Elbow Joint
Committee members:
Ed Chadwick
Brendan McCormack
A C Nicol
Bo Peterson, Chairman
Victor Waide
March 1996
This is a note with some remarks about making a kinematical description of the elbow joint, the definition of the bone coordinate systems and how to define the attitudes for the bones.
Some requirements.
In order to make a kinematical description of the human body useful and practical, one has to consider some important points,such as the following:
 That it be based on established anatomical observations;
 That it be based on practical experimental methods to establish the geometrical ingredients to the level of approximation needed for the specific purpose;
 That it contain as few parameters as possible;
 That it allows for an easy description and visualisation of the attitude of the various bones;
 That it allows for an easy description of the kinematic relations to different levels of approximation needed for a specific purpose;
 That it be suited to theequations of motion.

Anatomical observations.
There seems to be a general consensus that:
Glenohumeral Joint
 The humeral head is partially spherical;
 The glenohumeral joint is a ball joint;
Elbow Joint
 The humeroulnar joint is a hinge joint;
 The centres of the trochlea and the capitulum are on the hinge axis for the humeroulnar joint;
 Both the contact surfaces at the humeroradial joint are partially spherical and the joint itself is a ball joint;
 The two contact surfaces for the radioulnar joint at the proximal end are partially cylindrical;
Wrist Joint
 The two contact surfaces for the radioulnar joint at the distal end are partially rotationally symmetric;
 The radioulnar joint is a hinge joint. The centre of the capitulum and the axis of the two radioulnar joints are on the joint axis.
The above is a reasonable approximation for many applications and allows for refinements. For example the following may be considered: the humeral head as gliding in a larger spherical surface (cavitus glenoidalis); the motion of the hinge axis for the humeroulnar joint; the deviation from a cylindrical shape for the radioulnar joints; the axial sliding motion at the radioulnar joints.
Experimental methods to establish the geometry.
To establish the axis of rotation for a hinge joint, or the spherical centre of a ball joint, by means of bony landmarks is a poor method because there is no clear relation between such different quantities. Palpation gives very low accuracy. However there are more powerful methods, giving sufficient accuracy, such as:
 light stereo video with light reflective surface markers attached to the skin;
 xray static stereophotogrammetry  using geometrical construction;
 computer tomography;
 xray motion stereophotogrammetry with spherical tantalum markers fixed in the bones  using dynamic analysis with optimisation;
 light stereo video with spherical light reflective markers fixed tothe bones with screws  using dynamic analysis with optimisation;
 acoustical methods similar to 5.
The methods under 4, 5 and 6 allow for the highest degree of refinement by optimising to a specific geometrical model more accurately than one with hinge joints and ball joints. We want to follow the method outlined in the first two sections.
Coordinate systems.
The segmental local centre of mass does not appear to have a physica lmeaning. To use such a centre (if possible to define) would only involve more unnecessary parameters and complicate the description of the loadpoints (contradicting requirements 3, 4 and 5). We prefer to proceed as follows.
Requirement 3  concerning few parameters  leads to the interest inconnecting the various bone coordinate systems directly to the geometry giving rise to the kinematics. Spherical centres can be put in the origin or on a coordinate axis. Hinge joint axes can be put in coordinate planes. Requirement 4  concerning easy visualisation  leads to an interest in the ability to line up all coordinate systems as much as possible ( not strictly possible ). This point and requirement 3 leads to an interest in linking the coordinate systems together with the origin of the more distally one on a coordinate axis of the more proximal one.
One way to define the coordinate systems is as follows:
The humerus coordinate system has its origin at the geometrical centre of the caput humeri. The centre of the trochlea is on a coordinate axis. The humeroulnar hinge axis is in a coordinate plane. This use of the trochlea allows for an easy interpretation of old concepts such as for example carrying angle (and differs from Vander Helm).
The ulnar coordinate system has it origin at the centre of the trochlea. The centre of the ulnar head is on a coordinate axis. The humeroulnar hinge axis is in a coordinate plane. By the centre of the ulnar head we mean a point on the "rotation axis" where the radius is largest.
The radius coordinate system has its origin at the centre of the capitulum. The centre of the scaphoid contact surface of the radius is on a coordinate axis. The radioulnar hinge axis is in a coordinate plane. By the centre of the scaphoid contact surface we mean the centroid for this nearly flat surface. It can be estimated by using the plane containing the hinge axis and the styloid process (of the radius) and drawing a normal (to the plane) from the dorsal tubercle at the contact surface.
The various axes are numbered as follows. Each bone has its 1axis directed towards its distal point mentioned above. For the humerus the hinge axis isin the 13 plane. The 3axis has a direction such that a vector from the centre of the trochlea to the centre of the capitulum has a positive projection on the humerus 3axis. For the ulna the hinge axis is in the 13 plane. A vector from the centre of the trochlea to the centre of the capitulum hasa positive projection on the ulna 3axis. For the radius the radioulnar hinge axis is in the 13 plane. A vector from the centre of the capitulum to the centre of the ulnar head has a negative projection on the radius 3axis.
For the humerus the hinge axis makes an angle [[alpha]] withthe positive 3axis in the direction towards the negative humerus 1axis. For the ulna the hinge axis makes an angle [[beta]] with the positive 3axis in the direction towards the positive ulna1axis. When the ulna and the radius 13 planes are coinciding the elbow hinge axis makes an angle [[eta]] with the positive radius 3axis towardsthe positive radius 1axis.
Attitudes for the bones.
When one wants to represent the attitudes (absolute or relative) for the various bones there are some important facts/propertiesto consider.
 A representation of the rotation group in 3 dimensions needs more than 3 parameters in order to be unique. Thus euler angles (rotation around the same axis twice) or cardan angles (rotation around three different axes) gives at least one attitude which is not uniquely defined.
