Discussion

Consider a two-link spatial robotic arm with three degrees of freedom at the end of the terminal link.  If a unit force is applied to the end of the terminal link, the resulting acceleration of the link will vary in magnitude and direction with changes in the direction of the applied force and changes in the configuration of the robot.  A parametric plot of the acceleration vector over the direction space of the applied force yields the "ellipsoid of mobility" of the end of the terminal link.

This notebook uses two methods to find the ellipsoid of mobility.  First, the robotic arm is modeled with reference point coordinates giving it a total of 12 degrees of freedom (6 per body).  Constraints are applied to represent the joints of the robot leaving 3 unconstrained degrees of freedom, one per joint angle.  After defining mass properties for each of the links, the ellipsoid of mobility is found by applying a unit force to the end of the terminal link and measuring the resulting acceleration over the direction space of the applied force.  This technique may be applied to spatial models of arbitrary complexity and coupling.

The second method uses generaized coordinates to create a model with only three degrees of freedom, one for each free joint angle.  From this (much simpler) model, the generalized mass matrix is obtained and transformed into an inverse mass matrix with respect to the end of the terminal link, where is the Jacobian of end point velocity with respect to the generalized coordinates.  The ellipsoid of mobility can now be obtained ploting the acceleration vector .