Tracking Control for Multiple Trailer Systems by Adaptive Algorithmic Control

In recent years, a truck-trailer system is the most useful physical distribution system. The truck-trailer systems have more convenience than coastal services or freight trains. Meanwhile, problems of the traffic jam and the air pollution in an urban area have become serious, year after year. Therefore improvement and rationalization of the transport efficiency are social needs. There are many papers suggesting a platoon system of several trucks as a part of development of ITS (Intelligent Transport System). These platoon systems consist of several unmanned trucks automatically following a truck driven by a conductor, and it is commonly believed that it brings improvements of energy efficiency along with alleviation of the traffic jam. Moreover, there is a purpose of covering insufficient workforce of truck drivers who have to do severe labors, too. In the platoon, trucks are not physically connected to each other, and thus there is much flexibility. On the other hand, even if each vehicle is physically connected by mechanical linkage, this is not important restrictions, for transport robots which are operated in the factory, because moving range and action plan are limited. Moreover, the multiple trailer system is safer than platoon system, because if each vehicle is physically connected, there is no danger of collision among trailers. In this paper, we deal with a control method for a physically connected multiple trailer robot, which is a transport system in factories. The control method of connected vehicle has been studied for a long time (Laumond, 1986). In particular, there are many papers which studied controlling its backward motion with guaranteed stability (Sampei & Kobayashi, 1994). Moreover, kinematical model of a multiple trailer system is described by a nonholonomic system, and it is a controllable nonlinear system (Hermann & Krener, 1977). In theoretical field, it has been a hot subject of research, because asymptotic stabilization is impossible using one continuous time-invariant since the nonholonomic system does not satisfy the Brockett's necessary condition for stabilizability (Brockett, 1983). Therefore, the control problem of nonholonomic system is a theoretically difficult problem, thereupon various researches such as time-variant controller (M'Closkey & Murray, 1993) or hybrid control techniques (Matsune et al., 2005) are performed. We look at this issue from more practical point of view, then investigate a real-time control algorithm, which is based on the so called algorithmic control (Kobayashi et al., 2005a), (Imae et al., 2005) with a similar formulation of the model predictive control (MPC)


Introduction
In recent years, a truck-trailer system is the most useful physical distribution system. The truck-trailer systems have more convenience than coastal services or freight trains. Meanwhile, problems of the traffic jam and the air pollution in an urban area have become serious, year after year. Therefore improvement and rationalization of the transport efficiency are social needs. There are many papers suggesting a platoon system of several trucks as a part of development of ITS (Intelligent Transport System). These platoon systems consist of several unmanned trucks automatically following a truck driven by a conductor, and it is commonly believed that it brings improvements of energy efficiency along with alleviation of the traffic jam. Moreover, there is a purpose of covering insufficient workforce of truck drivers who have to do severe labors, too. In the platoon, trucks are not physically connected to each other, and thus there is much flexibility. On the other hand, even if each vehicle is physically connected by mechanical linkage, this is not important restrictions, for transport robots which are operated in the factory, because moving range and action plan are limited. Moreover, the multiple trailer system is safer than platoon system, because if each vehicle is physically connected, there is no danger of collision among trailers. In this paper, we deal with a control method for a physically connected multiple trailer robot, which is a transport system in factories. The control method of connected vehicle has been studied for a long time (Laumond, 1986). In particular, there are many papers which studied controlling its backward motion with guaranteed stability (Sampei & Kobayashi, 1994). Moreover, kinematical model of a multiple trailer system is described by a nonholonomic system, and it is a controllable nonlinear system (Hermann & Krener, 1977). In theoretical field, it has been a hot subject of research, because asymptotic stabilization is impossible using one continuous time-invariant since the nonholonomic system does not satisfy the Brockett's necessary condition for stabilizability (Brockett, 1983). Therefore, the control problem of nonholonomic system is a theoretically difficult problem, thereupon various researches such as time-variant controller (M'Closkey & Murray, 1993) or hybrid control techniques (Matsune et al., 2005) are performed. We look at this issue from more practical point of view, then investigate a real-time control algorithm, which is based on the so called algorithmic control (Kobayashi et al., 2005a),  with a similar formulation of the model predictive control (MPC) technique for nonlinear continuous time system. Our algorithmic design approach is a technique for ensuring robustness by adopting a numeric solution called Riccati Equation Based (REB) algorithm using quasi linearization that includes feedback solution. Moreover, though details are described later, the control technique by algorithmic design which we proposed is an effective method for nonholonomic systems because our method is switching and applying the control strategy on a short control interval and thus the controller is discontinuous time variant, which does not violate Brockett's theorem. We showed the effectiveness of proposed method applicable to nonholonomic systems through some simulations and an experiment with a differential-driven unicycle vehicle model (Kobayashi et al., 2005b). Then, we extend our design method by incorporating numerical robustness for disturbances and parameter uncertainties and, by focusing on the switching interval of control strategy on iterative process of algorithmic design (Kobayashi et al., 2006). We discussed about effectiveness of our approach for an unstable motion control of high order nonlinear system, in this paper. In the most of conventional research, the direct-hooked type model (Lee et al., 2001) is treated. The direct-hooked model can be transformed to a canonical form called chained form (Murray & Sastry, 1993). Then, control problem for the direct-hooked model can be reduced to a canonical problem. However, the direct-hooked model has a tracking error of follow-on trailers (Fig.1). Therefore, there are many suggestions for eliminating the tracking error by model constructions or mechanical linkage design. We pick up a off-hooked model (Lee et al., 2004) which has a most simple structure and cannot be converted to canonical form (Ishikawa, 1993). Therefore, proposed algorithmic design is considered as an effective strategy for the off-hooked trailer system, because our approach can treat the general nonlinear systems. The effectiveness is discussed through a numerical simulation result. The outline of this paper is as follows. In section 2, we describe the nonlinear optimal control problems and the Riccati Equation Based algorithm. In section 3, the algorithmic design method is described in detail. Also, we make an extension of our design method for robustness. The backward motion control problem of multiple trailer systems is formulated in section 4. In section 5, we show some simulation results in order to demonstrate the effectiveness of adaptive algorithmic design. Section 6 concludes the paper.  (2) where 0 t is initial time, 0 x is initial state given. Here, we denote the state variable by tt.
Based on the problem formulation (1) to (2), we describe our on-line computational design method, that is to say, algorithmic design method (Kobayashi et al., 2005a). It is known that whether or not the algorithmic design method succeeds depends on how effective the algorithm is to iteratively search the numerical solutions of optimal control problems. In this paper, we adopt one of the so-called Riccati-equation based algorithms (REB algorithms (Imae & Torisu, 1998)), which is known to be reliable and effective in searching numerical solutions. Details are given later.

