Trans. Japan Soc. Aero. Space Sci. Vol. 47, No. 157, pp. 209–215, 2004

Collision Avoidance Control Law for Aircraft under Uncertain Information By Hideaki SHIOIRI1Þ and Seiya U ENO2Þ 2Þ

1Þ Air Development Squadron 51, Japan Maritime Self Defense Force, Kanagawa, Japan Graduate School of Environment and Information Sciences, Yokohama National University, Kanagawa, Japan

(Received February 25th, 2004)

This paper describes a collision avoidance problem for aircraft. In a conventional avoidance problem, it is assumed that the target information is certain. However, information may not be always certain, and handling of uncertain information has not been discussed. Therefore, a new control law is proposed to deal with uncertain information and to obtain correct information. The uncertainty depending on position, which is defined in the inertial or relative coordinate system, is dealt with in this paper. To cover each coordinate system, the proposed control law is applied to the ‘corner’ and ‘infog’ problems. Several elements are defined to express uncertainty of target information. Simulation results show that severe avoidance caused by conventional law is improved to obtain satisfactory performance by dealing with uncertain information. Key Words:

1.

Guidance and Control, Collision Avoidance, Uncertain Information

Introduction

Decision-making is based on various information. This situation is the same for avoiding collisions between air traffic. Conventional avoidance problems assume that information about an avoidance object (intruder or other obstacle; called a target in this paper) is correct. Therefore, control law is designed based on certain information.1,2) However, all the information may not be correct and most is uncertain. Nevertheless, there has been no research on control law to deal with uncertain information. On the other hand, from the optimization viewpoint, most discussion focuses on improvement of economy and efficiency3–6) for flight time and fuel consumption. Safety is hardly discussed. Therefore, this paper proposes a new control law that can deal with uncertain information. New parameters quantifying uncertainty of information are defined for this purpose. The proposed control law provides new performance by enabling the aircraft to obtain target information and to check the certainty of the information. This paper deals with the problem that information depends on the evader’s position, and the amount of information varies with the evader’s maneuvering. Uncertainty of position dependence is defined by two coordinate systems: absolute (ground fixed coordinate), and relative (body fixed coordinate). Therefore, two problems are analyzed in this paper: the shadow zone created by an obstacle, and inability to predict beyond a certain distance. This paper shows that introduction of design procedures for uncertain information improves conventional avoidance performance. To evaluate the avoidance performance, evaluation functions are selected from the viewpoint of safety and/or comfort. Simulation results are evaluated Ó 2004 The Japan Society for Aeronautical and Space Sciences


A part of this paper was presented at 40th Aircraft Symposium, October 12, 2002.

using fuzzy logic. 2.

Introduction of Uncertainty

2.1. Uncertain parameters The following parameters quantifying uncertainty are defined to design the new 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. 2.1.1. Uncertainty of existence (1) IP (Information Probability) IP is a parameter describing the probability, possibility or likelihood of the target existence. IP takes a fixed value from 0 to 1 and is assigned by the decision maker before avoidance. When IP ¼ 0, there is no probability of existence. On the other hand, the target existence is certain when IP ¼ 1. (2) IC (Information Clarity) IC is a parameter describing the clarity of target existence. IC varies with the quantity of information. When IC ¼ 0, there is no information about existence. On the other hand, information on target existence becomes clear when IC ¼ 1. When the information has uncertainly, IC varies from 0 to 1 with relative distance between the target and evader. For example, IC varies as shown in Fig. 1(a) when visibility is obscured by fog. Information becomes clear (IC ¼ 1) at a certain distance and worsens gradually with further distance. No information is provided (IC ¼ 0) beyond a certain distance. Figure 1 shows an environment when visibility is just barely secured at 5000 m. The target is found definitely when the relative distance is less than 4000 m. As the relative distance increases, visibility worsens gradually and

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Trans. Japan Soc. Aero. Space Sci. 1

IC 0

4000

relative distance Rr [m]

5000

(a) Information Clearness IC. Assumes visibility of 4000 to 5000 m. Findable target when Rr ≤ 4000. Unable to find target when Rr ≥ 5000. Linear variation when 4000 < Rr < 5000. 1

IA

0

existence zone rE [m]

