WEBVTT 00:01.500 --> 00:05.660 Welcome back to the mini-projects here in the Python course and we are 00:05.660 --> 00:08.080 now looking at the second project. 00:08.740 --> 00:13.740 And here it's about model calibration of a one-track model of a 00:13.740 --> 00:14.060 vehicle. 00:16.380 --> 00:21.540 And yes, that comes in principle, or I used a bit of code that I also 00:21.540 --> 00:25.200 used in a publication last year. 00:26.220 --> 00:29.960 It was about validation of a vehicle one-track model. 00:31.280 --> 00:35.940 And yes, if you want to, you can read a bit of the background there in 00:35.940 --> 00:36.240 the paper. 00:36.380 --> 00:41.980 This is referenced in the bibtex file to the repository. 00:42.640 --> 00:45.360 And you can also use the code for the paper, which is also available 00:45.360 --> 00:46.000 here in the github. 00:48.220 --> 00:49.240 But you don't need that. 00:49.500 --> 00:53.420 So everything is prepared here so that you can start there. 00:53.420 --> 01:01.000 Yes, and the mini-project is about trying reference data that come 01:01.000 --> 01:02.000 from a two-track model. 01:02.520 --> 01:09.980 In this case I used a Simulink model and thus fitted a one-track 01:09.980 --> 01:10.400 model. 01:11.140 --> 01:16.740 And there we will adjust the transverse stiffness of the wheels with 01:16.740 --> 01:17.440 an optimizer. 01:17.440 --> 01:21.560 And yes, calibrating reference data from simulation models. 01:22.760 --> 01:25.700 This is a very typical task that occurs in practice. 01:26.820 --> 01:29.420 This is to settle a large field of mathematical optimization. 01:31.520 --> 01:34.600 Mathematical optimization is a very important topic. 01:35.220 --> 01:37.060 This occurs in all possible fields. 01:38.360 --> 01:41.980 This can also be in the direction of production, for example, that you 01:41.980 --> 01:43.220 want to optimize some production steps. 01:44.320 --> 01:48.640 Or in finance mathematics and of course also a lot in the field of 01:48.640 --> 01:48.900 engineering. 01:50.100 --> 01:53.880 And in essence, when optimizing, it is about trying to improve a 01:53.880 --> 01:58.180 problem with free parameters. 01:59.920 --> 02:03.580 And for this you set up a cost function and then there are different 02:03.580 --> 02:08.680 optimization procedures that change these parameters until this cost 02:08.680 --> 02:11.260 function has a better value than the starting value. 02:11.260 --> 02:15.000 This is perhaps on a very, very high level of flight, which is 02:15.000 --> 02:16.080 actually mathematical optimization. 02:17.440 --> 02:23.240 And in connection with models with free parameters or changeable 02:23.240 --> 02:28.340 parameters, which are also often called greybox models, the whole 02:28.340 --> 02:29.560 thing is called calibrating. 02:30.220 --> 02:34.640 This is not to be confused with calibrating measuring and testing 02:34.640 --> 02:34.820 devices. 02:35.780 --> 02:42.380 This is simply the term for these physical models with unknown 02:42.380 --> 02:46.240 parameters or parameters where you only roughly know the order of 02:46.240 --> 02:46.240 magnitude. 02:49.260 --> 02:51.060 I don't think I have to say much about the one-spot model. 02:53.400 --> 02:56.020 I assume you all have good knowledge of it. 02:57.440 --> 02:58.680 We'll just do that very briefly. 02:59.140 --> 03:01.880 Here on the right is an exemplary representation. 03:01.880 --> 03:09.060 What you do is you take a vehicle, which normally has four wheels, you 03:09.060 --> 03:14.060 pack these wheels together in axes on each single wheel. 03:16.280 --> 03:21.460 This has shown that up to about 4 meters per square second 03:21.460 --> 03:26.900 acceleration such a one-spot model can very well reproduce the 03:26.900 --> 03:29.020 behavior of a vehicle. 03:32.860 --> 03:36.440 Some dynamics are basically ignored in the one-spot model. 03:37.660 --> 03:40.280 This is pitch and roll dynamics, which then disappears. 03:41.300 --> 03:45.240 And the focus, for example, is assumed to be on the level of the 03:45.240 --> 03:45.240 runway. 03:46.040 --> 03:48.560 So that doesn't work for all vehicles. 03:48.860 --> 03:52.420 If they have very high buildings or something and then the cross 03:52.420 --> 03:56.540 -dynamics gets bigger, then these one-spot models become inaccurate. 03:56.540 --> 04:01.160 But for many, many applications this is actually good enough. 04:02.080 --> 04:03.140 And this is used quite often. 04:04.360 --> 04:08.980 And down here is shown the state-of-the-art form of the one-spot 04:08.980 --> 04:09.340 model. 04:10.240 --> 04:15.540 You can read the exact reference and so on here in the book by 04:15.540 --> 04:16.240 Schramm. 04:16.340 --> 04:18.980 Everything is described exactly there. 