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 Kinematic Chains and Robots II

Propagation of Motions along a Chain Open kinematic chain Frame {0} Frame {1} Frame {2} 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

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

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

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

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

Example: Propagation of Motions along a Chain

Screw Transformation Matrix Consider the case where the origin of frame {b} has translated a distance u 1 along the X-axis and rotated an angle of  about the X-axis: is along the positive X-axis: has components along the Y-axis and Z-axis: has components along the -Y-axis and Z-axis: Let frame {a}=(X, Y, Z) and frame {b}= X-axis Y-axis Z-axis “O”  u1u1 The displacement of frame {b} w.r.t. {0} is:

Screw Transformation Matrix X-axis Y-axisZ-axis “O”  u1u1 This situation is like a screw action along the X-axis. The rotation matrix is given as: The screw transformation matrix is

Screw Transformation Matrix The story so far:

Screw Transformation Matrix Next, consider the case where the origin of frame {b} has translated a distance u 2 along the Y-axis and rotated an angle of  about the Y-axis. The screw transformation matrix will be:

Screw Transformation Matrix Similarly, for the case where the origin of frame {b} has translated a distance u 3 along the Z-axis and rotated an angle of  about the Z-axis. The screw transformation matrix will be:

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

Transformations are not commutative Example: Notice that R 1 involves a 90° rotation about the X-axis and R 2 involves a 90° rotation about the Y-axis

Transformations are not commutative R 2 R 1 involves a 90° rotation about the X-axis follow by a 90° rotation about the Y-axis X-axis Y-axis Z-axis X AB X A B X-axis Y-axis Z-axis A B X-axis Y-axis Z-axis

Transformations are not commutative X-axis Y-axis Z-axis X AB R 1 R 2 involves a 90° rotation about the Y-axis follow by a 90° rotation about the X-axis A X-axis Y-axis Z-axis AB A B X-axis Y-axis Z-axis X

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 Summary