Uniting Attitude Estimation With Global Asymptotic Stability

This article approaches the problem of attitude estimation using biased gyro and body-referenced measurements of inertial vectors. First, a new observer for attitude estimation is proposed through introducing a family of synergistic potential functions with explicit expression of synergistic gap. Furthermore, it is convenient to adjust the design by adding additional state-dependent gains without adapting the switching rules. The proposed observer exhibits faster convergence for large attitude estimation errors than similar work in literature. Second, to achieve better steady-state performance, we unite this global observer with a traditional local observer, i.e., multiplicative extended Kalman filter (MEKF). The new uniting observer inherits the desired transient and steady-state performance from the two sub-observers, respectively. The performance of the proposed methods has been evaluated via simulation and experiments.


I. INTRODUCTION A. Background
A TTITUDE estimation is a crucial subject of robot navigation, guidance, and control and has therefore gathered considerable attention from both industry and scientific community [1], [2], [3], [4], [5], [6], [7]. The problem considered here is that of estimating attitude using angular rate and body-referenced measurement of known inertial vectors, which are accessible in sensors such as gyros, accelerometers, magnetometer, and camera. It is well-known that the topological obstructions which exist intrinsically in the state space of the rigid-body attitudes preclude a continuous attitude observer from obtaining global convergence [8]. Other obstacles are posed by the parametrization of an attitude: no three-component representations of attitude can be both global and nonsingular; unit quaternions do not have singularities, but can induce undesired unwinding phenomenon [9]. Recently, hybrid 4-D parameters are proposed in [10] to separate the pitch and roll angle estimation from the yaw estimation, but the singularity is still unavoidable. Particularly, this singularity can lead to critical dependence of the estimation performance on the initial conditions [11]. Finally, the steady-state performance of an attitude estimator-which refers to the accuracy while the estimation approaches the true value closely-can deteriorate due to the need for integration and the presence of gyros bias and noise. This article therefore focuses on introducing new approaches to achieve attitude estimation with global convergence and good steady-state performance.

B. Related Work
The earliest solution for attitude estimation is to algebraically resolve attitude by minimizing the Wahba cost function of two or more nonparallel body-referenced measurements of known inertial vectors, e.g., three-axis attitude determination (TRIAD) and quaternion estimator (QUEST). Since angular rate can be used to provide a priori estimate of the attitude, the dynamical approaches with implementation of gyros have received increased attention. The typical examples are Kalman filters (KFs); particularly, the extended KF (EKF) has been the workhorse of real-time attitude estimation [3], [5], [6], [7], [12]. For instance, multiplicative EKF (MEKF) in [13] and [2] computes an unconstrained estimation of attitude errors represented by 3-D vector and parametrizes the global attitude with a unit quaternion. EKF often yields near-optimal performance when the estimate is close to the true state [14]. Nevertheless, EKF is locally stable for most nonlinear systems. It implies that the filter can exhibit poor convergence or even diverge in some situations [15]. To deal with these shortcomings, the state-of-the-art KFs use assorted artifice, such as unscented KF [16], invariant KF [17], and adaptive tuning of the covariance matrices [12], [18], [19]. It is noteworthy that all these optimal stochastic estimators cannot ensure the global convergence in a deterministic context.
Recently, geometric nonlinear observers (NLOs) for attitude estimation have been extensively studied. Some NLOs (e.g., [20], [21], [22], [23], [24], [25], [26], [27], [28]) that were designed as the continuous system possess at best This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ guaranteed almost global asymptotic stable (GAS)-i.e., the system asymptotically converges to the origin from almost all the states except a set of Lebesgue measure zero-due to the aforementioned topological obstructions. However, this non-global property can give rise to slow convergence rate for large estimation errors. Some works (e.g., [23], [28]) use the additional state-dependent gains to accelerate the convergence, but this strategy shrinks the basin of attraction dramatically. Aiming at obtaining a global estimation, a discontinuous attitude observer [29] has been developed, placing the discontinuities at the largest estimation errors. However, it has been shown in [30] that these algorithms are susceptible to arbitrarily small adversarial disturbances. Recently, the synergistic control strategy has been implemented to develop the global NLOs [31], [32], [33], [34], [35]. The essential principle consists of designing a family of Lyapunov functions and gradient-dependent feedback pairs such that the Lyapunov functions always decrease along the solutions and a switch mechanism guarantees a strict decrease when the solutions reach local minima [36]. It should be noted that the aggressively tuned NLO achieves improved convergence rate but can have deteriorated steady-state performance by tradeoff [14].