04:19.900 --> 04:20.900 Here only very briefly. 04:21.120 --> 04:30.160 So what goes in as, in principle, driving entrance size, is the little 04:30.160 --> 04:32.400 tether back here. 04:33.060 --> 04:38.120 This is used for the angle at which the front wheel is slid in. 04:39.420 --> 04:42.040 Or described angle at which the front wheel is slid in. 04:42.420 --> 04:44.240 Then we have two state sizes. 04:44.240 --> 04:49.640 This is the gear rate, this psi dot, and the swimming angle, beta. 04:51.140 --> 04:57.300 And then here in the A-matrix of the state-of-the-art model you have 04:57.300 --> 04:59.560 quite a few constants. 05:00.380 --> 05:04.700 These are distances, i.e. 05:04.700 --> 05:08.860 distances between the center of gravity and the wheel axles. 05:08.960 --> 05:11.000 These are this L, V and LR. 05:11.640 --> 05:18.920 Then you have the cross-stripes for the axles and thus also for the 05:18.920 --> 05:19.220 tires. 05:19.500 --> 05:20.940 So here everything is summarized. 05:22.400 --> 05:26.080 This is called CF and CR here in the small model. 05:26.980 --> 05:28.680 And what else is in it is the speed. 05:29.160 --> 05:31.980 It is now constant here, with which the vehicle moves forward. 05:33.500 --> 05:38.800 And then there is such an inertia, a sluggishness around the high 05:38.800 --> 05:39.060 axle. 05:40.140 --> 05:41.700 This is then the large tether. 05:43.340 --> 05:49.200 And as I said, this tilt of the front wheel is then the model 05:49.200 --> 05:49.520 entrance. 05:54.640 --> 05:56.380 And the reference data. 05:56.900 --> 05:58.900 Normally you take measurement data from driving tests. 05:58.980 --> 06:03.620 But now here for the lecture I just took a reference model that has a 06:03.620 --> 06:04.200 different structure. 06:04.760 --> 06:06.300 This is a more complicated model. 06:06.400 --> 06:07.640 This is then a two-spoke model. 06:08.640 --> 06:15.400 This is simply available in Simulink in the Vehicle Dynamics Toolbox. 06:16.220 --> 06:19.180 You can simply use this and load it in. 06:19.240 --> 06:20.580 This is also available in various versions. 06:21.580 --> 06:24.440 In terms of the representation, you can already see that it has four 06:24.440 --> 06:24.700 wheels. 06:25.240 --> 06:28.260 So that also takes into account such roll dynamics. 06:31.460 --> 06:38.800 And what you can put in there for modeling is a steering angle that 06:38.800 --> 06:39.520 you put in. 06:39.640 --> 06:41.440 And here I made it relatively easy. 06:41.980 --> 06:43.880 A sinus wave comes in, i.e. 06:43.880 --> 06:48.240 a sinus maneuver for a certain period of time. 06:49.000 --> 06:53.140 And otherwise all other parameters have been kept constant. 06:54.180 --> 06:56.720 And the speed is also rather slow, i.e. 06:56.740 --> 06:57.780 10 meters per second. 06:58.400 --> 07:02.760 This is simply there so that the cross-dynamics does not get so high 07:02.760 --> 07:07.280 and that the entrance model is still halfway there. 07:07.520 --> 07:16.140 And the two sizes that we look at, the gear ratio and the swimming 07:16.140 --> 07:19.500 angle, that the entrance model can still depict that quite well. 07:20.120 --> 07:22.660 And that has then simply been saved here as a mat file. 07:24.380 --> 07:27.660 The model is also in the Propository as a Simulink file. 07:27.660 --> 07:31.400 If you have the corresponding toolbox, you can also run it. 07:31.820 --> 07:39.080 And then there is such a small m-file that converts the mat files into 07:39.080 --> 07:40.000 CSV files. 07:42.060 --> 07:46.660 And you can also find the CSV files here under mini-projects. 07:47.400 --> 07:51.560 They are then simply tied into the repository here. 07:51.560 --> 07:56.480 And these are then simply time and the corresponding value. 07:56.660 --> 08:01.640 So the first line is the time and the second line here would be the 08:01.640 --> 08:02.120 swimming angle. 08:04.140 --> 08:06.600 And here would be the same for the gear ratio. 08:06.740 --> 08:08.540 This is then in this CSV. 08:11.220 --> 08:15.400 And then there is the model entrance, i.e. 08:15.460 --> 08:18.080 the inclination of the front wheel as an angle. 08:24.040 --> 08:28.160 In order to process this with Python and to optimize such a model, we 08:28.160 --> 08:29.800 need a few steps first. 08:30.980 --> 08:32.940 First we load a few libraries. 08:34.100 --> 08:42.640 I already said optimization and there is this Optimize module at 08:42.640 --> 08:42.640 SciPy. 08:45.000 --> 08:48.400 Otherwise we take a second library, which is called Control. 08:48.400 --> 08:57.480 This is a nice library with which you can handle control techniques in 08:57.480 --> 08:59.260 Python. 