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
Lagrangian Formulation Space Biomedical Engineerin Life Support Each column vector is given by G-b, xri-lci (revolute jt) Position vector of centroid of link i wrt inboard link (prismatic jt) coordinate frame b (revolute jt) bi-1= 3x1 unit vector along joint (prismatic jt) axIs T=∑(mvva+0o)(25) Where m=mass of link i I. =3x 3 inertia tensor at the centroid wrt base coord frame Note: I. varies with the orientation of the link wrt the base coord frame
Lagrangian Formulation • Each column vector is given by: = × = − − − − 0 b J b b r J ( ) 1 1 1 1, ( ) i j Aj j j j ci i Lj (revolute jt) (prismatic jt) (revolute jt) (prismatic jt) rj −1,ci = Position vector of centroid of link i wrt inboard link coordinate frame b j −1 = 3x1 unit vector along joint axis j-1 (2-4) ∑ ( = = + n i i i T ci i T m i ci 1 2 1 2 T 1 v v ω I ω (2-5) where inertia tensor at the centroid, wrt base coord frame mass of link = 3 × 3 = i i m i I Note: wrt the base coord frame i I ) varies with the orientation of the link
Lagrangian Formulation Space Biomedical Engineerin Life Support T=∑4gy+qyg1J=4面26 where H=∑mg+19)=smh(27 (H is symmetric positive definite) Note:I can be obtained from i. the inertia tensor defined relative to the coord frame fixed to the link, using RLR 2-8)
Lagrangian Formulation Note: tensor defined relative to the coord frame fixed to the link, using ( ( ( ) 1 ( ) ( ) ( ) ( ) 2 1 1 ( ) ( ) ( ) ( ) 2 1 is symmetric positive definite system inertia tensor (n n) where H H J J J I J T q J J q q J I J q q Hq = + = × = + = ∑ ∑ = = n i i i A T i A i L T i i L T n i i i A T i A i T L T i L T i m m � � � � � � i I i I T i i i i 0 0 I = R I R (2-6) (2-7) (2-8) can be obtained from , the inertia ) )
Lagrangian Formulation Space Biomedical Engineerin Life Support Potential Energy U=∑mgna(2.9 Generalized forces Q=T+JF(2-10) t=joint torques JF = external forces and moments Lagrange's Equations of Motion(see Asada Slotine for derivation) ∑H4,+∑∑+G=Q(=1…)/%OHn,, j=l k= Inertia Centrifugal Gravity Generalized (2-11) G, 2m, Ji torques Coriolis trqs torque Forces
Lagrangian Formulation Potential Energy ∑ = = n i ci T m i 1 0, U g r (2-9) Generalized Forces ext T Q = τ + J F τ = joint torques = external forces and moments ext T J F Lagrange’s Equations of Motion (see Asada & Slotine for derivation) H q h q qk Gi Qi (i n n j j n k j ijk n j ij 1 , , 1 1 ∑ CC +∑∑ C C + = = l = = i jk k ij ijk q H q H h ∂ ∂ − ∂ ∂ = 2 1 (2-11) (2-10) Inertia torques Centrifugal/ Coriolis trqs Gravity torque Generalized Forces ( ) 1 j Li n j T G i ∑mjg J = = ) =1
Example: 2 dof planar arm Space Biomedical Engineerin Life Support Velocities of centroids c, and c sin e o cos0 O 7sn1-l2Sin(1+62)-l2sn(1+2) Z cos0,+le cos(8,+0,) le cos(8,+8, These 2X2 matrices are the Ji Jo associated with the angular velocities are 1X2 row vectors in this planar case O1=1=[0j ,τ1 O2=61+2=[11
Example: 2 dof planar arm • V elocities of centroids c1 and c2 0 x 0 y 1l 2l 1 1 θ ,τ 2 2 θ ,τ c1 l c 2 l 1 1 I , m 2 2 I , m v q� − = cos 0 sin 0 1 1 1 1 1 θ θ c c c l l v q� + + + − − + − + = cos cos( ) cos( ) sin sin( ) sin( ) 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 2 θ θ θ θ θ θ θ θ θ θ c c c c c l l l l l l • These 2x2 matrices are the • associated with the angular velocities are 1x2 row vectors in this planar case (i ) J L (i ) J A q q � � � � � [ 1 1 ] [ 1 0 ] 2 1 2 1 1 = + = = = ω θ θ ω θ
Example: 2 dof planar arm Space Biomedical Engineerin Life Support Substituting the linear and angular Jacobians into eqn(2-7) gives H +1+m2(42+12+212c082)+12m212cos2+m2+ 6,+m2l,2+ Centrifugal Coriolis term coefficients hn1=0,h2=-m2l2in2,hn2+h21=-2m22in2 h21=m2lina2,h2=0,h2+h21=0 Gravity terms Ju+mJ (2 20L1 +m Substituting the above into(2-11) gives H16+H22+h262+(h12+h21)062+G1= H262+H21+h21 +g
Example: 2 dof planar arm • Substituting the linear and angular Jacobians into eqn (2-7) gives H qC + + + + + + + + + + = 2 2 2 2 2 2 2 1 2 2 2 2 2 2 1 2 2 2 2 1 2 2 2 2 2 2 2 1 2 1 2 1 1 cos ( 2 cos ) cos m l l m l I m l I m l I m l l l l I m l l m l I c c c c c c c c θ θ θ sin , 0, 0 0 , sin , 2 sin 211 2 1 2 2 222 212 221 111 122 2 1 2 2 112 121 2 1 2 2 = = + = = = − + = − h m l l h h h h h m l l h h m l l c c c θ θ θ • Centrifugal / Coriolis term coefficients • Gravity ter ms • Substituting the above into (2-11) gives [ ] [ ] ( 2) 2 2 ( 1) 2 1 2 ( 2) 2 1 ( 1) 1 1 1 L L T L L T G m m G m m g J J g J J = + = + 2 2 2 22 2 12 1 211 1 112 121 1 2 1 1 2 11 1 12 2 122 2 ( ) θ θ θ τ θ θ θ θ θ τ + + + = + + + + + = H H h G H H h h h G CC CC C CC CC C C C