5000

2.2. Application to control law From the viewpoint of complexity and difficulty, it is wrong to design a new control law adopting uncertainty. Therefore, we propose designing a control law to deal with uncertainty easily by introducing a new technique into the conventional control law with only minor modification. 2.2.1. Uncertainty coefficient As a first step in designing a control law to deal with uncertainty, we consider an uncertainty coefficient k based on the uncertainty parameters of existence and location (IP , IC , IT and IA ). k is a function of these parameters expressed as k ¼ f ðIP ; IC ; IT ; IA Þ

(b) Information Location Accuracy IA. Large existence zone when rE ≥ 5000, and linear variation when 0 < rE < 5000. Fig. 1. Uncertain parameters.

the target cannot be found when the relative distance exceeds 5000 m. (3) IT (Information Truth) IT is a parameter describing the target existence. The value determines whether the target exists or not. It takes a value of either 0 or 1. When IT ¼ 0, there is no target. On the other hand, when IT ¼ 1, there is a target. 2.1.2. Uncertainty of location (1) IA (Information Location Accuracy) IA is a measure of the area in which the target exists. For example, in a 2D model, IA is a circle with a radius of arbitrary length. As shown in Fig. 1(b), it is assumed that IA depends on the radius of the zone containing the target. IA 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. (2) IL (Information Localization) IL represents the ratio of cleared area (SC ) to focused area (SE ). SE is the area of the domain where safety is required. Therefore, SE is determined from the course, velocity and avoidance ability of the evader. Figure 2 shows a cleared domain in part of a focused area with area SC . The shaded part of the focused area is not cleared yet. IL is given by IL ¼ SC =SE

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As shown in Fig. 3, 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. We assume that k has three components as follows, depending on relative distance: part based on original estimate (corresponding to IC ¼ 0), part based on information that gradually becomes clear (corresponding to 0 < IC < 1), and part based on clear information (corresponding to IC ¼ 1). In the situation in Fig. 1(a), when each uncertain parameter is given, the value of k becomes as shown in Fig. 4 explained as follows. k takes a constant value depending on IP and IA when Rr > 5000, because the amount of information quantity does not vary yet. Information gradually becomes clear when 4000 < Rr < 5000. Variation of IC reflects variation of k. k takes either 0 or 1 depending on only IT when Rr < 4000, because the information becomes certain. IT is finally decided whether the target exists or not. However, if it is assumed that IT approaches a true value gradually as the information becomes clear, k is determined in real time. In such a circumstance, control input (angular velocity) !uncert takes a value between target existence and target absence. Therefore !uncert is expressed by the follow-

ω avo (target existence)

ð1Þ

IL takes a value from 0 to 1. When IL ¼ 0, there is no cleared part. On the other hand, the focused area is fully localized when IL ¼ 1. The temporary target location is assumed to be the geometrical center of the unclear part until the target location becomes clear.

ð2Þ

ω uncert (uncertain)

ω notavo (target absence)

evader target

Fig. 3. Flight path and angular velocity with uncertain information.

cleared part SC

1 IP*IA2

k

0

existence zone

remarkable area SE

Fig. 2. Information Localization IL .

4000

5000

relative distance Rr [m]

Fig. 4. Uncertainty coefficient k. Solid line signifies IT ¼ 1 (target existence); dashed line signifies IT ¼ 0 (target absence).

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Shadow Zone v target

T

unclear

cleared vA

(a) IR close to 0.

(b) IR close to 1. evade

Fig. 5. Concept of Information Acquisition Requirement IR .

r

Fig. 6. ‘‘Corner problem.’’

ing equation using k ð3Þ

remarkable area

where !avo and !notavo are the control input to the evader for target existence and target absence, respectively. 2.2.2. Information acquisition requirement value As the second step in constructing a control law to deal with uncertainty, a new parameter IR (Information Acquisition Requirement) is defined. IR is the required value for amount of IL . In other words, IR represents the degree of need to obtain information on the focused area. IR is determined by the decision maker and takes a value from 0 to 1. When IR ¼ 0, there is no requirement. On the other hand, all information is required when IR ¼ 1. As shown in Fig. 5, the information-clear part in the focused area becomes small as IR approaches 0. In contrast, as IR 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 value, IR , design of a new control law to deal with uncertainty is comparatively easy.

CT w

L3T

Shadow Zone corner

x

w

w

L2A CA

y

R2A

evader initial position

Fig. 7. Condition of corner problem.