09:00.040 --> 09:05.060 For example, you can generate and simulate such state-of-the-art 09:05.060 --> 09:05.060 models. 09:05.800 --> 09:07.920 But you can also generate controllers. 09:09.640 --> 09:13.940 And yes, it is also worth taking a look at this control library. 09:14.360 --> 09:15.680 It is also very nicely documented. 09:15.680 --> 09:20.020 And the other two imports are standard NumPy. 09:20.240 --> 09:22.180 As I said, you need that in almost every script. 09:22.480 --> 09:26.000 And then for plotting, so that we can look at a few results, the 09:26.000 --> 09:26.580 matplotlib. 09:28.880 --> 09:32.960 Then there is a small code block here in the notebook. 09:33.660 --> 09:39.980 This is simply because the notebooks are also tested on Travis CI with 09:39.980 --> 09:40.820 continuous integration. 09:41.620 --> 09:54.180 This is just a bit of a workaround to find these CSV files from 09:54.180 --> 09:54.180 different paths. 09:54.780 --> 09:57.040 This is not so important here at first. 09:57.980 --> 09:59.680 You can certainly solve this a bit more elegantly. 10:00.180 --> 10:04.320 If you have a suggestion, I would be interested in how to do this 10:04.320 --> 10:04.840 well. 10:04.840 --> 10:11.180 It's just that Travis starts all the things from a different folder 10:11.180 --> 10:13.720 than if you run it locally. 10:14.660 --> 10:16.260 But that's not so important now. 10:17.100 --> 10:22.980 What we still need is a function so that we can read the CSV files in 10:22.980 --> 10:23.300 Python. 10:23.580 --> 10:27.400 And there is already a nicely pre-made function from NumPy. 10:28.060 --> 10:30.840 It's called genfrom.txt. 10:31.780 --> 10:34.000 And then you can very easily read the things. 10:35.780 --> 10:38.060 This is shown here with the three lines. 10:38.700 --> 10:41.640 So two-dimensional arrays would be generated each. 10:43.800 --> 10:46.300 And you always assign the file here. 10:46.960 --> 10:52.540 And as a separator between the individual values, you can also assign 10:52.540 --> 10:52.540 it. 10:52.620 --> 10:56.120 In this case, the file is a comma. 10:56.120 --> 11:00.500 So it's relatively easy to read CSV files in Python. 11:05.170 --> 11:10.350 And then, of course, it makes sense for us to display the written data 11:10.350 --> 11:15.630 in the CSV files and later also the model outputs and simulations. 11:16.810 --> 11:17.410 And then we can look at plots. 11:19.130 --> 11:23.290 And then we do a common plotting function. 11:24.070 --> 11:26.790 And here it becomes relatively easy. 11:27.230 --> 11:34.690 So the arguments, that's the gear ratio of the swimming angle and the 11:34.690 --> 11:34.970 title. 11:35.210 --> 11:38.930 Just so that you can interactively overwrite the diagram a bit. 11:39.330 --> 11:41.410 And then this is plotted, so to speak. 11:42.790 --> 11:45.530 You can see these three plot commands here. 11:45.850 --> 11:47.470 That's very easy. 11:47.610 --> 11:48.610 And then a legend. 11:51.070 --> 11:54.690 And then let's just take a look at what it looks like for the two 11:54.690 --> 11:58.730 -track model that was simulated in Simulink. 11:59.750 --> 12:06.870 And then you can just call up the function and ultimately pass on the 12:06.870 --> 12:08.190 correct values. 12:09.350 --> 12:16.670 And you can see here that it took 30 seconds. 12:16.670 --> 12:21.470 I just see a mistake down here. 12:21.830 --> 12:23.750 The legend description is not correct. 12:24.270 --> 12:31.290 This is simply because the data was tested with 100 Hz. 12:31.610 --> 12:33.510 And that's why it says 3000 there. 12:33.790 --> 12:36.990 But that's not the time in seconds, it's actually the samples. 12:37.570 --> 12:38.970 We'd have to change that again. 12:40.150 --> 12:42.230 And yes, there you can now see the sequences. 12:42.370 --> 12:48.950 So here this sinusoidal movement of the angle of the front wheel. 12:49.710 --> 12:56.310 And then how the gear speed, the gear ratio and the swimming angle 12:56.310 --> 12:57.690 behave with it. 12:58.450 --> 13:01.470 And in the end the maneuver is over and then everything goes back to 13:01.470 --> 13:01.670 zero. 13:02.070 --> 13:03.690 So that would be the reference data now. 13:04.550 --> 13:10.190 And then we're going to build up the one-track model and see how it 13:10.190 --> 13:12.190 behaves with the same stimulation. 13:12.610 --> 13:15.410 So the task would be here now, the same steering angle goes in. 13:16.170 --> 13:21.290 And then the question is, how well can the one-track model then 13:21.290 --> 13:22.710 reproduce these processes? 13:24.110 --> 13:28.470 This is a simpler model of the structure compared to the reference. 