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The '''Fundamental Equation of Constrained Motion'''<ref>Udwadia, F.E.; Kalaba, R.E. (1996). ''Analytical Dynamics: A New Approach.'' Cambridge University Press. ISBN 0-521-04833-8</ref> 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 <math>S</math> is, in general, completely described by <math>n</math> [[generalized coordinates]] so that its generalized coordinate <math>n</math>-vector is given by 
 
:<math>q:=[q_1,q_2,\ldots,q_n]^T.</math>
 
Using Newtonian or [[Lagrangian dynamics]], the unconstrained equations of motion of the system <math>S</math> under study can be derived as
 
:<math> M(q,t)\ddot{q}(t)=Q(q,\dot{q},t),</math>
 
where it is assumed that the initial conditions <math>q(0)</math> and <math>\dot{q}(0)</math> are known. We call the system <math>S</math> unconstrained because <math>\dot{q}(0)</math> may be arbitrarily assigned. Here, the dots represent derivatives with respect to time. The <math>n</math> by <math>n</math> matrix <math> M </math> is [[symmetric matrix|symmetric]], and it can be [[positive definite matrix|positive definite]] <math>(M > 0)</math> or semi-positive definite <math>(M \geq 0)</math>. Typically, it is assumed that <math>M</math> is positive definite; however, it is not uncommon to derive the unconstrained equations of motion of the system <math>S</math> such that <math>M</math> is only semi-positive definite; i.e., the mass matrix may be singular.<ref name=UdwadiaPho>{{Cite journal | doi = 10.1098/rspa.2006.1662 | last1 = Udwadia | first1 = F.E. | last2 = Phohomsiri | first2 = P. | year = 2006 | title = Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics | url = http://ae-www.usc.edu/bio/udwadia/papers/Explicit%20equations%20of%20motion%20singular%20mass%20matrices.pdf | journal = Proceedings of the Royal Society of London, Series A | volume = 462 | issue = 2071| pages = 2097–2117 }}</ref><ref name=UdwadiaSch>{{Cite journal | doi = 10.1007/s00707-009-0272-2 | last1 = Udwadia | first1 = F.E. | last2 = Schutte | first2 = A.D. | year = 2010 | title = Equations of motion for general constrained systems in Lagrangian mechanics | url = http://ae-www.usc.edu/bio/udwadia/papers/ActaMech.pdf | journal = Acta Mechanica | volume = 213 | issue = 1| pages = 111–129 }}</ref>  The <math>n</math>-vector <math>Q</math> denotes the total [[generalized force]] impressed on the system; it can be expressible as the summation of all the [[conservative force]]s with the non-conservative forces.
 
=== Constraints ===
 
We now assume that the unconstrained system <math>S</math> is subjected to a set of <math> m </math> consistent equality constraints given by
:<math> A(q,\dot{q},t)\ddot{q} = b(q,\dot{q},t), </math>
where <math>A</math> is a known ''m'' by ''n'' matrix of rank ''r'' and <math>b</math> is a known ''m''-vector.  We note that this set of constraint equations encompass a very general variety of [[Holonomic constraints|holonomic]] and [[nonholonomic system|non-holonomic]] equality constraints. For example, holonomic constraints of the form
:<math> \varphi(q,t) = 0 </math>
can be differentiated twice with respect to time while non-holonomic constraints of the form
:<math>\psi(q,\dot{q},t) = 0</math>
can be differentiated once with respect to time to obtain the <math>m</math> by <math>n</math> matrix <math>A</math> and the <math>m</math>-vector <math>b</math>.  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 <math>S</math>, an additional force is conceptualized to arise, namely, the force of constraint. Therefore, the constrained system <math>S_c</math> becomes
:<math>M\ddot{q}=Q+Q^{c}(q,\dot{q},t),</math>
where  <math>Q^{c}</math>—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 <math>S</math>,
 
2.  given the generalized displacement <math>q(t)</math> and the generalized velocity <math>\dot{q}(t)</math> of the constrained system <math>S_c</math> at time <math>t</math>, and
 
