/// Initialize the bodies, anchors, axis, and reference angle using the world
/// anchor and world axis.
public void Initialize(Body b1, Body b2, Vector2 anchor, Vector2 axis)
{
bodyA = b1;
bodyB = b2;
localAnchorA = bodyA.GetLocalPoint(anchor);
localAnchorB = bodyB.GetLocalPoint(anchor);
localAxisA = bodyA.GetLocalVector(axis);
}
// Linear constraint (point-to-line)
// d = p2 - p1 = x2 + r2 - x1 - r1
// C = dot(perp, d)
// Cdot = dot(d, cross(w1, perp)) + dot(perp, v2 + cross(w2, r2) - v1 - cross(w1, r1))
// = -dot(perp, v1) - dot(cross(d + r1, perp), w1) + dot(perp, v2) + dot(cross(r2, perp), v2)
// J = [-perp, -cross(d + r1, perp), perp, cross(r2,perp)]
//
// Angular constraint
// C = a2 - a1 + a_initial
// Cdot = w2 - w1
// J = [0 0 -1 0 0 1]
//
// K = J * invM * JT
//
// J = [-a -s1 a s2]
// [0 -1 0 1]
// a = perp
// s1 = cross(d + r1, a) = cross(p2 - x1, a)
// s2 = cross(r2, a) = cross(p2 - x2, a)
// Motor/Limit linear constraint
// C = dot(ax1, d)
// Cdot = = -dot(ax1, v1) - dot(cross(d + r1, ax1), w1) + dot(ax1, v2) + dot(cross(r2, ax1), v2)
// J = [-ax1 -cross(d+r1,ax1) ax1 cross(r2,ax1)]
// Block Solver
// We develop a block solver that includes the joint limit. This makes the limit stiff (inelastic) even
// when the mass has poor distribution (leading to large torques about the joint anchor points).
//
// The Jacobian has 3 rows:
// J = [-uT -s1 uT s2] // linear
// [0 -1 0 1] // angular
// [-vT -a1 vT a2] // limit
//
// u = perp
// v = axis
// s1 = cross(d + r1, u), s2 = cross(r2, u)
// a1 = cross(d + r1, v), a2 = cross(r2, v)
// M * (v2 - v1) = JT * df
// J * v2 = bias
//
// v2 = v1 + invM * JT * df
// J * (v1 + invM * JT * df) = bias
// K * df = bias - J * v1 = -Cdot
// K = J * invM * JT
// Cdot = J * v1 - bias
//
// Now solve for f2.
// df = f2 - f1
// K * (f2 - f1) = -Cdot
// f2 = invK * (-Cdot) + f1
//
// Clamp accumulated limit impulse.
// lower: f2(3) = max(f2(3), 0)
// upper: f2(3) = min(f2(3), 0)
//
// Solve for correct f2(1:2)
// K(1:2, 1:2) * f2(1:2) = -Cdot(1:2) - K(1:2,3) * f2(3) + K(1:2,1:3) * f1
// = -Cdot(1:2) - K(1:2,3) * f2(3) + K(1:2,1:2) * f1(1:2) + K(1:2,3) * f1(3)
// K(1:2, 1:2) * f2(1:2) = -Cdot(1:2) - K(1:2,3) * (f2(3) - f1(3)) + K(1:2,1:2) * f1(1:2)
// f2(1:2) = invK(1:2,1:2) * (-Cdot(1:2) - K(1:2,3) * (f2(3) - f1(3))) + f1(1:2)
//
// Now compute impulse to be applied:
// df = f2 - f1
/// <summary>
/// Initialize the bodies, anchors, axis, and reference angle using the world
/// anchor and world axis.
/// </summary>
/// <param name="b1"></param>
/// <param name="b2"></param>
/// <param name="anchor"></param>
/// <param name="axis"></param>
public void Initialize(Body b1, Body b2, Vector2 anchor, Vector2 axis)
{
bodyA = b1;
bodyB = b2;
localAnchorA = bodyA.GetLocalPoint(anchor);
localAnchorB = bodyB.GetLocalPoint(anchor);
localAxis1 = bodyA.GetLocalVector(axis);
referenceAngle = bodyB.GetAngle() - bodyA.GetAngle();
}
