SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization

Gill, Philip E.; Murray, Walter; Saunders, Michael A.

doi:10.1137/s1052623499350013

Cited by 1,431 publications

(680 citation statements)

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“…In this section, we empirically compare LBmpcIPM with two dense active set solvers (LSSOL v1.05-4 and qpOASES v3.0beta) (Gill et al, 1986;Ferreau, Bock, and Diehl, 2008). We begin with simulations on a model of a quadrotor helicopter.…”

Section: Experimental and Simulation Resultsmentioning

confidence: 99%

“…The QP optimization central to the LBMPC technique can be numerically solved using a (2011) seek to leverage this sparsity. In Chapter 5 we explore this direction, but in the present chapter we describe results we have obtained using LSSOL (Gill et al, 1986). LSSOL is a dense, two-phase active-set solver implemented in FORTRAN 77.…”

Section: Methodsmentioning

confidence: 99%

“…However, the number of entries in the table can grow exponentially with horizon, input, and state dimensions. An alternative approach is to use a QP optimization solver (e.g., (Gill et al, 1986;Gill, Murray, and Saunders, 2002;Ferreau, Bock, and Diehl, 2008)) at each time step. Specialized solvers that can exploit the sparsity of QP-MPC have computational complexity that scales linearly in the prediction horizon of the QP-MPC (Wang and Boyd, 2010;Rao, Wright, and Rawlings, 1998), as opposed to non-sparse solvers that scale cubicly.…”

Section: Introductionmentioning

confidence: 99%

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On-board Model Predictive Control of a Quadrotor Helicopter: Design, Implementation, and Experiments

Bouffard¹

2012

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Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. REPORT DATE DEC 20122. REPORT TYPE 3. DATES COVERED 00-00-2012 to 00-00-2012 Copyright © 2012, by the author(s). TITLE AND SUBTITLE On-board Model Predictive Control of a QuadrotorAll rights reserved.Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. AcknowledgementThe This report describes work in applying model predictive control (MPC) techniques to the control of quadrotor helicopters, a type of micro aerial vehicle (MAV) platform that has gained great popularity in recent years both in research and commercial/military settings. MPC is a form of optimal control which is attractive in part because it allows engineering requirements to be addressed directly in the design of the controller in terms of costs to be minimized and constraints to be satisfied in an optimization problem. Furthermore, for many engineering problems of interest, the optimization to be performed is convex, meaning that a global optimum can be efficiently computed. MPC first found broad early application in the process industry, where the typically longer time scales were compatible with the time necessary to solve the optimization problem. More recently with both the exponential increase in available computing power and the development of more efficient solution techniques, MPC has become an option for control of systems with faster dynamics, such as quadrotors.We bring together results from our application of two distinct variants of MPC. The common thread is that we seek advanced control algorithms that can be applied to an autonomous MAV like the quadrotor, ideally without requiring any external resources, i.e. we aim to perform all computations required for real-time closed-loop control on-board the vehicle.The first variant is known as explicit MPC, where in a sense the heavy numerical work of solving optimization problems is done a priori and off-line, such that the on-line implementat...

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Section: Experimental and Simulation Resultsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On-board Model Predictive Control of a Quadrotor Helicopter: Design, Implementation, and Experiments

Bouffard¹

2012

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“…Optimization was performed with SNOPT [51,52], a robust constrained nonlinear optimization program; we used optimality and feasibility tolerances of 10 −6 . The parametrizedleg-force biped model used at piece-wise linear forces with at least 14 segments to simulate each step of walking.…”

Section: (Iii) Computational Solutionmentioning

confidence: 99%

Walking on a moving surface: energy-optimal walking motions on a shaky bridge and a shaking treadmill can reduce energy costs below normal

Joshi

Srinivasan

2015

Proc. R. Soc. A.

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Understanding how humans walk on a surface that can move might provide insights into, for instance, whether walking humans prioritize energy use or stability. Here, motivated by the famous human-driven oscillations observed in the London Millennium Bridge, we introduce a minimal mathematical model of a biped, walking on a platform (bridge or treadmill) capable of lateral movement. This biped model consists of a pointmass upper body with legs that can exert force and perform mechanical work on the upper body. Using numerical optimization, we obtain energyoptimal walking motions for this biped, deriving the periodic body and platform motions that minimize a simple metabolic energy cost. When the platform has an externally imposed sinusoidal displacement of appropriate frequency and amplitude, we predict that body motion entrained to platform motion consumes less energy than walking on a fixed surface. When the platform has finite inertia, a massspring-damper with similar parameters to the Millennium Bridge, we show that the optimal biped walking motion sustains a large lateral platform oscillation when sufficiently many people walk on the bridge. Here, the biped model reduces walking metabolic cost by storing and recovering energy from the platform, demonstrating energy benefits for two features observed for walking on the Millennium Bridge: crowd synchrony and large lateral oscillations.

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“…In a second-derivative (Newton) NLP solver, the first derivatives of a quasi-Newton method are used together with an accurate approximation of the Lagrangian Hessian. Examples of commonly-used, first-derivative NLP solvers include NPSOL [26] and SNOPT [27,28], whereas well-known, second-derivative NLP solvers include IPOPT [29] and KNITRO [30].…”

mentioning

confidence: 99%

Exploiting Sparsity in Direct Collocation Pseudospectral Methods for Solving Optimal Control Problems

Patterson¹,

Rao²

2012

Journal of Spacecraft and Rockets

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In a direct collocation pseudospectral method, a continuous-time optimal control problem is transcribed to a finitedimensional nonlinear programming problem. Solving this nonlinear programming problem as efficiently as possible requires that sparsity at both the first-and second-derivative levels be exploited. In this paper, a computationally efficient method is developed for computing the first and second derivatives of the nonlinear programming problem functions arising from a pseudospectral discretization of a continuous-time optimal control problem. Specifically, in this paper, expressions are derived for the objective function gradient, constraint Jacobian, and Lagrangian Hessian arising from the previously developed Radau pseudospectral method. It is shown that the computation of these derivative functions can be reduced to computing the first and second derivatives of the functions in the continuous-time optimal control problem. As a result, the method derived in this paper reduces significantly the amount of computation required to obtain the first and second derivatives required by a nonlinear programming problem solver. The approach derived in this paper is demonstrated on an example where it is found that significant computational benefits are obtained when compared against direct differentiation of the nonlinear programming problem functions. The approach developed in this paper improves the computational efficiency of solving nonlinear programming problems arising from pseudospectral discretizations of continuous-time optimal control problems.

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SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization

Cited by 1,431 publications

References 67 publications

On-board Model Predictive Control of a Quadrotor Helicopter: Design, Implementation, and Experiments

On-board Model Predictive Control of a Quadrotor Helicopter: Design, Implementation, and Experiments

Walking on a moving surface: energy-optimal walking motions on a shaky bridge and a shaking treadmill can reduce energy costs below normal

Exploiting Sparsity in Direct Collocation Pseudospectral Methods for Solving Optimal Control Problems

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