《太空生物工程与生命支撑sbe_dynam_lec》由会员分享，可在线阅读，更多相关《太空生物工程与生命支撑sbe_dynam_lec（10页珍藏版）》请在金锄头文库上搜索。
1、Musculoskeletal Dynamics Grant Schaffner ATA Engineering, Inc. Part II Musculoskeletal Dynamics Free Body Diagram of Link i 0x 0y 0z 0O 1iO iO Joint i Joint i+1 gim Link i ii , 1r cii, rcii , 1r ii , 1N 1, + iiNii , 1f 1, + iifciv i iI Lagrangian Formulation of Equations of Motion Describes the beha
2、vior of a dynamic system in terms of work and energy stored in the system Constraint forces are automatically eliminated (an advantage and a disadvantage), called the “closed form” dynamic equations Equations are derived systematically (easier to use) ii i ii ii n qQ niQ q L q L dt d qqL qq coorddge
3、neralizetoingcorrespondforcedgeneralize :fromderivedthenaremotionofEquations :LagrangianDefine energypotentialtotal energykinetictotal systemdynamicaofscoordinatedgeneralize , 1 ),( ,1 = = = = = = m D D m UT U T (2-1) Lagrangian Formulation Compute the velocity and angular velocity of an individual
4、link i (think of the link as an end effector with coord sys at the link c.m.) qJJJ qJJJv DDlD DDlD )()( 1 )( 1 )()( 1 )( 1 i Ai i Ai i Aci i Li i Li i Lci qq qq =+= =+= where j-th column vectors of the 3xn Jacobian matrices link i, i.e., )(i LjJ )(i AjJ )(i LJ )(i AJ )()( 1 )( )()( 1 )( 00JJJ 00JJJ
5、hh hh i Ai i A i A i Li i L i L = = Note: tion of link i depends on only joints 1 through i, the column vectors are set to zero for j (2-2) (2-3) are the and for the linear and angular velocities of and Since the mo iLagrangian Formulation Each column vector is given by: = = 0 bJ b rb J 1)( 1 , 11)(
6、 ji Aj j cijji Lj (revolute jt) (prismatic jt) (revolute jt) (prismatic jt) = cij , 1r Position vector of centroid of link i wrt inboard link coordinate frame = 1jb 3x1 unit vector along joint axis j-1 (2-4) ( = += n i ii T ici T ciim 1 2 1 2 1 IvvT (2-5) where framecoordbasewrtcentroid,theattensori
7、nertia linkofmass 33= = i i im I Note: wrt the base coord frameiI )varies with the orientation of the linkLagrangian Formulation Note: tensor defined relative to the coord frame fixed to the link, using ( ( )( 1 )()()()( 2 1 1 )()()()( 2 1 definitepositivesymmetricis n)(ntensorinertiasystem where H
8、JIJJJH qHqqJIJqqJJqT =+= =+= = = n i i Ai Ti A i L Ti Li T n i i Ai Ti A Ti L Ti L T i m m iI iI T iiii 00 RIRI = (2-6) (2-7) (2-8) , the inertia can be obtained from ) ) Lagrangian Formulation Potential Energy = = n i ci T im 1 , 0rgU (2-9) Generalized Forces ext T FJQ += torquesjoint= momentsandfo
9、rcesexternal= ext T FJ Lagranges Equations of Motion (see Asada & Slotine for derivation) ( niQGqqhqH iik n j j n k ijkj n j ij , 1 1 1 lCCC C =+ = = i jk k ij ijk q H q Hh = 2 1 (2-11) (2-10) Inertia torques Centrifugal/ Coriolis trqs Gravity torque Generalized Forces )( 1 j Li n j T ji mG Jg = = )
10、1=Example: 2 dof planar arm Velocities of centroids c1 and c2 0x 0y 1l 2l 11 , 22 , 1cl 2cl 11 , mI 22 , mI qv = 0cos 0sin 11 11 1 c c c l l qv + + = )cos()cos(cos )sin()sin(sin 21221211 21221211 2 cc cc c lll lll These 2x2 matrices are the associated with the angular velocities are 1x2 row vectors
11、in this planar case )(i LJ )(i AJ q q 11 01 212 11 =+= = Example: 2 dof planar arm Substituting the linear and angular Jacobians into eqn (2-7) gives qH C + + = 2 2 222 2 222212 2 2 2222122221 2 2 2 121 2 11 cos cos)cos2( IlmIlml lm Ilml lmIl lllmIlm ccc ccccc 0, 0,sin sin2,sin, 0 2212122222212211 22121211122212122111 =+= =+= hhhl lmh l lmhhl lmhh c cc Centrifugal / Coriolis term coefficients Gravity terms Substituting the above into (2-11) gives )2( 22 )1( 212 )2( 12 )1( 111 LL T LL T mmG mmG JJg JJg += += 22 2 1211112222 1121121112 2 2122212111 )( =+ =+ GhHH GhhhHH CC CC C CCCC CC C
