Relative Orbit Determination Using Only Intersatellite Range Measurements

Open AccessEngineering NotesRelative Orbit Determination Using Only Intersatellite Range MeasurementsTong Qin, Dong Qiao and Malcolm MacdonaldTong QinBeijing Institute of Technology, 100081 Beijing, People’s Republic of China*Ph.D. Candidate, School of Aerospace Engineering; .Search for more papers by this author, Dong QiaoBeijing Institute of Technology, 100081 Beijing, People’s Republic of China†Professor, School of Aerospace Engineering; (Corresponding Author).Search for more papers by this author and Malcolm MacdonaldUniversity of Strathclyde, Glasgow, Scotland G1 1XJ, United Kingdom‡Professor, Department of Mechanical and Aerospace Engineering; .Search for more papers by this authorPublished Online:11 Jan 2019https://doi.org/10.2514/1.G003819SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail AboutI. IntroductionThe autonomous, relative orbit determination (OD) of a spacecraft constellation using intersatellite range measurements has been widely investigated because it is an imperative approach to enable a reduction in ground-based tracking requirements, and hence cost. One fundamental question in relative OD is which orbit elements can be obtained using only intersatellite range measurements.With respect to this question, prior work can be classified into two groups according to the dynamic environment of the constellation. In the multibody system, it has been shown that, owing to the nonsymmetric gravity from the noncentral gravitational body, all the orbit elements of two spacecraft can be obtained using only intersatellite range measurements [1,2]. In two-body dynamics, it has been shown that intersatellite range measurements are not adequate to obtain all orbit elements because the rotation of the orbits with respect to the inertial reference system cannot be determined [3–9]. Particularly, Liu found it was only possible to obtain the orbital shape and size among the Keplerian orbit elements [10]. Furthermore, Hill and Born found element combinations describing the relative orientation of the two orbits [1]. Therefore, a conclusion can be drawn by prior work that the shape, size, and relative orientation of two orbits can be obtained using only intersatellite range measurements, which reveals the basic performance of relative OD in two-body dynamics.However, the conclusion of relative OD in two-body dynamics does not hold in some cases. To be specific, the obtainable orbit elements deviate from the prior conclusion under certain orbital geometric configurations. This Note aims to dig out these special configurations and further find out the obtainable elements in each special configuration by means of an observability analysis. The relative OD system is linearized so that the observability can be analytically revealed, although the linearization can hide the properties of the system to some extent. The remaining parts of this Note are organized as follows:The state equation, the observation equation, and the observability matrix (OM) for relative OD using only intersatellite ranges are given in Sec. II. Subsequently, the observable state and state combinations under various geometric configurations are revealed analytically by analyzing the linear relationship among the columns of the observability matrix. Finally, numerical simulations are performed to validate the analysis. The investigation in this Note can be a basic