On Modeling and Control of a Holonomic Tricopter

Abstract

Most holonomic Unmanned Aerial Vehicles (HUAVs) (usually small-scale), employ at least four high-power thrusting actuators, resulting in increased mass, inertia, and energy consumption. In this work, a novel design for a HUAV using only three high-power thrusting actuators is developed. The pointing dynamics of the high rpm motor-propeller assemblies are included in the analysis, resulting in the exposure of oscillatory gyroscopic dynamics. These are studied analytically to yield vectoring oscillation frequency estimates, useful in selecting actuators. A geometric, singularity free, control framework comprised of a new control-allocation scheme and a new vectoring controller is developed. Stability proofs are included and controller robustness is addressed. Simulations (MATLAB/GAZEBO) demonstrating the effectiveness of the developed design and control framework are provided; a HUAV with independent position/attitude regulation (full pose control) using only three thrusting actuators is showcased.

Appendices

Appendix A: Supplementary Material

1.1 A.1 Mappings & Kinematics used in Section 2.2

Cross product map, first used in Eq. (1)

$$ \begin{array}{@{}rcl@{}} &&\!\!\!\!\! S:\mathbb{R}^{3}\to\mathfrak{so}(3),\text{ with }\\ && S(\mathbf{r}){=}[{0},{-r_{3}},{r_{2}};{r_{3}},{0},{-r_{1}};{-r_{2}},{r_{1}},0], {S^{-1}}(S(\mathbf{r})){=}\mathbf{r}\\ \end{array} $$

(A1)

where $\mathfrak {so}(3)$ is the set of all 3 × 3 skew symmetric matrices.

Velocity and acceleration of the CM of the i^th-TA, first used in Eq. (1)

$$ \begin{array}{@{}rcl@{}} {~}^{I }\mathbf{v}_{i}&=&{~}^{I }\mathbf{v}_{0}+{~}^{I}\dot{\mathbf{R}}_{0}{~}^{0}\mathbf{r}_{i}-{~}^{I}\dot{\mathbf{R}}_{i}{~}^{i}\mathbf{p}_{i } \end{array} $$

(A2a)

$$ \begin{array}{@{}rcl@{}} {~}^{I }\dot{\mathbf{v}}_{i}&=&{~}^{I}\dot{\mathbf{v}}_{0}+{~}^{I}\ddot{\mathbf{R}}_{0}{~}^{0}\mathbf{r}_{i}-{~}^{I}\ddot{\mathbf{R}}_{i}{~}^{i}\mathbf{p}_{i } \end{array} $$

(A2b)

1.2 A.2 Terms of Eq. (8) used in Section 2.2

Control input matrix, $\mathbf {B}\in \mathbb {R}^{15{\times }9}$, and mass matrix, $\mathbf {M}\in \mathbb {R}^{15\times 15}$, from Eq. (8) given by

$$ \begin{array}{@{}rcl@{}} \mathbf{B}=\begin{bmatrix} \mathbf{0}_{3\times3}&\mathbf{0}_{3\times3}&\mathbf{0}_{3\times3}\\ -{~}^{0}\mathbf{R}_{1}&-{~}^{0}\mathbf{R}_{2}&-{~}^{0}\mathbf{R}_{3}\\ \mathbf{1}_{3\times3}&\mathbf{0}_{3\times3}&\mathbf{0}_{3\times3}\\ \mathbf{0}_{3\times3}&\mathbf{1}_{3\times3}&\mathbf{0}_{3\times3}\\ \mathbf{0}_{3\times3}&\mathbf{0}_{3\times3}&\mathbf{1}_{3\times3} \end{bmatrix} , \mathbf{M}=\begin{bmatrix} M_{1,1}&M_{1,2}&M_{1,3}&M_{1,4}&M_{1,5}\\ M_{2,1}&M_{2,2}&M_{2,3}&M_{2,4}&M_{2,5}\\ M_{3,1}&M_{3,2}&M_{3,3}&0&0\\ M_{4,1}&M_{4,2}&0&M_{4,4}&0\\ M_{5,1}&M_{5,2}&0&0&M_{5,5} \end{bmatrix} \end{array} $$

(A3)

where $M_{i,j}\in \mathbb {R}^{3\times 3}$ are given by

$$ \begin{array}{@{}rcl@{}} M_{1,1}&=&\sum\limits_{i=0}^{3}m_{i}\mathbf{1}, M_{1,2}={-}\sum\limits_{i=1}^{3}m_{i}{~}^{I}\mathbf{R}_{0}S\left( {~}^{0}\mathbf{r}_{i}\right),\\ M_{1,i+2}&=&m_{i}{~}^{I}\mathbf{R}_{i}S\left( {~}^{i}\mathbf{p}_{i}\right) ,i=1-3 \end{array} $$

(A4a)

$$ \begin{array}{@{}rcl@{}} M_{2,1}&=&\sum\limits_{i=1}^{3}S\left( {~}^{0}\mathbf{r}_{i}\right)m_{i}{~}^{0}\mathbf{R}_{I},\\ M_{2,2}&=&{~}^{0}\mathbf{J}_{0}{-}\sum\limits_{i=1}^{3}S\left( {~}^{0}\mathbf{r}_{i}\right)m_{i}S\left( {~}^{0}\mathbf{r}_{i}\right) \end{array} $$

(A4b)

$$ \begin{array}{@{}rcl@{}} M_{2,i+2}&=&S\left( {~}^{0}\mathbf{r}_{i}\right){~}^{0}\mathbf{R}_{i}m_{i}S\left( {~}^{i}\mathbf{p}_{i}\right),\\ M_{i+2,1}&=&-S\left( {~}^{i}\mathbf{p}_{i}\right)m_{i}{~}^{i}\mathbf{R}_{I},i=1-3 \end{array} $$

(A4c)

$$ \begin{array}{@{}rcl@{}} M_{i+2,2}&=&S\left( {~}^{i}\mathbf{p}_{i}\right)m_{i}{~}^{i}\mathbf{R}_{0}S\left( {~}^{0}\mathbf{r}_{i}\right),\\ M_{i+2,i+2}&=&{~}^{i}\mathbf{J}_{i}-S\left( {~}^{i}\mathbf{p}_{i}\right)m_{i}S\left( {~}^{i}\mathbf{p}_{i}\right),i=1-3 \end{array} $$

(A4d)

The components of $\boldsymbol {\Gamma }=[{{\varGamma }}_{1},{{\varGamma }}_{2},{{\varGamma }}_{3},{{\varGamma }}_{4},{{\varGamma }}_{5}]^{T}\in \mathbb {R}^{15{\times }1}$ from Eq. (8) are given by

$$ \begin{array}{@{}rcl@{}} {{\varGamma}}_{1}& = &-\sum\limits_{i=1}^{3}m_{i}{~}^{I}\boldsymbol{\Phi}_{i},\\ {{\varGamma}}_{2}& = & -\sum\limits_{i=1}^{3}S\left( {~}^{0}\mathbf{r}_{i}\right){~}^{0}\mathbf{R}_{I}m_{i}{~}^{I}\boldsymbol{\Phi}_{i}-S\left( {~}^{0}\boldsymbol{\omega}_{0}\right){~}^{0}\mathbf{J}_{0}{~}^{0}\boldsymbol{\omega}_{0} \end{array} $$

(A5a)

$$ \begin{array}{@{}rcl@{}} {{\varGamma}}_{i+2}& = &S\left( {~}^{i}\mathbf{p}_{i}\right)m_{i}{~}^{i}\mathbf{R}_{I}{~}^{I}\boldsymbol{\Phi}_{i}-S\left( {~}^{i}\boldsymbol{\omega}_{i}\right){~}^{i}\mathbf{J}_{i}{~}^{i}\boldsymbol{\omega}_{i},i=1-3\\ \end{array} $$

(A5b)

where ${~}^{I}\boldsymbol {\Phi }_{i}\in \mathbb {R}^{3\times 3}$ is given by

$$ \begin{array}{@{}rcl@{}} {~}^{I}\boldsymbol{\Phi}_{i}={~}^{I}\mathbf{R}_{0}S\left( {~}^{0}\boldsymbol{\omega}_{0}\right)S\left( {~}^{0}\boldsymbol{\omega}_{0}\right){~}^{0}\mathbf{r}_{i}-{~}^{I}\mathbf{R}_{i}S\left( {~}^{i}\boldsymbol{\omega}_{i}\right)S\left( {~}^{i}\boldsymbol{\omega}_{i}\right){~}^{i}\mathbf{p}_{i} \end{array} $$

(A6a)

The components of $\mathbf {G}=[G_{1},G_{2},G_{3},G_{4},G_{5}]^{T}\in \mathbb {R}^{15{\times }1}$ from Eq. (8) are given by

$$ \begin{array}{@{}rcl@{}} G_{1}&=&\sum\limits_{i=0}^{3}{~}^{I}\mathbf{g}_{i},G_{2}=\sum\limits_{i=1}^{3}S\left( {~}^{0}\mathbf{r}_{i}\right){~}^{0}\mathbf{R}_{I}{~}^{I}\mathbf{g}_{i},\\ G_{i+2}&=&-S\left( {~}^{i}\mathbf{p}_{i}\right){~}^{i}\mathbf{R}_{I}{~}^{I}\mathbf{g}_{i},i=1-3 \end{array} $$

(A7)

