Kinematics & Dynamics Adam Finkelstein Princeton University COS 426, Spring 2005 Overview • Kinematics " Considers only motion " Determined by positions, velocities, accelerations • Dynamics " Considers underlying forces " Compute motion from initial conditions and physics Example: 2-Link Structure • Two links connected by rotational joints ! 1 ! 2 X = (x,y) l 2 l 1 (0,0) “End-Effector” Forward Kinematics • Animator specifies joint angles: ! 1 and ! 2 • Computer finds positions of end-effector: X )) sin( sin ), cos( cos ( 2 1 2 1 1 2 1 2 1 1 ! + ! + ! ! + ! + ! = l l l l X ! 1 ! 2 X = (x,y) l 2 l 1 (0,0) Forward Kinematics • Joint motions can be specified by spline curves ! 1 ! 2 X = (x,y) l 2 l 1 (0,0) ! 2 ! 1 t Forward Kinematics • Joint motions can be specified by initial conditions and velocities ! 1 ! 2 X = (x,y) l 2 l 1 (0,0) 1 . 0 2 . 1 250 ) 0 ( 60 ) 0 ( 2 1 2 1 ! = " = " = " = " dt d dt d o o
8
Embed
Kinematics & Dynamics - 18- kinematics . · PDF fileSummary of Kinematics ¥Forward kinematics "Specify conditions (joint angles) "Compute positions of end-effectors...
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Kinematics & Dynamics
Adam Finkelstein
Princeton University
COS 426, Spring 2005
Overview
• Kinematics" Considers only motion
" Determined by positions, velocities, accelerations
• Dynamics" Considers underlying forces
" Compute motion from initial conditions and physics
Example: 2-Link Structure
• Two links connected by rotational joints
!1
!2
X = (x,y)
l2
l1
(0,0)
“End-Effector”
Forward Kinematics
• Animator specifies joint angles: !1 and !2
• Computer finds positions of end-effector: X
))sin(sin),cos(cos( 2121121211 !+!+!!+!+!= llllX
!1
!2
X = (x,y)
l2
l1
(0,0)
Forward Kinematics
• Joint motions can be specified by spline curves
!1
!2
X = (x,y)
l2
l1
(0,0)
!2
!1
t
Forward Kinematics
• Joint motions can be specified by initial conditionsand velocities
!1
!2
X = (x,y)
l2
l1
(0,0)
1.02.1
250)0(60)0(
21
21
!="
="
="="
dt
d
dt
d
oo
Example: 2-Link Structure
• What if animator knows position of “end-effector”
!1
!2
X = (x,y)
l2
l1
(0,0)
“End-Effector”
Inverse Kinematics
• Animator specifies end-effector positions: X
• Computer finds joint angles: !1 and !2:
xllyl
yllxl
))cos(())sin((
))cos(()sin((
22122
221221
!++!
!++!"=!
!1
!2
X = (x,y)l2
l1
(0,0)
!!"
#$$%
& ''+=( '
21
2
2
2
1
22
1
2cos
ll
llxx
2
Inverse Kinematics
• End-effector postions specified by spline curves
!1
!2
X = (x,y)
l2
l1
(0,0)
y
x
t
Inverse Kinematics
• Problem for more complex structures" System of equations is usually under-defined
" Multiple solutions
!1
!2
l2
l1
(0,0)
X = (x,y)
l3
!3
Three unknowns: !1, !2 , !3
Two equations: x, y
Inverse Kinematics
• Solution for more complex structures:" Find best solution (e.g., minimize energy in motion)