EE 570 Location and Navigation Theory Practice Navigation

EE 570: Location and Navigation: Theory & Practice Navigation Mathematics Tuesday 5 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 1 of 12

Navigation Mathematics : Earth surface and Gravity - Earth Modeling • The Earth can be modeled as an oblate spheroid § A circular cross section when view from the polar axis (top view) § An elliptical cross-section when viewed perpendicular to the polar axis (side view) Ratio exaggerated • This ellipsoid (i. e. oblate spheroid) is Max variation betw. ellipsoid and geoid is +3 to -51 meters. an approximation to the “geoid” www. nrcan. gc. ca • The geoid is a gravitational equipotential surface which “best” fits (in a least square sense) the mean sea level Tuesday 5 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 2 of 12

Navigation Mathematics : Earth surface and Gravity - Earth Modeling • Equator Tuesday 5 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 3 of 12

Navigation Mathematics : Earth surface and Gravity - Earth Modeling • We can define a position “near” the Earth’s surface in terms of latitude, longitude, and height § Geocentric latitude intersects the center of mass of the Earth § Geodetic latitude (L) is the angle between the normal to the ellipsoid and the equatorial plane Reference Ellipsoid Equatorial Plane Tuesday 5 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Surface Normal G La eod titu etic de c tri n e oc de e G titu La Slide 4 of 12

Navigation Mathematics : Earth surface and Gravity - Earth Modeling Equatorial Plane Tuesday 5 Feb 2013 G La eod titu etic de Ge o he det igh ic t • Reference Ellipsoid NMT EE 570: Location and Navigation: Theory & Practice Slide 5 of 12

Navigation Mathematics : Earth surface and Gravity - Earth Modeling • Transverse radius of curvature ze § The radius of curvature for east-west motion h b • The meridian radius of curvature Lb RE ye Xe/Ye plane Disk of constant Latitude (Lb) xe RE = Transverse radius of curvature Tuesday 29 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 6 of 12

Navigation Mathematics : Earth surface and Gravity - Earth Modeling E +h b )Co s(L b) ye (R ze (RE+hb)Cos(Lb) (RE(1 -e 2)+hb)Sin(Lb) • Curvilinear to ECEF coordinates 2 -e R ) h b (1 E Lb RE Xe/Ye plane (RE+hb)Cos(Lb)Sin( b) xe (RE+hb)Cos(Lb)Cos( b) Disk of constant Latitude (Lb) RE = Transverse radius of curvature Tuesday 29 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 7 of 12

Navigation Mathematics : Earth surface and Gravity - Gravity Models • Tuesday 5 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 8 of 12

Navigation Mathematics : Earth surface and Gravity - Gravity Models • Relationship between specific force, inertial acceleration, and gravitational attraction • When stationary on the surface of the Earth § Recall case 1: A fixed point in a rotating frame o Considering frame {0} to be the {i} frame, {1} = {e}, and {2} ={b} gives o Coordinatizing in the e-frame gives Tuesday 5 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 9 of 12

Navigation Mathematics : Earth surface and Gravity - Gravity Models • Thus, when stationary on the surface of the Earth the acceleration is due to Centrifugal force • Therefore, the acceleration due to gravity is Tuesday 5 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 10 of 12

Navigation Mathematics : Earth surface and Gravity - Gravity Models • Tuesday 5 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 11 of 12

Navigation Mathematics : Earth surface and Gravity - Gravity Models • On March 17, 2002 NASA launched the Gravity Recovery and Climate Experiment (GRACE) which led to the development of some of the most precise Earth gravity models NASA's Grace Gravity Model Tuesday 5 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 12 of 12