Department of Mechanical Engineering MECE 551 Final examination Winter 2008 April 16, 9:00 11:30. Question Value Mark

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1 Deprtment of Mechnicl Engineering MECE 55 Finl exmintion Winter 8 April 6, 9: :3 Notes: You my hve your text book nd one pge formul sheet Electronic devices re not llowed except n pproved clcultor NAME: ID#: Question Vlue Mrk Totl - -

2 - -. A schemtic of Scr mnipultor is shown followed by the Jcobin mtrix: = ) cos( ) cos( cos ) sin( ) sin( sin J ) [] The end-effector of the robot is supposed to grb kg pylod nd move it long x direction with n initil ccelertion of 4 m/s. If the corresponding joint vlues re =, =, d 3 =.5m, 4 =3, find the required joint force/torque ssuming ll the links re mssless b) [5] The forth nd fifth rows of the Jcobin re zero. Wht does this imply in terms of the force/moment genertion by this mnipultor.?

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4 ) [] Using the potentil field method, find the ttrctive nd repulsive forces pplied to the moving point t X. Assume, η=α= nd ρ =4 m where the unit squre in the figure is m-by- m. X X f - 4 -

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6 3) [] In the following figure, pth plnning problem is shown in which rectngulr moving body is supposed to be moved without collision with the obstcle. Show the point tht will represent the moving body fter it is shrunk s well s the ugmented obstcle clerly on the figure. Moving body Obstcle - 6 -

7 4) A plnr Crtesin mnipultor is supposed to execute motion long stright line s shown in the following figure. Since the pth is reltively simple, we wnt to use pth prmeteriztion method to design smooth trjectory. It is lso known tht: The mximum ccelertion nd decelertion of ech ctutor is 3m/s nd m/s, respectively Mximum velocity of ech ctutor is 4m/s It is desired to strt nd end the motion with mximum ccelertion nd decelertion, respectively The cruise speed should be t the mximum vilble The velocities t the beginning nd end of the motion re zero x( s) ) [5] Find the prmeterized pth function, i.e. X ( s) = where s is the trveled y( s) distnce (pth prmeter) b) [5] Find the mximum ccelertion, decelertion nd cruise velocity vilble long the pth c) [5] Sketch the ccelertion curve of the end-effector, & s&(t ), using the results of prt b s well s the following vribles: t : when the ccelertion period ends t d : when the decelertion strts t f : Totl time of the motion d) [5] Find t nd decelertion period: t f -t d c) [5] Find smooth trjectory, s(t), by integrting the ccelertion curve seprtely for periods -t, t -t d nd t d -t f. d) [5] Find s f (the totl length of the pth) nd t f X f d d X s - 7 -

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10 5) Let s ssume the Crtesin mnipultor of the lst problem is implemented using liner motors s ctutors for both joints. Liner ctutors hve governing equtions very similr to DC rotry motors, i.e.: F = ki V V V m b b = kx& = di L + Ri dt where F m is the generted force, V nd V b re the input nd bck EMF voltges, respectively, i is the motor current, x is the position of the motor nd k, L, R re the constnts similr to those in rotry motors. Let: Mss of the motor shft: M m =.5 kg Viscous friction t the liner berings of the motor B m = N.s/m k= R= Ohm L=.5 V.s/A ) [5] For n independent joint control, write the dynmic equtions of the second link nd its ctutor. Assume the mss of the second link long with the end-effector is.5kg. Consider term for the disturbnce force due to the dynmic couplings b) [8] Using Lplce trnsform, derive the block digrm model of the second link where the input is the voltge nd the disturbnce force nd the output is the position of the joint. Also neglect the dynmics of the electricl systems in your finl model. c) [5] Assuming zero disturbnce, sketch the closed-loop block digrm ssuming unit feedbck nd proportionl gin of 9. Find the dmping rtio of the response to unit step input. Explin wht trnsient response you expect. d) [7] Add derivtive compenstor nd find the gin vlue for desired dmping rtio of. Also show tht your finl system is stble. - -

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