Compound Movements inMulti-joint Systems

Compound movements in multi-joint systems almost never havea fixed center of rotation, with no translation. In fact, the principal advantage of compound movements isoften that they convert rotations into translations. One can reach out using rotations in the shoulder andelbow. By their nature, jointsconvert small rotatory movements within the physical joint into large excursionsof parts of the bones at some distance from the joint.  That allows for substantialamplification in anatomical movements. However, the action that is needed is not a rotation about a fixedpoint. By spacing joints outbetween multiple links, one may produce quite large, complex, and nuancedmovements of a wide variety of types. We now turn to a consideration of several simple systems that producecompound movements.

Two Link Systems/Linear Movement

A Basic System

Consider a simple system with two links that rotate about acentral joint. Such a simplesystem can already produce linear movement, rotatory movement, and combinationsof both. However, to create alinear movement, there must be a precise coordination of the actions in thejoints.

In the above figure, a two link armature has beenconstructed and compelled to move the distal end of the armature along astraight line extending away from its proximal endpoint.  The arrangements of the links areillustrated for successive 30° steps in the orientation of the distal link, .  Note that the central joint iscompelled to move along a circular arc, because it is of fixed length with afixed center of rotation at its proximal end. In fact, it can move on a spherical surface, but thesituation has been simplified here by constraining it to a plane, because doingso does not fundamentally alter the analysis that that follows.  In life, the joint might follow aspherical trajectory. For instance,when drawing back a bowstring in preparation to shoot an arrow, one’s elbowrotates laterally as the string is pulled back. That rotation occurs in the shoulder and it effectivelyrotates the whole system as a unit.

One can readily see in the figure that the relationshipsbetween the locations of the joint and the distal end of the distal link arenot simple and the relationships between the angles and areclearly non-linear. In fact, ifthe distal extremity is drawn further proximally, the joint will reverse thedirection of its movement and begin to swing up again.  That may be why human arms and forearmsare approximately the same length and why their thighs and legs are about thesame length.

Describing the System

The first step towards a deep understanding a process isoften to describe it as precisely as possible. Therefore, let’s consider how such a two link system mightbe described.

To start with, let the proximal end of the proximal process,, be fixed. Thelocation of the joint between the links, , and the distal end of the distal link, , are then expressed in terms of and therotations of the links at and . The vectorfrom to is and the vectorfrom to is .

The orientations of the proximal and distal links canalso be written down.

If the ratio of the initial orientations is , then the expression for the orientation of the distalsegment can be written in terms of the orientation of the proximal segment.

We can combine these expressions and obtain the descriptionfor the distal end of the distal segment in terms of the initial conditions forthe armature.

However, the location of the distal endpoint may also bedescribed as lying a given distance in a direction relative to , therefore we can write down its location irrespective ofthe details of the linkage.

Setting aside rotations of the system as a whole about theline of movement, the linkage lies in a plane defined by the two links.  If the distal endpoint moves along astraight line, then the change in its orientation must be about an axisperpendicular to the plane that contains the two armatures.  Consequently, the vector of therotation that changes the orientation of the distal endpoint is the vector ofthe ratio of the links.

The angle of the quaternion of the rotation, , is the change in angle between the links.

We can write down the expression for the distal endpointorientation in term of these quantities.

These alternative descriptions lead directly to a pair ofequations involving the rotation quaternions.

These equations express the relationships between therotations at and . Clearly theydepend on the lengths of the links and the direction of the movement.  If the direction of one link changes,then the direction of the other link must change and the vectors of therotation quaternions must all lie in the same direction, perpendicular to theplane that contains the two links. Both of the component rotations must be unitary rotations, because thelinks to not change length, therefore they must be in opposite directions inorder to keep the distal endpoint, , on a line through the proximal endpoint, .

We can write out the rotation quaternions in trigonometricform as follows.

This means that we can rewrite the expression for therelationship the locations as follows.

We were able to simplify the expression because the axisof rotation is perpendicular to the moving link. Consequently, the expression uses twice the angle in theoriginal expression.

A Calculation

Now, let us consider some calculations to illustrate theforegoing points. Start with asituation like that in the following figure.

The two links are of equal length (1.0) and the initialpositions for the two links are along the i-axis.

The proximal link is rotated 30°, 45°, and 60° to generate , respectively. That means that the interior angle at is 60°, 90°, and 120°, respectively.  Note that if the angle that the firstlink swings through is , then the following relationships hold.

It is straightforward to compute the values of .

The distal endpoints may be similarly computed.

Tempo

Because of the geometry of the described situation, it isclear that the distal endpoint of the armature traces a straight line thatpasses through as itmoves. The dynamics of the proximallink are fairly simple, but the dynamics of the distal link are substantiallymore complex.

If the angular velocity is uniform at , then the angular velocity at is twice as fast and the linear velocity along the line ofmovement is sinusoidal with the fastest movement at the beginning of themovement. To make the movementuniform along the line of movement the angular excursions must follow a time courselike the arcsine of time. Normalanatomical movements follow neither trajectory. They tend to accelerate to a maximum, hold that value formost of the excursion, and then decelerate to a finish.

Division Between Rotation and Translation

It is clear that the component movements are both rotationsand yet the compound movement contains a large translation component.  There must also be a rotationcomponent, because the orientation of the distal endpoint changes.  Translation does not changeorientation, therefore there must be a rotation. We can compute the magnitude of the rotation by taking theratio of the orientations. In thiscase, we know that the rotation is entirely in the plane of the armature andthe distal link goes from being directly vertical (-i) to being directlyhorizontal (j), therefore therotation component has an angular excursion of 90° about the –k axis. The total translation is i +j, a distance of . The distancetraveled during the rotation is radians or,since the radius is 1.0, a distance of 1.57. Consequently, there are comparable amounts of rotation andtranslation.

