DynamicsThe branch of physics that treats the action of force onbodies in motion or at rest; kinetics, kinematics, andstatics, collectively. — Websters dictionaryOutline• Conservation of Momentum• Inertia Tensors - translation a n d ro tat io n• Dynamics– Newton/Euler Dynamics– Lagrangian Dynamics– State Space Form• Simulation and Feedforward Compensation• Linear Analysis:the Dynamic Manipulabili ty Ellipsoid1 Copyrightc2013 Roderic GrupenNewton’s Laws1. a particle will remain i n a state of constant rectilinear motionunless acted on by an external force;2. the time-rate-of-change in the momentum (mv) of the particleis proportional to the externally appli ed fo rce s, F =ddt(mv);and3. any force imposed on body A by body B is reciprocated by anequal and opposite reaction f orc e o n bod y B by body A.Conservation of MomentumLinear:F =ddt[m ˙x] = m¨x[N] =kg msec2Angular:τ =ddthI˙θi= I¨θ[Nm] =kg m2sec22 Copyrightc2013 Roderic GrupenInertia TensorArdvxyzMASS MOMENTS MASS PRODU C TSOF INERTIA OF INERTIAIxx=R R R(y2+ z2)ρdv Ixy=R R RxyρdvIyy=R R R(x2+ z2)ρdv Ixz=R R RxzρdvIzz=R R R(x2+ y2)ρdv Iyz=R R RyzρdvI =Ixx−Ixy−Ixz−IyxIyy−Iyz−Izx−IzyIzz3 Copyrightc2013 Roderic GrupenEXAMPLE: Inertia Tensor ofan Eccentric Rectangular PrismxyzlhwAIxx=Zh0Zl0Zw0(y2+ z2)ρdxdydz=Zh0Zl0(y2+ z2)wρdydz=Zh0(y33+ z2y)l0wρdz=Zh0l33+ z2lwρdz=l3z3+lz33h0(wρ)=l3h3+lh33wρor, since the mass of the rectangle m = (wlh)ρ,Ixx=m3(l2+ h2).4 Copyrightc2013 Roderic GrupenEXAMPLE: Inertia Tensor ofan Eccentric Rectangular Prism...completing the other moments and products of in erti a y i el d s:AI =m3(l2+ h2)m4wlm4hwm4wlm3(w2+ h2)m4hlm4hwm4hlm3(l2+ w2)5 Copyrightc2013 Roderic GrupenParallel Axis Theor em -Transla ti ng the Inertia Tenso rxyzACM(x,y,z)ACMthe moments o f i n erti a l ook l i ke:AIzz=CMIzz+ m(Ax2CM+Ay2CM),and the products of inertia are:AIxy=CMIxy+ m(AxCMAyCM).6 Copyrightc2013 Roderic GrupenEXAMPLE:The Symmetric Recta ngula r PrismxyzlhwACMCMIzz=AIzz− m(Ax2CM+Ay2CM)=m3(l2+ w2) −m4(l2+ w2)=m12(l2+ w2)CMIxy=AIxy− m(AxCMAyCM)=m4(wl) −m4(wl) = 0.moving the axes of rotation to the center of mass results in adiagonalized inertia tens orCMI =m12(l2+ h2) 0 00 (w2+ h2) 00 0 (l2+ w2)7 Copyrightc2013 Roderic GrupenRotating the Inert i a Tensorangular momentu m L0= I0ω about frame 0 in a vector quantitythat is conserved.we can exp re ss it r el ati ve to frame 1 asL1=1R0L0orI1ω1= R(I0ω0)= RI0RTRω0and therefore,I1= RI0RT.8 Copyrightc2013 Roderic GrupenRotating Coordinate SystemsDefinition (Inertial Frame)the frame where the absolute state of motion iscompletely knownLet frame A be an in erti a l frame. Frame B has an absol u te veloc-ity, ωBB(written in terms of frame B coord in a tes ).rA(t) =ARB(t)rB(t)˙rA(t) =ARB(t)ddt[rB(t)] +ddt[ARB(t) ]rB(t)yxzωxωyωzωzωyωzωxωyωxTo evaluate the second ter m on the right,consider how theˆx,ˆy, andˆz, basis vectorsfor frame B change by virtue of ωB.˙ˆx = ωzˆy −ωyˆz˙ˆy = −ωzˆx −ωxˆz˙ˆz = ωzˆy −ωyˆzsoddt[ARB(t) ]rB(t) =0 ωz−ωy−ωz0 ωxωy−ωx0rxryrz= ω × r9 Copyrightc2013 Roderic GrupenRotating Coordinate SystemsTherefore,˙rA(t) =ARB(t)ddt[rB(t)] +ddt[ARB(t) ]rB(t)=ARB˙rB+ (ωBB× rB) and, in fact, all vector quantitie s expressed in local frame s thatare moving relative to an inertial frame are differe ntiated in thiswayddt[ARB(t)(·)B] =ARBddt(·)B+ (ωB× (·)B)10 Copyrightc2013 Roderic GrupenRotating Coordinate Systems:Low Pressure Systemslarge sc al e atmosp he ri c flows converge at l ow pressure regions. Anonrotating planet, this would result in flow lines directed radiallyinward.−( v)LωLωvbut the earth rotates...consider a stationary inertial fra me Aand a rotat in g frame B attached tothe earthvA=ARB(t)vB˙vA=ARB[˙vB+ (ω × vB)]so that to an observer that travelswith frame B:˙vB=BRA[˙vA] − (ω × vB)a convergent flow and a rotating system, therefore, leads to acounterc lockwise flow in the northern hemisphere and a clockwiserotation in the southern hemisp h e re.11 Copyrightc2013 Roderic GrupenNewton/Euler MethodO1O2012l1l20ωv = v = = 0 00 0O11v 1v 1ω1N1F1O22v 2v 2F2N2ω2O33F3N3v 3v 3ω3v = v + f ( ) ω ω +v = v + g ( ) 1 0O11O101 0O111=v = v + f ( ) ω ω +v = v + g ( ) OOO=333 222 33333ω ω +OOi iv = v + f ( ) O=iii i−1 ii−1i−1 iv = v + g ( ) i3FN3O33l32F2N21N1F13f4η4η3f33−f3−ηf2η22−η2−ffη11v = v + f ( ) ω ω +v = v + g ( ) OOO=222 111 222 22 outwarditeration inwarditeration12 Copyrightc2013 Roderic GrupenNewton/Euler Equatio nsFNvωω+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 a n d N are the net force and torque ve cto rs on link i writ-ten in in erti al coordinates, and Riis the rotati on matrix relatin gframe i to the inertial frame, and ωiis the total angular velocityof link i written in link i coordin a tes .13 Copyrightc2013 Roderic GrupenNewton/Euler Equatio nsFNvωω+iziyiθxiIf F and N are written in the local coordinate frame for li n k i,thenmI300 M˙v˙ω+ω × mvω × Mω=FN= WW ∈ R6is the generalized force or wrench consisting of forcesand torques acting on link i w ritte n i n link i coordinates....if we can account for the fu ll state of motion, (ω,˙ω,˙v), then wecan compute the total load, W, acting on the center of mass anddefine the equation of motion f o r li nk i.14 Copyrightc2013 Roderic GrupenRecursive Newton/Euler Eq ua ti onsFNvωω+iziyiθxiPropagate the absolute state of motion, (ω,˙ω,˙v)iat frame i toframe (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˙ωi15 Copyrightc2013 Roderic GrupenRecursive Newton/Euler Eq ua ti onsLinear Acceleration:˙vw10pQ110y0x0z0y1x1z101vpQ0Q01t0pQ=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)