Safe Proximity Operation to Rotating Non-Cooperative Spacecraft with Complex Shape Using Gaussian Mixture Model-Based Fixed-Time Control

: This paper studies the safety control problem for rotating spacecraft proximity maneuver in presence of complex shaped obstacles. First, considering the attitude change of the target spacecraft, a dynamic model of close-range relative motion in a body-ﬁxed coordinate system is derived using a novel approach. Then, the Gaussian mixture model (GMM) is utilized to reconstruct the complex shape of the spacecraft, and a novel GMM-based artiﬁcial potential function (APF) is proposed to represent the collision avoidance requirement. By combining GMM-based APF with ﬁxed-time stability methodology, a ﬁxed-time control (FTC) is designed for close-range proximity operation to a rotating spacecraft having a complex shape. The presented GMM-FTC scheme can guarantee the convergence of relative state errors, and ensure that no collision occurs. Finally, simulation results are provided to illustrate the feasibility of the proposed control approach. spacecraft and a target spacecraft with complex shape. The GMM is used to design a novel artificial potential function (APF) to depict the complex shape obstacle. Numerical simulations of GMM-based APF and GMM-FTC are implemented. The former results show that the potential field distribution of GMM-based APF can well meet the requirements of collision avoidance and proximity operation. The latter results show the high precision of convergence, small amplitude control

(1) A novel GMM-based potential function is proposed, which can depict obstacle constraints of arbitrary targets with complex shape and provide attraction force to the desired position and repulsive force to avoid collision to the components of target spacecraft. (2) A compound controller is designed for ultra-close-range safe proximity based on GMM and FTC method, which can guarantee not only collision avoidance, but also fixed-time convergence to the desired state.
The rest of this paper is organized as follows: in the next section, the novel dynamic model in body-fixed frame is derived and the required definition is given. The main methodologies are presented in Section 3, in which the GMM-based APF is stated, and a GMM-based FTC is developed to achieve collision avoidance and fixed-time convergence. Finally, simulation results and conclusions are given in Sections 4 and 5, respectively.

Mathematical Preliminaries
The Euclidean norm of a vector or the induced norm of a matrix is represented by · . The n × n identity matrix is denoted by I n . For the given vector x = x 1 , x 2 , x 2 T ∈ 3 and a constant ζ ∈ , we define x ζ = x ζ 1 , x ζ 1 , x ζ 1 T , sig ζ (x) = |x 1 | ζ sgn(x 1 ), |x 2 | ζ sgn(x 2 ), |x 3 | ζ sgn(x 3 ) T in which sgn(·) represents the sign function, and the skew symmetric matrix S(x) ∈ 3×3 is defined as Lemma 1 ([31]). If there exists a continuous radially unbound function V: n → + ∪ {0} , any solution x(t) of V satisfies the inequality .

Coordinate Frames
The proximity operation mission scenario is described in Figure 1. There are two spacecrafts, the servicer and target. The target is non-cooperative, which has a pair of solar panels and antennas, rotating and running on an elliptical orbit. To describe the relative motion of spacecraft proximity operation process, the frame system should be established first. As shown in Figure 1, there are three types of coordinate frames, including Earth-centered inertial (ECI) frame, local vertical local horizontal (LVLH) frame, and body-fixed frame. They are denoted as F i = O i , i i , j i , k i , F l = O l , i l , j l , k l , and F b = O b , i b , j b , k b , respectively. F i , F l , and F b are all right-handed system. O i is located in the center of the Earth; i i is along the direction of the vernal equinox; k i points toward the north pole; j i is perpendicular to i i and k i . O l and O b are located at the target; i l points along the instantaneous vector from Earth center to the target, j l is perpendicular to i l , pointing along the instantaneous tangential velocity; k l completes the triad. i b , j b and k b point to the three body axes of the target, respectively.

Coordinate Frames
The proximity operation mission scenario is described in Figure 1. There are two spacecrafts, the servicer and target. The target is non-cooperative, which has a pair of solar panels and antennas, rotating and running on an elliptical orbit. To describe the relative motion of spacecraft proximity operation process, the frame system should be established first. As shown in Figure 1, there are three types of coordinate frames, including Earth-centered inertial (ECI) frame, local vertical local horizontal (LVLH) frame, and body-fixed frame. They are denoted as ℱ = { , , , } , ℱ = { , , , }, and ℱ = { , , , }, respectively. ℱ , ℱ , and ℱ are all right-handed system. is located in the center of the Earth; is along the direction of the vernal equinox; points toward the north pole; is perpendicular to and . and are located at the target; points along the instantaneous vector from Earth center to the target, is perpendicular to , pointing along the instantaneous tangential velocity; completes the triad. , and point to the three body axes of the target, respectively.

