Assignment No 2

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University of Calgary *

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650

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Jan 9, 2024

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1 Department of Mechanical and Manufacturing Engineering Schulich School of Engineering, University of Calgary ENME 650 Mobile Robotics Instructor: Dr. A. Ramirez-Serrano Assignment # 2 (30 points) Topic: Estimation and basic concepts Due: Friday, March 3, 2023 Part I: Deeper understanding of Scientific Papers Question No. 1 (10 Points) : For this question (Question # 1) you are required to read the following paper and answer the following questions: J. Cox, "Blanche - An Experiment in Guidance and Navigation of an Autonomous Robot Vehicle", IEEE Transactions on Robotics and Automation, Volume 7, No. 2, pp. 193-204, April, 1991. a) What does the equation in Section IV:B – Sensor Data tells you? Describe the parts (x, y, C, R, r and  ) by a schematic drawing of the robot and its sensor system. (2 points) b) Summarize the three steps of the iterative algorithm, referred to as the Cox matching algorithm, described in the paper. What are the main ideas and results of the different steps? What error (distance) is minimized in the final step (illustrate this with a figure)? (2 points) c) When is the algorithm terminated, i.e., when do the iterations stop? (1 point) d) In the end of the description of the algorithm the author mentions something about rejecting outliers, what does this mean? Why is this important? What would happen if the outliers where not rejected? Also suggest one way how you can reject such outliers. (2 points) e) The algorithm also, together with a result, calculates a co-variance matrix, which represents the error of the match result, i.e. how reliable the result is in x, y and  . (3 points) Question No. 2 (3 Points) : For this Question you are required to read the following paper which you can find using the following URL address http://www.frc.ri.cmu.edu/~hpm/project.archive/robot.papers/2003/CACM.2003.html and answer the following questions: a) In his article “ Robots, After All” , Dr. Hans Moravec talks about robots having a “ sense of space ”. Very briefly explain what Dr. Moravec says about such topic.

2 Part II: Understanding of the topics of the course Question No. 3 (15 Points) : For this problem you will use the kinematic model of the mobile robot developed in Assignment No. 1 (provided below as Equation 1a). You can also use Equation 1a to determine to what extent you correctly obtained the model of the robot in Assignment # 1. The parameters for the robot such as wheel radius, etc. to be used in the Assignment are shown in Figure 1 (for illustration purposes only). Figure 1: Robot’s parameters. The parameters used in the Motion model are described in Figure 2 (based on the wheels type used). For this problem (per the configuration of the robot) we will consider the angular wheel parameters  ,  , and  as follows (see Figure 2):  1 = 0  ,  2 = 120  ,  3 = 240  ,  1 =  2 =  3 = 0  , and  = 45  .

3 Figure 2: Wheel parameters ( X R and Y R denote the robot’s frame of reference) . Using the kinematic (Motion) model (Equation 1a) for the 3-Mecanum-wheel omnidirectional mobile robot illustrated in Figure 1, and the Measurement model (Equation 1b) estimate the location of the robot after it is commanded to move using the input commands provided below. That is, you are required to compute the state variables (estimate) where the robot is after applying each set of input commands u t = (ω t1 , ω t2 , ω t3 ) provided in Table 1 to the robot and using the provided measurement model (Equation 1b), where ω 1 , ω 2 , and ω 3 is the set of rotational speed for each of the three wheels. Use the Bayes estimation formulation/procedure to compute where the robot is and the uncertainly about its state State = (x t , y t , θ t ) after each set of inputs are given to the robot and the corresponding measurements are obtained. Equation(s) 1: The state of the robot, x t , include the following three parameters: State: x t = (x R , y R , θ R ) The inputs to the system are to be considered as the three independent rotational speeds of the wheels. Please note that the inputs are stated with respect to the corresponding wheel ’ s frame of reference. Input: u t = (ω t1 , ω t2 , ω t3 ) The motion model is provided as: x t = A x t -1 + B u t +  : (Equation 1a) Where (refer to Figures 1 and 2): c( ) and s( ) represent the cos( ) and sin( ) functions, respectively. Δ i =  R + α i + γ for i = 1 to 3 δ = 3 l c (γ)  = G(0, 2) is the noise represented as a Gaussian distribution with zero (0) mean and variance 2.

