source: code/branches/ode/ode-0.9/ode/src/quickstep.cpp @ 216

/*************************************************************************
 *                                                                       *
 * Open Dynamics Engine, Copyright (C) 2001-2003 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 <ode/misc.h>
#include "lcp.h"
#include "util.h"

#define ALLOCA dALLOCA16

typedef const dReal *dRealPtr;
typedef dReal *dRealMutablePtr;
#define dRealArray(name,n) dReal name[n];
#define dRealAllocaArray(name,n) dReal *name = (dReal*) ALLOCA ((n)*sizeof(dReal));

//***************************************************************************
// configuration

// for the SOR and CG methods:
// uncomment the following line to use warm starting. this definitely
// help for motor-driven joints. unfortunately it appears to hurt
// with high-friction contacts using the SOR method. use with care

//#define WARM_STARTING 1


// for the SOR method:
// uncomment the following line to determine a new constraint-solving
// order for each iteration. however, the qsort per iteration is expensive,
// and the optimal order is somewhat problem dependent.
// @@@ try the leaf->root ordering.

//#define REORDER_CONSTRAINTS 1


// for the SOR method:
// uncomment the following line to randomly reorder constraint rows
// during the solution. depending on the situation, this can help a lot
// or hardly at all, but it doesn't seem to hurt.

#define RANDOMLY_REORDER_CONSTRAINTS 1

//****************************************************************************
// special matrix multipliers

// multiply block of B matrix (q x 6) with 12 dReal per row with C vektor (q)
static void Multiply1_12q1 (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*12] * C[k];
  A[0] = sum;
  sum = 0;
  for (k=0; k<q; k++) sum += B[1+k*12] * C[k];
  A[1] = sum;
  sum = 0;
  for (k=0; k<q; k++) sum += B[2+k*12] * C[k];
  A[2] = sum;
  sum = 0;
  for (k=0; k<q; k++) sum += B[3+k*12] * C[k];
  A[3] = sum;
  sum = 0;
  for (k=0; k<q; k++) sum += B[4+k*12] * C[k];
  A[4] = sum;
  sum = 0;
  for (k=0; k<q; k++) sum += B[5+k*12] * C[k];
  A[5] = sum;
}

//***************************************************************************
// testing stuff

#ifdef TIMING
#define IFTIMING(x) x
#else
#define IFTIMING(x) /* */
#endif

//***************************************************************************
// various common computations involving the matrix J

// compute iMJ = inv(M)*J'

static void compute_invM_JT (int m, dRealMutablePtr J, dRealMutablePtr iMJ, int *jb,
        dxBody * const *body, dRealPtr invI)
{
        int i,j;
        dRealMutablePtr iMJ_ptr = iMJ;
        dRealMutablePtr J_ptr = J;
        for (i=0; i<m; i++) {
                int b1 = jb[i*2];
                int b2 = jb[i*2+1];
                dReal k = body[b1]->invMass;
                for (j=0; j<3; j++) iMJ_ptr[j] = k*J_ptr[j];
                dMULTIPLY0_331 (iMJ_ptr + 3, invI + 12*b1, J_ptr + 3);
                if (b2 >= 0) {
                        k = body[b2]->invMass;
                        for (j=0; j<3; j++) iMJ_ptr[j+6] = k*J_ptr[j+6];
                        dMULTIPLY0_331 (iMJ_ptr + 9, invI + 12*b2, J_ptr + 9);
                }
                J_ptr += 12;
                iMJ_ptr += 12;
        }
}


// compute out = inv(M)*J'*in.