Translational Relative Dynamics
In LVLH frame, the relative position vector between the servicer and target is denoted as , , then the relative dynamics can be described as [41] c d where,

Translational Relative Dynamics
In LVLH frame, the relative position vector between the servicer and target is denoted as r = [x, y, z] T , then the relative dynamics can be described as [41] .. where, (2) µ = 3.986 × 10 14 m 2 /s 2 is gravitational constant of the Earth. r t is orbit radius of target spacecraft. f c , is the controlled acceleration applied to the service spacecraft. f d is the external disturbance. ω e and . ω e represent the angular velocity and angular acceleration of target spacecraft, which can be obtained by .
where, e is the eccentricity of the target, a is the semi-major axis, and f is the true anomaly. Equation (1) is established by Tschauner and Hempel (TH) [41], which offers better accuracy than the equation established by Clohessy and Wiltshire (CW) [42] in slightly eccentric orbits. The above analysis establishes the dynamic model of the two spacecraft in the frame F l , without considering the attitude change of the target spacecraft. When the target spacecraft rotates on the orbit, its attitude change will cause a collision risk in the proximity operation process. To remedy this problem, the dynamic model in body-fixed frame F b needs to be considered, so as to depict the translational motion of servicer in rotating frame. Assuming the angle velocity of the target relative to the frame F l is ω b , the vector conversion formulas between the two coordinate systems, F l and F b , are introduced as Then, substituting Equations (4)-(6) into Equation (1) yields Equation (7) can be written in the form of state equation as where, X = r T , Equation (8) is the relative dynamic model in target body-fixed frame F b . Note that all the subsequent parameters are expressed in the frame F b .
In the proximity operation mission, the desired state is denoted as . Define e = X − X d as the relative error matrix, then the relative error dynamic model can be written as where, r d . When the system reaches steady state, the relative error matrix converges to zero, i.e., e = 0, . e = 0. Then, if the external disturbance γ is ignored, u = −u eq can be obtained from Equation (9).

Main Methodologies
The main methodologies are summarized in Figure 2. The details of GMM-APF and GMM-FTC design are expanded in Sections 3.1 and 3.2, respectively.

GMM-Based APF Design
When performing the proximity operation to the target, the complex shape of the target, as shown in Figure 1, must be considered for the requirement of collision avoidance. Therefore, GMM is applied, which can give an analytical expression and more accurate depiction of the target outline. The GMM can be regarded as a model composed of K single Gaussian probability density functions [28], as shown in Equation (10). These K sub-models are hidden variables of the mixed model. The GMM is used here because the Gaussian distribution has good mathematical properties and good computing performance.
, and j R π ∈ are the expectation, covariance matrix, and weight of jth gaussian component, respectively.
Meanwhile, j π satisfies the constrain To describe the outline by GMM, the three key parameters should be solved first. Utilizing servicer's camera or other sensing equipment, a series of feature points of the target's surface can be obtained in space, such as point-cloud model [26], which is recorded as Then, using the EM algorithm, the outline, described by the feature points, can be fitted by GMM. The steps of solving GMM by EM algorithm are shown below [43].
(2) E-step: Calculate the probability ij γ that each data i r comes from sub-model j, based on above parameters.
(3) M-step: Calculate the next generation parameters as follows,

