b2ContactSolver.cpp

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/*
* Copyright (c) 2006-2011 Erin Catto http://www.box2d.org
*
* This software is provided 'as-is', without any express or implied
* warranty.  In no event will the authors be held liable for any damages
* arising from the use of this software.
* Permission is granted to anyone to use this software for any purpose,
* including commercial applications, and to alter it and redistribute it
* freely, subject to the following restrictions:
* 1. The origin of this software must not be misrepresented; you must not
* claim that you wrote the original software. If you use this software
* in a product, an acknowledgment in the product documentation would be
* appreciated but is not required.
* 2. Altered source versions must be plainly marked as such, and must not be
* misrepresented as being the original software.
* 3. This notice may not be removed or altered from any source distribution.
*/

#include <Box2D/Dynamics/Contacts/b2ContactSolver.h>

#include <Box2D/Dynamics/Contacts/b2Contact.h>
#include <Box2D/Dynamics/b2Body.h>
#include <Box2D/Dynamics/b2Fixture.h>
#include <Box2D/Dynamics/b2World.h>
#include <Box2D/Common/b2StackAllocator.h>

#define B2_DEBUG_SOLVER 0

bool g_blockSolve = true;

struct b2ContactPositionConstraint
{
        b2Vec2 localPoints[b2_maxManifoldPoints];
        b2Vec2 localNormal;
        b2Vec2 localPoint;
        int32 indexA;
        int32 indexB;
        float32 invMassA, invMassB;
        b2Vec2 localCenterA, localCenterB;
        float32 invIA, invIB;
        b2Manifold::Type type;
        float32 radiusA, radiusB;
        int32 pointCount;
};

b2ContactSolver::b2ContactSolver(b2ContactSolverDef* def)
{
        m_step = def->step;
        m_allocator = def->allocator;
        m_count = def->count;
        m_positionConstraints = (b2ContactPositionConstraint*)m_allocator->Allocate(m_count * sizeof(b2ContactPositionConstraint));
        m_velocityConstraints = (b2ContactVelocityConstraint*)m_allocator->Allocate(m_count * sizeof(b2ContactVelocityConstraint));
        m_positions = def->positions;
        m_velocities = def->velocities;
        m_contacts = def->contacts;

        // Initialize position independent portions of the constraints.
        for (int32 i = 0; i < m_count; ++i)
        {
                b2Contact* contact = m_contacts[i];

                b2Fixture* fixtureA = contact->m_fixtureA;
                b2Fixture* fixtureB = contact->m_fixtureB;
                b2Shape* shapeA = fixtureA->GetShape();
                b2Shape* shapeB = fixtureB->GetShape();
                float32 radiusA = shapeA->m_radius;
                float32 radiusB = shapeB->m_radius;
                b2Body* bodyA = fixtureA->GetBody();
                b2Body* bodyB = fixtureB->GetBody();
                b2Manifold* manifold = contact->GetManifold();

                int32 pointCount = manifold->pointCount;
                b2Assert(pointCount > 0);

                b2ContactVelocityConstraint* vc = m_velocityConstraints + i;
                vc->friction = contact->m_friction;
                vc->restitution = contact->m_restitution;
                vc->tangentSpeed = contact->m_tangentSpeed;
                vc->indexA = bodyA->m_islandIndex;
                vc->indexB = bodyB->m_islandIndex;
                vc->invMassA = bodyA->m_invMass;
                vc->invMassB = bodyB->m_invMass;
                vc->invIA = bodyA->m_invI;
                vc->invIB = bodyB->m_invI;
                vc->contactIndex = i;
                vc->pointCount = pointCount;
                vc->K.SetZero();
                vc->normalMass.SetZero();

