Udwadia–Kalaba formulation

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.{{cite journal

| last1 = Zhao

| first1 = Xiao

| last2 = Chen

| first2 = Ye-Hwa

| last3 = Zhao

| first3 = Han

| last4 = Dong

| first4 = Fang-Fang

| year = 2018

| title = Udwadia–Kalaba equation for constrained mechanical systems:Formulation and applications

| language = en

| journal = Chinese Journal of Mechanical Engineering

| volume = 31

| issue = 1

| pages = 106–120

| doi = 10.1186/s10033-018-0310-x

| bibcode = 2018ChJME..31..106Z

| doi-access = free

}}

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

:$\backslash mathbf\{q\}:=[q\_1,q\_2,\backslash ldots,q\_n]^\backslash mathrm\{T\}.$

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):

{{Equation box 1

|title=Udwadia–Kalaba equations of motion (Unconstrained)

|indent=:

|equation= $\backslash mathbf\{M\}(q,t)\backslash ddot\{\backslash mathbf\{q\}\}(t)=\backslash mathbf\{Q\}(q,\backslash dot\{q\},t)\backslash ,,$

|cellpadding = 6 |border = 1 |border colour = black |background colour=white}}

where the dots represent derivatives with respect to time:

:$\backslash dot\{q\}\_i\; =\; \backslash frac\{dq\_i\}\{dt\}\backslash ,.$

It is assumed that the initial conditions q(0) and $\backslash dot\{\backslash mathbf\{q\}\}(0)$ are known. We call the system S unconstrained because $\backslash dot\{\backslash mathbf\{q\}\}(0)$ 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 $(\backslash mathbf\{M\}\; >\; 0)$ or semi-positive definite $(\backslash mathbf\{M\}\; \backslash geq\; 0)$. 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 (it has no inverse matrix).{{Cite journal|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://ruk.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|bibcode=2006RSPSA.462.2097U|doi=10.1098/rspa.2006.1662|s2cid=38437}}{{Cite journal|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://ruk.usc.edu/bio/udwadia/papers/ActaMechanica.pdf|journal=Acta Mechanica|volume=213|issue=1|pages=111–129|doi=10.1007/s00707-009-0272-2|s2cid=7432252}}

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

:$\backslash mathbf\{A\}(q,\backslash dot\{q\},t)\backslash ddot\{\backslash mathbf\{q\}\}\; =\; \backslash mathbf\{b\}(q,\backslash dot\{q\},t),$

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

:$\backslash varphi(q,t)\; =\; 0$

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

:$\backslash psi(q,\backslash dot\{q\},t)\; =\; 0$

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

{{Equation box 1

|title=Udwadia–Kalaba equations of motion (Constrained)

|indent=:

|equation= $\backslash mathbf\{M\}\backslash ddot\{\backslash mathbf\{q\}\}=\backslash mathbf\{Q\}+\backslash mathbf\{Q\}\_\{c\}(q,\backslash dot\{q\},t),$

|cellpadding = 6 |border = 1 |border colour = black |background colour=white}}

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 $\backslash dot\{q\}(t)$ of the constrained system S_c at time t, and
3. given the constraints in the form $\backslash mathbf\{A\}\backslash ddot\{q\}=\backslash mathbf\{b\}$ 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.

Below, for positive definite {{tmath|\mathbf{M} }}, {{tmath|\mathbf{M}^{-1/2} }} denotes the inverse of its square root, defined as

:$\backslash mathbf\{M\}^\{-1/2\}\; =\; \backslash mathbf\{W\}\backslash mathbf\{\backslash Lambda\}^\{-1/2\}\backslash mathbf\{W\}^T$,

where {{tmath|\mathbf{W} }} is the orthogonal matrix arising from eigendecomposition (whose rows consist of suitably selected eigenvectors of {{tmath|\mathbf{M} }}), and {{tmath|\mathbf{\Lambda}^{-1/2} }} is the diagonal matrix whose diagonal elements are the inverse square roots of the eigenvalues corresponding to the eigenvectors in {{tmath|\mathbf{W} }}.

