ECE569 Exam 1 October 28, 2015 1 Name: Score: /100 This exam is closed-book. You must show ALL of your work for full credit. Please read the questions carefully. Please check your answers carefully. Calculators may NOT be used. Please leave fractions as fractions, but simplify them, etc. I do not want the decimal equivalents. Cell phones and other electronic communication devices must be turned off and stowed under your desk. Please do not write on the backs of the exam or additional pages. The instructor will grade only one side of each page. Extra paper is available from the instructor. Please write your name on every page that you would like graded. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
ECE569 Exam 1 October 28, 2015 2 1. (10 points) Find the H matrix corresponding to the following transformation: First rotate by π/2 about the fixed y-axis. Call the new frame F 1. Then translate 7 cm along the current y-axis. Call the new frame F 2. Then rotate by π/4 about the current z-axis. Call the new frame F 3. Finally translate 5 cm along the fixed z-axis. Call this new frame F 4. Please first evaluate the expression symbolically (i.e. in terms of variables) and then evaluate the resulting expression by substituting the values of the variables.
ECE569 Exam 1 October 28, 2015 3 2. (5 points) If joint Ji is positioned at joint angle π/4, and link Li has link length 110 mm, and link twist π/2, what is A i?
ECE569 Exam 1 October 28, 2015 4 3. (5 points) Identify the DH parameters for the matrix 1 0 0 2 0 2 1 1 A i = 2 0 1 0 2 2 1 8. Assume that all angle measurements are between 0 and 2π. a i = α i = d i = θ i =
ECE569 Exam 1 October 28, 2015 5 4. (5 points) Give the expression for T 3 5 in terms of the appropriate A i. 5. (5 points) Give the expression for T 5 3 in terms of the appropriate A i.
ECE569 Exam 1 October 28, 2015 6 6. (10 points) Using the A i and the H you just found in problem (1), 3 determine the following. If p 2 = 2, what are the coordinates of 1 p with respect to the fixed frame?
ECE569 Exam 1 October 28, 2015 7 7. (5 points) Forward kinematics determines the position and orientation of the end effector from which of the following: (Circle all that apply.) forces on links joint angles joint torques link geometry link moment of inertia 8. (5 points) What is the definition of a skew symmetric matrix?
ECE569 Exam 1 October 28, 2015 8 9. (10 points) Please select frames of reference F 0, F 1, and F 2 and generate the table of DH parameters. Please place F 0 on the solid surface below the cylinder. If you find you need variables not listed in the diagram, please define and use them.
ECE569 Exam 1 October 28, 2015 9 10. (5 points) Starting from Frame F 0, rotate first by θ 1 about the fixed x-axis and then by θ 2 about the current z-axis to orient Frame F 1. Find the rotation matrix representing this transformation. 11. (5 points) Starting from Frame F 0, rotate first by θ 1 about the fixed x-axis and then by θ 2 about the fixed z-axis to orient Frame F 1. Find the rotation matrix representing this transformation.
ECE569 Exam 1 October 28, 2015 10 12. (10 points) A homogeneous transformation H describes a rotation R and a translation d. (a) Give the expression for H in terms of R and d. (b) Give the expression for H 1 in terms of R and d.
ECE569 Exam 1 October 28, 2015 11 13. (5 points) Determine the singularities of the system whose Jacobian is c 1 d 3 0 s 1 J 11 = s 1 d 3 0 c 1. 0 1 0 14. (5 points) What is the origin of frame F 1 with respect to frame F 0 if 1 0 0 6 0 2 1 1 A 1 = 2 0 1 0 2 2 1 5?
ECE569 Exam 1 October 28, 2015 12 15. (10 points) Where J v = [J v1 J v2... J vn ] and J ω = [J ω1 J ω2... J ωn ], give the expressions for J vi and J ωi.
ECE569 Exam 1 October 28, 2015 13 Formula Sheet A set of Basic Homogeneous Transformations that generate SE(3) Trans x,a = Trans y,b = Trans z,c = 1 0 0 a 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 b 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 c Rot x,α = Rot y,β = Rot z,γ = 1 0 0 0 0 c α s α 0 0 s α c α 0 c β 0 s β 0 0 1 0 0 s β 0 c β 0 c γ s γ 0 0 s γ c γ 0 0 0 0 1 0 In the Denavit-Hartenberg (DH) convention, A i is the product of four basic transformations, A i = Rot z,θi Trans z,di Trans x,ai Rot x,αi c θi s θi 0 0 1 0 0 0 = s θi c θi 0 0 0 1 0 0 0 0 1 0 0 0 1 d i = c θi s θi c αi s θi s αi a i c θi s θi c θi c αi c θi s αi a i s θi 0 s αi c αi d i 1 0 0 a i 0 1 0 0 0 0 1 0 1 0 0 0 0 c αi s αi 0 0 s αi c αi 0