The Anatomical Geometry of Muscles Sets

This chapter will consider a concept that will be called muscleset and the way that muscles controlmovements through muscle set. Forpresent purposes, muscle set will be the lengths and directions of the musclesthat bind together a collection of bones. Strictly speaking, there need not be bones. For instance, the muscles of facial expression are a set ofinterdependent muscles that move skin and fascia. However, we will concentrate upon muscles that attach to bonesand move those bones upon other bones, in joints between the bones. We start with the observation thatmuscles connect bones and move them relative to each other.

Muscular attachments to bones are consistent from body tobody, being particular to the muscle and the bone. Therefore, muscles appear to be definite anatomicalentities, rather than simply mental constructs that help us to describe theiranatomy. This consistency suggeststhat their particular arrangement serves a purpose, which is implicit in theiranatomy.

Since bones are rigid, the locations of the muscularattachment sites upon bone remain fixed relative to the placement of the bone.Consequently, as location and orientation of a bone change, so do the locationsof its muscular attachments. Iftwo bones are linked through a joint or a series of joints, then movement in ajoint (or joints) will change the locations of the muscle attachments in apredictable way. That will changethe lengths and directions of the muscles. If we know the locations of themuscle attachments relative to the bones and the placements of the bones, thenit is possible to compute the set of muscle lengths and their directions. Therewill be a unique set of muscle lengths for each and every placement of the bones.

The muscle set for all the extrinsic eye muscles versus placement (gaze direction).

The set of muscle lengths can be plotted against the bonyplacements. The result is asurface in an abstract space. Anexample of such a surface is illustrated above for the extrinsic eye musclesthat move the eyeball in the orbit. Such a surface has the dimensions of the bone’s, or in this case theeyeball’s, placement plus the muscle’s lengths. In the illustrated surface, lengths of the eye muscles areplotted against gaze direction.

As with many movement systems, the internal constraints ofthe anatomy and functional restrictions dictated by physiology reduce thenumber of placement dimensions, from a possible theoretical number of six(three for location and three for orientation), down to two (up/down andmedial/lateral). There is ananatomical dimension for each muscle length, each of which is plotted aseparate section of the surface in the illustration. The number of dimensions for the muscle set surface is thenumber of dimensions for placement, in this case two dimensions.  In other systems the surface may havethree or more dimensions. That surface is embedded in a space with a dimensionfor each muscle plus the number of placement dimensions.  For example, the muscle set for theextrinsic eye muscles forms a two-dimensional surface in an eight dimensionalspace. A similar surface may begenerated for any muscle set.

In this chapter we will examine how the muscle set dependsupon bony placement and vice versa. In particular, we wish to examine howthe disposition of the muscles affects movements that occur between bones andhow the movements between bones lead to interdependency between the componentsof the muscle set.

A Simple Muscle Set

The following figure illustrates a simple anatomicalarrangement that will serve as a model system. It involves two bones, Aand B, linked by a set of musclesnamed for their attachments. Thebones have three processes (V, L, R)that are directed symmetrically away from the center of the bone (C). The centers of the bones are linked by processes that extend into ajoint between the bones (J).  The lengths of the processes to thejoint are not necessarily equal. In fact, one of the parameters that we will vary will be the relativedistances from the bone centers to the joint (and ). Muscleattachment may be situated on any of the processes or the center of thebone.

A musculoskeletal system is defined by the geometry ofthe component bones and the manner in which its muscles attach to thosebones. The distribution of themuscles relative to the joint determine their actions.

Some sample muscles are illustrated.  For instance, muscles may extend betweenthe end of the vertical process () of bone A andthe ends of any of the processes of bone B(). The musclebetween the vertical processes is called , which tells us that it connects the apex of the verticalprocess of A () to the apex of the vertical process of B (). Similarly,the other two muscles are designated by the names and . A muscle fromthe center of A, , to an attachment part way up the shaft of the verticalprocess, , is designated by the name .

Geometrical Anatomy of a System

The origins and insertions of each muscle and thearrangement of its bones define a musculoskeletal anatomical system.  Such a system may be as small a pair ofbones with their muscles or a complex assembly of bones and muscles.  Often the system will contain otherelements, such as ligaments, joint capsules, and/or fascial sheets.

Both the origins and insertions may be referred to asinsertions, because either can be fixed with the other moving or both may bemoving. When we wish to explicitlystate that one end of the muscle is fixed, then it will be called theorigin. This is at variance withstandard anatomical practice, which places the origin nearer to the bodymidline, irrespective of its functional role.

Muscle attachments will usually be expressed as extensionvectors in framed vectors of bones. Other possible extensions of a bone may be its joints and any fulcra formuscles that may exist. The set offramed vectors that contain this information form the mathematical descriptionof an anatomical system, its basis. The basis and a set of muscle lengths determine the geometrical anatomyof the system. With thatinformation, we can compute the consequences of muscle actions.

Joints

In most anatomical systems a joint may reasonably assumed tohave a single instantaneous axis of rotation, which may, however, shift itslocation relative to the participating bones as a function of joint angle.  For instance, the instantaneous axis ofrotation for the ulna-humeral joint rotates as a function of joint angle.   That is illustrated by notingthat the ulna is parallel to the humerus when the joint is fully flexed, but itgenerally lies at an angle to the humerus when the joint is fully extended.  The angle in full extension is calledthe carrying angle.

It should be noted that not all joints have a singleinstantaneous axis of rotation. For instance, a saddle joint will have two distinct axes of rotation,which are mutually orthogonal and on opposite sides of the joint surface.  There are also joints in which thejoint is divided into two or more parts, which are separated by a disc withinthe joint. The sternoclavicularjoint and the temporomandibular joints are instances of such joints. Anarticular disc may interact with the bones on each side of the joint in muchthe same way or in rather different ways. In fact, such discs are often saddle shaped which means that the twosides of the joint have different axes of rotation. Muscles acting across such a joint may have differenteffects depending on the state of each component joint.  In effect, we have two interdependentjoints, each with its own muscle moments. However, since such joints are usually small comparedwith the muscles acting upon them, it is often feasible to collapse thecompound joint into a single complex joint.

Anatomical Joints versus FunctionalJoints

It is necessary to differentiate between anatomical jointsand functional joints. An anatomicaljoint is the physical structure, usually acleft between bones, with articular surfaces where they abut and/or they aretethered together by ligaments and/or joint capsules. The anatomical joint is interesting and important tounderstanding the movements between the bones. However, because the articular surfaces are nearly alwayscurved surfaces, the axes of rotation are usually outside the anatomicaljoint. The placement of the axisof rotation defines a functional joint. In the lower cervicalspine, the axes of rotation for a joint may be several vertebrae caudal to theanatomical joint, which may place the functional joint entirely outside thebones of the neck (see chapters on the cervical spine).

Muscle Moment

The muscle moment, , is the ratio of the muscle’s insertions (A, B)relative to the functional joint (J)across which they are acting (). It is aquaternion, with a vector that passes through the joint, perpendicular to theplane that contains the joint and both insertions. Where an attachment has been designated as the origin, theratio will usually be the origin over the insertion, because that is thedirection in which the muscle is rotating the moving bone.

A muscle’s moment is the ratio of it origin to itsinsertion, where both are defined relative to the joint that it is operatingacross.

The angle of the moment ( or ) is the angular excursion from B to A,viewed from J.  The vector of the moment isperpendicular to the plane of the vectors and . The tensor ofthe moment ( or ) is the ratio of the distance from J to Ato the distance from J to B. Wewill often throw away the tensor by using the norm of the musclemoment, the unitary muscle moment, . Often, we willbe primarily interested in the vector of the muscle moment, .

Muscle moments reflect the turning ability of the muscleacting at the joint. It may be easily seen that the angle of the muscle momentis a function of the muscle’s location relative to the joint (see next figure,below). For instance, a musclethat lies close to the joint will produce greater amounts of angular excursionthan the same length muscle are a greater distance. However, the more distant muscle will move the joint morereadily with the same effort, as dictated by the geometry of levers (Force xDistance = Constant = Work).

The same length muscle is shifted between threepositions along a single direction. The red version (AB) is symmetrical relative to the joint,the green version (A´B´) is positioned so that the A end isdirectly opposite the joint, and the blue version (A´´B´´) issome distance away from the joint. All three versions are a distance daway from the joint in a direction that is perpendicular to the direction ofthe muscle. The muscle has alength of l and the ABmuscle is shifted a distance d fromthe perpendicular to the muscle though the joint.

The location of the muscle relative to the jointdetermines the tensor of its moment in a non-linear fashion.

The placement of the muscle relative to the joint along theaxis of the muscle may also affect the muscle moment. If the muscle is symmetrical with respect to the joint, thenthe lengths of the insertion vectors will be equal and the moment will be aunit quaternion. If the muscle isaligned so that one insertion is as close as possible to the joint, then therelative lengths of the insertion vectors will be as great or small aspossible. If both insertions aredistant from the joint, then the tensor of the movement will also beapproximately unity.

The tensor of the muscle moment is the ratio of the lengthsof the vectors to the muscle insertions.

If we assume a muscle length of 1.0 () and vary the distance to the line of action of the muscle () and the displacement of the origin of the muscle (d), then the tensor varies in the mannerillustrated in the following figure. For the muscle close to the joint, , the behavior is much more non-linear than for the musclemoderately distant () or distant (). Therelationship is most sensitive for situations where the muscle extends toeither side of the joint ().

The tensor of the muscle moment is plotted versus theoffset for a range of offsets and distances to the line of action from thejoint. The tensor is highlysensitive to the location of the muscle relative to the joint, especially whenthe muscle lies close to the joint.

The tensor of the muscle moment will generally not be usefulin what follows and it will be common practice to compute and use the unitquaternion for the muscle moment. For most muscles the muscle passes quite close to the joint and itattaches to either side of the joint in a highly asymmetrical fashion.  That is when the tensor is apt to beapproximately maximal or minimal.

