I will start with the recording. Welcome everybody to this week's seminar at the Chair of Dynamics,

Control and Numerics at FOU Erlangen. We are very happy to have today's speaker here,

Frau Professor Katrin Vlaskamp from the Saarland University and she will give a talk about optimal

control meets graph-based methods, autonomous driving on turnpikes. The stage is yours, we are

looking forward to your talk. Yeah, thank you Tobias and thank you Enrique for the kind invitation.

I'm very happy to present you some of our works today in the seminar and as I said in the

beginning I chose this title and this project since it gives me the opportunity to combine

some very different pieces of work of me, some of them are quite old, so this topic dates back to

my PhD thesis times but also some much newer or recent results. Okay, so let's see.

Oh that was one too much. There is a little delay but let's see, I think it will work.

So why do I choose first of all autonomous driving as an application? Let me motivate this.

So the application of autonomous driving is a very interesting field for us as

applied mathematicians or engineers I would say because between the challenges that occur from

obtaining information from the sensors and the perception phase on the very beginning and the

question of how to actually perform the actuation on the car, there's lots of interesting tasks or

challenges which is typically called the decision making step and this is where we can contribute

methods I would say. So in most of the works in literature the decision making itself is split up

into different topics, so there's planning first where route planning, the behavior layer and motion

planning is mentioned and then it is said that the control then afterwards takes care of bringing the

and then it is said that the control then afterwards takes care of bringing the

computed inputs into the control unit and takes care of feedback control. However,

some modern methods do not fit into the strict distinction of tasks if you think of

learning-based or AI-based methods and also the approach I'm going to present you in the next

say half an hour I would say it basically covers several of these tasks at the same time and

it's interesting to investigate further for that reason. Also a motivation for us is that we have

a research project currently going on within the Schwerpunkt program of the DFG Cooperative

Interacting Automobiles, so our project there is called GROKO Plan for Graph-Based Optimal

Cooperative Planning of Cars and I will at the very end of my talk show you some first results

we obtain in this project. The mathematical basis for my presentation is called Motion Planning

with Motion Primitives and it is based off a foundational work of Emilio Frazuli which you

can see here and his approach to motion planning is to split up the task into two steps and in the

first step you design a library of motion primitives and these motion primitives have special

properties they should be symmetry exploiting so this is something I will explain in detail

on the next slides. They are special types of motion primitives called the trim primitives which

are of particular interest since they are nice properties and another type of motion primitives

are maneuvers which you need in order to sequence these primitives and this is actually what you do

in the second step then so given a specific planning task like this car should go from here

to there the planning problem is then reduced to finding the optimal sequence of the pre-computed

primitives and this allows you to split up the task into an offline phase where this library

or later I will show you that it has an automaton structure can be computed and the online phase

where the graph-based planning have to take place and well to have it as an ongoing example I will

mostly talk about autonomous driving but in fact the method has been developed in a general fashion

and can be applied to other systems so autonomous systems for in particular as well originally

Frasuli presented helicopter examples in my previous institution in Bremen I had a collaboration on

computing motion primitives planning for ships and harbors for example and you could well assume

that other fields of application let it be astrodynamics or some automation tasks for robots

can be treated in a similar way so please don't feel restricted by this autonomous driving application

okay so let me give you a rough outline of my talk and also the introduce a problem setting

so I will keep it quite general consider a time invariant non-linear system of this form

and so let's just assume that whenever I fixed to a specific control signal I have the well-defined