static void multiply_invM_JT (int m, int nb, dRealMutablePtr iMJ, int *jb,
        dRealMutablePtr in, dRealMutablePtr out)
{
        int i,j;
        dSetZero (out,6*nb);
        dRealPtr iMJ_ptr = iMJ;
        for (i=0; i<m; i++) {
                int b1 = jb[i*2];
                int b2 = jb[i*2+1];
                dRealMutablePtr out_ptr = out + b1*6;
                for (j=0; j<6; j++) out_ptr[j] += iMJ_ptr[j] * in[i];
                iMJ_ptr += 6;
                if (b2 >= 0) {
                        out_ptr = out + b2*6;
                        for (j=0; j<6; j++) out_ptr[j] += iMJ_ptr[j] * in[i];
                }
                iMJ_ptr += 6;
        }
}


// compute out = J*in.

static void multiply_J (int m, dRealMutablePtr J, int *jb,
        dRealMutablePtr in, dRealMutablePtr out)
{
        int i,j;
        dRealPtr J_ptr = J;
        for (i=0; i<m; i++) {
                int b1 = jb[i*2];
                int b2 = jb[i*2+1];
                dReal sum = 0;
                dRealMutablePtr in_ptr = in + b1*6;
                for (j=0; j<6; j++) sum += J_ptr[j] * in_ptr[j];
                J_ptr += 6;
                if (b2 >= 0) {
                        in_ptr = in + b2*6;
                        for (j=0; j<6; j++) sum += J_ptr[j] * in_ptr[j];
                }
                J_ptr += 6;
                out[i] = sum;
        }
}


// compute out = (J*inv(M)*J' + cfm)*in.
// use z as an nb*6 temporary.

static void multiply_J_invM_JT (int m, int nb, dRealMutablePtr J, dRealMutablePtr iMJ, int *jb,
        dRealPtr cfm, dRealMutablePtr z, dRealMutablePtr in, dRealMutablePtr out)
{
        multiply_invM_JT (m,nb,iMJ,jb,in,z);
        multiply_J (m,J,jb,z,out);

        // add cfm
        for (int i=0; i<m; i++) out[i] += cfm[i] * in[i];
}

//***************************************************************************
// conjugate gradient method with jacobi preconditioner
// THIS IS EXPERIMENTAL CODE that doesn't work too well, so it is ifdefed out.
//
// adding CFM seems to be critically important to this method.

#if 0

static inline dReal dot (int n, dRealPtr x, dRealPtr y)
{
        dReal sum=0;
        for (int i=0; i<n; i++) sum += x[i]*y[i];
        return sum;
}


// x = y + z*alpha

static inline void add (int n, dRealMutablePtr x, dRealPtr y, dRealPtr z, dReal alpha)
{
        for (int i=0; i<n; i++) x[i] = y[i] + z[i]*alpha;
}


static void CG_LCP (int m, int nb, dRealMutablePtr J, int *jb, dxBody * const *body,
        dRealPtr invI, dRealMutablePtr lambda, dRealMutablePtr fc, dRealMutablePtr b,
        dRealMutablePtr lo, dRealMutablePtr hi, dRealPtr cfm, int *findex,
        dxQuickStepParameters *qs)
{
        int i,j;
        const int num_iterations = qs->num_iterations;

        // precompute iMJ = inv(M)*J'
        dRealAllocaArray (iMJ,m*12);
        compute_invM_JT (m,J,iMJ,jb,body,invI);

        dReal last_rho = 0;
        dRealAllocaArray (r,m);
        dRealAllocaArray (z,m);
        dRealAllocaArray (p,m);
        dRealAllocaArray (q,m);

        // precompute 1 / diagonals of A
        dRealAllocaArray (Ad,m);
        dRealPtr iMJ_ptr = iMJ;
        dRealPtr J_ptr = J;
        for (i=0; i<m; i++) {
                dReal sum = 0;
                for (j=0; j<6; j++) sum += iMJ_ptr[j] * J_ptr[j];
                if (jb[i*2+1] >= 0) {
                        for (j=6; j<12; j++) sum += iMJ_ptr[j] * J_ptr[j];
                }
                iMJ_ptr += 12;
                J_ptr += 12;
                Ad[i] = REAL(1.0) / (sum + cfm[i]);
        }

