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Velocity Analysis Jacobian

Velocity Analysis Jacobian. University of Bridgeport. Introduction to ROBOTICS. 1. Kinematic relations. X=FK( θ). θ = IK(X). Task Space. Joint Space. Location of the tool can be specified using a joint space or a cartesian space description. Velocity relations.

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Velocity Analysis Jacobian

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  1. Velocity AnalysisJacobian University of Bridgeport Introduction to ROBOTICS 1

  2. Kinematic relations X=FK(θ) θ =IK(X) Task Space Joint Space Location of the tool can be specified using a joint space or a cartesian space description

  3. Velocity relations Relation between joint velocity and cartesian velocity. JACOBIAN matrix J(θ) Task Space Joint Space

  4. Jacobian • Suppose a position and orientation vector of a manipulator is a function of 6 joint variables: (from forward kinematics) X = h(q)

  5. Jacobian Matrix Forward kinematics

  6. Jacobian Matrix Jacobian is a function of q, it is not a constant!

  7. Linear velocity Angular velocity Jacobian Matrix The Jacbian Equation

  8. (x , y) 2 l2 l1 1 Example • 2-DOF planar robot arm • Given l1, l2 ,Find: Jacobian

  9. Singularities • The inverse of the jacobian matrix cannot be calculated when det [J(θ)] = 0 • Singular points are such values of θ that cause the determinant of the Jacobian to be zero

  10. V (x , y) l2 Y 2 =0 l1 1 x • Find the singularity configuration of the 2-DOF planar robot arm determinant(J)=0 Det(J)=0

  11. Jacobian Matrix • Pseudoinverse • Let A be an mxn matrix, and let be the pseudoinverse of A. If A is of full rank, then can be computed as:

  12. Jacobian Matrix Example: Find X s.t. Matlab Command: pinv(A) to calculate A+

  13. Jacobian Matrix • Inverse Jacobian • Singularity • Jacobian is non-invertable • Boundary Singularities: occur when the tool tip is on the surface of the work envelop. • Interior Singularities: occur inside the work envelope when two or more of the axes of the robot form a straight line, i.e., collinear

  14. Jacobian Matrix • If • Then the cross product

  15. Remember DH parmeter • The transformation matrix T

  16. Jacobian Matrix

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