The angle of the muscle moment quaternion is the differencebetween the two insertions vectors.

Determining the Perpendicular Distance from a Joint to a Muscle

Given a muscle that has a direction and a musclemoment

,

it is possible to determine the nearest approach of themuscle line of action to the joint, . We compute theunit vector of each parameter to determine the directions of the muscle and ofthe turning vector, which we know is perpendicular to the plane that containsthe two insertions and the joint.

The unit vector in the direction of the perpendicular tothe line of muscle action through the joint is mutually perpendicular to thealready computed vectors.

We can specify the point of intersection between theperpendicular and the line of action in two different ways.  It is the point A plus some multiple of the unit vector in thedirection of the muscle action, , and it is the joint location plus some multiple of the unitvector in the direction of the perpendicular through the joint to the line ofaction, . On the otherhand the insertion A is thejoint plus the vector to the insertion.

If we write out the three component equations for the threeorthogonal basis vectors, then there are three equations with two unknowns,therefore we can determine the value of , which is the distance from the joint to the line of actionof the muscle, .

Muscle Set Surfaces as a Function of Joint Anatomy

Let us envision a generalized joint, as illustrated in theabove figure. It is a universaljoint with a spherical joint surface, so that all types of movements arepotentially possible, rather like the shoulder and hip joints.  Most joints are not that free.  In fact, the hip and shoulder jointsare not as free to move. However,we can consider most joints as special cases of this general joint.

We introduce a muscle that connects two bony offsets, one oneach bone (O_A onbone A and O_B on bone B). The offsets are attached to the bones are S_A and S_B, respectively.  The functional joint, J,is located at the center of the spherical facet. The value of each of these points is a variable and specificranges of values are characteristic of a particular type of joint. The geometryof a joint determines the special features of that joint.  A large part of the fascination instudying joint lies in seeing how their geometry determines their functionalcharacter. A detailedconsideration of the many possible variants is not appropriate here, becauserelationships between joint anatomy and joint function are complex, but a few specificexamples of different types of joints will be considered below.

Our present objective is to illustrate how a muscle setsurface may be computed for a set of muscles crossing a joint.  To start, we need to describe theanatomy with a set of framed vectors. The first defines the three points that are illustrated on bone A along with a frame of referencefor that bone. It contains alocation for the bone, L_A,and an attachment site, S_A,of the offset, O_A.   For Bone B we will use the location of the functional joint, J, as the location of the bone.

Muscle length is . It is thedependent variable of the muscle set surface. As described earlier in this chapter, the muscle moment isthe ratio of the origin to the insertion, relative to the joint.  In this instance, the muscle moment andits unit vector are as follows.

The axis of rotation for the muscle at the joint, , is the second element of the frame of reference for themuscle relative to the joint. Thefirst element is the unit vectorin the direction of the muscle, . Thethird element of the frame of reference, the perpendicular to the muscle fromthe joint, is the ratio of these two unit vectors.

Frame of Reference for a Bone Relative to a Muscle Attachment

The frame of the bone relative to a muscle (M) is an ordered set of three unitvectors: 1.) in the direction ofthe bone shaft (a),  2.) in the direction of the axis ofrotation from the bone shaft to the muscle attachment (b), and 3.)in the direction of the perpendicular from the bone shaft to the muscleattachment (g).

In this system, let us define the direction of a bone as theunit vector in the direction of the root of the offset.  In a multi-muscle system, we would haveto decide on a common direction for all the muscles, but any direction that ispicked may be expressed as a simple ratio to a particular direction and theroot of the offset has been chosen to lie on the ‘shaft’ of the bone in themodel that we are considering here. The placement of a bone will be its location and its orientation, whichwill be set to include the direction of the bone, the axis of rotation from thebone to the offset, and the perpendicular direction from the shaft of the boneto the offset. In the currentmodel, the frame of reference for a bone will be three ordered vectors.

The first component is the direction of the bone, the secondis the plane of the muscle attachment relative to the bone, and the third isthe perpendicular direction of the muscle attachment relative to the bone.

The Muscle Set Surface is an Invariant

The muscle set surface is an invariant for the geometry of ajoint/muscle system. It may beexpressed in a form that does not depend upon the particular placement of thesystem because one can always rotate and translate the joint configuration sothat one bone, say bone A, is in astandard location and orientation and then the same transformation applied tothe moving bone, bone B, will bringit along in its original relation to bone A.  All possible configurations of bones A and Bcan be realized in such a standardized musculoskeletal system.  It is a canonical image of the system.

In such a canonical system, a muscle set surface can beexpressed with complete generality as function of joint movements relative to aneutral placement. Muscle lengthis not affected by moving the system into a standard form, because internalspatial relationships are unchanged by a rotation of the system as awhole. On the other hand, musclelength varies as a function of movements from a neutral placement of the joint,irrespective of the orientation of the musculoskeletal system as a whole.

Since the movement will be the same for all muscles crossingthe joint, the complete muscle set surface may be computed and plotted againstthe same independent variables. The complete muscle set surface is the combination of the collection ofindividual muscle set surfaces.

Note that muscle force is not an invariant.  The muscle set for a shoulder remainsthe same irrespective of the orientation of the shoulder, but the forces neededto hold shoulder in a configuration may be very different, depending upon theorientation of the shoulder. Theset of muscle forces required to abduct one’s shoulder 90° in standing is quitedifferent from set of muscle forces needed to perform the same movement whenlying on one’s side or back. Theset of muscle forces is clearly not an invariant for the geometry of ajoint/muscle system.

The Calculation of Muscle Set Surfaces

We will now consider a small number of examples of thecalculation of a muscle set surface for a single muscle, plotted againstrelative bone placement in a joint that the muscle crosses.  We will consider two simple muscleconfigurations and then a more complex muscle configuration, with multiplecomponents that act differently, but in a coordinated fashion.

In each case the calculation is essentially the same.  The location of the muscle insertionfor neutral placement, O_I , relativeto the functional joint, J, iscomputed.

The muscle insertion is rotated about the longitudinal axisof bone B, through an angle , and then about a transverse axis perpendicular to thelongitudinal axis that will cause bone Bto flex or extend upon bone A,through an angle . Inkinesiological terminology, the moving bone spins about its longitudinal axisthrough an angle of and then swingsabout a transverse axis that moves with the bone, though an angle of . The initialtransverse axis is perpendicular to the plane of the muscle and the bone, the axis of the frame of reference for the bone relative to themuscle attachment.

The muscle vector, , is the difference between the new location of the muscleinsertion and the location of the muscle origin. It is the length of that is plottedversus the placement of bone B.

Rotating bone Babout its longitudinal axis once more, after the computed movement, will alterthe placement of bone B by changingits orientation. However, thatoption will not be used here. Therefore, the placement will have its orientation determined by thelocation and the orientation will have null spin relative to neutralplacement. The advantage for presentpurposes is that placement has only two dimensions, allowing us to plot themuscle set surfaces as two-dimensional surfaces in a three-dimensional space.

There are other options for creating an array of placementsof bone B. The one sketched herewill give an array that is like the lines of longitude and latitude on aglobe. In fact all of the surfacesplotted below are for a hemisphere of movement. This system seems to be a natural array for a universaljoint. Other, more restricted,joints might warrant a different type of array. In which case, the rotation quaternions might be constructeddifferently.

A Long Muscle With Its Insertion Near the Joint

For a first example, consider a muscle like that illustratedabove to illustrate the concept of a generalized universal joint, where theorigin is far from the joint and the insertion is near it.  In particular, let the origin be 0.2units from the proximal end of bone A,on an offset of 0.1 units off the axis of the bone, and let the insertion be0.1 units distal to the universal joint in bone Bwith an offset of 0.1 units in the same direction as the offset on bone A. The joint is constructed to have a radius of curvature of 0.1units. Consequently, in neutralconfiguration the muscle has a length of 1.0 units.

The muscle extends from an origin near the proximal endof bone A to an insertion near thejoint on bone B.  Both offsets are aligned in neutralposition.

The muscle set surface is fairly simple.  There is small concavity in the surfacecentered upon neutral position. Asbone B is laterally rotated through90° in either direction, there is a subtle lengthening of the muscle, so musclecontraction will have a modest tendency to bring the two offsets intoalignment. However, the much greatertendency with shortening of the muscle will be to flex the joint until it isbent about 130°, beyond which the muscle will become longer with furtherflexion and the moving bone will tend to roll laterally.  This behavior accords with ourintuitive impression of what such muscles do.

This arrangement seems to be well designed for situationswhere a large joint excursion into flexion or extension is required.  It gives large movements with modestamounts of muscle contraction. Allthe flexion movements converge on a common placement, just as the lines oflongitude converge upon the poles.

A Short Muscle With Its Origin and Insertion Near the Joint

In a second example, the muscle origin is moved distallyuntil it lies just proximal to the universal joint and the muscle insertionlies just distal to the joint and rotated laterally through 90°.  The proximal offset is 0.1 units fromthe axis of the bone and the distal offset is 0.2 units.  In words, the muscle wraps about aquarter of the way around the joint.

As one might expect, the muscle becomes longer as the jointis rotated so as to increase the angle between the offsets and it shortens whenthe angle is reduced until they are aligned. Flexing the joint also reduced the length of the muscle, butgenerally not as quickly. Musclecontraction that produces a rotation that brings the offsets into alignment andflexes the joint is the movement that causes the greatest shortening of themuscle. However, the extent towhich the muscle can bring about that movement is restricted by the gap betweenthe insertions when the offsets are aligned being smaller than the muscle canachieve. The maximal contractionof a muscle from greatest to shortest length is probably 50%.  That means that the muscle cannot moveinto the nearest corner of the surface in the illustration, because the gap ison the order of 40% of that in neutral position and the muscle must be able tobecome longer than it is in neutral configuration if the joint is able to turnlaterally in the direction that opens the angle between the offsets.  Consequently, only that part of thesurface that lies above 0.6 is likely to occur in a real system and the rangemay be substantially less.