#ifdef WARM_STARTING
        // compute residual r = b - A*lambda
        multiply_J_invM_JT (m,nb,J,iMJ,jb,cfm,fc,lambda,r);
        for (i=0; i<m; i++) r[i] = b[i] - r[i];
#else
        dSetZero (lambda,m);
        memcpy (r,b,m*sizeof(dReal));           // residual r = b - A*lambda
#endif

        for (int iteration=0; iteration < num_iterations; iteration++) {
                for (i=0; i<m; i++) z[i] = r[i]*Ad[i];  // z = inv(M)*r
                dReal rho = dot (m,r,z);                // rho = r'*z

                // @@@
                // we must check for convergence, otherwise rho will go to 0 if
                // we get an exact solution, which will introduce NaNs into the equations.
                if (rho < 1e-10) {
                        printf ("CG returned at iteration %d\n",iteration);
                        break;
                }

                if (iteration==0) {
                        memcpy (p,z,m*sizeof(dReal));   // p = z
                }
                else {
                        add (m,p,z,p,rho/last_rho);     // p = z + (rho/last_rho)*p
                }

                // compute q = (J*inv(M)*J')*p
                multiply_J_invM_JT (m,nb,J,iMJ,jb,cfm,fc,p,q);

                dReal alpha = rho/dot (m,p,q);          // alpha = rho/(p'*q)
                add (m,lambda,lambda,p,alpha);          // lambda = lambda + alpha*p
                add (m,r,r,q,-alpha);                   // r = r - alpha*q
                last_rho = rho;
        }

        // compute fc = inv(M)*J'*lambda
        multiply_invM_JT (m,nb,iMJ,jb,lambda,fc);

#if 0
        // measure solution error
        multiply_J_invM_JT (m,nb,J,iMJ,jb,cfm,fc,lambda,r);
        dReal error = 0;
        for (i=0; i<m; i++) error += dFabs(r[i] - b[i]);
        printf ("lambda error = %10.6e\n",error);
#endif
}

#endif

//***************************************************************************
// SOR-LCP method

// nb is the number of bodies in the body array.
// J is an m*12 matrix of constraint rows
// jb is an array of first and second body numbers for each constraint row
// invI is the global frame inverse inertia for each body (stacked 3x3 matrices)
//
// this returns lambda and fc (the constraint force).
// note: fc is returned as inv(M)*J'*lambda, the constraint force is actually J'*lambda
//
// b, lo and hi are modified on exit


struct IndexError {
        dReal error;            // error to sort on
        int findex;
        int index;              // row index
};


#ifdef REORDER_CONSTRAINTS

static int compare_index_error (const void *a, const void *b)
{
        const IndexError *i1 = (IndexError*) a;
        const IndexError *i2 = (IndexError*) b;
        if (i1->findex < 0 && i2->findex >= 0) return -1;
        if (i1->findex >= 0 && i2->findex < 0) return 1;
        if (i1->error < i2->error) return -1;
        if (i1->error > i2->error) return 1;
        return 0;
}

#endif


static void SOR_LCP (int m, int nb, dRealMutablePtr J, int *jb, dxBody * const *body,
        dRealPtr invI, dRealMutablePtr lambda, dRealMutablePtr fc, dRealMutablePtr b,
        dRealMutablePtr lo, dRealMutablePtr hi, dRealPtr cfm, int *findex,
        dxQuickStepParameters *qs)
{
        const int num_iterations = qs->num_iterations;
        const dReal sor_w = qs->w;              // SOR over-relaxation parameter

        int i,j;

#ifdef WARM_STARTING
        // for warm starting, this seems to be necessary to prevent
        // jerkiness in motor-driven joints. i have no idea why this works.
        for (i=0; i<m; i++) lambda[i] *= 0.9;
#else
        dSetZero (lambda,m);
#endif

#ifdef REORDER_CONSTRAINTS
        // the lambda computed at the previous iteration.
        // this is used to measure error for when we are reordering the indexes.
        dRealAllocaArray (last_lambda,m);
#endif

