data/src/engine/engine_collision_primitive.c

// 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 "engine/engine_collision_primitive.h"

#include <mujoco/mjdata.h>
#include <mujoco/mjmacro.h>
#include <mujoco/mjmodel.h>
#include "engine/engine_util_blas.h"
#include "engine/engine_util_misc.h"
#include "engine/engine_util_spatial.h"


//--------------------------- plane collisions -----------------------------------------------------

// raw plane : sphere
static int mjraw_PlaneSphere(mjContact* con, mjtNum margin,
                             const mjtNum* pos1, const mjtNum* mat1, const mjtNum* size1,
                             const mjtNum* pos2, const mjtNum* mat2, const mjtNum* size2) {
  // set normal
  con[0].frame[0] = mat1[2];
  con[0].frame[1] = mat1[5];
  con[0].frame[2] = mat1[8];

  // compute distance, return if too large
  mjtNum tmp[3] = {pos2[0] - pos1[0], pos2[1] - pos1[1], pos2[2] - pos1[2]};
  mjtNum cdist = mju_dot3(tmp, con[0].frame);
  if (cdist > margin + size2[0]) {
    return 0;
  }

  // depth and position
  con[0].dist = cdist - size2[0];
  mju_scl3(tmp, con[0].frame, -con[0].dist/2 - size2[0]);
  mju_add3(con[0].pos, pos2, tmp);

  mju_zero3(con[0].frame+3);
  return 1;
}



// plane : sphere
int mjc_PlaneSphere(const mjModel* m, const mjData* d,
                    mjContact* con, int g1, int g2, mjtNum margin) {
  mjGETINFO
  return mjraw_PlaneSphere(con, margin, pos1, mat1, size1, pos2, mat2, size2);
}



// plane : capsule
int mjc_PlaneCapsule(const mjModel* m, const mjData* d,
                     mjContact* con, int g1, int g2, mjtNum margin) {
  mjGETINFO

  // get capsule axis, segment = scaled axis
  mjtNum axis[3] = {mat2[2], mat2[5], mat2[8]};
  mjtNum segment[3] = {size2[1]*axis[0], size2[1]*axis[1], size2[1]*axis[2]};

  // get point 1, do sphere-plane test
  mjtNum pos[3];
  mju_add3(pos, pos2, segment);
  int n1 = mjraw_PlaneSphere(con, margin, pos1, mat1, size1, pos, mat2, size2);

  // get point 2, do sphere-plane test
  mju_sub3(pos, pos2, segment);
  int n2 = mjraw_PlaneSphere(con+n1, margin, pos1, mat1, size1, pos, mat2, size2);

  // align contact frames with capsule axis
  if (n1) {
    mju_copy3(con->frame+3, axis);
  }
  if (n2) {
    mju_copy3((con+n1)->frame+3, axis);
  }

  return n1+n2;
}



// plane : cylinder
int mjc_PlaneCylinder(const mjModel* m, const mjData* d,
                      mjContact* con, int g1, int g2, mjtNum margin) {
  mjGETINFO
  mjtNum normal[3] = {mat1[2], mat1[5], mat1[8]};
  mjtNum axis[3] = {mat2[2], mat2[5], mat2[8]};

  // project, make sure axis points towards plane
  mjtNum prjaxis = mju_dot3(normal, axis);
  if (prjaxis > 0) {
    mju_scl3(axis, axis, -1);
    prjaxis = -prjaxis;
  }

  // compute normal distance to cylinder center
  mjtNum vec[3] = {pos2[0] - pos1[0], pos2[1] - pos1[1], pos2[2] - pos1[2]};
  mjtNum dist0 = mju_dot3(vec, normal);

  // remove component of -normal along axis, compute length
  mju_scl3(vec, axis, prjaxis);
  mju_subFrom3(vec, normal);
  mjtNum len_sqr = mju_dot3(vec, vec);

  // general configuration: normalize vector, scale by radius
  if (len_sqr >= mjMINVAL*mjMINVAL) {
    mjtNum scl = size2[0]/mju_sqrt(len_sqr);
    vec[0] *= scl;
    vec[1] *= scl;
    vec[2] *= scl;
  }

