1. Background: Coordinate System Transformations

2. i j Derivation of the 2D Rotation Matrix (basis vectors) θ

3. i j Coordinate Transformations (2D) θ Find in global space:

4. i j Coordinate Transformations (2D) θ Find in global space:

5. Coordinate Transformations Note: The rows of the 2x2 rotation matrix from global to local space are composed of the unit vectors of the local coordinate system axis (in global space).

6. i j Derivation of the 3D Rotation Matrix (basis vectors) θ k K’

7. Z Rotation from Global to Local Note: The rows of the 3x3 rotation matrix from global to local space are composed of the unit vectors of the local coordinate system axis (in global space).

8. Translations of Bodies in 3D Y X Z

9. Y X Z X’ z’ y’ P P’ Coordinate System Transformations (Global to Local example)

10. Translations and Rotation (Example: Global to Local) Y X Z X Y Z For: and Ox=8, Oy=8, Oz=0 Find P in Local Space (P ’ )? O P P’P’

11. Coordinate System Transformations (Example: Global to Local) Y X Z X Y Z Transform global to local: P’P’

12. Rotation Matrices are orthonormal The rotation from Global to Local Space is the inverse of the rotation from Local to Global Space Rotation Matrices are orthonormal and thus the inverse is equal to the transpose

13. Coordinate System Transformations (Local to Global) Y X Z X’ z’ y’ P P’ R ’ is the rotational transformation from local to global space

14. Y X Z X Y Z For: and Ox=8, Oy=8, Oz=0 Find P in Global Space O P P’P’ Coordinate System Transformations (Example: Local to Global)

15. Y X Z X Y Z Transform local to global: Coordinate System Transformations (Example: Local to Global)

16. Coordinate System Transformations Global to Local: Local to Global: Where the rows of the rotation matrix from global to local space (R) are composed of the unit vectors of the local coordinate system axis (in global space).

17. Non-Optimal Pose Estimation

18. Pelvis

19. Anatomical Coordinate Systems (pelvis) How to calculate the anatomical coordinate systems 3) Create M/L Unit Vector i from step 2 unit vector = (i x, i y, i z ) 4) Find the vector from the Sacrum to Origin 1) Find midpt between LASIS & RASIS = Origin 5) Create Unit Vector v from step 4 unit vector = (v x, v, v z ) 6) Use i X v to get Inferior/Superior Unit Vector unit vector = (k x, k y, k z ) 7)Use k X i to get Anterior/Posterior Unit Vector unit vector = (j x, j y, j z ) X Z Y 2) Find the vector from Origin to right ASIS

20. Finding the Hip Center (in Pelvic Coordinate System) X Z Y Right Hip (Bell, Brand and Pederson): X = 0.36 * ASIS Distance Y = -0.19 * ASIS Distance Z = -0.30 * ASIS Distance

21. Pelvis Non-Optimal Calibration: Finding the local coordinates of the hip center (which are needed to calibrate the other segments) Anthropometrics, Graphics Scaling and Graphing the Calibration Trial

22. Pelvis: code is already part of GaitProject

23. Thigh

24. Anatomical Coordinate Systems (thigh) How to calculate the anatomical coordinate systems 2) Find the Knee center (Midpt of knee targets) 3) Find the vector from Knee to Hip 7) Find the the A/P axis, j, from k X v unit vector = (j x, j y, j z ) 8) Find the M/L axis, i, via j X k unit vector = (i x, i y, i z ) 1) Find the Hip center in Lab from Pelvis (tricky) 4) Find the Inferior/Superior Axis, k, from Unit vector of step 3 unit vector = (k x, k y, k z ) 5) Find the vector from medial to lateral Knee 6) Find the vector v which is unit vector of step 5 unit vector = (v x, v y, v z ) v Z X Y

25. Problem: Calculating the anatomical coordinate system requires a target (medial knee) not used during the walking trial

