Mujoco_WASM/src/user/user_util.cc

// Copyright 2021 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 "user/user_util.h"

#include <algorithm>
#include <cctype>
#include <cerrno>
#include <climits>
#include <cmath>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <optional>
#include <sstream>
#include <string>
#include <string_view>
#include <utility>
#include <vector>

#include <mujoco/mujoco.h>
#include "engine/engine_crossplatform.h"


// workaround with locale bug on some MacOS machines
#if defined (__APPLE__) && defined (__MACH__)
#include <xlocale.h>
#include <locale.h>

#define strtof(X, Y) strtof_l((X), (Y), _c_locale)
#define strtod(X, Y) strtod_l((X), (Y), _c_locale)
#endif

// check if numeric variable is defined
bool mjuu_defined(double num) {
  return !std::isnan(num);
}


// compute address of M[g1][g2] where M is triangular n-by-n
int mjuu_matadr(int g1, int g2, int n) {
  if (g1<0 || g2<0 || g1>=n || g2>=n) {
    return -1;
  }

  if (g1>g2) {
    int tmp = g1;
    g1 = g2;
    g2 = tmp;
  }

  return g1*n + g2;
}


// set 4D vector
void mjuu_setvec(double* dest, double x, double y, double z, double w) {
  dest[0] = x;
  dest[1] = y;
  dest[2] = z;
  dest[3] = w;
}
void mjuu_setvec(float* dest, double x, double y, double z, double w) {
  dest[0] = (float)x;
  dest[1] = (float)y;
  dest[2] = (float)z;
  dest[3] = (float)w;
}


// set 3D vector
void mjuu_setvec(double* dest, double x, double y, double z) {
  dest[0] = x;
  dest[1] = y;
  dest[2] = z;
}
void mjuu_setvec(float* dest, double x, double y, double z) {
  dest[0] = (float)x;
  dest[1] = (float)y;
  dest[2] = (float)z;
}


// set 2D vector
void mjuu_setvec(double* dest, double x, double y) {
  dest[0] = x;
  dest[1] = y;
}


// add to double array
void mjuu_addtovec(double* dest, const double* src, int n) {
  for (int i=0; i<n; i++) {
    dest[i] += src[i];
  }
}

// zero double array
void mjuu_zerovec(double* dest, int n) {
  for (int i=0; i<n; i++) {
    dest[i] = 0;
  }
}

// zero float array
void mjuu_zerovec(float* dest, int n) {
  for (int i=0; i<n; i++) {
    dest[i] = 0;
  }
}

// dot-product in 3D
double mjuu_dot3(const double* a, const double* b) {
  return a[0]*b[0] + a[1]*b[1] + a[2]*b[2];
}


// distance between 3D points
double mjuu_dist3(const double* a, const double* b) {
  return sqrt((a[0]-b[0])*(a[0]-b[0]) + (a[1]-b[1])*(a[1]-b[1]) + (a[2]-b[2])*(a[2]-b[2]));
}


// L1 norm between vectors
double mjuu_L1(const double* a, const double* b, int n) {
  double res = 0;
  for (int i=0; i<n; i++) {
    res += std::abs(a[i]-b[i]);
  }

  return res;
}


// normalize vector to unit length, return previous length
double mjuu_normvec(double* vec, const int n) {
  double nrm = 0;

  for (int i=0; i<n; i++) {
    nrm += vec[i]*vec[i];
  }
  if (nrm < mjEPS) {
    return 0;
  }

  nrm = sqrt(nrm);

  // don't normalize if nrm is within mjEPS of 1
  if (std::abs(nrm - 1) > mjEPS) {
    for (int i=0; i<n; i++) {
      vec[i] /= nrm;
    }
  }

  return nrm;
}


// normalize float vector to unit length, return previous length
float mjuu_normvec(float* vec, const int n) {
  float nrm = 0;

  for (int i=0; i<n; i++) {
    nrm += vec[i]*vec[i];
  }
  if (nrm < mjEPS) {
    return 0;
  }

  nrm = sqrt(nrm);

  // don't normalize if nrm is within mjEPS of 1
  if (std::abs(nrm - 1) > mjEPS) {
    for (int i=0; i<n; i++) {
      vec[i] /= nrm;
    }
  }

  return nrm;
}


// convert quaternion to rotation matrix
void mjuu_quat2mat(double* res, const double* quat) {
  // identity quat: identity mat
  if (quat[0] == 1 && quat[1] == 0 && quat[2] == 0 && quat[3] == 0) {
    res[0] = 1;
    res[1] = 0;
    res[2] = 0;
    res[3] = 0;
    res[4] = 1;
    res[5] = 0;
    res[6] = 0;
    res[7] = 0;
    res[8] = 1;
    return;
  }

  // regular processing
  double q00 = quat[0]*quat[0];
  double q01 = quat[0]*quat[1];
  double q02 = quat[0]*quat[2];
  double q03 = quat[0]*quat[3];
  double q11 = quat[1]*quat[1];
  double q12 = quat[1]*quat[2];
  double q13 = quat[1]*quat[3];
  double q22 = quat[2]*quat[2];
  double q23 = quat[2]*quat[3];
  double q33 = quat[3]*quat[3];

  res[0] = q00 + q11 - q22 - q33;
  res[4] = q00 - q11 + q22 - q33;
  res[8] = q00 - q11 - q22 + q33;

  res[1] = 2*(q12 - q03);
  res[2] = 2*(q13 + q02);
  res[3] = 2*(q12 + q03);
  res[5] = 2*(q23 - q01);
  res[6] = 2*(q13 - q02);
  res[7] = 2*(q23 + q01);

