Suboptimal obstacles avoidance control of spacecraft rendezvous

Abstract

Spacecraft on-orbital services and docking require their autonomous rendezvous control system to have obstacle avoidance capability. Motivated by this, a suboptimal velocity artificial potential function-based control scheme is presented. An ellipsoid model is applied to describe the outer envelopes of the service spacecraft and the obstacles via an eigenvalue algorithm. This has better description precision than the traditional methods. The potential sigmoid function is used to generate repulsive force to avoid obstacles collision. A velocity artificial potential function-based controller is finally developed to ensure that the relative speed of the service spacecraft is reduced to zero before reaching the outer envelops of obstacles. The shaping parameters of the attractive potential function are adaptively optimized. Numerical simulations are performed to demonstrate that the approach can achieve a safe and autonomous rendezvous with fuel cost saved.

Keywords

Autonomous rendezvous artificial potential function adaptive obstacle avoidance sigmoid potential function

Introduction

For on-orbital servicing spacecrafts, autonomous rendezvous to target has attracted extensive research interests (Hu et al., 2018, 2019; Gao et al., 2009; Mancini et al., 2020, Yang, 2019). Several projects have been started to investigating safe and autonomous rendezvous problem such as the NASA’s DART program (Timothy, 2003) and the Phoenix mission (Gunn et al., 2015), and so forth. Note that the space environment is getting harsh for future space rendezvous and operations. The problem of safe autonomous rendezvous control with the capability of avoiding obstacles should be addressed (Colin, 1993; Sun, 2019; Sun et al., 2017, 2019; Zappulla, 2018).

The existing autonomous rendezvous controllers can be categorized into two types. The first type mainly focuses on designing controller via the optimal theory (Xing and Pan, 2012). Its core is to change the problem of obstacle avoidance into an optimization issue with constrained conditions (Epenoy, 2011; Martinson, 2013; Tillerson, 2002; Xing and Pan, 2011), and Clark (2015)). Then, the optimal theory is applied to solve this constrained optimization problem. In Breger and Jonathan (2008) and Mueller and Larsson (2008), fuel cost is considered as an important performance index. The integer linear programming algorithm is employed to plan an optimal trajectory for rendezvous on-line. Although some optimization theory is available for autonomous rendezvous control, this problem is still open, because the optimization theory-based solution needs expensive calculation.

The other category of autonomous rendezvous is the analytical control. It is implemented with low complexity. The corresponding control has analytic expression (Clark, 2015). For this category, the artificial potential function-based control (APF) is widely seen (Khatib, 1986; Lopez and McInnes, 1995). It was first applied in Colin (1993) to solve the spacecraft rendezvous control problem. The APF was then extended to achieve satellite formation flying control (Dong et al., 2006) and on-orbital assembly (Badawy and Colin, 2006). In JohnOlcayto and Colin (2007), an APF controller was proposed for docking of the international space station in the V/R-bar directions. In Zhang et al. (2010), a controller for safe corridor rendezvous was presented by integrating APF and the ellipse cissoids. In Su (2012), an APF control design was discussed for electromagnetic satellite formation in the Earth gravity field.

It is worth mentioning that APF is always used to solve the autonomous rendezvous problem by combining with advanced control theories to satisfy different requirements imposed by missions. An adaptive APF control law was proposed to achieve avoidance of static obstacles (Munoz et al., 2010) and to improve precision (Gao and Luo, 2012). In Zhang (2010), a hybrid control method was developed by using APF and the fuzzy control theory for obstacles avoidance. In Lim et al. (2005), applying APF and the sliding mode control to achieve satellite formation mission was discussed. Employing linear quadratic regulator instead of the attractive potential in APF was reported in Mccamish et al. (2010). The convergence rate of the control algorithm for obstacles avoidance was improved. In Palacios et al. (2015), an LQR/APF control law was developed and verified for satellite formation on elliptical orbit.

Although the APF-based approaches are efficient to solve the autonomous rendezvous problem, they are not optimal with respect to fuel consumption and may contain local minima (Zappulla and Richard, 2016). Moreover, the shapes of space objects are required to be spherical. The outer envelope of these objects is considered with less accuracy. If the three-dimensional dimensions of space objects are different, then such simplified method results in redundancy of space description. Motivating by solving these two drawbacks, a new velocity artificial potential function (VAPF) control approach is presented in this paper. It can ensure that the spacecraft’s velocity is reduced to zero before reaching the outer envelope of obstacles. Then, an adaptive version of the VAPF controller is presented by using the optimal control theory. The parameters of the attractive APF are ensured to be optimal by applying an adaptive algorithm to add some degree of optimality. The main contributions of this study are listed as:

In comparison with the existing avoidance controllers applying the spherical shape to describe the outer envelope of spacecrafts and obstacles, the proposed approach applies the ellipsoid to describe the space and the obstacle. An innovative eigenvalue algorithm is derived to calculate the Euler distance between the ellipsoids for the judgment of collision. The description of the space and the obstacles are more accurate. The problem of the redundancy of the obstacle’s description and the decrease of the control accuracy in the conventional methods can be solved.

To the best knowledge of the authors, this paper is the first result of introducing the potential sigmoid function (PSF) as the repulsive force to avoid the obstacles. It has advantages to represent the potential field of arbitrarily shaped obstacles.

The proposed adaptive VAPF control approach can achieve the autonomous spacecraft rendezvous and obstacles avoidance with some optimized properties.

The rest of this paper is organized as follows. The relative dynamics for spacecraft rendezvous is established in Section 2. In Section 3, a new approach to describe the shortest Euler distance of an ellipsoid is presented. In Section 4, a new attractive potential function (APF) is designed by using the potential Sigmoid function. A velocity APF-based controller and an adaptive version are synthesized in Section 5 to achieve the main objectives. In Section 5, numerical simulations are performed to verify the effectiveness and the obstacles avoidance capability of the controller. The paper is ended with some conclusions given in Section 7.

Mathematical model and problem description

The relative dynamics of spacecraft rendezvous

Spacecraft rendezvous usually involves a leader spacecraft in an elliptical planar orbit and a service spacecraft in a desired relative orbit. The service spacecraft is to rendezvous with the leader spacecraft. Let $r_{c} \in ℜ$ be the radial distance between the center of an inertial frame and the center of the mass of the leader spacecraft, and its true anomaly be $θ \in ℜ$ . The leader spacecraft’s orbital motion can be mathematically described by

{\overset{\cdot\cdot}{r}}_{c} - r_{c} {\overset{\cdot}{θ}}^{2} + \frac{μ}{r_{c}^{2}} = 0

(1)

r_{c} \overset{\cdot\cdot}{θ} + 2 {\overset{\cdot}{r}}_{c} \overset{\cdot}{θ} = 0

(2)

where $μ \in ℜ$ is the gravitational constant of the Earth. $\overset{\cdot}{θ}$ and $\overset{\cdot\cdot}{θ}$ are the leader spacecraft’s angular rate and its acceleration of the true anomaly, respectively.