                b2ContactPositionConstraint* pc = m_positionConstraints + i;
                pc->indexA = bodyA->m_islandIndex;
                pc->indexB = bodyB->m_islandIndex;
                pc->invMassA = bodyA->m_invMass;
                pc->invMassB = bodyB->m_invMass;
                pc->localCenterA = bodyA->m_sweep.localCenter;
                pc->localCenterB = bodyB->m_sweep.localCenter;
                pc->invIA = bodyA->m_invI;
                pc->invIB = bodyB->m_invI;
                pc->localNormal = manifold->localNormal;
                pc->localPoint = manifold->localPoint;
                pc->pointCount = pointCount;
                pc->radiusA = radiusA;
                pc->radiusB = radiusB;
                pc->type = manifold->type;

                for (int32 j = 0; j < pointCount; ++j)
                {
                        b2ManifoldPoint* cp = manifold->points + j;
                        b2VelocityConstraintPoint* vcp = vc->points + j;
        
                        if (m_step.warmStarting)
                        {
                                vcp->normalImpulse = m_step.dtRatio * cp->normalImpulse;
                                vcp->tangentImpulse = m_step.dtRatio * cp->tangentImpulse;
                        }
                        else
                        {
                                vcp->normalImpulse = 0.0f;
                                vcp->tangentImpulse = 0.0f;
                        }

                        vcp->rA.SetZero();
                        vcp->rB.SetZero();
                        vcp->normalMass = 0.0f;
                        vcp->tangentMass = 0.0f;
                        vcp->velocityBias = 0.0f;

                        pc->localPoints[j] = cp->localPoint;
                }
        }
}

b2ContactSolver::~b2ContactSolver()
{
        m_allocator->Free(m_velocityConstraints);
        m_allocator->Free(m_positionConstraints);
}

// Initialize position dependent portions of the velocity constraints.
void b2ContactSolver::InitializeVelocityConstraints()
{
        for (int32 i = 0; i < m_count; ++i)
        {
                b2ContactVelocityConstraint* vc = m_velocityConstraints + i;
                b2ContactPositionConstraint* pc = m_positionConstraints + i;

                float32 radiusA = pc->radiusA;
                float32 radiusB = pc->radiusB;
                b2Manifold* manifold = m_contacts[vc->contactIndex]->GetManifold();

                int32 indexA = vc->indexA;
                int32 indexB = vc->indexB;

                float32 mA = vc->invMassA;
                float32 mB = vc->invMassB;
                float32 iA = vc->invIA;
                float32 iB = vc->invIB;
                b2Vec2 localCenterA = pc->localCenterA;
                b2Vec2 localCenterB = pc->localCenterB;

                b2Vec2 cA = m_positions[indexA].c;
                float32 aA = m_positions[indexA].a;
                b2Vec2 vA = m_velocities[indexA].v;
                float32 wA = m_velocities[indexA].w;

                b2Vec2 cB = m_positions[indexB].c;
                float32 aB = m_positions[indexB].a;
                b2Vec2 vB = m_velocities[indexB].v;
                float32 wB = m_velocities[indexB].w;

                b2Assert(manifold->pointCount > 0);

                b2Transform xfA, xfB;
                xfA.q.Set(aA);
                xfB.q.Set(aB);
                xfA.p = cA - b2Mul(xfA.q, localCenterA);
                xfB.p = cB - b2Mul(xfB.q, localCenterB);

                b2WorldManifold worldManifold;
                worldManifold.Initialize(manifold, xfA, radiusA, xfB, radiusB);

                vc->normal = worldManifold.normal;

                int32 pointCount = vc->pointCount;
                for (int32 j = 0; j < pointCount; ++j)
                {
                        b2VelocityConstraintPoint* vcp = vc->points + j;

                        vcp->rA = worldManifold.points[j] - cA;
                        vcp->rB = worldManifold.points[j] - cB;

                        float32 rnA = b2Cross(vcp->rA, vc->normal);
                        float32 rnB = b2Cross(vcp->rB, vc->normal);

                        float32 kNormal = mA + mB + iA * rnA * rnA + iB * rnB * rnB;

                        vcp->normalMass = kNormal > 0.0f ? 1.0f / kNormal : 0.0f;

