Massachustts Institut of Tchnolog Dpartmnt of Mchanical Enginring. Introduction to Robotics Mid-Trm Eamination Novmbr, 005 :0 pm 4:0 pm Clos-Book. Two shts of nots ar allowd. Show how ou arrivd at our answr. Do not lav multipl answrs. Indicat which on is our corrct answr. Problm (50 points + 5 points tra) Th MIT Mchanical Enginring is dvloping a dinosaur robot for a local scinc musum. Figur shows a schmatic of th robot, having thr dgrs of frdom. Th joint angls θ, θ and θ ar all masurd from th positiv ais fid to th floor, and th link lngths ar OA = AB = m. Th had of th dinosaur is at Point E T (, ), which is mtrs from Joint (Point B), and its orintation ϕ is masurd from th positiv ais to lin EH, th had cntrlin. Not that lin EH bnds down / from lin BE, as shown in th figur. Answr th following qustions. m m E C Link ϕ H Link Link : th whol bod Actuator Joint Actuator Blt-Pull Mass-lss Transmission Joint O B θ θ A Link 0 Link θ Joint Figur Ovrall structur of dinosaur robot Link 0 Actuator Joint Figur Actuators and lg mchanism
a). Obtain th position and orintation of th dinosaur had, functions of joint angls. p = ( ϕ ) T, as p = ϕ to joint b). Obtain th Jacobian matri rlating th had vlocit ( ) T T vlocitis: q = ( θ θ ). θ c). Obtain singular configurations using th abov Jacobian in part b). First, mathmaticall solv th singularit condition, intrprt th rsult, and sktch singular configurations. For th rst of th qustions, rfr to Figur along with Figur, dpicting th actuators and transmission mchanisms. Actuator gnrats torqu btwn link 0 and link. Not that th bod of Actuator is fid to link 0, whil its output shaft is connctd to link. Actuator is fid to Link, and its output torqu is transmittd to Joint, i.. th kn joint, through th mass-lss blt-pull sstm with a gar ratio of :. Actuator is fid to Link, whil its output shaft is c onnctd to Link. Al l actuator torqus,, and ar masurd in a right hand sns, as shown b th arrows in th figur. Displacmnts of th individual actuators ar dnotd,, and ar masurd in th sam dirction as th torqus. d). Th bod of th dinosaur wighs,000 kg. Th cntr of mass is at Point C, which is mtr from point B on lin BE, as shown in Figur. How m uch actuator torqus,, and ar ndd to bar th wight of th bod at th configuration of θ = / 4, θ = / 4, and θ = /. Ignor friction and th mass of th lg. ). If on uss th Jacobian matri rlating th had vlocit to actuator vlocitis for amining th singular configuration, ar th singular configurations diffrnt from th ons obtaind in part c)? Eplain wh. th Problm (50 points + 0 points tra) A rscu robot is working at a disastr sit moving a hav objct at th tip of th arm. To rduc th load th rscu robot rsts on a solid stationar wall at Point A, and slid th scond link on th wall, as shown in th figur. Th contact btwn th scond link and th wall at Point A is assumd to b friction-lss. Th coordinats of Point A ar A, A with rfrnc to th bas coordinat sstm. Similar to th. robot, this rscu rob ot has a two d.o.f. arm with both motors fid to th bas. (Th torqu of th scond motor is transmittd through a blt-pull mchanism to th scond joint. Th two pull diamtrs ar th sam.) Using th paramtrs and variabls shown in th figur, answr th following qustions.
Joint L θ ( < 0) Objct θ Link A A A Wall Friction-lss slid contact F Figur Rscu robot moving a hav objct a). Undr what conditions, can w us th lngth L btwn Joint and Point A as a gnralizd coordinat that locats th sstm uniqul, assuming that Link is in contact with Point A at all tims? b). Whn link slids on th wall at Point A, th robot motion has to satisf som gomtric constraint. Obtain constraint quations b writing th coordinats of Point A,, A A, as f unctions of th joint angls θ, θ and lngth L, i.. A = A( θ, θ, and = θ, θ,. A A( c ). Diffrntiating th functions, A = A( θ, θ, and A = A( θ, θ,, in part b), find th rlationship among virtual displacmnts δθ, δθ and δ L that satisf th gomtric constraints. d ). Obtain th virtual wor k don b th motor torqu s,,, and th vrtical ndpoint forc - F. ). Using th Principl of Virtual W ork, show that th joint torqus,, a nd th vrtical ndpoint forc F satisf th following form of condition whn th sstm is in quilibrium: F = a + b If tim prmits, obtain a and b b solving th quilibrium condition drivd from th Virtual Work Principl for givn joint angls θ, θ and lngth L. You will gt additional
0 points. (If ou gt stuck in computing a and b, procd for th nt qustion that can b solvd without knowing a and b.) f). DC motors ar usd for driving th two joints of th robot. Lt K m b th motor constant of th first motor, and K m b that of th scond motor. Show that th total powr loss at th two motors whn gnrating joint torqus, is givn b: P loss = + r K r K m m whr r, r ar gar ratios of th motors. g). Using paramtrs a and b in part ), obtain th o ptimal valus of th joint torqus, that minimiz th total powr loss in both motors, P loss, whil baring th vrtical ndpoint forc - F. 4
Class Avrag 75.8 Standard Dviation.40 Point distribution Problm -a 0 -b 0 -c 0 -d 0 + 5 tra - 0 subtotal 50 Problm -a 6 -b 6 -c 6 -d 6-0 + 0 tra -f 6 -g 0 subtotal 50 TOTAL 00 + 5 tra 5