BilateralLagrangianConstraint

This component belongs to the category of Constraint Laws used for the Lagrange constraint resolution and inherits from the PairInteractionConstraint. The BilateralLagrangianConstraint defines an holonomic constraint law between a pair of simulated body, i.e. the constraint defined between the pair of objects must have an equality form:

$Holonomic constraint law$

Such a constraint is suited for attachment cases or sliding joints. For an attachment case, if the vertex i of object 1 and the vertex j of object 2 are attached, the holonomic constraint law can be written as $Attachment law$ .

For a BilateralLagrangianConstraint, the constraint matrix $Constraint matrix$ (derivative of the constraint law) corresponds to:

$\begin{equation*} &\mathbf{H}_1 = \begin{bmatrix} \begin{bmatrix} 0 & 0 & 0\\\end{bmatrix}_0 & \hdots & \begin{bmatrix} 1 & 0 & 0\\\end{bmatrix}_i & \hdots & \begin{bmatrix} 0 & 0 & 0\\\end{bmatrix}_{N_1}\\ \begin{bmatrix} 0 & 0 & 0\\\end{bmatrix}_0 & \hdots & \begin{bmatrix} 0 & 1 & 0\\\end{bmatrix}_i & \hdots & \begin{bmatrix} 0 & 0 & 0\\\end{bmatrix}_{N_1}\\ \begin{bmatrix} 0 & 0 & 0\\\end{bmatrix}_0 & \hdots & \begin{bmatrix} 0 & 0 & 1\\\end{bmatrix}_i & \hdots & \begin{bmatrix} 0 & 0 & 0\\\end{bmatrix}_{N_1}\\ \end{bmatrix} \end{equation*}$ for object 1
$\begin{equation*} &\mathbf{H}_2 = \begin{bmatrix} \begin{bmatrix} 0 & 0 & 0\\\end{bmatrix}_0 & \hdots & \begin{bmatrix} -1 & 0 & 0\\\end{bmatrix}_j & \hdots & \begin{bmatrix} 0 & 0 & 0\\\end{bmatrix}_{N_2}\\ \begin{bmatrix} 0 & 0 & 0\\\end{bmatrix}_0 & \hdots & \begin{bmatrix} 0 & -1 & 0\\\end{bmatrix}_j & \hdots & \begin{bmatrix} 0 & 0 & 0\\\end{bmatrix}_{N_2}\\ \begin{bmatrix} 0 & 0 & 0\\\end{bmatrix}_0 & \hdots & \begin{bmatrix} 0 & 0 & -1\\\end{bmatrix}_j & \hdots & \begin{bmatrix} 0 & 0 & 0\\\end{bmatrix}_{N_2}\\ \end{bmatrix} \end{equation*}$ for object 2

As all constraint laws, the BilateralLagrangianConstraint will be called in the following functions and for the following steps:

getConstraintViolation(): project the free velocity in the constraint space and compute the free interpenetration $\begin{equation} &\dot{\delta}_1^{free}=\mathbf{H}_1v_1^{free} \end{equation}$
buildConstraintMatrix(): build the compliance made up of $Compliance of object 1$ and $Compliance of object 2$

Data

As a PairInteractionConstraint, the BilateralLagrangianConstraint requires the following Data:

object1: link towards the object 1 to constraint
object2: link towards the object 2 to constraint
first_point: index of the constraint on the first model (object 1)
second_point: index of the constraint on the second model (object 2)

Usage

The BilateralLagrangianConstraint can only be used in the context of Lagrange constraint resolution. The scene must therefore contain:

a FreeMotionAnimationLoop
a ConstraintSolver

Moreover, each constrained object must define in its node a ConstraintCorrection so that the corrective motion can be applied.

Example

This component is used as follows in XML format:

<BilateralLagrangianConstraint template="Vec3d" object1="@CUBE_2/Constraints/points" object2="@CUBE_4/Constraints/points" first_point="1" second_point="0" />

or using SofaPython3:

node.addObject('BilateralLagrangianConstraint', template='Vec3d' object1='@CUBE_2/Constraints/points' object2='@CUBE_4/Constraints/points' first_point='1' second_point='0')

An example scene involving a BilateralLagrangianConstraint is available in examples/Component/Constraint/Lagrangian/BilateralLagrangianConstraint_PGS.scn