source: code/branches/ode/ode-0.9/contrib/BreakableJoints/stepfast.cpp @ 216

/*************************************************************************
 *                                                                       *
 * Open Dynamics Engine, Copyright (C) 2001,2002 Russell L. Smith.       *
 * All rights reserved.  Email: russ@q12.org   Web: www.q12.org          *
 *                                                                       *
 * Fast iterative solver, David Whittaker. Email: david@csworkbench.com  *
 *                                                                       *
 * This library is free software; you can redistribute it and/or         *
 * modify it under the terms of EITHER:                                  *
 *   (1) The GNU Lesser General Public License as published by the Free  *
 *       Software Foundation; either version 2.1 of the License, or (at  *
 *       your option) any later version. The text of the GNU Lesser      *
 *       General Public License is included with this library in the     *
 *       file LICENSE.TXT.                                               *
 *   (2) The BSD-style license that is included with this library in     *
 *       the file LICENSE-BSD.TXT.                                       *
 *                                                                       *
 * This library is distributed in the hope that it will be useful,       *
 * but WITHOUT ANY WARRANTY; without even the implied warranty of        *
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the files    *
 * LICENSE.TXT and LICENSE-BSD.TXT for more details.                     *
 *                                                                       *
 *************************************************************************/

// This is the StepFast code by David Whittaker. This code is faster, but
// sometimes less stable than, the original "big matrix" code.
// Refer to the user's manual for more information.
// Note that this source file duplicates a lot of stuff from step.cpp,
// eventually we should move the common code to a third file.

#include "objects.h"
#include "joint.h"
#include <ode/config.h>
#include <ode/objects.h>
#include <ode/odemath.h>
#include <ode/rotation.h>
#include <ode/timer.h>
#include <ode/error.h>
#include <ode/matrix.h>
#include "lcp.h"
#include "step.h"


// misc defines

#define ALLOCA dALLOCA16

#define RANDOM_JOINT_ORDER
//#define FAST_FACTOR   //use a factorization approximation to the LCP solver (fast, theoretically less accurate)
#define SLOW_LCP      //use the old LCP solver
//#define NO_ISLANDS    //does not perform island creation code (3~4% of simulation time), body disabling doesn't work
//#define TIMING


static int autoEnableDepth = 2;

void dWorldSetAutoEnableDepthSF1 (dxWorld *world, int autodepth)
{
        if (autodepth > 0)
                autoEnableDepth = autodepth;
        else
                autoEnableDepth = 0;
}

int dWorldGetAutoEnableDepthSF1 (dxWorld *world)
{
        return autoEnableDepth;
}

//little bit of math.... the _sym_ functions assume the return matrix will be symmetric
static void
Multiply2_sym_p8p (dReal * A, dReal * B, dReal * C, int p, int Askip)
{
        int i, j;
        dReal sum, *aa, *ad, *bb, *cc;
        dIASSERT (p > 0 && A && B && C);
        bb = B;
        for (i = 0; i < p; i++)
        {
                //aa is going accross the matrix, ad down
                aa = ad = A;
                cc = C;
                for (j = i; j < p; j++)
                {
                        sum = bb[0] * cc[0];
                        sum += bb[1] * cc[1];
                        sum += bb[2] * cc[2];
                        sum += bb[4] * cc[4];
                        sum += bb[5] * cc[5];
                        sum += bb[6] * cc[6];
                        *(aa++) = *ad = sum;
                        ad += Askip;
                        cc += 8;
                }
                bb += 8;
                A += Askip + 1;
                C += 8;
        }
}

static void
MultiplyAdd2_sym_p8p (dReal * A, dReal * B, dReal * C, int p, int Askip)
{
        int i, j;
        dReal sum, *aa, *ad, *bb, *cc;
        dIASSERT (p > 0 && A && B && C);
        bb = B;
        for (i = 0; i < p; i++)
        {
                //aa is going accross the matrix, ad down
                aa = ad = A;
                cc = C;
                for (j = i; j < p; j++)
                {
                        sum = bb[0] * cc[0];
                        sum += bb[1] * cc[1];
                        sum += bb[2] * cc[2];
                        sum += bb[4] * cc[4];
                        sum += bb[5] * cc[5];
                        sum += bb[6] * cc[6];
                        *(aa++) += sum;
                        *ad += sum;
                        ad += Askip;
                        cc += 8;
                }
                bb += 8;
                A += Askip + 1;
                C += 8;
        }
}


// this assumes the 4th and 8th rows of B are zero.

static void
Multiply0_p81 (dReal * A, dReal * B, dReal * C, int p)
{
        int i;
        dIASSERT (p > 0 && A && B && C);
        dReal sum;
        for (i = p; i; i--)
        {
                sum = B[0] * C[0];
                sum += B[1] * C[1];
                sum += B[2] * C[2];
                sum += B[4] * C[4];
                sum += B[5] * C[5];
                sum += B[6] * C[6];
                *(A++) = sum;
                B += 8;
        }
}


// this assumes the 4th and 8th rows of B are zero.

static void
MultiplyAdd0_p81 (dReal * A, dReal * B, dReal * C, int p)
{
        int i;
        dIASSERT (p > 0 && A && B && C);
        dReal sum;
        for (i = p; i; i--)
        {
                sum = B[0] * C[0];
                sum += B[1] * C[1];
                sum += B[2] * C[2];
                sum += B[4] * C[4];
                sum += B[5] * C[5];
                sum += B[6] * C[6];
                *(A++) += sum;
                B += 8;
        }
}


