FROM THE EXPANSION OF THE SPACE OF THE
DISPLACEMENTS TOWARDS THE
TRANSFORMATIONS OF THE PLANE WITH THE
INVOLVEMENT OF THE LEBESGUE'S THEOREM
AND THE LIMIT

Edern Ollivier

<figure><img src="image_1.png" /></figure>

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Edern Ollivier. FROM THE EXPANSION OF THE SPACE OF THE DISPLACEMENTS TOWARDS THE TRANSFORMATIONS OF THE PLANE WITH THE INVOLVEMENT OF THE LEBESGUE'S THEOREM AND THE LIMIT. 2022. [https://hal.science/hal-03572778v2](https://hal.science/hal-03572778v2)

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Distributed under a Creative Commons Attribution 4.0 International License
FROM THE EXPANSION OF THE SPACE OF THE
DISPLACEMENTS TOWARDS THE TRANSFORMATIONS OF THE
PLANE WITH THE INVOLVEMENT OF THE LEBESGUE'S
THEOREM AND THE LIMIT*

EDERN OLLIVIER†

**Abstract.** The odometric model gives the place to a space of displacements. I have been resolving to simulate, integrate, explain and then to validate the odometric model. The model of resolution is to be programmed in the Maple language, with the help of a solver for Maple 2021.

**Key words.** Odometry, Vehicle location and navigation systems, Mobile positioning systems, Trajectory, Convergence of a model, Control of a vehicle.

**AMS subject classifications.** 65N30/35R11

**1. Introduction.** As one introduction, I could say that the odometric model is well known now and that everything is done for the space of displacements of a model of automation to be improved with the odometric model.

**2. Main results.** I have done the approximation of the computed odometric model then I can understand the purpose of the space of the displacements : There is the convergence of the odometric model with the absolute convergence until an hypergeometric function with a gamma, and moreover all this space of displacements has a structure K-completed of a group, with the neutral element that is the rotation of an angle of $2*Pi$ modulo $Pi$. The formula below is used to be the steering angle of the mathematical function of the computed odometric model.

$$ \alpha(t) = \tan(t) $$

The formula below is used to be the steering angle integrated of the mathematical function of the computed odometric model.

$$ \theta(t) = -\ln(\cos(t)) $$

Moreover the formula below is used to be the value to integer of the mathematical terminal of the computed odometric model.

$$ \beta(t) = \int_{0}^{t} \cos(x)^{I} dx $$

It deals with the value of the steering angle at the value of $Pi/2$. Normally the speed value of the wheels of the vehicle should always be tangential to the vehicle. Indeed the wheels are to be rolling in the direction of the vehicle. And when the vehicle is turning then the direction of the vehicle is changing of the driving direction.

Here I state our main result as Theorem 2.1; The proof is deferred to Proof 3.

**THEOREM 2.1** (*theta* Integration Value [3]). *There is the absolute convergence to the function integrated of the steering angle at the value of integration *t* :*

$$ \theta(t) = -\ln(\cos(t)) $$

<footnote>*Written and completed 16/03/2022.</footnote>

<footnote>**Funding:** This work was not funded by the IEEE</footnote>

<footnote>† (edernolli@gmail.com, http://www.maintenance-elec.fr).</footnote>
**THEOREM 2.2 (Terminal Point Theorem).** There is the integration of the insta-bility terminal point of the computed odometric model at the value of integration Pi/2

39

$$
\beta(\Pi/2) = \int_{0}^{\Pi/2} \cos(x)^I dx = \Gamma(1/2 + I/2) \cdot \sqrt{(\Pi)/(2 \cdot \Gamma(1 + I/2))}
$$

40

41

In other words, the terminal value is computed to a Gamma.

42

$$
\beta(\Pi/2) = \Gamma(1/2 + I/2) \cdot \sqrt{(\Pi)/(2 \cdot \Gamma(1 + I/2))}
$$

43

Observe that Theorems 2.1 and 2.2 correctly mix definitions of the value of integration.

44

45

COROLLARY 2.3. Let $f(x)$ be continuous and differentiable everywhere. If $f(x)$ has at least two terminal points, then $\int_{0}^{\Pi/2} f(x)^{I} dx$ must have at least one value.