The components of $\mathbf {P}=[P_{1},P_{2},P_{3},P_{4},P_{5}]^{T}\in \mathbb {R}^{15{\times }1}$ from Eq. (8) are given by

$$ \begin{array}{@{}rcl@{}} P_{1}&=&\sum\limits_{i=1}^{3}{~}^{I}\mathbf{R}_{i}{~}^{i}\mathbf{f}_{p,i},P_{2}=\sum\limits_{i=1}^{3}S\left( {~}^{0}\mathbf{r}_{i}\right){~}^{0}\mathbf{R}_{i}{~}^{i}\mathbf{f}_{p,i},\\ P_{i+2}&=&{~}^{i}\mathbf{n}_{p,i}-S\left( {~}^{i}\mathbf{p}_{i}\right){~}^{i}\mathbf{f}_{p,i},i=1-3 \end{array} $$

(A8)

The components of $\mathbf {D}=[D_{1},D_{2},D_{3},D_{4},D_{5}]^{T}\in \mathbb {R}^{15{\times }1}$ from Eq. (8) are given by

$$ \begin{array}{@{}rcl@{}} D_{1}&=&\sum\limits_{i=0}^{3}{~}^{I}\mathbf{f}_{w,i},D_{2}=\sum\limits_{i=1}^{3}S\left( {~}^{0}\mathbf{r}_{i}\right){~}^{0}\mathbf{R}_{I}{~}^{I}\mathbf{f}_{w,i}+{~}^{0}\mathbf{n}_{w,0} \end{array} $$

(A9a)

$$ \begin{array}{@{}rcl@{}} D_{i+2}&=&{~}^{i}\mathbf{n}_{w,i}-S\left( {~}^{i}\mathbf{p}_{i}\right){~}^{i}\mathbf{R}_{I}{~}^{I}\mathbf{f}_{w,i},i=1-3 \end{array} $$

(A9b)

1.3 A.3 Vectoring Controller Terms used in Section 4.1

Pointing dynamics terms, first used in Eq. (9)

$$ \begin{array}{@{}rcl@{}} {~}^{i}\boldsymbol{\mathcal{J}}_{i}&{=}&{~}^{i}\mathbf{J}_{i}-m_{i}\left( S\left( {~}^{i}\mathbf{p}_{i}\right)\right)^{2} \end{array} $$

(A10a)

$$ \begin{array}{@{}rcl@{}} {~}^{i}\mathbf{Z}_{i}&{=}&S\left( {~}^{i}\mathbf{p}_{i}\right){{~}^{i}\mathbf{R}_{I}}m_{i}{~}^{I }\boldsymbol{\Xi}_{i}-{~}^{i}\mathbf{J}_{i} {{~}^{i}\mathbf{R}_{0}} \end{array} $$

(A10b)

$$ \begin{array}{@{}rcl@{}} {~}^{I }\boldsymbol{\Xi}_{i}&{=}&{~}^{I}\mathbf{R}_{i}S\left( {~}^{i}\mathbf{p}_{i}\right){{~}^{i}\mathbf{R}_{0}}-{~}^{I}\mathbf{R}_{0}S\left( {~}^{0}\mathbf{r}_{i}\right) \end{array} $$

(A10c)

$$ \begin{array}{@{}rcl@{}} {~}^{i}\mathbf{n}_{r,i}&{=}&{~}^{i}\mathbf{J}_{i}S\left( {~}^{i}\boldsymbol{\omega}^{~}_{i/0}\right){{~}^{i}\mathbf{R}_{0}}{~}^{0}\boldsymbol{\omega}_{0}-S\left( {~}^{i}\boldsymbol{\omega}_{i }\right){~}^{i}\mathbf{J}_{i} {~}^{i}\boldsymbol{\omega}_{i}\\&&{+}S\left( {~}^{i}\mathbf{p}_{i}\right){{~}^{i}\mathbf{R}_{I}}\Big(m_{i}\left( {~}^{I }\mathbf{k}_{i}{-}{{~}^{I}\mathbf{R}_{i}}\left( S({~}^{i}\boldsymbol{\omega}^{~}_{i/0})\right)^{2}{~}^{i}\mathbf{p}_{i }\right){-}{~}^{I}\mathbf{g}_{i}\Big)\\ &&+\left( {-}\text{sgn}\left( {~}^{i}{\omega}_{i_{3}}\right)b_{m}\mathbf{1}-S\left( {~}^{i}\mathbf{p}_{i}\right)\right){~}^{i}\mathbf{f}_{p,i} \end{array} $$

(A10d)

where the term ${~}^{I }\mathbf {k}_{i}\in \mathbb {R}^{3{\times }1}$ used above and throughout the manuscript is given by

$$ \begin{array}{@{}rcl@{}} {~}^{I }\mathbf{k}_{i}&=&{~}^{I}\mathbf{R}_{0}\left( S\left( {~}^{0}\boldsymbol{\omega}_{0}\right)\right)^{2}{~}^{0}\mathbf{r}_{i}{-}{~}^{I}\mathbf{R}_{i}S\left( {~}^{i}\mathbf{p}_{i}\right)S\left( {~}^{i}\boldsymbol{\omega}^{~}_{i/0 }\right){{~}^{i}\mathbf{R}_{0}}{~}^{0}\boldsymbol{\omega}_{0}\\ &&-{~}^{I}\mathbf{R}_{i}\left( \left( S\left( {{~}^{i}\mathbf{R}_{0}}{~}^{0}\boldsymbol{\omega}_{0}\right)\right)^{2}+S\left( {{~}^{i}\mathbf{R}_{0}}{~}^{0}\boldsymbol{\omega}_{0 }\right)S\left( {~}^{i}\boldsymbol{\omega}^{~}_{i/0}\right)\right.\\ &&\left.+S\left( {~}^{i}\boldsymbol{\omega}^{~}_{i/0}\right)S\left( {{~}^{i}\mathbf{R}_{0}}{~}^{0}\boldsymbol{\omega}_{0 }\right)\vphantom{\left( S\left( {{~}^{i}\mathbf{R}_{0}}{~}^{0}\boldsymbol{\omega}_{0}\right)\right)^{2}}\!\right){~}^{i}\mathbf{p}_{i } \end{array} $$

(A11)

1.4 A.4 Feedback terms used in Section 4.2

Attitude error function bounding inequality, [30], first used in Eq. (C4)

$$ \lVert\mathbf{e}_{R}\rVert^{2}\leq{{\varPsi}}_{0}\leq 2\lVert\mathbf{e}_{R}\rVert^{2} $$

(A12)

Error function (attitude error vector) derivatives $\dot {{{\varPsi }}}_{0}$ ($\dot {\mathbf {e}}_{R}$), [30], first used in Eq. (39)

$$ \begin{array}{@{}rcl@{}} \dot{{{\varPsi}}}_{0}&{=}&\mathbf{e}_{R}^{T}\mathbf{e}_{\omega} \end{array} $$

(A13a)

$$ \begin{array}{@{}rcl@{}} \dot{\mathbf{e}}_{R}&{=}&{~}^{0}\mathbf{E} \mathbf{e}_{\omega} \end{array} $$

(A13b)

$$ \begin{array}{@{}rcl@{}} {~}^{0}\mathbf{E}&{=}&\frac{\left\{\text{tr} \left[ {{~}^{0}\mathbf{R}_{I}}{~}^{I}\mathbf{R}^{0_{d}}_{0} \right] \mathbf{1}{-}{{~}^{0}\mathbf{R}_{I}}{~}^{I}\mathbf{R}^{0_{d}}_{0}{+}2\mathbf{e}_{R}\mathbf{e}_{R}^{T}\right\}}{2\sqrt{1{+}tr\left[{{~}^{0}\mathbf{R}_{I}}{~}^{I}\mathbf{R}^{0_{d}}_{0}\right]}} \end{array} $$

(A13c)

Supporting term ${~}^{0}\mathbf {a}_{0}\in \mathbb {R}^{3{\times }1}$, first used in Eq. (39)

$$ {~}^{0}\mathbf{a}_{0}{=}S\left( {~}^{0}\boldsymbol{\omega}_{0}\right){{~}^{0}\mathbf{R}_{I}}{~}^{I}\mathbf{R}^{0_{d}}_{0} {~}^{0}\boldsymbol{\omega}_{0_{d}}{-}{{~}^{0}\mathbf{R}_{I}}{~}^{I}\mathbf{R}^{0_{d}}_{0} {~}^{0}\dot{\boldsymbol{\omega}}_{0_{d}} $$

(A14)

Vectors $\mathbf {h}_{p},\mathbf {h}_{g}\in \mathbb {R}^{6{\times }1}$, first used in used Eq. (37)

$$ \begin{array}{@{}rcl@{}} \mathbf{h}_{p}&=&b_{T}\sum\limits_{i=1}^{3}\left[{~}^{I}\mathbf{R}_{0}{~}^{0}\mathbf{h}_{p,i};\mathbf{A}_{i}{~}^{I}\mathbf{R}_{0}{~}^{0}\mathbf{h}_{p,i}\right] \end{array} $$

