resubmitting ode

1 files changed, 1795 insertions, 0 deletions

diff --git a/libraries/ode-0.9/ode/src/step.cpp b/libraries/ode-0.9/ode/src/step.cpp
new file mode 100644
index 0000000..19d0473
--- /dev/null
+++ b/libraries/ode-0.9/ode/src/step.cpp
@@ -0,0 +1,1795 @@
+/*************************************************************************
+ *                                                                       *
+ * Open Dynamics Engine, Copyright (C) 2001,2002 Russell L. Smith.       *
+ * All rights reserved.  Email: russ@q12.org   Web: www.q12.org          *
+ *                                                                       *
+ * 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.                     *
+ *                                                                       *
+ *************************************************************************/
+
+#include "objects.h"
+#include "joint.h"
+#include <ode/config.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 "util.h"
+
+//****************************************************************************
+// misc defines
+
+#define FAST_FACTOR
+//#define TIMING
+
+// memory allocation system
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+unsigned int dMemoryFlag;
+#define REPORT_OUT_OF_MEMORY fprintf(stderr, "Insufficient memory to complete rigid body simulation.  Results will not be accurate.\n")
+
+#define ALLOCA(t,v,s) t* v=(t*)malloc(s)
+#define UNALLOCA(t)  free(t)
+
+#else
+#define ALLOCA(t,v,s) t* v=(t*)dALLOCA16(s)
+#define UNALLOCA(t)  /* nothing */
+#endif
+
+
+//****************************************************************************
+// debugging - comparison of various vectors and matrices produced by the
+// slow and fast versions of the stepper.
+
+//#define COMPARE_METHODS
+
+#ifdef COMPARE_METHODS
+#include "testing.h"
+dMatrixComparison comparator;
+#endif
+
+// undef to use the fast decomposition
+#define DIRECT_CHOLESKY
+#undef REPORT_ERROR
+
+//****************************************************************************
+// special matrix multipliers
+
+// this assumes the 4th and 8th rows of B and C are zero.
+
+static void Multiply2_p8r (dReal *A, dReal *B, dReal *C,
+                           int p, int r, int Askip)
+{
+  int i,j;
+  dReal sum,*bb,*cc;
+  dIASSERT (p>0 && r>0 && A && B && C);
+  bb = B;
+  for (i=p; i; i--) {
+    cc = C;
+    for (j=r; j; 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];
+      *(A++) = sum; 
+      cc += 8;
+    }
+    A += Askip - r;
+    bb += 8;
+  }
+}
+
+
+// this assumes the 4th and 8th rows of B and C are zero.
+
+static void MultiplyAdd2_p8r (dReal *A, dReal *B, dReal *C,
+                              int p, int r, int Askip)
+{
+  int i,j;
+  dReal sum,*bb,*cc;
+  dIASSERT (p>0 && r>0 && A && B && C);
+  bb = B;
+  for (i=p; i; i--) {
+    cc = C;
+    for (j=r; j; 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];
+      *(A++) += sum; 
+      cc += 8;
+    }
+    A += Askip - r;
+    bb += 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 MultiplyAdd1_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;
+}
+
+
+// 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;
+}
+
+//****************************************************************************
+// the slow, but sure way
+// note that this does not do any joint feedback!
+
+// given lists of bodies and joints that form an island, perform a first
+// order timestep.
+//
+// `body' is the body array, `nb' is the size of the array.
+// `_joint' is the body array, `nj' is the size of the array.
+
+void dInternalStepIsland_x1 (dxWorld *world, dxBody * const *body, int nb,
+                             dxJoint * const *_joint, int nj, dReal stepsize)
+{
+  int i,j,k;
+  int n6 = 6*nb;
+
+#ifdef TIMING
+  dTimerStart("preprocessing");
+#endif
+
+  // number all bodies in the body list - set their tag values
+  for (i=0; i<nb; i++) body[i]->tag = i;
+
+  // 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).
+  ALLOCA(dxJoint*,joint,nj*sizeof(dxJoint*));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (joint == NULL) {
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+  memcpy (joint,_joint,nj * sizeof(dxJoint*));
+
+  // 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.
+  // @@@ check computation of rotational force.
+  ALLOCA(dReal,I,3*nb*4*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (I == NULL) {
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+  ALLOCA(dReal,invI,3*nb*4*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (invI == NULL) {
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+
+  //dSetZero (I,3*nb*4);
+  //dSetZero (invI,3*nb*4);
+  for (i=0; i<nb; i++) {
+    dReal tmp[12];
+    // compute inertia tensor in global frame
+    dMULTIPLY2_333 (tmp,body[i]->mass.I,body[i]->posr.R);
+    dMULTIPLY0_333 (I+i*12,body[i]->posr.R,tmp);
+    // compute inverse inertia tensor in global frame
+    dMULTIPLY2_333 (tmp,body[i]->invI,body[i]->posr.R);
+    dMULTIPLY0_333 (invI+i*12,body[i]->posr.R,tmp);
+    // compute rotational force
+    dMULTIPLY0_331 (tmp,I+i*12,body[i]->avel);
+    dCROSS (body[i]->tacc,-=,body[i]->avel,tmp);
+  }
+
+  // add the gravity force to all bodies
+  for (i=0; i<nb; i++) {
+    if ((body[i]->flags & dxBodyNoGravity)==0) {
+      body[i]->facc[0] += body[i]->mass.mass * world->gravity[0];
+      body[i]->facc[1] += body[i]->mass.mass * world->gravity[1];
+      body[i]->facc[2] += body[i]->mass.mass * world->gravity[2];
+    }
+  }
+
+  // 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.