C. Contributions and Organization
The main contributions of this article are twofold. First, a new attitude and gyro bias NLO is proposed with guaranteed global and robust asymptotic stability. It is based on the synergistic control and incorporates additional state-dependent gains but without changing the switching conditions. This design not only eliminates the non-robustness of the continuous and discontinuous NLOs [20], [21], [22], [23], [24], [25], [26], [27], [28], [29] but also yields an improved convergence rate. In comparison to the existing synergistic NLOs [31], [33], [37], the proposed NLO uses a new family of synergistic functions that possess the explicit expression of the synergistic gap parameter and avoid the complicated warping angle function. In addition, the NLO can be implemented without reconstructing the attitude from body-referenced vector measurements, enabling reduced computation cost in practice. Second, a uniting observer is introduced to combine the proposed NLO with an adaptive implementation based on the estimation error. The uniting observer therefore achieves fast convergence from the aggressively tuned NLO and better steady-state performance from MEKF when the covariance information is available. Moreover, the uniting design could be adapted easily with other global or local attitude observers.
The rest of this article is organized as follows. Mathematical preliminary and problem formulation are briefly presented in Section II. Section III derives the global observer scheme and a new observer. Section IV introduce the uniting observer. The simulation and experimental results are presented in Section V. Finally, conclusions and future perspectives are discussed in Section VI.

II. PRELIMINARIES AND PROBLEM FORMULATION
This section first introduces the notations used in the rest of the article, then formulates the problem of attitude estimation using angular velocity and vector measurements, and finally recalls the synergistic potential functions that play an important role in the proposed observers.

A. Notation
Let R ≥0 := [0, +∞), N be the set of nonnegative integers, and S n = {x ∈ R n+1 : |x| = 1} be the unit n-sphere, where | · | stands for the standard Euclidean norm. The 3-D rotation group is denoted by SO(3) = {R ∈ R 3×3 : R ⊤ R = I 3 , det(R) = 1}, and the space of the 3-by-3 skew-symmetric matrices is denoted by so(3) = { ∈ R 3×3 : ⊤ = − }. Given two matrices A, B ∈ R m×n , their Euclidean inner product is defined as ⟨A, B⟩ = tr(A ⊤ B). The operator (·) ∧ is the skew-symmetric operator such that x ∧ y = x × y, for x, y ∈ R 3 . The inverse operator is (·) ∨ such that (x ∧ ) ∨ = x. Let us define the anti-symmetric projection P a : R 3×3 → so(3) such that P a (X ) = (X − X ⊤ )/2 and define composition map ψ : R 3×3 → R 3 such that ψ(X ) = (P a (X )) ∨ . For each A ∈ R n×n , we define λ(A), λ max (·), λ min (·), and E(A) as the set, maximum, and minimum of eigenvalues of A as well as the unit eigenvectors of A, respectively. The orientation of the body-fixed frame relative to the spatial frame is described by R ∈ SO(3). Given a rotation angle θ ∈ R and an axis u ∈ S 2 , one can recover the attitude by R a (θ, u) = I + u ∧ sin θ + (u ∧ ) 2 (1 − cos θ ). Rotation matrices are also parametrized by the Gibbs vector through the map R g :

