Observer-based Geometric Impedance Control of a Fully-Actuated Hexarotor for Physical Sliding Interaction with Unknown Generic Surfaces

Abstract

Aerial physical interaction is a promising field for unmanned aerial vehicles in future applications. This paper presents a novel paradigm for automatic aerial contact-based sliding interaction (inspection/cleaning) tasks in aerial robotics allowing a 3D force with a constant norm to be applied on generic surfaces with unknown geometry. The interaction task is achieved by a fully-actuated hexarotor equipped with a rigidly attached end-effector under a passivity-based geometric impedance controller and a new sliding-mode extended state observer to estimate the interaction wrench. In order to increase the observer performance and reduce the estimation chattering phenomenon, the observer is innovatively incorporated with a super-twisting algorithm and a sigmoid function with a switching gain being adaptively updated by a fuzzy logic system. A detailed stability analysis for the observer is presented based on the Lyapunov stability theory. The proposed control approach is validated in several simulations in which we try to accomplish the aerial physical sliding interaction task with different types of objects under various sliding speeds.

Appendix

In this appendix, the proof of Theorem 1 is provided.

Firstly, we consider the error dynamics of FASSTESO by,

$$ \left\{ {\begin{array}{*{20}{c}} \dot{\tilde{\xi}}_{1}= {\tilde{\xi} }_{2} - {\alpha_{1}}{{({{\tilde{\xi} }_{1}})}^{\frac{1}{2}}}sign({{\tilde{\xi} }_{1}})\\ {}{{\dot{\tilde{\xi} }_{2}} = \delta - {\bar{\alpha }_{2}}sig({{\tilde{\xi} }_{1}})} \end{array}} \right. $$

(21)

where ${\bar {\alpha }_{2}}{\text { = }}({\mu _{1}} + {\mu _{2}}{K_{f}}){\alpha _{2}}$ is defined for proof simplification. Meanwhile we introduce a new vector $\boldsymbol {\zeta } = {[\begin {array}{*{20}{c}} {{\zeta _{1}}}&{{\zeta _{2}}} \end {array}]^{T}}\in {\mathbb {R}^{2}}$, in which ${\zeta _{1}} = {{\left | {{{\tilde {\xi } }_{1}}} \right |}^{\frac {1}{2}}}sign({{\tilde {\xi } }_{1}}),{\zeta _{2}} = {{\tilde {\xi } }_{2}}$.

Choose the positive definite Lyapunov function V as,

$$ V = {\boldsymbol{\zeta}^{T}}\boldsymbol{{\varLambda}} \boldsymbol{\zeta} $$

(22)

where $\boldsymbol {{\varLambda }} {\text { = }}\left [ {\begin {array}{*{20}{c}} {{\alpha _{1}}^{2} + 1}&{ - {\alpha _{1}}}\\ { - {\alpha _{1}}}&1 \end {array}} \right ]\in {\mathbb {R}^{2 \times 2}}$ is a positive definite matrix.

Take time derivative of V we can obtain,

$$ \begin{array}{lll} \dot{V} &=& 2{\boldsymbol{\zeta}^{T}}\boldsymbol{{\varLambda}} \dot{\boldsymbol{\zeta}} \\ &=&2({\alpha_{1}}^{2} + 1){\zeta_{1}}{\dot{\zeta }_{1}} - 2{\alpha_{1}}{\dot{\zeta }_{1}}{\zeta_{2}} + 2{\dot{\zeta }_{2}}({\zeta_{2}} - {\alpha_{1}}{\zeta_{1}}) \end{array} $$

(23)

$$ \left\{ {\begin{array}{*{20}{c}} {}{{\zeta_{1}} = {{\left| {{{\tilde{\xi} }_{1}}} \right|}^{\frac{1}{2}}}sign({{\tilde{\xi} }_{1}})}\\ {}{{\zeta_{2}} = {{\tilde{\xi} }_{2}}}\\ {{\dot{\zeta }_{1}} = \frac{1}{2}{{\left| {{{\tilde{\xi} }_{1}}} \right|}^{- \frac{1}{2}}}(- {\alpha_{1}}{{\left| {{{\tilde{\xi} }_{1}}} \right|}^{\frac{1}{2}}}sign({{\tilde{\xi} }_{1}}) + {{\tilde{\xi} }_{2}})}\\ {}{{\dot{\zeta }_{2}} = \delta - {\bar{\alpha }_{2}}sig({{\tilde{\xi} }_{1}})} \end{array}} \right. $$

(24)