// this assumes the 4th and 8th rows of B are zero.

static void
Multiply1_8q1 (dReal * A, dReal * B, dReal * C, int q)
{
        int k;
        dReal sum;
        dIASSERT (q > 0 && A && B && C);
        sum = 0;
        for (k = 0; k < q; k++)
                sum += B[k * 8] * C[k];
        A[0] = sum;
        sum = 0;
        for (k = 0; k < q; k++)
                sum += B[1 + k * 8] * C[k];
        A[1] = sum;
        sum = 0;
        for (k = 0; k < q; k++)
                sum += B[2 + k * 8] * C[k];
        A[2] = sum;
        sum = 0;
        for (k = 0; k < q; k++)
                sum += B[4 + k * 8] * C[k];
        A[4] = sum;
        sum = 0;
        for (k = 0; k < q; k++)
                sum += B[5 + k * 8] * C[k];
        A[5] = sum;
        sum = 0;
        for (k = 0; k < q; k++)
                sum += B[6 + k * 8] * C[k];
        A[6] = sum;
}

//****************************************************************************
// body rotation

// return sin(x)/x. this has a singularity at 0 so special handling is needed
// for small arguments.

static inline dReal
sinc (dReal x)
{
        // if |x| < 1e-4 then use a taylor series expansion. this two term expansion
        // is actually accurate to one LS bit within this range if double precision
        // is being used - so don't worry!
        if (dFabs (x) < 1.0e-4)
                return REAL (1.0) - x * x * REAL (0.166666666666666666667);
        else
                return dSin (x) / x;
}


// given a body b, apply its linear and angular rotation over the time
// interval h, thereby adjusting its position and orientation.

static inline void
moveAndRotateBody (dxBody * b, dReal h)
{
        int j;

        // handle linear velocity
        for (j = 0; j < 3; j++)
                b->pos[j] += h * b->lvel[j];

        if (b->flags & dxBodyFlagFiniteRotation)
        {
                dVector3 irv;                   // infitesimal rotation vector
                dQuaternion q;                  // quaternion for finite rotation

                if (b->flags & dxBodyFlagFiniteRotationAxis)
                {
                        // split the angular velocity vector into a component along the finite
                        // rotation axis, and a component orthogonal to it.
                        dVector3 frv, irv;      // finite rotation vector
                        dReal k = dDOT (b->finite_rot_axis, b->avel);
                        frv[0] = b->finite_rot_axis[0] * k;
                        frv[1] = b->finite_rot_axis[1] * k;
                        frv[2] = b->finite_rot_axis[2] * k;
                        irv[0] = b->avel[0] - frv[0];
                        irv[1] = b->avel[1] - frv[1];
                        irv[2] = b->avel[2] - frv[2];

                        // make a rotation quaternion q that corresponds to frv * h.
                        // compare this with the full-finite-rotation case below.
                        h *= REAL (0.5);
                        dReal theta = k * h;
                        q[0] = dCos (theta);
                        dReal s = sinc (theta) * h;
                        q[1] = frv[0] * s;
                        q[2] = frv[1] * s;
                        q[3] = frv[2] * s;
                }
                else
                {
                        // make a rotation quaternion q that corresponds to w * h
                        dReal wlen = dSqrt (b->avel[0] * b->avel[0] + b->avel[1] * b->avel[1] + b->avel[2] * b->avel[2]);
                        h *= REAL (0.5);
                        dReal theta = wlen * h;
                        q[0] = dCos (theta);
                        dReal s = sinc (theta) * h;
                        q[1] = b->avel[0] * s;
                        q[2] = b->avel[1] * s;
                        q[3] = b->avel[2] * s;
                }

                // do the finite rotation
                dQuaternion q2;
                dQMultiply0 (q2, q, b->q);
                for (j = 0; j < 4; j++)
                        b->q[j] = q2[j];

                // do the infitesimal rotation if required
                if (b->flags & dxBodyFlagFiniteRotationAxis)
                {
                        dReal dq[4];
                        dWtoDQ (irv, b->q, dq);
                        for (j = 0; j < 4; j++)
                                b->q[j] += h * dq[j];
                }
        }
        else
        {
                // the normal way - do an infitesimal rotation
                dReal dq[4];
                dWtoDQ (b->avel, b->q, dq);
                for (j = 0; j < 4; j++)
                        b->q[j] += h * dq[j];
        }

        // normalize the quaternion and convert it to a rotation matrix
        dNormalize4 (b->q);
        dQtoR (b->q, b->R);

        // notify all attached geoms that this body has moved
        for (dxGeom * geom = b->geom; geom; geom = dGeomGetBodyNext (geom))
                dGeomMoved (geom);
}

//****************************************************************************
//This is an implementation of the iterated/relaxation algorithm.
//Here is a quick overview of the algorithm per Sergi Valverde's posts to the
//mailing list:
//
//  for i=0..N-1 do
//      for c = 0..C-1 do
//          Solve constraint c-th
//          Apply forces to constraint bodies
//      next
//  next
//  Integrate bodies

void
dInternalStepFast (dxWorld * world, dxBody * body[2], dReal * GI[2], dReal * GinvI[2], dxJoint * joint, dxJoint::Info1 info, dxJoint::Info2 Jinfo, dReal stepsize)
{
        int i, j, k;
# ifdef TIMING
        dTimerNow ("constraint preprocessing");
# endif

        dReal stepsize1 = dRecip (stepsize);

        int m = info.m;
        // nothing to do if no constraints.
        if (m <= 0)
                return;

        int nub = 0;
        if (info.nub == info.m)
                nub = m;

