👨🏽‍🔧 🧜 👩‍👩‍👦‍👦 Cuaterniones. Solución de un problema de navegación 🚥 🕌 📎

Historia

Hace algún tiempo me enfrenté a un problema interesante relacionado con la navegación por satélite. Usando el frente de fase de la señal, el objeto de navegación (NVO) mide las coordenadas de los satélites de navegación (NS) en su sistema de coordenadas (sistema local, LSK). Además, el ONV recibe los valores de las posiciones del NS en el sistema de coordenadas global (GCS), y mide el tiempo de recepción de la señal del NS (Fig. 1). Se requería calcular las coordenadas de la ONV en el GSK y la hora del sistema, es decir, solucionar el problema de navegación.

El problema era interesante porque su solución teóricamente permite reducir el número de NN en comparación con el número de NN que se requieren en los métodos implementados en los sistemas de navegación por satélite. En ese momento, me dediqué principalmente al estudio de la calidad de las mediciones del frente de fase y la obtención de ecuaciones de navegación para coordenadas y tiempo, asumiendo que el cálculo de la orientación y coordenadas del NVG no causaría ningún problema especial. Además, en un avión, el problema se resolvió rápida y fácilmente.

, , , . , . - , .

, , . , . , , , .

, :

$\ mathbf R_i = \ begin {bmatrix} x_i & y_i & z_i \ end {bmatrix} ^ T$ : - - , i = 1, 2, 3 : ,

$\ mathbf R_i ^ {'} = \ begin {bmatrix} x_i ^ {'} & y_i ^ {'} & z_i ^ {'} \ end {bmatrix} ^ T$ : - - ; $\ mathbf R_i$ $\ mathbf R_i ^ {'}$ ; , : 3- E ^ 3 $(\ mathbf e_x, \ mathbf e_y, \ mathbf e_z)$ $(\ mathbf e_x ^ {'}, \ mathbf e_y ^ {'}, \ mathbf e_z ^ {'})$ ; . , - , , .

$\ mathbf R_a = \ begin {bmatrix} x_a & y_a & z_a \ end {bmatrix} ^ T$ : - ,

$\ mathbf M$ : , " " (. 2)

$\ mathbf R_i$ $\ mathbf R_i ^ {'}$ : $\ mathbf R_i = x_i \ mathbf e_x + y_i \ mathbf e_y + z_i \ mathbf e_z, \ quad \ mathbf R_i ^ {'} = x_i ^ {'} \ mathbf e_x ^ {'} + y_i ^ {i} \ mathbf e_y ^ {'} + z_i ^ {'} \ mathbf e_z ^ {'}$ , i = 1,2,3 - .

$\begin{cases} \mathbf e_x^{'} = a_1^1 \mathbf e_x + a_1^2 \mathbf e_y + a_1^3 \mathbf e_z,\\ \mathbf e_y^{'} = a_2^1 \mathbf e_x + a_2^2 \mathbf e_y + a_2^3 \mathbf e_z,\\ \mathbf e_z^{'} = a_3^1 \mathbf e_x + a_3^2 \mathbf e_y + a_3^3 \mathbf e_z, \end{cases}$

$\lbrace a_j^i: a_j^i \in \mathcal R \rbrace$ , $\mathcal R$ - , i = 1, 2, 3 .

$\mathbf M = \begin{bmatrix} a_1^1 & a_2^1 & a_3^1\\ a_1^2 & a_2^2 & a_3^2\\ a_1^3 & a_2^3 & a_3^3 \end{bmatrix},$

$\mathbf e_x^{'} = \mathbf M \mathbf e_x,\; \mathbf e_y^{'} = \mathbf M \mathbf e_y,\; \mathbf e_z^{'} = \mathbf M \mathbf e_z,\;$

, $\mathbf e_x = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}^T$ , $\mathbf e_y = \begin{bmatrix} 0 & 1 & 0 \end{bmatrix}^T$ , $\mathbf e_z = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix}^T$ . , $\mathbf R_i^{'}$ $\mathbf R_i^{''} = \mathbf M \mathbf R_i^{'}$ . , $\mathbf M$ ( " "), , $E^3 \to E^3$ , . .