reference for the configuration design of a spacecraft constellation.II. Relative Orbit Determination ModelsA. State ModelThe aim of relative OD is to obtain the size, shape, true anomaly, and relative orientation of two spacecraft. Consider two orbits, with the Keplerian orbit elements E1=[a1,e1,n1,Ω1,i1,ω1](1)and E2=[a2,e2,n2,Ω2,i2,ω2](2)where the parameters in each vector represent the semimajor axis, the eccentricity, the true anomaly, the longitude of the ascending node, the inclination, and the argument of periapsis, respectively. Among the 12 Keplerian orbit elements in Eqs. (1) and (2), a1 and a2 describe the orbital sizes; e1 and e2 describe the orbital shapes. By projecting two elliptic orbits onto a celestial sphere, and generating two great circles, the relative orientation of the two orbits can be expressed in celestial sphere, as shown in Fig. 1.Fig. 1 Relative orientation elements of two elliptic orbits projected onto a celestial sphere.The three elements describing the relative orientation are θ, which is the angle between the two orbit planes; ϕ1, which is the angular distance along the first orbit from the periapse to one of the two intersections of the orbits (this angle is positive in the direction of orbital motion.); and ϕ2, which is the angular distance along the second orbit from the periapse to the same intersection of the orbits [1]. For a circular orbit, in which the periapse is not defined, the argument of periapse is assumed to be zero. From spherical trigonometry, the equations for all three relative orientation elements with respect to the Keplerian elements are θ=cos−1(cos(i1)cos(i2)+sin(i1)sin(i2)cos(ΔΩ))(3)ϕ1=tan−1(sin(ΔΩ)sin(i1)cot(i2)−cos(i1)cos(ΔΩ))−ω1(4)and ϕ2=tan−1(sin(ΔΩ)−sin(i2)cot(i1)+cos(i2)cos(ΔΩ))−ω2(5)where ΔΩ=Ω2−Ω1 is the difference of longitude of the ascending nodes. Equations (4) and (5) are only valid with a nonzero denominator, requiring that the two orbit planes do not coincide with each other. For coplanar orbits, ϕ1 and ϕ2 can be taken as ω1 and ω2, respectively.Combining the elements for relative orbits, the state vector to be estimated is X=[a1,e1,n1,a2,e2,n2,θ,ϕ1,ϕ2]T(6)Neglecting all orbit perturbations, n1 and n2 are time-varying states in the state vector, for which the state equations are n˙1=μa1(1−e12)(1+e1cos(n1))2a1(1−e12)(7)n˙2=μa2(1−e22)(1+e2cos(n2))2a2(1−e22)(8)where μ is the gravitational parameter of the central body. The state model is then given as X˙=f(X)=[0,0,n˙1,0,0,n˙2,0,0,0]T(9)B. Observation ModelTo express the distance of two spacecraft using the elements in Eq. (6), an orbit reference frame system for each orbit is used, with its origin located at the mass center of the central body; the x axis directed toward the periapse; the y axis along the semilatus rectum in the orbit plane; and the z axis along the orbit angular momentum vector, completing the right-handed system. The relative distance between two spacecraft is given as ρ=‖r2−T21r1‖(10)where T21 is the transformation matrix from the orbit 1 reference frame to the orbit 2 reference frame; and r2 and r1 are, in their own orbit reference systems, the position vectors of two spacecraft: r1=[r1cos(n1),r1sin(n1),0]T(11)r2=[r2cos(n2),r2sin(n2),0]T(12)r1=a1(1−e12)1+e1cos(n1)(13)r2=a2(1−e22)1+e2cos(n2)(14)By rotating the orbit 1 reference frame through ϕ1 along its z axis, then through θ along the x axis of the once rotated system, and finally −ϕ2 along the z axis of the twice-rotated