/// <summary>Initialize position dependent portions of the velocity constraints.</summary>
public void InitializeVelocityConstraints()
{
for (int i = 0; i < count; ++i)
{
ContactVelocityConstraint vc = VelocityConstraints[i];
ContactPositionConstraint pc = positionConstraints[i];
float radiusA = pc.RadiusA;
float radiusB = pc.RadiusB;
Manifold manifold = contacts[vc.ContactIndex].Manifold;
int indexA = vc.IndexA;
int indexB = vc.IndexB;
float mA = vc.InvMassA;
float mB = vc.InvMassB;
float iA = vc.InvIa;
float iB = vc.InvIb;
Vector2 localCenterA = pc.LocalCenterA;
Vector2 localCenterB = pc.LocalCenterB;
Vector2 cA = positions[indexA].C;
float aA = positions[indexA].A;
Vector2 vA = velocities[indexA].V;
float wA = velocities[indexA].W;
Vector2 cB = positions[indexB].C;
float aB = positions[indexB].A;
Vector2 vB = velocities[indexB].V;
float wB = velocities[indexB].W;
Debug.Assert(manifold.PointCount > 0);
Transform xfA = new Transform();
Transform xfB = new Transform();
xfA.Q.Set(aA);
xfB.Q.Set(aB);
xfA.P = cA - MathUtils.Mul(xfA.Q, localCenterA);
xfB.P = cB - MathUtils.Mul(xfB.Q, localCenterB);
WorldManifold.Initialize(ref manifold, ref xfA, radiusA, ref xfB, radiusB, out Vector2 normal,
out FixedArray2 <Vector2> points, out _);
vc.Normal = normal;
int pointCount = vc.PointCount;
for (int j = 0; j < pointCount; ++j)
{
VelocityConstraintPoint vcp = vc.Points[j];
vcp.RA = points[j] - cA;
vcp.RB = points[j] - cB;
float rnA = MathUtils.Cross(vcp.RA, vc.Normal);
float rnB = MathUtils.Cross(vcp.RB, vc.Normal);
float kNormal = mA + mB + iA * rnA * rnA + iB * rnB * rnB;
vcp.NormalMass = kNormal > 0.0f ? 1.0f / kNormal : 0.0f;
Vector2 tangent = MathUtils.Cross(vc.Normal, 1.0f);
float rtA = MathUtils.Cross(vcp.RA, tangent);
float rtB = MathUtils.Cross(vcp.RB, tangent);
float kTangent = mA + mB + iA * rtA * rtA + iB * rtB * rtB;
vcp.TangentMass = kTangent > 0.0f ? 1.0f / kTangent : 0.0f;
// Setup a velocity bias for restitution.
vcp.VelocityBias = 0.0f;
float vRel = MathUtils.Dot(vc.Normal,
vB + MathUtils.Cross(wB, vcp.RB) - vA - MathUtils.Cross(wA, vcp.RA));
if (vRel < -vc.Threshold)
{
vcp.VelocityBias = -vc.Restitution * vRel;
}
}
// If we have two points, then prepare the block solver.
if (vc.PointCount == 2 && Settings.BlockSolve)
{
VelocityConstraintPoint vcp1 = vc.Points[0];
VelocityConstraintPoint vcp2 = vc.Points[1];
float rn1A = MathUtils.Cross(vcp1.RA, vc.Normal);
float rn1B = MathUtils.Cross(vcp1.RB, vc.Normal);
float rn2A = MathUtils.Cross(vcp2.RA, vc.Normal);
float rn2B = MathUtils.Cross(vcp2.RB, vc.Normal);
float k11 = mA + mB + iA * rn1A * rn1A + iB * rn1B * rn1B;
float k22 = mA + mB + iA * rn2A * rn2A + iB * rn2B * rn2B;
float k12 = mA + mB + iA * rn1A * rn2A + iB * rn1B * rn2B;
// Ensure a reasonable condition number.
const float kMaxConditionNumber = 1000.0f;
if (k11 * k11 < kMaxConditionNumber * (k11 * k22 - k12 * k12))
{
// K is safe to invert.
vc.K.Ex = new Vector2(k11, k12);
vc.K.Ey = new Vector2(k12, k22);
vc.NormalMass = vc.K.Inverse;
}
else
{
// The constraints are redundant, just use one.