$\mathbf R_a = \mathbf R_i - \mathbf M \mathbf R_i^{'}.$ (1)

(1) $\mathbf R_a$ $\mathbf M$ , .

$\mathbf R = \mathbf R_a + \mathbf M \mathbf R^{'},\qquad (2)$

$\mathbf R = \begin{bmatrix} x_1 & x_2 & x_3\\ y_1 & y_2 & y_3\\ z_1 & z_2 & z_3 \end{bmatrix},\quad \mathbf R_1 = \begin{bmatrix} x_1^{'} & x_2^{'} & x_3^{'}\\ y_1^{'} & y_2^{'} & y_3^{'}\\ z_1^{'} & z_2^{'} & z_3 ^ {'} \end{bmatrix} -$

, $\mathbf R_a$ $\mathbf M$ . (1) (2) , $\mathbf R_a$ :

$\mathbf R_a = \mathbf R_i - \mathbf M \mathbf R_i^{'} = \mathbf R_k - \mathbf M \mathbf R_k^{'},\quad i \ne k;\quad i,k=1,2,3,$

$(x_k^{'} - x_i^{'}) \mathbf M \mathbf e_x + (y_k^{'} - y_i^{'}) \mathbf M \mathbf e_y + (z_k^{'} - z_i^{'}) \mathbf M \mathbf e_z =\\ = (x_k - x_i) \mathbf M \mathbf e_x + (y_k - y_i) \mathbf M \mathbf e_y + (z_k - z_i) \mathbf M \mathbf e_z,$

$\mathbf M \mathbf R_{ki}^{'} = \mathbf R_{ki}, \qquad (3)$

$\mathbf R_{ki}^{'} = \mathbf R_k^{'} - \mathbf R_i^{'}$ .

- $\mathbf M$ (3) (1), . , (3) (2), .

$\begin{cases} \mathbf M \mathbf R_{12}^{'} = \mathbf R_{12},\\ \mathbf M \mathbf R_{13}^{'} = \mathbf R_{13} , \end{cases} \qquad (4)$

, .

, $\mathbf M$ , (3) (4) . , .

, , , .

, , $\dot q = q^0 + iq^1 + jq^2 + kq^3 = q^0 + \dot{ \mathbf q}$ , q^0 , q^1 , q^2 , q^3 $\in \mathcal R$ , q^0 - ( ), $\dot{\mathbf q}$ - ; 1, i, j, k - :

$\begin{split} 1^2 = 1,\quad 1i = i1,\quad 1j = j1,\quad 1k = k1,\quad i^2 = j^2 = k^2 = -1,\\ ij = -ji = k,\quad jk = -kj = i,\quad ki = -ik = j. \end{split}$

(), , . $\dot q = q^0 + \dot{\mathbf q} = q^0 + iq^1 + jq^2 + kq^3$ , , $\Vert \dot q \Vert = 1$ , , $\Vert \dot q \Vert = q^{(0)^2} + \Vert \dot {\mathbf q } \Vert = 1$ , $\Vert \dot q \Vert = \cos^2 \gamma + \sin^2 \gamma = 1$ , $\cos^2 \gamma = q^{(0)^2}$ , $\sin^2 \gamma = \Vert \dot { \mathbf q} \Vert = q^{(1)^2} + q^{(2)^2} + q^{(3)^2}$ . , - . $\dot { \mathbf q}$

$\mathbf{ \dot q} = \frac{1}{\Vert \dot {\mathbf q} \Vert} (q^{(1)^2} + q^{(2)^2} + q^{(3)^2}) \Vert \dot {\mathbf q} \Vert = (q^{(x)^2} + q^{(y)^2} + q^{(z)^2}) \Vert \dot {\mathbf q} \Vert = (q^{(x)^2} + q^{(y)^2} + q^{(z)^2}) \sin^2 \gamma,$

$q^x = \frac{q^1}{\vert \dot {\mathbf q} \vert}$ , $q^y = \frac{q^2}{\vert \dot {\mathbf q} \vert}$ , $q^z = \frac{q^3}{\vert \dot {\mathbf q} \vert}$ , $\dot q$ :

$\dot q = \cos \gamma + \dot {\mathbf q}_n \sin \gamma,$

$\dot {\mathbf q}_n = iq^x + jq^y + kq^z$ . $\dot R$ $\dot {\mathbf R}$ ,

$\dot R^{'} = \dot q \circ \dot R \circ \dot q^{-1} \qquad (5)$

$\dot{\mathbf R}^{'}$ , $\dot {\mathbf R}$ , $2 \gamma$ , $\dot {\mathbf q}_n$ . " $\dot q$ $\dot R$ ", , , $\dot q$ $\dot{\mathbf R}$ $\dot{\mathbf R}^{'}$ (5).