(A15a)

$$ \begin{array}{@{}rcl@{}} {~}^{0}\mathbf{h}_{p,i}&=&{~}^{0}\mathbf{q}_{i}\left( 2\left( {~}^{i}\mathbf{e}^{T}_{3}{{~}^{i}\boldsymbol{\omega}^{~}_{i/0}}\right)\left( \left( {{~}^{i}\mathbf{R}_{0}}{~}^{0}\boldsymbol{\omega}_{0}\right)^{T}{~}^{i}\mathbf{e}_{3}\right)+\left( \left( {{~}^{i}\mathbf{R}_{0}}{~}^{0}\boldsymbol{\omega}_{0}\right)^{T}{~}^{i}\mathbf{e}_{3}\right)^{2}\right)\\ &&-{~}^{0}\mathbf{q}_{i_{d}}\left( 2\left( {~}^{i}\mathbf{e}^{T}_{3}{{~}^{i}\boldsymbol{\omega}^{~}_{i_{d}/0}}\right)\left( \left( {~}^{i}\mathbf{R}^{i_{d}}_{0} {~}^{0}\boldsymbol{\omega}_{0}\right)^{T}{~}^{i}\mathbf{e}_{3}\right)\right.\\ &&\left.+\left( \left( {~}^{i}\mathbf{R}^{i_{d}}_{0} {~}^{0}\boldsymbol{\omega}_{0}\right)^{T}{~}^{i}\mathbf{e}_{3}\right)^{2}\right) \end{array} $$

(A15b)

$$ \begin{array}{@{}rcl@{}} \mathbf{h}_{g}&=&\left[{~}^{I }\mathbf{h}_{g_{1}};{~}^{0}\mathbf{h}_{g_{2}}\right] \end{array} $$

(A15c)

$$ \begin{array}{@{}rcl@{}} {~}^{I }\mathbf{h}_{g_{1}}&=&-{~}^{I}\mathbf{g}_{0}+\sum\limits_{i=1}^{3}\left( m_{i}{~}^{I }\mathbf{k}_{i}-{~}^{I}\mathbf{g}_{i}\right) \end{array} $$

(A15d)

$$ \begin{array}{@{}rcl@{}} {~}^{0}\mathbf{h}_{g_{2}}&=&{~}^{0}\boldsymbol{\omega}_{0 }{\times}{~}^{0}\mathbf{J}_{0 }{~}^{0}\boldsymbol{\omega}_{0 }{-}\sum\limits_{i=1}^{3}{{~}^{0}\mathbf{R}_{I}}{~}^{I}\mathbf{R}_{i}{~}^{i}\mathbf{J}_{i }S\left( {~}^{i}\boldsymbol{\omega}^{~}_{i/0 }\right){{~}^{i}\mathbf{R}_{0}}{~}^{0}\boldsymbol{\omega}_{0 }\\ &&{-}\sum\limits_{i=1}^{3}{~}^{0}\boldsymbol{\Theta}_{i}{{~}^{i}\mathbf{R}_{I}}\left( m_{i}{~}^{I}\mathbf{k}_{i}{-}{~}^{I}\mathbf{g}_{i}\right){+}\sum\limits_{i=1}^{3}{{~}^{0}\mathbf{R}_{I}}{~}^{I}\mathbf{R}_{i}{~}^{i}\mathbf{y}_{i} \end{array} $$

(A15e)

where the terms ${~}^{0}\boldsymbol {\Theta }_{i}\in \mathbb {R}^{3{\times }3}, {~}^{i}\mathbf {y}_{i }\in \mathbb {R}^{3{\times }1}$, $\mathbf {A}_{i}\in \mathbb {R}^{3\times 3}$ used above and throughout the manuscript are given by

$$ \begin{array}{@{}rcl@{}} {~}^{0}\boldsymbol{\Theta}_{i}&=&{{~}^{0}\mathbf{R}_{I}}{~}^{I}\mathbf{R}_{i}S\left( {~}^{i}\mathbf{p}_{i}\right)-S\left( {~}^{0}\mathbf{r}_{i}\right){{~}^{0}\mathbf{R}_{I}}{~}^{I}\mathbf{R}_{i} \end{array} $$

(A16a)

$$ \begin{array}{@{}rcl@{}} {~}^{i}\mathbf{y}_{i}&=&S\left( {{~}^{i}\mathbf{R}_{0}}{~}^{0}\boldsymbol{\omega}_{0}\right){~}^{i}\mathbf{J}_{i}{{~}^{i}\mathbf{R}_{0}}{~}^{0}\boldsymbol{\omega}_{0}+S\left( {{~}^{i}\mathbf{R}_{0}}{~}^{0}\boldsymbol{\omega}_{0}\right){~}^{i}\mathbf{J}_{i}{~}^{i}\boldsymbol{\omega}^{~}_{i/0 }\\ &&+S\left( {~}^{i}\boldsymbol{\omega}^{~}_{i/0}\right){~}^{i}\mathbf{J}_{i}{{~}^{i}\mathbf{R}_{0}}{~}^{0}\boldsymbol{\omega}_{0} \end{array} $$

(A16b)

$$ \begin{array}{@{}rcl@{}} \mathbf{A}_{i}&=&{-}\text{sgn}\left( {~}^{i}{\omega}_{{3}}\right)b_{m}{{~}^{0}\mathbf{R}_{I}}-{~}^{0}\boldsymbol{\Theta}_{i}{{~}^{i}\mathbf{R}_{I}} \end{array} $$

(A16c)

and for ${~}^{I }\mathbf {k}_{i}\in \mathbb {R}^{3{\times }1}$, see Eq. (A11).

Matrix term $\mathbf {C}\in \mathbb {R}^{6{\times }6}$, first used in Eq. (37)

$$ \begin{array}{@{}rcl@{}} \mathbf{C}&{=}&\begin{bmatrix} \left( m_{0}+{\sum}_{i=1}^{3}m_{i}\right)\mathbf{1}&{\sum}_{i=1}^{3}m_{i}{~}^{I}\boldsymbol{\Xi}_{i}\\ -{\sum}_{i=1}^{3}{~}^{0}\boldsymbol{\Theta}_{i}{{~}^{i}\mathbf{R}_{I}}m_{i}&C_{2,2} \end{bmatrix} \end{array} $$

(A17a)

$$ \begin{array}{@{}rcl@{}} C_{2,2}&{=}&{~}^{0}\mathbf{J}_{0}{+}\sum\limits_{i=1}^{3}\left( {{~}^{0}\mathbf{R}_{I}}{~}^{I}\mathbf{R}_{i}{~}^{i}\mathbf{J}_{i} {{~}^{i}\mathbf{R}_{0}}{-}{~}^{0}\boldsymbol{\Theta}_{i}{{~}^{i}\mathbf{R}_{I}}m_{i}{~}^{I }\boldsymbol{\Xi}_{i}\right) \end{array} $$

(A17b)

where ${~}^{I}\boldsymbol {\Xi }_{i},{~}^{0}\boldsymbol {\Theta }_{i}\in \mathbb {R}^{3\times 3}$ are given in Eqs. (A10c), (A16a).

Allocation matrix $\mathbf {A}\in \mathbb {R}^{6\times 9}$, first used in Eq. 39

$$ \mathbf{A}{=}[\mathbf{1}_{3\times3},\mathbf{1}_{3\times3},\mathbf{1}_{3\times3};\mathbf{A}_{1},\mathbf{A}_{2},\mathbf{A}_{3}] $$

(A18)

and $\mathbf {A}_{i}\in \mathbb {R}^{3\times 3}$ is given in Eq. (A16a).

1.5 A.5 Disturbance modeling term used in Section 5.1.2

Exponential map using the Rodrigues formulation [29], first used in Eq. (45)

$$ \begin{array}{@{}rcl@{}} &&\!\!\!\!\! \exp:\mathbb{R}\times\mathbb{R}^{3}\to\text{SO}(3),\text{ with }\\ && \exp(a,\boldsymbol{\xi}){=}\mathbf{1}{+}S(\boldsymbol{\xi})\sin a{+}S(\boldsymbol{\xi})^{2}(1{-}\cos a) \end{array} $$

(A19)

1.6 A.6 Figure “R” pose trajectory used in Section 5.2

Way-points and pose continuity constraints used with minimum-snap polynomials for trajectory generation, employed in Section 5.2

