In the study of the dynamics of mechanical systems, the configuration of a given system [S] is, in general, completely described by [n] generalized coordinatesso that its generalized coordinate [n] -vector is given by

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

Using Newtonian orLagrangian 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] issymmetric, and it can bepositive 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.[3][4]The [n] -vector [Q] denotes the totalgeneralized forceimpressed on the system; it can be expressible as the summation of all theconservative forceswith the non-conservative forces.

Constraints[edit]

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 knownmbynmatrix of rankrand [b] is a knownm-vector. We note that this set of constraint equations encompass a very general variety ofholonomicandnon-holonomicequality 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 theconstrainedsystem—the acceleration—at timet, which is in accordance with the agreed upon principles of analytical dynamics.

The Fundamental Equation of the Constrained Motion[edit]

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[5][6]

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

where the '+' symbol denotes theMoore-Penrose inverseof 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] ).[3][4]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[4]

[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[4]

[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 definiteorpositive 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).]