. : $\dot {\mathbf q} \dot {\mathbf r}$ $\dot {\mathbf q} \cdot \dot {\mathbf r}$ : , $\dot q \circ \dot r$ : , $\dot {\mathbf q} \times \dot {\mathbf r}$ : .

$\mathbf R_{ki}^{'}$ $\mathbf R_{ki}$ . , . k = 1 , i = 2.

-, , , .

-, . , , (3) $\mathbf M$ .

, -, $\vert \mathbf R_{12}^{'} \vert = \vert \mathbf R_{12} \vert$ . , $\mathbf M$ - , $\mathbf R_{12}^{'}$ , (3) , $\mathbf R_{12}^{'}$ $\mathbf R_{12}$ .

T (3) , $\mathbf R_{12}^{'}$ $\mathbf R_{12}$ :

$\mathbf r_1 \equiv \frac{\mathbf R_{12}}{\vert \mathbf R_{12} \vert} = \begin{bmatrix} r_1^1 & r_1^2 & r_1^3 \end{bmatrix}^T, \qquad \mathbf p_1 \equiv \frac{\mathbf R_{12}^{'}}{\vert \mathbf R_{12}^{'} \vert} = \begin{bmatrix} p_1^1 & p_1^2 & p_1^3 \end{bmatrix}^T,$

$\dot r = 0 + \mathbf{\dot r}_1 = 0 + ir_1^1 + jr_1^2 + kr_1^3, \quad \dot p = 0 + \mathbf{\dot p}_1 = 0 + ip_1^1 + jp_1^2 + kp_1^3.$

(3) :

$\dot q \circ \dot r_1 \circ \dot q^{-1} = \dot p_1,\qquad (6)$

$\dot q$ - , $\mathbf M$ , (1) :

$\dot R_a = \dot R_1 - \dot q \circ \dot R_1 \circ \dot q^{-1}, \qquad (7)$

$\dot R_a = 0 + ix_a + jy_a + kz_a$ , $\dot R_1 = 0 + ix_1 + jy_1 + kz_1$ .

$\dot {\mathbf r}_1$ $\dot {\mathbf p}_1$ , , , $\dot {\mathbf r}_1$ $\dot {\mathbf p}_1$ $\gamma_1$ , ( $\lambda_1$ ). , $\dot {\mathbf r}_1$ $\dot {\mathbf p}_1$ (. 3), , $\gamma_1$ .

(6), (3), - : $\dot q$ , (6). , , , $\dot {\mathbf r}_1$ $\dot {\mathbf p}_1$ . . 3 , $\dot {\mathbf d}_1$ , $\lambda_1$ . , $\gamma_{d_1}\ = \gamma_1$ .

. 4, ) ) , $\dot {\mathbf q}_1$ $\tau_1$ $\dot {\mathbf r}_1$ $\dot {\mathbf p}_1$ . , , $\dot {\mathbf r}_1$ $\dot {\mathbf p}_1$ , ( $\lambda_1$ ), $\gamma_{q_1}$ , $\tau_1$ , $\gamma_1$ , $\lambda_1$ .

$\dot q$ , $\dot r_2 = \dot{R}_3^{'} - \dot{R}_1^{'}$ $\dot p_2 = \dot R_3 - \dot R_1$ , $\mathbf R_3$ $\mathbf R_1$ . (6), (4),

$\begin{cases} \dot q \circ \dot r_1 \circ \dot q^{-1} = \dot p_1,\\ \dot q \circ \dot r_2\circ \dot q^{-1} = \dot p_2. \end{cases} \qquad (8)$

, $\dot q$ , (8), , , . , (7) R_a - .

(8), - , , - .

