Parameters and constants for the simulation.

## 1. Introduction

The collision avoidance control has been one of the key technology for future transportation. Recently, many unmanned systems are developed in shapes of robots, cars, ships, aircraft, etc. In these environments, proper navigation and control systems including collision avoidance is needed. This paper is on collision avoidance control law for air vehicles under uncertain information. The control law uses information amount as one of the physical parameter for control system.

In the field of guidance, navigation, and control, collision avoidance of automated transportation system has been one of main interest of researchers. Many researches started from collision avoidance of ships (Ciletti et al., 1997) where collision avoidance has been one of the problems due to the increasing demand for the naval transportations. Wide varieties of studies on collision avoidance are treated in fields of robots (Fukuda & Kubota, 1999), cars (Hiraoka et al., 2009a, 2009b) and satellites. Some of these researches treat avoidance problems with the formation control which requires the cooperative information control (Slater et al., 2006; Stipanovic et al., 2007).

In the field of aeronautics, the Traffic alert and Collision Avoidance System (TCAS) has been one of the references for the collision avoidance. TCAS exchanges the information of aircrafts and advises the aircraft to avoid in vertical direction. For the conflicts in collision avoidance control, Frazzoli et al. (2001) have shown feasible strategy to treat the conflict problem. Gates (2009) has proposed rule-based collision avoidance control strategies for real-time online collision avoidance. Miele et al. (2010) has proposed collision avoidance control for case of abort landing with low computational load which can be calculated by on-board computer.

Conventional avoidance problems assume that all information about avoidance (intruders and environments) is certain. Therefore, control law is designed based on certain information. However in real cases, all information may not be correct and most of it is uncertain. These uncertainty of information differes by the relative position of the evader and intruder or the absolute position of intruder. There has been no research on control law to deal with uncertain information. This paper proposes control law that treats uncertain information. New parameters quantifying information amount are defined for this purpose. The proposed control law provides new performance by enabling the aircraft to obtain information and to check the certainty of the information.

Two different cases of numerical simulations are used to investigate the usage of the information amount. The first case defines the problem as the uncertainty of the information changes by the relative position of the evader and the target. The problem treats the case where the amount of infomation changes by relative position, for example, the flight in fog or smoke. The information is clearer as the evader gets closer to the fog. These uncertainties are quantified and used as parameters for collision avoidance control law. The second case defines the problem as the uncertainty of the infomation is given as absolute position. The infomation can or cannot be obtained by the position itself, for example, the flight around urban buildings or moutains. In both cases, the information amount is obtained from focused area assigned by the user. Using the information amount, the control law is designed for safer flight of the air vehicles.

## 2. Information amount

In this study, amounts of information are treated as parameters for the control law. First, the focused area : S_{E} is treated as the region of the area that the user focuses. This area can be large if the vehicle is moving fast or very small if the vehicle is in urban area moving very slowly. The cleared area : S_{C} is the area where the information are certain. In the cleared area, all of the infomation is available, meaning if there is an intruder in that region, the evader can obtain all the infomation of the intruder. In the other hand, the blurred area : S_{B} is the area where infomation is uncertain.

From these parameters of the areas, the infomation amount is derived qunatatively as physical value to be used. One of the important factor used in this paper is information localization : *I*_{L} which is calculated from the amount of cleared area out of the focused area. The *I*_{L} is,

and the schematic image of this areas are shown in Fig.1.

Another important factor for the information amount is information acquisition requirement : *I*_{R} which is requirement of the *I*_{L} for safe flight. The evader selects the proper amount for the vechicle to obtain. The higher the value is, more infomation have to be obtained. In the other hand, evader have to move more when the value is high. This amount can be changed by the requirement of the user.

The infomation amounts can be changed according to the users request and experience. If the evader is moving fast, the *S*_{E} might be large but the *I*_{R} could be small. If the evader is trying to turn the corner, *S*_{E} might be small but the *I*_{R} could be very large. These amounts are similar to that of human sence of avoiding the dangers, which makes this amount unique and useful for colision avoidance.

## 3. Collision avoidance law

The total system of collision avoidance law in this paper consists of three types of control laws. They are actual collision avoidance, information gathering, and cource keeping. They are switched by the risk of collision and amount of infomation obtained. All of the simulation are in 2-dimensions and either acceralation or angular velocity of the vehicle is used as input variables. Fig.2 shows the basic definition of variables and constants used in this paper.