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4 For this question you are required to use the following Measurement Model p( y t | x t ) represented as a 5 x 5 matrix: 0.01 0.2 0.3 0.2 0.2 0.2 0.5 0.78 0.5 0.5 0.1 0.02 0.01 0.1 03 0.001 0.001 0.01 0.001 0.001 0.001 0.1 0.3 0.1 0.1 (Equation 1b) Assume that the input (wheel) commands will occur at time steps: t 0 , t 1 , and t 2 and the corresponding control command will remain unchanged until the next set of control commands at time t i+1 . That is, once the robot has received a given control command, the mobile robot will continue using the same wheel motions in such a manner until a different set of control commands is communicated. Table 1: Input commands to the robot Time: Wheel No. 1 speed: ω 1 Wheel No. 2 speed: ω 2 Wheel No. 3 speed: ω 3 t 0 (0 seconds) 4.5 -10 5 t 1 (0.1 seconds) 4.5 10 -7.5 t 2 (0.2 seconds) 4.5 -15 -7.5 At the start, the robot is positioned in the middle of a 2-dimensional square surface divided in a grid cell map of size n x n = 100 x 100 cells (Figure 3). 1x1 1,2 … Y R 1, n 1x2 2x2 X R . 1x n 3x n … n x n Figure 3: Terrain where the robot operates and the initial position (x = 0, y = 0) of the robot. The initial position of the robot will be considered to be in the middle of the terrain at grid coordinates x = 0, y = 0 (you can also select a grid cell (e.g., n/(2-1), n/(2-1)) where the robot will start but that information must be clearly identified in your assignment). Each grid cell comprising the map will be of size 1 x 1cm. Furthermore, it will be assumed that at the start (e.g., after the robot has been turn ON), the robot has no idea where it is located in the surface/terrain, thus, at the start it is assumed that the robot can be at any location within the map with the same probability. Thus, each location is initially assumed to have the same probability (1/ n 2 ) for the robot to be at such location (where n is the number of grid cells on each side of the square terrain/surface). The robot has three Mecanum wheels arranged around the robot as illustrated in Figure 1, each having passive rollers oriented at 45 degrees w.r.t the wheel’s axis of rotation as shown in Figure 1. The position of the wheels on the robot are as shown in Figure 1. Additional specific information about the robot that you will need when solving the assignment is provided below:

5  The distance between each two wheel s’ axis of rotation is: L=15 cm  The radius of each wheel is: r = 10 cm  The wheels are positioned around the vehicle at 120 degrees apart and the wheel ’ s axis f rotation is considered to be perpendicular to the vehicle ’ s chassis (see Fig. 1),  Each wheel is powered by an electric motor directly attached to the wheel’s rotating hub/shaft.  The outputs of the model will be the position, ( x R , y R ), and heading, θ R , of the vehicle at the corresponding time step (see Equation 1a). The measurements model is assumed to represent a GPS sensor that is not very precise but has an error represented as a Gaussian distribution over the area of interest (as provided in the Measurement Model: Equation 1b). For this problem you are required to:  Show a history of the state evolution showing the (estimated) state of the mobile robot at times t 0 , t 1 , and t 2 vs the real position (as per the Motion model with no noise/disturbances) of the robot. You don’t need to show the entire 100 x 100 grid cell map. You can only provide the region of the map that has been updated clearly showing which is the region being displayed and the location of the robot.  You must also provide all the calculations performed when computing last state estimation (after the robot completed the commands received at time t 2 . It is recommended that you create a MATLAB file (not mandatory though) to perform your computations ( and submit the Matlab via email as part of the assignment ). The MATLAB source code must be well documented (e.g., describing the names of the variables used, the purpose of each piece of code, etc.) for anybody to be able to understand it, mark it, and potentially modify it. Part III: Understanding of basic topics discussed in lectures Question No. 4 (1 Point) : How do you find the set of Instantaneous Centre of Rotation (ICR) of a robot? Provide at least two ways. Question No. 5 (1 Point) : Using the methods described in Question 4, find ALL the ICR of the 2 mobile robots shown below (show your response graphically and theoretically) and justify it in writing.

6 A synchro drive mobile *************** END OF THE ASSIGNMENT ***************

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