. , , $\dot {\mathbf r}_1$ $\dot {\mathbf p}_1$ , $\dot {\mathbf r}_2$ $\dot {\mathbf p}_2$ . , $\dot {\mathbf r}_1$ ( $\dot {\mathbf p}_1$ ) $\dot {\mathbf r}_2$ ( $\dot {\mathbf p}_2$ ), $\tau_1$ , $\tau_2$ . , $\dot {\mathbf q}$ . $\dot {\mathbf r}_i$ $\dot {\mathbf p}_i$ , i = 1, 2, , $\gamma_{q_i}$ , $\tau_i$ , $\gamma_i$ (. 5).

1. $\dot {\mathbf b}_1$ , $\dot {\mathbf r}_1$ $\dot {\mathbf p}_1$ , $\pi$ (. 4, ), )).

2. , $\dot {\mathbf r}_i$ $\dot {\mathbf p}_i$ , i = 1, 2, , , $\dot {\mathbf d}_i$ $\dot {\mathbf b}_i$ ( $\delta_i$ ), . 6). .

$\tau_i$ , , $\dot {\mathbf r}_i$ ( $\dot {\mathbf p}_i$ ), $\delta_i$ .

, , , $\dot {\mathbf r}_i$ $\dot {\mathbf p}_i$ , i = 1, 2, $\delta_i$ , . $\dot {\mathbf q}$ $\delta_i$ (. 7).

, $\dot {\mathbf d}_i$ . , $\beta_1$ $\dot {\mathbf d}_1$ , $\beta_2$ $\dot {\mathbf d}_2$ (. 8). $\beta_1 \ne \beta_2$ .

. 8. $\dot e_{d_1}$ , $\dot {\mathbf r}_1$ $\pi$ $\lambda_1$ , ..

$\dot e_{d_1} = 0 + \dot {\mathbf s}_1, \qquad (9)$

$\beta_1$ $\delta_1$ ,

$\dot e_{q_1}(\beta_i) = \dot h_1(\beta_1) \circ \dot e_{d_1} \circ \dot h_1^{-1}(\beta_1) \qquad (10)$

$\dot {\mathbf e}_{q_1}(\beta_1)$ , $\pi$ $\dot {\mathbf r}_1$ , $\dot {\mathbf r}_1$ . $\dot {\mathbf e}_{d_1}$ $\dot {\mathbf e}_{q_1}$ $\dot h_1(\beta_1)$ , $\delta_1$ .

2 , $\dot {\mathbf r}_1$ $\dot {\mathbf p}_1$ $\dot {\mathbf e}_{q_1}(\beta_1)$ $\gamma_{q_1}$ , , , $\beta_1$ . , $\beta_1$ $\gamma_{q_1} = f(\beta_1)$ , $\dot {\mathbf e}_{q_i}$ $\dot q$ , .

$\gamma_{q_1} = f(\beta_1)$ .

$\beta_1$ . $\dot {\mathbf q}$ $\dot {\mathbf h}_1$ $\dot {\mathbf h}_2$ , $\dot {\mathbf h}_1(\beta_1) \times \dot {\mathbf h}_2(\beta_2)$ $\dot {\mathbf q}$ .

$\dot {\mathbf q}_e \equiv \dot {\mathbf h}_1(\beta_1) \times \dot {\mathbf h}_2(\beta_2) \qquad (11)$

: $\dot {\mathbf e}_{q_1}(\beta_1) \dot {\mathbf q}_e$ . , $\dot {\mathbf q}_e$ $\beta_1$ , $\beta_2$ . $\dot {\mathbf q}_e$ $\dot h_1 = \dot h_1 (\beta_1) \vert _{\beta_1 = \pi}$ $\dot h_2 = \dot h_2 (\beta_2) \vert _{\beta_2 = \pi}$ , , $\vert \dot {\mathbf h}_i \vert = 1$ . , $\beta_1$ , ,

$\dot {\mathbf e}_{q_1}(\beta_1) \frac{\dot {\mathbf q}_e}{ \vert \dot {\mathbf q}_e \vert } = \dot {\mathbf e}_{q_1}(\beta_1) \dot{\mathbf q}_{e_{n}} = 1, \qquad (12)$

, $\vert \dot {\mathbf e}_{q_1} (\beta_1) \vert = 1$ . , $\beta_1$ , $\gamma_{q_1} = f(\beta_1)$ $\dot {\mathbf e}_{q_1}(\beta_1)$ , $\dot {\mathbf q}$ :

$\dot {\mathbf q} = \cos \frac{\gamma_{q_1}}{2} + \dot {\mathbf e}_{q_1} \sin \frac{\gamma_{q_1}}{2}. \qquad (13)$

, . , $\dot e_{d_2 }$ , $\dot e_{q_2}$ , $\dot h_2$ , $\beta_2$ , $\gamma_2$ , $\gamma_{q_2}$ . "1" "2" " i ", , i = 1, 2 .