$$ \begin{array}{@{}rcl@{}} \mathbf{wp}_{0.0}\lvert_{t=0}:\mathbf{x}_{0}&=&[0,0,0.0965]^{T} [\text{m}],\dot{\mathbf{x}}_{0}=\mathbf{0} [\text{m}/\text{s}],\\ \ddot{\mathbf{x}}_{0}&=&\mathbf{0} [\text{m}/\text{s}^{2}],\dddot{\mathbf{x}}_{0}=\mathbf{0} [\text{m}/\text{s}^{3}]\\ \mathbf{h}&=&[0,0,0]^{T} [\text{deg}],\dot{\mathbf{h}}_{~}=\mathbf{0} [\text{deg}/\text{s}],\\ \ddot{\mathbf{h}}_{~}&=&\mathbf{0} [\text{deg}/\text{s}^{2}],\dddot{\mathbf{h}}_{~}=\mathbf{0} [\text{deg}/\text{s}^{3}]\\ \mathbf{wp}_{1}\lvert_{t=10}:\mathbf{x}_{0}&=&[1,1,2]^{T} [\text{m}],\dot{\mathbf{x}}_{0}=\mathbf{0} [\text{m}/\text{s}],\\ \ddot{\mathbf{x}}_{0}&=&\mathbf{0} [\text{m}/\text{s}^{2}],\dddot{\mathbf{x}}_{0}=\mathbf{0} [\text{m}/\text{s}^{3}]\\ \mathbf{h}&=&[-180,-64,0]^{T} [\text{deg}],\dot{\mathbf{h}}_{~}=\mathbf{0} [\text{deg}/\text{s}],\\ \ddot{\mathbf{h}}_{~}&=&\mathbf{0} [\text{deg}/\text{s}^{2}],\dddot{\mathbf{h}}_{~}=\mathbf{0} [\text{deg}/\text{s}^{3}]\\ \mathbf{wp}_{2}\lvert_{t=20}:\mathbf{x}_{0}&=&[2,2,2]^{T} [\text{m}],\dot{\mathbf{x}}_{0}=\mathbf{0} [\text{m}/\text{s}],\ddot{\mathbf{x}}_{0}=\mathbf{0} [\text{m}/\text{s}^{2}],\\ \dddot{\mathbf{x}}_{0}&=&\mathbf{0} [\text{m}/\text{s}^{3}]\\ \mathbf{h}&=&[-270,-64,0]^{T} [\text{deg}],\dot{\mathbf{h}}_{~}=\mathbf{0} [\text{deg}/\text{s}],\\ \ddot{\mathbf{h}}_{~}&=&\mathbf{0} [\text{deg}/\text{s}^{2}],\dddot{\mathbf{h}}_{~}=\mathbf{0} [\text{deg}/\text{s}^{3}]\\ \mathbf{wp}_{3}\lvert_{t=30}:\mathbf{x}_{0}&=&[3,1,2]^{T} [\text{m}],\dot{\mathbf{x}}_{0}=\mathbf{0} [\text{m}/\text{s}],\ddot{\mathbf{x}}_{0}=\mathbf{0} [\text{m}/\text{s}^{2}],\\ \dddot{\mathbf{x}}_{0}&=&\mathbf{0} [\text{m}/\text{s}^{3}]\\ \mathbf{h}&=&[-360,-64,0]^{T} [\text{deg}],\dot{\mathbf{h}}_{~}=\mathbf{0} [\text{deg}/\text{s}],\\ \ddot{\mathbf{h}}_{~}&=&\mathbf{0} [\text{deg}/\text{s}^{2}],\dddot{\mathbf{h}}_{~}=\mathbf{0} [\text{deg}/\text{s}^{3}]\\ \mathbf{wp}_{4}\lvert_{t=40}:\mathbf{x}_{0}&=&[2,0,2]^{T} [\text{m}],\dot{\mathbf{x}}_{0}=\mathbf{0} [\text{m}/\text{s}],\ddot{\mathbf{x}}_{0}=\mathbf{0} [\text{m}/\text{s}^{2}],\\ \dddot{\mathbf{x}}_{0}&=&\mathbf{0} [\text{m}/\text{s}^{3}]\\ \mathbf{h}&=&[-450,-64,0]^{T} [\text{deg}],\dot{\mathbf{h}}_{~}=\mathbf{0} [\text{deg}/\text{s}],\\ \ddot{\mathbf{h}}_{~}&=&\mathbf{0} [\text{deg}/\text{s}^{2}],\dddot{\mathbf{h}}_{~}=\mathbf{0} [\text{deg}/\text{s}^{3}]\\ \mathbf{wp}_{5}\lvert_{t=50}:\mathbf{x}_{0}&=&[1,1,2]^{T} [\text{m}],\dot{\mathbf{x}}_{0}=\mathbf{0} [\text{m}/\text{s}],\ddot{\mathbf{x}}_{0}=\mathbf{0} [\text{m}/\text{s}^{2}],\\ \dddot{\mathbf{x}}_{0}&=&\mathbf{0} [\text{m}/\text{s}^{3}]\\ \mathbf{h}&=&[-540,-64,0]^{T} [\text{deg}],\dot{\mathbf{h}}_{~}=\mathbf{0} [\text{deg}/\text{s}],\\ \ddot{\mathbf{h}}_{~}&=&\mathbf{0} [\text{deg}/\text{s}^{2}],\dddot{\mathbf{h}}_{~}=\mathbf{0} [\text{deg}/\text{s}^{3}]\\ \mathbf{wp}_{6}\lvert_{t=60}:\mathbf{x}_{0}&=&[0,1,2]^{T} [\text{m}],\dot{\mathbf{x}}_{0}=\mathbf{0} [\text{m}/\text{s}],\ddot{\mathbf{x}}_{0}=\mathbf{0} [\text{m}/\text{s}^{2}],\\ \dddot{\mathbf{x}}_{0}&=&\mathbf{0} [\text{m}/\text{s}^{3}]\\ \mathbf{h}&=&[-540,-45,0]^{T} [\text{deg}],\dot{\mathbf{h}}_{~}=\mathbf{0} [\text{deg}/\text{s}],\\ \ddot{\mathbf{h}}_{~}&=&\mathbf{0} [\text{deg}/\text{s}^{2}],\dddot{\mathbf{h}}_{~}=\mathbf{0} [\text{deg}/\text{s}^{3}] \end{array} $$

where vector h components contain the desired yaw, pitch and roll commands.

Appendix B: Proof of Proposition 1

The following Lyapunov functions are defined

$$ \begin{array}{@{}rcl@{}} V_{i}\left( {{\varPsi}}_{i},{~}^{i}\mathbf{e}_{q,i},{~}^{i}\mathbf{e}_{\omega,i}\right)&=&\frac{1}{2}{~}^{i}\mathbf{s}_{i }^{T}{~}^{i}\mathbf{s}_{i}+\alpha_{i}{{\varPsi}}_{i}, i{=}1,2,3 \end{array} $$

(B1a)

$$ \begin{array}{@{}rcl@{}} \alpha_{i}&=&2\lambda_{\mathcal{J}_{i}}k_{q}k_{\omega}(\gamma_{2}+\gamma_{3}) \end{array} $$

(B1b)

Note that Ψ_i and the error vectors appear explicitly in the Lyapunov function. Thus control design is similar to nonlinear control design in Euclidean spaces [38]. Via Eq. (25), it holds

$$ \lambda_{min}(\boldsymbol{{\varPi}}_{1_{i}})\lVert\mathbf{z}_{i}\rVert^{2}\leq\mathbf{z}_{i}^{T}\boldsymbol{{\varPi}}_{1_{i}}\mathbf{z}_{i}\leq V_{i}\leq\mathbf{z}_{i}^{T}\boldsymbol{{\varPi}}_{2_{i}}\mathbf{z}_{i}\leq\lambda_{max}(\boldsymbol{{\varPi}}_{2_{i}})\lVert\mathbf{z}_{i}\rVert^{2} $$

(B2)

where $\mathbf {z}_{i}=[\lVert {~}^{i}\mathbf {e}_{q,i}\rVert ;\lVert {~}^{i}\mathbf {e}_{\omega ,i}\rVert ]$ and $\boldsymbol {{\varPi }}_{1_{i}},\boldsymbol {{\varPi }}_{2_{i}}\in \mathbb {R}^{2\times 2}$ are given by

$$ \boldsymbol{{\varPi}}_{1_{i}}=\begin{bmatrix} \frac{(k_{q}+{{\varPsi}}_{i})^{2}+\alpha_{i}}{2}&-\frac{(k_{q}+{{\varPsi}}_{i})k_{\omega}}{2}\\ -\frac{(k_{q}+{{\varPsi}}_{i})k_{\omega}}{2}&{k_{\omega}^{2}}/{2} \end{bmatrix}, \boldsymbol{{\varPi}}_{2_{i}}=\begin{bmatrix} \frac{(k_{q}+{{\varPsi}}_{i})^{2}}{2}+\frac{\alpha_{i}}{2-\psi_{i}}&\frac{(k_{q}+{{\varPsi}}_{i})k_{\omega}}{2}\\ \frac{(k_{q}+{{\varPsi}}_{i})k_{\omega}}{2}&{k_{\omega}^{2}}/{2} \end{bmatrix} $$

(B3)

The derivative of Eq. (28b) is needed to proceed. It is given by

$$ ^{i}\dot{\mathbf{s}}_{i }=\dot{{{\varPsi}}}_{i}{~}^{i}\mathbf{e}_{q,i}{+}(k_{q}+{{\varPsi}}_{i}){~}^{i}\dot{\mathbf{e}}_{q,i}{+}k_{\omega}{~}^{i}\dot{\boldsymbol{\omega}}^{~}_{i/0}{+}k_{\omega}{~}^{i}\mathbf{a}_{i } $$

(B4)

Differentiating Eq. (B1a) and using Eqs. (B4), (26a), (26c) results in

$$ \dot{V}_{i}{=}{~}^{i}\mathbf{s}_{i}^{T}\left( \dot{{{\varPsi}}}_{i}{~}^{i}\mathbf{e}_{q,i}{+}(k_{q}+{{\varPsi}}_{i}){~}^{i}\dot{\mathbf{e}}_{q,i}{+}k_{\omega}{~}^{i}\dot{\boldsymbol{\omega}}^{~}_{i/0 }{+}k_{\omega}{~}^{i}\mathbf{a}_{i}\right)+\alpha_{i}\dot{{{\varPsi}}}_{i} $$

(B5)