system, the orbit 1 reference frame will coincide with that of orbit 2. Defining Tx(⋅) and Tz(⋅) as the rotation matrix functions of the rotation angle along the x and z axes, respectively, the rotation matrix from orbit 1 to orbit 2 is T21=Tz(−ϕ2)Tx(θ)Tz(ϕ1)=[cos(ϕ2)−sin(ϕ2)0sin(ϕ2)cos(ϕ2)0001][1000cos(θ)sin(θ)0−sin(θ)cos(θ)]⁢[cos(ϕ1)sin(ϕ1)0−sin(ϕ1)cos(ϕ1)0001](15)Left multiplying r2 and T21r1 by Tx(−(1/2)θ)Tz(−ϕ2), the observation model is then given in a symmetric formation as D=ρ+υ=‖Tx(−12θ)Tz(ϕ2)r2−Tx(12θ)Tz(ϕ1)r1‖+υ(16)where υ is the measurement noise.C. Observability MatrixThe observability matrix measures the feasibility of a navigation system, and a theoretical analysis can be conducted based on the OM to show the observable states or state combinations [11–13]. The observations can be related to the states with a partial differential matrix at time ti, which is Hi=[∂ρ∂X]i=[∂ρ∂a1,∂ρ∂e1,∂ρ∂n1,∂ρ∂a2,∂ρ∂e2,∂ρ∂n2,∂ρ∂θ,∂ρ∂ϕ1,∂ρ∂ϕ2]i(17)The matrix should be mapped to the initial epoch t0 through the state transformation matrix (STM) [14] as H˜i=Hiϕ(ti,t0)(18)where ϕ(ti,t0) is the STM from t0 to ti. The differential equation of the STM is given by {ϕ˙(ti,t0)=[∂f(X)X]iϕ(ti,t0)ϕ(t0,t0)=I9×9(19)For the state model in Eq. (9), the STM has the following format: ϕ(ti,t0)=[A1000A2000I3](20)Ak=[100010ϕaknk(ti,t0)ϕeknk(ti,t0)ϕnknk(ti,t0)]k=1,2(21)ϕ˙aknk(ti,t0)=∂n˙k∂ak+∂n˙k∂nkϕaknk(ti,t0),ϕaknk(t0,t0)=0,k=1,2(22)ϕ˙eknk(ti,t0)=∂n˙k∂ek+∂n˙k∂nkϕ˙eknk(ti,t0),ϕ˙eknk(t0,t0)=0,k=1,2(23)ϕ˙nknk(ti,t0)=∂n˙k∂nkϕnknk(ti,t0),ϕnknk(t0,t0)=1,k=1,2(24)With measurements collected from t0 to ti, the OM is given as N=[H˜0⋮H˜i](25)To analyze the observable states analytically, each column of H˜i is derived according to Eqs. (16–24) as {H˜i1=[∂ρ∂a1]i+[∂ρ∂n1]iϕa1n1(ti,t0),H˜i2=[∂ρ∂e1]i+[∂ρ∂n1]iϕe1n1(ti,t0),H˜i3=[∂ρ∂n1]iϕn1n1(ti,t0)H˜i4=[∂ρ∂a2]i+[∂ρ∂n2]iϕa2n2(ti,t0),H˜i5=[∂ρ∂e2]i+[∂ρ∂n2]iϕe2n2(ti,t0),H˜i6=[∂ρ∂n2]iϕn2n2(ti,t0)H˜i7=[∂ρ∂θ]i,H˜i8=[∂ρ∂ϕ1]i,H˜i9=[∂ρ∂ϕ2]i(26)The nine items in Eq. (26) correspond to the nine states in Eq. (6), respectively. If the OM has a full rank, which means the columns are linearly independent from each other, the whole system is observable and all states can be estimated by a navigation filter. If there are linearly dependent columns, the corresponding states are unobservable. If there exists a linear combination of the dependent columns, which is independent from other columns, the corresponding linear state combination is observable, and the total amount of observable states and state combinations is equal to the rank of the OM [15].III. Observability AnalysisIn this section, the linear relationship among the columns in the OM under various geometric configurations is analyzed to reveal the observable states and state combinations. The geometric configurations are classified into coplanar and non-coplanar situations, and each situation is further divided into five cases according to orbit shape and size.A. Non-Coplanar OrbitsConsider k as an integer. The elements of two non-coplanar orbits should satisfy both Eqs. (27) and (28): ΔΩ≠2kπori1≠i2(27)ΔΩ≠(2k−1)πori1≠−i2(28)1. General Case (e1≠0, e2≠0, e1≠e2)In the general case, there are two elliptical orbits in different shapes and planes. The rank of the observability matrix is calculated as nine, and all the states in Eq. (6) are observable, as previously determined in Ref. [1]. Next, four special cases with particular restriction on orbit elements are discussed to