Kinematically Coupled Relative Spacecraft Motion

Kinematically Coupled Relative Spacecraft Motion Control Using the State-Dependent Riccati Equation Method

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Daero Lee 1; Hyochoong Bang 2; Eric A. Butcher 3; and Amit K. Sanyal 4

Abstract: This paper presents kinematically coupled relative spacecraft motion control in the close proximity of a tumbling target using the state-dependent Riccati equation method for proximity operation mission. In general, a rigid-body dynamics can be expressed as both translation and rotation about the center of mass. However, a kinematic coupling between the rotational and translational dynamics occurs when it is not expressed about the center of mass. Thus, kinematically coupled relative spacecraft motion model is derived to describe the relative motion about the selected arbitrary points on both the target and spacecraft. Then, spacecraft relative motion is represented by combining the relative translational and rotational dynamics of arbitrary points on the spacecraft. The spacecraft is required to achieve the desired position and attitude to track a tumbling spacecraft quickly in the effect of kinematic translation and rotation coupling. The state-dependent Riccati equation method is implemented to design a nonlinear controller in six degrees of freedom. Numerical simulation results validate kinematically coupled relative spacecraft motion control with respect to a tumbling target. DOI: 10.1061/(ASCE)AS.1943-5525.0000436. © 2014 American Society of Civil Engineers. Author keywords: Coupled translational dynamics; State-dependent Riccati equation; Kinematic coupling; Feature point; Tumbling target.

Introduction Precise position and attitude maneuvers are a critical portion of many space missions such as autonomous rendezvous, docking with other spacecraft, removal of space debris, reorientation of satellites, and observation. Recent space programs, such as NASA’s Demonstration of Autonomous Rendezvous Technology (DART) program, the U.S. Air Force’s XSS series (Davis and Melanson 2004), Defense Advanced Research Projects Agency’s (DARPA) Orbital Express program (Gottselig 2002), European Space Agency’s (ESA) Automatic Transfer Vehicle (ATV) program (Gonnaud and Pascal 2000; Pinard et al. 2007), and the Hubble Robotic and Deorbit Mission (HRSDM) (Zimpfer et al. 2005) have been proposed and developed to demonstrate many of the technologies required for space exploration. A core technology in these missions is the capability of autonomous rendezvous and capturing, which requires precise position and attitude control. Such technology can lead to significant developments in the area of autonomous rendezvous and capture with applications to supply and repair the International Space Station (ISS) and for exploration of the Moon, Mars, and beyond. 1

Postdoctoral Research Associate, Dept. of Aerospace Engineering, Korea Advanced Institute of Science and Technology, 291 Daerhak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea (corresponding author). E-mail: [email protected] 2 Professor, Dept. of Aerospace Engineering, Korea Advanced Institute of Science and Technology, 291 Daerhak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea. E-mail: [email protected] 3 Dwight and Aubrey Chapman Professor, Dept. of Aerospace and Mechanical Engineering, Univ. of Arizona, Tucson, AZ 85721. E-mail: [email protected] 4 Assistant professor, Dept. of Mechanical and Aerospace Engineering, New Mexico State Univ., Las Cruces, NM 88003. E-mail: asanyal@nmsu .edu Note. This manuscript was submitted on October 10, 2013; approved on May 22, 2014; published online on August 4, 2014. Discussion period open until January 4, 2015; separate discussions must be submitted for individual papers. This paper is part of the Journal of Aerospace Engineering, © ASCE, ISSN 0893-1321/04014099(13)/$25.00. © ASCE

In this paper, we consider the problem of spacecraft synchronizing its position and attitude with respect to a tumbling target whose attitude is varying quickly with large angle variations. Thus, a spacecraft is required to achieve close-in proximity maneuvering and attitude alignment with a tumbling target. However, this is still a challenging task, especially in the case of a tumbling target (Ma et al. 2007; Terui 1998; Wei et al. 2011; Xin and Pan 2011). When two spacecraft have no relative translational and rotational motions, eventually it will be possible to perform service operations, including space debris removal, and inspection and repair of malfunctioning satellite with a general docking or capturing mechanism. In order to ensure synchronization of attitude and translational motion with a tumbling target, accurate relative motion modeling as well as nonlinear control design for the highly nonlinear kinematics and dynamics is necessary. In general, rigidbody fixed dynamics can be represented as both the translation and rotation about the center of mass (CM). However, since relative translational and rotational dynamics can also be described about arbitrary points, also called feature points, spacecraft relative motion needs to be composed by combining both types of dynamics. When spacecraft are close to each other—such as in the final phase of rendezvous, the docking phase, or capture phase—they can no longer be treated as point masses because the shape and size of spacecraft affects the relative translation between the off-CM points. This effect is accentuated as the distance between the two spacecraft becomes closer (Segal and Gurfil 2009; Alfriend et al. 2010). When a feature point on a spacecraft does not coincide with spacecraft’s CM, a kinematic coupling between the translational and rotational dynamics is generated. In most recent studies, there are two different kinds of couplings. One is incurred by external moments such as the gravity-gradient moment (which is the most obvious example) and depends on the altitude and attitude. The other is incurred regardless of external perturbation moment due to the use of off-CM points for rendezvous and docking. Segal and Gurfil (2009) derived the kinematically coupled relative spacecraft motion model in the assumption that the gravitygradient moment is neglected in both the target and spacecraft. They quantitatively compared the propagated trajectories using

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the kinematically coupled relative equation of motion and the Clohessy-Wiltshire (CW) relative equation of motion, respectively. The kinematic coupling effect is a key for high precision modeling of tight spacecraft formation flying, rendezvous, and docking. For this reason, the linear CW relative equation model can result in considerable errors in the modeling of relative spacecraft translational motion. Thus, a linear control design for complicated spacecraft maneuvers may not be suitable and may lead to a substantially higher fuel cost, especially when long-term maneuvers are required during external disturbances, or when large position or angle maneuvers are required. Various nonlinear control techniques have been investigated in the past to address this problem. Among the many control techniques, the state-dependent Riccati equation (SDRE) method (Chang et al. 2013; Cimen 2010, 2012; Cloutier and Stansbery 2002; Felicetti and Palmerini 2013; Massari and Zamaro 2014; Stansbery and Cloutier 2000) has emerged since the mid-90 s as a general design method that provides systematic and effective means of designing nonlinear controllers, observers, and filters. It provides a very effective algorithm. This method entails a factorization or parameterization of the nonlinear system into the product of the state vector and the matrix valued function that depends on the state vector itself. Thus, the nonlinear system is transformed into a linear structure that has state-dependent coefficient (SDC) matrices and minimizes the nonlinear performance index. Then, the SDRE method can fully capture the nonlinearities of a system. An algebraic Riccati equation (ARE) using the SDC matrices is then solved at each step to provide the suboptimal control law. From a computational standpoint, nonlinear parameterization for SDRE control provides a numerically efficient method that involves only ARE, where a very general set of problems is considered by retaining a feedback control structure. This method has also been widely used for aerospace engineering projects that require real-time control systems. For instance, Cimen (2010, 2012) described systematic and effective design of nonlinear feedback controllers and showed the survey of the SDRE in nonlinear optimal feedback control synthesis. Stansbery and Cloutier (2000) proposed an SDRE nonlinear method of nonlinear regulation to perform tracking of the position and attitude of a spacecraft in the proximity of a tumbling target. Menon et al. (2002) demonstrated the feasibility of implementing the SDRE technique in real-time using commercial, off-the-shelf computers. In this paper, a new kinematically coupled relative spacecraft motion model expressed in the target local-vertical-local-horizontal (LVLH) frame about two arbitrary feature points located on the target and the servicing spacecraft without the assumption used in Alfriend et al. (2010) and Segal and Gurfil (2009) will be derived. Thus, the relative motion will be described about the feature points on the target and servicing spacecraft bodies. Then, the Lˆ t x Lˆ t y Lˆ t z

problem of the driving the servicing spacecraft to a fixed position in the target LVLH frame and orienting the servicing spacecraft along with the tumbling target attitude in the presence of the kinematic coupling is formulated as an optimal tracking problem through a unified SDRE control. The kinematic perturbations that result from the coupling to the rotational dynamics are handled to avoid the singularity in the state SDC parameterization. Numerical simulation results using a six-degree-of-freedom target and the spacecraft show that this method achieves good tracking performances in the presence of the kinematic coupling.