  // disk parallel to plane: pick x-axis of cylinder, scale by radius
  else {
    vec[0] = mat2[0]*size2[0];
    vec[1] = mat2[3]*size2[0];
    vec[2] = mat2[6]*size2[0];
  }

  // project vector on normal
  mjtNum prjvec = mju_dot3(vec, normal);

  // scale axis by half-length
  mju_scl3(axis, axis, size2[1]);
  prjaxis *= size2[1];

  // check first point, construct contact
  int cnt = 0;
  if (dist0 + prjaxis + prjvec <= margin) {
    con[cnt].dist = dist0 + prjaxis + prjvec;
    mju_add3(con[cnt].pos, pos2, vec);
    mju_addTo3(con[cnt].pos, axis);
    mju_addToScl3(con[cnt].pos, normal, -con[cnt].dist*0.5);
    mju_copy3(con[cnt].frame, normal);
    mju_zero3(con[cnt].frame+3);
    cnt++;
  } else {
    return 0;  // nearest point is above margin: no contacts
  }

  // check second point, construct contact
  if (dist0 - prjaxis + prjvec <= margin) {
    con[cnt].dist = dist0 - prjaxis + prjvec;
    mju_add3(con[cnt].pos, pos2, vec);
    mju_subFrom3(con[cnt].pos, axis);
    mju_addToScl3(con[cnt].pos, normal, -con[cnt].dist*0.5);
    mju_copy3(con[cnt].frame, normal);
    mju_zero3(con[cnt].frame+3);
    cnt++;
  }

  // try to add triangle points on side closer to plane
  mjtNum prjvec1 = -prjvec*0.5;
  if (dist0 + prjaxis + prjvec1 <= margin) {
    // compute sideways vector: vec1
    mjtNum vec1[3];
    mju_cross(vec1, vec, axis);
    mju_normalize3(vec1);
    mju_scl3(vec1, vec1, size2[0]*mju_sqrt(3.0)/2);

    // add point A
    con[cnt].dist = dist0 + prjaxis + prjvec1;
    mju_add3(con[cnt].pos, pos2, vec1);
    mju_addTo3(con[cnt].pos, axis);
    mju_addToScl3(con[cnt].pos, vec, -0.5);
    mju_addToScl3(con[cnt].pos, normal, -con[cnt].dist*0.5);
    mju_copy3(con[cnt].frame, normal);
    mju_zero3(con[cnt].frame+3);
    cnt++;

    // add point B
    con[cnt].dist = dist0 + prjaxis + prjvec1;
    mju_sub3(con[cnt].pos, pos2, vec1);
    mju_addTo3(con[cnt].pos, axis);
    mju_addToScl3(con[cnt].pos, vec, -0.5);
    mju_addToScl3(con[cnt].pos, normal, -con[cnt].dist*0.5);
    mju_copy3(con[cnt].frame, normal);
    mju_zero3(con[cnt].frame+3);
    cnt++;
  }

  return cnt;
}



// plane : box
int mjc_PlaneBox(const mjModel* m, const mjData* d,
                 mjContact* con, int g1, int g2, mjtNum margin) {
  mjGETINFO

  // get normal, difference between centers, normal distance
  mjtNum norm[3] = {mat1[2], mat1[5], mat1[8]};
  mjtNum dif[3] = {pos2[0] - pos1[0], pos2[1] - pos1[1], pos2[2] - pos1[2]};
  mjtNum dist = mju_dot3(dif, norm);

  // test all corners, pick bottom 4
  int cnt = 0;
  for (int i=0; i < 8; i++) {
    // get corner in local coordinates
    mjtNum vec[3];
    vec[0] = (i&1 ? size2[0] : -size2[0]);
    vec[1] = (i&2 ? size2[1] : -size2[1]);
    vec[2] = (i&4 ? size2[2] : -size2[2]);

    // get corner in global coordinates relative to box center
    mjtNum corner[3];
    mju_mulMatVec3(corner, mat2, vec);