}


// multiply two unit quaternions
void mjuu_mulquat(double* res, const double* qa, const double* qb) {
  double tmp[4];
  tmp[0] = qa[0]*qb[0] - qa[1]*qb[1] - qa[2]*qb[2] - qa[3]*qb[3];
  tmp[1] = qa[0]*qb[1] + qa[1]*qb[0] + qa[2]*qb[3] - qa[3]*qb[2];
  tmp[2] = qa[0]*qb[2] - qa[1]*qb[3] + qa[2]*qb[0] + qa[3]*qb[1];
  tmp[3] = qa[0]*qb[3] + qa[1]*qb[2] - qa[2]*qb[1] + qa[3]*qb[0];
  mjuu_normvec(tmp, 4);
  mjuu_copyvec(res, tmp, 4);
}


// multiply matrix by vector, 3-by-3
void mjuu_mulvecmat(double* res, const double* vec, const double* mat) {
  double tmp[3] = {
    mat[0]*vec[0] + mat[1]*vec[1] + mat[2]*vec[2],
    mat[3]*vec[0] + mat[4]*vec[1] + mat[5]*vec[2],
    mat[6]*vec[0] + mat[7]*vec[1] + mat[8]*vec[2]
  };
  res[0] = tmp[0];
  res[1] = tmp[1];
  res[2] = tmp[2];
}


// multiply transposed matrix by vector, 3-by-3
void mjuu_mulvecmatT(double* res, const double* vec, const double* mat) {
  double tmp[3] = {
    mat[0]*vec[0] + mat[3]*vec[1] + mat[6]*vec[2],
    mat[1]*vec[0] + mat[4]*vec[1] + mat[7]*vec[2],
    mat[2]*vec[0] + mat[5]*vec[1] + mat[8]*vec[2]
  };
  res[0] = tmp[0];
  res[1] = tmp[1];
  res[2] = tmp[2];
}

// compute res = R * M * R'
void mjuu_mulRMRT(double* res, const double* R, const double* M) {
  double tmp[9];

  // tmp = R*M
  tmp[0] = R[0]*M[0] + R[1]*M[3] + R[2]*M[6];
  tmp[1] = R[0]*M[1] + R[1]*M[4] + R[2]*M[7];
  tmp[2] = R[0]*M[2] + R[1]*M[5] + R[2]*M[8];
  tmp[3] = R[3]*M[0] + R[4]*M[3] + R[5]*M[6];
  tmp[4] = R[3]*M[1] + R[4]*M[4] + R[5]*M[7];
  tmp[5] = R[3]*M[2] + R[4]*M[5] + R[5]*M[8];
  tmp[6] = R[6]*M[0] + R[7]*M[3] + R[8]*M[6];
  tmp[7] = R[6]*M[1] + R[7]*M[4] + R[8]*M[7];
  tmp[8] = R[6]*M[2] + R[7]*M[5] + R[8]*M[8];

  // res = tmp*R'
  res[0] = tmp[0]*R[0] + tmp[1]*R[1] + tmp[2]*R[2];
  res[1] = tmp[0]*R[3] + tmp[1]*R[4] + tmp[2]*R[5];
  res[2] = tmp[0]*R[6] + tmp[1]*R[7] + tmp[2]*R[8];
  res[3] = tmp[3]*R[0] + tmp[4]*R[1] + tmp[5]*R[2];
  res[4] = tmp[3]*R[3] + tmp[4]*R[4] + tmp[5]*R[5];
  res[5] = tmp[3]*R[6] + tmp[4]*R[7] + tmp[5]*R[8];
  res[6] = tmp[6]*R[0] + tmp[7]*R[1] + tmp[8]*R[2];
  res[7] = tmp[6]*R[3] + tmp[7]*R[4] + tmp[8]*R[5];
  res[8] = tmp[6]*R[6] + tmp[7]*R[7] + tmp[8]*R[8];
}


// multiply two matrices, all 3-by-3
void mjuu_mulmat(double* res, const double* A, const double* B) {
  double tmp[9];
  tmp[0] = A[0]*B[0] + A[1]*B[3] + A[2]*B[6];
  tmp[1] = A[0]*B[1] + A[1]*B[4] + A[2]*B[7];
  tmp[2] = A[0]*B[2] + A[1]*B[5] + A[2]*B[8];

  tmp[3] = A[3]*B[0] + A[4]*B[3] + A[5]*B[6];
  tmp[4] = A[3]*B[1] + A[4]*B[4] + A[5]*B[7];
  tmp[5] = A[3]*B[2] + A[4]*B[5] + A[5]*B[8];

  tmp[6] = A[6]*B[0] + A[7]*B[3] + A[8]*B[6];
  tmp[7] = A[6]*B[1] + A[7]*B[4] + A[8]*B[7];
  tmp[8] = A[6]*B[2] + A[7]*B[5] + A[8]*B[8];
  mjuu_copyvec(res, tmp, 9);
}