Let $F_{L}$ denote the local-vertical-local-horizontal frame fixed at the center of the leader spacecraft, and $r = [x y z]^{T} \in ℜ^{3}$ represent the relative position from the leader to the service spacecraft and expressed in $F_{L}$ . Then, the relative dynamics for governing rendezvous is given by Cao et al. (2013) and Cao et al. (2014b)

\overset{\cdot}{X} = A (X) X + B (X) u

(3)

\begin{matrix} with X = [r \overset{\cdot}{r}]^{T}, A (X) = [\begin{matrix} 0_{3 \times 3} & I_{3 \times 3} \\ A_{21} & A_{22} \end{matrix}], \\ B (X) = [\begin{matrix} 0_{3 \times 3} \\ I_{3 \times 3} \end{matrix}], A_{22} = [\begin{matrix} 0 & 2 \overset{\cdot}{θ} & 0 \\ - 2 \overset{\cdot}{θ} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}], and \end{matrix}

A_{21} = [\begin{matrix} {\overset{\cdot}{θ}}^{2} - \frac{μ}{r_{c}^{3}} + \frac{\frac{μ}{r_{t}^{2}} - \frac{μ}{r_{c}^{3}} r_{t}^{2}}{r^{T} r} x & \overset{\cdot\cdot}{θ} + \frac{\frac{μ}{r_{t}^{2}} - \frac{μ}{r_{c}^{3}} r_{t}^{2}}{r^{T} r} y & \frac{\frac{μ}{r_{t}^{2}} - \frac{μ}{r_{c}^{3}} r_{t}^{2}}{r^{T} r} z \\ - \overset{\cdot\cdot}{θ} & {\overset{\cdot}{θ}}^{2} - \frac{μ}{r_{c}^{3}} & 0 \\ 0 & 0 & - \frac{μ}{r_{c}^{3}} \end{matrix}]

(4)

where $r_{t} = \sqrt{{(r_{c} + x)}^{2} + y^{2} + z^{2}}$ . $u = [u_{x} u_{y} u_{z}]^{T} \in ℜ^{3}$ is thrust control input for rendezvous.

Problem formulation

Let $r_{f} \in ℜ^{3}$ be the desired position, then the control objective of the paper is to design a control law for be $u$ to ensure that the relative position $r$ between the leader and the service spacecraft can asymptotically converge to $r_{f}$ , i.e., $lim_{t \to \infty} r = r_{f}$ , while the obstacles are avoided.

New method to describe the shortest Euler distance

If the relative distance can be estimated accurately, then less orbital maneuvering of the spacecraft is needed. This can improve control and maneuvering efficiency. Therefore, the mathematical model of the shortest Euler distance should be established with the location and the geometry size of the leader/service spacecrafts and the obstacles considered. Motivated by this, an ellipsoid modeling method but not the spherical modeling is presented. For this method, the ellipsoid is adopted to describe the outer envelope of obstacles and the spacecraft. The lengths of the ellipsoid’s three axes are equal to the largest size of the obstacles in the corresponding direction from the center of mass, respectively. Let $E_{o}$ and $E_{c}$ be the outer envelope ellipsoid of the obstacle and the spacecraft, respectively. Then, these two ellipsoids are such that

(r - r_{i})^{T} M_{i} (r - r_{i}) = 1

(5)

where $i = s$ , $o$ , denotes the service spacecraft and the obstacle. $r_{i} = [x_{i} y_{i} z_{i}]^{T} \in ℜ^{3}$ is the center of mass of the ellipsoid. The matrix $M_{i}$ determines the dimensional size of the ellipsoid. If three axes of the ellipsoid are the same as the frame $F_{L}$ , then $M_{i}$ can be denoted as $M_{i} = diag ([a_{i} b_{i} c_{i}]^{T})$ , where $a_{i}$ , $b_{i}$ , and $c_{i}$ are the size of semi-major on the main axis of the ellipsoid.

It is well-known that there is no direct algorithm to compute the distance between two ellipsoids. In this study, an eigenvalue algorithm (Walter et al., 1989) is designed to calculate the shortest Euler distance between two ellipsoids. The algorithm can provide an analytical solution. Moreover, it is characterized by fast calculation and high computational precision in comparison with the iterative method.

Suppose that the distance between the point $r_{1}^{*}$ and the point $r_{2}^{*}$ is the shortest Euler distance of two ellipsoids in Figure 1. The coordinate position of $r_{1}^{*}$ can be obtained by solving the eigenvalue. The detailed procedures to achieve this are described as follows.

Figure 1.

The distance between two ellipsoids.

Firstly, the ellipsoid of the service spacecraft should be converted into a unit sphere by transforming coordinate. Then, the distance calculation in the transformed coordinate changes into obtaining the minimum value of $(\bar{r} - {\bar{r}}_{c})^{T} {\bar{M}}_{c} (\bar{r} - {\bar{r}}_{c})$ (Ren et al., 2007), where ${\bar{M}}_{c} = M_{o}^{- 0.5} M_{c} M_{o}^{- 0.5}$ , ${\bar{r}}_{c} = M_{o}^{0.5} (r - r_{o})$ , and $\bar{r} = M_{o}^{0.5} (r - r_{o})$ .

According to the Lagrange multiplier rule, the coordinate position of the point $r_{1}^{*}$ satisfies

r_{1}^{*} = λ_{1} {\bar{M}}_{c} (r_{1}^{*} - {\bar{r}}_{c})

(6)

where $λ_{1} \in ℜ$ is the Lagrange multiplier.

