Alternatie non-linear predictie control under constraints applied to a to-heeled nonholonomic mobile robot Gaspar Fontineli Dantas Júnior *, João Ricardo Taares Gadelha *, Carlor Eduardo Trabuco Dórea * Departamento de Engenharia da Computação e Automação Uniersidade Federal do Rio Grande do Norte Natal, Brazil gsatnad@yahoo.com *, joaoricardotg@gmail.com *, cetdorea@dca.ufrn.br * Abstract This paper presents a noel algorithm for Model Predictie Control (MPC) of a nonlinear class that can be applied to systems subject to linear state and control constraints. A pieceise linear representation of the nonlinear system is obtained, by fixing the sequence of future inputs. A quadratic optimization problem is then repeatedly soled until conergence is reached. In this case, it is proed that no prediction error occurs and, consequently, the state and control constraints are satisfied by the original nonlinear system. The proposed technique is applied to a to-heeled nonholonomic mobile robot, knon to be difficult to control ith linear techniques. Keyords Model Predictie Control; Nonlinear Model Predictie Control, Control Under Constraints, Nonlinear Systems, Nonholonomic To Wheeled Robot. I. INTRODUCTION Model predictie control (MPC) has a long history in the field of control engineering. It is one of the fe areas that has receied on-going interest from researchers in both the industrial and academic communities. Four major aspects of model predictie control make the design methodology attractie to both practitioners and academics. The first aspect is the design formulation, hich uses a completely multiariable system frameork here the performance parameters of the multiariable control system are related to the engineering aspects of the system; hence, they can be understood and tuned by engineers. The second aspect is the ability of the method to handle both soft and hard constraints in a multiariable control frameork. This is particularly attractie to industry here tight profit margins and limits on the process operation are ineitably present. The third aspect is the ability to perform on-line process optimization. The fourth aspect is the simplicity of the design frameork in handling all these complex issues. []. A heeled mobile robot (WMR) is defined as a heeled ehicle that can moe autonomously ithout assistance from external human operator. The WMR is equipped ith a set of motorized actuators and, sometimes, ith an array of sensors, hich help it to carry out useful ork []. Wheeled mobile robots (WMRs) are increasingly present in industrial and serice robotics, particularly hen flexible motion capabilities are required on reasonably smooth grounds and surfaces []. The problem of autonomous control of WMRs has attracted the interest of researchers in ie of its theoretical challenges. In the absence of orkspace obstacles, the basic motion tasks assigned to a WMR may be reduced to folloing a gien trajectory or reaching a gien destination. From a control iepoint, the second problem is easier than the first. A schematic diagram of the robot is presented on figure. The configuration is represented by its position on the Cartesian space (x and y, that is the position of the robotbody center ith relation to a referential frame fixed on the orkspace), and by its orientation (angle beteen the robot orientation ector and the reference axis, fixed on the orkspace). Figure : Schematic diagram of a to-heeled nonholonomic robot. In this paper is proposed to make a to-heeled differentially drien nonholonomic mobile robot reach a gien destination. A pieceise linear representation of the nonlinear system is obtained, by initially fixing the sequence of future inputs. A quadratic optimization problem is then repeatedly soled until conergence is reached so that, the first element of the optimized input ector can be used in the system. In this case, it is proed that no prediction error occurs and, consequently, the state and control constraints are satisfied by the original nonlinear system.