The muscle set surface for a muscle that crosses theuniversal joint diagonally between an origin and insertion where both are closeto the joint.

As a result of these considerations, such muscles will tendto be important for laterally rotating a joint. They are most stretched and shortened by such movements andthey are comparatively insensitive to flexion and extension.  In this particular geometry, the amountof flexion is comparable to the amount of lateral rotation with muscleshortening.

A Deltoid-like Muscle

Next, consider a muscle that is in many ways like the deltoidmuscle of the shoulder. We willconsider the muscle in terms of three component muscle descriptions thatrepresent different aspects of the muscle. The first component, the middle component, extends from anoffset that directly overhangs the joint to an insertion some distance down theshaft of bone B.  The joint is considered to be inneutral position when bone B isextended 90° relative to bone A.  The other two components of the musclediffer in having their origins anterior and posterior to the joint as well asproximal to the joint. One mightimagine the offset from bone A tohave the shape of a horseshoe lying in a horizontal plane above the joint.  The insertion for all three componentsof the muscle will be at the same point on bone B.

A deltoid-like muscle takes its origin from an offsetring above the joint and it inserts into the shaft of the moving bone.  Three muscle components are drawn: oneat the apex of the offset that runs directly down, one that takes originanteriorly and one that takes origin posteriorly. They have a common insertion.

Bone B is rotatedabout its long axis through a series of angular excursions from -90° to +90°and then about an axis perpendicular to the plane that contains the shaft ofthe bone and the offset, again through a series of angular excursions from -90°to +90°. In each location bone B could be again rotated about its shaft,to give a variety of orientations, but that movement tends to have only a smalleffect on muscle length, so it is not been explored here.

In the following calculations, if the functional joint istaken to be the origin of the coordinate system, then the origins and insertionof the three components are taken to be at the following locations.

The valuesare approximations from actual shoulder joints, where the radius of curvaturefor the spherical facet is set equal to 0.1 units. One usually obtains the best results when using the valuesapproximately equal to actual anatomical values, because they usually give thebest compromise of all the possible values. By choosing values that differ from the anatomical values,one can often discover why the anatomical values are what they are.

The following figure shows the geometrical relations of thefirst component of the muscle. Itresembles the first illustration of this section, that for a long muscle thatjust crosses the joint, but it is different in a number of interesting ways.

Rotation about the long axis of bone B in a pendant position leads to minor lengthening of themuscle so contraction of the muscle will tend to move the bones towards neutralconfiguration. That effect is morepronounced as bone B is abducted(moving towards the left in the illustrated surface). The trend is reversed when there is more that 150° ofextension, however, anatomical joints would not normally support that muchextension. In actualglenohumeral joints, the range ofabduction is on the order of 60° to 90° of abduction and the range of medialand lateral rotation are usually less than 90° from neutral placement(Kapandji).

Muscle set surface for the middle component of adeltoid-like muscle where the neutral point is chosen with the arm pendant.

The more pronounced geometrical relationship is the changein muscle length as the bone B iselevated. The return oncontraction becomes less as the joint approaches 150° of elevation, but itremains the dominant consequence of muscle contraction throughout the entirephysiological range. Consequently,the contraction of the central component of this deltoid-like muscle, workingalone, tends to lift the arm directly laterally. All the contraction vectors are directed towards the centralmeridian through neutral placement and elevation.

That raises the question of what one gains by having theanterior and posterior components of the muscle. Because their arrangement is symmetrical with respect to thebones their muscle set surfaces are also symmetrical.

Muscle set surfaces for the anterior and posteriorcomponents of the deltoid-like muscle constrain the ability of a bone to movein the opposite direction and pull it towards the same direction at the origin.These two surfaces are mirror reflections of each other in the coronal planethrough neutral placement.

The anterior and posterior components are more directed atbringing bone B forwards and backwards. Rotation of the bone about its long axis will moderately lengthen amuscle for rotation in one direction and shorten it for rotation in theopposite direction. When the boneis rotated so as to lengthen the muscle, the muscle is not very effective inelevating the bone until it is already elevated about 50°.  On the other hand, when the bone isrotated so as to shorten the muscle length, the further shortening of themuscle will work to further rotate it about its axis in the same direction andto elevate it. Since portions ofthe muscle take origin anteriorly and portions take origin posteriorly, thereis an effective aid to elevation in all directions. The portion of the muscle that is contracting will act topull the moving bone further in the same direction. By resisting lengthening, an eccentric part of the musclemay prevent movements from moving into a substantial portion of the potentialmovement range. Consequently,these components may act as brakes upon movements in the direction away fromtheir original placement.

The three component muscle set surfaces consideredabove are plotted together and viewed from a different viewpoint.

The muscle set surface for a deltoid like muscle iscomplex. In the above figure thethree surfaces that were considered individually are plotted in the samecoordinate system to illustrate their differences and how they might workcooperatively in the control of the joint. In fact, the complete muscle set surface for thedeltoid-like muscle is a stack of such surfaces, a sheave, as it were.  In the figure, we see two end sheetsand the median sheet in the sheave. In addition, as mentioned above, there are other options for orientationat each location of the bone, so each of these surfaces would extend into athird placement dimension, rotation about the axis of the bone.  So the true surfaces are actuallyvolumes in a four-dimensional space. Those volumes would have a three-dimensional mesh, like the lines drawnon the surfaces that are illustrated here, so that one would follow definitepaths through the volume surface as the placement changed along meridians oflocation and orientation.

Such surfaces are complex. In most instances, three or more dimensions of placement aswell as the muscle lengths for several muscles or muscle components.  They are usually beyond our ability toreadily visualize or comprehend in their entirety. However, with judicious simplification, one can often learninteresting things about how a muscle functions, how its geometrical anatomyinfluences its functioning, and why the muscle takes the form that itdoes. There is not space here todelve deeply into these ideas, but they warrant a separate consideration atgreater length, elsewhere.

Muscle Actions

Frame of Reference for a Muscle Relative to a Joint

In the course of the derivation in the last section, wecomputed the three components of a frame of reference for the muscle relativeto the joint. The three vectors , , and form a set ofmutually perpendicular unit vectors that are related to the direction of themuscle with respect to the joint. The unit vector is the directionof the muscle, its line of action. Theunit vector is the directionfrom the joint to the line of action of the muscle. The unit vector is the turningaxis of the muscle about the joint, theaxis of rotation for the muscle pulling across the joint. These are all important directions forthe interactions between the muscle and the joint. Consequently, the frame of the muscle can be written as the ordered set of these threeunit vectors.

The Turning Index of a Muscle Relative to a Joint

The distance between a muscle and the joint that it isacting across will affect the turning angle that it can produce.  More distal muscles will cause lessrotation with the same amount of contraction. However, they are able to do the turning more readilybecause the product of the force and the distance is a constant, namely work.  If one wishes to bring two ribstogether, then the intercostals muscles, which are comparatively distant fromthe vertebral spine, will do so with comparatively little effort, but they donot move the ribs a great distance. A force applied to a rib near its joint with the spine is capable ofmuch more movement of the rib with comparatively little contraction, but muchmore effort is required.

Muscles more distant from a joint will have smallerangles for the same muscle length.

Contraction Moments

A muscle that is initially of length experiences asmall contraction , which moves the Bend to B´.  The altered muscle AB´ has a length and a direction .  If the contraction is small, then onemay generally use the vector of the turning quaternion as a goodapproximation of its axis of rotation. The actual turning moment is the ratio of the old muscle terminus to thenew muscle terminus.

The instantaneous axis of rotation is .

A muscle contracts from Bto B´, a distance of dl. The vectors to the insertion change from V_Bto V_B´ .

The reason for making the distinction is that the shorteningof the muscle may move the new muscle insertion out of line with the originalmuscle direction. For instance, ifthe result of a muscle contraction is to rotate bone B, then the muscle insertion on B is also rotated and theinstantaneous axis of rotation may be quite different from the turning vectorfor the muscle.

We can define the contraction in terms of an effort, , and an angle, , where is the anglebetween the line of action for the muscle, V_AB, and the armature from thejoint to the insertion, V_B. The normed muscle moment times the effort may be taken to bea new entity that will be called the contraction moment of the muscle.

The contraction moment is a function of the effort ofthe muscle contraction and the angle between the muscle’s line of action andthe vector from the joint to the muscle insertion. The resolution of the muscle pull into a radial and rotatorycomponent is drawn offset from the insertion.

While the muscle moment and the contraction moment look likethey should be variants of the same concept, they are quite different.  The muscle moment is a geometricaldescription of a muscle that encapsulates its turning vector, its scope ofaction, and the relative locations of its insertions relative to the joint ofinterest. We extract the turningvector and use it as the basis of the contraction moment, which is anexpression of the distribution of forces for the contraction.

Note that the vector component, , is perpendicular to the plane of the muscle and joint.  It is a torque, rather than aforce. Its magnitude is themagnitude of the vector that completes the parallelogram, r, but it is perpendicular to theplane of the parallelogram. The vector rindicates the force that acts to rotate the armature.

The scalar, , gives the relative effort in the direction of the armature,that is, compression or distraction of the joint. If the angle, , is less than 90°, then the action is distraction of thejoint, that is, ispositive. If is greater than 90°, then the action is a compression of thejoint, that is, is negative. One could equally well choose the complementaryangle, , in which case compression of the lever arm would bepositive effort and distraction would be negative effort.  The convention is set so that the forceis positive when tensile and negative when compressive.

The effort is divided into a component directed along thelever arm from the joint to the insertion, which will be called the radial,compressive, or tensile impetus, and an effort in the direction of the line ofaction of the muscle, which will be called the rotatory impetus.

The scalar component times a unit vector in the direction ofthe lever arm is the radial impetus. It is a force directed along the lever arm.

