[1][2] The method was first described by Anatolii Fedorovich Vereshchagin[3][4] 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 method generalizes to constraint forces that do not obey D'Alembert's principle.
[6][7][8] The Udwadia–Kalaba equation was developed in 1992 and describes the motion of a constrained mechanical system that is subjected to equality constraints.
[5] 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.
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
The n-by-n matrix M is symmetric, and it can be 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).
[10][11] 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 m-vector.
For example, holonomic constraints of the form can be differentiated twice with respect to time while non-holonomic constraints of the form can be differentiated once with respect to time to obtain the m-by-n matrix A and the m-vector b.
The central problem of constrained motion is now stated as follows: find the equations of motion for the constrained system—the acceleration—at time t, which is in accordance with the agreed upon principles of analytical dynamics.
denotes the inverse of its square root, defined as where
is the orthogonal matrix arising from eigendecomposition (whose rows consist of suitably selected eigenvectors of
When the matrix M is positive definite, the equation of motion of the constrained system Sc, at each instant of time, is[5][12] where the '+' symbol denotes the pseudoinverse of the matrix
The force of constraint is thus given explicitly as and since the matrix M is positive definite the generalized acceleration of the constrained system Sc is determined explicitly by In the case that the matrix M is semi-positive definite
[10][11] But since the observed accelerations of mechanical systems in nature are always unique, this rank condition is a necessary and sufficient condition for obtaining the uniquely defined generalized accelerations of the constrained system Sc at each instant of time.
has full rank, the equations of motion of the constrained system Sc at each instant of time are uniquely determined by (1) creating the auxiliary unconstrained system[11] and by (2) applying the fundamental equation of constrained motion to this auxiliary unconstrained system so that the auxiliary constrained equations of motion are explicitly given by[11] Moreover, when the matrix
This yields, explicitly, the generalized accelerations of the constrained system Sc as This equation is valid when the matrix M is either positive definite or positive semi-definite.
If the displacement is irreversible, then it performs virtual work.
The Udwadia–Kalaba equation is modified by an additional non-ideal constraint term to The method can solve the inverse Kepler problem of determining the force law that corresponds to the orbits that are conic sections.
[13] We take there to be no external forces (not even gravity) and instead constrain the particle motion to follow orbits of the form where
Differentiating twice with respect to time and rearranging slightly gives a constraint We assume the body has a simple, constant mass.
We also assume that angular momentum about the focus is conserved as with time derivative We can combine these two constraints into the matrix equation The constraint matrix has inverse The force of constraint is therefore the expected, central inverse square law Consider a small block of constant mass on an inclined plane at an angle
The constraint that the block lie on the plane can be written as After taking two time derivatives, we can put this into a standard constraint matrix equation form The constraint matrix has pseudoinverse We allow there to be sliding friction between the block and the inclined plane.
Therefore, the virtual work associated with a virtual displacement will depend on C. We may summarize the three forces (external, ideal constraint, and non-ideal constraint) as follows: Combining the above, we find that the equations of motion are This is like a constant downward acceleration due to gravity with a slight modification.
If the block is moving up the inclined plane, then the friction increases the downward acceleration.
If the block is moving down the inclined plane, then the friction reduces the downward acceleration.