                        b2Vec2 tangent = b2Cross(vc->normal, 1.0f);

                        float32 rtA = b2Cross(vcp->rA, tangent);
                        float32 rtB = b2Cross(vcp->rB, tangent);

                        float32 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;
                        float32 vRel = b2Dot(vc->normal, vB + b2Cross(wB, vcp->rB) - vA - b2Cross(wA, vcp->rA));
                        if (vRel < -b2_velocityThreshold)
                        {
                                vcp->velocityBias = -vc->restitution * vRel;
                        }
                }

                // If we have two points, then prepare the block solver.
                if (vc->pointCount == 2 && g_blockSolve)
                {
                        b2VelocityConstraintPoint* vcp1 = vc->points + 0;
                        b2VelocityConstraintPoint* vcp2 = vc->points + 1;

                        float32 rn1A = b2Cross(vcp1->rA, vc->normal);
                        float32 rn1B = b2Cross(vcp1->rB, vc->normal);
                        float32 rn2A = b2Cross(vcp2->rA, vc->normal);
                        float32 rn2B = b2Cross(vcp2->rB, vc->normal);

                        float32 k11 = mA + mB + iA * rn1A * rn1A + iB * rn1B * rn1B;
                        float32 k22 = mA + mB + iA * rn2A * rn2A + iB * rn2B * rn2B;
                        float32 k12 = mA + mB + iA * rn1A * rn2A + iB * rn1B * rn2B;

                        // Ensure a reasonable condition number.
                        const float32 k_maxConditionNumber = 1000.0f;
                        if (k11 * k11 < k_maxConditionNumber * (k11 * k22 - k12 * k12))
                        {
                                // K is safe to invert.
                                vc->K.ex.Set(k11, k12);
                                vc->K.ey.Set(k12, k22);
                                vc->normalMass = vc->K.GetInverse();
                        }
                        else
                        {
                                // The constraints are redundant, just use one.
                                // TODO_ERIN use deepest?
                                vc->pointCount = 1;
                        }
                }
        }
}

void b2ContactSolver::WarmStart()
{
        // Warm start.
        for (int32 i = 0; i < m_count; ++i)
        {
                b2ContactVelocityConstraint* vc = m_velocityConstraints + i;

                int32 indexA = vc->indexA;
                int32 indexB = vc->indexB;
                float32 mA = vc->invMassA;
                float32 iA = vc->invIA;
                float32 mB = vc->invMassB;
                float32 iB = vc->invIB;
                int32 pointCount = vc->pointCount;

                b2Vec2 vA = m_velocities[indexA].v;
                float32 wA = m_velocities[indexA].w;
                b2Vec2 vB = m_velocities[indexB].v;
                float32 wB = m_velocities[indexB].w;

                b2Vec2 normal = vc->normal;
                b2Vec2 tangent = b2Cross(normal, 1.0f);

                for (int32 j = 0; j < pointCount; ++j)
                {
                        b2VelocityConstraintPoint* vcp = vc->points + j;
                        b2Vec2 P = vcp->normalImpulse * normal + vcp->tangentImpulse * tangent;
                        wA -= iA * b2Cross(vcp->rA, P);
                        vA -= mA * P;
                        wB += iB * b2Cross(vcp->rB, P);
                        vB += mB * P;
                }

                m_velocities[indexA].v = vA;
                m_velocities[indexA].w = wA;
                m_velocities[indexB].v = vB;
                m_velocities[indexB].w = wB;
        }
}

void b2ContactSolver::SolveVelocityConstraints()
{
        for (int32 i = 0; i < m_count; ++i)
        {
                b2ContactVelocityConstraint* vc = m_velocityConstraints + i;

                int32 indexA = vc->indexA;
                int32 indexB = vc->indexB;
                float32 mA = vc->invMassA;
                float32 iA = vc->invIA;
                float32 mB = vc->invMassB;
                float32 iB = vc->invIB;
                int32 pointCount = vc->pointCount;

                b2Vec2 vA = m_velocities[indexA].v;
                float32 wA = m_velocities[indexA].w;
                b2Vec2 vB = m_velocities[indexB].v;
                float32 wB = m_velocities[indexB].w;

                b2Vec2 normal = vc->normal;
                b2Vec2 tangent = b2Cross(normal, 1.0f);
                float32 friction = vc->friction;

                b2Assert(pointCount == 1 || pointCount == 2);

