// A remaining part of phase 2 that requires external forces
void ForwardDynamicsABM::calcABMPhase2Part2()
{
    const LinkTraverse& traverse = body->linkTraverse();
    const int n = traverse.numLinks();

    for(int i = n-1; i >= 0; --i){
        DyLink* link = static_cast<DyLink*>(traverse[i]);

        link->pf()   -= link->f_ext();
        link->ptau() -= link->tau_ext();

        for(DyLink* child = link->child(); child; child = child->sibling()){
            link->pf()   += child->pf();
            link->ptau() += child->ptau();

            if(!child->isFixedJoint()){
                const double uu_dd = child->uu() / child->dd();
                link->pf()   += uu_dd * child->hhv();
                link->ptau() += uu_dd * child->hhw();
            }
        }

        if(i > 0){
            if(!link->isFixedJoint()){
                link->uu() += link->u() - (link->sv().dot(link->pf()) + link->sw().dot(link->ptau()));
            }
        }
    }
}
void ForwardDynamicsABM::calcMotionWithRungeKuttaMethod()
{
    DyLink* root = body->rootLink();

    if(!root->isFixedJoint()){
        T0 = root->T();
        vo0 = root->vo();
        w0  = root->w();
    }

    vo.setZero();
    w.setZero();
    dvo.setZero();
    dw.setZero();

    int n = body->numLinks();
    for(int i=1; i < n; ++i){
        DyLink* link = body->link(i);
        q0[i]  = link->q();
        dq0[i] = link->dq();
        dq[i]  = 0.0;
        ddq[i] = 0.0;
    }

    calcABMLastHalf();

    if(!sensorHelper.forceSensors().empty()){
        updateForceSensors();
    }

    integrateRungeKuttaOneStep(1.0 / 6.0, timeStep / 2.0);
    calcABMPhase1();
    calcABMPhase2();
    calcABMPhase3();

    integrateRungeKuttaOneStep(2.0 / 6.0, timeStep / 2.0);
    calcABMPhase1();
    calcABMPhase2();
    calcABMPhase3();

    integrateRungeKuttaOneStep(2.0 / 6.0, timeStep);
    calcABMPhase1();
    calcABMPhase2();
    calcABMPhase3();

    if(!root->isFixedJoint()){
        SE3exp(root->T(), T0, w0, vo0, timeStep);
        root->vo() = vo0 + (dvo + root->dvo() / 6.0) * timeStep;
        root->w()  = w0  + (dw  + root->dw()  / 6.0) * timeStep;
    }

    for(int i=1; i < n; ++i){
        DyLink* link = body->link(i);
        link->q()  =  q0[i] + ( dq[i] + link->dq()  / 6.0) * timeStep;
        link->dq() = dq0[i] + (ddq[i] + link->ddq() / 6.0) * timeStep;
    }
}
void ForwardDynamicsABM::calcABMPhase2()
{
    const LinkTraverse& traverse = body->linkTraverse();
    const int n = traverse.numLinks();

    for(int i = n-1; i >= 0; --i){
        DyLink* link = static_cast<DyLink*>(traverse[i]);

        link->pf()   -= link->f_ext();
        link->ptau() -= link->tau_ext();

        // compute articulated inertia (Eq.(6.48) of Kajita's textbook)
        for(DyLink* child = link->child(); child; child = child->sibling()){

            if(child->isFixedJoint()){
                link->Ivv() += child->Ivv();
                link->Iwv() += child->Iwv();
                link->Iww() += child->Iww();

            }else{
                const Vector3 hhv_dd = child->hhv() / child->dd();
                link->Ivv().noalias() += child->Ivv() - child->hhv() * hhv_dd.transpose();
                link->Iwv().noalias() += child->Iwv() - child->hhw() * hhv_dd.transpose();
                link->Iww().noalias() += child->Iww() - child->hhw() * (child->hhw() / child->dd()).transpose();
            }

            link->pf()  .noalias() += child->Ivv() * child->cv() + child->Iwv().transpose() * child->cw() + child->pf();
            link->ptau().noalias() += child->Iwv() * child->cv() + child->Iww() * child->cw() + child->ptau();

            if(!child->isFixedJoint()){
                const double uu_dd = child->uu() / child->dd();
                link->pf()   += uu_dd * child->hhv();
                link->ptau() += uu_dd * child->hhw();
            }
        }

        if(i > 0){
            if(!link->isFixedJoint()){
                // hh = Ia * s
                link->hhv().noalias() = link->Ivv() * link->sv() + link->Iwv().transpose() * link->sw();
                link->hhw().noalias() = link->Iwv() * link->sv() + link->Iww() * link->sw();
                // dd = Ia * s * s^T
                link->dd() = link->sv().dot(link->hhv()) + link->sw().dot(link->hhw()) + link->Jm2();
                // uu = u - hh^T*c + s^T*pp
                link->uu() = link->u() -
                    (link->hhv().dot(link->cv()) + link->hhw().dot(link->cw()) +
                     link->sv().dot(link->pf()) + link->sw().dot(link->ptau()));
            }
        }
    }
}
void ForwardDynamicsABM::calcMotionWithEulerMethod()
{
    calcABMLastHalf();

