• Segmental Angles• Joint Angles: Euler• Joint Angles: Helical• Exporting Data from GaitProject to Visual3d

Tonight’s Lecture

Segmental Angles

X Rotation from Global to Local

𝑅𝑥 = อ1 0 00 cos𝜃𝑥 sin𝜃𝑥0 −sin𝜃𝑥 cos𝜃𝑥อ

X

Y

X’

Y’

P

Z

Z’

X Rotation Local to Global

𝑅𝑥 = อ1 0 00 cos𝜃𝑥 −sin𝜃𝑥0 sin𝜃𝑥 cos𝜃𝑥 อ

X

Y

X’

Y’

P

Z

Z’t

Y Rotation from Global to Local

𝑅𝑦 = อcos𝜃 0 −sin𝜃0 1 0sin𝜃 0 cos𝜃 อ

X

Y

X’

Y’

P

Z

Z’

Y Rotation from Local to Global

𝑅𝑦 = อcos𝜃 0 sin𝜃0 1 0−sin𝜃 0 cos𝜃อ

X

Y

X’

Y’

P

Z

Z’

t

X

Y

X’

Y’

qz

P

ZZ’ 𝑅𝑧 = อ

cos𝜃 sin𝜃 0−sin𝜃 cos𝜃 00 0 1อ

Z Rotation from Global to Local

X

Y

X’

Y’

qz

P

ZZ’ 𝑅𝑧 = อ

cos𝜃 −sin𝜃 0sin𝜃 cos𝜃 00 0 1อ

Z Rotation from Local to Global

t

Y

XZ

Y

XZ

About the x axis

About the y axis

About the z axis

Orientation of Bodies in 3D

The final orientation dependson the rotation sequence

About the x axis

About the y axis

About the y axis

About the x axis

Y

XZ

Rotation Matrices 3D(𝜃𝑥 ,𝜃𝑦 ,𝜃𝑧 ) 𝑅= 𝑅𝑧𝑅𝑦𝑅𝑥

• Rotation’s depend on the order they are applied

𝑅𝑦𝑅𝑥 = ቮcos𝜃𝑦 0 −sin𝜃𝑦0 1 0sin𝜃𝑦 0 cos𝜃𝑦 ቮอ1 0 00 cos𝜃𝑥 sin𝜃𝑥0 −sin𝜃𝑥 cos𝜃𝑥อ

Rotation Matrices 3D

𝑅𝑦𝑅𝑥 = ቮcos𝜃𝑦 sin𝜃𝑦sin𝜃𝑥 −sin𝜃𝑦cos𝜃𝑥0 cos𝜃𝑥 sin𝜃𝑥sin𝜃𝑦 −cos𝜃𝑦sin𝜃𝑥 cos𝜃𝑦cos𝜃𝑥 ቮ

𝑅= 𝑅𝑧𝑅𝑦𝑅𝑥

Rotation Matrices 3D𝑅= 𝑅𝑧𝑅𝑦𝑅𝑥

𝑅𝑧𝑅𝑦𝑅𝑥 = อcos𝜃𝑧 sin𝜃𝑧 0−sin𝜃𝑧 cos𝜃𝑧 00 0 1อቮcos𝜃𝑦 sin𝜃𝑦sin𝜃𝑥 −sin𝜃𝑦cos𝜃𝑥0 cos𝜃𝑥 sin𝜃𝑥sin𝜃𝑦 −cos𝜃𝑦sin𝜃𝑥 cos𝜃𝑦cos𝜃𝑥 ቮ

𝑅𝑧𝑅𝑦𝑅𝑥 = ቮcos𝜃𝑧cos𝜃𝑦 cos𝜃𝑧sin𝜃𝑦sin𝜃𝑥 + sin𝜃𝑧cos𝜃𝑥 −cos𝜃𝑧sin𝜃𝑦cos𝜃𝑥 + sin𝜃𝑧sin𝜃𝑥−sin𝜃𝑧cos𝜃𝑦 −sin𝜃𝑧sin𝜃𝑦sin𝜃𝑥 + cos𝜃𝑧cos𝜃𝑥 sin𝜃𝑧sin𝜃𝑦cos𝜃𝑥 + cos𝜃𝑧sin𝜃𝑥sin𝜃𝑦 −cos𝜃𝑦sin𝜃𝑥 cos𝜃𝑦cos𝜃𝑥 ቮ

𝜃𝑧 = 𝜃𝑥 = 𝑎𝑠𝑖𝑛(− 𝑅32𝑐𝑜𝑠𝜃𝑦)

Euler Angles: Simple Method

R = อ𝑅11 𝑅12 𝑅13𝑅21 𝑅22 𝑅23𝑅31 𝑅32 𝑅33อ

𝜃𝑦 = 𝑎𝑠𝑖𝑛(𝑅31 )

𝑅𝑧𝑅𝑦𝑅𝑥 = ቮcos𝜃𝑧cos𝜃𝑦 cos𝜃𝑧sin𝜃𝑦 sin𝜃𝑥 + sin𝜃𝑧cos𝜃𝑥 −cos𝜃𝑧sin𝜃𝑦cos𝜃𝑥 + sin𝜃𝑧sin𝜃𝑥−sin𝜃𝑧cos𝜃𝑦 −sin𝜃𝑧sin𝜃𝑦sin𝜃𝑥 + cos𝜃𝑧cos𝜃𝑥 sin𝜃𝑧sin𝜃𝑦cos𝜃𝑥 + cos𝜃𝑧sin𝜃𝑥sin𝜃𝑦 −cos𝜃𝑦sin𝜃𝑥 cos𝜃𝑦cos𝜃𝑥 ቮ

𝑎𝑠𝑖𝑛(− 𝑅21𝑐𝑜𝑠𝜃𝑦)