,

, :

$\dot d_i$ : $\dot{\mathbf r}_i$ $\dot{\mathbf p}_i$ ,
$\dot b_i$ : $\dot{\mathbf r}_i$ $\dot{\mathbf p}_i$ ; $\dot{\mathbf d}_i$ $\dot{\mathbf b}_i$ $\delta_i$ , $\dot q$ ,
$\dot e_{d_i}$ : $\dot d_i$ ; $\dot e_{d_i}$ 1,
$\dot h_i(\beta_i)$ : $\dot{\mathbf e}_{d_i}$ $\delta_i$ $\beta_i$ ; $\beta_i$ ,
$\gamma_{q_i} = f(\beta_i)$ : $\gamma_{q_i}$ $\dot q$ $\dot e_{q_i}$ ,
$\dot e_{q_i} (\beta_i)$ : $\dot{\mathbf e}_{d_i}$ $\delta_i$ $\beta_i$ ; $\dot e_{q_i}$ ,
$\dot q_e$ : $\beta_i$ ,
, , $\dot q$ : $\dot e_{q_i}$ , $\beta_i$ $\gamma_{q_i}$ .

$\dot d_i$

$\dot d_i$ , $\dot {\mathbf r}_i$ $\dot {\mathbf p}_i$ . $\dot {\mathbf r}_i$ $\lambda_i$ , $\lambda_i$ (. 3). $\dot { \mathbf r}_i \times \dot {\mathbf p}_i$ . $\vert \dot {\mathbf r}_i \vert = \vert \dot {\mathbf p}_i \vert = 1$ , $\vert \dot {\mathbf r}_i \times \dot {\mathbf p}_i \vert = \vert \dot {\mathbf r}_i \vert \vert \dot {\mathbf p}_i \vert \sin \gamma_i$ , :

$\dot {\mathbf r}_i \times \dot {\mathbf p}_i = \frac{ \dot {\mathbf r}_i \times \dot {\mathbf p}_i }{ \vert \dot {\mathbf r}_i \times \dot {\mathbf p}_i \vert } \vert \dot {\mathbf r}_i \times \dot {\mathbf p}_i \vert = \dot {\mathbf s}_i \sin \gamma_i, \qquad (14)$

$\dot {\mathbf s}_i$ - ,

$\dot {\mathbf s}_i = \frac{1}{\sin \gamma_i} \begin{vmatrix} i & j & k \\ r_i^1 & r_i^2 & r_i^3 \\ p_i^1 & p_i^2 & p_i^3 \end{vmatrix} = is_i^x + js_i^y + ks_i^z, \qquad (15)$

$s_i^x = \frac{r_i^2 p_i^3 - r_i^3 p_i^2}{\sin \gamma_i}, \quad s_i^y = \frac{r_i^3 p_i^1 - r_i^1 p_i^3}{\sin \gamma_i}, \quad s_i^z = \frac{r_1^1 p_1^2 - r_i^2 p_i^1}{\sin \gamma_i}. \qquad (16)$

$\dot {\mathbf s}_i \sin \gamma_i$ $\dot s_i$ , $\dot {\mathbf r}_i$ $2 \gamma_i$ $\lambda_i$ : $\dot s_1 = \cos \gamma_i + \dot {\mathbf s}_i \sin \gamma_i$ . $\dot {\mathbf d}_i$ $\gamma_i$

$\dot d_i = \cos \frac{\gamma_i}{2} + \dot {\mathbf s}_i \sin \frac{\gamma_i}{2} \qquad (17)$

$\dot p_i = \dot d_i \circ \dot r_i \circ \dot d_i^{-1}.$

$\dot b_i$

$\dot b_i$ .

