NM EE 589 & UNM ME 48/58 ROBO ENGINEERING Dr. Stephen Bruder NM EE 589 & UNM ME 48/58
5. Robot Dynamics 5. he Microbot Arm Dynamic Model A Second Dynamic Model Example: he Microbot Robot Arm he Denavit-Hartenberg table: i D-H params. i- a i- di i f z e z g g z h (t) z h 9 (t) x e (t) y, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide /
Link 5. Robot Dynamics 5. he Microbot Arm Dynamic Model Step : Compute the Inertia Matrices (in center of mass coordinate frame) Link # A cylindrical shell Inertia Matrix: m r h mr m r h Pose of the link s center of mass c h h ẑ zˆc ẑ r ŷ ˆx yˆc xˆc ŷ ˆx h / h /, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide /
Link 5. Robot Dynamics 5. he Microbot Arm Dynamic Model Link # A conical Shell Inertia Matrix: mr m r m e 4 8 m r m e 4 8 Pose of the link s center of mass e ˆx xˆc zˆc yˆc c e r ẑ ŷ e /, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide 4 /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model Link # - A slender rod Inertia Matrix: mf mf Pose of the link s center of mass ŷ f ẑ ŷ ˆx ẑ f ˆx c f, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide 5 /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model Step : Forward Kinematics Construct the D-H table and i s and c i s. C S S C h c h C S S C h c c, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide 6 /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model Construct the D-H table and i s and c i s. C S S C CC CS S SC SS C S C h c e CC CS S CC e S C S S C S C e S C Se h c c, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide 7 /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model Construct the D-H table and i s and c i s. C S e S C CC CS S CC e SC SS C SC e S C Se h c f CC CS S C f CeC S C S S C C f C e S S C S f Se h c c, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide 8 /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model Step : Compute Velocities Compute the linear velocities for each center of mass. v d d ( Pc ) dt dt h c v c, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide 9 /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model CC e d d v ( P ) S c Ce dt dt S e h c e (S C S C ) e (C C S S ) C e v c C f C e C d d v ( P ) C C S c f e dt dt S f S e h c C f CeS S ( ) f S ec S ( ) f S es C f CeC v C ( ) f C e c, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model hen, the angular velocity for each link ωi are computed as: ẑ ( R) C S S C h c C S S C h, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model hen, the angular velocity for each link ωi are computed as: S ˆ z C S ( ) R C CC CS S SC SS C S C h CC CS S CC e S C S S C S C e S C Se h c, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model hen, the angular velocity for each link ωi are computed as: S ( ) ˆ C ( ) z S ( ) R C CC CS S CC e SC SS C SC e S C Se h CC CS S C f CeC S C S S C C f C e S S C S f Se h c c, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model Step 4: Kinetic & Potential Energy Noting that gravity acts in the negative z direction gives: g [ g] where g 9.8 m/s. he potential energy of each link (υ i = m i g Pc i ) is: υ υ υ m gh m g S e h m g S f S e h, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide 4 /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model he link kinetic energy of each link ( ( К m v v ) ) is: К К К mr i i c c i i i i i C e r r m r e m 8 4 8 m C e C fc e C f f (S S C C ) ef e 4 4 f (S S C C ) ef f m f C 4 4, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide 5 /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model Step 5: Euler-Lagrange Eqn Apply the Euler-Lagrange equations. Let s simplify the eqns. Set: e =, f =, r =, m =, m =, m = 6 he Lagrangian becomes: KE PE К К К υ υ υ Applying the Euler-Lagrange Eqn. d i i dt i 9 4 C CC 4C C (8 C ) g S h 6 g(s S h) gh, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide 6 /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model 9 4 C CC 4C C (8 C ) g S h 6 g(s S h) gh ( 9C S S C C S 8C S ) 8gC 6 g(c C ) ( C S 8C S ) S S 6gC 9 C CC 4C 4 C (8 C ) (8 C ) 8, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide 7 /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model 9 C CC 4C d 9C 4CC 8C 6 ( 4CS 58CS 6CS 4SC ) dt ( 4CS 6CS ) 4 C (8 C ) d (4 4C ) C S 4S 8 dt d (8 C ) S 8 dt (8 C ) 8, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide 8 /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model Putting this all together. d i i dt i 9C 4C C 8C 6 4 C S ( ) 58 C S 6 C S ( ) 4 S C (4 4C ) (8 C ) ( 9C S S C C S 8C S ) 8gC 6 g(c C ) S 4S ( C S 8C S ) S 6gC (8 C ) 8, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide 9 /
5. Robot Dynamics 5. he Microbot Arm Dynamic Model Step 6: Generalized Form Rewriting in the generalized matrix form D( q) q C( q, q) q G( q) 9C 4C C 8C 6 4 4C 8 C 8 C 8 4S 6S C 58C S 4S C 6S C 9CS S 8SC 4S S (C 8C )S S g(c 6C ) 6gC, Dr. Stephen Bruder ME 48/58: Robotics Engineering hursday 8th Nov Slide /