GMM-Based APF Design
When performing the proximity operation to the target, the complex shape of the target, as shown in Figure 1, must be considered for the requirement of collision avoidance. Therefore, GMM is applied, which can give an analytical expression and more accurate depiction of the target outline. The GMM can be regarded as a model composed of K single Gaussian probability density functions [28], as shown in Equation (10). These K sub-models are hidden variables of the mixed model. The GMM is used here because the Gaussian distribution has good mathematical properties and good computing performance.
where, Φ r µ j ,Σ j means the GMM's jth gaussian component. µ j ∈ R 3 , Σ j ∈ R 3×3 , and π j ∈ R are the expectation, covariance matrix, and weight of jth gaussian component, respectively. Meanwhile, π j satisfies the constrain K j=1 π j = 1. D is the dimension of position vector r. Θ = θ j = π j , µ j , Σ j contains three key parameters, which determine the GMM. To describe the outline by GMM, the three key parameters should be solved first. Utilizing servicer's camera or other sensing equipment, a series of feature points of the target's surface can be obtained in space, such as point-cloud model [26], which is recorded as R = r i = [x i , y i , z i ] T , i = 1, 2, · · · , N . Then, using the EM algorithm, the outline, described by the feature points, can be fitted by GMM. The steps of solving GMM by EM algorithm are shown below [43].
(2) E-step: Calculate the probability γ ij that each data r i comes from sub-model j, based on above parameters.
(3) M-step: Calculate the next generation parameters as follows, (4) Iterate the E-step and M-step until satisfying the condition in Equation (15). ε is small positive constant, and the superscript k means the kth calculation.
In actual calculation process, the initial parameters greatly affect the calculation results, and the E-step often takes a long time. To improve the computational efficiency, in other words, to optimize initial parameters µ j , the K-means clustering algorithm is employed. The main effect of K-means is to divide a set of data into K clusters according to Euclidean distance, the steps of which are as follows [44].
(1) Set the initial K means.
(2) Calculate the Euclidean distance between r i and each mean µ j (j = 1, 2, . . . , K), then, dividing r i to the nearest cluster center, i.e., mean and recording as r ij . (3) Recalculate each cluster center by the data in same cluster.
(4) Iterate the steps (2) and (3) The basic principle of the APF method is to construct an analytical scalar function related to states, usually composed of two types of fields: attractive potential field and repulsive potential field. Traditionally, the attractive potential field is a bowl-shaped function, while the repulsive potential field is a hill-shaped function around the restricted area. Therefore, when the object moves along the negative gradient of the potential field, it can reach the desired position without violating the path constraints.
In this paper, a novel artificial potential function based on GMM is proposed. The whole potential function is expressed as where, φ att and φ rep are the attractive potential function and repulsive potential function, respectively. For the repulsive potential function φ rep , it should be designed to satisfy the following properties: (1) φ rep equals zero when the service spacecraft reaches the desired position; (2) φ rep tends to infinity when the service is close to the target structural components. Therefore, based on the GMM, the repulsive φ rep is designed as where, M is a positive definite symmetric matrix. The GMM-based repulsive potential function, as shown in Equation (18), can provide a potential field for collision avoidance constrained by the complete shape of the target spacecraft without ignoring any details. It is worth noting that the GMM-based repulsive potential function is suitable for arbitrarily complex shape.
The attractive function φ att is designed in the classic form [19] where, P is a positive definite symmetric matrix.
With the novel GMM-based potential function, as shown above, service spacecraft is able to avoid collision with the target's complex shape structural components, such as solar panel and antenna, as well as to implement proximity operation.

GMM-Based Fixed-Time Controller Design
In this section, a novel nonlinear feedback controller is proposed for safe proximity operation to spacecraft with complex shape, by combining the GMM with fixed-time control method.
By defining y 1 = e ∈ 3 and y 2 = . e ∈ 3 , and ignoring the external disturbance f d , Equation (8) can be reduced to the block form where, Introduce the nonlinear coordinate transformation s = Φ(y) [31]: where, α 1 , β 1 , k s ∈ 3×3 , ϕ 1 = k s ∇ r φ is related to the GMM-based artificial function in Equation (17). Then, through the transformation, Equation (20) can be transformed into where, M is a positive definite symmetric matrix. The GMM-based repulsive potential function, as shown in Equation (18), can provide a potential field for collision avoidance constrained by the complete shape of the target spacecraft without ignoring any details. It is worth noting that the GMMbased repulsive potential function is suitable for arbitrarily complex shape. The attractive function att φ is designed in the classic form [19] ( ) ( ) 1 2 where, P is a positive definite symmetric matrix.
With the novel GMM-based potential function, as shown above, service spacecraft is able to avoid collision with the target's complex shape structural components, such as solar panel and antenna, as well as to implement proximity operation.

