5- Question 4 In the state estimation problem, the Kalman filter is very popular. The key...

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5- Question 4 In the state estimation problem, the Kalman filter is very popular. The key equations for the Kalman filter are as below (mainly to test your understanding of the Kalman filter based on the simple sensor inputs from GNSS and odometer. Note that the state below can be initialized with zero values in position): Zty = (1) XFX1+ BUt (2) PFPFT + Q (3) AZZ-HX7 = (4) St HPHT + R = (5) K = PHTS1 = (6) XX+K+AZt (7) P = (I-KH)P Given a GNSS/odometer integration problem, the GNSS provides the position measurements at a time (epoch) t as ZGNSS = [ZCNSS ZGNSS ZENSST. The ZNSS, ZCNSS and ZGNSS denotes the absolute position measurement. The covariance matrix (R) associated with the GNSS measurement and process noise (Q) associated with the odometer is as below: [12 0 0 R = 0 12 [22 0 0 0,Q= 0 22 0 0 0 22 0 0 22 The odometer provides the relative position between two consecutive epochs as Zodom = [zodom Zt,x zodom zodom. The time difference between two consecutive epochs is denoted as At which equals 1 second. Be noted that the odometer only provides the relative motion. The state that we wish to estimate is the position X = [x y z]T. The initial covariance of the state is given as below: [100 0 0 Po = 0 100 0 0 0 1002 The relative measurement from the odometer between epoch 0 and epoch 1 is denoted as Zodom = = [10 10 -10]. The GNSS position measurement at epoch 1 is denoted as ZGNSS [99 -12]. The state of the system at epoch 0 is as X = [000]. Given those measurements from epoch 0 and epoch 1, a) Calculate the matrixes F and B based on the relative motion from the odometer. (3 marks) b) Calculate the equation (2) of the Kalman filter based on the given process noise Q. (3 marks) c) Calculate the observation matrix H in equation (3) based on the observation from GNSS ZENSS (3 marks) d) Calculate the equation (4) based on the covariance matrix associated with the GNSS position measurement R. e) Calculate the equation (5) and (7). (4 marks) (3 marks) f) If only the pseudorange measurements are provided to integrate with the odometer using the Kalman filter, the state that we wish to estimate is the position X = [Pr.tx'Prty' Pr.tz/rt]". We also need to estimate the receiver clock bias as well. Describe how can this be achieved. (10 marks) .../5 5- Question 4 In the state estimation problem, the Kalman filter is very popular. The key equations for the Kalman filter are as below (mainly to test your understanding of the Kalman filter based on the simple sensor inputs from GNSS and odometer. Note that the state below can be initialized with zero values in position): Zty = (1) XFX1+ BUt (2) PFPFT + Q (3) AZZ-HX7 = (4) St HPHT + R = (5) K = PHTS1 = (6) XX+K+AZt (7) P = (I-KH)P Given a GNSS/odometer integration problem, the GNSS provides the position measurements at a time (epoch) t as ZGNSS = [ZCNSS ZGNSS ZENSST. The ZNSS, ZCNSS and ZGNSS denotes the absolute position measurement. The covariance matrix (R) associated with the GNSS measurement and process noise (Q) associated with the odometer is as below: [12 0 0 R = 0 12 [22 0 0 0,Q= 0 22 0 0 0 22 0 0 22 The odometer provides the relative position between two consecutive epochs as Zodom = [zodom Zt,x zodom zodom. The time difference between two consecutive epochs is denoted as At which equals 1 second. Be noted that the odometer only provides the relative motion. The state that we wish to estimate is the position X = [x y z]T. The initial covariance of the state is given as below: [100 0 0 Po = 0 100 0 0 0 1002 The relative measurement from the odometer between epoch 0 and epoch 1 is denoted as Zodom = = [10 10 -10]. The GNSS position measurement at epoch 1 is denoted as ZGNSS [99 -12]. The state of the system at epoch 0 is as X = [000]. Given those measurements from epoch 0 and epoch 1, a) Calculate the matrixes F and B based on the relative motion from the odometer. (3 marks) b) Calculate the equation (2) of the Kalman filter based on the given process noise Q. (3 marks) c) Calculate the observation matrix H in equation (3) based on the observation from GNSS ZENSS (3 marks) d) Calculate the equation (4) based on the covariance matrix associated with the GNSS position measurement R. e) Calculate the equation (5) and (7). (4 marks) (3 marks) f) If only the pseudorange measurements are provided to integrate with the odometer using the Kalman filter, the state that we wish to estimate is the position X = [Pr.tx'Prty' Pr.tz/rt]". We also need to estimate the receiver clock bias as well. Describe how can this be achieved. (10 marks) .../5