Coordinate systems

For a meaningful output, positioning and navigation results are ex- pressed based on a common reference system, which defines the origin and the orientation of the axes of the system, as well as the mathematical and physical models. The reference frame is the realisation of a reference system through observations and measurements. Orthogonal reference systems are most commonly seen in positioning which has six degrees of freedom, including the position of the origin o, the orientation of the axes x, y and z. Reference systems commonly apply the orthogonal right-handed

convention, where the three axes are always oriented in such a way that when the thumb and first two fingers of the right hand are extended per- pendicularly, the thumb is the x-axis, the index finger is the y-axis and the middle finger is the z-axis.

Two fundamental reference systems are commonly applied in naviga- tion problems and are specified here, i.e. the space-fixed celestial reference system and the Earth-centred Earth-fixed (ECEF) terrestrial reference sys- tem. The celestial reference system represents an approximation to an inertial system which describes the motion of the Earth and other bodies in space. It is not strictly an inertial system because it is affected by the annual revolution. We will only introduce the ECEF reference system here as it rotates with the Earth and is commonly used to describe motions on the Earth, as shown in Figure 2.1. This is a three dimensional geocentric coordinate system which is realised by the International Terrestrial Refer- ence Frame (ITRF) that is maintained by the International Earth Rotation Service (IERS) (Seeber, 1993; Torge and Muller, 2012). The system ori- entation changes with respect to Earth’s solid body as well as time. The

2.1. Indoor positioning

system origin is in the Earth’s centre of mass, Z-axis directed towards a conventional mean North pole, X- and Y-axes lies on the mean equatorial plane that is perpendicular to Z-axis. The XZ-plane is generated by the mean meridian plane of Greenwich. Y-axis is directed so the system is a right-handed system.

Figure 2.1: Earth-fixed terrestrial system (Source: Torge and Muller (2012))

To describe positions and locate geographical features in a reference system, coordinate reference systems (CRS) are defined that is coordinate- based regional or global systems which defines a specific map projection and the transformation between different reference systems. ECEF coordin- ates may be expressed by Cartesian coordinates (X, Y, Z) or ellipsoidal coordinates (ϕ, λ, r), which represent points in a three-dimensional space. The relationship between the two coordinates is as shown in Figure 2.2. ϕ and λ are the latitude and longitude from the ellipsoid and r is the ellipsoidal height. A note here is that as the Earth is an ellipsoid in reality, thus the centre of the ellipsoidal coordinates will not lie on the origin of the Cartesian coordinates.

Figure 2.2: Cartesian and ellipsoidal coordinates (Source: Torge and Muller (2012))

Different reference frames are implemented for different positioning and mapping purposes. The World Geodetic System 1984 (WGS 84) is a geocentric terrestrial reference system used for GPS that was developed by the U.S. Department of Defence. It is globally consistent and consists of a standard coordinate system for the Earth, a standard spheroidal reference surface for altitude, and the geoid which defines the nominal sea level. GPS related position data are defined in the WGS 84 reference frame. The refined WGS 84 frame introduced in 2002 agrees with ITRF2000 at centimetre level. Local reference frames refers to a coordinate system that defines a consistent reference over a small region within the global coordinate system. The Ordnance Survey national grid reference system is a geographic grid reference used in Great Britain. The grid is based on the OSGB36 datum which is a coordinate system and set of reference points that is the regional best fit for Great Britain.

2.1.1.1

Inertial coordinate frames

To describe a navigation problem, at least two reference frames are usually applied: an object frame that describes the motion of the moving body and a reference frame that describes a known body relative to the moving body. To integrate positioning results from different systems, results must be expressed in the same reference frame and coordinates. Several common reference frames are listed here (Rogers, 2007).

• Earth-Centred Inertial frame (i-frame): the i-frame is a space fixed reference frame, centred at the Earth’s centre of mass and axes are

2.1. Indoor positioning

non-rotating with respect to the fixed stars, defined by the axes Oxi,Oyi, Ozi, with Ozicoincident with the Earth’s polar axis. x− and

y−axes lie within the equatorial plane, but do not rote with the Earth. • Earth-Centred Earth-fixed frame (e-frame): origin at the centre of

the Earth and axes are fixed with respect to the Earth, defined by Oxe, Oye, Oze, with Ozealong the Earth’s polar axis, Oxepoints from

the centre to the intersection of the plane of the Greenwich meridian with the Earth’s equatorial plane. The e-frame rotates with respect to the i-frame following the Earth’s rotation Ω about the axis Ozi, axes

are shown in brown in Figure 2.3.

• Local Navigation frame (n-frame): the n-frame’s origin is located at the navigation solution point P, i.e. navigation system or user etc. The down(D) axis is the local vertical which follows the ellipsoid normal pointing towards the Earth. The north(N) axis is the projection in the plane orthogonal to the D-axis of the line from P to the north pole. East(E) axis completes the orthogonal set by pointing East. The n-frame might rotate with respect to the Earth-fixed frame at a rate of ωen, which is governed by the motion of P with respect

to the Earth. This frame is important as it is useful in defining the users’ attitude. Another common set of axes used in this frame is east-north-up (ENU). The relationship of n-frame to e-frame is shown in Figure 2.3.

Figure 2.3: Relationship between the local navigation frame and body frame

igation frame and the orthogonal axis set remains fixed with respect to the body of the system. It is used to define the relative attitude of the object with respect to the location navigation frame. The x−axis commonly points towards the direction of travel, z-axis aligns with the direction of gravity and y-axis completes the orthogonal set. To describe angular motions of the body, the axes are also known as roll, pitch and yaw. The axes and its relationship to the local navigation frame is shown in Figure 2.3.

Usually, different systems will give results in different reference frames. For example, GNSS positioning results are expressed in the ECEF (WGS 84) frame by longitude λ, latitude ϕ and altitude h. IMU measurements are normally expressed with respect to the body frame. Terrestrial positioning and navigation results are usually given in the local navigation frame by ENU coordinates. To compare or integrate measurements from different systems requires the results to be converted to the same reference frame first. The positioning results throughout the work in this thesis will be given in a local reference frame of the experimental environment, refer- enced to the Ordnance Survey National Grid, expressed in ENU Cartesian coordinates.