// transpose 3-by-3 matrix
void mjuu_transposemat(double* res, const double* mat) {
  double tmp[9] = {mat[0], mat[3], mat[6],
                   mat[1], mat[4], mat[7],
                   mat[2], mat[5], mat[8]};
  mjuu_copyvec(res, tmp, 9);
}


// convert global to local axis relative to given frame
void mjuu_localaxis(double* al, const double* ag, const double* quat) {
  double mat[9];
  double qneg[4] = {quat[0], -quat[1], -quat[2], -quat[3]};
  mjuu_quat2mat(mat, qneg);
  mjuu_mulvecmat(al, ag, mat);
}


// convert global to local position relative to given frame
void mjuu_localpos(double* pl, const double* pg, const double* pos, const double* quat) {
  double a[3] = {pg[0]-pos[0], pg[1]-pos[1], pg[2]-pos[2]};
  mjuu_localaxis(pl, a, quat);
}


// compute quaternion rotation from parent to child
void mjuu_localquat(double* local, const double* child, const double* parent) {
  double pneg[4] = {parent[0], -parent[1], -parent[2], -parent[3]};
  mjuu_mulquat(local, pneg, child);
}


// compute vector cross-product a = b x c
void mjuu_crossvec(double* a, const double* b, const double* c) {
  a[0] = b[1]*c[2] - b[2]*c[1];
  a[1] = b[2]*c[0] - b[0]*c[2];
  a[2] = b[0]*c[1] - b[1]*c[0];
}


// compute normal vector to given triangle, return length
double mjuu_makenormal(double* normal, const float* a, const float* b, const float* c) {
  double v1[3] = {b[0]-a[0], b[1]-a[1], b[2]-a[2]};
  double v2[3] = {c[0]-a[0], c[1]-a[1], c[2]-a[2]};
  double res;

  mjuu_crossvec(normal, v1, v2);
  if ((res=mjuu_normvec(normal, 3)) < mjEPS) {
    normal[0] = normal[1] = 0;
    normal[2] = 1;
  }

  return res;
}


// compute quaternion as minimal rotation from [0;0;1] to vec
void mjuu_z2quat(double* quat, const double* vec) {
  double z[3] = {0, 0, 1};
  mjuu_crossvec(quat+1, z, vec);
  double s = mjuu_normvec(quat+1, 3);
  if (s<1E-10) {
    quat[1] = 1;
    quat[2] = quat[3] = 0;
  }
  double ang = atan2(s, vec[2]);
  quat[0] = cos(ang/2);
  quat[1] *= sin(ang/2);
  quat[2] *= sin(ang/2);
  quat[3] *= sin(ang/2);
}


// compute quaternion given frame (axes are in matrix columns)
void mjuu_frame2quat(double* quat, const double* x, const double* y, const double* z) {
  const double* mat[3] = {x, y, z};  // mat[c][r] indexing

  // q0 largest
  if (mat[0][0]+mat[1][1]+mat[2][2]>0) {
    quat[0] = 0.5 * sqrt(1 + mat[0][0] + mat[1][1] + mat[2][2]);
    quat[1] = 0.25 * (mat[1][2] - mat[2][1]) / quat[0];
    quat[2] = 0.25 * (mat[2][0] - mat[0][2]) / quat[0];
    quat[3] = 0.25 * (mat[0][1] - mat[1][0]) / quat[0];
  }

  // q1 largest
  else if (mat[0][0]>mat[1][1] && mat[0][0]>mat[2][2]) {
    quat[1] = 0.5 * sqrt(1 + mat[0][0] - mat[1][1] - mat[2][2]);
    quat[0] = 0.25 * (mat[1][2] - mat[2][1]) / quat[1];
    quat[2] = 0.25 * (mat[1][0] + mat[0][1]) / quat[1];
    quat[3] = 0.25 * (mat[2][0] + mat[0][2]) / quat[1];
  }

  // q2 largest
  else if (mat[1][1]>mat[2][2]) {
    quat[2] = 0.5 * sqrt(1 - mat[0][0] + mat[1][1] - mat[2][2]);
    quat[0] = 0.25 * (mat[2][0] - mat[0][2]) / quat[2];
    quat[1] = 0.25 * (mat[1][0] + mat[0][1]) / quat[2];
    quat[3] = 0.25 * (mat[2][1] + mat[1][2]) / quat[2];
  }

  // q3 largest
  else {
    quat[3] = 0.5 * sqrt(1 - mat[0][0] - mat[1][1] + mat[2][2]);
    quat[0] = 0.25 * (mat[0][1] - mat[1][0]) / quat[3];
    quat[1] = 0.25 * (mat[2][0] + mat[0][2]) / quat[3];
    quat[2] = 0.25 * (mat[2][1] + mat[1][2]) / quat[3];
  }

  mjuu_normvec(quat, 4);
}


// invert frame transformation
void mjuu_frameinvert(double newpos[3], double newquat[4],
                      const double oldpos[3], const double oldquat[4]) {
  // position
  mjuu_localaxis(newpos, oldpos, oldquat);
  newpos[0] = -newpos[0];
  newpos[1] = -newpos[1];
  newpos[2] = -newpos[2];

