EE 570 Location and Navigation Theory Practice Navigation
- Slides: 15
EE 570: Location and Navigation: Theory & Practice Navigation Mathematics Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 1 of 15
Navigation Mathematics : Coordinate Frame Transformations • Determine the detailed kinematic relationships between the 4 major frames of interest § The Earth-Centered Inertial (ECI) Coordinate Frame (iframe) § The Earth-Centered Earth-Fixed (ECEF) Coordinate Frame (e -frame) § The Local Navigation (Nav) Coordinate Frame (n-frame) § The Body Coordinate Frame (b-frame) Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 2 of 15
Navigation Mathematics : Coordinate Frame Transformations - ECI/ECEF • Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 3 of 15
Navigation Mathematics : Coordinate Frame Transformations - ECI/ECEF • The angular velocity and acceleration are: • The angle of rotation is GMST: Greenwich Mean Sidereal time • The orientation of frame {e} wrt frame {i} becomes Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 4 of 15
Navigation Mathematics : Coordinate Frame Transformations - ECEF/Nav • Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 5 of 15
Navigation Mathematics : Coordinate Frame Transformations - ECEF/Nav § Angular velocity of the n-frame wrt the e-frame resolved in the e-frame as a Skew-Symmetric matrix § The angular velocity vector Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 6 of 15
Navigation Mathematics : Coordinate Frame Transformations - ECI/ECEF/Nav § Hence the orientation of the n-frame wrt the i-frame becomes Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 7 of 15
Navigation Mathematics : Coordinate Frame Transformations - ECI/ECEF/Nav § The angular velocity of the n-frame wrt the i-frame resolved in the i-frame is: Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 8 of 15
Navigation Mathematics : Coordinate Frame Transformations - ECEF/Nav • The vector from origin of the e-frame to the n-frame org resolved in the e-frame (from the last lecture) Origins of the {n} and {b} frames are the same • The velocity of the n-frame wrt the e-frame resolved in the e-frame (see handout#1 for proof) Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 9 of 15
Navigation Mathematics : Coordinate Frame Transformations - ECEF/Nav • Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 10 of 15
Navigation Mathematics : Coordinate Frame Transformations - ECEF/Nav • Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 11 of 15
Navigation Mathematics : Coordinate Frame Transformations – Body Frame • Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 12 of 15
Navigation Mathematics : Coordinate Frame Transformations – Body Frame • The angular velocity of the b-frame wrt the i-frame resolved/coordinatized in the i-frame Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 13 of 15
Navigation Mathematics : Coordinate Frame Transformations – Body Frame • Position vectors to the origin of the body frame § The origins of the body and Nav frames are co-incident § The origins of the ECI and ECEF frames are co-incident § Velocity of the b-frame wrt the i-frame resolved in the iframe o Case #2: A moving point in a rotating frame Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 14 of 15
Navigation Mathematics : Coordinate Frame Transformations – Body Frame § Acceleration of the b-frame wrt the i-frame resolved in the i-frame o Case #2: A moving point in a rotating frame Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 15 of 15
