figure 1: obtained from the original paper

## Overview:

1. When comparing two morphologically similar(i.e. geometrically similar) species that may differ in size it’s natural to ask under what circumstances their gaits might be similar. Given that the differences in size might be important-consider for example the difference in size between a cat and a rhinoceros-a dimensionless analysis is necessary.

2. 130 years ago, it was William Froude that introduced a dimensionless parameter that proved to be an important criterion for dynamic similarity when comparing boats of different hull lengths. In particular, dimensionless analysis using $$Fr$$ proved very useful in understanding why the Great Eastern, the largest ship in the world at the time, was a massive failure. In fact, the ship couldn’t earn enough to pay for its fuel.

3. Essentially, Froude found that large and small models of geometrically similar hulls produced similar wave patterns when their Froude numbers $$Fr$$ were equal. To be precise, $$Fr$$ is equal to:

$$Fr = \frac{\lVert v \rVert^2}{gL}$$

where $$v$$ is the velocity, $$g$$ is the gravitational acceleration and $$L$$ is the characteristic length.

4. While Froude concentrated on the movement of ships it was D’Arcy Wentworth Thompson who first recognised the connection between the Froude number and animal locomotion. On page 23 of On Growth and Form, Thompson notes:

In two similar and closely related animals, as is also in two steam engines, the law is bound to hold that the rate of working must tend to vary with the square of the linear dimension, according to Froude’s Law of steamship comparison.

5. Despite the popularity of Thompson’s work, the importance of the Froude number as a tool for analysing locomotion wasn’t fully appreciated until Robert Alexander, a Zoology professor at Leeds, empirically demonstrated that the movement of animals of geometrically similar form but different size would be dynamically similar when they moved with the same Froude number.

## Alexander’s dynamic similarity criteria:

figure 2: African rhino in its natural habitat

One of Alexander’s most striking observations was that the galloping movements of cats and rhinoceroses are remarkably similar even though the rhino is three orders of magnitude larger. After much empirical analysis, Alexander postulated five dynamic similarity criteria in [3]:

1. Each leg has the same phase relationship.
2. Corresponding feet have equal duty factors (% of cycle in ground contact).
3. Relative (i.e. dimensionless) stride lengths are equal.
4. Forces on corresponding feet are equal multiples of body weight.
5. Power outputs are proportional to body weight times speed.

Alexander hypothesised, and provided the necessary experimental evidence to demonstrate that animals meet these five criteria when they travel at speeds that translate to equal values of $$Fr$$. This important work by Alexander indicates that although the Froude number may appear to oversimplify complex problems in biomechanics, it has empirically proved to be an important factor in the dimensionless analysis of dynamic similarity. At this stage we may marvel at the fact that the Froude number, which emerged from a problem in hydrodynamics, should also play a key role in the comparative analysis of terrestrial locomotion.

While this theoretical issue isn’t addressed in [1], I have attempted to show the mathematical connection between the Froude number as it occurs in hydrodynamics and the Froude number as it occurs in biomechanics.

## Analysis of the Froude number:

figure 3: The inverted pendulum as a model for bipedal walking

In this section I shall demonstrate that in both the cases of a surface water wave and a bipedal walker, the Froude number provides a similar description. In fact, if we note that a surface water wave is approximately a transverse wave and the walking motion of a biped is approximately a longitudinal wave then the Froude number is simply the force magnitude responsible for linear displacement divided by the magnitude of the gravitational force.

### Surface water waves:

Given a surface water wave moving through a medium with density $$\rho$$ with constant velocity $$\lVert v \rVert = \frac{L}{T}$$(i.e. longitudinal displacement $$L$$ within a period $$T$$), the magnitude of the inertial force required to halt the motion of a volume $$L^3$$ is given by:

\begin{split} \lVert F_i \rVert = \text{mass}*\text{acceleration} & = \rho L^3 \cdot \frac{\lVert \Delta v \rVert}{\Delta t }
& = \rho L^3 \cdot \frac{\lVert v \rVert}{T}
& = \rho L^2 \cdot \lVert v \rVert \cdot \frac{L}{T}
& = \rho L^2 \cdot \lVert v \rVert^2
\end{split}

On the other hand, the magnitude of the gravitational force acting on this volume is given by:

$$\lVert F_g \rVert = \text{mass}*\text{gravitational acceleration} = \rho L^3 \cdot \lVert g \rVert$$

and we may define the Froude number in terms of the ratio of these force magnitudes:

$$Fr = \frac{\lVert F_i \rVert}{\lVert F_g \rVert} = \frac{\rho L^2 \cdot \lVert v \rVert^2}{\rho L^3 \cdot \lVert g \rVert } = \frac{\lVert v \rVert^2}{Lg}$$

and this number describes the stability of the flowing wave as shown in this video. In particular, when $$Fr < 1$$ the shallow water wave is stable and the motion of the wave is dominated by gravitational forces so surface waves generated by downstream disturbances can travel upstream but when $$Fr > 1$$ this is impossible.

### Bipedal walkers:

The inverted pendulum is a useful model for analysing bipedal walking as a leg forms the radius of an arc and the motion of the biped may then be approximated by a longitudinal wave. Furthermore, if we make reasonable modelling assumptions we may infer the speed limits on a bipedal walker.

In particular, we make the following assumptions:

1. We neglect air resistance.
2. We assume that the legs are rigid and interact with a single point on the ground.
3. We neglect any pelvic motion.
4. We neglect the inertial role of arm motions.

Given these assumptions, note that the force magnitude associated with motion in a circular arc is given by:

$$\lVert F \rVert = \frac{M \lVert v \rVert^2}{L}$$

where $$M$$ is the mass of the biped, $$\frac{\lVert v \rVert^2}{L}$$ is the inward acceleration of the mass, $$v$$ is the tangential velocity and $$L$$ is the radius of the circular orbit(i.e. the limb length).

Note further that during walking, the inward acceleration due to normal forces on the foot shouldn’t exceed the gravitational acceleration:

$$g > \frac{\lVert v \rVert^2}{L}$$

Meanwhile, if we consider the ratio of the centripetal force magnitude to the ratio of the gravitational force magnitude we have:

$$Fr = \frac{\lVert F_c \rVert}{\lVert F_g \rVert} = \frac{\frac{M \lVert v \rVert^2}{L}}{Mg}=\frac{\lVert v \rVert^2}{gL}$$

and in order to allow stable walking we must have:

$$Fr < 1 \implies \lVert v \rVert < \sqrt{gL}$$

so the maximum walking speed of a biped is proportional to the square root of $$L$$, the length of its legs. Likewise, when $$Fr > 1$$ running is necessary. It follows that both in the case of a shallow water wave and bipedal walkers, $$Fr \approx 1$$ defines the boundary between fundamentally different dynamics.

## Open questions:

1. Does the implication of dynamic similarity via equal Froude number hold for nonlinear motions?
2. The assumption of constant velocity appears to require steady-state assumptions. Can the Froude number be generalised to handle the case of intermittent locomotion?

These questions might have been answered by researchers in the robotics and biomechanics community but at this point I myself don’t have satisfactory answers.

# References:

1. Froude and the contribution of naval architecture to our understanding of bipedal locomotion. C. Vaughan & M. Malley. 2004.
2. On Growth and Form. D’Arcy Wentworth Thompson. 1917.
3. A dynamic similarity hypothesis for the gaits of quadrupedal mammals. Alexander RM, Jayes AS. 1983.