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Modeling and control for aerial manipulation based on center of inertia on SE(3)

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Abstract

The multipropeller aerial manipulation robot (MAMR) is composed of a multirotor aircraft and a manipulator. It has strong nonlinear coupling and underactuated characteristics, and its dynamics and control problems pose great challenges for researchers. In this paper, the dynamic model of MAMR based on the center of inertia (COI) is established, and the COI is expressed by exponential coordinates on the special Euclidean group SE(3). This kind of model based on COI considers the rotorcraft and the manipulator as a single system. The model can describe the motion of the MAMR by considering the change in the MAMR’s COI. The multi-rigid body system of a separated rotorcraft and manipulator is regarded as a single rigid system, which solves the strong coupling problem and realizes the pose integrated motion description. Then, based on the COI dynamics, a cascade trajectory linearization controller is designed to stabilize the MAMR. The cascade control architecture is very suitable for real-world experiments because of its convenience of parameter adjustment. The stability of the controller is proved by the Lyapunov method. Furthermore, the stability of the system is verified by simulation. Finally, a prototype is built, and outdoor experiments are carried out to prove the feasibility and effectiveness of our proposed method.

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Acknowledgements

The authors would like to thank professor Jianying Yang for his valuable discussion on the revised version of this paper. This work was supported by the National Natural Science Foundation of China (NSFC) (Nos. 91748201, T2121003 and 62073002).

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Authors and Affiliations

  1. Robotics Institute, School of Mechanical Engineering and Automation, Beihang University, Beijing, China

    Pin Guo, Kun Xu, Huichao Deng, Haoyuan Liu & Xilun Ding

  2. Research and Development Department, China Academy of Launch Vehicle Technology, Beijing, China

    Pin Guo

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  1. Pin Guo
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  2. Kun Xu
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  3. Huichao Deng
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Correspondence to Huichao Deng.

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Appendix

Appendix

1.1 A. Derivation of the COI expression

The generalized kinetic energy of the system can be expressed as

$$\begin{aligned} E = \frac{1}{2}{\left( {{{\varvec{\zeta }}}_C^B} \right) ^\mathrm{{T}}}{{\varvec{M}}}_C^B{{\varvec{\zeta }}}_C^B, \end{aligned}$$
(A.1)

where \({{\varvec{\zeta }}}_C^B = \mathrm{{A}}{\mathrm{{d}}_{{{{\varvec{g}}}_\mathrm{BC}}}}{{\varvec{\zeta }}}_C^C\) has been defined in Formula (1), and thus, we obtain

$$\begin{aligned}&E = \frac{1}{2}{\left( {\mathrm{{A}}{\mathrm{{d}}_{{{{\varvec{g}}}_\mathrm{BC}}}}{{\varvec{\zeta }}}_C^C} \right) ^\mathrm{{T}}}{{\varvec{M}}}_C^B\mathrm{{A}}{\mathrm{{d}}_{{{{\varvec{g}}}_\mathrm{BC}}}}{{\varvec{\zeta }}}_C^C \nonumber \\&\quad \mathrm{{ = }}\frac{1}{2}{\left( {{{\varvec{\zeta }}}_C^C} \right) ^\mathrm{{T}}}\mathrm{{Ad}}_{{{{\varvec{g}}}_\mathrm{BC}}}^\mathrm{{T}}{{\varvec{M}}}_C^B\mathrm{{A}}{\mathrm{{d}}_{{{{\varvec{g}}}_\mathrm{BC}}}}{{\varvec{\zeta }}}_C^C \nonumber \\&\quad \mathrm{{ = }}\frac{1}{2}{\left( {{{\varvec{\zeta }}}_C^C} \right) ^\mathrm{{T}}}\left( {\mathrm{{Ad}}_{{{{\varvec{g}}}_\mathrm{BC}}}^\mathrm{{T}}{{\varvec{M}}}_C^B\mathrm{{A}}{\mathrm{{d}}_{{{{\varvec{g}}}_\mathrm{BC}}}}} \right) {{\varvec{\zeta }}}_C^C \end{aligned}$$
(A.2)

From the physical meaning of the kinetic energy, we also obtain

$$\begin{aligned} E = \frac{1}{2}{\left( {{{\varvec{\zeta }}}_C^C} \right) ^\mathrm{{T}}}{{\varvec{M}}}_C^C{{\varvec{\zeta }}}_C^C. \end{aligned}$$
(A.3)

Comparing Eqs. (A.2) and (A.3), we obtain Formula (4) as

$$\begin{aligned} {{\varvec{M}}}_C^C = \mathrm{{Ad}}_{{{{\varvec{g}}}_\mathrm{BC}}}^T{{\varvec{M}}}_C^B\mathrm{{A}}{\mathrm{{d}}_{{{{\varvec{g}}}_\mathrm{BC}}}} \end{aligned}$$
(A.4)

