1. Introduction
@Recently the formation Flying has become very common and its keeping strategies are now of the great interest in astrodynamics these days. It may be simply true that the optimal station keeping is realized only when every spacecraft information is shared by every member in the formation. This is the case called gfull informedh state and the strategy taken result in a centralized control. However, in case the number of member constituting a formation becomes divergent and the states uncertainty may deform the relative geometry before the entire information is exchanged.
Instead of this centralized control law, it is practical to apply a localized control law that requests the relative motion among a few spacecraft adjacent to each other. In this case, the strategy will be a decentralized one. In the decentralized control, as the information is not shared by every member, the optimal control is impossible and non-optimal transition behavior is observed. The simplest example of this non-optimal transition behavior is the traffic jam phenomenon in highways on the ground. When the head car of platoon rises/slowdown in the speed, the density change phenomenon like compressional wave is observed. This is because that the member cannot use the full information. The partially informed control makes the transition behavior not uniformly converged to the goal and the settle time increases. In this paper, the formation behavior is dealt via a z-transformation method first. This z-formation expression can easily shows the uniform transition behavior of the partially informed formation. Then a uniformly convergent strategy is proposed. Since this strategy is highly flexible, it is applicable to the actual traffic control system and we show to apply this strategy to the traffic control.
2. Formation Expression with Z-transformation
A Z-transformation is a classical analysis strategy for discrete time systems to represent a series of impulses in time domain. The same method is revised and used to express the formation flying information that is inevitably defined as discrete node.
Fig. 1 shows the typical one-dimensional formation such as cars on the highway. The most fundamental expression rule in z-transformation is in impulses. The z-transformation of a single impulse is unit, 1. And here, shifting an impulse backward is done by an operator of 1/z, while forward shifting is by z. Therefore, when a series of constant impulses from the top position at zero is given by
(1)
where, y* denotes the intensity of the impulses that is the interval between the members.
3. One-Dimensional Formation
3.1 First Order System
The most fundamental formation control is the control of the distance intervals between the cars on a highway. Here denote be the interval between the n-1th car and n-th car and be the control input of n-th car. The equation of motion of first order system is
(2)
With z-transformation, this equation can be expression as follows;
(3)
The most straightforward control strategy to keep the distance may be found as
(4)
This is what usual drivers do. The equation of motion can be written as follows;
(5)
The solution of this equation is
(6)
where, shows the initial condition. This solution includes z in the exponential term that can be expanded as follows;
(7)
This equation means that the time constants of members are different and the uniformly convergence is impossible.
Fig.2 presents a numerical simulation in case the target interval is taken zero. This shows how traffic jam grows and the intervals become short gradually. This is not the uniform convergence. The simplest control law to avoid this traffic jam phenomenon is
(8)
Since the closed loop gain portion excuses z variable, the solution is simply
(9)
This transition proceeds uniformly and is ideal. The eq. (8) has the structure of
(10)
This is the structure that the control input of car ahead is added to the control input which is obtained from its own observation.
3.2 Second Order System
Here we consider the second order system. The equation of motion is
(11)
When the distance between the car is y, the equation of motion about the distance can be written as follows;
(12)
Here I introduce the control law which is what usual drivers do,
(13)
This is the control law that uses the relative distance and absolute velocity. The closed loop characteristic is obtained by
(14)
Here I assume and is small as it is driverfs sensitivity to the velocity. Then the following solution is obtained;
(15)
where,
(16)
Since the eq. (15) includes variable, convergence transition behavior is not uniform. The distance between n-th car and (n-1)-th is the coefficient of and expressed as follows;
(17)
This is the function of n and it means that response of car depends on its position in the car train. The eq. (15) represents the structure that propagates the amplitude,
(18)
Fig. 3 shows the histories of distance between cars. This is like compressional wave and the amplitude of backward car is very large. The simplest way to prevent this compressional phenomenon is which implies the distance control is to be relaxed. However it is not the fundamental solution and it means that the compressional phenomenon is inevitable.
Here we introduce the new control law that is like mentioned in previous session.
(19)
The equation of motion is
(20)
Eq. (14) has z in its second term and it propagates the compressional wave like eq. (18). Eq. (20) can be interpreted as
(21)
This means that the control input from its distance error is added to each control input. Note as log as , the stability is concluded guaranteed. Even for the non-linear cases, as long as is satisfied. This means the uniform stability is assured regardless of each carfs skill and performance.
Fig. 4 and Fig. 5 show the traffic control example. Distance between 1st and 2nd cars abruptly extends to 160m and the distance between 5th and 6th cars suddenly becomes 150m. Here is assumed the driving skill is common. Fig.4 shows the result obtained by the new control law show in eq. (21). As this figure shows, the Dramatic improvement is obtained. Small localized oscillation is found but is damped quickly.