Derivation of Kinematically Coupled Relative Spacecraft Motion Model This section is devoted to deriving kinematically coupled relative spacecraft motion model. The relative equation of motion about arbitrary selected feature points on the target and spacecraft bodies in the LVLH frame is derived. In order to describe the relative equations of motion about arbitrary selected feature points on their bodies, four coordinate systems need to be defined as shown in Figs. 1 and 2. Inertial frame is represented by the Earth-Centered Inertial (ECI) frame fIg ¼ f I^ x I^ y I^ z g. The I^ x axis points toward the vernal equinox; the I^ axis points perpendicular to the y

East in the equatorial plane; the I^ z axis lies perpendicular to the Earth’s equatorial plane through the north pole. The LVLH frame, ^t ; L ^t ; L ^ t g, centered at the CM of the target, has fLt g ¼ f L x y z ^ t axis is the ^ Ltx axis along the target radius vector from the Earth; L z ^ t completes the triad. Bodydirection of the orbital normal and L y

fixed frames of the target and the servicing spacecraft are defined as fBs g ¼ f Bsx ; Bsy ; Bsz g and fBt g ¼ f Btx ; Bty ; Btz g, respectively. Both the target and spacecraft are assumed to be rigid bodies. Arbitrary feature points on the target and spacecraft rigid bodies are considered to describe the relative equations of motion about these points, as shown in Fig. 2. Let PjBt be a point on the j

target body-fixed. Then, PjBt ¼ ½ PBtx

PjBt

y

PjBt T is a position z

vector in the target body-fixed frame fBt g. Similarly, let PjBs be a

j j j T point on the spacecraft body-fixed. Thus, PjBs ¼ ½ PBsx PBsy PBsz is a position vector in the spacecraft body-fixed frame Bs . Let ρij denote the relative position vector between the feature point j on the target body-fixed and the feature point i on the servicing spacecraft body-fixed, whereas ρ is the relative position vector between the CMs of the two bodies in the target LVLH frame. From Fig. 2, one can note that the following geometric relation in the vector form is valid:

PjBt þ ρij ¼ ρ þ PiBs

ð1Þ

LVLH frame{Lt}

rt

Iˆ z

Iˆ z

Iˆ y Inertial frame{I}

Fig. 1. Inertial and LVLH frames © ASCE

Fig. 2. Two rigid-body-fixed spacecraft with feature points in the LVLH and body-fixed-fixed frames 04014099-2

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where T LI t is obtained using the 3-1-3 rotation sequence

Thus, the relative position vector ρij is ρij ¼ ρ þ PiBs − PjBt

ð2Þ

Resolving Eq. (2) in the target LVLH frame using transformation matrices, we can express the relative position as


Lt ρ

ij

¼ Lt ρ þ T LBts PiBs − T LBtt PjBt

ð3Þ

where the superscript Lt in the vector is used to specify that the vector components are taken along the unit directions of the target LVLH frame Lt . The definitions of transformation matrices used in these derivations are listed in the notation list. Note that T LBtt is expressed as the following the successive transformation matrices T LBtt ¼ T LI t T IBt

2

ð4Þ

q2t1 − q2t2 − q2t3 þ q2t4

6 T IBt ¼ 4 2ðqt1 qt2 − qt3 qt4 Þ

2ðqt1 qt3 þ qt2 qt4 Þ

T LI t ¼ T 3 ðθt ÞT 1 ðit ÞT 3 ðΩt Þ

ð5Þ

where θt , it , and Ωt are the argument of latitude, the inclination, and the right ascension of right ascending node of the target orbit, respectively. T IBt is expressed in terms of the target quaternions whose components are defined as qt ¼ ½ qt1 qt2 qt3 qt4 T . The four elements of the target quaternion are defined by qti ¼ eˆ sinðϑ=2Þ;

i ¼ 1; 2; 3

ð6aÞ

qt4 ¼ cosðϑ=2Þ

ð6bÞ

where eˆ = Euler axis of rotation and ϑ = Euler-axis rotation angle. T IBt is given by

3


2ðqt1 qt3 − qt2 qt4 Þ

−q2t1 þ q2t2 − q2t3 þ q2t4



−q2t1 − q2t2 þ q2t3 þ q2t4

7 5

ð7Þ

Note that T LBts is expressed as the following successive transformation matrices: T LBts ¼ T LI t T IBs where the transformation matrix T IBs ½ qs1 qs2 qs3 qs4 T 2

is expressed as spacecraft quaternion whose components are defined as qs ¼

q2s1 − q2s2 − q2s3 þ q2s4

6 T IBs ¼ 4 2ðqs1 qs2 − qs3 qs4 Þ

2ðqs1 qs3 þ qs2 qs4 Þ


2ðqs1 qs3 − qs2 qs4 Þ

−q2s1 þ q2s2 − q2s3 þ q2s4


2ðqs2 qs3 − qs1 qs4 Þ

ρij ¼ Lt ρ þ ðT LI t T IBs ÞPiBs þ ð−T LI t T IBt ÞPjBt

Lt

Lt d Lt d Lt ð ρij Þ ¼ ð ρÞ þ T˙ LBts PiBs þ T LBts P˙ iBs − T˙ LBtt PiBt − T LBtt P˙ jBt dt dt ð11aÞ

Lt d

dt

ðLt ρÞ þ T˙ LBts PiBs þ T LBts P˙ iBs − T˙ LBtt PiBt − T LBtt P˙ jBt

ω×

ð10Þ

Taking time derivative of Eq. (3) is

ρij0 ¼

ð11bÞ

¼

þ

ω1 ω2 ω3

q2s3

#

þ

" ¼

T FF12 ωF2 Þ× T FF12 ¼ −ω×F1 =F2 T FF12 ,

7 5

ð9Þ

q2s4

0 ω3 −ω2

−ω3 0 ω1

ω2 −ω1 0

# ð12Þ

where

ωF1 =F2

is

equal

to

T FF12 ωF2 Þ.

ðωF1 − We can express the time derivatives of T˙ LBts and T˙ LBtt using the above Lemma as

t

Lemma The time derivative of the transformation matrix that transforms from the frame F2 to the frame F1 can be obtained using the intermediate frame that is the inertial frame and the time derivative of transformation matrix ðd=dtÞðT BI Þ ¼ −ω×B=I T BI (Schaub and Junkins 2003) where the operator ðÞ× converts three-dimensional vector into the skew-symmetric matrix defined by

−

q2s2

3

T BI is a transformation matrix from the inertial frame I to body-fixed frame B, and ωB=I is the angular velocity of the frame B relative to the inertial frame I. Then, T˙ FF12 ¼ −ðωF1 −

where the definition of time derivative in the target LVLH frame is represented by ðLt d=dtÞðÞ ¼ ðÞ 0. The time derivatives of T˙ LBts and T˙ LBt are obtained using the following Lemma.

