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 fontbumper 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 statevector 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) where 
(1) 
F , (2)
v(k) is the process noise, and the process noise covariance is given by
Q (3)
The radar measures the distance, given by 

z(k) = Hx(k) + w(k) 
(4) 
where
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
 Initial state is x(0) = [0, 2, 2]^{T}
 Use ˜q = .001
Plot the following on the same axis:
 The true distance x(k) between the vehicles
 The measured distance z(k)
Plot the following on separate axes:
 Velocity ˙x(k)
 Acceleration ¨x(k)
Question 2 (Filter initialization)
Implement the folloiwng two approaches to initialize the Kalman filter (then compare their performance in KF implementations for each question)
 Three point initialization (of position, velocity and accelration)
 Random initialization (that could be far away from the true value)
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)
 Position (true vs. KF)
 Velocity (true vs. KF)
 Position variance
 Velocity variance
 NIS (along with its performance limits on the same axis)
 NEES (along with its performance limits on the same axis)
Question 4 (Kalman filter vs. RLS filter)
 Develop an RLS filter to estimate x(k).
 Use the simulated data above to compare the RLS filter against KF
 Discuss your comparison results
Question 5 (Model mismatch analysis)
Explain how modelmismatch can be spotted in a Kalman filter. Implement the following model mismatched filters to demonstrate your analysis
 A filter consisting of 2states (using the WNA model)
 A filter with incorrect knowledge of the covariance matrices
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.
 Rewrite the statespace model (1)(4) such that the state x(k) represents the true state of vehicle B
 Demonstrate a Kalman filter based assuming the parameters used in Question 1 for WNA model
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.
 Assuming that A has a “measurement” of its true velocity (e.g., through the rpm of its wheels) develop a statespace model that it can use to estimate its true state
 Assuming the same radar installed in A, develop a new state space model by which A can estimate the state of the vehicle B
 Demonstrate the above through a simulation (i.e., first, generate the data by making assumptions and then employ a Kalman filter to estimate the states)
 Your solution to the above was likely in the form of a “joint state estimation” model. Can you compare the performance of this approach to the following
 Relative to an alternate approach (e.g., estimate own state first and use a method similar to earlier ones)
 Relative to the true state
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
 x(kk) vs x(kn)
 P(kk) vs P(kn) for different types of initializations (three point vs. random)
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