# Artificial Potential Function Safety and Obstacle Avoidance Guidance for Autonomous Rendezvous and Docking with Noncooperative Target

Authors: Jianghui Liu; Haiyang Li
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The problem of artificial potential function (APF) safety and obstacle avoidance guidance for autonomous rendezvous and docking of chaser spacecraft with noncooperative spacecraft is studied. The relative motion equation of the chaser and the target is established based on the line-of-sight coordinate system, the reference state is designed, and the corresponding state error is deduced. The attitude motion equation of the noncooperative target spacecraft in space is established. The safety and obstacle avoidance guidance problem of autonomous rendezvous and docking with noncooperative target is transformed into a path planning problem in a dynamic environment. The attractive potential function is designed according to the state error. In order to ensure that the chaser can safely approach the noncooperative target spacecraft, a safe corridor with ellipse cissoid is designed in the final approaching stage of autonomous rendezvous and docking. The obstacle is assumed to be a sphere with a certain radius to avoid its influence in the approach, and the obstacle potential function is designed based on the Gaussian function method. The total potential function of the system is designed according to the attractive potential function, the safe potential function, and the obstacle potential function. The total potential function of the system is modified to ensure that the reference state is the minimum of the total potential function of the system. The stability of the system is proven according to the Lyapunov stability principle, and the conditions for satisfying the monotonic decrease in the total potential function of the system are deduced. Finally, the effectiveness of the proposed method is verified by three sets of numerical simulations.


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## Body

## 1. Introduction

In recent years, with the rapid development of space technology, extensive attention has been drawn to autonomous on-orbit service technology, such as on-orbit assembly, fuel injection, component replacement, and on-orbit repair for uncontrolled spacecraft [1–6]. Compared with the cooperative target spacecraft with stable attitude in space, the uncontrolled target spacecraft generally performs free-tumbling attitude motion, showing strong noncooperative characteristics. Unlike the cooperative situations, when approaching the noncooperative target, the desired position of the chaser varies with time. To fulfill these tasks, autonomous rendezvous and docking with noncooperative target lays as the cornerstone for unmanned chaser spacecraft [7–9]. However, the space environment is complex because of space debris and accessories of noncooperative targets such as antennas, which are overwhelming obstacles for autonomous rendezvous and docking. Therefore, a guidance system is crucial to avoid collision with space debris and ensure chaser’s safe approach to the noncooperative target spacecraft.Some typical robust adaptive control methods are applied to the autonomous rendezvous and docking problem. Aiming at the measurement uncertainty and input saturation of spacecraft’s rendezvous and docking, Singla et al. designed an adaptive output feedback controller [10]; Lee and Vukovich proposed a robust adaptive terminal sliding mode control with finite-time convergence [11]; Xia and Huo designed a robust adaptive backstepping neural network controller [12]; Sun et al. proposed a robust saturation control strategy [13]; Kasaeian et al. conceived sliding mode prediction guidance method [14]; Park et al. and Zhu et al. [15, 16] proposed the model predictive control method; and Yan et al. used the inverse push scheme to design the control law [17]. Considering the total coupling uncertainty dynamics and thrust eccentricity of the rendezvous and docking process, Sun designed an adaptive sliding mode controller [18, 19]. In addition, some traditional optimization methods are also applied in this process. Harris studied the minimum rendezvous time between multiple spacecraft based on the differential drag method [20]. Boyarko et al. studied the minimum time and minimum energy optimal trajectory for the rendezvous and docking of two spacecraft in a circular orbit [21]. Oghim et al. studied the minimum velocity change and wait time for noncoplanar and general orbits to achieve optimal rendezvous [22]. Duan et al. used fuel consumption as a constraint and designed a fuel optimal controller for rendezvous and docking of noncooperative targets [23]. Ciarcià et al. designed a trajectory planner based on inverse dynamics with minimum energy as optimization objective [24]. Huang and Jia studied the finite-time rendezvous and docking problem with noncooperative targets [25]. Breger and How proposed an active safety method, which regarded the safety of rendezvous and docking process as a constraint and guaranteed safety without additional fuel consumption [26]. Jacobsen et al. used genetic algorithms to optimize the safety and fuel, and since the safety is only considered as a goal rather than a constraint, its rendezvous trajectory is actually unsafe [27].The artificial potential function (APF) guidance method and trajectory curvature guidance method [28] show superior obstacle avoidance performance in path planning applications. The APF guidance method derived from robot path planning has the advantages of theoretically proven stability and high computational efficiency. It has successfully been applied to the spacecraft autonomous rendezvous and docking and autonomous path planning problems. Cao et al. conceived a suboptimal APF sliding mode control method [29], and Feng et al. designed a new sliding mode control strategy based on APF [30]. Dong et al. proposed a real-time adaptive control law, which introduced the potential function method to constrain the path of the chaser [31] and then drew into a special artificial potential field method based on double quaternion to constrain the track of the chaser [32]. Chen et al. used the artificial potential field method to perform on-orbit assembly of four flexible spacecraft, and the proposed control strategy can protect the chaser spacecraft against collision before reaching the desired position [33]. Bloise designed the obstacle avoidance guidance algorithm by combining artificial potential field theory with sliding mode technology [34]. In terms of safety research on rendezvous and docking with noncooperative target, Zhang et al. concentrated on obstacle avoidance using artificial potential field guidance [35].However, the above research is not without flaws. First, most relative motion equations are based on the circular orbits of the Clohessy–Wiltshire (CW) equation, and the relative motion between spacecraft is described by the orbital coordinate system, which requires high-precision telemetry and filtering on the ground. Those methods often result in poor real-time performance and large position error. Secondly, the relevant research focuses more on the cooperative target but less on the noncooperative target. Besides, risky debris as the cause of collisions in the space is often ignored since research concerning rendezvous and docking safety centers on avoiding collisions with the accessories of the target.Therefore, based on the above considerations, the relative motion equation is established in line-of-sight coordinate system in this paper, which is applicable to any orbit. The relative motion between the chaser and the target mainly depends on the sensor of the chaser itself, which delivers better real-time performance and smaller position error. Secondly, the method proposed in this paper is aimed at noncooperative target and is more widely used. Finally, the method proposed here can not only achieve safe rendezvous and docking but also avoid the influence of space debris.The rest sections are as follows:Section 2 introduces the relative motion equation between the chaser and the target and deduces the attitude motion equation of the target. Section 3 establishes the attractive potential function, the safe potential function, and the obstacle potential function, respectively. The time derivative of each potential function is deduced, and the total potential function of the system is established and modified. The stability of the system is analyzed by using Lyapunov theory, and the control acceleration of the chaser is deduced in Section 4. Simulation studies performed on autonomous rendezvous and docking with noncooperative target are presented in Section 5. Section 6 draws conclusions.


