altos: Add functions to init quaternions from vector pairs and euler angles

Our low sampling rate means that the "cheap" hack for
integrating quaternion rotations by using sin(x) ≃ x doesn't work, so
instead we have to compute the partial rotation the hard way.

Signed-off-by: Keith Packard <keithp@keithp.com>

diff --git a/src/core/ao_quaternion.h b/src/core/ao_quaternion.h
index 6a701a1877aca163decc06ccd58578b5f8952873..6c8855003f59e9219bae63d7e9bfa45ff58828e0 100644 (file)
--- a/src/core/ao_quaternion.h
+++ b/src/core/ao_quaternion.h
@@ -110,17 +110,68 @@ static inline void ao_quaternion_normalize(struct ao_quaternion *r,
 }
 
 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)
 {
@@ -146,4 +197,44 @@ static inline void ao_quaternion_init_zero_rotation(struct ao_quaternion *r)
        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_ */