// Copyright 2022 DeepMind Technologies Limited // // Licensed under the Apache License, Version 2.0 (the "License"); // you may not use this file except in compliance with the License. // You may obtain a copy of the License at // // http://www.apache.org/licenses/LICENSE-2.0 // // Unless required by applicable law or agreed to in writing, software // distributed under the License is distributed on an "AS IS" BASIS, // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. // See the License for the specific language governing permissions and // limitations under the License. #include "engine/engine_derivative.h" #include #include #include // IWYU pragma: keep #include "engine/engine_core_constraint.h" #include "engine/engine_crossplatform.h" #include "engine/engine_io.h" #include "engine/engine_passive.h" #include "engine/engine_support.h" #include "engine/engine_util_blas.h" #include "engine/engine_util_errmem.h" #include "engine/engine_util_misc.h" #include "engine/engine_util_spatial.h" #include "engine/engine_util_sparse.h" //------------------------- derivatives of spatial algebra ----------------------------------------- // derivatives of cross product, Da and Db are 3x3 static void mjd_cross(const mjtNum a[3], const mjtNum b[3], mjtNum* restrict Da, mjtNum* restrict Db) { // derivative w.r.t a if (Da) { mju_zero(Da, 9); Da[1] = b[2]; Da[2] = -b[1]; Da[3] = -b[2]; Da[5] = b[0]; Da[6] = b[1]; Da[7] = -b[0]; } // derivative w.r.t b if (Db) { mju_zero(Db, 9); Db[1] = -a[2]; Db[2] = a[1]; Db[3] = a[2]; Db[5] = -a[0]; Db[6] = -a[1]; Db[7] = a[0]; } } // derivative of mju_crossMotion w.r.t velocity static void mjd_crossMotion_vel(mjtNum D[36], const mjtNum v[6]) { mju_zero(D, 36); // res[0] = -vel[2]*v[1] + vel[1]*v[2] D[0 + 2] = -v[1]; D[0 + 1] = v[2]; // res[1] = vel[2]*v[0] - vel[0]*v[2] D[6 + 2] = v[0]; D[6 + 0] = -v[2]; // res[2] = -vel[1]*v[0] + vel[0]*v[1] D[12 + 1] = -v[0]; D[12 + 0] = v[1]; // res[3] = -vel[2]*v[4] + vel[1]*v[5] - vel[5]*v[1] + vel[4]*v[2] D[18 + 2] = -v[4]; D[18 + 1] = v[5]; D[18 + 5] = -v[1]; D[18 + 4] = v[2]; // res[4] = vel[2]*v[3] - vel[0]*v[5] + vel[5]*v[0] - vel[3]*v[2] D[24 + 2] = v[3]; D[24 + 0] = -v[5]; D[24 + 5] = v[0]; D[24 + 3] = -v[2]; // res[5] = -vel[1]*v[3] + vel[0]*v[4] - vel[4]*v[0] + vel[3]*v[1] D[30 + 1] = -v[3]; D[30 + 0] = v[4]; D[30 + 4] = -v[0]; D[30 + 3] = v[1]; } // derivative of mju_crossForce w.r.t. velocity static void mjd_crossForce_vel(mjtNum D[36], const mjtNum f[6]) { mju_zero(D, 36); // res[0] = -vel[2]*f[1] + vel[1]*f[2] - vel[5]*f[4] + vel[4]*f[5] D[0 + 2] = -f[1]; D[0 + 1] = f[2]; D[0 + 5] = -f[4]; D[0 + 4] = f[5]; // res[1] = vel[2]*f[0] - vel[0]*f[2] + vel[5]*f[3] - vel[3]*f[5] D[6 + 2] = f[0]; D[6 + 0] = -f[2]; D[6 + 5] = f[3]; D[6 + 3] = -f[5]; // res[2] = -vel[1]*f[0] + vel[0]*f[1] - vel[4]*f[3] + vel[3]*f[4] D[12 + 1] = -f[0]; D[12 + 0] = f[1]; D[12 + 4] = -f[3]; D[12 + 3] = f[4]; // res[3] = -vel[2]*f[4] + vel[1]*f[5] D[18 + 2] = -f[4]; D[18 + 1] = f[5]; // res[4] = vel[2]*f[3] - vel[0]*f[5] D[24 + 2] = f[3]; D[24 + 0] = -f[5]; // res[5] = -vel[1]*f[3] + vel[0]*f[4] D[30 + 1] = -f[3]; D[30 + 0] = f[4]; } // derivative of mju_crossForce w.r.t. force static void mjd_crossForce_frc(mjtNum D[36], const mjtNum vel[6]) { mju_zero(D, 36); // res[0] = -vel[2]*f[1] + vel[1]*f[2] - vel[5]*f[4] + vel[4]*f[5] D[0 + 1] = -vel[2]; D[0 + 2] = vel[1]; D[0 + 4] = -vel[5]; D[0 + 5] = vel[4]; // res[1] = vel[2]*f[0] - vel[0]*f[2] + vel[5]*f[3] - vel[3]*f[5] D[6 + 0] = vel[2]; D[6 + 2] = -vel[0]; D[6 + 3] = vel[5]; D[6 + 5] = -vel[3]; // res[2] = -vel[1]*f[0] + vel[0]*f[1] - vel[4]*f[3] + vel[3]*f[4] D[12 + 0] = -vel[1]; D[12 + 1] = vel[0]; D[12 + 3] = -vel[4]; D[12 + 4] = vel[3]; // res[3] = -vel[2]*f[4] + vel[1]*f[5] D[18 + 4] = -vel[2]; D[18 + 5] = vel[1]; // res[4] = vel[2]*f[3] - vel[0]*f[5] D[24 + 3] = vel[2]; D[24 + 5] = -vel[0]; // res[5] = -vel[1]*f[3] + vel[0]*f[4] D[30 + 3] = -vel[1]; D[30 + 4] = vel[0]; } // derivative of mju_mulInertVec w.r.t vel static void mjd_mulInertVec_vel(mjtNum D[36], const mjtNum i[10]) { mju_zero(D, 36); // res[0] = i[0]*v[0] + i[3]*v[1] + i[4]*v[2] - i[8]*v[4] + i[7]*v[5] D[0 + 0] = i[0]; D[0 + 1] = i[3]; D[0 + 2] = i[4]; D[0 + 4] = -i[8]; D[0 + 5] = i[7]; // res[1] = i[3]*v[0] + i[1]*v[1] + i[5]*v[2] + i[8]*v[3] - i[6]*v[5] D[6 + 0] = i[3]; D[6 + 1] = i[1]; D[6 + 2] = i[5]; D[6 + 3] = i[8]; D[6 + 5] = -i[6]; // res[2] = i[4]*v[0] + i[5]*v[1] + i[2]*v[2] - i[7]*v[3] + i[6]*v[4] D[12 + 0] = i[4]; D[12 + 1] = i[5]; D[12 + 2] = i[2]; D[12 + 3] = -i[7]; D[12 + 4] = i[6]; // res[3] = i[8]*v[1] - i[7]*v[2] + i[9]*v[3] D[18 + 1] = i[8]; D[18 + 2] = -i[7]; D[18 + 3] = i[9]; // res[4] = i[6]*v[2] - i[8]*v[0] + i[9]*v[4] D[24 + 2] = i[6]; D[24 + 0] = -i[8]; D[24 + 4] = i[9]; // res[5] = i[7]*v[0] - i[6]*v[1] + i[9]*v[5] D[30 + 0] = i[7]; D[30 + 1] = -i[6]; D[30 + 5] = i[9]; } // derivative of mju_subQuat w.r.t inputs void mjd_subQuat(const mjtNum qa[4], const mjtNum qb[4], mjtNum Da[9], mjtNum Db[9]) { // no outputs, quick return if (!Da && !Db) { return; } // compute axis-angle quaternion difference mjtNum axis[3]; mju_subQuat(axis, qa, qb); // normalize axis, get half-angle mjtNum half_angle = 0.5 * mju_normalize3(axis); // identity mjtNum Da_tmp[9] = { 1, 0, 0, 0, 1, 0, 0, 0, 1 }; // add term linear in cross product matrix K mjtNum K[9] = { 0, -axis[2], axis[1], axis[2], 0, -axis[0], -axis[1], axis[0], 0 }; mju_addToScl(Da_tmp, K, half_angle, 9); // add term linear in K * K mjtNum KK[9]; mju_mulMatMat3(KK, K, K); mjtNum coef = 1.0 - (half_angle < 6e-8 ? 