Define a matrix as

D_{c} = [\begin{matrix} {\tilde{M}}_{c} & - I \\ - {\tilde{r}}_{c} {\tilde{r}}_{c}^{T} & {\tilde{M}}_{c} \end{matrix}]

(7)

where ${\tilde{M}}_{c} = {\bar{M}}_{c}^{- 1}$ and ${\tilde{r}}_{c} = M_{o}^{- 0.5} {\bar{r}}_{c}$ . Let $λ_{1}$ be the minimum eigenvalue of $D_{c}$ , then it follows from (6) that

r_{1}^{*} = r_{o} + λ_{1} M_{o}^{- 0.5} (λ_{1} I - {\tilde{M}}_{c})^{- 1} {\bar{r}}_{c}

(8)

Using the same solution to obtaining the position of the point $r_{1}^{*}$ , the position of the point $r_{2}^{*}$ on the outer envelope of the service spacecraft ellipsoid can also be calculated as

r_{2}^{*} = r_{1}^{*} + λ_{2} (λ_{2} I - {\tilde{M}}_{o})^{- 1} {\bar{r}}_{o}

(9)

where ${\bar{r}}_{o} = r_{o} - r_{1}^{*}$ . $λ_{2} \in ℜ$ is the minimum eigenvalue of $D_{o}$ .

D_{o} = [\begin{matrix} {\tilde{M}}_{c} & - I \\ - {\tilde{r}}_{c} {\tilde{r}}_{c}^{T} & {\tilde{M}}_{c} \end{matrix}]

(10)

where ${\bar{r}}_{o} = r_{o} - r_{1}^{*}$ , ${\tilde{r}}_{o} = M_{o}^{- 0.5} {\bar{r}}_{o}$ , and ${\bar{r}}_{o} = r_{o} - r_{1}^{*}$ .

Based on (8)-(9), the shortest Euler distance $d \in ℜ$ between that two ellipsoids can be calculated as

d = | | r_{1}^{*} - r_{2}^{*} | |

(11)

Development of new VAPF

Because the APF method is a powerful solution to the obstacleâ€™s avoidance problem, the main problem stated in this work will be solved by using APF. More specifically, a novel APF is presented as

ϕ = k_{a} ϕ_{a} + k_{r} \sum_{i = 1}^{m} ϕ_{r_{i}}

(12)

where $k_{a} > 0$ and $k_{r} > 0$ are two positive gains. $m \in ℜ$ is the number of obstacles. $ϕ \in ℜ$ is the total APF. $ϕ_{a} \in ℜ$ is the attractive potential function. $ϕ_{r_{i}} \in ℜ$ is the repulsive potential function. These two potentials are designed as follows.

Attractive potential function design

The attractive potential function $ϕ_{a}$ in (12) is chosen as a quadratic function

ϕ_{a} = 0.5 (r - r_{f})^{T} P (r - r_{f})

(13)

where $P \in ℜ^{3 \times 3}$ is a positive semi-definite matrix and the shaping parameters determined by the designer.

Repulsive potential function design

Invoking the ellipsoid described in (5), the relative position of the center of mass between the service spacecraft and the obstacle can be formulated as

S_{i} (r, r_{oi}) = 1 - (r - r_{oi})^{T} M_{oi} (r - r_{oi}), i = 1, 2, . . ., m

(14)

where $r_{oi} \in ℜ^{3}$ and $M_{oi} \in ℜ^{3 \times 3}$ is the center of mass and the dimensional size of the ellipsoid for the ith obstacle. $S_{i} (r, r_{oi}) > 0$ is that the center of mass of the service spacecraft falls within the ellipsoid envelope of the $i$ th obstacle. It represents that the center of mass of the service spacecraft is on the boundary of the ellipsoid envelope the $i$ th obstacle.

Unlike the traditional repulsive potential functions, based on the shortest Euler distance, a novel repulsive potential function is designed by using the generalized potential sigmoid function (Munoz, 2011; Ren et al., 2004), which is specified by

ϕ_{ri} = {\begin{matrix} 0, if d > d_{s} and d_{ct} < d_{ot} \\ \frac{1}{1 + e^{- γ (1 - {(r - r_{oi})}^{T} M_{oi} (r - r_{oi}))}}, otherwise \end{matrix}

(15)

where $γ \in ℜ$ is applied to adjust the $i$ th obstacle affecting the area. $d_{ct}$ is the distance between the center of mass of the service spacecraft and the desired position. $d_{ot}$ is the distance between the center of mass of the obstacle and the desired position. $d_{s}$ denotes the minimum stop distance as

d_{s} = \frac{v_{co}^{2}}{2 a_{max}}

(16)

with the maximum acceleration $a_{max}$ and the relative parallel velocity $v_{co}$

v_{co} = ({\overset{\cdot}{r}}_{c} - {\overset{\cdot}{r}}_{o})^{T} \frac{r_{1}^{*} - r_{1}^{*}}{| | r_{1}^{*} - r_{1}^{*} | |}

(17)

To the authors’ best knowledge, this work is the first result of using this generalized potential sigmoid function to the spacecraft rendezvous problem. Due to the application of the generalized potential sigmoid function, the proposed APF (12) including the potentials (13) and (15) has the following advantages in comparison with the traditional APFs:

The developed APF (12) can accurately describe the two-dimensional and the three-dimensional structure of the obstacles. Moreover, the calculation burden is small.

The APF (12) is the first and the higher-order continuous differentiable. It ensures the generated force to be continuous.

The amplitude of the potential field (15) can be modified by adjusting the correlation parameters. It lets the proposed APF be suitable for the whole system.

APF-based control

Using the synthesized APF (12), a VAPF-based controller is first presented in this section. Applying this controller, the control objective of autonomous spacecraft rendezvous is achieved with obstacle avoidance guaranteed. An adaptive VAPF is then presented by introducing the optimal theory to adaptively optimize the shaping parameters of the attractive potential function. Some degree of optimality is ensured.

VAPF-based control

The relative dynamics (1)–(4) is typically a nonlinear system. The Lyapunov theory can be applied to synthesize a controller for avoiding obstacles. To ensure that the service spacecraft does not collide with the obstacles during the rendezvous maneuvering, the velocity of service spacecraft must be reduced to zero before reaching the outer envelope of the obstacles.

