Appendix
1.1 Q-Law Thrust Vector Partial Derivatives
First the partials derivatives of \(\frac{\partial Q}{\partial \text{\oe}}\) w.r.t. to the gains are needed. Taking the partials of Eq. 20:
$$\begin{aligned} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_a}= {} W_{P}\frac{\partial }{\partial W_a} \frac{\partial P}{\partial \text{\oe}}\varvec{W}^T\varvec{V} + \frac{\partial }{\partial W_a}(1 + W_{P}P)\varvec{W}^T\frac{\partial \varvec{V}}{\partial \text{\oe}} \end{aligned}$$
(62)
$$\begin{aligned} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_a}= {} W_{P}\frac{\partial P}{\partial \text{\oe}} \frac{\partial \varvec{W}^T}{\partial W_a}\varvec{V} + (1 + W_{P}P) \frac{\partial \varvec{W}^T}{\partial W_a}\frac{\partial \varvec{V}}{\partial \text{\oe}} \end{aligned}$$
(63)
$$\begin{aligned} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_a}= {} W_{P}\frac{\partial P}{\partial \text{\oe}}[1, 0, 0, 0, 0] \varvec{V} + (1 + W_{P}P)[1, 0, 0, 0, 0]\frac{\partial \varvec{V}}{\partial \text{\oe}} \end{aligned}$$
(64)
$$\begin{aligned} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_a}= {} W_{P}\frac{\partial P}{\partial \text{\oe}}{S}_\text {a}\left[ \frac{d(\text {a},\text {a}_{T})}{\dot{\text {a}}_{xx}}\right] ^2+ (1 + W_{P}P)\frac{\partial V_a}{\partial \text{\oe}} \end{aligned}$$
(65)
The partials derivatives w.r.t. the other gains are derived in a similar fashion.
$$\begin{aligned} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_e}= {} W_{P}\frac{\partial P}{\partial \text{\oe}}{S}_\text {e}\left[ \frac{d(\text {e},\text {e}_{T})}{\dot{\text {e}}_{xx}}\right] ^2+ (1 + W_{P}P)\frac{\partial V_e}{\partial \text{\oe}} \end{aligned}$$
(66)
$$\begin{aligned} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_i}= {} W_{P}\frac{\partial P}{\partial \text{\oe}}{S}_\text {i}\left[ \frac{d(\text {i},\text {i}_{T})}{\dot{\text {i}}_{xx}}\right] ^2+ (1 + W_{P}P)\frac{\partial V_i}{\partial \text{\oe}} \end{aligned}$$
(67)
$$\begin{aligned} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_{\omega }}= {} W_{P}\frac{\partial P}{\partial \text{\oe}}{S}_{\omega }\left[ \frac{d({\omega },{\omega }_{T})}{\dot{{\omega }}_{xx}}\right] ^2+ (1 + W_{P}P)\frac{\partial V_{\omega }}{\partial \text{\oe}} \end{aligned}$$
(68)
$$\begin{aligned} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_{\Omega }}= {} W_{P}\frac{\partial P}{\partial \text{\oe}}{S}_{\Omega }\left[ \frac{d({\Omega },{\Omega }_{T})}{\dot{{\Omega }}_{xx}}\right] ^2+ (1 + W_{P}P)\frac{\partial V_{\Omega }}{\partial \text{\oe}} \end{aligned}$$
(69)
Now, using \(\frac{\partial ^2 Q}{\partial \text{\oe} \partial W_{\text{\oe}}}\), the \(D_1\), \(D_2\), and \(D_3\) coefficient derivatives can be found, leading to the thrust vector gain partial derivatives.
