};
static inline void ao_quaternion_multiply(struct ao_quaternion *r,
- struct ao_quaternion *a,
- struct ao_quaternion *b)
+ 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);
}
static inline void ao_quaternion_conjugate(struct ao_quaternion *r,
- struct ao_quaternion *a)
+ const struct ao_quaternion *a)
{
r->r = a->r;
r->x = -a->x;
r->z = -a->z;
}
-static inline float ao_quaternion_normal(struct ao_quaternion *a)
+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);
}
static inline void ao_quaternion_scale(struct ao_quaternion *r,
- struct ao_quaternion *a,
+ const struct ao_quaternion *a,
float b)
{
r->r = a->r * b;
}
static inline void ao_quaternion_normalize(struct ao_quaternion *r,
- struct ao_quaternion *a)
+ const struct ao_quaternion *a)
{
float n = ao_quaternion_normal(a);
*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,
- struct ao_quaternion *a,
- struct ao_quaternion *b)
+ const struct ao_quaternion *a,
+ const struct ao_quaternion *b)
{
struct ao_quaternion c;
struct ao_quaternion t;
- ao_quaternion_conjugate(&c, b);
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->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_ */
projects / fw / altos / blobdiff