II. CONTROL STRATEG A. Model Predictie Control applied to a Linear System In the case here the system is in linear format, e hae the folloing representation in discrete time: +=+ = Where R is the state ector, R is the input ector, R is the output ector and N is the sampling instant. To find the prediction equation in the absence of disturbances and noise hen the measurement of the full state ector is aailable [,8], e hae: + + + + = + + + + + + Where is the prediction horizon, is the control horizon, is the estimated state ector and is the estimated difference of the control ariable. Thus, the prediction of the system output () is gien in matrix form by: = Ψ+ Υ + Θ Ψ= + Θ = + Υ = Where the tracking error is gien by: εk=τk Ψ Υ 7 In the eent of constraints in,, and, respectiely, they are treated as follos: 8 9 Suppose: =[,,,,], here f is the last column of F = [Γ,], g is the last column of G = [P,], p is the last column of P = [,], is the last column of W These inequations can be reritten in dependency ith Δ, according to equation () in the folloing matrix form: + ΓΘ Γ[Ψ+ Υ ] Δ + The predictie control algorithms utilize cost functions in order to penalize deiations of the output from the reference. Based on this error, one can obtain: = + + + Δu+ Where is the performance index (cost function) of finite horizon to be optimized, >, is the predicted controlled output, is the reference trajectory, Q is the eighting matrix of the tracking error, R is the eighting matrix of the control effort (Q and R are positie definite matrices). Equation () can be ritten in the folloing matrix form: = + Δ + = + + = +
Δ= Δu 7 Δu+ Subject to the eighting matrices gien by: Q = + 8 R = 9 It is possible to rite the cost function as follos: = Δ + Δ Δ = Θ QEk = Θ Θ + Ek is the tracking error ector. To find the optimal Δ, it is needed to find the gradient of and set it to zero. So the optimal set of future input moes can be found by: Δ= Since the matrix H is usually ill-conditioned, the best ay to find the optimal U, respecting the constraints described in the inequation (), is using Quadratic Programming. B. Alternatie Nonlinear Model Predictie Control For the application of the proposed control method, the nonlinear system represented in discrete time must be ritten in the folloing form [,7]: +=,+, = Where R is the state ector, R is the input ector, R is the output ector and N is the sampling instant. Considering the prediction equation of state steps ahead, one has: ++=+,+++ +,++ It is possible to notice that if + and + ere fixed oer the horizon =,,, it ould be possible to find the equation of state in a pieceise linear system described by: ++=,++,+,=+,+,=+,+7 Hoeer, + and + are not knon oer the horizon since + depends on the future inputs +. Neertheless, it is possible to describe an algorithm that conerges to the correct predicted alues, as follos: Differently from the linear case, here the matrices A, B and C are constant, based on equation (), one can find: = Ψ + Υ + Θ 8 Ψ = 9 + Υ = + + Θ = + + The proposed controller is explained in the folloing steps:. Take a guess and fix hat could be the future inputs +, =,,. Use it to calculate +, =,, ith the equation ().. Substitute + and + in equation (7) to obtain a pieceise linear representation of the nonlinear system for each : =,,.. Formulate the equation (8).. Similarly to the Linear MPC, Use equation () to find the optimal U.. Repeat steps, and until the norm ector beteen the ne + and the old + is ithin a predetermined tolerance.. Once the norm ector reaches a desired alue, the actual control input becomes the first sample of. 7. Repeat steps to for each sampling instant of the algorithm. There ill be an error beteen the predicted states and the states that ould be generated by the nonlinear model due to the folloing reason: in step, the future alues of the states are calculated from the application of optimal control sequence calculated in step and the non-linear model. In step, these future alues are used to form the pieceise linear model in the optimization of the next iteration. Hoeer, this pieceise linear model only corresponds to the non-linear model if the future alues of the states did not change from one iteration to another, i.e., this ould occur if the optimal sequence + did not modify from one iteration to the next. Thus, if this algorithm conerges, the final alues of + and + predicted by the pieceise linear model are equal to those predicted by the original nonlinear model. Thus, if the optimization problem has a feasible solution, it ensures that constraints () are met along the
