Analysis of traversable pits model to make intelligent wheeled vehicles

In this paper, the issue of passing wheeled vehicles from pits is discussed. The issue is modeled by defining the limits of passing wheeled vehicles. The proposed model has been studied based on changes in the effective parameters. Finally, in order to describe the problem, the proposed model has been solved for wheeled vehicles based on the effective parameters by using one of the numerical methods.


Introduction
Today, almost a mathematical model is first determined for solving any problem.Then, the model is tried to solve using the existing methods or developed approaches.[1][2][3][4]6] About passing the ground vehicles from pits, the failure of the vehicle in contacting with two kinds of barrier should be considered.A failure mode is HUF and it typically occurs when a vehicle wants to pass a barrier that causes contacting with the ground (barrier).Another type of failure is NIF and it typically occurs when a vehicle falls in a pit and its front contacts with the ground.Therefore, in order to pass the vehicle from the pit, these two types of failure of must be removed.
Production and hosting by ISPACS GmbH.http://www.ispacs.com/journals/jsca/2017/jsca-00100/International Scientific Publications and Consulting Services In this study, the stages to obtain traversable pits model of wheeled vehicles has been presented.Importance and necessity of doing this study include: 1-Excavation and construction projects of mountainous areas roads: According to the facilities of wheeled vehicles in the project of excavation and construction of roads, understanding the pits of the path and creating the traversable pits can speed up the implementation of the project.

Militaryissues:
By recognizing the facilities of wheeled vehicles of the enemy, insurmountable pits can be designed for their vehicles.3.For private vehicles, it can be determined that what pits are traversable and what pits are in surmountable.
The proposed model has been studied based on changes in effective parameters.Finally, in order to describe the above mentioned problem, the proposed model has been solved for wheeled vehicles based on the effective parameters using one of numerical methods.This study has been performed regardless of power (force) of wheeled vehicles.Therefore, the future research, this subject (detection of traversable pits) will be studied by creating a team including researchers in the fields of electronics, mechanics and computers, manufacturing sensors to make intelligent devices.The rest of this paper is organized as follows.Section 2 briefly reviews the basic definitions and concepts related to the subject.Section 3 shows the proposed traversable pits model with the analysis.Section 4 applied the proposed issue in the actual examples.Finally, Section 5 concludes the paper.

Preliminaries and notations
This section reviews some basic definitions and concepts related to the subject as follows.

Definition 2.1 [7] Bisection (or interval halving) method
The Bisection (or interval halving) method is an algorithm for locating the real roots of a function.
The objective is to find two values of x, say 1 x and 2 x , so that x , we can find () m fx .Then, the following decisions are made: 1.If () m fx and 1 () fx have the same sign, their product will be positive, that is, This indicates that m x and 1 x are on the left side of the x−axis crossing as shown in Figure 1.
In this case, we replace 1 x with m x .2. If () m fx and 1 () fx have opposite signs, their product will be negative, that is, This indicates that m x and 2 x are on the right side of the x−axis crossing as in Figure 2. In this case, we replace 2 x with m x .After making the appropriate substitution, the above process is repeated until the root we are seeking has a specified tolerance.To terminate the iterations, we either: a. specify a number of iterations b. specify a tolerance on the error of () fx Bekker [5] classifies two failures: hung up failures (HUF) and nose in failures (NIF).The first one occurs when the obstacle collides with the belly of the vehicle when it is trying to climb over it.NIF happens when the nose of the vehicle touches the ground, before the wheels.Placing the wheels in front of the nose, makes a NIF impossible.

Definition 2.2. Hang-up failure (HUF)
HUF is defined contacting the front and rear axles of the vehicles with the ground (barrier).

Definition 2.3. Nose-in failure (NIF) NIF is defined contacting the front of the vehicles with the ground (barrier).
3 A model of the traversable pits with the analysis

