Add example for a multiphase problem.

[moorepants]

I realized that we can probably do multiphase problems with something like this. But I am currently limited to the duration of each phase having to be equal. @Peter230655 any ideas on how to let this block collide with the wall at any time? My intuition is that I need to move the collision equation from the instance constraints into the DAE somehow.

[Peter230655]

I realized that we can probably do multiphase problems with something like this. But I am currently limited to the duration of each phase having to be equal. @Peter230655 any ideas on how to let this block collide with the wall at any time? My intuition is that I need to move the collision equation from the instance constraints into the DAE somehow.

Let me think about it. :-)
(As I understand, you want it with Newtonian collission, so we have different eoms before and after the impact?)

[moorepants]

There is essentially just 1 eom in this case because the block is always sliding:

$$ m\dot{v} = -\textrm{sgn}({v})\mu m g + F $$

but at $x=1$ the impact occurs and you apply a mometum balance for before and after impact:

$$ v_\textrm{after} = e v_\textrm{before} $$

If you break the EoM into the moving right and moving left portions, then you can remove the dicontinuity in the EoM. So then you do have 2 EoMs, one for before and one after.

[Peter230655]

There is essentially just 1 eom in this case because the block is always sliding:

m v ˙ = − s g n ( v ) μ m g + F

As I understood the equations in your code, F = 0 after the impact, the equation above seems to imply F is present always?

but at x = 1 the impact occurs and you apply a mometum balance for before and after impact:

v a f t e r = e v b e f o r e

If you break the EoM into the moving right and moving left portions, then you can remove the dicontinuity in the EoM. So then you do have 2 EoMs, one for before and one after.

I guess with Newtonian impact, by definition the speed will change discontinuously - something opty does not like.
So, two eoms are likely necessary.
Is Newtonian collision important, or would Hunt-Crossley also be o.k.?

I am asking, so I know in which direction to think.

[moorepants]

As I understood the equations in your code, F = 0 after the impact, the equation above seems to imply F is present always?

I wrote one equation in this comment, but I split it into two in the opty formulation.

[moorepants]

Is Newtonian collision important, or would Hunt-Crossley also be o.k.?

The point of this example is to avoid discontinuities when transitioning between phases of a problem. I can solve this with an impact force, but that's not the purpose. Sometimes you want to model transitions from one eom to another, e.g. standing on the ground, jumping off, and landing but not have to model the contact forces with things such as Hunt-Crossely. Newton's momentum exchange is a perfectly good model for impact also.

[Peter230655]

Is Newtonian collision important, or would Hunt-Crossley also be o.k.?

The point of this example is to avoid discontinuities when transitioning between phases of a problem. I can solve this with an impact force, but that's not the purpose. Sometimes you want to model transitions from one eom to another, e.g. standing on the ground, jumping off, and landing but not have to model the contact forces with things such as Hunt-Crossely. Newton's momentum exchange is a perfectly good model for impact also.

All clear now to me, I think.

[Peter230655]

Could this be done as a two stage problem, similar to PR #262 ?

In your example, splitting it up will simply result in using maximum force until impact, and in the second phase there is no control left for opty to do anything with. It will just slide.

In PR #262 the second phase is depending on the first one, through the amount of fuel left. (the third phase is like your second phase: nothing left to be controlled)
When I did it, I had to optimize the the first and second phases separately - and surely this must be worse (maximally equal) to optimizing them combined. I do not know, how Betts optimized them, but the results of PR #262 and of Betts are quite close.

So, I guess the goal must be to do it in one stage (splitting it up is 'primitive') - let me think along these lines.

[moorepants]

When I did it, I had to optimize the the first and second phases separately - and surely this must be worse (maximally equal) to optimizing them combined.

Yes, I want to solve these kinds of problems where all phases are solve simultaneously. My example here may be able to be solved by separating into multiple optimal control problems but, in general, we'd like to solve problems this way that cannot be solved separately. That's what I'm trying to do, but there may not be a "hack" to get opty to do this. It may be something we have to build into opty more carefully.

[Peter230655]

When I did it, I had to optimize the the first and second phases separately - and surely this must be worse (maximally equal) to optimizing them combined.

Yes, I want to solve these kinds of problems where all phases are solve simultaneously. My example here may be able to be solved by separating into multiple optimal control problems but, in general, we'd like to solve problems this way that cannot be solved separately. That's what I'm trying to do, but there may not be a "hack" to get opty to do this. It may be something we have to build into opty more carefully.

I do not know, but I am getting clearer in the direction of thinking.
Would it be of any help to ask John Betts whether he solved #262 simultaneously or one after another?