+  //
+  // joints with m=0 are inactive and are removed from the joints array
+  // entirely, so that the code that follows does not consider them.
+  int m = 0;
+  ALLOCA(dxJoint::Info1,info,nj*sizeof(dxJoint::Info1));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (info == NULL) {
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+
+  ALLOCA(int,ofs,nj*sizeof(int));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (ofs == NULL) {
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+
+  for (i=0, j=0; j<nj; j++) {   // i=dest, j=src
+    joint[j]->vtable->getInfo1 (joint[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) {
+      joint[i] = joint[j];
+      i++;
+    }
+  }
+  nj = i;
+
+  // the purely unbounded constraints
+  for (i=0; i<nj; i++) if (info[i].nub == info[i].m) {
+    ofs[i] = m;
+    m += info[i].m;
+  }
+  int nub = m;
+  // the mixed unbounded + LCP constraints
+  for (i=0; i<nj; i++) if (info[i].nub > 0 && info[i].nub < info[i].m) {
+    ofs[i] = m;
+    m += info[i].m;
+  }
+  // the purely LCP constraints
+  for (i=0; i<nj; i++) if (info[i].nub == 0) {
+    ofs[i] = m;
+    m += info[i].m;
+  }
+
+  // create (6*nb,6*nb) inverse mass matrix `invM', and fill it with mass
+  // parameters
+#ifdef TIMING
+  dTimerNow ("create mass matrix");
+#endif
+  int nskip = dPAD (n6);
+  ALLOCA(dReal, invM, n6*nskip*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (invM == NULL) {
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+
+  dSetZero (invM,n6*nskip);
+  for (i=0; i<nb; i++) {
+    dReal *MM = invM+(i*6)*nskip+(i*6);
+    MM[0] = body[i]->invMass;
+    MM[nskip+1] = body[i]->invMass;
+    MM[2*nskip+2] = body[i]->invMass;
+    MM += 3*nskip+3;
+    for (j=0; j<3; j++) for (k=0; k<3; k++) {
+      MM[j*nskip+k] = invI[i*12+j*4+k];
+    }
+  }
+
+  // assemble some body vectors: fe = external forces, v = velocities
+  ALLOCA(dReal,fe,n6*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (fe == NULL) {
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+
+  ALLOCA(dReal,v,n6*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (v == NULL) {
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+
+  //dSetZero (fe,n6);
+  //dSetZero (v,n6);
+  for (i=0; i<nb; i++) {
+    for (j=0; j<3; j++) fe[i*6+j] = body[i]->facc[j];
+    for (j=0; j<3; j++) fe[i*6+3+j] = body[i]->tacc[j];
+    for (j=0; j<3; j++) v[i*6+j] = body[i]->lvel[j];
+    for (j=0; j<3; j++) v[i*6+3+j] = body[i]->avel[j];
+  }
+
+  // this will be set to the velocity update
+  ALLOCA(dReal,vnew,n6*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (vnew == NULL) {
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+  dSetZero (vnew,n6);
+
+  // if there are constraints, compute cforce
+  if (m > 0) {
+    // 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.
+    ALLOCA(dReal,c,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (c == NULL) {
+      UNALLOCA(vnew);
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    ALLOCA(dReal,cfm,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (cfm == NULL) {
+      UNALLOCA(c);
+      UNALLOCA(vnew);
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    ALLOCA(dReal,lo,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (lo == NULL) {
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(vnew);
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    ALLOCA(dReal,hi,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (hi == NULL) {
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(vnew);
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    ALLOCA(int,findex,m*sizeof(int));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (findex == NULL) {
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(vnew);
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    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;
+
+    // create (m,6*nb) jacobian mass matrix `J', and fill it with constraint
+    // data. also fill the c vector.