        // a copy of the 'hi' vector in case findex[] is being used
        dRealAllocaArray (hicopy,m);
        memcpy (hicopy,hi,m*sizeof(dReal));

        // precompute iMJ = inv(M)*J'
        dRealAllocaArray (iMJ,m*12);
        compute_invM_JT (m,J,iMJ,jb,body,invI);

        // compute fc=(inv(M)*J')*lambda. we will incrementally maintain fc
        // as we change lambda.
#ifdef WARM_STARTING
        multiply_invM_JT (m,nb,iMJ,jb,lambda,fc);
#else
        dSetZero (fc,nb*6);
#endif

        // precompute 1 / diagonals of A
        dRealAllocaArray (Ad,m);
        dRealPtr iMJ_ptr = iMJ;
        dRealMutablePtr J_ptr = J;
        for (i=0; i<m; i++) {
                dReal sum = 0;
                for (j=0; j<6; j++) sum += iMJ_ptr[j] * J_ptr[j];
                if (jb[i*2+1] >= 0) {
                        for (j=6; j<12; j++) sum += iMJ_ptr[j] * J_ptr[j];
                }
                iMJ_ptr += 12;
                J_ptr += 12;
                Ad[i] = sor_w / (sum + cfm[i]);
        }

        // scale J and b by Ad
        J_ptr = J;
        for (i=0; i<m; i++) {
                for (j=0; j<12; j++) {
                        J_ptr[0] *= Ad[i];
                        J_ptr++;
                }
                b[i] *= Ad[i];

                // scale Ad by CFM. N.B. this should be done last since it is used above
                Ad[i] *= cfm[i];
        }

        // order to solve constraint rows in
        IndexError *order = (IndexError*) ALLOCA (m*sizeof(IndexError));

#ifndef REORDER_CONSTRAINTS
        // make sure constraints with findex < 0 come first.
        j=0;
        int k=1;

        // Fill the array from both ends
        for (i=0; i<m; i++)
                if (findex[i] < 0)
                        order[j++].index = i; // Place them at the front
                else
                        order[m-k++].index = i; // Place them at the end

        dIASSERT ((j+k-1)==m); // -1 since k was started at 1 and not 0
#endif

        for (int iteration=0; iteration < num_iterations; iteration++) {

#ifdef REORDER_CONSTRAINTS
                // constraints with findex < 0 always come first.
                if (iteration < 2) {
                        // for the first two iterations, solve the constraints in
                        // the given order
                        for (i=0; i<m; i++) {
                                order[i].error = i;
                                order[i].findex = findex[i];
                                order[i].index = i;
                        }
                }
                else {
                        // sort the constraints so that the ones converging slowest
                        // get solved last. use the absolute (not relative) error.
                        for (i=0; i<m; i++) {
                                dReal v1 = dFabs (lambda[i]);
                                dReal v2 = dFabs (last_lambda[i]);
                                dReal max = (v1 > v2) ? v1 : v2;
                                if (max > 0) {
                                        //@@@ relative error: order[i].error = dFabs(lambda[i]-last_lambda[i])/max;
                                        order[i].error = dFabs(lambda[i]-last_lambda[i]);
                                }
                                else {
                                        order[i].error = dInfinity;
                                }
                                order[i].findex = findex[i];
                                order[i].index = i;
                        }
                }
                qsort (order,m,sizeof(IndexError),&compare_index_error);

                //@@@ potential optimization: swap lambda and last_lambda pointers rather
                //    than copying the data. we must make sure lambda is properly
                //    returned to the caller
                memcpy (last_lambda,lambda,m*sizeof(dReal));
#endif
#ifdef RANDOMLY_REORDER_CONSTRAINTS
                if ((iteration & 7) == 0) {
                        for (i=1; i<m; ++i) {
                                IndexError tmp = order[i];
                                int swapi = dRandInt(i+1);
                                order[i] = order[swapi];
                                order[swapi] = tmp;
                        }
                }
#endif

                for (int i=0; i<m; i++) {
                        // @@@ potential optimization: we could pre-sort J and iMJ, thereby
                        //     linearizing access to those arrays. hmmm, this does not seem
                        //     like a win, but we should think carefully about our memory
                        //     access pattern.