The vector component of the contraction moment is thecontraction moment torque times the lever arm. It is an impetus in the direction perpendicular to the leverarm and in the plane of the muscle and the joint. It causes a rotation about the axis of rotation of thejoint, so, the force is in the plane of the muscle and the joint and in thedirection of the right thumb as the fingers of the right hand curl from therotation axis to the lever arm.

The term impetus is used because the quantity isproportional to the length of the lever arm, so that the same force exerted ata greater distance will more readily rotate the mobile bone.  Shortly, we will render it into a forceby choosing a standard length lever arm.

The vectors and are alwaysperpendicular, because it was part of the definition of the muscle moment that is perpendicularto the plane determined by the two insertions of the muscle, A and B, and the joint, J.  Because and areperpendicular, their product will always be a vector, in this case , rather than a full quaternion. Normally, the product of two vectors is a quaternion andvectors are by definition quaternions, quaternions with null scalars, but theproduct being specifically a vector is consistent with the product behaving asa force.

The radial and rotatory impetuses are also alwaysperpendicular to each other, so their sum, M,is the sum of the two orthogonal vectors and it is aligned with the line ofaction of the muscle. Thecontraction moment describes the effort or a potential effort acting about ajoint that a muscle is capable of producing. Potential efforts are included because, when muscle actionsare combined, the actual movement produced may be different from thecontraction moments of any of the competing muscles. We wish to examine the consequences of multiple muscleacting about the same joint, however, to do so, it is necessary to define onemore concept.

Contraction Momentum

The contraction moment acts at an insertion to give a forcepair that is the product of the contraction moment and the lever arm of theinsertion relative to the joint. Let us call that product the momentum ofthe insertion relative to the joint, . It is a pairof vectors, expressed as a sum.

Clearly, is the same as V_B in the previousexpressions, but we adopt it as a more general expression of the vector to themoving end of the muscle. Then weextract the direction of that vector, , for the final expression of the momentum of muscle n. The order of the product is important because we are multiplyingquaternions. The given order inthe definition of the momentum ensues that the axis of rotation is in thecorrect direction. The momentum isthe sum of a force and a torque.

The momentum is a quaternion with the scalar being the forceof compression or distraction operating at the functional joint and the vectorcomponent is the rotating force or the torque, also operating at thejoint. The momentum is the unitvector in the direction of the insertion times the muscle contraction moment.

Moving the Contraction Momentum to the Joint

The radial force effectively acts at the functional joint, J, so we can move it to the jointand see how it would move the swinging bone upon the stationary bone.  In the following figure, in panel B,the muscle pulls with a force Mthat is resolved into a radial force, , and rotatory force, . In panel C,the radial force, , has been redrawn as originating at the joint and it isapparent that it will tend to move bone 2towards bone 1 and to the leftto roughly equal extents.

The rotatory force, r,acts differently. It will tend torotate bone 2 about the functionaljoint. The force is applied at the joint as well, but it is a product of avector aligned with the axis of rotation for the joint, , and the lever arm, . However, we can more readily appreciate the force if we visualizeit as pushing on a standard lever arm in the plane of the axis ofrotation. We draw a ray thatstarts in the joint and extends directly away. Any ray will do, however, in this case it has been drawndown the center of bone 2. Vector indicates the direction of the ray. At a distance of oneunit in the direction of we place avector, , that extends perpendicular to the ray and which has amagnitude equal to the magnitude of the radial torque times the lever arm. Thevector indicates the direction of . Because the attachment site for the muscle is less than aunit distant from the functional joint, the vector extending perpendicular tothe ray is a bit shorter than the vector at the insertion.  If the insertion were more than a unitaway then the vector would be longer. In general, for the n’th muscle, the normalized rotatory force is -

The unit vector is directedalong the radial axis through the joint and the unit vector is perpendicularto it in the plane of the torque. In the illustration is in the directionof the purple line and is in thedirection of the vector .

A muscle pulls on a tuberosity on Bone 2 to move it upon Bone 1. A. the functional joint,the origin, and the insertion are illustrated along with the vectors from thefunctional joint to the origin and to the insertion. The muscle moment is the ratio of the vector from the jointto the origin to the vector from the joint to the insertion.  B. The muscle effort is resolved into radial and rotatory forcecomponents. C.  The force components of the muscleeffort are replaced with an equivalent pair of vectors, which are the musclemomentum evaluated at the joint.

The normalized rotatory force expresses the tendency of bone2 to rotate about the functional joint,that is, about the vector of the moment of the muscle, . It is a scalarequal to the ratio of the length of the rotatory force vector to the length ofthe lever arm times the unit vector in the direction of the axis of rotation.

Combining Muscle Momenta

The combination of muscle actions requires that themuscle momenta be recast so that they are all operating at the joint.  The radial forces add according to thestandard rules of vector addition. The rotatory components also add by vector addition, but they are allexpressed as multiples of vectors in the same plane.

Now we can consider the situation where we have severalmuscles acting across a joint, each with an instantaneous axis of rotation, , an effort, , and an angle of action . We can thewrite a contraction moment for each.

The resultant of all the muscles actions will be the sumof the muscle contraction momenta.

We can break the combined muscle momentum into two parts,the radial and rotatory components, both of which operate at the joint, but indifferent ways.

Normally, the movement will change the axis of the musclemoments and a new expression for the movement will have to be computed.Therefore, the momentum is a continuously changing entity that is contingentupon the current muscle set and determining future muscle sets in amusculoskeletal system.

Problems With Muscles Working on Swing Joints

Most of the examples that will be considered here will bequite simple. We will start withexamples in which the muscles are in direct opposition and then move on to morecomplex situations where combined contractions of muscles change their turningvectors.

Axes of Rotation for Anatomical Joints

For many joints there is a single axis of rotation that isenforced by the anatomy of the joint’s articular surfaces and the tethering ofthe ligaments about the joint. Theknee joint between the femur and the tibia is set by the two condyles, whichmake the joint surface essentially cylindrical, even though the individualcondyles are ovoid. The movementis constrained by the collateral and cruciate ligaments.  The humero-ulnar joint in the elbow isalso cylindrical, because of the helical articular surface, which is locallysaddle-like. Ligaments about thejoint and the radio-humeral joint prevent much rotation about the trochlea ofthe ulna. Normally, saddle jointspermit rotation about two separate axes of rotation, which lay on oppositesides of the articular surface, but mediolateral swing is blocked by thestructure of the elbow joint.

The gleno-humeral joint has a nearly spherical surface, asdoes the trochanter of the femur. Therefore there is a great deal more variation in the axis of rotation,which depends upon the placement of the humeral head relative to the glenoidfossa or the trochanteric head relative to the acetabulum.  Since both are universal joints, therange of the axis of rotation is largely constrained by ligamentous tetheringand abutments with nearby structures.

A Simple Example: One Muscle at a Substantial Distance from the Joint

Now that we have laid out a few definitions andrelationships, it is time to consider a concrete example, to see how thesefactors come into play. The firstexample is very simple. Assume asingle muscle, , the one between the apices of the vertical processes of thetwo bones in the second illustration of this chapter. Let the joint linkage of bone Abe 2.0 units and the joint linkage of bone Bbe 1.0 unit. The verticalprocesses are 1.0 unit long and each insertion is at the apex of itsprocess. The framed vector forbone A in neutral position will beas follows.

The framed vector for bone B will be as follows.

The muscle direction is the difference between its insertions,. If the musclecontracts, then its moment is the ratio of its origin to its insertion.

The muscle is attached at some distance from the joint,but with a long lever arm. Thiscombination gives modest movement with moderate contraction and effort.

For a small contraction, the new arrangement of bone B can be computed from the muscle momentand the framed vector of the bone. Let the excursion be 10°.

If you recalculate the muscle moment, you will find that itnow has an angle of 98.435° about the +kaxis. The muscle has contractedfrom a length of 3.0 to a length of 2.82. We obtained a 9.25% change in the angle of the muscle moment with a6.15% shortening of the muscle.

Note that the muscle must contract to a third of itsoriginal length in order to bring the link armature of B to a 90° angle with the link armature of A. That is very non-physiological. That is why muscles that attach in this manner do not produce largechanges in the angle of a joint. Muscles can in theory contract to about half their length if they startstretched to the point where the sacromere elements are just about to disengageand contract to the point where they are maximally overlapping.   Muscles normally work in a muchsmaller range. Let us consider amore natural arrangement of the muscle with respect to the bones and the joint.

The momentum of the muscle working at the joint is readilycalculated. The lever arm is at a45° angle to the joint linkage for bone B.  Therefore, and . The muscle isat an angle of 135° to the lever arm. That leads to the following momentum.

The radial force at the joint is compression at a 45° angleaway from the muscle insertion and a rotatory force about the axis of themuscle moment. The radial force isabout 0.7 times the rotatory force.

A More Natural, Still Simple Example: Muscle Crosses Joint Near the Joint

Let the attachment site on bone A be about 0.1 units alongthe vertical process and the attachment site upon bone B be about 0.1 unitsbeyond the joint on the link armature. The framed vector for bone Ain neutral position will be as follows.

The framed vector for bone B will be as follows.

The direction of the muscle is the difference between itsinsertions, . If the musclecontracts, then its moment is the ratio of its origin to its insertion.

The great majority of the muscle, about 20/21, is on theA side of the joint. The musclesubtends almost 180° relative to the joint and the axis of rotation is in thepositive k direction.

The muscle is attached in such a ways as to operateclose to the joint and to have a long belly. This combination produces large movements with smallcontractions.

For a small contraction, the new arrangement of bone B can be computed from the muscle momentand the framed vector of the bone. Let the excursion be 10°, as is the previous example.

If we do the calculation, then the result is the followingframed vector for bone B.