46

47

Proof. Let $a$ and $b$ be two distinct terminal points of $f$. And also with $g$ the limited development of the $f$ function on the interval $[a, b]$. By Theorem 2.2, there exists a number $c$ such that the serie converges normally by Lebesgue :

48

49

$$
\int_a^b f(x)^I dx \le \sum_{n=0}^{\infty} \int_a^c g^n(x)^I dx.
$$

50

51

For example let $f(x)$ to be the $\cos(x)$ function to be the following function to be evaluated :

52

53

$f(x) = \cos(x)$

54

And let $g(x)$, the evaluation is to be done normally around zero but here we can approximate the function in $c$ with $c \le b$ by a polynom of order $2n$ by Taylor :

55

$$
g^n(x) = (-x)^n \cdot \frac{x^2}{(2 \cdot n)!} \quad \square
$$

56

57

COROLLARY 2.4. Let theta(t) be defined most of everywhere. If theta(t) has at least a limit point, then $\lim_{t \to \Pi/2} \theta(t) = \infty$.

58

59

$$
\textbf{Proof.} \quad \text{Trivially, the limit is given by the expression } \lim_{t \to \frac{\Pi}{2}} -\ln(\cos(t)) = \infty \quad \square
$$

60

Proof. On one hand the space of displacements is a subgroup of the space of the transformations of the plane implies that each element of the space of displacements is the limit of a series of elements of this space of displacements. And another hand for each element of this space of displacements Proof 3 there exists a bounded sequence for any displacements, cf. The work of Guillaume Artus on the slavings to be given to the computer to make a robot go from one position to another in the plane [1], in the space of the displacements such that the limit belongs to this space of the displacements of the plane implies that the space of the displacements is completed in the body of the transformations of the plane.

62

63

64

65

66

67

68

69

Note about the explanations on the first page of calculus : The theorem has been fully proven so that now I can access the datas from the first series with the linearized functions in the matrix of state and measured, given by the chinese team in a paper published by the IEEE editor, I have done the primary effort to the state matrix of functions, nevertheless they did the last one to ensure the compat- ibility of the odometric model with the mathematical background of mine and also

70

72

73

74
75 to get the values with the vehicle [2]. All the datas have been computed in a ta-
76 ble by a program under Ubuntu 20.04 with the GCC and also the GLIBC software,
77 it used to be accessed on the DataPort of the IEEE website with the odometric key
78 word : https://ieee-dataport.org/open-access/odometric-datas-simulation-odometric-
79 model to give an example from an linear approximation of the odometric model
80 with GCC. For the simulation of the odometric model please see at : https://ieee-
81 dataport.org/documents/new-figure-simulation-odometric-model to give an example
82 from a simulation of the odometric model with Scilab.

83 Note about the example on the first page of calculus : The best effort example
84 is the half turn like maneuver. In the space of the displacements there are two arcs
85 of circle to do for instance, instead of a rotation of Pi in the space of the transfor-
86 mations in the plane, and usually its can be done for more figures of maneuver with
87 the homothety, or even with the translation. It is usually appointed the transverse
88 function in the space of the displacements.
89 □

89 3. **Algorithm**. Our analysis leads to the algorithm in Algorithm 3.1.

<caption>**Algorithm 3.1 Build spiral**</caption>

Define croquette := proc(n)
Define locali, R, x, z
Define R := [[0, 0]]
Define i := 1
while i < n/2 do

Define x := i/n * Pi
Define z := sin(x) * hypergeom([1/2, 1/2 - I/2], [3/2], sin(x)²)
Update z := evalf(z)
Update R := [op(R), [Re(z), Im(z)]]
Update i := i + 1

end while
return R
Define end

4. **Experimental results.** Figure 1 shows an example result of the integration.
Figure 2 shows again an example result of the integration.
Figure 3 shows again an example result of the integration.

5. Discussion of $z = [Re(z), Im(z)]$. As one discussion, I could say that the values of the integrated computed odometric model is computed in the given figures with a procedure recording the complex values at every steps of the procedure given by a volunteer from the Inria of Saclay in France.

6. **Conclusions.** As one conclusion, I could say that the odometric model is well known now and that everything is done for the commands of a model of automation to be validated with the odometric model [3]. I should also emphasize that a thesis can be done with the first issue of mine about the so called linearization of the equations of the odometric model given back by the issue of the chinese team of researchers [1].

