NONLINEAR METHOD OF VEHICLE VELOCITY DETERMINATION BASED ON INVERSE SYSTEM AND TENSOR PRODUCT OF LEGENDRE POLYNOMIALS-INTERMEDIATE CLASS

Presented paper discusses two different nonlinear approaches to precrash velocity determination for vehicles from Intermediate Car Class. Data that was used to perform analyses introduced in this paper was taken from National Highway Traffic Safety Administration (NHTSA) database. Such database is comprised of substantial number of crash cases and main focus was put on frontal impacts. Hitherto used energy methods are based on linear model which proves to be inaccurate and producing significant errors. Presented considerations concern the inverse system and tensor product of Legendre polynomials. The focus of those methods is to establish the value of nonlinear coefficient bk which is the slope factor of precrash velocity Vt and deformation ratio Cs function.


Introduction
In recent years the development of methods for precrash velocity determination seemed to reach a plateau.Linear models, however simple and showing great ease of use, unfortunately produce significant inaccuracies, reaching up to 30% [39].Authors focus on developing superior methods in terms of precision, based on nonlinear models, which already have shown promising results in the recent past [18][19][20][21][22][23][24][25][26].
Once the above parameters are established, the deformation work can be defined.This quantity is then directly used to find the value of EES from the following equation: The In recent years the development of methods for precrash velocity determination seemed to reach a plateau.Linear models, however simple and showing great ease of use, unfortunately produce significant inaccuracies, reaching up to 30% [39].Authors focus on developing superior methods in terms of precision, based on nonlinear models, which already have shown promising results in the recent past [18][19][20][21][22][23][24][25][26].
Once the above parameters are established, the deformation work can be defined.This quantity is then directly used to find the value of EES from the following equation: (1) where: − Equivalent Energy Speed, − deformation work, − vehicle mass.
The EES [6,39,39] defines the amount of energy to deform the vehicle [7,24,38,32,33] while impacting a rigid obstacle, i.e.only plastic deformations occur and the entire kinetic energy accumulated by the vehicle is transferred to deformation work.This is then used as a baseline to find the actual velocity of a vehicle in a real life accident and asses if, for example, collision avoiding maneuvers has been applied [2,23] or if the vehicle speed was properly adjusted according to traffic conditions [34].Analyzed set of cases focuses on frontal impacts [35,36].
The first method of EES determination is based on a simple observation, that coefficient dependent on behaves similarly to function .This led Authors to consideration of inverse systems: Application of the least square approximation gives much better results than other methods used in this field.
where: EES Equivalent Energy Speed, W deformation work, m vehicle mass.
The EES [6,39,39] defines the amount of energy to deform the vehicle [7,24,38,32,33] while impacting a rigid obstacle, i.e.only plastic deformations occur and the entire kinetic energy accumulated by the vehicle is transferred to deformation work.This is then used as a baseline to find the actual velocity of a vehicle in a real life accident and asses if, for example, collision avoiding maneuvers has been applied [2,23] or if the vehicle speed was properly adjusted according to traffic conditions [34].Analyzed set of cases focuses on frontal impacts [35,36].
The first method of EES determination is based on a simple observation, that coefficient b k dependent on C s behaves similarly to function .This led Authors to consideration of inverse systems: 1, , , .Application of the least square approximation gives much better results than other methods used in this field.
The second method focuses on Legendre polynomials, which are orthogonal function in the domain of [-1,1].For the purpose of Octave implementation, the set of Legendre polynomials was rescaled and renumbered.Finally, the Legendre polynomials tensor product is considered and the least square method is applied.

Inverse system method description
Assuming that there are given points (x n ,y n ) N n=1 and a function set (f m ) M m=1 .The goal is to find coefficients (a m ) M m=1 , that minimize the value of: The Archives of Automotive Engineering -Archiwum Motoryzacji Vol.83, No. 1, 2019 3 The second method focuses on Legendre polynomials, which are orthogonal function in the domain of .For the purpose of Octave implementation, the set of Legendre polynomials was rescaled and renumbered.Finally, the Legendre polynomials tensor product is considered and the least square method is applied.