Substituting into Eqs. (B5), (9), (28a), (26b), (27c), in the given order, after some manipulations

$$ \begin{array}{@{}rcl@{}} \dot{V}_{i}&{=}&{~}^{i}\mathbf{s}_{i }^{T}\Big({{\varDelta}}\mathcal{J}_{i} \Big(\left( {~}^{i}\mathbf{e}_{q,i}{~}^{i}\mathbf{e}_{q,i}^{T}\right){~}^{i}\mathbf{e}_{\omega,i}\\ &&{+} \left( k_{q}{+}{{\varPsi}}_{i}\right){~}^{i}\mathcal{E}_{i } {~}^{i}\mathbf{e}_{\omega,i} {-}k_{\omega}S\left( {{~}^{i}\mathbf{R}_{0}}{~}^{0}\mathbf{R}^{i_{d}}_{i} {~}^{i}\boldsymbol{\omega}^{~}_{i_{d}/0}\right){~}^{i}{\mathbf{e}}_{\boldsymbol{\omega},i} \\ &&{-}k_{\omega}{{~}^{i}\mathbf{R}_{0}}{~}^{0}\mathbf{R}^{i_{d}}_{i} {~}^{i}\dot{\boldsymbol{\omega}}^{~}_{i_{d}/0} \Big) {-}\gamma {~}^{i }\boldsymbol{\mathcal{J}}_{i }^{-1}\widehat{{~}^{i }\boldsymbol{\mathcal{J}}_{i }} {~}^{i}\mathbf{s}_{i}\\ &&+k_{\omega}{~}^{i }\boldsymbol{\mathcal{J}}_{i}^{-1} \Big\{ S\left( {~}^{i}\mathbf{p}_{i}\right){{~}^{i}\mathbf{R}_{I}}m_{i}\left( {~}^{I }\dot{\mathbf{v}}_{0}{-}\widehat{{~}^{I}\dot{\mathbf{v}}_{0}}\right)\\ &&{+}\left( {~}^{i}\mathbf{Z}_{i}{~}^{0}\dot{\boldsymbol{\omega}}_{0 }{-}\widehat{{~}^{i}\mathbf{Z}_{i}}\widehat{{~}^{0}\dot{\boldsymbol{\omega}}_{0}}\right) {+}\left( {~}^{i}\mathbf{n}_{r,i}{-}\widehat{{~}^{i}\mathbf{n}_{r,i}}\right)\\ &&+S\left( {~}^{i}\mathbf{p}_{i}\right){{~}^{i}\mathbf{R}_{I}}\left( {-}{~}^{I}\mathbf{f}_{w,i}\right){+}{~}^{i}\mathbf{n}_{w,i}\Big\}\Big){+}\alpha_{i}{~}^{i}\mathbf{e}_{q,i}^{T}{~}^{i}\mathbf{e}_{\omega,i} \end{array} $$

(B6a)

$$ \begin{array}{@{}rcl@{}} {{\varDelta}}\mathcal{J}_{i}&{=}&\mathbf{1}{-}{~}^{i }\boldsymbol{\mathcal{J}}_{i }^{-1}\widehat{{~}^{i }\boldsymbol{\mathcal{J}}_{i}} \end{array} $$

(B6b)

After some rearranging, the above equation is given by

$$ \begin{array}{@{}rcl@{}} \dot{V}_{i}&{=}&{~}^{i}\mathbf{s}_{i }^{T}\left( {~}^{i}\mathbf{N}_{i }{~}^{i}\mathbf{e}_{\omega,i} +{~}^{i}\mathbf{n}_{d,i} {-}\gamma {~}^{i }\boldsymbol{\mathcal{J}}_{i}^{-1}\widehat{{~}^{i }\boldsymbol{\mathcal{J}}_{i }} {~}^{i}\mathbf{s}_{i }\right){+}\alpha_{i}{~}^{i}\mathbf{e}_{q,i}^{T}{~}^{i}\mathbf{e}_{\omega,i} \end{array} $$

(B7a)

$$ \begin{array}{@{}rcl@{}} {~}^{i}\mathbf{N}_{i}&=&{~}^{i}\mathbf{N}_{n,i}-k_{\omega}{~}^{i}\mathbf{N}_{\omega,i} \end{array} $$

(B7b)

$$ \begin{array}{@{}rcl@{}} {~}^{i}\mathbf{n}_{d,i}&=&k_{\omega}{~}^{i }\boldsymbol{\mathcal{J}}_{i}^{-1} \Big\{ S\left( {~}^{i}\mathbf{p}_{i }\right){{~}^{i}\mathbf{R}_{I}}m_{i}\left( {~}^{I}\dot{\mathbf{v}}_{0}{-}\widehat{{~}^{I}\dot{\mathbf{v}}_{0}}\right) {+}\left( {~}^{i}\mathbf{Z}_{i}{~}^{0}\dot{\boldsymbol{\omega}}_{0 }{-}\widehat{{~}^{i}\mathbf{Z}_{i}}\widehat{{~}^{0}\dot{\boldsymbol{\omega}}_{0 }}\right) \\ &&{+}\left( {~}^{i}\mathbf{n}_{r,i}{-}\widehat{{~}^{i}\mathbf{n}_{r,i}}\right) {+}S\left( {~}^{i}\mathbf{p}_{i }\right){{~}^{i}\mathbf{R}_{I}}\left( {-}{~}^{I }\mathbf{f}_{w,i}\right){+}{~}^{i}\mathbf{n}_{w,i}\Big\}\\ &&-{{\varDelta}}\mathcal{J}_{i}k_{\omega}{{~}^{i}\mathbf{R}_{0}}{~}^{0}\mathbf{R}^{i_{d}}_{i} {~}^{i}\dot{\boldsymbol{\omega}}^{~}_{i_{d}/0} \end{array} $$

(B7c)

where ${~}^{i}\mathbf {n}_{d,i}$ depends on the estimation errors, the disturbances and the desired angular acceleration and ${~}^{i}\mathbf {N}_{n,i}$, ${~}^{i}\mathbf {N}_{\omega ,i}$, are given by

$$ \begin{array}{@{}rcl@{}} {~}^{i}\mathbf{N}_{n,i}&{=}&{{\varDelta}}\mathcal{J}_{i}\left( {~}^{i}\mathbf{e}_{q,i}{~}^{i}\mathbf{e}_{q,i}^{T}{+}(k_{q}{+}{{\varPsi}}_{i}){~}^{i}\mathcal{E}_{i }\right) \end{array} $$

(B8a)

$$ \begin{array}{@{}rcl@{}} {~}^{i}\mathbf{N}_{\omega,i}&{=}&{{\varDelta}}\mathcal{J}_{i}S\left( {{~}^{i}\mathbf{R}_{0}}{~}^{0}\mathbf{R}^{i_{d}}_{i}{~}^{i}\boldsymbol{\omega}^{~}_{i_{d}/0}\right) \end{array} $$

(B8b)

Inspecting the term ${~}^{i}\mathcal {E}_{i}$ given in Eq. (27a), the magnitude of Eq. (B8) above depends on the desired trajectory. Additionally Eqs. (7), (32), (30a), (30b) imply that

$$ \begin{array}{@{}rcl@{}} \exists N_{n,i_{max}}, N_{\omega,i_{max}}, n_{d,i_{max}}\in(0,\infty)\big|\lVert{~}^{i}\mathbf{N}_{n,i}\rVert&\leq& N_{n,i_{max}}\\ \lVert{~}^{i}\mathbf{N}_{\omega,i}\rVert&\leq& N_{\omega,i_{max}}\\ \lVert{~}^{i}\mathbf{n}_{d,i}\rVert&\leq& n_{d,i_{max}} \end{array} $$

(B9)

Employing Eq. (B9), the Lyapunov derivative in Eq. (B7a) is expanded to

$$ \begin{array}{@{}rcl@{}} \dot{V}_{i}&{\leq}&-\gamma\lambda_{\mathcal{J}_{i}}{~}^{i}\mathbf{s}_{i}^{T}{~}^{i}\mathbf{s}_{i }+{\varUpsilon}_{i}+\mathcal{N}_{i}\lVert{~}^{i}\mathbf{e}_{\omega,i}\rVert\\ &&+\left( k_{\omega}N_{n,i_{max}}{+}k_{\omega}^{2}N_{\omega,i_{max}}\right)\lVert{~}^{i}\mathbf{e}_{\omega,i}\rVert^{2}{+}\alpha_{i}{~}^{i}\mathbf{e}_{q,i}^{T}{~}^{i}\mathbf{e}_{\omega,i} \end{array} $$

(B10a)

$$ \begin{array}{@{}rcl@{}} {\varUpsilon}_{i}&{=}&(k_{q}{+}{{\varPsi}}_{i})n_{d,i_{max}} \end{array} $$

(B10b)

$$ \begin{array}{@{}rcl@{}} \mathcal{N}_{i}&{=}&k_{\omega}n_{d,i_{max}}{+}(k_{q}{+}{{\varPsi}}_{i})\left( N_{n,i_{max}}{+}k_{\omega}N_{\omega,i_{max}}\right) \end{array} $$

(B10c)

Employing Eq. (28c) and rearranging

$$ \begin{array}{@{}rcl@{}} \dot{V}_{i}&{\leq}&-\gamma_{1}\lambda_{\mathcal{J}_{i}}{~}^{i}\mathbf{s}_{i }^{T}{~}^{i}\mathbf{s}_{i} -\mathbf{z}_{i}^{T}\boldsymbol{{\varPi}}_{3_{i}}\mathbf{z}_{i} -\gamma_{3}\lambda_{\mathcal{J}_{i}}{~}^{i}\mathbf{s}_{i}^{T}{~}^{i}\mathbf{s}_{i } +{\Upsilon}_{i}+\mathcal{N}_{i}\lVert{~}^{i}\mathbf{e}_{\omega,i}\rVert\\ &&+\left( k_{\omega}N_{n,i_{max}}{+}k_{\omega}^{2}N_{\omega,i_{max}}\right)\lVert{~}^{i}\mathbf{e}_{\omega,i}\rVert^{2}{+}2\lambda_{\mathcal{J}_{i}}k_{q}k_{\omega}\gamma_{3}{~}^{i}\mathbf{e}_{q,i}^{T}{~}^{i}\mathbf{e}_{\omega,i} \end{array} $$

