McGill Dept of Mechanical Engineering MECH572A Introduction to robotics Lecture 9
MECH572A Introduction To Robotics Lecture 9 Dept. Of Mechanical Engineering
Review Velocity, Acceleration and Static Analysis Mapping between Cartesian and joint space Linear transformation J6=t J-W=T Joint rate+ Cartesian rate EE Wrench Joint force/torque Rate analysis Static analysis Jacobian matrix- General form for n-R manipulator e e xrl e2 xr2 en xrn Special case: 6R Decoupled Manipulator(PUMA) 311 o J
Review • Velocity, Acceleration and Static Analysis Mapping between Cartesian and joint space (Linear transformation) Jacobian Matrix – General form for n-R manipulator Special case: 6R Decoupled Manipulator (PUMA) Joint Rate Cartesian rate EE Wrench Joint force/torque Rate analysis Static analysis
Review Velocity, Acceleration and Static Analysis 6R Decoupled manipulator - Solve two linear systems of equations 3 equations 3 unknowns) Rate problem J110a +J120u =w J210a=c b J1ba=心 ju,8 11 Static problem =7 Jif=Ta-iinur
Review • Velocity, Acceleration and Static Analysis 6R Decoupled Manipulator -> Solve two linear systems of equations (3 equations 3 unknowns) Rate problem Static problem
Review Singularity analysis Based on the anal ysis of Jacobian matrix Singularity analysis of 6r decoupled manipulator General concept Conditioning analysis of J 12& J21 Acceleration Analysis 6=t-J6 6=J-l(t-J)
Review • Singularity Analysis - Based on the analysis of Jacobian Matrix - Singularity analysis of 6R decoupled manipulator - General concept – Conditioning analysis of J12 & J21 • Acceleration Analysis
Planar Manipulator 3-Revolute Planar Manipulator Properties 1)en/e2//e 2=03 3)ⅹ1,X2andx coplanar 八中 4)bl=b2=b3=0 5)al, az, and a3 none zero (Link length) O A1 XI
Planar Manipulator • 3-Revolute Planar Manipulator Properties: 1) e1 // e2 // e3 2) 1 = 2 = 3 3) X1, X2 and X3 coplanar 4) b1 = b2 = b3=0 5) a1, a2, and a3 none zero (Link length) e1 e2 e3 X1 X2 X3
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