By combining (24), then (23) could be rewritten as,

$$ \dot V = \mathrm{M} + 2{\bar{\alpha }_{2}}{{\tilde{\xi} }_{2}}(sign({{\tilde{\xi} }_{1}}) - sig({{\tilde{\xi} }_{1}})) + 2{\alpha_{1}}{\bar{\alpha }_{2}}sig({{\tilde{\xi} }_{1}}){\left| {{{\tilde{\xi} }_{1}}} \right|^{\frac{1}{2}}}sign({{\tilde{\xi} }_{1}}) $$

(25)

where

$$ \begin{array}{l} \mathrm{M}{\text{ = }} - ({\alpha_{1}}^{2} + 1){\alpha_{1}}{\left| {{{\tilde{\xi} }_{1}}} \right|^{\frac{1}{2}}} + ({\alpha_{1}}^{2} + 1 - 2{\bar{\alpha }_{2}}){{\tilde{\xi} }_{2}}sign({{\tilde{\xi} }_{1}})\\ {\kern20pt}- 2{\alpha_{1}}(\frac{1}{2}{\left| {{{\tilde{\xi} }_{1}}} \right|^{- \frac{1}{2}}}(- {\alpha_{1}}{\left| {{{\tilde{\xi} }_{1}}} \right|^{\frac{1}{2}}}sign({{\tilde{\xi} }_{1}}) + {{\tilde{\xi} }_{2}})){{\tilde{\xi} }_{2}}\\ {\kern19pt}+ 2\delta ({{\tilde{\xi} }_{2}} - {\alpha_{1}}{\left| {{{\tilde{\xi} }_{1}}} \right|^{\frac{1}{2}}}sign({{\tilde{\xi} }_{1}})) \end{array} $$

(26)

$$ \left\{ {\begin{array}{*{20}{c}} {sign({{\tilde{\xi} }_{1}}) - sig({{\tilde{\xi} }_{1}}) \le \left| {sign({{\tilde{\xi} }_{1}}) - \frac{{1 - {e^{- \beta {{\tilde{\xi} }_{1}}}}}}{{1 + {e^{- \beta {{\tilde{\xi} }_{1}}}}}}} \right| \le {\beta^{- \frac{1}{2}}}{{\left| {{{\tilde{\xi} }_{1}}} \right|}^{- \frac{1}{2}}}}\\ {}{sig({{\tilde{\xi} }_{1}})sign({{\tilde{\xi} }_{1}}) \le 1} \end{array}} \right. $$

(27)

By employing inequality (27), Eq. 25 can be reconstructed as,

$$ \dot{V} \le \mathrm{M} + 2{\bar{\alpha }_{2}}{\beta^{- \frac{1}{2}}}{{\tilde{\xi} }_{2}}{\left| {{{\tilde{\xi} }_{1}}} \right|^{- \frac{1}{2}}} + 2{\alpha_{1}}{\bar{\alpha }_{2}}{\left| {{{\tilde{\xi} }_{1}}} \right|^{\frac{1}{2}}} $$

(28)

$$ {{{\tilde{\xi} }_{2}} \le \left| {{{\tilde{\xi} }_{2}}} \right| \le {{\tilde{\xi} }_{2}}^{2} + \frac{1}{4}} $$

(29)

By help of inequality (29), Eq. 28 can be rewritten as,

$$ \begin{array}{l} \dot{V} \le \mathrm{M} + 2{\bar{\alpha }_{2}}{\beta^{- \frac{1}{2}}}{\left| {{{\tilde{\xi} }_{1}}} \right|^{- \frac{1}{2}}}({{\tilde{\xi} }_{2}}^{2} + \frac{1}{4}) + 2{\alpha_{1}}{\bar{\alpha }_{2}}{\left| {{{\tilde{\xi} }_{1}}} \right|^{\frac{1}{2}}}\\ {\kern8pt}= - {\left| {{{\tilde{\xi} }_{1}}} \right|^{- \frac{1}{2}}}(- {\left| {{{\tilde{\xi} }_{1}}} \right|^{\frac{1}{2}}}\mathrm{M} - 2{\bar{\alpha }_{2}}{\beta^{- \frac{1}{2}}}{{\tilde{\xi} }_{2}}^{2} - 2{\alpha_{1}}{\bar{\alpha }_{2}}\left| {{{\tilde{\xi} }_{1}}} \right| \\{\kern20pt}- \frac{1}{2}{\bar{\alpha }_{2}}{\beta^{- \frac{1}{2}}}) \end{array} $$

(30)

$$ {\zeta_{1}}{\zeta_{2}} \le \left| {{\zeta_{1}}} \right|\left| {{\zeta_{2}}} \right| \le \frac{1}{2}{\zeta_{1}}^{2} + \frac{1}{2}{\zeta_{2}}^{2} $$

(31)