13:29.130 --> 13:32.890 And that's actually quite often the case that when you model, that you 13:32.890 --> 13:38.630 have simpler models of the structure than the reference data or the 13:38.630 --> 13:40.870 systems with which you have recorded the reference data. 13:41.670 --> 13:47.370 That's the main reason why you even simulate to simplify things and 13:47.370 --> 13:48.090 then also to predict. 13:52.640 --> 13:57.480 So we take the Control Systems library from Python. 13:57.480 --> 14:01.620 Here again the model structure is simply given in the state space 14:01.620 --> 14:02.100 representation. 14:02.920 --> 14:07.220 And now let's take a quick look at what it looks like, what features 14:07.220 --> 14:07.560 are there. 14:08.080 --> 14:13.460 So you can model with this Python Control Systems library LTE systems, 14:13.640 --> 14:14.080 for example. 14:15.700 --> 14:17.580 You can generate time series. 14:19.160 --> 14:21.120 What else do you have? 14:21.940 --> 14:26.880 Yes, such modules with which you can do frequency analysis, time 14:26.880 --> 14:33.020 series, also state space models can be generated and modeled. 14:34.040 --> 14:39.100 So this is really a very, very large library with which you can work. 14:39.860 --> 14:43.880 And as I said, there is also a very nice documentary about it. 14:47.650 --> 14:54.130 And such a state space model can now be defined with the library as 14:54.130 --> 14:54.610 follows. 14:55.330 --> 14:59.070 Up here, that's basically a function that I wrote myself. 14:59.550 --> 15:01.930 A few arguments are passed on. 15:02.590 --> 15:07.210 These are then the transverse stiffnesses, so basically all the 15:07.210 --> 15:09.430 parameters that the reference model has. 15:10.130 --> 15:12.870 These are then the distances between the center of gravity and the 15:12.870 --> 15:21.150 front and rear axle, the vehicle mass and then the sluggishness around 15:21.150 --> 15:23.390 the high axle and the speed. 15:24.130 --> 15:27.730 And then the individual elements in this A-matrix are simply 15:27.730 --> 15:27.810 calculated. 15:27.950 --> 15:31.750 These are these A11, A12 entries. 15:31.810 --> 15:39.250 So A11 stands for this first term and A12 is the second term. 15:39.250 --> 15:46.130 So this is basically where this A-matrix is built up, the transition 15:46.130 --> 15:47.730 matrix of the state space model. 15:48.670 --> 15:51.650 And here it is simply put together as a numpy array, as a two 15:51.650 --> 15:52.450 -dimensional array. 15:54.110 --> 15:56.450 We do the same with this B-vector. 15:56.810 --> 16:01.190 So that's exactly one-to-one with the mathematical formulation on the 16:01.190 --> 16:01.750 slide before. 16:02.290 --> 16:05.790 And the C and D-matrix are basically only available as standard in the 16:05.790 --> 16:06.130 model. 16:08.790 --> 16:12.970 So we just do that with standard values. 16:14.550 --> 16:19.390 And then you can use this command CTRL-SS, which stands for state 16:19.390 --> 16:22.070 space model, to generate such an object. 16:23.290 --> 16:28.250 And when we have generated it, then you can also simulate it with the 16:28.250 --> 16:28.630 library. 16:34.260 --> 16:38.940 And first of all, we take initial values, which we also used from the 16:38.940 --> 16:40.840 Simulink model, i.e. 16:40.860 --> 16:41.420 transferred one-to-one. 16:42.940 --> 16:45.140 Here you can see that the model is being built up. 16:45.440 --> 16:48.600 And here you can now see values for the transverse stiffness in front 16:48.600 --> 16:51.520 and back and then a few distances in meters and a vehicle mass. 16:52.520 --> 16:58.500 And with that, such a model is generated as an object. 16:59.500 --> 17:05.660 And with the command from the CTRL-Systems library, CTRL-Force 17:05.660 --> 17:10.260 -Response, you can then simulate the state space model. 17:10.260 --> 17:13.780 And of course the model will be transferred. 17:19.520 --> 17:21.480 A time vector will be transferred. 17:22.040 --> 17:25.320 And then this U here, which is basically the steering wheel angle. 17:25.540 --> 17:29.580 So that's the driving input into the model. 17:29.860 --> 17:32.520 And we also have a few starting conditions that can be specified. 17:33.500 --> 17:36.240 And then you get different values. 17:36.560 --> 17:37.940 You can also have the time vector output. 17:38.820 --> 17:41.460 I have now let the conditions out, so to speak. 17:43.300 --> 17:45.520 And then you can basically plot them. 17:52.310 --> 17:54.710 This is done here in the plot. 17:54.810 --> 17:56.570 Then we use the same plot function again. 17:57.190 --> 17:59.830 As I said, sorry, the x-axis labels down here. 17:59.950 --> 18:02.570 So that's basically hundredths of a second. 18:03.650 --> 18:05.950 And then we are like this. 18:06.810 --> 18:11.310 It fits qualitatively, but it looks a bit different in all the 18:11.310 --> 18:15.970 projections than the two-track model that we had as a reference. 