1.0 : half_angle / mju_tan(half_angle)); mju_addToScl(Da_tmp, KK, coef, 9); if (Da) { mju_copy(Da, Da_tmp, 9); } if (Db) { // Db = -Da^T mju_transpose(Db, Da_tmp, 3, 3); mju_scl(Db, Db, -1.0, 9); } } // derivative of mju_quatIntegrate w.r.t scaled velocity // reference: https://arxiv.org/abs/1711.02508, Eq. 183 void mjd_quatIntegrate(const mjtNum vel[3], mjtNum scale, mjtNum Dquat[9], mjtNum Dvel[9], mjtNum Dscale[3]) { // scaled velocity mjtNum s[3] = {scale*vel[0], scale*vel[1], scale*vel[2]}; // 3 basis matrices mjtNum eye[9] = { 1, 0, 0, 0, 1, 0, 0, 0, 1 }; mjtNum cross[9] = { 0, s[2], -s[1], -s[2], 0, s[0], s[1], -s[0], 0 }; mjtNum outer[9] = { s[0]*s[0], s[0]*s[1], s[0]*s[2], s[1]*s[0], s[1]*s[1], s[1]*s[2], s[2]*s[0], s[2]*s[1], s[2]*s[2] }; // squared norm, norm of s mjtNum xx = mju_dot3(s, s); mjtNum x = mju_sqrt(xx); // 4 coefficients: a=cos(x), b=sin(x)/x, c=(1-cos(x))/x^2, d=(x-sin(x))/x^3} mjtNum a = mju_cos(x); mjtNum b, c, d; // x is not small: use full expressions if (mju_abs(x) > 1.0/32) { b = mju_sin(x) / x; c = (1.0 - a) / xx; d = (1.0 - b) / xx; } // |x| <= 1/32: use 6th order Taylor expansion (Horner form) else { b = 1 + xx/6 * (xx/20 * (1 - xx/42) - 1); c = (1 + xx/12 * (xx/30 * (1 - xx/56) - 1)) / 2; d = (1 + xx/20 * (xx/42 * (1 - xx/72) - 1)) / 6; } // derivatives mjtNum Dvel_[9]; for (int i=0; i < 9; i++) { if (Dquat) Dquat[i] = a*eye[i] + b*cross[i] + c*outer[i]; if (Dvel || Dscale) Dvel_[i] = b*eye[i] + c*cross[i] + d*outer[i]; } if (Dvel) mju_copy(Dvel, Dvel_, 9); if (Dscale) mju_mulMatVec3(Dscale, Dvel_, vel); } //------------------------- dense derivatives of component functions ------------------------------- // no longer used, except in tests // derivative of cvel, cdof_dot w.r.t qvel (dense version) static void mjd_comVel_vel_dense(const mjModel* m, mjData* d, mjtNum* Dcvel, mjtNum* Dcdofdot) { int nv = m->nv, nbody = m->nbody; mjtNum mat[36]; // clear Dcvel mju_zero(Dcvel, nbody*6*nv); // forward pass over bodies: accumulate Dcvel, set Dcdofdot for (int i=1; i < nbody; i++) { // Dcvel = Dcvel_parent mju_copy(Dcvel+i*6*nv, Dcvel+m->body_parentid[i]*6*nv, 6*nv); // Dcvel += D(cdof * qvel), Dcdofdot = D(cvel x cdof) for (int j=m->body_dofadr[i]; j < m->body_dofadr[i]+m->body_dofnum[i]; j++) { switch ((mjtJoint) m->jnt_type[m->dof_jntid[j]]) { case mjJNT_FREE: // Dcdofdot = 0 mju_zero(Dcdofdot+j*6*nv, 18*nv); // Dcvel += cdof * (D qvel) for (int k=0; k < 6; k++) { Dcvel[i*6*nv + k*nv + j+0] += d->cdof[(j+0)*6 + k]; Dcvel[i*6*nv + k*nv + j+1] += d->cdof[(j+1)*6 + k]; Dcvel[i*6*nv + k*nv + j+2] += d->cdof[(j+2)*6 + k]; } // continue with rotations j += 3; mjFALLTHROUGH; case mjJNT_BALL: // Dcdofdot = D crossMotion(cvel, cdof) for (int k=0; k < 3; k++) { mjd_crossMotion_vel(mat, d->cdof+6*(j+k)); mju_mulMatMat(Dcdofdot+(j+k)*6*nv, mat, Dcvel+i*6*nv, 6, 6, nv); } // Dcvel += cdof * (D qvel) for (int k=0; k < 6; k++) { Dcvel[i*6*nv + k*nv + j+0] += d->cdof[(j+0)*6 + k]; Dcvel[i*6*nv + k*nv + j+1] += d->cdof[(j+1)*6 + k]; Dcvel[i*6*nv + k*nv + j+2] += d->cdof[(j+2)*6 + k]; } // adjust for 3-dof joint j += 2; break; default: // Dcdofdot = D crossMotion(cvel, cdof) * Dcvel mjd_crossMotion_vel(mat, d->cdof+6*j); mju_mulMatMat(Dcdofdot+j*6*nv, mat, Dcvel+i*6*nv, 6, 6, nv); // Dcvel += cdof * (D qvel) for (int k=0; k < 6; k++) { Dcvel[i*6*nv + k*nv + j] += d->cdof[j*6 + k]; } } } } } // subtract (d qfrc_bias / d qvel) from qDeriv (dense version) void mjd_rne_vel_dense(const mjModel* m, mjData* d) { int nv = m->nv, nbody = m->nbody; mjtNum mat[36], mat1[36], mat2[36], dmul[36], tmp[6]; mj_markStack(d); mjtNum* Dcvel = mjSTACKALLOC(d, nbody*6*nv, mjtNum); mjtNum* Dcdofdot = mjSTACKALLOC(d, nv*6*nv, mjtNum); mjtNum* Dcacc = mjSTACKALLOC(d, nbody*6*nv, mjtNum); mjtNum* Dcfrcbody = mjSTACKALLOC(d, nbody*6*nv, mjtNum); mjtNum* row = mjSTACKALLOC(d, nv, mjtNum); // compute Dcvel and Dcdofdot mjd_comVel_vel_dense(m, d, Dcvel, Dcdofdot); // clear Dcacc mju_zero(Dcacc, nbody*6*nv); // forward pass over bodies: accumulate Dcacc, set Dcfrcbody for (int i=1; i < nbody; i++) { // Dcacc = Dcacc_parent mju_copy(Dcacc + i*6*nv, Dcacc + m->body_parentid[i]*6*nv, 6*nv); // Dcacc += D(cdofdot * qvel) for (int j=m->body_dofadr[i]; j < m->body_dofadr[i]+m->body_dofnum[i]; j++) { // Dcacc += cdofdot * (D qvel) for (int k=0; k < 6; k++) { Dcacc[i*6*nv + k*nv + j] += d->cdof_dot[j*6 + k]; } // Dcacc += (D cdofdot) * qvel mju_addToScl(Dcacc+i*6*nv, Dcdofdot+j*6*nv, d->qvel[j], 6*nv); } //---------- Dcfrcbody = D(cinert * cacc + cvel x (cinert * cvel)) // Dcfrcbody = (D mul / D cacc) * Dcacc mjd_mulInertVec_vel(dmul, d->cinert+10*i); mju_mulMatMat(Dcfrcbody+i*6*nv, dmul, Dcacc+i*6*nv, 6, 6, nv); // mat = (D cross / D cvel) + (D cross / D mul) * (D mul / D