B. Problem Formulation
The attitude kinematics are given bẏ where ω ∈ R 3 is the angular velocity expressed in the bodyfixed frame. We use the gyro dynamics error model in [13] whereω ∈ R 3 is the gyros' output, ω ∈ R 3 is the true angular rate, b ∈ R 3 is the gyro bias, and ν v and ν u are the independent zero-mean white-noise processes. We also assume there exists a set of n ≥ 2 noncollinear unit vectors r i , i = 1, . . . , n, known in F a , such as gravity and geomagnetic field. The error model is whereȳ i is the body-reference measurement, and ν y i is the independent zero-mean white-noise process.
LetR (respectivelyb) be the estimate of R (respectively b). The attitude estimate evolves along the trajectory ofṘ =Rω ∧ . The estimation errors are defined bỹ Our objective is to obtain good estimation (R,b) using the measurementω andȳ i , i = 1, . . . , n.

C. Synergistic Potential Functions
We will reformulate the synergistic potential function following from [37] and [33].
Definition 1: A continuously differentiable function V : Let Q ⊂ N be a finite index set and V q is a potential function where ∇ R denotes the gradient with respect to R. Definition 2: The set of potential functions {V (R, q) : q ∈ Q} is centrally synergistic relative to A := {I } × Q with gap exceeding δ if there exists δ > 0 such that Define Y a = [r 1 , . . . , r n ] ∈ R 3×n and Y = [y 1 , . . . , y n ] ∈ R 3×n , for n > 2. If there only exist two inertial vectors, we use r 3 = r 1 ×r 2 and y 3 = y 1 ×y 2 . Then, it is easy to see Y = R ⊤ Y a . Also, denote the measurement of Y a byȲ . Then, the Wahba cost function J 0 : is the positive diagonal matrix of the weight factors for the measured vectors. Interestingly enough, this cost function has been broadly applied in literature [20], [21], [22], [23], [24], [25] in a simpler manner, where M ∈ R 3×3 is symmetric and positive definite. It is clear that 0 with respect to R is a potential function. Let M a = Y a W Y ⊤ a and G a = tr(M a )I − M a . A trivial verification shows that J 0 (R, Y ) = 0 (R, M a ). According to [33,Lemma 2], the set of critical points of 0 is {I } ∪ {R a (π, E(M a ))} and the maximum of 0 is 2λ max (G a ).

III. GLOBAL ATTITUDE OBSERVER
This section presents the general form of the global NLO for attitude estimation and states its global asymptotic stability. Then, it details how to use a family of synergistic functions to exemplify the proposed observer. The readers who would like to focus on the implementation of the proposed global NLO can skip Section III-A.

A. Global Observer Design
Assumption 1: The set of potential functions V (R, q), q ∈ Q is synergistic with gap exceeding δ > 0 as in Definition 2.
Remark 1: Such functions fulfilling Assumption 1 will be given in Proposition 1. Assumption 2 is standard in [14] and [20] and is a mild condition in practice. Define the variableω :=ω −b −ω. Then, the following attitude and gyro bias NLO is proposed: where k ω , k b , and D are positive scalars, the jump map is defined as , and the flow and jump set are defined as respectively. Note that the jump map only depends onR and the logic variable q in state. The flowchart of (6) is shown in Fig. 1.
Theorem 1: Let Assumption 1 and 2 hold. Then, the set of A a is globally and robustly asymptotically stable for the hybrid closed-loop system by (1), (2), and (6).
Remark 2: If we dropped out the switching mechanism in (6), the best that the observer (6) for each logical index q ∈ Q can achieve is almost GAS. The underlying issue is consistent with the fact that attitude estimates with rotational degrees of freedom cannot be globally asymptotically stabilized to the true values in a continuous manner [8]. Furthermore, an arbitrarily small piecewise-constant noise signal can destroy the asymptotic stability; see [30] and [35].
Theorem 2: Suppose that there exists a continuously differentiable function α : [0, a) → [0, ∞) such that α(0) = 0 and 0 < α ′ (·) < r for some r ∈ R ≥0 , and a family of centrally synergistic potential functions {V 0 (R, q) : q ∈ Q}. The composite function V 1 := α • V 0 is centrally synergistic. In addition, consider the hybrid observer (6) such that the flow map (6) is applied with V 1 while the jump map, flow, and jump set are defined with V 0 . Then, the set of A a is globally and robustly asymptotically stable for the closed loop.
Remark 3: Theorem 2 proved in Appendix B provides a simple way to add state-dependent gain. Although V 1 is applicable to (6), one can directly enjoy the state-dependent gain α ′ (V 0 ) on the switch rules of the original observer used with V 0 . This avoids calculating a new synergistic gap.

