ELEC 8900 Special Topics: Estimation, Filtering, and Tracking
Project: Implementation of Kalman Filter
Figure 1 shows two automobiles in a car following model where the objective of the follower vehicle A is to estimate the distance to the vehicle B in front based on radar measurements. The vehicle A continuously measures the distance between the B and itself with the help of a font-bumper mounted radar. Based on the estimated distance, the vehicle adjusts its controls to maintain a constant distance.
z(𝑘) = 𝑥(𝑘) + 𝑤(𝑘) ∀ 𝑘 = 1, … , 𝑛
Figure 1: Measured distance using a radar. The true distance at time k is denoted x(k), corresponding measurement and the measurement noise are denoted z(k) and w(k), respectively.
The state-vector x(k) = [x(k), x˙(k), x¨(k)]T, consisting of (relative) position, velocity and acceleration, is assumed to undergo the following process model
x(k + 1) = Fx(k) + v(k)
F , (2)
v(k) is the process noise, and the process noise covariance is given by
The radar measures the distance, given by
z(k) = Hx(k) + w(k)
H = [1 0 0] (5)
is the measurement matrix.
Question 1 (Data simulation)
Let us assume that the radar measurements are taken at a sampling rate of 10Hz. Simulate
100 measurement samples, i..e, k = 1,2,,...,100, for the following assumptions
Plot the following on the same axis:
Plot the following on separate axes:
Question 2 (Filter initialization)
Implement the folloiwng two approaches to initialize the Kalman filter (then compare their performance in KF implementations for each question)
Question 3 (The Kalman filter implementation)
Implement a Kalman filter Plot the following quantities against time (see Figure 5.3.2.-1 on page 220 of the textbook for hint)
Question 4 (Kalman filter vs. RLS filter)
Question 5 (Model mismatch analysis)
Explain how model-mismatch can be spotted in a Kalman filter. Implement the following model mismatched filters to demonstrate your analysis
Question 6 (Modelling relative vs. true vehicle state — optional)
The model described in (1)-(4) models the relative (position, velocity and acceleration) of B w.r.t. A. In order to find the real values, these quantities need to be translated. For example, the relative velocity xˆ˙(k) = 3 km/h needs to be translated (baed on the true velocity of A). The objective of this question is to develop a new model such that the state x(k) contains the true state of the vehicle B based on the same measurement z(k) which is either the relative distance or relative velocity of the vehicle B w.r.t. A.
Question 7 (Joint estimation of both vehicles’ states — optional)
In reality, the vehicle A does not know its true position or velocity — all it has is an estimate of these quantities.
Question 8 (Kalman smoother — optional)
Implement a Kalman smoother for the WNA problem (i.e., data generate using WNA and
KS implemented for WNA). Plot the following for comparison no the same axis
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