3. given the constraints in the form <math>A\ddot{q}=b</math> 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 <math>M</math> is positive definite, the equation of motion of the constrained system <math>S_c</math>, at each instant of time, is<ref>{{Cite journal | doi = 10.1098/rspa.1992.0158 | last1 = Udwadia | first1 = F.E. | last2 = Kalaba | first2 = R.E. | year = 1992 | title = A new perspective on constrained motion | url = http://ae-www.usc.edu/bio/udwadia/papers/A_new_perspective.pdf | journal = Proceedings of the Royal Society of London, Series A | volume = 439 | issue = 1906| pages = 407–410 }}</ref><ref>{{Cite journal | doi = 10.1016/0016-0032(93)90099-G | last1 = Udwadia | first1 = F.E. | last2 = Kalaba | first2 = R.E. | year = 1993 | title = On motion | url = http://ae-www.usc.edu/bio/udwadia/papers/On_Motion.pdf | journal = Journal of the Franklin Institute | volume = 330 | issue = 3| pages = 571–577 }}</ref>
 
:<math> M\ddot{q} = Q + M^{1/2}\left(AM^{-1/2}\right)^+(b-AM^{-1}Q), </math>
 
where the '+' symbol denotes the [[Moore-Penrose pseudoinverse|Moore-Penrose inverse]] of the matrix <math>AM^{-1/2}</math>. The force of constraint is thus given explicitly as
 
:<math> Q^{c} = M^{1/2}\left(AM^{-1/2}\right)^+(b-AM^{-1}Q), </math>
 
and since the matrix <math>M</math> is positive definite the generalized acceleration of the constrained system <math>S_c</math> is determined explicitly by
 
:<math> \ddot{q} = M^{-1}Q + M^{-1/2}\left(AM^{-1/2}\right)^+(b-AM^{-1}Q). </math>
 
In the case that the matrix <math>M</math> is semi-positive definite <math>(M \geq 0)</math>, the above equation cannot be used directly because <math>M</math> may be singular.  Furthermore, the generalized accelerations may not be unique unless the <math>n+m</math> by <math>n</math> matrix
 
:<math> \hat{M} = \left[\begin{array}{c} M \\ A \end{array}\right] </math>
 
has full rank (rank = <math>n</math>).<ref name=UdwadiaPho/><ref name=UdwadiaSch/> 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 <math>S_c</math> at each instant of time. Thus, when <math>\hat{M}</math> has full rank, the equations of motion of the constrained system <math>S_c</math> at each instant of time are uniquely determined by (1) creating the auxiliary unconstrained system<ref name=UdwadiaSch/>
 
:<math> M_A \ddot{q}:=(M+A^+A)\ddot{q} = Q + A^+b := Q_b, </math>
 
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<ref name=UdwadiaSch/>
 
:<math> M_A \ddot{q} = Q_b + M_A^{1/2}(AM_A^{-1/2})^+(b-AM_A^{-1}Q_b). </math>
 
Moreover, when the matrix <math>\hat{M}</math> has full rank, the matrix <math>M_A</math> is always positive definite. This yields, explicitly, the generalized accelerations of the constrained system <math> S_c </math> as
 
:<math> \ddot{q} = M_A^{-1}Q_b + M_A^{-1/2}(AM_A^{-1/2})^+(b-AM_A^{-1}Q_b). </math>
 
This equation is valid when the matrix <math>M</math> is either positive definite ''or'' positive semi-definite!  Additionally, the force of constraint that causes the constrained system <math>S_c</math>—a system that may have a singular mass matrix <math>M</math>—to satisfy the imposed constraints is explicitly given by
 
:<math> Q^{c} =  M_A^{1/2}(AM_A^{-1/2})^+(b-AM_A^{-1}Q_b). </math>
 
== References ==
<references/>
 
[[Category:Dynamics]]

Latest revision as of 22:06, 12 January 2015

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