B. Synergistic Functions
The global stability of the proposed scheme relies on a synergistic family of functions. Now, we will introduce angular warping for inducing such a synergistic family where q ∈ Q, {u q ∈ S 2 : q ∈ Q} is a set of unit vectors, . The next proposition proved in Appendix C states that it is possible to generate a synergistic family from the composition of (7) and (5).
Remark 4: Compared with the work in [33] and [32], our result does not require to specify the warping angle function by the inverse trigonometric function about 0 , and the explicit expression of the gap (8) is computed which was not given in [31].
Remark 5: It is necessary to determine u for the implementation of Proposition 1, which is equivalent to finding u such that (v i , u) > 0 for i = 1, 2, 3. One possible solution can be found in [33,Proposition 2].
It is enough to apply V 0 proposed in Proposition 1 to construct the global NLO. Nevertheless, since the gradient of 0 (R, M a ) vanishes at its critical points, the observer might suffer from slow convergence when the initial attitude errors are large. As such, in light of Theorem 2, the following new synergistic family of potential functions is introduced: where c > 2λ max (G a ) is a constant. Hence, a state-dependent gain is added in the feedback to improve convergence rates.
Finally, we summarize the explicit form of the proposed global NLO as follows: the gains k ω , k b , and D are defined in (6), 0 < k < (2λ max (G)) −1 , and u q and synergistic gap can be obtained from Proposition 1. In practical implementation, one has to replace the ideal value y i with the measurementȳ i . Clearly, the proposed observer operates in two modes, represented by the logical variable q.

IV. UNITING OBSERVER
This section presents the uniting observer that would improve the steady-state performance of NLO. First, the MEKF is recalled as a local observer, and it provides a second-order approximation and performs quite well when the estimation is close to the nominal truth value. Next, the global NLO from Section III-B is combined with the MEKF in a uniting framework.

A. Local Observer
The main idea of MEKF is to use an unconstrained estimate of the three-component parametrization of attitude error to reset the attitude estimate. The state of MEKF is defined as whereg ∈ R 3 is the Gibbs vector ofR ⊤ R. Of note, (R,b) are the implicit states of MEKF, because they need to be propagated too. Then, the linear approximation of the dynamics of x is given bẏ The observation equation is derived from (3), given bȳ , and H t denotes the new matrix by concatenating matrices [2ŷ ∧ i 0 3 ], i = 1, . . . , n, in a vertical direction. Then, the discrete-time of MEKF proceeds in three steps: time propagation (12a) and (12b), measurement update (12c)-(12e), and reset (12f) K is the Kalman gain, and F and G are defined as When a reset is performed after each measurement update, the term x t+1 is identically zero, and thus priori estimate x − t+2 is identically zero. It follows that the propagating of x in (12a) can be avoided.
Remark 6: MEKF possesses a robust steady-state performance, but it often exhibits poor convergence properties when the initial attitude estimation errors are large with an underestimated covariance; see the first case of simulations in Section V. In addition, the undesired stationary points of
Define an overall state ξ := (ζ 0 , ζ 1 , z 0 , z 1 , τ, s), where z i ∈ R ≥0 is the error-norm estimator, τ ∈ R is a temporal regularization, and s ∈ {0, 1} is the mode index. Also, denote the overall state space by W u = W 0 × W 1 × R 3 ≥0 × {0, 1}. Following the hybrid framework in [38], we proposed the attitude uniting observer (see Fig. 2) where the flow and jump maps are given by The error-norm estimator z i gives an estimate of discrepancy between the observer estimate and the current system state. The error norm estimate is calculated by the Wahba cost function, ρ(ȳ,ŷ) = J 0 (R,Ȳ ). The function Θ(ζ 1 ) := (ζ 1 , P 0 ) maps the global observer variable ζ 1 to the local observer variable ζ 0 , where P 0 is any positive definite symmetric covariance matrix that ensures the convergence of MEKF. The flow and jump sets are defined as F u = F 1 ∪ F 2 and This observer operates in two modes denoted by the logical variable s: local mode and global mode. When s = 0, the local mode is activated. The estimate is given by the local observer when its error-norm estimator z 0 is bounded by c 0 . When z 0 ≥ c 0 , we conjecture that the local observer is not converging to the truth or at a slow rate. Then, in that case, the global mode is switched on, in the sense that a jump is imposed as shown in Fig. 2. Of note, mode switch from global to local is allowed when the global observer has a small error norm and work for at least a T 1 units of time. The second condition helps avoid unnecessary multiple consecutive jumps in the scheme.