Fig.6 shows the other traffic control example. 1st car suddenly speed up 30[m/s] and 2nd car pull in front of 3rd car. These lines indicate Interval Numbers. And vertical axis indicates travel distance[m], abscissa axis indicates the time[s]. The result is locally vibrated but as a result, this carfs behaviors converge.
4. Two-Dimensional Formation
Here we consider two-dimensional formation keeping problem expressed by the Hillfs equation.
Two-dimensional Hillfs equation is;
(22)
This can be converted as the Fig.6 shows;
(23)
where
(24)
The relative motion can be expressed as;
(25)
where . This can be converted via z-transformation as;
(26)
The new control law like previous session is
(27)
This is decomposed by the inverse z-transform.
(28)
This equation shows that the new control law has the structure in which the control input is propagated backward.
Fig.8 shows the trajectories of satellites that are positioned at the origin. New control guarantees uniform convergence and the entire formation is controlled with little delay.
5. Actual Application of New Control Law
In this section, we consider the problems when the new control law is applied to car control.
5.1 Accumulation of Acceleration
Here we show the how the new control law is implemented. As mentioned above, the acceleration is accumulated and relayed backward in the new control law. However, theoretically, the control law using distance from the top car has same performance. In this case, the distance between the cars are measured and accumulated and relayed backward. The important reasons to use not distance information but the acceleration are;
1. Distance measurement is not popular and is not implemented easily.
2. Distance controls are left for human part instead of machine part.
The latter means that the control law can avoid the conflict between the driver and control system.
5.2 Problems in Actual Application
When the Distance controls are left for human part instead of machine part, we should consider the following problems;
1. Each driver has different control gain.
2. In eq. (21), the acceleration that relayed backward contains control effort only for distance maintenance and it is hardly directly available.
Here we consider the stability of the system when the control gains of members are not uniform. The first-order system cam be written as
(29)
The closed-loop system can be written in matrix from as follows;
(30)
where and are velocity error feedback gain and distance error feedback gain, respectively. When we assume that this system is a linear time-invariant system, the characteristic equation is
(31)
This represent that the stability of this system is assured if the each member has stable controller and The stability is little affected by the drivers skills dispersion.
Next we show the strategy that separate the control input for velocity error and distance error. In order to separate the control effort for velocity, the model gain is introduced. Suppose the driverfs whole effort is expressed by , here is given an alternative form in replacing ;
(32)
is the model gain. The detail proof is omitted here, but it is verified that regardless of , the system is stable. This means is selectable by either each driver or existing auto-cruise function.
5.3 Summary of New Control Law
In this section, We suppose is the control input by human and is the relayed acceleration.
(33)
(34)
represent the switch gain of new control law and auto-cruise function, respectively. means that the new control law is available. We can choose arbitrary value between 0 and 1 as and this means that the user can choose the degree of machine control. Letting 0 means that the auto-cruise function is not available.
5.4 Numerical Simulation
We has already shown the simulation result when the driving skill is common in Fig.4 and Fig.5.
Fig.9 shows the control result when . In this case, we assume that the driving skill is common. This figure shows that the satisfactory response is assured even if the is not equal to 1. Fig.10 shows the result when the driverfs skill are not uniform. The stability is little affected by the driverfs skills dispersion and the new control law is very effective. Fig.11 shows the result when the auto-cruise function is invalid. Intermittent distance fluctuation takes place for the low acceleration car. However, the oscillation is damped out.
6. Experiment with Radio-Control Cars
In order to demonstrate the effectiveness of new control law, we conduct experiments with radio control cars.
Fig.12 shows the experimental setup. Each car is equipped with communication device to relay the acceleration. The controller with distance information is equipped instead of driver. The ground station can give the input like disturbance to each car. In this experiment, the rail is placed in a circular pattern. Only distance control is performed and steering control is not performed.
Fig.13 shows the result with conventional control law. The oscillation of distance between the 2nd-3rd car is inspired by the 1st -2nd carfs motion and it is expanded. Fig.14 shows the result with new control law. In this case, the 3rd carfs motion is independent from the motion of 2nd car.
7. Summary
The paper presented the decentralized and localized formation keeping control strategies for both one and two dimensional systems. A new z-transformation approach was developed and successfully derived the relayed and decentralized control strategies. The strategies effectively improve the performance of formation control. This control strategy is easily applicable to the actual traffic control system because the complicated devices such as ranging system are unnecessary.
References
[1]Zhang and Petros, gAutonomous Intelligent Cruise Control Using Front and Back Information for Tight Vehicle Following Maneuvers,h IEEE Trans, Vol48,No.1,1999
[2]Maziar and Edward, gA Decentralized Lateral-Longitudinal Controller for a platoon of Vehicle Operating on a Plane,h NSERC No.A4396,2005.