                // Solve tangent constraints first because non-penetration is more important
                // than friction.
                for (int32 j = 0; j < pointCount; ++j)
                {
                        b2VelocityConstraintPoint* vcp = vc->points + j;

                        // Relative velocity at contact
                        b2Vec2 dv = vB + b2Cross(wB, vcp->rB) - vA - b2Cross(wA, vcp->rA);

                        // Compute tangent force
                        float32 vt = b2Dot(dv, tangent) - vc->tangentSpeed;
                        float32 lambda = vcp->tangentMass * (-vt);

                        // b2Clamp the accumulated force
                        float32 maxFriction = friction * vcp->normalImpulse;
                        float32 newImpulse = b2Clamp(vcp->tangentImpulse + lambda, -maxFriction, maxFriction);
                        lambda = newImpulse - vcp->tangentImpulse;
                        vcp->tangentImpulse = newImpulse;

                        // Apply contact impulse
                        b2Vec2 P = lambda * tangent;

                        vA -= mA * P;
                        wA -= iA * b2Cross(vcp->rA, P);

                        vB += mB * P;
                        wB += iB * b2Cross(vcp->rB, P);
                }

                // Solve normal constraints
                if (pointCount == 1 || g_blockSolve == false)
                {
                        for (int32 i = 0; i < pointCount; ++i)
                        {
                                b2VelocityConstraintPoint* vcp = vc->points + i;

                                // Relative velocity at contact
                                b2Vec2 dv = vB + b2Cross(wB, vcp->rB) - vA - b2Cross(wA, vcp->rA);

                                // Compute normal impulse
                                float32 vn = b2Dot(dv, normal);
                                float32 lambda = -vcp->normalMass * (vn - vcp->velocityBias);

                                // b2Clamp the accumulated impulse
                                float32 newImpulse = b2Max(vcp->normalImpulse + lambda, 0.0f);
                                lambda = newImpulse - vcp->normalImpulse;
                                vcp->normalImpulse = newImpulse;

                                // Apply contact impulse
                                b2Vec2 P = lambda * normal;
                                vA -= mA * P;
                                wA -= iA * b2Cross(vcp->rA, P);

                                vB += mB * P;
                                wB += iB * b2Cross(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, , 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;

                        b2VelocityConstraintPoint* cp1 = vc->points + 0;
                        b2VelocityConstraintPoint* cp2 = vc->points + 1;

                        b2Vec2 a(cp1->normalImpulse, cp2->normalImpulse);
                        b2Assert(a.x >= 0.0f && a.y >= 0.0f);

                        // Relative velocity at contact
                        b2Vec2 dv1 = vB + b2Cross(wB, cp1->rB) - vA - b2Cross(wA, cp1->rA);
                        b2Vec2 dv2 = vB + b2Cross(wB, cp2->rB) - vA - b2Cross(wA, cp2->rA);

                        // Compute normal velocity
                        float32 vn1 = b2Dot(dv1, normal);
                        float32 vn2 = b2Dot(dv2, normal);

                        b2Vec2 b;
                        b.x = vn1 - cp1->velocityBias;
                        b.y = vn2 - cp2->velocityBias;

                        // Compute b'
                        b -= b2Mul(vc->K, a);

                        const float32 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'
                                //
                                b2Vec2 x = - b2Mul(vc->normalMass, b);

                                if (x.x >= 0.0f && x.y >= 0.0f)
                                {
                                        // Get the incremental impulse
                                        b2Vec2 d = x - a;

