Model Formulation Over Lie Groups and Numerical Methods to Simulate the Motion of Gyrostats and Quadrotors

Fiori, Simone

doi:10.3390/math7100935

Cited by 13 publications

(14 citation statements)

References 35 publications

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“…The present section summarises from the essay [19] a mathematical model of a gyrostat satellite and a model of a quadrotor drone , both formulated within the framework of Lie‐group system theory.…”

Section: Mathematical Model Of a Satellite And A Dronementioning

confidence: 99%

“…Assuming that the centre of a gyrostat is fixed in space, one may attach an inertial reference frame

F_{E}

to the centre and refer the position of each point of the platform to a platform‐fixed reference frame

F_{P}

(we shall refer the reader to [19, 20] for details about this mathematical model). The mathematical model of a gyrostat satellite includes a number of constant physical terms, which are summarised in the following matrices [19]:

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ \{\begin{matrix} left1em4pt \\ D : = diag (\sqrt{\frac{J_{y} J_{z}}{J_{x}}}, \sqrt{\frac{J_{x} J_{z}}{J_{y}}}, \sqrt{\frac{J_{x} J_{y}}{J_{z}}}), \\ {\overset{false^}{J}}_{normalg} : = \frac{1}{2} [\begin{matrix} 1em4pt \\ J_{y} - J_{x} + J_{z} & 0 & 0 \\ 0 & J_{x} - J_{y} + J_{z} & 0 \\ 0 & 0 & J_{x} + J_{y} - J_{z} \end{matrix}], \\ P : = \frac{1}{2} [\begin{matrix} 1em4pt \\ γ & 0 & 0 \\ 0 & γ & 0 \\ 0 & 0 & - γ \end{matrix}], \end{matrix} \end{matrix}

where

J_{x} > 0

J_{y} > 0

and

J_{z} > 0

denote principal moments of inertia and

γ ⩾ 0

is a friction coefficient.…”

Section: Mathematical Model Of a Satellite And A Dronementioning

confidence: 99%

“…The mathematical model of a gyrostat satellite includes a number of constant physical terms, which are summarised in the following matrices [19]:

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ \{\begin{matrix} left1em4pt \\ D : = diag (\sqrt{\frac{J_{y} J_{z}}{J_{x}}}, \sqrt{\frac{J_{x} J_{z}}{J_{y}}}, \sqrt{\frac{J_{x} J_{y}}{J_{z}}}), \\ {\overset{false^}{J}}_{normalg} : = \frac{1}{2} [\begin{matrix} 1em4pt \\ J_{y} - J_{x} + J_{z} & 0 & 0 \\ 0 & J_{x} - J_{y} + J_{z} & 0 \\ 0 & 0 & J_{x} + J_{y} - J_{z} \end{matrix}], \\ P : = \frac{1}{2} [\begin{matrix} 1em4pt \\ γ & 0 & 0 \\ 0 & γ & 0 \\ 0 & 0 & - γ \end{matrix}], \end{matrix} \end{matrix}

where

J_{x} > 0

J_{y} > 0

and

J_{z} > 0

denote principal moments of inertia and

γ ⩾ 0

is a friction coefficient. The equations of a gyrostat satellite may be cast in the language of Lie groups as [19]

({, \begin{array}{l} \begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em.5em \\ C : = & [κ_{1} (ω_{r} - ω_{x}) + κ_{2} (ω_{r}^{<...>} \end{matrix} \end{array})

…”

Section: Mathematical Model Of a Satellite And A Dronementioning

confidence: 99%

“…Second‐order dynamical systems on Lie groups are described by differential equations formulated on trivialised tangent bundles, which play the role of phase spaces for these systems. Rotational dynamical systems are prototypical in Lie‐group‐system theory [19, 20]. In addition, models that insist on the space of symmetric, positive‐definite matrices are able to model human motion across video frames [21].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Extension of a PID control theory to Lie groups applied to synchronising satellites and drones

