public void SolveVelocityConstraints()
{
// Here be dragons
for (var i = 0; i < _contactCount; ++i)
{
var velocityConstraint = _velocityConstraints[i];
var indexA = velocityConstraint.IndexA;
var indexB = velocityConstraint.IndexB;
var mA = velocityConstraint.InvMassA;
var iA = velocityConstraint.InvIA;
var mB = velocityConstraint.InvMassB;
var iB = velocityConstraint.InvIB;
var pointCount = velocityConstraint.PointCount;
var vA = _linearVelocities[indexA];
var wA = _angularVelocities[indexA];
var vB = _linearVelocities[indexB];
var wB = _angularVelocities[indexB];
var normal = velocityConstraint.Normal;
var tangent = Vector2.Cross(normal, 1.0f);
var friction = velocityConstraint.Friction;
DebugTools.Assert(pointCount == 1 || pointCount == 2);
// Solve tangent constraints first because non-penetration is more important
// than friction.
for (var j = 0; j < pointCount; ++j)
{
VelocityConstraintPoint velConstraintPoint = velocityConstraint.Points[j];
// Relative velocity at contact
var dv = vB + Vector2.Cross(wB, velConstraintPoint.RelativeVelocityB) - vA - Vector2.Cross(wA, velConstraintPoint.RelativeVelocityA);
// Compute tangent force
float vt = Vector2.Dot(dv, tangent) - velocityConstraint.TangentSpeed;
float lambda = velConstraintPoint.TangentMass * (-vt);
// b2Clamp the accumulated force
var maxFriction = friction * velConstraintPoint.NormalImpulse;
var newImpulse = Math.Clamp(velConstraintPoint.TangentImpulse + lambda, -maxFriction, maxFriction);
lambda = newImpulse - velConstraintPoint.TangentImpulse;
velConstraintPoint.TangentImpulse = newImpulse;
// Apply contact impulse
Vector2 P = tangent * lambda;
vA -= P * mA;
wA -= iA * Vector2.Cross(velConstraintPoint.RelativeVelocityA, P);
vB += P * mB;
wB += iB * Vector2.Cross(velConstraintPoint.RelativeVelocityB, P);
}
// Solve normal constraints
if (velocityConstraint.PointCount == 1)
{
VelocityConstraintPoint vcp = velocityConstraint.Points[0];
// Relative velocity at contact
Vector2 dv = vB + Vector2.Cross(wB, vcp.RelativeVelocityB) - vA - Vector2.Cross(wA, vcp.RelativeVelocityA);
// Compute normal impulse
float vn = Vector2.Dot(dv, normal);
float lambda = -vcp.NormalMass * (vn - vcp.VelocityBias);
// b2Clamp the accumulated impulse
float newImpulse = Math.Max(vcp.NormalImpulse + lambda, 0.0f);
lambda = newImpulse - vcp.NormalImpulse;
vcp.NormalImpulse = newImpulse;
// Apply contact impulse
Vector2 P = normal * lambda;
vA -= P * mA;
wA -= iA * Vector2.Cross(vcp.RelativeVelocityA, P);
vB += P * mB;
wB += iB * Vector2.Cross(vcp.RelativeVelocityB, P);
}
else
{
// Block solver developed in collaboration with Dirk Gregorius (back in 01/07 on Box2D_Lite).
// Build the mini LCP for this contact patch
//
// vn = A * x + b, vn >= 0, , vn >= 0, x >= 0 and vn_i * x_i = 0 with i = 1..2
//
// A = J * W * JT and J = ( -n, -r1 x n, n, r2 x n )
// b = vn0 - velocityBias
//
// The system is solved using the "Total enumeration method" (s. Murty). The complementary constraint vn_i * x_i
// implies that we must have in any solution either vn_i = 0 or x_i = 0. So for the 2D contact problem the cases
// vn1 = 0 and vn2 = 0, x1 = 0 and x2 = 0, x1 = 0 and vn2 = 0, x2 = 0 and vn1 = 0 need to be tested. The first valid
// solution that satisfies the problem is chosen.
//
// In order to account of the accumulated impulse 'a' (because of the iterative nature of the solver which only requires
// that the accumulated impulse is clamped and not the incremental impulse) we change the impulse variable (x_i).