cvel) mju_mulInertVec(tmp, d->cinert+10*i, d->cvel+i*6); mjd_crossForce_vel(mat, tmp); mjd_crossForce_frc(mat1, d->cvel+i*6); mju_mulMatMat(mat2, mat1, dmul, 6, 6, 6); mju_addTo(mat, mat2, 36); // Dcfrcbody += mat * Dcvel (use body 0 as temp) mju_mulMatMat(Dcfrcbody, mat, Dcvel+i*6*nv, 6, 6, nv); mju_addTo(Dcfrcbody+i*6*nv, Dcfrcbody, 6*nv); } // clear world Dcfrcbody, for style mju_zero(Dcfrcbody, 6*nv); // backward pass over bodies: accumulate Dcfrcbody for (int i=nbody-1; i > 0; i--) { if (m->body_parentid[i]) { mju_addTo(Dcfrcbody+m->body_parentid[i]*6*nv, Dcfrcbody+i*6*nv, 6*nv); } } // qDeriv -= D(cdof * cfrc_body) for (int i=0; i < nv; i++) { for (int k=0; k < 6; k++) { // compute D(cdof * cfrc_body), store in row mju_scl(row, Dcfrcbody + (m->dof_bodyid[i]*6+k)*nv, d->cdof[i*6+k], nv); // dense to sparse: qDeriv -= row int end = d->D_rowadr[i] + d->D_rownnz[i]; for (int adr=d->D_rowadr[i]; adr < end; adr++) { d->qDeriv[adr] -= row[d->D_colind[adr]]; } } } mj_freeStack(d); } //------------------------- sparse derivatives of component functions ------------------------------ // internal sparse format: dense body/dof x sparse dof x 6 (inner size is 6) // copy sparse B-row from parent, shared ancestors only static void copyFromParent(const mjModel* m, mjData* d, mjtNum* mat, int n) { // return if this is world or parent is world if (n == 0 || m->body_weldid[m->body_parentid[n]] == 0) { return; } // count dofs in ancestors int ndof = 0; int np = m->body_weldid[m->body_parentid[n]]; while (np > 0) { // add self dofs ndof += m->body_dofnum[np]; // advance to parent np = m->body_weldid[m->body_parentid[np]]; } // copy: guaranteed to be at beginning of sparse array, due to sorting mju_copy(mat + 6*d->B_rowadr[n], mat + 6*d->B_rowadr[m->body_parentid[n]], 6*ndof); } // add sparse B-row to parent, all overlapping nonzeros static void addToParent(const mjModel* m, mjData* d, mjtNum* mat, int n) { // return if this is world or parent is world if (n == 0 || m->body_weldid[m->body_parentid[n]] == 0) { return; } // find matching nonzeros int np = m->body_parentid[n]; int i = 0, ip = 0; while (i < d->B_rownnz[n] && ip < d->B_rownnz[np]) { // columns match if (d->B_colind[d->B_rowadr[n] + i] == d->B_colind[d->B_rowadr[np] + ip]) { mju_addTo(mat + 6*(d->B_rowadr[np] + ip), mat + 6*(d->B_rowadr[n] + i), 6); // advance both i++; ip++; } // mismatch columns: advance parent else if (d->B_colind[d->B_rowadr[n] + i] > d->B_colind[d->B_rowadr[np] + ip]) { ip++; } // child nonzeroes must be subset of parent; SHOULD NOT OCCUR else { mjERROR("child nonzeroes must be subset of parent"); } } } // derivative of cvel, cdof_dot w.r.t qvel static void mjd_comVel_vel(const mjModel* m, mjData* d, mjtNum* Dcvel, mjtNum* Dcdofdot) { int nv = m->nv, nbody = m->nbody; int* Badr = d->B_rowadr, * Dadr = d->D_rowadr; mjtNum mat[36], matT[36]; // 6x6 matrices // forward pass over bodies: accumulate Dcvel, set Dcdofdot for (int i = 1; i < nbody; i++) { // Dcvel = Dcvel_parent copyFromParent(m, d, Dcvel, i); // process all dofs of this body int doflast = m->body_dofadr[i] + m->body_dofnum[i]; for (int j = m->body_dofadr[i]; j < doflast; j++) { // number of dof ancestors of dof j int Jadr = (j < nv - 1 ? m->dof_Madr[j + 1] : m->nM) - (m->dof_Madr[j] + 1); // Dcvel += D(cdof * qvel), Dcdofdot = D(cvel x cdof) switch ((mjtJoint) m->jnt_type[m->dof_jntid[j]]) { case mjJNT_FREE: // Dcdofdot = 0 (already cleared) // Dcvel += cdof * D(qvel) mju_addTo(Dcvel + 6*(Badr[i] + Jadr + 0), d->cdof + 6*(j + 0), 6); mju_addTo(Dcvel + 6*(Badr[i] + Jadr + 1), d->cdof + 6*(j + 1), 6); mju_addTo(Dcvel + 6*(Badr[i] + Jadr + 2), d->cdof + 6*(j + 2), 6); // continue with rotations j += 3; Jadr += 3; mjFALLTHROUGH; case mjJNT_BALL: // Dcdofdot = Dcvel * D crossMotion(cvel, cdof) for (int dj=0; dj < 3; dj++) { mjd_crossMotion_vel(mat, d->cdof + 6 * (j + dj)); mju_transpose(matT, mat, 6, 6); mju_mulMatMat(Dcdofdot + 6*Dadr[j + dj], Dcvel + 6*Badr[i], matT, Jadr + dj, 6, 6); } // Dcvel += cdof * (D qvel) mju_addTo(Dcvel + 6*(Badr[i] + Jadr + 0), d->cdof + 6*(j + 0), 6); mju_addTo(Dcvel + 6*(Badr[i] + Jadr + 1), d->cdof + 6*(j + 1), 6); mju_addTo(Dcvel + 6*(Badr[i] + Jadr + 2), d->cdof + 6*(j + 2), 6); // adjust for 3-dof joint j += 2; break; case mjJNT_HINGE: case mjJNT_SLIDE: // Dcdofdot = D crossMotion(cvel, cdof) * Dcvel mjd_crossMotion_vel(mat, d->cdof + 6 * j); mju_transpose(matT, mat, 6, 6); mju_mulMatMat(Dcdofdot + 6*Dadr[j], Dcvel + 6*Badr[i], matT, Jadr, 6, 6); // Dcvel += cdof * (D qvel) mju_addTo(Dcvel + 6*(Badr[i] + Jadr), d->cdof + 6*j, 6); break; default: mjERROR("unknown joint type"); } } } } // subtract d qfrc_bias / d qvel from qDeriv static void mjd_rne_vel(const mjModel* m, mjData* d) { int nv = m->nv, nbody = m->nbody; const int* Badr = d->B_rowadr; const int* Dadr = d->D_rowadr; const int* Bnnz = d->B_rownnz; mjtNum mat[36], mat1[36], mat2[36], dmul[36], tmp[6]; mj_markStack(d); mjtNum* Dcdofdot = mjSTACKALLOC(d, 6*m->nD, mjtNum); mjtNum* Dcvel = mjSTACKALLOC(d, 6*m->nB, mjtNum); mjtNum* Dcacc = mjSTACKALLOC(d, 6*m->nB, mjtNum); mjtNum* Dcfrcbody = mjSTACKALLOC(d, 6*m->nB, mjtNum); mjtNum* row = mjSTACKALLOC(d, nv, mjtNum); // clear