  // orientation
  newquat[0] = oldquat[0];
  newquat[1] = -oldquat[1];
  newquat[2] = -oldquat[2];
  newquat[3] = -oldquat[3];
}


// accumulate frame transformations (forward kinematics)
void mjuu_frameaccum(double pos[3], double quat[4],
                     const double childpos[3], const double childquat[4]) {
  double mat[9], vec[3], qtmp[4];
  mjuu_quat2mat(mat, quat);
  mjuu_mulvecmat(vec, childpos, mat);
  pos[0] += vec[0];
  pos[1] += vec[1];
  pos[2] += vec[2];
  mjuu_mulquat(qtmp, quat, childquat);
  mjuu_copyvec(quat, qtmp, 4);
}


// accumulate frame transformation in second frame
void mjuu_frameaccumChild(const double pos[3], const double quat[4],
                          double childpos[3], double childquat[4]) {
  double p[] = {pos[0], pos[1], pos[2]};
  double q[] = {quat[0], quat[1], quat[2], quat[3]};
  mjuu_frameaccum(p, q, childpos, childquat);
  mjuu_copyvec(childpos, p, 3);
  mjuu_copyvec(childquat, q, 4);
}


// invert frame accumulation
void mjuu_frameaccuminv(double pos[3], double quat[4],
                        const double childpos[3], const double childquat[4]) {
  double mat[9], vec[3], qtmp[4];
  double qneg[4] = {childquat[0], -childquat[1], -childquat[2], -childquat[3]};
  mjuu_mulquat(qtmp, quat, qneg);
  mjuu_copyvec(quat, qtmp, 4);
  mjuu_quat2mat(mat, quat);
  mjuu_mulvecmat(vec, childpos, mat);
  pos[0] -= vec[0];
  pos[1] -= vec[1];
  pos[2] -= vec[2];
}


// convert local_inertia[3] to global_inertia[6]
void mjuu_globalinertia(double* global, const double* local, const double* quat) {
  double mat[9];
  mjuu_quat2mat(mat, quat);

  double tmp[9] = {
    mat[0]*local[0], mat[3]*local[0], mat[6]*local[0],
    mat[1]*local[1], mat[4]*local[1], mat[7]*local[1],
    mat[2]*local[2], mat[5]*local[2], mat[8]*local[2]
  };

  global[0] = mat[0]*tmp[0] + mat[1]*tmp[3] + mat[2]*tmp[6];
  global[1] = mat[3]*tmp[1] + mat[4]*tmp[4] + mat[5]*tmp[7];
  global[2] = mat[6]*tmp[2] + mat[7]*tmp[5] + mat[8]*tmp[8];
  global[3] = mat[0]*tmp[1] + mat[1]*tmp[4] + mat[2]*tmp[7];
  global[4] = mat[0]*tmp[2] + mat[1]*tmp[5] + mat[2]*tmp[8];
  global[5] = mat[3]*tmp[2] + mat[4]*tmp[5] + mat[5]*tmp[8];
}


// compute off-center correction to inertia matrix
//  mass * [y^2+z^2, -x*y, -x*z;  -x*y, x^2+z^2, -y*z;  -x*z, -y*z, x^2+y^2]
void mjuu_offcenter(double* res, const double mass, const double* vec) {
  res[0] = mass*(vec[1]*vec[1] + vec[2]*vec[2]);
  res[1] = mass*(vec[0]*vec[0] + vec[2]*vec[2]);
  res[2] = mass*(vec[0]*vec[0] + vec[1]*vec[1]);
  res[3] = -mass*vec[0]*vec[1];
  res[4] = -mass*vec[0]*vec[2];
  res[5] = -mass*vec[1]*vec[2];
}


// compute viscosity coefficients from mass and inertia
void mjuu_visccoef(double* visccoef, double mass, const double* inertia, double scl) {
  // compute equivalent box
  double ebox[3];
  ebox[0] = sqrt(mjMAX(mjEPS, (inertia[1] + inertia[2] - inertia[0])) / mass * 6.0);
  ebox[1] = sqrt(mjMAX(mjEPS, (inertia[0] + inertia[2] - inertia[1])) / mass * 6.0);
  ebox[2] = sqrt(mjMAX(mjEPS, (inertia[0] + inertia[1] - inertia[2])) / mass * 6.0);

  // apply formula for box (or rather cross) viscosity

  //  torque components
  visccoef[0] = scl * 4.0 / 3.0 * ebox[0] * (ebox[1]*ebox[1]*ebox[1] + ebox[2]*ebox[2]*ebox[2]);
  visccoef[1] = scl * 4.0 / 3.0 * ebox[1] * (ebox[0]*ebox[0]*ebox[0] + ebox[2]*ebox[2]*ebox[2]);
  visccoef[2] = scl * 4.0 / 3.0 * ebox[2] * (ebox[0]*ebox[0]*ebox[0] + ebox[1]*ebox[1]*ebox[1]);

  //  force components
  visccoef[3] = scl * 4*ebox[1]*ebox[2];
  visccoef[4] = scl * 4*ebox[0]*ebox[2];
  visccoef[5] = scl * 4*ebox[0]*ebox[1];
}