Inverse system method description
Assuming that there are given points and a function set .The goal is to find coefficients , that minimize the value of: ( This problem is called the least-square function approximation and is very well covered in the literature [2].The first author of this solution was not established, since both Carl Gauss and Adriena-Marie Legendre were working on the topic. In order to find the minimum, the function has to be differentiated with respect to obtaining a series of equations (3) It is convenient to use the matrix notation: (4) It is worth noticing that matrix M on LHS (2) is symmetric in sense, that for arbitrary .This obvious observation allows to speed up the calculations almost twice.In that stage, the solution of least square problem is down to solving of the matrix and making following observations, on condition that it is inverse: (5) Returning to the original problem of finding the as a function of , the set of points is adjacent to , where is the number of crash tests.As shown in the preceding sections, the following is true: This problem is called the least-square function approximation and is very well covered in the literature [2].The first author of this solution was not established, since both Carl Gauss and Adriena-Marie Legendre were working on the topic.
In order to find the minimum, the function has to be differentiated with respect to a k , k = 1, …, M obtaining a series of equations The second method focuses on Legendre polynomials, which are orthogonal function in the domain of .For the purpose of Octave implementation, the set of Legendre polynomials was rescaled and renumbered.Finally, the Legendre polynomials tensor product is considered and the least square method is applied.

Inverse system method description
Assuming that there are given points and a function set .The goal is to find coefficients , that minimize the value of: ( This problem is called the least-square function approximation and is very well covered in the literature [2].The first author of this solution was not established, since both Carl Gauss and Adriena-Marie Legendre were working on the topic.
In order to find the minimum, the function has to be differentiated with respect to obtaining a series of equations It is convenient to use the matrix notation: (4) It is worth noticing that matrix M on LHS ( 2) is symmetric in sense, that for arbitrary .This obvious observation allows to speed up the calculations almost twice.In that stage, the solution of least square problem is down to solving of the matrix and making following observations, on condition that it is inverse: (5) Returning to the original problem of finding the as a function of , the set of points is adjacent to , where is the number of crash tests.As shown in the preceding sections, the following is true:

It is convenient to use the matrix notation:
The Archives of Automotive Engineering -Archiwum Motoryzacji Vol.83, No. 1,2019 The second method focuses on Legendre polynomials, which are orthogonal function in the domain of .For the purpose of Octave implementation, the set of Legendre polynomials was rescaled and renumbered.Finally, the Legendre polynomials tensor product is considered and the least square method is applied.

Inverse system method description
Assuming that there are given points and a function set .The goal is to find coefficients , that minimize the value of: ( This problem is called the least-square function approximation and is very well covered in the literature [2].The first author of this solution was not established, since both Carl Gauss and Adriena-Marie Legendre were working on the topic.
In order to find the minimum, the function has to be differentiated with respect to obtaining a series of equations It is convenient to use the matrix notation: (4) It is worth noticing that matrix M on LHS ( 2) is symmetric in sense, that for arbitrary .This obvious observation allows to speed up the calculations almost twice.In that stage, the solution of least square problem is down to solving of the matrix and making following observations, on condition that it is inverse: (5) Returning to the original problem of finding the as a function of , the set of points is adjacent to , where is the number of crash tests.As shown in the preceding sections, the following is true: for arbitrary k, m = 1, …M.This obvious observation allows to speed up the calculations almost twice.In that stage, the solution of least square problem is down to solving of the M matrix and making following observations, on condition that it is inverse: The Archives of Automotive Engineering -Archiwum Motoryzacji Vol.83, No. 1,2019 The second method focuses on Legendre polynomials, which are orthogonal function in the domain of .For the purpose of Octave implementation, the set of Legendre polynomials was rescaled and renumbered.Finally, the Legendre polynomials tensor product is considered and the least square method is applied.

Inverse system method description
Assuming that there are given points and a function set .The goal is to find coefficients , that minimize the value of: ( This problem is called the least-square function approximation and is very well covered in the literature [2].The first author of this solution was not established, since both Carl Gauss and Adriena-Marie Legendre were working on the topic.
In order to find the minimum, the function has to be differentiated with respect to obtaining a series of equations It is convenient to use the matrix notation: (4) It is worth noticing that matrix M on LHS (2) is symmetric in sense, that for arbitrary .This obvious observation allows to speed up the calculations almost twice.In that stage, the solution of least square problem is down to solving of the matrix and making following observations, on condition that it is inverse: (5) Returning to the original problem of finding the as a function of , the set of points is adjacent to , where is the number of crash tests.As shown in the preceding sections, the following is true: Returning to the original problem of finding the b k as a function of C s , the set of points , where n is the number of crash tests.As shown in the preceding sections, the following is true: The Archives of Automotive Engineering -Archiwum Motoryzacji Vol.83, No. 1, 2019 4 . ( The solution has the following form: Whereas the function, that best approximates given points is of following form: The rest of the procedure of precrash velocity determination is standard and was already discussed in [22], [24].As a quick reminder, the subsequent coefficients are defined (with emphasis on crash test number dependence): ( The solution has the following form: Whereas the function, that best approximates given points is of following form: The rest of the procedure of precrash velocity determination is standard and was already discussed in [22], [24].As a quick reminder, the subsequent coefficients are defined (with emphasis on crash test number dependence): ( The solution has the following form: Whereas the function, that best approximates given points is of following form: The rest of the procedure of precrash velocity determination is standard and was already discussed in [22], [24].As a quick reminder, the subsequent coefficients are defined (with emphasis on crash test number dependence): The rest of the procedure of precrash velocity determination is standard and was already discussed in [22], [24].As a quick reminder, the subsequent coefficients are defined (with emphasis on crash test number n dependence): The Archives of Automotive Engineering -Archiwum Motoryzacji Vol.83, No. 1, 2019 4 . ( The solution has the following form: Whereas the function, that best approximates given points is of following form: The rest of the procedure of precrash velocity determination is standard and was already discussed in [22], [24].As a quick reminder, the subsequent coefficients are defined (with emphasis on crash test number dependence): where m is the vehicle mass, b gs = 3.05 is the speed limit and L t is the width of dent zone.This yields following relations: The Archives of Automotive Engineering -Archiwum Motoryzacji Vol.83, No. 1, 2019 5 where is the vehicle mass, is the speed limit and is the width of dent zone.This yields following relations: ( 14) and ( 15) The relative error is then calculated by following formula: (16)