+#   ifdef TIMING
+    dTimerNow ("create J");
+#   endif
+    ALLOCA(dReal,J,m*nskip*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (J == NULL) {
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(vnew);
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    dSetZero (J,m*nskip);
+    dxJoint::Info2 Jinfo;
+    Jinfo.rowskip = nskip;
+    Jinfo.fps = dRecip(stepsize);
+    Jinfo.erp = world->global_erp;
+    for (i=0; i<nj; i++) {
+      Jinfo.J1l = J + nskip*ofs[i] + 6*joint[i]->node[0].body->tag;
+      Jinfo.J1a = Jinfo.J1l + 3;
+      if (joint[i]->node[1].body) {
+        Jinfo.J2l = J + nskip*ofs[i] + 6*joint[i]->node[1].body->tag;
+        Jinfo.J2a = Jinfo.J2l + 3;
+      }
+      else {
+        Jinfo.J2l = 0;
+        Jinfo.J2a = 0;
+      }
+      Jinfo.c = c + ofs[i];
+      Jinfo.cfm = cfm + ofs[i];
+      Jinfo.lo = lo + ofs[i];
+      Jinfo.hi = hi + ofs[i];
+      Jinfo.findex = findex + ofs[i];
+      joint[i]->vtable->getInfo2 (joint[i],&Jinfo);
+      // adjust returned findex values for global index numbering
+      for (j=0; j<info[i].m; j++) {
+        if (findex[ofs[i] + j] >= 0) findex[ofs[i] + j] += ofs[i];
+      }
+    }
+
+    // compute A = J*invM*J'
+#   ifdef TIMING
+    dTimerNow ("compute A");
+#   endif
+    ALLOCA(dReal,JinvM,m*nskip*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (JinvM == NULL) {
+      UNALLOCA(J);
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(vnew);
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    //dSetZero (JinvM,m*nskip);
+    dMultiply0 (JinvM,J,invM,m,n6,n6);
+    int mskip = dPAD(m);
+    ALLOCA(dReal,A,m*mskip*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (A == NULL) {
+      UNALLOCA(JinvM);
+      UNALLOCA(J);
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(vnew);
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    //dSetZero (A,m*mskip);
+    dMultiply2 (A,JinvM,J,m,n6,m);
+
+    // add cfm to the diagonal of A
+    for (i=0; i<m; i++) A[i*mskip+i] += cfm[i] * Jinfo.fps;
+
+#   ifdef COMPARE_METHODS
+    comparator.nextMatrix (A,m,m,1,"A");
+#   endif
+
+    // compute `rhs', the right hand side of the equation J*a=c
+#   ifdef TIMING
+    dTimerNow ("compute rhs");
+#   endif
+    ALLOCA(dReal,tmp1,n6*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (tmp1 == NULL) {
+      UNALLOCA(A);
+      UNALLOCA(JinvM);
+      UNALLOCA(J);
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(vnew);
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    //dSetZero (tmp1,n6);
+    dMultiply0 (tmp1,invM,fe,n6,n6,1);
+    for (i=0; i<n6; i++) tmp1[i] += v[i]/stepsize;
+    ALLOCA(dReal,rhs,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (rhs == NULL) {
+      UNALLOCA(tmp1);
+      UNALLOCA(A);
+      UNALLOCA(JinvM);
+      UNALLOCA(J);
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(vnew);
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    //dSetZero (rhs,m);
+    dMultiply0 (rhs,J,tmp1,m,n6,1);
+    for (i=0; i<m; i++) rhs[i] = c[i]/stepsize - rhs[i];
+
+#   ifdef COMPARE_METHODS
+    comparator.nextMatrix (c,m,1,0,"c");
+    comparator.nextMatrix (rhs,m,1,0,"rhs");
+#   endif
+
+
+
+ 
+
+#ifndef DIRECT_CHOLESKY
+    // solve the LCP problem and get lambda.
+    // this will destroy A but that's okay
+#   ifdef TIMING
+    dTimerNow ("solving LCP problem");
+#   endif
+    ALLOCA(dReal,lambda,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (lambda == NULL) {
+      UNALLOCA(rhs);
+      UNALLOCA(tmp1);
+      UNALLOCA(A);
+      UNALLOCA(JinvM);
+      UNALLOCA(J);
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(vnew);
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    ALLOCA(dReal,residual,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (residual == NULL) {
+      UNALLOCA(lambda);
+      UNALLOCA(rhs);
+      UNALLOCA(tmp1);
+      UNALLOCA(A);
+      UNALLOCA(JinvM);
+      UNALLOCA(J);
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(vnew);
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    dSolveLCP (m,A,lambda,rhs,residual,nub,lo,hi,findex);
+    UNALLOCA(residual);
+    UNALLOCA(lambda);
+
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (dMemoryFlag == d_MEMORY_OUT_OF_MEMORY)
+      return;
+#endif
+
+
+#else
+
+    // OLD WAY - direct factor and solve
+
+    // factorize A (L*L'=A)
+#   ifdef TIMING
+    dTimerNow ("factorize A");
+#   endif
+    ALLOCA(dReal,L,m*mskip*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (L == NULL) {
+      UNALLOCA(rhs);
+      UNALLOCA(tmp1);
+      UNALLOCA(A);
+      UNALLOCA(JinvM);
+      UNALLOCA(J);
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(vnew);
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    memcpy (L,A,m*mskip*sizeof(dReal));
+    if (dFactorCholesky (L,m)==0) dDebug (0,"A is not positive definite");
+
+    // compute lambda
+#   ifdef TIMING
+    dTimerNow ("compute lambda");
+#   endif
+    ALLOCA(dReal,lambda,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (lambda == NULL) {
+      UNALLOCA(L);
+      UNALLOCA(rhs);
+      UNALLOCA(tmp1);
+      UNALLOCA(A);
+      UNALLOCA(JinvM);
+      UNALLOCA(J);
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(vnew);
+      UNALLOCA(v);
+      UNALLOCA(fe);
+      UNALLOCA(invM);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    memcpy (lambda,rhs,m * sizeof(dReal));
+    dSolveCholesky (L,lambda,m);
+#endif
+
+#   ifdef COMPARE_METHODS
+    comparator.nextMatrix (lambda,m,1,0,"lambda");
+#   endif
+
+    // compute the velocity update `vnew'
+#   ifdef TIMING
+    dTimerNow ("compute velocity update");
+#   endif
+    dMultiply1 (tmp1,J,lambda,n6,m,1);
+    for (i=0; i<n6; i++) tmp1[i] += fe[i];
+    dMultiply0 (vnew,invM,tmp1,n6,n6,1);
+    for (i=0; i<n6; i++) vnew[i] = v[i] + stepsize*vnew[i];
+
+#ifdef REPORT_ERROR
+    // see if the constraint has worked: compute J*vnew and make sure it equals
+    // `c' (to within a certain tolerance).