## 2. Relative Dynamics and Attitude Motion

### 2.1. Relative Dynamics

The relative motion between the chaser and the target is shown in Figure1, where FI=oIxIyIzI is the Earth-centered inertial coordinate system, FS=oSxSySzS is the line-of-sight coordinate system, and FB=oBxByBzB is the body coordinate system of the target. rcI∈ℝ3 and rtI∈ℝ3 are the position vectors of the chaser’s centroid and target’s centroid in the frame FI, respectively. rctI=xctI,yctI,zctIT is the position vector of the target relative chaser described in the frame FI, and the relative distance between the two spacecraft is r=rctI. qε∈−π/2,π/2 is the azimuth angle, qβ∈−π,π is the altitude angle, and qε∈ℝ and qβ∈ℝ are collectively referred to as position angles.Figure 1
Relative motion coordinate system.The relative motion equation between two spacecraft described in the frameFS is [36](1)r¨−rq˙ε2+q˙β2cos2qε=atx−acx+Δax,rq¨ε2+2r˙q˙ε+rq˙β2sinqεcosqε=aty−acy+Δay,−rq¨βcosqε+2rq˙εq˙βsinqε−2r˙q˙βcosqε=atz−acz+Δaz,where atx,aty,atzT and acx,acy,aczT are the acceleration vectors of the target and the chaser in the frame FI, respectively. Δax,Δay,ΔazT is the gravitational difference between the two spacecraft. When r∈ℝ is small, Δax,Δay,ΔazT can be ignored. It can be seen from equation (1) that the relative motion equation is only related to the relative position r, the azimuth angle qε, and the altitude angle qβ and is independent of the target’s orbit, so the relative motion equation is applicable to any orbit. The attitude transfer matrix from the frame FI to the frame FS is MIS∈ℝ3×3, and its expression is(2)MIS=cosqεcosqβsinqε−cosqεsinqβ−sinqεcosqβcosqεsinqεsinqβsinqβ0cosqβ.Equation (1) shows that there is a strong coupling between r∈ℝ and cosqε, and both r∈ℝ and cosqε are positive numbers. The state variables are defined as x1=r, x2=qε, x3=qβ, x4=r˙, x5=rq˙ε, and x6=rq˙βcosqε. Then, equation (1) yields(3)x˙1=x4,x˙2=x5x1,x˙3=x6x1cosx2,x˙4=x52+x62x1+atx−acx+Δax,x˙5=−x4x5x1−x62sinx2x1cosx2+aty−acy+Δay,x˙6=−x4x6x1+x5x6sinx2x1cosx2−atz+acz−Δaz.The reference values of relative distance and position angle are set asrd, qεd, qβd, r˙d, q˙εd, and q˙βd. Then, the error variables can be defined as(4a)e1=r−rd=x1−rd,(4b)e2=qε−qεd=x2−qεd,(4c)e3=qβ−qβd=x3−qβd,(4d)e4=r˙−r˙d=x4−r˙d,(4e)e5=rq˙ε−q˙εd=x5−x1q˙εd,(4f)e6=rq˙β−q˙βdcosqε=x6−x1q˙βdcosx2.The error vectors are set asep=e1,e2,e3T and ev=e4,e5,e6T. The current state are xp=x1,x2,x3T and xv=x4,x5,x6T, and the reference state is xpd=rd,qεd,qβdT. Then, the following expression can be obtained:(5)ep=xp−xpd,(6)ev=x4−r˙d,x5−x1q˙εd,x6−x1q˙βdcosx2T.The time derivatives of equations (5) and (6) give(7)e˙p=P1xev,(8)e˙v=x˙v−P2x,where(9)P1x=diag1,1x1,1x1cosx2,P2x=r¨dx˙1q˙εd+x1q¨εdx˙1q˙βdcosx2−x1x2q˙βdsinx2+x1q¨βdcosx2.


### 2.2. Attitude Motion

The attitude motion model of the frameFB relative to the frame FI of the uncontrolled tumbling target spacecraft under the condition of no external torque is [11](10)Jtω˙B+SωBJtωB=0,where Jt∈ℝ3×3 is the inertial matrix of target and ωB∈ℝ3 is the angular velocity vector of the target in the frame FB. S⋅ represents the skew-symmetric matrix of the vector’s cross multiplication. In this paper, for any vector ξ=ξ1,ξ2,ξ3+, there is(11)Sξ=0−ξ3ξ2ξ30−ξ1−ξ2ξ10T.In order to facilitate the analysis, the quaternion is used to describe the attitude of spacecraft. The attitude kinematics equation described by the quaternion is(12)q˙=120−ωBTωB−SωBq,where q=q0,q¯TT and q¯=q1,q2,q3T. The attitude transfer matrix MIB∈ℝ3×3 from the frame FI to the frame FB is(13)MIBq=q02+q12−q22−q322q1q2+2q0q32q1q3−2q0q22q1q2−2q0q3q02−q12+q22−q322q0q1+2q2q32q1q3+2q0q22q2q3−2q0q1q02−q12−q22+q32.The docking position isDB∈ℝ3 in the frame FB, and then the desired position DCd∈ℝ3 of chaser in the frame FI is(14)DCd=MIBq−1⋅−DB.Since the target is in an uncontrolled tumbling state in space, the desired positionDCd is continuously changing, and the qεd, qβd, q˙εd, and q˙βd are also continuously changing.