mju_zero(Dcdofdot, 6*m->nD); mju_zero(Dcvel, 6*m->nB); mju_zero(Dcacc, 6*m->nB); mju_zero(Dcfrcbody, 6*m->nB); // compute Dcvel and Dcdofdot mjd_comVel_vel(m, d, Dcvel, Dcdofdot); // forward pass over bodies: accumulate Dcacc, set Dcfrcbody for (int i=1; i < nbody; i++) { // Dcacc = Dcacc_parent copyFromParent(m, d, Dcacc, i); // process all dofs of this body int doflast = m->body_dofadr[i] + m->body_dofnum[i]; for (int j=m->body_dofadr[i]; j < doflast; j++) { // number of dof ancestors of dof j int Jadr = (j < nv - 1 ? m->dof_Madr[j + 1] : m->nM) - (m->dof_Madr[j] + 1); // Dcacc += cdofdot * (D qvel) mju_addTo(Dcacc + 6*(Badr[i] + Jadr), d->cdof_dot + 6*j, 6); // Dcacc += (D cdofdot) * qvel // Dcacc[row i] and Dcdofdot[row j] have identical sparsity mju_addToScl(Dcacc + 6*Badr[i], Dcdofdot + 6*Dadr[j], d->qvel[j], 6*Bnnz[i]); } //---------- Dcfrcbody = D(cinert * cacc + cvel x (cinert * cvel)) // Dcfrcbody = (D mul / D cacc) * Dcacc mjd_mulInertVec_vel(dmul, d->cinert + 10*i); mju_transpose(mat1, dmul, 6, 6); mju_mulMatMat(Dcfrcbody + 6*Badr[i], Dcacc + 6*Badr[i], mat1, Bnnz[i], 6, 6); // mat = (D cross / D cvel) + (D cross / D mul) * (D mul / D cvel) mju_mulInertVec(tmp, d->cinert + 10*i, d->cvel + i*6); mjd_crossForce_vel(mat, tmp); mjd_crossForce_frc(mat1, d->cvel + i*6); mju_mulMatMat(mat2, mat1, dmul, 6, 6, 6); mju_addTo(mat, mat2, 36); // Dcfrcbody += mat * Dcvel (use worldbody as temp) mju_transpose(mat1, mat, 6, 6); mju_mulMatMat(Dcfrcbody, Dcvel + 6*Badr[i], mat1, Bnnz[i], 6, 6); mju_addTo(Dcfrcbody + 6*Badr[i], Dcfrcbody, 6*Bnnz[i]); } // clear worldbody Dcfrcbody mju_zero(Dcfrcbody, 6*Bnnz[0]); // backward pass over bodies: accumulate Dcfrcbody for (int i=m->nbody-1; i > 0; i--) { addToParent(m, d, Dcfrcbody, i); } // process all dofs, update qDeriv for (int j=0; j < nv; j++) { // get body index int i = m->dof_bodyid[j]; // qDeriv -= D(cdof * cfrc_body) mju_mulMatVec(row, Dcfrcbody + 6*Badr[i], d->cdof + 6*j, Bnnz[i], 6); mju_subFrom(d->qDeriv + Dadr[j], row, Bnnz[i]); } mj_freeStack(d); } //--------------------- utility functions for (d force / d vel) Jacobians -------------------------- // add J'*B*J to qDeriv static void addJTBJ(const mjModel* m, mjData* d, const mjtNum* J, const mjtNum* B, int n) { int nv = m->nv; // allocate dense row mj_markStack(d); mjtNum* row = mjSTACKALLOC(d, nv, mjtNum); // process non-zero elements of B for (int i=0; i < n; i++) { for (int j=0; j < n; j++) { if (!B[i*n+j]) { continue; } // process non-zero elements of J(i,:) for (int k=0; k < nv; k++) { if (J[i*nv+k]) { // row = J(i,k)*B(i,j)*J(j,:) mju_scl(row, J+j*nv, J[i*nv+k] * B[i*n+j], nv); // add row to qDeriv(k,:) int rownnz_k = d->D_rownnz[k]; for (int s=0; s < rownnz_k; s++) { int adr = d->D_rowadr[k] + s; d->qDeriv[adr] += row[d->D_colind[adr]]; } } } } } mj_freeStack(d); } // add J'*B*J to qDeriv, sparse version static void addJTBJSparse( const mjModel* m, mjData* d, const mjtNum* J, const mjtNum* B, int n, int offset, const int* J_rownnz, const int* J_rowadr, const int* J_colind) { // compute qDeriv(k,p) += sum_{i,j} ( J(i,k)*B(i,j)*J(j,p) ) for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { if (!B[i*n+j]) { continue; } // loop over non-zero elements of J(i,:) int nnz_i = J_rownnz[offset+i]; int adr_i = J_rowadr[offset+i]; int nnz_j = J_rownnz[offset+j]; int adr_j = J_rowadr[offset+j]; for (int k = 0; k < nnz_i; k++) { int ik = adr_i + k; int colik = J_colind[ik]; // qDeriv(k,:) += J(j,:) * J(i,k)*B(i,j) mju_addToSclSparseInc(d->qDeriv + d->D_rowadr[colik], J + adr_j, d->D_rownnz[colik], d->D_colind + d->D_rowadr[colik], nnz_j, J_colind + adr_j, J[ik]*B[i*n+j]); } } } } //----------------------------- derivatives of actuator forces ------------------------------------- // derivative of mju_muscleGain w.r.t velocity static mjtNum mjd_muscleGain_vel(mjtNum len, mjtNum vel, const mjtNum lengthrange[2], mjtNum acc0, const mjtNum prm[9]) { // unpack parameters mjtNum range[2] = {prm[0], prm[1]}; mjtNum force = prm[2]; mjtNum scale = prm[3]; mjtNum lmin = prm[4]; mjtNum lmax = prm[5]; mjtNum vmax = prm[6]; mjtNum fvmax = prm[8]; // scale force if negative if (force < 0) { force = scale / mju_max(mjMINVAL, acc0); } // optimum length mjtNum L0 = (lengthrange[1]-lengthrange[0]) / mju_max(mjMINVAL, range[1]-range[0]); // normalized length and velocity mjtNum L = range[0] + (len-lengthrange[0]) / mju_max(mjMINVAL, L0); mjtNum V = vel / mju_max(mjMINVAL, L0*vmax); // length curve mjtNum FL = mju_muscleGainLength(L, lmin, lmax); // velocity curve mjtNum dFV; mjtNum y = fvmax-1; if (V <= -1) { // FV = 0 dFV = 0; } else if (V <= 0) { // FV = (V+1)*(V+1) dFV = 2*V + 2; } else if (V <= y) { // FV = fvmax - (y-V)*(y-V) / mju_max(mjMINVAL, y) dFV = (-2*V + 2*y) / mju_max(mjMINVAL, y); } else { // FV = fvmax dFV = 0; } // compute FVL and scale, make it negative return -force*FL*dFV/mju_max(mjMINVAL, L0*vmax); } // add (d qfrc_actuator / d qvel) to qDeriv void mjd_actuator_vel(const mjModel* m, mjData* d) { int nu = m->nu; // disabled: nothing to add if (mjDISABLED(mjDSBL_ACTUATION)) { return; } // process actuators for (int i=0; i < nu; i++) { // skip if disabled if (mj_actuatorDisabled(m, i)) { continue; } mjtNum bias_vel = 0, gain_vel = 0; // affine bias if (m->actuator_biastype[i] == mjBIAS_AFFINE) { // extract bias info: prm = [const, kp, kv] bias_vel = (m->actuator_biasprm + mjNBIAS*i)[2]; } // affine gain if (m->actuator_gaintype[i] == mjGAIN_AFFINE) { // extract bias info: prm = [const, kp, kv] gain_vel = (m->actuator_gainprm + mjNGAIN*i)[2]; } // muscle gain else if (m->actuator_gaintype[i] == mjGAIN_MUSCLE) { gain_vel = mjd_muscleGain_vel(d->actuator_length[i], d->actuator_velocity[i], m->actuator_lengthrange+2*i, m->actuator_acc0[i], m->actuator_gainprm + mjNGAIN*i); } // force = gain .