(B11a)

$$ \begin{array}{@{}rcl@{}} \boldsymbol{{\varPi}}_{3_{i}}&{=}&\begin{bmatrix} \gamma_{2}\lambda_{\mathcal{J}_{i}}(k_{q}+{{\varPsi}}_{i})^{2}&-\gamma_{2}\lambda_{\mathcal{J}_{i}}{{\varPsi}}_{i} k_{\omega}\\ -\gamma_{2}\lambda_{\mathcal{J}_{i}}{{\varPsi}}_{i} k_{\omega}&\gamma_{2}\lambda_{\mathcal{J}_{i}}k_{\omega}^{2} \end{bmatrix} \end{array} $$

(B11b)

Via $\gamma _{3}={\sum }_{i=4}^{5}\gamma _{i},\gamma _{i},\in \mathbb {R}^{+}$ after some manipulations

$$ \begin{array}{@{}rcl@{}} \dot{V}_{i}&{\leq}&-\gamma_{1}\lambda_{\mathcal{J}_{i}}{~}^{i}\mathbf{s}_{i}^{T}{~}^{i}\mathbf{s}_{i } -\mathbf{z}_{i}^{T}\boldsymbol{{\varPi}}_{3_{i}}\mathbf{z}_{i}\\ &&-\mathbf{z}_{i}^{T}\boldsymbol{{\varPi}}_{4_{i}}\mathbf{z}_{i} +{\Upsilon}_{i}+\mathcal{\phi}\left( \lVert{~}^{i}\mathbf{e}_{\omega,i}\rVert\right) \end{array} $$

(B12a)

$$ \begin{array}{@{}rcl@{}} \mathcal{\phi}\left( \lVert{~}^{i}\mathbf{e}_{\omega,i}\rVert\right)&=&\mathcal{N}_{i}\lVert{~}^{i}\mathbf{e}_{\omega,i}\rVert\\ &&-\left( (\gamma_{5}\lambda_{\mathcal{J}_{i}}{-}N_{\omega,i_{max}})k_{\omega}^{2}{-}k_{\omega}N_{n,i_{max}}\right)\lVert{~}^{i}\mathbf{e}_{\omega,i}\rVert^{2} \end{array} $$

(B12b)

$$ \begin{array}{@{}rcl@{}} \boldsymbol{{\varPi}}_{4_{i}}&{=}&\begin{bmatrix} \gamma_{3}\lambda_{\mathcal{J}_{i}}(k_{q}+{{\varPsi}}_{i})^{2}&-\gamma_{3}\lambda_{\mathcal{J}_{i}}{{\varPsi}}_{i} k_{\omega}\\ -\gamma_{3}\lambda_{\mathcal{J}_{i}}{{\varPsi}}_{i} k_{\omega}&\gamma_{4}\lambda_{\mathcal{J}_{i}}k_{\omega}^{2} \end{bmatrix} \end{array} $$

(B12c)

The condition in Eq. (28f) ensures that matrix $\boldsymbol {{\varPi }}_{4_{i}}$ is positive definite. Conditions in Eqs. (28d), (28e), ensure that the last term of Eq. (B12b) is non-positive. For an angular velocity error vector, ${~}^{i}\mathbf {e}_{\omega ,i}$, such that

$$ \lVert{~}^{i}\mathbf{e}_{\omega,i}\rVert>\frac{\mathcal{N}_{i}}{\left( \gamma_{5}\lambda_{\mathcal{J}_{i}}{-}N_{\omega,i_{max}}\right)k_{\omega}^{2}{-}k_{\omega}N_{n,i_{max}}} $$

(B13)

then Eq. (B12b) is non-positive. Additionally for $\lVert {~}^{i}\mathbf {e}_{\omega ,i}\rVert$ less or equal to Eqs. (B13), (B12b) is bounded by

$$ \mathcal{\phi}\left( \lVert{~}^{i}\mathbf{e}_{\omega,i}\rVert\right)\leq\frac{\mathcal{N}^{2}_{i}}{4\left( \gamma_{5}\lambda_{\mathcal{J}_{i}}{-}N_{\omega,i_{max}}\right)k_{\omega}^{2}{-}4k_{\omega}N_{n,i_{max}}},\forall t $$

(B14)

Thus the following inequality is valid

$$ \begin{array}{@{}rcl@{}} \dot{V}_{i}&{\leq}&-\gamma_{1}\lambda_{\mathcal{J}_{i}}{~}^{i}\mathbf{s}_{i}^{T}{~}^{i}\mathbf{s}_{i} -\lambda_{min}(\boldsymbol{{\varPi}}_{3_{i}})\lVert\mathbf{z}_{i}\rVert^{2} -\lambda_{min}(\boldsymbol{{\varPi}}_{4_{i}})\lVert\mathbf{z}_{i}\rVert^{2} +{\delta}_{i} \end{array} $$

(B15a)

$$ \begin{array}{@{}rcl@{}} {\delta}_{i}&{=}&{{\varUpsilon}}_{i}+\frac{\mathcal{N}^{2}_{i}}{4(\gamma_{5}\lambda_{\mathcal{J}_{i}}{-}N_{\omega,i_{max}})k_{\omega}^{2}{-}4k_{\omega}N_{n,i_{max}}} \end{array} $$

(B15b)

Concluding, for z_i such that

$$ \lVert\mathbf{z}_{i}\rVert\geq\sqrt{\frac{{\delta}_{i}}{\lambda_{min}(\boldsymbol{{\varPi}}_{3_{i}})}} $$

(B16)

then

$$ \dot{V}_{i}{\leq}-\gamma_{1}\lambda_{\mathcal{J}_{i}}{~}^{i}\mathbf{s}_{i}^{T}{~}^{i}\mathbf{s}_{i} -\lambda_{min}(\boldsymbol{{\varPi}}_{4_{i}})\lVert\mathbf{z}_{i}\rVert^{2}{\leq}-\lambda_{min}(\boldsymbol{{\varPi}}_{4_{i}})\lVert\mathbf{z}_{i}\rVert^{2} $$

(B17)

The first term in Eq. (B17) ensures the desired sliding behavior. Finally using Eq. (B2)

$$ \dot{V}_{i}{\leq}-({\lambda_{min}(\boldsymbol{{\varPi}}_{4_{i}})}/{\lambda_{max}(\boldsymbol{{\varPi}}_{2_{i}})}){V}_{i} $$

(B18)

Boundedness: Using Eq. (B16), the following sets are defined

$$ \begin{array}{@{}rcl@{}} \text{L}_{\sigma}&{=}&\left\{\left( {~}^{i}\mathbf{e}_{q,i},{~}^{i}\mathbf{e}_{\omega,i}\right)\in\mathbb{R}^{3}{\times}\mathbb{R}^{3}\mid \text{Eq.} \mathrm{(B16)}\right\} \end{array} $$

(B19a)

$$ \begin{array}{@{}rcl@{}} \text{L}_{\phi}&{=}&\left\{\left( {~}^{i}\mathbf{e}_{q,i},{~}^{i}\mathbf{e}_{\omega,i}\right)\in\mathbb{R}^{3}{\times}\mathbb{R}^{3}\mid {{\varPsi}}_{i}<2\right\} \end{array} $$

(B19b)

For a gain γ such that, Eq. (33) is satisfied, then $\text {L}^{c}_{\sigma }{\subset }\text {L}_{\phi }$ ((.)^c is the complement set of (.)). Hence, Eq. (B18) implies that for all initial states beginning in L_ϕ, it holds that

$$ \alpha_{i}{{\varPsi}}_{i}{\leq}{V}_{i}(t){\leq}{V}_{i}(0)e^{-\frac{\lambda_{min}(\boldsymbol{{\varPi}}_{4_{i}})}{\lambda_{max}(\boldsymbol{{\varPi}}_{2_{i}})}t} $$

(B20)

Thus Ψ_i exponentially decreases and the pointing tracking errors exponentially converge to $\text {L}^{c}_{\sigma }$ and are UUB despite the presence of parametric uncertainties, disturbances and signal estimation errors. An estimate of the ultimate bound is given in Eq. (34).

Inspecting Eqs. (B15b), (B11b), if γ₂ is increased, $\lambda _{min}(\boldsymbol {{\varPi }}_{3_{i}})$ increases also. Since both Eqs. (B10b), (B10b) (they comprise δ_i), are a function of the estimation errors, the disturbances and the attitude errors which are all bounded, for suitable γ₂, γ₅ (γ₅ reduces Eq. (B15b)) the conditions of Eq. (33) are satisfied. However depending on the actuator/thrust constraints the value of γ₂, γ₅ is limited. In this case Eq. (33) is satisfied for less aggressive trajectories, smaller estimation errors and smaller disturbances.