GMM-Based Fixed-Time Controller Design
In this section, a novel nonlinear feedback controller is proposed for safe proximity operation to spacecraft with complex shape, by combining the GMM with fixed-time control method.
By defining where, 11 Introduce the nonlinear coordinate transformation ( ) = Φ s y [31]: where, where, ( ) ( ) where, r φ ∇ and rr φ ∇ are the gradient and double gradient of GMM-based artificial potential function with respect to 1 y , shown as where, 8 of 20 ymmetric matrix. The GMM-based repulsive potential function, as ovide a potential field for collision avoidance constrained by the ecraft without ignoring any details. It is worth noting that the GMMn is suitable for arbitrarily complex shape. is designed in the classic form [19] ( ) ( ) mmetric matrix. d potential function, as shown above, service spacecraft is able to s complex shape structural components, such as solar panel and proximity operation.
oller Design near feedback controller is proposed for safe proximity operation to y combining the GMM with fixed-time control method. d ) ( ) ( ) ( ) ( ) ( ) where, ∇ r φ and ∇ rr φ are the gradient and double gradient of GMM-based artificial potential function with respect to y 1 , shown as where, where, η j and θ j are written as Substituting Equation (20) into Equation (22) yields In this section, a novel nonlinear feedback controller is proposed for safe proximity operation to spacecraft with complex shape, by combining the GMM with fixed-time control method.
where, 11 3 = A 0 , 12 23 3 Introduce the nonlinear coordinate transformation ( ) = Φ s y [31]: where, where, where, r φ ∇ and rr φ ∇ are the gradient and double gradient of GMM-based artificial potential function with respect to 1 y , shown as Introduce the asymptotic law of the switching surface as where, 0 < ζ 1 < 1, ζ 2 > 1 and λ i ∈ R 3×3 are positive-definite diagonal matrix. Combing Equations (29) and (30), the control input u of GMM-FTC is expressed as u = k s otential function is suitable for arbitrarily complex shape. e function att φ is designed in the classic form [19] ( ) ( ) tive definite symmetric matrix. vel GMM-based potential function, as shown above, service spacecraft is able to ith the target's complex shape structural components, such as solar panel and s to implement proximity operation.

ixed-Time Controller Design
n, a novel nonlinear feedback controller is proposed for safe proximity operation to mplex shape, by combining the GMM with fixed-time control method.
) ( ) rr φ ∇ are the gradient and double gradient of GMM-based artificial potential ect to 1 y , shown as The GMM-FTC in Equation (31) is expressed in frame F b . However, the common controller is usually expressed in LVLH or inertial frame. Based on the vector transformation relation in Equation (6), the controller u l , expressed in frame F l for relative acceleration, differs from the u expressed in frame F b by the additional converted acceleration a e and Coriolis acceleration a k , as shown below.
where, r b and v b are the relative position vector and velocity vector between servicer and target, respectively.

Theorem 1.
For the two spacecraft relative dynamic system as shown in Equation (8), if GMM-FTC is provided by Equation (31), then the trajectory of the closed-loop system converges to an arbitrary vicinity of the origin in a fixed time. In other words, the trajectory of the servicer's proximity is practical fixed-time stable.

Proof. Choose a Lyapunov function candidate
From Lemma 2, and substituting Equation (30) into the time derivative of V yields where λ i−min is the minimum diagonal element among the λ i .
Substituting Equation (33) into Equation (34) yields Therefore, from Corollary 1, it can be concluded that the system Equation (22) with GMM-FTC is stable in fixed-time.
The convergence time T max is given as

Numerical Simulation
To demonstrate the effectiveness of the proposed GMM-based artificial potential function and GMM-FTC, numerical simulations are presented in this section.

GMM-Based Artificial Potential Function Simulation
Suppose the main body of the target spacecraft is a cube of 8 cubic meters, which has two solar panels with a length of 6 m and a width of 2 m, and two antennas with a length of 2 m, as shown in Figure 3. By using equal division method, obtain 40,000 feature points data. Then, combining the K-means and EM method, calculate the key parameters of GMM. The initial parameters are listed in Table 1 and the calculation results are listed in Table 2. Because of the uniform sampling points and the regular shape of target spacecraft, the initial key parameters in Table 1 are very similar to the calculation results in Table 2.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 20 Therefore, from Corollary 1, it can be concluded that the system Equation (22) with GMM-FTC is stable in fixed-time. □ The convergence time Tmax is given as

Numerical Simulation
To demonstrate the effectiveness of the proposed GMM-based artificial potential function and GMM-FTC, numerical simulations are presented in this section.

