Data for Velocity, acceleration of car in three dimensions for at least 6 equal time intervals?

Question

I am working on Kalman Filter for my IB Math HL internal assessment.

I need 6 sets of data mentioned in the question to generate updated state matrices which will be compared with already measured exact value. I also need percentage of errors in the these estimates.

Further more, to generate a complete estimate i also need the values of the cars position and velocity in the three dimensions with respect to each of the individual set of value and i also need the error percentage.

Hence i can make good estimates by using the measured values as well as values from the Kalman filter

Answer 1

Since the time derivative of the velocity function is acceleration, d dt v ( t ) = a (t ), we can take the indefinite integral of both sides, finding ∫ d dt v (t )dt = ∫ a( t )dt + C 1, where C 1 is a constant of integration. Since ∫ d dt v (t ) dt = v (t ), the velocity is given by v (t ) = ∫ a( t )dt + C 1. Similarly, the time derivative of the position function is the velocity function, d dt x ( t ) = v (t ). Thus, we can use the same mathematical manipulations we just used and find x (t ) = ∫ v ( t )dt + C 2, where C 2 is a second constant of integration. We can derive the kinematic equations for a constant acceleration using these integrals. With a(t ) = a a constant, and doing the integration in (Figure) , we find v (t ) = ∫ adt + C 1 = at + C 1. If the initial velocity is v (0) = v 0, then v 0 = 0 + C1 . Then, C1 = v 0 and v (t ) = v 0 + at ,

Answer 2

6 sets of data:

-measured value

-kalman filter input

-percentage of errors

-position and velocity in three dimensions with respect to each individual value

-error percentage

..............................................................

Answer 3

The following are example of what such data might look like:

Time Interval 1:

Velocity (m/s) = (10, 20, 5)

Acceleration (m/s^2) = (2, 0, 3)

Time Interval 2:

Velocity (m/s) = (12, 20, 8)

Acceleration (m/s^2) = (-1, 1, 1)

Time Interval 3:

Velocity (m/s) = (11, 21, 9)

Acceleration (m/s^2) = (0, -2, 1)

Time Interval 4:

Velocity (m/s) = (11, 19, 10)

Acceleration (m/s^2) = (2, 1, -1)

Time Interval 5:

Velocity (m/s) = (13, 20, 9)

Acceleration (m/s^2) = (-1, -1, 2)

Time Interval 6:

Velocity (m/s) = (12, 18, 11)

Acceleration (m/s^2) = (0, 2, -1)

Answer 4

In this example, the first row represents the initial state of the car at time t=0. The second row represents the state of the car after one second, and so on.

We can calculate the velocity in each dimension by taking the difference between the position at the current time and the position at the previous time, divided by the time interval. For example, the velocity in the x dimension at time t=1 is:

Velocity x (t=1) = (Position x (t=1) - Position x (t=0)) / (Time interval) = (10 - 0) / 1 = 10 m/s

Similarly, we can calculate the acceleration in each dimension by taking the difference between the velocity at the current time and the velocity at the previous time, divided by the time interval. For example, the acceleration in the x dimension at time t=1 is:

Acceleration x (t=1) = (Velocity x (t=1) - Velocity x (t=0)) / (Time interval) = (10 - 0) / 1 = 10 m/s^2

We can repeat this process for each dimension and each time interval to obtain a complete set of data for velocity and acceleration in three dimensions for the car.