Investigation of the negative speed’s effect on the base of the braitenberg vehicle

The Braitenberg vehicle has been applied to various areas. However, it’s noticeable that Braitenberg assumes the vehicle’s speed always remain positive, which implies that certain behaviors, such as moving backward is unlikely to be achieved. The purpose of this paper is to investigate the combination of positive speed and negative speed on the basis of the Braitenberg vehicle and analyze its effects on the vehicle’s behaviors. Three functions are used: a simple non-continuous function, a linear function and an inverse proportional function. Trajectories resulting from each are categorized into certain behaviors and discussed.


Introduction
The Braitenberg vehicle was first introduced as a method for simulating animal behaviors [1].Though Braitenberg vehicles are often treated as teaching tools to attract students into robotics, it's been proven that they could benefit not only robotics but also other fields such as neuroscience [2].They could be used to solve real-world problems as well as for theoretical analysis [3][4].The fourth has also been analyzed and showed to follow certain regularities [5].All these studies shown the value and potential of the Braitenberg vehicle.However, Braitenberg vehicle's speed always remains positive.This means the vehicle always moves forward.Even when it simulates animals' fearful behaviors, like vehicle 2, it achieves it through turning instead of retreating.However, many creatures are known to be capable of walking backward, such as for rapid turning [6].Unfortunately, such behavior is unlikely to be simulated with robots that are only capable of moving forward.In contrast, by simulating vehicles with negative speed, we may achieve more complex and more diverse behaviors with relatively simple implementation.And hopefully, the results could also give some inspiration to more advanced applications and research in different fields.
Indeed, more advanced simulations have been developed to simulate more complex behaviors and it may seem unnecessary to keep researching it on the basis of the Braitenberg vehicle.However, one advantage of Braitenberg vehicle is its "simple internal structures" [1].It could be easily implemented using relatively simple functions in various software by people without advanced knowledge of engineering or programming.Therefore, such low requirements allow it to be understood and adopted as a facility for research in various areas easily.Thus, it is valuable to investigate more behaviors while keeping this simplicity as much as possible.Therefore, it is necessary to investigate vehicles' retreating behavior with negative speed.
However, if the speed remains negative, the vehicle only moves backward, displaying a fearful behavior all the time.This is similar to the case where speed is always positive.According to Braitenberg's book and other existing research, in this case, corresponding behaviors are more predictable.Moreover, as the vehicle leaves the stimulus, the interactions between the vehicle and the stimulus may be less preferable as in the real world, obstacles are more likely to appear between the vehicle and the stimulus, which may interfere with the reception of stimulation.Thus, this paper will investigate the combination of positive and negative speed as well as the transition phrase where the speed changes from positive to negative.Since the speed's dependence on the stimulation doesn't follow "the more, the more" or "the more, the less", it's considered a special extension of Braitenberg vehicle 4 [1].

Method
This section introduces and explains the method used in the experiment, including the implementation environment, and functions used to interpret the influence of the stimulus and analyzation strategy.