 If the euler or cardan angles are defined for axes fixed in the body then the same attitude is obtained for arotation of the same angles but now around the space fixed axes and in the opposite order.
 The distance concept for attitudes given by euler or cardan angles is essentially useless because of its complexity. Thus we cannot compare different attitudes to see how close they are.
Euler and cardan angles are, after appropriate choice, suitable for providing a visual interpretation. This is why we want to use them in the following. We need a laboratory system with the 3axis vertical. We put the"model subject" with a sagital plane in the 23 plane of the laboratory system. Other systems, for example abody fixed (thorax) system, together with the humerus system are introducedand lined up with the laboratory system for the position above. The coordinatesystems for the right arm will be right handed and we concentrate on theright side in the following. We introduce the rotation Ri([[phi]]) for frames by the angle [[phi]], in positive sense, around the iaxis. The humerus, the ulna and the radius systems are expressed as a rotation of the type R1([[gamma]]h)R2([[beta]]h)R3([[alpha]]h) of the thorax system, starting with R3([[alpha]]h). This corresponds to the rotation R3([[alpha]]h)R2([[beta]]h)R1([[gamma]]h) around the body fixed system, starting with R1([[gamma]]h). This has the attractive resemblance to the longitude and latitude coordinate system of the earth with the addition where one either rotates around the space fixed 1axis first or rotates around the body fixed long axis last. A similar approach is indicated (but not completed) on page 31 in Joint motion methods of measuring and recording, American Academy of Orthopaedic Surgeons 1965. The non unique position for the cardan type of attitude representation used above will appear for [[beta]]h = +/90deg. for the humerus i.e. vertically up or down. The former is outside the range of motion, as is also for many subjects the latter.
The ulna system can be expressed as a rotation R2([[beta]])R3(180[[phi]])R2([[alpha]]) of the humerus system using the elbow angle [[phi]]. Here, for essentially a straight arm the elbow angle[[phi]] = 180deg. The radius system can be expressed as a rotation R2(([[epsilon]]+[[eta]][[beta]]))R1(90deg.+[[gamma]])R2([[epsilon]]) of the ulna system using the lower arm twist angle [[gamma]]. Here [[epsilon]] is the angle between the ulna 1axis and the radialulnar hinge axis (smaller than 90deg.). The 90deg. is introduced in order to make the ulnar and radial 13 planes orthogonal when [[gamma]] = 0deg. This is in the middle of the working range. The angle [[gamma]] is used in the (normal) positive sense for the direction from the capitulum to the ulnar head. Using the bone lengths it is now a straightforward matter to obtain the space coordinates for the end point (for example the ulnar head). In order to solve the inverse kinematic problem one of course needs subsidiary conditions.
Final remarks.
Interpretation of the angles
The lower arm twist angle [[gamma]] is roughly in the interval +/90deg.. Supination corresponds to positive [[gamma]] and pronation to negative. The elbow angle [[phi]] is roughly in the interval(25deg., 170deg.) but some subjects can obtain 190deg.. The humerus twist angle [[gamma]]h is roughly in the interval (60deg., 90deg.). Outward rotation of the humerus correspond to positive [[gamma]]h and inward rotation to negative. Motion of the humerus in a vertical plane corresponds to fixed [[alpha]]h and varying [[beta]]h. Motion of the humerus in the horizontal plane corresponds to [[beta]]h =0deg. and varying [[alpha]]h.
Specific positions
The medical normal position is obtained as follows.
Lower arm twist angle [[gamma]] = 0deg.,
elbow angle [[phi]] = 180deg.,
humerus angles: [[alpha]]h = 0deg., [[beta]]h = 90deg., [[gamma]]h = 0deg..
As mentioned above this position for the humerus is non unique. Thus the medical normal position is not suitable as the reference position for stereometric investigations using the above cardan angles. It is not very good because of another reason also  it is difficult to distinguish between the twist angles of the lower arm and the humerus. A position much better as a reference position is as follows.
Lower arm twist angle [[gamma]] = 0deg., elbow angle [[phi]] =90deg.,
humerus angles: [[alpha]]h = 0deg., [[beta]]h = 0deg., [[gamma]]h = 0deg..
This position is easily obtained by the experimentalist using only visual references.
It is recommended to use the
elbow angle [[phi]]=90deg. (or averaging around this) while determining the elbow joint axis relative the various bones.
Continuation to the wrist
The wrist joint has two degrees of freedom, but we have not been able to study this joint enough to give a detailed description in this note.
Cardan angles suggested for bodies which are long and slender in the direction of the 1axis. The rotation of frames is given by R1([[gamma]])R2([[beta]])R3([[alpha]]) starting with [[alpha]] and the rotations are around body fixed axis. The same rotation is given by R3([[alpha]])R2([[beta]])R1([[gamma]]) starting with [[gamma]]  but now the rotations are around space fixed axis.  
Figure showing the humerus and its coordinate system as well as the elbow joint axis 0  0. The points [[Omega]]h, [[Omega]]u and[[Omega]]r are the origins of the humerus, the ulna and the radius coordinate systems.  
Figure showing the ulna and the radius when the rotation angle [[gamma]] around the axis [[Omega]]r  [[Omega]]a is 90deg..  
Figure showing the lower arm geometry when the rotation angle [[gamma]] around the axis [[Omega]]r [[Omega]]a is 90deg.. Theulna 1axis is in the humerus 13 plane when the elbow angle [[phi]] is 180deg.. The radius 1 and 3axes are in the ulna 13 plane when the angle [[gamma]] around the [[Omega]]r  [[Omega]]a axis (i.e. arm long axis) is 90deg.. 