V. SIMULATION AND EXPERIMENT
In this section, the simulation and experiment results are shown to demonstrate the performance of the proposed global NLO and uniting observer in comparison to the existing methods. For brevity, the uniting observer is termed as UO, and the global NLO (10) is termed as H-NLO1. Two almost global NLOs are given. One is the continuous observer (6) applied with 1 , i.e., the non-global version of H-NLO1 without angular warping, and is termed as NLO1, which is a state-dependent complementary filter analogous to the filters in [23] and [28]. The other is NLO0 in [22], which is in fact the continuous observer (6) applied with 0 . The MEKF is implemented with (12), which differs slightly from the ones in [13] and [2] in that the attitude estimate herein is represented by the rotation matrices instead of quaternions. Note that the MEKF, NLO1, and NLO0 are the local estimators and that the proposed UO and H-NLO1 are the global estimators.

A. Simulation
Numerical examples are presented to validate our theoretical results. The attitude trajectories are generated by ω(t) = [1.0 sin(0.9t) 0.6 cos(0.7t) 0.2 cos(0.2t)] ⊤ rad/s and initial attitude R(0) = I . There exist three known inertial vectors In the first case, no noise is added in measurements to observe the behaviors of the observers around the undesired critical points. The initial conditions are set to be close to the undesired critical point of the estimators. Therefore, it can be shown in Fig. 3 that the estimate errors of the local estimators have been nondecreasing for a long time. Although the errors of MEKF and NLO1 finally decrease to zero, it is because the undesired critical points are unstable equilibria, and thus the system can be rendered to leave those equilibria by the accumulated computation errors. On the other hand, the poor convergence did not exist in the proposed global observer H-NLO1, because H-NLO1 can make use of the hybrid switching mechanism to choose a nonvanishing feedback at the whole state space except from the desired equilibrium; see the switching event denoted by q. The proposed UO also inherited improved convergence from H-NLO1 via the switching mechanism: it operated in the global mode (s = 1) before 5 s, thereby behaving in full accord with H-NLO1; when the estimation errors were decreased to a small value below, the local mode (s = 0) was switched on and then the steady-state errors were maintained at a better level by the second-order approximation of MEKF.
The second case involves the aforementioned noised measurements with two initial conditions. The first initial condition is the same as the preceding case that involves no noise, and the results are depicted in Fig. 4(a), (c), and (e). It can be seen that the slow convergence remains in the local stable estimators, although NLO1 and NLO0 escaped the trap of unstable equilibria earlier and the MEKF behaved similar to the first case. This is because the second-order filter (MEKF) is more robust to the disturbances than the first-order observer (NLO1 and NLO0) when they are placed at an equilibrium. NLO1 converged faster than NLO0 from large initial errors, since the additional state-dependent gain increases as the state error gets large. In contrast, the proposed H-NLO1 and UO did not suffer from the slow convergence intrinsically, since they did not possess other equilibria except the desired equilibrium under the guarantee of the hybrid switching mechanism.
The second initial condition of the second case is set as large estimation errors but not the equilibria, and the results are depicted in Fig. 4(b), (d), and (f). It is easy to see that all the NLOs including H-NLO1, NLO1, and NLO0 obtained a faster convergence than MEKF. This is consistent with the existing observation in [15] that the EKF can exhibit poor convergence or even diverge in some situation. Moreover, it could be seen in Fig. 4(d) that the proposed H-NLO1 and UO converged faster than NLO1 and NLO0, since the additional state-dependent gain in H-NLO1 enlarged the state feedback by the norm of the estimation errors and H-NLO1 always chooses the feedback (indexed by q) that decreases the errors faster via synergistic strategy.
Finally, the UO achieved the same steady-state performance as MEKF: the root-mean-square (rms) errors are 6.5×10 −3 rad for attitude and 2.4 × 10 −3 rad/s for bias. The rms errors of H-NLO1, NLO1, and NLO0 are 9.6 × 10 −3 rad for attitude and 2.9 × 10 −3 rad/s for bias. The UO achieved a smaller rms error than H-NLO1 in this case, because the local observer MEKF is the optimal second-order estimator for the linear Gaussian measurements.