1.1.1
\(\frac{\partial {u}}{\partial W_a}\)
$$\begin{aligned} \frac{\partial D_1}{\partial W_a}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_a}\frac{\partial \dot{\text{\oe}}}{\partial f_{\theta }} \end{aligned}$$
(70)
$$\begin{aligned} \frac{\partial D_2}{\partial W_a}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_a}\frac{\partial \dot{\text{\oe}}}{\partial f_{r}} \end{aligned}$$
(71)
$$\begin{aligned} \frac{\partial D_3}{\partial W_a}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_a}\frac{\partial \dot{\text{\oe}}}{\partial f_{h}} \end{aligned}$$
(72)
$$\begin{aligned} \frac{\partial u_r}{\partial W_a}= {} \frac{D_2(D_1\frac{\partial D_1}{\partial W_a}+D_2\frac{\partial D_2}{\partial W_a} + D_3\frac{\partial D_3}{\partial W_a})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_2}{\partial W_a} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(73)
$$\begin{aligned} \frac{\partial u_{\theta }}{\partial W_a}= {} \frac{D_1(D_1\frac{\partial D_1}{\partial W_a}+D_2\frac{\partial D_2}{\partial W_a} + D_3\frac{\partial D_3}{\partial W_a})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_1}{\partial W_a} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(74)
$$\begin{aligned} \frac{\partial u_h}{\partial W_a}= {} \frac{D_3(D_1\frac{\partial D_1}{\partial W_a}+D_2\frac{\partial D_2}{\partial W_a} + D_3\frac{\partial D_3}{\partial W_a})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_3}{\partial W_a} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(75)
1.1.2
\(\frac{\partial {u}}{\partial W_e}\)
$$\begin{aligned} \frac{\partial D_1}{\partial W_e}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_e}\frac{\partial \dot{\text{\oe}}}{\partial f_{\theta }} \end{aligned}$$
(76)
$$\begin{aligned} \frac{\partial D_2}{\partial W_e}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_e}\frac{\partial \dot{\text{\oe}}}{\partial f_{r}} \end{aligned}$$
(77)
$$\begin{aligned} \frac{\partial D_3}{\partial W_e}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_e}\frac{\partial \dot{\text{\oe}}}{\partial f_{h}} \end{aligned}$$
(78)
$$\begin{aligned} \frac{\partial u_r}{\partial W_e}= {} \frac{D_2(D_1\frac{\partial D_1}{\partial W_e}+D_2\frac{\partial D_2}{\partial W_e} + D_3\frac{\partial D_3}{\partial W_e})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_2}{\partial W_e} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(79)
$$\begin{aligned} \frac{\partial u_{\theta }}{\partial W_e}= {} \frac{D_1(D_1\frac{\partial D_1}{\partial W_e}+D_2\frac{\partial D_2}{\partial W_e} + D_3\frac{\partial D_3}{\partial W_e})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_1}{\partial W_e} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(80)
$$\begin{aligned} \frac{\partial u_h}{\partial W_e}= {} \frac{D_3(D_1\frac{\partial D_1}{\partial W_e}+D_2\frac{\partial D_2}{\partial W_e} + D_3\frac{\partial D_3}{\partial W_e})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_3}{\partial W_e} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(81)
1.1.3
\(\frac{\partial {u}}{\partial W_i}\)
$$\begin{aligned} \frac{\partial D_1}{\partial W_i}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_i}\frac{\partial \dot{\text{\oe}}}{\partial f_{\theta }} \end{aligned}$$
(82)
$$\begin{aligned} \frac{\partial D_2}{\partial W_i}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_i}\frac{\partial \dot{\text{\oe}}}{\partial f_{r}} \end{aligned}$$
(83)
$$\begin{aligned} \frac{\partial D_3}{\partial W_i}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_i}\frac{\partial \dot{\text{\oe}}}{\partial f_{h}} \end{aligned}$$
(84)
$$\begin{aligned} \frac{\partial u_r}{\partial W_i}= {} \frac{D_2(D_1\frac{\partial D_1}{\partial W_i}+D_2\frac{\partial D_2}{\partial W_i} + D_3\frac{\partial D_3}{\partial W_i})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_2}{\partial W_i} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(85)
$$\begin{aligned} \frac{\partial u_{\theta }}{\partial W_i}= {} \frac{D_1(D_1\frac{\partial D_1}{\partial W_i}+D_2\frac{\partial D_2}{\partial W_i} + D_3\frac{\partial D_3}{\partial W_i})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_1}{\partial W_i} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(86)
