In classical mechanics, the Udwadia–Kalaba formulation is a method for deriving the equations of motion of a constrained mechanical system. The method was first described by Vereshchagin for the particular case of robotic arms, and later generalized to all mechanical systems by Firdaus E. Udwadia and Robert E. Kalaba in 1992. The approach is based on Gauss's principle of least constraint. The Udwadia–Kalaba method applies to both holonomic constraints and nonholonomic constraints, as long as they are linear with respect to the accelerations. The method generalizes to constraint forces that do not obey D'Alembert's principle. Background The Udwadia–Kalaba equation was developed in 1992 and describes the motion of a constrained mechanical system that is subjected to equality constraints. This differs from the Lagrangian formalism, which uses the Lagrange multipliers to describe the motion of constrained mechanical systems, and other similar approaches such as the Gibbs-Appell approach. The physical interpretation of the equation has applications in areas beyond theoretical physics, such as the control of highly nonlinear general dynamical systems. The central problem of constrained motion In the study of the dynamics of mechanical systems, the configuration of a given system S is, in general, completely described by n generalized coordinates so that its generalized coordinate n-vector is given by where T denotes matrix transpose. Using Newtonian or Lagrangian dynamics, the unconstrained equations of motion of the system S under study can be derived as a matrix equation (see matrix multiplication): where the dots represent derivatives with respect to time: It is assumed that the initial conditions q(0) and are known. We call the system S unconstrained because may be arbitrarily assigned. The n-vector Q denotes the total generalized force acted on the system by some external influence; it can be expressed as the sum of all the conservative forces as well as non-conservative forces. The n-by-n matrix M is symmetric, and it can be positive definite or semi-positive definite . Typically, it is assumed that M is positive definite; however, it is not uncommon to derive the unconstrained equations of motion of the system S such that M is only semi-positive definite; i.e., the mass matrix may be singular (it has no inverse matrix). Constraints We now assume that the unconstrained system S is subjected to a set of m consistent equality constraints given by where A is a known m-by-n matrix of rank r and b is a known