetic field signals from magnetic sensors, i.e., magnetometers.
In order to determine the orientation of the sensor module for rotation representation and transformation between the l-frame and the b-frame, the Euler angles must be determined.
In the swing phases, the Euler angles are calculated through integration of the angular velocity over time assuming that the original orientation of the sensor module is known. However, the original orientation is usually unknown in reality. In addition, the gyroscope signal summed during integration is perturbed by gyroscope bias. Thus the accuracy of the orientation will decay overtime. Therefore, Huang et al (2010) use magnetometers to detect the magnetic field of the earth. Using data from magnetometers, together with gyroscopes and accelerometers, the orientation in the beginning of each human walk can be found. Similarly, the orientation of the sensor module in stance phase, when the sensor module is in a near-static status, can be updated in each gait. With the updates of orientation in stance phases, the integral drifts of rotated angles, which increase linearly over time, are greatly reduced.
To find the near-static orientation, the Euler angles , θ and φ must be calculated. Because gyroscopes cannot be used in a near-static status, signals from accelerometers and magnetometers have to be used to calculate the orientations of the sensor module. However, these angles cannot be obtained directly. To resolve this problem, the tilts and heading of the sensor module need to be investigated.
where the two-argument function, atan2, is a variation of the arctangent function. For any real arguments x and y, which are not both equal to zero, atan2(y, x) is the angle in radians between the positive x-axis of a plane and the point given by the coordinates (x, y) on it.
The heading of a pedestrian, thus that of the sensor module on him/her, can be calculated using the x-axis and y-axis magnetic field outputs if the sensor module is level in a horizontal plane. As shown in Fig. 6, only the x and y components of the Earth's magnetic field, are used when determining the heading of sensor module. The heading is defined by: yx atan 2( , ) hh   where h x , h y and h z are components of magnetic field on x, y, z axis respectively. No matter whether a person is standing or walking, the sensor module attached to the human body is not confined to a flat and level plane. This makes it more difficult to determine the heading direction, because the tilt angles of the x-axis and the y-axis are always changing and the sensor module cannot stay horizontal to the earth's surface. Errors caused by tilt angles can be large if not compensated.
To compensate the tilt of the sensor module when it is in an arbitrary position, the l-frame is transformed into the b-frame through three rotations with Euler angles , θ and  in sequence. The DCMs of the rotations are denoted as R 1 ( , z), R 2 (θ, y) and R 3 (ϕ, x) respectively. These Euler angles are related to the tilt but they are not all the tilts of the sensor module. The algorithm (Huang et al 2010) is as follows.
Rotations of the gravity vector from the l-frame to the b-frame are given by where g b = (g x g y g z ] T and g l = (0 0 1] T . Therefore, we have This equation is actually the same as equation (10) if the sensor module is not moving. Using equation (13) and (14), we obtain   atan 2 sign(cos ), sign(cos ) This equation is different from equation (11). Hence the  angle of roll is not the y-tilt angle β. y-tilt should not be used as roll for orientation calculation and magnetometer compensation.
After both pitch θand roll  are found from acceleration data, the attitude, or yaw , can be found from magnetic field data by solving the following simultaneous equation:   (17) As illustrated in Fig. 6, h x , h y and h z are the projections of magnetic field on the horizontal plane and the z-axis (pointing downward) respectively, i.e., h b = [h x h y h z ] T =(acos , -asin ] T . Then the compensated magnetic field can be obtained as below: The angle calculated in equation (21) is actually a compass reading, i.e., the heading r e l a t i v e t o m a g n e t i c n o r t h . T o get the true heading, denoted as T here, the magnetic declination D must be deduced from the compass reading : where D is the difference between magnetic north and true north. This difference is caused by the tilt of the earth's magnetic field generator relative to the earth's spin axis.

Experimental study
To implement the methods introduced in the Section 4,. A commercial sensor module shall be carefully selected And then mounted to on a person rigidly. The result of the on-site pedestrian tracking is promising.

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The Application of Magnetic Sensors in Self-Contained Local Positioning 79

Sensor module selection
Many factors, including size, cost, dynamic range, sampling rate and bias, have to be considered in choosing an appropriate sensor module. Firstly, the size of the sensor module should be small enough to be installed unobtrusively on footwear. Consequently, all 3D gyroscopes, 3D accelerometers and 3D magnetometers should be integrated into, actually orthogonally mounted in, this inertial/magnetic sensor module. Secondly, the price of this sensor module should be low enough to meet budget constraints for the targeted applications. Thirdly, the dynamic range of the sensor module should meet the requirements of the applications. Fourthly, the sampling rate should be high enough to avoid sampling errors. The sampling rate of the sensors should be at least 100 Hz. Finally, because of the sensitivity of sensors to temperature, each sensor in a module should be individually compensated for bias, which is a function of temperature.
With all these considerations an inertial measurement unit (IMU) was selected to build the prototype for testing. The IMU provides serial digital outputs of 3D acceleration, 3D angular velocity, and 3D magnetic field.

Sensor module mounting
In the experiments, the sensor module is mounted on the arch, i.e., the upper of closed footwear, as shown in Fig 7. The installation of the sensor should ensure that the relative movement between the sensor module and the foot-ware is minimized.

Field experiment
Using the methods introduced in Section 4 and the sensor module selected in Section 5.1, several experiments have been conducted at different paces in different places and different times. Most of these experiments are two-dimensional (2D) tests.

