# Predicting the vertical/horizontal/mediolateral force of an animal walking underwater

#### mgranatosky

Hello,

This may be an incredibly simple question, but I am looking for some help to clarify things for me.

I am a researcher interested in animal locomotion. Currently, I am working on a project collecting single limb force data (vertical, horizontal, and mediolateral substrate reaction force in a single limb) from a lungfish (a fish that walks along the bottom of rivers similar to a salamander).

I am hoping to do some modelling before we collect any data. I want to be able to predict approximately how much vertical, horizontal, and mediolateral force will be on each limb.

I believe that the static models I provided in the pictures attached should be some representation of the expected forces when the animal is walking on the ground. Essentially, use the static models to predict force across the stride. In the horizontal and mediolateral model you'll notice that there is an angle measurement involved. I believe that this angle should be relative to the center of mass, but is it actually at the hip/shoulder joint?

To make matters a little more complicated (maybe not) these animals only walk in the water. In our experiments we are planning to collect data while animals walking in a fish tank on a desk. This fish does not float (it sinks to the bottom), but does buoyancy play a factor in this scenario, and if so, how will this change calculations?

I appreciate your help so much, and welcome any edits or better suggestions.

Thank you so much

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#### mgranatosky

If it helps, I am basing these models on what to expect from previously established models (see attached images).

I also believe that this system may be more easily modeled as a table (more images below)

Thoughts?

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#### Woody

The difference in the Weight due to buoyancy is equal to the Weight of the volume of water displaced.

What is the volume of the animal? It will displace that amount of water.
Handily 1litre of water masses 1kg (this is built into the original SI definition of mass).

The vertical load on the "limbs" will be reduced by that amount, relative the load that would be exerted in air.

#### mgranatosky

So in the models provided above weight should be adjusted to:

W' = W - buoyancy?

Thank you so much

#### studiot

So in the models provided above weight should be adjusted to:

W' = W - buoyancy?

Thank you so much
Yes and no.

The bouyancy force only acts through the animal centre of mass in symmetrical situations.

The bouyancy force acts at the centre of bouyancy which is usually different from the Centre of mass of the fish.

This brings other forces into play. These will be significant in the case of a submerged fish.

http://www.cns.gatech.edu/PHYS-4421/lautrup/7.6/buoyancy.pdf

#### mgranatosky

Thank you for the PDF.

This is very helpful. I see that the center of buoyancy must always be above the COM, but how will this change the "effective weight" of the submerged animal.

For example let's say the fish weighed 2 kg, and has a volume of .00065 m3, what would that mean for the effective weight" of the submerged animal.

Thank you so much for your help

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#### THERMO Spoken Here

Sir Mgranatosky,

My thought is previous models scarcely apply. Density of a dog ~ 1100kg/m^3
density of air ~ 1.2kg/m^3. (ratio ~ 1000) The lungfish in water is in a near float / near sink
equilibrium. With its air-bladder (most fish have) it can "lift" any leg without
musculature. So how big are these fish? Density is near that of water I suppose (ratio ~ 1). The
value of a "dog" model is questionable. So does the lungfish "ambulate or what ???

Need... "vertical, horizontal, and mediolateral substrate reaction force in a single limb."
(1) I suppose the "vertical reaction" force (equilibrium condition) is something like... Fv = Fg - Fbuoy.
(2) "... horizontal reaction?" This is transient, hard even to approximate.
(3) "...mediolateral substrate reaction." Say what? I guess this force is measured in situ by an electronic
needle or something. Is this force got by mechanics? Sounds not like a boundary force.

Your "problem statement" did not tell why you want to know. Models are usually couched in
the why as well as the what.

So where to go; I don't know. Good luck and good day...

JP

#### mgranatosky

Hello JP,

"My thought is previous models scarcely apply. Density of a dog ~ 1100kg/m^3
density of air ~ 1.2kg/m^3. (ratio ~ 1000) The lungfish in water is in a near float / near sink
equilibrium. With its air-bladder (most fish have) it can "lift" any leg without
musculature.

So how big are these fish? Density is near that of water I suppose (ratio ~ 1). The
value of a "dog" model is questionable. So does the lungfish "ambulate or what ???

These are not particularly heavy fish, although they can get quite large. Regardless of size, these animals are applying force as they walk along the substrate, and are using muscles to propel (and potentially support) the body.

See:

A small step for lungfish, a big step for the evolution of walking - UChicago Medicine

Need... "vertical, horizontal, and mediolateral substrate reaction force in a single limb."
(1) I suppose the "vertical reaction" force (equilibrium condition) is something like... Fv = Fg - Fbuoy.
(2) "... horizontal reaction?" This is transient, hard even to approximate.
(3) "...mediolateral substrate reaction." Say what? I guess this force is measured in situ by an electronic
needle or something. Is this force got by mechanics? Sounds not like a boundary force.

(1) I agree with this interpretation
(2) As long as a limb is away from it's COM it will produce some sort of horizontal force
(3) Because the limbs are parasagittal (not directly underneath the body) in a coronal view they will transfer force to prevent the COM from collapsing

Your "problem statement" did not tell why you want to know. Models are usually couched in
the why as well as the what.

These fish are very interesting from an evolutionary point of view, as they are the sister-group to all tetrapods. Interestingly, they move their limbs in an ambulatory fashion to locomote. This means that the neuromuscular mechanisms for locomotion evolved prior to the water-to-land transition. These fish are also interested because they cannot support their body out of the water.

I am planning to collect single-limb force data from lungfish walking under the water to see if the basic force patterns observed during quadrupedal walking in tetrapods is also evident in the lungfish. I am hoping to use a model to predict what the expected forces walking under the water will be . It will be interesting to know whether lungfish follow this model or not. If they violate this model it would be interesting to understand what mechanisms they use to do so, and why.

Thank you so much for your interest