Affine Body Prismatic Joint Limit

#669 AffineBodyPrismaticJointLimit

This constitution is an InterAffineBody extra constitution defined on a prismatic-joint geometry. It contributes a unilateral cubic barrier-like penalty to the joint scalar coordinate.

The coordinate is denoted by \(x\) (unit: length), with lower/upper bounds \(l, u\) and penalty coefficient \(s\):

\[ E(x)= \begin{cases} 0, & l \le x \le u \\ s(x-u)^3, & x>u \\ s(l-x)^3, & x<l \end{cases} \]

Interpretation of each term: - \(x\): current prismatic relative coordinate of the joint. - \(l\): admissible lower bound. - \(u\): admissible upper bound. - \(s\): penalty strength (energy scale).

The scalar coordinate is evaluated in incremental form:

\[ x = \theta_0 + \delta, $$ $$ \theta_0 = \Delta \Theta(\mathbf{q}^{t-1}, \mathbf{q}_{ref}), \quad \delta = \Delta \Theta(\mathbf{q}, \mathbf{q}^{t-1}), \]

where: - \(\mathbf{q}\) is the current affine-body DOF, - \(\mathbf{q}^{t-1}\) is the previous-step DOF, - \(\mathbf{q}_{ref}\) is the reference DOF captured at initialization.

So \(\theta_0\) measures accumulated offset from the reference state, and \(\delta\) is the optimization-time increment in the current step.

First and second derivatives with respect to \(x\):

\[ \frac{dE}{dx}= \begin{cases} 0, & l \le x \le u \\ 3s(x-u)^2, & x>u \\ -3s(l-x)^2, & x<l \end{cases} \]

\[ \frac{d^2E}{dx^2}= \begin{cases} 0, & l \le x \le u \\ 6s(x-u), & x>u \\ 6s(l-x), & x<l \end{cases} \]

Boundary points (\(x=l\) or \(x=u\)) are treated as in-range; energy and derivatives are zero there.

Conceptual Requirement

This limit term is meaningful only on a geometry that already represents a prismatic joint relation (base joint UID = 20). The limit does not replace that base relation; it augments it as an extra constitution term.

Stored Attributes

Per joint edge: - limit/lower - limit/upper - limit/strength

Geometry metadata: - extra constitution UID 669 in meta.extra_constitution_uids.