Sensor fusion
The use of higher quality inertial/magnetic sensor modules will improve positioning accuracy but the cost of such a solution is often much higher. An alternative solution is to use various positioning data from multiple sources such as radio navigation aids and additional sensors onboard. Through the control applied to the drift error, a positioning system using multiple data sources "fused" together is able to outperform a system using a single data source. Complementary filtering techniques were used to integrate data from different positioning aids in the past. Nowadays, Kalman filtering is popular in combination of two estimates of a variable. For example, IPS data can be integrated with GPS data to provide enforced positioning ability in both indoor and outdoor environment.

Kalman filter
Although dynamical behavior of a linear system is often expressed in the form of continuous differential equations, in practice, the measurements are provided at discrete intervals of time. The form of difference equations are used as below: where the n vector, x k , is called the state of the system at time k, u k-1 is a p vector of deterministic inputs and w k-1 is the system noise at time k-1. F k-1 is an n×n state transition matrix from time k-1 to time k, known as the system matrix, and B k-1 is an n×p system input matrix. F k-1 and B k-1 are constant or time varying matrices. The system (or process) noise w k is normally distributed with zero mean and a power spectral density of Q k-1 .
The vector x k contains all the information regarding the present state of the system, but cannot be measured directly. Instead, only the system's m vector, z k , is available. This measurement vector is a linear combination of the state, x k , that is corrupted by the measurement noise v k . This can be expressed in terms of the system state by the following equation: where H k is an m×n measurement matrix. The measurement (or observation) noise, v k , has zero-mean and is normally distributed, with power spectral density R k .
The Kalman filter for the system described here seeks to provide the best estimates of the states, x k , using the measurements, z k , model of the system provided by the matrices F k , B k and H k , and knowledge of the system and measurement statistics given in the matrices Q k and R k .
From (23) and (24), we can recognize that Kalman filter is a recursive filter: only the state of last time step and the current measurement are used. The estimation is conceptualized as two distinct phases, "Predict" and "Update", described by two distinct set of equations. In the first phase, the prediction is based on previous best estimate. The result is called a priori state estimate. In the second phase, the updating is done based on the a priori state estimate with new measurement. The result is a posteriori state estimate. The two set of distinct equations are as below:

The prediction process
Predicted (a priori) state estimate is the best estimate of the state at time k-1, denoted as x k-1/k-1 . Since the process noise, w k-1 , is normally distributed with zero-mean, the best prediction of is given by while the predicted (a priori) estimate covariance at time k predicted at time k-1, is given by The measurement update On the arrival of new measurement z k , at time k, it is compared with the a priori state estimate of the measurement. The measurement is then used to update the prediction to generate a best estimate, i.e., the a posteriori state estimate at time k-1: The relevant covariance is The suboptimal Kalman gain is The state estimate is updated as The relevant estimate covariance is also updated as   // 1 kk k k kk   PI K H P (40)

Application examples
Kalman filter, sometimes with other techniques, have already been used for data fusion and positioning. Recently, Zhao & Wang (2011) use EKF for the fusion of data from inertial sensors, ultrasonic sensors, and magnetic sensor. A 3D magnetic sensor and a 3D accelerometer are combined to measure the gravity and the earth's magnetic field for the static orientation. A 3D gyroscope is used to obtain the dynamic orientation. The integral drift of accelerometer is periodically calibrated by an ultrasonic sensor. All these data are fused by EKF. The measurements are the position obtained by ultrasonic sensor and the orientation obtained by magnetic sensors and accelerometers. The states incorporated the position, velocity, and orientation. The experimental results demonstrated that the uncertainty of orientation and position is lowered.
In another research, Shen et al (2011) apply Kalman filter, together with Fast Orthogonal Search (FOS) in the 2D navigation for land vehicles. Kalman filter is used for data fusion of GPS and inertial sensors, i.e. the integration of GPS, IMU and vehicle built-in odometer. FOS is a nonlinear error identification technique. It is used to model non-stationary stochastic sensor errors and non-linear inertial errors in the study. The road test trajectories confirm that a module with both Kalman filtering and FOS technique outperforms that with Kalman filtering only.
It can be concluded that improvement of positioning accuracy will be possible if data from multiple sensors, including magnetic sensors, are fused by Kalman filter. Further improvement is achievable if Kalman filtering is combined with other techniques.

Summary
In this chapter, we firstly defined local positioning systems and introduced their applications. Then we presented strap-down inertial positioning techniques covering not only the concepts of dead reckoning and inertial position, but also the algorithms, especially for the rotation of reference frames and the tracking of the orientation of a moving object. We also evaluated the performances of inertial sensors, i.e., gyroscopes and accelerometers and pointed out that the increasing integral drift over time is the main problem affecting the accuracy of positioning. As an example among the efforts to solve this problem, we proposed an approach to integrate pedestrian dead reckoning with inertial positioning in precise tracking, where the characteristics of human gaits are utilized. Magnetic sensors have to be incorporated to update the orientation of moving objects and the relevant algorithm is provided. The experimental study was also included to demonstrate the implementation and the result of positioning. Finally, among existing solutions to enhance the accuracy of local positioning, Kalman filter, EKF and their algorithms are introduced and the examples of their applications are also presented. Magnetic sensors are important resources of data to be fused through Kalman filtering.