Riccati-equation based algorithm
Under the problem formulation (1) to (3), we describe an iterative algorithm for the numerical solutions of optimal control problems, based on Riccati differential equations. In this respect, the algorithm falls in the category of optimal control algorithms, as presented in (Nedeljkovic, 1981), (Imae et al., 1992), and so on.
[ Assumptions ] where the vertical bars are used to denote Euclidean norms for vectors. Now, we make some assumptions. ii. For each compact set for all 1 t ∈ℜ , n x ∈ℜ and uU ∈ . www.intechopen.com
STEP A7 Set 1 ii =+, and go to Step A2. Repeat Step A2 to Step A7 until the performance index J converges. Here, the integer i represents the number of iterations.

Real time control technique
In this section, we describe the outline of the algorithmic design for real time control of nonlinear system. See , (Kobayashi et al., 2005a) for more details. The basic idea of this real-time control design is the control strategy N u is executed one by one through N iterations of the above-mentioned REB algorithm from Step A2 to Step A7. In this design method, the controller is not needed in an explicit expression, and the control strategy is decided repeatedly by the REB algorithm. After the actual states are observed, the states of the next T Δ seconds from now are predicted by the state equation (1). Then, with the predicted states set as initial states, we obtain the next control strategy N u by N iterations of the REB algorithm from Step A2 to Step A7. Through sufficiently large number of iterations N , it could be expected to eventually reach the possible optimal solutions. However, the value of N should be decided for the iterative processing to end in the T Δ [sec]. We here describe how the algorithmic controller works. See also figure 1. Here, the feedback structure of the solution in  and ( . Then, we obtain the next control strategy N k u 1 + . If the rate of the value of performance index is less than a sufficiently small value γ , that is if following inequalities are satisfied, stop the iteration because it seems that the optimal solution was obtained.
, and go to Step B2.

States
x a1 x a2 Predicted Actual x p1 x  In our previous works, we verified the effectiveness of our algorithmic approach by applying to various nonlinear systems. For example, we tried a swing-up problem of inverted pendulum, or the obstacle avoidance problem for a unicycle robot. As a result, our approach gave the effective solution for these problems. The backward motion control problem for the multiple trailer system that we treat in this paper is a more difficult problem, because the system is a higher order nonlinear system. In spite of these difficulties, we confirmed the effectiveness of our algorithmic approach for such a complex problem through some numerical simulations. However, it is necessary to select carefully ΔT and N that are the design parameters of this algorithm. In the case of including disturbance, the feasibility of the algorithm depends on the combination of ΔT and N. For reducing the complexity of the method of deciding these design parameters, a simple way of computational artifice is shown in the next section. The simulation result is described in section 6.

Algorithmic design incorporating computational time
In this section, a simple computational artifice of the above-mentioned algorithmic design is pointed out. First, we describe the key notes here. In the above-mentioned algorithm, the interval of time T Δ to apply one control strategy N k u is called "switching time". And the maximum number of the iteration executed in a switching time N is called "maximum www.intechopen.com iteration". When the state was predicted, the obtained state trajectory is called "predictive trajectory" and actual trajectory is called "trajectory". In our algorithmic design, the computation of maximum iteration should be done in switching interval. The search process of the optimal solution is executed in this algorithm, and the required computation time depends on the state. Therefore, it was necessary to give some margin to the switching interval. If the maximum iteration is sufficiently large, it may obtain an optimal solution in each switching interval. However, the switching interval has to set to large, because long computation time is required. Because the feedback effect is obtained by observing each switching interval, it seems that if the switching interval is as short as possible, the performance of robustness is better. The key idea of the algorithm which we propose here is to treat the switching interval as varying. It increases the maximum iteration when time is required for searching the optimal solution, and the switching interval is increased along with it. On the other hand, when long time is not required to find the optimal solution, reduce the maximum number of iteration and the switching interval for improving the robustness. The maximum iteration is decided based on Fig.2   x .