                                        // Apply incremental impulse
                                        b2Vec2 P1 = d.x * normal;
                                        b2Vec2 P2 = d.y * normal;
                                        vA -= mA * (P1 + P2);
                                        wA -= iA * (b2Cross(cp1->rA, P1) + b2Cross(cp2->rA, P2));

                                        vB += mB * (P1 + P2);
                                        wB += iB * (b2Cross(cp1->rB, P1) + b2Cross(cp2->rB, P2));

                                        // Accumulate
                                        cp1->normalImpulse = x.x;
                                        cp2->normalImpulse = x.y;

#if B2_DEBUG_SOLVER == 1
                                        // Postconditions
                                        dv1 = vB + b2Cross(wB, cp1->rB) - vA - b2Cross(wA, cp1->rA);
                                        dv2 = vB + b2Cross(wB, cp2->rB) - vA - b2Cross(wA, cp2->rA);

                                        // Compute normal velocity
                                        vn1 = b2Dot(dv1, normal);
                                        vn2 = b2Dot(dv2, normal);

                                        b2Assert(b2Abs(vn1 - cp1->velocityBias) < k_errorTol);
                                        b2Assert(b2Abs(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
                                        b2Vec2 d = x - a;

                                        // Apply incremental impulse
                                        b2Vec2 P1 = d.x * normal;
                                        b2Vec2 P2 = d.y * normal;
                                        vA -= mA * (P1 + P2);
                                        wA -= iA * (b2Cross(cp1->rA, P1) + b2Cross(cp2->rA, P2));

                                        vB += mB * (P1 + P2);
                                        wB += iB * (b2Cross(cp1->rB, P1) + b2Cross(cp2->rB, P2));

                                        // Accumulate
                                        cp1->normalImpulse = x.x;
                                        cp2->normalImpulse = x.y;

#if B2_DEBUG_SOLVER == 1
                                        // Postconditions
                                        dv1 = vB + b2Cross(wB, cp1->rB) - vA - b2Cross(wA, cp1->rA);

                                        // Compute normal velocity
                                        vn1 = b2Dot(dv1, normal);

                                        b2Assert(b2Abs(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
                                        b2Vec2 d = x - a;

                                        // Apply incremental impulse
                                        b2Vec2 P1 = d.x * normal;
                                        b2Vec2 P2 = d.y * normal;
                                        vA -= mA * (P1 + P2);
                                        wA -= iA * (b2Cross(cp1->rA, P1) + b2Cross(cp2->rA, P2));

                                        vB += mB * (P1 + P2);
                                        wB += iB * (b2Cross(cp1->rB, P1) + b2Cross(cp2->rB, P2));

                                        // Accumulate
                                        cp1->normalImpulse = x.x;
                                        cp2->normalImpulse = x.y;

#if B2_DEBUG_SOLVER == 1
                                        // Postconditions
                                        dv2 = vB + b2Cross(wB, cp2->rB) - vA - b2Cross(wA, cp2->rA);

                                        // Compute normal velocity
                                        vn2 = b2Dot(dv2, normal);

                                        b2Assert(b2Abs(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
                                        b2Vec2 d = x - a;

                                        // Apply incremental impulse
                                        b2Vec2 P1 = d.x * normal;
                                        b2Vec2 P2 = d.y * normal;
                                        vA -= mA * (P1 + P2);
                                        wA -= iA * (b2Cross(cp1->rA, P1) + b2Cross(cp2->rA, P2));

                                        vB += mB * (P1 + P2);
                                        wB += iB * (b2Cross(cp1->rB, P1) + b2Cross(cp2->rB, 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;
                        }
                }

                m_velocities[indexA].v = vA;
                m_velocities[indexA].w = wA;
                m_velocities[indexB].v = vB;
                m_velocities[indexB].w = wB;
        }
}

void b2ContactSolver::StoreImpulses()
{
        for (int32 i = 0; i < m_count; ++i)
        {
                b2ContactVelocityConstraint* vc = m_velocityConstraints + i;
                b2Manifold* manifold = m_contacts[vc->contactIndex]->GetManifold();

