Robotics 1 Dynamic analysis of a 2-DOF planer robot using Newton-Euler recursive method Link # iθ id ia iα 1 1θ 0 1a 0 2 2θ 0 2a 0 −=10000100001111111110SaCSCaSCT −=10000100002122222221SaCSCaSCT OA = a1 AB = a2Robotics 2Determine all rotation matrices −=10000111110CSSCR −=10000222221CSSCR −=100001212121220CSSCR −=10000111101CSSCR −=10000222212CSSCR Determine the radial joint distances =0111110SaCar =0222221SaCar ()−=002/1111acr ()−=002/2222acr Determine inertia matrices =1000100001221111amI =1000100001222222amI Forward computation of velocities and Accelerations (a) Angular Velocity propagation () ( )[]iiioiiiioiR θωω&1111−−−−+= For a revolute joint () ( )[]111 −−−=nonnnnonR ωω For a prismatic joint joint) (revolute 1=i () ()[]10000111θωω&+=ooRRobotics 3()−==111111001110010000θ&&CSSCRoθω ()=11100θ&ωo joint) (revolute 2=i () ()+=21111222θωω&ooR ()+−=21222222000010000θθ&&CSSCoω ()+=)(002122θθ&&ωo (b) Angular Acceleration propagation joint) (revolute 1=i () ()( )[]101000000111θθωαα&&&+×+=oooR ()−=11111110010000θ&&CSSCoα ()=11100θ&&αo joint) (revolute 2=i () ()( )[]21211110112212θθωαα&&&+×+=oRRobotics 4()+−=+×+=212222221112202001000000000000θθθθθθ&&&&&&&&&&CSSCRα ()+=)(0021202θθ&&&&α (c) Linear Velocity propagation joint) (revolute 1=i () ()( )100110100001101rωvv RR ×+= ()×=−×=00000100000011111111111101aSaCaCSSCθθ&&v ()=0011101θ&av joint) (revolute 2=i () ()( )211220210112202rωvv RR ×+= ()−×++−−=010000)(0000100002222222221112222202SaCaCSSCaCSSCθθθ&&&v ()×++=00)(000221121121202aCaSaθθθθ&&&&v ()++=0)(00212121121202θθθθ&&&&aCaSavRobotics 5()++=0)(212121121202θθθθ&&&&aCaSav (c) Linear Acceleration propagation joint) (revolute 1=i ()()() ()()()()() ()××+×+××+××+×+=101010101010100001000000010000000011012 rθθrθrθωrωωrαaa&&&&&R ()()[]101010101001101rθθrθa ××+×=&&&&R ()××+×−=0000000010000111111111111111101SaCaSaCaθCSSCθθ&&&&a ()−×+−−=00001000011111111111111111101θθθ&&&&&&&CaSaθCaθSaCSSCa ()−−+−−=0010000211121111111111111101θθ&&&&&&SaCaθCaθSaCSSCa ()()()()()−+−−−−+−−=012111111121111111211111112111111101CSaθCaSCaθSaSSaθCaCCaθSaθθθθ&&&&&&&&&&&&a ()−=011211101θaa&&&θaRobotics 6joint) (revolute 2=i ()()() ()()()()() ()××+×+××+××+×+=212121212121211012110110121101101122022 rθθrθrθωrωωrαaa&&&&&R Consider individual components ()−−=01000011211222210112θaaCSSCR&&&θa ()++−=01212121121212110112θCaSaθSaCaR&&&&&&θθa (I) ()×−=×000100002222122222110112SaCaCSSCRθ&&rα ()−−=×01000012212222222110112θθ&&&&CaSaCSSCR rα ()=×00122110112θ&&aR rα (II) ()()××−=××000001000022221122222110110112SaCaCSSCRθθ&&rωω ()()−×−=××00010000122122122222110110112θθθ&&&CaSaCSSCR rωωRobotics 7()()−−−=××0100002122212222222110110112θθ&&SaCaCSSCR rωω ()()()−=××002122110110112θ&aR rωω (III) ()( )××=××000002222222112212110112SaCaRRθθ&&&rθω ()( )−×=××00022222222112212110112θθθ&&&&CaSaRR rθω ()( )−−−=××01000022212221222222212110112θθθθ&&&&&SaCaCSSCR rθω ()( )−=××0022212212110112θθ&&&aR rθω (IV)