                        int index = order[i].index;
                        J_ptr = J + index*12;
                        iMJ_ptr = iMJ + index*12;

                        // set the limits for this constraint. note that 'hicopy' is used.
                        // this is the place where the QuickStep method differs from the
                        // direct LCP solving method, since that method only performs this
                        // limit adjustment once per time step, whereas this method performs
                        // once per iteration per constraint row.
                        // the constraints are ordered so that all lambda[] values needed have
                        // already been computed.
                        if (findex[index] >= 0) {
                                hi[index] = dFabs (hicopy[index] * lambda[findex[index]]);
                                lo[index] = -hi[index];
                        }

                        int b1 = jb[index*2];
                        int b2 = jb[index*2+1];
                        dReal delta = b[index] - lambda[index]*Ad[index];
                        dRealMutablePtr fc_ptr = fc + 6*b1;

                        // @@@ potential optimization: SIMD-ize this and the b2 >= 0 case
                        delta -=fc_ptr[0] * J_ptr[0] + fc_ptr[1] * J_ptr[1] +
                                fc_ptr[2] * J_ptr[2] + fc_ptr[3] * J_ptr[3] +
                                fc_ptr[4] * J_ptr[4] + fc_ptr[5] * J_ptr[5];
                        // @@@ potential optimization: handle 1-body constraints in a separate
                        //     loop to avoid the cost of test & jump?
                        if (b2 >= 0) {
                                fc_ptr = fc + 6*b2;
                                delta -=fc_ptr[0] * J_ptr[6] + fc_ptr[1] * J_ptr[7] +
                                        fc_ptr[2] * J_ptr[8] + fc_ptr[3] * J_ptr[9] +
                                        fc_ptr[4] * J_ptr[10] + fc_ptr[5] * J_ptr[11];
                        }

                        // compute lambda and clamp it to [lo,hi].
                        // @@@ potential optimization: does SSE have clamping instructions
                        //     to save test+jump penalties here?
                        dReal new_lambda = lambda[index] + delta;
                        if (new_lambda < lo[index]) {
                                delta = lo[index]-lambda[index];
                                lambda[index] = lo[index];
                        }
                        else if (new_lambda > hi[index]) {
                                delta = hi[index]-lambda[index];
                                lambda[index] = hi[index];
                        }
                        else {
                                lambda[index] = new_lambda;
                        }

                        //@@@ a trick that may or may not help
                        //dReal ramp = (1-((dReal)(iteration+1)/(dReal)num_iterations));
                        //delta *= ramp;

                        // update fc.
                        // @@@ potential optimization: SIMD for this and the b2 >= 0 case
                        fc_ptr = fc + 6*b1;
                        fc_ptr[0] += delta * iMJ_ptr[0];
                        fc_ptr[1] += delta * iMJ_ptr[1];
                        fc_ptr[2] += delta * iMJ_ptr[2];
                        fc_ptr[3] += delta * iMJ_ptr[3];
                        fc_ptr[4] += delta * iMJ_ptr[4];
                        fc_ptr[5] += delta * iMJ_ptr[5];
                        // @@@ potential optimization: handle 1-body constraints in a separate
                        //     loop to avoid the cost of test & jump?
                        if (b2 >= 0) {
                                fc_ptr = fc + 6*b2;
                                fc_ptr[0] += delta * iMJ_ptr[6];
                                fc_ptr[1] += delta * iMJ_ptr[7];
                                fc_ptr[2] += delta * iMJ_ptr[8];
                                fc_ptr[3] += delta * iMJ_ptr[9];
                                fc_ptr[4] += delta * iMJ_ptr[10];
                                fc_ptr[5] += delta * iMJ_ptr[11];
                        }
                }
        }
}


void dxQuickStepper (dxWorld *world, dxBody * const *body, int nb,
                     dxJoint * const *_joint, int nj, dReal stepsize)
{
        int i,j;
        IFTIMING(dTimerStart("preprocessing");)

