Joint Velocity and the Jacobian How did I do that? Look! I can move!

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:

Link Differential Transformation Recall the general link transformation: Differentiating:

Differential Transformation (cont.) Using the formula for the inverse: Therefore:

Differential Transformation (cont.) The last expression can be written as: The angular and linear velocities are:

Velocity Propagation from link 2 link

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

Velocity Propagation: Prismatic Joint For the case i+1 is a prismatic joint:

An Example: V3 L2 L1

The Jacobian Jacobian = Multidimensional Derivative Example: 6 functions fi , i=1,…,6 6 variables xi , i=1,…,6 Write: y1 = f1(x1, x2, …, x6) y2 = f2(x1, x2, …, x6)  =  y6 = f6(x1, x2, …, x6) 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 non-invertible 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