# Rigid body simulation

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1 Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum Constraned sstem Performance s mportant collson detecton contact ponts and forces

2 Problems Partcle smulaton Y(t) = [ (t) v(t) ] Poston n phase space Control s everthng [ Ẏ(t) = v(t) f(t)/m ] Veloct n phase space Rgd bod concepts Poston and orentaton Translaton Poston Lnear veloct Mass tensor Lnear momentum orce Rotaton Orentaton Angular veloct Inerta tensor Angular momentum torque Translaton of the bod Rotaton of the bod (t) = R(t) = r r r r r r r r r (t) and R(t) are called spacal varables of a rgd bod

3 Bod space Poston and orentaton Bod space Bod space 0 A fed and unchanged space where the shape of a rgd bod s defned 0 R(t) 0 r0 r0 0 0 World space The geometrc center of the rgd bod les at the orgn of the bod space Poston and orentaton Gven a geometrc descrpton of the bod n bod space, we can use (t) and R(t) to transform the bod space descrpton nto world space (t) Let s assume the rgd bod has unform denst, what s the phscal meanng of (t)? 0 How do we compute the world coordnate of an arbtrar pont r0 on the bod? 0 (t) r (t) = (t) + R(t)r0 0 Poston and orentaton World space R(t) r0 r0 0 What s the phscal meanng of R(t)?

4 Poston and Orentaton Poston and orentaton Consder the -as n bod space, (1, 0, 0), what s the drecton of ths vector n world space at tme t? R(t) = r r r whch s the frst column of R(t) So (t) and R(t) defne the poston and the orentaton of the bod a tme t Net we need to defne how the poston and orentaton change over tme R(t) represents the drectons of,, and aes of the rgd bod n world space at tme t Lnear veloct Angular veloct Snce (t) s the poston of the center of mass n world space, ẋ(t) s the veloct of the center of mass n world space v(t) = ẋ(t) Imagne that we freee the poston of the COM n space, then an movement s due to the bod spnnng about some as that passes through the COM Otherwse, the COM would tself be movng

5 Angular veloct Angular veloct We descrbe that spn as a vector ω(t) Drecton of ω(t) Magntude of ω(t) How are R(t) and ω(t) related? Hnt: Consder a vector r(t) at tme t specfed n world space, how do we represent ṙ(t) n terms of ω(t) Lnear veloct and poston are related b v(t) = d dt (t) ω(t) ṙ(t) = b ω(t) = ω(t) b How are R(t) and ω(t) related? a (t) b ṙ(t)? r(t) ṙ(t) = ω(t) b = ω(t) b + ω(t) a ṙ(t) = ω(t) r(t) Angular veloct Angular veloct Gven the phscal meanng of R(t), what does each column of Ṙ(t) mean? At tme t, the drecton of -as of the rgd bod n world space s the frst column of R(t) r r r At tme t, what s the dervatve of the frst column of R(t)? (usng the cross product rule we just dscovered) Ṙ(t) = ω(t) r r r ω(t) r r r ω(t) r r r Ths s the relaton between angular veloct and the orentaton, but t s too cumbersome We can use a trck to smpl ths epresson

6 Angular veloct Angular veloct Consder two 3 b 1 vectors: a and b, the cross product of them s a b = a b b a a b + b a a b b a Ṙ(t) = ω(t) = ω(t) R(t) r r r r r r ω(t) r r r r r r r ω(t) ω(t) r r r r r Gven a, lets defne a* to be the matr then a b = 0 a a a 0 a a a 0 0 a a a 0 a a a 0 b b b = a b Vector relaton: Matr relaton: ṙ(t) = ω(t) r(t) Ṙ = ω(t) R(t) Perspectve of partcles Veloct of a partcle Imagne a rgd bod s composed of a large number of small partcles the partcles are ndeed from 1 to N each partcle has a constant locaton r 0 n bod space ṙ(t) = d dt r(t) = ω R(t)r 0 + v(t) = ω (R(t)r 0 + (t) (t)) + v(t) = ω (r (t) (t)) + v(t) the locaton of th partcle n world space at tme t s ṙ r (t) = (t) + R(t)r (t) = ω (r (t) (t)) + v(t) 0 angular component lnear component