$$\begin{aligned} \frac{\partial u_h}{\partial W_i}= {} \frac{D_3(D_1\frac{\partial D_1}{\partial W_i}+D_2\frac{\partial D_2}{\partial W_i} + D_3\frac{\partial D_3}{\partial W_i})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_3}{\partial W_i} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(87)
1.1.4
\(\frac{\partial {u}}{\partial W_{\omega} }\)
$$\begin{aligned} \frac{\partial D_1}{\partial W_{\omega }}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_{\omega }}\frac{\partial \dot{\text{\oe}}}{\partial f_{\theta }} \end{aligned}$$
(88)
$$\begin{aligned} \frac{\partial D_2}{\partial W_{\omega }}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_{\omega }}\frac{\partial \dot{\text{\oe}}}{\partial f_{r}} \end{aligned}$$
(89)
$$\begin{aligned} \frac{\partial D_3}{\partial W_{\omega }}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_{\omega }}\frac{\partial \dot{\text{\oe}}}{\partial f_{h}} \end{aligned}$$
(90)
$$\begin{aligned} \frac{\partial u_r}{\partial W_{\omega }}= {} \frac{D_2(D_1\frac{\partial D_1}{\partial W_{\omega }}+D_2\frac{\partial D_2}{\partial W_{\omega }} + D_3\frac{\partial D_3}{\partial W_{\omega }})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_2}{\partial W_{\omega }} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(91)
$$\begin{aligned} \frac{\partial u_{\theta }}{\partial W_{\omega }}= {} \frac{D_1(D_1\frac{\partial D_1}{\partial W_{\omega }}+D_2\frac{\partial D_2}{\partial W_{\omega }} + D_3\frac{\partial D_3}{\partial W_{\omega }})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_1}{\partial W_{\omega }} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(92)
$$\begin{aligned} \frac{\partial u_h}{\partial W_{\omega }}= {} \frac{D_3(D_1\frac{\partial D_1}{\partial W_{\omega }}+D_2\frac{\partial D_2}{\partial W_{\omega }} + D_3\frac{\partial D_3}{\partial W_{\omega }})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_3}{\partial W_{\omega }} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(93)
1.1.5
\(\frac{\partial {u}}{\partial W_{\omega} }\)
$$\begin{aligned} \frac{\partial D_1}{\partial W_{\Omega }}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_{\Omega }}\frac{\partial \dot{\text{\oe}}}{\partial f_{\theta }} \end{aligned}$$
(94)
$$\begin{aligned} \frac{\partial D_2}{\partial W_{\Omega }}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_{\Omega }}\frac{\partial \dot{\text{\oe}}}{\partial f_{r}} \end{aligned}$$
(95)
$$\begin{aligned} \frac{\partial D_3}{\partial W_{\Omega }}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial W_{\Omega }}\frac{\partial \dot{\text{\oe}}}{\partial f_{h}} \end{aligned}$$
(96)
$$\begin{aligned} \frac{\partial u_r}{\partial W_{\Omega }}= {} \frac{D_2(D_1\frac{\partial D_1}{\partial W_{\Omega }}+D_2\frac{\partial D_2}{\partial W_{\Omega }} + D_3\frac{\partial D_3}{\partial W_{\Omega }})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_2}{\partial W_{\Omega }} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(97)
$$\begin{aligned} \frac{\partial u_{\theta }}{\partial W_{\Omega }}= & {} \frac{D_1(D_1\frac{\partial D_1}{\partial W_{\Omega }}+D_2\frac{\partial D_2}{\partial W_{\Omega }} + D_3\frac{\partial D_3}{\partial W_{\Omega }})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_1}{\partial W_{\Omega }} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(98)
$$\begin{aligned} \frac{\partial u_h}{\partial W_{\Omega }}= {} \frac{D_3(D_1\frac{\partial D_1}{\partial W_{\Omega }}+D_2\frac{\partial D_2}{\partial W_{\Omega }} + D_3\frac{\partial D_3}{\partial W_{\Omega }})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_3}{\partial W_{\Omega }} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(99)
Next, we focus on taking the thrust vector partial derivatives w.r.t. the spacecraft state. First, we take the partial derivatives of Eqs. 16 to 18. In this step, symbolic derivatives were used to find expressions for the hessian elements of Q.