Theorem 1: For the relative dynamics (1)–(4), based on the APF (12), let its controller be designed as

u = - (H (ϕ) \overset{\cdot}{r} + K (\nabla ϕ + \overset{\cdot}{r}) + f (X))

(18)

where $f (X) = [A_{21} A_{22}] X$ . $K$ is a positive-definite and constant matrix. $\nabla ϕ$ is the gradient of potential function $ϕ$ with respect to $r$ , that is

\nabla ϕ = k_{a} P (r - r_{f}) - 2 k_{r} \sum_{i = 1}^{m} \frac{γ e^{- γ S_{i} (r, r_{oi})}}{{(1 + e^{- γ S_{i} (r, r_{oi})})}^{2}} M_{oi} (r - r_{oi})

(19)

Moreover, $H (ϕ)$ is the Hessian matrix of the potential $ϕ$

H (ϕ) = k_{a} P + k_{r} \sum_{i = 1}^{m} {\begin{matrix} \frac{2 γ M_{oi} + 4 γ^{2} M_{oi} (r - r_{oi}) (r - r_{oi})^{T} M_{oi}}{{(1 + e^{- γ S_{i} (r, r_{oi})})}^{2}} \\ - \frac{8 γ^{2} e^{- 2 γ S_{i} (r, r_{oi})} M_{oi} (r - r_{oi}) (r - r_{oi})^{T} M_{oi}}{{(1 + e^{- γ S_{i} (r, r_{oi})})}^{3}} \end{matrix}}

(20)

Then, the system’s velocity is governed to be equal to the negative gradient of the potential function (12). The control objective of autonomous rendezvous with obstacles avoidance capability is achieved.

Proof: Choose a Lyapunov function as follows for the relative dynamics (1)–(4)

V_{1} = 0.5 (\nabla ϕ + \overset{\cdot}{r})^{T} (\nabla ϕ + \overset{\cdot}{r})

(21)

Differentiating $V_{1}$ leads to

{\overset{\cdot}{V}}_{1} = (\nabla ϕ + \overset{\cdot}{r})^{T} (H (ϕ) \overset{\cdot}{r} + \overset{\cdot\cdot}{r})

(22)

Inserting (3)and the controller (18) into (22), it leaves ${\overset{\cdot}{V}}_{1}$ as

{\overset{\cdot}{V}}_{1} = - (\nabla ϕ + \overset{\cdot}{r})^{T} K (\nabla ϕ + \overset{\cdot}{r}) \leq 0

(23)

Hence, $lim_{t \to \infty} \overset{\cdot}{r} = - \nabla ϕ$ can be proved from (23). This means that the velocity $\overset{\cdot}{r}$ globally asymptotically converges to $- \nabla ϕ$ .

To this end, applying the results in Lim et al. (2005) and Cao et al. (2014), it can be proved that $lim_{t \to \infty} r = r_{f}$ from $lim_{t \to \infty} \overset{\cdot}{r} = - \nabla ϕ$ . Moreover, the obstacles will be avoided during the whole rendezvous maneuvering.

Adaptive VAPF controller design

The design of the AVPF in (18) does not take the performance index into consideration. It can ensure that the system’s velocity converges to $- \nabla ϕ$ only. Hence, the AVPF (18) is without any optimality. To increase its optimality, the proposed AVPF (18) is modified by embedding a performance index in this subsection. An adaptive VAPF controller will be designed by using the optimal theory and the adaptive control theory. For this controller, $ϕ_{a}$ will adaptively change its shape to make $- \nabla ϕ_{a}$ match a prescribed velocity. Hence, the prescribed velocity planned by the designer should have some degree of optimality with respect to the considered function.

It is known from the above analysis that the prescribed velocity should be preliminarily planned. To accomplish this work, let an optimal performance index be chosen as

J = \frac{1}{2} \int_{0}^{\infty} (X^{T} QX + u^{T} Ru) dt

(24)

where $Q$ and $R$ are the weight matrices, and they are diagonal and positive-definite.

Applying the relative dynamics (3) and minimizing the performance index (24), it can be obtained that the optimal trajectory is the solution of the following equation

\begin{matrix} \overset{\cdot}{X} (t) = (A (X) - B (X) R {(X)}^{- 1} B {(X)}^{T} E (X)) X (t) \\ = (A (X) - B (X) K_{SDRE} (X)) X (t) \end{matrix}

(25)

where $K_{SDRE} (X) = R (X)^{- 1} B (X)^{T} E (X)$ , and for $X \neq 0$ the matrix $E (X)$ is such that

\begin{matrix} E (X) A (X) + A (X) E (X) + Q (X) \\ - E (X) B (X) R {(X)}^{- 1} B {(X)}^{T} E (X) = 0 \end{matrix}

(26)

Let the optimal velocity be denoted by ${\overset{\cdot}{r}}_{op}$ . From (25), it has

{\overset{\cdot\cdot}{r}}_{op} = f (y) - B (y) K_{SDRE} (y) y

(27)

where $y = [r_{op} {\overset{\cdot}{r}}_{op}]^{T}$ . Here, ${\overset{\cdot\cdot}{r}}_{op}$ is the optimal acceleration.

When planning the optimal velocity ${\overset{\cdot}{r}}_{op}$ and the optimal acceleration ${\overset{\cdot\cdot}{r}}_{op}$ , the problem of avoiding obstacles is not considered. To solve this challenge, an adaptive VAPF control law is developed with the weighting parameters of the APF adaptively updated, which governs $- \nabla ϕ_{a}$ to match and converge to the optimal velocity ${\overset{\cdot}{r}}_{op}$ . The detailed procedure to obtain that adaptive VAPF control law is given as follows.

Based on (13), applying the Cholesky factorization, the positive matrix $P$ can be decomposed as

ρ = [\begin{matrix} ρ_{11} & ρ_{12} & ρ_{13} \\ 0 & ρ_{22} & ρ_{23} \\ 0 & 0 & ρ_{33} \end{matrix}]

(28)

where $ρ$ is an upper triangular matrix with the shaping $ρ_{ij}$ being time-varying. Then, it leaves $P$ as

P = ρ^{T} ρ = [\begin{matrix} ρ_{11}^{2} & ρ_{11} ρ_{12} & ρ_{11} ρ_{13} \\ ρ_{11} ρ_{12} & ρ_{12}^{2} + ρ_{22}^{2} & ρ_{12} ρ_{13} + ρ_{22} ρ_{23} \\ ρ_{11} ρ_{13} & ρ_{12} ρ_{13} + ρ_{22} ρ_{23} & ρ_{13}^{2} + ρ_{23}^{2} + ρ_{33}^{2} \end{matrix}]

(29)

Let the error between $- \nabla ϕ_{a}$ and the optimal velocity ${\overset{\cdot}{r}}_{op}$ as

e_{op} = {\overset{\cdot}{r}}_{op} + \nabla ϕ_{a} = {\overset{\cdot}{r}}_{op} + P (r - r_{f})

(30)

It follows that

{\overset{\cdot}{e}}_{op} = {\overset{\cdot\cdot}{r}}_{op} + \overset{\cdot}{P} (r - r_{f}) + P (\overset{\cdot}{r} - {\overset{\cdot}{r}}_{f}) = {\overset{\cdot\cdot}{r}}_{op} + P (\overset{\cdot}{r} - {\overset{\cdot}{r}}_{f}) + S (r - r_{f})

