Low cost inertial navigation

With the advancement in navigation technology in the last century, many systems that were either originally designed or not designed for positioning may now be applied for everyday civilian navigation and po- sitioning. Inertial sensors, including both the gyroscope (or gyro) and the accelerometer, were used as guidance systems in rockets and aircrafts around the 1950s. Over the years, various high-grade and low-grade INS emerged for applications in a wide area of navigation for aircraft, vehicles and pedestrian navigation. The cost of inertial systems also vary greatly between high-grade and low-grade systems, as they are targeted at differ- ent users and give very different performance. Indoor positioning is mostly targeted at everyday civilian usage, therefore keeping the cost down has always been a big issue. For such applications, only low-grade inertial sensors can be applied.

Inertial navigation was originally applied by mounting inertial sensors onto a stable platform that is independent to the motion of the vehicle. This is still used in some systems where high accuracy is required. However

2.1. Indoor positioning

many systems have removed this complexity by attaching the inertial sensor on to the body of the moving object, so called the “strap down” system. This reduces cost, size and enhance reliability compared to equivalent plat- form systems (Titterton and Weston, 2004). Low-cost pedestrian inertial navigation systems are generally strapdown systems where we attach low cost sensors onto the user’s foot or waist, or any other body parts that can capture the motion of the pedestrian motion, such as attaching sensors onto the users’ knees (Rantakokko et al., 2014). This thesis will only discuss inertial measurements obtained from foot-trackers, i.e. a low-cost IMU sensor that has been attached to the pedestrian’s foot. The inertial measurements indicate the movement of the person’s foot and predict steps from such measurements.

(a) LN3-2A gimballed inertial platform (de- veloped by Litton Industries first equipped on Lockheed F-104 Starfighter)

(b) Optolink’s strapdown inertial system SINS-501

Figure 2.4: Inertial navigation systems

The classical laws of mechanics tell us that the motion of a moving body will continue to move uniformly in a straight line unless disturbed by an external force, which produces a proportional acceleration on the body. As a result, the change in velocity and position of the body could be worked out if the acceleration is known. Based on this concept, IMU measure the acceleration of the moving body using accelerometers and

gyros to navigate objects with respect to a local reference frame (Titterton and Weston, 2004). A typical INS consists of an IMU and a navigation processor to form a dead reckoning navigation system.

Figure 2.5: INS process

IMU will be introduced in more detail as it is the main measurement component in the system. A typical IMU is combined of three acceleromet- ers and three gyros to provide 3-dimensional navigation measurements. Each accelerometer measures the force and detects acceleration in a single direction, while gyros detect the rotation of the body and determine the changes in the orientation of the accelerometers. The working process is as shown in Figure 2.5. The measurements of the sensors define the trans- lational motion and rotational motion of the moving body at each epoch, which is then used to work out its current position relative to its previous position. Navigation solutions can be solved in any of the reference frames. Calculations below show how inertial measurements in the ECEF frame (Groves, 2013b), denoted by e, from time t − τ to t are used to update the attitude and positions with respect to the local navigation frame (n−frame) denoted by n, ˙vn e = f n − (2ωn ie+ ω n en) × v n e + g n l (2.1)

where the superscripts of the vector denote the axis set in which the coordinates are expressed and the subscripts denote the the frame it is ex- pressed with respect to. gn

l is the local gravity vector in n−frame. v n e is the

velocity with respect to the Earth expressed in n−frame, with components v_en=h νN νE νD

iT

(2.2)

2.1. Indoor positioning

n−frame. ωn

ie is the turn rate of the Earth expressed in n−frame and

ωn

enis the turn rate of n−frame with respect to the Earth-fixed frame, i.e.

the turn rate of the navigation system, which may be expressed by the rate of change of longitude and latitude,

ωn en=

h

˙ℓ cos L − ˙L − ˙ℓsin L iT (2.3) where L is the latitude, ℓ is the longitude. If the Earth is assumed to be perfectly spherical, the position of system in latitude, longitude and height is given by, ˙L = νN R0+ h (2.4) ˙ℓ = νEsec L R0+ h (2.5) ˙h = −νD (2.6)

where R0 is the radius of the Earth and h is the ellipsoidal height.

Eq.2.1 is known as the navigation equation because its first integral gives the velocity and the second integral gives the position of the system. Inertial navigation is commonly applied in the DR technique, which gives the user’s motion and position with respect to the environment from relative measurements in the body frame. Pedestrian dead reckoning (PDR) is a navigation solution to resolve pedestrian navigation in challenging environments usually using step detection. Motion measurements are generally obtained from IMU or just accelerometers.