Simulation

The proposed control law is applied to for two avoidance problems with differences coordinate in a definition of uncertainty. 3.1. Ground fixed coordinate system—‘corner problem’ The first example of an avoidance problem is uncertainty of information defined in the absolute coordinate (ground fixed) system and called the ‘corner problem’ in this paper. 3.1.1. Statement of problem It is assumed that although visibility is good, a target behind obstacles cannot be seen. As shown in Fig. 6, a two-dimensional model that the evader flies on a perpendicular course with constant velocity is considered. There is a blind area (shadow zone) beyond the corner. So the evader cannot see a target that could be in the shadow zone. The course has a constant width but there is no particular outer wall. The model assumes a constant area for the corner with focused area SE . The prospect area is equivalent to cleared part SC , and the shadow zone is equivalent to the unclear part. In this case, provision of target information or not is decided by whether there is a target or not in the shadow zone. Therefore, IC takes the crisp value of 0 or 1. Then, the target flying

safe distance RS

standard path

3.

w origin

target initial position

!uncert ¼ k!avo þ ð1  kÞ!notavo

Table 1. Simulation parameters for corner problem. Variable

Symbol

Value

Initial condition value

vA , vT

250 (m/s)

R2A

(2000, 20000)

Path width

w

5000 (m)

CA

(0, 20000)

Safe distance

Rs

22500 (m)

Velocity

Path origin Corner Information Required

IR

L2A (2000, 20000)

(0, 0)

CT

(5000, 5000)

L3T

(20000, 0) (20000, 3000)

0, 0.9

on a straight course with constant velocity appears from the shadow zone from the right to the left in Fig. 6. 3.1.2. Initial conditions and requirement The initial position of the evader and target are shown in Fig. 7. Detailed values and other constants are shown in Table 1. The focused area is a rectangular domain formed by a course beyond the corner. Safe distance, RS , is measured along two lines. The first is a line extending in the direction of the evader. When the distance between the present position and standard course is less than RS , the second line becomes the standard course. The distance of RS gives the end of the focused area and shadow zone. A safety distance of 3000 to 4000 m is required to avoid a near miss or collision.

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–20 proposed law (IR=0.9) L2A CA R2A

Y Axis [km]

–10

conventional law L2A

0

CA R2A

10

target L3T

20 –10

0

10 X Axis [km]

20

Fig. 8. Avoidance trajectory (corner problem).

5000

4000

Rr [m]

3000

2000 conventional law

1000 target L3T

0 60

proposed law (IR=0.9)

L2A

L2A

CA

CA

R2A

R2A

70

80

90

time [s]

Fig. 9. Relative distance (corner problem).

ω [deg/s]

1

0 proposed law (IR=0.9)

–1

conventional law

–2 0

60

120

evader L2A target L3T

180

240

300

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The avoidance trajectory for the proposed new control law in Fig. 8 shows that the evader finds the target earlier and avoids it smoothly. The evader makes a left turn before the corner to collect information in the shadow zone and to satisfy the required value of IR . On the other hand, in the case of the conventional control law the evaders from CA and L2A are forced to make a left turn with over avoidance. In particular, overshoot is very large in case CA because the evader starts avoiding the target after making a right turn for course tracking. Furthermore, the evader from R2A is forced to avoid the target in the narrow space between the target and wall, because the evader’s view area is worse before the corner. Therefore, appropriate avoidance is hardly achieved by the conventional control law. The relative distance shown in Fig. 9 always satisfies the required separation of 3000 to 4000 m when using the proposed new control law. On the other hand, danger occurs when using the conventional control law because the evader cannot satisfy the required separation and gets very close to the target. The angular velocity for avoidance in Fig. 10 shows that moderate avoidance is achieved using the proposed new control law while large angular velocity like an impulse is required for the conventional control law. 3.2. Body fixed coordinate system—‘in-fog problem’ The second example of an avoidance problem is uncertainty of information defined in the relative coordinate (body fixed) system and called the ‘‘in-fog problem’’ in this paper. 3.2.1. Statement of problem It is assumed that 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. 11. Visibility is defined as a function of relative distance from the evader. When the relative distance is smaller than a certain distance, for example 4000 m, 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 5000 m. Visibility changes gradually between these two areas. The information clearness IC is defined as depending on the relative distance to the target as for visibility (Fig. 1(a)). 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, IP , and the position is given as the target existence zone (circle of radius rE ).

time [s]

Fig. 10. Avoidance angular velocity (corner problem). vA

3.1.3. Simulation results Figures 8, 9 and 10 show the avoidance trajectory, relative distance and avoidance angular velocity, respectively. Solid lines show the results for the proposed new control law for IR ¼ 0:9; dashed lines show the results for the conventional control law.7)

standard course

evader

x y

truth position existence zone (radius rE)

Fig. 11. Condition of ‘‘in-fog problem.’’

