Generalized Attitude Determination with One Dominant Vector Observation

Transcription

1 Generalized Attitude Determination with One Dominant Vector Observation John L. Crassidis University at Buffalo, State University of New Yor, Amherst, New Yor, Yang Cheng Mississippi State University, Mississippi State, MS 3976 This paper derives an algorithm to determine the approximate attitude of a vehicle from both vector and arc-length observations, which are the most general types of attitude observations. It is assumed that one of the vector observations is more accurate than the other vector and arc-length observations. The solution is found by solving a quartic polynomial equation. Then the quaternion can be determined from the polynomial solution. The attitude errorcovariance is also derived using both an attitude perturbation approach and a constrained least squares approach. Both are shown to yield identical results. An optimality condition is also derived that compares the derived suboptimal error-covariance with the optimal one. Several special cases, such as a set of one direction observation and an arc-length observation, are shown. Simulation results using a Monte Carlo analysis are shown to verify the derived algorithm. I. Introduction Attitude determination can be accomplished using a number of sensors, such as star tracers, magnetometers, horizon sensors, magnetometers, and the Global Positioning System (GPS. Even though the sensing mechanism for each is different, most attitude observations can be expressed in vector form, with the exception of GPS observations. Vectorform observations can be used to determine the attitude by solving Wahba s problem. A unique solution is given when at least two non-collinear vector observations exist. If the quaternion 3 is used for the attitude parametrization then the associated loss function in Wahba s problem is quadratic in nature, and the attitude orthogonalization constraint reduces down to a unit-norm vector constraint in the quaternion. Many solutions to this problem, such as Davenport s q Method and the QUEST algorithm 4, can be used. The loss function using only GPS observations with the quaternion parametrization is quartic in nature. If at least three non-planar baselines or sightlines exist then the GPS observations can be expressed in vector form 5. A transformed loss function can be derived under these conditions, but it is still quartic in the quaternion because it is in fact equivalent to the non-transformed loss function. A suboptimal solution is presented in 5, which reduces the quartic loss function into a quadratic one so that any solution that solves Wahba s problem be employed. The error-attitude covariance is also derived, which can be compared with the maximum lielihood Cramér-Rao lower bound to see the efficiency of the suboptimal solution. The suboptimal solution can be used as a starting point for an iterative algorithm, such as a simple nonlinear least squares approach to determine the optimal solution. An approximate solution using only vector observations with one dominant observation is derived in 6. The algorithm is very computationally efficient, and does not require iterative calculation or transcendental functions. This paper expands upon the wor shown in 6 by deriving an approximate attitude solution involving an accurate line-of-sight (LOS vector observation and other types of attitude observations, which may include other LOS or arc-length observations. These types of observation represent the most general type used for most attitude determination system. The attitude parametrization chosen in this paper is the quaternion. The suboptimal covariance is derived, as well as a simple scalar expression that can be computed without the attitude solution to chec the accuracy of the approximate attitude solution. Samuel P. Capen Chair Professor, Department of Mechanical & Aerospace Engineering. johnc@buffalo.edu. Fellow AIAA. Associate Professor, Department of Aerospace Engineering. cheng@ae.msstate.edu. Associate Fellow AIAA.

