Background: Coordinate System Transformations

1. Background: Coordinate System Transformations

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

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

4. j j’ Coordinate Transformations (2D) i’ θ i 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. j Derivation of the 3D Rotation Matrix(basis vectors) j’ i’ i 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. Coordinate System Transformations (Global to Local example) P’ z’ P’ y’ X’ P Y X Z

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

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

12. 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 Rotation Matrices are orthonormal

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

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

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

16. Global to Local: Local to Global: Coordinate System Transformations 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. Z 1) Find midpt between LASIS & RASIS = Origin 2) Find the vector from Origin to right ASIS Y X 3) Create M/L Unit Vector i from step 2 unit vector = (ix, iy, iz) 4) Find the vector from the Sacrum to Origin Anatomical Coordinate Systems (pelvis) How to calculate the anatomical coordinate systems 5) Create Unit Vector v from step 4 unit vector = (vx, v, vz) 6) Use iX v to get Inferior/Superior Unit Vector unit vector = (kx, ky, kz) • Use k X i to get Anterior/Posterior • Unit Vector • unit vector = (jx, jy, jz)

20. Z Y X Finding the Hip Center (in Pelvic Coordinate System) 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. Z 1) Find the Hip center in Lab from Pelvis (tricky) 2) Find the Knee center (Midpt of knee targets) X 3) Find the vector from Knee to Hip Y 4) Find the Inferior/Superior Axis, k, from Unit vector of step 3 unit vector = (kx, ky, kz) 5) Find the vector from medial to lateral Knee Anatomical Coordinate Systems (thigh) How to calculate the anatomical coordinate systems 6) Find the vector v which is unit vector of step 5 unit vector = (vx, vy, vz) v 7) Find the the A/P axis, j, from k X v unit vector = (jx, jy, jz) 8) Find the M/L axis, i, via j X k unit vector = (ix, iy, iz)

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. Select one Point (Lateral Knee) as Origin 2) Find the unit vector from origin to the Hip unit vector = (ktx, kty, ktz) 3) Use the lateral knee and 3rd target to find a second vector v 4) Find the the “A/P” axis from k X v Step 1: Creating the temporary local coordinate system 5) Use the results of step 4 to find “A/P” unit vector j unit vector = (jtx, jty, jtz) 6) Find the third axis via j X k unit vector = (itx, ity, itz)

28. Start with the tracking based local • coordinate system (Rtemp) 2) Transform P into the tracking coordinate system (P’) using Step 2: Storing the virtual lateral knee target ‘ P 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. Create the tracking based local • coordinate system (Rtemp) 2) Recall the stored vector (P’) from origin to the calibration target (in tracking coordinate system) 3) Transform P’ into the global coordinate system (P) using: Step 3: Find the anatomical system during movement ‘ P 4) Use the tracking targets (including virtual) to find anatomically based coordinate system

31. Thigh: code is already part of GaitProject

32. Shank

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

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. Select one Point (Lateral Ankle) as Origin 2) Find the unit vector from origin to the Knee unit vector = (ktx, kty, ktz) 3) Use the lateral ankle and 3rd target to find a second vector v 4) Use the results of step 3 to find a second unit vector unit vector = (vx, vy, vz) Step 1: Creating the temporary local coordinate system 5) Find the the “A/P” axis from k X v unit vector = (jtx, jty, jtz) 6) Find the third axis, i, via j X k unit vector = (itx, ity, itz)

37. Start with the tracking based local • coordinate system (Rtemp) 2) Transform P into the tracking coordinate system (P’) using Step 2: Storing the virtual lateral ankle target ‘ P 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. Create the tracking based local • coordinate system (Rtemp) 2) Recall the stored vector (P’) from origin to the calibration target (in tracking coordinate system) 3) Transform P’ into the global coordinate system (P) using: Step 3: Find the anatomical system during movement ‘ P 4) Use the tracking targets (including virtual) to find anatomically based coordinate system

40. Foot

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

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