Stability
by
James Rooney,
D.V.M.
In earlier studies I have been concerned with the stability of the horse, static and dynamic, employing the techniques of mathematical stability analysis (Rooney and Robertson 1996). Those studies were abstract without an obvious relationship to immediate, practical problems. The work to be presented here derives from those earlier studies and, likewise, is not of immediate practical interest. I write this so that those of you interested only in the immediately practical “cut and cure” can be on your way to matters of more concern and interest to you. Those of you who wish to join me are welcome. The material is not necessarily easy but is, I hope, challenging and a stimulus for thought. I hasten to note that none of this is final, hard theory; it is work in progress.
Scientific investigations invariably take place within some form of coordinate system. The first stage of most scientific studies is the establishment of a coordinate system appropriate to the particular problem to be investigated. The human brain, it appears, needs boundaries and picket lines in order to begin evaluation of any phenomenon. Lack of such orientation is, as many of you may have experienced, unnerving, frightening, and certainly not conducive to rational, systematic thinking.
From that bit of philosophy we go to the horse placed in a coordinate system as in Figure 1. While that may be moderately interesting, we must proceed and develop a rather lengthy argument. Liénard, a French engineer, constructed a phase plane picture of a nonlinear differential equation representing a real physical system and used a graphical technique to determine the stability of the differential equation and, so, the stability of the physical system (Rooney and Robertson 1996 and Rooney 1998)[1]. My approach has been to go directly to a 2-D representation of the horse, considering that figure to be in a phase plane, without attempting the impossible task of first representing the horse by a differential equation. The phase plane figure can, then, be analyzed by Liénard’s method, giving trajectories whose behavior determines the stability of the figure.
 Figure 1 |
Figure 1: A phase plane[2] can be many things, only one of which is considered here. The usual phase plane in dynamics has displacement on the horizontal (y) axis and the rate (velocity) of change of that displacement on the vertical (y’} axis. A closely related phase plane may be called a pseudo phase plane in which the y’ axis is simply a shifted y axis, for example, when y is 1, y’ is 2, etc. The construction I have used is essentially a pseudo phase plane which simply indicates that every change along one axis is correlated with a change along the other axis. This gives mathematicians the horrors, I gather, but I persist because the results are logically consistent and appear to represent the actual physical situation.
In Figure 2 we see the trajectories of the movement of the horse in the phase plane, y,y. The right hand circle[3] is a limit cycle formed by the hind leg and the left hand circle the limit cycle for the foreleg, representing the standing, static horse. It is known that the standing horse, as any animal, oscillates when standing still and is never truly and completely static. Trajectories forming limit cycles are the manifestation in the phase plane of this controlled oscillation.
 Figure 2 |
Figure 2: Limit cycles formed respectively around the foreleg and the hind leg. They do not join in the standing horse, being separated at the center of mass – the black oblong.
When the two circles are connected, the horse is moving from left hind to left fore and, then, back to the right hind, right fore, and back to the right hind, etc. Figure 3, (for all gaits other than the gallop; that will be considered later). The trajectories of the hind leg pass by the center of mass, forming a saddle node, when the initial condition (the impulse of the hind leg mediated by neural action) is sufficiently large.
 Figure 3 |
Figure 3: The limit cycles join to form a new, larger limit cycle for the moving horse, usually beginning with the hind leg.
We construct other two-dimensional coordinate systems. The first is the x, y plane, Figure 4, which gives us a conventional representation of gaits other than the gallop.
 Figure 4 |
Figure 4: The “overhead” view in the x, y plane showing the four legs. At the walk and pace, for example, the sequence is LH-LF-RH-RF.
The next 2-D coordinate system is a Poincaré section, the x,y plane. This is not so readily understood. Basically, one is looking at a cross section of a 3-D phase space. Up to now we have been dealing with 2-D phase planes. At first one would suppose that this x,ý plane shows a head-on (or hind end) view of the horse, but since this is a phase space and not a “real” space, this is a Poincaré section.
Examining Poincaré sections simplifies the interpretation of the trajectories in the phase space by reducing the system to a phase plane. That is of great value when dealing with many dynamic systems and is useful for interpreting horse dynamics as we shall see now.
We know that the gaits are cyclical and so form limit cycles in the phase space. That tells us, for openers, that there should be a point in the Poincaré plane since such a point indicates that a limit cycle is present. The point appears when the trajectory of the foreleg breaks the x,ý plane going in one direction only. As that same trajectory comes back through the plane no point appears. Each gait of the horse, in three dimensions, appears to form two limit cycles as each foreleg is protracted through the Poincaré section, Figure 5. It is important to recognize, however, that when moving, even at the walk, the horse places the forefeet on or nearly on a line with the center of mass. The Poincaré section, then, gives only one point or perhaps two closely overlapping points; telling us clearly what we know from experience and observation: that the gaits of the horse are limit cycles – cyclical recurring movement in three dimensions.
 Figure 5 |
Figure 5: A Poincaré section with two points, representing a limit cycle for each foreleg. In fact, there is normally only, or almost only, one point as discussed in the text.
The gallop whether diagonal or round, left or right lead, represents a dramatic variation on the basic pattern shown above, Figure 6, only the Poincaré section remaining the same, showing that the gallop forms a limit cycle as do the other gaits.
Basically, as has been discussed in The Lame Horse, the footfall sequence is the same for all gaits other than the gallop. The transition from trot, pace, or walk to the diagonal gallop may be called a phase transition or an angular phase transition. The footfall sequence is shifted 900, Figure 6.
 Figure 6 |
Figure 6: Footfall patterns at walk, trot, pace, and diagonal gallop.
The round gallop is quite different, Figure 7:
 Figure 7 |
Figure 7: Round gallop sequence can be LH-RH-RF-LF-LH- etc. or RH-LH-LF-RF- etc.
Next we look at the phase space diagrams without the stick figure of the horse used to generate those diagrams, Figure 8.
 Figure 8 |
Figure 8: The phase space diagram for all gaits except the gallop: 1. movement from LH to LF. 2. LF to RH. 3. RH to RF. 4. RF to LH, etc. The intersection of 2 and 4 is known as a separatrix. The separatrix defines each leg in a separate part of the phase space. Only one trajectory, one pass, is shown.
 Figure 9 |
Figure 9: The phase space diagram for the diagonal gallop, RF lead.
Bibliography
Rooney J R and Robertson J L
(1996) Equine Pathology. Iowa State University Press. Ames, Chap. 10.
Rooney J R (1998) The Lame Horse. Meerdink. Neenah, WI. Chaps. 9 & 11.
[1] For those interested the
Liénard technique is described in Stoker, J J (1992) Nonlinear
Vibrations in Mechanical and Electrical Systems. Reprint Edition. Wiley,. New
York.
[2] Phase plane is by definition
2-D. Phase space, to come, is 3-D. [3] These are not true circles but more
nearly ellipses.
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