1.1.6
\(\frac{\partial {u}}{\partial a}\)
$$\begin{aligned} \frac{\partial D_1}{\partial a}= & {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial a}\frac{\partial \dot{\text{\oe}}}{\partial f_{\theta }} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{\theta } \partial a} \end{aligned}$$
(100)
$$\begin{aligned} \frac{\partial D_2}{\partial a}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial a}\frac{\partial \dot{\text{\oe}}}{\partial f_{r}} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{r} \partial a} \end{aligned}$$
(101)
$$\begin{aligned} \frac{\partial D_3}{\partial a}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial a}\frac{\partial \dot{\text{\oe}}}{\partial f_{h}} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{h} \partial a} \end{aligned}$$
(102)
$$\begin{aligned} \frac{\partial u_r}{\partial a}= {} \frac{D_2(D_1\frac{\partial D_1}{\partial a}+D_2\frac{\partial D_2}{\partial a} + D_3\frac{\partial D_3}{\partial a})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_2}{\partial a} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(103)
$$\begin{aligned} \frac{\partial u_{\theta }}{\partial a}= {} \frac{D_1(D_1\frac{\partial D_1}{\partial a}+D_2\frac{\partial D_2}{\partial a} + D_3\frac{\partial D_3}{\partial a})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_1}{\partial a} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(104)
$$\begin{aligned} \frac{\partial u_h}{\partial a}= {} \frac{D_3(D_1\frac{\partial D_1}{\partial a}+D_2\frac{\partial D_2}{\partial a} + D_3\frac{\partial D_3}{\partial a})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_3}{\partial a} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(105)
1.1.7
\(\frac{\partial {u}}{\partial e}\)
$$\begin{aligned} \frac{\partial D_1}{\partial e}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial e}\frac{\partial \dot{\text{\oe}}}{\partial f_{\theta }} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{\theta } \partial e} \end{aligned}$$
(106)
$$\begin{aligned} \frac{\partial D_2}{\partial e}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial e}\frac{\partial \dot{\text{\oe}}}{\partial f_{r}} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{r} \partial e} \end{aligned}$$
(107)
$$\begin{aligned} \frac{\partial D_3}{\partial e}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial e}\frac{\partial \dot{\text{\oe}}}{\partial f_{h}} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{h} \partial e} \end{aligned}$$
(108)
$$\begin{aligned} \frac{\partial u_r}{\partial e}= {} \frac{D_2(D_1\frac{\partial D_1}{\partial e}+D_2\frac{\partial D_2}{\partial e} + D_3\frac{\partial D_3}{\partial e})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_2}{\partial e} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(109)
$$\begin{aligned} \frac{\partial u_{\theta }}{\partial e}= {} \frac{D_1(D_1\frac{\partial D_1}{\partial e}+D_2\frac{\partial D_2}{\partial e} + D_3\frac{\partial D_3}{\partial e})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_1}{\partial e} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(110)
$$\begin{aligned} \frac{\partial u_h}{\partial e}= {} \frac{D_3(D_1\frac{\partial D_1}{\partial e}+D_2\frac{\partial D_2}{\partial e} + D_3\frac{\partial D_3}{\partial e})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_3}{\partial e} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(111)
1.1.8
\(\frac{\partial {u}}{\partial i}\)
$$\begin{aligned} \frac{\partial D_1}{\partial i}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial i}\frac{\partial \dot{\text{\oe}}}{\partial f_{\theta }} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{\theta } \partial i} \end{aligned}$$
(112)
$$\begin{aligned} \frac{\partial D_2}{\partial i}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial i}\frac{\partial \dot{\text{\oe}}}{\partial f_{r}} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{r} \partial i} \end{aligned}$$
(113)
$$\begin{aligned} \frac{\partial D_3}{\partial i}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial i}\frac{\partial \dot{\text{\oe}}}{\partial f_{h}} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{h} \partial i} \end{aligned}$$
