axmol/external/Box2D/Dynamics/Contacts/b2ContactSolver.cpp

/*
* 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

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->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)
        {
            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);
            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 (vc->pointCount == 1)
        {
            b2VelocityConstraintPoint* vcp = vc->points + 0;

            // 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 = pc->invMassB;
        float32 iB = pc->invIB;
        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;
}