26. Solution: Create a virtual medial knee target which can be located during the motion trials

27. Step 1: Creating the temporary local coordinate system 1)Select one Point (Lateral Knee) as Origin 2) Find the unit vector from origin to the Hip unit vector = (k t x, k t y, k t z ) 3) Use the lateral knee and 3rd target to find a second vector v 6) Find the third axis via j X k unit vector = (i t x, i t y, i t z ) 5) Use the results of step 4 to find “A/P” unit vector j unit vector = (j t x, j t y, j t z ) 4) Find the the “A/P” axis from k X v

28. Step 2: Storing the virtual lateral knee target 1)Start with the tracking based local coordinate system (R temp ) P 2) Transform P into the tracking coordinate system (P’) using ‘ Where P are the global coordinates of lat knee and O are the global coordinates of the medial knee

29. Thigh Non-Optimal Calibration: Finding the local coordinates of the medial knee target in the calibration trial Anthropometrics, Graphics Scaling and Graphing the Calibration Trial

30. 1)Create the tracking based local coordinate system (R temp ) 2) Recall the stored vector (P’) from origin to the calibration target (in tracking coordinate system) P 3) Transform P’ into the global coordinate system (P) using: ‘ 4) Use the tracking targets (including virtual) to find anatomically based coordinate system Step 3: Find the anatomical system during movement

31. Thigh: code is already part of GaitProject

32. Shank

33. Anatomical Coordinate Systems (shank) How to calculate the anatomical coordinate systems 2) Find the Ankle center 3) Find the vector from Ankle to Knee 6) Find the the “A/P axis from k X v 8) Find the M/L axis via j X k unit vector = (i x, i y, i z ) 1) Find the Knee center in Lab from Thigh (tricky) 4) Find the Inferior/Superior Axis from Unit vector of step 3 unit vector = (k x, k y, k z ) 5) Find the vector v from medial to lateral Ankle 7) Find the Unit A/P vector j unit vector = (j x, j y, j z ) v

34. Problem: Calculating the anatomical coordinate system requires a target (medial ankle) not used during the walking trial

35. Solution: Create a virtual medial ankle target which can be located during the motion trials

36. Step 1: Creating the temporary local coordinate system 1)Select one Point (Lateral Ankle) as Origin 2) Find the unit vector from origin to the Knee unit vector = (k t x, k t y, k t z ) 3) Use the lateral ankle and 3rd target to find a second vector v 6) Find the third axis, i, via j X k unit vector = (i t x, i t y, i t z ) 4) Use the results of step 3 to find a second unit vector unit vector = (v x, v y, v z ) 5) Find the the “A/P” axis from k X v unit vector = (j t x, j t y, j t z )

37. Step 2: Storing the virtual lateral ankle target 1)Start with the tracking based local coordinate system (R temp ) P 2) Transform P into the tracking coordinate system (P’) using ‘ Where P are the global coordinates of lat ankle and O are the global coordinates of the medial ankle

38. Shank Non-Optimal Calibration: Finding the local coordinates of the medial ankle target in the calibration trial Anthropometrics, Graphics Scaling and Graphing the Calibration Trial

39. 1)Create the tracking based local coordinate system (R temp ) 2) Recall the stored vector (P’) from origin to the calibration target (in tracking coordinate system) P 3) Transform P’ into the global coordinate system (P) using: ‘ 4) Use the tracking targets (including virtual) to find anatomically based coordinate system Step 3: Find the anatomical system during movement

40. Foot

41. Anatomical Coordinate Systems (foot) How to calculate the anatomical coordinate systems 3) Find Z Axis from Unit vector of step 1 unit vector = (k x, k y, k z ) 4) Find the vector from Origin to Heel v 7) Find the the Y axis, j, from k X i unit vector = (k x, k y, k z ) 6) Find the unit vector from step 2 unit vector = (i x, i y, i z ) 2) Find the Vector from Toe to Origin 5) Find the M/L axis via k X v Y Z X 1)Find midpoint of ankle targets = Origin (from Shank - tricky)

42. Foot Non-Optimal Calibration: Anthropometrics, Graphics Scaling and Graphing the Calibration Trial

43. Assignment #5 – Build Non Optimal Model in Gait Project for the right thigh, right shank and right foot Let’s Look at the Pelvis and Thigh Code right now