2 II. Algorithm Development The generalized attitude determination problem minimizes the following loss function: J(A σ b Ar + i j ij (φ ij c T i As j ( where A is the attitude matrix, which is a 3 3 proper orthogonal matrix, b is the LOS unit vector expressed in body-frame coordinates, r is the LOS unit vector expressed in reference-frame coordinates, σ is the standard deviation of the unit vector observation errors 7, and σ ij is the standard deviation of the arc-length observations. These arc-length observations, denoted by φ ij, may come from GPS attitude sensors or other sensors 8, where c i is a vector expressed in body-frame coordinates, and s j is a vector expressed in reference-frame coordinates. For GPS observations, c i is the ith baseline vector, and s j is the jth sightline vector. To date a non-iterative solution for the optimal attitude that minimizes Eq. ( is not available, but the optimal attitude can be found using an iterative NLS solution. A closed-form approximate solution for the attitude is now derived. Suppose that an accurate LOS observation is provided by r and b, whose associated standard deviation σ is much smaller than the other standard deviations. The above optimal attitude determination problem can be approximated by a suboptimal problem that minimizes the loss function subject to the constraint J(A b Ar + i j ij (φ ij c T i As j ( b Ar (3 Note that Eq. (3 contains no errors, which means it is assumed to be a noise-free measurement. The suboptimal problem can be solved by a computationally efficient algorithm, which is now developed. The attitude is subsequently parameterized by the quaternion q. The quaternion is a four-dimensional vector, defined as with q q :3 q 4 q :3 q q q 3 T e sin(ϑ/ (5a q 4 cos(ϑ/ where e is the unit Euler axis and ϑ is the rotation angle 3. A quaternion parameterizing an attitude satisfies a single constraint given by q. In terms of the quaternion, its associated attitude matrix is given by with (4 (5b A(q Ξ T (qψ(q (6 q 4 I 3 + q :3 q 4 I 3 q :3 Ξ(q q T, Ψ(q :3 q T :3 0 q 3 q q :3 q 3 0 q q q 0 where I 3 is a 3 3 identity matrix and the matrix q :3 is the standard cross-product matrix. Equation ( can be rewritten in terms of the quaternion as J(q q T (K + G q + α + i j ij (7a (7b ( q T C ij q (8

3 where α Ω(d K G i j Ω(b Γ(r (9a ij φ ij C ij σ + i j C ij Ω(c i Γ(s j d d d T, Γ(d 0 σij φ ij d d d T 0 (9b (9c (9d (9e where d is any 3 vector. Unless b r, the most general unit quaternion satisfying Eq. (3 is given by q cos(ψ/ q min + sin(ψ/ q 80 (0 with b r q min ( + b T r + b T r b + r q 80 ( + b T r 0 (a (b where ψ is an arbitrary parameter. When b r, q min and q 80 are indeterminate, but this condition can be avoided by solving for the attitude with respect to a reference coordinate frame related to the original reference frame by a 80 degree rotation about one of the coordinate axes, 4. Note that, in the generalized attitude determination problem, both the representations of r and those of s j are transformed by reference-frame rotations. Substituting Eq. (0 into q T (K + Gq leads to where q T (K + G q (µ cos ψ + ν sin ψ + κ/ ( µ q T min (K + G q min q T 80 (K + G q 80 (3a ν q T min (K + G q 80 κ q T min (K + G q min + q T 80 (K + G q 80 Substituting Eq. (0 into ( q T C ij q gives ( q T C ij q ( µij cos ψ + ν ij cos ψ + κ ij /4 (4 where (3b (3c µ ij q T min C ijq min q T 80 C ijq 80 (5a ν ij q T min C ijq 80 κ ij q T min C ijq min + q T 80 C ijq 80 (5b (5c Performing the multiplications in Eq. (4 yields i j ij ( q T C ij q (γ cos ψ + γ sin ψ + γ 3 sin ψ cos ψ + γ 4 cos ψ + γ 5 sin ψ + γ 6 (6 3

4 where γ 4 γ 4 γ 3 γ 4 γ 5 γ 6 4 i j i j i j i j i j i j ij µ ij (7a ij ν ij (7b ij µ ij ν ij (7c ij µ ij κ ij (7d ij ν ij κ ij (7e ij κ ij (7f Substituting Eqs. ( and (6 into Eq. (8 gives J(ψ γ cos ψ + γ sin ψ + γ 3 sin ψ cos ψ + (µ + γ 4 cos ψ + (ν + γ 5 sin ψ + (κ + α + γ 6 / (8 The necessary condition to minimize J(ψ is given by (µ + γ 4 sin ψ + (ν + γ 5 cos ψ (γ γ sin(ψ + γ 3 cos(ψ 0 (9 where the identities sin(ψ sin ψ cos ψ and cos(ψ cos ψ sin ψ have been used. Note that terms involving sin(ψ and cos(ψ now appear that are not in the formulation shown in 6. If only LOS observations exist then G 0, and all the γ terms are also zero. In this case, the solution reduces to the solution given by 6. For the general case multiple roots may exist now because of the extra terms. An iterative approach can be used to find all the roots. But, here a different approach is derived that does not involve an iterative solution. Define x sin ψ, which gives cos ψ ± x. Using these definitions, Eq. (9 can be written as γ 3 ( x (µ + γ 4 x ± x (γ γ x (ν + γ 5 (0 Squaring both sides of Eq. (0, and collecting terms gives where δ 4 x 4 + δ 3 x 3 + δ x + δ x + δ 0 0 ( δ 0 γ3 (ν + γ 5 δ (γ γ (ν + γ 5 γ 3 (µ + γ 4 δ (µ + γ 4 + (ν + γ 5 4 γ3 + (γ γ δ 3 4 γ 3 (µ + γ 4 (γ γ (ν + γ 5 δ 4 4 γ3 + (γ γ (a (b (c (d (e There are two possibilities for x. One is that they are all real, and the other is that two are real and the other two are complex conjugates. Descartes Rule of Signs can be used to determine the number of real roots. Once these roots have been determined, sin ψ, cos ψ, and ψ are given by sin ψ x (3a 4

