university of pennsylvania prof. mark yim robotics ... 36 characteristics at singular configurations...
TRANSCRIPT
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Prof. Mark YimUniversity of Pennsylvania
Week 7: Manipulator Jacobian
Robotics: Fundamentals
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KinematicsJoint space Cartesian space
Forward
Inverse
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Mapping spaces
θ1
θ2a1
a2
x0
y0
x
y
θ1
θ2
Joint Space Cartesian Space
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Manipulator Jacobian
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Jacobian Matrix
J
Jij
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Manipulator Jacobian
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Position Jacobian
Jv
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Position Jacobian
Jv
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Position Jacobian
Joint 1
Joint 2
Link 1
Link 2
x0
y0
x1
y1
x2
y2
θ1
θ2a1
a2
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Position Jacobian
Joint 1
Joint 2
Link 1
Link 2
x0
y0
x1
y1
x2
y2
θ1
θ2
a1
a2
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Position Jacobian
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Prismatic Joints
Link ai αi di θi1 0 -90 0 θ12 0 90 d2 θ23 0 0 d3 0
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Prismatic Joints
Link ai αi di θi1 0 -90 0 θ12 0 90 d2 θ23 0 0 d3 0
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Prismatic Joints
Link ai αi di θi1 0 -90 0 θ12 0 90 d2 θ23 0 0 d3 0
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Revolute Joints
Link ai αi di θi1 0 -90 0 θ12 0 90 d2 θ23 0 0 d3 0
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Orientation Jacobian
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Angular Velocity
ω is the angular velocity of j with respect to i expressed in frame k
kziy
ixiz
jzjy
jx
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Orientation Jacobian
x0
y0
x1
y1
x2
y2
θ1
θ2
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Orientation Jacobian
for revolute:
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Orientation Jacobian
for prismatic:
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Combining Linear and Angular Velocity
If Revolute:
If Prismatic:
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Jacobian Examples
x0
y0
x1
y1
x2y2
θ1
θ2
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2 Link Arm Jacobian
x0
y0
x1
y1
x2
y2
θ1
θ2
If Revolute:
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2 Link Arm Jacobian
x0
y0
x1
y1
x2
y2
θ1
θ2
a1
a2
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2 Link Arm Jacobian
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Stanford Arm JacobianLink ai αi di θi1 0 -90 0 θ12 0 90 d2 θ23 0 0 d3 0
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Link ai αi di θi1 0 -90 0 θ12 0 90 d2 θ23 0 0 d3 0
4 0 -90 0 θ45 0 90 0 θ56 0 0 d6 θ6
Stanford Arm
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Stanford Arm MATLABT01 = a1T02 = T01*a2T03 = T02*a3T04 = T03*a4T05 = T04*a5T06 = T05*a6
Z0 = [0;0;1]; P0 = [0;0;0]Z1 = T01(1:3, 3)Z2 = T02(1:3, 3)Z3 = T03(1:3, 3)Z4 = T04(1:3, 3)Z5 = T05(1:3, 3)Z6 = T06(1:3, 3)P1 = T01(1:3, 4)P2 = T02(1:3, 4)P3 = T03(1:3, 4)P4 = T04(1:3, 4)P5 = T05(1:3, 4)P6 = T06(1:3, 4)
Jp1 = cross(Z0,P6-P0)Jo1 = Z0Jp2 = cross(Z1,P6-P1)Jo2 = Z1Jp3 = Z2Jo3 = [0;0;0]Jp4 = cross(Z3,P6-P3)Jo4 = Z3Jp5 = cross(Z4,P6-P4)Jo5 = Z4Jp6 = cross(Z5,P6-P5)Jo6 = Z5
J = [Jp1 Jp2 Jp3 Jp4 Jp5 Jp6 ; Jo1 Jo2 Jo3 Jo4 Jo5 Jo6]
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Results
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ResultsJ11 - c1*d2 - d6*(s5*(c1*s4 + c2*c4*s1) + c5*s1*s2) - d3*s1*s2J12 c1*(c2*d3 + d6*(c2*c5 - c4*s2*s5))J13 c1*s2J14 d6*s1*s2*(c2*c5 - c4*s2*s5) - c2*d6*(s5*(c1*s4 + c2*c4*s1) + c5*s1*s2)J15 d6*(c1*c4 - c2*s1*s4)*(c2*c5 - c4*s2*s5) - d6*s2*s4*(s5*(c1*s4 + c2*c4*s1) + c5*s1*s2)J16 0
J21 c1*d3*s2 - d6*(s5*(s1*s4 - c1*c2*c4) - c1*c5*s2) - d2*s1J22 s1*(c2*d3 + d6*(c2*c5 - c4*s2*s5))J23 s1*s2J24 - c2*d6*(s5*(s1*s4 - c1*c2*c4) - c1*c5*s2) - c1*d6*s2*(c2*c5 - c4*s2*s5)J25 d6*(c4*s1 + c1*c2*s4)*(c2*c5 - c4*s2*s5) - d6*s2*s4*(s5*(s1*s4 - c1*c2*c4) - c1*c5*s2)J26 0
J31 0J32 c1*(d2*s1 + d6*(s5*(s1*s4 - c1*c2*c4) - c1*c5*s2) - c1*d3*s2) - s1*(c1*d2 + d6*(s5*(c1*s4 + c2*c4*s1) + c5*s1*s2) + d3*s1*s2)J33 c2J34 c1*d6*s2*(s5*(c1*s4 + c2*c4*s1) + c5*s1*s2) + d6*s1*s2*(s5*(s1*s4 - c1*c2*c4) - c1*c5*s2)J35 d6*(c1*c4 - c2*s1*s4)*(s5*(s1*s4 - c1*c2*c4) - c1*c5*s2) - d6*(s5*(c1*s4 + c2*c4*s1) + + c5*s1*s2)*(c4*s1 + c1*c2*s4)J36 0
J41 0J42 -s1J43 0J44 c1*s2J45 - c4*s1 - c1*c2*s4J46 c1*c5*s2 - s5*(s1*s4 - c1*c2*c4)
J61 1J62 0J63 0J64 c2J65 s2*s4J66 c2*c5 - c4*s2*s5
J51 0J52 c1J53 0J54 s1*s2J55 c1*c4 - c2*s1*s4J56 s5*(c1*s4 + c2*c4*s1) + c5*s1*s2
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Prof. Mark YimUniversity of Pennsylvania
Week 7: Singularities, Manipulability, Forces, Torques
Robotics: Fundamentals
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SingularitiesConfigurations for which the rank J(q) is less than its maximum value are called
singularities or singular configurations.
