Estimate Calories Burned from Arbitrary Limb Movement

I’m looking to create a model using a reasonable physics-based approach to estimate calories burned for arbitrary movement.

To keep the scope fairly limited and tractable, I am looking at the displacement of limbs over time, tracked in real time. Imagine, for many points in time, comparing and accumulating movement differences for ‘before and after’ of a skeletal frame. See my crude example below:

Imagine I know all of the following:

• Length of the limbs
• Time between movements
• Physical distance moved of body parts from t to t+1
• Height and mass of the person
• Time difference from t to t+1

Here’s my rough analysis plan so far, I’m hoping someone can critique and improve. Note that for my experiment I won’t have the perfect ability to measure everything, so my goal is to get a solid theoretical foundation and then try and approximate it as best I can.

1. Treat the limbs, located at halfway point of the limbs, as point masses.
2. Estimate the mass of the hands as mass of limbs, for displacement distance of limbs use halfway distance (assume evenly distributed mass). As well, use a single point mass in the center of torso to look at displacement of whole body.
3. Using https://exrx.net/Kinesiology/Segments, we see the following:

• mass of arm: 5.7% of body weight (assuming male)
• mass of leg: 16.7%
• obviously each limb is not concentrated at the tip (hand or foot), so I assumed mass is evenly distributed along a limb, and will divide by 2, for force quantities below
• assume I know the distance displacements of the limbs and center-of-torso in (x,y,z) space
• ^ though it seems I care most about the Z dimension, i.e. moving against gravity

4. Energy (in Joules) can be converted to calories, e.g. 1 calorie = 4184 J
5. One way of calculating Joules is Work = Force x Distance
6. For each limb, and the center-of-torso, Work = mass(of limb) x gravity x displacement

(Note – I’m not sure how delta_t factors into this yet, or if I need to. Intuitively it seems like it would, but maybe it just matters for total calorie count as it takes a certain number of time to perform X movements, so counting reps just makes sense. See jumping jack example below.)

Let’s do an example and see if it makes any sense…

Say a man is doing jumping jacks, they are 180cm, 80kg, arm length is 0.8m, leg length is 1.2m

Then there’s 4 point masses moving that we consider. Let’s start with the arms. Each hand displaces e.g. 1.5 meters (from down by side, to almost vertically up), but divide by 2 because we consider the limb to be a point mass at the halfway mark. Then the ‘arm’ moved 0.75m. The mass of each arm is estimated at 80kg * 5.7% = 4.56 kg

Similar for the legs, assume they move out 1m, or 0.5m for the point mass. They weigh 80kg * 16.7% = 13.36kg.

Finally I realized it’s probably good to use a center of body torso point mass as the person will bob up and down a bit. Assume this comprises the rest of the body mass, i.e. 80kg – 2*(arms from above) – 2*(legs from above). Let’s say on each jumping jack this movement is 0.3m. But note that it only matters on the way up for the jumping jack.

So the jumping jack does the following:

• arms out (burns calories, working against gravity)
• legs out (burns calories, working against gravity)
• torso drops (no calories for this, falls with gravity)
• arms in (no calories – assume user lets it fall via gravity, i.e. not working against)
• legs in (no calories – assume user lets it fall via gravity, i.e. not working against)
• torso up (burns calories)
• For the torso, let’s just assume the full user’s mass is bobbing up and down

Some numbers.

• Arms: W = [4.56kg * 0.75m * 9.81m/s2] x 2(there are 2 arms) = 67 J
• Legs: W = [13.36kg * 0.5m * 9.81m/s2] x 2(there are 2 legs) = 131 J
• Torso (on the way up): W = [80kg * 0.3m * 9.81m/s2] = 235 J

Total: 433 J for 1 jumping jack, or 0.1 calories.

Now comparing to results online of "how many calories does jumping jacks burn", it seems this is a decent estimate. A reference:

https://burned-calories.com/sport/jumping-jacks
"You will burn around 10 calories for every minute doing Jumping Jacks. Doing the math, this means one Jumping Jack equals 0.2 calories."

OK, seems we’re on the right order of magnitude at least!

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How would you change or improve this method of calculating calories burned? For experiments, I will measure weights, limb lengths, etc of people, and use video frames of them doing activities to determine relative displacements.

Thanks for reading, hope you find this interesting!