+#   ifdef TIMING
+    dTimerNow ("verify constraint equation");
+#   endif
+    dMultiply0 (tmp1,J,vnew,m,n6,1);
+    dReal err = 0;
+    for (i=0; i<m; i++) {
+                err += dFabs(tmp1[i]-c[i]);
+    }
+        printf ("total constraint error=%.6e\n",err);
+#endif
+
+    UNALLOCA(c);
+    UNALLOCA(cfm);
+    UNALLOCA(lo);
+    UNALLOCA(hi);
+    UNALLOCA(findex);
+    UNALLOCA(J);
+    UNALLOCA(JinvM);
+    UNALLOCA(A);
+    UNALLOCA(tmp1);
+    UNALLOCA(rhs);
+    UNALLOCA(lambda); 
+    UNALLOCA(L);
+  }
+  else {
+    // no constraints
+    dMultiply0 (vnew,invM,fe,n6,n6,1);
+    for (i=0; i<n6; i++) vnew[i] = v[i] + stepsize*vnew[i];
+  }
+
+#ifdef COMPARE_METHODS
+  comparator.nextMatrix (vnew,n6,1,0,"vnew");
+#endif
+
+  // apply the velocity update to the bodies
+#ifdef TIMING
+  dTimerNow ("update velocity");
+#endif
+  for (i=0; i<nb; i++) {
+    for (j=0; j<3; j++) body[i]->lvel[j] = vnew[i*6+j];
+    for (j=0; j<3; j++) body[i]->avel[j] = vnew[i*6+3+j];
+  }
+
+  // update the position and orientation from the new linear/angular velocity
+  // (over the given timestep)
+#ifdef TIMING
+  dTimerNow ("update position");
+#endif
+  for (i=0; i<nb; i++) dxStepBody (body[i],stepsize);
+
+#ifdef TIMING
+  dTimerNow ("tidy up");
+#endif
+
+  // zero all force accumulators
+  for (i=0; i<nb; i++) {
+    body[i]->facc[0] = 0;
+    body[i]->facc[1] = 0;
+    body[i]->facc[2] = 0;
+    body[i]->facc[3] = 0;
+    body[i]->tacc[0] = 0;
+    body[i]->tacc[1] = 0;
+    body[i]->tacc[2] = 0;
+    body[i]->tacc[3] = 0;
+  }
+
+#ifdef TIMING
+  dTimerEnd();
+  if (m > 0) dTimerReport (stdout,1);
+#endif
+
+  UNALLOCA(joint);
+  UNALLOCA(I);
+  UNALLOCA(invI);
+  UNALLOCA(info);
+  UNALLOCA(ofs);
+  UNALLOCA(invM);
+  UNALLOCA(fe);
+  UNALLOCA(v);
+  UNALLOCA(vnew);
+}
+
+//****************************************************************************
+// an optimized version of dInternalStepIsland1()
+
+void dInternalStepIsland_x2 (dxWorld *world, dxBody * const *body, int nb,
+                             dxJoint * const *_joint, int nj, dReal stepsize)
+{
+  int i,j,k;
+#ifdef TIMING
+  dTimerStart("preprocessing");
+#endif
+
+  dReal stepsize1 = dRecip(stepsize);
+
+  // number all bodies in the body list - set their tag values
+  for (i=0; i<nb; i++) body[i]->tag = i;
+
+  // 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).
+  ALLOCA(dxJoint*,joint,nj*sizeof(dxJoint*));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+  if (joint == NULL) {
+    dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+    return;
+  }
+#endif
+  memcpy (joint,_joint,nj * sizeof(dxJoint*));
+
+  // 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.