STEP C3
The maximum iteration is k N , and calculate the rate of the value of performance index in each iteration, similarly as the computation from Step A3 to Step A7 (1 , 2 ,, ) k iN = . STEP C4 If the rate of the value of performance index is larger than a sufficiently small value γ , that is if following inequalities are satisfied, it seems that the optimal solution was not obtained. where 0 γ > . Then, let 1 ii = + , and execute the computation from Step A3 to Step A7. Execute these iterative computations till maximum k iN = . If following inequalities are satisfied, discontinue the iteration because it seems that the optimal solution was obtained.
The computation time which was required to the above-mentioned computation is set to k τ .
Then, we obtain the next control strategy

Modeling
The kinematical model of the multiple trailer system which we treat is shown in Fig.4. The meaning of next equation (15) is the state equation of the first vehicle (autotruck) which is driven pulling the follow-on passive trailers.
The control input vector of this system is denoted by Here, 0 v and ω denotes the velocity and angular velocity of the first vehicle respectively. This model is a differentialdriven vehicle model which has nonholonomic constraint, and is regarded as one of the most typical nonholonomic systems. It is known that although this model has www.intechopen.com controllability, it can not be asymptotically stabilizable by any continuous time-invariant controller (Brockett, 1993). For this reason, there have been many references dealing with the stabilization problem for this model using various kinds of controllers. One successful approach is to convert it into the so-called chained form and then establish a time-varying controller. Although such an approach leads to asymptotical stabilization, it is applicable only for the case where the system's dimension is low (less than four).Since we deal with a multiple trailer system, whose dimension is obviously much larger than four, the approach of utilizing chained form with a time-varying controller can not be applied here, and more practical strategy is desirable. The most of conventional research have treated the direct-hooked type trailer model. This in Fig.4, and the kinematics of the th i trailer is as follows. Fig. 4 Mechanical linkage design of multiple trailer system.
Only the first vehicle (truck) is driven and the following vehicles (trailers) are passively pulled by the truck. i trailer respectively. The direct-hooked model can be transformed to a chained form. However, this model has a tracking error of follow-on trailers. Therefore, we deal with the off-hooked model ( ) which can eliminate the tracking error (Fig. 5). However, the model of offhooked trailer system cannot be transformed to canonical form. Fig. 4 shows a off-hooked model, and the following equation denotes the th i trailer's kinematics.

Problem formulation
Tracking control problem of the multiple trailer system is formulated as a nonlinear optimal control problem in this section. For simplicity of notation, we consider one truck and two trailers. Even if the number of the trailer increases, our control design can be extended very easily. In that case, increase of the computational cost is inevitable.
The performance index is given by where the state vector and input vector are denoted by ξ and u respectively. P , Q , R denote the weighting matrices. We set where, θ ilim (i=1,2) is an absolute value of limitation of the relative angle, and lim v and ω lim are the absolute value of the limitation of the control input.  Fig. 6 shows the permitted region of follow-on trailers. The weight parameters are set to ) 2 , 1 ( 001

Numerical simulation
The control strategy of our approach is obtained by processing the iterative calculation of the REB algorithm in each ΔT. Through sufficiently large number of iterations N, it could be expected to eventually reach the possible optimal solutions. Through some simulation results we can obtain the effective solution with roughly ΔT=100[msec] by the PC which we use. However, it is not necessarily the case that the effective solution is obtained, especially in the case of including a disturbance. The simple computational artifice described in section 3.2. partially reduces such a problem. The example of the simulation result of applying the algorithm to the case of including a disturbance is shown in the following. . However, real time feasibility is not guaranteed by these parameters, because the computation time varies according to running condition. Fig. 8 -Fig. 11 show the simulation result with the initial state Impulsive disturbances on θ 1 and θ 2 have been added in this simulation at 5, 10, 15 and 20[sec], whose magnitude is 0.5[rad]. www.intechopen.com

Conclusions
We discussed the real time control algorithm using the numerical solution called algorithmic control. Then, we improved the conventional algorithmic design for the numerical robustness via incorporating computation time. The key idea is to adjust the maximum number of iteration with the computational time. This approach was applied to a tracking control problem of the multiple trailer system. We showed through a numerical simulation that the proposed algorithm is executable in real time, and it has robustness against disturbances.

Acknowledgment
This research has been supported in part by the Japan Ministry of Education, Sciences and Culture under Grants-in-Aid for Scientific Research (B) 18760326.