© ASCE

−q2s1

"

Substituting Eqs. (4) and (5) into Eq. (3) gives the following relation for the relative position: Lt

ð8Þ

T˙ LBts ¼ −ω×Lt =Bs T LBts

ð13Þ

T˙ LBtt ¼ −ω×Lt =Bt T LBtt

ð14Þ

where angular velocity vectors ωLt =Bs and ωLt =Bt define the angular velocities of the target LVLH frame Lt relative to the spacecraft body frame Bs and the target body frame Bt , respectively. They are expressed in the target LVLH frame Lt using the transformation matrices T LBts and T LBtt , respectively

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ωLt =Bs ¼ ωLt − T LBts ωBs

ρij0 ¼ ρ 0 þ f−½ω×Lt − ðT LI t T IBs ωBs Þ× T LI t T IBs

ð15Þ

þ T LI t T IBs ½ω×Bs − ðT BI s T ILt ωLt Þ× gPiBs ωLt =Bt ¼ ωLt − T LBtt ωBt

ð16Þ

þ f½ω×Lt − ðT LBts ωBs Þ× T LI t T IBt − T LI t T IBt ½ω×Bt − ðT BI t T ILt ωLt Þ× gPjBt

where ω×Lt =Bs ¼ ω×Lt − ðT LBts ωBs Þ×


ω×Lt =Bt ¼ ω×Lt − ðT LBtt ωBt Þ×

The rotational dynamics related with the angular velocities ωBs and ωBt , and the definitions of ωLt will be explained in the next section. The time derivatives of PiBs and PjBt in the target LVLH frame are denoted as ðLt d=dtÞPiBs and ðLt d=dtÞPjBt , respectively. They are written using the transport theorem18. Since the target and spacecraft are assumed to be rigid bodies, the first time derivatives of feature points in the body-fixed frames are zero Bs d

dt

PiBs

¼

Bt d

dt

PiBt

¼0

Taking time derivative ρij0 in Eq. (28) is

ð17Þ ð18Þ

˙ ×Lt =Bs þ ðω×Lt =Bs Þ2 T LBts − 2ω×Lt =Bs T LBts ω×Bs =Lt ρij0 0 ¼ ρ 0 0 þ f½−ω ˙ ×Bs =Lt þ ðω×Bs =Lt Þ2 gPiBs þ f½ω ˙ ×Lt =Bt − ðω×Lt =Bt Þ2 T LBtt þ T LBts ½ω ˙ ×Bt =Lt þ ðω×Bt =Lt Þ2 gPjBt þ 2ω×Lt =Bt T LBtt ω×Bt =Lt − T LBtt ½ω

dt

PiBs ¼

Lt

Bs d

dt

PiBs þ ωBs =Lt × Pis ¼ ω×Bs =Lt Pis

Bt

d j d j P ¼ P þ ωBt =Lt × Pit ¼ ω×Bt =Lt Pit dt Bt dt Bt

ð19Þ

ð20Þ

ð21Þ

where angular velocity vectors ωBs =Lt and ωBt =Lt define the angular velocities of the spacecraft body frame Bs and the target body frame Bt relative to the target LVLH frame Lt , respectively ωBs =Lt ¼ ωBs − T BLts ωLt

ð30Þ

where ðÞ 0 0 ¼ ðLt d2 =dt2 ÞðÞ is the second time derivative in the ˙ ×Lt =Bs , ω ˙ ×Lt =Bt , ω ˙ ×Bs =Lt and ω ˙ ×Bt =Lt are computed target LVLH frame, ω by taking time derivatives of Eqs. (16), (17), (23), and (24), respectively

Calculating ðLt d=dtÞPiBs and ðLt d=dtÞPjBt results in (Schaub and Junkins 2003; Sidi 1997) Lt d

ð29Þ

˙ ×Lt =Bs ¼ ω ˙ Bs Þ × ˙ ×Lt þ ðω×Lt =Bs T LBts ωBs − T LBts ω ω

ð31Þ

˙ ×Lt =Bt ¼ ω ˙ ×Lt þ ðω×Lt =Bt T LBtt ωBt − T LBtt ω ˙ Bt Þ× ω

ð32Þ

˙ Lt Þ × ˙ ×Bs =Lt ¼ ω ˙ ×Bs þ ðω×Bs =Lt T BLts ωLt − T BLts ω ω

ð33Þ

˙ Lt Þ × ˙ ×Bt =Lt ¼ ω ˙ ×Bt þ ðω×Bt =Lt T BLtt ωLt − T BLtt ω ω

ð34Þ

Substituting Eqs. (4), (8), (16), (17), (22)–(25), and (31)–(34) into Eq. (30) gives the following relation for the relative acceleration: ˙ ×Lt þ f½ω×Lt − ðT LI t T IBs ωBs Þ× T LI t T IBs ωBs ρij0 0 ¼ ρ 0 0 þ ½−ðω ˙ Bs g× Þ þ ½ω×Lt − ðT LI t T IBs ωBs Þ× 2 T LI t T IBs − T LI t T IBs ω

ð22Þ

− 2½ω×Lt − ðT LI t T IBs ωBs Þ× T LI t T IBs ½ω×Bs − ðT BI s T ILt ωLt Þ× ωBt =Lt ¼ ωBt − T BLtt ωLt

˙ ×Bs þ f½ω×Bs − ðT BI s T ILt ωLt Þ× T BI s T ILt ωLt þ T LI t T IBs ½ðω

ð23Þ

˙ Lt g× Þ þ ½ω×Lt − ðT LI t T IBs ωBs Þ× 2 PiBs − T BI s T ILt ω

where ðT BLts ωLt Þ×

ð24Þ

ω×Bt =Lt ¼ ω×Bt − ðT BLtt ωLt Þ×

ð25Þ

ω×Bs =Lt

T BLts

¼

ω×Bs

−

˙ ×Lt þ f½ω×Lt − ðT LI t T IBs ωBs Þ× T LI t T IBs ωBs þ ½ðω ˙ Bs g× Þ − ½ω×Lt − ðT LI t T IBs ωBs Þ× 2 T LI t T IBt − T LI t T IBs ω þ 2½ω×Lt − ðT LI t T IBt ωBt Þ× T LI t T IBt ½ω×Bs − ðT BI s T ILt ωLt Þ× ˙ ×Bs þ f½ω×Bs − ðT BI s T ILt ωLt Þ× T BI s T ILt ωLt − T LI t T IBt ½ðω

T BLtt

˙ Lt g× Þ þ ½ω×Lt − ðT LI t T IBs ωBs Þ× 2 PjBt − T BLts ω

and are expressed as the following the successive transformation matrices: T BLts ¼ T BI s T ILt

ð26Þ

T BLtt ¼ T BI t T ILt

ð27Þ

ρij0 ¼ ρ 0 þ ð−ω×Lt =Bs T LBts þ T LBts ω×Bs =Lt ÞPiBs þ ðω×Lt =Bt T LBtt − T LBtt ω×Bt =Lt ÞPjBt

ð28Þ

In order to simply express ρij , ρij0 , and ρij0 0 using their components along the target LVLH frame Lt for the controller design in the next section, let the sum of two terms related with PiBs and PjBt in Eqs. (10), (29), and (35) be ρP , ρP0 , and ρP0 0 , respectively. ρP , ρP0 , and ρP0 0 are the position, velocity and acceleration variations generated by the selection of arbitrary feature points on both bodies instead of both CMs. They are expressed in terms of T IBs , T IBt , ˙ Bt , ω ˙ Bs , ω L t , ω ˙ Lt , their skew symmetric matrices T LI t , ωBt , ωBs , ω and feature points PiBs and PjBt . Thus,

Substituting Eqs. (4), (8), (15), (16), and (24)–(27) into Eq. (28) gives the following relation for the relative velocity: © ASCE

ð35Þ

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ρP ¼ ðT LI t T IBs ÞPiBs þ ð−T LI t T IBt ÞPjBt

ð36Þ J. Aerosp. Eng.