If you recalculate the muscle moment, you will find that itnow has an angle of 167.138° about the +kaxis. The muscle has contractedfrom a length of 2.1024 to a length of 2.1001, a difference of 0.1%. We obtained the same movement as in thefirst example with a much smaller contraction. A little thought will reveal that the linkage for bone B can be taken through 90° of rotationabout the joint by shortening the muscle to a length of 2.0, which is slightlyless than a 5% contraction. A 10%contraction will rotate bone Bthrough almost 180°.

In general, placing an insertion near a joint and anotherinsertion at some distance will produce large angular excursions with smallmuscle contractions. Being nearthe joint, such muscles must pull harder than more distant muscles to producethe same movement. We will seethat a large fraction of the effort is going into compressing the joint.  The relative magnitudes of the scalarand vector components of the contraction moment reflect the relative fractionsof the effort that is going into compressing or distracting the joint versusrotating the joint.

The momentum of the muscle working at the joint is againreadily calculated. The lever armis at a 0° angle to the joint linkage for bone B.  Therefore, and . The muscle isat an angle of 177.27° to the lever arm. That leads to the following momentum.

The radial force at the joint is compression along the axisof bone A and a small rotatory forceabout the axis of the muscle moment. The radial force is about 21 times the rotatory force.  This arrangement is not as efficientway to rotate bone B on bone A. Most of the effort goes into compressing the joint.

Insertion Offset From the Axis of the Bone

Let us consider a slight anatomical modification to the lastarrangement. Place the insertionon a tubercle that extends 0.1 units away from the shaft of the joint linkageprocess. The framed vector wouldbe as follows.

The efficiencies of the last arrangement are retained inthat small contractions produce large angular excursions.  Let us consider the momentum.  The expression is a mixture of theprevious two expressions. The lever arm is at a 45° angle to the joint linkagefor bone B.  Therefore, and . The muscle isat an angle of 135° to the lever arm. That leads to the following momentum.

The radial force at the joint is compression at a 45° angleto the axis of bone A and a smallrotatory force about the axis of the muscle moment. The radial force is about 7 times the rotatory force.  This arrangement is about three timesas efficient for rotating bone B onbone A when compared to the lastarrangement. Still, most of theeffort goes into compressing the joint, but we are getting a substantiallybetter proportion going into rotating the joint, which is normally the desiredconsequence of the muscular effort. If we move the tubercle closer to the joint, then the efficiency of themuscle in rotating the joint increases. If the tubercle is adjacent to the joint, then all the effort goes intorotating the joint. (). Consequently,we tend to get maximal efficiency in turning a bone in a joint if we place themuscle insertion near the functional joint, on the side of the muscle.

Contrary Efforts

Next let us consider two muscles that act at the same joint,but in opposite directions. Letthe first be the muscle that we have just considered and the second be itsreflection across the joint. Theframed vector for bone A in neutralposition will be as follows.

The framed vector for bone Bwill be as follows.

The muscles are then the difference between theirinsertions,

If the muscle contracts, then its moment is the ratio of itsorigin to its insertion. We mayreadily compute both muscle moments. The difference is the direction of theturning vector.

The combined muscle momentum is the weighted sum of the twocomponent muscle momenta.

If the efforts are equal, then the combined momentum iscompression along the shaft of bone A,because the component momenta cancel each other. The limb does not move, but the joint is compressed with aforce that is twice that generated by each muscle. If the efforts are not equal then the joint will rotate,because there is a non-zero rotatory force.

The momentum is composed of two components, a radialcomponent and a rotatory component. If one of the efforts is zero, then the combined momentum becomes equalto the remaining effort. When bothefforts are the same, the rotatory component becomes zero and the two radialcomponents add vectorially.

Opposing Forces

If and are the musclemoment axes of two muscles that act in direct opposition to each other, as inthe situation that we have just considered, then their combined action may beto hold the limb in a single placement or to move thelimb in the direction of one of the moments . If the twomoments are not in direct opposition, then there is a combined muscle momentthat is not aligned with either muscle moment vector .

Pairs of muscles may work cooperatively to move the bonesabout an intermediate axis; thereby creating a virtual muscle that behavesdifferently than either component muscle alone. The next few sections will consider a few musculoskeletalsystems that do not have the muscles in direct opposition and therefore areable to move the swinging bone about an axis different from the axis of anymuscle in the set. These are moreinteresting, because they allow the bones to move in a wide range ofdirections.

The simplest such systems are those that have threenon-coplanar muscles. In suchsystems pairs of muscles can act together to create virtual muscle with anyaxis of rotation in the plane of the two muscle axes that can be expressed asthe positive sum of the two axes of rotation. Muscles can only pull, so, negative coefficients of the axesof rotation are not permissible. The third muscle can oppose all virtual muscles formed by the othertwo. By taking different pairs ofmuscles, one can create axes of rotation in a full circle of directions.

We will find that three muscles generate geodesictrajectories, but the moving bone can attain only orientations that have nullspin relative to the starting orientation. In order to reach other trajectories, one must have at leastone more muscle.

Three Opposing Muscles with Muscle Moments in a Single Plane

Let us return to the original diagram of the musculoskeletalsystem and consider the muscle moments of the three muscles that link theapices of similar processes (V_AV_B,R_AR_B, and L_AL_B).  Let the processes be one unit long andthe linkage processes be one unit long as well. Then we can write down the frames for the two bones.  The muscle attachments are 120° apartand equally far from the center of the bones. Therefore, we can write the framed vectors by inspection.

The muscles that join similar vertices areillustrated. They are capable ofrotating bone B along geodesictrajectories from neutral position, which is shown here.  Equal co-contraction of all threemuscles will compress the joint.

The direction of all the muscles is , but the vectors of the axes of rotation are rotatedrelative to each other by 120°.

All the axes of rotation are in the same plane, a verticalplane through the joint. Clearly,any combination of actions in these three parallel muscles must also lie inthat plane. In a system of threeopposing muscles that have all of their muscle moment vectors in the same planethe system is stable in that the resultant muscle moment vector lies in thesame plane.

All of the trajectories that are generated by these musclesare great circle or geodesic arcs for the shaft of the bone.  The orientation of the moving bone isdetermined by the initial orientation and the direction of the arc.

The three lever arms are the hypotenuses of right triangles,so we can write down their directions and magnitudes and the angle between eachlever arm and its muscle is 135° in every case. That information is sufficientto allows us to write down the combined momentum.

We can now massage this equation,simplify, and re-arrange the terms to obtain a more useful form for ourpurposes.

This is a complicated formula, but some examination willreveal that it has the expected symmetries. Setting any two efforts equal to zero gives the momentum ofthe third muscle. If all threemuscles contract with equal force, then the bone will not move, because the sumof any two will be the opposite of the third. The joint will be compressed in the direction of the shaftof bone A with a force that is oneand a half times the contraction forces, .

The axis of rotation is always in the j,k-plane, perpendicular to the shaft of the jointlinkage for bone A.  Because all the axes of rotation areconfined to the same plane, all the rotations are geodesic trajectories thatretain null spin relative to the orientation of bone B in neutral position. The orientation of bone B isdetermined by its position and the neutral orientation.

If the muscles are not parallel to the joint linkages of thebones in neutral position, then the trajectories are not geodesic, because thethree axes of rotation are not is a single plane, but the orientation of bone B is still determined by the position ofthe bone. We will consider such a situation next.

Three or More Muscles with Muscle Moments Not in a Single Plane

Let us now consider a second set of muscles, the musclesthat extend from one process to the next process in a clockwise direction (V_AR_B, R_AL_B,and L_AV_B).  These muscles are illustrated in thefollowing figure. We could equallywell use the ones that are next in a counter-clockwise direction.  The framed vectors are the same as inthe last section. So, we canreadily compute the muscle directions.

First we define the muscles.

Three diagonal muscles are illustrated.  They connect each process to the nextprocess in a positive direction, if the rotation axis is directed towards thejoint. These muscle configurationswill rotate Bone B in an obliquetrajectory, when viewed end-on or from the side. Consequently, they do not produce geodesic trajectories ineither of those frames of reference and they impart a twist to the orientationof the bone. When all threemuscles co-contract with equal force, they cause bone B to rotate about an axis aligned with the linkage process tothe joint.

If we convert to contraction moments and sum the threemuscle contractions the result is as follows.

If each muscle makes the same effort (), then the contraction moment simplifies to an interestingexpression.

We can compute the muscle contraction momentum much as wedid in the last example. In fact,some of the parameters are the same. The lever arms are exactly the same as in the case with parallelmuscles. We have to recalculatethe angle between a lever arm and a muscle by taking the ratio of their vectors.

Each muscle independently will cause bone B to make an oblique rotation, but,working together, they cause the bone to spin on its axis.  The rotation is about the axis thatextends through the links and the joint.

The final expression is written in that form because the twoterms are describing two different forces. The first says that there is compression along the axis ofbone A equal to about 1.98 times theeffort of each muscle and the second term says that there is a rotation aboutthe axis of the bone with an effort about 1.13 times the effort of eachmuscle. Bone B rotates about the axis of the linkagewith bone A in the direction thatwill bring process R of bone B into alignment with process V of bone A.

This is a fairly difficult computation to obtain a resultthat we can obtain with a bit of intuition, however, the power of the approachbecomes apparent when we do not choose a situation with such symmetry ofeffort. It is not difficult towrite a program that incorporates the basic approach that has been illustratedhere and use it to compute the trajectory of the bones under any statedconditions. All that is requiredis a statement of the placement of the two bones with the extensions for themuscle attachments and the definition of the muscles in terms of theirattachments.

By spinning about its axis, the insertion sites upon bone B move into alignment with the insertionsites on bone A.  In all positions short of alignment,the muscle directions are tilted in three oblique directions, but the sum ofthe contractions always generate a contraction moment that is in the directionof the i axis. Eventually, the axes will becomealigned and the muscles will be parallel, as in the previous example, the axisof rotation will become the null vector, and the force of contraction will betotally compression of the joint.

