Improvements on formulations of dynamic models for robotic systems

Methodological advancements appropriate for robotic systems are presented. Rigid robots with lumped parameters are considered. Firstly, an effective recursive procedure is introduced to compute absolute angular velocities using relative angular velocities between consecutive reference frameworks. Based on this, the rotational kinetic energy is reformulated. Next, an improvement on a proposed methodology is detailed for deriving the symbolic dynamic model of the robot based on the Euler–Lagrange (E-L) equations. Assuming that the positions of the Centres of Mass (CM) do not depend explicitly on time, the proposed E-L modeling methodology takes advantage of the quadratic form of the kinetic energy to get the inertia matrix. After that, the inertia matrix is used to obtain the E-L equations based on a certain set of quadratic forms, and the gravity vector is obtained from the potential energy. These quadratic forms are equivalent to the Kronecker product used to get the E-L equations of motion. Furthermore, from these foundational results, Hamilton’s equations of motion and a linearized Euler–Lagrange model are derived using quadratic forms. Finally, the effectiveness of these methodologies is demonstrated through applications to a Double Pendulum, a spherical robot, and a Robot Maker 110, showcasing their practical utility in robotic system analysis and design.