According to inequality (31), Eq. 30 could be reconstructed as,

$$ \begin{array}{l} \dot{V} \le - {\left| {{{\tilde{\xi} }_{1}}} \right|^{- \frac{1}{2}}}({\vartheta_{1}}{\zeta_{1}}^{2} + {\vartheta_{2}}{\zeta_{2}}^{2} + {\vartheta_{3}}{\zeta_{1}}{\zeta_{2}} - \frac{1}{2}{\bar{\alpha }_{2}}{\beta^{- \frac{1}{2}}})\\ {\kern8pt}= - {\left| {{{\tilde{\xi} }_{1}}} \right|^{- \frac{1}{2}}}(\boldsymbol{{\zeta^{T}} {\varXi} \zeta} - \frac{1}{2}{\bar{\alpha }_{2}}{\beta^{- \frac{1}{2}}}) \end{array} $$

(32)

where

$$ \left\{ \begin{array}{l} \begin{array}{*{20}{c}} {{\vartheta_{1}} = {\alpha_{1}}^{3} - 2{\alpha_{1}}{\bar{\alpha }_{2}} - 2\delta {\alpha_{1}} + {\alpha_{1}} - \delta }\\ {}{{\vartheta_{2}} = - 2{\bar{\alpha }_{2}}{\beta^{- \frac{1}{2}}} + {\alpha_{1}} - \delta }\\ {}{{\vartheta_{3}} = 2{\bar{\alpha }_{2}} - 2{\alpha_{1}}^{2} - 1} \end{array}\\ {\kern4pt}\boldsymbol{{\varXi}} = \left[ {\begin{array}{*{20}{c}} {{\vartheta_{1}}}&{{\vartheta_{3}}/2}\\ {{\vartheta_{3}}/2}&{{\vartheta_{2}}} \end{array}} \right] \end{array} \right. $$

(33)

where several parameters should be adjusted to make $\boldsymbol {{\varXi }}\in {\mathbb {R}^{2 \times 2}}$ a positive definite matrix. In that case, we could obtain,

$$ \begin{array}{*{20}{l}} {\dot{V} \le - {\left| {{{\tilde{\xi} }_{1}}} \right|^{- \frac{1}{2}}}(\boldsymbol{{\zeta^{T}} {\varXi} \zeta} - \frac{1}{2}{\bar{\alpha }_{2}}{\beta^{- \frac{1}{2}}})}\\ {\kern11pt}{ \le - {\left| {{{\tilde{\xi} }_{1}}} \right|^{- \frac{1}{2}}}({\lambda_{\min }}(\boldsymbol{{\varXi}} ){{\left\| \boldsymbol{\zeta} \right\|}^{2}} - \frac{{({\mu_{1}} + {\mu_{2}}{K_{f}}){\alpha_{2}}}}{{2\sqrt \beta }})} \end{array} $$

(34)

where ${\lambda _{{\min \limits } }}(\boldsymbol {{\varXi }} )$ is the minimum eigenvalue of the positive definite matrix Ξ. Then we can get that $\dot {V} \le 0$ when ${\left \| \boldsymbol {\zeta } \right \|^{2}} \ge \frac {{({\mu _{1}} + {\mu _{2}}{K_{f}}){\alpha _{2}}}}{{2{\lambda _{{\min \limits } }}(\boldsymbol {{\varXi }} )\sqrt \beta }}$. Therefore, the upper bound of the estimated error $\left \| \boldsymbol {\zeta } \right \|$ would be constrained by the bounded ball ${\boldsymbol {B_{\zeta }} } = \left \{ {\left . \boldsymbol {\zeta } \right |{{\left \| \boldsymbol {\zeta } \right \|}^{2}} \le \frac {{({\mu _{1}} + {\mu _{2}}{K_{f}}){\alpha _{2}}}}{{2{\lambda _{{\min \limits } }}(\boldsymbol {{\varXi }} )\sqrt \beta }}} \right \}$.

As for the parameter selection, to make Ξ positive definite, according to Eq. 33, we can get,

$$ \left\{ {\begin{array}{*{20}{c}} {{\vartheta_{1}} > 0}\\ {{\vartheta_{1}}{\vartheta_{2}} - {\vartheta_{3}}^{2}/4 > 0} \end{array}} \right. $$

(35)

when conducting a real aerial interaction, once α₁, α₂, μ₁, μ₂ are initially selected, in addition to $\delta \in \left [ {\begin {array}{*{20}{c}} { - {\delta ^ + }}&{{\delta ^ + }} \end {array}} \right ]$, according to Eq. 35, we can obtain a range which is used to constrain the fuzzy logic output K_f to guarantee the convergence of the whole system.