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The purpose of this paper is to associate the multibody dynamics procedures with a geometrical picture involving the concepts of configuration manifolds, tangent vector spaces, and orthogonality of constraint reactions to the constraint surfaces. An unconstrained mechanical system is assigned a free configuration manifold and is treated as a generalized particle on the manifold. The system dynamics is then formulated in the local tangent space to the manifold at the system representation point. Imposed constraints on the system, the tangent space splits into the velocity restricted and velocity admissible subspaces, while the system configuration manifold confines to the holonomic constraint manifold. Based on these geometrical concepts, a uniform vector‐matrix formulation is developed. Both holonomic and nonholonomic systems are treated in a unified way, and the dynamic equations are expressible either in generalized velocities or in quasi‐velocities. Using a geometrically grounded projection method, compact schemes for obtaining different types of equations of motion and for determination of constraint reactions are provided. Some fresh contributions to the theory of constrained systems are reported. A relationship between the present formulation and the other classical methods of analytical dynamics is shown.

This paper deals with a class of controlled mechanical systems in which the number of control inputs, equal to the number of desired system outputs, is smaller than the number of degrees of freedom. The related inverse dynamics control problem, i.e., the determination of control input strategy that force the underactuated system to complete the partly specified motion, is a challenging task. In the present formulation, the desired system outputs, expressed in terms of the system states, are treated as servo-constraints on the system, and the problem is viewed from the constrained motion perspective. Mixed orthogonal-tangent realization of the constraints by the available control reactions is stated, and a specialized methodology for solving the "singular" control problem is developed. The governing equations are manipulated to index three differential-algebraic equations, and a simple numerical code for solving the equations is proposed. The feedforward control law obtained as a solution to these equations can then be enhanced by a closed-loop control strategy with feedback of the actual servo-constraint violations to provide stable tracking of the reference motion in the presence of perturbations and modeling uncertainties. An overhead trolley crane executing a load-prescribed motion serves as an illustration. Some results of numerical simulations are reported.

Abstract. Underactuated multibody systems have fewer control inputs than degrees of freedom. In trajectory tracking control of such systems an accurate and efficient feedforward control is often necessary. For multibody systems feedforward control by model inversion can be designed using servo-constraints. So far servo-constraints have been mostly applied to differentially flat underactuated mechanical systems. Differentially flat systems can be inverted purely by algebraic manipulations and using a finite number of differentiations of the desired output trajectory. However, such algebraic solutions are often hard to find and therefore the servo-constraint approach provides an efficient and practical solution method. Recently first results on servo-constraint problems of non-flat underactuated multibody systems have been reported. Hereby additional dynamics arise, so-called internal dynamics, yielding a dynamical system as inverse model. In this paper the servo-constraint problem is analyzed for both, differentially flat and non-flat systems. Different arising important phenomena are demonstrated using two illustrative examples. Also strategies for the numerical solution of servo-constraint problems are discussed.

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