Inverse system method results
The database used in this considerations consists of 465 crash tests.Authors have used 80% of the database, and then the results are validated.For 80% the following values are given: (17) therefore, (18) Figure 1 presents the approximation of the inverse method For comparison, Figure 2 presents the same data range, but with linear approximation.
There is no doubt that the nonlinear model is the superior one.Figures 3 and 4 present the relative error for nonlinear and linear method respectively.where is the vehicle mass, is the speed limit and is the width of dent zone.This yields following relations: ( 14) and ( 15) The relative error is then calculated by following formula: (16)

Inverse system method results
The database used in this considerations consists of 465 crash tests.Authors have used 80% of the database, and then the results are validated.For 80% the following values are given: (17) therefore, (18) Figure 1 presents the approximation of the inverse method For comparison, Figure 2 presents the same data range, but with linear approximation.
There is no doubt that the nonlinear model is the superior one.Figures 3 and 4 present the relative error for nonlinear and linear method respectively.

The relative error is then calculated by following formula:
The Archives of Automotive Engineering -Archiwum Motoryzacji Vol.83, No. 1, 2019 5 where is the vehicle mass, is the speed limit and is the width of dent zone.This yields following relations: ( 14) and ( 15) The relative error is then calculated by following formula: (16)

Inverse system method results
The database used in this considerations consists of 465 crash tests.Authors have used 80% of the database, and then the results are validated.For 80% the following values are given: (17) therefore, (18) Figure 1 presents the approximation of the inverse method For comparison, Figure 2 presents the same data range, but with linear approximation.( There is no doubt that the nonlinear model is the superior one.Figures 3 and 4 present the relative error for nonlinear and linear method respectively.

Inverse system method results
The database used in this considerations consists of 465 crash tests.Authors have used 80% of the database, and then the results are validated.For 80% the following values are given: The Archives of Automotive Engineering -Archiwum Motoryzacji Vol.83, No. 1, 2019 5 where is the vehicle mass, is the speed limit and is the width of dent zone.This yields following relations: ( 14) and ( 15) The relative error is then calculated by following formula: (16)

Inverse system method results
The database used in this considerations consists of 465 crash tests.Authors have used 80% of the database, and then the results are validated.For 80% the following values are given: (17) therefore, (18) Figure 1 presents the approximation of the inverse method For comparison, Figure 2 presents the same data range, but with linear approximation.( There is no doubt that the nonlinear model is the superior one.Figures 3 and 4 present the relative error for nonlinear and linear method respectively.

therefore,
The Archives of Automotive Engineering -Archiwum Motoryzacji Vol.83, No. 1, 2019 5 where is the vehicle mass, is the speed limit and is the width of dent zone.This yields following relations: ( 14) and ( 15) The relative error is then calculated by following formula: (16)

Inverse system method results
The database used in this considerations consists of 465 crash tests.Authors have used 80% of the database, and then the results are validated.For 80% the following values are given: (17) therefore, (18) Figure 1 presents the approximation of the inverse method For comparison, Figure 2 presents the same data range, but with linear approximation.( There is no doubt that the nonlinear model is the superior one.Figures 3 and 4 present the relative error for nonlinear and linear method respectively.For comparison, Figure 2 presents the same data range, but with linear approximation.