As bone B isrotating through the 120° to bring it into alignment with bone A, the muscles that were parallel arestretched as they move into a diagonal alignment, the opposite of the alignmentof the diagonal muscles that we started with. Consequently, by a suitable choice of the lengths of theparallel and diagonal muscles one may obtain any orientation in a 120° arc.

It Takes Three Muscle Pairs or Six Axes of Rotation to Reach All Placements

With three muscle moments, there are placements that onecannot reach. For the situationwith the parallel muscles it was possible to reach only those placements thatwere on great circles from the initial placement. In fact, most placements are impossible to achieve with onlythree muscles. In addition, sincemuscles cannot exert negative effort, there are many locations that cannot bereached.

All the muscles are illustrated with bone B in neutral position.  With all six muscles, it is possible tocontrol the location and orientation of bone B.  Shortening the diagonal set of muscleswill lengthen the parallel set, so that with 60° of rotation one has twodiagonal sets of muscles with opposite sense. With 120° of rotation, the diagonal set becomes a parallelset and the parallel set becomes a diagonal set with the opposite sense.

Muscles cannot have negative moment vectors.  Therefore, one often sees threeopposing pairs of muscles to produce rotations in all directions.  Three opposing pairs or six vectors aresufficient to generate all placements. More generally, six non-coplanar vectors allow for full independentcontrol of placement (location and orientation). For instance, the bone may be rotated until it has thedesired orientation and then moved along a geodesic until it is at the desiredlocation. By pairing muscles onemay create parallel muscle equivalents.

Muscle Sets that are Able to Determine Placement

A bone can rotated to a location by two coplanar unit momentvectors and then a third moment vector that travels with the bone can be usedto rotate about its axis to obtain the desired rotation.  The process can be reversed by settingorientation and then location.

Most Movements Continually Change Muscle Moments and thus ContractionMoments

As the bones rotate, the muscle insertions rotate relativeto each other, which means that their moments change.

For instance, if the diagonal muscles shorten then they rotatebone B upon bone A and make the parallel muscles shift from coplanar totetrahedral while the diagonal muscles shift from tetrahedral to coplanar.  By partially rotating the bonesrelative to each other, we can effectively create parallel muscles by combiningpairs of muscles. Because of thisphenomenon, it is possible to rotate bone B until it is spin neutral to itsfinal placement, then use the virtual parallel muscles to rotate it along ageodesic trajectory to the final location.

Adjacent muscles will have directions on opposite sides ofthe geodesic plane. Thereforeadding them in correct weightings will produce a vector in the geodesic plane.

The Geometry of Axes of Rotation Determines the Nature of PermittedMovements

Parallel Muscles Produce Geodesic Movements; Non-aligned Muscles ProduceConical Rotations

For any combination of three non-aligned muscles that havean effort greater than zero in every muscle, the resultant contraction momentvector is not in the plane of any pairs of muscle moment vectors.  If the three muscles are aligned, thatis parallel, then their contraction moment vectors must be coplanar, whichmeans that the trajectories that result from their co-contraction are geodesicsfor that rotation plane. If themuscles are not aligned or, equivalently, they have non-coplanar muscle momentvectors, then the trajectories are not geodesic for any of the planes of anypair of muscle moment vectors, unless one muscle is not contracting.  There will be a twist in orientation asthe bone moves. In kinesiologicalvocabulary, there will be a non-pure swing. If one of the three muscles is not contracting, then therange of possible trajectories is restricted to those that have axes ofrotation in the plane of the two remaining muscle moment vectors, between thevectors, that is, the vectors that may be expressed as a positive vector sum ofthe two muscle moment vectors.

This is equivalent to stating that three independent vectorsform the basis of a three-dimensional space, except that, since musclescontract, only positive sums can exist. Consequently, only the convex space bounded by the planes determined byeach pair of vectors contains possible moment vectors.

Spin and Swing are Relative Quantities of Movement

It should be noted that any rotation in which a referencepoint on the moving object remains in a plane may be expressed as a geodesicrotation by choosing another reference point relative to the object.  For example, as the earth rotates, thecity of Saskatoon, Saskatchewan, traces out a conical rotation.  From the point of view of Saskatoon theEarth is experiencing a conical rotation. However, a city on the equator is experiencing a pure swing and thenorth and south poles are experiencing pure spin. In our current situation, we can truthfully and meaningfullysay that bone B will experience aconical rotation under the stated conditions, but we should always keep thethought in the back of our mind that there is a context in which the movementmight be seen as a pure swing or a pure spin. The implication for our present considerations is that themoving bone will experience a concurrent twisting rotation as it moves.  That is why the wording is so carefuland complex in the previous paragraph.

Any conical rotation can be expressed as the product of apure spin and a pure swing. Elsewhere, we illustrated a protocol that will always give two componentmovements that combine to give the same outcome as the conical rotation(Transformations of Orientation: Revisiting Swing and Spin).  Similarly, we can argue that it takessix independent muscle pairs to guarantee that a muscle set will give anyplacement within a given range. The pairs need not actually be paired, but it should be possible togenerate six independent movements. The reason for pairs is that muscles can only contract, so, if one is tomove in both directions along a line of action, then it is necessary to havemuscles to pull in both directions.

In the chapter about the movements of the eyeball(Transformations of Orientation in a Universal Joint) there were six muscles,therefore the eyeball could potentially be placed in all placements in acontinuous region of the placement space. It was noteworthy that functional constraints reduced the actual rangeto the placements with null spin relative to neutral gaze.

Co-contraction of Muscles Will Produce Intermediate Actions

The two muscle moment vectors, V₁and V₂, determine a plane and a perpendicular vector P₁₂.  Any reference point, R₁₂,on P₁₂ will experience geodesic rotations when rotatedabout V₁, V₂or any combination of both vectors. Muscles may combine to form virtual muscles that have muscle momentvectors that are positive linear combinations of the muscle moments vectors, aV₁ + bV₂.

By the co-contraction of pairs of muscles one may obtainvirtual muscle alignments that are intermediate to the alignments of themuscles. In the case of the threeparallel muscles, we were able to obtain any trajectory that was a geodesicthrough neutral position. The planefor the three parallel muscles is the geodesic plane for neutral position.  That is to say that if the location ofthe bone in neutral position is taken as the reference point then all therotation that are generated by combinations of the muscle actions will be greatcircle that passes through the bone’s location in neutral position.

More generally, if we have two vectors, they define a planeand if we take a reference point on a line that passes through the center ofrotation in the direction of the ratio of the vectors or its negative, that is,upon a line perpendicular to the plane defined by the vectors, then themovements produced by those muscles will produce geodesic movements of thereference point. Those movementswill have an axis of rotation that may be expressed a positive sum of themuscle moment vectors. In the caseof the parallel muscles, the three muscle moment vectors were all in the sameplane and each pair was able to produce a sector. The three sectors comprise the entire set of possibledirections, therefore the three parallel muscles are able to generate geodesicmovements through neutral position in all possible directions.

The axis of rotation for three muscles pulling togetherlies in the region between the axes of rotation for the individual muscles andit is a weighted sum of the individual axes of rotation.

When the three muscle moment vectors are not coplanar, thenthe possible combined contraction moments vectors will lie in the sector thatis defined by the three directions. There is no single reference point, other than the center of rotationthat is on the perpendiculars to any two muscle planes, therefore, there is noreference point that will experience geodesic rotations for two differentcombinations of the muscle contractions. Consequently, at a deep level, the rotations produced by muscles withnon-coplanar muscle moments are conical, non-geodesic, rotations.

Four Turning Vectors Reach All Locations, Six Reach All Placements

It turns out that if we have four non-coplanar muscle momentvectors, then we can take them in triplets to generate four sectors thatcompletely exhaust the possible muscle moment vectors.  One can see this by allowing the threemuscle moment vectors to point to the vertices of a tetrahedron.  The movement vectors that are positivesums of the triplet will pass though the face defined by the three elements ofthe triplet. Each triplet definesa face of the tetrahedron and the four faces of the tetrahedron form a completeclosed surface. Therefore, allpossible muscle moment vectors can be expressed as a linear positive sum of thefour muscle moment vectors.

In practice, we generally do not need to be able to move toall possible locations, only locations in a sector of space, therefore, theoretically,can make due with three non-coplanar axes of rotation.

All possible rotations have been accounted for so fourunaligned muscles would be sufficient to move any anatomical object from anylocation to any other location on the surface of a sphere.  They would not be sufficient to movethe anatomical object to a particular location with a particularorientation.

Four muscle moments determine two rotation axes.  All rotations are geodesic rotationsfrom neutral position. Orientationis always null spin relative to the orientation in neutral position.

Generally, we wish to move through a sector, therefore a setof three muscles will be sufficient to move an anatomical object to anylocation in that sector. However,it is possible to choose two pairs of muscles such that each pair has oppositemuscle moment vectors and the different pairs have differently oriented momentvectors. The four recti of theeyeball are an example of such an arrangement. Since we have only two effective axes of rotation, allmovements in such a system will be geodesic, with null spin relative to neutralposition.

For rotations that change location, there are only twodimensions (up/down, right/left). The third dimension would be depth or radial distance and it is not possibleto change it with a rotation in a single joint alone. In a system with two joints, radial distance may be avariable. If we addanother axis of rotation, aligned with the vector of the plane of the other twoaxes, then we can change orientation.

In a one joint system, that is all the control that ispossible for the modification of orientation. Consequently, a single joint system has a maximum of threeindependent dimensions of placement. With a two joint system, one may obtain the full six dimensions ofplacement. An eyeball in its orbitis a single joint system, so it needs only six muscles.  If we view eye movements in the contextof a moving head, then there are two joints and location and orientation eachhave three dimensions.