        // compute A = J*invM*J'. first compute JinvM = J*invM. this has the same
        // format as J so we just go through the constraints in J multiplying by
        // the appropriate scalars and matrices.
#   ifdef TIMING
        dTimerNow ("compute A");
#   endif
        dReal JinvM[2 * 6 * 8];
        //dSetZero (JinvM, 2 * m * 8);

        dReal *Jsrc = Jinfo.J1l;
        dReal *Jdst = JinvM;
        if (body[0])
        {
                for (j = m - 1; j >= 0; j--)
                {
                        for (k = 0; k < 3; k++)
                                Jdst[k] = Jsrc[k] * body[0]->invMass;
                        dMULTIPLY0_133 (Jdst + 4, Jsrc + 4, GinvI[0]);
                        Jsrc += 8;
                        Jdst += 8;
                }
        }
        if (body[1])
        {
                Jsrc = Jinfo.J2l;
                Jdst = JinvM + 8 * m;
                for (j = m - 1; j >= 0; j--)
                {
                        for (k = 0; k < 3; k++)
                                Jdst[k] = Jsrc[k] * body[1]->invMass;
                        dMULTIPLY0_133 (Jdst + 4, Jsrc + 4, GinvI[1]);
                        Jsrc += 8;
                        Jdst += 8;
                }
        }


        // now compute A = JinvM * J'.
        int mskip = dPAD (m);
        dReal A[6 * 8];
        //dSetZero (A, 6 * 8);

        if (body[0])
                Multiply2_sym_p8p (A, JinvM, Jinfo.J1l, m, mskip);
        if (body[1])
                MultiplyAdd2_sym_p8p (A, JinvM + 8 * m, Jinfo.J2l, m, mskip);

        // add cfm to the diagonal of A
        for (i = 0; i < m; i++)
                A[i * mskip + i] += Jinfo.cfm[i] * stepsize1;

        // compute the right hand side `rhs'
#   ifdef TIMING
        dTimerNow ("compute rhs");
#   endif
        dReal tmp1[16];
        //dSetZero (tmp1, 16);
        // put v/h + invM*fe into tmp1
        for (i = 0; i < 2; i++)
        {
                if (!body[i])
                        continue;
                for (j = 0; j < 3; j++)
                        tmp1[i * 8 + j] = body[i]->facc[j] * body[i]->invMass + body[i]->lvel[j] * stepsize1;
                dMULTIPLY0_331 (tmp1 + i * 8 + 4, GinvI[i], body[i]->tacc);
                for (j = 0; j < 3; j++)
                        tmp1[i * 8 + 4 + j] += body[i]->avel[j] * stepsize1;
        }
        // put J*tmp1 into rhs
        dReal rhs[6];
        //dSetZero (rhs, 6);

        if (body[0])
                Multiply0_p81 (rhs, Jinfo.J1l, tmp1, m);
        if (body[1])
                MultiplyAdd0_p81 (rhs, Jinfo.J2l, tmp1 + 8, m);

        // complete rhs
        for (i = 0; i < m; i++)
                rhs[i] = Jinfo.c[i] * stepsize1 - rhs[i];

#ifdef SLOW_LCP
        // solve the LCP problem and get lambda.
        // this will destroy A but that's okay
#       ifdef TIMING
        dTimerNow ("solving LCP problem");
#       endif
        dReal *lambda = (dReal *) ALLOCA (m * sizeof (dReal));
        dReal *residual = (dReal *) ALLOCA (m * sizeof (dReal));
        dReal lo[6], hi[6];
        memcpy (lo, Jinfo.lo, m * sizeof (dReal));
        memcpy (hi, Jinfo.hi, m * sizeof (dReal));
        dSolveLCP (m, A, lambda, rhs, residual, nub, lo, hi, Jinfo.findex);
#endif

        // LCP Solver replacement:
        // This algorithm goes like this:
        // Do a straightforward LDLT factorization of the matrix A, solving for
        // A*x = rhs
        // For each x[i] that is outside of the bounds of lo[i] and hi[i],
        //    clamp x[i] into that range.
        //    Substitute into A the now known x's
        //    subtract the residual away from the rhs.
        //    Remove row and column i from L, updating the factorization
        //    place the known x's at the end of the array, keeping up with location in p
        // Repeat until all constraints have been clamped or all are within bounds
        //
        // This is probably only faster in the single joint case where only one repeat is
        // the norm.

#ifdef FAST_FACTOR
        // factorize A (L*D*L'=A)
#       ifdef TIMING
        dTimerNow ("factorize A");
#       endif
        dReal d[6];
        dReal L[6 * 8];
        memcpy (L, A, m * mskip * sizeof (dReal));
        dFactorLDLT (L, d, m, mskip);