ρP0 ¼ f−½ω×Lt − ðT LI t T IBs ωBs Þ× T LI t T IBs

x 0 0 − 2f˙ t y 0 0 − f¨ t y − f˙ 2t x ¼ −

þ T LI t T IBs ½ω×Bs − ðT BI s T ILt ωLt Þ× gPiBs

ð42aÞ

þ f½ω×Lt − ðT LBts ωBs Þ× T LI t T IBt − T LI t T IBt ½ω×Bt − ðT BI t T ILt ωLt Þ× gPjBt

˙ ×Lt þ f½ω×Lt − ðT LI t T IBs ωBs Þ× T LI t T IBs ωBs ρP0 0 ¼ ½−ðω Downloaded from ascelibrary.org by ARIZONA,UNIVERSITY OF on 09/09/14. Copyright ASCE. For personal use only; all rights reserved.

y 0 0 þ 2f˙ t x 0 þ f¨ t x − f˙ 2t y ¼ −

ð37Þ

z00 ¼ −

˙ Bs g × Þ − T LI t T IBs ω

þ ½ω×Lt − ðT LI t T IBs ωBs Þ× 2 T LI t T IBs − 2½ω×Lt − ðT LI t T IBs ωBs Þ× T LI t T IBs ½ω×Bs − ðT BI s T ILt ωLt Þ× ˙ ×Bs þ f½ω×Bs − ðT BI s T ILt ωLt Þ× T BI s T ILt ωLt þ T LI t T IBs ½ðω ˙ Lt g× Þ þ ½ω×Lt − ðT LI t T IBs ωBs Þ× 2 PiBs − T BI s T ILt ω ˙ ×Lt þ f½ω×Lt − ðT LI t T IBs ωBs Þ× T LI t T IBs ωBs − T LI t T IBs ω ˙ Bs g × Þ þ ½ðω

− ðT LI t T IBs ωBs Þ× 2 PjBt

00

z þ

ð45Þ

The orbital angular acceleration of the target is given by ˙ Lt ¼ ½ 0 ω

0 f¨ t T

ð46Þ

We can derive a more general model for the case where PiBs and are not zero vectors. This model is obtained by substituting Eqs. (39)–(41) into Eq. (42). Then it produces the following general description of the translational motion between any arbitrary points Pjt and Pis in the absence of perturbing forces: PiBt

xij0 0 − ρP0 0x − 2f˙ t ðyij0 − ρP0 y Þ − f¨ t ðyij − ρPy Þ − f˙ 2t ðxij − ρPx Þ ¼−

μðrt þ xij − ρPx Þ μ þ 2 r3ij rt

ð47aÞ

yij0 0 − ρP0 0y þ 2f˙ t ðxij0 − ρP0 x Þ þ f¨ t ðxij − ρPx Þ − f˙ 2t ðyij − ρPy Þ ¼− ð40Þ

z 0 þ ρP0 z

and the relative acceleration components 2 00 3 2 x00 þ ρ00 3 xij Px 6 y00 7 6 y00 þ ρ00 7 4 ij 5 ¼ 4 Py 5

ð44Þ

r˙ f¨ t ¼ −2 t f˙ t rt

z þ ρ Pz

The relative velocity components 2 0 3 2 x0 þ ρ0 3 xij Px 6 y0 7 6 y0 þ ρ0 7 4 ij 5 ¼ 4 Py 5

0 f˙ t T

The orbital angular acceleration satisfies

ð38Þ

These terms in Eq. (38) result from the kinematic coupling to the rotational dynamics about the selected feature points and can be processed as a kinematic perturbation (Alfriend et al. 2010; Segal and Gurfil 2009). This perturbation always exists, regardless of the orbit altitude and external perturbations. It is known that this perturbation is accentuated as the relative distance becomes small. Expressing the vectors using theirs components along the target LVLH frame Lt , ρ ¼ ½ x y z T , ρ ¼ ½ xij yij zij T , ρP ¼ ½ ρPx ρPy ρPz T , ρP0 ¼ ½ ρP0 x ρP0 y ρP0 z T , ρP00 ¼ ½ ρP00x ρP0 0y ρP0 0z T gives the following the relative position components: 2 3 2 3 x þ ρ Px xij 6y 7 6y þ ρ 7 ð39Þ 4 ij 5 ¼ 4 Py 5

zij0 0

ð42cÞ

2

˙ Lt ¼ ½ 0 ω

˙ L t g× Þ ˙ ×Bs þf½ω×Bs − ðT BI s T ILt ωLt Þ× T BI s T ILt ωLt −T BLts ω − T LI t T IBt ½ðω

zij0

μz ½ðrt þ xÞ þ y2 þ z2 3=2

ð42bÞ

The orbital angular velocity of the target is given by

þ 2½ω×Lt − ðT LI t T IBt ωBt Þ× T LI t T IBt ½ω×Bs − ðT BI s T ILt ωLt Þ×

zij

μy ½ðrt þ xÞ2 þ y2 þ z2 3=2

where μ is the gravitational parameter, f represents the true anomaly of the target, and rt is the current orbit radius of the target. The true anomaly rate or the orbital angular speed of the target is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ f˙ t ¼ ð43Þ ð1 þ et cos ft Þ2 3 at ð1 − e2t Þ3

− ½ω×Lt − ðT LI t T IBs ωBs Þ× 2 T LI t T IBt

þ ½ω×Lt

μðrt þ xÞ μ þ 2 2 2 2 3=2 rt ½ðrt þ xÞ þ y þ z

μðyij − ρPy Þ r3ij

ð47bÞ

μðzij − ρPz Þ ð47cÞ r3ij qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where rij ¼ ðrt þ xij − ρPx Þ2 þ ðyij − ρPy Þ2 þ ðzij − ρPz Þ2 . These equations are coupled with the target and spacecraft rotational dynamics that will be explained in the next section through the components of angular velocities and angular accelerations ˙ Bs and ω ˙ Bt ), and the orbital angular velocity (ωBs and ωBt ) and (ω ˙ Lt . and acceleration of the target ωLt and ω zij0 0 − ρP0 0z ¼ −

ð41Þ

ρP0 0z

Kinematically Coupled Relative Spacecraft Motion Dynamics Rotational Dynamics The translational motion model describing the relative motion model between the target and spacecraft CMs ρ¨ is a model for the case where PiBs ¼ PiBt ¼ 0. A complete derivation of the relative equations of motion with no disturbances for eccentric orbits is found in Alfriend et al. (2010) and Schaub and Junkins (2003). It is given by (Alfriend et al. 2010; Segal and Gurfil 2009) © ASCE