    // compute distance to plane, skip if too far or pointing up
    mjtNum ldist = mju_dot3(norm, corner);
    if (dist + ldist > margin || ldist > 0) {
      continue;
    }

    // construct contact
    con[cnt].dist = dist + ldist;
    mju_copy3(con[cnt].frame, norm);
    mju_zero3(con[cnt].frame+3);
    mju_addTo3(corner, pos2);
    mju_scl3(vec, norm, -con[cnt].dist/2);
    mju_add3(con[cnt].pos, corner, vec);

    // count; max is 4
    if (++cnt >= 4) {
      return 4;
    }
  }

  return cnt;
}



//--------------------------- sphere and capsule collisions ----------------------------------------

// sphere : sphere (actual implementation, can be called with modified parameters)
static int mjraw_SphereSphere(mjContact* con, mjtNum margin,
                              const mjtNum* pos1, const mjtNum* mat1, const mjtNum* size1,
                              const mjtNum* pos2, const mjtNum* mat2, const mjtNum* size2) {
  // check bounding spheres (this is called from other functions)
  mjtNum dif[3] = {pos1[0] - pos2[0], pos1[1] - pos2[1], pos1[2] - pos2[2]};
  mjtNum cdist_sqr = mju_dot3(dif, dif);
  mjtNum min_dist = margin + size1[0] + size2[0];
  if (cdist_sqr > min_dist*min_dist) {
    return 0;
  }

  // depth and normal
  con[0].dist = mju_sqrt(cdist_sqr) - size1[0] - size2[0];
  mju_sub3(con[0].frame, pos2, pos1);
  mjtNum len = mju_normalize3(con[0].frame);

  // if centers are the same, norm = cross-product of z axes
  //  if z axes are parallel, norm = [1;0;0]
  if (len < mjMINVAL) {
    mjtNum axis1[3] = {mat1[2], mat1[5], mat1[8]};
    mjtNum axis2[3] = {mat2[2], mat2[5], mat2[8]};
    mju_cross(con[0].frame, axis1, axis2);
    mju_normalize3(con[0].frame);
  }

  // position
  mju_scl3(con[0].pos, con[0].frame, size1[0] + con[0].dist/2);
  mju_addTo3(con[0].pos, pos1);

  mju_zero3(con[0].frame+3);
  return 1;
}



// sphere : sphere
int mjc_SphereSphere(const mjModel* m, const mjData* d,
                     mjContact* con, int g1, int g2, mjtNum margin) {
  mjGETINFO
  return mjraw_SphereSphere(con, margin, pos1, mat1, size1, pos2, mat2, size2);
}



// raw sphere : capsule
int mjraw_SphereCapsule(mjContact* con, mjtNum margin,
                        const mjtNum* pos1, const mjtNum* mat1, const mjtNum* size1,
                        const mjtNum* pos2, const mjtNum* mat2, const mjtNum* size2) {
  // get capsule length and axis
  mjtNum len = size2[1];
  mjtNum axis[3] = {mat2[2], mat2[5], mat2[8]};

  // find projection, clip to segment
  mjtNum vec[3] = {pos1[0] - pos2[0], pos1[1] - pos2[1], pos1[2] - pos2[2]};
  mjtNum x = mju_clip(mju_dot3(axis, vec), -len, len);

  // find nearest point on segment, do sphere-sphere test
  mju_scl3(vec, axis, x);
  mju_addTo3(vec, pos2);
  return mjraw_SphereSphere(con, margin, pos1, mat1, size1, vec, mat2, size2);
}



// sphere : capsule
int mjc_SphereCapsule(const mjModel* m, const mjData* d,
                      mjContact* con, int g1, int g2, mjtNum margin) {
  mjGETINFO
  return mjraw_SphereCapsule(con, margin, pos1, mat1, size1, pos2, mat2, size2);
}



// sphere : cylinder
int mjc_SphereCylinder(const mjModel* m, const mjData* d,
                       mjContact* con, int g1, int g2, mjtNum margin) {
  mjGETINFO