Environments
The vehicle was implemented in the MATLAB Robotics Playground, which is an easy and convenient tool for robotics simulation.It includes sufficient basic functionalities for the experiments.
The object environment is chosen for the experiment.Although other supported environments could also be used, the object environment provides an advantage.It supports a so-called object sensor that only detects the object in the environment, allowing the vehicle to focus on the desired target and ignore any disturbance from the surroundings.This would simplify implementation by simulating the robots in an ideal environment where target identification would never be impeded by other subjects.On the contrary, other environments, though they may produce a more realistic simulation, will increase the unnecessary difficulty of implementation at the same time.For instance, the obstacle environment only supports sensors that detect both obstacles and the walls, allowing the vehicle to respond to the existence of both the target and the environment.Although certain algorithms may be applied to eliminate the disturbance of the walls, it's unnecessary for this experiment.The paper focuses on the behavior of the vehicle with a combination of positive and negative speed and thus, the simpler the environment, the easier the effect can be observed.Hence, environmental inference, which may complicate the analysis, is not one of the paper's major concerns.Figure 1 shows the setup of the environment.A 6 x 6 arena is created and is divided into four quadrants by the red dash lines.The object is placed at the center of the arena, which is also where the two dash lines intersect.The vehicle has two object sensors facing different directions.Indeed, other research has shown one sensor could be used as well [4].Their strategy is to only reflect the intensity of stimulation received from one sensor on one wheel while keep another's speed constant during the experiment.In this case, the other wheel is considered "extremely dull to stimulus" [4].However, since two sensors do not add extra burdens, this experiment chose to use two sensors and each detection result will be reflected on one wheel.Because of the object sensor's property, it gives a null value when no object is detected within the range.Since it's not this paper's chief purpose to discuss how to handle the null value, each sensor is given a 360degree range of detection.
There are several adjustable variables in this environment: the size of the arena, the position of the object, and that of the vehicle.However, size is considered less important as long as the vehicle can move around without too much disturbance from the environment.The position of the object is determined to be the center of the arena.This is also to reduce disturbance from the environment during the movement.If the object is placed near a corner, when the vehicle approaches, it's more likely that the vehicle may hit the wall or get stuck between the object and the wall.This could interfere with the vehicle's movement and increase the difficulty of observing the vehicle's behavior.Therefore, in this case, the only variable that will be adjusted during the experiment is the initial position of the vehicle.There are mainly two types of positions considered: one is on the axis and the other is in a quadrant.Since the sensor has detection range of 360 degrees, the first and fourth quadrants are considered symmetrical to the second and third quadrants.Thus, only one of them needs to be chosen for placing the vehicle.Moreover, since the only difference between placing the vehicle in the second and third quadrants is which sensor receives stronger stimulation at the beginning, it's redundant to test both.Thus, only one of them will be picked for the experiment.The left part of the horizontal axis, which is also the x-axis, is considered special since when a vehicle is placed on it, it faces the object, and the intensity of stimulation received at the beginning by two sensors will be the same.Therefore, the two locations to place the vehicle will be the third quadrant and the left x-axis.

Function
This section introduces the functions used to reflect the dependence of speed on the intensity of stimulation.two variables only determine the speed the vehicle could have, the experiment is more interested in the last variable that determines when the vehicle experiences the speed reverse.
As it can be seen from Figure 2, the speed remains as a positive constant when the intensity of stimulation is smaller than the transition value m and changes immediately to a negative constant speed when the intensity passes m.This is a relatively simple function.The vehicle won't continuously change its speed according to the stimulation it receives but will only reverse the direction of movement when the stimulation reaches a certain degree.However, since other changes are avoided using this function, the effect of reversing the wheel could be directly observed without disturbance.That's the reason for using this as the beginning of the investigation.

Linear function.
The second function used is a linear function.The equation used is shown below.The variable y means the speed and the variable x represents the intensity of stimulation.There are two adjustable variables: value a which determines the slope of the function and value b which determines the intersection on the y-axis.These two variables together determine when the vehicle's speed reverses.
As it can be observed from the following graph, the speed will continue to decrease as the intensity of stimulation increases, and once it passes the transition point, which is the intersection of the function with the x-axis (marked as a red dot in the graph), the wheels will move in a reversed direction.As the stimulation continues to intensify, the absolute value of the speed will increase.The formula used for the function is shown below.The variable y means the speed and variable x represents the intensity of stimulation.The function has two adjustable variables: the value m that determines the transition point and Vi that determines the acceleration.
The following graph shows the speed dependency on the intensity of stimulation.The speed will first increase when the stimulation is less than that at the transition point.When it passes the transition point, the speed will become negative immediately and then the absolute value will decrease as the intensity of stimulation continues to increase.

Measure and analysis
The intensity of the stimulation in the experiment is represented by the distance between the vehicle and the object.The closer the vehicle gets to the object, the stronger the stimulation.Therefore, the data received from the object sensor will be used to calculate the corresponding speed.
Each function introduced earlier will be tested by adjusting those variables, and a distinct trajectory will be recorded and analyzed.Since the chief goal of this paper is to research the vehicle's behavior, especially when it experiences the reverse of its moving direction, the transition point is set to always be less than the furthest distance the vehicle could achieve from the object.This is because, if the value does not fall into the given range, the resulting dependence of the speed will remain positive all the time.In this case, the vehicle will never experience the desired reverse.Thus, to avoid wasting time on barren trials, the adjustment will be made according to the size of the arena.

