The physics of falling particularly related to rough ground appears to be one of those areas which for some reason hasn’t had a lot of research. This article sets out the various issues and is an initial stab at trying to get to grips with this subject. Comments are welcome.

Note: no toddlers were injured or killed during the making of this article.

For reasons which are too complex to explain here,  I wanted to create a model explain how the roughness of terrain affects human locomotion. After a great deal of reading & research to try to find already available work, I found there was very little research into human movement on rough ground and nothing that could answer the specific question of how the roughness of the ground affects humans.
After failing to find research, I then considered how I might find suitable data, but realising there are big issues for this research, I then tried to work out whether it would be possible to create a theoretical model so as to predict the behaviour of humans on rough terrain and derive some useful information that way.
That then led me to the question: “what is it about rough terrain that is a problem to humans on foot and how does this affect us?” And my suggestion is that the key issue relates to “falling over” as well as energy needed.
Previous authors have attempted to model the effect of terrain using a “terrain factor”. For example see: Pandolf et al. “Prediction Modeling of Physiological Responses and Soldier Performance in the Heat”
These authors derive an equation similar to that below for the metabolic energy usage:
M = 1.5W+2.0(W+L)(L/W) ²+n(W+L)[1.5(V_w)²+0.35GV_W]

where M -metabolic rate, (watt)
W ex-external work, (watt)
W -nude body weight,(kg)
L -clothing and equipment weight,(kg)
n -terrain factor
V -walking velocity, (ms – w )
G -grade, (%)

The terrain factor is a number varying from 1 for ideal conditions upwards.

The above equation includes factors for height gained (like Naismith’s formula) and load being carried, but the “terrain factor” is only a number. And  whilst useful research, except for the small formula related to snow depth, it doesn’t relate to any parameters to do with the ground itself. Instead it is a qualitative judgement. I wanted to know the scale of roughness that affects walking. So, after thinking about this further, I realised that the above terrain factor comprised several unrelated factors.
Increased distance factor
The simplest factor is that in terrains like bush, it is not possible to walk “as the crow flies”. As such whilst the speed of locomotion can remain the same, because the person is constantly walking around obstacles to navigate from one point to another the total length walked increases such that the time is given by:

Time = distance x terrain_complexity_factor / speed

Where the terrain complexity factor is the length actually walked in the terrain divided by the map distance.
Unable to plan route
When walking we naturally plan our route ahead in order to minimise the distance and/or “cost” of walking. But when the terrain is wooded, bad weather or night restrict our vision or when the terrain drops away from us (δ²h/δx² <0), we cannot see where we are going and so cannot plan the best route to minimise the distance/cost of travel. As such the route we take tends to be longer than strictly required.
Walking “uphill” on the flat factor
The next factor works in a different way. It is caused by the way some surfaces give way when stepped on. So, e.g. if snow compresses under the body weight by 0.1m at each step, then we have to step up 0.1m to the next footstep and so it is as if we are walking uphill by 0.1m for every step. This does not increase the length walked, but instead will have a similar fatigue factor to that encountered walking uphill.
The unknown surface factor
However, snow will be more tiring than just walking uphill by the same amount, because the snow hides the real sub-surface and so we cannot so easily anticipate the extent any step will compress the snow. As such we cannot judge exactly how much forward momentum will be converted into potential energy gained leaving a potential mismatch.
To explain this, each step, we play a betting game, estimating how much momentum will be converted to potential energy. When we get the bet right, the body centre of gravity is used to cause us to topple forward on one leg increasing speed, until we plant that leg down and then we use the speed gained to push ourselves up onto that leg, converting kinetic energy to potential energy.
If however we get that “bet” wrong, then if don’t lose as much kinetic energy as we expect, we are moving faster than needed and risk toppling forward, but if we lose too much kinetic energy, we slow too much and can potentially fall backwards.
On normal surfaces we can very accurately estimate just how much kinetic energy is changed into potential – to be reused in the next step, so the gait is relatively energy efficient.
But the problem with unseen surfaces like snow covered ground, is we cannot estimate exactly how much energy will remain at the end of each step, so even if the ground beneath the snow is flat, we have to over-estimate the energy we need for each step just in case there an unexpected dip of rise. The result is we use more energy than needed.
The slipperiness factor
On any surface one which we walk with a coefficient of friction λ, there is an angle equal to tan^-1 λ at which if we were to lean against a wall at more than this angle, our feet will give way and we will fall. Likewise, if we step forward and the force is directly down our legs, if the legs drop beneath tan^-1 λ, then we will slip and likely fall. Typical low coefficient surfaces include, ice, snow covered rocks, wet algae covered rocks, loose rock & scree and wet grass.
As such when walking on surfaces of low friction, we cannot take such long steps and we often walk “conservatively” so that we have reserves of movement to try to recover in case we start to slip.