                for (int32 j = 0; j < vc->pointCount; ++j)
                {
                        manifold->points[j].normalImpulse = vc->points[j].normalImpulse;
                        manifold->points[j].tangentImpulse = vc->points[j].tangentImpulse;
                }
        }
}

struct b2PositionSolverManifold
{
        void Initialize(b2ContactPositionConstraint* pc, const b2Transform& xfA, const b2Transform& xfB, int32 index)
        {
                b2Assert(pc->pointCount > 0);

                switch (pc->type)
                {
                case b2Manifold::e_circles:
                        {
                                b2Vec2 pointA = b2Mul(xfA, pc->localPoint);
                                b2Vec2 pointB = b2Mul(xfB, pc->localPoints[0]);
                                normal = pointB - pointA;
                                normal.Normalize();
                                point = 0.5f * (pointA + pointB);
                                separation = b2Dot(pointB - pointA, normal) - pc->radiusA - pc->radiusB;
                        }
                        break;

                case b2Manifold::e_faceA:
                        {
                                normal = b2Mul(xfA.q, pc->localNormal);
                                b2Vec2 planePoint = b2Mul(xfA, pc->localPoint);

                                b2Vec2 clipPoint = b2Mul(xfB, pc->localPoints[index]);
                                separation = b2Dot(clipPoint - planePoint, normal) - pc->radiusA - pc->radiusB;
                                point = clipPoint;
                        }
                        break;

                case b2Manifold::e_faceB:
                        {
                                normal = b2Mul(xfB.q, pc->localNormal);
                                b2Vec2 planePoint = b2Mul(xfB, pc->localPoint);

                                b2Vec2 clipPoint = b2Mul(xfA, pc->localPoints[index]);
                                separation = b2Dot(clipPoint - planePoint, normal) - pc->radiusA - pc->radiusB;
                                point = clipPoint;

                                // Ensure normal points from A to B
                                normal = -normal;
                        }
                        break;
                }
        }

        b2Vec2 normal;
        b2Vec2 point;
        float32 separation;
};

// Sequential solver.
bool b2ContactSolver::SolvePositionConstraints()
{
        float32 minSeparation = 0.0f;

        for (int32 i = 0; i < m_count; ++i)
        {
                b2ContactPositionConstraint* pc = m_positionConstraints + i;

                int32 indexA = pc->indexA;
                int32 indexB = pc->indexB;
                b2Vec2 localCenterA = pc->localCenterA;
                float32 mA = pc->invMassA;
                float32 iA = pc->invIA;
                b2Vec2 localCenterB = pc->localCenterB;
                float32 mB = pc->invMassB;
                float32 iB = pc->invIB;
                int32 pointCount = pc->pointCount;

                b2Vec2 cA = m_positions[indexA].c;
                float32 aA = m_positions[indexA].a;

                b2Vec2 cB = m_positions[indexB].c;
                float32 aB = m_positions[indexB].a;

                // Solve normal constraints
                for (int32 j = 0; j < pointCount; ++j)
                {
                        b2Transform xfA, xfB;
                        xfA.q.Set(aA);
                        xfB.q.Set(aB);
                        xfA.p = cA - b2Mul(xfA.q, localCenterA);
                        xfB.p = cB - b2Mul(xfB.q, localCenterB);

                        b2PositionSolverManifold psm;
                        psm.Initialize(pc, xfA, xfB, j);
                        b2Vec2 normal = psm.normal;

                        b2Vec2 point = psm.point;
                        float32 separation = psm.separation;

                        b2Vec2 rA = point - cA;
                        b2Vec2 rB = point - cB;

                        // Track max constraint error.
                        minSeparation = b2Min(minSeparation, separation);

                        // Prevent large corrections and allow slop.
                        float32 C = b2Clamp(b2_baumgarte * (separation + b2_linearSlop), -b2_maxLinearCorrection, 0.0f);

