Planar manipulator Displacement analysis From geometry a1C1+ a2C12=a a2心12=x-a11 4.101a) a181+a2812=y a2812=y-a18 (4.101b) (4.103a)2+(4.103b)2 C:=a2+a2+2a1we1+2a1ys1-(x2+y2)=0 (4.102) Also C: C+s1=1 Solution depends on the relative position between line L and circle c aL intersects with C 2 roots b)tangent to C. I root c)Ldoes not intersect with C. No root
Planar Manipulator • Displacement Analysis From geometry (4.103a)² + (4.103b)² Also Solution depends on the relative position between line L and circle C a) L intersects with C: 2 roots b) L tangent to C: 1 root c) L does not intersect with C: No root
Planar Manipulator Displacement analysis The case of two real roots ez (
Planar Manipulator • Displacement Analysis The case of two real roots:
Planar Manipulator Velocity analysis Jo=t 61 e b=2 (4.103b) e1xrte2xr2e3×r 0 e1=e2=e3=e三0 r 0 WP JP ,103 2-D cross i Es product ei xri=ai matrix si≡[cv
Planar Manipulator • Velocity Analysis 2-D crossproduct matrix
Planar Manipulator Velocity analysis Es Es esa ta/a> p (4.105) Es1 Es2 Es3 (4.106) 6k3 Mapping rates between joint and Cartesian space
Planar Manipulator • Velocity Analysis Mapping rates between joint and Cartesian space
Planar Manipulator Acceleration Anal ysis J0+J0=t J0=t-Je 6≡62,t 1, 63 p s3=(61+的2+63)Ea 的2=如2+=(61+Ea2+k s1=a1+52=仇Fa1+岛
Planar Manipulator • Acceleration Analysis