// convert axisAngle to quaternion
static void mjuu_axisAngle2Quat(double res[4], const double axis[3], double angle) {
  // zero angle: identity quat
  if (angle == 0) {
    res[0] = 1;
    res[1] = 0;
    res[2] = 0;
    res[3] = 0;
  }

  // regular processing
  else {
    double s = sin(angle*0.5);
    res[0] = cos(angle*0.5);
    res[1] = axis[0]*s;
    res[2] = axis[1]*s;
    res[3] = axis[2]*s;
  }
}

// rotate vector by quaternion
void mjuu_rotVecQuat(double res[3], const double vec[3], const double quat[4]) {
  // zero vec: zero res
  if (vec[0] == 0 && vec[1] == 0 && vec[2] == 0) {
    res[0] = res[1] = res[2] = 0;
  }

  // null quat: copy vec
  else if (quat[0] == 1 && quat[1] == 0 && quat[2] == 0 && quat[3] == 0) {
    mjuu_copyvec(res, vec, 3);
  }

  // regular processing
  else {
    // tmp = q_w * v + cross(q_xyz, v)
    double tmp[3] = {
      quat[0]*vec[0] + quat[2]*vec[2] - quat[3]*vec[1],
      quat[0]*vec[1] + quat[3]*vec[0] - quat[1]*vec[2],
      quat[0]*vec[2] + quat[1]*vec[1] - quat[2]*vec[0]
    };

    // res = v + 2 * cross(q_xyz, t)
    res[0] = vec[0] + 2 * (quat[2]*tmp[2] - quat[3]*tmp[1]);
    res[1] = vec[1] + 2 * (quat[3]*tmp[0] - quat[1]*tmp[2]);
    res[2] = vec[2] + 2 * (quat[1]*tmp[1] - quat[2]*tmp[0]);
  }
}

// update moving frame along a curve or initialize it, returns edge length
//   inputs:
//     normal    - normal vector computed by a previous call to the function
//     edge      - edge vector (non-unit tangent vector)
//     tprv      - unit tangent vector of previous body
//     tnxt      - unit tangent vector of next body
//     first     - 1 if the frame requires initialization
//   outputs:
//     quat      - frame orientation
//     normal    - unit normal vector
double mjuu_updateFrame(double quat[4], double normal[3], const double edge[3],
                       const double tprv[3], const double tnxt[3], int first) {
  double tangent[3], binormal[3];

  // normalize tangent
  mjuu_copyvec(tangent, edge, 3);
  mjuu_normvec(tangent, 3);

  // compute moving frame
  if (first) {
    // use the first vertex binormal for the first edge
    mjuu_crossvec(binormal, tangent, tnxt);
    mjuu_normvec(binormal, 3);

    // compute edge normal given tangent and binormal
    mjuu_crossvec(normal, binormal, tangent);
    mjuu_normvec(normal, 3);
  } else {
    double darboux[4];

    // rotate edge normal about the vertex binormal
    mjuu_crossvec(binormal, tprv, tangent);
    double angle = atan2(mjuu_normvec(binormal, 3), mjuu_dot3(tprv, tangent));
    mjuu_axisAngle2Quat(darboux, binormal, angle);
    mjuu_rotVecQuat(normal, normal, darboux);
    mjuu_normvec(normal, 3);

    // compute edge binormal given tangent and normal
    mjuu_crossvec(binormal, tangent, normal);
    mjuu_normvec(binormal, 3);
  }
  // global orientation of the frame
  mjuu_frame2quat(quat, tangent, normal, binormal);

  // return edge length
  return sqrt(mjuu_dot3(edge, edge));
}


// eigenvalue decomposition of symmetric 3x3 matrix
static const double kEigEPS = 1E-12;
int mjuu_eig3(double eigval[3], double eigvec[9], double quat[4], const double mat[9]) {
  double D[9], tmp[9], tmp2[9];
  double tau, t, c;
  int iter, rk, ck, rotk;

  // initialize with unit quaternion
  quat[0] = 1;
  quat[1] = quat[2] = quat[3] = 0;

  // Jacobi iteration
  for (iter=0; iter < 500; iter++) {
    // make quaternion matrix eigvec, compute D = eigvec'*mat*eigvec
    mjuu_quat2mat(eigvec, quat);
    mjuu_transposemat(tmp2, eigvec);
    mjuu_mulmat(tmp, tmp2, mat);
    mjuu_mulmat(D, tmp, eigvec);

    // assign eigenvalues
    eigval[0] = D[0];
    eigval[1] = D[4];
    eigval[2] = D[8];

    // find max off-diagonal element, set indices
    if (std::abs(D[1]) > std::abs(D[2]) && std::abs(D[1]) > std::abs(D[5])) {
      rk = 0;     // row
      ck = 1;     // column
      rotk = 2;   // rotation axis
    } else if (std::abs(D[2]) > std::abs(D[5])) {
      rk = 0;
      ck = 2;
      rotk = 1;
    } else {
      rk = 1;
      ck = 2;
      rotk = 0;
    }

    // terminate if max off-diagonal element too small
    if (std::abs(D[3*rk+ck]) < kEigEPS) {
      break;
    }

