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/*
* Copyright © 2013 Keith Packard <keithp@keithp.com>
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; version 2 of the License.
*
* This program is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* General Public License for more details.
*
* You should have received a copy of the GNU General Public License along
* with this program; if not, write to the Free Software Foundation, Inc.,
* 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA.
*/
#ifndef _AO_QUATERNION_H_
#define _AO_QUATERNION_H_
#include <math.h>
struct ao_quaternion {
float r; /* real bit */
float x, y, z; /* imaginary bits */
};
static inline void ao_quaternion_multiply(struct ao_quaternion *r,
const struct ao_quaternion *a,
const struct ao_quaternion *b)
{
struct ao_quaternion t;
#define T(_a,_b) (((a)->_a) * ((b)->_b))
/*
* Quaternions
*
* ii = jj = kk = ijk = -1;
*
* kji = 1;
*
* ij = k; ji = -k;
* kj = -i; jk = i;
* ik = -j; ki = j;
*
* Multiplication p * q:
*
* (pr + ipx + jpy + kpz) (qr + iqx + jqy + kqz) =
*
* ( pr * qr + pr * iqx + pr * jqy + pr * kqz) +
* (ipx * qr + ipx * iqx + ipx * jqy + ipx * kqz) +
* (jpy * qr + jpy * iqx + jpy * jqy + jpy * kqz) +
* (kpz * qr + kpz * iqx + kpz * jqy + kpz * kqz) =
*
*
* (pr * qr) + i(pr * qx) + j(pr * qy) + k(pr * qz) +
* i(px * qr) - (px * qx) + k(px * qy) - j(px * qz) +
* j(py * qr) - k(py * qx) - (py * qy) + i(py * qz) +
* k(pz * qr) + j(pz * qx) - i(pz * qy) - (pz * qz) =
*
* 1 * ( (pr * qr) - (px * qx) - (py * qy) - (pz * qz) ) +
* i * ( (pr * qx) + (px * qr) + (py * qz) - (pz * qy) ) +
* j * ( (pr * qy) - (px * qz) + (py * qr) + (pz * qx) ) +
* k * ( (pr * qz) + (px * qy) - (py * qx) + (pz * qr);
*/
t.r = T(r,r) - T(x,x) - T(y,y) - T(z,z);
t.x = T(r,x) + T(x,r) + T(y,z) - T(z,y);
t.y = T(r,y) - T(x,z) + T(y,r) + T(z,x);
t.z = T(r,z) + T(x,y) - T(y,x) + T(z,r);
#undef T
*r = t;
}
static inline void ao_quaternion_conjugate(struct ao_quaternion *r,
const struct ao_quaternion *a)
{
r->r = a->r;
r->x = -a->x;
r->y = -a->y;
r->z = -a->z;
}
static inline float ao_quaternion_normal(const struct ao_quaternion *a)
{
#define S(_a) (((a)->_a) * ((a)->_a))
return S(r) + S(x) + S(y) + S(z);
#undef S
}
static inline void ao_quaternion_scale(struct ao_quaternion *r,
const struct ao_quaternion *a,
float b)
{
r->r = a->r * b;
r->x = a->x * b;
r->y = a->y * b;
r->z = a->z * b;
}
static inline void ao_quaternion_normalize(struct ao_quaternion *r,
const struct ao_quaternion *a)
{
float n = ao_quaternion_normal(a);
if (n > 0)
ao_quaternion_scale(r, a, 1/sqrtf(n));
else
*r = *a;
}
static inline float ao_quaternion_dot(const struct ao_quaternion *a,
const struct ao_quaternion *b)
{
#define T(_a) (((a)->_a) * ((b)->_a))
return T(r) + T(x) + T(y) + T(z);
#undef T
}
static inline void ao_quaternion_rotate(struct ao_quaternion *r,
const struct ao_quaternion *a,
const struct ao_quaternion *b)
{
struct ao_quaternion c;
struct ao_quaternion t;
ao_quaternion_multiply(&t, b, a);
ao_quaternion_conjugate(&c, b);
ao_quaternion_multiply(r, &t, &c);
}
/*
* Compute a rotation quaternion between two vectors
*
* cos(θ) + u * sin(θ)
*
* where θ is the angle between the two vectors and u
* is a unit vector axis of rotation
*/
static inline void ao_quaternion_vectors_to_rotation(struct ao_quaternion *r,
const struct ao_quaternion *a,
const struct ao_quaternion *b)
{
/*
* The cross product will point orthogonally to the two
* vectors, forming our rotation axis. The length will be
* sin(θ), so these values are already multiplied by that.
*/
float x = a->y * b->z - a->z * b->y;
float y = a->z * b->x - a->x * b->z;
float z = a->x * b->y - a->y * b->x;
float s_2 = x*x + y*y + z*z;
float s = sqrtf(s_2);
/* cos(θ) = a · b / (|a| |b|).
*
* a and b are both unit vectors, so the divisor is one
*/
float c = a->x*b->x + a->y*b->y + a->z*b->z;
float c_half = sqrtf ((1 + c) / 2);
float s_half = sqrtf ((1 - c) / 2);
/*
* Divide out the sine factor from the
* cross product, then multiply in the
* half sine factor needed for the quaternion
*/
float s_scale = s_half / s;
r->x = x * s_scale;
r->y = y * s_scale;
r->z = z * s_scale;
r->r = c_half;
ao_quaternion_normalize(r, r);
}
static inline void ao_quaternion_init_vector(struct ao_quaternion *r,
float x, float y, float z)
{
r->r = 0;
r->x = x;
r->y = y;
r->z = z;
}
static inline void ao_quaternion_init_rotation(struct ao_quaternion *r,
float x, float y, float z,
float s, float c)
{
r->r = c;
r->x = s * x;
r->y = s * y;
r->z = s * z;
}
static inline void ao_quaternion_init_zero_rotation(struct ao_quaternion *r)
{
r->r = 1;
r->x = r->y = r->z = 0;
}
/*
* The sincosf from newlib just calls sinf and cosf. This is a bit
* faster, if slightly less precise
*/
static inline void
ao_sincosf(float a, float *s, float *c) {
float _s = sinf(a);
*s = _s;
*c = sqrtf(1 - _s*_s);
}
/*
* Initialize a quaternion from 1/2 euler rotation angles (in radians).
*
* Yes, it would be nicer if there were a faster way, but because we
* sample the gyros at only 100Hz, we end up getting angles too large
* to take advantage of sin(x) ≃ x.
*
* We might be able to use just a couple of elements of the sin taylor
* series though, instead of the whole sin function?
*/
static inline void ao_quaternion_init_half_euler(struct ao_quaternion *r,
float x, float y, float z)
{
float s_x, c_x;
float s_y, c_y;
float s_z, c_z;
ao_sincosf(x, &s_x, &c_x);
ao_sincosf(y, &s_y, &c_y);
ao_sincosf(z, &s_z, &c_z);
r->r = c_x * c_y * c_z + s_x * s_y * s_z;
r->x = s_x * c_y * c_z - c_x * s_y * s_z;
r->y = c_x * s_y * c_z + s_x * c_y * s_z;
r->z = c_x * c_y * s_z - s_x * s_y * c_z;
}
#endif /* _AO_QUATERNION_H_ */
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