reveal the obtainable states in the non-coplanar situation. Note that there is no restriction on orbit elements that are not referred to in each case.2. Circular Orbit and Elliptic Orbit (e1=0, e2≠0)In this case, spacecraft 1 moves in a circular orbit and spacecraft 2 moves in an elliptic orbit. Because r1=a1, the relative range of the two spacecraft can be written as ρ=‖Tx(−12θ)Tz(ϕ2)r2−Tx(12θ)a1[cos(n1−ϕ1),sin(n1−ϕ1),0]T‖(29)From Eq. (29), an equivalence of partial derivative in matrix Hi can be obtained as ∂ρ∂n1=−∂ρ∂ϕ1(30)For the circular orbit, the state equation of the true anomaly is simplified as n˙1=μa11a1(31)Combining Eq. (31) with Eq. (24), the following element in the STM can be obtained: ϕn1n1(ti,t0)=1+∫t0ti0 dt=1(32)Then, considering Eqs. (32) and (26), the linearly dependent two columns in the OM are H˜i3=[∂ρ∂n1]iϕn1n1(ti,t0)=[∂ρ∂n1]i=[−∂ρ∂ϕ1]i=−H˜i8(33)Therefore, the rank of the observability matrix decreases to eight. The sum of H˜i3 and −H˜i8 is linearly independent from other columns. Thus, the observable states and state combinations are X^=[a1,e1,n1−ϕ1,a2,e2,n2,θ,ϕ2](34)The unobservability of n1 and ϕ1 is actually a property of the circular orbit. Because the periapse is not defined, ω1 is arbitrary. Thus, n1 can be arbitrary, and ϕ1 can be arbitrary according to Eq. (4). However, the change of n1 and ϕ1 does not affect the relative range between the two spacecraft as long as n1−ϕ1 is fixed. Therefore, the observable element is not n1 or ϕ1 but n1−ϕ1.It should be noted that a strictly circular orbit is, of course, unlikely. However, the significance of the observability analysis is to state (as might be expected) that, when an orbit is close to circular, it is hard to estimate the true anomaly and the angular distance along the that orbit from its periapse, which is poorly defined, to one of the two intersections of the two orbits on the celestial sphere.3. Two Circular Orbits (e1=0, e2=0)Compared with the previous case, this case with one more circular orbit brings in another pair of linearly dependent columns in OM. Now, applying the analysis for orbit 1 in the previous case to orbit 2, it is found (as might be expected) that the additional pair of linearly dependent columns is the two columns corresponding to the true anomaly and the angular distance of orbit 2. Therefore, the two pairs of linearly dependent columns in this subcase are H˜i3=−H˜i8,H˜i6=−H˜i9(35)Seven states and state combinations are observable, which are X^=[a1,e1,n1−ϕ1,a2,e2,n2−ϕ2,θ](36)4. Two Symmetric Elliptic Orbits (a1=a2, e1=e2≠0, n1=n2, ϕ1=ϕ2)In this case, two spacecraft move in two symmetric elliptic orbits and have the same true anomaly. Strictly, such relative geometry is ill-advised because the two spacecraft trajectories will directly intersect.The intersatellite range in this case is given as ρ=‖Tx(−12θ)Tz(ϕ2)r2−Tx(12θ)Tz(ϕ1)r1‖=‖r1sinθ2sin(n1−ϕ1)+r2sinθ2sin(n2−ϕ2)‖(37)Because the two orbits are symmetric, the columns corresponding to the same elements of the two orbits must be the same. This is found by noting that ∂ρ∂a1=1ρ(r1sinθ2sin(n1−ϕ1)+r2sinθ2sin(n2−ϕ2))∂r1∂a1(38)∂ρ∂a2=1ρ(r1sinθ2sin(n1−ϕ1)+r2sinθ2sin(n2−ϕ2))∂r2∂a2(39)∂r1∂a1=1−e121+e1cos(n1)=1−e221+e2cos(n2)=∂r2∂a2(40)Combining Eqs. (38–40) leads to ∂ρ∂a1=∂ρ∂a2(41)Likewise, the following equation can be also obtained: ∂ρ∂e1=∂ρ∂e2,∂ρ∂n1=∂ρ∂n2,∂ρ∂ϕ1=∂ρ∂ϕ2(42)Considering the equivalences a1=a2, e1=e2≠0, and n1=n2, the following partial derivatives can be