Analytical Solutions to a Class of Feedback Systems on SO( n )

Markdahl, Johan; Hu, Xiaoming

doi:10.3182/20140824-6-za-1003.02051

Cited by 5 publications

(15 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Relevant literature includes works concerning stabilization [19], synchronization [7], and estimation [8] on SO(n). It also includes the previous work [13,14] of the authors. Note that work on SO(n) for n ≥ 4 is not only of theoretical concern; it also finds applications in the visualization of high-dimensional data [34].…”

Section: Introductionmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

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].…”

mentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

See 3 more Smart Citations

A geodesic feedback law to decouple the full and reduced attitude

Markdahl

Hoppe

Wang

et al. 2017

Systems & Control Letters

Self Cite

View full text Add to dashboard Cite

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 .

show abstract

Section: Introductionmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

A geodesic feedback law to decouple the full and reduced attitude

Markdahl

Hoppe

Wang

et al. 2017

Systems & Control Letters

Self Cite

View full text Add to dashboard Cite

show abstract

“…This would also allow for the error order of the Runge-Kutta method to be evaluated numerically. While it is true that closed-form solutions of ODEs on nonlinear spaces cannot be obtained in general, the solutions are available in special cases, as shown in [47].…”

Section: Discussionmentioning

confidence: 99%

Model Formulation Over Lie Groups and Numerical Methods to Simulate the Motion of Gyrostats and Quadrotors

Fiori

2019

Mathematics

View full text Add to dashboard Cite

The present paper recalls a formulation of non-conservative system dynamics through the Lagrange–d’Alembert principle expressed through a generalized Euler–Poincaré form of the system equation on a Lie group. The paper illustrates applications of the generalized Euler–Poincaré equations on the rotation groups to a gyrostat satellite and a quadcopter drone. The numerical solution of the dynamical equations on the rotation groups is tackled via a generalized forward Euler method and an explicit Runge–Kutta integration method tailored to Lie groups.

show abstract

Exact solutions to a class of feedback systems on SO(n)

Markdahl

2016

Automatica

Self Cite

View full text Add to dashboard Cite

Analytical Solutions to a Class of Feedback Systems on SO( n )

Cited by 5 publications

References 18 publications

A geodesic feedback law to decouple the full and reduced attitude

A geodesic feedback law to decouple the full and reduced attitude

Model Formulation Over Lie Groups and Numerical Methods to Simulate the Motion of Gyrostats and Quadrotors

Exact solutions to a class of feedback systems on SO(n)

Contact Info

Product

Resources

About