* [ctrl/act] if (gain_vel != 0) { if (m->actuator_dyntype[i] == mjDYN_NONE) { bias_vel += gain_vel * d->ctrl[i]; } else { int act_first = m->actuator_actadr[i]; int act_last = act_first + m->actuator_actnum[i] - 1; bias_vel += gain_vel * d->act[act_last]; } } // add if (bias_vel != 0) { addJTBJSparse(m, d, d->actuator_moment, &bias_vel, 1, i, d->moment_rownnz, d->moment_rowadr, d->moment_colind); } } } //----------------- utilities for ellipsoid-based fluid force derivatives -------------------------- static inline mjtNum pow2(const mjtNum val) { return val*val; } static inline mjtNum ellipsoid_max_moment(const mjtNum size[3], const int dir) { const mjtNum d0 = size[dir]; const mjtNum d1 = size[(dir+1) % 3]; const mjtNum d2 = size[(dir+2) % 3]; return 8.0/15.0 * mjPI * d0 * pow2(pow2(mju_max(d1, d2))); } // add 3x3 matrix D to one of the four quadrants of the 6x6 matrix B // row_quad and col_quad should be either 0 or 1 (not checked) static void addToQuadrant(mjtNum* restrict B, const mjtNum D[9], int col_quad, int row_quad) { int r = 3*row_quad, c = 3*col_quad; B[6*(c+0) + r+0] += D[0]; B[6*(c+0) + r+1] += D[1]; B[6*(c+0) + r+2] += D[2]; B[6*(c+1) + r+0] += D[3]; B[6*(c+1) + r+1] += D[4]; B[6*(c+1) + r+2] += D[5]; B[6*(c+2) + r+0] += D[6]; B[6*(c+2) + r+1] += D[7]; B[6*(c+2) + r+2] += D[8]; } //----------------- components of ellipsoid-based fluid force derivatives -------------------------- // forces due to fluid mass moving with the body, B is 6x6 static void mjd_addedMassForces( mjtNum* restrict B, const mjtNum local_vels[6], const mjtNum fluid_density, const mjtNum virtual_mass[3], const mjtNum virtual_inertia[3]) { const mjtNum lin_vel[3] = {local_vels[3], local_vels[4], local_vels[5]}; const mjtNum ang_vel[3] = {local_vels[0], local_vels[1], local_vels[2]}; const mjtNum virtual_lin_mom[3] = { fluid_density * virtual_mass[0] * lin_vel[0], fluid_density * virtual_mass[1] * lin_vel[1], fluid_density * virtual_mass[2] * lin_vel[2] }; const mjtNum virtual_ang_mom[3] = { fluid_density * virtual_inertia[0] * ang_vel[0], fluid_density * virtual_inertia[1] * ang_vel[1], fluid_density * virtual_inertia[2] * ang_vel[2] }; mjtNum Da[9]; mjtNum Db[9]; // force[:3] += cross(virtual_ang_mom, ang_vel) mjd_cross(virtual_ang_mom, ang_vel, Da, Db); addToQuadrant(B, Db, 0, 0); for (int i=0; i < 9; ++i) { Da[i] *= fluid_density * virtual_inertia[i % 3]; } addToQuadrant(B, Da, 0, 0); // force[:3] += cross(virtual_lin_mom, lin_vel) mjd_cross(virtual_lin_mom, lin_vel, Da, Db); addToQuadrant(B, Db, 0, 1); for (int i=0; i < 9; ++i) { Da[i] *= fluid_density * virtual_mass[i % 3]; } addToQuadrant(B, Da, 0, 1); // force[3:] += cross(virtual_lin_mom, ang_vel) mjd_cross(virtual_lin_mom, ang_vel, Da, Db); addToQuadrant(B, Db, 1, 0); for (int i=0; i < 9; ++i) { Da[i] *= fluid_density * virtual_mass[i % 3]; } addToQuadrant(B, Da, 1, 1); } // torque due to motion in the fluid, D is 3x3 static inline void mjd_viscous_torque( mjtNum* restrict D, const mjtNum lvel[6], const mjtNum fluid_density, const mjtNum fluid_viscosity, const mjtNum size[3], const mjtNum slender_drag_coef, const mjtNum ang_drag_coef) { const mjtNum d_max = mju_max(mju_max(size[0], size[1]), size[2]); const mjtNum d_min = mju_min(mju_min(size[0], size[1]), size[2]); const mjtNum d_mid = size[0] + size[1] + size[2] - d_max - d_min; // viscous force and torque in Stokes flow, analytical for spherical bodies const mjtNum eq_sphere_D = 2.0/3.0 * (size[0] + size[1] + size[2]); const mjtNum lin_visc_torq_coef = mjPI * eq_sphere_D*eq_sphere_D*eq_sphere_D; // moments of inertia used to compute angular quadratic drag const mjtNum I_max = 8.0/15.0 * mjPI * d_mid * (d_max*d_max)*(d_max*d_max); const mjtNum II[3] = { ellipsoid_max_moment(size, 0), ellipsoid_max_moment(size, 1), ellipsoid_max_moment(size, 2) }; const mjtNum x = lvel[0], y = lvel[1], z = lvel[2]; const mjtNum mom_coef[3] = { ang_drag_coef*II[0] + slender_drag_coef*(I_max - II[0]), ang_drag_coef*II[1] + slender_drag_coef*(I_max - II[1]), ang_drag_coef*II[2] + slender_drag_coef*(I_max - II[2]) }; const mjtNum mom_visc[3] = { x * mom_coef[0], y * mom_coef[1], z * mom_coef[2] }; const mjtNum density = fluid_density / mju_max(mjMINVAL, mju_norm3(mom_visc)); // -density * [x, y, z] * mom_coef^2 const mjtNum mom_sq[3] = { -density * x * mom_coef[0] * mom_coef[0], -density * y * mom_coef[1] * mom_coef[1], -density * z * mom_coef[2] * mom_coef[2] }; const mjtNum lin_coef = fluid_viscosity * lin_visc_torq_coef; // initialize mju_zero(D, 