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ME427MTS - Isik University Faculty of the College of...

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Isik University Faculty of the College of Engineering Mechanical Engineering Department ME 427 INTRODUCTION TO ROBOTICS MIDTERM EXAMINATION Date: Saturday, April 23, 2011 Time Allowed: 120 Minutes Student’s Name:_ _______________________________________________ Student’s No.___________________________________________________ Scoring Box Question Number Maximum Score Obtained Score Q1 20 Points Q2 15 Points Q3 10 Points Q4 10 Points Q5 10 Points Q6 15 Points Q7 20 Points Total Score 100 points
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Q1. A) Write the matrix product that will give the resulting rotation matrix (do not perform the matrix multiplication). Consider the following sequence of rotations: (a) Rotate by φ about the world x-axis. (b) Rotate by θ about the current z-axis. (c) Rotate by ψ about the current x-axis. (d) Rotate by about the world z-axis. Solution Rule: For fixed or world frame pre-multiply and for current axes post multiply , , , , x z x z R R R R R B) Write the matrix product that will give the resulting rotation matrix (do not perform the matrix multiplication). Consider the following sequence of rotations: (a) Rotate by φ about the world x-axis. (b) Rotate by θ about the world z-axis. (c) Rotate by ψ about the current x-axis. (d) Rotate by about the world z-axis. Solution Rule: For fixed or world frame pre-multiply and for current axes post multiply , , , , x x z z R R R R R C) Write the matrix product that will give the resulting rotation matrix (do not perform the matrix multiplication). Find the rotation matrix representing a roll of 45° followed by a yaw of 45° followed by a pitch of 45°. Solution Rule: fixed axis rotation with order yaw about x, pitch about y, and roll about z 45 , 45 , 45 , x y z R R R R
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D) Suppose that three coordinate frames {1}, {2}, and {3} are given, and suppose 0 0 1 0 1 0 1 0 0 R and 2 1 2 3 0 2 3 2 1 0 0 0 1 R 1 3 1 2 Find R 3 2 0 2 3 2 1 0 2 1 2 3 1 0 0 0 0 1 0 1 0 1 0 0 2 1 2 3 0 2 3 2 1 0 0 0 1 Therefore, 2 1 2 3 0 2 3 2 1 0 0 0 1 where Solution 2 3 1 2 1 1 2 2 1 1 3 2 1 1 3 1 2 2 3 R R R R R R R R R T T
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E) Let 90 and 1) 1, , 1 ( 3 1 k . Find R(k, θ)
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