The solution to this central problem is given by the Udwadia–Kalaba equation. When the matrix M is positive definite, the equation of motion of the constrained system S_c, at each instant of time, is{{Cite journal|last1=Udwadia|first1=F.E.|last2=Kalaba|first2=R.E.|year=1993|title=On motion|url=http://ruk.usc.edu/bio/udwadia/papers/On_Motion.pdf|journal=Journal of the Franklin Institute|volume=330|issue=3|pages=571–577|doi=10.1016/0016-0032(93)90099-G}}

:$\backslash mathbf\{M\}\backslash ddot\{\backslash mathbf\{q\}\}\; =\; \backslash mathbf\{Q\}\; +\; \backslash mathbf\{M\}^\{1/2\}\backslash left(\backslash mathbf\{A\}\backslash mathbf\{M\}^\{-1/2\}\backslash right)^+(\backslash mathbf\{b\}-\backslash mathbf\{A\}\backslash mathbf\{M\}^\{-1\}\backslash mathbf\{Q\}),$

where the '+' symbol denotes the pseudoinverse of the matrix $\backslash mathbf\{A\}\backslash mathbf\{M\}^\{-1/2\}$. The force of constraint is thus given explicitly as

:$\backslash mathbf\{Q\}\_c\; =\; \backslash mathbf\{M\}^\{1/2\}\backslash left(\backslash mathbf\{A\}\backslash mathbf\{M\}^\{-1/2\}\backslash right)^+(\backslash mathbf\{b\}-\backslash mathbf\{A\}\; \backslash mathbf\{M\}^\{-1\}\backslash mathbf\{Q\}),$

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

:$\backslash ddot\{\backslash mathbf\{q\}\}\; =\; \backslash mathbf\{M\}^\{-1\}\backslash mathbf\{Q\}\; +\; \backslash mathbf\{M\}^\{-1/2\}\; \backslash left(\backslash mathbf\{A\}\backslash mathbf\{M\}^\{-1/2\}\backslash right)^+(\backslash mathbf\{b\}-\backslash mathbf\{A\}\backslash mathbf\{M\}^\{-1\}\backslash mathbf\{Q\}).$

In the case that the matrix M is semi-positive definite $(\backslash mathbf\{M\}\; \backslash geq\; 0)$, 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

:$\backslash hat\{\backslash mathbf\{M\}\}\; =\; \backslash left[\backslash begin\{array\}\{c\}\; \backslash mathbf\{M\}\; \backslash \backslash \; \backslash mathbf\{A\}\; \backslash end\{array\}\backslash right]$

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 $\backslash hat\{\backslash mathbf\{M\}\}$ 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

:$\backslash mathbf\{M\}\_\backslash mathbf\{A\}\; \backslash ddot\{\backslash mathbf\{q\}\}:=(\backslash mathbf\{M\}+\backslash mathbf\{A\}^+\backslash mathbf\{A\})\backslash ddot\{\backslash mathbf\{q\}\}\; =\; \backslash mathbf\{Q\}\; +\; \backslash mathbf\{A\}^+\backslash mathbf\{b\}\; :=\; \backslash mathbf\{Q\}\_\backslash mathbf\{b\},$

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

:$\backslash mathbf\{M\}\_\backslash mathbf\{A\}\; \backslash ddot\{\backslash mathbf\{q\}\}\; =\; \backslash mathbf\{Q\}\_\backslash mathbf\{b\}\; +\; \backslash mathbf\{M\}\_\backslash mathbf\{A\}^\{1/2\}(\backslash mathbf\{A\}\backslash mathbf\{M\}\_\backslash mathbf\{A\}^\{-1/2\})^+(\backslash mathbf\{b\}-\backslash mathbf\{A\}\backslash mathbf\{M\}\_\backslash mathbf\{A\}^\{-1\}\backslash mathbf\{Q\}\_\backslash mathbf\{b\}).$

