Overview

The direct kinematics model (DK) provides the relative location of segments and their landmarks for specific joint angles defined by the user.

The model

The DK model was developed consisting of a specific set of articulated segments. The DK model is described by an articulation chain of segments and the individual joints between them. For each joint a coordinate system (JCS) is defined that connects two adjacent segments and describes the joint rotation axes and their rotation sequence. Each segment is a rigid body that represents one or more bones (see following table). The DK model is positioned by changing the angles of these joints. Local coordinate systems (LCS) of the segments, were essentially constructed using anatomical landmarks, following the ISB recommendations [Wu 2002; Wu 2005], and they are used to define the relative angles of the joints. The following tables summarize the segments and the joints defined into the DK model and the anatomical landmarks employed for the definition of the local coordinate systems.

Structure

The segments are articulated to each other following a tree structure, with the pelvis being the root of this structure (see following figure). Three rotational degrees of freedom (DOF) are allowed for each joint, with the exception of the joint connecting the pelvis with the environment that three additional translational DOF are allowed. The orientation of each segment is computed relative to its predecessor in the tree structure, taking into account the rotational angles of the joint connecting the segments.

Structure tree of the segments and the joints connecting them.

Examples

The following figure demonstares two examples of positioning the reference model, by using joint angles defined by the user. Initially, the DK model is constructed by calling the function "CreatePiperModel.m". The model structure in .txt format and anatomical landmarks coordinates in .csv format serve as inputs for this function. This function computes LCS using the anatomical landmarks coordinates, articulates the segments using the model structure and computes the local coordinate systems (figure bellow, gray skeleton). The function provides the dimensions of the articulated model and the anatomical landmarks coordinates both in global coordinate system and in LCS. The function "RepositionSkeletonByDK.m" is used to re-position the skeleton. The dimensions and the structure of the articulated model and the joint angles corresponding the desired posture serve as input for this function. The function computes the new position of each segment using the position of its predecessor in the tree structure and the angles of the joint connecting the segments as defined by the user. The function provides the anatomical landmarks coordinates in the new posture, expressed in both the global coordinate system and in LCS (figure bellow, red skeleton). For the first example the flexion angles of the right hip, right knee, right glenohumeral and right elbow were increased 60, -80, 80 and -60° respectively. These flexion angles resulted in the posture represented with the red skeleton (figure bellow left, red skeleton). Although only flexion angles were defined, it can be noticed that the forearm is also internally rotated relative to the spine. This occurred because the computation of the position of the forearm segment is based on the position and orientation of the arm and the angles defined for the elbow joint. As the arm is inverted the elbow flexion causes the forearm to move internally relatively to the spine. For the second example, the joint angles of the driving posture were employed (figure bellow right).

Re-positioning of the reference model (gray skeleton). 1. Right hip, knee, glenohumeral and elbow flexion. 2. Joint angles of driving posture.

Description of the program scripts

Workflow and respective scripts

Step	Action	Script
1	Symmetrize skeleton	SymLMs
2	Compute joint angles of the driving posture (target posture)	CreatePersonalizedPiperModel
3	Compute joint coordinate systems and express anatomical landmarks with repect to LCS for source skeletons	CreatePersonalizedPiperModel
4	Compute the coordinates of the anatomical landmarks on the target posture	RepositionSkeletonByDK

Description of the main scripts of the kinematic model

Script	Input	Output	Description
CreatePersonalizedPiperModel(sbj)	Piper.Model.txt (sbj)_LMsImport.csv ZeroJointAngles.txt	(sbj).ScannedPosture.LMs.csv (sbj).ScannedPosture.JointAngles.csv (sbj).JointCenters.csv (sbj).Dimensions.mat (sbj).Body.mat	Computes local coordinate systems for each segment and computes the positions of the anatomical landmarks with respect to the GCS and LCS
RepositionSkeletonByDK(sbj)	Piper.Model.txt (sbj).Dimensions.mat (sbj).Body.mat DrivingPosture.JointAngles.csv	(sbj).DrivingPosture.LMs.csv	Positions the articulated model based on joint angles defined by the user and computes the positions of the anatomical landmarks in the target posture with respect to the GCS and LCS
SymLMs	(sbj)_lm.csv	(sbj)_LMsSym.csv	i) Creates symmetrized to the left extremities anatomical landmarks on the respective right extrimities ii) Modifies the input landmarks set such as the headers are capitalized and landmarks are registered to segments instead of body parts (Alias table 1) ex. Styloid_process_of_left_radius Left_radius Ulna_L Left_ischial_tuberosity Left_hip_bone Pelvis
Segment_(segment name)	Previous segment in the kinematic chain, anatomical landmarks	LCS A structure variable including the the segment, body parts, LCS, axes rotation sequence, distal and proximal joints	Defines i) the LCS of the (segment name) ii) the proximal and the distal joints of the (segment name)

Description of batch process scripts

Script	Description
BatchProcess4ComputingLMsInScannedAndDrivingPostures	Executes the CreatePersonalizedPiperModel and RepositionSkeletonByDK scripts for all the selected sbj
BatchProcess4SymLMs	Executes the SymLMs script for all the selected sbj