    // 2x2 symmetric Schur decomposition
    tau = (D[4*ck]-D[4*rk])/(2*D[3*rk+ck]);
    if (tau >= 0) {
      t = 1.0/(tau + sqrt(1 + tau*tau));
    } else {
      t = -1.0/(-tau + sqrt(1 + tau*tau));
    }
    c = 1.0/sqrt(1 + t*t);

    // terminate if cosine too close to 1
    if (c > 1.0-kEigEPS) {
      break;
    }

    // express rotation as quaternion
    tmp[1] = tmp[2] = tmp[3] = 0;
    tmp[rotk+1] = (tau >= 0 ? -sqrt(0.5-0.5*c) : sqrt(0.5-0.5*c));
    if (rotk == 1) {
      tmp[rotk+1] = -tmp[rotk+1];
    }
    tmp[0] = sqrt(1.0 - tmp[rotk+1]*tmp[rotk+1]);
    mjuu_normvec(tmp, 4);

    // accumulate quaternion rotation
    mjuu_mulquat(quat, quat, tmp);
    mjuu_normvec(quat, 4);
  }

  // sort eigenvalues in decreasing order (bubblesort: 0, 1, 0)
  for (int j=0; j < 3; j++) {
    int j1 = j%2;       // lead index

    // only swap if the eigenvalues are different
    if (eigval[j1]+kEigEPS < eigval[j1+1]) {
      // swap eigenvalues
      t = eigval[j1];
      eigval[j1] = eigval[j1+1];
      eigval[j1+1] = t;

      // rotate quaternion
      tmp[0] = 0.707106781186548;     // cos(pi/4) = sin(pi/4)
      tmp[1] = tmp[2] = tmp[3] = 0;
      tmp[(j1+2)%3+1] = tmp[0];
      mjuu_mulquat(quat, quat, tmp);
      mjuu_normvec(quat, 4);
    }
  }

  // recompute eigvec
  mjuu_quat2mat(eigvec, quat);

  return iter;
}

// transform vector by pose
void mjuu_trnVecPose(double res[3], const double pos[3], const double quat[4],
                     const double vec[3]) {
  // res = quat*vec + pos
  mjuu_rotVecQuat(res, vec, quat);
  res[0] += pos[0];
  res[1] += pos[1];
  res[2] += pos[2];
}

// strip directory from filename
std::string mjuu_strippath(std::string filename) {
  // find last pathsymbol
  size_t start = filename.find_last_of("/\\");

  // no path found: return original
  if (start==std::string::npos) {
    return filename;
  }

  // return name without path
  else {
    return filename.substr(start+1, filename.size()-start-1);
  }
}


// compute frame quat and diagonal inertia from full inertia matrix, return error if any
const char* mjuu_fullInertia(double quat[4], double inertia[3], const double fullinertia[6]) {
  if (!mjuu_defined(fullinertia[0])) {
    return nullptr;
  }

  double eigval[3], eigvec[9], quattmp[4];
  double full[9] = {
    fullinertia[0], fullinertia[3], fullinertia[4],
    fullinertia[3], fullinertia[1], fullinertia[5],
    fullinertia[4], fullinertia[5], fullinertia[2]
  };

  mjuu_eig3(eigval, eigvec, quattmp, full);

  // check mimimal eigenvalue
  if (eigval[2]<mjEPS) {
    return "inertia must have positive eigenvalues";
  }

  // copy
  if (quat) {
    mjuu_copyvec(quat, quattmp, 4);
  }

  if (inertia) {
    mjuu_copyvec(inertia, eigval, 3);
  }

  return nullptr;
}


// strip extension
std::string mjuu_stripext(std::string filename) {
  // find last dot
  size_t end = filename.find_last_of('.');

  // no path found: return original
  if (end == std::string::npos) {
    return filename;
  }

  // return name without extension
  return filename.substr(0, end);
}

std::string mjuu_getext(std::string_view filename) {
  size_t dot = filename.find_last_of('.');

  if (dot == std::string::npos) {
    return "";
  }
  return std::string(filename.substr(dot, filename.size() - dot));
}


// is directory path absolute
bool mjuu_isabspath(std::string path) {
  // empty: not absolute
  if (path.empty()) {
    return false;
  }

  // path is scheme:filename which we consider an absolute path
  // e.g. file URI's are always absolute paths
  if (mjp_getResourceProvider(path.c_str()) != nullptr) {
    return true;
  }

  // check first char
  const char* str = path.c_str();
  if (str[0]=='\\' || str[0]=='/') {
    return true;
  }

  // find ":/" or ":\"
  if (path.find(":/") != std::string::npos ||
      path.find(":\\") != std::string::npos) {
    return true;
  }

  return false;
}



// assemble two file paths
std::string mjuu_combinePaths(const std::string& path1, const std::string& path2) {
  // path2 has absolute path
  if (mjuu_isabspath(path2)) {
    return path2;
  }

  std::size_t n = path1.size();
  if (n > 0 && path1[n - 1] != '\\' && path1[n - 1] != '/') {
    return path1 + "/" + path2;
  }
  return path1 + path2;
}



// assemble three file paths
std::string mjuu_combinePaths(const std::string& path1, const std::string& path2,
                              const std::string& path3) {
  return mjuu_combinePaths(path1, mjuu_combinePaths(path2, path3));
}



