EEE 187: Take Home Test #2

Transcription

1 EEE 187: Take Home Test #2 Date: 11/30/2017 Due : 12/06/2017 at 5pm 1 Please read. Two versions of the exam are proposed. You need to solve one only. Version A: Four problems, Python is required for some questions. Version B: Five problems, no specific programming language is required Please clearly mention which version you solved. You can discuss the exam with one student consultant. You need to provide the name of your consultant on the first page of your paper. You and your consultant cannot discuss the exam with a third party. The instructor is not considered a third party. You cannot divide work between you and your consultant. Every student should work on every question and problem. Reports are individual. Your consultant cannot contribute to writing your report. Team members can solve different versions. However, they can contribute to the common problems only. In other words you cannot work with your consultant on problems that are not included in your version. Show your work for partial credit. Attach your code. Two points are reserved to the organization in each problem. This is version B. PROBLEM 1: ACCELEROMETER LEAST SQUARES FILTERING (10 PTS) The EEE 187 class decided to send Chris in an exploratory mission to an unknown planet. Chris took with him an accelerometer, two apples, a wig, and the energy drink shown in figure 1a. While there is a chance that Chris will never return to earth, he still accepted the mission. After a few days on the planet, Chris was able to send the acceleration data shown in figure 1b. Chris is very proud and is asking us to estimate the acceleration of gravity of the new planet. He suggested to use the recursive least squares filtering method for acceleration estimation. This filter is characterized by the following equations where G(k + 1) = P (k)h T [ HP (k)h T + R(k) ] 1 ˆx(k + 1) = ˆx(k) + G(k)[y(k) H ˆx(k)] (2) P (k + 1) = [I G(k)H] P (k)[i G(k)H] T + [G(k)]R(k)[G(k)] T (3) ˆx is the estimated acceleration P is the covariance matrix of the estimated acceleration y is the measured acceleration R is the measurement covariance matrix. It characterizes the goodness of the measurement. H is a matrix that gives the relationship between ˆx and y. That is y = H ˆx. The value of H needs to be determined in this problem. G is the filter s gain. I is the identity matrix. k is the iteration index and [ ] T is the transpose matrix. The goal is to determine the estimated acceleration ˆx and its covariance matrix P at each iteration when a new measurement is obtained. The data points can be obtained from the instructor s website. 1) The planet that Chris visited does not have a name, suggest a meaningful name for it. 2) Draw Chris on the planet. (1)

2 Meassured acceleration m/s (a) Energy drink Figure 1: Figures for problem Time (s) (b) Measured acceleration Right Wheel Wheel rotation (rad/s) Left Wheel Time in seconds Figure 2: Wheels angular speed in radians per seconds 3) Chris claimed that planet earth seen from the new planet is flat. What is your scientific interpretation of the claim? 4) Determine matrix H knowing that y and ˆx represent the measured and estimated value of the same quantity. 5) Write code to implement the recursive least squares method to obtain the acceleration of gravity of the planet. Include your code in the report. The measurement data can be retrieved from the instructor s website. The measurement covariance is R = Feel free to pick reasonable initial guess values for ˆx and P. For example, you can take ˆx(t = 0) = 9.81m/s 2, P (t = 0) = R. 6) On the same graph, plot the measured acceleration y and the estimated acceleration ˆx as a function of time. 7) What is your estimation of the acceleration of gravity of the new planet? Feel free to use the last value of ˆx. 8) What is the variance of the estimation? Feel free to use the last value of P. 9) Compare between the covariance of the measurement and the covariance of the estimation and discuss the effectiveness of the least squares filter. 10) Assuming the acceleration of gravity of the new planet is 26 in SI unit, by how much the height of Chris will change when he returns to earth? Note: Please do not spend two days on questions 1), 2), 3) and 10). PROBLEM 2: ORIENTATION ANGLE FROM ROTATIONAL SPEED (10 PTS). The wheels rotation (in rad/s) for a differential drive is shown in figure 2 in the time interval [0, 120s]. The radius of the wheels and the distance between them are r = 0.1m and L = 0.3m, respectively. The data for the angular speed of the wheels can be retrieved from the instructor s website. 1) Plot the angular velocity ω(t) of the robot in the time interval [0, 120s]

