Affine Body Revolute Joint Limit

#670 AffineBodyRevoluteJointLimit

Affine Body Revolute Joint Limit restricts the rotation angle of a Revolute Joint to a specified range. When the joint angle goes beyond the lower or upper bound, a cubic penalty is applied. It is an InterAffineBody extra constitution defined on revolute-joint geometry.

Energy

The limit adds an energy \(E(x)\) where \(x\) is the scalar joint angle (its exact meaning is given in Meaning of \(x\)). Let \(l,u\) be lower/upper limits and \(s\) be limit/strength.

For normal range width (\(u>l\)):

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

The normalized gaps are dimensionless, so changing \((l,u)\) does not require retuning \(s\) just because the interval width changed.

For degenerate limit (\(u=l\)), use fallback cubic:

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

Boundary points are treated as in-range.

Meaning of \(x\)

The revolute coordinate is evaluated in incremental form:

\[ x=\theta^t+\delta, \]

\[ \theta^t=\Delta\Theta(\mathbf{q}^{t},\mathbf{q}_{ref}), \quad \delta=\Delta\Theta(\mathbf{q},\mathbf{q}^{t}). \]

Here:

• \(\mathbf{q}\) is the current affine-body DOF.
• \(\mathbf{q}^{t}\) is the previous-time-step DOF.
• \(\mathbf{q}_{ref}\) is the reference DOF captured at initialization.
• \(\theta^t\) is the accumulated revolute angle from the previous step.

Note that \(\mathbf{q}\) is the concatenation of the DOF of the two affine bodies connected by the prismatic joint.

\[ \mathbf{q} = \begin{bmatrix} \mathbf{q}_i \\ \mathbf{q}_j \end{bmatrix}, \]

The Revolute Joint geometry is represented as an edge with two endpoints. The joint axis direction is defined by the order of the two endpoints giving \(+\hat{\mathbf{t}}\).

• \(x=0\): current relative revolute angle equals the reference angle.
• \(x>0\): positive rotation around \(+\hat{\mathbf{t}}\) (right-hand rule; counterclockwise when viewed along \(+\hat{\mathbf{t}}\)).
• \(x<0\): negative rotation around \(+\hat{\mathbf{t}}\) (clockwise when viewed along \(+\hat{\mathbf{t}}\)).

For Revolute Joints:

\[ \Delta\Theta(\mathbf{q}_a,\mathbf{q}_b)= \operatorname{atan2} \left( \sin\theta_a\cos\theta_b-\cos\theta_a\sin\theta_b,\; \cos\theta_a\cos\theta_b+\sin\theta_a\sin\theta_b \right), \]

\[ \cos\theta=\frac{\hat{\mathbf{b}}_i\cdot\hat{\mathbf{b}}_j+\hat{\mathbf{n}}_i\cdot\hat{\mathbf{n}}_j}{2}, \quad \sin\theta=\frac{\hat{\mathbf{n}}_i\cdot\hat{\mathbf{b}}_j-\hat{\mathbf{b}}_i\cdot\hat{\mathbf{n}}_j}{2}. \]

The sign of \(x\) follows the sign of \(\sin\theta\) under this convention. The angle branch is \((-\pi,\pi]\).

Requirement

This limit term is meaningful only on a geometry that already represents a Revolute Joint (UID=18). The limit augments that base relation as an extra constitution term.

Attributes

On edges:

• limit/lower: \(l\) in the energy above
• limit/upper: \(u\) in the energy above
• limit/strength: \(s\) in the energy above