Joint Velocity and the Jacobian Look Im moving

Joint Velocity and the Jacobian Look! I’m moving!

Chapter Objectives By the end of the Chapter, you should be able to: • Characterize frame velocity • Compute linear and rotational velocity • Compute Jacobian and robot singularities Bibliography: • Craig’s book • Handout

Velocity of a Point The position of a point in frame B in terms of frame A is Point velocity in A = derivative with respect to time: When differentiating, two frames come into play: • The frame with respect to which we differentiate • The frame in which result is expressed, e. g. :

Rotational Velocity Suppose now that B is rotating w. r. t. A: Differentiating: A trick to get an economic representation: It can be shown that:

Rotational Velocity (cont. ) We write: Wedge operator Where: Rotational Velocity A “cool” expression of velocity due to time varying rotation:

Linear + Rotational Velocity If we have simultaneous time varying rotation & translat. : Using Homogeneous Coordinates, we can show that:

Velocity Propagation

Velocity Propagation (cont. ) Rotational velocities may be added as vectors: Where: Also: With respect to the linear velocity:

An Example: V 3 L 2 L 1

The Jacobian • Jacobian = Multidimensional Derivative • Example: – 6 functions fi , i=1, …, 6 – 6 variables xi , i=1, …, 6 Write: y 1 = f 1(x 1, x 2, …, x 6) y 2 = f 2(x 1, x 2, …, x 6) = y 6 = f 6(x 1, x 2, …, x 6) – In vector form: Y = F(X)

The Jacobian (cont. ) • Taking derivatives: Jacobian • Dividing by t on both sides: • The Jacobian is a time varying transformation mapping velocities to velocities

Jacobian for a Manipulator • Robot kinematics give: frame of EE = F(joint variables) • Using the Jacobian: velocity of EE = J joint variable derivatives • If all joints rotational, and calling: then we write:

Singularities of a Robot • If J is invertible, we can compute joint velocities given Cartesian velocities: • Important relationship: shows how to design joint velocities to achieve Cartesian ones • Most robots have joint values for which J is noninvertible • Such points are called singularities of the robot.

Singularities (cont. ) • Two classes of singularities: – Workspace boundary singularities – Workspace interior singularities • Robot in singular configuration: it has lost one or more degrees of freedom in Cartesian space