Mec-B12 — Robot Mechanics
Robot components (sensors, actuators, and end effectors, and their selection criteria); basic categories of robots (serial and parallel manipulators, mobile robots); mobility/constraint analysis; workspace analysis; rigid body kinematics (homogeneous transformation, angle and axis of rotation, Euler angles, cylindrical and spherical coordinates); manipulator kinematics and motion trajectories (displacement and velocity analyses, differential relations, Jacobian matrix); non-redundant and redundant sensing/actuation of manipulators; manipulator statics (force and stiffness); singularities; and manipulator dynamics.
Subject outline
Mec-B12 Robot Mechanics is a Mechanical Engineering technical exam (PEO National Examination 16-Mec-B12). Its official syllabus covers robot components (sensors, actuators and end effectors, and their selection criteria), the basic categories of robots (serial and parallel manipulators, mobile robots), mobility, constraint and workspace analysis, rigid-body kinematics, manipulator kinematics, the Jacobian and motion trajectories, redundant sensing and actuation, manipulator statics (force and stiffness), singularities and manipulator dynamics. The two papers solved here examine the kinematics of serial manipulators, and that is what this course teaches: frames and rotation matrices and how to test one for validity; homogeneous transformations, their inverses and chains of them; Z-Y-Z Euler angles and fixed-angle sets, and the two solutions every extraction has; Denavit–Hartenberg parameters and the link transform; forward kinematics of prismatic and revolute chains; inverse kinematics, closed-form solutions and kinematic decoupling at a spherical wrist; the manipulator Jacobian by differentiation and by columns; static forces through the Jacobian transpose; singularities and the reachable and dexterous workspace; and trajectory generation with cubic splines through via points. It does not teach robot components, parallel or mobile robots, mobility and constraint analysis, the angle-and-axis representation, redundant sensing and actuation, manipulator stiffness or manipulator dynamics.
The paper is 3 hours and closed book, and it is one of the PEO papers that allows an aid sheet: one 8.5 × 11″ formula sheet, written on both sides, holding no worked examples, which must be handed in with the paper. The May 2016 paper permits a non-programmable approved calculator; the December 2018 paper, an approved Casio or Sharp calculator. Both sittings set seven questions; answer any five — five constitute a complete paper, only the first five as they appear in the answer book are marked, and each question is of equal value. Every question is worth the same 20 marks — the May paper prints the (20) beside each question, and most questions on both papers print a split across their parts (the vector-transformation question of each sitting is 12 + 8) — so treat the parts of every question as sharing its 20 and never leave one blank.
The same syllabus is also listed as 08-Mfg-B8 — the Manufacturing Engineering listing carries it word for word; the question papers in this course are Mechanical papers, May 2016 headed 07-Mec-B12 (the earlier code) and December 2018 headed 16-Mec-B12. Same syllabus, two faculty codes.
What this course gives you
Worked solutions — all fourteen questions from May 2016 (seven) and December 2018 (seven), fully retypeset as live maths — every matrix, every transform — with the final answers pulled out ahead of the working, and every published answer checked, with corrections flagged at the exact step that goes wrong. These two papers carry an unusual number of transcription slips, and the largest change the answer: both published velocity answers (May Q1(b) and December Q2(b)) carry the translation column through a free vector, so they are recomputed with w = 0, giving (−1.34, 22.32, −15) and (−9.82, −22.99, 5); the May Q2 product of three transforms is never evaluated in the original and several of its intermediate matrices drift from the values the question gives, so it is recomputed from the question's own matrices; the May Q5 Z-Y-Z Euler extraction substitutes elements its own formulas do not call for — the matrix as printed gives (210°, 45°, 90°), not (90°, 45°, 45°), verified by rebuilding the matrix from the angles; May Q7(b) is not answered at all in the published solution and is worked here, the arm being singular where sin θ₂ = 0; and the December Q3(b) product boxed in the original cannot be a homogeneous transform — its bottom row is [0, 0, −0.999, −0.999] and its (3,4) entry is −2372 — so the three matrices are multiplied out as the question states them, giving −28.32 there and an orthonormal rotation block. Four questions were set on manipulator drawings that the published solutions misread, and they are re-solved here from the figures as printed: May Q3, Q4 and Q6 treat Figure 1 as a cylindrical arm (and the published Q6 Jacobian belongs to yet another, planar arm), whereas the arm drawn slides sideways, turns about a vertical axis and then lifts — so its DH table carries two twists, its inverse kinematics has two branches and det J = cos q₂; and December Q4's joint 3 turns about a horizontal axis, so α₂ = −90° and the rotation block no longer collapses to θ₁ + θ₃. Further headed notes correct smaller slips in the published working (a √3 that should be √2, a −6.5 that should be −0.5, a 45 s that should be 4 s) or supply what the original leaves out — the second inverse-kinematics branch, the missing frame diagram. Where a published figure stands it is used; where it does not, the original is shown and a flagged correction of its own — headed, and set into the working at the exact step that fails — explains what went wrong and gives the corrected value. Nothing has been silently overwritten. Two manipulators followed end to end — the P–R–P arm of May Q3, Q4 and Q6 is taken from frame assignment and its DH table through the forward transform, closed-form inverse kinematics, the Jacobian and its reachable and dexterous workspace; the two-link planar arm of May Q7 and December Q6 from joint torques through the Jacobian to its singular configurations — the same arm the notes use to draw the Jacobian, statics and workspace. Study notes — ten condensed topics: frames, rotation matrices and their properties; homogeneous transformations; composing chains of transforms; Euler angles and fixed-angle sets; Denavit–Hartenberg parameters; forward kinematics; inverse kinematics; the Jacobian and velocity; statics, singularities and workspace; and trajectory generation — each pointing back at the exam questions it answers. Formula sheet — 31 relations in five groups: rotations; homogeneous transforms; Euler and fixed angles; DH parameters and forward kinematics; and the Jacobian, statics and trajectories — every relation these two papers use. This exam allows one 8.5 × 11″ formula sheet written on both sides, so print this, check it against your own text and copy what you need onto your own sheet in your own hand — with no worked examples on it, because the sheet is handed in with the paper. Reference links — ten searches, each with a note on what it is good for: two for reading (a web search for Craig's Introduction to Robotics, the textbook these questions are adapted from, and a search of MIT OpenCourseWare) and eight YouTube searches, each for the topic it names — homogeneous transforms, DH parameters, the Z-Y-Z extraction, inverse kinematics and decoupling, the Jacobian, singularities, static forces and torques, and trajectory generation. Printable resources — four blank exam stationery templates: a ruled answer sheet matching the answer-book layout, dot-grid rough-work paper, graph paper and isometric paper — plus the formula sheet, all printing straight from the page. Practice papers — the two question papers on their own, unmarked, for a timed run: May 2016 and December 2018 (5 pages each, questions 1 to 7) — the same two sittings that are solved above.
Throughout, 54 figures were drawn for this pack and placed 61 times — 32 in the solutions and 29 in the notes — alongside 7 scanned images (the question figures and the published solver's own sketches), so the frames, the Euler sequences turning, the DH parameters, the two inverse-kinematics branches, the Jacobian columns, the workspace and the spline through its via point are shown rather than described.