        // compute lambda
#       ifdef TIMING
        dTimerNow ("compute lambda");
#       endif

        int left = m;                           //constraints left to solve.
        int remove[6];
        dReal lambda[6];
        dReal x[6];
        int p[6];
        for (i = 0; i < 6; i++)
                p[i] = i;
        while (true)
        {
                memcpy (x, rhs, left * sizeof (dReal));
                dSolveLDLT (L, d, x, left, mskip);

                int fixed = 0;
                for (i = 0; i < left; i++)
                {
                        j = p[i];
                        remove[i] = false;
                        // This isn't the exact same use of findex as dSolveLCP.... since x[findex]
                        // may change after I've already clamped x[i], but it should be close
                        if (Jinfo.findex[j] > -1)
                        {
                                dReal f = fabs (Jinfo.hi[j] * x[p[Jinfo.findex[j]]]);
                                if (x[i] > f)
                                        x[i] = f;
                                else if (x[i] < -f)
                                        x[i] = -f;
                                else
                                        continue;
                        }
                        else
                        {
                                if (x[i] > Jinfo.hi[j])
                                        x[i] = Jinfo.hi[j];
                                else if (x[i] < Jinfo.lo[j])
                                        x[i] = Jinfo.lo[j];
                                else
                                        continue;
                        }
                        remove[i] = true;
                        fixed++;
                }
                if (fixed == 0 || fixed == left)        //no change or all constraints solved
                        break;

                for (i = 0; i < left; i++)      //sub in to right hand side.
                        if (remove[i])
                                for (j = 0; j < left; j++)
                                        if (!remove[j])
                                                rhs[j] -= A[j * mskip + i] * x[i];

                for (int r = left - 1; r >= 0; r--)     //eliminate row/col for fixed variables
                {
                        if (remove[r])
                        {
                                //dRemoveLDLT adapted for use without row pointers.
                                if (r == left - 1)
                                {
                                        left--;
                                        continue;       // deleting last row/col is easy
                                }
                                else if (r == 0)
                                {
                                        dReal a[6];
                                        for (i = 0; i < left; i++)
                                                a[i] = -A[i * mskip];
                                        a[0] += REAL (1.0);
                                        dLDLTAddTL (L, d, a, left, mskip);
                                }
                                else
                                {
                                        dReal t[6];
                                        dReal a[6];
                                        for (i = 0; i < r; i++)
                                                t[i] = L[r * mskip + i] / d[i];
                                        for (i = 0; i < left - r; i++)
                                                a[i] = dDot (L + (r + i) * mskip, t, r) - A[(r + i) * mskip + r];
                                        a[0] += REAL (1.0);
                                        dLDLTAddTL (L + r * mskip + r, d + r, a, left - r, mskip);
                                }

                                dRemoveRowCol (L, left, mskip, r);
                                //end dRemoveLDLT

                                left--;
                                if (r < (left - 1))
                                {
                                        dReal tx = x[r];
                                        memmove (d + r, d + r + 1, (left - r) * sizeof (dReal));
                                        memmove (rhs + r, rhs + r + 1, (left - r) * sizeof (dReal));
                                        //x will get written over by rhs anyway, no need to move it around
                                        //just store the fixed value we just discovered in it.
                                        x[left] = tx;
                                        for (i = 0; i < m; i++)
                                                if (p[i] > r && p[i] <= left)
                                                        p[i]--;
                                        p[r] = left;
                                }
                        }
                }
        }

        for (i = 0; i < m; i++)
                lambda[i] = x[p[i]];
#       endif
        // compute the constraint force `cforce'
#       ifdef TIMING
        dTimerNow ("compute constraint force");
#endif

        // compute cforce = J'*lambda
        dJointFeedback *fb = joint->feedback;
        dReal cforce[16];
        //dSetZero (cforce, 16);