+#ifdef dGYROSCOPIC  
+  ALLOCA(dReal,I,3*nb*4*sizeof(dReal));
+# ifdef dUSE_MALLOC_FOR_ALLOCA
+  if (I == NULL) {
+    UNALLOCA(joint);
+    dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+    return;
+  }
+# endif
+#else
+  dReal *I = NULL;
+#endif // for dGYROSCOPIC
+
+  ALLOCA(dReal,invI,3*nb*4*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+  if (invI == NULL) {
+    UNALLOCA(I);
+    UNALLOCA(joint);
+    dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+    return;
+  }
+#endif
+
+  //dSetZero (I,3*nb*4);
+  //dSetZero (invI,3*nb*4);
+  for (i=0; i<nb; i++) {
+    dReal tmp[12];
+
+    // compute inverse inertia tensor in global frame
+    dMULTIPLY2_333 (tmp,body[i]->invI,body[i]->posr.R);
+    dMULTIPLY0_333 (invI+i*12,body[i]->posr.R,tmp);
+#ifdef dGYROSCOPIC
+    // compute inertia tensor in global frame
+                dMULTIPLY2_333 (tmp,body[i]->mass.I,body[i]->posr.R);
+                dMULTIPLY0_333 (I+i*12,body[i]->posr.R,tmp);
+
+    // compute rotational force
+    dMULTIPLY0_331 (tmp,I+i*12,body[i]->avel);
+    dCROSS (body[i]->tacc,-=,body[i]->avel,tmp);
+#endif
+  }
+
+  // add the gravity force to all bodies
+  for (i=0; i<nb; i++) {
+    if ((body[i]->flags & dxBodyNoGravity)==0) {
+      body[i]->facc[0] += body[i]->mass.mass * world->gravity[0];
+      body[i]->facc[1] += body[i]->mass.mass * world->gravity[1];
+      body[i]->facc[2] += body[i]->mass.mass * world->gravity[2];
+    }
+  }
+
+  // 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;
+  ALLOCA(dxJoint::Info1,info,nj*sizeof(dxJoint::Info1));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+  if (info == NULL) {
+    UNALLOCA(invI);
+    UNALLOCA(I);
+    UNALLOCA(joint);
+    dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+    return;
+  }
+#endif
+  ALLOCA(int,ofs,nj*sizeof(int));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+  if (ofs == NULL) {
+    UNALLOCA(info);
+    UNALLOCA(invI);
+    UNALLOCA(I);
+    UNALLOCA(joint);
+    dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+    return;
+  }
+#endif
+  for (i=0, j=0; j<nj; j++) {   // i=dest, j=src
+    joint[j]->vtable->getInfo1 (joint[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) {
+      joint[i] = joint[j];
+      joint[i]->tag = i;
+      i++;
+    }
+    else {
+      joint[j]->tag = -1;
+    }
+  }
+  nj = i;
+
+  // the purely unbounded constraints
+  for (i=0; i<nj; i++) if (info[i].nub == info[i].m) {
+    ofs[i] = m;
+    m += info[i].m;
+  }
+  int nub = m;
+  // the mixed unbounded + LCP constraints
+  for (i=0; i<nj; i++) if (info[i].nub > 0 && info[i].nub < info[i].m) {
+    ofs[i] = m;
+    m += info[i].m;
+  }
+  // the purely LCP constraints
+  for (i=0; i<nj; i++) if (info[i].nub == 0) {
+    ofs[i] = m;
+    m += info[i].m;
+  }
+
+  // this will be set to the force due to the constraints
+  ALLOCA(dReal,cforce,nb*8*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+  if (cforce == NULL) {
+    UNALLOCA(ofs);
+    UNALLOCA(info);
+    UNALLOCA(invI);
+    UNALLOCA(I);
+    UNALLOCA(joint);
+    dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+    return;
+  }
+#endif
+  dSetZero (cforce,nb*8);
+
+  // if there are constraints, compute cforce
+  if (m > 0) {
+    // 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.
+    ALLOCA(dReal,c,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (c == NULL) {
+      UNALLOCA(cforce);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    ALLOCA(dReal,cfm,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (cfm == NULL) {
+      UNALLOCA(c);
+      UNALLOCA(cforce);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    ALLOCA(dReal,lo,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (lo == NULL) {
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(cforce);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    ALLOCA(dReal,hi,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (hi == NULL) {
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(cforce);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    ALLOCA(int,findex,m*sizeof(int));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (findex == NULL) {
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(cforce);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    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
+    ALLOCA(dReal,J,2*m*8*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (J == NULL) {
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(cforce);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    dSetZero (J,2*m*8);
+    dxJoint::Info2 Jinfo;
+    Jinfo.rowskip = 8;
+    Jinfo.fps = stepsize1;
+    Jinfo.erp = world->global_erp;
+    for (i=0; i<nj; i++) {
+      Jinfo.J1l = J + 2*8*ofs[i];
+      Jinfo.J1a = Jinfo.J1l + 4;
+      Jinfo.J2l = Jinfo.J1l + 8*info[i].m;
+      Jinfo.J2a = Jinfo.J2l + 4;
+      Jinfo.c = c + ofs[i];
+      Jinfo.cfm = cfm + ofs[i];
+      Jinfo.lo = lo + ofs[i];
+      Jinfo.hi = hi + ofs[i];
+      Jinfo.findex = findex + ofs[i];
+      joint[i]->vtable->getInfo2 (joint[i],&Jinfo);
+      // adjust returned findex values for global index numbering
+      for (j=0; j<info[i].m; j++) {
+        if (findex[ofs[i] + j] >= 0) findex[ofs[i] + j] += ofs[i];
+      }
+    }
+
+    // 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
+    ALLOCA(dReal,JinvM,2*m*8*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (JinvM == NULL) {
+      UNALLOCA(J);
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(cforce);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    dSetZero (JinvM,2*m*8);
+    for (i=0; i<nj; i++) {
+      int b = joint[i]->node[0].body->tag;
+      dReal body_invMass = body[b]->invMass;
+      dReal *body_invI = invI + b*12;
+      dReal *Jsrc = J + 2*8*ofs[i];
+      dReal *Jdst = JinvM + 2*8*ofs[i];
+      for (j=info[i].m-1; j>=0; j--) {
+        for (k=0; k<3; k++) Jdst[k] = Jsrc[k] * body_invMass;
+        dMULTIPLY0_133 (Jdst+4,Jsrc+4,body_invI);
+        Jsrc += 8;
+        Jdst += 8;
+      }
+      if (joint[i]->node[1].body) {
+        b = joint[i]->node[1].body->tag;
+        body_invMass = body[b]->invMass;
+        body_invI = invI + b*12;
+        for (j=info[i].m-1; j>=0; j--) {
+          for (k=0; k<3; k++) Jdst[k] = Jsrc[k] * body_invMass;
+          dMULTIPLY0_133 (Jdst+4,Jsrc+4,body_invI);
+          Jsrc += 8;
+          Jdst += 8;
+        }
+      }
+    }
+
+    // now compute A = JinvM * J'. A's rows and columns are grouped by joint,
+    // i.e. in the same way as the rows of J. block (i,j) of A is only nonzero
+    // if joints i and j have at least one body in common. this fact suggests
+    // the algorithm used to fill A:
+    //
+    //    for b = all bodies
+    //      n = number of joints attached to body b
+    //      for i = 1..n
+    //        for j = i+1..n
+    //          ii = actual joint number for i
+    //          jj = actual joint number for j
+    //          // (ii,jj) will be set to all pairs of joints around body b
+    //          compute blockwise: A(ii,jj) += JinvM(ii) * J(jj)'
+    //
+    // this algorithm catches all pairs of joints that have at least one body
+    // in common. it does not compute the diagonal blocks of A however -
+    // another similar algorithm does that.
