This rucking calorie calculator estimates the energy cost of walking under load using the Pandolf equation, with three published corrections applied on top of it. This page sets out every equation, every coefficient, the ranges each was validated in, and the places where the model is known to be weak. If a number on this site ever looks wrong to you, everything you need to check it is here.
Nothing on this page is proprietary. The equations are published research, the coefficients come from measured data, and the calculator prints its own working for any set of inputs you give it. That is deliberate. An estimate you cannot audit is worth very little.
The short version
- The base equation is Pandolf, Givoni and Goldman (1977), developed at the US Army Research Institute of Environmental Medicine.
- On descents, a correction from Santee and colleagues is applied, because the base equation returns physically impossible values downhill.
- A load-ratio correction addresses the equation’s documented tendency to under-predict for contemporary loads.
- Terrain coefficients are the measured values from Soule and Goldman (1972).
- A separate walking equation supplies the “compared with walking it unloaded” figure. It is never mixed into the ruck estimate.
- Every result carries an error band, and inputs outside a validated range are flagged where you can see them.
Step one: the base equation
M = 1.5W + 2.0(W + L)(L/W)² + η(W + L)(1.5V² + 0.35VG)
- M
- Metabolic rate, watts (gross — it includes the resting component)
- W
- Body mass, kilograms
- L
- Load carried, kilograms
- V
- Velocity, metres per second
- G
- Grade, percent
- η
- Terrain coefficient (1.0 on blacktop)
Pandolf KB, Givoni B, Goldman RF. Predicting energy expenditure with loads while standing or walking very slowly. Journal of Applied Physiology 1977;43(4):577-581. PMID 908672
Pandolf and colleagues had test subjects walk on treadmills carrying loads while their oxygen consumption was measured, then fitted an equation to the results. It has three parts. The first is what your body costs to run at rest. The second is the penalty for carrying weight at all, which grows with the square of the load-to-body-weight ratio — this is why an extra ten pounds hurts far more at 60 lb than at 20 lb. The third is the cost of moving, which scales with the square of your speed and linearly with grade.
One detail that gets misreported elsewhere: the output is gross metabolic rate. It includes the resting component. When this site shows a net figure it subtracts basal metabolism explicitly and labels it as net.
Where it was validated
- Loads up to 70 kg (about 154 lb)
- Speeds up to roughly 1.97 m/s, or 4.4 mph
- Positive grades
Push past any of these and the calculator still gives you a number, but it says so on screen and widens the error band. Above about 4.4 mph most people stop walking and start a march-run, at which point the squared speed term overstates the cost.
Step two: descents
The base equation breaks downhill. Take an 80 kg person carrying 20 kg at 1.4 m/s down a 10% grade: it returns minus 63.5 watts. A negative metabolic rate is not a small inaccuracy, it is a physical impossibility, and it is a well-known limitation rather than a secret.
CF = (G(W+L)V)/3.5 − ((W+L)(G+6)²)/W + (25 − V²)
- CF
- Correction subtracted from M when the grade is negative
Santee WR, Blanchard LA, Small MG, Gonzalez JA, Blanchard LA. Load carriage model development and testing with field data. USARIEM Technical Report T01-11 2001;T01-11.
The correction is subtracted from the base rate whenever grade is negative. Applied to the same case it returns about 333 watts, which is a sensible figure for loaded downhill walking.
Something worth understanding here: as descents get steeper, predicted cost falls, bottoms out at around minus 11%, then climbs again. That is not a glitch. Walking cost has a minimum near minus 10% grade and rises on steeper descents, because your legs shift from propelling you to braking you, and eccentric muscle work costs energy. Anyone who has come down a steep trail under a heavy pack already knows this in their quadriceps.
The correction was validated at 4 km/h and above, with loads to 27 kg, on grades to minus 12%. Steeper than that and the shape of the curve is still right, but the size of the rise is not established. Those results get flagged.
Step three: the load-ratio correction
The base equation was fitted in the 1970s. Modern loads and load-carriage systems are different, and researchers have measured the gap. Drain and colleagues found the equation under-predicted actual metabolic rate by 12 to 17% at 2.8 mph, and by 21 to 33% at 4 mph, with a 22.7 kg load.
The calculator applies a multiplier that scales with both speed and the load-to-body-weight ratio. You can see its exact value for your own inputs in the “Show the math” panel on any calculator page — it is displayed rather than hidden inside the result.
That multiplier is also tapered at light loads, and the trace shows the taper as its own line. The reason: Drain measured under-prediction while subjects carried 22.7 kg. Applied flat, the correction fired even with an empty pack — inflating unloaded walking by about 24% for a correction whose entire justification is load carriage. It is now scaled by your load ratio relative to the ratio Drain tested, so at that ratio nothing changes and at zero load the correction disappears.
Be clear about what that taper is: a reasonable assumption, not a measured relationship. Nobody has published how the under-prediction varies with load ratio, so the engine scales it linearly. When this was introduced, figures at light and moderate loads dropped by roughly 3 to 11%; heavy loads did not move. The change is recorded in the changelog.
Where this correction is weakest
At 4 mph the multiplier returns about 32.5%, comfortably inside the measured band. At 2.8 mph it returns about 24%, against a measured 12 to 17%. That is an overshoot, and it happens because the base multiplier applies regardless of speed.
