An Overview on Constraint Enforced Formulations of Variational Dynamics

A couple of weeks ago a very interesting paper entered my hard drive: Review of classical approaches for constraint enforcement in multibody systems.
It is a fairly clarifying overview on solid grounds of certain methodologies to solve multibody systems dynamics.
These methodologies differ from the ones I have encountered in the spreaded dynamics engines in their apparently more robust and simple formulation.
Here is a synthesis of this paper:

Multibody systems present two distinguishable features:

1. Bodies undergo finite relative rotations, which introduce nonlinearities
2. Bodies are connected by mechanical joints that impose restrictions, which mean a set of governing equations that combine differential and algebraic equations (ODEs and DAEs, respectively).

Lagrange's equation of the first kind has the following aspect:

Where

• M=M(q,t) is the mass matrix
• q is the generalized coordinates vector
• B is the constraint matrix
• λ is the array of Lagrange multipliers
• F is the dynamic externally applied forces

If all the constraints are holonomic (velocity independent), B is called the Jacobian matrix, and the generalized coordinates, q, are linked by m algebraic constraints.
Lagrange's equations of the first kind form a set of (m+n) Differential Algebraic Equations.
The approach of the following methods is to use algebraic procedures in order to eliminate Lagrange's multipliers and then obtain a set of ODEs.

• Maggi's formulation: implies the creation of a vector containing the so called kinematic parameters, generalized speeds or independent quasi-velocities by the analyst in order to obtain a Γ matrix. This matrix spans the null space of the constraint matrix and allows for the elimination of the Lagrange multipliers.
• Index-1 formulation: requires that initial condition of the problem be subjected to the constraint conditions. Then, it is possible to obtain a system of 2nd order ODEs that is solvable by rearranging the previous equation, extracting the Lagrange multipliers and hence obtaining:

• Null space formulation: this method solves the system of second order ODEs by premultiplying the first part of the equation by the transposed null space matrix thus eliminating the Lagrange multipliers.
• Udwadia-Kalaba formulation: this method represents a more compact and general form of solving the DAEs by means of the Moore-Penrose generalized inverse. It is based on Gauss' Principle of Minimum Constraint, which establishes that the explicit equations of motion be expressed as the solution of a quadratic minimization problem subjected to constraints, but at the acceleration level.

All these formulations transform the (2n+m) first order DAEs into ODEs by eliminating Lagrange multipliers.
Maggi's formulation yields (2n-m) first order ODEs.
Index-1, null space and Udwadia-Kalaba form sets of (n) second order ODEs which could be alternatively recast into (2n) first order ODEs for the n generalized coordinates and the n generalized velocities.
The main advantage of these methodologies is not so much the reduction in the number of equations but rather in the change from DAEs to ODEs.
There is a warning on the constraint drift phenomenon, for which these method will be more affected (not so much in Maggi's formulation), and that would require constraint stabilization techniques.

My conclusion is that it seems the way to go, not only for the claims of more stable and quick numerical techniques available to get them working, but also for an apparently more clear approach in the theoretical field.
Particularly, I have done some research into the Udwadia-Kalaba formulation, and definitively is a very promising one.