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4.1 Generalized Predictive Control The basic idea of GPC is to calculate a sequence of future control signals in such a way that it minimizes a cost function defined over a prediction horizon [1], [2]. The index to be optimized is the expectation of a quadratic function measuring the distance between the predicted system output and some predicted reference sequence over the horizon plus a quadratic function measuring the control effort. 122 ˆ ()11p cH HJk y r ukjkj kjjN jλ =−+∆ ∑∑ ++ +−== where: N1- minimum costing horizons Hp – Prediction horizon Hc≤Hp≥1 Control horizon ∆uK - Control action increment, ∆uk=uk-uk-1 λk - control energy weight j ky+ˆ - Prediction of the system output j k r + - reference predictive trajectory The system model can be presented in the ARMAX form [1] 111 11) ( ) () ( ) ( ) ( ) (−−− −−+ − =zt z C T t u z B t y z A eξ (22) Where 1() Az−, 1() Bz−and 1() Cz−are the matrix parameters of transfer function () H z . To compute the output predictions is necessary to know the system model that must be controlled (Fig. 7). The parameters used by the GPC are obtained from the configuration shown in Fig. 6. Fig. 6. Manipulator system block diagram controlled by vision. The transfer, () H z , function is given by: 1 () 11() ()*( ) 2 1cT pz z a Hz JFzJ cp zzz− +==− (23) The parameters of this function are used in the predictive controller implementation. 4.2 System Identification The robot model is obtained by identifying each of the joints dynamics to obtain a six order linear model: 54321543210654321543210() ibz bz bz bz bz bFzzazazazazaza−−−−−−−−−−−+++++=++++++ (24) In the identification procedure a PRBS is used as input signal. A prediction error method (PEM) is used to identify the robot dynamics. The noise model 1() Cz− of order 1 was selected. In this approach the identification of () H z (Fig. 6) is performed around a reference condition. Since the robot is controlled in velocity and the dynamics depend mainly on the first joints, assuming small displacements is possible to linearize the system around the position q. It was also necessary to consider a diagonal inertia matrix. Under these conditions, the Jacobian matrix is constant as well as H(z). This procedure is valid at low velocities. This means that the cross coupled terms are neglected. 4.3 Model Predictive Controller In this approach the model is formulated in a space state form: ⎩⎨⎧== + = +) ( ) () 0 ( ), ( ) ( ) 1 ( 0t Cx t yx x t Bu t Ax t x (25) where x(t) is the state, u(t) is the control input and y(t) is the output. Fig. 7 State space manipulator scheme. The state space manipulator dynamics controlled in velocity is represented in Fig. 7. The parameters values of At, Bt and Ct are obtained from the identification algorithm PEM. The matrices in equation (25) are computed through the image Jacobian matrix by the following definitions: 1tttB BJ C CJ A A −= == (26) The predictive control algorithm is (Clarke et al., 1987): * p p At ∫ Ct J-1 J Bt Jc ∫ GPC ZOH Robot velocity control p J-1 q p ∆ * p F(z) Vision (Z-1) H(z) ∗q (21) 1. At time t predict the output from the system, ) / ( ˆ t k t y + , where k=N1,N1+1,…,N2.These outputs will depend on the future control signals, ) / ( ˆ t j t u + , j=0,1,…,N3 and on the measured state vectors at time t. 2. Choose a criterion based on these variables and optimise with respect to 3 ,..., 1 , 0 ), / ( ˆ N j t j t u = + . 3. Apply ) / ( ˆ ) ( t t u t u = . 4. At time t+1 go to 1 and repeat. 5. SIMULATION PROCEDURE 5.1 System configuration The implemented Visual Servoing package allows the simulation of different kind of cameras. In this particular case, it was chosen a Costar camera placed in the end-effector and positioned according with oz axis. Its target of eight coplanar points was created which will serve as control reference. The accuracy of the camera position control in the world coordinate system was increased by the use of redundant features [11]. The centre of the target corresponds to the point with coordinates (0,0) and the remaining points are placed symmetrically in relation to this point. The target pose is referenced to the robot base frame. In the case of servoing a trajectory, the target is remained fixed and the desired point is variable. As the primitive of the target points is obtained it is possible to estimate the operational coordinates of the camera position point. Visual Servoing with a PI controller. In 2D Visual Servoing the image characteristics are used to control the robot. Images acquired by the camera are function of the end effector’s position, since the camera is fixed on the end effector of the robot. They are compared with the corresponding desired images. In the present case the image characteristics are the centroids of the target points. Fig. 8 represents the model simulation of the implemented 2D visual servoing architecture. In this case CT is a PI controller.