## 2.1. Relative Dynamics

The relative motion between the chaser and the target is shown in Figure1, where FI=oIxIyIzI is the Earth-centered inertial coordinate system, FS=oSxSySzS is the line-of-sight coordinate system, and FB=oBxByBzB is the body coordinate system of the target. rcI∈ℝ3 and rtI∈ℝ3 are the position vectors of the chaser’s centroid and target’s centroid in the frame FI, respectively. rctI=xctI,yctI,zctIT is the position vector of the target relative chaser described in the frame FI, and the relative distance between the two spacecraft is r=rctI. qε∈−π/2,π/2 is the azimuth angle, qβ∈−π,π is the altitude angle, and qε∈ℝ and qβ∈ℝ are collectively referred to as position angles.Figure 1
Relative motion coordinate system.The relative motion equation between two spacecraft described in the frameFS is [36](1)r¨−rq˙ε2+q˙β2cos2qε=atx−acx+Δax,rq¨ε2+2r˙q˙ε+rq˙β2sinqεcosqε=aty−acy+Δay,−rq¨βcosqε+2rq˙εq˙βsinqε−2r˙q˙βcosqε=atz−acz+Δaz,where atx,aty,atzT and acx,acy,aczT are the acceleration vectors of the target and the chaser in the frame FI, respectively. Δax,Δay,ΔazT is the gravitational difference between the two spacecraft. When r∈ℝ is small, Δax,Δay,ΔazT can be ignored. It can be seen from equation (1) that the relative motion equation is only related to the relative position r, the azimuth angle qε, and the altitude angle qβ and is independent of the target’s orbit, so the relative motion equation is applicable to any orbit. The attitude transfer matrix from the frame FI to the frame FS is MIS∈ℝ3×3, and its expression is(2)MIS=cosqεcosqβsinqε−cosqεsinqβ−sinqεcosqβcosqεsinqεsinqβsinqβ0cosqβ.Equation (1) shows that there is a strong coupling between r∈ℝ and cosqε, and both r∈ℝ and cosqε are positive numbers. The state variables are defined as x1=r, x2=qε, x3=qβ, x4=r˙, x5=rq˙ε, and x6=rq˙βcosqε. Then, equation (1) yields(3)x˙1=x4,x˙2=x5x1,x˙3=x6x1cosx2,x˙4=x52+x62x1+atx−acx+Δax,x˙5=−x4x5x1−x62sinx2x1cosx2+aty−acy+Δay,x˙6=−x4x6x1+x5x6sinx2x1cosx2−atz+acz−Δaz.The reference values of relative distance and position angle are set asrd, qεd, qβd, r˙d, q˙εd, and q˙βd. Then, the error variables can be defined as(4a)e1=r−rd=x1−rd,(4b)e2=qε−qεd=x2−qεd,(4c)e3=qβ−qβd=x3−qβd,(4d)e4=r˙−r˙d=x4−r˙d,(4e)e5=rq˙ε−q˙εd=x5−x1q˙εd,(4f)e6=rq˙β−q˙βdcosqε=x6−x1q˙βdcosx2.The error vectors are set asep=e1,e2,e3T and ev=e4,e5,e6T. The current state are xp=x1,x2,x3T and xv=x4,x5,x6T, and the reference state is xpd=rd,qεd,qβdT. Then, the following expression can be obtained:(5)ep=xp−xpd,(6)ev=x4−r˙d,x5−x1q˙εd,x6−x1q˙βdcosx2T.The time derivatives of equations (5) and (6) give(7)e˙p=P1xev,(8)e˙v=x˙v−P2x,where(9)P1x=diag1,1x1,1x1cosx2,P2x=r¨dx˙1q˙εd+x1q¨εdx˙1q˙βdcosx2−x1x2q˙βdsinx2+x1q¨βdcosx2.


## 2.2. Attitude Motion

The attitude motion model of the frameFB relative to the frame FI of the uncontrolled tumbling target spacecraft under the condition of no external torque is [11](10)Jtω˙B+SωBJtωB=0,where Jt∈ℝ3×3 is the inertial matrix of target and ωB∈ℝ3 is the angular velocity vector of the target in the frame FB. S⋅ represents the skew-symmetric matrix of the vector’s cross multiplication. In this paper, for any vector ξ=ξ1,ξ2,ξ3+, there is(11)Sξ=0−ξ3ξ2ξ30−ξ1−ξ2ξ10T.In order to facilitate the analysis, the quaternion is used to describe the attitude of spacecraft. The attitude kinematics equation described by the quaternion is(12)q˙=120−ωBTωB−SωBq,where q=q0,q¯TT and q¯=q1,q2,q3T. The attitude transfer matrix MIB∈ℝ3×3 from the frame FI to the frame FB is(13)MIBq=q02+q12−q22−q322q1q2+2q0q32q1q3−2q0q22q1q2−2q0q3q02−q12+q22−q322q0q1+2q2q32q1q3+2q0q22q2q3−2q0q1q02−q12−q22+q32.The docking position isDB∈ℝ3 in the frame FB, and then the desired position DCd∈ℝ3 of chaser in the frame FI is(14)DCd=MIBq−1⋅−DB.Since the target is in an uncontrolled tumbling state in space, the desired positionDCd is continuously changing, and the qεd, qβd, q˙εd, and q˙βd are also continuously changing.


## 3. Artificial Potential Function

The artificial potential function (APF) developed in recent years was originally used for path planning of robots. APF has theoretically been proven of its stability and high computational efficiency [37, 38], which can be applied to path planning of autonomous rendezvous and docking process [28–34]. The principle of APF guidance is to define an APF reflecting global motion of the chaser. In this paper, the APF is the total potential function of the system, which is composed of the attractive potential function, the safe potential function, and the obstacle potential function. Represented by the attractive potential function, the reference position applies gravity to the chaser. The APF has a global minimum at the reference position. Represented by the safe potential function, the safe corridor exerts repulsive force on the chaser. The obstacle potential function is used to represent the obstacle exerting repulsive force on the chaser. Safe corridor and obstacles constitute the path restricted area of the chaser. The APF has a global large value in the path restricted area. The Lyapunov stability theory is used to analyze the APF, and the appropriate control law is found to be guaranteeing the time derivative of the APF is negative, thus keeping the APF monotonously decreasing. The chaser can be ensured to effectively avoid the path restricted area while reaching the reference position, thus achieving the desired guidance.


