Notes_08_11 Two-Dimensional Inverse Dynamics Kinematically driven Must use centroidal coordinate frames ! r qi i i r qi i i F Q T on i on i on i mi Mi 0 0 0 mi 0 0 0 J Gi Single body Mi qi Qon i ALL Q on i ALL Q on i APPLIED Q on i CONSTRAINT 1 of 9 Notes_08_11 System of multiple bodies q 2 q q3 q 4 q2 q q3 q 4 M 2 M 03x 3 0 3x 3 M 2 0 3x 3 0 3x 3 03x 3 03x 3 M 3 03x 3 03x 3 M 4 0 3x 3 0 3x 3 q 2 Q on 2 ALL M 3 0 3x 3 q 3 Q on 3 ALL 0 3x 3 M 4 q 4 Q on 4 ALL Qon 2 ALL QALL Qon 3 ALL Q on 4 ALL Mq QALL QALL QAPPLIED QCONSTRAINT CONSTRAINT KINEMATIC DRIVER QCONSTRAINT QKINEMATIC QDRIVER 2 of 9 Notes_08_11 3 of 9 Virtual work Mq QALL 0 q T Mq QALL 0 QALL QAPPLIED QCONSTRAINT q T QCONSTRAINT 0 q 0 for kinematic consistency q T Mq QAPPLIED 0 q q 0 subject to q Virtual work for one revolute q T QCONSTRAINT ? r q r q i i j i F 0 j j Fon j REV P on i REV P Fon i PREV P P T Fon i REV Bi s i ' Q P on i REV q i T Q on i PREV q j T Q on j PREV ri i T Q P on j REV Fon j PREV P P T Fon j REV B j s j ' ? 0 T Fon i PREV Fon j PREV rj ? 0 P P P T P T B s ' F B s ' F j j j on i REV on j REV i i ri T Fon i PREV i Bi s i ' P T Fon i PREV rj T Fon j PREV j B j s j ' P T Fon j PREV r r B s ' ri P ri i Bi s i ' P r F P T i P on i REV P rj P j F T P on j REV j ? 0 j j OK j P ri P rj P ? 0 Notes_08_11 4 of 9 Lagrange multiplier theorem bT x 0 general problem bT x T Ax 0 using Lagrange multipliers for any arbitrary x q T Mq QAPPLIED 0 virtual work b Mq QAPPLIED x q Mq QAPPLIEDT q T q q 0 Mq Q APPLIED Ax 0 subject to subject to q 0 q A q for arbitrary size but kinematically consistent q q 0 q T T Mq q T QAPPLIED each row in is multiplied times corresponding column in q T q each row in corresponds to matching row in q and Lagrange multipliers Mq q T QAPPLIED Mq QAPPLIED q T Mq QALL QAPPLIED QCONSTRAINTS QCONSTRAINTS q T Qon 2 ALL QALL Qon 3 ALL Q on 4 ALL QCONSTRAINTS Qon 2 CONSTRAINTS Qon 3 CONSTRAINTS Q on 4 CONSTRAINTS T Notes_08_11 5 of 9 Equations of motion (EOM) Mq q T QAPPLIED q q q q M nq x 1 nq x 1 nq x nq QAPPLIED nq x 1 nc x 1 nc x 1 nc x 1 q nc x nq nq = number of generalized coordinates nk = number of kinematic constraints nd = number of driver constraints nc = total number of constraints (nc = nk + nd) Inverse dynamics – kinematically driven must have full rank q q 1 solve kinematics compute constraint forces KINEMATIC DRIVER q nc = nq q T QAPPLIED Mq 1 KINEMATIC DRIVER Inverse dynamics – simultaneous EOM matrix Mq q T QAPPLIED M nq x nq q nc x nq q Q T q nq x nc T 0 1 nc x 1 Q q q APPLIED nq x 1 nq x 1 0 nc x nc nc x 1 q M q q and APPLIED M EOM q T q 0 nc nq x nc nq Notes_08_11 6 of 9 Statics q 0 Mq q T QAPPLIED q T QAPPLIED 1 Lagrange multipliers for specific constraints F r i T F r j T on i CONSTRAINT on i CONSTRAINT on j CONSTRAINT on j CONSTRAINT T CONSTRAINT CONSTRAINT B s ' P T i T r i CONSTRAINT i T CONSTRAINT P T T r j CONSTRAINT j T CONSTRAINT CONSTRAINT CONSTRAINT B s ' j i j T CONSTRAINT CONSTRAINT Revolute REV rj P ri P 0 2x1 F REV T 0 F REV T 0 on i REV on i REV on j REV on j REV check body i I 2 check body j I 2 P r i REV P r j REV P i REV P j REV B i s i ' P B j s j ' P OK OK Notes_08_11 7 of 9 Double revolute REV _ REV dij dij C2 0 T F on i REV _ REV on i CONSTRAINT on j REV _ REV d r r for a i ri Q ri P P ij P j i 2dij REV _ REV T F for 0 2d ij REV _ REV T on j CONSTRAINT 0 Parallel vectors a i parallel to a j PARALLEL a i R a j 0 T T F 02x1 T norma i norma j PARALLEL F 02x1 T norma i norma j PARALLEL on i PARALLEL on i PARALLEL on j PARALLEL on j PARALLEL and a r r and a i ri Q ri P Q j j P j Pin-in-slot a i parallel to d ij PIN _ SLOT a i R dij 0 T F on i PIN _ SLOT T on j PIN _ SLOT for d r r P ij R a i PIN _ SLOT on i CONSTRAINT F T norma i normdij PIN _ SLOT R a i PIN _ SLOT j P i Notes_08_11 T on j PIN _ SLOT 8 of 9 0 Relative angle (including driver) ANGLE j i C f ( t ) 0 F 0 2x1 T ANGLE F 0 2x1 T ANGLE on i ANGLE on i ANGLE on j ANGLE on j ANGLE Gear pair GEAR j i C 0 F on i GEAR on i CONSTRAINT on j GEAR internal/external pair 0 02 x1 T F external pair 0 GEAR 02 x1 T on j CONSTRAINT GEAR Gear pair on rotating link k GEAR _ ON _ K j k i k C 0 F on i GEAR T 02 x1 on i CONSTRAINT F on j GEAR GEAR 02 x1 external pair 0 internal/external pair 0 Notes_08_11 T on j CONSTRAINT F on k GEAR T GEAR 02x1 on j CONSTRAINT 1 GEAR 9 of 9