    if(!sensorHelper.forceSensors().empty()){
        updateForceSensors();
    }

    DyLink* root = body->rootLink();

    if(!root->isFixedJoint()){
        root->dv() =
            root->dvo() - root->p().cross(root->dw())
            + root->w().cross(root->vo() + root->w().cross(root->p()));
        Position T;
        SE3exp(T, root->T(), root->w(), root->vo(), timeStep);
        root->T() = T;
        root->vo() += root->dvo() * timeStep;
        root->w()  += root->dw()  * timeStep;
    }

    int n = body->numLinks();
    for(int i=1; i < n; ++i){
        DyLink* link = body->link(i);
        link->q()  += link->dq()  * timeStep;
        link->dq() += link->ddq() * timeStep;
    }
}
void ForwardDynamicsABM::integrateRungeKuttaOneStep(double r, double dt)
{
    DyLink* root = body->rootLink();

    if(!root->isFixedJoint()){

        SE3exp(root->T(), T0, root->w(), root->vo(), dt);
        root->vo().noalias() = vo0 + root->dvo() * dt;
        root->w().noalias()  = w0  + root->dw()  * dt;

        vo  += r * root->vo();
        w   += r * root->w();
        dvo += r * root->dvo();
        dw  += r * root->dw();
    }

    int n = body->numLinks();
    for(int i=1; i < n; ++i){

        DyLink* link = body->link(i);

        link->q()  =  q0[i] + dt * link->dq();
        link->dq() = dq0[i] + dt * link->ddq();

        dq[i]  += r * link->dq();
        ddq[i] += r * link->ddq();
    }
}
// A part of phase 2 (inbound loop) that can be calculated before external forces are given
void ForwardDynamicsABM::calcABMPhase2Part1()
{
    const LinkTraverse& traverse = body->linkTraverse();
    const int n = traverse.numLinks();

    for(int i = n-1; i >= 0; --i){
        DyLink* link = static_cast<DyLink*>(traverse[i]);

        for(DyLink* child = link->child(); child; child = child->sibling()){

            if(child->isFixedJoint()){
                link->Ivv() += child->Ivv();
                link->Iwv() += child->Iwv();
                link->Iww() += child->Iww();

            }else{
                const Vector3 hhv_dd = child->hhv() / child->dd();
                link->Ivv().noalias() += child->Ivv() - child->hhv() * hhv_dd.transpose();
                link->Iwv().noalias() += child->Iwv() - child->hhw() * hhv_dd.transpose();
                link->Iww().noalias() += child->Iww() - child->hhw() * (child->hhw() / child->dd()).transpose();
            }

            link->pf()  .noalias() += child->Ivv() * child->cv() + child->Iwv().transpose() * child->cw();
            link->ptau().noalias() += child->Iwv() * child->cv() + child->Iww() * child->cw();
        }

        if(i > 0){
            if(!link->isFixedJoint()){
                link->hhv().noalias() = link->Ivv() * link->sv() + link->Iwv().transpose() * link->sw();
                link->hhw().noalias() = link->Iwv() * link->sv() + link->Iww() * link->sw();
                link->dd() = link->sv().dot(link->hhv()) + link->sw().dot(link->hhw()) + link->Jm2();
                link->uu() = -(link->hhv().dot(link->cv()) + link->hhw().dot(link->cw()));
            }
        }
    }
}
void ForwardDynamicsABM::calcABMPhase3()
{
    const LinkTraverse& traverse = body->linkTraverse();

    DyLink* root = static_cast<DyLink*>(traverse[0]);

    if(root->isFreeJoint()){

        // - | Ivv  trans(Iwv) | * | dvo | = | pf   |
        //   | Iwv     Iww     |   | dw  |   | ptau |

        Eigen::Matrix<double, 6, 6> M;
        M << root->Ivv(), root->Iwv().transpose(),
            root->Iwv(), root->Iww();
        
        Eigen::Matrix<double, 6, 1> f;
        f << root->pf(),
            root->ptau();
        f *= -1.0;

        Eigen::Matrix<double, 6, 1> a(M.colPivHouseholderQr().solve(f));

        root->dvo() = a.head<3>();
        root->dw() = a.tail<3>();

    } else {
        root->dvo().setZero();
        root->dw().setZero();
    }

    const int n = traverse.numLinks();
    for(int i=1; i < n; ++i){
        DyLink* link = static_cast<DyLink*>(traverse[i]);
        const DyLink* parent = link->parent();
        if(!link->isFixedJoint()){
            link->ddq() = (link->uu() - (link->hhv().dot(parent->dvo()) + link->hhw().dot(parent->dw()))) / link->dd();
            link->dvo().noalias() = parent->dvo() + link->cv() + link->sv() * link->ddq();
            link->dw().noalias()  = parent->dw()  + link->cw() + link->sw() * link->ddq();
        }else{
            link->ddq() = 0.0;
            link->dvo() = parent->dvo();
            link->dw()  = parent->dw(); 
        }
    }
}