        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).
        //@@@ do we really need to do this? we'll be sorting constraint rows individually, not joints
        dxJoint **joint = (dxJoint**) ALLOCA (nj * sizeof(dxJoint*));
        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 a vertical stack of 3x4 matrices, one per body.
        //dRealAllocaArray (I,3*4*nb);  // need to remember all I's for feedback purposes only
        dRealAllocaArray (invI,3*4*nb);
        for (i=0; i<nb; i++) {
                dMatrix3 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);
#ifdef dGYROSCOPIC
                dMatrix3 I;
                // 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);
                dMULTIPLY0_333 (I,body[i]->posr.R,tmp);
                // compute rotational force
                //dMULTIPLY0_331 (tmp,I+i*12,body[i]->avel);
                dMULTIPLY0_331 (tmp,I,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 joint information (m = total constraint dimension, nub = number of unbounded variables).
        // joints with m=0 are inactive and are removed from the joints array
        // entirely, so that the code that follows does not consider them.
        //@@@ do we really need to save all the info1's
        dxJoint::Info1 *info = (dxJoint::Info1*) ALLOCA (nj*sizeof(dxJoint::Info1));
        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;

        // create the row offset array
        int m = 0;
        int *ofs = (int*) ALLOCA (nj*sizeof(int));
        for (i=0; i<nj; i++) {
                ofs[i] = m;
                m += info[i].m;
        }

        // if there are constraints, compute the constraint force
        dRealAllocaArray (J,m*12);
        int *jb = (int*) ALLOCA (m*2*sizeof(int));
        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.
                dRealAllocaArray (c,m);
                dRealAllocaArray (cfm,m);
                dRealAllocaArray (lo,m);
                dRealAllocaArray (hi,m);
                int *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. an m*12 matrix will be created
                // to store the two jacobian blocks from each constraint. it has this
                // format:
                //
                //   l1 l1 l1 a1 a1 a1 l2 l2 l2 a2 a2 a2 \    .
                //   l1 l1 l1 a1 a1 a1 l2 l2 l2 a2 a2 a2  }-- jacobian for joint 0, body 1 and body 2 (3 rows)
                //   l1 l1 l1 a1 a1 a1 l2 l2 l2 a2 a2 a2 /
                //   l1 l1 l1 a1 a1 a1 l2 l2 l2 a2 a2 a2 }--- jacobian for joint 1, body 1 and body 2 (3 rows)
                //   etc...
                //
                //   (lll) = linear jacobian data
                //   (aaa) = angular jacobian data
                //
                IFTIMING (dTimerNow ("create J");)
                dSetZero (J,m*12);
                dxJoint::Info2 Jinfo;
                Jinfo.rowskip = 12;
                Jinfo.fps = stepsize1;
                Jinfo.erp = world->global_erp;
                int mfb = 0; // number of rows of Jacobian we will have to save for joint feedback
                for (i=0; i<nj; i++) {
                        Jinfo.J1l = J + ofs[i]*12;
                        Jinfo.J1a = Jinfo.J1l + 3;
                        Jinfo.J2l = Jinfo.J1l + 6;
                        Jinfo.J2a = Jinfo.J1l + 9;
                        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];
                        }
                        if (joint[i]->feedback)
                                mfb += info[i].m;
                }

                // we need a copy of Jacobian for joint feedbacks
                // because it gets destroyed by SOR solver
                // instead of saving all Jacobian, we can save just rows
                // for joints, that requested feedback (which is normaly much less)
                dRealAllocaArray (Jcopy,mfb*12);
                if (mfb > 0) {
                        mfb = 0;
                        for (i=0; i<nj; i++)
                                if (joint[i]->feedback) {
                                        memcpy(Jcopy+mfb*12, J+ofs[i]*12, info[i].m*12*sizeof(dReal));
                                        mfb += info[i].m;
                                }
                }


