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 Title: Udwadia–Kalaba equation Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

The Fundamental Equation of Constrained Motion is a mathematical method for derivation of the equations of motion of a constrained system. This equation was originally developed by Firdaus E. Udwadia and Robert E. Kalaba in a series of papers, beginning in 1992. This equation has application in the field of analytical dynamics.

## 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

q:=[q_1,q_2,\ldots,q_n]^T.

Using Newtonian or Lagrangian dynamics, the unconstrained equations of motion of the system S under study can be derived as

M(q,t)\ddot{q}(t)=Q(q,\dot{q},t),

where it is assumed that the initial conditions q(0) and \dot{q}(0) are known. We call the system S unconstrained because \dot{q}(0) may be arbitrarily assigned. Here, the dots represent derivatives with respect to time. The n by n matrix M is symmetric, and it can be positive definite (M > 0) or semi-positive definite (M \geq 0). 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. The n-vector Q denotes the total generalized force impressed on the system; it can be expressible as the summation of all the conservative forces with the non-conservative forces.

### Constraints

We now assume that the unconstrained system S is subjected to a set of m consistent equality constraints given by

A(q,\dot{q},t)\ddot{q} = b(q,\dot{q},t),

where A is a known m by n matrix of rank r and b is a known m-vector. We note that this set of constraint equations encompass a very general variety of holonomic and non-holonomic equality constraints. For example, holonomic constraints of the form

\varphi(q,t) = 0

can be differentiated twice with respect to time while non-holonomic constraints of the form

\psi(q,\dot{q},t) = 0

can be differentiated once with respect to time to obtain the m by n matrix A and the m-vector b. In short, constraints may be specified that are (1) nonlinear functions of displacement and velocity, (2) explicitly dependent on time, and (3) functionally dependent.

As a consequence of subjecting these constraints to the unconstrained system S, an additional force is conceptualized to arise, namely, the force of constraint. Therefore, the constrained system S_c becomes

M\ddot{q}=Q+Q^{c}(q,\dot{q},t),

where Q^{c}—the constraint force—is the additional force needed to satisfy the imposed constraints. The central problem of constrained motion is now stated as follows:

1. given the unconstrained equations of motion of the system S,

2. given the generalized displacement q(t) and the generalized velocity \dot{q}(t) of the constrained system S_c at time t, and

3. given the constraints in the form A\ddot{q}=b as stated above,

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.

## The Fundamental Equation of Constrained Motion

The solution to this central problem is given by the fundamental equation of constrained motion. When the matrix M is positive definite, the equation of motion of the constrained system S_c, at each instant of time, is

M\ddot{q} = Q + M^{1/2}\left(AM^{-1/2}\right)^+(b-AM^{-1}Q),

where the '+' symbol denotes the Moore-Penrose inverse of the matrix AM^{-1/2}. The force of constraint is thus given explicitly as

Q^{c} = M^{1/2}\left(AM^{-1/2}\right)^+(b-AM^{-1}Q),

and since the matrix M is positive definite the generalized acceleration of the constrained system S_c is determined explicitly by

\ddot{q} = M^{-1}Q + M^{-1/2}\left(AM^{-1/2}\right)^+(b-AM^{-1}Q).

In the case that the matrix M is semi-positive definite (M \geq 0), the above equation cannot be used directly because M may be singular. Furthermore, the generalized accelerations may not be unique unless the n+m by n matrix

\hat{M} = \left[\begin{array}{c} M \\ A \end{array}\right]

has full rank (rank = n). 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 S_c at each instant of time. Thus, when \hat{M} has full rank, the equations of motion of the constrained system S_c at each instant of time are uniquely determined by (1) creating the auxiliary unconstrained system

M_A \ddot{q}:=(M+A^+A)\ddot{q} = Q + A^+b := Q_b,

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

M_A \ddot{q} = Q_b + M_A^{1/2}(AM_A^{-1/2})^+(b-AM_A^{-1}Q_b).

Moreover, when the matrix \hat{M} has full rank, the matrix M_A is always positive definite. This yields, explicitly, the generalized accelerations of the constrained system S_c as

\ddot{q} = M_A^{-1}Q_b + M_A^{-1/2}(AM_A^{-1/2})^+(b-AM_A^{-1}Q_b).

This equation is valid when the matrix M is either positive definite or positive semi-definite! Additionally, the force of constraint that causes the constrained system S_c—a system that may have a singular mass matrix M—to satisfy the imposed constraints is explicitly given by

Q^{c} = M_A^{1/2}(AM_A^{-1/2})^+(b-AM_A^{-1}Q_b).