Optimal Control on Lie Groups: The Projection Operator Approach

Saccon, Alessandro; Hauser, John; Aguiar, A. Pedro

doi:10.1109/tac.2013.2258817

Cited by 58 publications

(81 citation statements)

References 30 publications

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“…Here, Hess1 V (ĝ(t), t) : Tĝ (t) G → T * g(t) G is the Hessian operator, see Section II. Since ∂ ∂t V : G × R → R satisfies the HJB equation (23), by straightforward application of the chain rule we obtain…”

Section: The Left-trivialized Hjb Equation and The Structure Of Thmentioning

confidence: 99%

“…Here, we follow the approach detailed in, e.g., [27], [23] and select the product group structure, that is, for (R, X) and (S, Y) ∈ SO(3)×R 3 , we define the group product as…”

Section: A Derivation Of the Filter Equationmentioning

confidence: 99%

“…First, as done in [27], [23], we make use of the symmetric Cartan-Schouten (0)-connection, characterized by the connection function…”

Section: A Derivation Of the Filter Equationmentioning

confidence: 99%

See 2 more Smart Citations

Second-Order-Optimal Minimum-Energy Filters on Lie Groups

Saccon

Trumpf

Mahony

et al. 2016

IEEE Trans. Automat. Contr.

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Abstract-Systems on Lie groups naturally appear as models for physical systems with full symmetry. We consider the state estimation problem for such systems where both input and output measurements are corrupted by unknown disturbances. We provide an explicit formula for the second-order-optimal nonlinear filter on a general Lie group where optimality is with respect to a deterministic cost measuring the cumulative energy in the unknown system disturbances (minimum-energy filtering). The resulting filter depends on the choice of affine connection which encodes the nonlinear geometry of the state space. As an example, we look at attitude estimation, where we are given a second order mechanical system on the tangent bundle of the special orthogonal group SO(3), namely the rigid body kinematics together with the Euler equation. When we choose the symmetric Cartan-Schouten (0)-connection, the resulting filter has the familiar form of a gradient observer combined with a perturbed matrix Riccati differential equation that updates the filter gain. This example demonstrates how to construct a matrix representation of the abstract general filter formula.

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Section: The Left-trivialized Hjb Equation and The Structure Of Thmentioning

confidence: 99%

“…Here, we follow the approach detailed in, e.g., [27], [23] and select the product group structure, that is, for (R, X) and (S, Y) ∈ SO(3)×R 3 , we define the group product as…”

Section: A Derivation Of the Filter Equationmentioning

confidence: 99%

See 1 more Smart Citation

Second-Order-Optimal Minimum-Energy Filters on Lie Groups

Saccon

Trumpf

Mahony

et al. 2016

IEEE Trans. Automat. Contr.

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“…This involves trajectory optimization over manifolds such as the SE( 3) Lie group, instead of just vector spaces of the form R n . We accomplish this by iteratively optimizing over increments to the trajectory, defined in terms of the corresponding Lie algebra-se( 3) in our case (Saccon et al, 2013). We applied this extension of TrajOpt to two real-world clinical applications.…”

Section: Introductionmentioning

confidence: 99%

Motion planning with sequential convex optimization and convex collision checking

Schulman

Duan

et al. 2014

The International Journal of Robotics Research

652

472

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“…Different approaches for LQR on SO(3), SE(3) or coverings of these groups exists. Saccon et al [7] derive a LQR controller on SO(3) through Pontryagin's Maximum Principle. Marinho et alin [8] use a dual-quaternion representation to derive a LQR tracking controller.…”

Section: Introductionmentioning

confidence: 99%

Learning task-space synergies using Riemannian geometry

Zeestraten

Havoutis

Calinon

et al. 2017

2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)

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Abstract-In the context of robotic control, synergies can form elementary units of behavior. By specifying taskdependent coordination behaviors at a low control level, one can achieve task-specific disturbance rejection. In this work we present an approach to learn the parameters of such lowlevel controllers by demonstration. We identify a synergy by extracting covariance information from demonstration data. The extracted synergy is used to derive a time-invariant state feedback controller through optimal control. To cope with the non-Euclidean nature of robot poses, we utilize Riemannian geometry, where both estimation of the covariance and the associated controller take into account the geometry of the pose manifold. We demonstrate the efficacy of the approach experimentally in a bimanual manipulation task.

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Optimal Control on Lie Groups: The Projection Operator Approach

Cited by 58 publications

References 30 publications

Second-Order-Optimal Minimum-Energy Filters on Lie Groups

Second-Order-Optimal Minimum-Energy Filters on Lie Groups

Motion planning with sequential convex optimization and convex collision checking

Learning task-space synergies using Riemannian geometry

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