Local Coordinate Systems From Motion Capture Data

Transcription

1 Local Coordinate Systems From Motion Capture Data

2 Anatomical Coordinate Systems (thigh) How to calculate the anatomical coordinate systems 1) Find the Hip center 2) Find the Knee center 3) Find the Inferior/Superior Axis unit vector = (l 1, m 1, n 1 ) 4) Find the the A/P axis (using the Hip, Knee and Lateral Knee Targets) unit vector = (l 2, m 2, n 3 ) 5) Find the M/L axis via X product unit vector = (l 3, m 3, n s ) R tig = l 1 l 2 l 3 m 1 m 2 m 3 n 1 n 2 n 3

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

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

5 Step 1: Creating the temporary local coordinate system 1) Select one Point (Lateral Knee) as Origin 2) Find the unit vector from origin to a second tracking target (hip) unit vector = (l t 1, m t 1, n t 1) 3) Use the 3 targets to find a plane and the second axis unit vector = (l t 2, m t 2, n t 3) 4) Find the third axis via X product unit vector = (l t 3, m t 3, n t s) R temp = l t 1 l t 2 l t 3 m t 1 m t 2 m t 3 n t 1 n t 2 n t 3

6 Step 2: Storing the virtual lateral knee target 1) Start with the tracking based local coordinate system (Rtemp) 2) Find the vector (P) from origin to the medial knee calibration Target* (in global coordinate System) 3) Transform P into the tracking coordinate system (P ) using P P = R temp t P

7 Step 3: Find the anatomical system during movement 1) Create the tracking based local coordinate system (Rtemp) 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: P = R temp P + O 4) Use the tracking targets (including virtual) to find anatomically based coordinate system

8 Anatomical Coordinate Systems (shank) How to calculate the anatomical coordinate systems 1) Find the Knee center 2) Find the Ankle center 3) Find the Inferior/Superior Axis unit vector = (l 1, m 1, n 1 ) 4) Find the the A/P axis (using the knee, ankle and lateral ankle targets) unit vector = (l 2, m 2, n 3 ) 5) Find the M/L axis via X product unit vector = (l 3, m 3, n s ) l 1 l 2 l 3 m 1 m 2 m 3 n 1 n 2 n 3

9 Problem: Again calculating the anatomical coordinate system requires a target (medial ankle) not used during the walking trial

10 Solution: Again we create a virtual medial knee target which can be located during the motion trials

11 Step 1: Creating the temporary local coordinate system 1) Select one Point (lateral ankle) as Origin 2) Find the vector from origin to a second tracking target (knee) unit vector = (l t 1, m t 1, n t 1) 3) Use the 3 targets to find a plane and the second axis unit vector = (l t 2, m t 2, n t 3) 4) Find the third axis via X product unit vector = (l t 3, m t 3, n t s) R temp = l t 1 l t 2 l t 3 m t 1 m t 2 m t 3 n t 1 n t 2 n t 3

12 Step 2: Storing the virtual lateral knee target 1) Start with the tracking based local coordinate system (Rtemp) 2) Find the vector (P) from origin to the medial ankle calibration Target* (in global coordinate System) 3) Transform P into the tracking coordinate system (P ) using P P = R t temp P

13 Step 3: Find the anatomical system during movement 1) Create the tracking based local coordinate system 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: P = R temp P + O 4) Use the tracking targets (including virtual) to find anatomically based coordinate system

14 Anatomical Coordinate Systems (foot) How to calculate the anatomical coordinate systems 1) Find the Ankle center 2) Find the A/P Axis unit vector = (l 2, m 2, n 2 ) 3) Find the the M/L axis (using the ankle, heel and toe targets) unit vector = (l 3, m 3, n 3 ) 4) Find the Vertical axis via X product unit vector = (l 1, m 1, n 1 ) R foot = l 1 l 2 l 3 m 1 m 2 m 3 n 1 n 2 n 3

15 Orientation of Bodies in 3D About the x axis About the y axis About the z axis Y Y Z X Z X

16 The final orientation depends on the rotation sequence About the x axis About the y axis About the y axis About the x axis Y Z X

17 Rotation Matrices 3D (θ x, θ y, θ z ) R = R z R y R x Rotation s depend on the order they are applied

18 Rotation Matrices 3D R = R z R y R x R y R x = cos θ y 0 sin θ y sin θ y 0 cos θ y cos θ x sin θ x 0 sin θ x cos θ x R y R x = cos θ y sin θ y sin θ x sin θ y cos θ x 0 cos θ x sin θ x sin θ y cos θ y sin θ x cos θ y cos θ x

19 Rotation Matrices 3D R = R z R y R x R z R y R x = cos θ z sin θ z 0 sin θ z cos θ z cos θ y sin θ y sin θ x sin θ y cos θ x 0 cos θ x sin θ x sin θ y cos θ y sin θ x cos θ y cos θ x R z R y R x = cos θ z cos θ y cos θ z sin θ y sin θ x sin θ z cos θ x cos θ z sin θ y cos θ x + sin θ z sin θ x sin θ z cos θ y sin θ z sin θ y sin θ x + cos θ z cos θ x sin θ z sin θ y cos θ x cos θ z sin θ x sin θ y cos θ y sin θ x cos θ y cos θ x

20 Euler Angles R z R y R x = cos θ z cos θ y cos θ z sin θ y sin θ x sin θ z cos θ x cos θ z sin θ y cos θ x + sin θ z sin θ x sin θ z cos θ y sin θ z sin θ y sin θ x + cos θ z cos θ x sin θ z sin θ y cos θ x cos θ z sin θ x sin θ y cos θ y sin θ x cos θ y cos θ x R = l 1 l 2 l 3 m 1 m 2 m 3 n 1 n 2 n 3 θ x = Atan2(n 2, n 3 ) θ z = Atan2(m1, l1) θ y = Atan2( n1, l m 1 3 ) Atan2(x,y) is the arc tangent of x/y in the interval [-pi,+pi] radians