Abstract
This study analyzes the optimal transfer trajectory of a spacecraft propelled by a spin-stabilized electric solar wind sail (E-sail) with a single conducting tether and a spin axis with a fixed direction in an inertial (heliocentric) reference frame. The approach proposed in this study is useful for rapidly analyzing the optimal transfer trajectories of the current generation of small spacecraft designed to obtain in-situ evidence of the E-sail propulsion concept. In this context, starting with the recently proposed thrust model for a single-tether E-sail, this study discusses the optimal control law and performance in a typical two-dimensional interplanetary transfer by considering the (binary) state of the onboard electron emitter as the single control parameter. The resulting spacecraft heliocentric trajectory is a succession of Keplerian arcs alternated with propelled arcs, that is, the phases in which the electron emitter is switched on. In particular, numerical simulations demonstrated that a single-tether E-sail with an inertially fixed spin axis can perform a classical mission scenario as a circle-to-circle two-dimensional transfer by suitably varying a single control parameter.
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Abbreviations
- a :
-
propulsive acceleration vector (mm/s2)
- a c :
-
characteristic acceleration (mm/s2)
- C :
-
spacecraft center of mass
- \(\hat{\boldsymbol{d}}\) :
-
auxiliary unit vector
- e :
-
error vector
- E :
-
endpoint Lagrangian
- f :
-
dynamics vector
- h :
-
orbital angular momentum magnitude (km2/s)
- H :
-
Hamiltonian function
- \(\cal{H}\) :
-
maximized Hamiltonian
- H c :
-
part of H that depends on the control
- \(\{\hat{\boldsymbol{i}},\hat{\boldsymbol{j}},\hat{\boldsymbol{k}}\}\) :
-
unit vectors of \(\cal{T}\)
- J :
-
cost function
- O :
-
Sun’s center of mass
- r :
-
orbital radius (au)
- \(\hat{\boldsymbol{r}}\) :
-
radial unit vector
- t :
-
time (years)
- \(\hat{\boldsymbol{t}}\) :
-
transverse unit vector
- \(\cal{T}\) :
-
body reference frame
- \(\cal{T}_{\odot}\) :
-
heliocentric reference frame
- u :
-
radial component of spacecraft velocity (km/s)
- x :
-
spacecraft state vector
- {x, y, z}:
-
axes of \(\cal{T}\)
- α d :
-
angle between \(\hat{\boldsymbol{d}}\) and \(\hat{\boldsymbol{r}}\) (rad)
- α λ :
-
primer vector angle (rad)
- α 0 :
-
auxiliary angle (rad)
- β :
-
dimensionless form of ac
- θ :
-
polar angle (rad)
- λ :
-
costate vector
- λ p :
-
primer vector with λp = ∥λp∥
- {λ r, λ θ, λ u, λ h}:
-
costates
- μ ⊙ :
-
Sun’s gravitational parameter (km3/s2)
- τ :
-
dimensionless control parameter
- ω :
-
spacecraft spin rate (rad/s)
- f:
-
final
- 0:
-
initial, at time t0 ≜ 0
- ⊕:
-
at 1 au from the Sun
- ·:
-
time derivative
- −:
-
mean value
- ∧:
-
unit vector
- ⋆:
-
optimal
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The authors have no competing interests to declare that are relevant to the content of this article. The author Alessandro A. Quarta is the Associate Editor of this journal.
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Alessandro A. Quarta received his Ph.D. degree in aerospace engineering from the University of Pisa in 2005 and he currently is a professor of flight mechanics at the Department of Civil and Industrial Engineering of the University of Pisa. His main research areas include spaceflight simulation, spacecraft mission analysis and design, low-thrust trajectory optimization, and solar sail and E-sail dynamics and control. E-mail: a.quarta@ing.unipi.it
Marco Bassetto received his Ph.D. degree in industrial engineering at the Department of Civil and Industrial Engineering of the University of Pisa. After receiving his Ph.D. degree, at the same department he was first the holder of a scholarship entitled “Dynamic analysis and control of an E-sail” and then held the position of research assistant in spaceflight mechanics. From January 2023 he holds the position of assistant professor of space systems at the Department of Civil and Industrial Engineering of the University of Pisa. His research activity focuses on trajectory design and attitude control of spacecraft propelled with low-thrust propulsion systems, such as solar sails and electric solar wind sails. E-mail: marco.bassetto@unipi.it
Giovanni Mengali received his Doctor Engineer degree in aeronautical engineering in 1989 from the University of Pisa. Since 1990, he has been with the Department of Aerospace Engineering (now Department of Civil and Industrial Engineering) of the University of Pisa, first as a Ph.D. student, then as an assistant and an associate professor. Currently, he is a professor of spaceflight mechanics. His main research areas include spacecraft mission analysis, trajectory optimization, solar sails, electric sails, and aircraft flight dynamics and control. E-mail: g.mengali@ing.unipi.it
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Quarta, A.A., Bassetto, M. & Mengali, G. Optimal orbit transfer of single-tether E-sail with inertially fixed spin axis. Astrodyn (2024). https://doi.org/10.1007/s42064-023-0194-0
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DOI: https://doi.org/10.1007/s42064-023-0194-0