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Saturday November 18, 2017
26
May

Gait Transitions


Written by James Rooney, D.V.M.
Category: Hoof Mechanics and Physics
Hits: 4061

The question of how and why horses change gaits has been debated in the literature for a number of years. That literature which was reviewed in Rooney 1999. I do not wish to discourage the reader, medicine but the material in this essay is not trivial and will require attention and thought.

I blundered into this area with some work published in the Online Journal of Veterinary Research in 1999. Since then, pharmacy further work has been done on the same data with some interesting results and suggestions.

The thrust of that earlier paper was that the trot to gallop transition in horses occurs when the actual (damped) frequency of the gait is at or near the natural (undamped) frequency. When actual and natural frequencies correspond in this way, ask resonance occurs. A characteristic of resonance is large amplitude vibrations which can be destructive. My thesis was, and is, that the “appreciation” of approaching, possibly dangerous, resonance is the trigger for the change of gait. This was put in physical terms by measuring the actual (damped) frequency and calculating the natural frequency. That calculation was based on changes in coffin joint angles as the leg was loaded. The two frequencies were then compared and fundamental resonance was identified when the two frequencies were the same or nearly so. The method is summarized next below, and the original paper can be consulted for the full story, Rooney 1999.

Methods

The dynamic responses of a stationary or moving mechanical system may be evaluated in terms of the undamped natural frequencies (Fn) and the damped natural frequencies (Fd). The damped natural frequencies are those frequencies actually measurable in the standing and moving horse. The undamped natural frequencies cannot be measured in the horse, and we resort to the following calculations to approximate Fn.

The undamped natural frequencies are obtained at each of a series of static positions, and this series of statically obtained frequencies are considered approximations to the dynamic undamped natural frequencies. "When a machine is mounted on springs, there will be an initial deflection in the springs caused by the weight of the machine." Grosjean and Longmore 1974. Because the static deflection and the natural frequency both depend on the mass and spring stiffness, they can be related by:

gait_transitions_formula

where Fn is the undamped natural frequency, g is the acceleration of gravity and ds is the deflection. We take the horse as analogous to such a machine mounted on springs, Rooney 1969. When the leg is not loaded, the coffin bone (distal phalanx) is approximately at right angles to the axis of the middle and proximal phalanges. When the foot is on the ground and bearing weight, the angle formed by the axis of the phalanges and the ground will be an angle less than 90°; on average in the standing horse about 50°.

The rotation from 90 to 50° was calculated as linear displacement (deflection) of the fetlock joint toward the ground.

gait_transitions_formula2

where s is the linear distance from fetlock to ground, a is the palmar angle of the coffin joint, and r is the linear distance from the center of rotation of the fetlock joint to the center of the coffin joint, ~15 cm. (Figure 1)

gait_transitions_figure1
Figure 1: The measurements used to
determine undamped natural frequencies.

The deflection then is:

gait_transitions_formula3

The value of ds is entered into the first equation to obtain Fn for the standing horse.

The deflection of the fetlock is taken at 1° intervals from 50 to 10° coffin angle. The assumption, as indicated, is that the undamped natural frequencies at each position can be taken as approximations to the dynamic undamped natural frequencies.

The two sets of data, Fd and Fn were compared and the two frequencies were nearly identical at 1.8 Hz which is at the trot/gallop transition as found, for example, by McMahon 1975,1984. (Table, below)

It is important to note that gait transitions are not carved in stone but can be modified by fatigue, bumping, and size of the animal. This is dealt with at length in the original paper.

New Studies

The present study is based on the average gait characteristics of the same four Thoroughbred horses moving steadily from standstill to full gallop on a treadmill. Different ways of plotting the same data from Thompson et al 1989 are helpful in both locating and better understanding the sites of gait transitions. Reexamination of this data suggests that a resonance condition could be related to the walk/trot transition as well as to the trot/gallop transition.

In Figure 2 the damped frequency is plotted on the horizontal axis and the stride length on the vertical axis. The areas of walk/trot and trot/gallop transition are indicated. In Fig.2 the damped frequency on the horizontal axis is plotted versus the first derivative of that frequency on the vertical axis, i.e., in the phase plane. The transition areas are clearly evident. The large variation of the amplitude of the trajectory in the phase plane near and at the transitions is notable and suggests that frequency is a major determinant of gait transition.

gait_transitions_stridelength
Figure 2: Damped frequency and stride length for average of 4 horses. The walk is to the left of the left vertical bar, the trot between bars, and the gallop to the right of the right bar.

gait_transitions_phaseplane
Figure 3: The phase plane representation shows large amplitude changes in the rate of change of damped frequency related to the transition from walk to trot, left arrow (~0.75 Hz), and trot to gallop (~1.75 Hz), right arrow..

Velocity and stride length were plotted versus time as well as in the phase plane, but none of these plots provided the information of Figures 2 and 3.