3 3 2) Plot the orientation angle θ(t) of the robot in the time interval [0, 120s]. Note that at time t = 0, the robot s orientation is π/3rad. 3) Plot the angular acceleration α(t) of the robot in the time interval [1, 120s]. We know that ω(0) = 0rad/s. 4) What is the final value (at time t = 120s) of the orientation angle in rad/s? Hint: Note that θ = ω and ω = α. PROBLEM 3: GPS PSEUDORANGE EQUATIONS. (10 PTS) We want to determine the time shift and the coordinates of a GPS receiver using trilateration. The location of the satellites is summarized in the table below. Satellite 1 Satellite 2 Satellite 3 Satellite 4 x y (4) z P SR Table 2: Normalized coordinates of the satellites and pseudorange values. 1) What is the minimum number of GPS satellites needed to determine the time and the coordinates of the receiver? 2) Determine the time shift and the coordinates of the GPS receiver. PROBLEM 4: CALIBRATION OF SHARP IR SENSOR (10 PTS) In this problem we want to determine the calibration curve of the sharp sensor shown in figures 3 and 4. It is clear that the relationship between the input (distance) and output (voltage) is nonlinear. In order to have one to one relationship, we ignore any distance that is less than 3cm. The data points are shown in table 3. To obtain the calibration curve, we propose to use polynomial regression model as follows where d i is the distance in centimeters and v i is the voltage in volts. a 0, a 1,..., a k are the coefficients of the polynomial, to be determined. k is the degree of the polynomial n is the number of data points. For n measurements, system (5) can be written under matrix form as follows d 0 1 v 1 v k 1 a 0 d 1. = 1 v 2 v2 k a d n 1 v n vn k a k System (7) can be written under matrix form as follows where and d i = a 0 + a 1 v i + a 2 v 2 i a k v k i (5) i = 1,..., n (6) D = V A (8) D = [ ] T d 0 d 1 d n (9) 1 v 1 v k 1 1 v 2 v k 2 V =..... (10). 1 v n vn k A = [ a 0 a 1 a k ] T (11) (7)

4 4 Figure 3: Sensor input/output data from datasheet Voltage (V) Distance (cm) Figure 4: Sensor input/output data in the interval [3, 40]cm The least squares method can be used to obtain the coefficient of the polynomials as follows A = ( V T V ) 1 V T D (12) This equation is called the modified Pseudo-Cash equation. 1) Use Matlab, Python or similar tools to determine the coefficients when a polynomial of degree six (k = 6) is used. Use equation (12). 2) Use Matlab, Python or similar tools to determine the coefficients when a polynomial of degree four(k = 4) is used. Use equation (12). 3) Plot the polynomials on the same graph as the data points shown in figure 4. (These are the points shown in table

5 5 3. They can be obtained from the instructor s website) 4) Use your polynomials to compute the absolute error for the points shown in table 4 and complete the table. 5) From the polynomial of degree six, what is the distance if the voltage reads 0.85V? Distance(cm) V oltage(v ) Table 3: Data for Sharp sensor. The distance The voltage Error in distance Error in distance in in using 4th degree using 6th degree centimeters volts polynomial polynomial Table 4: Specific calibration values for comparison (13) PROBLEM 5: DIFFERENTIAL DRIVE AND PROPORTIONAL CONTROL (10 PTS) We propose to use a simple proportional law to plan the path of a differential drive robot to reach a given configuration. The initial and final configurations are shown in table 1 below. Initial Goal x 1 15 y 1 15 θ π/2 π/4 Table 1: Initial and desired configurations The configurations are shown in figure 5. The robot s motion is given by (14) ẋ = v cos(θ) (15) ẏ = v sin(θ) (16) θ = ω (17)

6 6 Figure 5: Initial and desired configurations where v and ω are the linear and angular speeds of the robot. The maximum speed of the robot is 5m/s. The odometry equations are given by x(k + 1) = T v(k) cos(θ(k)) + x(k) (18) y(k + 1) = T v(k) sin(θ(k)) + y(k) (19) θ(k + 1) = T ω(k) + θ(k) (20) The proposed closed loop law is given by v(k) = K 1 ρ(k) (21) ω(k) = K 2 (θ(k) θ Goal ) + K 3 (λ(k) θ Goal ) (22) with and K 1 = 0.1, K 2 = 0.5, K 3 = 0.6 (23) λ(k) = atan2(y G (k) y(k), x G (k) x(k)), ρ(k) = (y G (k) y(k)) 2 + (x G (k) x(k)) 2 (24) where (x G, y G ) and (x, y) represent the coordinates of the destination point (goal) and the robot, respectively. Note that the values of the gains are chosen to satisfy stability. We want to use the odometry equations with a sampling time of T = 0.1s 1) Write code to simulate the robot s path and show its trajectory. You can represent the robot by a geometric point. 2) Plot the speed of the robot as a function of time. 3) Plot the orientation angle of the robot as a function of time. 4) Write code to create a video of the simulation. Send the video to the instructor by . Video taken by a camera does not count and will not be considered.