Coordinate Projection Method

Coordinate projection is applied after an integration step is completed. Considering a multibody system in its index-1 form (5.1.3a),(5.1.3b),(5.1.4b), a typical integration step results in values Pn,Vn, and An being the solution of the nonlinear system 

These values are then projected orthogonally back onto the manifolds given by the constraints on position and velocity level, i.e. the projected values are defined as the solution of the nonlinear constrained least squares problem 

The projected values, which now fulfill also position and velocity constraints, are then used to advance the solution. 

This projection method can be applied to any discretization method. It has first been proposed for differential equations with invariants in [Sha86], where also convergence for one step methods has been proven. Convergence for multistep methods was shown in [Eich93]. 

The numerical solution of the projection step can be computed iteratively by a Gaufi-Newton method, see also Ch. 7.2.2. There, a nonlinear constrained least squares problem is solved iteratively. The nonlinear functions are linearized about the current iterate and the problem is reduced to a sequence of linear constrained least squares problems. 

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The task to solve an underdetermined linear system of equations occurs frequently in multibody dynamics, e.g. when using coordinate projection methods to stabilize the numerical integration of the index reduced system of equations of motion, see Sec. 5.3.1. 

Both projection methods differ by the quantities which are projected. The coordinate projection method projects the solution of the discretized equation orthogonally onto the manifold M = x ip x) — (p xo) = 0 while the derivative projection determines Xn in such a way that the residual Sh xn) — fn is orthogonal to the tangent on M in xq. 

Eich93] Eich E. (1993) Convergence results for a coordinate projection method applied to constrained mechanical systems. SIAM J. Numer. Anal, 30(5) 1467-1482. 

The desired permutational symmetry can be accounted for in the projection method. Consider a system of N particles invariant under the action of a group, G, represented by a set, Pa c, of N x N permutation matrices. A model potential represented as an expansion in k is a function of the n = N — 1 component vectors of r, the relative coordinates. The permutation Pa acting on the N particle coordinates induces a transformation on the center of mass and relative coordinates given by... 

Since the analyte is often present in quite a complex matrix comprising other elements or substances that may or may not interfere with any projected method of determination, and may indeed be quite a minor constituent of the whole sample, these considerations will effectively dominate the working details of any analytical procedure. There can therefore be no case for prescribing a single definitive method for (say) nickel that would be universally applicable to its determination in alloys see Alloys), organometallic compounds see Organometallic Complexes), inorganic salts see Ionic Bonds), or coordination compounds see Coordination Complexes), to its presence as a trace constituent see Trace Element) in maize or other plant products, or in natural waters or effluents. 

Reaction coordinate-based methods identify TSE structures as those conformations residing at the top of the free energy barrier projected onto a given reaction coordinate q (see Figure 13). Several different reaction coordinates have been introduced to extract TSE conformations from simula-tions/ " with the most common coordinate being the number of native contacts A serious concern with reaction coordinate-based... 

From the projection algorithm above note that the vectors wj, are coordinates of the latent variables on the original variable axes. The choice of these weightings defines the various methods. Principal components represent the special case of the general projection method defined above where wj, and normalized loadings p are identical. Principal components can thus be found by iterating steps 1-3 until = p. ... 

We pointed out earlier that for equations of motion of constrained mechanical systems written in index-1 form the position and velocity constraints form integral invariants, see (5.1.16). Thus the coordinate projection and the implicit state space method introduced in the previous section can be viewed as numerical methods for ensuring that the numerical solution satisfies these invariants. 

Figure 5.9 Projection methods coordinate projection (left) and derivative projection (right)...

PLS is a projection method and thus has a simple geometric interpretation as a projection of the X matrix (a swarm of points in a /(-dimensional space) onto an A-dimensional (hyper-)plane in such a way that the coordinates of the projection (ta,fl= 1,2.A) are good predictors of Y (see... 

USCT IT. The US block forms beam data by the mirror-shadow method and ensures simultaneously precise measurement of coordinates of sensors. It consists of two multichannel blocks, namely tomographic (USTB) for multiangle collection of projection data and coordinate (USCB) on surfaces waves for coordinates measurement of US sensors. 

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