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

:$\backslash ddot\{\backslash mathbf\{q\}\}\; =\; \backslash mathbf\{M\}\_\backslash mathbf\{A\}^\{-1\}\backslash mathbf\{Q\}\_\backslash mathbf\{b\}\; +\; \backslash mathbf\{M\}\_\backslash mathbf\{A\}^\{-1/2\}(\backslash mathbf\{A\}\backslash mathbf\{M\}\_\backslash mathbf\{A\}^\{-1/2\})^+(\backslash mathbf\{b\}-\backslash mathbf\{A\}\backslash mathbf\{M\}\_\backslash mathbf\{A\}^\{-1\}\backslash mathbf\{Q\}\_\backslash mathbf\{b\}).$

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

:$\backslash mathbf\{Q\}\_\{c\}\; =\; \backslash mathbf\{M\}\_\backslash mathbf\{A\}^\{1/2\}(\backslash mathbf\{A\}\backslash mathbf\{M\}\_\backslash mathbf\{A\}^\{-1/2\})^+(\backslash mathbf\{b\}-\backslash mathbf\{A\}\backslash mathbf\{M\}\_\backslash mathbf\{A\}^\{-1\}\backslash mathbf\{Q\}\_\backslash mathbf\{b\}).$

At any time during the motion we may consider perturbing the system by a virtual displacement δr consistent with the constraints of the system. The displacement is allowed to be either reversible or irreversible. If the displacement is irreversible, then it performs virtual work. We may write the virtual work of the displacement as

:$W\_c(t)=\backslash mathbf\{C\}^\{\backslash mathrm\{T\}\}(q,\backslash dot\{q\},t)\backslash delta\backslash mathbf\{r\}(t)$

The vector $\backslash mathbf\{C\}(q,\backslash dot\{q\},t)$ describes the non-ideality of the virtual work and may be related, for example, to friction or drag forces (such forces have velocity dependence). This is a generalized D'Alembert's principle, where the usual form of the principle has vanishing virtual work with $\backslash mathbf\{C\}(q,\backslash dot\{q\},t)=0$.

The Udwadia–Kalaba equation is modified by an additional non-ideal constraint term to

:$\backslash mathbf\{M\}\backslash ddot\{\backslash mathbf\{q\}\}\; =\; \backslash mathbf\{Q\}\; +\; \backslash mathbf\{M\}^\{1/2\}\; \backslash left(\backslash mathbf\{A\}\; \backslash mathbf\{M\}^\{-1/2\}\; \backslash right)^\; +\; (\backslash mathbf\{b\}-\backslash mathbf\{A\}\; \backslash mathbf\{M\}^\{-1\}\; \backslash mathbf\{Q\})\; +\; \backslash mathbf\{M\}^\{1/2\}\backslash left[\backslash mathbf\{I\}-\backslash left(\backslash mathbf\{A\}\; \backslash mathbf\{M\}^\{-1/2\}\; \backslash right)^+\backslash mathbf\{A\}\backslash mathbf\{M\}^\{-1/2\}\backslash right]\backslash mathbf\{M\}^\{-1/2\}\backslash mathbf\{C\}$

The method can solve the inverse Kepler problem of determining the force law that corresponds to the orbits that are conic sections.{{cite journal | last1 = Zhang | first1 = Bingzhan | last2 = Zhen | first2 = Shengchao | last3 = Zhao | first3 = Han | last4 = Huang | first4 = Kang | last5 = Deng | first5 = Bin | last6 = Chen | first6 = Ye-Hwa | title = A novel study on Kepler's law and inverse square law of gravitation | journal = Eur. J. Phys. | volume = 36 | issue = 3 | pages = 035018 | date = 2015 | doi = 10.1088/0143-0807/36/3/035018|bibcode = 2015EJPh...36c5018Z | s2cid = 119566554 }} We take there to be no external forces (not even gravity) and instead constrain the particle motion to follow orbits of the form

:$r=\backslash varepsilon\; x\; +\; \backslash ell$

where $r=\backslash sqrt\{x^2+y^2\}$, $\backslash varepsilon$ is the eccentricity, and