How can I find distance traveled with a gyroscope and accelerometer?

I want to build an app that calculates accurate Distance travelled by iPhone (not long distance) using Gyro+Accelerometer. No need for GPS here.

How should I approach this problem?

Solution 1:

Basic calculus behind this problem is in the expression

(and similar expressions for displacements in y and z) and basic geometry is the Pythagorean theorem

So, once you have your accelerometer signals passed through a low-pass filter and binned in time with sampling interval dt, you can find the displacement in x as (pardon my C...)

float dx=0.0f;
float vx=0.0f;
for (int i=1; i<n; i++)
 {
   vx+=(acceleration_x[i-1] + acceleration_x[i])/2.0f*dt;
   dx+=vx*dt;
 }

and similarly for dy and dz. Here

float acceleration_x[n];

contains x-acceleration values from start to end of measurement at times 0, dt, 2*dt, 3*dt, ... (n-1)*dt.

To find the total displacement, you just do

dl=sqrt(dx*dx + dy*dy + dz*dz);

Gyroscope is not necessary for this, but if you are measuring linear distances, you can use the gyroscope reading to control that rotation of the device was not too large. If rotation was too strong, make the user re-do the measurement.

Solution 2:

You get position by integrating the linear acceleration twice but the error is horrible. It is useless in practice.

Here is an explanation why (Google Tech Talk) at 23:20. I highly recommend this video.

Similar questions:

• track small movements of iphone with no GPS

• What is the real world accuracy of phone accelerometers when used for positioning?

• how to calculate phone's movement in the vertical direction from rest?

• iOS: Movement Precision in 3D Space

• How to use Accelerometer to measure distance for Android Application Development

• Distance moved by Accelerometer

Update (24 Feb 2013): @Simon Yes, if you know more about the movement, for example a person walking and the sensor is on his foot, then you can do a lot more. These are called

     domain specific assumptions.

They break miserably if the assumptions do not hold and can be quite cumbersome to implement. Nevertheless, if they work, you can do fun things. See the links in my answer Android accelerometer accuracy (Inertial navigation) at indoor positioning.

Solution 3:

You should use the Core Motion interface like described in Simple iPhone motion detect. Especially all rotations can be tracked very accurately. If you plan to do something related to linear movements this is very hard stuff. Have a look at Getting displacement from accelerometer data with Core Motion.