Joint Angles Calculation Lei Zhou and Xiaolin Li October 28, 2014

Outline Joint Coordinate System (JCS) Euler’s Angle Helical Method Concept Example Euler’s Angle Helical Method

Joint Coordinate System ----concept The joint coordinate system(JCS) is defined by two independent body-fixed axes and the common perpendicular.

Joint Coordinate System ----concept The Joint Coordinate System (JCS) was proposed by Grood and Suntay (1983) to encourage the use of clinically relevant models.

Joint Coordinate System ----concept Joint Coordinate System is composed of the two body fixed axes, e1 and e3 and their mutual perpendicular, e2. Click View then Header and Footer to change this footer

Example---- Hip joint Anatomical landmarks used ASIS: anterior superior iliac spine (Nomina anatomica: Spina iliaca anterior superior). PSIS: posterior superior iliac spine (Spina iliaca posterior superior). FE: femoral epicondyle (Epicondylus femoris medialis, Epicondylus femoris lateralis).

Example Where is the “O” The common origin of both axis systems is the point of reference for the linear translation occurring in the joint, at its initial neutral position.

Example Pelvic coordinate system—XYZ O: The origin coincident with the right hip center of rotation. Z: The line parallel to a line connecting the right and left ASISs, and pointing to the right. X: The line parallel to a line lying in the plane defined by the two ASISs and the midpoint of the two PSISs, orthogonal to the Z-axis, and pointing anteriorly. Y: The line perpendicular to both X and Z, pointing cranially.

Example Femoral coordinate system—xyz o: The origin coincident with the right hip center of rotation, coincident with that of the pelvic coordinate system (O) in the neutral configuration. y: The line joining the midpoint between the medial and lateral FEs and the origin, and pointing cranially. z: The line perpendicular to the y-axis, lying in the plane defined by the origin and the two FEs, pointing to the right. x: The line perpendicular to both y- and z-axis, pointing anteriorly

Example JCS and motion for the right hip joint e1: The axis fixed to the pelvis and coincident with the Z-axis of the pelvic coordinate system. Rotation (a): flexion or extension. Displacement (q1): mediolateral translation. e3: The axis fixed to the femur and coincident with the y-axis of the right femur coordinate system. Rotation (g): internal or external rotation. Displacement (q3): proximo-distal translation.

Example JCS and motion for the right hip joint e2: The floating axis, the common axis perpendicular to e1 and e3. Rotation (b): adduction or abduction. Displacement (q2): antero-posterior translation.

Euler’s Angle Definition and Function Describe the orientation of a rigid body Used to represent the orientation of a frame of reference relative to another. Represent a sequence of three elemental rotations The elemental rotations can either occur about the axes of the fixed coordinate system (global coordinate system) or about the axes of a rotating coordinate system (local coordinate system).

Euler’s Angles Characters Specific orders, the sequence is not actual path of motion taken to arrive at that position Different rotation orders can lead to different orientations ex: book on a desk; The book will no doubt be oriented in space differently after each set of rotation Twelve possible sequences of rotation axes Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y) Cardan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z)

Euler angles in biomechanics Purpose A method used to describe three dimension motion of a joint General sequence The first rotation is defined relative to an axis oriented in the global coordinate system. The third is defined with regard to an axis fixed within the rotating body. (Local coordinate system). The second is performed relative to the floating axis, which is always orthogonal to both the first and third axis.

A gymnast performation Precession The first rotation takes place relative to an axis defined in the global reference system. Tilt Floating axis: The axis of tilt is not fixed with regard to both the global reference frame and the local reference frame. Spin Rotates around its longitudinal axis: fixed in the body.

Common Sequence in biomechanics studies XYZ sequence X is the flexion/extension in sagittal plane Y is the abduction/adduction Z is the axial (internal/external) rotation The second and the third rotations are about local axes transformed by previous rotations. Ex. Xy’x’’ Why do we use this sequence? XYZ sequence is associated with minimal planar crosstalk and as such its use is encouraged. Sinclair J, Taylor P J, Edmundson C J, et al. Influence of the helical and six available Cardan sequences on 3D ankle joint kinematic parameters[J]. Sports Biomechanics, 2012, 11(3): 430-437.

About an axis of GCS flexion About an axis of LCS abduction About an axis of LCS External rotation a) 45° about the z-axis b) -30°about the x-axis c) -45°about the y-axis The leg in the final position with Euler angles (45° -30°-45°)

Helical angles Also a method to describe three dimension motions A screw axis/helical axis: a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles’ Theorem: any rigid-body motion can be obtained as the rotation around an axis, and a translation parallel to the screw.

Six parameters to define a helical motion: Two coordinates of the piercing point of the helical axis with any one of the three coordinate planes Two direction cosines of the helical axis The translation along and the rotation about the helical axis

When using helical method Large displacement of the glenohumeral joint Arm translation-arm rotation Euler angles are not specific

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