/******************** breakable joint contribution ***********************/
        // this saves us a few dereferences
    dxJointBreakInfo *jBI = joint->breakInfo;
    // we need joint feedback if the joint is breakable or if the user
    // requested feedback.
        if (jBI||fb) {
                // we need feedback on the amount of force that this joint is
                // applying to the bodies. we use a slightly slower computation
                // that splits out the force components and puts them in the
                // feedback structure.
                dJointFeedback temp_fb; // temporary storage for joint feedback
                dReal data1[8],data2[8];
                if (body[0])
                {
                        Multiply1_8q1 (data1, Jinfo.J1l, lambda, m);
                        dReal *cf1 = cforce;
                        cf1[0] = (temp_fb.f1[0] = data1[0]);
                        cf1[1] = (temp_fb.f1[1] = data1[1]);
                        cf1[2] = (temp_fb.f1[2] = data1[2]);
                        cf1[4] = (temp_fb.t1[0] = data1[4]);
                        cf1[5] = (temp_fb.t1[1] = data1[5]);
                        cf1[6] = (temp_fb.t1[2] = data1[6]);
                }
                if (body[1])
                {
                        Multiply1_8q1 (data2, Jinfo.J2l, lambda, m);
                        dReal *cf2 = cforce + 8;
                        cf2[0] = (temp_fb.f2[0] = data2[0]);
                        cf2[1] = (temp_fb.f2[1] = data2[1]);
                        cf2[2] = (temp_fb.f2[2] = data2[2]);
                        cf2[4] = (temp_fb.t2[0] = data2[4]);
                        cf2[5] = (temp_fb.t2[1] = data2[5]);
                        cf2[6] = (temp_fb.t2[2] = data2[6]);
                }
                // if the user requested so we must copy the feedback information to
                // the feedback struct that the user suplied.
                if (fb) {
                        // copy temp_fb to fb
                        fb->f1[0] = temp_fb.f1[0];
                        fb->f1[1] = temp_fb.f1[1];
                        fb->f1[2] = temp_fb.f1[2];
                        fb->t1[0] = temp_fb.t1[0];
                        fb->t1[1] = temp_fb.t1[1];
                        fb->t1[2] = temp_fb.t1[2];
                        if (body[1]) {
                                fb->f2[0] = temp_fb.f2[0];
                                fb->f2[1] = temp_fb.f2[1];
                                fb->f2[2] = temp_fb.f2[2];
                                fb->t2[0] = temp_fb.t2[0];
                                fb->t2[1] = temp_fb.t2[1];
                                fb->t2[2] = temp_fb.t2[2];
                        }
                }
                // if the joint is breakable we need to check the breaking conditions
                if (jBI) {
                        dReal relCF1[3];
                        dReal relCT1[3];
                        // multiply the force and torque vectors by the rotation matrix of body 1
                        dMULTIPLY1_331 (&relCF1[0],body[0]->R,&temp_fb.f1[0]);
                        dMULTIPLY1_331 (&relCT1[0],body[0]->R,&temp_fb.t1[0]);
                        if (jBI->flags & dJOINT_BREAK_AT_B1_FORCE) {
                                // check if the force is to high
                                for (int i = 0; i < 3; i++) {
                                        if (relCF1[i] > jBI->b1MaxF[i]) {
                                                jBI->flags |= dJOINT_BROKEN;
                                                goto doneCheckingBreaks;
                                        }
                                }
                        }
                        if (jBI->flags & dJOINT_BREAK_AT_B1_TORQUE) {
                                // check if the torque is to high
                                for (int i = 0; i < 3; i++) {
                                        if (relCT1[i] > jBI->b1MaxT[i]) {
                                                jBI->flags |= dJOINT_BROKEN;
                                                goto doneCheckingBreaks;
                                        }
                                }
                        }
                        if (body[1]) {
                                dReal relCF2[3];
                                dReal relCT2[3];
                                // multiply the force and torque vectors by the rotation matrix of body 2
                                dMULTIPLY1_331 (&relCF2[0],body[1]->R,&temp_fb.f2[0]);
                                dMULTIPLY1_331 (&relCT2[0],body[1]->R,&temp_fb.t2[0]);
                                if (jBI->flags & dJOINT_BREAK_AT_B2_FORCE) {
                                        // check if the force is to high
                                        for (int i = 0; i < 3; i++) {
                                                if (relCF2[i] > jBI->b2MaxF[i]) {
                                                        jBI->flags |= dJOINT_BROKEN;
                                                        goto doneCheckingBreaks;
                                                }
                                        }
                                }
                                if (jBI->flags & dJOINT_BREAK_AT_B2_TORQUE) {
                                        // check if the torque is to high
                                        for (int i = 0; i < 3; i++) {
                                                if (relCT2[i] > jBI->b2MaxT[i]) {
                                                        jBI->flags |= dJOINT_BROKEN;
                                                        goto doneCheckingBreaks;
                                                }
                                        }
                                }
                        }
                        doneCheckingBreaks:
                        ;
                }
        }
/*************************************************************************/
        else
        {
                // no feedback is required, let's compute cforce the faster way
                if (body[0])
                        Multiply1_8q1 (cforce, Jinfo.J1l, lambda, m);
                if (body[1])
                        Multiply1_8q1 (cforce + 8, Jinfo.J2l, lambda, m);
        }

        for (i = 0; i < 2; i++)
        {
                if (!body[i])
                        continue;
                for (j = 0; j < 3; j++)
                {
                        body[i]->facc[j] += cforce[i * 8 + j];
                        body[i]->tacc[j] += cforce[i * 8 + 4 + j];
                }
        }
}

void
dInternalStepIslandFast (dxWorld * world, dxBody * const *bodies, int nb, dxJoint * const *_joints, int nj, dReal stepsize, int maxiterations)
{
#   ifdef TIMING
        dTimerNow ("preprocessing");
#   endif
        dxBody *bodyPair[2], *body;
        dReal *GIPair[2], *GinvIPair[2];
        dxJoint *joint;
        int iter, b, j, i;
        dReal ministep = stepsize / maxiterations;

        // make a local copy of the joint array, because we might want to modify it.
        // (the "dxJoint *const*" declaration says we're allowed to modify the joints
        // but not the joint array, because the caller might need it unchanged).
        dxJoint **joints = (dxJoint **) ALLOCA (nj * sizeof (dxJoint *));
        memcpy (joints, _joints, nj * sizeof (dxJoint *));

        // get m = total constraint dimension, nub = number of unbounded variables.
        // create constraint offset array and number-of-rows array for all joints.
        // the constraints are re-ordered as follows: the purely unbounded
        // constraints, the mixed unbounded + LCP constraints, and last the purely
        // LCP constraints. this assists the LCP solver to put all unbounded
        // variables at the start for a quick factorization.
        //
        // joints with m=0 are inactive and are removed from the joints array
        // entirely, so that the code that follows does not consider them.
        // also number all active joints in the joint list (set their tag values).
        // inactive joints receive a tag value of -1.

        int m = 0;
        dxJoint::Info1 * info = (dxJoint::Info1 *) ALLOCA (nj * sizeof (dxJoint::Info1));
        int *ofs = (int *) ALLOCA (nj * sizeof (int));
        for (i = 0, j = 0; j < nj; j++)
        {       // i=dest, j=src
                joints[j]->vtable->getInfo1 (joints[j], info + i);
                dIASSERT (info[i].m >= 0 && info[i].m <= 6 && info[i].nub >= 0 && info[i].nub <= info[i].m);
                if (info[i].m > 0)
                {
                        joints[i] = joints[j];
                        joints[i]->tag = i;
                        i++;
                }
                else
                {
                        joints[j]->tag = -1;
                }
        }
        nj = i;

