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184.108.40.206 Reaction Forces from Springs
The reaction force generated by a spring acting on rigid body is given by where is a gap function that represents the vector distance between the joint origins, is the (optional) Lagrange multiplier used when invoking the augmented Lagrangian method, and is a penalty parameter that represents the spring stiffness. The tensor is a projection that limits the reaction force to the directions that are not free to move. In general, there are three possible options for : For example, in a spherical or revolute joint, whereas in an unconstrained prismatic joint, with representing the axis of motion; and in an unconstrained planar joint, with representing the normal to the plane. Note that in all cases we choose to define the axis in the basis of rigid body . If the constraint is enforced properly, then should be the same as , within a user-defined numerical tolerance. In practice, we let .
The linearization of in (220.127.116.11-1) is given by The linearization of the gap function produces whereas the linearization of the three possible projections yields where Therefore, in the expression for in (7.9.1-6), we have