B. Experiment
The methods were also evaluated using a inertial measurement unit (IMU) (MTi-3, Xsens Inc., The Netherlands). It provides the measurements of gyro, accelerometer, and magnetometer and outputs an attitude estimate for reference. The sensor component specifications are given in Table I and the orientation performance is 0.5 • rms for roll/pitch and 2 • rms for yaw. The experiment was also equipped with a vision camera, MicronTracker Hx40 from Claronav Inc., for real-time optical measurement of the pose. The specifications of H × 40 are given in Table II. The experimental setup is shown in Fig. 5. A marker was attached to the IMU to allow the MicronTracker software to detect the IMU's 6-degree-offreedom (DoF) pose. Since the inertial referenced frame of  MicronTracker is usually different from that of IMU, we attach another marker to the stationary platform for a referenced constant attitude, and then compensate the constant difference [see Fig. 5(b)].
The inertial vectors are normalized from gravity, local magnetic field, and their cross-product, given by  (0.659, 0.192, 0.149). The parameters for H-NLO1 and During the experiments, the sensor stayed stationary for the first 80 s, was moved and rotated randomly within the field of view of the camera from 80 to 150 s, and finally returned to being stationary. The results for two initial conditions are plotted in Figs. 6 and 7. Since no reference exists for gyro bias, the norm of bias estimates is given instead of the bias estimation errors. We also adopt the outputs from the MTi and MicronTracker as the ground truth, respectively, to evaluate the attitude estimation errors of different methods. Note that MTi provides the attitude from the inertial information and that MicronTracker provides the attitude from the visual information.
In the first case, the initial conditions are set such that the attitude error is close to the undesired equilibria and the bias estimation is zero. As illustrated in Fig. 6, the proposed H-NLO1 and UO exhibit better convergence than other local estimators. It is because H-NLO1 incorporates the additional state-dependent gain and also benefits from the synergistic strategy so that the desired equilibrium is globally stable. On the other hand, the poor convergence of the local estimators seems to be less serious than in the simulation results, since we could not set the initial conditions at exactly the unwanted equilibria in practice due to the errors in the inertial vectors. Particularly, the vector that is associated with the magnetic field is susceptible to disturbances. In addition, the UO achieves the steady state faster than H-NLO1, because the local observer was activated when the estimation errors became sufficiently small; see the switching event of variable s in Fig. 6(d).
In the second case, the initial conditions are set such that the attitude error is not far from the desired equilibrium. Therefore, the estimation errors of all the estimators decrease to zero quickly, as illustrated in Fig. 7. However, by the enlarged plots, it can still be noted that the MEKF produces the poorest convergence, because it has the additional state covariance to be estimated in comparison to other estimators. Moreover, it is shown in Fig. 7(b) that the UO quickly stabilizes the bias estimation when the local observer is activated. It is because the UO has reduced the estimation error to a small neighborhood of zero, from which region the optimal second-order approximation allows rapid convergence.
The attitude rms errors in the two cases are listed in Table III. It can be seen that the estimation errors of all the algorithms are smaller when compared with the reference from MTi-3 than the reference from MicronTracker. The reason might lie in that the attitude from MTi-3 is also extracted from inertial information, and therefore its error source is similar to that of those algorithms. In contrast, the lower sampling period and the time lag of the MicronTracker can amplify the estimation errors, so it cannot distinguish the steady-state performance of those algorithms. Referring to MTi-3 as the ground truth, one can conclude from Table III that the UO and MEKF exhibit better steady-state performance than the others. Note that the UO and MEKF achieve similar rms errors because MEKF is the local observer of UO. The factors that affect the accuracy of the attitude estimation can be divided into two categories. The first category is related to sensor specifications, including noise density, bias instability, nonlinearity, and calibration errors. The second category mainly contains the measurement disturbances and environmental noises. For example, the accelerometer measures specific force composed of the gravity and the translation acceleration, and the magnetometer measurement is affected by the nearby ferromagnetic materials. Consequently, the estimation errors in the dynamic state are larger than the errors in the static state; see Figs. 6 and 7.
Finally, the execution time in the two cases is given in Table IV. The computation was carried out by using MAT-LAB2022b on a laptop with Intel i5-10210U 1.60 GHz CPU. The proposed H-NLO1 has a lower time complexity than NLO1 and the NLO0, since it possesses the additional hybrid switching mechanism to compute at each iteration. The MEKF needs more computation cost in comparison to other NLOs, in view of the fact that the Kalman gain contains the matrix  TABLE III  RMS ERRORS WITH TWO REFERENCED GROUND TRUTH   TABLE IV  TIME COMPLEXITY: EXECUTION TIME inverse operation. The proposed UO that achieves improved convergence and steady-state performance from H-NLO1 and MEKF, respectively, however requires more computation cost by tradeoff.

