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22-Mec-B12 Robotics: May 2016

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

  1. Question 1 Transforming a position and a velocity between frames (20 marks)
  2. Question 2 Frame diagram and composition of transformations (20 marks)
  3. Question 3 Axis assignment, DH table and forward kinematics (20 marks)
  4. Question 4 Inverse position kinematics — all closed-form solutions (20 marks)
  5. Question 5 Inverse orientation for a spherical wrist — Z–Y–Z Euler angles (20 marks)
  6. Question 6 Jacobian, reachable workspace and dexterous workspace (20 marks)
  7. Question 7 Static joint torques and singular configurations of a two-link arm (20 marks)

Start with Question 1 →

Paper format. National Exams, May 2016 — 07-Mec-B12 Robot Mechanics, 3 hours. CLOSED BOOK, with one 8.5″×11″ formula sheet (both sides) and an approved non-programmable calculator permitted. FIVE questions constitute a complete paper and each question is of equal value (20 marks); the first five as they appear in the answer book are marked. All seven questions are worked here.

Reference texts. J. J. Craig, Introduction to Robotics: Mechanics and Control, 4th ed. (the exam's leading-super/subscript notation and the Z–Y–Z Euler convention follow this text directly); M. W. Spong, S. Hutchinson and M. Vidyasagar, Robot Modeling and Control, 2nd ed.; S. B. Niku, Introduction to Robotics: Analysis, Control, Applications, 3rd ed.; K. M. Lynch and F. C. Park, Modern Robotics, 1st ed.

Notation (from the exam's own nomenclature page). A leading superscript names the frame a quantity is referenced to, so $^{A}V$ is the vector $V$ expressed in frame $\{A\}$. A leading subscript together with a leading superscript names a transformation: $^{B}_{A}T$ describes frame $\{A\}$ relative to frame $\{B\}$, and therefore maps coordinates the other way, $^{B}P = {}^{B}_{A}T\,{}^{A}P$. Keeping that direction straight is the single most common source of lost marks on this paper, so every question below states explicitly which way the given transform points.

Check: sign convention for the revolute joint. Figure 1 shows the direction of positive motion with a curved arrow but does not label a sense on the base triad. Throughout Questions 3–6 the base frame is taken as $x_{0}$ along the horizontal arm, $z_{0}$ vertically up and $y_{0}$ completing the right-handed set, with positive $q_{2}$ following the right-hand rule about $+x_{0}$. Every result below is stated in that frame; a candidate who assigns the opposite sense obtains the same magnitudes with $q_{2}\rightarrow-q_{2}$.