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KINEMATIC CHAINS AND ROBOTS (II)

KINEMATIC CHAINS AND ROBOTS (II). Kinematic Chains and Robots II. Many machines can be viewed as an assemblage of rigid bodies called kinematic chains. This lecture continues the discussion on the analysis of kinematic chains with focus on robots.

elijah-guerra
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KINEMATIC CHAINS AND ROBOTS (II)

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  1. KINEMATIC CHAINS AND ROBOTS (II)

  2. Kinematic Chains and Robots II • Many machines can be viewed as an assemblage of rigid bodies called kinematic chains. This lecture continues the discussion on the analysis of kinematic chains with focus on robots. • After this lecture, the student should be able to: • Understand the screw transformation matrix • Perform a combination of transformation matrices to describe a series of rigid body motions • Have the foundation for kinematic analysis of robotic systems

  3. Frame {1} Frame {2} Frame {0} Open kinematic chain Propagation of Motions along a Chain Let us consider the following open kinematic chain Let = Transformation Matrix of frame {1} w.r.t. frame {0} = Transformation Matrix of frame {2} w.r.t. frame {1} = Transformation Matrix of frame {1} w.r.t. frame {0} is given as

  4. Propagation of Motions along a Chain In general for an open kinematic chain with “n” links, the overall transformation matrix is given as: Given Its inverse is:

  5. Example: Propagation of Motions along a Chain Given Find and its’ inverse

  6. Reduces to: Example: Propagation of Motions along a Chain Useful formulas

  7. Example: Propagation of Motions along a Chain to give where and

  8. Example: Propagation of Motions along a Chain

  9. Example: Propagation of Motions along a Chain

  10. Z-axis has components along the Y-axis and Z-axis: u1 The displacement of frame {b} w.r.t. {0} is: is along the positive X-axis: “O” has components along the -Y-axis and Z-axis: Y-axis  X-axis Screw Transformation Matrix Let frame {a}=(X, Y, Z) and frame {b}= Consider the case where the origin of frame {b} has translated a distance u1 along the X-axis and rotated an angle of  about the X-axis:

  11. Z-axis u1 “O” Y-axis  X-axis Screw Transformation Matrix This situation is like a screw action along the X-axis. The rotation matrix is given as: The screw transformation matrix is

  12. Screw Transformation Matrix The story so far:

  13. Screw Transformation Matrix Next, consider the case where the origin of frame {b} has translated a distance u2 along the Y-axis and rotated an angle of  about the Y-axis. The screw transformation matrix will be:

  14. Screw Transformation Matrix Similarly, for the case where the origin of frame {b} has translated a distance u3 along the Z-axis and rotated an angle of  about the Z-axis. The screw transformation matrix will be:

  15. Transformations are not commutative Remember, generally given two or more transformations, i.e. pure rotations or general transformations, Transformations are NOT commutative!

  16. Transformations are not commutative Example: Notice that R1 involves a 90° rotation about the X-axis and R2 involves a 90° rotation about the Y-axis

  17. Z-axis X Z-axis Y-axis Z-axis A B B X-axis X Y-axis A Y-axis A B X-axis X-axis Transformations are not commutative R2 R1 involves a 90° rotation about the X-axis follow by a 90° rotation about the Y-axis

  18. Z-axis Z-axis A B Y-axis A X X-axis Y-axis Z-axis A B X-axis X Y-axis A B X-axis Transformations are not commutative R1 R2 involves a 90° rotation about the Y-axis follow by a 90° rotation about the X-axis

  19. Summary • Many machines can be viewed as an assemblage of rigid bodies called kinematic chains. This lecture continues the discussion on the analysis of kinematic chains with focus on robots. • The following were covered: • Combination of transformation matrices to describe a series of rigid body motions • Screw transformation matrix • Foundation for kinematic analysis of robotic systems

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