// return true if the text is in a valid content type format:
// {type}/{subtype}[;{parameter}={value}]
static bool mjuu_isValidContentType(std::string_view text) {
  // find a forward slash that's not the last character
  size_t n = text.find('/');
  if (n == std::string::npos || n == text.size() - 1) {
    return false;
  }

  size_t m = text.find(';');
  if (m == std::string::npos) {
    return true;
  }

  if (m + 1 <= n) {
    return false;
  }

  // just check if there's an equal sign; this isn't robust enough for general
  // validation, but works for our scope, hence this is a private helper
  // function
  size_t s = text.find('=');
  if (s == std::string::npos || s + 1 <= m) {
    return false;
  }

  return true;
}



// return type from content_type format {type}/{subtype}[;{parameter}={value}]
// return empty string on invalid format
std::optional<std::string_view> mjuu_parseContentTypeAttrType(std::string_view text) {
  if (!mjuu_isValidContentType(text)) {
    return std::nullopt;
  }

  return { text.substr(0, text.find('/')) };
}



// return subtype from content_type format {type}/{subtype}[;{parameter}={value}]
// return empty string on invalid format
std::optional<std::string_view> mjuu_parseContentTypeAttrSubtype(std::string_view text) {
  if (!mjuu_isValidContentType(text)) {
    return std::nullopt;
  }

  size_t n = text.find('/');
  size_t m = text.find(';', n + 1);
  if (m == std::string::npos) {
    return { text.substr(n+1) };
  }

  return { text.substr(n + 1, m - n - 1) };
}



// convert filename extension to content type; return empty string if not found
std::string mjuu_extToContentType(std::string_view filename) {
  std::string ext = mjuu_getext(filename);

  if (!strcasecmp(ext.c_str(), ".stl")) {
    return "model/stl";
  } else if (!strcasecmp(ext.c_str(), ".obj")) {
    return "model/obj";
  } else if (!strcasecmp(ext.c_str(), ".ply")) {
    return "model/ply";
  } else if (!strcasecmp(ext.c_str(), ".msh")) {
    return "model/vnd.mujoco.msh";
  } else if (!strcasecmp(ext.c_str(), ".png")) {
    return "image/png";
  } else {
    return "";
  }
}

// get the length of the dirname portion of a given path
int mjuu_dirnamelen(const char* path) {
  if (!path) {
    return 0;
  }

  int pos = -1;
  for (int i = 0; path[i]; ++i) {
    if (path[i] == '/' || path[i] == '\\') {
      pos = i;
    }
  }

  return pos + 1;
}

namespace mujoco::user {

std::string FilePath::Combine(const std::string& s1, const std::string& s2) {
  // str2 has absolute path
  if (!AbsPrefix(s2).empty()) {
    return s2;
  }

  std::size_t n = s1.size();
  if (n > 0 && s1[n - 1] != '\\' && s1[n - 1] != '/') {
    return s1 + "/" + s2;
  }
  return s1 + s2;
}

std::string FilePath::PathReduce(const std::string& str) {
  std::vector<std::string> dirs;
  std::string abs_prefix = AbsPrefix(str);

  int j = abs_prefix.size();

  for (int i = j; i < str.size(); ++i) {
    if (IsSeperator(str[i])) {
      std::string temp = str.substr(j, i - j);
      j = i + 1;
      if (temp == ".." && !dirs.empty()) {
        dirs.pop_back();
      } else if (temp != ".") {
        dirs.push_back(std::move(temp));
      }
    }
  }

  // push the rest of the string
  dirs.push_back(str.substr(j, str.size() - j));

  // join the path
  std::stringstream path;
  auto it = dirs.begin();
  path << abs_prefix << *it++;
  for (; it != dirs.end(); ++it) {
    path << "/" << *it;
  }
  return path.str();
}

FilePath FilePath::operator+(const FilePath& path) const {
  return FilePath(path_, path.path_);
}

std::string FilePath::Ext() const {
  std::size_t n = path_.find_last_of('.');

  if (n == std::string::npos) {
    return "";
  }
  return path_.substr(n, path_.size() - n);
}

FilePath FilePath::StripExt() const {
  size_t n = path_.find_last_of('.');

  // no extension
  if (n == std::string::npos) {
    return FilePathFast(path_);
  }

  // return path without extension
  return FilePathFast(path_.substr(0, n));
}

// is directory absolute path
std::string FilePath::AbsPrefix(const std::string& str) {
  // empty: not absolute
  if (str.empty()) {
    return "";
  }

  // path is scheme:filename which we consider an absolute path
  // e.g. file URI's are always absolute paths
  const mjpResourceProvider* provider = mjp_getResourceProvider(str.c_str());
  if (provider != nullptr) {
    std::size_t n = std::strlen(provider->prefix);
    return str.substr(0, n + 1);
  }

  // check first char
  if (str[0] == '\\' || str[0] == '/') {
    return str.substr(0, 1);
  }

  // find ":/" or ":\"
  std::size_t pos = str.find(":/");
  if (pos != std::string::npos) {
    return str.substr(0, pos + 2);
  }

  pos = str.find(":\\");
  if (pos != std::string::npos) {
    return str.substr(0, pos + 2);
  }

  return "";
}

FilePath FilePath::StripPath() const {
  // find last path symbol
  std::size_t n = path_.find_last_of("/\\");

