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Unformatted text preview: Massachusetts institute of Technology Depariment of Mechanical Engineering 2.12 Introduction to Robotics Exercise problems and Solutions for the meter. i _ End-offerm Exam Figure 1 shows a two d.o.F. planer robot with n parallelogram mechanism. Note rlmtlinlrs l andfiesweil as2andemepsrafleitoeschother,andthstjoi11taxesl and: are aligned. Links I end 2 are doves by independent DC motors fixed to the base. Using the notations shown in the figure, answer the following questions. 1'1] thein the leeobisn Instr-ix relating endpoint VEiDC-itiflfl v: and v); to joint velocities r and do. [l-Zl Obtain tbejoint torques, n and 1?, required for bearing endpoint forces, E: and F}. [1-3] Let K; be e M feedback gain matrix relating joint torques :- = {r}, tsfto joint position error :58 = 153:, 5:92]: that is. T = Kp-Efl Obtain the gain matrix K,» first generates the desired endpoint compliance specified in Figure 2. Narnely, the desire eompiiseee is Ci and C2 in the two directions of principal mos, xi: and ye, whiie the axis are issngleafromthexexis. one Figure 2 Desired compliance Figure l Planar robot with parslleiogram mmhsnisrn [1 “1]. Obtain the moment of inertia reflected to Joint I when Joint 2 is fitted. that is, the ]—| element ofthc inertia matrix H = 151'le associated With generatlzcd coordinates :15". E32. Wlttcit particuiar motion oftlte arm links is the H” element associated with? [1—5}. Explain why H” and H21, ofthe inertia ntatria do not vary. although the arm configuration varies. i.e. configuration-invariant. [1-15]. Show that the ofildiagonal element oflhe inertia matrix is given by HI: = (‘ifljflfit'fi _ ’ndftft'dJGfls{at "'93)- [l—i']. We want to dcconpie and linearize the arm dynamics by redistributing mass among the tour links. Obtain conditions for the mass parameters to make the arm dynamics totally decoupled and linearized For all arm configurations. J—‘tlso explain why that happens. Problem 1 s robot is unscrewing s hexarhead bolt by using a wrench, as-shown in the figure below. Obtain both natural and artificial constraints to perform this task by using Mason's hybrid positiontforee control. Define an appropflate Cnft‘fifl'lc and parameters necessarjtr for descaihtng the constraints, and show your notations in a sketch. Assume that the process is quasi—static and no friction. Mao assume that there is no gap between the head ofthe bolt and the wrench. Massachusetts Institute of Technology Department of Mechanics! Engineering PIPE-15%;?! (In!) Ename'wT' sssrfi'flsfii rem, éw'mee by - I'l’lfimiéigl +54 age/91-x] g; érd‘fflfii; these}; ‘31-!) [airfinflr , jf'nffl-Wéa {53; fly 1:2: $4419: ; *14 we; 1;; gAV—x‘ I (1-2). Em Jaw-Q ’Z= J'TF. (Tr):[—-firfr‘fl§: fifimfi? Fa, TI em: 4+ng F, {1:3} oil/t AF) Wspb {a Miner's/1‘ 531,1 Title"? 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