Direct trajectory optimization using nonlinear programming and collocation

Hargraves, C. R.; Paris, Stephen

doi:10.2514/6.1986-2000

Cited by 123 publications

(162 citation statements)

References 9 publications

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“…In direct methods, the nonlinear optimization simultaneously searches over parameterizations of u(t) and x(t); here no simulation is required and instead the dynamics are imposed as a set of optimization constraints, typically evaluated at a selection of collocation points [9]. Mixtures of shooting and direct methods are also possible, and fall under the umbrella of multiple shooting.…”

Section: Introductionmentioning

confidence: 99%

Direct Trajectory Optimization of Rigid Body Dynamical Systems through Contact

Posa

Tedrake

2013

Springer Tracts in Advanced Robotics

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Direct methods for trajectory optimization are widely used for planning locally optimal trajectories of robotic systems. Most state-of-the-art techniques treat the discontinuous dynamics of contact as discrete modes and restrict the search for a complete path to a specified sequence through these modes. Here we present a novel method for trajectory planning through contact that eliminates the requirement for an a priori mode ordering. Motivated by the formulation of multi-contact dynamics as a Linear Complementarity Problem (LCP) for forward simulation, the proposed algorithm leverages Sequential Quadratic Programming (SQP) to naturally resolve contact constraint forces while simultaneously optimizing a trajectory and satisfying nonlinear complementarity constraints. The method scales well to high dimensional systems with large numbers of possible modes. We demonstrate the approach using three increasingly complex systems: rotating a pinned object with a finger, planar walking with the Spring Flamingo robot, and high speed bipedal running on the FastRunner platform.

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Section: Introductionmentioning

confidence: 99%

Direct Trajectory Optimization of Rigid Body Dynamical Systems through Contact

Posa

Tedrake

2013

Springer Tracts in Advanced Robotics

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“…Pytlak solved a state constrained optimal control problem using a gradient algorithm and applied it for some problems (see [8], [9]). Hargraves and Paris [10] reintroduced the direct transcription approach, by discretising the dynamic equations using a collocation method. A cubic polynomial is used to approximate the state variables and linear interpolation for the control variables.…”

Section: Methodsmentioning

confidence: 99%

Trajectory Shaping of Surface-to-Surface Missile With Terminal Impact Angle Constraint

Subchan¹

2010

MST

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This paper presents trajectory shaping of a surface-to-surface missile attacking a fixed with terminal impact angle constraint. The missile must hit the target from above, subject to the missile dynamics and path constraints. The problem is reinterpreted using optimal control theory resulting in the formulation of minimum integrated altitude. The formulation entails nonlinear, two-dimensional missile flight dynamics, boundary conditions and path constraints. The generic shape of optimal trajectory is: level flight, climbing, diving; this combination of the three flight phases is called the bunt manoeuvre. The numerical solution of optimal control problem is solved by a direct collocation method. The computational results is used to reveal the structure of optimal solution which is composed of several arcs, each of which can be identified by the corresponding manoeuvre executed and constraints active.

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“…Often, the cost functions used exhibits the Bellman optimality principle that allows for a dynamic programming solution. However, work on optimal control has also considered more general trajectory optimization problems [25] that define cost functions over the trajectories of a continuous system subject to constraints. A standard approach to trajectory optimization reformulates the problem as a non-linear optimization, and uses numerical optimization techniques [8], [9].…”

Section: ) Direct Multiple Shootingmentioning

confidence: 99%

A trajectory splicing approach to concretizing counterexamples for hybrid systems

Zutshi

Sankaranarayanan

Deshmukh³

et al. 2013

52nd IEEE Conference on Decision and Control

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Abstract-This paper examines techniques for finding falsifying trajectories of hybrid systems using an approach that we call trajectory splicing. Many formal verification techniques for hybrid systems, including flowpipe construction, can identify plausible abstract counterexamples for property violations. However, there is often a gap between the reported abstract counterexamples and the concrete system trajectories. Our approach starts with a candidate sequence of disconnected trajectory segments, each segment lying inside a discrete mode. However, such disconnected segments do not form concrete violations due to the gaps that exist between the ending state of one segment and the starting state of the subsequent segment. Therefore, trajectory splicing uses local optimization to minimize the gap between these segments, effectively splicing them together to form a concrete trajectory.We demonstrate the use of our approach for falsifying safety properties of hybrid systems using standard optimization techniques. As such, our approach is not restricted to linear systems. We compare our approach with other falsification approaches including uniform random sampling and a robustness guided falsification approach used in the tool S-Taliro. Our preliminary evaluation clearly shows the potential of our approach to search for candidate trajectory segments and use them to find concrete property violations.

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Direct trajectory optimization using nonlinear programming and collocation

Cited by 123 publications

References 9 publications

Direct Trajectory Optimization of Rigid Body Dynamical Systems through Contact

Direct Trajectory Optimization of Rigid Body Dynamical Systems through Contact

Trajectory Shaping of Surface-to-Surface Missile With Terminal Impact Angle Constraint

A trajectory splicing approach to concretizing counterexamples for hybrid systems

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