a) In Figure Q2-1, if the initial value of the projection of the longitudinal velocity ve on the X axis (i.e., Vx(O)) is

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A In Figure Q2 1 If The Initial Value Of The Projection Of The Longitudinal Velocity Ve On The X Axis I E Vx O Is 1 (55.06 KiB) Viewed 72 times

A In Figure Q2 1 If The Initial Value Of The Projection Of The Longitudinal Velocity Ve On The X Axis I E Vx O Is 2 (54.88 KiB) Viewed 72 times

A In Figure Q2 1 If The Initial Value Of The Projection Of The Longitudinal Velocity Ve On The X Axis I E Vx O Is 3 (38.18 KiB) Viewed 72 times

a) In Figure Q2-1, if the initial value of the projection of the longitudinal velocity ve on the X axis (i.e., Vx(O)) is 1 m/s, plot the variation of the projections of v. on both the X and Y axis (i.e., Vx and V) along one full rotation of the robot. Y(m) X (m) Figure Q2-1 A two-wheel robot travels along a circle [4 marks] b) Explain Bayes' rule of the probability theory. Discuss the importance of the probability theory and Bayes' rule to mapping? Using the Bayes' rule, answer the following question: You have installed a sensor on your robot that measures the 1-D distance L from a reflector with a constant accuracy of £2 m. You have fixed the reflector at an accurate distance of 100 m from your reference point (i.e.. f.(100) = 1). You want to use the sensor to estimate the location of your robot from the readings of the distance of your robot from the reflector (i.e., Pr (x|L)). First, you locate your robot at the positions of (-2,-1,0,1,2) m where you measure several times the distance L from the reflector. The distributions of the measurements (i.e. fx(LX)) are normal with the mean of respectively (102,101,100,99,98} m and the variance of 1 m. You locate the robot randomly with uniform distribution at xe[-2.2] (i.e.. fx(x) =) and measure the distance L from the reflector using your sensor.

If your sensor measures L = 100 m, calculate the probability that your robot locates at -1 $*$ 1 (i.e., Pr (-15 * $ 1)2 = 100)). The required values of the normal distribution are as follows: f.(1001x = 0) = N(100; u = 100, 0 = 1) = 0.5; f(1001x = 1) = f (100|x = -1) = N(100; x = 99,0 = 1) = 0.32; f.(100|x = 2) = f (100|x = -2) = N(100; = 98,0 = 1) = 0.074; Hint: You will use the Bayes' rule to calculate the conditional density of fx(x|L) at xe{-1,0,1) and L. = 100. Then using your calculated values and Figure Q2-2, you will estimate the area under the curve by approximating the red-hatched polygon. The integral of the distribution function of a random variable x between any two values (a, b) is equivalent to the cumulative probability that the random variable takes a value between [a, b] fx(x\L = 100) Figure Q2-2 Approximating the probability of x between [a, b] by calculating the area under the distribution function. [10 marks) c) What is the main problem with the odometrical localisation? Particle filtering (PF) is a technique to solve the simultaneous localisation and mapping (SLAM) problem and is explained in Figure Q2-3. Fill the gaps with the required words to make the sentences meaningful.

1. Particles will be spread randomly over the determined Initiate random particles at (xy) 2. Particles share the same as robots that is used to predict their new location after applying the control commands. Move particles with applying the same control command as the robot 3. The measurement is not accurate and involves Scanning map me by the robot 4. Likelihood estimates how probable the particles location are the same as the _'s location. Calculating the likelihood of particles Choose the particle with the maximum likelihood Resampling 5. Resampling helps to get a better statistical distribution with a number of particles 6. Suggest a simple way to estimate the location of the robot & from the location of the particles (x): Estimate location of the robot & from the locations of particles X 8 Figure Q2-3 A flowchart that represents a PF algorithm for estimating the location of a robot from the scanned map.