**Appendix A. An appendix for the modulo $P_i$.**

LEMMA A.1. *Idem with the 3\*Pi/2 value modulo Pi.*

**Acknowledgments.** I would like to acknowledge the assistance of a volunteer of the Inria of Saclay in France in solving together these figures.
$$ > \theta := -\log(\cos(t)); \qquad \theta := -\ln(\cos(t)) \tag{1} $$

$$ > \text{zinfly} := \text{int}\left(\cos(t)', t = 0 - \frac{\text{Pi}}{2}\right); \qquad \text{zinfly} := \frac{\Gamma\left(\frac{1}{2} + \frac{1}{2}\right) \sqrt{\pi}}{2\Gamma\left(1 + \frac{1}{2}\right)} \tag{2} $$

$$ > \text{zinfly} := [[\text{evalf}(\text{zinfly})], \text{Im}(\text{evalf}(\text{zinfly}))]]; \qquad \text{zinfly} := [[1.049478078, -0.5824562520]] \tag{3} $$

$$ > \text{int}\left(\cos(t)', t = 0 - \frac{7 \cdot \text{Pi}}{100}\right); \qquad \text{sin}\left(\frac{7 \pi}{100}\right) \text{hypergeom}\left(\left[\frac{1}{2}, \frac{1}{2} - \frac{1}{2}\right], \left[\frac{3}{2}\right], \sin\left(\frac{7 \pi}{100}\right)^2\right) \tag{4} $$

```R
> croquette:=proc(n)
  local i,R,z;
  R:=[];
  i:=1;
  while i<n/2 do
    x:=i/n*Pi;
    z:=sin(x)*hypergeom({1/2,1/2-1/2},[3/2],sin(x)^2);
    z:=evalf(z);
    R:=[op(R), [Re(z),Im(z)]];
    i:=i+1;
  od;
  R;
end;
```

```R
> C3000 := croquette(3000);
> display([plot(C3000), plot(zinfy, style = point, color = black)]);
```

<figure><img src="image_3.png" /></figure>

```R
> kroquette:=proc(n,rho)
```
```R
local i,R,x,z;
R:={[0,0]};
i:=2;
while i<n do
  x:=Pi/2-rho*i+Pi/2;
  z:=sin(x)*hypereom([1/2,1/2-1/2],[3/2],sin(x)^2);
  z:=qval(z);
  R:=[op(R),[Re(z),Im(z)]];
  i:=1+i;
od;
R:
end:
```

```R
> K1000 := kroquette(200, 7/8);
```

```R
> display([plot(K1000), plot(zinfy, style = point, color = black)]);
```

<figure><img src="image_4.png" /></figure>

```R
> with(plots);
```

```R
[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal,
  conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, densityplot, display,
  dualaxisplot, fieldplot, fieldplot3d, gradplot, gradplot3d, implicitplot, implicitplot3d, inequal,
  interactive, interactiveparams, intersectplot, listcontplot, listcontplot3d, listdensityplot, listplot,
  listplot3d, loglogplot, logplot, matrixplot, multiple, odeplot, pareto, plotcompare, pointplot,
  pointplot3d, polarplot, polygonplot, polygonplot3d, polyhedra_supported, polyhedraplot,
  rootlocus, semilogplot, setcolors, setoptions, setoptions3d, shadebetween, spacecurve,
  sparsematrixplot, surfdata, textplot, textplot3d, tubeplot]
```
<figure><img src="image_5.png" /></figure>
[1] Guillaume Artus, Pascal Morin, Claude Samson. Control of a maneuvering mobile robot by the transverse function approach : control design and simulation results. [Research Report] RR-5155, INRIA. 2004, pp.16. (inria-00077042);

[2] Xiaocong Su, Chaoyong Zhang, Duanfeng Chu, Shidong Liu, Deng, and Chaozhong Wu, Trajectory Planning and Tracking for Autonomous Vehicle Based On State Lattice And Model Predictive Lattice, IEEE Intelligent Transportation Systems Magazine (Volume 11, Issue 2, Summer 2019 );

[3] Edern Ollivier. Simulation, integration and explanation of the odometric model for the purpose of the numerical analysis: A pool of trajectories for the engineers of the automation of the fair and automated vehicle. 2019. (hal-02162501v10).