18:16.890 --> 18:20.370 And yes, that's simply because the one-track model has a simpler 18:20.370 --> 18:20.870 structure. 18:21.470 --> 18:29.110 And although we have now passed the exact values, the reference data 18:29.110 --> 18:29.770 cannot follow that closely. 18:29.770 --> 18:34.310 Yes, and a typical task is to tune such models. 18:34.810 --> 18:37.750 And yes, then there are different tasks. 18:38.070 --> 18:41.830 So one task would be to use a model to determine such unknown 18:41.830 --> 18:42.550 parameters. 18:43.910 --> 18:46.850 We don't want to do that here now, but we want to tune the parameters 18:46.850 --> 18:51.630 in such a way that the output of the model fits better with the 18:51.630 --> 18:52.310 reference data. 18:53.310 --> 18:58.610 And yes, that's basically a step in the work that you take to improve 18:58.610 --> 19:01.030 the prediction of the model. 19:01.890 --> 19:07.610 This can lead to the fact that the parameters are no longer in the 19:07.610 --> 19:08.810 actual size order. 19:09.930 --> 19:16.130 So you don't have the identification of the parameters as a goal, but 19:16.130 --> 19:19.730 rather that the model output is well connected to the reference data. 19:19.730 --> 19:25.290 And depending on how different the model structures are, this can vary 19:25.290 --> 19:25.970 significantly. 19:26.790 --> 19:33.230 So if you have reference data from a system or a reference model that 19:33.230 --> 19:37.750 is quite complicated and has a lot of dynamics in it, and you then 19:37.750 --> 19:41.850 take a very, very simple model and want to simulate it, then it may be 19:41.850 --> 19:47.650 that the parameter values ​​then deviate a whole corner from this 19:47.650 --> 19:48.510 reference model. 19:48.970 --> 19:52.190 So that's relatively normal. 19:52.850 --> 19:56.330 This is simply due to this simplification that you have in the 19:56.330 --> 19:57.190 simulation model. 19:57.790 --> 20:00.790 Certain physical effects are completely suppressed there. 20:00.790 --> 20:04.950 And then it is actually not surprising that the parameters can no 20:04.950 --> 20:10.730 longer match exactly with the reference parameters. 20:12.110 --> 20:18.210 But such simple models can be used very well to make predictions and 20:18.210 --> 20:19.390 to model the system behavior. 20:20.770 --> 20:23.290 And the simple models still have the charm that they also calculate 20:23.290 --> 20:24.210 very, very quickly. 20:25.070 --> 20:28.530 So if you have a very simple model structure, it can be simulated 20:28.530 --> 20:32.150 much, much faster than a very large, complicated model. 20:32.810 --> 20:35.710 Here in the example, the difference in simulation time is not that 20:35.710 --> 20:36.010 big. 20:36.110 --> 20:40.750 But if you think of larger system simulations, it can do a lot more 20:40.750 --> 20:41.590 very quickly. 20:43.170 --> 20:47.610 So that's just a bit of a background on why we want to tune the model 20:47.610 --> 20:55.250 parameters a bit and want to push the simple model into the same 20:55.250 --> 21:00.990 starting behavior with the optimization as the more complicated two 21:00.990 --> 21:01.530 -spot model. 21:05.940 --> 21:08.640 Let's take a look at a direct comparison again. 21:09.440 --> 21:15.560 Here the reference data from the two-spot model and the simulated data 21:15.560 --> 21:16.860 from the one-spot model are shown. 21:16.860 --> 21:23.340 And here you can see that, for example, the swimming angle of the one 21:23.340 --> 21:25.460 -spot model is significantly larger. 21:26.200 --> 21:27.180 So that's shown here in red. 21:28.500 --> 21:31.300 And the reference, that's this two-spot model. 21:31.880 --> 21:36.440 So you can see that it fits quite well in terms of quality, but not in 21:36.440 --> 21:36.920 terms of quantity. 21:38.360 --> 21:45.740 The one-spot model can quite well depict the characteristic behavior, 21:46.520 --> 21:49.520 but it doesn't quite fit into the peaks. 21:49.640 --> 21:52.880 And also in terms of dynamics, it looks a bit different. 21:53.660 --> 21:56.180 For example, this is a bit sluggish here. 21:56.380 --> 21:58.160 There's something going on behind it. 22:01.560 --> 22:05.840 What we need now for mathematical optimization is a so-called cost 22:05.840 --> 22:07.820 function or objective function. 22:07.820 --> 22:13.620 And you can always set it up yourself or you have to set it up 22:13.620 --> 22:16.240 yourself for mathematical optimization and calibrating. 22:16.700 --> 22:24.040 And what we want to do then is to take both states into account in the 22:24.040 --> 22:25.580 cost function. 