1.2 B. Derivation of the generalized velocity error

According to the tracking error defined on SE(3) and its exponential mapping, we further define

$$\begin{aligned}&{{{{\varvec{\hat{\zeta }}}}}_{C,\mathrm{err}}} = {{\varvec{g}}}_{C,\mathrm{err}}^{ - 1}{{{\dot{{\varvec{g}}}}}_{C,\mathrm{err}}}\nonumber \\&\qquad \quad \mathrm{{ = (}}{{\varvec{g}}}_C^{ - 1}{{{\varvec{g}}}_{C,\mathrm{com}}}{\mathrm{{)}}^{ - 1}}({\dot{{\varvec{g}}}}_C^{ - 1}{{{\varvec{g}}}_{C,\mathrm{com}}} + {{\varvec{g}}}_C^{ - 1}{{{\dot{{\varvec{g}}}}}_{C,\mathrm{com}}}) \nonumber \\&\qquad \quad \mathrm{{ = }}{{\varvec{g}}}_{C,\mathrm{com}}^{ - 1}{{{\varvec{g}}}_C}({\dot{{\varvec{g}}}}_C^{ - 1}{{{\varvec{g}}}_{C,\mathrm{com}}} + {{\varvec{g}}}_C^{ - 1}{{{\dot{{\varvec{g}}}}}_{C,\mathrm{com}}}) \nonumber \\&\qquad \quad \mathrm{{ = }}{{\varvec{g}}}_{C,\mathrm{com}}^{ - 1}{{{\varvec{g}}}_C}{\dot{{\varvec{g}}}}_C^{ - 1}{{{\varvec{g}}}_{C,\mathrm{com}}} + {{\varvec{g}}}_{C.\mathrm{com}}^{ - 1}{{{\dot{{\varvec{g}}}}}_{C,\mathrm{com}}} \nonumber \\ \end{aligned}$$
(B.1)

According to the definition, the second term can be written as \({{{\varvec{\hat{\zeta }}}}_{C,\mathrm{com}}}\), and the inverse derivative of the matrix in the first term can be derived as follows:

$$\begin{aligned}&{{{\varvec{g}}}_C}{{\varvec{g}}}_C^{ - 1} = {{\varvec{I}}} \nonumber \\&\frac{\mathrm{d}}{{\mathrm{d}t}}({{{\varvec{g}}}_C}{{\varvec{g}}}_C^{ - 1}) = 0\nonumber \\&{{{\dot{{\varvec{g}}}}}_C}{{\varvec{g}}}_C^{ - 1} + {{{\varvec{g}}}_C}{\dot{{\varvec{g}}}}_C^{ - 1} = 0\nonumber \\&{\dot{{\varvec{g}}}}_C^{ - 1} = - {{\varvec{g}}}_C^{ - 1}{{{\dot{{\varvec{g}}}}}_C}{{\varvec{g}}}_C^{ - 1} \end{aligned}$$
(B.2)

Substituting (B.1) into (B.2), we obtain

$$\begin{aligned}&{{{{\varvec{\hat{\zeta }}}}}_{C,\mathrm{err}}} = {{{{\varvec{\hat{\zeta }}}}}_{C,\mathrm{com}}} - {{\varvec{g}}}_{C.\mathrm{com}}^{ - 1}{{{\varvec{g}}}_C}{{\varvec{g}}}_C^{ - 1}{{{\dot{{\varvec{g}}}}}_C}{{\varvec{g}}}_C^{ - 1}{{{\varvec{g}}}_{C,\mathrm{com}}} \nonumber \\&\qquad \quad \mathrm{{ =\, }}{{{{\varvec{\hat{\zeta }}}}}_{C,\mathrm{com}}} - ({{\varvec{g}}}_{C.\mathrm{com}}^{ - 1}{{{\varvec{g}}}_C})({{\varvec{g}}}_C^{ - 1}{{{\dot{{\varvec{g}}}}}_C})({{\varvec{g}}}_C^{ - 1}{{{\varvec{g}}}_{C,\mathrm{com}}})\nonumber \\&\qquad \quad \mathrm{{ =\, }}{{{{\varvec{\hat{\zeta }}}}}_{C,\mathrm{com}}} - {{\varvec{g}}}_{C.err}^{ - 1}{{{{\varvec{\hat{\zeta }}}}}_C}{{{\varvec{g}}}_{C,\mathrm{err}}} \end{aligned}$$
(B.3)

1.3 C. Control gain derived from the PD spectrum theory

We define a closed-loop matrix \({{{\varvec{C}}}_1} \in {{\mathbb {R}}^{12 \times 12}}\) as

$$\begin{aligned} {{{\varvec{C}}}_1} = \left( {\begin{array}{*{20}{c}} {{{{\varvec{0}}}_{6 \times 6}}}&{}{{{{\varvec{E}}}_6}}\\ {{{{\varvec{C}}}_{11}}}&{}{{{{\varvec{C}}}_{12}}} \end{array}} \right) , \end{aligned}$$
(C.1)

where \({{{\varvec{C}}}_{11}} = diag\big ( - {\lambda _{11}}, - {\lambda _{21}}, - {\lambda _{31}}, - {\lambda _{41}}, - {\lambda _{51}}, - {\lambda _{61}}\big )\), and \({{{\varvec{C}}}_{12}} = diag\big ( - {\lambda _{12}}, - {\lambda _{22}}, - {\lambda _{32}}, - {\lambda _{42}}, - {\lambda _{52}}, - {\lambda _{62}}\big )\). The coefficient \({\lambda _{ij}}\) is a function of the bandwidth frequency, \({\Omega _i}\), and damping ratio, \({\varsigma _i}\). The expression is \({\lambda _{i1}} = \Omega _i^2\), \({\lambda _{i2}} = 2{\varsigma _i}{\Omega _i} - \frac{{{{\dot{\Omega }}_i}}}{{{\Omega _i}}}\), and \(i = 1,2,3\) represents the three axes.

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Guo, P., Xu, K., Deng, H. et al. Modeling and control for aerial manipulation based on center of inertia on SE(3). Nonlinear Dyn 111, 369–389 (2023). https://doi.org/10.1007/s11071-022-07858-w

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