//
// Substitute:
//
// x = a + d
//
// a := old total impulse
// x := new total impulse
// d := incremental impulse
//
// For the current iteration we extend the formula for the incremental impulse
// to compute the new total impulse:
//
// vn = A * d + b
// = A * (x - a) + b
// = A * x + b - A * a
// = A * x + b'
// b' = b - A * a;
VelocityConstraintPoint cp1 = velocityConstraint.Points[0];
VelocityConstraintPoint cp2 = velocityConstraint.Points[1];
Vector2 a = new Vector2(cp1.NormalImpulse, cp2.NormalImpulse);
DebugTools.Assert(a.X >= 0.0f && a.Y >= 0.0f);
// Relative velocity at contact
Vector2 dv1 = vB + Vector2.Cross(wB, cp1.RelativeVelocityB) - vA - Vector2.Cross(wA, cp1.RelativeVelocityA);
Vector2 dv2 = vB + Vector2.Cross(wB, cp2.RelativeVelocityB) - vA - Vector2.Cross(wA, cp2.RelativeVelocityA);
// Compute normal velocity
float vn1 = Vector2.Dot(dv1, normal);
float vn2 = Vector2.Dot(dv2, normal);
Vector2 b = new Vector2
{
X = vn1 - cp1.VelocityBias,
Y = vn2 - cp2.VelocityBias
};
// Compute b'
b -= Transform.Mul(velocityConstraint.K, a);
//const float k_errorTol = 1e-3f;
//B2_NOT_USED(k_errorTol);
for (; ;)
{
//
// Case 1: vn = 0
//
// 0 = A * x + b'
//
// Solve for x:
//
// x = - inv(A) * b'
//
Vector2 x = -Transform.Mul(velocityConstraint.NormalMass, b);
if (x.X >= 0.0f && x.Y >= 0.0f)
{
// Get the incremental impulse
Vector2 d = x - a;
// Apply incremental impulse
Vector2 P1 = normal * d.X;
Vector2 P2 = normal * d.Y;
vA -= (P1 + P2) * mA;
wA -= iA * (Vector2.Cross(cp1.RelativeVelocityA, P1) + Vector2.Cross(cp2.RelativeVelocityA, P2));
vB += (P1 + P2) * mB;
wB += iB * (Vector2.Cross(cp1.RelativeVelocityB, P1) + Vector2.Cross(cp2.RelativeVelocityB, P2));
// Accumulate
cp1.NormalImpulse = x.X;
cp2.NormalImpulse = x.Y;
break;
}
//
// Case 2: vn1 = 0 and x2 = 0
//
// 0 = a11 * x1 + a12 * 0 + b1'
// vn2 = a21 * x1 + a22 * 0 + b2'
//
x.X = -cp1.NormalMass * b.X;
x.Y = 0.0f;
vn1 = 0.0f;
vn2 = velocityConstraint.K[0].Y * x.X + b.Y;
if (x.X >= 0.0f && vn2 >= 0.0f)
{
// Get the incremental impulse
Vector2 d = x - a;
// Apply incremental impulse
Vector2 P1 = normal * d.X;
Vector2 P2 = normal * d.Y;
vA -= (P1 + P2) * mA;
wA -= iA * (Vector2.Cross(cp1.RelativeVelocityA, P1) + Vector2.Cross(cp2.RelativeVelocityA, P2));
vB += (P1 + P2) * mB;
wB += iB * (Vector2.Cross(cp1.RelativeVelocityB, P1) + Vector2.Cross(cp2.RelativeVelocityB, P2));
// Accumulate
cp1.NormalImpulse = x.X;
cp2.NormalImpulse = x.Y;
break;
}
//
// Case 3: vn2 = 0 and x1 = 0
//
// vn1 = a11 * 0 + a12 * x2 + b1'
// 0 = a21 * 0 + a22 * x2 + b2'
//
x.X = 0.0f;
x.Y = -cp2.NormalMass * b.Y;
vn1 = velocityConstraint.K[1].X * x.Y + b.X;
vn2 = 0.0f;
if (x.Y >= 0.0f && vn1 >= 0.0f)
{
// Resubstitute for the incremental impulse
Vector2 d = x - a;
// Apply incremental impulse
Vector2 P1 = normal * d.X;
Vector2 P2 = normal * d.Y;
vA -= (P1 + P2) * mA;
wA -= iA * (Vector2.Cross(cp1.RelativeVelocityA, P1) + Vector2.Cross(cp2.RelativeVelocityA, P2));
vB += (P1 + P2) * mB;
wB += iB * (Vector2.Cross(cp1.RelativeVelocityB, P1) + Vector2.Cross(cp2.RelativeVelocityB, P2));
// Accumulate
cp1.NormalImpulse = x.X;
cp2.NormalImpulse = x.Y;
break;
}
//
// Case 4: x1 = 0 and x2 = 0
//
// vn1 = b1
// vn2 = b2;
x.X = 0.0f;
x.Y = 0.0f;
vn1 = b.X;
vn2 = b.Y;
if (vn1 >= 0.0f && vn2 >= 0.0f)
{
// Resubstitute for the incremental impulse
Vector2 d = x - a;
// Apply incremental impulse
Vector2 P1 = normal * d.X;
Vector2 P2 = normal * d.Y;
vA -= (P1 + P2) * mA;
wA -= iA * (Vector2.Cross(cp1.RelativeVelocityA, P1) + Vector2.Cross(cp2.RelativeVelocityA, P2));
vB += (P1 + P2) * mB;
wB += iB * (Vector2.Cross(cp1.RelativeVelocityB, P1) + Vector2.Cross(cp2.RelativeVelocityB, P2));
// Accumulate
cp1.NormalImpulse = x.X;
cp2.NormalImpulse = x.Y;
break;
}
// No solution, give up. This is hit sometimes, but it doesn't seem to matter.