                        // Compute the effective mass.
                        float32 rnA = b2Cross(rA, normal);
                        float32 rnB = b2Cross(rB, normal);
                        float32 K = mA + mB + iA * rnA * rnA + iB * rnB * rnB;

                        // Compute normal impulse
                        float32 impulse = K > 0.0f ? - C / K : 0.0f;

                        b2Vec2 P = impulse * normal;

                        cA -= mA * P;
                        aA -= iA * b2Cross(rA, P);

                        cB += mB * P;
                        aB += iB * b2Cross(rB, P);
                }

                m_positions[indexA].c = cA;
                m_positions[indexA].a = aA;

                m_positions[indexB].c = cB;
                m_positions[indexB].a = aB;
        }

        // We can't expect minSpeparation >= -b2_linearSlop because we don't
        // push the separation above -b2_linearSlop.
        return minSeparation >= -3.0f * b2_linearSlop;
}

// Sequential position solver for position constraints.
bool b2ContactSolver::SolveTOIPositionConstraints(int32 toiIndexA, int32 toiIndexB)
{
        float32 minSeparation = 0.0f;

        for (int32 i = 0; i < m_count; ++i)
        {
                b2ContactPositionConstraint* pc = m_positionConstraints + i;

                int32 indexA = pc->indexA;
                int32 indexB = pc->indexB;
                b2Vec2 localCenterA = pc->localCenterA;
                b2Vec2 localCenterB = pc->localCenterB;
                int32 pointCount = pc->pointCount;

                float32 mA = 0.0f;
                float32 iA = 0.0f;
                if (indexA == toiIndexA || indexA == toiIndexB)
                {
                        mA = pc->invMassA;
                        iA = pc->invIA;
                }

                float32 mB = 0.0f;
                float32 iB = 0.;
                if (indexB == toiIndexA || indexB == toiIndexB)
                {
                        mB = pc->invMassB;
                        iB = pc->invIB;
                }

                b2Vec2 cA = m_positions[indexA].c;
                float32 aA = m_positions[indexA].a;

                b2Vec2 cB = m_positions[indexB].c;
                float32 aB = m_positions[indexB].a;

                // Solve normal constraints
                for (int32 j = 0; j < pointCount; ++j)
                {
                        b2Transform xfA, xfB;
                        xfA.q.Set(aA);
                        xfB.q.Set(aB);
                        xfA.p = cA - b2Mul(xfA.q, localCenterA);
                        xfB.p = cB - b2Mul(xfB.q, localCenterB);

                        b2PositionSolverManifold psm;
                        psm.Initialize(pc, xfA, xfB, j);
                        b2Vec2 normal = psm.normal;

                        b2Vec2 point = psm.point;
                        float32 separation = psm.separation;

                        b2Vec2 rA = point - cA;
                        b2Vec2 rB = point - cB;

                        // Track max constraint error.
                        minSeparation = b2Min(minSeparation, separation);

                        // Prevent large corrections and allow slop.
                        float32 C = b2Clamp(b2_toiBaugarte * (separation + b2_linearSlop), -b2_maxLinearCorrection, 0.0f);

                        // Compute the effective mass.
                        float32 rnA = b2Cross(rA, normal);
                        float32 rnB = b2Cross(rB, normal);
                        float32 K = mA + mB + iA * rnA * rnA + iB * rnB * rnB;

                        // Compute normal impulse
                        float32 impulse = K > 0.0f ? - C / K : 0.0f;

                        b2Vec2 P = impulse * normal;

                        cA -= mA * P;
                        aA -= iA * b2Cross(rA, P);

                        cB += mB * P;
                        aB += iB * b2Cross(rB, P);
                }

                m_positions[indexA].c = cA;
                m_positions[indexA].a = aA;

                m_positions[indexB].c = cB;
                m_positions[indexB].a = aB;
        }

        // We can't expect minSpeparation >= -b2_linearSlop because we don't
        // push the separation above -b2_linearSlop.
        return minSeparation >= -1.5f * b2_linearSlop;
}