// TODO_ERIN use deepest?
vc.PointCount = 1;
}
}
}
}
/// <summary>
/// Solves the velocity constraints
/// </summary>
public void SolveVelocityConstraints()
{
for (int i = 0; i < count; ++i)
{
ContactVelocityConstraint vc = VelocityConstraints[i];
int indexA = vc.IndexA;
int indexB = vc.IndexB;
float mA = vc.InvMassA;
float iA = vc.InvIa;
float mB = vc.InvMassB;
float iB = vc.InvIb;
int pointCount = vc.PointCount;
Vector2 vA = velocities[indexA].V;
float wA = velocities[indexA].W;
Vector2 vB = velocities[indexB].V;
float wB = velocities[indexB].W;
Vector2 normal = vc.Normal;
Vector2 tangent = MathUtils.Cross(normal, 1.0f);
float friction = vc.Friction;
Debug.Assert(pointCount == 1 || pointCount == 2);
// Solve tangent constraints first because non-penetration is more important
// than friction.
for (int j = 0; j < pointCount; ++j)
{
VelocityConstraintPoint vcp = vc.Points[j];
// Relative velocity at contact
Vector2 dv = vB + MathUtils.Cross(wB, vcp.RB) - vA - MathUtils.Cross(wA, vcp.RA);
// Compute tangent force
float vt = Vector2.Dot(dv, tangent) - vc.TangentSpeed;
float lambda = vcp.TangentMass * -vt;
// b2Clamp the accumulated force
float maxFriction = friction * vcp.NormalImpulse;
float newImpulse = MathUtils.Clamp(vcp.TangentImpulse + lambda, -maxFriction, maxFriction);
lambda = newImpulse - vcp.TangentImpulse;
vcp.TangentImpulse = newImpulse;
// Apply contact impulse
Vector2 p = lambda * tangent;
vA -= mA * p;
wA -= iA * MathUtils.Cross(vcp.RA, p);
vB += mB * p;
wB += iB * MathUtils.Cross(vcp.RB, p);
}
// Solve normal constraints
if (pointCount == 1 || !Settings.BlockSolve)
{
for (int j = 0; j < pointCount; ++j)
{
VelocityConstraintPoint vcp = vc.Points[j];
// Relative velocity at contact
Vector2 dv = vB + MathUtils.Cross(wB, vcp.RB) - vA - MathUtils.Cross(wA, vcp.RA);
// 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 = lambda * normal;
vA -= mA * p;
wA -= iA * MathUtils.Cross(vcp.RA, p);
vB += mB * p;
wB += iB * MathUtils.Cross(vcp.RB, 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, 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 = vc.Points[0];
VelocityConstraintPoint cp2 = vc.Points[1];
Vector2 a = new Vector2(cp1.NormalImpulse, cp2.NormalImpulse);
Debug.Assert(a.X >= 0.0f && a.Y >= 0.0f);
// Relative velocity at contact
Vector2 dv1 = vB + MathUtils.Cross(wB, cp1.RB) - vA - MathUtils.Cross(wA, cp1.RA);
Vector2 dv2 = vB + MathUtils.Cross(wB, cp2.RB) - vA - MathUtils.Cross(wA, cp2.RA);
// Compute normal velocity
float vn1 = Vector2.Dot(dv1, normal);
float vn2 = Vector2.Dot(dv2, normal);
Vector2 b = Vector2.Zero;
b.X = vn1 - cp1.VelocityBias;
b.Y = vn2 - cp2.VelocityBias;
// Compute b'
b -= MathUtils.Mul(ref vc.K, a);
for (;;)