This section shows the attitude kinematics and dynamics of the target and spacecraft, respectively. They are used to describe the kinematically coupled relative spacecraft motion model as well as the spacecraft attitude control. The attitude kinematics and dynamics of the target are given by

04014099-5

J. Aerosp. Eng.

J. Aerosp. Eng.


1 q˙ t ¼ Ωt ðωBt Þqt 2

ð48Þ

˙ Bt þ ω ~ Bt J t ωBt ¼ MBt Jt ω

ð49Þ

where J t , ωBt ¼ ½ ωBtx ωBty ωBtz T and MBt are the moment of inertia, angular velocity, and the Earth gravity-gradient moment of the target in the target body-fixed frame, respectively. ω×Bt and ΩðωBt Þ are defined as 2 3 0 −ωBtz ωBty 7 ~ Bt ¼ 6 ω ð50Þ 4 ωBt 0 −ωBt 5 z

−ωBty

x

ωBtx

2

0

q2t1 − q2t2 − q2t3 þ q2t4

6 T BI t ¼ 4 2ðqt1 qt2 − qt3 qt4 Þ


Ωt ðωBt Þ ¼



ð55Þ

where J s , ωBs ¼ ½ ωBsx ωBsy ωBsz T , MBs , ΓBs are the spacecraft moment of inertia, spacecraft angular velocity, the Earth gravity-gradient moment of spacecraft, and the control moment, respectively, in the spacecraft body-fixed frame. ω×Bs and ΩðωBs Þ are defined as 3 2 ω Bs y 0 −ωBsz 7 6 ω×Bs ¼ 4 ωBs ð56Þ 0 −ωBs 5 x

2

0

q2t1 − q2t2 − q2t3 þ q2t4

6 T BI t ¼ 4 2ðqt1 qt2 − qt3 qt4 Þ


−q2t1

−

q2t2

þ

q2t3

þ

Ωs ðωBs Þ ¼

3 7 5

−ω×

Bs

−ωTBs

ð53Þ

ωBs 0

MBs ¼ 3ðμ=r5s Þ½ðT BI s rs Þ × J s ðT BI s rs Þ

ð57Þ

ð58Þ

where rs is the spacecraft position vector and T BI s is the transformation matrix from the inertial frame to the target body-fixed frame and is defined in terms of the target quaternions qt :


−q2t1 þ q2t2 − q2t3 þ q2t4



−q2t1

−

q2t2

þ

q2t3

þ

3 7 5

ð59Þ

q2t4

x˙ ¼ fðxÞ þ BðxÞu

with respect to the state x and control u subject to the nonlinear differential constraints

ð52Þ

q2t4


Let us consider the autonomous, infinite-horizon, general nonlinear regulator problem for minimizing the cost function Z 1 ∞ T J¼ ½x QðxÞx þ uT RðxÞudt ð60Þ 2 0

ð51Þ

and the gravity gradient moment about the CM of the target is given by

Summary of the SDRE Method

© ASCE

0

where rt is the target position vector in the inertial frame, and T BI t is the transformation matrix from the inertial frame to the target body-fixed frame and is defined in terms of the target quaternions qt :

−q2t1 þ q2t2 − q2t3 þ q2t4

˙ Bs þ ω~ Bs J s ωBs ¼ MBs þ ΓBs Js ω

ωBt

MBt ¼ 3ðμ=r5t Þ½ðT BI t rt Þ × J t ðT BI t rt Þ


ð54Þ

ωBsx

−ωTBt


1 q˙ s ¼ Ωs ðωBs Þqs 2

−ωBsy

Bt

and the gravity gradient moment about the CM of the target is given by

The attitude kinematics and dynamics of spacecraft are given by

z

−ω×

ð61Þ

where QðxÞ ≥ 0 and RðxÞ > 0 for all x and fð0Þ ¼ 0. The state and input weighting matrices, which are design parameters, are assumed to be state-dependent. The details of the SDRE method can be found in Chang et al. (2013), Cimen (2010, 2012), Cloutier and Stansbery (2002), Felicetti and Palmerini (2013), Massari and Zamaro (2014), and Stansbery and Cloutier (2000). The SDRE design method consists of the following steps. First, use the direct

04014099-6

J. Aerosp. Eng.

J. Aerosp. Eng.

parameter method to bring Eq. (62) into the SDC form, which is a linear-like structure

Cloutier and Stansbery 2002; Stansbery and Cloutier 2000). The modified integral servomechanism is given by

x˙ ¼ AðxÞx þ BðxÞu

uðxÞ ¼ −RðxÞ−1 BðxÞT PðxÞ½xðtÞ − xr ðtÞ

ð62aÞ

where xr ðtÞ is the reference state to be tracked.

where


fðxÞ ¼ AðxÞx

ð62bÞ

Although the SDC parameterization is unique in the case of scalar x for all x ≠ 0, it is not unique in the multivariable case such that the SDC parameterization AðxÞ itself can be parameterized as Aðx; αÞ, where α is a vector of free design parameters13-16. The introduction of α creates extra degrees of freedom that are not available in traditional methods. These additional degrees of freedom provided by the non-uniqueness of the SDC parameterization can be used not only to improve controller performance, but also to avoid singularities or loss of controllability, as well as effect tradeoffs between performance, optimality, stability, robustness, and disturbance rejection, thus offering a more flexible nonlinear optimal policy. To obtain a valid solution of the SDRE, the pair fAðxÞ; BðxÞg has to be pointwise stabilized in the linear sense so that for all x in the domain of interest, a feasible (i.e., positive definite) solution may be obtained. Second, solve the SDRE PðxÞAðxÞ þ AðxÞT PðxÞ − PðxÞBðxÞRðxÞ−1 BðxÞT PðxÞ þ QðxÞ ¼ 0 ð63Þ where PðxÞ is state-dependent and positive definite for x ≠ 0. Third, construct the nonlinear feedback controller equation uðxÞ ¼ −RðxÞ−1 BðxÞT PðxÞxðtÞ ¼ −KðxÞxðtÞ

ð64Þ

where the gain KðxÞ ¼ RðxÞ−1 BðxÞT PðxÞ. In order to perform tracking control or reference following, the SDRE controller can be implemented as a modified integral servomechanism to achieve the time-varying reference state in a simple form (Lee et al. 2012). However, the integral servomechanism includes the time-integral of position vector in the state vector (Cimen 2010, 2012;

2

4 − 3 cos nt

6 6 6 6 6ðsin nt − ntÞ 6 6 6 Φðt; t0 Þ ¼ 6 0 6 6 6 3n sin nt 6 6 4 −6nð1 − cos ntÞ 0