In this instance, it makes most sense to view the movementas a translation that sweeps along the circular arc centered on the proximalend of the proximal segment as the rotation occurs about that traveling centerof rotation. Curiously, theangular excursion of the rotation is 90°, but the joint at opens from 0° to 180°. So the joint between the links experiences a 180° rotation.  This situation shows how the way youview a structure can determine what is seen.

Transverse Movements

Clearly, as set up, it is possible to make movements thatare perpendicular to the reaching movement that we have just considered.  However, when we look at examples ofjointed systems like that considered here, they do not make such movements inthe middle joint. In both ourupper and lower extremities, the central joint in the limb is essentially constrainedto move in a single plane, that is, they are hinge joints.  When we look closely, there is a smallamount of play, but that is mostly to make the joint more effective by lockingthe knee or allowing rotation at the wrist. This observation leads one to ask why the system is soconstrained.

Both the hip and shoulder joints are ball and socketjoints. They allow a wide varietyof trajectories and thereby set the plane in which the knee or elbow joint willrotate. The ankle/foot and thewrist/hand are also able to move in multiple planes to set the orientation ofthe end of the limb, the part that usually engages the rest of the world.

When we examine the musculature, it becomes apparent whythis arrangement is used. Theintrinsic instability of the proximal joint requires a great deal of musculatureto control the limb’s orientation. In the shoulder we find the large masses of the latissimus dorsi, thetrapezius, rhomboid and levator scapulae muscles to the scapula and thesupraspinatus, infraspinatus, and teres major, subscapularis, and minor musclesfrom the scapula to the humerus to control the posterior aspect of the jointand the pectoralis major and minor, coracobrachialis, and biceps to control theanterior aspect. The serratusanterior acts through the scapula and the deltoid acts both anteriorly andposteriorly. In the hip, we haveall the glutei, the piriformis, the gemmeli, obturators, tensor fascia lata,the quadriceps muscles, and iliopsoas, the hamstrings and several adductors operatingaround the joint. Because of themasses of these muscles, it is necessary to keep them close to the axis of thebody, to reduce the angular momentum of inertia of the limb.  By making the middle joint a singleaxis joint, it is possible to make do with two sets of muscles, the flexors andextensors. They are still massive,but that mass in closer to the body axis than the joint.  Compared to the mass of muscles neededto control the proximal joint, they are small.

It is possible to allow the distal joint to move aboutmultiple axes, because the mass that needs to be moved is comparatively small,the hand or foot. The muscle canbe placed proximal to the joint and the muscles can be relatively small. Mostof the muscles that control the fingers are in the proximal forearm andsimilarly for the toes and the calf. In animals that need to move very fast, the wrist and hand are elongatedand there is minimal muscular mass, giving a long lever arm with acomparatively low angular moment of inertia.

It is necessary to have a greater freedom of movement in thedistal end of the armature to partially compensate for the shifts oforientation that are enforced by the arrangement of the more proximaljoints. That is especially so inapes and monkeys, which are able to grip objects with their hands.  It is not sufficient to get the hand tothe right place, but it also has to be correctly oriented.  One could do without.  A person with a fused wrist can stilldo most things, but the movement to obtain the correct rotation must occur inthe proximal joint, often by expending substantially more energy to lift theentire upper limb.

One way around that would to be to allow the middle joint torotate about the axis of the distal link. In fact that is approximately what happens. The movement in the jointthat moves the forearm is about a single axis, but there is an axis of rotationfor the wrist that passes through the distal end of the ulna and the proximalend of the radius. Thephysiologically relevant movement is the rotation of the distal end of the radiusabout the distal end of the ulna. It is complemented by rotations about an axis parallel with the longaxis of the radial facet (abduction/adduction) and an axis parallel with thelong axis of the radial facet (flexion/extension). The wrist cannot move about intermediate axes because of theoblong shape of the facet. Movingabout an oblique axis, intermediary to the flexion/extension and theabduction/adduction axes, would force the joint to separate, which isrestrained from happening by the ligaments about the joint.  As a consequence, there are threeseparate axes of rotation without a ball and socket joint.  That arrangement is more stable whileallowing for a great deal of control of orientation with much less muscle mass.

The proximal end of the radius lies in the elbow joint,where it is essentially a ball and socket joint, but the annular ligamenteffectively restrains movements in that joint to rotation about an axis nearlyparallel with the ulna shaft. Itis actually not necessary to the wrist joint. It can be surgically removed by cutting off the proximal endof the radius with minimal consequences for wrist movement.  The principal loss is probably a lossof stability when punching with the hand. Even then, the fascial ligament between the two bones and the distalannular ligament are strong restraints on that type of movement unless aconsiderable force is placed behind the punch.

In summation, even though there are theoreticalpossibilities of generating transverse movements by movements in the centraljoint of the armature, there are mechanical reasons why it is not a goodsolution to the generation of such movements. The normal anatomical means of generating such movements isto rotate the proximal link in the proximal joint and allow the flexion/extension in the central joint to move the distal end of the distal link totrace out a circular arc.

Movements in the distal joint are made more stable byfractionating the movement into several axes that travel with the bones and arefixed relative the local landmarks. So, the abduction/adduction axis for the wrist in anatomical positionallows the wrist to move in a parasagittal plane when the distal end of theradius is rotated 90° about the ulna. Each component joint is essentially a hinge joint, therefore requiresmuch less muscle to control its movements, but, added together, we can obtain aconsiderable amount of movement in multiple directions.