                // create an array of body numbers for each joint row
                int *jb_ptr = jb;
                for (i=0; i<nj; i++) {
                        int b1 = (joint[i]->node[0].body) ? (joint[i]->node[0].body->tag) : -1;
                        int b2 = (joint[i]->node[1].body) ? (joint[i]->node[1].body->tag) : -1;
                        for (j=0; j<info[i].m; j++) {
                                jb_ptr[0] = b1;
                                jb_ptr[1] = b2;
                                jb_ptr += 2;
                        }
                }
                dIASSERT (jb_ptr == jb+2*m);

                // compute the right hand side `rhs'
                IFTIMING (dTimerNow ("compute rhs");)
                dRealAllocaArray (tmp1,nb*6);
                // put v/h + invM*fe into tmp1
                for (i=0; i<nb; i++) {
                        dReal body_invMass = body[i]->invMass;
                        for (j=0; j<3; j++) tmp1[i*6+j] = body[i]->facc[j] * body_invMass + body[i]->lvel[j] * stepsize1;
                        dMULTIPLY0_331 (tmp1 + i*6 + 3,invI + i*12,body[i]->tacc);
                        for (j=0; j<3; j++) tmp1[i*6+3+j] += body[i]->avel[j] * stepsize1;
                }

                // put J*tmp1 into rhs
                dRealAllocaArray (rhs,m);
                multiply_J (m,J,jb,tmp1,rhs);

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

                // scale CFM
                for (i=0; i<m; i++) cfm[i] *= stepsize1;

                // load lambda from the value saved on the previous iteration
                dRealAllocaArray (lambda,m);
#ifdef WARM_STARTING
                dSetZero (lambda,m);    //@@@ shouldn't be necessary
                for (i=0; i<nj; i++) {
                        memcpy (lambda+ofs[i],joint[i]->lambda,info[i].m * sizeof(dReal));
                }
#endif

                // solve the LCP problem and get lambda and invM*constraint_force
                IFTIMING (dTimerNow ("solving LCP problem");)
                dRealAllocaArray (cforce,nb*6);
                SOR_LCP (m,nb,J,jb,body,invI,lambda,cforce,rhs,lo,hi,cfm,findex,&world->qs);

#ifdef WARM_STARTING
                // save lambda for the next iteration
                //@@@ note that this doesn't work for contact joints yet, as they are
                // recreated every iteration
                for (i=0; i<nj; i++) {
                        memcpy (joint[i]->lambda,lambda+ofs[i],info[i].m * sizeof(dReal));
                }
#endif

                // note that the SOR method overwrites rhs and J at this point, so
                // they should not be used again.

                // add stepsize * cforce to the body velocity
                for (i=0; i<nb; i++) {
                        for (j=0; j<3; j++) body[i]->lvel[j] += stepsize * cforce[i*6+j];
                        for (j=0; j<3; j++) body[i]->avel[j] += stepsize * cforce[i*6+3+j];
                }