        // the purely unbounded constraints
        for (i = 0; i < nj; i++)
        {
                ofs[i] = m;
                m += info[i].m;
        }
        dReal *c = NULL;
        dReal *cfm = NULL;
        dReal *lo = NULL;
        dReal *hi = NULL;
        int *findex = NULL;

        dReal *J = NULL;
        dxJoint::Info2 * Jinfo = NULL;

        if (m)
        {
        // create a constraint equation right hand side vector `c', a constraint
        // force mixing vector `cfm', and LCP low and high bound vectors, and an
        // 'findex' vector.
                c = (dReal *) ALLOCA (m * sizeof (dReal));
                cfm = (dReal *) ALLOCA (m * sizeof (dReal));
                lo = (dReal *) ALLOCA (m * sizeof (dReal));
                hi = (dReal *) ALLOCA (m * sizeof (dReal));
                findex = (int *) ALLOCA (m * sizeof (int));
        dSetZero (c, m);
        dSetValue (cfm, m, world->global_cfm);
        dSetValue (lo, m, -dInfinity);
        dSetValue (hi, m, dInfinity);
        for (i = 0; i < m; i++)
                findex[i] = -1;

        // get jacobian data from constraints. a (2*m)x8 matrix will be created
        // to store the two jacobian blocks from each constraint. it has this
        // format:
        //
        //   l l l 0 a a a 0  \    .
        //   l l l 0 a a a 0   }-- jacobian body 1 block for joint 0 (3 rows)
        //   l l l 0 a a a 0  /
        //   l l l 0 a a a 0  \    .
        //   l l l 0 a a a 0   }-- jacobian body 2 block for joint 0 (3 rows)
        //   l l l 0 a a a 0  /
        //   l l l 0 a a a 0  }--- jacobian body 1 block for joint 1 (1 row)
        //   l l l 0 a a a 0  }--- jacobian body 2 block for joint 1 (1 row)
        //   etc...
        //
        //   (lll) = linear jacobian data
        //   (aaa) = angular jacobian data
        //
#   ifdef TIMING
        dTimerNow ("create J");
#   endif
                J = (dReal *) ALLOCA (2 * m * 8 * sizeof (dReal));
                dSetZero (J, 2 * m * 8);
                Jinfo = (dxJoint::Info2 *) ALLOCA (nj * sizeof (dxJoint::Info2));
        for (i = 0; i < nj; i++)
        {
                Jinfo[i].rowskip = 8;
                Jinfo[i].fps = dRecip (stepsize);
                Jinfo[i].erp = world->global_erp;
                Jinfo[i].J1l = J + 2 * 8 * ofs[i];
                Jinfo[i].J1a = Jinfo[i].J1l + 4;
                Jinfo[i].J2l = Jinfo[i].J1l + 8 * info[i].m;
                Jinfo[i].J2a = Jinfo[i].J2l + 4;
                Jinfo[i].c = c + ofs[i];
                Jinfo[i].cfm = cfm + ofs[i];
                Jinfo[i].lo = lo + ofs[i];
                Jinfo[i].hi = hi + ofs[i];
                Jinfo[i].findex = findex + ofs[i];
                //joints[i]->vtable->getInfo2 (joints[i], Jinfo+i);
        }

        }

        dReal *saveFacc = (dReal *) ALLOCA (nb * 4 * sizeof (dReal));
        dReal *saveTacc = (dReal *) ALLOCA (nb * 4 * sizeof (dReal));
        dReal *globalI = (dReal *) ALLOCA (nb * 12 * sizeof (dReal));
        dReal *globalInvI = (dReal *) ALLOCA (nb * 12 * sizeof (dReal));
        for (b = 0; b < nb; b++)
        {
                for (i = 0; i < 4; i++)
                {
                        saveFacc[b * 4 + i] = bodies[b]->facc[i];
                        saveTacc[b * 4 + i] = bodies[b]->tacc[i];
                        bodies[b]->tag = b;
                }
        }

        for (iter = 0; iter < maxiterations; iter++)
        {
#       ifdef TIMING
                dTimerNow ("applying inertia and gravity");
#       endif
                dReal tmp[12] = { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 };

                for (b = 0; b < nb; b++)
                {
                        body = bodies[b];

                        // for all bodies, compute the inertia tensor and its inverse in the global
                        // frame, and compute the rotational force and add it to the torque
                        // accumulator. I and invI are vertically stacked 3x4 matrices, one per body.
                        // @@@ check computation of rotational force.

                        // compute inertia tensor in global frame
                        dMULTIPLY2_333 (tmp, body->mass.I, body->R);
                        dMULTIPLY0_333 (globalI + b * 12, body->R, tmp);
                        // compute inverse inertia tensor in global frame
                        dMULTIPLY2_333 (tmp, body->invI, body->R);
                        dMULTIPLY0_333 (globalInvI + b * 12, body->R, tmp);

                        for (i = 0; i < 4; i++)
                                body->tacc[i] = saveTacc[b * 4 + i];
                        // compute rotational force
                        dMULTIPLY0_331 (tmp, globalI + b * 12, body->avel);
                        dCROSS (body->tacc, -=, body->avel, tmp);

                        // add the gravity force to all bodies
                        if ((body->flags & dxBodyNoGravity) == 0)
                        {
                                body->facc[0] = saveFacc[b * 4 + 0] + body->mass.mass * world->gravity[0];
                                body->facc[1] = saveFacc[b * 4 + 1] + body->mass.mass * world->gravity[1];
                                body->facc[2] = saveFacc[b * 4 + 2] + body->mass.mass * world->gravity[2];
                                body->facc[3] = 0;
                        }