Appendix C: Proof of Proposition 2

1.1 C.1 Body-0 Attitude Error Dynamics

First the derivative of Eq. (36b) is calculated. Then Eqs. (6b), (6c), are substituted in the derivative followed by Eqs. (1c), (3). Finally Eqs. (A2b) and (5), are substituted into the resulting expression to get

$$ \dot{\mathbf{e}}_{\mathbf{v},\boldsymbol{\omega}}=\mathbf{C}^{-1}\left( \sum\limits_{i=1}^{3}\begin{bmatrix} {~}^{I}\mathbf{R}_{i}{~}^{i}\mathbf{f}_{p,i}\\ \mathbf{A}_{i}{~}^{I}\mathbf{R}_{i}{~}^{i}\mathbf{f}_{p,i} \end{bmatrix}{-}\mathbf{d}{-}\mathbf{h}_{g}\right){+}\begin{bmatrix} -{~}^{I}\dot{\mathbf{v}}_{d}\\ {~}^{0}\mathbf{a}_{0} \end{bmatrix} $$

(C1)

where $\mathbf {A}_{i}\in \mathbb {R}^{3{\times }3}$ is given in Eq. (A16a). The vector $\mathbf {d}=[d_{1};d_{2}]\in \mathbb {R}^{6{\times }1}$ is given by

$$ \begin{array}{@{}rcl@{}} d_{1}&=&\sum\limits_{i=1}^{3}\Big(m_{i}{~}^{I}\mathbf{R}_{i}S\left( {~}^{i}\mathbf{p}_{i }\right){~}^{i}\dot{\boldsymbol{\omega}}^{~}_{i/0 }{-}{~}^{I}\mathbf{f}_{w,i}\\ &&-m_{i}{{~}^{I}\mathbf{R}_{i}}\left( S\left( {~}^{i}\boldsymbol{\omega}^{~}_{i/0 }\right)\right)^{2}{~}^{i}\mathbf{p}_{i}\Big){-}{~}^{I }\mathbf{f}_{w,0} \end{array} $$

(C2a)

$$ \begin{array}{@{}rcl@{}} d_{2}&=&\sum\limits_{i=1}^{3}{{~}^{0}\mathbf{R}_{I}}{~}^{I}\mathbf{R}_{i}\left\{{~}^{i}\mathbf{J}_{i } {~}^{i}\dot{\boldsymbol{\omega}}^{~}_{i/0 }+S\left( {~}^{i}\boldsymbol{\omega}^{~}_{i/0 }\right){~}^{i}\mathbf{J}_{i} {~}^{i}\boldsymbol{\omega}^{~}_{i/0 }-{~}^{i}\mathbf{n}_{w,i}\right\}{-}{~}^{0}\mathbf{n}_{w,0}\\ &&-\sum\limits_{i=1}^{3}{~}^{0}\boldsymbol{\Theta}_{i}{{~}^{i}\mathbf{R}_{I}}\Big(m_{i}{~}^{I}\mathbf{R}_{i}S\left( {~}^{i}\mathbf{p}_{i}\right){~}^{i}\dot{\boldsymbol{\omega}}^{~}_{i/0 }{-}{~}^{I }\mathbf{f}_{w,i}\\ &&-m_{i}{{~}^{I}\mathbf{R}_{i}}\left( S\left( {~}^{i}\boldsymbol{\omega}^{~}_{i/0}\right)\right)^{2}{~}^{i}\mathbf{p}_{i}\Big) \end{array} $$

(C2b)

where the matrix ${~}^{0}\boldsymbol {\Theta }_{i}\in \mathbb {R}^{3{\times }3}$ is given in Eq. (A16a). Vector d contains bounded disturbance terms (see Eq. (7)); under the action of Eq. (28a), the relative angular velocity and acceleration terms, ${~}^{i}{\boldsymbol {\omega }}^{~}_{i/0 }$, ${~}^{i}\dot {\boldsymbol {\omega }}^{~}_{i/0}$, of each TA with respect to body-0 are stable and bounded (see proof of Proposition 1 or/and see Section 4.4.1). Thus vector d is bounded and small; the small mass-inertia characteristics and geometry of each TA contributes to this also.

1.2 C.2 Lyapunov Analysis

For $\zeta _{1},\zeta _{2}\in \mathbb {R}^{+}$ such that Eq. (37b) is satisfied, the following Lyapunov function is defined

$$ V=\frac{1}{2}\mathbf{z}^{T}\mathbf{z}+\zeta_{1}K_{x}K_{v}\mathbf{e}_{x}^{T}\mathbf{e}_{x}+2\zeta_{1}K_{{{\varOmega}}}K_{R}{{\varPsi}}_{0} $$

(C3)

Defining $\mathbf {z}_{c}=[ \lVert \mathbf {e}_{x}\rVert ;\lVert \mathbf {e}_{v}\rVert ;\lVert \mathbf {e}_{R}\rVert ;\lVert \mathbf {e}_{\omega }\rVert ]\in \mathbb {R}^{4}$ and via Eq. (A12) the Lyapunov function is bounded by

$$ \begin{array}{@{}rcl@{}} &&\lambda_{min}(\mathcal{W}_{1})\lVert\mathbf{z}_{c}\rVert^{2}\!\leq\! \mathbf{z}_{c}^{T}\mathcal{W}_{1}\mathbf{z}_{c}\!\leq\! V \!\leq\!\mathbf{z}_{c}^{T}\mathcal{W}_{2}\mathbf{z}_{c}\leq\lambda_{max}(\mathcal{W}_{2})\lVert\mathbf{z}_{c}\rVert^{2} \end{array} $$

(C4a)

$$ \begin{array}{@{}rcl@{}} \mathcal{W}_{1}&{ = }&\begin{bmatrix} \frac{{K_{x}^{2}}}{2}{+}\zeta_{1}K_{x}K_{v}&-\frac{K_{x}K_{v}}{2}&0&0\\ -\frac{K_{x}K_{v}}{2}&\frac{{K_{v}^{2}}}{2}&0&0\\ 0&0&\frac{(K_{R}{+}{{\varPsi}}_{0})^{2}}{2}{+}2\zeta_{1}K_{{{\varOmega}}}K_{R}&-\frac{(K_{R}{+}{{\varPsi}}_{0})K_{{{\varOmega}}}}{2}\\ 0&0&-\frac{(K_{R}{+}{{\varPsi}}_{0})K_{{{\varOmega}}}}{2}&\frac{K_{{{\varOmega}}}^{2}}{2} \end{bmatrix} \end{array} $$

(C4b)

$$ \begin{array}{@{}rcl@{}} \mathcal{W}_{2}&{ = }&\begin{bmatrix} \frac{{K_{x}^{2}}}{2}{+}\zeta_{1}K_{x}K_{v}&\frac{K_{x}K_{v}}{2}&0&0\\ \frac{K_{x}K_{v}}{2}&\frac{{K_{v}^{2}}}{2}&0&0\\ 0&0&\frac{(K_{R}{+}{{\varPsi}}_{0})^{2}}{2}{+}4\zeta_{1}K_{{{\varOmega}}}K_{R}&\frac{(K_{R}{+}{{\varPsi}}_{0})K_{{{\varOmega}}}}{2}\\ 0&0&\frac{(K_{R}{+}{{\varPsi}}_{0})K_{{{\varOmega}}}}{2}&\frac{K_{{{\varOmega}}}^{2}}{2} \end{bmatrix} \end{array} $$

(C4c)

The time derivative of the Lyapunov function is given by,

$$ \begin{array}{@{}rcl@{}} \dot{V}&=&\mathbf{z}^{T}\left( \begin{bmatrix} K_{v}\mathbf{1}&0_{3{\times}3}\\ 0_{3{\times}3}&K_{{{\varOmega}}}\mathbf{1} \end{bmatrix}\dot{\mathbf{e}}_{\mathbf{v},\boldsymbol{\omega}}{+}\begin{bmatrix} K_{x}\mathbf{e}_{v}\\ (K_{R}{+}{{\varPsi}}_{0})\dot{\mathbf{e}}_{R}{+}\dot{{{\varPsi}}}_{0}\mathbf{e}_{R} \end{bmatrix}\right)\\ &&+2\zeta_{1}K_{x}K_{v}\mathbf{e}_{x}^{T}\mathbf{e}_{v}+2\zeta_{1}K_{{{\varOmega}}}K_{R}\mathbf{e}_{R}^{T}\mathbf{e}_{\omega} \end{array} $$

(C5)

The thrust tracking error of each TA in ${\mathscr{I}}_{0}$, is given by the following algebraic relation

$$ {{~}^{0}\mathbf{R}_{I}}\left( {~}^{I}\mathbf{R}_{i}{~}^{i}\mathbf{f}_{p,i}-{~}^{I}\mathbf{R}^{i_{d}}_{i} {~}^{i}\mathbf{f}_{{p,i}_{d}}\right)=b_{T}{~}^{0}\mathbf{q}_{i}\left( {{~}^{i}\boldsymbol{\omega}_{i }^{T}}{~}^{i}\mathbf{e}_{3}\right)^{2}-b_{T}{~}^{0}\mathbf{q}_{i_{d}}\left( {{~}^{i}\boldsymbol{\omega}_{i_{d}}^{T}}{~}^{i}\mathbf{e}_{3}\right)^{2} $$

(C6)

Using Eq. (C6) together with Eqs. (20c), (22c) the i^th TA generated thrust can be expressed by the following algebraic relation

$$ {~}^{I}\mathbf{R}_{i}{~}^{i}\mathbf{f}_{p,i}={{~}^{I}\mathbf{R}_{0}}\left( {~}^{0}\mathbf{f}_{{p,i}_{e}}+b_{T}{~}^{0}\mathbf{h}_{p,i}\right)+{~}^{I}\mathbf{R}^{i_{d}}_{i} {~}^{i}\mathbf{f}_{{p,i}_{d}} $$