GMM-Based Artificial Potential Function Simulation
Suppose the main body of the target spacecraft is a cube of 8 cubic meters, which has two solar panels with a length of 6 m and a width of 2 m, and two antennas with a length of 2 m, as shown in Figure 3. By using equal division method, obtain 40,000 feature points data. Then, combining the Kmeans and EM method, calculate the key parameters of GMM. The initial parameters are listed in Table 1 and the calculation results are listed in Table 2. Because of the uniform sampling points and the regular shape of target spacecraft, the initial key parameters in Table 1 are very similar to the calculation results in Table 2.      Based on the calculation results, the GMM-based reconstructed point-cloud and potential distribution are illustrated in Figure 4. Figure 4a gives the reconstructed point-cloud with additional red contours for better appearance. Figure 4b-d are the projection of GMM potential distribution in x-O-y, x-O-z, and y-O-z planes, respectively, where the brighter color means the higher potential. It is obvious that the GMM can give an accurate depiction of the complex shape of spacecraft with an analytical expression in Equation (10).
Based on the calculation results, the GMM-based reconstructed point-cloud and potential distribution are illustrated in Figure 4. Figure 4a gives the reconstructed point-cloud with additional red contours for better appearance. Figure 4b-d are the projection of GMM potential distribution in x-O-y, x-O-z, and y-O-z planes, respectively, where the brighter color means the higher potential. It is obvious that the GMM can give an accurate depiction of the complex shape of spacecraft with an analytical expression in Equation (10).  . Then, the simulation of GMM-based artificial potential function in Equation (17) is obtained, as shown in Figure 5. The three figures show the potential of GMM-based APF in x-y, xz and y-z 2D coordinates, respectively. It is obvious that the surface of main body and two solar panels have high potential value, as well as the two antennas. The desired position has the globally lowest potential value, equaling to zero, which means that the spatial positions away from the desired position have higher potential value. So that the service spacecraft, moving in the potential space determined by GMM-based APF, will be attracted to the desired position, as well as avoiding collision with the target spacecraft. Assume that the desired position is r f = −2, 2, 2 T , and M = diag 2 2 2 , P = diag 2 2 2 . Then, the simulation of GMM-based artificial potential function in Equation (17) is obtained, as shown in Figure 5. The three figures show the potential of GMM-based APF in x-y, x-z and y-z 2D coordinates, respectively. It is obvious that the surface of main body and two solar panels have high potential value, as well as the two antennas. The desired position has the globally lowest potential value, equaling to zero, which means that the spatial positions away from the desired position have higher potential value. So that the service spacecraft, moving in the potential space determined by GMM-based APF, will be attracted to the desired position, as well as avoiding collision with the target spacecraft.
z and y-z 2D coordinates, respectively. It is obvious that the surface of main body and two solar panels have high potential value, as well as the two antennas. The desired position has the globally lowest potential value, equaling to zero, which means that the spatial positions away from the desired position have higher potential value. So that the service spacecraft, moving in the potential space determined by GMM-based APF, will be attracted to the desired position, as well as avoiding collision with the target spacecraft.

GMM-FTC Simulation
To implement the simulation of GMM-FTC, parameters of the controller are given in Table 3 and the orbit elements of target spacecraft are given in Table 4. The initial relative motion parameters are given in Table 5. The simulation step size is set to h = 0.1. The thruster output error is set to ±5%.

GMM-FTC Simulation
To implement the simulation of GMM-FTC, parameters of the controller are given in Table 3 and the orbit elements of target spacecraft are given in Table 4. The initial relative motion parameters are given in Table 5. The simulation step size is set to h = 0.1. The thruster output error is set to ±5%. Table 3. Parameters of Gaussian mixture model-fixed-time control (GMM-FTC).

Parameters
Value Table 4. Orbit elements of the target spacecraft.

Elements Value Units
Semi-major axis, a 6878.14 km Eccentricity, e 0.01 Inclination, i 45 deg Longitude of the ascending node, Ω 0 deg Argument of perigee, ω 0 deg Initial true anomaly, σ 0 0 deg Table 5. Initial relative motion parameters expressed in frame F b .

Parameters Value Units
Initial relative position, r(0)   Table 5. Initial relative motion parameters expressed in frame ℱ .

