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Special Issue: Aeroelasticity
Open AccessArticle

Optimization Provenance of Whiplash Compensation for Flexible Space Robotics

Department of Mechanical Engineering (CVN), Columbia University, New York, NY 10027, USA
Aerospace 2019, 6(9), 93; https://doi.org/10.3390/aerospace6090093
Received: 16 July 2019 / Revised: 13 August 2019 / Accepted: 16 August 2019 / Published: 30 August 2019
(This article belongs to the Special Issue Control and Optimization Problems in Aerospace Engineering)
Automatic controls refer to the application of control theory to regulate systems or processes without human intervention, and the notion is often usefully applied to space applications. A key part of controlling flexible space robotics is the control-structures interaction of a light, flexible structure whose first resonant modes lie within the bandwidth of the controller. In this instance, the designed-control excites the problematic resonances of the highly flexible structure. This manuscript reveals a novel compensator capable of minimum-time performance of an in-plane maneuver with zero residual vibration (ZV) and zero residual vibration-derivative (ZVD) at the end of the maneuver. The novel compensator has a whiplash nature of first commanding maneuver states in the opposite direction of the desired end state. For a flexible spacecraft simulator (FSS) free-floating planar robotic arm, this paper will first derive the model of the flexible system in detail from first principles. Hamilton’s principle is augmented with the adjoint equation to produce the Euler–Lagrange equation which is manipulated to prove equivalence with Newton’s law. Extensive efforts are expended modeling the free–free vibration equations of the flexible system, and this extensive modeling yields an unexpected control profile—a whiplash compensator. Equations of motion are derived using both the Euler–Lagrange method and Newton’s law as validation. Variables are then scaled for efficient computation. Next, general purposed pseudospectral optimization software is used to seek an optimal control, proceeding afterwards to validate optimality via six theoretical optimization necessary conditions: (1) Hamiltonian minimization condition; (2) adjoint equations; (3) terminal transversality condition; (4) Hamiltonian final value condition; (5) Hamiltonian evolution equation; and lastly (6) Bellman’s principle. The results are novel and unique in that they initially command full control in the opposite direction from the desired end state, while no such results are seen using classical control methods including classical methods augmented with structural filters typically employed for controlling highly flexible multi-body systems. The manuscript also opens an interesting question of what to declare when the six optimality necessary conditions are not necessarily in agreement (we choose here not to declare finding the optimal control, instead calling it suboptimal). View Full-Text
Keywords: space flight control problems; active dynamic control; structural mechanics control problems; vibration suppression; zero residual vibration (ZV); zero residual vibration-derivative (ZVD); pseudospectral; time optimal; flexible control; multi-body; space robotics; structural filters space flight control problems; active dynamic control; structural mechanics control problems; vibration suppression; zero residual vibration (ZV); zero residual vibration-derivative (ZVD); pseudospectral; time optimal; flexible control; multi-body; space robotics; structural filters
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Sands, T. Optimization Provenance of Whiplash Compensation for Flexible Space Robotics. Aerospace 2019, 6, 93.

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