Overcoming Nonsmoothness and Control Chattering in Noncon...

What’s Happening

Alright so With some hints for good numerics The post Overcoming Nonsmoothness and Control Chattering in Nonconvex Optimal Control Problems appeared first on Towards Data Science.

Introduction One might encounter a number of frustrating difficulties when trying to numerically solve a difficult nonlinear and nonconvex optimal control problem. In this article I will consider such a difficult problem, that of finding the shortest path between two points through an obstacle field for a well-known model of a wheeled robot. (we’re not making this up)

I will examine common issues that arise when trying to solve such a problem numerically (in particular, nonsmoothness of the cost and chattering in the control) and how to address them.

The Details

Examples help to clarify the concepts. 1 Outline First, Ill introduce the car model that well study throughout the article.

Then, Ill state the optimal control problem in all its detail. The next section then exposes all the numerical difficulties that arise, finishing with a sensible nonlinear programme that attempts to deal with them.

Why This Matters

Ill then present the details of a homotopy method , which helps in guiding the solver towards a good solution. Ill then show you some numerical experiments to clarify everything, and finish off with references for further reading. A car model Well consider the following equations of motion, [ \begin(align) \dot x_1(t) = u_1(t)\cos(x_3(t)), \ \dot x_2(t) = u_1(t)\sin(x_3(t)), \ \dot x_3(t) = u_2(t), \end(align) ] where (t \geq 0) denotes time, (x_1\in\mathbb(R)) and (x_2\in\mathbb(R)) denote the cars position, (x_3\in\mathbb(R)) denotes its orientation, (u_1\in\mathbb(R)) its velocity and (u_2\in\mathbb(R)) its rate of turning.

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Key Takeaways

• This is a common model of a differential-drive robot , which consists of two wheels that can turn independently.
• This allows it to drive forwards and backwards, rotate when stationary and perform other elaborate driving manoeuvres.
• Note that, because (u_1) can be 0, the model allows the car to stop instantaneously.
• A differential drive robot, as modelled of motion.

The Bottom Line

Note that, because (u_1) can be 0, the model allows the car to stop instantaneously. A differential drive robot, as modelled of motion.

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