 Terrain roughness

I wanted to know how roughness affects human locomotion. As such I needed a definition of “roughness” which for simplicity I thought could be defined as the average particle size of a well sorted and flat surface.
Using this approach we can immediately divide the size range:

1. Particles much smaller than foot size such as sand act like a flat surface. There may be some give from loose particles, but otherwise it can be characterised with macro parameters such as coefficient of friction, slope and “flatness” of the overall surface.
2. Particles much larger than body size. Rather than walking over such “particles” the characteristic “gait” is one of scrambling between the boulders. As such movement on this type of terrain is strongly influenced by whether there are small particles between the larger boulders on which to stand and walk. If there are, the main impact of the boulders is to increase the length of route and the amount of climbing descending on the rough ground.
3. Particles of about foot size. Too large to be ignored, too small to have to walk around. Each one is identifiable either as a place to put a foot or a potential trip hazard.

However, there is much more to a surface than simple average “particle” size. Because most surfaces are a mixture of particles. Also whether the particles are free to move or fixed is important. So a more typical surface is like this one they used for the filming of the lunar landings (joke)
Here, there are a variety of particle sizes. Some are “embedded” and unable to move whilst others can be moved under pressure. Also, larger stones protrude from the general surface, and the general surface itself is undulating with various slopes.
However, unless or until I can model the effect of a simple surface on humans, such a terrain is too complex to model.

Foot sized particles

The issues of locomotion over/between small/Large particles can be treated relatively simply, either by using a macro model of behaviour for small particles, or by treating them as an obstacle in the path as in the case of larger particles (aka boulders).
However, there appears to be no easy way to assess the impact of foot-sized particles. An extensive review of the literature has reveal no research at all. So, there is no indication even how to assess or measure the impact let alone model it.

