# What's the best method of measuring displacement and speed without resorting to GPS?

Is it possible to determine the displacement, speed and acceleration of someone while running, by using methods like dead reckoning, sensor fusion or other, and sensors like accelerometers, gyroscopes, magnetometers, altimeters or others?

It should be a standalone solution (no GPS or RF).

In the affirmative case what's the ballpark precision and accuracy for distances:

• below 50cm;
• between 100 to 400m.

Could an inertial measuring unit (IMU) be a good solution for this case?

Measuring velocity and displacement directly requires referencing to an external system (Galilean relativity). But it sounds like you've ruled that out. So you can only measure acceleration and rotation, which can be done with accelerometers and gyros (maybe combined into an IMU).

Displacement is the integral of velocity, and velocity is the integral of acceleration. So the displacement sensitivity depends on how long constant velocity is maintained. Integrating up the initial acceleration will give you a velocity estimate with some error $\sigma_V$, but then if there is no change in velocity for a time $T$, the displacement error will be $\sigma_D = \sigma_V T$.

Suppose a runner starts from a standstill and accelerates to 2 m/s in the course of 1 s. (I'm not a runner, so I could be off by a factor of order unity on the numbers here.) That's about $0.2 g$. A typical accelerometer like the ADXL335 has noise of maybe 200 $\mu g$ over that 1 s interval. So you can estimate the velocity to about 0.1%, $\sigma_V \approx 2$ mm/s. The 500 m total run length takes 250 s, so at the end, you're off by $\sigma_D =$ 50 cm. Shorter distances will have correspondingly lower error.

However, the accelerometer had better be on when you start your run! Otherwise you don't know your velocity and can't estimate the displacement at all.

• Do I only need an accelerometer or accuracy can be improved by combining a gyro in an IMU, before integrating aceleration? – Rui Lima Jan 30 '15 at 14:17
• A purely accelerometer based solution will only give that sort of accuracy if the object accelerating is on a rail and the accelerometers are fixed relative to the direction of travel. In practice this isn't the case. A gyro lets you correct for changes in your accelerometer's orientation (assuming it's a 3 axis one). All of these measurements have error, which over time results in "drift" between the calculated and actual location. Military aircraft tend to have an inertial navigation system which can give reasonable accuracy over a few hours, but with COTS components you'll get a lot less. – LeoR Jan 30 '15 at 14:30
• I think your error is optimistic: you used the velocity error for 1 s to find the displacement error for 250 s. The accelerometer doesn't care whether you're at a constant velocity or not - the noise is always present. – Greg d'Eon Jan 31 '15 at 0:16
• @Kynit, LeoR, fair points. There are obviously several sources of error that I didn't discuss. A lot could be written on this topic - my aim was to get equidna thinking along the right lines. – Dave Kielpinski Jan 31 '15 at 3:25
• That's fine - it's pretty clear that this type of setup works, as long as accuracy isn't a huge concern – Greg d'Eon Jan 31 '15 at 3:31