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Spacecraft Coupled Roto-translational Time-Optimal Control for Rendezvous Missions

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Abstract

This paper addresses the time-optimal roto-translational orbital rendezvous maneuver for an inertially asymmetric rigid spacecraft in an all-thrusters configuration. To begin with, the target and chaser relative rotational and translational dynamics are driven. Then, in the presence of torque and force constraints, the simultaneously time-optimal attitude and position control problem is numerically solved using the pseudo-spectral method. The costates are then computed to establish the first-order optimality of the obtained solutions, which is confirmed by satisfying Pontryagin's minimum principle. It is demonstrated via simulation that the obtained control forces and moments are basically “bang-bang,” which is the most natural and convenient form for on–off thrusters.

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Data Availability

Some or all data, models, or code generated or used during the study are available in a repository or online in accordance with funder data retention policies.

Abbreviations

\({\left[{{\varvec{v}}}_{B}^{L}\right]}^{L}\) :

Velocity vector of the chaser's body center of mass (\(B\)) with respect to the target's LVLH frame (\(L\)), expressed in the target's LVLH coordinates (\(L\))

\({\left[{{\varvec{s}}}_{BL}\right]}^{L}\) :

Position vector of the chaser's body center of mass (\(B\)) with respect to the target's LVLH frame origin (\(L\)), expressed in the target's LVLH coordinates (\(L\))

\({D}^{Y}{\varvec{x}}\) :

Time derivative of vector \({\varvec{x}}\) with respect to frame \(Y\)

\({\left[{\varvec{f}}\right]}^{L}\) :

Control force vector, expressed in the target's LVLH coordinate system (\(L\))

\({\left[{\varvec{f}}\right]}^{B}\) :

Control force vector, expressed in the chaser's body coordinate system (\(B\))

\({m}_{B}\) :

Chaser’s mass

\({\omega }_{n}\) :

Orbital mean angular rate of the target

\({{\varvec{I}}}_{B}\) :

Inertial matrix of the chaser's body

\({\left[{{\varvec{\omega}}}^{\alpha \beta }\right]}^{\gamma }\) :

Angular velocity vector of frame \(\alpha\) with respect to frame \(\beta\), expressed in coordinate system \(\gamma\)

\({\left[{\varvec{m}}\right]}^{B}\) :

Control torque vector, expressed in the chaser's body coordinate system (\(B\))

\({[{\varvec{\sigma}}]}^{BL}\) :

MRP vector representing the orientation of the target's LVLH frame w.r.t. the chaser's body frame

\({e}_{x}\) , \({e}_{y}\) , \({e}_{z}\) :

Components of Euler axis

\(\Phi\) :

Principal rotation angle

\({[{\varvec{T}}]}^{LB}\) :

Transformation matrix from the chaser's body coordinates to the target's LVLH coordinates

\([{{\varvec{\Sigma}}]}^{BL}\) :

Skew-symmetric form of \({[{\varvec{\sigma}}]}^{BL}\)

\(J\) :

Cost function

\(\phi\) :

Scalar boundary constraint in cost function

\(g\) :

Integrand function in cost function

\({\varvec{x}}\) :

State vector

\({\varvec{u}}\) :

Control input vector

\({\varvec{f}}\) :

Dynamic constraints

\(\boldsymbol{\varphi }\) :

Boundary conditions

\({\varvec{C}}\) :

Inequality path constraints

\(t\) :

Time

\(\tau\) :

Transformed time

\({P}_{N}\) :

\({N}^{th}\)-Degree Legendre polynomial

\({\varvec{X}}\) :

Approximated state trajectory

\({\varvec{U}}\) :

Approximated control trajectory

\({L}_{i}\) :

Lagrange polynomials \((i=0,\dots ,N)\)

\({L}_{i}^{*}\) :

Lagrange polynomials \((i=1,\dots ,N)\)

\(w\) :

Gauss weights

\({\varvec{D}}\) :

Differential approximation matrix

\({\mathcal{L}}^{\left(1\right)}\) , \({\mathcal{L}}^{\left({\varvec{P}}\right)}\) , \({\mathcal{L}}_{({\varvec{r}})}^{({\varvec{r}}+1)}\) :

Initial constraints for the first phase, terminal constraints for the last phase (\(P\)), and continuity constraints for the interior phases

\(H\) :

Hamiltonian

\({\varvec{\lambda}}\) :

Costate vector

\({\varvec{\mu}}\) :

Lagrange multiplier associated with the path constraint

\({\varvec{\nu}}\) :

Lagrange multiplier associated with the the boundary condition

\({\varvec{\Lambda}}\) :

Estimated costates in LG points

\(\widetilde{{\varvec{\Lambda}}}\) :

KKT multipliers

\({n}_{I}\) :

Number of phases

\({n}_{LG}\) :

Number of LG points in each phase

\({{\varvec{B}}}_{u}\), \({{\varvec{B}}}_{{\varvec{l}}}\) :

Upper and lower bound of states and controls values

\({{\varvec{S}}}_{0}\) :

Initial guess for states and controls values in LG points

\(x,y,z\) :

Components of a vector in arbitrary coordinate system

\(0\) :

Initial time

\(f\) :

Terminal time

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  1. Amirkabir University of Technology, Tehran, Iran

    Sayyed Mohammad Mousavi, Sayyed Majid Esmailifar & Mohammad Chiniforoushan

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  1. Sayyed Mohammad Mousavi
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Mousavi, S.M., Esmailifar, S.M. & Chiniforoushan, M. Spacecraft Coupled Roto-translational Time-Optimal Control for Rendezvous Missions. J Astronaut Sci 70, 23 (2023). https://doi.org/10.1007/s40295-023-00390-y

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