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Comment by fxtentacle | Hacker News Reader

Think of numerical optimization (e.g. gradient decent) as if you're letting a marble run down a hill. Only for some hill shapes can you be sure that the marble will reach the bottom. For other shapes, it'll get stuck on a small flat or in a small hole. Those would be called zero gradient or local minima failures.

If your hill has a convex shape, then the marble will always roll up the bottom eventually.

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It's generally understood that motion planning for nontrivial problems is very non-convex, and it's basically impossible to predict if a given initial condition can be optimized to the global optima. Even simple convex 3D geometry in the environment projects to very complex non-convex geometry in the configuration space of the robot (the space you're performing optimization in).

While that is true, the critical question is whether it will be locally smooth once you get close enough to your goal. Long-distance planning is generally bad, even for humans. Short-distance planning tends to work well with A-star

The problem is that "distance" here is not necessarily in any intuitive or useful space, and knowing if you are close to the goal may be as hard as finding the full solution. You can be quite "close" to a solution in, say, Euclidean distance over joint angles, while constraints like joint limits and collisions mean that the actual solution path will be quite long.

I apologize, I hadn't thought about robot arms. All the planning stuff that I was involved with so far was either quadcopters or slowfliers or buggies, so only cases where the Euclidean norm is useful.

this was so apt and it makes so much sense. a bulb just went on in my head. thanks so much for your simple and precise explanation.

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