5 cos ψ sign γ 3 ( x (µ + γ 4 x sign (γ γ x (ν + γ 5 x (3b ψ ATAN(sin ψ, cos ψ where sign( returns the sign of the argument, and ATAN is the four-quadrant inverse tangent function. The loss function given by Eq. (8 is evaluated to find the minimizing ψ. Note that ψ itself is not actually required to compute the loss function because Eqs. (3a and (3b can directly be used in Eq. (8. Finally, the minimizing quaternion is calculated using Eq. (0, and using the identities cos(ψ/ ± ( + cos ψ/ and sin(ψ/ ± ( cos ψ/. If ψ π, π, then ψ/ π/, π/, and cos(ψ/ and sin(ψ/ have the same sign when sin ψ > 0 and the opposite sign when sin ψ < 0. Then, the minimizing quaternion is given by + cos ψ cos ψ q q min + sign(sin ψ q 80 (4 The minimizing quaternion can also be determined without computing transcendental functions by using Eqs. (3a and (3b in Eq. (4. The algorithm (without reference frame rotations is summarized in Algorithm. Algorithm Generalized Quaternion Estimation with One Dominant Vector : Compute matrices K, G, and C ij and scalar α Eqs. (9 : Compute µ, ν, κ Eqs. (3c 3: Compute γ through γ 6 Eqs. (5 and (7 4: Compute δ 0 through δ 4 Eqs. ( 5: Solve for real x ( sin ψ Eq. ( 6: Compute all cos ψ using real values of x Eq. (3b 7: Select the sin ψ and cos ψ that minimize the loss function Eq. (8 8: Construct q min and q 80 Eq. ( 9: Construct the quaternion estimate q Eq. (4 Now special cases of the coefficients of the quartic equation are discussed. Consider the case when three orthonormal c i s exist with σc j σj σj σ3j. Then, the double summation on the right side of Eq. (8 is given by i j ij ( q T C ij q 3 i j j j c j st j AT c i c T i As j c j st j AT As j Thus, this term is now independent of the attitude matrix. Therefore, Eq. (8 is equivalent to solving Wahba s problem 5. The same is true when three orthonormal s j s exist with σis σi σ i σ i3, which leads to i j ij ( q T C ij q c j 3 i j i is ct i As js T j AT c i is ct i AAT c i This again is independent of the attitude matrix. In both cases, since the quartic terms in the loss function given by Eq. (8 vanish, then µ ij ν ij κ ij 0 and thus γ γ γ 3 γ 3 γ 5 0. The quartic equation reduces to a quadratic equation (µ + ν x ν 0. i is (3c (5 (6 5

