EE 570 Location and Navigation Theory Practice Navigation

EE 570: Location and Navigation: Theory & Practice Navigation Mathematics Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 1 of 20

Navigation Mathematics : Kinematics – Rotation Matrices • The rotation matrix (C) is a 3× 3 matrix requiring 9 parameters to describe an orientation. • It can be shown that any orientation can be uniquely described using only 3 -parameters. • Some examples of 3 -parameter descriptions include: § Fixed axis rotations, § Relative (or Euler) axis rotations, and § Angle-axis rotations Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 2 of 20

Navigation Mathematics : Kinematics – Rotation Matrices • Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 3 of 20

Navigation Mathematics : Kinematics – Rotation Matrices • Alternatively, we can derive the same result by noting that Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 4 of 20

Navigation Mathematics : Kinematics – Rotation Matrices • Recalling that • Similarly, it can be shown that Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 5 of 20

Navigation Mathematics : Kinematics – Rotation Matrices • In general, Rotations do NOT commute § Consider Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 6 of 20

Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations Relative Axis Rotation Rotate frame {1} about z 0/z 1 by yaw ( ) Fixed axis rotation Rotate frame {1} about z 0/z 1 by yaw ( ) Initially frames {0} and {1} are both identically aligned Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 7 of 20

Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations Relative Axis Rotation Fixed axis rotation Rotate frame {2} about y 1 by pitch ( ) Rotate frame {2} about y 0 by pitch ( ) Initially frames {1} and {2} are both identically aligned Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 8 of 20

Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations Relative Axis Rotation Fixed axis rotation Rotate frame {3} about x 0 by roll ( ) Rotate frame {3} about x 2 by roll ( ) Initially frames {2} and {3} are both identically aligned Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 9 of 20

Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 10 of 20

Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations • A Rotation matrix can be interpreted in two distinct ways: 1. IT DESCRIBES THE ORIENTATION OF ONE COORDINATE FRAME WRT ANOTHER COORDINATE FRAME. Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 11 of 20

Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations • For the case of the Relative axis Yaw, Pitch, Roll rotation sequence: This is the same result as Eqn. (2. 15) pp. 28 in the textbook Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 12 of 20

Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations • For the case of the Fixed axis rotations it is intuitive to think in terms of interpretation number 3. for rotational matrices. o The first “yaw” rotation “rotates” the frame {0} basis vectors to become the frame {1} basis vectors o The second “pitch” rotation “rotates” the frame {1} basis vectors to become the frame {2} basis vectors o The third “roll” rotation “rotates” the frame {2} basis vectors to become the frame {3} basis vectors Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 13 of 20

Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations • For the case of the Fixed axis Yaw, Pitch, Roll rotation sequence: Recall that, in general, rotations do not commute Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 14 of 20

Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations • Fixed Axis Rotations § Multiply on the LEFT!!! § RFinal = Rn … R 2 R 1 You can mix the • Relative (i. e. Euler type) Axis Rotations two types of rotations!! § Multiply on the RIGHT!!! § RFinal = R 1 R 2 … Rn Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 15 of 20

Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations: An Example • Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 16 of 20

Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations: An Example • Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 17 of 20

Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations: An Example • A gyro triad example Side View 45 Top View 30 Frame {3} wrt {0}? Frame {1} wrt {0}? Frame {2} wrt {0}? Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 18 of 20

Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations: An Example • gyro_triad_example. m Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 19 of 20

Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations: An Example • Tuesday 22 Jan 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 20 of 20