Structured linearization of discrete mechanical systems on Lie groups: A synthesis of analysis and control

Fan, Taosha; Murphey, Todd D.

doi:10.1109/cdc.2015.7402357

Cited by 10 publications

(8 citation statements)

References 23 publications

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“…Linearization of constrained mechanical systems in discrete time has been proposed by [27] and [28]. However, in these methods the linear models incorporate the constraints directly and they are not explicitly retained.…”

Section: B Linearization Of Mechanically Constrained Systemsmentioning

confidence: 99%

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Linear-Quadratic Optimal Control in Maximal Coordinates

Brüdigam,

Manchester

2020

Preprint

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The Linear-quadratic regulator (LQR) is an efficient control method for linear and linearized systems. Typically, LQR is implemented in minimal coordinates (also called generalized or "joint" coordinates). However, recent research suggests that there may be numerical and control-theoretic advantages when using higher-dimensional non-minimal state parameterizations for dynamical systems. One such parameterization is maximal coordinates, in which each link in a multi-body system is parameterized by its full SE(3) pose and joints between links are modeled with algebraic constraints. Such constraints can also account for closed kinematic loops or contact with the environment. This paper presents an extension to the standard LQR control method to handle dynamical systems with explicit physical constraints. Experiments comparing the basins of attraction and tracking performance of minimal-and maximal-coordinate LQR controllers suggest that maximal-coordinate LQR achieves greater robustness and improved tracking compared to minimalcoordinate LQR when applied to nonlinear systems.

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Section: B Linearization Of Mechanically Constrained Systemsmentioning

confidence: 99%

“…Discrete-time LQR for systems with mechanical constraints has been proposed by [27] and [28]. However, as mentioned before, the constraints are directly incorporated in the linear system and not explicitly retained.…”

Section: Linear-quadratic Regulationmentioning

confidence: 99%

Linear-Quadratic Optimal Control in Maximal Coordinates

Brüdigam,

Manchester

2020

Preprint

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“…By the least action principle with Equation ( 9), (10), and ( 14), we can derive the DEL equation for a single rigid body in SE(3), which is the well known discrete reduced Euler-Poincaré equations [11,16]:…”

Section: Variational Integrators In Se(3)mentioning

confidence: 99%

A Linear-Time Variational Integrator for Multibody Systems

Lee¹,

Liu²,

Park³

et al. 2016

Preprint

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We present an efficient variational integrator for simulating multibody systems. Variational integrators reformulate the equations of motion for multibody systems as discrete Euler-Lagrange (DEL) equation, transforming forward integration into a root-finding problem for the DEL equation. Variational integrators have been shown to be more robust and accurate in preserving fundamental properties of systems, such as momentum and energy, than many frequently used numerical integrators. However, state-of-the-art algorithms suffer from O(n 3 ) complexity, which is prohibitive for articulated multibody systems with a large number of degrees of freedom, n, in generalized coordinates. Our key contribution is to derive a quasi-Newton algorithm that solves the root-finding problem for the DEL equation in O(n), which scales up well for complex multibody systems such as humanoid robots. Our key insight is that the evaluation of DEL equation can be cast into a discrete inverse dynamic problem while the approximation of inverse Jacobian can be cast into a continuous forward dynamic problem. Inspired by Recursive Newton-Euler Algorithm (RNEA) and Articulated Body Algorithm (ABA), we formulate the DEL equation individually for each body rather than for the entire system, such that both inverse and forward dynamic problems can be solved efficiently in O(n). We demonstrate scalability and efficiency of the variational integrator through several case studies.

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“…Variational integrators conserve symplectic form, constraints and energetic quantities [1][2][3][4][5][6]. As a result, variational integrators generally outperform the other types of integrators with respect to numerical accuracy and stability, thus permitting large time steps in simulation and trajectory optimization, which is useful for complex robotic systems [1][2][3][4][5][6]. Moreover, variational integrators can also be regularized for collisions and friction by leveraging the linear complementarity problem (LCP) formulation [7,8].…”

Section: Introductionmentioning

confidence: 99%

Efficient Computation of Higher-Order Variational Integrators in Robotic Simulation and Trajectory Optimization

Fan¹,

Schultz²,

Murphey³

2019

Preprint

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This paper addresses the problem of efficiently computing higherorder variational integrators in simulation and trajectory optimization of mechanical systems as those often found in robotic applications. We develop O(n) algorithms to evaluate the discrete Euler-Lagrange (DEL) equations and compute the Newton direction for solving the DEL equations, which results in linear-time variational integrators of arbitrarily high order. To our knowledge, no linear-time higher-order variational or even implicit integrators have been developed before. Moreover, an O(n 2 ) algorithm to linearize the DEL equations is presented, which is useful for trajectory optimization. These proposed algorithms eliminate the bottleneck of implementing higher-order variational integrators in simulation and trajectory optimization of complex robotic systems. The efficacy of this paper is validated through comparison with existing methods, and implementation on various robotic systems-including trajectory optimization of the Spring Flamingo robot, the LittleDog robot and the Atlas robot. The results illustrate that the same integrator can be used for simulation and trajectory optimization in robotics, preserving mechanical properties while achieving good scalability and accuracy.

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Structured linearization of discrete mechanical systems on Lie groups: A synthesis of analysis and control

Cited by 10 publications

References 23 publications

Linear-Quadratic Optimal Control in Maximal Coordinates

Linear-Quadratic Optimal Control in Maximal Coordinates

A Linear-Time Variational Integrator for Multibody Systems

Efficient Computation of Higher-Order Variational Integrators in Robotic Simulation and Trajectory Optimization

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