(31)

where $S = [\begin{matrix} g_{11} & g_{12} & g_{13} \\ g_{21} & g_{22} & g_{23} \\ g_{31} & g_{32} & g_{33} \end{matrix}]$ is a matrix with $g_{11} = 2 {\overset{\cdot}{ρ}}_{11} ρ_{11}$ , $g_{12} = {\overset{\cdot}{ρ}}_{11} ρ_{12} + ρ_{11} {\overset{\cdot}{ρ}}_{12}$ , $g_{13} = {\overset{\cdot}{ρ}}_{11} ρ_{13} + ρ_{11} {\overset{\cdot}{ρ}}_{13}$ , $g_{21} = {\overset{\cdot}{ρ}}_{11} ρ_{12} + ρ_{11} {\overset{\cdot}{ρ}}_{12}$ , $g_{22} = 2 {\overset{\cdot}{ρ}}_{12} ρ_{12} + 2 {\overset{\cdot}{ρ}}_{22} ρ_{22}$ , $g_{23} =$

\begin{matrix} {\overset{\cdot}{ρ}}_{12} ρ_{13} + ρ_{12} {\overset{\cdot}{ρ}}_{13} + {\overset{\cdot}{ρ}}_{22} ρ_{23} + ρ_{22} {\overset{\cdot}{ρ}}_{23}, \\ g_{31} = {\overset{\cdot}{ρ}}_{11} ρ_{13} + ρ_{11} {\overset{\cdot}{ρ}}_{13}, \\ g_{32} = {\overset{\cdot}{ρ}}_{12} ρ_{13} + ρ_{12} {\overset{\cdot}{ρ}}_{13} + {\overset{\cdot}{ρ}}_{22} ρ_{23} + ρ_{22} {\overset{\cdot}{ρ}}_{23}, \\ and g_{33} = 2 {\overset{\cdot}{ρ}}_{13} ρ_{13} + 2 {\overset{\cdot}{ρ}}_{23} ρ_{23} + 2 {\overset{\cdot}{ρ}}_{33} ρ_{33} . \end{matrix}

Additionally, (31) can be rewritten as

{\overset{\cdot}{e}}_{op} = {\overset{\cdot\cdot}{r}}_{op} + N \overset{\cdot}{ρ} + P (\overset{\cdot}{r} - {\overset{\cdot}{r}}_{f})

(32)

with $ρ = {[{\overset{\cdot}{ρ}}_{11} {\overset{\cdot}{ρ}}_{12} {\overset{\cdot}{ρ}}_{13} {\overset{\cdot}{ρ}}_{22} {\overset{\cdot}{ρ}}_{23} {\overset{\cdot}{ρ}}_{33}]}^{T}$ . The matrix $N$ is

N = [\begin{matrix} n_{11} & n_{12} & n_{13} & n_{14} & n_{15} & n_{16} \\ n_{21} & n_{12} & n_{23} & n_{24} & n_{25} & n_{26} \\ n_{31} & n_{32} & n_{33} & n_{34} & n_{35} & n_{36} \end{matrix}]

(33)

where $n_{11} = 2 ρ_{11} (r_{1} - r_{f 1}) + ρ_{12} (r_{2} - r_{f 2}) + ρ_{13} (r_{3} - r_{f 3})$ , $n_{12} = ρ_{11} (r_{2} - r_{f 2})$ , $n_{13} = ρ_{11} (r_{3} - r_{f 3})$ , $n_{14} = n_{15} = n_{16} = 0$ , $n_{21} = ρ_{12} (r_{1} - r_{f 1})$ , $n_{22} = ρ_{11} (r_{1} - r_{f 1}) + 2 ρ_{12} (r_{2} - r_{f 2}) + ρ_{13} (r_{3} - r_{f 3})$ , $n_{23} = ρ_{12} (r_{3} - r_{f 3})$ , $n_{24} = 2 ρ_{22} (r_{2} - r_{f 2}) + ρ_{23} (r_{3} - r_{f 3})$ , $n_{25} = ρ_{22} (r_{3} - r_{f 3})$ , $n_{26} = 0$ , $n_{31} = ρ_{13} (r_{1} - r_{f 1})$ , $n_{32} = ρ_{13} (r_{2} - r_{f 2})$ , $n_{33} = ρ_{11} (r_{1} - r_{f 1}) + ρ_{12} (r_{2} - r_{f 2}) + 2 ρ_{13} (r_{3} - r_{f 3})$ , $n_{34} = ρ_{23} (r_{2} - r_{f 2})$ , $n_{35} = 2 ρ_{23} (r_{3} - r_{f 3}) + ρ_{22} (r_{2} - r_{f 2})$ , and $n_{36} = 2 ρ_{33} (r_{3} - r_{f 3})$ .

Theorem 2: Consider the spacecraft rendezvous problem with its relative dynamics described by (1)–(4), using the APF (12), the optimal velocity obtained from (25), and the controller (18) with the weighting parameter matrix $P$ updated by

\overset{\cdot}{ρ} = N^{T} (N N^{T})^{- 1} (- {\overset{\cdot\cdot}{r}}_{op} - P (\overset{\cdot}{r} - {\overset{\cdot}{r}}_{f}) - K_{M} e_{op})

(34)

where $K_{M}$ is a positive scalar. Then, it follows that $lim_{t \to \infty} {\overset{\cdot}{r}}_{op} = - \nabla ϕ_{a}$ and $lim_{t \to \infty} \overset{\cdot}{r} = - \nabla ϕ$ . $- \nabla ϕ_{a}$ matches the pre-planned optimal velocity ${\overset{\cdot}{r}}_{op}$ . Moreover, The control objective of rendezvous with obstacles avoidance capability and some degree of optimality is achieved.

Proof: Applying (34), it can be obtained from (32) that

{\overset{\cdot}{e}}_{op} = - K_{M} e_{op}

(35)

Choose a Lyapunov function as $V = V_{1} + 0.5 e_{ep}^{T} e_{op}$ , then differentiating $V$ and inserting (18) result in

\overset{\cdot}{V} = {\overset{\cdot}{V}}_{1} + e_{ep}^{T} {\overset{\cdot}{e}}_{op} = - (\nabla ϕ + \overset{\cdot}{r})^{T} K (\nabla ϕ + \overset{\cdot}{r}) + e_{ep}^{T} {\overset{\cdot}{e}}_{op}

(36)

Substituting (35) into (36) yields

\overset{\cdot}{V} = - (\nabla ϕ + \overset{\cdot}{r})^{T} K (\nabla ϕ + \overset{\cdot}{r}) - K_{M} e_{ep}^{T} e_{op}

(37)

Because $K$ is positive and definite $K_{M} > 0$ , $lim_{t \to \infty} \overset{\cdot}{r} = - \nabla ϕ$ and $lim_{t \to \infty} e_{op} = 0$ , i.e., $lim_{t \to \infty} {\overset{\cdot}{r}}_{op} = - \nabla ϕ_{a}$ be proved from (37). Hence, according to the analysis in Theorem 1, the conclusion in Theorem 2 can be proved.