break;
}
}
_linearVelocities[indexA] = vA;
_angularVelocities[indexA] = wA;
_linearVelocities[indexB] = vB;
_angularVelocities[indexB] = wB;
}
}
public void InitializeVelocityConstraints()
{
Span <Vector2> points = stackalloc Vector2[2];
for (var i = 0; i < _contactCount; ++i)
{
var velocityConstraint = _velocityConstraints[i];
var positionConstraint = _positionConstraints[i];
var radiusA = positionConstraint.RadiusA;
var radiusB = positionConstraint.RadiusB;
var manifold = _contacts[velocityConstraint.ContactIndex].Manifold;
var indexA = velocityConstraint.IndexA;
var indexB = velocityConstraint.IndexB;
var invMassA = velocityConstraint.InvMassA;
var invMassB = velocityConstraint.InvMassB;
var invIA = velocityConstraint.InvIA;
var invIB = velocityConstraint.InvIB;
var localCenterA = positionConstraint.LocalCenterA;
var localCenterB = positionConstraint.LocalCenterB;
var centerA = _positions[indexA];
var angleA = _angles[indexA];
var linVelocityA = _linearVelocities[indexA];
var angVelocityA = _angularVelocities[indexA];
var centerB = _positions[indexB];
var angleB = _angles[indexB];
var linVelocityB = _linearVelocities[indexB];
var angVelocityB = _angularVelocities[indexB];
DebugTools.Assert(manifold.PointCount > 0);
Transform xfA = new Transform(angleA);
Transform xfB = new Transform(angleB);
xfA.Position = centerA - Transform.Mul(xfA.Quaternion2D, localCenterA);
xfB.Position = centerB - Transform.Mul(xfB.Quaternion2D, localCenterB);
Vector2 normal;
InitializeManifold(ref manifold, xfA, xfB, radiusA, radiusB, out normal, points);
velocityConstraint.Normal = normal;
int pointCount = velocityConstraint.PointCount;
for (int j = 0; j < pointCount; ++j)
{
VelocityConstraintPoint vcp = velocityConstraint.Points[j];
vcp.RelativeVelocityA = points[j] - centerA;
vcp.RelativeVelocityB = points[j] - centerB;
float rnA = Vector2.Cross(vcp.RelativeVelocityA, velocityConstraint.Normal);
float rnB = Vector2.Cross(vcp.RelativeVelocityB, velocityConstraint.Normal);
float kNormal = invMassA + invMassB + invIA * rnA * rnA + invIB * rnB * rnB;
vcp.NormalMass = kNormal > 0.0f ? 1.0f / kNormal : 0.0f;
Vector2 tangent = Vector2.Cross(velocityConstraint.Normal, 1.0f);
float rtA = Vector2.Cross(vcp.RelativeVelocityA, tangent);
float rtB = Vector2.Cross(vcp.RelativeVelocityB, tangent);
float kTangent = invMassA + invMassB + invIA * rtA * rtA + invIB * rtB * rtB;
vcp.TangentMass = kTangent > 0.0f ? 1.0f / kTangent : 0.0f;
// Setup a velocity bias for restitution.
vcp.VelocityBias = 0.0f;
float vRel = Vector2.Dot(velocityConstraint.Normal, linVelocityB + Vector2.Cross(angVelocityB, vcp.RelativeVelocityB) - linVelocityA - Vector2.Cross(angVelocityA, vcp.RelativeVelocityA));
if (vRel < -_velocityThreshold)
{
vcp.VelocityBias = -velocityConstraint.Restitution * vRel;
}
}
// If we have two points, then prepare the block solver.
if (velocityConstraint.PointCount == 2)
{
var vcp1 = velocityConstraint.Points[0];
var vcp2 = velocityConstraint.Points[1];
var rn1A = Vector2.Cross(vcp1.RelativeVelocityA, velocityConstraint.Normal);
var rn1B = Vector2.Cross(vcp1.RelativeVelocityB, velocityConstraint.Normal);
var rn2A = Vector2.Cross(vcp2.RelativeVelocityA, velocityConstraint.Normal);
var rn2B = Vector2.Cross(vcp2.RelativeVelocityB, velocityConstraint.Normal);
var k11 = invMassA + invMassB + invIA * rn1A * rn1A + invIB * rn1B * rn1B;
var k22 = invMassA + invMassB + invIA * rn2A * rn2A + invIB * rn2B * rn2B;
var k12 = invMassA + invMassB + invIA * rn1A * rn2A + invIB * rn1B * rn2B;
// Ensure a reasonable condition number.
const float k_maxConditionNumber = 1000.0f;
if (k11 * k11 < k_maxConditionNumber * (k11 * k22 - k12 * k12))
{
// K is safe to invert.
velocityConstraint.K[0] = new Vector2(k11, k12);
velocityConstraint.K[1] = new Vector2(k12, k22);
velocityConstraint.NormalMass = velocityConstraint.K.Inverse();
}
else
{
// The constraints are redundant, just use one.
// TODO_ERIN use deepest?
velocityConstraint.PointCount = 1;
}
}
}
}