obtained: ∂n˙1∂a1=∂n˙2∂a2,∂n˙1∂e1=∂n˙2∂e2,∂n˙1∂n1=∂n˙2∂n2(43)Substituting Eq. (43) into Eqs. (22–24) leads to ϕa1n1(ti,t0)=ϕa2n2(ti,t0),ϕe1n1(ti,t0)=ϕe2n2(ti,t0),ϕn1n1(ti,t0)=ϕn2n2(ti,t0)(44)Combining Eqs. (41–44) with Eqs. (26), the linearly dependent columns of the OM can be obtained as H˜i1=H˜i4,H˜i2=H˜i5,H˜i3=H˜i6,H˜i8=H˜i9(45)The rank of the OM decreases to five, and the observable states and state combinations are given as X^=[a1+a2,e1+e2,n1+n2,θ,ϕ1+ϕ2](46)5. Two Symmetric Circular Orbits (a1=a2, e1=e2=0, n1=n2, ϕ1=ϕ2)The observability analyses in the previous two cases are valid in this case. Thus, the linearly dependent columns are a combination of those, which are given by H˜i1=H˜i4,H˜i2=H˜i5,H˜i3=H˜i6=−H˜i8=−H˜i9(47)Therefore, the rank of the observability decreases to four. The observable states and state combinations are X^=[a1+a2,e1+e2,n1+n2−ϕ1−ϕ2,θ](48)B. Coplanar OrbitsThe elements of two coplanar orbits satisfy Eq. (49) or Eq. (50): i1=i2andΔΩ=2kπ(49)i1=−i2andΔΩ=(2k−1)π(50)The following five cases are discussed to reveal the observability in the coplanar situation.1. Two Different Coplanar Elliptic Orbits (e1≠0, e2≠0, e1≠e2)In this case, the partial derivative of the intersatellite range to the angle between orbit planes is given by ∂ρ∂θ=1ρ(Tz(ϕ2)r2−Tz(ϕ1)r1)T(∂Tx(θ)∂θTz(ϕ2)r2−∂Tx(θ)∂θTz(ϕ1)r1)=1ρ[r2cos(n2−ϕ2)−r1cos(n1−ϕ1)r2sin(n2−ϕ2)−r1sin(n1−ϕ1)0]T⁢[00r2sin(n2−ϕ2)−r1sin(n1−ϕ1)]=0(51)From Eq. (26), it is obtained that H˜i7=0(52)The partial derivatives of the intersatellite range to ϕ1 and ϕ2 can be obtained as ∂ρ∂ϕ1=1ρ[r1cos(n1−ϕ1)−r2cos(n2−ϕ2)r1sin(n1−ϕ1)−r2sin(n2−ϕ2)0]T[r1sin(n1−ϕ1)−r1cos(n1−ϕ1)0](53)∂ρ∂ϕ2=1ρ[r2cos(n2−ϕ2)−r1cos(n1−ϕ1)r2sin(n2−ϕ2)−r1sin(n1−ϕ1)0]T[r2sin(n2−ϕ2)−r2cos(n2−ϕ2)0](54)Expanding Eqs. (53) and (54), it can be found that the two partial derivations are opposite to each other. Considering the last two items of Eq. (26), two linearly dependent columns can be obtained as H˜i8=−H˜i9(55)The linearly dependent columns in Eqs. (52) and (55) decrease the rank of the OM to seven. Therefore, seven states and state combinations are observable, which are X^=[a1,e1,n1,a2,e2,n2,ϕ1−ϕ2](56)2. Coplanar Elliptic Orbit and Circular Orbit (e1=0, e2≠0)Compared with the geometric configuration with non-coplanar elliptic and circular orbits, the coplanar geometric configuration involves another two groups of linearly dependent columns: the previous case. Thus, the linearly dependent columns in this subcase are H˜i3=−H˜i8=H˜i9,H˜i7=0(57)The rank of the OM decreases to six. The corresponding observable states and state combinations are X^=[a1,e1,n1−(ϕ1−ϕ2),a2,e2,n2](58)3. Two Coplanar Circular Orbits with Different Sizes (e1=0, e2=0, a1≠a2)The linearly dependent columns in this case are a combination of those in prior cases, which are given by H˜i3=−H˜i6=−H˜i8=H˜i9,H˜i7=0(59)There are five observable states and state combinations, which are given as X^=[a1,e1,n1−n2−ϕ1+ϕ2,a2,e2](60)4. Two Coplanar Elliptic Orbits with the Same Size and Shape (a1=a2, e1=e2≠0)Compared with the prior case, the extra constraints a1=a2 and e1=e2 in this case generate no more linearly dependent columns in the OM. Therefore, the observability in this case is given by Eq. (56). This case is separately discussed because, geometrically, the configuration with two orbits is degraded to the configuration with one orbit.5. Two Coplanar Circular Orbits with the Same Size (a1=a2, e1=e2=0)In this case, two spacecraft move in the same