                // if joint feedback is requested, compute the constraint force.
                // BUT: cforce is inv(M)*J'*lambda, whereas we want just J'*lambda,
                // so we must compute M*cforce.
                // @@@ if any joint has a feedback request we compute the entire
                //     adjusted cforce, which is not the most efficient way to do it.
                /*for (j=0; j<nj; j++) {
                        if (joint[j]->feedback) {
                                // compute adjusted cforce
                                for (i=0; i<nb; i++) {
                                        dReal k = body[i]->mass.mass;
                                        cforce [i*6+0] *= k;
                                        cforce [i*6+1] *= k;
                                        cforce [i*6+2] *= k;
                                        dVector3 tmp;
                                        dMULTIPLY0_331 (tmp, I + 12*i, cforce + i*6 + 3);
                                        cforce [i*6+3] = tmp[0];
                                        cforce [i*6+4] = tmp[1];
                                        cforce [i*6+5] = tmp[2];
                                }
                                // compute feedback for this and all remaining joints
                                for (; j<nj; j++) {
                                        dJointFeedback *fb = joint[j]->feedback;
                                        if (fb) {
                                                int b1 = joint[j]->node[0].body->tag;
                                                memcpy (fb->f1,cforce+b1*6,3*sizeof(dReal));
                                                memcpy (fb->t1,cforce+b1*6+3,3*sizeof(dReal));
                                                if (joint[j]->node[1].body) {
                                                        int b2 = joint[j]->node[1].body->tag;
                                                        memcpy (fb->f2,cforce+b2*6,3*sizeof(dReal));
                                                        memcpy (fb->t2,cforce+b2*6+3,3*sizeof(dReal));
                                                }
                                        }
                                }
                        }
                }*/

                if (mfb > 0) {
                        // straightforward computation of joint constraint forces:
                        // multiply related lambdas with respective J' block for joints
                        // where feedback was requested
                        mfb = 0;
                        for (i=0; i<nj; i++) {
                                if (joint[i]->feedback) {
                                        dJointFeedback *fb = joint[i]->feedback;
                                        dReal data[6];
                                        Multiply1_12q1 (data, Jcopy+mfb*12, lambda+ofs[i], info[i].m);
                                        fb->f1[0] = data[0];
                                        fb->f1[1] = data[1];
                                        fb->f1[2] = data[2];
                                        fb->t1[0] = data[3];
                                        fb->t1[1] = data[4];
                                        fb->t1[2] = data[5];
                                        if (joint[i]->node[1].body)
                                        {
                                                Multiply1_12q1 (data, Jcopy+mfb*12+6, lambda+ofs[i], info[i].m);
                                                fb->f2[0] = data[0];
                                                fb->f2[1] = data[1];
                                                fb->f2[2] = data[2];
                                                fb->t2[0] = data[3];
                                                fb->t2[1] = data[4];
                                                fb->t2[2] = data[5];
                                        }
                                        mfb += info[i].m;
                                }
                        }
                }
        }

        // compute the velocity update:
        // add stepsize * invM * fe to the body velocity

        IFTIMING (dTimerNow ("compute velocity update");)
        for (i=0; i<nb; i++) {
                dReal body_invMass = body[i]->invMass;
                for (j=0; j<3; j++) body[i]->lvel[j] += stepsize * body_invMass * body[i]->facc[j];
                for (j=0; j<3; j++) body[i]->tacc[j] *= stepsize;
                dMULTIPLYADD0_331 (body[i]->avel,invI + i*12,body[i]->tacc);
        }

#if 0
        // check that the updated velocity obeys the constraint (this check needs unmodified J)
        dRealAllocaArray (vel,nb*6);
        for (i=0; i<nb; i++) {
                for (j=0; j<3; j++) vel[i*6+j] = body[i]->lvel[j];
                for (j=0; j<3; j++) vel[i*6+3+j] = body[i]->avel[j];
        }
        dRealAllocaArray (tmp,m);
        multiply_J (m,J,jb,vel,tmp);
        dReal error = 0;
        for (i=0; i<m; i++) error += dFabs(tmp[i]);
        printf ("velocity error = %10.6e\n",error);
#endif

        // update the position and orientation from the new linear/angular velocity
        // (over the given timestep)
        IFTIMING (dTimerNow ("update position");)
        for (i=0; i<nb; i++) dxStepBody (body[i],stepsize);

        IFTIMING (dTimerNow ("tidy up");)

        // zero all force accumulators
        for (i=0; i<nb; i++) {
                dSetZero (body[i]->facc,3);
                dSetZero (body[i]->tacc,3);
        }

        IFTIMING (dTimerEnd();)
        IFTIMING (if (m > 0) dTimerReport (stdout,1);)
}