$\dot b_i = \dot d_{b_i} \circ \dot r_i \circ \dot d_{b_i}^{-1}, \qquad (18)$

$\dot d_{b_i}$ - , $\dot {\mathbf r}_i$ $\frac{\gamma_i}{2}$ . (17),

$\dot d_{b_i} = \cos \frac{\gamma_i}{4} + \dot {\mathbf s}_1 \sin \frac{\gamma_a}{4}. \qquad (19)$

$\dot d_{b_i}$ (19) (18), , (14), :

$\begin{split} \dot b_i = \dot {\mathbf r}_i \cos^2 \frac{\gamma_i}{4} - \dot {\mathbf r}_i \circ \dot {\mathbf s}_i \sin \frac{\gamma_i}{4} \cos \frac{\gamma_i}{4} + \dot {\mathbf s}_i \circ \dot {\mathbf r}_i \sin \frac{\gamma_i}{4} \cos \frac{\gamma_i}{4} - \dot {\mathbf s}_i \circ \dot {\mathbf r}_i \circ \dot {\mathbf s}_i \sin^2 \frac{\gamma_i}{4} = \\ = \dot {\mathbf r}_i \cos^2 \frac{\gamma_i}{4} - \dot {\mathbf r}_i \times (\dot {\mathbf r}_i \times \dot {\mathbf p}_i) \frac{\sin \frac{\gamma_i}{2}}{\sin \gamma_i} - ((\dot {\mathbf r}_i \times \dot {\mathbf p}_i) \times \dot {\mathbf r}_i ) \times (\dot {\mathbf r}_i \times \dot {\mathbf p}_i ) \frac{\sin^2 \frac{\gamma_i}{4}}{\sin^2 \gamma_i}. \end{split}$

" " ,

$\dot b_i =0 + (\dot {\mathbf r}_i + \dot {\mathbf p}_i) \frac{\sin \frac{\gamma_i}{2}}{\sin \gamma_i}. \qquad (20)$

, $\vert \dot {\mathbf b}_i \vert = 1$ .

$\gamma_{q_1} = f(\beta_1)$

$\gamma_{q_1} = f(\beta_1)$ . $\tau_1$ . 6, (. 9). $\alpha_i = \frac{\pi}{2} - \beta_1$ .

$\gamma_{q_1}$ , $\dot {\mathbf q}_{e_1}$ $\dot {\mathbf r}_1$ $\dot {\mathbf p}_1$ , $\beta_1$ , $\dot {\mathbf q}_{e_1}$ $\dot {\mathbf r}_1$ , $\dot {\mathbf p}_1$ (.. $\lambda_1$ ). , $\beta_1 = 0$ $\mathbf{\dot d}_1$ .

$\begin{split} \vert RR^{'} \vert = \vert \dot {\mathbf r} \vert \sin \frac{\gamma_1}{2}, \quad \vert \dot {\mathbf b}_1 \vert = \vert \dot {\mathbf r} \vert \cos \frac{\gamma_i}{2}, \quad \vert R^{'}O^{'} \vert = \vert \dot {\mathbf b}_1 \vert \sin \alpha_1, \end{split}$

$\vert R^{'}O^{'} \vert = \vert \dot {\mathbf r}_1 \vert \cos \frac{\gamma_1}{2} \sin \alpha_1$ . $\tan \frac{\gamma_{q_1}}{2} = \frac{\vert RR^{'} \vert }{\vert R^{'}O^{'} \vert}$ ),

$\tan \frac{\gamma_{q_1} }{2} = \vert \dot {\mathbf r}_1 \vert \sin \frac{\gamma_1}{2} \frac{1}{\cos \frac{\gamma_1}{2} \sin \alpha_1} = \frac{1}{\sin \alpha_1} \tan \frac{\gamma_i}{2},$

$\gamma_{q_1} = 2 \arctan (\frac{1}{\cos \beta_1} \tan \frac{\gamma_1}{2}). \qquad (21)$

(21) , $\beta_1 = 0$ $\gamma_{q_1} = \gamma_1$ , $\beta_1 = \frac{\pi}{2}$ $\gamma_{q_1} = \pi$ , 1. , $\gamma_{q_1}$ , $\gamma_i$ .