C. Discussion
In the preceding simulation and experiments, the proposed H-NLO1 achieves global convergence by synergistic control strategy and that the UO not only inherits the desired convergence from H-NLO1 but also maintains a better steadystate performance. The global convergence is significant in practice because it intrinsically excludes the ill-defined singularities and unwanted equilibria that is non-robust to arbitrarily small disturbances. This is of particular importance in the robotics field with increasing human-robot integration during which safety will be a basic requirement. Moreover, a uniting observer has been proposed to improve the steady-state performance of the global observer by integrating a high-performance local observer as we showed in the experiments. However, there still exist several limitations with the proposed approaches. First, H-NLO1 and UO demand more computation cost than the existing approaches. Second, MEKF may not be a desired local observer in other scenarios, for example, the measurement covariance is not Gaussian. If no other variant of KF could have better rms errors than H-NLO1, H-NLO1 is recommended instead of UO.

VI. CONCLUSION
This article proposes a general scheme for designing global attitude observer and a uniting observer. The new scheme is directly perceived through the variational principle in classical mechanics. Under the scheme, a global observer is developed with faster convergence. Since MEKF has a better steady-state performance when noise process is Gaussian and known, then we design a uniting observer that inherits the desired attributes from MEKF and the new global observer. In the future, possible directions of this research include: improving the steady-performance for non-Gaussian noise, the case of only one inertial vector for measurement, and the case of intermittent measurements.
Finally, appealing to [36,Th. 3.26], one can deduce that asymptotic stability of A a is robust, since A a is a globally asymptotically stable compact set for closed-loop H a .
Let V 1 = T (ω,b)+ V 1 (R, q) be a Lyapunov function candidate of the closed loop. By use of the result above, V 1 is strictly decreasing over the jumps J a,1 = {x ∈ X : µ V 0 (R, q) ≥ δ}. As in the proof of Theorem 1, the same reasoning enables us to assert A a globally and robustly asymptotically stable for the closed loop.

ACKNOWLEDGMENT
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