9); // set diagonal D[0] = D[4] = D[8] = x*mom_sq[0] + y*mom_sq[1] + z*mom_sq[2] - lin_coef; // add outer product mju_addToScl3(D, mom_sq, x); mju_addToScl3(D+3, mom_sq, y); mju_addToScl3(D+6, mom_sq, z); } // drag due to motion in the fluid, D is 3x3 static inline void mjd_viscous_drag( mjtNum* restrict D, const mjtNum lvel[6], const mjtNum fluid_density, const mjtNum fluid_viscosity, const mjtNum size[3], const mjtNum blunt_drag_coef, const mjtNum slender_drag_coef) { const mjtNum d_max = mju_max(mju_max(size[0], size[1]), size[2]); const mjtNum d_min = mju_min(mju_min(size[0], size[1]), size[2]); const mjtNum d_mid = size[0] + size[1] + size[2] - d_max - d_min; // viscous force and torque in Stokes flow, analytical for spherical bodies const mjtNum eq_sphere_D = 2.0/3.0 * (size[0] + size[1] + size[2]); const mjtNum A_max = mjPI * d_max * d_mid; const mjtNum a = pow2(size[1] * size[2]); const mjtNum b = pow2(size[2] * size[0]); const mjtNum c = pow2(size[0] * size[1]); const mjtNum aa = a*a, bb = b*b, cc = c*c; const mjtNum x = lvel[3], y = lvel[4], z = lvel[5]; const mjtNum xx = x*x, yy = y*y, zz = z*z, xy=x*y, yz=y*z, xz=x*z; const mjtNum proj_denom = aa*xx + bb*yy + cc*zz; const mjtNum proj_num = a*xx + b*yy + c*zz; const mjtNum dA_coef = mjPI / mju_max(mjMINVAL, mju_sqrt(proj_num*proj_num*proj_num * proj_denom)); const mjtNum A_proj = mjPI * mju_sqrt(proj_denom/mju_max(mjMINVAL, proj_num)); const mjtNum norm = mju_sqrt(xx + yy + zz); const mjtNum inv_norm = 1.0 / mju_max(mjMINVAL, norm); const mjtNum lin_coef = fluid_viscosity * 3.0 * mjPI * eq_sphere_D; const mjtNum quad_coef = fluid_density * ( A_proj*blunt_drag_coef + slender_drag_coef*(A_max - A_proj)); const mjtNum Aproj_coef = fluid_density * norm * (blunt_drag_coef - slender_drag_coef); const mjtNum dAproj_dv[3] = { Aproj_coef * dA_coef * a * x * (b * yy * (a - b) + c * zz * (a - c)), Aproj_coef * dA_coef * b * y * (a * xx * (b - a) + c * zz * (b - c)), Aproj_coef * dA_coef * c * z * (a * xx * (c - a) + b * yy * (c - b)) }; // outer product D[0] = xx; D[1] = xy; D[2] = xz; D[3] = xy; D[4] = yy; D[5] = yz; D[6] = xz; D[7] = yz; D[8] = zz; // diag(D) += dot([x y z], [x y z]) mjtNum inner = xx + yy + zz; D[0] += inner; D[4] += inner; D[8] += inner; // scale by -quad_coef*inv_norm mju_scl(D, D, -quad_coef*inv_norm, 9); // D += outer_product(-[x y z], dAproj_dv) mju_addToScl3(D+0, dAproj_dv, -x); mju_addToScl3(D+3, dAproj_dv, -y); mju_addToScl3(D+6, dAproj_dv, -z); // diag(D) -= lin_coef D[0] -= lin_coef; D[4] -= lin_coef; D[8] -= lin_coef; } // Kutta lift due to motion in the fluid, D is 3x3 static inline void mjd_kutta_lift( mjtNum* restrict D, const mjtNum lvel[6], const mjtNum fluid_density, const mjtNum size[3], const mjtNum kutta_lift_coef) { const mjtNum a = pow2(size[1] * size[2]); const mjtNum b = pow2(size[2] * size[0]); const mjtNum c = pow2(size[0] * size[1]); const mjtNum aa = a*a, bb = b*b, cc = c*c; const mjtNum x = lvel[3], y = lvel[4], z = lvel[5]; const mjtNum xx = x*x, yy = y*y, zz = z*z, xy=x*y, yz=y*z, xz=x*z; const mjtNum proj_denom = aa * xx + bb * yy + cc * zz; const mjtNum proj_num = a * xx + b * yy + c * zz; const mjtNum norm2 = xx + yy + zz; const mjtNum df_denom = mjPI * kutta_lift_coef * fluid_density / mju_max( mjMINVAL, mju_sqrt(proj_denom * proj_num * norm2)); const mjtNum dfx_coef = yy * (a - b) + zz * (a - c); const mjtNum dfy_coef = xx * (b - a) + zz * (b - c); const mjtNum dfz_coef = xx * (c - a) + yy * (c - b); const mjtNum proj_term = proj_num / mju_max(mjMINVAL, proj_denom); const mjtNum cos_term = proj_num / mju_max(mjMINVAL, norm2); // cosA = proj_num/(norm*proj_denom), A_proj = pi*sqrt(proj_denom/proj_num) // F = cosA * A_proj * (([a,b,c] * vel) \times vel) \times vel // derivative obtained with SymPy D[0] = a-a; D[1] = b-a; D[2] = c-a; D[3] = a-b; D[4] = b-b; D[5] = c-b; D[6] = a-c; D[7] = b-c; D[8] = c-c; mju_scl(D, D, 2 * proj_num, 9); const mjtNum inner_term[3] = { aa * proj_term - a + cos_term, bb * proj_term - b + cos_term, cc * proj_term - c + cos_term }; mju_addToScl3(D + 0, inner_term, dfx_coef); mju_addToScl3(D + 3, inner_term, dfy_coef); mju_addToScl3(D + 6, inner_term, dfz_coef); D[0] *= xx; D[1] *= xy; D[2] *= xz; D[3] *= xy; D[4] *= yy; D[5] *= yz; D[6] *= xz; D[7] *= yz; D[8] *= zz; D[0] -= dfx_coef * proj_num; D[4] -= dfy_coef * proj_num; D[8] -= dfz_coef * proj_num; mju_scl(D, D, df_denom, 9); } // Magnus force due to motion in the fluid, B is 6x6 static inline void mjd_magnus_force( mjtNum* restrict B, const mjtNum lvel[6], const mjtNum fluid_density, const mjtNum size[3], const mjtNum magnus_lift_coef) { const mjtNum volume = 4.0/3.0 * mjPI * size[0] * size[1] * size[2]; // magnus_coef = magnus_lift_coef * fluid_density * volume const mjtNum magnus_coef = magnus_lift_coef * fluid_density * volume; mjtNum D_lin[9], D_ang[9]; // premultiply by magnus_coef const mjtNum lin_vel[3] = { magnus_coef * lvel[3], magnus_coef * lvel[4], magnus_coef * lvel[5] }; const mjtNum ang_vel[3] = { magnus_coef * lvel[0], magnus_coef * lvel[1], magnus_coef * lvel[2] }; // force[3:] += magnus_coef * cross(ang_vel, lin_vel) mjd_cross(ang_vel, lin_vel, D_ang, D_lin); addToQuadrant(B, D_ang, 