6 The attitude solution is nonunique if and only if the loss function in ψ is independent of ψ. The condition is γ γ, γ 3 µ + γ 4 ν + γ 5 0 (7 In that case, the loss function becomes J(ψ (γ + κ + α + γ 6 / and the coefficients of the quartic equation become δ δ δ 3 δ 4 0. III. Covariance Analysis This section derives the covariance of the attitude errors for the suboptimal algorithm. A simple expression is also derived that can be used to chec the accuracy of the suboptimal algorithm as compared to the optimal solution derived from a maximum lielihood analysis. Two methods are used to derive the relationship between the attitude error and the measurement noise. The first is based on a perturbation approach, while the second is based on a constrained least-squares approach. Both will be shown to yield identical results. Under the small-error assumption, the attitude matrix is approximated by A (I 3 δϑ A true (8 where δϑ is the vector of the attitude errors. The error-quaternion is approximated by δq δϑt matrix associated with the attitude estimate is T. The covariance P sub E{δϑ δϑ T } (9 Let b true A true r, which is the true body-vector, and let φ true ij denote the true arc-length. The models for the body and arc-length observations are respectively given by b b true + b (30a φ ij φ true ij + φ ij (30b where b is the body-vector error whose covariance is given by the QUEST Measurement Model (QMM 7, and φ ij is the arc-length error whose variance is given by σ ij. The optimal covariance, denoted by P opt, is given by the inverse of the combination of the Fisher information matrix (FIM associated with the LOS observation plus the FIM associated with other observations: P opt F F I 3 b true T + + i j ij Atrue s j c i c T i Atrue s j T I 3 b true T (3a (3b where σ is the standard deviation associated with the error in b, and F is the FIM. Also, define F F 3 b true T (3a σeff T F b true (3b where F is the portion of the FIM associated with the remaining observations outside of b true, and σeff is an effective error-variance. A. Perturbation Approach For this approach, the error-quaternion is given by δϑ δq δq min + ˆψ δq 80 (33 6

7 where δq min 4 + T b δq T b b true b + T b b true + b 0 (34a (34b Also, the notation ˆψ is used to denote ψ with error due to measurement noise. So, δϑ ˆψ b true b true b. Now an expression for ˆψ in terms of the noise is derived. The necessary condition that ˆψ satisfies is approximated by Then ˆψ is given by After significant algebra it can be shown that where { σ btrue b true d { (µ + γ 4 ˆψ + (ν + γ 5 (γ γ ( ˆψ + γ 3 0 (35 ˆψ ν + γ 3 + γ 5 µ + γ + γ + γ 4 (36 ˆψ ( d T b + ij φ ij + T b i j σ (btrue Substituting Eq (37 into the δϑ expression gives (37 σ btrue b true (38a ij σij ct i Atrue s j b true (38b } ( b true + σ ij i j } T b true b true + i j ( b true T δϑ b + d i j Thus, the covariance is given by P sub P v + P a + P, where Using the following identities: P σ ( P v P a b true c T i Atrue s j b true ij ( ij b true φ ij + b true d σ T i j d σ ij ij d T + btrue d ( b true A true s j c i (38c c T i Atrue s j b true T + btrue d (38d b (39 b true T (40a ( b true T (40b b true T + btrue d T (40c d T F b true σeff (4a P v + P g σeff btrue T (4b 7

8 then gives ( b true T + btrue d b true I 3 σeff btrue T F (4c P sub σeff btrue T + σ I3 σeff btrue T F I 3 σeff btrue T F T It is important to note that Eq. (4 is a general expression for any F. Some special cases for F will be shown later. Also, the matrix I 3 σeff btrue T F is a projection matrix onto the plane perpendicular to b true, so I3 σeff btrue T F b true 0. In addition, substituting Eq. (3a into Eq. (4 shows that P sub remains unchanged using F in place of F. (4 B. Constrained Least Squares Approach The following measurement model is assumed for the constrained least-squares problem 9: y H x true + y y H x true (43a (43b where y is the zero-mean measurement noise with the covariance matrix being the identity matrix. The optimal estimate x minimizes the loss function J(x y H x (44 subject to y H x The optimality condition is given by H T H H T x H 0 λ H T y y where λ is the vector of Lagrange multipliers. For this attitude determination problem, the linearized measurement model is (45 To apply a constrained least-square estimation algorithm, define b btrue δϑ σ σ (46a φ ij ct i Atrue s j δϑ σ ij σ ij (46b x δϑ (47a H b true y T σ b T σ,..., bt n σ n, φ σ,..., φ N M σ N M,..., btrue n T σ n, Atrue s T c σ T (47b,..., Atrue s M T c N σ N M T (47c y M b σ (47d H M h h T (47e where h is any vector perpendicular to b true, and h b true h. This reduced form for H is chosen because the full form leads to a non-invertible matrix. It still provides the correct properties for the constraint because it can be verified that Mb true 0 (48a 8