circular orbit. The linearly dependent columns in Eq. (59) are also valid. Moreover, the relative range of the two spacecraft can be simplified as ρ=r1(2−2cos(n1−n2−ϕ1+ϕ2))(61)For two circular orbits with the same semimajor axis, it can be obtained that n˙1=n˙2. Thus, n1−n2 is a constant, which means the intersatellite range is also a constant. The partial derivatives of intersatellite range to semimajor axes can be also obtained as ∂ρ∂a1=(2−2cos(n1−n2−ϕ1+ϕ2))=∂ρ∂a2=c1(62)where c1 is a constant. Consider the following STM component: ϕnknk(ti,t0)=1+∫t0ti0 dt=1,k=1,2(63)Combining Eq. (63) with Eq. (26), it can be obtained that H˜i3=[∂ρ∂n1]i=a1sin(n1−n2−ϕ1+ϕ2)(2−2cos(n1−n2−ϕ1+ϕ2))=−[∂ρ∂n2]i=−H˜i6=c2(64)where c2 is also a constant. Substituting Eqs. (62–64) into Eq. (26) leads to H˜i1+H˜i4=[∂ρ∂a1]i+H˜i3ϕa1n1(ti,t0)+[∂ρ∂a2]i+H˜i6ϕa2n2(ti,t0)=2c1(65)Combining Eqs. (59), (64), and (65), the additional linearly dependent relation in this case as compared with the case from Sec. III.B.3 is 2c2(Hi1+Hi4)−c1(Hi3−Hi6−Hi8+Hi9)=0(66)The rank of the OM decreases to four, and the corresponding observable states and state combinations are then obtained as X^=[e1,e2,a1−a2,c1(a1+a2)+2c2(n1−n2−ϕ1+ϕ2)](67)The observability of the relative orbit states is summarized in Tables 1 and 2.IV. SimulationA. Observability TestThe most rigorous situation for relative OD with two spacecraft in the same circular orbits is simulated to verify the observability analysis. The nominal orbit elements are given in Table 3. Suppose that the two spacecraft tracked each other, where possible, with a 10 s measurement rate. The radio ranging measurement error in Eq. (16) is assumed to be Gaussian white noise with a standard deviation of 1 m. The unscented Kalman filter is adopted to solve the nonlinear estimation problem, and the observability of each state and state combination is tested by the state covariance matrix, for which the diagonal elements represent the error covariance of the corresponding states [15]. The error covariance of the linear state combinations can be obtained from Cov(aXi+bXj)=a2P(i,i)+b2P(j,j)+2abP(i,j)(68)where Xi is the ith element of the state vector, and P(i,j) is the element of the error covariance matrix at the ith row and jth column. The error covariance of an observable state or state combination can converge as estimation progresses. As the square root of covariance, the STD has the same convergence tendency with the covariance and reflects the estimation accuracy more intuitively. The STD profiles of the nine basic relative orbit states are depicted in Fig. 2a. The solid lines represent the observable states, whereas the dashed lines are the unobservable states. It was shown that only the eccentricity can be estimated accurately in Sec. III. The STD profiles of the observable state combinations in this case are depicted in Fig. 2b. It is illustrated that both state combinations converge, validating the observability analysis. The detailed final convergence results are given in Tables 4 and 5. The columns of the unobservable states are shown for a comparison.Fig. 2 STDs in geometric configurations with the same circular orbits.B. Relative Orbit Determination AccuracyMonte Carlo simulations are performed to show the relative orbit determination accuracy expressed by the position and velocity in the orbit reference system. The relative position vector has been given in Eq. (9) as r2−T21r1. Likewise, the relative velocity