Parameters Value Units
Initial relative position, ( )  Figure 6 shows the trajectory of target spacecraft from initial position (blue point), to desired position (red point) under the GMM-FTC, and the spatial relationship between target spacecraft shape and trajectory. Figure 6b is the projection of Figure 6a in x-O-y plane. It can be seen that under the GMM-FTC, the service can implement the proximity operation process while avoiding collision with any structural component of the target.  respectively. It can be seen that these parameters converge to small regions containing zero in 150 s. The stable precision of position error is better than 1 × 10 −4 m, as shown in partially enlarged view in Figure 7. From Figure 8, the amplitude of velocity is less than 0.15 m/s, which means the service keeps very low speed during proximity, which is good for improving the security. The precision is better than 1 × 10 −4 m/s. From Figure 9, the amplitude of control force is less than 0.06 m/s 2 . Note that the Figures 7-9 are the time response of position error y 1 , velocity error y 2 , and control force u, respectively. It can be seen that these parameters converge to small regions containing zero in 150 s. The stable precision of position error is better than 1 × 10 −4 m, as shown in partially enlarged view in Figure 7. From Figure 8, the amplitude of velocity is less than 0.15 m/s, which means the service keeps very low speed during proximity, which is good for improving the security. The precision is better than 1 × 10 −4 m/s. From Figure 9, the amplitude of control force is less than 0.06 m/s 2 . Note that the control force u is stable at a constant, because the non-zero desired position vector contributes the constant force u eq as shown in Equation (9). To sum up in conclusion, under the effect of GMM-FTC, it can obtain high precision convergence, small amplitude control force, and enhanced security in the proximity operation process.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 20 control force u is stable at a constant, because the non-zero desired position vector contributes the constant force eq u as shown in Equation (9). To sum up in conclusion, under the effect of GMM-FTC, it can obtain high precision convergence, small amplitude control force, and enhanced security in the proximity operation process.    control force u is stable at a constant, because the non-zero desired position vector contributes the constant force eq u as shown in Equation (9). To sum up in conclusion, under the effect of GMM-FTC, it can obtain high precision convergence, small amplitude control force, and enhanced security in the proximity operation process.      Figure 12 illustrates the trajectory of servicer and the attitude change of target in the frame F l . The figures are projections in x-O-y plane. The red curve is the trajectory of the servicer, the blue and green points are the initial and time-variant desired positions, respectively. Note that one antenna is depicted by red straight line in order to better show the attitude changes of the spacecraft, i.e., counterclockwise rotation. From Figure 12, it can be seen that servicer can approach to the desired position without any collision. After 150 s, the servicer has reached the desired position and moved to the following phase. It can be seen that the servicer can follow the rotating target and keep it in the desired position well.    Figure 12 illustrates the trajectory of servicer and the attitude change of target in the frame ℱ . The figures are projections in x-O-y plane. The red curve is the trajectory of the servicer, the blue and green points are the initial and time-variant desired positions, respectively. Note that one antenna is depicted by red straight line in order to better show the attitude changes of the spacecraft, i.e., counterclockwise rotation. From Figure 12, it can be seen that servicer can approach to the desired position without any collision. After 150 s, the servicer has reached the desired position and moved to the following phase. It can be seen that the servicer can follow the rotating target and keep it in the desired position well.   Figure 12 illustrates the trajectory of servicer and the attitude change of target in the frame ℱ . The figures are projections in x-O-y plane. The red curve is the trajectory of the servicer, the blue and green points are the initial and time-variant desired positions, respectively. Note that one antenna is depicted by red straight line in order to better show the attitude changes of the spacecraft, i.e., counterclockwise rotation. From Figure 12, it can be seen that servicer can approach to the desired position without any collision. After 150 s, the servicer has reached the desired position and moved to the following phase. It can be seen that the servicer can follow the rotating target and keep it in the desired position well.

Conclusions
In this paper, a body-fixed dynamic model is derived, and the Gaussian mixture model-based fixed time controller (GMM-FTC) is proposed to solve the proximity operation problem between a service spacecraft and a target spacecraft with complex shape. The GMM is used to design a novel artificial potential function (APF) to depict the complex shape obstacle. Numerical simulations of

Conclusions
In this paper, a body-fixed dynamic model is derived, and the Gaussian mixture model-based fixed time controller (GMM-FTC) is proposed to solve the proximity operation problem between a service spacecraft and a target spacecraft with complex shape. The GMM is used to design a novel artificial potential function (APF) to depict the complex shape obstacle. Numerical simulations of GMM-based APF and GMM-FTC are implemented. The former results show that the potential field distribution of GMM-based APF can well meet the requirements of collision avoidance and proximity operation. The latter results show the high precision of convergence, small amplitude control force, and strengthened safety in proximity operation process.