### 3.1. Attractive Potential Function

The attractive potential function representing reference position is designed as(15)Ωd=12epTQpep+12evTQvev,where Qp∈ℝ3×3 and Qv∈ℝ3×3 are positive definite symmetric matrices. Equation (15) shows that when ep=0 and ev=0, Ωd=0, otherwise Ωd>0. The reference position (ep=0 and ev=0) is the minimum value of the attractive potential function.


### 3.2. Safe Potential Function

During autonomous rendezvous and docking with noncooperative targets, the chaser must avoid colliding with accessories such as solar sails and antennas. Therefore, it is necessary to construct a safe corridor to restrict the chaser’s path to the entire corridor. In this paper, the safe corridor is defined as the conical interior formed by rotating around the docking axis of the target at a safe angleφs∈0,π/2. Within the safe distance of the target, the angle between the chaser’s line-of-sight and the docking axis of the target must satisfy the following condition:(16)α≤φs.In order to leave a safe margin, the elliptical cissoid surface rotated by the elliptical vine curve shown in Figure2 around the docking axis of the target is used as the path restricted of the chaser during the rendezvous and docking process. The elliptic vine curve equation is as follows:(17)y2=x3tan2φ2ρ−x,where ρ∈ℝ can be considered as the representation of the safe distance, while φ represents the safe angle. In order to facilitate the analysis, the x-axis of the frame FB is used as the docking axis. The elliptical cissoid surface is obtained by rotating the elliptical vine curve around the x-axis, and the inside of the surface is the safe corridor. As shown in Figure 3, the surface equation is(18)fx,y,z=y2+z2−x3tan2φ2ρ−x.Figure 2
Elliptical vine curve.Figure 3
Elliptical cissoid surface.The Gaussian function is used to represent the path restriction area. Therefore, the expression of the safe potential functionΩs0 based on the elliptical cissoid surface is [39](19)Ωs0=k1sexp−12k2s2frtcB,where k1s>0 indicates the height of the Gaussian function and k2s>0 indicates the width of the Gaussian function. rtcB=xtcB,ytcB,ztcBT is the position vector of the chaser relative to the target described in the frame FB , and the expression of rtcB∈ℝ3 is(20)rtcB=−MIBq⋅rctI.Since the attitudeq∈ℝ4 of the target is continuously changing, the safe corridor surface fixed to the target is also continuously changing, which is a dynamic safe corridor. At the same time, rtcB∈ℝ3 changes with the change in rctI∈ℝ3. As can be seen from equation (19), when the chaser is away from the curved surface, frtcB is larger and Ωs0 is smaller. When the chaser approaches the surface, frtcB is smaller and Ωs0 is larger. The appropriate k1s is set to ensure that the chaser is constrained to move inside the curved surface to satisfy the requirements of safe rendezvous and docking.The time derivative of equation (19) gives(21)Ω˙s0q,xp=Ωs0q,xp⋅−12k2s2∂frtcB∂rtcBT∂rtcB∂rctT∂rct∂xpT∂xp∂t+∂rtcB∂qT∂q∂t.Since the design requires that the chaser cannot pass through the point of the highest safe potential function, i.e.,frtcB≠0 is required, when calculating the time derivative of the potential function Ωs0 in equation (19), frtcB cannot take absolute value, i.e.,(22)∂frtcB∂rtcBT=∂frtcB∂rtcBT=−2xtcB2⋅3ρ−xtcBtan2φ2ρ−xtcB2,2ytcB,2ztcB.The other partial time derivatives in equation (21) are solved as follows:(23)∂rtcB∂rctT=−MIBq⋅diag1,1,1=−MIBq.r
c
t
I
=
M
I
S
T
r
c
t
S and rctS=r,0,0T yield(24)rctI=rcosqεcosqβ,rsinqε,−rcosqεsinqβT,∂rctI∂xpT=cosx2cosx3−x1sinx2cosx3−x1cosx2sinx3sinx2x1cosx20−cosx2sinx3x1sinx2sinx3−x1cosx2cosx3.Equations (13) and (20) yield(25)rtcB=−xctIq02+q12−q22−q32+yctI2q1q2+2q0q3+zctI2q1q3−2q0q2xctI2q1q2−2q0q3+yctIq02−q12+q22−q32+zctI2q0q1+2q2q3xctI2q1q3+2q0q2+yctI2q2q3−2q0q1+zctIq02−q12−q22+q32,∂rtcB∂qT=−C1,C2,where(26)C1=2q0xctI+2q3yctI−2q2zctI2q1xctI+2q2yctI+2q3zctI−2q3xctI+2q0yctI+2q1zctI2q2xctI−2q1yctI+2q0zctI2q2xctI−2q1yctI+2q0zctI2q3xctI−2q0yctI−2q1zctI,C2=−2q2xctI+2q1yctI−2q0zctI−2q3xctI+2q0yctI+2q1zctI2q1xctI+2q2yctI+2q3zctI−2q0xctI−2q3yctI+2q2zctI2q0xctI+2q3yctI−2q2zctI2q1xctI+2q2yctI+2q3zctI.


