The dynamic behavior of a -Degrees of Freedom (DoF) robot manipulator can be derived from the Euler-Lagrange equations of motion
where is the Lagrangian and is the potential energy. is the inertia matrix and are the joint velocities and positions, respectively. In compact form, these equations can be written as
where is the joint acceleration vector; is the Coriolis and centrifugal effects matrix; is the gravitational force vector, and is a generalized force vector.
It is well-known that (1) enjoys the following fundamental linearity property: for all , dynamics (1) can be written as
where is a regressor matrix of known functions and is a constant vector that is a function of the manipulator physical parameters (link masses, moments of inertia, etc.) .
In the document attached, ""Regressor_3DoF.pdf"", is used the linearity in the parameters property (2) and presents such factorisation for a 3-DoF planar manipulator. The document starts by presenting the joint space model and then such model is used to obtain the operational space model. Finally, the document reports the corresponding regressor matrix and parameters vector pair for this particular 3-DoF case.
 M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control. John Wiley & Sons, 2006.
If this information is useful to you and want to cite it please reference it as:
Aldana, C.; Nuno, Emmanuel; Basanez, L., "Bilateral teleoperation of cooperative manipulators," Robotics and Automation (ICRA), 2012 IEEE International Conference on , vol., no., pp.4274,4279, 14-18 May 2012.