0

Φðt; t0 Þ ¼

MðtÞ NðtÞ SðtÞ

TðtÞ

Reference States The reference positions and velocities for the translational maneuver are generated using the CW guidance scheme Eqs. (20) and (21) to derive a desired position in the target LVLH frame. The target quaternions and angular velocities are assumed to be available through onboard navigation and are generated using the target rotational dynamics in Eqs. (49) and (50), leading to the attitude of a tumbling rigid body-fixed. Reference Position and Velocity The state transition matrix obtained from the CW equation is used to generate a reference trajectory that will allow the servicing spacecraft to approach the desired point relative to the target. The CW’s state transition matrix (STM) is obtained from the analytical solution of the CW equation. Using the CW’s STM, the initial velocity to intercept the target or the desired position can be determined (Clohessy and Wiltshire 1960; Prussing and Conway 1993). Then, the time varying reference positions and velocities to are propagated from the initial position and velocity to achieve the desired position. If a relative state vector is defined as ρr xr ðtÞ ¼ ð66Þ ¼ ½ xr yr zr xr0 yr0 zr0 T ρr0 then the relative position and velocity are propagated using the STM as ρr ðt0 Þ ρr ðtÞ ¼ Φðt; t0 Þ 0 ð67Þ ρr0 ðtÞ ρr ðt0 Þ where

sin nt n

0

− n2 ð1

2 n ð1

− cos ntÞ

4 sin nt − 3nt − cos ntÞ n

1

0

0

cos nt

0

0

0

0

cos nt

2 sin nt

0

0

−2 sin nt

4 cos nt − 3

0

−n sin nt

0

0

where Φðt; t0 Þ is the state transition matrix obtained from the CW equations and n is the mean motion of the target. The state transition matrix Φðt; t0 Þ is partitioned into four 3 × 3 partitions as

© ASCE

ð65Þ

ð69Þ

0

3

7 7 7 0 7 7 7 sin nt 7 7 n 7 7 0 7 7 7 0 5

ð68Þ

cos nt

The necessary initial relative velocity to intercept the target at the final time is then obtained as ρr0 ðt0 Þ ¼ N −1 ðtf Þ½rd ðtf Þ − Mðtf Þρr ðt0 Þ

ð70Þ

where rd ðtf Þ is the desired final position, t0 is the initial time, and tf is the final flight time. The reference positions and velocities are obtained to intercept the desired final position rd ðtf Þ by

04014099-7

J. Aerosp. Eng.

J. Aerosp. Eng.

propagating the initial reference state from the initial time to the final flight time.

zij0 0 − ρP0 0z ¼ − where


SDRE Control Design for Coupled Position and Attitude Maneuver

ρij0T

qTs

xij ∈ ℝ9 × S3

ωTs T ;

ð71Þ control variables ∶ u ¼ ½ FTLt

ΓTBs T ;

u ∈ ℝ6

ð72Þ

xij0 0 − ρP0 0x − 2f˙ t ðyij0 − ρP0 y Þ − f¨ t ðyij − ρPy Þ − f˙ 2t ðxij − ρPx Þ μðrt þ xij − ρPx Þ FLt x þ ms r3ij

ð73aÞ

× −1 −1 ˙ Bs ¼ −J −1 ω s ω Bs J s ω s þ J s M Bs J s Γ Bs

yij0 0 − ρP0 0y þ 2f˙ t ðxij0 − ρP0 x Þ þ f¨ t ðxij − ρPx Þ − f˙ 2t ðyij − ρPy Þ ¼−

μðyij − ρPy Þ r3ij

þ

F Lt y ms

2 0 3×3 6a 6 4∶6×1∶3 6 6 6 01×3 6 6 6 6 01×3 6 6 6 6 01×3 6 Aðxij Þ ¼ 6 6 6 01×3 6 6 6 6 01×3 6 6 6 6 01×3 6 6 4 01×3

03×3

03×1

03×1

03×1

03×1

03×1

03×1

a4∶6×4∶6

03×1 εω 2

03×1

03×1

03×1

0

0

0

0

0 0

03×1 qs4 2 qs3 2 −qs2 2 −qs1 2

03×1 qs − 3 2 qs4 2 qs1 2 −qs2 2

0

0

0

0

0

0

01×3 01×3

0

εω 2

01×3

0

0

εω 2

01×3

0

0

0

01×3

a11;7 qs1 ðqTs qs Þ a12;7 qs1 ðqTs qs Þ a13;7 qs1 ðqTs qs Þ



01×3 01×3

2

a4∶6×1∶3

μ f˙ 2 − 3 6 rc 6 6 ð−f¨ t Þ ¼6 6 6 4 0

ð75Þ

For the SDRE formulation, the particular Aðxij Þ is selected for the nonlinear state equation using Eqs. (73)–(75) through the SDC parameterization

ð73bÞ

where 0m×n is a m × n null matrix

© ASCE

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrt þ xij − ρPx Þ2 þ ðyij − ρPy Þ2 þ ðzij − ρPz Þ2

However, the spacecraft attitude kinematics in Eq. (74) is modified from Eq. (54) since no solution exists in the algebraic Riccati equation when the quaternion kinematics in Eq. (54) is used. For this reason, a small constant εω ¼ −0.0001 is added to the spacecraft quaternion kinematics to obtain the solution of the algebraic Riccati equation (Stansbery and Cloutier 2000; Terui 1998). Notice that the addition of εω correction is only an artifact since the quaternions parameters represent only three independent parameters. Therefore generating the numerical issue. The servicing spacecraft attitude dynamics for the SDC parameterization is given by

The state equations can be written using Eqs. (47), (54), and (55) as

¼−

ð73cÞ

and vector components ρP ¼ ½ ρPx ρPy ρPz T , ρP0 ¼ ½ ρP0 x ρP0 y ρP0 z T , and ρP0 0 ¼ ½ ρP0 0x ρP0 0y ρP0 0z T are already defined in Eqs. (39)–(41), and FLt x , FLt y , and FLt z are the control force elements in the target LVLH frame, and ms is the spacecraft mass 3 2 εωt qs1 þ qs4 ωs1 − qs3 ωs2 þ qs3 ωs3 7 6 1 6 εωt qs2 þ qs3 ωs1 þ qs4 ωs2 − qs1 ωs3 7 7 q˙ s ¼ 6 ð74Þ 7 26 4 εωt qs3 − qs2 ωs1 þ qs1 ωs2 þ qs4 ωs3 5 εωt qs4 − qs1 ωs1 − qs2 ωs2 − qs3 ωs3

The objectives of the controller design are to have the spacecraft, the relative position, and velocity ρij and ρij0 track the reference relative position and velocity ρr and ρ˙ r at a certain distance from the target—located at the origin of the Hill frame—in the presence of the kinematic coupling. It is also desirable to have the spacecraft body-fixed frame Bc f bcx ; bcy ; bcz g align with the target bodyfixed frame Bt f btx ; bty ; btz g. The state and control variables for this problem are chosen to be state variables∶ xij ¼ ½ ρTij

rij ¼

μðzij − ρPz Þ FLt z þ ms r3ij

f˙ 2t

μ f˙ 2 − 3 rij 0

04014099-8

J. Aerosp. Eng.

εω 2 a11;7 qs4 ðqTs qs Þ a12;7 qs4 ðqTs qs Þ a12;7 qs4 ðqTs qs Þ

0

03×1 3 03×1 7 7 qs2 7 7 7 2 7 −qs1 7 7 7 2 7 qs4 7 7 7 2 7 −qs3 7 7 7 2 7 7 7 0 7 7 7 7 0 7 7 7 5 0

ð76aÞ

3

7 7 7 7 0 7 7 μ 5 − 3 rij

ð76bÞ

J. Aerosp. Eng.

8 > > > > > > > >
r2t r3c > > > > > μ μ > 0 0T 0 > x − r =ðρ ρ Þ : t ij ij r2t r3c