+
+    int mskip = dPAD(m);
+    ALLOCA(dReal,A,m*mskip*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (A == NULL) {
+      UNALLOCA(JinvM);
+      UNALLOCA(J);
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(cforce);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    dSetZero (A,m*mskip);
+    for (i=0; i<nb; i++) {
+      for (dxJointNode *n1=body[i]->firstjoint; n1; n1=n1->next) {
+        for (dxJointNode *n2=n1->next; n2; n2=n2->next) {
+          // get joint numbers and ensure ofs[j1] >= ofs[j2]
+          int j1 = n1->joint->tag;
+          int j2 = n2->joint->tag;
+          if (ofs[j1] < ofs[j2]) {
+            int tmp = j1;
+            j1 = j2;
+            j2 = tmp;
+          }
+
+          // if either joint was tagged as -1 then it is an inactive (m=0)
+          // joint that should not be considered
+          if (j1==-1 || j2==-1) continue;
+
+          // determine if body i is the 1st or 2nd body of joints j1 and j2
+          int jb1 = (joint[j1]->node[1].body == body[i]);
+          int jb2 = (joint[j2]->node[1].body == body[i]);
+          // jb1/jb2 must be 0 for joints with only one body
+          dIASSERT(joint[j1]->node[1].body || jb1==0);
+          dIASSERT(joint[j2]->node[1].body || jb2==0);
+
+          // set block of A
+          MultiplyAdd2_p8r (A + ofs[j1]*mskip + ofs[j2],
+                            JinvM + 2*8*ofs[j1] + jb1*8*info[j1].m,
+                            J     + 2*8*ofs[j2] + jb2*8*info[j2].m,
+                            info[j1].m,info[j2].m, mskip);
+        }
+      }
+    }
+    // compute diagonal blocks of A
+    for (i=0; i<nj; i++) {
+      Multiply2_p8r (A + ofs[i]*(mskip+1),
+                     JinvM + 2*8*ofs[i],
+                     J + 2*8*ofs[i],
+                     info[i].m,info[i].m, mskip);
+      if (joint[i]->node[1].body) {
+        MultiplyAdd2_p8r (A + ofs[i]*(mskip+1),
+                          JinvM + 2*8*ofs[i] + 8*info[i].m,
+                          J + 2*8*ofs[i] + 8*info[i].m,
+                          info[i].m,info[i].m, mskip);
+      }
+    }
+
+    // add cfm to the diagonal of A
+    for (i=0; i<m; i++) A[i*mskip+i] += cfm[i] * stepsize1;
+
+#   ifdef COMPARE_METHODS
+    comparator.nextMatrix (A,m,m,1,"A");
+#   endif
+
+    // compute the right hand side `rhs'
+#   ifdef TIMING
+    dTimerNow ("compute rhs");
+#   endif
+    ALLOCA(dReal,tmp1,nb*8*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (tmp1 == NULL) {
+      UNALLOCA(A);
+      UNALLOCA(JinvM);
+      UNALLOCA(J);
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(cforce);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    //dSetZero (tmp1,nb*8);
+    // put v/h + invM*fe into tmp1
+    for (i=0; i<nb; i++) {
+      dReal body_invMass = body[i]->invMass;
+      dReal *body_invI = invI + i*12;
+      for (j=0; j<3; j++) tmp1[i*8+j] = body[i]->facc[j] * body_invMass +
+                            body[i]->lvel[j] * stepsize1;
+      dMULTIPLY0_331 (tmp1 + i*8 + 4,body_invI,body[i]->tacc);
+      for (j=0; j<3; j++) tmp1[i*8+4+j] += body[i]->avel[j] * stepsize1;
+    }
+    // put J*tmp1 into rhs
+    ALLOCA(dReal,rhs,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (rhs == NULL) {
+      UNALLOCA(tmp1);
+      UNALLOCA(A);
+      UNALLOCA(JinvM);
+      UNALLOCA(J);
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(cforce);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    //dSetZero (rhs,m);
+    for (i=0; i<nj; i++) {
+      dReal *JJ = J + 2*8*ofs[i];
+      Multiply0_p81 (rhs+ofs[i],JJ,
+                     tmp1 + 8*joint[i]->node[0].body->tag, info[i].m);
+      if (joint[i]->node[1].body) {
+        MultiplyAdd0_p81 (rhs+ofs[i],JJ + 8*info[i].m,
+                          tmp1 + 8*joint[i]->node[1].body->tag, info[i].m);
+      }
+    }
+    // complete rhs
+    for (i=0; i<m; i++) rhs[i] = c[i]*stepsize1 - rhs[i];
+
+#   ifdef COMPARE_METHODS
+    comparator.nextMatrix (c,m,1,0,"c");
+    comparator.nextMatrix (rhs,m,1,0,"rhs");
+#   endif
+
+    // solve the LCP problem and get lambda.
