1. EE 495 Modern Navigation Systems Navigation Mathematics Rotation Matrices – Part II Wednesday, Jan 14 EE 495 Modern Navigation Systems Slide 1 of 21

2. Navigation Mathematics: Kinematics – Rotation Matrices Wednesday, Jan 14 EE 495 Modern Navigation Systems 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 Slide 2 of 16

3. Navigation Mathematics : Kinematics – Rotation Matrices Wednesday, Jan 14 EE 495 Modern Navigation Systems Slide 3 of 16

4. Navigation Mathematics : Kinematics – Rotation Matrices Wednesday, Jan 14 EE 495 Modern Navigation Systems We can derive the same result by noting that Slide 4 of 16

5. Navigation Mathematics : Kinematics – Rotation Matrices Wednesday, Jan 14 EE 495 Modern Navigation Systems Similarly, it can be shown that Slide 5 of 16

6. Navigation Mathematics : Kinematics – Rotation Matrices Wednesday, Jan 14 EE 495 Modern Navigation Systems In general, Rotations do NOT commute  Consider Slide 6 of 16

7. Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations Wednesday, Jan 14 EE 495 Modern Navigation Systems Fixed Axis Rotation  Conduct rotations about the original (i.e. fixed) x, y, or z axes Relative axis rotation  Conduct rotations about the current location (i.e. relative) of the x, y, or z axes o Sometimes referred to as Euler rotations Let’s compare the two:  Step 1: Rotate about the z-axis, then  Step 2: Rotate about the y-axis, then  Step 3: Rotate about the x-axis. Slide 7 of 16

8. Relative Axis RotationFixed axis rotation Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations Wednesday, Jan 14 EE 495 Modern Navigation Systems Rotate frame {1} about z 0 /z 1 by yaw (  ) Initially frames {0} and {1} are both identically aligned     Slide 8 of 9

9. Fixed axis rotation Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations Wednesday, Jan 14 EE 495 Modern Navigation Systems 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    Relative Axis RotationFixed axis rotation Slide 9 of 9

10. Relative Axis RotationFixed axis rotation Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations Wednesday, Jan 14 EE 495 Modern Navigation Systems Rotate frame {3} about x 2 by roll (  ) Rotate frame {3} about x 0 by roll (  ) Initially frames {2} and {3} are both identically aligned   Slide 10 of 9

11. Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations Wednesday, Jan 14 EE 495 Modern Navigation Systems Slide 11 of 16

12. Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations Wednesday, Jan 14 EE 495 Modern Navigation Systems Slide 12 of 16

13. Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations Wednesday, Jan 14 EE 495 Modern Navigation Systems For the case of the Relative axis Yaw, Pitch, Roll rotation sequence: This is the same result as Eqn. (2.24) pp. 38 in the textbook 123 Slide 13 of 16

14. Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations Wednesday, Jan 14 EE 495 Modern Navigation Systems 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 Slide 14 of 16

15. Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations Wednesday, Jan 14 EE 495 Modern Navigation Systems For the case of the Fixed axis Yaw, Pitch, Roll rotation sequence: Recall that, in general, rotations do not commute 1 2 3 Slide 15 of 16

16. Navigation Mathematics : Kinematics – Fixed vs Relative axis rotations Fixed Axis Rotations  Multiply on the LEFT!!!  R Final = R n … R 2 R 1 Relative (i.e. Euler type) Axis Rotations  Multiply on the RIGHT!!!  R Final = R 1 R 2 … R n You can mix the two types of rotations!! Wednesday, Jan 14 EE 495 Modern Navigation Systems Slide 16 of 16