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In this paper we study the modeling and control of robot manipulators with elastic joints. We first derive a simple model to represent the dynamics of elastic joint manipulators. The model is derived under two assumptions regarding dynamic coupling between the actuators and the links, and is useful for cases where the elasticity in the joints is of greater significance than gyroscopic interactions between the motors and links. In the limit as the joint stiffness tends to infinity, our model reduces to the usual rigid model found in the literature, showing the reasonableness of our modeling assumptions. We show that our model is significantly more tractable with regard to controller design than previous nonlinear models that have been used to model elastic joint manipulators. Specifically, the nonlinear equations of motion that we derive are shown to be globally linearizable by diffeomorphic coordinate transformation and nonlinear static state feedback, a result that does not hold for previously derived models of elastic joint manipulators. We also detail an alternate approach to nonlinear control based on a singular perturbation formulation of the equations of motion and the concept of integral manifold. We show that by a suitable nonlinear feedback, the manifold in state space which describes the dynamics of the rigid manipulator, that is, the manipulator without joint elasticity, can be made invariant under solutions of the elastic joint system. The implications of this result for the control of elastic joint robots are discussed.

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In this paper, we consider the application of a new formulation of passivity-based control (PBC), known as interconnection and damping (IDA) assignment to the problem of stabilization of underactuated mechanical systems, which requires the modification of both the potential and the kinetic energies. Our main contribution is the characterization of a class of systems for which IDA-PBC yields a smooth asymptotically stabilizing controller with a guaranteed domain of attraction. The class is given in terms of solvability of certain partial differential equations. One important feature of IDA-PBC, stemming from its Hamiltonian (as opposed to the more classical Lagrangian) formulation, is that it provides new degrees of freedom for the solution of these equations. Using this additional freedom, we are able to show that the method of "controlled Lagrangians"-in its original formulation-may be viewed as a special case of our approach. As illustrations we design asymptotically stabilizing IDA-PBCs for the classical ball and beam system and a novel inertia wheel pendulum. For the former, we prove that for all initial conditions (except a set of zero measure) we drive the beam to the right orientation. Also, we define a domain of attraction for the zero equilibrium that ensures that the ball remains within the bar. For the inertia wheel, we prove that it is possible to swing up and balance the pendulum without switching between separately derived swing up and balance controllers and without measurement of velocities.

In this paper, we give a tutorial account of several of the most recent adaptive control results for rigid robot manipulators.Our intent is to lend some perspective to the growing list of adaptive control results for manipulators by providing a unified framework for comparison of those adaptive control algorithms which have been shown to be globally convergent. In most cases we are able to simplify the derivations and proofs of these results as well.