prediction horizon. This is the largest adantage of the proposed method compared to those based on approximations of the model that cannot guarantee the satisfaction of constraints in the states. Unlike strategies that focus on the direct resolution of the non-conex optimization problem, the strategy proposed here does not guarantee optimality. Proides hoeer, solutions that improe at each iteration of the algorithm. III. ROBOTIC APPLICATION The proposed robot is described by the folloing kinematic model: =cos =sen = Where is the forard elocity, is the steering elocity, ; is the position of the mass center of the robot moing in the plane and denotes its heading angle from the horizontal axis. It can be transformed into the folloing state space representation: cos + Where is the system output and it as decided to be used to control, independently, all three state ariables. The dynamic model is deried from the actuators dynamics and the robot dynamics parameters, like mass, inertia momentum and friction coefficients []. The suggested model indicates that the problem is truly nonlinear based on the matrix B, thus linear control is ineffectie, een locally, and innoatie design techniques are needed []. Here, the elocities and are taken as the inputs and are subject to the folloing constraints: Approximating the deriaties for finite differences using a sampling period equal to T, one obtains the folloing discrete-time model: + Tcos + + + Basically, the proposed controller is used to steer the robot from a starting position, ith heading angle to a, heading angle, according to Figure : same: Figure : Control objectie IV. RESULTS In all cases, the folloing conditions ill be the,,,,,,,,,,. A. Verifying constraints applied to the states ariables Keeping the heading angle constant,78 rad, and,,, figures, and shos the results hen no constraint is applied to the state ariables and figures,7 and 8 shos results hen it is, as follos: Figure : 7 8 9..8... Figure : -. 7 8 9
Figure : Robot trajectory The constraint applied to state ariables as: + It is possible to notice that once one of the state ariables reaches the imposed constraint, the robot stops (figure 8) according ith the imposed constraint...... Figure : B. Verifying the behaior hen the heading angle aries Gien ( ),,9 rad, rad, figures 9, and shos results for,,, hile figures, and, for,,, as follos: Figure 9:... 7 8 9.9.8.7 Figure 7:..8 Figure :......... 7 8 9 Figure 8: Robot trajectory -. 8 Figure : Robot trajectory 8.....
Figure : of inputs, the robot is not capable to reach the desired reference hen heading angle aries. A proposal for future ork ould be to proide a set of coordinates for a predetermined trajectory to be folloed as a reference, or een the use of a more elaborated model that ill allo the proposed controller sho its full capabilities. Once it is done, comparison of results ith other nonlinear methods of control ould be motiating...8... Figure : Figure : Robot trajectory -. 8 REFERENCES [] WANG, L. Model Predictie Control System Design and Implementation. Adances in Industrial Control. Springer, 8. [] SIDEC, N. Dynamic Modeling and Control of Nonholonomic Wheeled Mobile Robot Subjected to Wheel Slip. PhD Thesis, Vanderbilt Uniersity, USA, 8. [] SCHRAFT, R. D., SCHMIERER, G. Sericeroboter. Springer. Berlin,998 [] LUCA, A.; ORIOLO, G.; VENDITTELLI, M. Control of Wheeled Mobile Robots: An Experimental Oerie. Uniersita degli Studi di Roma, Italy. [] CAMACHO, E. F.; BORDONS, C. Model Predictie Control. Springer-Verlag Ltd,. [] FONTES, A.; DÓREA, C.; GARCIA, M. An iteratie algorithm for constrained mpc ith stability of bilinear systems, Proc. th Mediterranean Conference on Control andautomation, Ajaccio, pp., 8. [7] KOCH, S.; DURAISKI, R.; FERNANDES, P.; TRIERWEILER, J. NMPC ith statespace models obtained through linearization on equilibrium manifold. Proc. Int. Symp. on Adanced Control of Chemical Process, pp.,. [8] MACIEJOWSKI, J. M. Predictie Control ith Constraints. Person Education Limited,. [9] MATLAB. ersion 7.. (Ra), The MathWorks Inc., Natick, Massachusetts, EUA,. 8 8 8 It is possible to notice that once the reference as changed to,, the robot is unable to reach it (figure ) as expected. V. CONCLUSION After analyzing the results, it as possible to isualize the satisfaction of the imposed constraints hich is one of the adantages for the use of the proposed MPC, but because of the nonholonomic restrictions, constant reference and the number of the state ariables higher than the number