Melvin Leok’s Professional Blog

Controlled Lagrangian and matching techniques are developed for the stabilization of relative equilibria and equilibria of discrete mechanical systems with symmetry as well as broken symmetry. Interesting new phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. In particular, to make the discrete theory effective, one can make an appropriate selection of momentum levels or, alternatively, introduce a new parameter into the controlled Lagrangian to complete the kinetic matching procedure. Specifically, new terms in the controlled shape equation that are necessary for potential matching in the discrete setting are introduced. The theory is illustrated with the problem of stabilization of the cart-pendulum system on an incline. The paper also discusses digital and model predictive controllers.

arxiv:0704.3875

Celestial Mechanics and Dynamical Astronomy, 98(2), 121-144, 2007
Published online: 12 April 2007
doi:10.1007/s10569-007-9073-x

Equations of motion, referred to as full body models, are developed to describe the dynamics of rigid bodies acting under their mutual gravitational potential. Continuous equations of motion and discrete equations of motion are derived using Hamilton’s principle. These equations are expressed in an inertial frame and in relative coordinates. The discrete equations of motion, referred to as a Lie group variational integrator, provide a geometrically exact and numerically efficient computational method for simulating full body dynamics in orbital mechanics; they are symplectic and momentum preserving, and they exhibit good energy behavior for exponentially long time periods. They are also efficient in only requiring a single evaluation of the gravity forces and moments per time step. The Lie group variational integrator also preserves the group structure without the use of local charts, reprojection, or constraints. Computational results are given for the dynamics of two rigid dumbbell bodies acting under their mutual gravity; these computational results demonstrate the superiority of the Lie group variational integrator compared with integrators that are not symplectic or do not preserve the Lie group structure.

Computer Methods in Applied Mechanics and Engineering, 196(29-30), 2907-2924, 2007
Available online: 7 March 2007
doi:10.1016/j.cma.2007.01.017

We develop the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity. The equations are derived using a variational approach where variations are defined on the Lie group of rigid body configurations. Both continuous equations of motion and variational integrators are developed in Lagrangian and Hamiltonian forms, and the reduction from the inertial frame to a relative coordinate system is also carried out. The Lie group variational integrators are shown to be symplectic, to preserve conserved quantities, and to guarantee exact evolution on the configuration space. One of these variational integrators is used to simulate the dynamics of two rigid dumbbell bodies.