One may see how to control both location and orientation ina single joint system by imagining a system in which one may move a bone byrotating it about its shaft until one reaches the desired orientation and thenmoving the bone along a geodesic trajectory. Alternatively, one may move the bone to the correct locationan then rotate it about its shaft to obtain the desired orientation.  With such a system one can obtain allpossible placements within the bounds set by the muscle and bone geometry.  One might imagine a set of four alignedmuscle that move the bone up and down and from side to side and two musclesthat rotate it about its axis.

Note that it is not necessary for the muscle moment vectorsto be orthogonal or parallel. Itonly helps to see the actions of the system when such is the case.  For instance, the six muscles thatcontrol the eyeball are not orthogonal. Each muscle contributes to varying degrees to both swing and spin of theeye, depending upon the placement of the eye. The details of these interactions are complex, but not inprinciple different from the system of mutually orthogonal muscle pairsconsidered here.

Joint Action Spaces Generally Have Reduced Action Dimensions and FixedRelationships Between Location and Orientation

Even though the eyeball has the potential for a relativeindependence between the orientation and the location of the eye, it does notuse that freedom, because there is a functional constraint upon the eye tomaintain the visual image of the world upright upon the retina.  Consequently, if we know gazedirection, we know gaze orientation. In brief, orientation is spin neutral relative to neutral gaze.  A system capable of three degrees offreedom uses only two. In fact,there is reason to think that the eye can use all three degrees of freedom whentrying to view a world tilted relative to vertical. The eye is rotated about its line of sight to partlycompensate for the tilt. The sametype of compensatory correction occurs if one tilts one’s head when looking at ascene. However, that is a wrinklethat we will not address here.

Many musculoskeletal systems reduce their potential degreesof freedom when performing their normal actions. For instance, most bones do not have independent control oflocation and orientation. Often,as with the eyeball, there is a functional covariation of location andorientation, where knowing the location gives a good estimate of orientation.Another way in which anatomical movements may be more restricted is whencertain directions of movement are prevented by bone architecture.  For instance, in the ulnar-humeraljoint, the trochlear facet restricts the movements to essentially onedimension. The ulna follows ashallow helical trajectory, which is responsible for the phenomena of carryingangle. In isolation, theulnar-humeral joint does have the potential to rotate the ulna about its longaxis, but that does not happen because of ligamentous tethering and abutmentswith other bones.

In the temporomandibular joint there is also normally asingle dimension of movement, essentially back and forth along a bony ridge,but the jaw has a complex trajectory with respect to location and orientation,because changing location leads to an intricate sequence of orientations thatrotate the jaw. It is furthercomplicated by the joint having two widely spaced facets, on either side of themouth.

In fact, there are probably very few joints where there isrelative independence of location and orientation, such as the gleno-humeraljoint of the shoulder and the trochanteric hip joint. Even those joints tend to have a certain orientationassociated with a particular location, but we can move some distance away fromthose usual placements, if we so choose. In a therapeutic setting, it is common to move the joint into the unusedparts of its potential movement space, in order to stretch muscles and/orconnective tissue. Sometimes ajoint moves into an abnormal parts of its movement space as a result of externallyimposed forces.

Neural Control of Placement

Because of the geometry of many joints, there is not asimple relationship between location and orientation and we seldom move along asingle dimension in natural movement. When we reach for a cup, our shoulder is changing its direction andorientation together in a complex manner. The intricacy of such movements is good evidence that we do notrepresent movements in terms of location or orientation, but in terms ofplacement. The most efficientmovements are probably ones that move on smooth, comparatively simple,trajectories in placement space.

In our analysis of the movements of the eyeball, the mostefficient movements were not the shortest trajectories in location space ororientation space, but they are conical rotations that change both at the sametime in a well defined manner.

Non-muscular Forces

Inertial Forces

Not all the forces acting across a joint are due to musclecontractions. We will brieflyconsider two other types of forces. The first are the inertial forces, due to accelerations.  These come in two basic varieties thosedue to gravity, the weight of the limbs, and those due to acceleration of partsof the musculoskeletal system relative to other parts.  The inertial forces are usuallyexpressed as the mass of the part concentrated at the center of mass or thecenter of inertia. Other impressedforces may be expressed as a force applied at a particular point.

Bone A articulateswith bone B in joint J.  A force Fis applied to bone B at point C. The force will cause a compression at the joint in a direction oppositeto the vector from the joint to the point of contact, L = C – J, and rotation about an axis thatis mutually perpendicular to both Land F.

The force Fis applied at a point C onbone B. The point of application is separated from the joint, J, by a vector L. We can compute a unit vector in the direction that is theratio of F to L.

We can write down the momentum of the applied force as wasdone for the muscle momenta.

Tethering and Abutment

The second type of non-muscular force is the resistive forceassociated with tethering and abutment. Joints are usually stabilized and constrained by ligaments that tetherthe joint, not allowing two points on different bones to separate by more thana fixed distance (). There may besome stretch in the ligament, but usually very little, because the purpose of aligament is to restrict movement in certain directions.

Abutment is often associated with tethering, because thetethering forces articular surface together by restricting rotation.  Two surfaces come into abutment whenthey are forced together and they cannot move any closer.  Two articular surfaces may not occupythe same region of space (). Once again,there is often some give, but generally not much.

Tethering occurs when points on two bones becomeseparated by a distance equal to the length of the tether.  Abutment occurs when two points ondifferent bones have the same location.

Both of the situations create a passive force in a structureor structures that resists further movement. The force generated is the sum of the active forces frommuscle contractions and the inertial forces from the weight of the limbs andthe loads being moved as they apply at the attachments of the ligament or thecontact in an abutment. The twoanatomical structures work in similar and different ways.

Both do not enter into consideration until a condition ismet. For the tether, there is noforce until the ligament attachments on different bones move a certain distanceapart. At that point furtherseparation is not longer an option. Depending upon the organization of the ligament with respect to thejoint, the movement may be limited in certain directions, but not others.  There are always some options forfurther movement. Generally,ligaments allow the bone to swing in an arc about each other.  However, movement in certain directionsmay be blocked by abutment and the joint becomes fixed and cannot bend furtherin that direction.

In the knee, the arrangement of the lateral and medialligaments and the cruciate ligaments is such that one can move into aclose-packed position at the end of knee extension with a fillip of rotation tolock it in place. The ligamentsrestrict the direction of movement and abutment prevents the knee fromextending beyond a certain point. However, reversing the terminal rotation allows the knee to bebent.

The lumbar intervertebral discs are constructed so that theintervertebral ligaments allow a certain amount of rotation and rocking betweenthe vertebrae, but they firmly stop all movements beyond that measure.  The facet joints operate primarily byabutment. The two together controlmovements between vertebrae.

While abutments tend to compress the joint by forcing thearticular surfaces closer together, they also tend to open joints, by formingan alternative articulation or fulcrum. The functional joint may suddenly move to a new location, which may bedangerous, if unanticipated. Themuscle forces are arranged about the joint so that they are in balance for aparticular fulcrum. If the fulcrumsuddenly changes, the forces that the muscles must bear may suddenly change anda muscle will be overstrained before it can react to protect itself. Otherstructures about the joint may be forced to assume strains that they were notmeant to deal with, leading to tearing of their fabric.  Many joints seem to combine ligamentsand potential abutments so that the ligament is in place to prevent theabutment or to take the strain if and when it occurs. Muscles are not good candidates for that role, because theytake time to react to strain and they may be excessively stretched before thecompensatory response can be mounted.

Abutments also experience a passive strain that depends uponthe various forces operating about the joint. The forces can be calculated by computing the combinedmomentum with the join at the point of abutment, rather than at the usualfunctional joint.

The following chapter considers the manner in whichligaments and abutments restrict movement. To consider those points here would take us away from thegeneral themes developed in this chapter and make this chapter too long.  It is too long already, so we will wrapup and consider a new set of ideas in a fresh chapter.

Pulling It Together

It should be obvious that only certain combinations ofmuscle lengths can occur in any given musculoskeletal system.  As some muscles shorten, others mustshorten or lengthen as well. Theserelationships are implicit in the rigidity of the bones and the constancy ofthe muscle insertions relative to the bones. That relationship between the lengths of the muscles and thebones that they link across joints is the muscle set.

A little thought will reveal that the muscle set is asurface. For every value of theplacement of the bones, there is a unique set of muscle lengths.  Generally, the surface will be morecomplex than our usual experience of two dimensional surfaces in a threedimensional space. The independentvariable is placement, which may be one dimensional, as in a hinge joint, threedimensional, as in a universal joint, or six dimensional, as in linkeduniversal joints. Placement mayhave even more dimensions, when we are considering a system with multiplejoints. For instance, the lowercervical spine has six independently moveable sets of joints () with two independent axes of rotation.  Each bone can change its location andits orientation relative to its neighbors.

Each muscle in the muscle set adds a dimension.  For instance, the eyeball has sixextrinsic muscles, therefore six dependent dimensions.  It has two placement dimensions,therefore the muscle set surface is a two dimensional surface in an eightdimensional space. The number ofplacement dimensions is the number of dimensions for the muscle set surface andthe sum of the placement dimensions and the muscle dimensions is the muscle setspace.

We are using independent and dependent in a formal sense,because, in practice, we manipulate the muscle lengths to move the bones.  Placement is a function of the musclelengths. However, changing theplacement of the bones changes the muscle lengths. Each reflects the other through the relationships that weare calling muscle set.

The muscle set is the surface of permissible muscle lengthcombinations plotted against the bone’s placements. It is generally not possible to visualize such a space, butthere are tricks that allow us to get some insight into the structure of such aspace. They compromise some of thefeatures and preserve others.

Contour Maps of the Muscle Set Space

One such trick is to construct a contour map.  When we want to illustrate ageographical region in terms of all of its ups and downs, we often resort to acontour map. The procedure usuallyinvolves computing the curves that have a constant height in the landscape, aset of concentric lines on a flat sheet of paper. On occasion, we want to illustrate how the elevation variesalong vertical slices through the landscape and we stack the boundaries of theslices in a horizontal projection upon a flat vertical surface.