5
For comparison, Figure 2 presents the same data range, but with linear approximation.
There is no doubt that the nonlinear model is the superior one.Figures 3 and 4 present the relative error for nonlinear and linear method respectively.
There is no doubt that the nonlinear model is the superior one.Figures 3 and 4 present the relative error for nonlinear and linear method respectively.The average error of the inverse method equals to 6.3355% whereas, the linear model produces average error of 7.2675%.
Finally, a plot comparing the nonlinear and linear models is presented in Figure 5.The detailed numerical data is enclosed in the Table 1.

Tensor product method description
Let us assume that there are given points (x n ,y n, z n ) N n=1 and function family (h m ) M m=1 (functions of two variables).Again, the objective is to obtain the coefficients (a m ) M m=1 , which minimize its value.Let us assume that there are given points and function family (functions of two variables).Again, the objective is to obtain the coefficients , which minimize its value.(20) Similarly to section 2, the issue of least square approximation is reduced to a linear solution: Similarly to section 2, the issue of least square approximation is reduced to a linear solution: The Archives of Automotive Engineering -Archiwum Motoryzacji Vol.83, No. 1, 2019 (21) As for the choice of function , Legendre polynomial product tensors are chosen.Consideration involves a sequence of polynomials defined by a iterative formula: Linear error Nonlinear error

Result of tensor product method
The database consists of 465 crash tests.Model is created upon 80% of records and then validated.The program returns following values: 9 ,

Result of tensor product method
The database consists of 465 crash tests.Model is created upon 80% of records and then validated.The program returns following values: The plot of Legendre polynomials tensor product approximation is presented in Figure 6.It is clearly visible that it has big advantage over the linear model presented in Figure 7.The average value of relative error is as presented in Figure 8.Compared to result from section 3, this still shows an improvement in accuracy.Moreover, it is a better result than the linear model, where the relative error was as shown in Figure 9.The plot of Legendre polynomials tensor product approximation is presented in Figure 6.It is clearly visible that it has big advantage over the linear model presented in Figure 7.The average value of relative error is 0.058355 ≈ 5.8355% as presented in Figure 8.Compared to result from section 3, this still shows an improvement in accuracy.Moreover, it is a better result than the linear model, where the relative error was 0.069791 ≈ 6.9791% as shown in Figure 9.  Finally, a comparison of linear and Legendre approach is presented in Figure 10.

Conclusions
As the preceding sections clearly demonstrate, the nonlinear approach shows promising results.The advantage is best visible in the figures 6 and 7, where Authors put emphasis on the data structure itself.That is the crucial aspect when analyzing data and shows evident advantage of the nonlinear over the linear approach.
When comparing these methods to the other papers done by the Authors, it does not show such an extensive improvement.This may indicate, that this particular approach might need some further development in order to achieve comparable or even lower error values.Nevertheless, the improvement is clearly visible, especially when considering the whole spectrum of examined cases.This is due to the fact of applying spline functions, which estimate the EES speed with larger error, although the error is distributed more evenly throughout the analyzed vehicles.
The solution has the following form:The Archives of Automotive Engineering -Archiwum Motoryzacji Vol.83, No. 1, 2019 4 .
Whereas the function, that best approximates given points is of following form:The Archives of Automotive Engineering -Archiwum Motoryzacji Vol.83, No. 1, 2019 4 .

Figure 1 .
Figure 1.Inverse system least square approximation of .

Figure 2 .
Figure 2. Linear least square approximation of .

Figure 1 .
Figure 1.Inverse system least square approximation of .

Figure 2 .
Figure 2. Linear least square approximation of .

Figure 1 .
Figure 1.Inverse system least square approximation of .

Figure 2 .
Figure 2. Linear least square approximation of .

Figure 1 .
Figure 1.Inverse system least square approximation of .

Figure 2 .
Figure 2. Linear least square approximation of .

Figure 1 .
Figure 1.Inverse system least square approximation of .

Figure 2 .
Figure 2. Linear least square approximation of .

Figure 1
Figure 1 presents the approximation of the inverse method.

Figure 1 .
Figure 1.Inverse system least square approximation of b k [ ].

Figure 2 .
Figure 2. Linear least square approximation of .

Figure 4 .
Figure 4. Relative error of linear model.

Figure 5 .
Figure 5. Performance of linear and nonlinear models (inverse system).

Figure 8 .
Figure 8. Value of relative error in nonlinear model.

Figure 8 .
Figure 8. Value of relative error in nonlinear model.

Figure 9 .
Figure 9. Value of relative error in linear model.

Figure 10 .
Figure 10.Performance of linear and nonlinear models (Legendre tensor product).

Table 1 .
Detailed numerical values for inverse system method.

Table 2
presents detailed data of Legendre approach.

Table 2 .
Detailed numerical values of the inverse method.