Newton/Euler EquationsFNvωω+iziyiθxiNewton’s EquationF =ddt[Ri(mivi)] = Ri(mi˙vi) +˙Ri(mivi)= Ri[mi˙vi+ (ωi× mivi)]Euler’s EquationN =ddt[Ri(Miωi)]= Ri[Mi˙ωi+ (ωi× Miωi)]where F and N are the net force and torque vectors on link i writtenin inertial coordinates, and Riis the rotation matrix relating frame i tothe inertial frame, and ωiis the total angular velocity of link i writtenin link i coordinates.12 Copyrightc"2007 by Roderic GrupenNewton/Euler EquationsFNvωω+iziyiθxiIf F and N are written in the local coordinate frame for link i, then!mI300 M"!˙v˙ω"+!ω × mvω × Mω"=!FN"= WW ∈ R6is the generalized force or wrench consist i ng o f forces andtorques acting on link i written in link i coor dinates ....if we can account for the full state of motion, (ω,˙ω,˙v), then we cancompute the total load, W, acting on the center of mass and definethe equation of motion for link i.13 Copyrightc"2007 by Roderic GrupenRecursive Newton/Euler EquationsFNvωω+iziyiθxiPropagate the absolute state of motion, (ω,˙ω,˙v)iat frame i to frame(i + 1).Angular Velocity: ωREV OLUT E :i+1ωi+1=i+1Riiωi+˙θi+1ˆzi+1P RISMAT IC :i+1ωi+1=i+1RiiωiAngular Acceleration :˙ωREV OLUT E :i+1˙ωi+1=i+1Rii˙ωi+ (i+1Riiωi×˙θi+1ˆzi+1)+¨θi+1ˆzi+1P RISMAT IC :i+1˙ωi+1=i+1Rii˙ωi14 Copyrightc"2007 by Roderic GrupenRecursive Newton/Euler Equations: cont.Linear Acceleration:˙vQt10pQ1pQ0w010z1y11v01x1y0x0z00pQ=0R11pQ+0t10vQ=0R11˙pQ+ (0ω1×0R11pQ) +0v10˙vQ=ddt#0R11˙pQ$+ (0˙ω1×0R11pQ) + (0ω1×ddt#0R11pQ$)+0˙v1=0R11¨pQ+ (0ω1×0R11˙pQ) + (0˙ω1×0R11pQ)+(0ω1×0R11˙pQ) + (0ω1×0ω1×0R11pQ) +0˙v115 Copyrightc"2007 by Roderic GrupenRecursive Newton/Euler Equations: cont.Linear Acceleration:˙vNow, s ubstitute:frame 0 ⇔ frame (i − 1)frame 1 ⇔ frame iframe 2 ⇔ frame (i + 1)i+1˙vi+1=i+1Ri−1#i−1Rii¨pi+1+ 2(i−1ωi×i−1Rii˙pi+1)+(i−1˙ωi×i−1Riipi+1)+(i−1ωi×i−1ωi×i−1Riipi+1) +i−1˙vi$=i+1Ri#i¨pi+1+ 2(iωi×i˙pi+1) + (i˙ωi×ipi+1)+(iωi×iωi×ipi+1) +i˙vi$REV OLUT E :ipi+1= const,i˙pi+1=i¨pi+1= 0i+1˙vi+1=i+1Ri#i˙vi+ (i˙ωi×ipi+1)+(iωi×iωi×ipi+1)$P RISMAT IC :ipi+1= diˆxi,i˙pi+1=˙diˆxi,i¨pi+1=¨diˆxii+1˙vi+1=i+1Ri%i˙vi+¨diˆxi+ 2(iωi×˙diˆxi)+(i˙ωi× diˆxi)+(iωi×iωi× diˆxi)$16 Copyrightc"2007 by Roderic GrupenRecursive Newton/Euler Equations: cont.Now, refer the translational acceleration to the center of mass:i+1˙vcm,(i+1)= (i+1˙ωi+1×i+1pcm)+(i+1ωi+1×i+1ωi+1×i+1pcm)+i+1˙vi+1and we may w rite the Newton-Euler equation of motion:i+1Fi+1= mi+1i+1˙vcm i+1i+1Ni+1= Mi+1i+1˙ωi+1+ (i+1ωi+1× Mi+1i+1ωi+1)17 Copyrightc"2007 by Roderic GrupenForces in Open Kinematic Chainslink ifiηiηi+1fi+1FiixiyiNiixi+1yi+1&F orces =iFi=ifi−iRi+1i+1fi+1, orifi=iFi+iRi+1i+1fi+1&T orques =iNi=iηi−iηi+1−(ipcm×ifi)−((ipi+1−ipcm)×ifi+1,but,ifi=iFi+iRi+1i+1fi+1, so that,iNi=iηi−iηi+1− (ipcm×iFi) − (ipi+1×ifi+1)or,iηi=iNi+iRi+1i+1ηi+1+ (ipcm×iFi) + (ipi+1×iRi+1i+1fi+1)18 Copyrightc"2007 by Roderic