22:26.260 --> 22:34.200 And what we also do is to normalize the values first, so that each of 22:34.200 --> 22:37.340 the two states is equally important in the cost function. 22:38.540 --> 22:40.120 And that's done here now. 22:41.040 --> 22:44.540 So a function is written, which is simply called error. 22:45.020 --> 22:47.160 You could have called it cost function, for example. 22:47.160 --> 22:57.420 And now the absolute error between the reference data and the one-spot 22:57.420 --> 23:01.800 model is calculated here with this error 0 and error 1. 23:02.400 --> 23:04.660 And then it is scaled again with the two lines. 23:06.880 --> 23:10.860 And here in the last line, the cost function is then built together. 23:11.800 --> 23:19.420 And that is always the quadratic error of the gear rate and the 23:19.420 --> 23:20.040 swimming angle. 23:20.280 --> 23:21.740 And both are simply summarized with a plus. 23:23.040 --> 23:27.660 You could also put a weighting factor in between here. 23:27.660 --> 23:33.680 Then you can basically tune the model either only to the gear rate or 23:33.680 --> 23:37.960 only to the swimming angle or some mixtures of them. 23:37.960 --> 23:41.580 So here in the cost function, you can basically influence and define 23:41.580 --> 23:44.120 your optimization problem very strongly. 23:46.640 --> 23:53.340 And also this normalization that I did there is basically optional. 23:53.340 --> 23:57.680 But it is important that each of the two states is equally important, 23:57.680 --> 23:58.700 as it is written here. 24:00.280 --> 24:05.460 But you can also use the notebook later and try to build a weighting 24:05.460 --> 24:06.340 factor in here. 24:06.340 --> 24:12.860 And then switch out one of the two states and see what the optimizer 24:12.860 --> 24:14.260 does with it in the end. 24:18.620 --> 24:24.380 And what we do first with this initially defined one-spot model is 24:24.380 --> 24:27.440 that we simply let the value of this cost function be spent. 24:28.860 --> 24:31.900 Since everything is now normalized, it no longer has a unit. 24:33.920 --> 24:36.200 You can't really interpret it that well. 24:36.200 --> 24:38.840 What you can only say is, the smaller the better. 24:39.080 --> 24:43.900 So the better the one-spot model is in the initial behavior with the 24:43.900 --> 24:45.100 two -spot model. 24:47.160 --> 24:51.780 And we start here with a value of about 3082. 24:53.220 --> 24:56.420 And now the optimizer comes into play, which then changes the 24:56.420 --> 24:59.220 parameters and, so to speak, should improve this value. 25:01.520 --> 25:05.260 What we then use is the minimize function from SciPy Optimize. 25:07.640 --> 25:10.700 And we're going to optimize two parameters now. 25:12.100 --> 25:15.140 This is the front and rear side stiffness. 25:16.880 --> 25:22.400 And for that we first need an optimizer and have to build it up. 25:23.420 --> 25:25.840 And you can see that here in principle. 25:28.620 --> 25:32.500 So first the cost function is built up. 25:32.680 --> 25:35.260 For this you need a simulation again and again. 25:35.980 --> 25:47.500 So what is changed is then the value for the front stiffness and the 25:47.500 --> 25:48.860 value for the rear stiffness. 25:48.860 --> 25:50.640 This is transferred here as an argument. 25:51.880 --> 25:54.020 And that's just an array again. 25:54.720 --> 25:58.200 And the first entry basically stands for the front, the second for the 25:58.200 --> 25:58.400 rear. 25:58.880 --> 26:01.700 And the other values are now left to the default values. 26:02.940 --> 26:06.940 So we don't turn around in this optimization problem. 26:07.000 --> 26:08.600 But you can also change that if you want. 26:09.700 --> 26:16.080 So if you now add x2 and x3 here and simply increase the vector of the 26:16.080 --> 26:17.020 parameters to be optimized. 26:17.020 --> 26:20.580 Then the optimization problem will be a little bigger again. 26:20.840 --> 26:25.480 But the optimizer then has more freedom to tune the model. 26:26.540 --> 26:34.020 I didn't do that here because we leave these geometric sizes to the 26:34.020 --> 26:34.820 default values. 26:35.620 --> 26:42.100 And then the model is basically simulated with this set of parameters. 26:42.100 --> 26:47.660 And in the end the cost function then brings out these error values. 26:48.420 --> 26:53.320 So the optimizer needs that to see whether a change here in this x0 26:53.320 --> 26:58.640 and x1 goes in the right direction or leads to greater costs. 27:02.830 --> 27:05.130 We'll take out an optimizer now. 27:06.470 --> 27:08.450 It is now chosen a bit randomly. 27:09.110 --> 27:13.130 In the documentation of the toolbox there are many optimizers. 27:13.210 --> 27:15.690 You can take a look and try out a few others. 