                }

#ifdef RANDOM_JOINT_ORDER
#ifdef TIMING
                dTimerNow ("randomizing joint order");
#endif
                //randomize the order of the joints by looping through the array
                //and swapping the current joint pointer with a random one before it.
                for (j = 0; j < nj; j++)
                {
                        joint = joints[j];
                        dxJoint::Info1 i1 = info[j];
                        dxJoint::Info2 i2 = Jinfo[j];
                        int r = rand () % (j + 1);
                        joints[j] = joints[r];
                        info[j] = info[r];
                        Jinfo[j] = Jinfo[r];
                        joints[r] = joint;
                        info[r] = i1;
                        Jinfo[r] = i2;
                }
#endif

                //now iterate through the random ordered joint array we created.
                for (j = 0; j < nj; j++)
                {
#ifdef TIMING
                        dTimerNow ("setting up joint");
#endif
                        joint = joints[j];
                        bodyPair[0] = joint->node[0].body;
                        bodyPair[1] = joint->node[1].body;

                        if (bodyPair[0] && (bodyPair[0]->flags & dxBodyDisabled))
                                bodyPair[0] = 0;
                        if (bodyPair[1] && (bodyPair[1]->flags & dxBodyDisabled))
                                bodyPair[1] = 0;
                        
                        //if this joint is not connected to any enabled bodies, skip it.
                        if (!bodyPair[0] && !bodyPair[1])
                                continue;
                        
                        if (bodyPair[0])
                        {
                                GIPair[0] = globalI + bodyPair[0]->tag * 12;
                                GinvIPair[0] = globalInvI + bodyPair[0]->tag * 12;
                        }
                        if (bodyPair[1])
                        {
                                GIPair[1] = globalI + bodyPair[1]->tag * 12;
                                GinvIPair[1] = globalInvI + bodyPair[1]->tag * 12;
                        }

                        joints[j]->vtable->getInfo2 (joints[j], Jinfo + j);

                        //dInternalStepIslandFast is an exact copy of the old routine with one
                        //modification: the calculated forces are added back to the facc and tacc
                        //vectors instead of applying them to the bodies and moving them.
                        if (info[j].m > 0)
                        {
                        dInternalStepFast (world, bodyPair, GIPair, GinvIPair, joint, info[j], Jinfo[j], ministep);
                        }               
                }
                //  }
#       ifdef TIMING
                dTimerNow ("moving bodies");
#       endif
                //Now we can simulate all the free floating bodies, and move them.
                for (b = 0; b < nb; b++)
                {
                        body = bodies[b];

                        for (i = 0; i < 4; i++)
                        {
                                body->facc[i] *= ministep;
                                body->tacc[i] *= ministep;
                        }

                        //apply torque
                        dMULTIPLYADD0_331 (body->avel, globalInvI + b * 12, body->tacc);

                        //apply force
                        for (i = 0; i < 3; i++)
                                body->lvel[i] += body->invMass * body->facc[i];

                        //move It!
                        moveAndRotateBody (body, ministep);
                }
        }
        for (b = 0; b < nb; b++)
                for (j = 0; j < 4; j++)
                        bodies[b]->facc[j] = bodies[b]->tacc[j] = 0;
}


#ifdef NO_ISLANDS

// Since the iterative algorithm doesn't care about islands of bodies, this is a
// faster algorithm that just sends it all the joints and bodies in one array.
// It's downfall is it's inability to handle disabled bodies as well as the old one.
static void
processIslandsFast (dxWorld * world, dReal stepsize, int maxiterations)
{
        // nothing to do if no bodies
        if (world->nb <= 0)
                return;

#       ifdef TIMING
        dTimerStart ("creating joint and body arrays");
#       endif
        dxBody **bodies, *body;
        dxJoint **joints, *joint;
        joints = (dxJoint **) ALLOCA (world->nj * sizeof (dxJoint *));
        bodies = (dxBody **) ALLOCA (world->nb * sizeof (dxBody *));

        int nj = 0;
        for (joint = world->firstjoint; joint; joint = (dxJoint *) joint->next)
                joints[nj++] = joint;

        int nb = 0;
        for (body = world->firstbody; body; body = (dxBody *) body->next)
                bodies[nb++] = body;

        dInternalStepIslandFast (world, bodies, nb, joints, nj, stepsize, maxiterations);
#       ifdef TIMING
        dTimerEnd ();
        dTimerReport (stdout, 1);
#       endif
}

#else

//****************************************************************************
// island processing

// this groups all joints and bodies in a world into islands. all objects
// in an island are reachable by going through connected bodies and joints.
// each island can be simulated separately.
// note that joints that are not attached to anything will not be included
// in any island, an so they do not affect the simulation.
//
// this function starts new island from unvisited bodies. however, it will
// never start a new islands from a disabled body. thus islands of disabled
// bodies will not be included in the simulation. disabled bodies are
// re-enabled if they are found to be part of an active island.

static void
processIslandsFast (dxWorld * world, dReal stepsize, int maxiterations)
{
#ifdef TIMING
        dTimerStart ("Island Setup");
#endif
        dxBody *b, *bb, **body;
        dxJoint *j, **joint;

        // nothing to do if no bodies
        if (world->nb <= 0)
                return;

        // make arrays for body and joint lists (for a single island) to go into
        body = (dxBody **) ALLOCA (world->nb * sizeof (dxBody *));
        joint = (dxJoint **) ALLOCA (world->nj * sizeof (dxJoint *));
        int bcount = 0;                         // number of bodies in `body'
        int jcount = 0;                         // number of joints in `joint'
        int tbcount = 0;
        int tjcount = 0;
        