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Table 2. Simulation parameters for in-fog problem. Variable

Symbol

Velocity

Value

vA

Uncertain element

Value

250 (m/s)

IP

0, 0.25, 0.5, 0.75, 1

(0, 0)

IC

same as Fig. 1(a)

Initial position Existence zone radius

rE

1000 (m)

IT

1

Center of existence zone

(25000, 2000)

IA

0.8

Target true position

(25500, 2500)

IR

0.2, 0.5, 0.9

3.2.2. Initial conditions and requirement The initial position of the evader and target existence zone are shown in Fig. 11. Other constants are shown in Table 2. Figure 1(a) is used for IC . Also the required separation is set from 3000 m to 4000 m. 3.2.3. Simulation results Figures 12 and 13 show the avoidance trajectory and angular velocity, respectively. Solid lines represent the results for the proposed new 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. 12 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 new control law depend on the value of IR , because IR indicates the degree of

–1.2

conventional law (certain information)

proposed law

Y Axis [km]

(IR=0.2) (IR=0.9)

(incorrect information)

–0.8

–0.4

0.0 10

20 30 X Axis [km]

40

Fig. 12. Avoidance trajectory (in-fog problem) IP ¼ 0:25.

2 proposed law

1

(IP=0.25)

need to obtain information. When IR is larger, the evader must get closer to the target and does not take early avoidance. Therefore, two-stage avoidance occurs when IR ¼ 0:9. The first stage is avoidance based on information and information acquisition; the second stage is avoidance after finding the target. Figure 13 shows the angular velocities for avoidance. Introduction of information uncertainty reduces the sudden and severe avoidance that occurs using the conventional control law with incorrect information. 4.

Evaluation

Fuzzy evaluation is used to confirm the performance and features of the proposed new control law. The evaluation method7) was modified slightly for this paper. 4.1. Evaluation method Evaluation parameters are classified into upper class and lower class and membership function defined for each parameter. The grade of the membership function is called the evaluation grade. As a general rule, simulation results are evaluated using the upper class evaluation grade. However, when there is no clear difference in the upper class evaluation grade, the lower class evaluation grade is referenced. The weight for the lower class depends on the upper class evaluation grade. 4.2. Evaluation parameters Minimum relative distance (Rrmin ) was selected for the upper class evaluation parameter because relative distance to the target is most important in collision avoidance from the safety viewpoint. On the other hand, the maximum angular velocity (j!jmax ) was selected as the lower class evaluation parameter from the viewpoint of flight comfort and moderate avoidance. These membership functions are defined in Fig. 14. 4.3. Evaluation results 4.3.1. Corner problem Figure 15 shows the evaluation result for the corner problem. (a) IR ¼ 0 corresponds to the conventional control law

ω [deg/s]

0 1

1

–1 –2 (certain information)

0

(incorrect information) –3 conventional law

0

3000

4000 Rrmin [m]

60

120

0

1.3

5.0

|ω |max [deg/s]

180

time [s]

Fig. 13. Avoidance angular velocity (in-fog problem) IR ¼ 0:5.

(a) Upper class

(b) Lower class

Fig. 14. Membership function of evaluation parameters.

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Trans. Japan Soc. Aero. Space Sci. target initial position L3T Rrmin

1.0

target initial position C T

0.8 |ω |max

evaluation grade

evaluation grade

1.0

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0.6 0.4

0.8 0.6 |ω |max

0.4

Rrmin

0.2

0.2

0.0

0.0 L2A

CA

R2 A

L2A

CA

0.00

R2A

evader initial position

(a) IR = 0

target initial position CT |ω |max

target initial position L3T

0.8 0.6 0.4 0.2 0.0 L2A

CA

R2A

1.00

Fig. 16. Evaluation results for in-fog problem (IT ¼ 1).