(C7)

where ${~}^{0}\mathbf {h}_{p,i}\in \mathbb {R}^{3}$ is found in Eq. (A15b) and ${~}^{0}\mathbf {f}_{{p,i}_{e}}$ is the thrust tracking error of each TA relative to body-0. It is given by

$$ {~}^{0}\mathbf{f}_{{p,i}_{e}}=b_{T}{~}^{0}\mathbf{q}_{i}\left( {{~}^{i}\boldsymbol{\omega}^{T}_{i/0}}{~}^{i}\mathbf{e}_{3}\right)^{2}-b_{T}{~}^{0}\mathbf{q}_{i_{d}}\left( {{~}^{i}\boldsymbol{\omega}^{T}_{i_{d}/0}}{~}^{i}\mathbf{e}_{3}\right)^{2} $$

(C8)

Due to the action of the vectoring controller, ${~}^{0}\mathbf {f}_{{p,i}_{e}}$ is small and bounded; because ${~}^{i}\boldsymbol {\omega }^{~}_{i/0}$ is bounded (see proof of Proposition 1 or/and see Section 4.4.1). Substituting Eqs. (C1) into (C5) followed by Eq. (C7), results in

$$ \begin{array}{@{}rcl@{}} \dot{V}&{=}&\mathbf{z}^{T}\Big\{ \begin{bmatrix} K_{v}\mathbf{1}&0_{3{\times}3}\\ 0_{3{\times}3}&K_{{{\varOmega}}}\mathbf{1} \end{bmatrix}\mathbf{C}^{-1}\Bigg(\sum\limits_{i=1}^{3}\Big(\begin{bmatrix} {~}^{I}\mathbf{R}^{i_{d}}_{i} {~}^{i}\mathbf{f}_{p,i_{d}}\\ \mathbf{A}_{i}{~}^{I}\mathbf{R}^{i_{d}}_{i} {~}^{i}\mathbf{f}_{p,i_{d}} \end{bmatrix}\\ &&+\begin{bmatrix} {~}^{I}\mathbf{R}_{0}{~}^{0}\mathbf{f}_{p,i_{e}}\\ \mathbf{A}_{i}{~}^{I}\mathbf{R}_{0}{~}^{0}\mathbf{f}_{p,i_{e}} \end{bmatrix}\Big){-}\mathbf{d}{-}\mathbf{h}_{g}{+}\mathbf{h}_{p}\Bigg)\\ &&{+}\begin{bmatrix} -K_{v}{~}^{~}\dot{\mathbf{v}}_{d}{+}K_{x}\mathbf{e}_{v}\\ K_{{{\varOmega}}}{~}^{0}\mathbf{a}_{0 }{+}(K_{R}{+}{{\varPsi}}_{0})\dot{\mathbf{e}}_{R}{+}\dot{{{\varPsi}}}_{0}\mathbf{e}_{R} \end{bmatrix}\Big\}\\ &&+2\zeta_{1}K_{x}K_{v}\mathbf{e}_{x}^{T}\mathbf{e}_{v}+2\zeta_{1}K_{{{\varOmega}}}K_{R}\mathbf{e}_{R}^{T}\mathbf{e}_{\omega} \end{array} $$

(C9)

By substituting Eqs. (39) in (C9), followed by Eq. (37), after considerable manipulations

$$ \begin{array}{@{}rcl@{}} \dot{V}&{=}&-\zeta\mathbf{z}^{T}\mathbf{z}+\mathbf{z}^{T}\boldsymbol{\epsilon}{+}2\zeta_{1}K_{x}K_{v}\mathbf{e}_{x}^{T}\mathbf{e}_{v}+2\zeta_{1}K_{{{\varOmega}}}K_{R}\mathbf{e}_{R}^{T}\mathbf{e}_{\omega} \end{array} $$

(C10a)

$$ \begin{array}{@{}rcl@{}} \boldsymbol{\epsilon}&{=}&\begin{bmatrix} K_{v}\mathbf{1}&0_{3{\times}3}\\ 0_{3{\times}3}&K_{{{\varOmega}}}\mathbf{1} \end{bmatrix}\mathbf{C}^{-1}\left( \sum\limits_{i=1}^{3}\begin{bmatrix} {~}^{I}\mathbf{R}_{0}{~}^{0}\mathbf{f}_{p,i_{e}}\\ \mathbf{A}_{i}{~}^{I}\mathbf{R}_{0}{~}^{0}\mathbf{f}_{p,i_{e}} \end{bmatrix}-\mathbf{d}\right) \end{array} $$

(C10b)

$$ \begin{array}{@{}rcl@{}} \lVert\boldsymbol{\epsilon}\rVert&{\leq}&\epsilon_{_{max}} \end{array} $$

(C10c)

Note that vector 𝜖 contains bounded quantities like disturbance terms. Due to space limitations the bound, $\epsilon _{_{max}}$, is not given explicitly but a conservative estimate can be found using Eqs. (7), (B16) and $\lVert {~}^{i}\mathbf {e}_{q,i}\rVert {\leq }1$. Employing that ζ = ζ₁ + ζ₂, Eq. (C10a) is rearranged to

$$ \dot{V}{=}-\zeta_{1}\mathbf{z}^{T}\mathbf{z}-\zeta_{2}\mathbf{z}^{T}\mathbf{z}+\mathbf{z}^{T}\boldsymbol{\epsilon}{+}2\zeta_{1}K_{x}K_{v}\mathbf{e}_{x}^{T}\mathbf{e}_{v}+2\zeta_{1}K_{{{\varOmega}}}K_{R}\mathbf{e}_{R}^{T}\mathbf{e}_{\omega} $$

(C11)

Expanding the first term, after several manipulations

$$ \begin{array}{@{}rcl@{}} \dot{V}&{\leq}&-\mathbf{z}_{c}^{T}\mathcal{W}_{3}\mathbf{z}_{c}-\zeta_{2}\mathbf{z}^{T}\mathbf{z}+\mathbf{z}^{T}\boldsymbol{\epsilon} \end{array} $$

(C12a)

$$ \begin{array}{@{}rcl@{}} \mathcal{W}_{3}&{=}&\begin{bmatrix} \zeta_{1}{K_{x}^{2}}&0&0&0\\ 0&\zeta_{1}{K_{v}^{2}}&0&0\\ 0&0&\zeta_{1}(K_{R}{+}{{\varPsi}}_{0})^{2}&-\zeta_{1}K_{{{\varOmega}}}{{\varPsi}}_{0}\\ 0&0&-\zeta_{1}K_{{{\varOmega}}}{{\varPsi}}_{0}&\zeta_{1}{K_{{{\varOmega}}}^{2}} \end{bmatrix} \end{array} $$

(C12b)

Using the Cauchy-Schwarz inequality on Eq. (C12a)

$$ \begin{array}{@{}rcl@{}} \dot{V}&\leq&{-}\lambda_{min}(\mathcal{W}_{3})\lVert\mathbf{z}_{c}\rVert^{2}{-}\zeta_{2}\lVert\mathbf{z}\rVert^{2}{+}\lVert\mathbf{z}\rVert\epsilon_{_{max}} {\implies}\\ {\forall}\lVert\mathbf{z}\rVert &\geq&\frac{\epsilon_{_{max}}}{\zeta_{2}}{\implies}\dot{V}{<}{-}\lambda_{min}(\mathcal{W}_{3})\lVert\mathbf{z}_{c}\rVert^{2} \end{array} $$

(C13)

Furthermore via Eq. (C4a) the following holds

$$ \dot{V}{\leq}{-}\frac{\lambda_{min}(\mathcal{W}_{3})}{\lambda_{max}(\mathcal{W}_{2})} V $$

(C14)

Concluding, via the comparison lemma, $\lVert \mathbf {z}_{c}\rVert$ exponentially decreases with a rate $e^{-\frac {\lambda _{min}(\mathcal {W}_{3})}{\lambda _{max}(\mathcal {W}_{2})} t}$ into an envelope of radius $\frac {\epsilon _{_{max}}}{\zeta _{2}}$ of the origin and it is maintained there.

Boundedness: Based on the analysis above the following sets are defined,

$$ \begin{array}{@{}rcl@{}} \text{L}_{{\varsigma}}&{=}&\left\{(\mathbf{e}_{\mathbf{x},R},\mathbf{e}_{\mathbf{v},\boldsymbol{\omega}})\in\mathbb{R}^{6}{\times}\mathbb{R}^{6}\mid\lVert\mathbf{z}\rVert{\geq}\frac{\epsilon_{_{max}}}{\zeta_{2}}\right\} \end{array} $$

(C15a)

$$ \begin{array}{@{}rcl@{}} \text{L}_{{\varSigma}}&{=}&\left\{(\mathbf{e}_{\mathbf{x},R},\mathbf{e}_{\mathbf{v},\boldsymbol{\omega}})\in\mathbb{R}^{6}{\times}\mathbb{R}^{6}\mid\lVert\mathbf{z}\rVert{<}\sqrt{2\lambda_{max}(\mathcal{W}_{2})}\right\} \end{array} $$

(C15b)

For a sufficiently large gain ζ such that Eq. (43) is satisfied then $\text {L}^{c}_{{\varsigma }}{\subset }\text {L}_{{\varSigma }}$ ((.)^c the complement set of (.)). Hence, for all initial system states in L_Σ, the tracking errors (e_x, e_v, e_R, e_ω) exponentially converge to $\text {L}^{c}_{{\varsigma }}$ and are UUB despite the presence of disturbances and thrust tracking errors. The boundary of the set $\text {L}^{c}_{{\varsigma }}$ is an estimate of the ultimate bound. Also Eqs. (C13), (C14), and (C15a) indicate that as ζ is increased, the error states converge faster to the desired equilibrium, while the ultimate bound set $\text {L}^{c}_{{\varsigma }}$ shrinks.