{
//
// Case 1: vn = 0
//
// 0 = A * x + b'
//
// Solve for x:
//
// x = - inv(A) * b'
//
Vector2 x = -MathUtils.Mul(ref vc.NormalMass, b);
if (x.X >= 0.0f && x.Y >= 0.0f)
{
// Get the incremental impulse
Vector2 d = x - a;
// Apply incremental impulse
Vector2 p1 = d.X * normal;
Vector2 p2 = d.Y * normal;
vA -= mA * (p1 + p2);
wA -= iA * (MathUtils.Cross(cp1.RA, p1) + MathUtils.Cross(cp2.RA, p2));
vB += mB * (p1 + p2);
wB += iB * (MathUtils.Cross(cp1.RB, p1) + MathUtils.Cross(cp2.RB, p2));
// Accumulate
cp1.NormalImpulse = x.X;
cp2.NormalImpulse = x.Y;
#if B2_DEBUG_SOLVER
// Postconditions
dv1 = vB + MathUtils.Cross(wB, cp1.rB) - vA - MathUtils.Cross(wA, cp1.rA);
dv2 = vB + MathUtils.Cross(wB, cp2.rB) - vA - MathUtils.Cross(wA, cp2.rA);
// Compute normal velocity
vn1 = Vector2.Dot(dv1, normal);
vn2 = Vector2.Dot(dv2, normal);
Debug.Assert(Math.Abs(vn1 - cp1.VelocityBias) < k_errorTol);
Debug.Assert(Math.Abs(vn2 - cp2.VelocityBias) < k_errorTol);
#endif
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 = vc.K.Ex.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 = d.X * normal;
Vector2 p2 = d.Y * normal;
vA -= mA * (p1 + p2);
wA -= iA * (MathUtils.Cross(cp1.RA, p1) + MathUtils.Cross(cp2.RA, p2));
vB += mB * (p1 + p2);
wB += iB * (MathUtils.Cross(cp1.RB, p1) + MathUtils.Cross(cp2.RB, p2));
// Accumulate
cp1.NormalImpulse = x.X;
cp2.NormalImpulse = x.Y;
#if B2_DEBUG_SOLVER
// Postconditions
dv1 = vB + MathUtils.Cross(wB, cp1.rB) - vA - MathUtils.Cross(wA, cp1.rA);
// Compute normal velocity
vn1 = Vector2.Dot(dv1, normal);
Debug.Assert(Math.Abs(vn1 - cp1.VelocityBias) < k_errorTol);
#endif
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 = vc.K.Ey.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 = d.X * normal;
Vector2 p2 = d.Y * normal;
vA -= mA * (p1 + p2);
wA -= iA * (MathUtils.Cross(cp1.RA, p1) + MathUtils.Cross(cp2.RA, p2));
vB += mB * (p1 + p2);
wB += iB * (MathUtils.Cross(cp1.RB, p1) + MathUtils.Cross(cp2.RB, p2));
// Accumulate
cp1.NormalImpulse = x.X;
cp2.NormalImpulse = x.Y;
#if B2_DEBUG_SOLVER
// Postconditions
dv2 = vB + MathUtils.Cross(wB, cp2.rB) - vA - MathUtils.Cross(wA, cp2.rA);
// Compute normal velocity
vn2 = Vector2.Dot(dv2, normal);
Debug.Assert(Math.Abs(vn2 - cp2.VelocityBias) < k_errorTol);
#endif
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 = d.X * normal;
Vector2 p2 = d.Y * normal;
vA -= mA * (p1 + p2);
wA -= iA * (MathUtils.Cross(cp1.RA, p1) + MathUtils.Cross(cp2.RA, p2));
vB += mB * (p1 + p2);
wB += iB * (MathUtils.Cross(cp1.RB, p1) + MathUtils.Cross(cp2.RB, p2));
// Accumulate
cp1.NormalImpulse = x.X;
cp2.NormalImpulse = x.Y;
}
// No solution, give up. This is hit sometimes, but it doesn't seem to matter.