Fiori

Cervigni

Ippoliti

et al. 2020

IET Control Theory & Appl

Self Cite

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Section: Mathematical Model Of a Satellite And A Dronementioning

confidence: 99%

“…Assuming that the centre of a gyrostat is fixed in space, one may attach an inertial reference frame

F_{E}

to the centre and refer the position of each point of the platform to a platform‐fixed reference frame

F_{P}

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ \{\begin{matrix} left1em4pt \\ D : = diag (\sqrt{\frac{J_{y} J_{z}}{J_{x}}}, \sqrt{\frac{J_{x} J_{z}}{J_{y}}}, \sqrt{\frac{J_{x} J_{y}}{J_{z}}}), \\ {\overset{false^}{J}}_{normalg} : = \frac{1}{2} [\begin{matrix} 1em4pt \\ J_{y} - J_{x} + J_{z} & 0 & 0 \\ 0 & J_{x} - J_{y} + J_{z} & 0 \\ 0 & 0 & J_{x} + J_{y} - J_{z} \end{matrix}], \\ P : = \frac{1}{2} [\begin{matrix} 1em4pt \\ γ & 0 & 0 \\ 0 & γ & 0 \\ 0 & 0 & - γ \end{matrix}], \end{matrix} \end{matrix}

where

J_{x} > 0

J_{y} > 0

and

J_{z} > 0

denote principal moments of inertia and

γ ⩾ 0

is a friction coefficient.…”

Section: Mathematical Model Of a Satellite And A Dronementioning

confidence: 99%

“…The mathematical model of a gyrostat satellite includes a number of constant physical terms, which are summarised in the following matrices [19]:

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ \{\begin{matrix} left1em4pt \\ D : = diag (\sqrt{\frac{J_{y} J_{z}}{J_{x}}}, \sqrt{\frac{J_{x} J_{z}}{J_{y}}}, \sqrt{\frac{J_{x} J_{y}}{J_{z}}}), \\ {\overset{false^}{J}}_{normalg} : = \frac{1}{2} [\begin{matrix} 1em4pt \\ J_{y} - J_{x} + J_{z} & 0 & 0 \\ 0 & J_{x} - J_{y} + J_{z} & 0 \\ 0 & 0 & J_{x} + J_{y} - J_{z} \end{matrix}], \\ P : = \frac{1}{2} [\begin{matrix} 1em4pt \\ γ & 0 & 0 \\ 0 & γ & 0 \\ 0 & 0 & - γ \end{matrix}], \end{matrix} \end{matrix}

where

J_{x} > 0

J_{y} > 0

and

J_{z} > 0

denote principal moments of inertia and

γ ⩾ 0

is a friction coefficient. The equations of a gyrostat satellite may be cast in the language of Lie groups as [19]

({, \begin{array}{l} \begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em.5em \\ C : = & [κ_{1} (ω_{r} - ω_{x}) + κ_{2} (ω_{r}^{<...>} \end{matrix} \end{array})

…”

Section: Mathematical Model Of a Satellite And A Dronementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Extension of a PID control theory to Lie groups applied to synchronising satellites and drones

Fiori

Cervigni

Ippoliti

et al. 2020

IET Control Theory & Appl

Self Cite

View full text Add to dashboard Cite

“…These applications motivated the development of mean and covariance propagation techniques to approximate the solution of diffusion equations in the special Euclidean group SE(N) and more generally, in unimodular groups [16,17]. Other numerical techniques to solve differential equations on Lie groups, through a generalisation of Euler and Runge-Kutta schemes have been developed in [18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group

Jayaraman

Campolo

Chirikjian

2020

Entropy

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The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker–Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

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Modeling, simulation and control of a spacecraft: Automated reorientation under directional constraints

Fiori,

Sabatini,

Rachiglia

et al. 2024

Acta Astronautica

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Model Formulation Over Lie Groups and Numerical Methods to Simulate the Motion of Gyrostats and Quadrotors

Cited by 13 publications

References 35 publications

Extension of a PID control theory to Lie groups applied to synchronising satellites and drones

Extension of a PID control theory to Lie groups applied to synchronising satellites and drones

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group

Modeling, simulation and control of a spacecraft: Automated reorientation under directional constraints

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