Guaranteed Performance of Nonlinear Attitude Filters on the Special Orthogonal Group SO(3)

Abstract

This paper proposes two novel nonlinear attitude filters evolved directly on the Special Orthogonal Group SO (3) able to ensure prescribed measures of transient and steadystate performance. The tracking performance of the normalized Euclidean distance of attitude error is trapped to initially start within a large set and converge systematically and asymptotically to the origin from almost any initial condition. The convergence rate is guaranteed to be less than the prescribed value and the steady-state error does not exceed a predefined small value. The first filter uses a set of vectorial measurements with the need for attitude reconstruction. The second filter does not require attitude reconstruction and instead uses only a rate gyroscope measurement and two or more vectorial measurements. These filters provide good attitude estimates with superior convergence properties and can be applied to measurements obtained from low cost inertial measurement units (IMUs). Simulation results illustrate the robustness and effectiveness of the proposed attitude filters with guaranteed performance considering high level of uncertainty in angular velocity along with body-frame vector measurements.

Systematic convergence of nonlinear stochastic estimators on the Special Orthogonal Group SO(3)

“…where Ω ∈ {B} denotes the true angular velocity. Considering the normalized Euclidean distance of R in (2) and the identity in (9), the kinematics in (14) can be expressed in terms of normalized Euclidean distance as d dt…”

“…Define R y as a reconstructed attitude of the rotational matrix R. Consider the following estimator Proof (21), (22), and (39), respectively. From (14) and (41), the error kinematics can be translated into…”

“…[Ω] × ] =R [Ω] × + [Ω] ×R is the Lie bracket as defined in (6). In view of (14) and (15), the error dynamics in (45) can be expressed in the sense of normalized Euclidean distance as…”

“…Proof (21), (22), and (39), respectively. From (14) and (63), the attitude error kinematics are analogous to (45). From (57), it can be found thaṫ…”

“…44) withR = R yR , Υ(R) = vex(P a (R)), E being defined in(29), γ 1 , γ 2 , and k w being positive constants, ||R|| I = Tr{I 3 −R} being defined in (2), ξ denoting PPF defined in(23), andb andσ being the estimates of b and σ, respectively. Consider the attitude dynamics in(14), vector measurements in(12), and angular velocity measurements in (16) coupled with the estimator in (41), (42), (43), and (44). Assume that at least two body-frame non-collinear vectors are available for measurements.…”