vector can be given as v2−T21v1, where v1 and v2 are, respectively, the velocity vectors of spacecraft 1 and 2 in their own orbit reference frame. The Monte Carlo simulations are to analyze the estimation accuracy of the relative position and velocity vectors. The relative accuracy can be also regarded as a reference for absolute OD accuracy of other spacecraft when the absolute orbit of a certain spacecraft is known precisely. The orbits in Table 3 are used for Monte Carlos simulations. To show the influence of the geometric configuration on navigation accuracy, another pair of orbits is obtained by decreasing the eccentricity of spacecraft 2 from 0.4 to 0.01, and the relative OD of this pair of orbits is simulated for comparison.The position and velocity errors of the 500-case simulations and the 3-sigma bounds are depicted in Fig. 3. The errors in the last 4 h are magnified and depicted in the subwindows. Although both simulations in Figs. 3a and 3b are conducted in general geometric configurations, the navigation accuracy in Fig. 3b is much lower than that in Fig. 3a due to the existence of a nearly circular orbit. Therefore, the geometric configuration of a constellation should be far away from the special geometric configurations in Sec. III. In Fig. 3a, the initial 3-sigma position errors are over 50 km and the velocity errors are over 10 m/s. After 24 h, the triaxial position errors converge to around 200 m and the triaxial velocity errors converge to around 0.05 m/s.Fig. 3 Relative position and velocity errors of 500-case simulations in general geometric configuration.V. ConclusionsThis Note provides a detailed assessment of the relative orbit determination using only intersatellite range measurements for spacecraft constellations. The observable relative states of two spacecraft are revealed in various relative geometric configurations, which are classified into 10 cases according to the shape and relative orientation of two orbits. Nine states, at most, are observable for two non-coplanar and nonsymmetric elliptic orbits, whereas four are at least observable for two circular orbits with the same size. Numerical simulations are performed to validate the observability analysis and analyze the relative orbit determination accuracy. 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Google ScholarTablesTable 1 Observability under non-coplanar geometric configurationsCaseGeometry descriptionObservable elements1General geometric configurationa1,e1,a2,e2,n1,n2,ϕ1,ϕ2,θ2Non-coplanar circular orbit and elliptic orbita1,e1,a2,e2,n1−ϕ1,n2,ϕ2,θ3Two non-coplanar circular orbitsa1,e1,a2,e2,n1−ϕ1,n2−ϕ2,θ4Two symmetric elliptic orbits with the same phasea1+a2,e1+e2,n1+n2,ϕ1+ϕ2,θ5Two symmetric circular orbits with the same phasea1+a2,e1+e2,n1+n2−ϕ1−ϕ2,θTable 2 Observability under coplanar geometric configurationsCaseGeometry descriptionShape and size1Two different elliptic orbitsa1,e1,a2,e2,n1,n2,ϕ1−ϕ22One circular orbit and one elliptic orbita1,e1,n1−ϕ1+ϕ2,a2,e2,n23Two different circular orbitsa1,e1,a2,e2,n1−n24Same elliptic orbita1,e1,a2,e2,n1,n2,ϕ1−ϕ25Same circular orbite1,e2,a1−a2,c1(a1+a2)+2c2(n1−n2−ϕ1+ϕ2)Table 3 Nominal orbit elements of two circular orbits with the same sizeSpacecrafta, kmei, degΩ, degω, degn, deg111,397.2060000211,397.20600040Table 4 Convergence of nine basic relative states for two coplanar circular orbits with the same size a1ae1n1aae2aaϕ1aaInitial STDs10 km0.010.1 deg0.1 km0.010.1 deg0.1 deg0.1 