Walk/Trot Transition

In my 1999 paper only the trot/gallop transition was considered. As shown in Figures 2 and 3 it is apparent that the area of the walk/trot transition can also be found. It is not immediately clear, however, that frequency resonance can be related to that transition as it was for the trot/gallop transition.

According to Burton 1994, superharmonic resonance of order 3 occurs when the driving (damped) frequency is near 1/3 the natural frequency. This is different, of course, from the fundamental resonance when the damped and natural frequencies are the same or nearly so. In the Table the natural frequency is in the left column, the damped frequency in the middle, and 3 times the damped frequency is in the right column.. This is done, of course, to compare the damped frequency with the natural frequency at order 3.

gait_transitions_chart

In the yellow highlighted row Fn and the superharmonic frequency Fd *3 are almost the same, and this appears to be superharmonic resonance order 3 at the walk/trot transition. The green highlights the fundamental resonance at the trot/gallop transition. It is rewarding to observe that the value of 0.79Hz for Fd found here for superharmonic resonance is almost the same as that shown for the actual data in Figures 1and 2 for the walk/trot transition. The third column will be discussed below.

Resonance

The overall concept of resonance as a significant factor in gait transitions is compatible with the sophisticated analyses of Schöner et al 1990. Resonance can imply loss of stability, and the detection of resonance permits the central gait processor to change the gait in order to avoid macroscopic instability.

Standardbred race horses provide interesting examples of instability. These horses are said to "make a break" when they shift suddenly from trot or pace to the gallop. There are several reasons for this transition, including interference of one leg with another and bumping particularly on turns. The animal becomes macroscopically unstable and switches to the gallop, the more stable gait at higher velocities. There, of course, needs to be a reason why the gallop is more stable than the trot or pace at higher velocities. While not worked out in detail, the following may be suggested. There is one fly period (no feet on the ground) per stride at the usual diagonal or round gallop while there are two fly periods per stride at the trot or pace. It is reasonable that the fly periods are at least potentially unstable, so that the gallop has less opportunity for instability per stride than the trot or pace.

There is no data available to directly relate such "breaks" to a resonance transition. While it may be simply a matter of over-ride, one might speculate that the misstep (bump or interference) overloads one or two legs, so that the coffin joint(s) rotates into the trot/gallop frequency transition area. This assumes that coffin rotation at trot/pace racing speeds is less than at the trot/gallop transition and gallop as can be seen by examination of the plates in Muybridge 1957.

Harness horses may break for other reasons such as inadequate training and fatigue. Fatigue has been discussed in my earlier paper. A basic premise is that training involves teaching a horse to allow the human brain to supercede the horse’s brain, to allow the human to determine what the horse does. The training of a Standardbred to race at the trot or pace, then, includes convincing an already genetically selected horse to override the usual response of the trot/gallop transition mechanism described here. With inadequate training that mechanism is inadequately or incompletely over-ridden and breaks occur: the trot to gallop transition occurs.

Self-excitation

The fourth column is the damping constant , E, and is clear evidence for considering the horse as a self-excited system. The complete series for E is not shown since it continues negative as the damped frequency (velocity) increases to the limit. As described in the earlier report, positive damping means energy is being dissipated from a self-excited system while negative damping means energy is begin supplied to that system. The energy supply is, of course, active muscle work and passive recovery of tendon/ligament/fascial strain. The energy loss is of several types, but we are concerned here primarily with the loss as a result of the braking action of the feet on the ground.

There is evidence that braking is greatest at the slow walk and increases with increase in velocity. This is quite plausible, certainly, since the opposite – slowing down – must mean increasing braking. It is well-known (though I need reference) that the efficiency of movement increases with velocity and that is evident in the Table as the positive damping decreases until it becomes negative.

Bibliography

Burton, T D Introduction to Dynamic Systems Analysis. McGraw-Hill. New York. 1994. p.661.

Grosjean, J and Longmore D K Fundamentals of Vibration in Petrusewicz, S A and Longmore, D K. Eds. Noise and Vibration Control for Industrialists, Elek Science, London, 1974.

McMahon, T A. Using body size to understand the structural design of animals: quadrupedal locomotion. Journal of Applied Physiology 39: 619-627, 1975.

McMahon, T A Muscles, Reflexes, and Locomotion. Princeton University Press, Princeton, 1984.

Rooney, J R. Biomechanics of Lameness. Williams and Wilkins, Baltimore, 1969.

Rooney, J R The Lame Horse. Meerdink, 2nd Ed, Neenah, WI, 1998.

Rooney, J R Gait transitions in horses. Online Journal of Veterinary Research. 4: 64-72, 1999.

Schöner, G, Jiang W Y, and Kelso, J A S A synergetic theory of quadrupedal gaits and gait transitions. Journal of Theoretical Biology 142: 359-361, 1990.

Thompson, K N, Rooney, J R and Shapiro R. Equine Locomotor Patterns During Gait Transitions. Proceedings Equine Nutrition and Physiology Symposium 11:27-28, 1989.

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