A model of the traversable pits
About passing the ground vehicles from pits, the failure of the vehicle in contacting with two kinds of barrier should be considered.A failure mode is HUF and it typically occurs when a vehicle wants to pass a barrier that causes contacting with the ground (barrier).Another type of failure is NIF and it typically occurs when a vehicle falls in a pit and its front contacts with the ground.Therefore, in order to pass the vehicle from the pit, these two types of failure of must be removed.Consider the following hypothetical pit: Obviously, if  = 90 0 and  1 = 0 0 , then, HUF and NIF occur respectively.The appropriate pit means passing from NIF and HUF that arises by an appropriate reduction of from 90 0 and an appropriate increase of angle of  1 from 0 0 .
In the problem of traversable pit, the target is calculation of the appropriate angle of with having the angle of larger than 0 0 .
The following figure shows the NIF components of a vehicle.We assume that is the maximum angle of the vehicle when it is in a pit, if is the maximum angle in which HUF does not occur.In this case, the components of NIF are applied in the following equation: 2) The above equation can be as follows: where, The proposed mathematical model of a problem is applicable when an solution can be provided for the model in the numbers domain.Hence, by using the following theorem, we show that the proposed equation has an solution.According to the theorem, nonlinear equation of the proposed model gives approximation of using the numerical method, that ultimately, we consider  larger than  .

Analysis of the proposed traversable pits model
For each vehicle, the proposed model has certain parameters of L, D, h and 1  that the parameters of h, D and 1  are changeable.In this section, the problem sensitivity corresponding to the effective parameters is presented theoretically.

Changes of D:
If D changes to , the changes of coefficients of Equation (3.3) will be as follows: where, ( 1). 2 The following theorem is provided to applicable look into equation (3.8).

Theorem 3.2. (Sensitivity to changes of D
and Lack of HUF, we conclude that, To prove the (ii), It is enough if we show that, ˆ( ) 0 D f   .Because, in this case, we have, ˆ( ) 0 , (0) 0 Suppose for the sake of contradiction that, ˆ( ) 0

Changes of h:
If h changes to hh   , the changes of coefficients of Equation (3.3) will be as follows: http://www.ispacs.com/journals/jsca/2017/jsca-00100/International Scientific Publications and Consulting Services where, Similar to the above, the following theorem is provided to applicable look into equation (3.13).

Theorem 3.3. (Sensitivity to changes of h)
i) Proof.Similar to the proof of the theorem (3.2).

Changes of h, D:
Similar to the above, the following theorem is provided to applicable look into equation (3.17).
Similar to the above cases, the sensitivity of the parameters of

Actual examples
In this section, the proposed model for several samples is presented and the appropriate angle is obtained using the numerical method of bisection (Using MATLAB) in the following modes: 1.The desired mode, 2. The sensitive mode (based on changes of effective parameters).It should be noted that, the nonlinear equation of the proposed model gives approximation of using the numerical method, that ultimately, we consider  larger than  .

Conclusion and future research
In this paper, the issue of passing the wheeled vehicles through the pits has been presented.A bout passing the ground vehicles through the pits, the failure of the vehicle in contacting with two kinds of barrier has been considered: A failure mode is hang-up failure (HUF) and it typically occurs when a vehicle wants to pass a barrier that causes contacting with the ground (barrier).Another type of failure is nose-in failure (NIF) and it typically occurs when a vehicle falls in a pit and its front contacts with the ground.Therefore, in order to pass the vehicle through the pit, these two types of failure of must be removed that leads to a nonlinear equation.The proposed model has been studied based on changes in effective parameters.Based on the changes of fundamental parameters, in the case of reducing the radius of the wheel, reducing the height of the pit, reducing the radius of the wheel and pit height at the same time, increasing the radius of the wheel and decreasing the height of the pit simultaneously, and increasing the slope angle in front of the pit, the maximum angle of the vehicle in a pit is decreased.In other cases including increasing the radius of the wheel and reducing the height of the pit simultaneously, and reducing the slope angle in front of the pit, the maximum angle of the vehicle in a pit is increased.The obtained information make more efficient the use of wheeled vehicles in excavation and construction projects of mountainous areas roads and military issues.Finally, in order to describe the above mentioned problem, the proposed model has been solved for wheeled vehicles based on the effective parameters using one of numerical methods.This study has been performed regard less of power (force) of wheeled vehicles.Therefore, the future research, this subject (detection of traversable pits) will be studied by creating a team including researchers in the fields of electronics, mechanics and computers, manufacturing sensors to make intelligent devices.

Figure 1 :
Figure 1: Sketches to illustrate the bisection method when and have same sign

Figure 2 :
Figure 2: Sketches to illustrate the bisection method when and have opposite signs

Figure 4 :
Figure 4: The NIF components of wheeled vehicle on the hypothetical pit
If h and D changes to hh   and respectively, the changes of coefficients of Equation (3.3) will be as follows: D D   http://www.ispacs.com/journals/jsca/2017/jsca-00100/International Scientific Publications and Consulting Services