deg0.1 degFinal STDs4.5, km2.0e−50.54 deg4.5 km1.7e−50.52 deg0.070 deg0.07 deg0.077 degConvergence ratio, %54.999.8−43854.999.8−41930.023.223.2aUnobservable states are shown for comparison.Table 5 Convergence of observable state combinations for two coplanar circular orbits with the same size a1−a2(a1+a2)+2c2/c1(n1−n2−ϕ1+ϕ2)Initial STDs, km14.1110.2Final STDs, km0.0380.36Convergence ratio, %99.799.7 Previous article FiguresReferencesRelatedDetailsCited byFully Autonomous Orbit Determination and Synchronization for Satellite Navigation and Communication Systems in Halo Orbits21 February 2023 | Remote Sensing, Vol. 15, No. 5Angle-Only Cooperative Orbit Determination Considering Attitude Uncertainty8 January 2023 | Sensors, Vol. 23, No. 2Autonomous Navigation Based on the Earth-Shadow Observation near the Sun–Earth L2 Point28 November 2022 | Applied Sciences, Vol. 12, No. 23Asteroid Approaching Orbit Optimization Considering Optical Navigation ObservabilityIEEE Transactions on Aerospace and Electronic Systems, Vol. 58, No. 6Performance analysis of crosslink radiometric measurement based autonomous orbit determination for cislunar small satellite formationsAdvances in Space Research, Vol. 40Configuration Stability Analysis for Geocentric Space Gravitational-Wave Observatories17 September 2022 | Aerospace, Vol. 9, No. 9Multi-Spacecraft Tracking and Data Association Based on Uncertainty Propagation29 July 2022 | Applied Sciences, Vol. 12, No. 15Maneuvering Spacecraft Orbit Determination Using Polynomial Representation10 May 2022 | Aerospace, Vol. 9, No. 5Autonomous navigation for deep space small satellites: Scientific and technological advancesActa Astronautica, Vol. 193Observability Analysis and Improvement Approach for Cooperative Optical Orbit Determination18 March 2022 | Aerospace, Vol. 9, No. 3Comparison of Autonomous Orbit Determination for Satellite Pairs in Lunar Halo and Distant Retrograde Orbits27 June 2022 | NAVIGATION: Journal of the Institute of Navigation, Vol. 69, No. 2Fully Decentralized Cooperative Navigation for Spacecraft ConstellationsIEEE Transactions on Aerospace and Electronic Systems, Vol. 57, No. 4Relative Orbit Determination for Unconnected Spacecraft Within a ConstellationTong Qin, Dong Qiao and Malcolm Macdonald21 December 2020 | Journal of Guidance, Control, and Dynamics, Vol. 44, No. 3Three-spacecraft autonomous orbit determination and observability analysis with inertial angles-only measurementsActa Astronautica, Vol. 170 What's Popular Volume 42, Number 3March 2019 CrossmarkInformationCopyright © 2018 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the ISSN 0731-5090 (print) or 1533-3884 (online) to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. TopicsAerospace SciencesAstrodynamicsAstronauticsControl TheoryGuidance, Navigation, and Control SystemsKalman FilterOrbital ManeuversOrbital PropertySatellite Navigation SystemsSpace OrbitSpacecraft GuidanceSpacecraft Guidance and Control KeywordsOrbit DeterminationConstellationsKeplerian OrbitInertial Reference SystemUnscented Kalman FilterNumerical SimulationSpacecraft TrajectoriesMonte Carlo SimulationNearly Circular OrbitGaussian White NoiseAcknowledgmentsThis Note is sponsored by the National Natural Science Fund (grant nos.11572038 and 61703040), the Chang Jiang Scholars Program, and the Discipline Innovative Engineering Plan (111 project, B16003).PDF Received10 May 2018Accepted30 October 2018Published online11 January 2019