Remark 2: The matrix $N N^{T}$ is seen to be singular for $r = r_{f}$ . However, in practice, the service spacecraft position $r$ will not converge to the desired position $r_{f}$ accurately due to the perturbation and control errors. Hence, the singularity of $N N^{T}$ does not exist.

Remark 3: The VAPF controller presented in Theorem 2 can optimize the attractive potential only without optimizing the repulsive APF. The whole APF is not optimal. Hence, that controller is actually a partially optimized control law. It is named as the suboptimal velocity artificial potential function-based control (SVAPF). Although this method is not perfect enough, it is still of significant improvement over the traditional method. Moreover, in comparison with the result in Cao et al. (2014), the proposed adaptive control law is suitable for the nonlinear system of spacecraft rendezvous with obstacle avoidance. However, the solution in Cao et al. (2014) is only effective to the obstacle avoidance problem of a two-dimensional plane for a linear system. This is also one of the main contributions of the paper.

It is seen in Theorem 2 that the shaping parameters of the attractive APF are optimized. When the service spacecraft is far from the obstacles, the adaptive $ϕ_{a}$ governs the service spacecraft to follow the optimal velocity ${\overset{\cdot}{r}}_{op}$ . When ${\overset{\cdot}{r}}_{op}$ drives the service spacecraft nearby any obstacle, the repulsive potential $ϕ_{r}$ will push the service spacecraft to avoid the collision. Consequently, the velocity will re-converge to the optimal velocity ${\overset{\cdot}{r}}_{op}$ . Based on this, the proposed approach in this subsection has some degree of optimality.

Simulation results

In this section, the effectiveness of the VAPF and SVAPF controllers is numerically validated for a spacecraft rendezvous maneuvering. The relative dynamics (1)–(4) are applied. The orbital parameters of the leader spacecraft are with its inclination 30 degrees, the longitude of ascending node 60 degrees, the semi-major axis 6978 km, the eccentricity 0.05, the argument of the perigee 0 deg/sec, and the standard gravitational parameter 398600 ${km}^{3}$ / $\sec^{2}$ . The service spacecraft starts the rendezvous maneuvering with the initial relative position $r (0) = [100 300 200]^{T}$ m and the velocity $r (0) = [\begin{matrix} 0 & 0 & 0 \end{matrix}]^{T}$ m/sec. The ellipsoid describing the dimension of the service spacecraft is with $M_{s} = diag ([3 2 4]^{T})$ . Moreover, three obstacles are considered with their parameters listed in Table 1.

Table 1.

The parameters of the obstacles.

The obstacle number	The relative position (m)	The relative velocity (m/sec)	The ellipsoid envelope parameters (m)
#1	$[79 250 155]^{T}$	$[0 0 0]^{T}$	$a_{1} = 4, b_{1} = 3, c_{1} = 2$
#2	$[62 186 124]^{T}$	$[0 0 0]^{T}$	$a_{1} = 6, b_{1} = 5, c_{1} = 7$
#3	$[55 140 100]^{T}$	$[0 0 0]^{T}$	$a_{1} = 5, b_{1} = 6, c_{1} = 4$

To accomplish the planned rendezvous task, the desired position is determined as $r_{f} = [\begin{matrix} 0 & 0 & 0 \end{matrix}]^{T}$ m and ${\overset{\cdot}{r}}_{f} = [\begin{matrix} 0 & 0 & 0 \end{matrix}]^{T}$ m/sec. The disturbance force acting on the system is considered. It includes the gravitational perturbations, the atmospheric drag, the solar radiation pressure, and other disturbances. More specifically, this disturbance is assumed as

F_{d} = [\begin{matrix} 5 \sin (0.01 t) \\ 5 \cos (0.005 t) \\ 5 \sin (0.005 t) \end{matrix}] \times 10^{- 5}

(38)

When implementing the developed VAPF and SVAPF in simulation, their control gains are chosen as $k_{a} = 1$ , $k_{r} = 20$ , $γ = 0.01$ , $K_{M} = 0.1$ , $a_{max} = 5 \times 10^{- 2}$ , $R = diag ([\begin{matrix} 3 \times 10^{9} 3 \times 10^{9} & 3 \times 10^{9} \end{matrix}]^{T})$ , $Q = diag ([3 \times 10^{1} 3 \times 10^{1} 3 \times 10^{1} 3 \times 10^{1} 3 \times 10^{6} 3 \times 10^{6} 3 \times 10^{6}]^{T})$ , and

\begin{matrix} P = diag ([1.5 \times 10^{1} 1.5 \times 10^{1} 1.5 \times 10^{1} \\ 1.5 \times 10^{6} 1.5 \times 10^{6} 1.5 \times 10^{6}]^{T}) \end{matrix}

(39)

With the application of the VAPF and the SVAPF controller to perform the rendezvous maneuver, the resulted rendezvous behavior is shown in Figures 2–3, respectively. No collision between the service spacecraft and any obstacle is observed for both the VAPF and the SVAPF. The objective of obstacle avoidance is achieved by the VAPF and the SVAPF. The desired position is rendezvoused. However, comparing Figure 2 with Figure 2, it is found that these two control laws lead to different rendezvous trajectories. That is because the SVAPF is improved and modified from the VAPF by integrating an optimal strategy and adding an adaptive law. This optimal strategy can update the shaping parameters of the attractive potential function to add some degree of optimality. Hence, the SVAPF ensured a larger safe distance between the service spacecraft and the obstacles than the VAPF. Moreover, the obstacles and the service spacecraft in Figures 2–3 are described by an ellipsoid with their actual sizes and labeled by the blue and green borders, respectively. The ellipsoid is more accurate and suitable to describe the obstacles and the service spacecraft in comparison with the conventional spherical. This is the highlight research in this work, and it can be found in Figures 2–3.

Figure 2.

The obstacle avoidance trajectory obtained from the VAPF.