9 and H T H F. The noise vector satisfies The optimal estimates are E { H T y y MM T I (48b M T M I 3 b true T (48c } 0, E δϑ λ H T y y H T y y T F M T H T y M 0 y F σ I (49 (50 Note that F may be singular. Let F M T M 0 where the following quantities can be easily derived: B C C T D (5 B btrue T T F b true C M M FB D M FC T σeff btrue T (5a (5b (5c Note that Eq. (5a follows from MB 0 3. The error is given by δϑ B C T H T y y (53 which is equivalent to Eq. (39. Then the covariance is F P sub B C T 0 0 σ I B T C T B FB T + σ (54 It can be shown that B FB T σeff btrue T (55a CC T I 3 σeff btrue T F I 3 σeff btrue T F T Hence, substituting the B and C expressions from Eq. (5 into Eq. (54 gives bac exactly the same suboptimal covariance in Eq. (4. (55b C. Optimality Condition Assume P sub and F are nonsingular and σeff > 0. A condition required to mae P sub F close to the identity matrix is now derived. Carrying out the multiplication gives P sub F σeff btrue T F + σ F σeff σ b true T FF + F b true T F σeff btrue T F b true T F (56 Solving Eq. (3a for F, and substituting the resulting expression into Eq. (56 leads to P sub F σeff btrue T F b true T σeff btrue T F I 3 b true T + I 3 + σ I3 σeff btrue T F F I 3 σeff btrue T F (57 9

10 It is straightforward to show that the first three terms on the right side Eq. (57 reduce down to zero. Therefore, P sub F I 3 + σ I3 σeff btrue T F F I 3 σeff btrue T F (58 Clearly, P sub F I 3 only when the second term vanishes, but P sub is close to the optimal covariance F when a dominant vector exists, that is, σ σ and σ σ ij. Taing the trace of P sub F gives Tr(P sub F 3 + σ Tr { I 3 σeff btrue T F F } (59 where the idempotent property of the projection matrix has been used. Then a normalized condition to ensure that a good solution is provided is given by ɛ 3 σ Tr { I 3 σeff btrue T F F } << (60 The trace in Eq. (60 can be shown to be greater than or equal to zero. Since σeff /Trbtrue simply required to show T F > 0, then it is Trb true T F Tr( F Trb true T F 0 (6 If A and B are positive semi-definite matrices, then 0 Tr(A B Tr(ATr(B 0. Defining A b true T F and B F shows that the inequality in Eq. (6 is valid. This proves ɛ 0. Clearly, P sub is not optimal if ɛ 0 or equivalently ɛ T F b true Tr( F T F b true 0 (6 Note also that ɛ 0 is not a sufficient condition for P sub to be optimal. D. Special Cases As stated previously, Eq. (4 is valid for any F. If the QMM model with unit vectors b true i is used for F, then Then σeff is given by F σ I3 b true T { σeff σ I3 T b true σ btrue b true σ btrue (63 T b true } (64 where the identity a b a b (a T b for any 3 vectors a and b has been used. This is equivalent to the σ eff given in 6. Define w σ eff F b true. Substituting Eq. (63 into w gives w σ eff σ eff σ btrue b true b true b true This is equivalent to the w expression given in 6. Therefore, P sub is equivalent to the expression derived in 6. Define c l T. Using l Tr( F 0 (65 (66a