With a muscle set, it is often convenient to slice thesurface so that it is projected upon a surface perpendicular to the axis for amuscle. In the instance of theextrinsic eye muscles it was convenient to slice the surface in the directionsof the individual muscles. It gaveus a set of six contours that could be plotted against placement, which hasonly two dimensions for the eyeball. Consequently, we could plot the projection of the surface for eachmuscle, a stack of six curvilinear planes (see the initial figure in thischapter). However, what is shownis a contour map of a two dimensional surface in an eight dimensional space.

Dimensionality of Muscle Set Spaces

In general, the dimension of the surface is the number ofindependent placement dimensions. In the instance of the eyeball, or any universal joint, there areeffectively three degrees of freedom, medial/lateral, elevation/depression, androtation about the line of gaze. The first two give the location or gaze direction and the third give theorientation. However, because offunctional constraints embedded in Donder’s and Listing’s laws, orientation isdetermined by gaze direction. Consequently, there are only two independent placement dimensions.  It is usual for the anatomy of a jointto introduce constraints on the number of independent placement dimensions fora system.

In general there is a dimension for the muscle set space foreach muscle, but that too can be altered by the anatomy.  If we start with a strict hinge joint,then the placement has a single dimension, joint angle, and there is a muscleor set of muscles the moves it one way and another muscle or set of musclesthat moves it in the opposite direction, but if we know the value for one, thenwe can calculate the length of the other muscle. The muscle set space has two dimension, one for joint angleand one for the length of one muscle or set of muscles.  If we have multiple muscles pulling oneor both directions, that would not fundamentally change the situation.

That does not mean that there is not interesting anatomy insuch a joint. For instance, onemuscle might be attached so as to be efficient is getting the joint moving atan extreme position. A secondmight be arranged so as to give power when the joint is in anotherplacement. The supraspinatus anddeltoid muscles operate in this manner when abducting the arm at thegleno-humeral joint in the shoulder.

Different Dimensional Muscle Set Surfaces

As statedabove, another way in which muscle set surfaces differ from our common image ofthem is that they may be one dimensional, as in the example of a hinge jointthat we just considered. They maybe two dimensional, as in the case of the extrinsic eye muscles.  They will usually be three dimensionalfor universal joints and they may be higher dimensional where the placement hasmore independent dimensions.

If we wished to consider the placement of a hand whilereaching from a fixed body, then the muscle set surface would be sixdimensional. We are warranted incalling a six dimensional surface a surface because it has fewer dimensionsthan the muscle set space in which it is embedded. In the hand placement example, there might be dozens ofmuscle dimensions. There couldeasily be a dozens muscles involved in the shoulder alone.

Collapsing Muscle Set Spaces

A little thought will lead one to realize that in the lastexample one of the remarks made at the start of this section no longerholds. The muscle set surface isno longer unique to the placement. We know that if you grip a stationary pole, thereby fixing the placementof your hand, and without moving your body, you are able to move your arm andforearm through a substantial excursion. Those movements change the lengths of many of the muscles in the muscleset, because the muscle attachments are moving relative to each other as thebones move relative to each other. It is obvious that the muscle set surface is not unique to the placementof the hand. That is because wehave collapsed the full muscle set space. If we introduce the placement of the scapula, humerus, radius, ulna, andthe carpal bones, then the muscle set surface is unique.  The space is also much larger.

We can collapse muscle set spaces by projecting the fullsurface into a smaller number of dimensions, much as sunlight shining throughshear curtains will create shadows where multiple layers of curtains exist atthe same point in the shadow.

Collapsing muscle set spaces is not a bad thing.  It reflects reality.  We frequently have an array of ways toaccomplish an action and we can choose among those options for the one thatbest suits the constraints on our movement. Sometimes the constraints are externally imposed, as whenreaching into a confined space. Sometimesthe constraints are internally imposed as in getting a grip for twisting thepole or pulling oneself towards the pole. Usually it is a combination of external and internal constraints thatthe nervous system has to address in generating the movement.  Consequently, it may be fortunate thatthere are many options from which to choose. We may choose the one that is optimal to the occasion.  We may change our choice if our musclesbecome tired in one configuration. We may choose an optimum after some experimenting, as a precisionathlete does when training and practicing to perform a jump or twist or adancer does when finding a graceful movement.

Elegance and Grace of Movement

There are movements that are judged to be elegant solutionsof the problem just considered. Wecommonly find ways of moving more efficiently to achieve a particularoutcome. When a person movesefficiently, we say that they are graceful. When a movement lacks efficiency then it is said to beclumsy. It is probable that elegant or graceful movement will turn out to beefficient in the sense that it traces a geodesic trajectory in the muscle setsurface, that is a shortest path between two states. However, the most graceful trajectory may be an optimal pathsubject to constraints.

When considering eye movements we found that thetrajectories that will move the eye between gazes and ensure that it iscorrectly oriented when it arrived is a particular type of geodesic in themuscle set surface for the extrinsic eye muscles. In that case, it was not the shortest possible path, but theshortest path subject to the constraint of automatically correctingorientation. When that path is notfollowed, the eye movement is inefficient and clumsy in that it requires acorrection movement to achieve its goal. Taking a shorter path would force one out of the muscle set surface.

Movement Depends Upon a Combination of Muscle Set and Combined MuscleMomenta

Muscle actions are specified by a combination of the muscleset as expressed in the muscle set surface and the interactions of the musclemomenta. The muscle set surfaceconstrains the possibilities and the geometry of the musculoskeletal system todirect the trajectory taken in that surface. Clearly the muscle set is determined by the geometry of thesystem, but there is still a multitude of ways that the muscles may co-contractto move the system in that surface. When muscles contract or relax, they change their axis of the movementand produce results that none alone is capable of producing in isolation.  The muscle set surface means that assome muscles contract others must also shorten or lengthen; which ones changeand how determines how the system will move or be stabilized.  All these factors are combined in thecombined muscle momentum, which expresses the direction of movement in themuscle set surface. Tethering and abutmentsmay also prohibit many configurations.

The muscle set determines the placement of the bones andmuscles, which are expressed as a point in that surface.  The various muscles working across thejoint create a combined muscle momentum that has a vector component thatspecifies the direction of movement for the next bit of the trajectory.  The new placement changes theconfiguration of muscle and bones, which starts the process over again.  In fact the process is continuous andthe result is a trajectory in the muscle set surface.

This analysis raises the obvious question of how thetrajectory is determined by the nervous system. As discussed above, it is likely that the chosen trajectoryis a geodesic in the muscle set surface, subject to certain constraints.  The nervous system in some sensefollows a path of least effort.

Invariants and Muscle Set Surfaces

In science, we often look for invariants, that is,parameters that remain the same during apparent change.  In physics, it was discovered thatenergy was constant in closed systems, collisions between perfectly elasticobjects conserve linear momentum, and a spinning body conserves angularmomentum. In chemistry, we expectthe mass of the products of a reaction to be the same as the mass of theconstituents that went into it and we expect the numbers of atoms of eachelement to be the same before and after the chemical transformation.

Similarly, when we scan our world, we find that it is filledwith many entities that remain essentially the same whether we move or theobject moves, such as chairs and tables. Even a friend is perceived as being a continuing unity despite manychanges in their location and their postures, how we are viewing them, and wherewe are. We are able to recognize achair whether it is near or far, upright or tilted on its side, in bright lightand dim light. It is the samechair if we view it face on or see it out of the corner of our eye.  These mental constructs are invariantsof a sort. They remain essentiallythe same through great changes in their apparent appearance.  Extracting such invariants from theclutter and chaos of our experience is something that brains do very well andcomputers do very poorly. It is thought that the basis of such discriminationis a continual conversation between the elements of communities of neurons andbetween such communities, which are feeding information forward and backward inmassive networks.

In addition to such perceptual invariants, there are motorinvariants. We do not consider allthe details of our muscles and joints when we reach for a cup of coffee.  The reaching is, in a sense, aninvariant that encapsulates the task that needs to be done.  We use multiple levels of neurons totranslate that concept into the actual muscle contractions and joint movementnecessary to place our hand at the correct location with the correctorientation to grasp the cup. Ifthe cup is nearer or further, if we have to reach around an obstacle, if we arewearing a heavy coat or are bare armed, if we are sitting upright or lying downthe task is much the same at the level of the reaching.

You may have noted that forces are not a central concern inthe analysis that we have been considering. That is because they are highly contingent upon the specificcondition in which the movement is occurring. They are critical to successful accomplishment of amovement, but we would never represent the movement in terms of the individualmuscle forces. The movement thatwe wish to accomplish is much more efficiently stored as a trajectory in amuscle set surface. That isespecially true if we need only enter the starting point and the finishingpoint and the surface automatically generates the trajectory.  Then that trajectory can be translatedinto actual muscle forces by low level circuits that lie in the brainstem andspinal gray matter. Automaticcompensation can be made for the weight of the arm and the posture of the body.

It should be clear from the analysis that we have beenengaged in that the control of movement is an incredibly difficultcomputational task. Far beyondwhat the human nervous system is capable of in real time.  Obviously the nervous system does notcare about quaternions. They are aconvenient way for us to describe and analyze movements, but not a way toorganize the nervous system. However, it is fairly straightforward to create neural networks thatembed the muscle set surfaces in their connectivity. It is largely a matter of practicing until we are able toperform the task to criterion in a wide range of situations. Basically the waythat one learns to skate, shoot baskets in basketball or to drive a car.  The problem with such a mechanism isthat it is very difficult to understand in detail, because the task is adistributed property of a number of collections of neurons that massivelyinteract with each other. It maybe impossible to understand what is happening by examining the neuronsone-by-one. In any case, we willnot try to deal with the problem at the level of individual neurons.  We will work with the invariants forthe movement, the muscle set surface for the musculoskeletal system.