INTRODUCTION TO DYNAMICS ANALYSIS OF ROBOTS (Part 2)

This lecture continues the discussion on the analysis of the instantaneous motion of a rigid body, i.e. the velocities and accelerations associated with a rigid body as it moves from one configuration to another. After this lecture, the student should be able to: Derive the principles of relative motion between bodies in terms of relative velocity Introduction to Dynamics Analysis of Robots (2)

Summary of previous lecture The sliding velocity ISA parallel to Axis of rotation passes through the point The rate of rotation If a point on the rigid body is fixed, i.e. The ISA has to pass through this fixed point.

Moving FORs X-axis Y-axis Z-axis Frame {a} e 1 -axis e 2 -axis e 3 -axis Q Frame {b} Consider a point “Q” on a body and two FORs as follow: If the two frames are only translated and “inline” as shown, then

If the two frames have undergone a rotation X-axis Y-axis Z-axis Frame {a} e 1 -axis e 2 -axis e 3 -axis Q Frame {b} We have to rotate frame {b} back to be “in-line” with frame {a} before adding, i.e. Moving FORs

Example: The 3 DOF RRR Robot: Y 0, Y 1 X 0, X 1 Z 0, Z 1 Z2Z2 X2X2 Y2Y2 Z3Z3 X3X3 Y3Y3 A=3 B=2C=1 P What is the position of point “P” after 1 second if all the joints are rotating at Example: Moving FORs

From the previous lecture, we know that for this robot: At t=1, Example: Moving FORs

Similarly: At t=1, Example: Moving FORs

Similarly: At t=1, Example: Moving FORs

Given We need to find

Example: Moving FORs We should get the same answer if we use transformation matrix method.

Example: Moving FORs

The answer is the same as that obtained earlier:

Velocity and moving FORs Consider the general case where = rotation of frame {b} w.r.t. frame {a} = position of point “Q” w.r.t. frame {b} = position of point “Q” relative to frame {b} w.r.t. frame {a} = origin of frame {b} w.r.t. frame {a} = Absolute position of point “Q” w.r.t. frame {a}

Velocity and moving FORs To get the instantaneous linear velocity of point “Q” w.r.t. frame {a}, we have to differentiate its absolute position where

Example: Velocity and moving FORs Example: The 3 DOF RRR Robot: Y 0, Y 1 X 0, X 1 Z 0, Z 1 Z2Z2 X2X2 Y2Y2 Z3Z3 X3X3 Y3Y3 A=3 B=2C=1 P What is the velocity of point “P” after 1 second if all the joints are rotating at

Example: Velocity and moving FORs At t=1,

Example: Velocity and moving FORs At t=1,

Example: Velocity and moving FORs At t=1,

Example: Velocity and moving FORs Given Find

Example: Velocity and moving FORs

There is no translation velocity between frames {3} and {2} and no translation velocity of point “P” in frame {3}

Example: Velocity and moving FORs There is no translation velocity between frames {2} and {1}

Example: Velocity and moving FORs There is no translation velocity between frames {1} and {0}

We should get the same answer if we use transformation matrix method. We know that Example: Velocity and moving FORs But

Example: Velocity and moving FORs where

Example: Velocity and moving FORs The answer is the same as that obtained earlier:

Summary This lecture continues the discussion on the analysis of the instantaneous motion of a rigid body, i.e. the velocities and accelerations associated with a rigid body as it moves from one configuration to another. The following were covered: The principles of relative motion between bodies in terms of relative velocity