So: below roughly 3 mph, treat the figure as an upper bound. The calculator flags this on screen when it applies to your inputs. Fixing it properly means making the multiplier speed-dependent, which requires field data this site does not yet have. When that changes, it will appear in the changelog.
Terrain
The terrain coefficient multiplies the movement term. Blacktop is 1.00 by definition; every other surface is measured against it.
| Surface | Coefficient | Notes |
|---|---|---|
| Blacktop or paved road | 1.00 | Baseline surface in the original studies. |
| Dirt road or hard-packed trail | 1.10 | |
| Light brush or packed grass | 1.20 | |
| Hard-packed snow | 1.30 | |
| Heavy brush or loose dirt | 1.50 | |
| Wet or muddy trail | 1.65 | Highly variable, 1.5 to 1.8 depending on depth. |
| Soft snow, ankle deep | 1.60 | Rises steeply with depth. |
| Loose sand | 2.10 | The most costly common surface, and the one most tools get wrong. |
These are the measured Soule and Goldman values, and they are frequently misattributed to Pandolf’s 1977 paper, which used blacktop. The number that matters most is loose sand at 2.10. A great many rucking calculators use something around 1.2 for sand, which under-reports a beach ruck by roughly 35%. This site used to be one of them. It was wrong, and it has been corrected.
The unloaded comparison
Ẇ = 1.44 + 1.94S^0.43 + 0.24S⁴
- Ẇ
- Metabolic rate per kilogram of body mass, watts
- S
- Speed, metres per second
Looney DP, Santee WR, Hansen EO, Bonventre PJ, Chalmers CR, Potter AW. Metabolic costs of standing and walking in healthy military-age adults: a meta-regression. Medicine & Science in Sports & Exercise 2019;51(2):346-351. PMID 30253591
To tell you how much extra a ruck costs over walking the same route empty, the calculator needs a good unloaded figure. It uses the Load Carriage Decision Aid walking equation from Looney and colleagues (2019), a meta-regression across many studies of standing and walking energy cost.
This equation feeds the comparison line only. It is never folded into the ruck estimate. Mixing two models and reporting a single number would make the result impossible to audit, which would defeat the point of this page.
Cross-checks
Alongside the primary estimate you can see a MET-based figure and, if you enter heart rate data, an estimate from the Keytel equations. These exist so you can see the spread rather than trust one number. When methods disagree by more than 20% the calculator says so and explains why. MET values in particular are coarse and were never designed for load carriage, so where they diverge, the Pandolf figure is the better estimate.
Adjustments that are not from the research
There are optional BMI and training-status modifiers under “Advanced adjustments”. They are our own heuristics, not derived from any validated load-carriage equation, and they exist because gait efficiency plausibly varies between people.
Because they are not evidence-based, they are off by default, excluded from the headline number unless you switch them on, and they widen the error band when you do. An earlier version of this site applied them silently, which gave them the same apparent authority as the peer-reviewed equations. That was a mistake.
How accurate is this?
With every input inside its validated range, expect roughly plus or minus 10%. One input outside range widens that to about 20%. Two or more, and you are looking at an order-of-magnitude estimate at about 25%, which the calculator will tell you.
Individual variation is real and it is not small. Gait economy, training history, pack fit, load distribution, heat, humidity and footwear all move the true figure. No predictive equation captures any of that. Treat the number as a well-founded estimate for planning and comparison, not a measurement of what your body actually did.
What we know is wrong with it
Load-ratio correction still overshoots at slow speeds
Against Drain et al. (2017), which measured 12-17% under-prediction at 2.8 mph, this engine returns more than that at slow walking speeds. At 4 mph it returns about 32% against a measured 21-33%, which is within range. Slow-speed estimates should be read as an upper bound until field data allows the correction to be made speed-dependent as well as load-dependent.
The correction is tapered at light loads, and that taper is an assumption
Drain measured under-prediction while carrying 22.7 kg. Applied flat, the correction fired even with an empty pack, inflating unloaded walking by about 24% for a correction that only has a justification under load. It is now scaled by load ratio relative to the ratio Drain tested, so it is unchanged at that ratio and tapers to nothing at zero load. The shape of that taper is a reasonable assumption rather than a measured relationship — nobody has published how the under-prediction varies with load ratio. Figures at light loads moved down by roughly 3 to 11% when this was introduced.
BMI and fitness modifiers are heuristic, not validated
These adjustments are not drawn from a validated load-carriage equation. They are off by default and excluded from the headline figure unless deliberately enabled.
Descents steeper than -12% are extrapolated
The Santee correction resolves the original equation returning sub-basal values on descent. Below about -11% the predicted cost starts rising again, which is the expected shape: walking cost reaches a minimum near -10% grade and climbs on steeper descents as eccentric braking takes over. That much matches the literature. What is not established is whether the magnitude of the rise is right beyond -12%, because that is outside the range the correction was validated in, so those results are flagged.
Sources
Every paper the engine depends on, with what it establishes and where it is used, is listed in the research library. Every entry resolves to a PubMed ID or DOI. They are re-checked against the primary source each quarter, because citation drift is silent and this site has been caught by it before.
Changes to the engine are recorded in the changelog, so a figure you saw six months ago can be traced to the version that produced it.