11 gives This equation can be rewritten as T F b true ɛ ɛ T F b true l l σ 4 ( c + σ 4 ( c + σ σ l ( c (66b ( c c l + c c l c l σ σ l ( c + c l c c l c l l+ l+ σ σ l (c + c l c c l c l (66c (67 σ l ( c c l (68 Clearly, ɛ 0 unless all c 0, which in turn requires that all b true be coaligned with b true, a degenerate case with nonunique solutions. Hence, P sub is not optimal for Wahba s problem. When F is constructed from arc-lengths only, it taes the form F N σp b true p p p T (69 where b true p is a unit vector, σ p is the associated standard deviation, and MN is the number of arc-lengths. It can be shown that N N ɛ σpσ q (c p c pc q c pq (70 p q When there is only one arc-length, that is, MN, then ɛ 0. In other cases, ɛ 0 and P sub is not optimal. It is now shown that P sub F I 3 when the attitude is determined from a direction and an arc-length 8. Without loss in generality it is assumed that this arc-length is given by φ. The FIM for this case is simply given by F I 3 b true T + F (7 with F Atrue s c c T Atrue s T (7 Also, P sub is given by Eq. (4 with F F. Replacing F with F in second term of the right side of Eq. (58 leads to the following requirement: I3 σeff btrue T F F I3 σeff btrue T F 03 (73 Since the projection matrix and F are both singular, then the validation of this requirement must be done using a brute-force approach. Substituting σeff (btrue T F b true and Eq. (7 into Eq. (73, and after some simple algebraic manipulations leads to the following condition: { (b true T A true s c c T Atrue s T b true } A true s c c T Atrue s T A true s c c T Atrue s T b true T A true s c c T Atrue s T (74 The right side of Eq. (74 can be rewritten as A true s c c T Atrue s T b true T A true s c c T Atrue s T { T A true s c } A true s c c T Atrue s T (75 Since { T A true } s c (b true T A true s c c T Atrue s T b true then Eq. (74 is satisfied. Therefore, P sub is the optimal covariance in this case. Conditions to obtain a deterministic attitude solution for this case are discussed in 8.

12 IV. Simulation Results A simulation is performed using the following vectors for the LOS and GPS observations with no other LOS observations: b true ( / 0 T c ( /0 T, c 0 0 T, c T s ( 3/3 T, s ( /0 T Note that the baselines are co-planer, which leads to an indeterminant solution using the approximate approach in 5. As in the simulation example shown in 6 5,000 test cases with uniformly distributed random attitudes are generated. The true body-vector b true is corrupted by Gaussian random noise with standard deviation of 0.0 degree per axis, which simulates a fine Sun sensor. The true GPS observations are corrupted by Gaussian random noise with a normalized standard deviation of 0.00, corresponding to an attitude error of about 0.5 degrees 5. (a Attitude errors using the approximate solution (b Attitude errors using the optimal solution Run Number (c Trace condition Fig. GPS simulation results with fine Sun sensor. A plot of the attitude errors using the approximate solution in Algorithm, along with their respective 3σ bounds, for the 5,000 test runs is shown in Fig. (a. The 3σ bounds change with each run because the GPS covariance is a function of the attitude. Clearly, the attitude errors are consistent with the 3σ bounds derived from P sub. A plot of the attitude errors using a NLS algorithm that minimizes the optimal loss function in Eq. (, along with their respective 3σ bounds, for the 5,000 test runs is shown in Fig. (b. Clearly, the attitude errors are consistent with the 3σ bounds derived from P opt. Comparing Fig. (a to Fig. (b shows that the errors are nearly identical. A plot of the trace condition given by Eq. (60 is shown in Fig. (c. This indicates that the attitude solution is very close to being optimal

13 Table Number of real roots Case Two Real Roots Four Real Roots Fine Sun Sensor Case 4, Coarse Sun Sensor Case 4, for all random attitudes. Table shows the number of root solutions for the polynomial in Eq. ( that give two real roots and four real roots for this case. A large number of cases involve only two roots. (a Attitude errors using the approximate solution (b Attitude errors using the optimal solution Run Number (c Trace condition Fig. GPS Simulation results with coarse Sun sensor. The simulation is rerun where the true body-vector b true is corrupted by Gaussian random noise with standard deviation of 0. degree per axis, which simulates a coarser Sun sensor. The standard deviation of the GPS observations is the same as before. A plot of the attitude errors using the approximate solution in Algorithm, along with their respective 3σ bounds, for the 5,000 test runs is shown in Fig. (a. Clearly, the attitude errors are again consistent with the 3σ bounds derived from P sub. But the errors are much larger than the previous case, as seen by comparing Fig. (a with Fig. (a. This intuitively is correct because a coarse Sun sensor is used in place of a fine Sun sensor in this case. A plot of the attitude errors using a NLS algorithm that minimizes the optimal loss function in Eq. (, along with their respective 3σ bounds, for the 5,000 test runs is shown in Fig. (b. Clearly, the attitude errors are consistent with the 3σ bounds derived from P opt. The errors are larger than those shown in Fig. (b because of the use of a coarse Sun sensor in this case. Comparing Fig. (a to Fig. (b shows that the errors are larger using the approximate solution than 3

