Real-time Control of an Autonomous Vehicle : a Neural Network Approach to the Path following Problem

A neural-network based approach to the control of non-linear dynamical systems such as wheeled mobile robots is presented. A general framework for the training of neural controllers is outlined, and applied to the lateral control of a vehicle for the path following and trajectory servoing problems. Simulation as well as experimental results on a four-wheel drive vehicle equipped with actuators and sensors are shown. Key-words : autonomously guided vehicles (AGVs), mobile robots, model reference control, nonlinear control, optimal control, path following, recurrent neural networks, trajectory servoing. Résumé : Nous décrivons une approche neuronale de la commande de processus dynamiques non-linéaires tels que les robots mobiles à roues. Nous esquissons les grands traits d'un cadre général pour l'apprentissage de correcteurs neuronaux, et appliquons nos méthodes à la commande de la direction d'un véhicule pour son asservissement sur trajectoire. Nous illustrons notre propos à l'aide de simulations, et présentons les résultats expérimentaux obtenus sur un véhicule tout-terrain équipé des capteurs de navigation et des actionneurs nécessaires au pilotage. Mots-Clés : asservissement sur trajectoire, commande avec modèle de référence, commande non-linéaire, commande optimale, robots mobiles, réseaux neuronaux bouclés, suivi de trajectoire, véhicules autonomes. 1. INTRODUCTION We address the lateral control of an autonomous vehicle along a predefined trajectory using neural networks. Experimental results on a full-scale outdoor robot, a standard four-wheel drive car equipped with the sensors and actuators needed for navigation and control, demonstrate that neural techniques can be applied to real-world problems in robotics. Classical control theory provides many design techniques to achieve performances specified in terms of rise and settling-time, gain and phase margin, bandwidth… These methodologies are well suited to the design of linear controllers for linear systems, with guaranteed stability and robustness. They are, however, of restricted use for control problems involving nonlinear dynamic processes with inequality constraints on state and control variables, such as wheeled mobile robots with actuator limitations. Optimal control theory has been widely used to solve such non-linear, constrained problems. But the conventional scheme of optimal control has its own drawbacks. Finding the control trajectory that minimizes the performance measure often requires the solution of non-linear differential equations. This can be achieved by using iterative numerical methods (quasilinearization, steepest descent…) which are time consuming, or by dynamic programming ; both miss closed-form expressions for the feedback control laws. Neural networks offer an alternative to the usual formulations and solutions of constrained optimization problems. Their approximation capabilities make them of possible use as models of the process to be controlled, as well as suitable controllers parameterizing non-linear optimal feedback control laws. In addition, the performance measure which, when minimized, corresponds to the optimal behaviour, can be defined with respect to a reference model. Finally, generic algorithms using a gradient-based approach to achieve the minimization of the performance measure regardless of model and controller complexity have been established. In the second part of this paper, we present a general framework for the training of neural networks for control purposes. Part 3 is devoted to our application : the lateral control of a vehicle for the path following problem. We first present our test-vehicle and its neural model. We subsequently apply the control scheme developed in part 2 using two different approaches of the path following problem. Simulation and experimental results are shown in both cases. They are discussed in part 4. 2. TRAINING OF RECURRENT NEURAL NETWORKS FOR NON-LINEAR CONTROL We assume that the process to be controlled is described by the following discrete-time model : { = f(S(k), U(k)) Y(k) = g(S(k)) where S(k), Y(k) and U(k) are the state, output and control vectors at time k respectively, and f and g are unknown non-linear functions. The control system consists of the following components : a neural network predictor model with state Sm and output Ym, is first trained : { = fNN(Xm(k), U(k)) Ym(k) = gNN(Sm(k)) where fNN and gNN are non-linear functions implemented by neural networks, and where the state input Xm may take different values depending on the particular predictor choice. For a recursive predictor, Xm(k) = Sm(k) ; for a non-recursive predictor, Xm(k) is often taken to be equal to the process state Xp(k). The rationale of this choice has been discussed in [NER92a]. a reference model (possibly a simple delay) is designed, which generates the desired output sequence {Yr(k)} from the setpoint sequence {R(k)} : { = fr(Sr(k),R(k)) Yr(k) = gr(Sr(k)) where Sr(k) is the state of the reference model at time k, Yr(k) its output, R(k) the setpoint vector, and fr and gr are known (possibly linear) functions. the neural controller, with weight matrix C, computes the control sequence {U(k)} from the setpoint sequence and the model state input, and can therefore implement any suitable non-linear state-feedback control law U(k) = y(Xm(k), R(k)) . The aim of the training is to compute the weights of the neural controller, either adaptatively or non adaptively, so that the output of the process becomes as close as possible to the output of the reference model. We restrict the scope of our presentation to a non-adaptive context in which the process itself is not taken into account during the training of the controller (for a general presentation, including adaptive control schemes, see [NER 93a]). This approach of neural control is well suited to the off-line validation of controller structures, provided the neural model of the process is accurate enough, and is often encountered in the literature (e. g. [NAR 91]). 2.1 Training phase The training algorithms aim at minimizing a cost-function J by a gradient-based technique. J involves the squared difference Em between the output of the model and the output of the reference model over a time horizon of length Nc : J = 1 2 ∑ k=Nt-Nc+1 Nt »Em(k)»2 = 1 2 ∑ k=Nt-Nc+1 Nt [Yr(k) Ym(k)]T W [Yr(k) Ym(k)] where W is a weighting matrix, Nc≤Nt (for example Nc=1 is chosen if a desired value exists for the final output only) and Nt is the number of time steps used for the evaluation of the gradient of the cost-function ([NER92b][NER93b]). The weights will be modified iteratively in the direction opposite to that of the gradient : ∆Cij = μ ∂J ∂Cij = μ ∂ ∂Cij ( 1 2 ∑ k=Nt-Nc+1 Nt »Em(k)»2 ) where μ is the gradient step. The value of the cost-function J, hence of its gradient, depends on the NN predictor model used for the computation of Ym. The NN predictor model is either recursive or non-recursive, each case leading to a specific training algorithm. a) The UD control algorithm (" UnDirected " algorithm, see figure 1) If the NN predictor model is recursive, the state input Xm(k) takes the values : Xm(0) arbitrary, and Xm(k) = Sm(k) ¢ k>0. Reference model Trained NN controller NN Predictor model Backprop. Backprop. R