The advantage of inertial navigation is that it is completely self-contained hence do not rely on signals from external systems once initialised. How- ever, such navigation errors are cumulative. Therefore, INS requires the correct knowledge of an initial position as well as periodic measurement corrections and aiding to prevent measurement error from accumulating. Due to the continuous demand for low cost and lightweight features in new sensors and systems, current low-cost inertial sensors looks into micro- electromechanical system (MEMS) sensors. MEMS has been adapted to making small mechanical structures using silicon or quartz, with properties such as small size, low weight, low power consumption, low cost and low maintenance, etc. Although the performance from MEMS inertial sensors is less stable than high-end INS, but its measurement error is reasonable as a low-cost sensor with approximately 1◦_{/h for gyros and}

Weston, 2004). It enables the mass production of inertial sensors to be implemented on less accuracy-demanding applications, such as mobile devices. However, the heading drift of low-cost inertial units can be so severe that it could accumulate up to hundreds of metres within a few seconds after initialisation. To compensate for this disadvantage, corrections must be implemented to provide accurate positioning results. While INS outputs a relative positioning result, it can be integrated with GNSS or some other sensor that provides absolute position solutions to enhance positioning accuracy (Grewal et al., 2013; Kempe, 2011).

2.1.2.1

IMU errors

Although IMU comes in different sizes and costs, from high-grade per- formance sensors that are used in military ships, spacecrafts and missiles to low-grade sensors that could be bought for $10, but this is not a perfect world and there will always be errors in measurements from all types of sensors, such as bias, scale factor, cross-coupling error or random noise. Despite the differences in hardware, all errors have some similar charac- teristics. Some main types of IMU sensor errors are explained and given below to illustrate a general idea of IMU performance.

System errors of any sensor generally consist of four types: a fixed bias, a temperature-dependent variation, a run-to-run variation and an in-run variation. The fixed component and temperature-dependent component can be calibrated and corrected in laboratory before put into actual util- isation. The run-to-run variation error remains the same throughout each run but varies between different runs. Therefore it should be calibrated each time the sensor is used. In-run variation error changes throughout each run and is very hard to observe. Usually, users hope to mitigate errors by calibrating the sensor before each run and also process the data by integrating with other sensors.

Bias is a constant error found in inertial sensors that is unaffected by the outside force or angular rate, which are also known as acceleration (or g) -independent bias, denoted as ba = (bax, bay, baz) and bg = (bgx, bgy, bgz)

respectively for accelerometers and gyros. Accelerometer biases are de- scribed in the unit of milli-g (mg), gyro biases are described by degree per hour (◦_{/hr) or degree per second for low-grade sensors. When describing a}

gyro bias, sometimes the term drift is used.

Scale factor errors relates the change in the output signal to a change in the input acceleration or rate, denoted as sa = (sax, say, saz) and sg =

2.1. Indoor positioning

Cross-coupling errors, also known as misalignment errors, are results of misalignment of the sensitive axes of the sensor to the orthogonal axes of the body frame. The accelerometer cross-coupling coefficient of β-axis specific force sensed by the α-axis accelerometer is denoted as ma,αβ, while

mg,αβ denotes the coefficient of β-axis angular rate sensed by the α-axis

gyro. Both scale factor and cross-coupling error are expressed as parts per million (ppm) or a percentage.

Random noise, also known as random walk, come from various sources such electric noise which varies in inverse proportion to the square root of the averaging time, denoted as wa = (wax, way, waz) and wg = (wgx, wgy, wgz)

for accelerometers and gyros respectively.

Some other errors such as scale factor nonlinearity, anisoinertia error, acceleration-dependent bias and anisoelastic bias can be found in MEMS sensors which are affected by the applied force or angular rate. However, these errors are very small compared to the ones mentioned above. There- fore they are not of great concern when working with low-cost MEMS IMU sensors, hence will not be discussed.

2.1.2.2

Corrections

Errors in INS solutions can be categorised into three types: initialisa- tion error, IMU measurement error and processing errors. Initialisation error can be reduced by integrating accurate sensors or providing external information during initialisation. Processing error is mainly due to the lim- itations of system iteration rate. This thesis will mainly focus on methods to reduce measurement errors, especially the heading bias which contributes to the position error cumulatively if uncorrected.

Zero Velocity Updates (ZUPT) has been used extensively in previous literature to correct the user’s velocity as well as restrict position errors and estimate the sensor bias when wearing the IMU is worn on the user’s foot (Foxlin, 2005; Godha and Lachapelle, 2008). The ZUPT is performed during the period when the foot is stationary on the ground. During this period, the velocity is assumed to be zero hence the force along the vertical direction should be approximately the negative gravity constant. Any measurements that does not agree with this can be assumed to be errors and thus corrected. Therefore, applying the ZUPT correction restricts the measurement error and improve navigation performance.

However, heading drifts cannot be completely eliminated even by ap- plying ZUPT. Heading drifts has to be eliminated by external measurement corrections or sensors. The Cardinal Heading Aided for Inertial Navigation

(CHAIN) was proposed to restrict heading drifts by estimating headings from the knowledge of building orientations (Abdulrahim et al., 2011). The idea is based on the assumption that most buildings and the rooms within them are constructed in rectangular shapes and building layout information must be available.

Inertial measurements have become widely available in mobile devices therefore have been widely applied for indoor positioning solutions. Iner- tial measurements from low cost IMUs will provide the basic user position propagation model in this thesis. A number of external sensors and meas- urements are applied to correct the inertial sensor errors and produce more reliable and accurate positioning results.