Possible Approaches

Speed
The most obvious metric is to assess the impact of rough terrain on speed. This appears to be obvious until one starts to consider the experimental set up.
Consider a typical man-made structure such as those shown right. The aim of this type of pavement appears to be to create an uneven surface. One might describe it only semi-jokingly as an “ankle breaking” surface.
Having walked on such surfaces, I can testify it is quite possible to walk over such a surface and indeed to do so at speed, but that there is an increased risk of injury or falling. So, I’m not convinced the main effect would be to physically slow someone down as much as create an injury hazard if attempted at normal speed. It could be attempted, but with increased injury risk.
Indeed, there is some research that suggests it is easier to run over rough surfaces than to walk! Presumably, because we do not try to balance at each step when running and therefore, so long as we can obtain a firm landing (not obvious on the surface above), we can run at speed over some rough ground more easily than walk.
But, realistically without a great deal of care, it would be difficult to take random people off the street and ask them to walk over this potentially ankle breaking surfaces such as the one above knowing that it may cause injury.
However, one is perfectly entitled to watch any idiots who choose of their own volition  to walk over such ground. The problem is finding somewhere when this occurs regularly enough on suitable ground to make it worthwhile setting up and observation point so as to take suitable measurements. But where?
Initially I considered finding a pebble beech. It has with well sorted particles and people do walk along them. However, people don’t “walk” along beaches but instead “saunter” … they take their time. And indeed, their speed will vary considerably on their interest in the area over which they are walking. Any difference in speed could easily have nothing to do with the surface, but instead whether their dog was playing in the water.
My next idea was to find a mountain with lots of rocks. I know there are lots of mountains with extensive areas of rocks. However, I soon realised that precisely for the reason that rocks slow progress, paths tend to avoid such areas. And, indeed, if there is a path that goes through a rocky area – and if it is used enough to produce a viable statistical count on any particular day – the footfall has invariably broken down and/or compacted the rocks so that it is no longer anywhere as difficult to walk on. However, an even greater problem is people tend to walk up and and down hills (yes it perplexes me as well – so conventional to just walk up a hill to then walk down!) Therefore any section going over rock or scree is usually only crossed where it is necessary as part of the climb. As such the walkers are not only crossing rock – but ascending or descending and this will hugely affect their speed.
What I really need is a pebble beach which is part of a well used footpath. Here people will be “walking with a purpose” and as such I could compare their speed over pebbles with that over an area of firm sand. But I am as yet unaware of such a location.
Metabolic rate
But the fundamental problem is that speed on its own is not a good indicator of the affect of terrain, because often we try to walk at a constant rate. What however can vary dramatically is the energy needed to cross any particular type of terrain. But without suitable equipment this is even more problematic than speed.
Rate of Injury
Another metric that then sprung to mind was the rate of injury. In other words, we take 100 school children, tell them to run across ankle breaking cobbles and use the figure of the number of injuries as a metric of the “problems” crossing that terrain. More sensibly, (if available) there might just be metric associating number of mountaineering injuries by terrain type.
Rate of Falls
Which leads nicely onto the easier to measure metric “the rate of fall”. In this kind of setup, we would need to assess the number of times people fall over when crossing terrain of a particular type. But why have people? Surely falling over is just a matter of simple physics?

Physics of falling

All that needs to occur for something to fall is for the centre of gravity to move so that it acts outside the base area! Isn’t that simple! Surely a few equations about how roughness affects tilt angle (as suggested in diagram above) and it will all be sorted.
However, human walking is not so simple. Take the simple instance in this video:

Toddler falling at half speed (trip)

The problem is that at each and every step, the infant could potentially fall over. That is because there is a dynamic instability in walking in that as we lift one foot we topple forward onto the next. Walking is a process of constantly falling over – it is just that (usually) we control this fall so that we move forward.
This however means the normal definition of “falling” that would be applied to objects does not suit humans when walking.
In the above case, the child appears to catch their right foot and this prevents them moving it forward to catch themselves. But they were actually “falling” in a sense before this point because it was only the lack of moving the foot that caused the fall – thus they were already falling and the foot was needed to control the normal tumbling movement that is part of walking.
So, falling in humans is not just a simple case of the centre of gravity moving beyond the “base”, otherwise we’d all fall as soon as we lift one leg.
Definition of falling
The simple definition of “centre of gravity moving outside the base” does not work with humans. Instead, having watched many instances of people falling over, I would suggest something along the lines of the following criteria:

That a person falls, when they begin to topple at a rate fast enough that the time for them to tumble to the ground is less than the time for them to find a foothold to resist and so control that movement.

Thus a “trip” is a normal movement whereby the foot is restricted so that it takes longer to move the foot than it takes to topple.
We “fall” into an unseen hole – not because we topple – because that is part of walking – but because we are unable to place our feet so as to resist the normal toppling movement of walking.
And children often fall because they allow their centre of gravity to go out to the side beyond their feet:

 Toddler Falling due to centre of gravity moving beyond their left leg at half speed

Thus I can now suggest various causes of falling:

