Musculoskeletal Dynamics s pace Slomerdlcal Engineerin Grant schaffner ATA Engineering, Inc Harvard Medical CHU AEDIC School OF TEC
Musculoskeletal Dynamics Grant Schaffner ATA Engineering, Inc
Space Biomedical Engineerin Life Support Part ll Musculoskeletal Dynamics
Part II Musculoskeletal Dynamics
Free Body Diagram of Link i Space Biomedical Engineerin Life Support -1 i,i+1 Link O mig Joint Joint i+1 Jo
Free Body Diagram of Link i 0 x 0 y 0 z O 0 O i−1 O i Joint i Joint i+1 m i g Link i i −1,i r i, ci r i −1,ci r Ni −1,i − Ni, i+1 i −1,i f − i, i+1 f vci ωi i I
Lagrangian formulation of equations Space Biomedical Engineerin of motion Life Support Describes the behavior 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) qu,, qn=generalized coordinates of a dynamic system T=total kinetic energy U=total potential energy Define Lagrangian: L(a 9=T-U Equations of motion are then derived from d dl dL dt di, da e 1…,n(2-1) Q =generalized force corresponding to generalized coord q
Lagrangian Formulation of Equations of Motion • Describes the behavior 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) i i i i i i i n Q q Q i n q L q L dt d L q q q q generalize d force corresponding to generalize d coord Equations of motion are then derived from : Define Lagrangian : total potential energy total kinetic energy generalize d coordinate s of a dynamic system 1 , , ( , ) , , 1 = = = ∂ ∂ − ∂ ∂ = − = = = m D D m T U U T (2-1)
Lagrangian Formulation Space Biomedical Engineerin Life Support Compute the velocity and angular velocity of an individual link i ( think of the link as an end effector with coord sys at the link c. m) Ja,+.,+J q1 0=h+…+J=Jq (2-2) where Ju? and JU are the j-th column vectors of the 3xn Jacobian matrices J and Jw for the linear and angular velocities of link i, e J….J0….0 J000 (2-3) Note: Since the motion of link i depends on only joints 1 through i, the column vectors are set to zero for i>
Lagrangian Formulation • Compute the velocity and angular velocity of an individual link i (think of the link as an end effector with coord sys a t the link c.m.) ω J J J q v J J J q D l D D D l D D ( ) ( ) 1 ( ) 1 ( ) ( ) 1 ( ) 1 i i A i Ai i ci A i i L i Li i ci L q q q q = + + = = + + = where j-th column vectors of the 3xn Jacobian matrices link i, i.e., (i ) Lj J (i ) Aj J (i ) J L (i ) J A [ ] [ ] ( ) ( ) 1 ( ) ( ) ( ) 1 ( ) J J J 0 0 J J J 0 0 h h h h 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) and are the and for the linear and angular velocities of Since the mo i