𝜃𝑥 = 𝑎𝑡𝑎𝑛2(−𝑅32,𝑅33 )

𝑎𝑡𝑎𝑛2(𝑅31,ට𝑅112 + 𝑅122 ) 𝜃𝑦 = 𝜃𝑧 = 𝑎𝑡𝑎𝑛2(−𝑅21,𝑅11)

Euler Angles: Robust Method(handles angles from -180 to +180)

R = อ𝑅11 𝑅12 𝑅13𝑅21 𝑅22 𝑅23𝑅31 𝑅32 𝑅33อ

Atan2(x,y) is the arc tangent of x/y in the interval [-pi,+pi] radians

𝑅𝑧𝑅𝑦𝑅𝑥 = ቮcos𝜃𝑧cos𝜃𝑦 cos𝜃𝑧sin𝜃𝑦 sin𝜃𝑥 + sin𝜃𝑧cos𝜃𝑥 −cos𝜃𝑧sin𝜃𝑦cos𝜃𝑥 + sin𝜃𝑧sin𝜃𝑥−sin𝜃𝑧cos𝜃𝑦 −sin𝜃𝑧sin𝜃𝑦sin𝜃𝑥 + cos𝜃𝑧cos𝜃𝑥 sin𝜃𝑧sin𝜃𝑦 cos𝜃𝑥 + cos𝜃𝑧sin𝜃𝑥sin𝜃𝑦 −cos𝜃𝑦sin𝜃𝑥 cos𝜃𝑦cos𝜃𝑥 ቮ

Joint Angles

Relative Motion: Joint Angles

Rthigh

Rshank

Joint Angles

Rthigh

Rshank

Rjoint*Rthigh = Rshank

Rjoint = Rshank*Rthigh’Rjoint

General form:

Rjoint = Rdistal*Rproximal’

𝜃𝑧 = 𝑎𝑠𝑖𝑛(− 𝑅21𝑐𝑜𝑠𝜃𝑦)

Joint Angles

𝜃𝑥 =

R = อ𝑅11 𝑅12 𝑅13𝑅21 𝑅22 𝑅23𝑅31 𝑅32 𝑅33อ

𝜃𝑦 = flexionabduction

Internal/external

𝑎𝑠𝑖𝑛(− 𝑅32𝑐𝑜𝑠𝜃𝑦) 𝑎𝑠𝑖𝑛(𝑅31 )

𝑅𝑗𝑜𝑖𝑛𝑡 = ቮcos𝜃𝑧cos𝜃𝑦 cos𝜃𝑧sin𝜃𝑦sin𝜃𝑥 + sin𝜃𝑧cos𝜃𝑥 −cos𝜃𝑧sin𝜃𝑦cos𝜃𝑥 + sin𝜃𝑧sin𝜃𝑥−sin𝜃𝑧cos𝜃𝑦 −sin𝜃𝑧sin𝜃𝑦sin𝜃𝑥 + cos𝜃𝑧cos𝜃𝑥 sin𝜃𝑧sin𝜃𝑦cos𝜃𝑥 + cos𝜃𝑧sin𝜃𝑥sin𝜃𝑦 −cos𝜃𝑦sin𝜃𝑥 cos𝜃𝑦cos𝜃𝑥 ቮ

Rjoint = Rdistal*Rproximal’

Lower Extremity: Joint Coordinate System

Red – flexion extension (thigh)

Green – ab/adduction (float)

Blue – internal/external (shank)

Joint Angles: Gimbal Lock

𝜃𝑥 =

R = อ𝑅11 𝑅12 𝑅13𝑅21 𝑅22 𝑅23𝑅31 𝑅32 𝑅33อ

𝜃𝑦 = 𝑎𝑠𝑖𝑛(−𝑅31 )

𝜃𝑧 = flexionab/adduction

Internal/external

What happens if the ab/aduction angle approaches 90 degrees?

Rjoint = Rdistal*Rproximal’

𝑎𝑠𝑖𝑛(− 𝑅32𝑐𝑜𝑠𝜃𝑦)

𝑎𝑠𝑖𝑛(− 𝑅21𝑐𝑜𝑠𝜃𝑦)

Joint Angles: Gimbal Lock

Solutions:

• Use alternative Euler Sequence (for example ZYZ is the ISB recommended sequence for the shoulder)

• Helical Angles

Helical Angles*

*From: Kinzel, Hall and Hillberry, J Biomech, 5, 1972

“The displacement of a moving body from position 1 to position 2 can always berepresented as a rotation about, and a translation along, a unique axis locatedSomewhere in the fixed body”.

Y

XZ

Helical Angles*

*From: Kinzel, Hall and Hillberry, J Biomech, 5, 1972

“The displacement of the moving body is completely defined if the locationand inclination of the screw axis, the rotation angle , and the translationmagnitude K are know ”.

Y

XZ

f

Helical Angles*

*From: Kinzel, Hall and Hillberry, J Biomech, 5, 1972

“The displacement of the moving body is completely defined if the locationand inclination of the screw axis, the rotation angle , and the translationmagnitude K are know ”.

Y

XZ

f

𝑅𝑚𝑜𝑡𝑖𝑜𝑛 = อ𝑟11 𝑟12 𝑟13𝑟21 𝑟22 𝑟23𝑟31 𝑟32 𝑟33อ Rotation:

Exporting Data from GaitProject to Visual3d

Assignment #7

• Compute and Export the Joint Angles for the Ankle, Knee and Hip using theNon-Optimal Method (3 files)

• Compute and Export the Joint Angles for the Ankle, Knee and Hip using theOptimal Method (3 files)

• Import All Exported Joint Angles (6 files) into Visual3d

• Run Visual3d IK

• Create Reports Comparing the joint angles obtained by all three methods