Joint Coordinate System

Coordinate Systems Coordinate systems are generally: Purpose: Cartesian Orthogonal Right-Handed Purpose: To quantitatively define the position of a particular point or rigid body

Cartesian Coordinate Systems Purpose: used to establish a Frame of Reference Generally, this system is defined by 2 things: An origin: 2-D coordinates (0,0) or 3-D location in space (0,0,0) A set of 2 or 3 mutually perpendicular lines with a common intersection point Example of coordinates: 2-D: (3,4) – along the x and y axes 3-D: (3,2,5) – along all 3 axes

Orthogonal Definition: Refers to axes that are perpendicular (at 90°) to one another at the point of intersection

Right-Handed Rule Coordinate systems tend to follow the right-hand rule This rule creates an orientation for a coordinate system Thumb, index finger, and middle finger X-axis = principal horizontal direction (thumb) Y-axis = orthogonal to x-axis (index) Z-axis = right orthogonal to the xy plane (middle)

Global Coordinate Systems A reference system for an entire system. When labelling the axes of the system, upper case (X, Y, Z) may be useful in a GCS Example – a landmark from a joint in the body (lateral condyle of the femur for the knee joint) Within a global coordinate system, the origin is of utmost importance Using a global coordinate system, the relative orientation and position of a rigid body can be defined. Not only a single point.

Local Coordinate Systems A reference system within the larger reference system (i.e. LCS is within the GCS) This system holds its own origin and axes, which are attached to the body in question Additional information: Must define a specific point on or within the body Must define the orientation to the global system Origin and orientation= secondary frame of reference (or LCS)

Joint Coordinate Systems A reference system for joints of the body in relation to larger GCS(the whole body) and to other body segments (LCS) Purpose To be able to define the relative position between 2 bodies. Relative position change = description of motion Orientation Origin Could be the centre of mass of a body segment (ex. The thigh) Could be the distal and proximal ends of bones

Human Movement Biomechanics Lab Joint Angles Methods Used Within Biomechanics Euler/Cardan Angles Joint Coordinate System Helical Axes Each method has specific advantages and disadvantages and the best method to use for a project depends on numerous factors Human Movement Biomechanics Lab

Human Movement Biomechanics Lab Euler’s Angles Leonhard Euler (1707-1783) 3D finite rotations are non-commutative They must be performed in specific ORDER Ex: book on desk The order of rotations is precisely described in biomechanics depending on the application 12 possible sequences of rotations First rotation defined relative to a GLOBAL axis Third rotation defined about an axis in rotating body (LOCAL) Second rotation defined about a floating axis in the second body Ex: (Xglobal, Ylocal, Xlocal) When the terminal rotation is the same it is known as an EULER ROTATIONS (6) When the terminal rotations are NOT the same these are considered CARDAN ROTATIONS (6) http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Euler.html http://www.strubi.ox.ac.uk/strubi/fuller/docs/spider2003/euler.gif Human Movement Biomechanics Lab

Euler Angles Purpose: A method used to describe 3-dimensional motion of a joint `Represent three sequential rotations about anatomical axes` Important to note about Euler angles is that they are dependent upon sequence of rotation Classified into two or three axes

Standard Euler Angles and Euler Angle of JCS Sequence dependency differs depending on which system is being looked at in order to describe 3-dimensional rotation about axes Standard Euler Angles: Dependent upon the order in which rotations occur Classified into rotations about 2 or 3 axes Euler Angle in a Joint Coordinate Systems: Independent upon the order in which rotations occur All angles are due to rotations about all 3 axes

Common Cardan Sequence in biomechanics studies Xyz sequence Rotation about medially-directed X axis (Global CS) Rotation about anteriorly-directed y axis (Local CS) Rotation about vertical axis (Local CS) See Fig 2.12 in text This sequence chosen to represent joint angles and recommended within biomechanics (Cole et al., 1993) Rotations occur about: flexion-extension axis, ab/adduction axis, and axial rotation Major Disadvantage: Gimbal Lock  when middle rotation equals π/2 it results in mathematical singularity and causes computational problems Human Movement Biomechanics Lab

Cardan Sequence Application Movement of a joint is defined as the motion of the distal (far) segment to the proximal segment (near) Ex (knee): thigh (proximal segment) Shank (distal segment) Find TTS Decompose rotation matrix into the three Cardan angles of flexion-extension, ab-adduction, axial rotation Human Movement Biomechanics Lab

Joint Coordinate System (JCS) Grood & Suntay (1983) Describe the motion of the knee joint Purpose: to insure that all three rotations had functional meaning for the knee How is it different than an Euler/Cardan rotation? NOT an orthogonal system Two segment-fixed axes and a FLOATING axis Essentially we must define the anatomical axes of interest from bony markers, the clinical axes of rotation, and the origin of the joint coordinate system for a complete analysis of motion Human Movement Biomechanics Lab

Human Movement Biomechanics Lab Helical Angles Woltring (1985, 1991) Another method to describe the orientation (both rotation & translation) between two reference systems Any two reference systems can be “matched” up through a single rotation and a translation about a single axi This axis does not necessarily have to line up with one of the axis of the local CS Good for joints that are hinge-like i.e. talocrural joint Human Movement Biomechanics Lab