        // set all body/joint tags to 0
        for (b = world->firstbody; b; b = (dxBody *) b->next)
                b->tag = 0;
        for (j = world->firstjoint; j; j = (dxJoint *) j->next)
                j->tag = 0;

        // allocate a stack of unvisited bodies in the island. the maximum size of
        // the stack can be the lesser of the number of bodies or joints, because
        // new bodies are only ever added to the stack by going through untagged
        // joints. all the bodies in the stack must be tagged!
        int stackalloc = (world->nj < world->nb) ? world->nj : world->nb;
        dxBody **stack = (dxBody **) ALLOCA (stackalloc * sizeof (dxBody *));
        int *autostack = (int *) ALLOCA (stackalloc * sizeof (int));

        for (bb = world->firstbody; bb; bb = (dxBody *) bb->next)
        {
#ifdef TIMING
                dTimerNow ("Island Processing");
#endif
                // get bb = the next enabled, untagged body, and tag it
                if (bb->tag || (bb->flags & dxBodyDisabled))
                        continue;
                bb->tag = 1;

                // tag all bodies and joints starting from bb.
                int stacksize = 0;
                int autoDepth = autoEnableDepth;
                b = bb;
                body[0] = bb;
                bcount = 1;
                jcount = 0;
                goto quickstart;
                while (stacksize > 0)
                {
                        b = stack[--stacksize]; // pop body off stack
                        autoDepth = autostack[stacksize];
                        body[bcount++] = b;     // put body on body list
                  quickstart:

                        // traverse and tag all body's joints, add untagged connected bodies
                        // to stack
                        for (dxJointNode * n = b->firstjoint; n; n = n->next)
                        {
                                if (!n->joint->tag)
                                {
                                        int thisDepth = autoEnableDepth;
                                        n->joint->tag = 1;
                                        joint[jcount++] = n->joint;
                                        if (n->body && !n->body->tag)
                                        {
                                                if (n->body->flags & dxBodyDisabled)
                                                        thisDepth = autoDepth - 1;
                                                if (thisDepth < 0)
                                                        continue;
                                                n->body->flags &= ~dxBodyDisabled;
                                                n->body->tag = 1;
                                                autostack[stacksize] = thisDepth;
                                                stack[stacksize++] = n->body;
                                        }
                                }
                        }
                        dIASSERT (stacksize <= world->nb);
                        dIASSERT (stacksize <= world->nj);
                }

                // now do something with body and joint lists
                dInternalStepIslandFast (world, body, bcount, joint, jcount, stepsize, maxiterations);

                // what we've just done may have altered the body/joint tag values.
                // we must make sure that these tags are nonzero.
                // also make sure all bodies are in the enabled state.
                int i;
                for (i = 0; i < bcount; i++)
                {
                        body[i]->tag = 1;
                        body[i]->flags &= ~dxBodyDisabled;
                }
                for (i = 0; i < jcount; i++)
                        joint[i]->tag = 1;
                
                tbcount += bcount;
                tjcount += jcount;
        }
        
#ifdef TIMING
        dMessage(0, "Total joints processed: %i, bodies: %i", tjcount, tbcount);
#endif

        // if debugging, check that all objects (except for disabled bodies,
        // unconnected joints, and joints that are connected to disabled bodies)
        // were tagged.
# ifndef dNODEBUG
        for (b = world->firstbody; b; b = (dxBody *) b->next)
        {
                if (b->flags & dxBodyDisabled)
                {
                        if (b->tag)
                                dDebug (0, "disabled body tagged");
                }
                else
                {
                        if (!b->tag)
                                dDebug (0, "enabled body not tagged");
                }
        }
        for (j = world->firstjoint; j; j = (dxJoint *) j->next)
        {
                if ((j->node[0].body && (j->node[0].body->flags & dxBodyDisabled) == 0) || (j->node[1].body && (j->node[1].body->flags & dxBodyDisabled) == 0))
                {
                        if (!j->tag)
                                dDebug (0, "attached enabled joint not tagged");
                }
                else
                {
                        if (j->tag)
                                dDebug (0, "unattached or disabled joint tagged");
                }
        }
# endif
  /******************** breakable joint contribution ***********************/
  dxJoint* nextJ;
  if (!world->firstjoint)
    nextJ = 0;
  else
    nextJ = (dxJoint*)world->firstjoint->next;
  for (j=world->firstjoint; j; j=nextJ) {
        nextJ = (dxJoint*)j->next;
        // check if joint is breakable and broken
    if (j->breakInfo && j->breakInfo->flags & dJOINT_BROKEN) {
                // detach (break) the joint
        dJointAttach (j, 0, 0);
                // call the callback function if it is set
                if (j->breakInfo->callback) j->breakInfo->callback (j);
                // finally destroy the joint if the dJOINT_DELETE_ON_BREAK is set
                if (j->breakInfo->flags & dJOINT_DELETE_ON_BREAK) dJointDestroy (j);
      }
  }
  /*************************************************************************/

#       ifdef TIMING
        dTimerEnd ();
        dTimerReport (stdout, 1);
#       endif
}

#endif


void dWorldStepFast1 (dWorldID w, dReal stepsize, int maxiterations)
{
        dUASSERT (w, "bad world argument");
        dUASSERT (stepsize > 0, "stepsize must be > 0");
        processIslandsFast (w, stepsize, maxiterations);
}