  // get cylinder sizes and axis
  mjtNum radius = size2[0];
  mjtNum height = size2[1];
  mjtNum axis[3] = {mat2[2], mat2[5], mat2[8]};

  // find sphere projection onto cylinder axis and plane
  mjtNum vec[3] = {pos1[0] - pos2[0], pos1[1] - pos2[1], pos1[2] - pos2[2]};
  mjtNum x = mju_dot3(axis, vec);
  mjtNum a_proj[3], p_proj[3];
  mju_scl3(a_proj, axis, x);
  mju_sub3(p_proj, vec, a_proj);
  mjtNum p_proj_sqr = mju_dot3(p_proj, p_proj);

  // get collision type
  int collide_side = mju_abs(x) < height;
  int collide_cap = p_proj_sqr < radius*radius;
  if (collide_side && collide_cap) {  // deep penetration (sphere origin inside cylinder)
    mjtNum dist_cap = height - mju_abs(x);
    mjtNum dist_radius = radius - mju_sqrt(p_proj_sqr);
    if (dist_cap < dist_radius) {  // disable one collision type
      collide_side = 0;
    } else {
      collide_cap = 0;
    }
  }

  // side collision: use sphere-sphere
  if (collide_side) {
    mju_addTo3(a_proj, pos2);
    return mjraw_SphereSphere(con, margin, pos1, mat1, size1, a_proj, mat2, size2);
  }

  // cap collision: use plane-sphere
  if (collide_cap) {
    const mjtNum flipmat[9] = {
      -mat2[0], mat2[1], -mat2[2],
      -mat2[3], mat2[4], -mat2[5],
      -mat2[6], mat2[7], -mat2[8]
    };
    const mjtNum* mat_cap;
    mjtNum pos_cap[3];
    if (x > 0) {  // top cap
      mju_addScl3(pos_cap, pos2, axis, height);
      mat_cap = mat2;
    } else {      // bottom cap
      mju_addScl3(pos_cap, pos2, axis, -height);
      mat_cap = flipmat;
    }
    int ncon = mjraw_PlaneSphere(con, margin, pos_cap, mat_cap, size2, pos1, mat1, size1);
    if (ncon) {
      // flip frame normal (because mjGEOM_PLANE < mjGEOM_SPHERE < mjGEOM_CYLINDER)
      mju_scl3(con->frame, con->frame, -1);
    }
    return ncon;
  }

  // otherwise corner collision: use sphere-sphere
  mju_scl3(p_proj, p_proj, size2[0] / mju_sqrt(p_proj_sqr));  // denominator cannot be 0
  mju_scl3(vec, axis, x > 0 ? height : -height);
  mju_addTo3(vec, p_proj);
  mju_addTo3(vec, pos2);

  // sphere-sphere with point sphere at the corner
  mjtNum size_zero[1] = {0};
  return mjraw_SphereSphere(con, margin, pos1, mat1, size1, vec, mat2, size_zero);
}



// raw capsule : capsule
int mjraw_CapsuleCapsule(mjContact* con, mjtNum margin,
                         const mjtNum* pos1, const mjtNum* mat1, const mjtNum* size1,
                         const mjtNum* pos2, const mjtNum* mat2, const mjtNum* size2) {
  // get capsule axes (scaled) and center difference
  mjtNum axis1[3] = {mat1[2] * size1[1], mat1[5] * size1[1], mat1[8] * size1[1]};
  mjtNum axis2[3] = {mat2[2] * size2[1], mat2[5] * size2[1], mat2[8] * size2[1]};
  mjtNum dif[3] = {pos1[0] - pos2[0], pos1[1] - pos2[1], pos1[2] - pos2[2]};

  // compute matrix coefficients and determinant
  mjtNum ma =  mju_dot3(axis1, axis1);
  mjtNum mb = -mju_dot3(axis1, axis2);
  mjtNum mc =  mju_dot3(axis2, axis2);
  mjtNum u  = -mju_dot3(axis1, dif);
  mjtNum v  =  mju_dot3(axis2, dif);
  mjtNum det = ma*mc - mb*mb;

