Analytical solutions to feedback systems on the special orthogonal group SO(n)

Markdahl, Johan; Thunberg, Johan; Hoppe, Jens; Hu, Xiaoming

doi:10.1109/cdc.2013.6760714

Cited by 5 publications

(25 citation statements)

References 18 publications

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“…Remark 3. The algorithm of [12] as well as the two algorithms of [14] are special cases of Algorithm 1. In [12], n = 3 and P = e 1 ⊗ e 1 .…”

Section: Control Designmentioning

confidence: 99%

“…In [12], n = 3 and P = e 1 ⊗ e 1 . In [14], n ∈ N, and P = I or P = I − e n ⊗ e n for the two respective algorithms. This paper explores the general case of n ∈ N and P ∈ {A ∈ R n×n | A 2 = A, A ⊤ = A}, with focus on projection matrices that satisfy rank P ≤ n − 2.…”

Section: Control Designmentioning

confidence: 99%

“…the second category. The closed-loop kinematics on SO(3) are solved for the states as functions of time and the initial conditions, providing precise knowledge of the workings of the transient dynamics.Recent work on the problem of finding exact solutions to closed-loop systems on SO(n) includes [13,14]. Related but somewhat different problems are addressed in [6,16,22].…”

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confidence: 99%

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A geodesic feedback law to decouple the full and reduced attitude

Markdahl

Hoppe

Wang

et al. 2017

Systems & Control Letters

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This paper presents a novel approach to the problem of almost global attitude stabilization. The reduced attitude is steered along a geodesic path on the n − 1-sphere. Meanwhile, the full attitude is stabilized on SO(n). This action, essentially two maneuvers in sequel, is fused into one smooth motion. Our algorithm is useful in applications where stabilization of the reduced attitude takes precedence over stabilization of the full attitude. A two parameter feedback gain affords further trade-offs between the full and reduced attitude convergence speed. The closed loop kinematics on SO(3) are solved for the states as functions of time and the initial conditions, providing precise knowledge of the transient dynamics. The exact solutions also help us to characterize the asymptotic behavior of the system such as establishing the region of attraction by straightforward evaluation of limits. The geometric flavor of these ideas is illustrated by a numerical example. the second category. The closed-loop kinematics on SO(3) are solved for the states as functions of time and the initial conditions, providing precise knowledge of the workings of the transient dynamics.Recent work on the problem of finding exact solutions to closed-loop systems on SO(n) includes [13,14]. Related but somewhat different problems are addressed in [6,16,22]. Earlier work [12] by the authors is strongly related but also underdeveloped; its scope is limited to the case of SO(3). This paper concerns a generalization of the equations studied in [12,14]. The results of [14] is also generalized in [13], partly towards application in model-predictive control and sampled control systems and without focus on the behavior of the reduced attitude. The work [15] addresses the problem of continuous actuation under discrete-time sampling. The exact solutions provide an alternative to the zero-order hold technique. The algorithm alternates in a fashion that is continuous in time between the closed-loop and open-loop versions of a single control law. The feedback law proposed in this paper can also be used in such applications by virtue of the exact solutions. PreliminariesLet A, B ∈ C n×n . The spectrum of A is written as σ(A). Denote the transpose of A by A ⊤ and the complex conjugate by A * . The inner product is defined by A, B = tr(A ⊤ B) and the Frobenius norm by A F = A, A 1 2 .

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“…Remark 3. The algorithm of [12] as well as the two algorithms of [14] are special cases of Algorithm 1. In [12], n = 3 and P = e 1 ⊗ e 1 .…”

Section: Control Designmentioning

confidence: 99%

Section: Control Designmentioning

confidence: 99%

mentioning

confidence: 99%

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A geodesic feedback law to decouple the full and reduced attitude

Markdahl

Hoppe

Wang

et al. 2017

Systems & Control Letters

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“…Sinclair, 2004] on SO(n). It also includes our previous paper [Markdahl et al, 2013], which we shall comment on shortly.…”

Section: Introductionmentioning

confidence: 99%

“…The main contribution of this paper is to provide analytical solutions to differential equations representing closed feedback loops on SO(n). Recent work on this problem include Markdahl et al [2012Markdahl et al [ , 2013. Other works such as those previously referenced by Elipe and Lanchares [2008], M.A.…”

Section: Introductionmentioning

confidence: 99%