Kinematic description of a general mechanical control system on SE(3) and aircraft modeling

Unlike simple mechanical control systems on a Lie group, a general mechanical control system on a Lie group includes a variable potential in configuration space, which makes it impossible for the Lagrangian of such a system to be left-invariant. This means, by definition, that the Euler-Poincaré equation cannot be applied directly to such a system. This paper first introduces the mathematical definitions of a general mechanical control system on matrix Lie group SE(3). Secondly, by redefining the Lagrangian of left-invariant kinematic energy minus potential energy, a modified Euler-Poincaré equation involving a potential function is deduced and obtained based on the continuous Lagrange-d'Alembert principle to describe the dynamics of the general mechanical control systems on SE(3). Finally, two applications of modeling an unmanned quadrotor and an unmanned airship are presented to verify the proposed approach.