University of Pennsylvania Prof. Mark Yim Robotics ... 36 Characteristics at Singular Configurations...

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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


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Prismatic Joints


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Revolute Joints


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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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x0

y0

x1

y1

x2

y2

θ1

θ2

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for revolute:

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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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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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Week 7: Singularities, Manipulability, Forces, Torques


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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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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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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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Manipulability

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Jacobian Transpose

R1=10

R2=20

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Principle of Virtual Work

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Static Forces

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Manipulability

x0

y0 x1y1

x2

y2

θ1 θ2a1 a2

x0

y0

y1

x2

y2

θ1

θ2

a1

a2

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Manipulability

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Week 7: Mobile Robot Jacobian


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Jacobian for Mobile Robots

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Mobility Ellipsoid

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Mobility Ellipsoid

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Mobility Ellipsoid

University of Pennsylvania Prof. Mark Yim Robotics ... 36 Characteristics at Singular Configurations...

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