27:17.170 --> 27:19.710 Here we just take this Powell optimizer. 27:20.650 --> 27:24.130 And what you have to give it is basically a start value. 27:24.530 --> 27:28.070 This is set here for the front and rear stiffness. 27:29.330 --> 27:35.670 And then you pass this objective function to it. 27:36.730 --> 27:45.370 Then you can simply call it up and get the optimized values displayed 27:45.370 --> 27:45.370 at the end. 27:51.920 --> 27:57.520 Here it is indicated again with a few additional output information. 27:58.460 --> 28:02.720 So how the value of the cost function is and how many iterations it 28:02.720 --> 28:03.120 has gone through. 28:04.240 --> 28:07.840 And how many times the function has been evaluated for it. 28:07.840 --> 28:10.920 And then we come to such a value at the end. 28:11.320 --> 28:13.420 So it is now smaller than the one before. 28:14.740 --> 28:18.140 And the optimizer has now found a set for these transverse 28:18.140 --> 28:18.800 stiffnesses. 28:18.860 --> 28:24.420 Where the input model simply delivers a result that is closer to the 28:24.420 --> 28:25.080 reference data. 28:26.660 --> 28:30.800 And of course you can also look at that again, that you plot that. 28:30.800 --> 28:35.700 And then I have now transferred the values ​​by hand from what the 28:35.700 --> 28:36.680 optimizer has found. 28:41.310 --> 28:47.650 And you can then first display it individually or in connection. 28:48.270 --> 28:52.710 And yes, you can now see that the input model has still not quite 28:52.710 --> 28:54.170 managed to map these reference data. 28:56.350 --> 29:01.190 But the overall value for the costs has already improved a bit. 29:02.510 --> 29:09.030 But yes, in principle, there is still a big discrepancy in the 29:09.030 --> 29:10.690 swimming angle that you can see there. 29:10.810 --> 29:13.990 That means there is still a bit of potential in how to improve that. 29:15.830 --> 29:20.130 And as I said, that's the mistake that came out after this 29:20.130 --> 29:21.370 optimization set. 29:21.370 --> 29:30.930 So that's the first step I took to adapt the model a bit better. 29:31.950 --> 29:36.210 And now the question is whether even better values ​​can ultimately be 29:36.210 --> 29:36.550 obtained. 29:37.690 --> 29:43.170 Or whether it is simply a limit case where the input model no longer 29:43.170 --> 29:47.050 makes sense to depict the dynamics of this two-stroke model. 29:48.590 --> 29:54.510 Of course, there will be some limits at some point, because the input 29:54.510 --> 29:58.150 model has a much simpler physics at its core. 30:00.410 --> 30:06.170 And that would be the task you can do here in this mini-tutorial. 30:07.230 --> 30:11.090 In this mini-project, in principle, you can now of course select other 30:11.090 --> 30:11.630 optimizers. 30:13.190 --> 30:17.730 You can also let other parameters flow into the optimization. 30:19.090 --> 30:25.030 So, for example, what I could imagine would be the sluggishness, i.e. 30:25.050 --> 30:27.490 the big tether of the input model. 30:27.890 --> 30:30.170 That you take that in again and then don't optimize over two 30:30.170 --> 30:31.890 parameters, but over three. 30:32.490 --> 30:34.370 And what you can also do is build an optimizer yourself as an 30:34.370 --> 30:35.430 alternative method. 30:37.430 --> 30:44.410 And the easiest way to look at optimization without going into 30:44.410 --> 30:49.230 complicated algorithms like genetic optimization. 30:49.950 --> 30:54.350 The easiest thing you can do is actually a random number search. 30:54.910 --> 31:02.570 It is also quite stable and delivers, or rather stable is perhaps 31:02.570 --> 31:05.870 mispronounced, of course, with every run, with such a random number 31:05.870 --> 31:08.130 search, a different value comes out. 31:08.490 --> 31:12.510 But where it is quite good is simply with cost functions that have 31:12.510 --> 31:14.530 several local optimizers. 31:16.310 --> 31:22.510 These procedures are usually able to get out of local optimizers. 31:24.310 --> 31:26.690 It's worth taking a quick look at it. 31:26.690 --> 31:32.710 And such a simple optimizer, which simply runs via Monte Carlo 31:32.710 --> 31:37.470 procedures and uses such random numbers, is shown here in pseudocode. 31:38.570 --> 31:39.510 So what do I mean by that? 31:39.810 --> 31:44.570 What you would have to set first with such an optimizer would be for 31:44.570 --> 31:48.450 the parameters to be optimized, min and max values that are somehow 31:48.450 --> 31:48.770 useful. 31:49.590 --> 31:53.190 Of course, we don't want to have negative values for the stiffness 31:53.190 --> 31:54.490 parameters, for example. 31:56.530 --> 31:58.830 That contradicts the whole physics. 31:59.410 --> 32:01.510 It makes sense to set such min and max values. 