Description of input/output files

File	Description
Piper.Model.txt	Defines the structure of the model: Joints, segments and containing body parts, anatomical landmarks to be used for LCS
(sbj)_lm.csv	Anatomical landmarks coordinates (name, body part, x, y, z)
(sbj)_LMsSym.csv	Anatomical landmarks coordinates (name, segment, x, y, z)
ZeroJointAngles.txt	Zero values for all the joint angles
(sbj).ScannedPosture.LMs.csv	Anatomical landmarks coordinates for the scan posture (name, segment, x, y, z, X, Y, Z)
(sbj).DrivingPosture.LMs.csv	Anatomical landmarks coordinates for the driving posture (name, segment, x, y, z, X, Y, Z)
(sbj).ScannedPosture.JointAngles.csv	Joint angles for the articulated model
(sbj).JointCenters.csv	Coordinates of the joint centers

Definitions of the joint coordinate systems

Joint	Definition	Anterior view	Lateral view
Pelvis-Environment	O: midpoint of left and right acetabular centers Z: the line crossing the right and left acetabular centers, pointing to the right Y: the line crossing the center of the fifth lumbar vertebral lower plate and the origin, pointing cranially X: the common line perpendicular to the Z and Y axes, pointing anteriorly
Hip	O: the acetabular center Y: the line connecting the acetabular with the knee center, pointing proximally X: the common line perpendicular to the line connecting the lateral and medial femoral epicondyles and the Y axis, pointing anteriorly Z: the common line perpendicular to the X and Y axes, pointing to the right
Knee	O: the midpoint of the femoral epicondyles Z: the line connecting the medial and lateral malleoli, pointing to the right X: the common line perpendicular to the line connecting the lateral malleolus with the midpoint of the lateral and medial tibial condyles and the line connecting the medial malleolus with the midpoint of the tibial condyles, pointing anteriorly Y: the common line perpendicular to the Z and X axes, pointing proximally
C2-C3	O: center of distal vertebral segment Y: the line connecting the centers of the proximal and distal vertebral plates, pointing cranially Z: the common perpendicular line to the line connecting the ventral and the dorsal points of the vertebral and the Y axis, pointing to the right X: the common line perpendicular to the Y and Z axes, pointing anteriorly
Sternoclavicular	O: the most ventral point of the clavicle Z: the line connecting the origin with the most dorsal point of the clavicle, pointing laterally X: the common line perpendicular to the Z axis and the line connecting the most anterior point of the superior plate of C7 and the most anterior point of the superior plate of T8, pointing anteriorly Y: the common line perpendicular to the Z and X axes, pointing caudally
Glenohumeral	O: the center of the humerus Y: the line connecting the center of the humerus with the elbow center, pointing cranially X: the common line perpendicular to the the Y axis and the line connecting the epicondyles of the humerus, pointing anteriorly Z: the common line perpendicular to the Y and X axes, pointing laterally
Skull	O: the center of the atlanto-occipital joint Z: the line connecting the left with the right porion, pointing to the right X: the line perpendicular to plane defined by the nasion, the left and the right porion, pointing caudally Y: the common line perpendicular to the Z and X axes, pointing anteriorly
Mandible	O: the midpoint of the right and the left jaw angles Z: the line connecting the left and the right jaw angles,pointing right X: the line perpendicular to plane defined by the incisive, the left and the right jaw angles, pointing caudally Y: the common line perpendicular to the Z and X axes, pointing anteriorly
Patella	O: the anterior center of the patella Z: the line connecting the most medial with the most lateral points of the patella, pointing laterally X: the common line perpendicular to the X axis and the line connecting the most proximal with most distal points on the anterior surface of the patella, pointing posteriorly Y: the common line perpendicular to the Z and X axes, pointing distally
Scapula	O: the most dorsal point of the left acromioclavicular joint Z: the line connecting the angulus acromialis with the trigonum spinae, pointing laterally X: the line perpenticular to plane defined by the angulus acromialis, trigonum spinae and the angulus inferior, pointing anteriorly Y: the common line perpendicular to the Z and X axes, pointing caudally
Sternum	O: the most ventral point of the left clavicle Z: the line connecting the points of the left and right sternoclavicular joints, pointing laterally X: the common line perpendicular to axis Z and the line connecting the xyphoid process and jugular notch, pointing anteriorly Y: the common line perpendicular to the Z and X axes, pointing caudally
RibCage	O: anterior midpoint of T4 Z: the line connecting the most ventral point of the the left sternoclavicular joint with the right point of the right sternoclavicular joint, pointing right X: the common line perpendicular to axis Z and the line connecting the anterior midpoints of T4 and T10, pointing posteriorly Y: the common line perpendicular to the X and Z axes, pointing cranially
Elbow	O: the midpoint of the epicondyles of the humerus Y: the line connecting the lateral epicondyle of the humerus with the most lateral point on the radial styloid, pointing proximally X: the common line perpendicular to the line connecting the most lateral point of the radial styloid with the most medial point of the ulnar styloid and the Y axis Z: the common line perpendicular to the X and Y axes

References

Wu G et al., 2002. ISB recommendation on definitions of joint coordinate system of various joints for the reporting of human joint motion - Part I: ankle, hip, and spine. Journal of Biomechanics 35: 543 - 548.
Wu G et al., 2005. ISB recommendation on definitions of joint coordinate systems of various joints for the reporting of human joint motion - Part II: shoulder, elbow, wrist and hand. Journal of Biomechanics 38: 981 - 992.