  // general configuration (non-parallel axes)
  if (mju_abs(det) >= mjMINVAL) {
    // find projections, clip to segments
    mjtNum x1 = (mc*u - mb*v) / det;
    mjtNum x2 = (ma*v - mb*u) / det;

    if (x1 > 1) {
      x1 = 1;
      x2 = (v - mb) / mc;
    } else if (x1 < -1) {
      x1 = -1;
      x2 = (v + mb) / mc;
    }
    if (x2 > 1) {
      x2 = 1;
      x1 = mju_clip((u - mb) / ma, -1, 1);
    } else if (x2 < -1) {
      x2 = -1;
      x1 = mju_clip((u + mb) / ma, -1, 1);
    }

    // find nearest points, do sphere-sphere test
    mjtNum vec1[3], vec2[3];
    mju_scl3(vec1, axis1, x1);
    mju_addTo3(vec1, pos1);
    mju_scl3(vec2, axis2, x2);
    mju_addTo3(vec2, pos2);

    return mjraw_SphereSphere(con, margin, vec1, mat1, size1, vec2, mat2, size2);
  }

  // parallel axes
  else {
    // x1 = 1
    mjtNum vec1[3];
    mju_add3(vec1, pos1, axis1);
    mjtNum x2 = mju_clip((v - mb) / mc, -1, 1);

    mjtNum vec2[3];
    mju_scl3(vec2, axis2, x2);
    mju_addTo3(vec2, pos2);
    int n1 = mjraw_SphereSphere(con, margin, vec1, mat1, size1, vec2, mat2, size2);

    // x1 = -1
    mju_sub3(vec1, pos1, axis1);
    x2 = mju_clip((v + mb) / mc, -1, 1);
    mju_scl3(vec2, axis2, x2);
    mju_addTo3(vec2, pos2);
    int n2 = mjraw_SphereSphere(con+n1, margin, vec1, mat1, size1, vec2, mat2, size2);

    // return if two contacts already found
    if (n1+n2 >= 2) {
      return n1+n2;
    }

    // x2 = 1
    mju_add3(vec2, pos2, axis2);
    mjtNum x1 = mju_clip((u - mb) / ma, -1, 1);
    mju_scl3(vec1, axis1, x1);
    mju_addTo3(vec1, pos1);
    int n3 = mjraw_SphereSphere(con+n1+n2, margin, vec1, mat1, size1, vec2, mat2, size2);

    // return if two contacts already found
    if (n1+n2+n3 >= 2) {
      return n1+n2+n3;
    }

    // x2 = -1
    mju_sub3(vec2, pos2, axis2);
    x1 = mju_clip((u + mb) / ma, -1, 1);
    mju_scl3(vec1, axis1, x1);
    mju_addTo3(vec1, pos1);
    int n4 = mjraw_SphereSphere(con+n1+n2+n3, margin, vec1, mat1, size1, vec2, mat2, size2);

    return n1+n2+n3+n4;
  }
}



// capsule : capsule
int mjc_CapsuleCapsule(const mjModel* m, const mjData* d,
                       mjContact* con, int g1, int g2, mjtNum margin) {
  mjGETINFO
  return mjraw_CapsuleCapsule(con, margin, pos1, mat1, size1, pos2, mat2, size2);
}



// sign of (signed) area of planar triangle
static mjtNum areaSign(const mjtNum p1[2], const mjtNum p2[2], const mjtNum p3[2]) {
  return mju_sign((p1[0]-p3[0])*(p2[1]-p3[1]) - (p2[0]-p3[0])*(p1[1]-p3[1]));
}



// find nearest point to p within line segment (u,v); return distance to p
static mjtNum pointSegment(mjtNum res[2], const mjtNum p[2],
                           const mjtNum u[2], const mjtNum v[2]) {
  // make u the origin
  mjtNum uv[2] = {v[0]-u[0], v[1]-u[1]};
  mjtNum up[2] = {p[0]-u[0], p[1]-u[1]};

  // project: find a s.t. uv is orthogonal to (up-a*uv)
  mjtNum a = mju_dot(uv, up, 2) / mju_max(mjMINVAL, mju_dot(uv, uv, 2));