Figure 3.

The obstacle avoidance trajectory obtained from the SVAPF.

Figure 4 shows the control input of the VAPF and the SVAPF. It is seen that the control amplitude of the SVAPF is smaller than the VAPF. This is owing to the optimal strategy incorporated in the SVAPF for reducing the fuel cost. Due to this optimal strategy, the shaping parameters of the designed attractive potential function $ϕ_{a}$ adjusted the trajectory of SVAPF indirectly. Moreover, the shaping parameters $P$ changed by the effect of obstacles avoidance. Therefore, it can add some degree of optimality into the attractive APF. This explains that the SVAPF has the capability of reducing the control consumption. The anticipation that large distance leads to less repulsive force and fuel cost is met. When the system converges to the desired position, the control input of the SVAPF and the VAPF are almost the same due to the external force.

Figure 4.

The control input force for the VAPF and the SVAPF.

The position tracking error $e = [e_{x} e_{y} e_{z}]^{T} = r - r_{f}$ and the velocity tracking error $\overset{\cdot}{e} = [e_{vx} e_{vy} e_{vz}]^{T} = \overset{\cdot}{r} - {\overset{\cdot}{r}}_{f}$ ensured by the VAPF control and the SVAPF control are illustrated in Figure 5–6, respectively. Because the control input of SVAPF is less than the VAPF, it is seen that the convergence rate of the position tracking error is slower than the rate ensured by the VAPF control. Due to the same reason, the amplitude of oscillations resulted by the SVAPF is also smaller than the VAPF at the beginning, as we can see in Figure 5. It is also observed in Figures 6 that the overshoot of the velocity tracking error from the SVAPF is smaller than the VAPF. However, it can be seen in Figures 5–6 that the tracking accuracy provided by the SVAPF control is a little bit inferior to the accuracy achieved by the VAPF. This also indicates that adding the optimization strategy will deteriorate the control performance. Nevertheless, the overall control accuracy is nearly consistent or very close to each other. The control accuracy obtained from the SVAPF control is desirable, acceptable, and enough high for spacecraft to accomplish the rendezvous maneuver.

Figure 5.

The position tracking from the VAPF and the SVAPF.

Figure 6.

The relative velocity performance from the VAPF and the SVAPF.

It is known that the service spacecraft™ lifecycle and mission-cycle are dependent on fuel consumption. The fuel consumption is an important performance index to evaluate the performance of the controllers. This is also one of the original motivations and the best highlight of this theoretical work. Hence, fuel costs are compared between VAPF and SVAPF. For the comparison, fuel consumption/cost index is mathematically given by (Cao et al. (2014) and Cao and Chen (2015))

C_{f} = \int_{0}^{t} \sqrt{u_{x}^{2} + u_{y}^{2} + u_{z}^{2}} dt

(40)

Figure 7 shows the fuel cost comparison result of the VAPF control and the SVAPF control. It can be seen that the SVAPF controller can save almost 50% energy than the VAPF control law. The conclusion that the scheme designed to optimize the shaping parameters can add more optimality is hence verified.

Figure 7.

The fuel cost of the VAPF and the SVAPF.

Conclusions

In this work, the spacecraft rendezvous problem in the presence of obstacles was investigated. The ellipsoid was introduced to describe the outer envelopes of the service spacecraft and the obstacles. This ellipsoid description was more accurate and suitable than the conventional spherical method. The potential sigmoid function was used to generate the repulsive force to avoid the obstacles. Based on this, a VAPF-based controller was presented. Moreover, an adaptive version of this controller was developed. Applying the proposed controller, the rendezvous maneuver was successfully performed with great obstacle avoidance capability. The proposed adaptive controller ensured better control performance and some degrees of optimality than the VAPF-based control when evaluating them by using the fuel cost index. It lets the adaptive controller be more suitable for the space mission application.

As one of future work, the obstacles avoidance control problem with fast rate should be conducted for spacecraft rendezvous with actautor constraint. Moreover, formation and cooperative control of spacecrafts in the presence of obstacles should be addressed further. This could be achieved based on the result in Du et al. (2019).

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Bing Xiao

References

Andrew

(2014) Continuous control artificial potential function methods and optimal control. Air Force Institute of Technology, PhD Thesis.

Badawy

Colin

(2006) On-orbit assembly using super-quadric potential fields. J. Guid. Control. Dyn. 31(1): 30–43.

Breger

Jonathan

(2008) Safe Trajectories for autonomous rendezvous of spacecraft. J. Guid. Control. Dyn. 31(5): 1478–1489.

Cao

Chen

(2015) Input-output linearization minimum sliding mode error feedback control for spacecraft formation with large perturbations. Proc. IMechE Part G: J Aerospace Eng. 229(2): 352–368.

Cao

Chen

Arun

(2014) Minimum sliding mode error feedback control for fault tolerant reconfigurable satellite formations with J2 perturbations. Acta Astronautica 96(1): 201–216.

Cao

Chen

Sheng

(2013) Fault tolerant small satellite attitude control using adaptive non-singular terminal sliding mode. Advances in Space Research 51(12): 2374–2393.

Cao

Chen

Zhao

(2014) Minimum sliding mode error feedback control for fault tolerant small satellite attitude control. Advances in Space Research 53(2): 309–324.

Clark

(2015) Model predictive control with ellipsoid obstacle constraints for spacecraft rendezvous. Clark Family Photography Collection 48(9): 257–262.

Colin (1993) Autonomous proximity maneuvering using artificial potential function. ESA J. 17(2): 159–169.

10.

Dong

Wang

Shin

(2006) Decentralized control of autonomous swarm systems using artificial potential functions: Analytical design guidelines. J. Intelligent Robust Syst. 45: 369–394.

11.

Wen

, et al. (2019) Distributed fixed-time consensus for nonlinear heterogeneous multi-agent systems. Automatica, in press. DOI: 10.1016/j.automatica.2019.108797.

12.

Epenoy

(2011) Fuel Optimization for continuous-thrust orbital rendezvous with collision avoidance constraint. J. Guid. Control. Dyn. 34(2): 493–503.

13.

Gao

Yang

Shi

(2009) Multi-objective robust H_∞ control of spacecraft rendezvous. IEEE Trans. Control Syst. Technol. 17(4): 794–802.

14.

Gao

Luo

(2012) Adaptive artificial potential function guidance for dynamic obstacle avoidance of spacecraft (in Chinese). Chinese Space Sci. Technol. 5: 1–8.

15.