× × a13;7 T ¼ J −1 s ð−ωBs J s ωBs þ MBs Þ

a12;7

03×3 61 6 I 3×3 6 Bðxij Þ ¼ 6 m 6 0 4 4×3 03×3

03×3

ð77Þ

J −1 s

104 I 3×3

6 6 03×3 Q¼6 6 4 04×3 03×3

Rðxij Þ ¼

03×3

03×4

03×3

03×4

04×3

103 I 4×4

03×3

03×4

03×3

3

7 03×3 7 7 7 04×3 5

ð78Þ

03×3

10−1 I 3×3

03×3

03×3

10−1 I 3×3

7 7 7 7 5

ρ˙ ij − ρ˙ r qs − qt

ð80Þ

Numerical Results and Analysis A six-degree-of-freedom simulation of a tumbling target and a spacecraft for the kinematically coupled motion is implemented to describe the relative motion about the selected feature points on both bodies using the SDRE control method. The spacecraft is required to achieve a desired position and attitude alignment with the target in the close proximity of a tumbling target. The simulation assumes the target CM is located at the origin of the inertial frame with zero velocity, as shown in Fig. 1. The target moment of the inertial matrix is 2 3 1.0 0 0 6 7 J t ¼ J t11 4 0 0.5 0 5 kg m2 ; J t11 ¼ 3000 ð81Þ 0

1.0

Spacecraft is assumed to have a constant mass of 3,000 kg with moment of inertia 2 3 3000 −300 −500 6 7 I s ¼ 4 −300 3000 −400 5 kg m2 ð82Þ −500

−400

3000

The target is orbiting the Earth with the orbital elements listed in Table 1. Two feature points on both bodies are selected as P1Bt ¼ ½ −2.0 1.0 −0.5 T m and P1Bs ¼ ½ 1.5 −1.5 0 T m, respectively. A special feature point on the target body-fixed is also selected as the target CM P0Bt ¼ ½ 0 0 0 T m to compare with nonzero feature point case. The relative positions and velocities about the CMs of the feature points (P1Bt ¼ ½ −2.0 1.0 −0.5 T m and P1Bs ¼ ½ 1.5 −1.5 0 T m), and (P0Bt ¼ ½ 0 0 0 T m and P1Bs ¼ ½ 1.5 −1.5 0 T m are denoted 0 ðtÞ, and ρ ðtÞ and ρ 0 ðtÞ, respecas ρðtÞ and ρ 0 ðtÞ, ρ11 ðtÞ and ρ11 10 10 tively. The initial conditions for feature points on the bodies that

ð79Þ

Table 1. Orbital Elements of the Target Orbital elements

Finally, the SDRE control is implemented as the modified integral servomechanism in Eq. (80) to track the time-varying reference positions and velocities, and the target’s quaternions and angular velocities. For the angular velocity tracking, the successive transformation matrices from the target body-fixed frame to the spacecraft body-fixed frame are used © ASCE

3

ωBs − T BI s T IBt ωBt

0

7 03×3 7 7 7 04×3 7 5

ð76cÞ

ρij − ρr

6 6 uðxij Þ ¼ −Rðxij Þ−1 Bðxij ÞT Pðxij Þ6 6 4

3

where I m×n is an m × n identity matrix. The satisfying state weighting matrix and the control weighting matrix for the cost function used in Eq. (60) are selected, respectively, as (Stansbery and Cloutier 2000; Xin and Pan 2011) 2

2

ð76dÞ

When the nonlinear system equations in Eqs. (73)–(75) are factored into a linear-like structure via the SDC parameterization, there is a violation due to the presence of state-independent terms or bias. During a translational maneuver, its velocity component can become zero, whereas its speed never becomes zero. Using the property that the speed of the spacecraft kρij0 k; never becomes zero, these can be multiplied and divided in Eq. (73) by the squared magnitude of the velocity, even though any component of spacecraft velocity v can become zero. This bias term bðtÞ can then be factored as bðtÞ ¼ ½bðtÞvT =ðvT vÞv (Cimen 2010, 2012; Stansbery and Cloutier 2000). The spacecraft rotational dynamics given in Eq. (75) cannot be directly factored into a linear form because of the first term, though there is no bias term. However, the rotational dynamics given in Eq. (75) can be factored into a linear form using the property that the square of the quaternion never goes to zero, thus leading to bðtÞ ¼ ½bðtÞqT =ðqT qÞq. As a result, the singularity in the SDC parameterization is avoided and the system matrix AðxÞ is obtained. The control distribution matrix is 2

9 μ μ > 0 0T 0 > z − r =ðρ ρ Þ > t ij ij > > r2t r3c > > > = μ μ 0 0T 0 z − r =ðρ ρ Þ ij ij 2 3 t > rt rc > > > > > μ μ > 0 0T 0 > z − r =ðρ ρ Þ ; ij ij 2 3 t rt rc

μ μ 0 0T 0 y − r =ðρ ρ Þ t ij ij r2t r3c μ μ 0 0T 0 y − r =ðρ ρ Þ t ij ij r2t r3c μ μ 0 0T 0 y − r =ðρ ρ Þ t ij ij r2t r3c

Semi-major axis Eccentricity Inclination Right ascension of ascending node Argument of perigee Mean anomaly

04014099-9

J. Aerosp. Eng.

Values 6,739.188 km 0.0005817 51.64 degrees 316.44 degrees 40.90 degrees 319.24 degrees J. Aerosp. Eng.

Table 2. Initial Conditions for Translational Maneuver Relative position (m)

Relative velocity (m=s)

ρðt0 Þ ¼ ½ 25 25 50T

ρ 0 ðt0 Þ ¼ ½ −0.188 −0.204 −0.4145T 0 ðt Þ ¼ ½−0.220 −0.189 −0.427 T ρ11 0

ρ11 ðt0 Þ ¼ ½ 28.50 22.72 51.60 T ρ10 ðt0 Þ ¼ ½ 26.50 23.81 51.35

0 ρ10 ðt0 Þ ¼ ½−0.202 −0.196 −0.425 T

T

Table 3. Initial Conditions for Rotational Maneuver Quaternions

Angular velocity (rad=s)

qt ðt0 Þ ¼ ½ 0 0

0

ωBt ðt0 Þ ¼ ½ 0 0.0100 −0.0273T

1 T

ωBt ðt0 Þ ¼ ½ 0

0

0 T

do not coincide with the CMs are calculated using Eqs. (10) and (28). Initial conditions for translational maneuver are listed in Table 2. In order to test the tracking capability of the controller, the relative position error ½ 5 5 5 T m is added. Initial conditions including quaternions and angular velocities of the target and spacecraft for rotational maneuver are listed in Table 3. From these given parameters and initial conditions including initial relative position error, the spacecraft is required to achieve the desired relative position rd ðtf Þ ¼ ½ 0 3.5 0 T m in the target −3

3 2 Mgx

Mgy

Mgz

1

Normal (m) Along-Track (m)

Earth Gravity−Gradient Moment (Nm)

Radial (m)

x 10

4

0 −1 −2 −3 −4

0

100

200

300

400

500

600

Time (sec)

40 20 00

100

200

300

ρref

30 20 10 0

0

100

200

0

100

200

400 ρ10

500

600

ρ11

300

400

500

600

300

400

500

600

50

0

Time (sec)

Fig. 3. Gravity-gradient moments of the target

Fig. 5. Reference and spacecraft relative positions −3

x 10

Mgx

Mgy

Radial (m)

3

Mgz

2

Normal (m) Along-Track (m)

