On the boundedness of the Hessian of the potential energy of robot manipulators

Several results on robot manipulator motion control require a uniform bound for the Hessian of the potential energy or equivalently the Jacobian of the gravity vector. Not all robot manipulators, however, ensure the existence of such a uniform bound. The first contribution of this article is the complete characterization of this class which is referred to as class BG J manipulators. The uniform bound of the Hessian is typically part of the control law expression and hence it plays an important role in controller gain synthesis. The second contribution of this article consists of deriving, for class BG J robot manipulators, an easy to compute explicit expression of the uniform bound in terms of kinematic and inertial link parameters. If for a particular robot manipulator the Hessian of potential energy is not uniformly bounded, a bound exists that is valid within the physical workspace of the manipulator. The third contribution of this article is the derivation of an explicit expression for the latter bound which is useful in the design and controller gain synthesis of control laws that are valid locally.