The risk of collision is described numerically for collision avoidance control law. Two values are introduced in this paper. One is Range to Closest Point to Approach: *R*_{CPA}, which is the minimum range between two vehicles when their velocities and directional angles are kept at present value. The closest point is shown in Fig.3. This value indicates the future risk of collision. The other is Time to Closest Point to Approach: *T*_{CPA}, which is time to the range between two vehicles is *R*_{CPA}. Even if *R*_{CPA} is small, it is not necessary to avoid quickly

when *T*_{CPA} is large, in the othere hand, when the *T*_{CPA} is small, the evader have to start the motion very quickly. The collision avoidance control law is derived from the combination of these two risk functions.

*R*_{CPA} and *T*_{CPA} are driven using the relative position and velocity in the body fixed coordinate shown in Fig.3. The *T*_{CPA} is derived as,

Then, the relative position and range at closest point of approach is given as,

The risk function is defined as the following equation.

where *R*_{0} represents the safety range. Collision will occur when *φ*=1. When the minimum range between two vehicle is equal to *R*_{0}, *φ* =0.5. The collision avoidance control law is designed to reduce *φ* less than 0.5 in a period of *T*_{CPA}. These parameters that show the risk of collisions are used for collision avoidance. When the risk is high, the direct collision avoidance control is activated to avoid the collision. One of the examples of collision avoidance control law is described in the following.

The collision avoidance control law satisfies the following equation.

*T*_{C} is time constant that is derived from the following requirement.

where *φ*_{0} is the initial value of risk function. The left hand side of Eq. (6) can be derived from the derivative of Eq. (5) and eliminating the effect of the angular velocity of the evader. The effect of the angular velocity is momentary, where they return to their original values after the avoidance. Let the absolute velocity and angle of direction of the evader be, *V*_{0} and *y*_{0}, and the intruder be, *V*_{1} and *y*_{1}, the relative velocity of the vehicles can be expressed as,

The first factor of the right hand side is the angular velocity of the evader, so by eliminating this factor, the following can be derived by taking the derivative of the relative velocity.

As total, the angular velocity for collision avoidance can be derived as,

where,

This collision avoidance control law activates when the risk is high. In other words, other 2 control laws, infomation gathering and course keeping laws are used when the risk is low. In the following 2 sections, the different types of infomation gathering control laws are introduced depending on the difference of the uncertainties up ahead. The course keeping control law is used to keep the original course, which is not important in this paper, so will not be explained in details.

## 4. Uncertainty depending on relative position

The information gathering control with uncertainty depending on relative position is introduced in this section. The uncertainty depending on relative position stands for the cases where the infomation that can be obtained are defined as function of relative distance to an uncertainty. This is applicable for the flights in the fog or smokes where the uncertainty differs by the distance, closer you are, clear infomation you can obtain. The control target is fixed wing aircraft and the control input is angular velocity. The control law is designed from fuzzy logic to realize the fuzziness of the infomation. First, the additional parameters of the uncertain infomation is explained in this section. Followed by the control law and control results.

### 4.1. Uncertain parameters

The control law uses additional parameters for infomation in this section. The basic parameters were *I*_{L} and *I*_{R} which was explained in section 2. The following parameters quantifying uncertainty are added to design the control law dealing with uncertain information. Uncertainty of information depends on the target existence and location. Therefore, first, uncertainty parameters are defined separately. Then, the uncertainty coefficient for the control law is obtained from the uncertainty parameters for existence and location.

#### 4.1.1. Information probability - *I*_{P}

Infomation probability is a parameter describing the probability, possibility or likelihood of the target existence. *I*_{P} takes a fixed value from 0 to 1 and is assigned by the user before avoidance. When *I*_{P} = 0, there is no probability of existence. On the other hand, the target existence is certain when *I*_{P} = 1.

#### 4.1.2. Information clarity - *I*_{C}

Infomation clarity is a parameter describing the clarity of target existence. *I*_{C} varies with the quantity of information. When *I*_{C} = 0, there is no information about existence. On the other hand, information on target existence becomes clear when *I*_{C} = 1. When the information has uncertainly, *I*_{C} varies from 0 to 1 with relative distance between the target and evader. For example, *I*_{C} varies as shown in Fig. 4 when visibility is obscured by fog. Information becomes clear (*I*_{C} = 1) at a certain distance and worsens gradually with further distance. No information is provided (*I*_{C} = 0) beyond a certain distance. Figure 1 shows an environment when visibility is just barely secured at 5000m. The target is found definitely when the relative distance is less than 4000m. As the relative distance increases, visibility worsens gradually and the target cannot be found when the relative distance exceeds 5000m.