1. Trips: whereby the leg toward the falling side is prevented from moving to where it can resist the fall
2. Lack of foothold: whereby the leg is free to move, but the ground does not provide the purchase
3. Slips: the ground is capable of resisting the movement perpendicular to the surface, but the angle of force is greater than the friction angle so that the point of force moves away from the necessary location. This causes the resisting force to move away from the point where it would control the movement and resist the normal topple. The result is a net force out of the plane of the topple to one side. This net force then induces another topple which must be controlled and if not the person will fall.
4. Forward kinetic energy too low for needed potential energy. In order to resist a topple, the forward motion must be judged so as to match that changed into potential energy during each “swing” of a step. If however we move forward without enough kinetic energy to “regain balance” we will fall backward. Such falls occur when we make a large step up.
5. Unable to move legs fast enough
   If however we move downhill and gain too much forward speed we may still be able to move our feet, but due to acceleration, we move faster and faster unable we are unable to move our feet fast enough forward to keep in control and then we tumble.
6. Centre of gravity moves outside of legs
   Whilst we have quite some leeway as to where the centre of gravity is in the forward direction, the same is not true if the centre of gravity moves beyond the leg that is already placed on the ground. Now, the recovering needed is to find a foothold beyond the outside of the (first) foot already on the ground. This might be achieved by the second foot crossing the first foot, or it may require placing the weight to be placed on the second foot allowing the first foot to be lifted and replaced well to the side. (This appears to be the cause of the second toddler falling)
7. Injury
   The final reason for falls, is that something occurs that causes injury to the foot so that whilst it is possible to locate it so as to resist the normal topple of walking, the foot is not able to take the force due to injury. A typical instance is a twisted ankle whereby, if weight could be applied the topple would be resisted as normal, but the placing of the ankle (perhaps twisted by being off centre on a stone) means the foot cannot hold the weight.

How rough ground relates to falling over

Having categorised various reason for falling over, I can now assess how each relates to the roughness of the ground:

• Trips
  On rough ground the legs must be lifted so as to clear obstacles. The larger they protrude from the surface, the higher the legs must be lifted. A misjudgement or unseen obstacle can result in a fall.
• Lack of foothold
  Rough surfaces will have hollows. If these hollows cause a person to “miss their footing” they may tumble
  
• Slips
  Rough surfaces will have surfaces that are not level. As such, it is all the easier to reach the critical resistance angle at which sidewards movement will result. In addition many rough surfaces have loose material which provides a rolling surface.
  
• Forward kinetic energy too low for needed potential energy.
  Even if the feet are lifted high enough on rough terrain, if the step is too high, there may not be sufficient forward momentum to bring the body to rest on the top of the step causing the person to fall backwards.
• Unable to move legs fast enough
  Usually only a problem with slopes, but may also affect those in water particularly where it is flowing and resisting movement.
• Centre of gravity moves outside of legs
  Most sober adults are experienced enough that they do not normally allow their centre of gravity to move to the side beyond their feet. However, on rough ground where there is less choice about the placement of feet as well as vertical movement and perhaps visual clues about verticality are confusing, it is relatively easy to misjudge the needed balance for the available footholds.
• Injury
  Foot injuries can occur due to loose or uneven surfaces, particularly where the foot is twisted to the side where the rock top is to one side, where the surface is not level, where there is a slip, or where objects move when being stood on.
  

Conclusion

I have identified 7 different ways in which the roughness of the ground can impede human gait. From these various are important are:

1. The height of obstacles
2. The depth of hollows
3. Coefficient of friction
4. Surface gradient (related to friction)
5. The limitation of foot placement
6. The balance between forward movement and potential energy to gain “balance”
7. Slopes
8. Impediments such as water (or vegetation) slowing down the movement of legs
9. Judgement of the ground height – necessary foot placement for control of balance
10. Loose, sloping or irregular footholds that can cause injury (particularly to ankles)

Addendum

I’m going to modify my definition of “falling over” to say that we are not falling over if:
A) The frictional torque available > gravitational torque.
Where frictional torque (flat ground) is sidewards friction force x vertical distance ground to centre of gravity, and gravitation torque is mass x gravitational acceleration x horizontal offset of centre of gravity from centre of upward force from feet.
Or …
B) Where time to fall over is > time to move feet to a position where the centre of gravity is between the feet, or at least close enough that there is then time to move again to another position where any remaining residual falling acceleration is low enough to allow time to move the next foot, etc..