### 3.3. Obstacle Potential Function

The obstacle is regarded as a sphere with a certain radius, and the expression of the obstacle potential functionΩo0 is(27)Ωo0=k1oexp−12k2o2rocIT⋅rocI,where k1o>0 represents the height of the potential function and k2o>0 can be considered as an approximate representation of the obstacle’s radius. rocI=xocI,yocI,zocI is the position vector of the chaser relative to the spherical obstacle described in the frame FI, and the expression of rocI∈ℝ3 is(28)rocI=−MIST⋅rctS+MIBqT⋅rtoB,where rctS=r,0,0T is the position vector of the target relative to the chaser described in the frame FS and rtoB=xtoB,ytoB,ztoB is the position vector of the spherical obstacle relative to the target described in the frame FB. When rtoB∈ℝ3 is unchanged, the obstacle remains hovering with the target. When rtoB∈ℝ3 changes, the obstacle is in relative motion with the target.The time derivative of equation (27) gives(29)Ω˙o0q,xp,rtoB=Ωo0q,xp,rtoB⋅−12k2o2⋅2rocIT⋅∂rocI∂xpT∂xp∂t+∂rocI∂qT∂q∂t+∂rocI∂rtoBT∂rtoB∂t,where(30)∂rocI∂xpT=−∂rct∂xpT=−cosx2cosx3−x1sinx2cosx3−x1cosx2sinx3sinx2x1cosx20−cosx2sinx3x1sinx2sinx3−x1cosx2cosx3,∂rocI∂rtoBT=MIBqT,∂rtoB∂t=x˙toB,y˙toB,z˙toBT.Equation (13) and the expression of rtoB∈ℝ3 yield(31)MIBqT⋅rtoB=xtoBq02+q12−q22−q32+ytoB2q1q2−2q0q3+ztoB2q1q3+2q0q2xtoB2q1q2+2q0q3+ytoBq02−q12+q22−q32+ztoB2q2q3−2q0q1xtoB2q1q3−2q0q2+ytoB2q0q1+2q2q3+ztoBq02−q12−q22+q32,∂rocI∂qT=C3,C4,where(32)C3=2q0xtoB−2q3ytoB+2q2ztoB2q1xtoB+2q2ytoB+2q3ztoB2q3xtoB+2q0ytoB−2q1ztoB2q2xtoB−2q1ytoB−2q0ztoB−2q2xtoB+2q1ytoB+2q0ztoB2q3xtoB+2q0ytoB−2q1ztoB,C4=−2q2xtoB+2q1ytoB+2q0ztoB−2q3xtoB−2q0ytoB+2q1ztoB2q1xtoB+2q2ytoB+2q3ztoB2q0xtoB−2q3ytoB+2q2ztoB−2q0xtoB+2q3ytoB−2q2ztoB2q1xtoB+2q2ytoB+2q3ztoB.


### 3.4. Total Potential Function

The total potential functionΩ0 is the sum of the attractive potential function Ωd, the safe potential function Ωs0, and the obstacle potential function Ωo0, and the expression of Ω0 is(33)Ω0=Ωd+Ωs0+Ωo0=12epTQpep+12evTQvev+k1sexp−12k2s2frtcB+k1oexp−12k2o2rocIT⋅rocI.It can be concluded from equation (33) that the minimum value of Ω0 is no longer precisely located in the reference position state. To avoid the above situation, the safe potential function Ωs0 of equation (19) and the obstacle potential function Ωo0 of equation (27) are redefined, each adding a quadratic term, which yields(34)Ωs=12epTQsepk1sexp−12k2s2frtcB,(35)Ωo=12epTQoepk1oexp−12k2o2rocIT⋅rocI,where Qs∈ℝ3×3 and Qo∈ℝ3×3 are positive definite symmetric matrices. Then, the modified system total potential function Ω expression is(36)Ω=Ωd+Ωs+Ωo=12epTQpep+12evTQvev+12epTQsepk1sexp−12k2s2frtcB+12epTQoepk1oexp−12k2o2rocIT⋅rocI.As can be seen from equation (36), the total potential function Ω has a global minimum at the reference position (ep=0 and ev=0). It should be pointed out that, in order to facilitate the analysis, equation (36) only considers the case of single obstacle. When there are multiple obstacles, it only needs to change Ωo in equation (35) to ∑iΩoi.


## 3.1. Attractive Potential Function

The attractive potential function representing reference position is designed as(15)Ωd=12epTQpep+12evTQvev,where Qp∈ℝ3×3 and Qv∈ℝ3×3 are positive definite symmetric matrices. Equation (15) shows that when ep=0 and ev=0, Ωd=0, otherwise Ωd>0. The reference position (ep=0 and ev=0) is the minimum value of the attractive potential function.


## 3.2. Safe Potential Function

During autonomous rendezvous and docking with noncooperative targets, the chaser must avoid colliding with accessories such as solar sails and antennas. Therefore, it is necessary to construct a safe corridor to restrict the chaser’s path to the entire corridor. In this paper, the safe corridor is defined as the conical interior formed by rotating around the docking axis of the target at a safe angleφs∈0,π/2. Within the safe distance of the target, the angle between the chaser’s line-of-sight and the docking axis of the target must satisfy the following condition:(16)α≤φs.In order to leave a safe margin, the elliptical cissoid surface rotated by the elliptical vine curve shown in Figure2 around the docking axis of the target is used as the path restricted of the chaser during the rendezvous and docking process. The elliptic vine curve equation is as follows:(17)y2=x3tan2φ2ρ−x,where ρ∈ℝ can be considered as the representation of the safe distance, while φ represents the safe angle. In order to facilitate the analysis, the x-axis of the frame FB is used as the docking axis. The elliptical cissoid surface is obtained by rotating the elliptical vine curve around the x-axis, and the inside of the surface is the safe corridor. As shown in Figure 3, the surface equation is(18)fx,y,z=y2+z2−x3tan2φ2ρ−x.Figure 2
Elliptical vine curve.Figure 3
Elliptical cissoid surface.The Gaussian function is used to represent the path restriction area. Therefore, the expression of the safe potential functionΩs0 based on the elliptical cissoid surface is [39](19)Ωs0=k1sexp−12k2s2frtcB,where k1s>0 indicates the height of the Gaussian function and k2s>0 indicates the width of the Gaussian function. rtcB=xtcB,ytcB,ztcBT is the position vector of the chaser relative to the target described in the frame FB , and the expression of rtcB∈ℝ3 is(20)rtcB=−MIBq⋅rctI.Since the attitudeq∈ℝ4 of the target is continuously changing, the safe corridor surface fixed to the target is also continuously changing, which is a dynamic safe corridor. At the same time, rtcB∈ℝ3 changes with the change in rctI∈ℝ3. As can be seen from equation (19), when the chaser is away from the curved surface, frtcB is larger and Ωs0 is smaller. When the chaser approaches the surface, frtcB is smaller and Ωs0 is larger. The appropriate k1s is set to ensure that the chaser is constrained to move inside the curved surface to satisfy the requirements of safe rendezvous and docking.The time derivative of equation (19) gives(21)Ω˙s0q,xp=Ωs0q,xp⋅−12k2s2∂frtcB∂rtcBT∂rtcB∂rctT∂rct∂xpT∂xp∂t+∂rtcB∂qT∂q∂t.Since the design requires that the chaser cannot pass through the point of the highest safe potential function, i.e.,frtcB≠0 is required, when calculating the time derivative of the potential function Ωs0 in equation (19), frtcB cannot take absolute value, i.e.,(22)∂frtcB∂rtcBT=∂frtcB∂rtcBT=−2xtcB2⋅3ρ−xtcBtan2φ2ρ−xtcB2,2ytcB,2ztcB.The other partial time derivatives in equation (21) are solved as follows:(23)∂rtcB∂rctT=−MIBq⋅diag1,1,1=−MIBq.r
c
t
I
=
M
I
S
T
r
c
t
S and rctS=r,0,0T yield(24)rctI=rcosqεcosqβ,rsinqε,−rcosqεsinqβT,∂rctI∂xpT=cosx2cosx3−x1sinx2cosx3−x1cosx2sinx3sinx2x1cosx20−cosx2sinx3x1sinx2sinx3−x1cosx2cosx3.Equations (13) and (20) yield(25)rtcB=−xctIq02+q12−q22−q32+yctI2q1q2+2q0q3+zctI2q1q3−2q0q2xctI2q1q2−2q0q3+yctIq02−q12+q22−q32+zctI2q0q1+2q2q3xctI2q1q3+2q0q2+yctI2q2q3−2q0q1+zctIq02−q12−q22+q32,∂rtcB∂qT=−C1,C2,where(26)C1=2q0xctI+2q3yctI−2q2zctI2q1xctI+2q2yctI+2q3zctI−2q3xctI+2q0yctI+2q1zctI2q2xctI−2q1yctI+2q0zctI2q2xctI−2q1yctI+2q0zctI2q3xctI−2q0yctI−2q1zctI,C2=−2q2xctI+2q1yctI−2q0zctI−2q3xctI+2q0yctI+2q1zctI2q1xctI+2q2yctI+2q3zctI−2q0xctI−2q3yctI+2q2zctI2q0xctI+2q3yctI−2q2zctI2q1xctI+2q2yctI+2q3zctI.