7 Veloct of a partcle Mass The mass of the th partcle s m r (t) = ω (r (t) (t)) + v(t) 0 Mass ω(t) M= r (t) 0 v(t) Center of mass n world space What about center of mass n bod space? Center of mass Inerta tensor Proof that the center of mass at tme t n word space s (t) m r (t) M 0 v(t) r (t) m =1 ω(t) (r (t) (t)) (t) N m r (t) = M Inerta tensor descrbes how the mass of a rgd bod s dstrbuted relatve to the center of mass m (r + r ) I= m r r m r r m r r (r + r ) m r r m r r m r r m (r + r ) r = r (t) (t) I(t) depends on the orentaton of a bod, but not the translaton = (t) or an actual mplementaton, we replace the fnte sum wth the ntegrals over a bod s volume n world space

8 Inerta tensor orce and torque B usng bod-space coordnates we can cheapl compute the nerta tensor for an orentaton b precomputng ntegrals n bod space m r m r r m r r m r r m r m r r I(t) = I(t) = m (rt r )1 = m ((R(t)r0 )T (R(t)r0 )1 (R(t)r0 )(R(t)r0 )T ) = m rt rt m r r m r r m r (t) denotes the total force from eternal forces actng on the th partcle at tme t (t) = 0 τ (t) = (r (t) (t)) (t) (t) r (t) (t) (t) r rt ) τ (t) = (r (t) (t)) (t) 0 0 m (R(t)(rT0 r0 )R(t)T 1 R(t)r0 rt0 R(t)T ) = R(t) " # m ((rt0 r0 )1 r0 rt0 ) R(t)T I(t) = R(t)Ibod R(t)T Ibod = m ((rt0 r0 )1 r0 rt0 ) orce and torque Torque dffers from force n that the torque on a partcle depends on the locaton of the partcle relatve to the center of mass Lnear momentum P(t) = = m r (t) m v(t) + ω(t) (t) conves no nformaton about where the varous forces acted on the bod, whle (t) contans the nformaton about the dstrbuton of the forces over the bod m (r (t) (t)) = M v(t) Total lnear moment of the rgd bod s the same as f the bod was smpl a partcle wth mass M and veloct v(t)

9 Angular momentum Dervatve of momentum Smlar to lnear momentum, angular momentum s defned as L(t) = I(t) ω(t) L(t) s ndependent of an translatonal effects, whle P(t) s ndependent of an rotatonal effects Change n lnear momentum s equvalent to the total forces actng on the rgd bod Ṗ(t) = M v(t) = (t) The relaton between angular momentum and the total torque s analogous to the lnear case L(t) = τ(t) Dervatve of momentum momentum vs. veloct Proof L(t) = τ(t) = r m r = m ( v ṙ ω r ω) = 0 r m ( v ṙ ω r ω) r = 0 ( ) ( ) m r ṙ ω m r r ω = τ m r r = m ((r T r )1 r r T ) = I(t) ( ) m r ṙ ω + I(t) ω = τ İ(t) = d dt m r r = m r ṙ m ṙ r Wh do we use momentum n the phase space nstead of veloct? Because the angular momentum s conserved when there s no eternal forces actng on the object We could let lnear veloct v(t) be a state varable, but usng lnear momentum P(t) s more consstent wth the wa we deal wth angular veloct and acceleraton İ(t)ω + I(t) ω = d dt (I(t)ω) = L(t) = τ

10 Equaton of moton Eample: Y(t) = (t) R(t) P(t) L(t) poston orentaton lnear momentum angular momentum d dt Y(t) = v(t) ω(t) R(t) (t) τ(t) ( 0, ) 0, 0 1. compute the I bod n bod space I bod = M Constants: M and I bod v(t) = P(t) M I(t) = R(t)I bod R(t) T ω(t) = I(t) 1 L(t) ( 0, 0, ) 0 Eample: Eample: 1. compute the I bod n bod space 1. compute the I bod n bod space. rotaton free movement. rotaton free movement 3. translaton free movement ( 3, 0, ) (3, 0, ) ( 3, 0, ) (3, 0, )

11 orce vs. torque pule Notes on mplementaton energ = 0 energ = 0 10 sec later energ = 1 MvT v Suppose a force acts on the block at the center of mass for 10 seconds. Snce there s no torque actng on the block, the bod wll onl acqure lnear veloct v after 10 seconds. The knetc energ wll be 1 MvT v Now, consder the same force actng off-center to the bod for 10 seconds. Snce t s the same force, the veloct of the center of mass after 10 seconds s the same v. However, the block wll also pck up some angular veloct ". The knetc energ wll be 1 MvT v + 1 ωt Iω If dentcal forces push the block n both cases, how can the energ of the block be dfferent? Usng quaternon nstead of transformaton matr more compact representaton less numercal drft

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