  // no path
  if (n == std::string::npos) {
    return FilePathFast(path_);
  }

  return FilePathFast(path_.substr(n + 1, path_.size() - (n + 1)));
}

std::string FilePath::StrLower() const {
  std::string str = path_;
  std::transform(str.begin(), str.end(), str.begin(),
                 [](unsigned char c) { return std::tolower(c); });
  return str;
}

// read file into memory buffer
std::vector<uint8_t> FileToMemory(const char* filename) {
  FILE* fp = fopen(filename, "rb");
  if (!fp) {
    return {};
  }

  // find size
  if (fseek(fp, 0, SEEK_END) != 0) {
    fclose(fp);
    mju_warning("Failed to calculate size for '%s'", filename);
    return {};
  }

  // ensure file size fits in int
  long long_filesize = ftell(fp);  // NOLINT(runtime/int)
  if (long_filesize > INT_MAX) {
    fclose(fp);
    mju_warning("File size over 2GB is not supported. File: '%s'", filename);
    return {};
  } else if (long_filesize < 0) {
    fclose(fp);
    mju_warning("Failed to calculate size for '%s'", filename);
    return {};
  }

  std::vector<uint8_t> buffer(long_filesize);

  // go back to start of file
  if (fseek(fp, 0, SEEK_SET) != 0) {
    fclose(fp);
    mju_warning("Read error while reading '%s'", filename);
    return {};
  }

  // allocate and read
  std::size_t bytes_read = fread(buffer.data(), 1, buffer.size(), fp);

  // check that read data matches file size
  if (bytes_read != buffer.size()) {  // SHOULD NOT OCCUR
    if (ferror(fp)) {
      fclose(fp);
      mju_warning("Read error while reading '%s'", filename);
      return {};
    } else if (feof(fp)) {
      buffer.resize(bytes_read);
    }
  }

  // close file, return contents
  fclose(fp);
  return buffer;
}

// convert vector to string separating elements by whitespace
template<typename T> std::string VectorToString(const std::vector<T>& v) {
  std::stringstream ss;

  for (const T& t : v) {
    ss << t << " ";
  }

  std::string s = ss.str();
  if (!s.empty()) s.pop_back();  // remove trailing space
  return s;
}

template std::string VectorToString(const std::vector<int>& v);
template std::string VectorToString(const std::vector<float>& v);
template std::string VectorToString(const std::vector<double>& v);
template std::string VectorToString(const std::vector<std::string>& v);

namespace {

template<typename T> T StrToNum(char* str, char** c);

template<> int StrToNum(char* str, char** c) {
  long n = std::strtol(str, c, 10);
  if (n < INT_MIN || n > INT_MAX) errno = ERANGE;
  return n;
}

template<> float StrToNum(char* str, char** c) {
  float f = strtof(str, c);
  if (std::isnan(f)) errno = EDOM;
  return f;
}

template<> double StrToNum(char* str, char** c) {
  double d = strtod(str, c);
  if (std::isnan(d)) errno = EDOM;
  return d;
}

template<> unsigned char StrToNum(char* str, char** c) {
  long n = std::strtol(str, c, 10);
  if (n < 0 || n > UCHAR_MAX) errno = ERANGE;
  return n;
}

inline bool IsNullOrSpace(char* c) {
  return std::isspace(static_cast<unsigned char>(*c)) || *c == '\0';
}

inline char* SkipSpace(char* c) {
  for (; *c != '\0'; c++) {
    if (!IsNullOrSpace(c)) {
      break;
    }
  }
  return c;
}
}  // namespace

template <typename T> std::vector<T> StringToVector(char* cs) {
  std::vector<T> v;
  char* ch = cs;

  errno = 0;
  // reserve worst case
  v.reserve((std::strlen(cs) >> 1) + 1);

  for (;;) {
    cs = SkipSpace(ch);                      // skip leading spaces
    if (*cs == '\0') break;                  // end of string
    T num = StrToNum<T>(cs, &ch);            // parse number
    if (!IsNullOrSpace(ch)) errno = EINVAL;  // invalid separator
    if (cs == ch) errno = EINVAL;            // failed to parse number
    if (errno && errno != EDOM) break;       // NaNs are quietly ignored
    v.push_back(num);
  }

  v.shrink_to_fit();
  return v;
}

template<> std::vector<std::string> StringToVector(const std::string& s) {
  std::vector<std::string> v;
  std::stringstream ss(s);
  std::string word;
  while (ss >> word) {
    v.push_back(word);
  }
  return v;
}

template std::vector<int>    StringToVector(char* cs);
template std::vector<float>  StringToVector(char* cs);
template std::vector<double> StringToVector(char* cs);


template <typename T> std::vector<T> StringToVector(const std::string& s) {
  return StringToVector<T>(const_cast<char*>(s.c_str()));
}
template std::vector<int>    StringToVector(const std::string& s);
template std::vector<float>  StringToVector(const std::string& s);
template std::vector<double> StringToVector(const std::string& s);
template std::vector<unsigned char> StringToVector(const std::string& s);

}  // namespace mujoco::user