## 3.3. Obstacle Potential Function

The obstacle is regarded as a sphere with a certain radius, and the expression of the obstacle potential functionΩo0 is(27)Ωo0=k1oexp−12k2o2rocIT⋅rocI,where k1o>0 represents the height of the potential function and k2o>0 can be considered as an approximate representation of the obstacle’s radius. rocI=xocI,yocI,zocI is the position vector of the chaser relative to the spherical obstacle described in the frame FI, and the expression of rocI∈ℝ3 is(28)rocI=−MIST⋅rctS+MIBqT⋅rtoB,where rctS=r,0,0T is the position vector of the target relative to the chaser described in the frame FS and rtoB=xtoB,ytoB,ztoB is the position vector of the spherical obstacle relative to the target described in the frame FB. When rtoB∈ℝ3 is unchanged, the obstacle remains hovering with the target. When rtoB∈ℝ3 changes, the obstacle is in relative motion with the target.The time derivative of equation (27) gives(29)Ω˙o0q,xp,rtoB=Ωo0q,xp,rtoB⋅−12k2o2⋅2rocIT⋅∂rocI∂xpT∂xp∂t+∂rocI∂qT∂q∂t+∂rocI∂rtoBT∂rtoB∂t,where(30)∂rocI∂xpT=−∂rct∂xpT=−cosx2cosx3−x1sinx2cosx3−x1cosx2sinx3sinx2x1cosx20−cosx2sinx3x1sinx2sinx3−x1cosx2cosx3,∂rocI∂rtoBT=MIBqT,∂rtoB∂t=x˙toB,y˙toB,z˙toBT.Equation (13) and the expression of rtoB∈ℝ3 yield(31)MIBqT⋅rtoB=xtoBq02+q12−q22−q32+ytoB2q1q2−2q0q3+ztoB2q1q3+2q0q2xtoB2q1q2+2q0q3+ytoBq02−q12+q22−q32+ztoB2q2q3−2q0q1xtoB2q1q3−2q0q2+ytoB2q0q1+2q2q3+ztoBq02−q12−q22+q32,∂rocI∂qT=C3,C4,where(32)C3=2q0xtoB−2q3ytoB+2q2ztoB2q1xtoB+2q2ytoB+2q3ztoB2q3xtoB+2q0ytoB−2q1ztoB2q2xtoB−2q1ytoB−2q0ztoB−2q2xtoB+2q1ytoB+2q0ztoB2q3xtoB+2q0ytoB−2q1ztoB,C4=−2q2xtoB+2q1ytoB+2q0ztoB−2q3xtoB−2q0ytoB+2q1ztoB2q1xtoB+2q2ytoB+2q3ztoB2q0xtoB−2q3ytoB+2q2ztoB−2q0xtoB+2q3ytoB−2q2ztoB2q1xtoB+2q2ytoB+2q3ztoB.


## 3.4. Total Potential Function

The total potential functionΩ0 is the sum of the attractive potential function Ωd, the safe potential function Ωs0, and the obstacle potential function Ωo0, and the expression of Ω0 is(33)Ω0=Ωd+Ωs0+Ωo0=12epTQpep+12evTQvev+k1sexp−12k2s2frtcB+k1oexp−12k2o2rocIT⋅rocI.It can be concluded from equation (33) that the minimum value of Ω0 is no longer precisely located in the reference position state. To avoid the above situation, the safe potential function Ωs0 of equation (19) and the obstacle potential function Ωo0 of equation (27) are redefined, each adding a quadratic term, which yields(34)Ωs=12epTQsepk1sexp−12k2s2frtcB,(35)Ωo=12epTQoepk1oexp−12k2o2rocIT⋅rocI,where Qs∈ℝ3×3 and Qo∈ℝ3×3 are positive definite symmetric matrices. Then, the modified system total potential function Ω expression is(36)Ω=Ωd+Ωs+Ωo=12epTQpep+12evTQvev+12epTQsepk1sexp−12k2s2frtcB+12epTQoepk1oexp−12k2o2rocIT⋅rocI.As can be seen from equation (36), the total potential function Ω has a global minimum at the reference position (ep=0 and ev=0). It should be pointed out that, in order to facilitate the analysis, equation (36) only considers the case of single obstacle. When there are multiple obstacles, it only needs to change Ωo in equation (35) to ∑iΩoi.