Result
Experiments have been conducted and results have been recorded.Analyzation will be discussed in this section.

Non-continuous function
Table 1 contains the data recorded from an experiment using a non-continuous function.The grey cells indicate the vehicle runs into the wall before the reverse could occur because the intensity of stimulation at the transition point is unable to be reached due to the size of the arena.By adjusting the initial position of the vehicle and the value at the transition point, two behaviors could be concluded.
First, when the vehicle does not receive a stimulation intensity that passes through a transition value during its movement, the speed changes will not be observed.As it can be seen from the chart, this occurs when the vehicle's initial position lies in the quadrant.Such behavior is because the vehicle is unable to reach the point where the intensity of stimulation reaches the transition value to experience the reverse.Therefore, during the entire movement, the vehicle only moves in a straight line, displaying an ignoring behavior.On the contrary, if the vehicle experienced a speed reverse, it would be observed to rapidly move back and forth in small increments.For example, when the vehicle faces the target at the beginning, it was observed to move either forward or backward, depending on the initial stimulation received from the object, to reach the transition point.And once it reaches that point, it continues moving rapidly back and forth.This result is not surprising.It can be observed from the Figure 5 that when the intensity of stimulation first passes the transition point, the direction of the wheel will reverse, resulting in the vehicle moving in a different direction.However, this will cause the intensity of the stimulation to pass the transition point again, resulting in the vehicle changing its direction once again and this will then occur repeatedly.In this case, the vehicle is unable to get closer or farther from the target enough to jump out of the recursion and thus, will have this so-called hesitation behavior.

Linear function
For this function, although similar movements were observed, no reversal of direction occurred.Trajectories were still categorized into two types depending on whether the intensity of stimulation reaches the transition point during the movement.When the intensity of stimulation doesn't reach the transition point, the vehicle is observed to move in a straight line until it hits the edge of the arena, displaying the same ignoring behavior.However, when such an approach occurs, the vehicle shows a slow and paused behavior.As the intensity of stimulation gets closer to the transition point, regardless of the direction of the In the quadrant (- In the quadrant (- In the quadrant (- The data collected from this experiment can be seen in the chart in Table 2. Still, the grey cells indicate the vehicle runs into the wall before further behaviors are observed because the intensity of stimulation at the transition point is unable to be reached due to the size of the arena.The final speed is the speed of the vehicle when the sensors eventually stop changing.In general, as the value of the transition point increases, the final speed follows a decreasing trend.
This function proves that, with smoother changes around the transition phase, the vehicle is unlikely to reverse its moving direction.Although, the constant speed is said earlier to be less important in the noncontinuous function, this result shows that it may be worth investigating how large the difference between the two values should be to observe the reverse.Therefore, additional trials are conducted, and the results were recorded in the following chart.The results are collected in Table 3.They indicate that the speed differences at the transition phase do not need to be zero to cause the pause behavior.When the speed difference equals 6 in the chart, the positive constant speed equals 3, which is close to the range of the final speed collected earlier from the linear function experiment.Indeed, in both experiments where the pause behavior is observed, the intensity of stimulation never passes the transition point, meaning the speed does not actually become zero.However, the speed has become so slow that the sensor could not reflect the minor changes in the stimulation received and, thus, the vehicle appears to pause.
Figure 8 shows the relationship between the speed of the vehicle and the time when the pause behavior is observed.It can be seen from the figure that the speed eventually becomes extremely close to zero but remains positive, indicating the reverse does not occur.