+    // this will destroy A but that's okay
+#   ifdef TIMING
+    dTimerNow ("solving LCP problem");
+#   endif
+    ALLOCA(dReal,lambda,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (lambda == NULL) {
+      UNALLOCA(rhs);
+      UNALLOCA(tmp1);
+      UNALLOCA(A);
+      UNALLOCA(JinvM);
+      UNALLOCA(J);
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(cforce);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    ALLOCA(dReal,residual,m*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (residual == NULL) {
+      UNALLOCA(lambda);
+      UNALLOCA(rhs);
+      UNALLOCA(tmp1);
+      UNALLOCA(A);
+      UNALLOCA(JinvM);
+      UNALLOCA(J);
+      UNALLOCA(findex);
+      UNALLOCA(hi);
+      UNALLOCA(lo);
+      UNALLOCA(cfm);
+      UNALLOCA(c);
+      UNALLOCA(cforce);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+    dSolveLCP (m,A,lambda,rhs,residual,nub,lo,hi,findex);
+
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (dMemoryFlag == d_MEMORY_OUT_OF_MEMORY)
+      return;
+#endif
+
+
+//  OLD WAY - direct factor and solve
+//
+//    // factorize A (L*L'=A)
+//#   ifdef TIMING
+//    dTimerNow ("factorize A");
+//#   endif
+//    dReal *L = (dReal*) ALLOCA (m*mskip*sizeof(dReal));
+//    memcpy (L,A,m*mskip*sizeof(dReal));
+//#   ifdef FAST_FACTOR
+//    dFastFactorCholesky (L,m);  // does not report non positive definiteness
+//#   else
+//    if (dFactorCholesky (L,m)==0) dDebug (0,"A is not positive definite");
+//#   endif
+//
+//    // compute lambda
+//#   ifdef TIMING
+//    dTimerNow ("compute lambda");
+//#   endif
+//    dReal *lambda = (dReal*) ALLOCA (m * sizeof(dReal));
+//    memcpy (lambda,rhs,m * sizeof(dReal));
+//    dSolveCholesky (L,lambda,m);
+
+#   ifdef COMPARE_METHODS
+    comparator.nextMatrix (lambda,m,1,0,"lambda");
+#   endif
+
+    // compute the constraint force `cforce'
+#   ifdef TIMING
+    dTimerNow ("compute constraint force");
+#   endif
+    // compute cforce = J'*lambda
+    for (i=0; i<nj; i++) {
+      dReal *JJ = J + 2*8*ofs[i];
+      dxBody* b1 = joint[i]->node[0].body;
+      dxBody* b2 = joint[i]->node[1].body;
+      dJointFeedback *fb = joint[i]->feedback;
+
+      if (fb) {
+        // the user has requested 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.
+        dReal data1[8],data2[8];
+        Multiply1_8q1 (data1, JJ, lambda+ofs[i], info[i].m);
+        dReal *cf1 = cforce + 8*b1->tag;
+        cf1[0] += (fb->f1[0] = data1[0]);
+        cf1[1] += (fb->f1[1] = data1[1]);
+        cf1[2] += (fb->f1[2] = data1[2]);
+        cf1[4] += (fb->t1[0] = data1[4]);
+        cf1[5] += (fb->t1[1] = data1[5]);
+        cf1[6] += (fb->t1[2] = data1[6]);
+        if (b2){
+          Multiply1_8q1 (data2, JJ + 8*info[i].m, lambda+ofs[i], info[i].m);
+          dReal *cf2 = cforce + 8*b2->tag;
+          cf2[0] += (fb->f2[0] = data2[0]);
+          cf2[1] += (fb->f2[1] = data2[1]);
+          cf2[2] += (fb->f2[2] = data2[2]);
+          cf2[4] += (fb->t2[0] = data2[4]);
+          cf2[5] += (fb->t2[1] = data2[5]);
+          cf2[6] += (fb->t2[2] = data2[6]);
+        }
+      }
+      else {
+        // no feedback is required, let's compute cforce the faster way
+        MultiplyAdd1_8q1 (cforce + 8*b1->tag,JJ, lambda+ofs[i], info[i].m);
+        if (b2) {
+          MultiplyAdd1_8q1 (cforce + 8*b2->tag,
+                            JJ + 8*info[i].m, lambda+ofs[i], info[i].m);
+        }
+      }
+    }
+    UNALLOCA(c);
+    UNALLOCA(cfm);
+    UNALLOCA(lo);
+    UNALLOCA(hi);
+    UNALLOCA(findex);
+    UNALLOCA(J);
+    UNALLOCA(JinvM);
+    UNALLOCA(A);
+    UNALLOCA(tmp1);
+    UNALLOCA(rhs);
+    UNALLOCA(lambda);
+    UNALLOCA(residual);
