Snakes, Swan’s Necks, andPuppy Dog Tails

It has been noted in passing that seven cervical vertebraein mammalian necks seems to be a threshold of sorts for movements (refs).  If you add an additional link, themovements take on a looser quality that does not seem quite ‘neck-like” and stiffening one linkage, as mighthappen with a fusion or arthritic restrictions, leads to a substantialreduction in movement. One mightspeculate that the number of vertebrae and the amount of movement in theintervertebral linkages might be in some way set by evolution to achieve theneeds of a mammalian neck (ref). With a very small number of exceptions (manatees, and two types ofsloth), all mammals have 7 cervical vertebrae.

A snake’s skeleton has a series of nearly identicalelements composed or a vertebra and two attached ribs.  The concatenation of many smallmovements between successive vertebrae allows the snake to take a wide varietyof sinuous shapes.

That leads to curiosity about the implications of chains ofmany similar linkages, such as occur in a cat’s tail, a snake’s body, or aswan’s neck. A swan’s neck mayhave up to 25 cervical vertebrae. Some of the aquatic reptiles of the Mesozoic, plesiosaurs, had up to 40cervical vertebrae. The cat’s tailmay have 21-23 caudal vertebrae. Snakes have largely identical series of vertebral segments, withattached ribs. All of theseanatomical structures have a great deal of mobility and a similar manner ofmoving.

In this chapter, we will consider the dynamics of amechanical system that has a high degree of repetition of concatenatedidentical elements. While we startwith the idea of a snake or a swan’s neck, we will be concerned with anabstract entity that we will call an artificial snake.  The results may not be directlyapplicable to actual snakes or swans necks or puppy dog tails, because a closeexamination will reveal that the elements in those structures are not in factidentical. Snake spines come closest.  Bird necks are most differentiated.

Artificial Snakes

We will talk about joints and muscles and how muscles movebones at joints. While we startwith the concepts of actual joints and muscles, we will again recast thoseentities into abstractions and consider the formal properties of groups of musclesand joints working in concert.

An artificial snake is created by linking a series ofidentical elements that can move about a central pivot point.  This mechanism can be describedabstractly by set of linked framed vectors.

We will start with a very simple toy.  A toy snake is constructed byconcatenating a number of identical elements []. Each has apivot point, which is taken to be the location [P]of the element. In the illustratedinstance all the pivot points have an axis of rotation [p] that is directed perpendicular to the referenceplane [p = r] and the pivotpoints are linked to each other in directions that extend parallel to thereference plane. The linkage [L] is an extension of the elementto the next pivot point and it defines a direction relative to the element [s]. The direction of the link and the direction of the pivotaxis define an orientation frame [O].  We will assume that the mutuallyperpendicular direction that your thumb points when the fingers of your righthand curl from the pivot axis to the linkage direction will the final axis ofthe orientation frame [t].

The frame vectors of a single element

In addition, we will have two armatures that extend awayfrom the pivot point in the plane of the pivot axis, that is, parallel with thereference plane. The two armatures[] will extend symmetrically to either side of the linkage soas to form a right angle between them. Their directions will be and . Theirlength will be ‘’. Therefore,each element of the snake may be written as a framed vector.

The abstract toy snake is expressed by a series oflinked elements composed of a pivot point at a particular location and twoextended vectors that arise from the pivot point and extend symmetrically toeither side of the axis of the element. The element is orientable. Each element can be described by a framed vector.

The rotation between elements is given by the quaternion

.

We can write down the expression for the snake from theseexpressions.

We can place it in space by specifying the location andorientation of the first element.

Since the axis of the pivot joint is fixed relative to theelement and it is aligned with the r axis,the calculation is very straight-forward. We can also use the full angle form of quaternion multiplication tocompute the new values of the frame elements.

The pivot axis is allowed to take any directionrelative to the element. Since thepivot axis is no longer perpendicular to the plane of the element, the r vector of the frame may be defined as the directionof the ratio of two of the extension vectors.

Having carried the analysis to this point in an essentiallytwo-dimensional situation, it is easy to write down the description for anarbitrary snake in three dimensions. The main difference is that the pivot axis can take on any direction,therefore, we need to use the half angle expression for rotation.

More generally, create a real or hypothetical verticalor lateral extension that is moved with the element.

The extension elements may be written as functions of theorientation frame of reference. Then, we need only keep track or the locations and orientations.