Gunn

Palmer

Leung

Ogilvie

(2015) DARPA phoenix payload orbital delivery system: Progress towards small satellite access to GEO. In: Annual AIAA/USU Conference on Small Satellites, Logan, USA, 8–13 August 2015. Reston, Virginia, USA: AIAA.

16.

Xie

Liu

(2018) Trajectory optimization for accompanying satellite obstacle avoidance. Aerosp. Sci. Technol. 82–83: 220–233.

17.

Chen

Guo

(2019) Fixed-time maneuver control of spacecraft autonomous rendezvous with a free-tumbling target. IEEE Tran. Aerosp. Electron. Syst. 55(2): 562–577.

18.

Dong

Zhang

(2019) Tracking control of spacecraft formation flying with collision avoidance. Aerosp. Sci. Technol. 42: 353–364.

19.

JohnOlcayto

Colin

(2007) Safety-critical autonomous spacecraft proximity operations via potential function guidance. In: AIAA InfoTech at Aerospace Conference, California, USA, 7–10 May 2007, pp. 1893–1902. Reston, Virginia, USA: AIAA.

20.

Khatib

(1986) Real time obstacle avoidance for manipulators and mobile robots. Int. J. Robotics Research 5(1): 90–98.

21.

Lim

Bang

Kim

(2005) Sliding mode control for the configuration of satellite formation flying using potential functions. Int. J. Aeronautical Space Sci. 6(2): 56–63.

22.

Lopez

McInnes

(1995) Autonomous rendezvous using artificial potential function guidance. J. Guid. Control. Dyn. 18(2): 237–241.

23.

Mancini

Bloise

Capello

Punta

(2020) Sliding mode control techniques and artificial potential field for dynamic collision avoidance in rendezvous maneuvers. IEEE Control Syst. Letters 4(2): 313–318.

24.

Martinson

(2013) Obstacle avoidance guidance and control algorithm for spacecraft maneuvers. In: AIAA Guidance, Navigation and Control Conference, Chicago, USA, 10–13 August 2009, pp. 1–9. Reston, Virginia, USA: AIAA.

25.

Mccamish

Romano

Nolet

Edwards

(2015) Flight testing of multiple-spacecraft control on SPHERES during close-proximity operation. Journal of Spacecraft Rocket 46(6): 1202–1213.

26.

Mueller

Larsson (2008) Collision avoidance maneuver planning with robust optimization. In: 7th International ESA Conference on Guidance, Navigation and Control Systems 18(2), Tralee, County Kerry, Ireland, 2–5 June 2008, pp. 237–241. Reston, Virginia, USA: AIAA.

27.

Munoz

(2011) Rapid path-planning algorithm for autonomous proximity operations of satellites. University of Florida, PhD Thesis.

28.

Munoz

Boyarko

Fitz-Coy

(2010) Rapid path-planning options for autonomous proximity operations of spacecraft. In: AAS Astrodynamics Specialist Conference, Toronto, Canada, 2–5 August 2010, pp. 1–24. Reston, Virginia, USA: AIAA.

29.

Palacios

Ceriotti

Radice

(2015) Close proximity formation flying via linear quadratic tracking controller and artificial potential function. Advances in Space Research 56(10): 2167–2176.

30.

Ren

McIsaac

Patel

(2007) A potential field model using generalized sigmoid functions. IEEE Trans. Syst. Man Cybern. B, Cybern. 37(22): 477–484.

31.

Ren

McIsaac

Huang

(2004) Motion planning for mobile robotics using the generalized sigmoid function. In: International Conference on Robotics Automation, New Orleans, LA, USA, 26 April–1 May 2004, pp. 2393–2398. New York, USA: IEEE.

32.

(2012) Artificial potential function method for satellite electromagnetic formation control (in Chinese). Journal of Beijing University of Aeronautics and Astronautics 38(2): 213–238.

33.

Sun

(2019) Adaptive fault tolerant constrained control of cooperative spacecraft rendezvous and docking. IEEE Trans. Ind. Electron. in press. DOI: 10.1109/TIE.2019.2913826.

34.

Sun

Huo

Jiao

(2017) Adaptive backstepping control of spacecraft rendezvous and proximity operations with input saturation and full state constraint. IEEE Trans. Ind. Electron. 64(1): 480–492.

35.

Sun

Luo

(2019) Uncertainty-dependent warning threshold for spacecraft rendezvous collision probability. IEEE Trans. Aerosp. Electron. Syst. 55(1): 2–16.

36.

Tillerson (2002) Tillerson

(2002) Coordination and control of multiple spacecraft using convex optimization techniques. Massachusetts Institute of Technology, PhD Thesis.

37.

Timothy

(2003) Demonstration of autonomous rendezvous technology (DART) project summary. In: Proceedings of the Society of Photo-Optical Instrumentation Engineers: Space Systems Technology and Operations, Orlando, USA, 24 April 2003, pp. 10–19. Washington, USA: NASA.

38.

Walter

Gene

Ursvon

(1989) A constrained eigenvalue problem. Linear Algebra and Its Applications 114–115: 815–839.

39.

Xing

Pan

(2011) Nonlinear optimal control of spacecraft approaching a tumbling target. Aerospace Science and Technology 15(2): 79–89.

40.

Xing

Pan

(2012) Indirect robust control of spacecraft via optimal control solution. IEEE Trans. Aeros. Electron. Syst. 48(2): 1798–1809.

41.

Yang

Cao

Gao

(2019) Sampled-data control for relative position holding of spacecraft rendezvous with thrust nonlinearity. IEEE Trans. Ind. Electron. 59(2): 1146–1153.

42.

Zappulla

Park

Virgili-LIop

Romano

(2018) Real-time autonomous spacecraft proximity maneuvers and docking using an adaptive artificial potential field approach. IEEE Trans. Control Syst. Technol. in press. DOI: 10.1109/TCST.2018.2866963.

43.

Zhang

(2010) Safe approaching to non-cooperative spacecraft using potential function guidance based fuzzy logic system. In: 8th World Congress on Intelligent Control and Automation, Jinan, China, 6–9 June 2010, pp. 3526–3531. New York, USA: IEEE.

44.

Zhang

Song

Pei

Duan

(2010) Ellipse cissoid-based potential function guidance for autonomous rendezvous and docking with non-cooperative target (in Chinese). Journal of Astronautics 31(10): 2259–2268.

45.

Zappulla Richard (2016) Experiments on autonomous spacecraft rendezvous and docking using an adaptive artificial potential field approach. The NPS Institutional Archive, Faculty and Researchers Collection: 4461–4478.