Earth Gravity−Gradient Moment (Nm)


qt ðt0 Þ ¼ ½0.2474 0 0 0.9689T

LVLH frame while tracking the reference trajectory for 120 s. Figs. 3 and 4 show the Earth gravity-gradient moments of acting on the target and spacecraft bodies, respectively. Fig. 5 shows that the relative position about nonzero, twofeature points tracks the reference trajectory leading to the desired position ½ 0 3.5 0 T m, expressed in the target LVLH frame. Fig. 6 shows the relative position deviations due to the kinematic coupling effect. These deviations are defined as Δρ10 ¼ ρ10 − ρ and Δρ11 ¼ ρ11 − ρ, respectively. The deviations in Fig. 6 obviously demonstrate that the relative motion about the target and spacecraft CMs does not coincide with the relative motion about feature points on the bodies. In addition, the deviations Δρ10 and Δρ11 show almost harmonic oscillations responses in each direction. The magnitude of these oscillations depends on the location of the feature points on the bodies and the angular velocities. Figs. 7 and 8 show the norms of the relative position and velocity tracking errors, respectively. They converge to less than 0.04 m and 0.003 m=s, respectively, in 30 s. The spacecraft achieves both the desired or reference position and velocity in about 30 s, even in the presence of 1 m position errors in each component. Because of the kinematic coupling, both tracking errors should have more time to converge to zero. However, these tracking errors can be reduced by commanding larger control force by adjusting the weighting matrices. Figs. 9 and 10 show the approach trajectories ρ11 about the feature points versus the reference trajectories ρref in

1

0

−1

−2

0

100

200

300

400

500

600

0 −5

0

100

200

300

∆ρ10

400

500

600

∆ρ11

5 0 −5

0

100

200

0

100

200

300

400

500

600

300

400

500

600

5 0 −5

Time (sec)

Time (sec)

Fig. 4. Gravity-gradient moments of the servicing spacecraft © ASCE

5

Fig. 6. Relative position deviations due to feature points 04014099-10

J. Aerosp. Eng.

J. Aerosp. Eng.

1

60

Start rho

40

ρref

0

0

5

10

15

20

25

btz (m)

11

30

Destination

20 0

0.04

−20 50 50

0.02 0

0

100

200

300

400

500

0 −50

bty (m)

600

−50

btx (m)

Time (sec)

Fig. 10. Approach trajectories in the target body-fixed frame

Fig. 7. Norm of position tracking error

1 0.3

Spacecraft

0.2 0.5 0.1 0

0

5

10

15

20

25

Quaternions

Norm of Velocity Tracking Error (m/s)

Target

30

−3

x 10 4

0

−0.5

2 0

−1 100

200

300

400

500

600

0

100

200

φ (deg) 30 20

0

100

200

11

θ (deg)

ρref Destination

0

400

500

600

100 0 −100

ψ (deg)

−10 100 30

50

300

Target (Reference) Spacecraft

ρ

10

20 10 0

600

0

Start

−50

500

200

−200

Cross-Track (m)

400

Fig. 11. Target and spacecraft quaternions

Fig. 8. Norm of velocity tracking error

0

300

Time (sec)

Time (sec)

Radial (m)


Norm of Position Tracking Error (m)

2

Along-Track (m)

0

100

200

0

100

200

300

400

500

600

300

400

500

600

100 0 −100

Time (sec)

Fig. 9. Approach trajectories in the LVLH frame Fig. 12. Euler angles

the target LVLH frames and the target body-fixed frames, respectively. The spacecraft can track the reference trajectory from the given initial position, including the position error as shown in Fig. 9. The approach trajectories in Fig. 10 vary arbitrary because they are expressed in quickly attitude-varying tumbling target bodyfixed frame. Fig. 11 shows that the quaternions of the servicing spacecraft track the target quaternions. Fig. 12 shows the Euler angles that are converted from the quaternions to illustrate the physics © ASCE

of the angular motion more obviously. The roll, pitch, and yaw in Euler angles are expressed as in 3-2-1 rotation sequence. All Euler angles in Fig. 12 converge to less than 0.3° in 60 s. Fig. 13 shows the quaternion errors between the tumbling target and the spacecraft. It converges to the desired quaternion error ½ 0 0 0 0 T . Fig. 14 shows the attitude tracking errors more intuitively through Euler angles errors between the target and the servicing spacecraft. Like the position tracking error, the attitude tracking error can also be reduced by commanding the

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0.12 ωx

Relative Angular Velocity (rad/s)

1

Quaternion Errors

0.8

q

1

q2

0.6

q

3

0.4

q

4

0.2 0

0.1

ωy ωz

0.08 0.06 0.04 0.02 0 −0.02

0

100

200

300

400

500

600

0

100

200

Time (sec)

300

400

500

600

Time (sec)

Fig. 13. Quaternion errors

Fig. 16. Angular velocity tracking errors

50

10

0 −10

Roll

Control Force (N)

Euler Angle Errors (deg)

0

Pitch −20

Yaw

−30 −40

F

F

x

F

y

z

−50

−100

−50 −150

−60 0

100

200

300

400

500

0

600

100

200

300

400

500

600

Time (sec)

Time (sec)

Fig. 17. Control force

Fig. 14. Euler angle tracking errors

50

0.12 Target (Reference) Spacecraft

0.1

40

0.08

30

τ

τ

x

0.06

y

τ

z

20

Nm

Angular Velocities (rad/s)


−0.2

0.04

10 0.02 0 0 −10

−0.02 −0.04

−20 0

100

200

300

400

500

0

600

100

200

300

400

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600

Time (sec)

Time (sec)

Fig. 15. Angular velocities

Fig. 18. Control torque

larger control moment. However, the compromise should be taken between the tracking error and the control effort for the control objective. Fig. 15 shows that the spacecraft angular velocity tracks the target spacecraft angular velocity in about 30 s, similar to the attitude tracking. Fig. 16 shows the relative angular velocity using the angular velocity tracking law in Eq. (80) and converges to zero in each component. Figs. 17 and 18 show the required control forces and control moments, respectively. They are initially larger

because the spacecraft is required to achieve the reference position while also required to coincide with the target attitude in a short time. In addition, higher control force and moment are required initially for the spacecraft to track quickly varying position and attitude of the tumbling target. However, these initially high transient control force and moment can be shrunk when the spacecraft is permitted to achieve this control objective in longer flight time.

© ASCE

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Conclusion In this paper, a kinematically coupled relative spacecraft motion model described in the target LVLH frame is useful. This model is combined with a tumbling target whose attitude varies quickly. A state-dependent Riccati equation control was used to perform kinematically coupled translational and rotational maneuvers simultaneously. A six-degrees-of freedom numerical simulation was used to demonstrate the kinematically coupled relative spacecraft motion control near the tumbling target using state-dependent Riccati equation method. The servicing spacecraft could achieve excellent tracking performance in the presence of kinematic coupling effect.

Notation The following symbols are used in this paper: T LI t = transformation matrix from the inertial (ECI) frame I to the target LVLH frame Lt ; T IBs = transformation matrix from the servicing spacecraft body-fixed frame Bs to the inertial (ECI) frame I; T LBts = transformation matrix from the servicing spacecraft body-fixed frame Bs to the target LVLH frame Lt ; T LBtt = transformation matrix from the target body-fixed frame Bt to the target LVLH frame Lt ; and T IBs = transformation matrix from spacecraft body-fixed frame Bs to the inertial (ECI) frame I.

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