#### 4.1.3. Information truth - *I*_{T}

Infomation truth is a parameter describing truth of the target existence. The value determines whether the target exists or not. It takes a value of either 0 or 1. When *I*_{T} = 0, there is no target. On the other hand, when *I*_{T} = 1, there is a target.

#### 4.1.4. Information location accuracy - *I*_{A}

Infomation location accuracy is a measure of the area in which the target exists. For example, in a 2D model, *I*_{A} is a circle with a radius of arbitrary length. As shown in Fig. 5, it is assumed that *I*_{A} depends on the radius of the zone containing the target. *I*_{A} takes a value between 0 (existence zone is vast) to 1 (existence zone is very small). In summary, the existence zone is the domain where the target may exist.

### 4.2. Application to control law

The control law using the uncertain infomation is introduced. From the viewpoint of complexity and difficulty, it is wrong to design a whole new control law adopting

uncertainty. Therefore, the control law to deal with uncertainty simply by introducing technique to the conventional control law with only minor modification is proposed.

#### 4.2.1. Uncertainty coefficient

As a first step in designing a control law to deal with uncertainty, the uncertainty coefficient *k* based on the uncertainty parameters of existence and location (*I*_{P}, *I*_{C} and *I*_{T}) is brought in. *k* is a function of these parameters expressed as,

As shown in Fig. 6, *k* is a coefficient introduced for the following reasons. Under the conventional control law, target information is certain and the evader flies a course for either target existence or target absence. On the other hand, when the information is uncertain, the evader flies somewhere between target existence and target absence. It is assumed that *k* has three components as follows, depending on relative distance: part based on original estimate (corresponding to *I*_{C} = 0), part based on information that gradually becomes clear (corresponding to 0 < *I*_{C} < 1), and part based on clear information (corresponding to *I*_{C} = 1).

In the situation in Fig. 4, when each uncertain parameter is given, the value of *k* becomes as shown in Fig. 7. *k* takes a constant value depending on *I*_{P} when *R*_{r} > 5000, because the amount of information quantity does not vary in this condition. Information gradually becomes clear when 4000 < *R*_{r} < 5000. Variation of *I*_{C} reflects variation of *k*. *k* takes either 0 or 1 depending on only *I*_{T} when *R*_{r} < 4000, because the information becomes certain. *I*_{T} is finally decided whether the target exists or not. However, if it is assumed that *I*_{T} approaches a true value gradually as the information becomes clear, *k* is determined in real time. In such a circumstance, control input (angular velocity) *ω*_{uncert} takes a value between target existence and target absence. Therefore *ω*_{uncert} is expressed by the following equation using *k,*

where *ω*_{avo} and *ω*_{notavo} are the control input to the evader for target existence and target absence, respectively. Avoidance control depends on the area where the target may exist, thus ω_{avo} is a function of *I*_{A}.

#### 4.2.2. Information acquisition requirement

The *I*_{R} explained in section 2 is now brought in for the control. As explained in section 2, the *I*_{R} is the requirement of *I*_{L} by the user. *I*_{R} represents the degree of need to obtain information on the focused area. *I*_{R} is determined by the user and takes a value from 0 to 1. When *I*_{R} = 0, there is no requirement. On the other hand, all information is required when *I*_{R} = 1. As shown in Fig. 8, the clear region in the focused area becomes small as *I*_{R} approaches 0. In contrast, as *I*_{R} approaches 1, the cleared part becomes large. If the conventional control law is modified by introduction of the uncertainty coefficient, *k*, and the information acquisition requirement, *I*_{R,} design of a new control law to deal with uncertainty is comparatively easy.

### 4.3. Simulation result of relative position - in-fog problem

An example of avoidance problem is uncertainty of information defined in the relative coordinate (body fixed) system is shown. The problem is assumed to be in-fog problem, where there is area where the information is uncertain upahead.