  // find nearest point to p, clamp to u or v if a is not in (0,1)
  if (a <= 0) {
    res[0] = u[0];
    res[1] = u[1];
  } else if (a >= 1) {
    res[0] = v[0];
    res[1] = v[1];
  } else {
    mju_addScl(res, u, uv, a, 2);
  }

  // compute distance
  return mju_sqrt((res[0]-p[0])*(res[0]-p[0]) + (res[1]-p[1])*(res[1]-p[1]));
}



// sphere : triangle with radius
int mjraw_SphereTriangle(mjContact* con, mjtNum margin,
                         const mjtNum* s, mjtNum rs,
                         const mjtNum* t1, const mjtNum* t2, const mjtNum* t3, mjtNum rt) {
  mjtNum rbound = margin + rs + rt;
  mjtNum X[3];

  // make t1 the origin: triangle is (O,A,B); sphere center is S
  mjtNum S[3] = { s[0]-t1[0],  s[1]-t1[1],  s[2]-t1[2]};
  mjtNum A[3] = {t2[0]-t1[0], t2[1]-t1[1], t2[2]-t1[2]};
  mjtNum B[3] = {t3[0]-t1[0], t3[1]-t1[1], t3[2]-t1[2]};

  // N is normal to triangle plane
  mjtNum N[3];
  mju_cross(N, A, B);
  mju_normalize3(N);

  // dstS is signed distance from S to plane; exit if too large
  mjtNum dstS = mju_dot3(N, S);
  if (mju_abs(dstS) > rbound) {
    return 0;
  }

  // P is projection of S in triangle plane
  mjtNum P[3];
  mju_addScl3(P, S, N, -dstS);

  // construct orthogonal axes (V1~A, V2) of triangle plane
  mjtNum V1[3], V2[3];
  mju_copy3(V1, A);
  mjtNum lenA = mju_normalize3(V1);
  mju_cross(V2, N, A);
  mju_normalize3(V2);

  // triangle is (o,a,b), sphere center is p
  mjtNum o[2] = {0, 0};
  mjtNum a[2] = {lenA, 0}; // equals {mju_dot3(V1, A), mju_dot3(V2, A)}
  mjtNum b[2] = {mju_dot3(V1, B), mju_dot3(V2, B)};
  mjtNum p[2] = {mju_dot3(V1, P), mju_dot3(V2, P)};

  // copmuted signs of areas of (p,o,a), (p,a,b), (p,b,o)
  mjtNum sign1 = areaSign(p, o, a);
  mjtNum sign2 = areaSign(p, a, b);
  mjtNum sign3 = areaSign(p, b, o);

  // p is inside triangle
  if (sign1 == sign2 && sign2 == sign3) {
    // P is nearest point to S within triangle
    mju_copy3(X, P);
  }

  // p is not inside triangle
  else {
    // find nearest point to p on triangle edges (o,a), (a,b), (b,o)
    mjtNum x[3][2], dstx[3];
    dstx[0] = pointSegment(x[0], p, o, a);
    dstx[1] = pointSegment(x[1], p, a, b);
    dstx[2] = pointSegment(x[2], p, b, o);

    // select minimum
    int best = (dstx[0] < dstx[1] && dstx[0] < dstx[2]) ? 0 : (dstx[1] < dstx[2] ? 1 : 2);

    // convert x[best] to 3D
    mju_scl3(X, V1, x[best][0]);
    mju_addToScl3(X, V2, x[best][1]);
  }

  // X is now the nearest point to S within the 3D triangle (O,A,B)
  // compute contact normal and distance
  mjtNum nrm[3] = {X[0]-S[0], X[1]-S[1], X[2]-S[2]};
  mjtNum dst = mju_normalize3(nrm);

  // exit if too far
  if (dst > rbound) {
    return 0;
  }

  // construct contact
  con[0].dist = dst - rs - rt;
  mju_addScl3(con[0].pos, s, nrm, rs + con[0].dist/2);
  mju_copy3(con[0].frame, nrm);
  mju_zero3(con[0].frame+3);

  return 1;
}