14 using a NLS solution that minimizes Eq. (, with the exception of the pitch errors in some runs. The sub-optimality for the coarse Sun sensor case is confirmed by the trace condition shown in Fig. (c. This is now much larger than, which indicates that the suboptimal covariance is not close to the optimal one. The suboptimal solution can be used as a starting guess for the NLS squares algorithm to determine the optimal one, which is how the optimal solution has been determined. Table shows the number of root solutions for the polynomial in Eq. ( that give two real roots and four real roots for this case. Once again, a large number of cases involve only two roots. V. Conclusions This paper presented a generalized attitude determination algorithm when one dominant vector observation is provided. The algorithm involves solving a quartic polynomial, which is nown to have closed-form solutions. It is also a non-iterative algorithm, and no transcendental functions are required by using the polynomial solution variable. A simple scalar expression was derived using the suboptimal covariance that can be computed without determining the attitude. This scalar quantity can be used to chec the accuracy of the derived attitude solution with respect to the optimal solution. When the case of only vector observations exists, then the approximate attitude solution simplifies greatly to a previously derived solution involving this case. For the case of one vector observation and one arc-length, then the approximate attitude solution was shown to be equivalent to the optimal attitude solution. Simulation results show that a good approximate attitude solution can be provided using the derived algorithm for a realistic scenario involving one fine Sun sensor observation and several Global Positioning System (GPS arc-length observations. The approximate attitude solution was shown to be worse than the optimal solution when a coarse Sun sensor was used in place of the fine Sun sensor. An iterative nonlinear least-squares (NLS algorithm needs to be employed to determine the optimal attitude, which may converge to a local minimum depending on the initial guess. Using the approximate solution for the initial guess in the NLS algorithm converged to the optimal solution for every case in the simulated trails, which demonstrates the usefulness of the approximate solution even when it does not approximate the optimal solution well. Acnowledgments The authors would lie to acnowledge many valuable conversations with F. Landis Marley. References Marley, F. L., and Crassidis, J. L., Fundamentals of Spacecraft Attitude Determination and Control, Springer, New Yor, NY, 04, pp. 3 4, 49. Chapters 4-5, Appendix A. Wahba, G., A Least Squares Estimate of Satellite Attitude, SIAM Review, Vol. 7, No. 3, 965, p doi:0.37/ Shuster, M., A Survey of Attitude Representations, Journal of the Astronautical Sciences, Vol. 4, No. 4, 993, pp Shuster, M. D., and Oh, S. D., Three-Axis Attitude Determination from Vector Observations, Journal of Guidance and Control, Vol. 4, No., 98, pp doi:0.54/ Crassidis, J. L., and Marley, F. L., New Algorithm for Attitude Determination Using Global Positioning System Signals, Journal of Guidance, Control, and Dynamics, Vol. 0, No. 5, 997, pp doi:0.54/ Marley, F. L., and Cheng, Y., Wahba s Problem with One Dominant Observation, Journal of Guidance, Control, and Dynamics, Vol. 4, No. 0, 08, pp doi:0.54/.g Shuster, M. D., Maximum Lielihood Estimation of Spacecraft Attitude, The Journal of the Astronautical Sciences, Vol. 37, No., 989, pp Shuster, M. D., Deterministic Three-Axis Attitude Determination, Journal of the Astronautical Sciences, Vol. 5, No. 3, 004, pp Crassidis, J. L., and Junins, J. L., Optimal Estimation of Dynamic Systems, nd ed., Chapman & Hall/CRC, Boca Raton, FL, 0, pp Uluö, Z., and Türmen, R., On Some Matrix Trace Inequalities, Journal of Inequalities and Applications, 00. doi:0.55/00/