break;
}
}
velocities[indexA].V = vA;
velocities[indexA].W = wA;
velocities[indexB].V = vB;
velocities[indexB].W = wB;
}
}
/// <summary>
/// Resets the step
/// </summary>
/// <param name="step">The step</param>
/// <param name="count">The count</param>
/// <param name="contacts">The contacts</param>
/// <param name="positions">The positions</param>
/// <param name="velocities">The velocities</param>
public void Reset(TimeStep step, int count, Contact[] contacts, Position[] positions, Velocity[] velocities)
{
this.step = step;
this.count = count;
this.positions = positions;
this.velocities = velocities;
this.contacts = contacts;
// grow the array
if (VelocityConstraints == null || VelocityConstraints.Length < count)
{
VelocityConstraints = new ContactVelocityConstraint[count * 2];
positionConstraints = new ContactPositionConstraint[count * 2];
for (int i = 0; i < VelocityConstraints.Length; i++)
{
VelocityConstraints[i] = new ContactVelocityConstraint();
}
for (int i = 0; i < positionConstraints.Length; i++)
{
positionConstraints[i] = new ContactPositionConstraint();
}
}
// Initialize position independent portions of the constraints.
for (int i = 0; i < this.count; ++i)
{
Contact contact = contacts[i];
Fixture fixtureA = contact.FixtureA;
Fixture fixtureB = contact.FixtureB;
Shape shapeA = fixtureA.Shape;
Shape shapeB = fixtureB.Shape;
float radiusA = shapeA.RadiusPrivate;
float radiusB = shapeB.RadiusPrivate;
Body bodyA = fixtureA.Body;
Body bodyB = fixtureB.Body;
Manifold manifold = contact.Manifold;
int pointCount = manifold.PointCount;
Debug.Assert(pointCount > 0);
ContactVelocityConstraint vc = VelocityConstraints[i];
vc.Friction = contact.Friction;
vc.Restitution = contact.Restitution;
vc.Threshold = contact.RestitutionThreshold;
vc.TangentSpeed = contact.TangentSpeed;
vc.IndexA = bodyA.IslandIndex;
vc.IndexB = bodyB.IslandIndex;
vc.InvMassA = bodyA.InvMass;
vc.InvMassB = bodyB.InvMass;
vc.InvIa = bodyA.InvI;
vc.InvIb = bodyB.InvI;
vc.ContactIndex = i;
vc.PointCount = pointCount;
vc.K.SetZero();
vc.NormalMass.SetZero();
ContactPositionConstraint pc = positionConstraints[i];
pc.IndexA = bodyA.IslandIndex;
pc.IndexB = bodyB.IslandIndex;
pc.InvMassA = bodyA.InvMass;
pc.InvMassB = bodyB.InvMass;
pc.LocalCenterA = bodyA.Sweep.LocalCenter;
pc.LocalCenterB = bodyB.Sweep.LocalCenter;
pc.InvIa = bodyA.InvI;
pc.InvIb = bodyB.InvI;
pc.LocalNormal = manifold.LocalNormal;
pc.LocalPoint = manifold.LocalPoint;
pc.PointCount = pointCount;
pc.RadiusA = radiusA;
pc.RadiusB = radiusB;
pc.Type = manifold.Type;
for (int j = 0; j < pointCount; ++j)
{
ManifoldPoint cp = manifold.Points[j];
VelocityConstraintPoint vcp = vc.Points[j];
if (step.WarmStarting)
{
vcp.NormalImpulse = this.step.DeltaTimeRatio * cp.NormalImpulse;
vcp.TangentImpulse = this.step.DeltaTimeRatio * cp.TangentImpulse;
}
else
{
vcp.NormalImpulse = 0.0f;
vcp.TangentImpulse = 0.0f;
}
vcp.RA = Vector2.Zero;
vcp.RB = Vector2.Zero;
vcp.NormalMass = 0.0f;
vcp.TangentMass = 0.0f;
vcp.VelocityBias = 0.0f;
pc.LocalPoints[j] = cp.LocalPoint;
}
}
}