Rrmin

evaluation grade

1.0

0.25 0.50 0.75 Information Probability (IP)

L2A

CA

R2A

evader initial position

(b) IR = 0.9 Fig. 15. Evaluation results for corner problem.

and (b) IR ¼ 0:9 corresponds to proposed new control law. The left and right sides indicate the avoidance evaluation grade from L3T and CT , respectively. No indication signifies zero evaluation. The evaluation grade of Rrmin , the upper class evaluation, shows that extremely high evaluation is provided by the proposed control law when the target approaches from L3T . In contrast, lower evaluations (zero evaluation except L2A ) are provided by the conventional control law for target approaches from L3T . On the other hand, the conventional control law achieves higher evaluation when the target approaches from CT . However, the proposed control law produces no low evaluations. We conclude that a satisfactory evaluation is provided by the proposed new control law because j!jmax values are excellent. Moreover, the results for the conventional control law are worse when the target’s initial position is closer to the wall. By contrast, the proposed control law is unaffected by the target’s initial position, and a stable high evaluation is provided from any initial position. From these evaluation results, when the target’s initial position is close to the wall, the proposed new control law provides good performance because the evader moves to a better position for obtaining information. Therefore, the proposed new control law provides stable avoidance performance with little impact from initial position. 4.3.2. In-fog problem Figure 16 shows the evaluation results for the in-fog problem. The case of IP ¼ 0 corresponds to results with incorrect information using the conventional control law. The case of IP ¼ 1 shows the results with clear information. If the probability used by the proposed control law is almost right, high evaluation that is equal to or better than the

conventional control law (IP ¼ 1) is provided by the evaluation grade of Rrmin . With the conventional control law, it is assumed that information on the target is correct. Therefore, evaluation is extremely low with incorrect information because sudden avoidance is required when the target appears suddenly without information (IP ¼ 0). However, with the proposed control law, satisfactory avoidance performance is achieved in all cases because it is possible to reflect the information probability (IP ) in the control law even when the probability is low. 5.

Conclusion

At present, there is no control law to deal with uncertain information, so we proposed a new control law for uncertain information. The proposed new control law allows the aircraft to move to obtain certain information. Two coordinate systems were considered and the following conclusions were drawn from the simulation and evaluation results. (1) Application to conventional control law The new control law can handle uncertainty without modification of the basic conventional control law. This is achieved by using an uncertainty coefficient obtained from uncertain parameters and an Information Acquisition Requirement value expressing the operator’s decision. In other words, a new control law dealing with uncertainty is achieved just by small modification of the conventional control law. (2) Reflection of individual operator’s demand An Information Acquisition Requirement value expressing the degree of the operator’s demand is proposed. Therefore, the avoidance trajectories differ with each operator and each operator can easily reflect his decision or demand in the new control law. (3) Higher avoidance performance For right information probability, the avoidance performance of the new control law is higher than that of the conventional control law. The probability is reflected in the new control law and provides better avoidance in all cases. (4) Improvement of severe avoidance (higher robustness for incorrect information) When the target appears suddenly from the shadow zone, the conventional control law cannot provide avoidance without dangerous and severe maneuvers. The new control law improves this problem, because it always offers two

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possibilities of the target appearing or not, providing satisfactory avoidance performance in all cases. References 1) Ishihara, K. and Ueno, S.: Collision Avoidance Control Law of Aircraft Based on Risk, Proc. JSASS 11th International Session in 35th Aircraft Symposium, 1997, pp. 615–618. 2) Suzuki, T.: A Note on the Collision Avoidance Problem, Proc. SICE 2001 Nagoya, 2001, No. 311-A-1. 3) Erzberger, H. and Lee, H.: Constrained Optimal Trajectories with

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Specified Range, J. Guid. Control Dynam., 3 (1980), pp. 78–85. 4) Chuang, C.-H. and Morimoto, H.: Periodic Optimal Cruise for a Hypersonic Vehicle with Constraints, J. Spacecraft Rockets, 34 (1997), pp. 165–171. 5) Betts, J. T.: Survey of Numerical Methods for Trajectory Optimization, J. Guid. Control Dynam., 19 (1998), pp. 193–207. 6) Speyer, J. L.: Periodic Optimal Flight, J. Guid. Control Dynam., 19 (1996), pp. 745–755. 7) Shioiri, H. and Ueno, S.: Three-Dimensional Collision Avoidance Control Law for Aircraft Using Risk Function and Fuzzy Logic, Trans. Jpn. Soc. Aeronaut. Space Sci., 46 (2004), pp. 253–261.