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On the Boundedness and Skew-Symmetric Properties of the Inertia and Coriolis Matrices for Vehicle-Manipulator Systems

Abstract This paper addresses the boundedness property of the inertia matrix and the skew-symmetric property of the Coriolis matrix for vehicle-manipulator systems. These properties are widely used in Lyapunov-based stability proofs and are therefore important to identify. For example, the skew-symmetric property does not depend on the system at hand, but on the choice of parameterisation of the Coriolis matrix, which is not unique. It is the authors' experience that many researchers take this assumption for granted without taking into account that there exist several parameterisations for which this is not true. In fact, most researchers refer to references that do not show this property for vehicle-manipulator systems, but for other systems such as single rigid bodies or manipulators on a fixed base. As a result, the otherwise rigorous stability proofs fall apart. In this paper we point out several references that are widely used, but that do not show this property and we refer to the correct references. As most references on this topics are not easily accessible, we also give the correct proofs for commonly used parameterisations of the Coriolis matrix and thus provide a proof for future reference. The same is the case for the boundedness property of the inertia matrix which for a bad choice of state variables will not necessarily hold. This can be solved by deriving the dynamics for vehicle-manipulator systems in terms of quasi-velocities, which allows us to describe the dynamics without the presence of the Euler angle singularities that normally arise in vehicle-manipulator dynamics. To the authors' best knowledge we derive for the first time the dynamic equations with both the skew-symmetric property of the Coriolis matrix and the boundedness property of the inertia matrix for vehicle-manipulator systems with non-Euclidean joints.