#### 4.3.1. Statement of problem

The evader cannot see the target beyond a certain distance because visibility is obscured by an obstacle like fog. The problem is defined as two-dimensional in the horizontal plane. The evader flies on a straight course with constant velocity towards a target that may exist in existence zone as shown in Fig. 9. Visibility is defined as a function of relative distance from the evader. When the relative distance is smaller than a certain distance, for example 4000m, the evader can see the target clearly. However, the evader cannot see the target when the relative distance is larger than a certain distance, for example 5000m. Visibility changes gradually between these two areas. The information clearness *I*_{C} is defined depending on the relative distance to the target as for visibility as explained in Fig.4. The target is close to evader’s course, but information about existence and position are uncertain. Therefore, the target existence is given as the information probability, *I*_{P}, and the position is given as the target existence zone (circlular region with radius of *r*_{E}).

#### 4.3.2. Initial conditions and requirement

The initial position of the evader and target existence zone are shown in Fig. 9. Other constants are shown in Table 1. Figure 4 is used for *I*_{C}. Also the required separation between the evader and the target is set from 3000m to 4000m.

#### 4.3.3. Initial conditions and requirement

Figures 10 and 11 show the avoidance trajectory and angular velocity, respectively. Solid lines represent the results for the proposed control law; dashed lines represent the results for the conventional control law. The figure shows two cases for the conventional control law: avoidance with correct information; and avoidance with incorrect information where target appears suddenly without information.

The avoidance trajectories in Fig. 10 show that avoidance using the conventional control law with incorrect information causes significant delay because the evader does not avoid until the target is found. On the other hand, the avoidance trajectories produced by the proposed

Parameters | Symbols | Values | Uncertain parameters | Values |

Velocity | vA | 250[m/s] | IP | 0.25 |

Initial position | (0, 0) | IC | Shown in Fig. 4 | |

Radius of existence zone | rE | 1000[m] | IT | 1 |

Center of existence zone | (25000, 2000) | IA | Shown in Fig.5 | |

True position of target | (25500, 2500) | IR | 0.2, 0.9 |

control law depend on the value of *I*_{R}, because *I*_{R} indicates the degree of necessary information. When *I*_{R} is large, the evader must fly closer to the target and does not take early avoidance. Therefore, two stage avoidance occurs when *I*_{R} = 0.9. The first stage is infomation gathering based on current infomation and information acquisition; the second stage is avoidance after finding the target. Figure 11 shows the angular velocities for avoidance. Information uncertainty reduces the sudden and severe avoidance that occurs using the conventional control law with incorrect information.

Both of the figures show that the proposed control law was able to increase the safetiness and reliability of the flight in uncertain information defined in relative position from the evader.

## 5. Uncertainty depending on absolute position

In this section, the uncertainty depending on absolute position is treated. Different from the uncertainty that differs by relative position as explained in section 4, the information does not change due to the environment. For example, when aircraft is going around a mountain or a helicopter going around the buildings, the information does not change due to relative position.

The information parameters *I*_{L} and *I*_{R} which was explained in section 2 is used in this section to see the effect of the information amount. The collision avoidance control explained in section 3 is brought in when the vehicle enters critical condition, otherwise the following either infomation gathering control or course keeping control takes place.

### 5.1. Design of information gathering control law

Design of information gathering control law is derived using a model in Fig.12 and Fig.13. Figure 12 is the vehicle in the ground fixed coordinate and Fig.13 shows the vehicle in body fixed coodinate. Focused area depends on the speed and direction of the vehicle, so the distance *x*_{P} can be set by the users descision. The angle *θ*_{P} also depends on the level of safety which can be chosen. The area of focused area is given as follows.

The shadow area cannot be seen from the vehicle. So, the area of shadow area is assumed as the following equation using the focused area.

where,

*x*_{A} and *y*_{A} are relative coordinate in the body fixed frame. The time derivatives of *x*_{A} and *y*_{A} are given as follows.

where *ω*_{0} and *V*_{0} are angular velocity and velocity of the vehicle, respectively. To derive the dynamical property of information amount, time derivative of the *I*_{L} is derived.

*S*_{B} and *S*_{E} are given in Eqs.(15) and (16). It is necessary to define the *x*_{P} and *θ*_{P} of the focused area in order to derive the time derivative. The distance of focused area *x*_{P} is proportional to the velocity of helicopter *V*_{0}.

where Δ*t* is a constant with the unit of time. This shows that the vehicle moves the distance of *x*_{P} in the period of Δ*t*. The angle *θ*_{P} is set as constant, thus the vehicle focuses wide area in case of high speed.