## 4. Proof of System Stability

Proof.
The modified total potential functionΩ is taken as Lyapunov function V, i.e.,(37)V=12epTQpep+12evTQvev+12epTQsepk1sexp−12k2s2frtcB+12epTQoepk1oexp−12k2o2rocIT⋅rocI.
As can be obtained from equation (37), V≥0. V=0 only when ep=0,0,0T and ev=0,0,0T, otherwise V>0. When ep=0,0,0T and ev=0,0,0T, the chaser has reached the desired position. The time derivative of equation (37) gives(38)V˙=epTQpe˙p+evTQve˙v+epTQse˙p⋅Ωs0q,xp+12epTQsep⋅Ω˙s0q,xp+epTQoe˙p⋅Ωo0q,xp,rtoB+12epTQoep⋅Ω˙o0q,xp,rtoB.
Substituting equations (7) and (8) into equation (38) yields(39)V˙=epTQpe˙p+P1−1xe˙pTQvx˙v−P2x+epTQse˙p⋅Ωs0q,xp+12epTQsep⋅Ω˙s0q,xp+epTQoe˙p⋅Ωo0q,xp,rtoB+12epTQoep⋅Ω˙o0q,xp,rtoB=epTQpe˙p+e˙pTP1−1xQvx˙v−P2x+epTQse˙p⋅Ωs0q,xp+12epTQsep⋅Ω˙s0q,xp+epTQoe˙p⋅Ωo0q,xp,rtoB+12epTQoep⋅Ω˙o0q,xp,rtoB,Let(40)P1−1xQvx˙v−P2x=−Qpep−Qle˙p−Qsep⋅Ωs0q,xp−12QnepTQsep⋅Ω˙s0q,xp−Qoep⋅Ωo0q,xp,rtoB−12QnepTQoep⋅Ω˙s0q,xp,rtoB,where Ql∈ℝ3×3 is positive definite symmetric matrix. Qn∈ℝ3×3 is the generalized inverse matrix of e˙pT, and the expression of Qn∈ℝ3×3 is(41)Qn=e˙Pe˙pT−1e˙P.
Equation (40) yields(42)x˙v=Qv−1P1x−Qpep−Qle˙p−Qsep⋅Ωs0q,xp−12QnepTQsep⋅Ω˙s0q,xp−Qoep⋅Ωo0q,xp,rtoB−12QnepTQoep⋅Ω˙s0q,xp,rtoB+P2x=P2x−Qv−1P1xQpep+QlP1xev+Qv−1P1x−Qsep⋅Ωs0q,xp−12QnepTQsep⋅Ω˙s0q,xp+Qv−1P1x−Qoep⋅Ωo0q,xp,rtoB−12QnepTQoep⋅Ω˙s0q,xp,rtoB=at+as+ao,where(43)at=P2x−Qv−1P1xQpep+QlP1xev,as=−Qv−1P1xQsep⋅Ωs0q,xp+12QnepTQsep⋅Ω˙s0q,xp,ao=−Qv−1P1xQoep⋅Ωo0q,xp,rtoB+12QnepTQoep⋅Ω˙s0q,xp,rtoB,where at∈ℝ3 represents the gravitational acceleration of the reference position potential field to the chaser, as∈ℝ3 represents the repulsive acceleration of the safe corridor potential field to the chaser, and ao∈ℝ3 represents the repulsive acceleration of the obstacle potential field to the chaser. Substituting equation (40) into (39) yields(44)V˙=−e˙pTQle˙p≤0.
This case completes the proof.
The control acceleration vector of the system in the frameFS is derived from equations (46) and (3):(45)a=diag−1,−1,1x˙v+x52+x62x1−x4x5x1−x62sinx2x1cosx2x4x6x1−x5x6sinx2x1cosx2.
Due to the limited thrust of the chaser, its control acceleration is limited, and the actual control acceleration in each direction is(46)ac=amax,a>amax,a,−amax≤a≤amax,−amax,a<−amax,where amax∈ℝ is the maximum acceleration that can be provided by the chaser in a single direction and a∈ℝ is the acceleration obtained by using equation (46).