(114)
$$\begin{aligned} \frac{\partial u_r}{\partial i}= {} \frac{D_2(D_1\frac{\partial D_1}{\partial i}+D_2\frac{\partial D_2}{\partial i} + D_3\frac{\partial D_3}{\partial i})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_2}{\partial i} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(115)
$$\begin{aligned} \frac{\partial u_{\theta }}{\partial i}= {} \frac{D_1(D_1\frac{\partial D_1}{\partial i}+D_2\frac{\partial D_2}{\partial i} + D_3\frac{\partial D_3}{\partial i})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_1}{\partial i} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(116)
$$\begin{aligned} \frac{\partial u_h}{\partial i}= & {} \frac{D_3(D_1\frac{\partial D_1}{\partial i}+D_2\frac{\partial D_2}{\partial i} + D_3\frac{\partial D_3}{\partial i})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_3}{\partial i} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(117)
1.1.9
\(\frac{\partial {u}}{\partial \omega }\)
$$\begin{aligned} \frac{\partial D_1}{\partial \omega }= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial \omega }\frac{\partial \dot{\text{\oe}}}{\partial f_{\theta }} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{\theta } \partial \omega } \end{aligned}$$
(118)
$$\begin{aligned} \frac{\partial D_2}{\partial \omega }= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial \omega }\frac{\partial \dot{\text{\oe}}}{\partial f_{r}} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{r} \partial \omega } \end{aligned}$$
(119)
$$\begin{aligned} \frac{\partial D_3}{\partial \omega }= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial \omega }\frac{\partial \dot{\text{\oe}}}{\partial f_{h}} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{h} \partial \omega } \end{aligned}$$
(120)
$$\begin{aligned} \frac{\partial u_r}{\partial \omega }= {} \frac{D_2(D_1\frac{\partial D_1}{\partial \omega }+D_2\frac{\partial D_2}{\partial \omega } + D_3\frac{\partial D_3}{\partial \omega })}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_2}{\partial \omega } }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(121)
$$\begin{aligned} \frac{\partial u_{\theta }}{\partial \omega }= {} \frac{D_1(D_1\frac{\partial D_1}{\partial \omega }+D_2\frac{\partial D_2}{\partial \omega } + D_3\frac{\partial D_3}{\partial \omega })}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_1}{\partial \omega } }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(122)
$$\begin{aligned} \frac{\partial u_h}{\partial \omega }= {} \frac{D_3(D_1\frac{\partial D_1}{\partial \omega }+D_2\frac{\partial D_2}{\partial \omega } + D_3\frac{\partial D_3}{\partial \omega })}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_3}{\partial \omega } }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(123)
1.1.10
\(\frac{\partial {u}}{\partial \Omega }\)
$$\begin{aligned} \frac{\partial D_1}{\partial \Omega }= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial \Omega }\frac{\partial \dot{\text{\oe}}}{\partial f_{\theta }} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{\theta } \partial \Omega } \end{aligned}$$
(124)
$$\begin{aligned} \frac{\partial D_2}{\partial \Omega }= & {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial \Omega }\frac{\partial \dot{\text{\oe}}}{\partial f_{r}} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{r} \partial \Omega } \end{aligned}$$
(125)
$$\begin{aligned} \frac{\partial D_3}{\partial \Omega }= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial \Omega }\frac{\partial \dot{\text{\oe}}}{\partial f_{h}} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{h} \partial \Omega } \end{aligned}$$
(126)
$$\begin{aligned} \frac{\partial u_r}{\partial \Omega }= {} \frac{D_2(D_1\frac{\partial D_1}{\partial \Omega }+D_2\frac{\partial D_2}{\partial \Omega } + D_3\frac{\partial D_3}{\partial \Omega })}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_2}{\partial \Omega } }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(127)
