# On ankle mechanics

Stand up on your toes. If you stay there for a while you’ll feel your calf muscles tension. Precisely how much force are the calf muscles exerting on your feet? This is an important question in designing humanoid robots, as the answer largely affects component and motor sizing of the robot.

The human foot, simplified to a single unarticulated body, can be seen as a mechanism that functions very similarly to a lever. While there is consensus to this function, whether the lever assembly formed by the foot actually multiples or divides the force exerted by the calf muscle has been a point of ongoing debate for quite some time. The aim of this post is to provide a concise proof based on statics as to the precise functioning of the foot-ankle mechanism as a lever.

#### Classes of Levers

There exist three classifications for levers. Class I levers are those that have the fulcrum in the middle. Levers of this type may require effort smaller or greater than the resistance depending on the ratio of lengths. Class II levers have the resistance at the middle. In this configuration, the effort is always less than the resistance. Class III levers have effort and resistance on the same side of the fulcrum, just like Class II, however the effort is closer to the fulcrum in this class. Thus the effort is always greater than the resistance. In this post we will be dealing with Class I and Class II levers. With respect to math, the only thing we will need to be aware of is the formula for calculating effort with respect to Class I levers:

$F = \frac{x_Q}{x_F} Q$

In the above formula, F signifies effort, Q signifies resistance and xF and xQ are distances from the fulcrum to the effort and resistance points correspondingly.

#### Ankle Forces

Imagine a person standing at the edge of a sidewalk, supported by their toes. In this configuration, the foot acts as a lever, using the force exerted by the gastrocnemius muscle (one of the calf muscles) to support the body on its toes. Even though there is agreement on the lever function in this configuration, there are conflicting views as to what type of lever (Class I or Class II) the foot functions like.

Let us momentarily assume that the function of the foot corresponds to a Class-II lever. Under this assumption, the fulcrum is at the point of ground contact and the figure above summarizes exerted forces. F is the force exerted by the gastrocnemius muscle. Q it the load on the ankle and R is the reaction of the ground. For the above system to be in static equilibrium, the sum of forces along the vertical and horizontal axes, as well as moments, should add up to zero. Under this assumption, it holds true that:

$F - Q + R = 0$

Or otherwise:

$R = Q - F$

However, as any bathroom scale will readily tell us, this cannot be true, as the ground reaction should be equal to body weight, thus one would expect that Q = R. This is an indication that something fishy is going on with Q in our assumptions.

In fact, what the above description does not take into account is that the gastrocnemius muscle that produces force F also produces a reaction on the body as well. This reaction is in the same direction as Q and complements the weight of the body, so that:

$Q = J + F$

Where J is the weight of the body, This takes into account the directions of forces in the figure above. Replacing the last in the first equation we get:

$R = Q - F \Rightarrow R = (J + F) - F \Rightarrow J = F$

Which is what we would expect according to our bathroom scale.

#### Redefining Fulcrum

Let us now throw equilibrium of moments in the mix as well. For this we will choose the ankle as center (point 2 above) and clockwise positive. Our new figure with forces and lengths is as follows:

Force equilibrium is the same as above, and moment equilibrium is as follows:

$F x_1 - R x_2 = 0 \Rightarrow F x_1 = R x_2$

Substituting previous formula into the above:

$F x_1 = (J - F) x_2 \Rightarrow F x_1 = Q x_2 + F x_2 - F x_2 \Rightarrow F x_1 = Q x_2$

The last formula above can be rewritten as:

$F = \frac{x_2}{x_1} Q$

or in other terms, the force exerted by the muscle is equal to the body mass times the ratio x2 over x1. Since x1 < x2, the muscle will exert force greater than the body weight to lift the body. This condition corresponds to functioning of the foot as a class-1 lever with the ankle as a fulcrum.

It is interesting to note that all of this is much easier to visualize if one considers the leg upside-down with the body as a frame of reference (instead of the ground). Here’s a figure outlining this:

#### Conclusion

This short post elaborated on the functioning of the human foot & ankle as a lever. In particular, we presented a concise argument in favor of the functioning of the ankle as a Class-I lever with the ankle as fulcrum. This is contrary to the commonly occurring fallacy that the ankle is a Class-II lever with the toes as a fulcrum.

#### Acknowledgements

I’d like to acknowledge the contribution of fellow roboticist Keith Gould who brought this topic to attention and shared many valuable views.

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##### 2 replies on “ On ankle mechanics ”
1. yconst says:

Thanks Keith for the great discussions 🙂

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