1, 0); addToQuadrant(B, D_lin, 1, 1); } //----------------- fluid force derivatives, ellipsoid and inertia-box models ---------------------- // fluid forces based on ellipsoid approximation void mjd_ellipsoidFluid(const mjModel* m, mjData* d, int bodyid) { mj_markStack(d); int nv = m->nv; int nnz = nv; int rownnz[6], rowadr[6]; mjtNum* J = mjSTACKALLOC(d, 6*nv, mjtNum); mjtNum* tmp = mjSTACKALLOC(d, 3*nv, mjtNum); int* colind = mjSTACKALLOC(d, 6*nv, int); int* colind_compressed = mjSTACKALLOC(d, 6*nv, int); mjtNum lvel[6], wind[6], lwind[6]; mjtNum geom_interaction_coef, magnus_lift_coef, kutta_lift_coef; mjtNum semiaxes[3], virtual_mass[3], virtual_inertia[3]; mjtNum blunt_drag_coef, slender_drag_coef, ang_drag_coef; if (mj_isSparse(m)) { // get sparse body Jacobian structure nnz = mj_bodyChain(m, bodyid, colind); // prepare rownnz, rowadr, colind for all 6 rows for (int i=0; i < 6; i++) { rownnz[i] = nnz; rowadr[i] = i == 0 ? 0 : rowadr[i-1] + nnz; for (int k=0; k < nnz; k++) { colind_compressed[i*nnz+k] = colind[k]; } } } for (int j=0; j < m->body_geomnum[bodyid]; j++) { const int geomid = m->body_geomadr[bodyid] + j; mju_geomSemiAxes(semiaxes, m->geom_size + 3*geomid, m->geom_type[geomid]); readFluidGeomInteraction( m->geom_fluid + mjNFLUID*geomid, &geom_interaction_coef, &blunt_drag_coef, &slender_drag_coef, &ang_drag_coef, &kutta_lift_coef, &magnus_lift_coef, virtual_mass, virtual_inertia); // scales all forces, read from MJCF as boolean (0.0 or 1.0) if (geom_interaction_coef == 0.0) { continue; } // map from CoM-centered to local body-centered 6D velocity mj_objectVelocity(m, d, mjOBJ_GEOM, geomid, lvel, 1); // compute wind in local coordinates mju_zero(wind, 6); mju_copy3(wind+3, m->opt.wind); mju_transformSpatial(lwind, wind, 0, d->geom_xpos + 3*geomid, // Frame of ref's origin. d->subtree_com + 3*m->body_rootid[bodyid], d->geom_xmat + 9*geomid); // Frame of ref's orientation. // subtract translational component from grom velocity mju_subFrom3(lvel+3, lwind+3); // get geom global Jacobian: rotation then translation if (mj_isSparse(m)) { mj_jacSparse(m, d, J+3*nnz, J, d->geom_xpos+3*geomid, m->geom_bodyid[geomid], nnz, colind); } else { mj_jacGeom(m, d, J+3*nv, J, geomid); } // rotate (compressed) Jacobian to local frame mju_mulMatTMat(tmp, d->geom_xmat+9*geomid, J, 3, 3, nnz); mju_copy(J, tmp, 3*nnz); mju_mulMatTMat(tmp, d->geom_xmat+9*geomid, J+3*nnz, 3, 3, nnz); mju_copy(J+3*nnz, tmp, 3*nnz); mjtNum B[36], D[9]; mju_zero(B, 36); mjd_magnus_force(B, lvel, m->opt.density, semiaxes, magnus_lift_coef); mjd_kutta_lift(D, lvel, m->opt.density, semiaxes, kutta_lift_coef); addToQuadrant(B, D, 1, 1); mjd_viscous_drag(D, lvel, m->opt.density, m->opt.viscosity, semiaxes, blunt_drag_coef, slender_drag_coef); addToQuadrant(B, D, 1, 1); mjd_viscous_torque(D, lvel, m->opt.density, m->opt.viscosity, semiaxes, slender_drag_coef, ang_drag_coef); addToQuadrant(B, D, 0, 0); mjd_addedMassForces(B, lvel, m->opt.density, virtual_mass, virtual_inertia); // make B symmetric if integrator is IMPLICITFAST if (m->opt.integrator == mjINT_IMPLICITFAST) { mju_symmetrize(B, B, 6); } if (mj_isSparse(m)) { addJTBJSparse(m, d, J, B, 6, 0, rownnz, rowadr, colind_compressed); } else { addJTBJ(m, d, J, B, 6); } } mj_freeStack(d); } // fluid forces based on inertia-box approximation void mjd_inertiaBoxFluid(const mjModel* m, mjData* d, int i) { mj_markStack(d); int nv = m->nv; int rownnz[6], rowadr[6]; mjtNum* J = mjSTACKALLOC(d, 6*nv, mjtNum); mjtNum* tmp = mjSTACKALLOC(d, 3*nv, mjtNum); int* colind = mjSTACKALLOC(d, 6*nv, int); mjtNum lvel[6], wind[6], lwind[6], box[3], B; mjtNum* inertia = m->body_inertia + 3*i; // equivalent inertia box box[0] = mju_sqrt(mju_max(mjMINVAL, (inertia[1] + inertia[2] - inertia[0])) / m->body_mass[i] * 6.0); box[1] = mju_sqrt(mju_max(mjMINVAL, (inertia[0] + inertia[2] - inertia[1])) / m->body_mass[i] * 6.0); box[2] = mju_sqrt(mju_max(mjMINVAL, (inertia[0] + inertia[1] - inertia[2])) / m->body_mass[i] * 6.0); // map from CoM-centered to local body-centered 6D velocity mj_objectVelocity(m, d, mjOBJ_BODY, i, lvel, 1); // compute wind in local coordinates mju_zero(wind, 6); mju_copy3(wind+3, m->opt.wind); mju_transformSpatial(lwind, wind, 0, d->xipos+3*i, d->subtree_com+3*m->body_rootid[i], d->ximat+9*i); // subtract translational component from body velocity mju_subFrom3(lvel+3, lwind+3); // init with dense int nnz = nv; // sparse Jacobian if (mj_isSparse(m)) { // get sparse body Jacobian structure nnz = mj_bodyChain(m, i, colind); // get sparse jacBodyCom mj_jacSparse(m, d, J+3*nnz, J, d->xipos+3*i, i, nnz, colind); // prepare rownnz, rowadr, colind for all 6 rows rownnz[0] = nnz; rowadr[0] = 0; for (int j=1; j < 6; j++) { rownnz[j] = nnz; rowadr[j] = rowadr[j-1] + nnz; for (int k=0; k < nnz; k++) { colind[j*nnz+k] = colind[k]; } } } // dense Jacobian else { mj_jacBodyCom(m, d, J+3*nv, J, i); } // rotate (compressed) Jacobian to local frame mju_mulMatTMat(tmp, d->ximat+9*i, J, 3, 3, nnz); mju_copy(J, tmp, 3*nnz); mju_mulMatTMat(tmp, d->ximat+9*i, J+3*nnz, 3, 3, nnz); mju_copy(J+3*nnz, tmp, 3*nnz); // add viscous force and