x0
y0
x1
y1
x2
y2
θ1
θ2a1
a2
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x0
y0
x1
y1
x2
y2
θ1
θ2
a1
a2
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x0
y0
x1
y1
x2 y2
θ1
θ2
a1
a2
det(J) = 0
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Characteristics at Singular Configurations
• Directions of motion may be lost
• Infinite joint velocities may be required for finite end-effector velocities
• Theoretically infinite end-effector forces may result from finite joint forces
• Often correspond to points on the boundary of the manipulator workspace.
• There may be no IK solution or there may be infinitely many IK solutions
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Using determinant
det(J) = 0
x0
y0
x1
y1
x2
y2θ1 θ2
a1 a2
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Decomposing 6DOF arms
Arm singularities Wrist singularities
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Decomposing 6DOF armsJ(q) is 6x6 and is singular
if and only ifdet (J) = 0
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Decomposing 6DOF arms
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Decomposing 6DOF arms
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Decomposing 6DOF arms
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Wrist Singularities
Singularities occur in the wrist:• if and only if joint axis are collinear (0 or π
radians)• unavoidable if moving through that point.
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Representational Singularities
By Juansempere - Own work, GFDL,
By Euler2.gif: Juansemperederivative work: Xavax - This file was derived
from Euler2.gif:, CC BY-SA 3.0,
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Arm Singularities
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Arm Singularities
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Stanford arm Singularities
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Manipulability
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Manipulability
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Manipulability
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Manipulability
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Manipulability
x0
y0 x1y1
x2
y2
θ1 θ2a1 a2
x0
y0
x1
y1
x2
y2
θ1
θ2a1a2x0
y0
y1
x2
y2
θ1
θ2
a1
a2
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Robo1x-1.7 53
Manipulability
x0
y0 x1y1
x2
y2
θ1 θ2a1 a2
x0
y0
x1
y1
x2
y2
θ1
θ2a1a2x0
y0
y1
x2
y2
θ1
θ2
a1
a2
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Robo1x-1.7 54
Manipulability
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Robo1x-1.7 55
Jacobian Transpose
R1=10
R2=20
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Robo1x-1.7 56
Principle of Virtual Work
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Robo1x-1.7 57
Static Forces
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Robo1x-1.7 58
Manipulability
x0
y0 x1y1
x2
y2
θ1 θ2a1 a2
x0
y0
y1
x2
y2
θ1
θ2
a1
a2
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Robo1x-1.7 59
Manipulability
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Robo1x-1.5 60
Prof. Mark YimUniversity of Pennsylvania
Week 7: Mobile Robot Jacobian
Robotics: Fundamentals
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Robo1x-1.7 61
Jacobian for Mobile Robots
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Robo1x-1.7 62
Jacobian for Mobile Robots
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Robo1x-1.7 63
Jacobian for Mobile Robots
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Robo1x-1.7 64
Jacobian for Mobile Robots
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Robo1x-1.7 65
Jacobian for Mobile Robots
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Robo1x-1.7 66
Jacobian for Mobile Robots
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Robo1x-1.7 67
Jacobian for Mobile Robots
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Robo1x-1.7 68
Jacobian for Mobile Robots
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Robo1x-1.7 69
Jacobian for Mobile Robots
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Robo1x-1.7 70
Jacobian for Mobile Robots
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Robo1x-1.7 71
Jacobian for Mobile Robots
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Robo1x-1.7 72
Jacobian for Mobile Robots
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Robo1x-1.7 73
Mobility Ellipsoid
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Robo1x-1.7 74
Mobility Ellipsoid
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Robo1x-1.7 75
Mobility Ellipsoid