$\dot h_i(\beta_i)$

. $\dot h_i$ . $\dot{\mathbf e}_{d_i}$ $\beta_i$ $\delta_i$ ,

$\dot h_i(\beta_i) = \cos \frac{\beta_i}{2} + \dot{\mathbf e}_{d_i} \times \dot{\mathbf b}_i \sin \frac{\beta_i}{2}, \qquad (22)$

, $\vert \dot{\mathbf e}_{d_i} \vert = 1$ , $\vert \dot{\mathbf b}_i \vert = 1$ . (9), (18), ,

$\begin{split} \dot{\mathbf e}_{d_i} \times \dot{\mathbf b}_i = \dot{\mathbf r}_i (\tan^{-2} \gamma_i \sin \frac{\gamma_i}{2} - \tan^{-1} \gamma_i \cos \frac{\gamma_i}{2} - \frac{\sin \frac{\gamma_i}{2}}{\sin^2 \gamma_i}) + \dot{\mathbf p}_i (\frac{\cos \frac{\gamma_i}{2}}{\sin \gamma_i} - \frac{\cos \gamma_i}{\sin^2 \gamma_i} \sin \frac{\gamma_i}{2} + \frac{\sin \frac{\gamma_i}{2}}{\sin^2 \gamma_i}). \end{split}$

, (22),

$\dot h_i(\beta_i) = \cos \frac{\beta_i}{2} + (\dot{\mathbf p}_i - \dot{\mathbf r}_i) \frac{\cos \frac{\gamma_i}{2}}{\sin \gamma_i} \sin \frac{\beta_i}{2}, \qquad (23)$

$\vert \dot h_i(\beta_i) \vert = 1$ .

$\dot e_{q_i}(\beta_i)$

$\dot e_{q_i}$ . (9), $\dot e_{q_i}$ . : $\dot e_{q_i}(\beta_i) = \dot h_i(\beta_i) \circ \dot e_{d_i} \circ \dot h_i^{-1}(\beta_i)$ . (23) (9), :

$\dot e_{q_i} = C(A^2 \dot{\mathbf r}_i \times \dot{\mathbf p}_i -2AB(\dot{\mathbf r}_i + \dot{\mathbf p}_i)(\cos \gamma_i -1) - 2B^2 \dot{\mathbf r}_i \times \dot{\mathbf p}_i (1-\cos \gamma_i)),$

$A \equiv \cos \frac{\beta_i}{2}$ , $B \equiv \frac{\cos \frac{\gamma_i}{2}}{\sin \gamma_i } \sin \frac{\beta_i}{2}$ , $C \equiv \frac{1}{\sin \frac{\gamma_i}{2}}$ . , :

$\dot e_{q_i} = \frac{1}{\sin \gamma_i} (\dot{\mathbf r}_i \times \dot{\mathbf p}_i \cos \beta_i + (\dot{\mathbf r}_i + \dot{\mathbf p}_i) \sin \frac{\gamma_i}{2} \sin \beta_i), \qquad (24)$

$\vert \dot e_{q_i} \vert = 1$ .

$\dot q_e$

$\dot q_e$ . , $\dot{\mathbf h}_1$ $\dot{\mathbf h}_2$ , $\dot {\mathbf q}_e \equiv \dot {\mathbf h}_i(\beta_i) \times \dot {\mathbf h}_i(\beta_i)$ ( (11)), $\vert \dot{\mathbf q}_e \vert \ne 1$ . , $\beta_i$ $\dot{\mathbf h}_i$ , $\beta_i$ , $\dot{\mathbf h}_i$ . (23) $\beta_i = \pi$ , :

$\dot h_1 = (\dot{\mathbf p}_1 - \dot{\mathbf r}_1) \frac{\cos \frac{\gamma_1}{2}}{\sin \gamma_1}, \quad \dot h_2 = (\dot{\mathbf p}_2 - \dot{\mathbf r}_2) \frac{\cos \frac{\gamma_2}{2}}{\sin \gamma_2}, \qquad (25)$

$\dot{\mathbf q}_e = (\dot{\mathbf p}_1 - \dot{\mathbf r}_1) \times (\dot{\mathbf p}_2 - \dot{\mathbf r}_2) \frac{\cos \frac{\gamma_1}{2}}{\sin \gamma_1} \frac{\cos \frac{\gamma_2}{2}}{\sin \gamma_2}. \qquad (26)$