torque if (m->opt.viscosity > 0) { // diameter of sphere approximation mjtNum diam = (box[0] + box[1] + box[2])/3.0; // mju_scl3(lfrc, lvel, -mjPI*diam*diam*diam*m->opt.viscosity) B = -mjPI*diam*diam*diam*m->opt.viscosity; for (int j=0; j < 3; j++) { if (mj_isSparse(m)) { addJTBJSparse(m, d, J, &B, 1, j, rownnz, rowadr, colind); } else { addJTBJ(m, d, J+j*nv, &B, 1); } } // mju_scl3(lfrc+3, lvel+3, -3.0*mjPI*diam*m->opt.viscosity); B = -3.0*mjPI*diam*m->opt.viscosity; for (int j=0; j < 3; j++) { if (mj_isSparse(m)) { addJTBJSparse(m, d, J, &B, 1, 3+j, rownnz, rowadr, colind); } else { addJTBJ(m, d, J+3*nv+j*nv, &B, 1); } } } // add lift and drag force and torque if (m->opt.density > 0) { // lfrc[0] -= m->opt.density*box[0]*(box[1]*box[1]*box[1]*box[1]+box[2]*box[2]*box[2]*box[2])* // mju_abs(lvel[0])*lvel[0]/64.0; B = -m->opt.density*box[0]*(box[1]*box[1]*box[1]*box[1]+box[2]*box[2]*box[2]*box[2])* 2*mju_abs(lvel[0])/64.0; if (mj_isSparse(m)) { addJTBJSparse(m, d, J, &B, 1, 0, rownnz, rowadr, colind); } else { addJTBJ(m, d, J, &B, 1); } // lfrc[1] -= m->opt.density*box[1]*(box[0]*box[0]*box[0]*box[0]+box[2]*box[2]*box[2]*box[2])* // mju_abs(lvel[1])*lvel[1]/64.0; B = -m->opt.density*box[1]*(box[0]*box[0]*box[0]*box[0]+box[2]*box[2]*box[2]*box[2])* 2*mju_abs(lvel[1])/64.0; if (mj_isSparse(m)) { addJTBJSparse(m, d, J, &B, 1, 1, rownnz, rowadr, colind); } else { addJTBJ(m, d, J+nv, &B, 1); } // lfrc[2] -= m->opt.density*box[2]*(box[0]*box[0]*box[0]*box[0]+box[1]*box[1]*box[1]*box[1])* // mju_abs(lvel[2])*lvel[2]/64.0; B = -m->opt.density*box[2]*(box[0]*box[0]*box[0]*box[0]+box[1]*box[1]*box[1]*box[1])* 2*mju_abs(lvel[2])/64.0; if (mj_isSparse(m)) { addJTBJSparse(m, d, J, &B, 1, 2, rownnz, rowadr, colind); } else { addJTBJ(m, d, J+2*nv, &B, 1); } // lfrc[3] -= 0.5*m->opt.density*box[1]*box[2]*mju_abs(lvel[3])*lvel[3]; B = -0.5*m->opt.density*box[1]*box[2]*2*mju_abs(lvel[3]); if (mj_isSparse(m)) { addJTBJSparse(m, d, J, &B, 1, 3, rownnz, rowadr, colind); } else { addJTBJ(m, d, J+3*nv, &B, 1); } // lfrc[4] -= 0.5*m->opt.density*box[0]*box[2]*mju_abs(lvel[4])*lvel[4]; B = -0.5*m->opt.density*box[0]*box[2]*2*mju_abs(lvel[4]); if (mj_isSparse(m)) { addJTBJSparse(m, d, J, &B, 1, 4, rownnz, rowadr, colind); } else { addJTBJ(m, d, J+4*nv, &B, 1); } // lfrc[5] -= 0.5*m->opt.density*box[0]*box[1]*mju_abs(lvel[5])*lvel[5]; B = -0.5*m->opt.density*box[0]*box[1]*2*mju_abs(lvel[5]); if (mj_isSparse(m)) { addJTBJSparse(m, d, J, &B, 1, 5, rownnz, rowadr, colind); } else { addJTBJ(m, d, J+5*nv, &B, 1); } } mj_freeStack(d); } //------------------------- derivatives of passive forces ------------------------------------------ // add (d qfrc_passive / d qvel) to qDeriv void mjd_passive_vel(const mjModel* m, mjData* d) { int nv = m->nv, nbody = m->nbody; // disabled: nothing to add if (mjDISABLED(mjDSBL_PASSIVE)) { return; } // dof damping for (int i=0; i < nv; i++) { int nnz_i = d->D_rownnz[i]; for (int j=0; j < nnz_i; j++) { int ij = d->D_rowadr[i] + j; // identify diagonal element if (d->D_colind[ij] == i) { d->qDeriv[ij] -= m->dof_damping[i]; break; } } } // flex edge damping for (int f=0; f < m->nflex; f++) { if (!m->flex_rigid[f] && m->flex_edgedamping[f]) { mjtNum B = -m->flex_edgedamping[f]; int flex_edgeadr = m->flex_edgeadr[f]; int flex_edgenum = m->flex_edgenum[f]; // process non-rigid edges of this flex for (int e=flex_edgeadr; e < flex_edgeadr+flex_edgenum; e++) { // skip rigid if (m->flexedge_rigid[e]) { continue; } // add sparse or dense if (mj_isSparse(m)) { addJTBJSparse(m, d, d->flexedge_J, &B, 1, e, d->flexedge_J_rownnz, d->flexedge_J_rowadr, d->flexedge_J_colind); } else { addJTBJ(m, d, d->flexedge_J+e*nv, &B, 1); } } } } // tendon damping for (int i=0; i < m->ntendon; i++) { if (m->tendon_damping[i] > 0) { mjtNum B = -m->tendon_damping[i]; // add sparse or dense if (mj_isSparse(m)) { addJTBJSparse(m, d, d->ten_J, &B, 1, i, d->ten_J_rownnz, d->ten_J_rowadr, d->ten_J_colind); } else { addJTBJ(m, d, d->ten_J+i*nv, &B, 1); } } } // fluid drag model, either body-level (inertia box) or geom-level (ellipsoid) if (m->opt.viscosity > 0 || m->opt.density > 0) { for (int i=1; i < nbody; i++) { if (m->body_mass[i] < mjMINVAL) { continue; } int use_ellipsoid_model = 0; // if any child geom uses the ellipsoid model, inertia-box model is disabled for parent body for (int j=0; j < m->body_geomnum[i] && use_ellipsoid_model == 0; j++) { const int geomid = m->body_geomadr[i] + j; use_ellipsoid_model += (m->geom_fluid[mjNFLUID*geomid] > 0); } if (use_ellipsoid_model) { mjd_ellipsoidFluid(m, d, i); } else { mjd_inertiaBoxFluid(m, d, i); } } } } //------------------------- main entry points ------------------------------------------------------ // analytical derivative of smooth forces w.r.t velocities: // d->qDeriv = d (qfrc_actuator + qfrc_passive - [qfrc_bias]) / d qvel void mjd_smooth_vel(const mjModel* m, mjData* d, int flg_bias) { // clear qDeriv mju_zero(d->qDeriv, m->nD); // qDeriv += d qfrc_actuator / d qvel mjd_actuator_vel(m, d); // qDeriv += d qfrc_passive / d qvel mjd_passive_vel(m, d); // qDeriv -= d qfrc_bias / d qvel; optional if (flg_bias) { mjd_rne_vel(m, d); } }