🐔 🤚🏿 😈 Quaternions. Solution of one navigation problem 🙃 🎈 🧑🏾‍🤝‍🧑🏻

History

Some time ago I was engaged in an interesting problem related to satellite navigation. Using the phase edge of the signal, the navigation object (NVO) measures the coordinates of navigation satellites (NS) in its coordinate system (local system, LSK). Also, the ONV receives the values of the positions of the NS in the global coordinate system (GCS), and measures the time of receiving the signal of the NS (Fig. 1). It was required to calculate the coordinates of the ONV in the GSK and the system time, that is, to solve the navigation problem.

The problem was interesting in that its solution theoretically makes it possible to reduce the number of NNs in comparison with how many NNs are required in the methods implemented in satellite navigation systems. At that time, I mainly paid my attention to the study of the quality of measurements of the phase front and obtaining navigation equations for coordinates and time, assuming that the calculation of the orientation and coordinates of the NVG would not cause any special problems. Moreover, on a plane the problem was solved quickly and easily.

, , , . , . - , .

, , . , . , , , .

, :

$\mathbf R_i = \begin{bmatrix} x_i & y_i & z_i \end{bmatrix}^T$ : - - , 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 , R_a .

$(x ^ {'}, y ^ {'}, z ^ {'})$ , , , , ( , "", , ). 1', 2' 3' 1, 2 3 (. 10, )). , $\ dot q$ (30) . . 10, ) 10 . ( - ) . . 10, ) , R_a , (, , ).

That's all for now. I will only note that in the near future I will try to work on one drawback of expression (30). When $\ gamma_1$ close to zero, that is, when the orientations of the HSC and LSC differ little, the quaternion is $\ dot q$ calculated with an error due to the multiplier $\ frac {1} {\ sin \ gamma_1}$ . This can lead to significant errors in calculating the orientation of the LCS and, as a result, errors in position determination R_a . More about this in the next article.

Quaternions. Solution of one navigation problem

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