32:02.490 --> 32:05.850 And then you just initialize the cost function. 32:06.070 --> 32:12.090 So that was this single value in front of us, the 3082, for example, 32:12.410 --> 32:13.330 which we had first. 32:13.330 --> 32:19.450 You initialize that first with a value. 32:20.110 --> 32:20.710 Ah, that's here. 32:20.850 --> 32:21.770 Sorry, I slipped a line. 32:23.090 --> 32:31.230 And then you start again with initial parameters and save that to a 32:31.230 --> 32:33.570 variable called best parameters. 32:34.650 --> 32:39.270 And what you can also do, if you make it a bit more complicated, which 32:39.270 --> 32:43.750 is not necessary here in the pseudocode below, is that you basically 32:43.750 --> 32:47.930 look at how the percentage change from one iteration from the 32:47.930 --> 32:49.570 optimizer to the next behaves. 32:49.610 --> 32:52.210 Then you can also build in such a stop criterion. 32:53.490 --> 32:57.810 But what the actual process is then in principle, here for the 32:57.810 --> 33:01.370 simplest form that I have come up with now, is that you make a pre 33:01.370 --> 33:05.570 -loop and then you simply iterate up to the maximum number of 33:05.570 --> 33:07.170 iterations you want, i.e. 33:07.190 --> 33:08.190 for example 1000 times. 33:09.110 --> 33:15.630 Then you throw random parameters, simply with an even distribution, 33:15.810 --> 33:20.010 for example, given the min and max values. 33:20.510 --> 33:22.990 So the parameters should be within these limits. 33:23.650 --> 33:28.790 Then the input model is simulated with these parameters and then the 33:28.790 --> 33:30.470 current costs are simply calculated. 33:30.470 --> 33:35.490 And then it is compared whether the current costs that this iteration 33:35.490 --> 33:38.970 loop has just brought out, whether they are better, i.e. 33:39.010 --> 33:42.150 smaller than the previous costs. 33:42.950 --> 33:44.710 And then you can basically save that for a short time. 33:45.890 --> 33:52.010 And that basically goes on until this maximum iteration is reached. 33:52.370 --> 33:57.770 And a few times it will probably jump into this if loop, if such a 33:57.770 --> 34:02.990 random parameter set has actually led to a simulation where the costs 34:02.990 --> 34:05.930 were lower than what was previously found. 34:06.710 --> 34:13.550 So here this loop runs for example 1000 times and maybe a few hundred 34:13.550 --> 34:18.890 times or in the lower hundred range it ends up here in this loop. 34:19.590 --> 34:24.990 And then the optimizer has ultimately found a parameter set that was 34:24.990 --> 34:29.050 better than the previous starting conditions. 34:29.490 --> 34:33.570 And then you can just try to improve the input model in this 34:33.570 --> 34:34.590 calibration task. 34:35.450 --> 34:37.690 As I said, you can also work a little bit on the cost function itself 34:39.110 --> 34:46.230 or on this error function, that you only try to optimize the swimming 34:46.230 --> 34:51.670 angle, for example, whether you can get better results and just gain a 34:51.670 --> 34:55.730 little experience how to solve such optimization problems with Python. 34:57.650 --> 35:02.070 The task setting is in principle such that the input model cannot 35:02.070 --> 35:05.530 completely follow the reference data of the two-stroke model. 35:06.850 --> 35:12.110 So it's not just a bit of a rush on the reference data, what such a 35:12.110 --> 35:15.170 model can compensate for, but here we really have to do with a 35:15.170 --> 35:17.050 different model structure. 35:18.270 --> 35:22.230 The reference data, as I said, are from a model with a more 35:22.230 --> 35:26.190 complicated physical structure and the input model is much easier from 35:26.190 --> 35:27.410 the structure. 35:28.210 --> 35:32.150 There are many physical effects completely neglected. 35:32.670 --> 35:37.590 So it won't be that you get a one-to-one match and the lines are 35:37.590 --> 35:38.310 exactly on top of each other. 35:38.630 --> 35:44.390 That's not really the goal here with the task, but that you deal with 35:44.390 --> 35:48.610 optimization a bit and play around with the cost functions a bit and 35:48.610 --> 35:54.630 write an optimizer yourself and then just see how relatively easy it 35:54.630 --> 35:58.690 is in Python and how you can handle such rather complicated questions 35:58.690 --> 36:01.350 with Python. 36:03.610 --> 36:06.870 Then I would say I'm looking forward to the results. 36:07.230 --> 36:08.870 Let's see where they come from. 36:08.870 --> 36:11.730 Surely they will find a few better settings. 36:13.050 --> 36:16.950 And then I would say thank you for listening and we'll see you again 36:16.950 --> 36:18.510 at the next mini-project.