Integrate the forward velocity to obtain displacement and compare with GPS displacement. We simplify the description of the motion by assuming that the vehicle is moving in a two- dimensional plane. Denote the position of the center-of-mass (CM) of the vehicle by (X,Y,0) and its rotation rate about the CM by (0,0,w). We denote the position of the inertial sensor in space by (x,y,0) and its position in the vehicle frame by (x,,0,0). Then the acceleration measured by the inertial sensor (i.e. its acceleration as sensed in the vehicle frame) is ?o?s = ? -@? - w?, 7x = ? +0? +0?? where all the quantities in these equations are evaluated in the vehicle frame. Assume that? = 0 (that is, the vehicle is not skidding sideways) and ignore the offset by setting ? = 0 (meaning that the IMU is on the center of mass of the vehicle, i.e. the point about which the car rotates). Then the first equation above reduces to ? = ?o?s - Integrate this to obtain ?. Compute ? and compare it to ?? How well do they agree? If there is a difference, what is it due to? Can you fix it? Plot all results. With the previous assumptions, use the heading from the magnetometer (or your complementary filter if you have better results) to rotate your previously estimated forward velocity.