+  }
+
+  // compute the velocity update
+#ifdef TIMING
+  dTimerNow ("compute velocity update");
+#endif
+
+  // add fe to cforce
+  for (i=0; i<nb; i++) {
+    for (j=0; j<3; j++) cforce[i*8+j] += body[i]->facc[j];
+    for (j=0; j<3; j++) cforce[i*8+4+j] += body[i]->tacc[j];
+  }
+  // multiply cforce by stepsize
+  for (i=0; i < nb*8; i++) cforce[i] *= stepsize;
+  // add invM * cforce to the body velocity
+  for (i=0; i<nb; i++) {
+    dReal body_invMass = body[i]->invMass;
+    dReal *body_invI = invI + i*12;
+    for (j=0; j<3; j++) body[i]->lvel[j] += body_invMass * cforce[i*8+j];
+    dMULTIPLYADD0_331 (body[i]->avel,body_invI,cforce+i*8+4);
+  }
+
+  // update the position and orientation from the new linear/angular velocity
+  // (over the given timestep)
+# ifdef TIMING
+  dTimerNow ("update position");
+# endif
+  for (i=0; i<nb; i++) dxStepBody (body[i],stepsize);
+
+#ifdef COMPARE_METHODS
+  ALLOCA(dReal,tmp, ALLOCA (nb*6*sizeof(dReal));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (tmp == NULL) {
+      UNALLOCA(cforce);
+      UNALLOCA(ofs);
+      UNALLOCA(info);
+      UNALLOCA(invI);
+      UNALLOCA(I);
+      UNALLOCA(joint);
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+  for (i=0; i<nb; i++) {
+    for (j=0; j<3; j++) tmp_vnew[i*6+j] = body[i]->lvel[j];
+    for (j=0; j<3; j++) tmp_vnew[i*6+3+j] = body[i]->avel[j];
+  }
+  comparator.nextMatrix (tmp_vnew,nb*6,1,0,"vnew");
+  UNALLOCA(tmp);
+#endif
+
+#ifdef TIMING
+  dTimerNow ("tidy up");
+#endif
+
+  // zero all force accumulators
+  for (i=0; i<nb; i++) {
+    body[i]->facc[0] = 0;
+    body[i]->facc[1] = 0;
+    body[i]->facc[2] = 0;
+    body[i]->facc[3] = 0;
+    body[i]->tacc[0] = 0;
+    body[i]->tacc[1] = 0;
+    body[i]->tacc[2] = 0;
+    body[i]->tacc[3] = 0;
+  }
+
+#ifdef TIMING
+  dTimerEnd();
+  if (m > 0) dTimerReport (stdout,1);
+#endif
+      
+  UNALLOCA(joint);
+  UNALLOCA(I);
+  UNALLOCA(invI);
+  UNALLOCA(info);
+  UNALLOCA(ofs);
+  UNALLOCA(cforce);
+}
+
+//****************************************************************************
+
+void dInternalStepIsland (dxWorld *world, dxBody * const *body, int nb,
+                          dxJoint * const *joint, int nj, dReal stepsize)
+{
+
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+  dMemoryFlag = d_MEMORY_OK;
+#endif
+
+#ifndef COMPARE_METHODS
+  dInternalStepIsland_x2 (world,body,nb,joint,nj,stepsize);
+
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (dMemoryFlag == d_MEMORY_OUT_OF_MEMORY) {
+      REPORT_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+
+#endif
+
+#ifdef COMPARE_METHODS
+  int i;
+
+  // save body state
+  ALLOCA(dxBody,state,nb*sizeof(dxBody));
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (state == NULL) {
+      dMemoryFlag = d_MEMORY_OUT_OF_MEMORY;
+      REPORT_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+  for (i=0; i<nb; i++) memcpy (state+i,body[i],sizeof(dxBody));
+
+  // take slow step
+  comparator.reset();
+  dInternalStepIsland_x1 (world,body,nb,joint,nj,stepsize);
+  comparator.end();
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+  if (dMemoryFlag == d_MEMORY_OUT_OF_MEMORY) {
+    UNALLOCA(state);
+    REPORT_OUT_OF_MEMORY;
+    return;
+  }
+#endif
+
+  // restore state
+  for (i=0; i<nb; i++) memcpy (body[i],state+i,sizeof(dxBody));
+
+  // take fast step
+  dInternalStepIsland_x2 (world,body,nb,joint,nj,stepsize);
+  comparator.end();
+#ifdef dUSE_MALLOC_FOR_ALLOCA
+    if (dMemoryFlag == d_MEMORY_OUT_OF_MEMORY) {
+      UNALLOCA(state);
+      REPORT_OUT_OF_MEMORY;
+      return;
+    }
+#endif
+
+  //comparator.dump();
+  //_exit (1);
+  UNALLOCA(state);
+#endif
+}
+
+