$$\begin{aligned} \frac{\partial u_{\theta }}{\partial \Omega }= {} \frac{D_1(D_1\frac{\partial D_1}{\partial \Omega }+D_2\frac{\partial D_2}{\partial \Omega } + D_3\frac{\partial D_3}{\partial \Omega })}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_1}{\partial \Omega } }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(128)
$$\begin{aligned} \frac{\partial u_h}{\partial \Omega }= & {} \frac{D_3(D_1\frac{\partial D_1}{\partial \Omega }+D_2\frac{\partial D_2}{\partial \Omega } + D_3\frac{\partial D_3}{\partial \Omega })}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_3}{\partial \Omega } }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(129)
1.1.11
\(\frac{\partial {u}}{\partial \theta }\)
$$\begin{aligned} \frac{\partial D_1}{\partial \theta }= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial \theta }\frac{\partial \dot{\text{\oe}}}{\partial f_{\theta }} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{\theta } \partial \theta } \end{aligned}$$
(130)
$$\begin{aligned} \frac{\partial D_2}{\partial \theta }= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial \theta }\frac{\partial \dot{\text{\oe}}}{\partial f_{r}} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{r} \partial \theta } \end{aligned}$$
(131)
$$\begin{aligned} \frac{\partial D_3}{\partial \theta }= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial \theta }\frac{\partial \dot{\text{\oe}}}{\partial f_{h}} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{h} \partial \theta } \end{aligned}$$
(132)
$$\begin{aligned} \frac{\partial u_r}{\partial \theta }= {} \frac{D_2(D_1\frac{\partial D_1}{\partial \theta }+D_2\frac{\partial D_2}{\partial \theta } + D_3\frac{\partial D_3}{\partial \theta })}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_2}{\partial \theta } }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(133)
$$\begin{aligned} \frac{\partial u_{\theta }}{\partial \theta }= {} \frac{D_1(D_1\frac{\partial D_1}{\partial \theta }+D_2\frac{\partial D_2}{\partial \theta } + D_3\frac{\partial D_3}{\partial \theta })}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_1}{\partial \theta } }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(134)
$$\begin{aligned} \frac{\partial u_h}{\partial \theta }= {} \frac{D_3(D_1\frac{\partial D_1}{\partial \theta }+D_2\frac{\partial D_2}{\partial \theta } + D_3\frac{\partial D_3}{\partial \theta })}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_3}{\partial \theta } }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(135)
1.1.12
\(\frac{\partial {u}}{\partial m}\)
$$\begin{aligned} \frac{\partial D_1}{\partial m}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial m}\frac{\partial \dot{\text{\oe}}}{\partial f_{\theta }} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{\theta } \partial m} \end{aligned}$$
(136)
$$\begin{aligned} \frac{\partial D_2}{\partial m}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial m}\frac{\partial \dot{\text{\oe}}}{\partial f_{r}} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{r} \partial m} \end{aligned}$$
(137)
$$\begin{aligned} \frac{\partial D_3}{\partial m}= {} \frac{\partial ^2 Q}{\partial \text{\oe} \partial m}\frac{\partial \dot{\text{\oe}}}{\partial f_{h}} + \frac{\partial Q}{\partial \text{\oe}}\frac{\partial ^2 \dot{\text{\oe}}}{\partial f_{h} \partial m} \end{aligned}$$
(138)
$$\begin{aligned} \frac{\partial u_r}{\partial m}= {} \frac{D_2(D_1\frac{\partial D_1}{\partial m}+D_2\frac{\partial D_2}{\partial m} + D_3\frac{\partial D_3}{\partial m})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_2}{\partial m} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(139)
$$\begin{aligned} \frac{\partial u_{\theta }}{\partial m}= {} \frac{D_1(D_1\frac{\partial D_1}{\partial m}+D_2\frac{\partial D_2}{\partial m} + D_3\frac{\partial D_3}{\partial m})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_1}{\partial m} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(140)
$$\begin{aligned} \frac{\partial u_h}{\partial m}= {} \frac{D_3(D_1\frac{\partial D_1}{\partial m}+D_2\frac{\partial D_2}{\partial m} + D_3\frac{\partial D_3}{\partial m})}{(D_1^2+D_2^2+D_3^2)^{3/2}} - \frac{\frac{\partial D_3}{\partial m} }{\sqrt{D_1^2+D_2^2+D_3^2}} \end{aligned}$$
(141)
