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当 $t_{f 1} \neq t_{f 2}$ 时, 计算例 7.2 。

   当 $t_{f 1} \neq t_{f 2}$ 时, 计算例 7.2 。

机器人学导论 introduction to robotics: mechanics and control

机器人学导论 introduction to robotics: mechanics and control

约翰 J.克雷格 4th Edition

Chapter 7, Problem 15

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当 $t_{f 1} \neq t_{f 2}$ 时, 计算例 7.2 。

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Key Concepts

Transversality Conditions

These are necessary optimality conditions that apply at the boundaries where the final time is free or where phase transitions occur. They ensure that the costate (or adjoint variable) appropriately accounts for variations in the terminal time or state, playing a crucial role in multi-phase and free-endpoint control problems.

Phase Matching Conditions

When dealing with multi-phase systems, it is important to impose continuity or matching conditions at the interface of different phases. These conditions ensure that state variables (and sometimes costate variables) transition smoothly from one phase to the next, even if the terminal times of the phases are not equal.

Multi-phase Optimal Control

This concept involves dividing a control problem into separate phases or segments, each possibly with distinct dynamics, constraints, or objectives. It is essential when the overall system behavior changes over different intervals, requiring tailored analysis and solution methods for each phase.

Variable Terminal Time (Free Final Time)

In optimal control, problems often allow the final time to vary rather than being fixed. This introduces additional conditions—such as transversality conditions—that must be satisfied at the endpoint, ensuring an optimal solution even when the end time is a decision variable.

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