Inverse proportional function
The behaviors observed follow a similar pattern but still have some variations because of the property of this function.
The trajectories observed are recorded in Table 4.The grey cells indicate the vehicle showed certain behavior but was interrupted by the object or the walls.Thus, they are less likely to reflect the actual trajectory.Similar ignoring and hesitation behaviors are observed for this experiment as well.The reasons for different behaviors are explained earlier but could also be more directly seen from the chart.When the distance from the object is greater than a certain value, which is related to that at the transition point, the vehicle will show an ignoring behavior.Otherwise, the vehicle will have hesitation behavior.This hesitation is a bit different from the previous one.Because the change in the speed is much greater when the reverse occurs in the inverse proportion function, the vehicle moves back and forth in a greater increment.On the axis (-1.4,0) 4 Hit the wall Figure 9 and Figure 10 show the speed and position of the vehicle when the hesitation behavior is observed.Compared to that resulting from a non-continuous function (see Figure 7), the differences between the peak and the bottom of the graph are significantly larger, and so do the speed variations.Also, because the vehicle experiences a greater range of hesitation, the time interval between adjacent peaks is greater since it takes longer for the vehicle to reverse its moving direction.

Conclusion
In conclusion, the experiment showed several behaviors with a combination of positive and negative speed: ignoring, hesitation and pause behaviors.
The ignoring behavior is partially due to the range of the sensor's detection.Since the sensors are given 360 degrees of detection, the differences between the two sensors received are too minor to reflect significant differences in the speed in this experiment.By adjusting the range of the sensor or using one sensor instead and making the speed of one wheel constant, the vehicle may be capable of more diverse trajectories.
The pause behavior could be considered a special kind of hesitation where the vehicle does not move in two directions repeatedly.And the hesitation behavior is thought to be more interesting and worth further investigation since this behavior is unlikely to be performed by the original Braitenberg vehicle.The range of the hesitation movement is shown to be related to the differences between the positive speed and negative speed when the reverse occurs.When the difference is small, the reversing behavior will be less likely to occur, and the range of hesitating movement will be close to zero.As the differences increase, the range of hesitating movement increases accordingly.
Moreover, as discussed in the method section, in this experiment, only one object is used and the sensors only respond to the object.Therefore, the results only reflect the vehicle's behavior in a unitary circumstance, which is very ideal and unlikely to occur in a real-world implementation.Thus, further

Figure 1 .
Figure 1.Environment setup for the experiment (drawn by the author).

Figure 2 .
Figure 2. Graph of the non-continuous function (drawn by the author).

2. 2 . 1 .
Non-continuous function.The first function is a simple non-continuous function.The function has three adjustable values: the positive constant speed Vp, the negative constant speed Vn, the value m at the transition point, which is the x-axis value where the red dotted line in Figure 2 intersects.Since the first Transition Point / value

Figure 3 .
Figure 3. Graph of the linear function (drawn by the author).
2.2.3.Inverse proportional function.The inverse proportional function is a combination of the previous two functions.It has a non-continuous transition, but the speed doesn't remain constant in other circumstances.

Figure 4 .
Figure 4. Graph of the inverse proportional function (drawn by the author).

VehicleFigure 5 :
Figure 5: Movement explanation for the non-continuous function (drawn by the author).

Figure 6 .
Figure 6.Horizontal position vs time when the vehicle displays hesitation behavior with non-continuous function (drawn by the author).

Figure 7 :
Figure 7: Speed vs time when the vehicle displays hesitation behavior with non-continuous function (drawn by the author).

Figure 6 and
Figure 6 and Figure 7 show the speed and position of the vehicle when the hesitation behavior is observed.The position graph shows the vehicle's movement discussed earlier: first moving forward towards the object and then moving back and forth in small increments.Figure 7 also shows the rapid changes in speed, which indicate the hesitation behavior.To clarify, the scale may seem different from the chart because the analysis software uses a different calibration.

Table 2 :
.1088/1742-6596/2634/1/012012 8 wheel, the vehicle will gradually slow down and the sensors show no changes in the intensity of stimulation.The position at which the vehicle eventually stops is determined by the transition point.The larger the value, the further the vehicle stops from the object.Results of linear function experiment.

Figure 8 .
Figure 8. Speed vs time when the vehicle displays pause behavior with linear function (drawn by the author).

Figure 9 .Figure 10 .
Figure 9. Horizontal position vs time when the vehicle displays hesitation behavior with inverse

Table 3 .
Results of additional investigation on when the pause behavior occurs with non-continuous function

Table 4
Results of the inverse proportional function experiment.