The time derivatives of the *S*_{B} and *S*_{E} are derived.

where,

*a*_{x} and *a*_{y} are horizontal acceleration in the body fixed frame. Substituting Eq. (21) into Eq.(19), the time derivative of *I*_{L} is given.

where,

Therefore, the information amount of safety can be changed by *a*_{x} and *a*_{y}. This shows that the direction of acceleration changes the information amount. The information gathering control law is required to keep the information amount higher than the specified value *I*_{R}. Thus the shortage of information amount is defined in the following equation.

In the case of *I*_{L} is less than *I*_{R}, the controller is required to increase *I*_{L}. On the other hand, in the case of *I*_{L} is greater than *I*_{R}, the high decreasing rate of *I*_{L} is not desired. Therefore the following control law satisfies the both cases.

where,

*k*_{E} is a feedback gain. The left hand side of Eq. (26) is given in Eq. (23). Equation (23) has two input variables, *a*_{x} and *a*_{y}. Thus the minimum norm of input vector is chosen.

where,

This control law uses the same feedback gain in the cases of *I*_{R}>*I*_{L} and *I*_{R} < *I*_{L}. The vehicle is required to keep the desired velocity and direction when the information amount satisfies the requirement. In this case, the following feedback law is used.

where *k*_{V} and *k*_{ψ} are feedback gains. Finally, the information gathering control law is given as follows.

The *İ*_{L} in the inequality conditions are given in Eq. (26). The control law is called Information Amount FeedBack (IAFB) for it feeds back the information amount as one of the parameters for information gathering.

### 5.2. Simulation result of absolute position

The simulation result using the collision avoidance control with IAFB is introduced. Two different cases of similation will be shown in this scetion. The first case is the case with the helicopters. The velocity of the vehicle can be changed directly by the control law. The second case is the case with the fixed wing aircraft. The input is given as the angular velocity and the velocity itself is kept as constant.

#### 5.2.1. Simulation result of helicopters

Figure 14 shows the initial condition of the evader and intruder. The intruder cannot be seen from evader at beginning of the control. The intruder is incoming from behind the obstacle with velocity of 10[m/s] and 20[m] away from the obstacle. The evader starts from 150[m] away from the obstacle with various position defined by *y*(0). Figure 15 shows the simulation result of the avoidance for different initial conditions. Figure 16 shows the case without IAFB for comparison.

In the first half of the control, the evader starts to obtain the information behind the obstacle. After the intruder is found approaching, the evader decides to evade either in front or back of the intruder depending on the estimated trajectory of the intruder passing in the way. In this case, the results were split into exactly two groups where evader decelerates and passes behind the intruder or accelerates and passes in front of intruder. In Fig.16, the case without IAFB, the first half of the information gathering does not occur, so the helicopter avoids the intruder after they find the incoming vehicle. Figure 17 is comparison of the minimum distance when the two vehicles pass each other. The result with IAFB shows higher level of avoidance due to the earlier motion of gathering information which leads to easier avoidance and faster recognition of the intruder.

#### 5.2.2. Simulation result of fixed wing aircraft

For fixed wing type aircraft, it is not easy and efficient to change the velocity so often. The control input for these types are changed to angular velocity input. Basic input is same as the one described in section 5.1. Most of the conditions are same as that of the case of the helicopters except that the cruising speed of evader and intruder is 100[m/s] and the results are compared with different *I*_{R} and different course of the intruder.

Figure 18 shows the trajectory of evader with different *I*_{R}, in the case of *I*_{R} =0 this is same as the case without IAFB, we can see that the trajectory changes by the amount of information required.

Figure 19 shows the time history of the relative distance of the vehicles. The results show that the relative distance decreases very quickly in the case without IAFB and the minimum distance between the vehicles are shorter than the others. This clearly shows the effectiveness of the IAFB.

Figure 20 shows the trajectory of evader when the intruder starts from different positions. The *I*_{R} is set as 0.9 for this simulation. In the cases when the intruder is far from the obstacle, the trajectory is smoother because the intruder is found quicker. The other two cases makes sharp turns due to the slower finding of the intruder.

## 6. Conclusion

Two cases of collision avoidance control is simulated to see the effect of the information amount as parameter for control. One was that uncertainty of the information changes by the relative position of the evader and the target and the other was that uncertainty of the infomation is given as absolute position. Both cases have shown smoother and safer trajectories than the conventional control laws. The simulation results have shown that the control laws using information amounts does not rely on the coodinates. The motion of the aircraft show similar trajectories to that of humans to obtain safe margin to gain information when they do not have enough information.