$\dot{\mathbf q}_{e_n} = \frac{ (\dot{\mathbf p}_1 - \dot{\mathbf r}_1) \times (\dot{\mathbf p}_2 - \dot{\mathbf r}_2) }{\vert (\dot{\mathbf p}_1 - \dot{\mathbf r}_1) \times (\dot{\mathbf p}_2 - \dot{\mathbf r}_2) \vert}. \qquad (27)$

$\beta_i$

(12) , $\beta_1$ :

$\frac{1}{\sin \gamma_1} (\dot{\mathbf r}_1 \times \dot{\mathbf p}_1 \cos \beta_1 + (\dot{\mathbf r}_1 + \dot{\mathbf p}_1) \sin \frac{\gamma_1}{2} \sin \beta_1) \frac{ (\dot{\mathbf p}_1 - \dot{\mathbf r}_1) \times (\dot{\mathbf p}_2 - \dot{\mathbf r}_2) }{\vert (\dot{\mathbf p}_1 - \dot{\mathbf r}_1) \times (\dot{\mathbf p}_2 - \dot{\mathbf r}_2) \vert} = 1. \qquad (28)$

$A \cos \beta_1 + B \sin \beta_1 = 1$ , :

$\beta_1 = \arcsin \frac{1}{\sqrt{A^2 + B^2}} - \arcsin \frac{A}{\sqrt{A^2 + B^2}}, \qquad (29)$

$\begin{split} A = \frac{\dot{\mathbf r}_1 \times \dot{\mathbf p}_1}{\sin \gamma_1} \frac{ (\dot{\mathbf p}_1 - \dot{\mathbf r}_1) \times (\dot{\mathbf p}_2 - \dot{\mathbf r}_2) }{\vert (\dot{\mathbf p}_1 - \dot{\mathbf r}_1) \times (\dot{\mathbf p}_2 - \dot{\mathbf r}_2) \vert}, \quad B = \frac{\sin \frac{\gamma_1}{2}}{\sin \gamma_1} (\dot{\mathbf r}_1 + \dot{\mathbf p}_1) \frac{ (\dot{\mathbf p}_1 - \dot{\mathbf r}_1) \times (\dot{\mathbf p}_2 - \dot{\mathbf r}_2) }{\vert (\dot{\mathbf p}_1 - \dot{\mathbf r}_1) \times (\dot{\mathbf p}_2 - \dot{\mathbf r}_2) \vert} \end{split}.$

$\beta_1$ , $\dot e_{q_1}$ (24), (13) :

$\dot q =\cos \frac{\gamma_{q_1}}{2} + \frac{1}{\sin \gamma_1} (\dot{\mathbf r}_1 \times \dot{\mathbf p}_1 \cos \beta_1 + (\dot{\mathbf r}_1 + \dot{\mathbf p}_1) \sin \frac{\gamma_1}{2} \sin \beta_1) \sin \frac{\gamma_{q_1}}{2}, \qquad (30)$

$\ gamma_ {q_1} = 2 \ arctan (\ frac {1} {\ cos \ beta_1} \ tan \ frac {\ gamma_1} {2}), \ qquad (31)$

$\ beta_1$ (29).

. (30), . , , . . 10 , Real academia de bellas artes .

$(x ^ {'}, y ^ {'}, z ^ {'})$ , , , , ( , "", , ). 1', 2' 3' 1, 2 3 (. 10, )). , $\ dot q$ (30) . . 10, ) 10 . ( - ) . . 10, ) , Real academia de bellas artes , (, , ).

Eso es todo por ahora. Solo señalaré que en un futuro próximo intentaré trabajar en un inconveniente de la expresión (30). Cuando está $\ gamma_1$ cerca de cero, es decir, cuando las orientaciones del HSC y LSC difieren poco, el cuaternión se $\ dot q$ calcula con un error debido al multiplicador $\ frac {1} {\ sin \ gamma_1}$ . Esto puede provocar errores importantes en el cálculo de la orientación del LCS y, como resultado, errores en la determinación de la posición Real academia de bellas artes . Más sobre esto en el próximo artículo.

Cuaterniones. Solución de un problema de navegación

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