Rigid Body Attitude Estimation- An Overview and Comparative Stud.pdf

The locally exponentially stable equilibrium point is

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The locally exponentially stable equilibrium point is ( R * e 1 , p * e 1 ) = ( I 3 , 0), and the other 3 equilibria are unstable. For instability proof of these equilibria, Chetaev-like arguments are provided. This observer has the advantage of being designed directly on the Special Euclidean group. The authors provide an interesting study of the application of Lie-algebra studies in the design of observers for special groups and their work can be regarded as a continuity to the previous works on the invariant observers. However, it is evident that while the invariant observers are observers with an invariant structure, the gradient and gradient-like observers are observers with non-invariant structures, but an invariant cost function from which the correction terms of the observer are derived. 4.5 Nonlinear Complementary Filters As discussed in section (4.3), linear complementary filters have been long known to be reliable tools for attitude estimations. Nonlinear complementary filters, however, are rela- tively new and have various di ff erences with the linear filters of this kind including di ff erent structure. In many of these filters, reconstructions of the attitude is required for the filter as in- puts and is then “filtered” to give better results. In such filters, observer-like structures with Lyapunov-based arguments are used to guarantee an ultimate convergence of the filter
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C hapter 4. D ynamic A ttitude F iltering and E stimation 71 output to the real attitude. By “measured attitude” we mean an estimation provided from numerical methods, such as QUEST, through vectorial measurements. Nonlinear complementary filters are designed based on the nonlinear structure of the system and therefore, give better results than linear approximative filters. With strong Lyapunov theory arguments, the estimated states are guaranteed to be closer to the actual system states. The earliest work in this field that captured the true nonlinear nature of the rotational dynamics was presented in [Salcudean, 1991]. The proposed filter has a nonlinear structure and stability proofs for nonlinear systems using Lyapunov analysis are provided. The work includes a globally convergent nonlinear filter for the attitude and angular velocity of a rigid body. The filter uses both the Inertia matrix I f and the vector of applied torque τ in the estimation law to estimate the angular velocity ˆ ω and the orientation I f ˙ ˆ ω = τ + 1 2 k p I - 1 f ˜ q sign(˜ q 0 ) , ˙ ˆ R = ˜ R T ( ˆ ω + k v I - 1 f ˜ q sign(˜ q 0 )) × ˆ R , (4.80) where ˜ Q = q 0 , ˜ q ) = Q ˆ Q - 1 is the quaternion error between the actual attitude quaternion and the estimated quaternion. k p and k v are positive coe ffi cients and [ . ] × is the equiva- lent representation of the skew-symmetric matrix. For the proof of filter convergence, a Lyapunov function V 1 is defined as V 1 = μ T μ + k p (1 - ˜ q 0 sign(˜ q 0 )) 2 + ˜ q T ˜ q , (4.81) where μ : = I f ω - I f ˆ ω . It is shown that the function is decreasing along the quaternion error trajectories and the quaternion error scalar ˜ q 0
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