## 5. Simulation Example

In this paper, the numerical simulation of the autonomous rendezvous and docking mission of the chaser spacecraft relative to the noncooperative target spacecraft in super close range is carried out. In order to visually analyze the effect of attractive potential function, safe potential function, and obstacle potential function, three sets of simulation scenarios are set up in this paper:(1)
The action of the attractive potential function alone is considered(2)
The actions of both the attractive potential function and safe potential function are considered(3)
The actions of all he attractive potential function, safe potential function, and obstacle potential function are consideredIt is assumed that the target spacecraft is in free-tumbling state without external torque in space, and there is no maneuver in the target. The initial states of the target and chaser in the frameFI are(47)rtIt0=−1535849.23,5181325.49,4056754.35Tm,vtIt0=−4902.795,−4476.845,3880.881Tm/s,rcIt0=−1535824.62,5181205.19,4056813.08Tm,vcIt0=−4902.895,−4476.765,3880.981Tm.The docking position in the frameFB is rdB=5,0,0Tm. The initial attitude quaternion and angular velocity of the target are qtt0=−0.0094,−0.5443,0.2292,−0.8069T and ωct0=−0.0035,0.0052,0.0035Trad/s. The maximum control acceleration in each direction provided by the chaser is amax=0.2m/s2. The parameters of the attractive potential function are Qv=diag1,1,1, Qp=diag100,100,100, and Qv=diag2500,2500,2500. The parameters of the safe potential function are Qs=diag0.001,0.001,0.001, k1s=105, and k2s=25. The parameters of the obstacle potential function are Qo=diag0.001,0.001,0.001, k1o=106, and k2o=6. The parameters of the elliptical vine curve function are ρ=20 and φ=10°. The safe distance of the target’s safe corridor is ρs=20m, and the safe angle is φs=10°. The moment of inertia of the target is(48)Jt=3336.3−135.4−154.2−135.43184.5−148.5−154.2−148.52423.7kg⋅m2The simulation time is 200 s, and the simulation step is 0.05 s.Scenario 1.
The action of the attractive potential function alone is considered.Qs and Qo in equation (42) are set as Qs=diag0,0,0 and Qo=diag0,0,0, respectively. The simulation results are presented in Figures 4–10.Figure 5
(a) Attitude and (b) angular velocity of the target.
(a)
(b)Figure 5
(a) Relative position and (b) relative velocity of the chaser and the desired position (Scenario 1).
(a)
(b)Figure 6
Errors of states (Scenario 1).
(a)
(b)
(c)
(d)
(e)
(f)Figure 7
Relative motion of the chaser and target in the frameFB (Scenario 1).Figure 8
Safe profile (Scenario 1).Figure 9
Control acceleration in the frameFS (Scenario 1).
(a)
(b)
(c)Figure 10
(a) Relative position and (b) relative velocity of the chaser and the desired position (Scenario 2).
(a)
(b)Figure4 shows the attitude and angular velocity of the target spacecraft, and the attitude and angular velocity of the target vary with time. Figure 5 shows the relative position and relative velocity of the chaser and the desired position. Figure 6 shows the errors of states. After 146 s, the chaser reaches the desired position, and the relative velocity of the chaser to the desired position is 0m/s, after which the relative position and relative velocity of the chaser to the desired position remain unchanged.Figure7 shows the relative motion of the chaser and target in the frame FB. Figure 8 shows the safe profile. From Figures 7 and 8, it can be conclude that the chaser crosses the safe corridor during the autonomous rendezvous and docking, and the chaser may collide with the accessories of the noncooperative target.Figure9 shows the control acceleration in the frame FS. In order to keep up with the desired position, the chaser’s line-of-sight direction (x-direction) always needs to provide a control acceleration, which changes back and forth between −0.2 m/s2 and 0.2 m/s2. The control acceleration in the y direction and the z direction changes dramatically at the initial time. After 81 s, the control acceleration in the y direction and the z direction maintained at an extremely small value of less than 0.002 m/s2.Scenario 1 shows that when the safe requirements are not considered, the autonomous rendezvous and docking with the noncooperative target by the chaser can be realized by the guidance of the attractive potential function. When considering the safe requirements, it is obvious that the attractive potential function alone cannot satisfy the requirements of safe rendezvous and docking.Scenario 2.
The action of the attractive potential function and safe potential function are both considered, and the other simulation conditions are the same as scenario 1.Qo in equation (42) is set as Qo=diag0,0,0. The simulation results are presented in Figures 10–14.It can be concluded from Figure10 that the relative position of the chaser and the desired position is 0m after 103 s, and the relative velocity is 0m/s. Figures 12 and 13 indicate that when considering the safe potential function, the chaser first approaches the target’s docking direction, and then slowly approaches the target along the docking direction without colliding with the safe corridor. Figure 14 shows that, after 69s, the control acceleration in the y direction and z direction tends to 0m/s2, and the control acceleration in the x direction changes back and forth between −0.2m/s2 and 0.2m/s2. Scenario 2 indicates that when considering the action of the safe potential function, the APF guidance can realize rendezvous and docking with noncooperative targets along the safe corridor. During the rendezvous and docking process, collision with the accessories of the target can be avoided, and the safe requirements can be guaranteed.Figure 11
Errors of states (Scenario 2).
(a)
(b)
(c)
(d)
(e)
(f)Figure 12
Relative motion of the chaser and target in the frameFB (Scenario 2).Figure 13
Safe profile (Scenario 2).Figure 14
Control acceleration in the frameFS (Scenario 2).
(a)
(b)
(c)Scenario 3.
Adding an obstacle to the path of autonomous rendezvous and docking in Scenario 2, the action of the attractive potential function, safe potential function, and obstacle potential function are all considered. The simulation results are presented in Figures15–19.Figure15 shows that the chaser has reached the desired position, the relative position is reduced to 0, and the relative speed is reduced to 0m/s2. Figures 17 and 18 indicate that the chaser can effectively avoid obstacle and achieve safe rendezvous and docking with noncooperative target when considering the action of the obstacle potential function. Figure 19 shows that, after a period of time, the control acceleration in the y direction and z direction tends to 0m/s2, and the control acceleration in the x direction changes back and forth between −0.2m/s2 and 0.2m/s2.Scenario 3 indicates that when considering the action of the obstacle potential function, the APF guidance can ensure the chaser effectively avoids the obstacles. The chaser approaches the noncooperative target along the safe corridor, which ensures the guidance requirements of safe rendezvous and docking and obstacle avoidance.Figure 15
Relative position and relative velocity of the chaser and the desired position (Scenario 3).
(a)
(b)Figure 16
Errors of states (Scenario 3).
(a)
(b)
(c)
(d)
(e)
(f)Figure 17
Relative motion of the chaser and target in the frameFB (Scenario 3).
(a)
(b)
(c)
(d)Figure 18
Safe profile (Scenario 3).Figure 19
Control acceleration in the frameFS (Scenario 3).
(a)
(b)
(c)


## 6. Conclusions

The problem of the APF safe approach and obstacle avoidance for autonomous rendezvous and docking of chaser to noncooperative target spacecraft is studied. The relative motion equation of the chaser relative to the noncooperative target spacecraft is established, and the attitude motion characteristics of the target spacecraft are analyzed. The attractive potential function, safe potential function, and obstacle potential function are designed. The time derivatives of the safe potential function and the obstacle potential function are obtained, respectively. The total potential function of the system is derived and modified. The stability of the system is analyzed according to Lyapunov theory, and the control acceleration of the chaser is derived. Finally, the numerical simulation is carried out, and the conclusions are as follows:(1)
The autonomous rendezvous and docking with the noncooperative target by the chaser can be realized by the guidance of the attractive potential function. When considering the safe requirements, it is obvious that the attractive potential function alone cannot satisfy the requirements of safe rendezvous and docking.(2)
When considering the action of the safe potential function, the APF guidance can realize rendezvous and docking with noncooperative targets along the safe corridor. During the rendezvous and docking process, collision with the accessories of the target can be avoided, and the safety requirements can be guaranteed.(3)
When considering the action of the obstacle potential function, the APF guidance can ensure the chaser effectively avoids the obstacles. The chaser approaches the noncooperative target along the safe corridor, which ensures the guidance requirements of safe rendezvous and docking and obstacle avoidance.



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*Source: 3451864-2019-08-21.xml*