WE propose the tracking trajectory control of a tricycle

Transcription

1 Proceedings of the International MultiConference of Engineers and Computer Scientists 7 Vol I, IMECS 7, March - 7, 7, Hong Kong Trajectory Tracking Controller Design for A Tricycle Robot Using Piecewise Multi-inear Models Tadanari Taniguchi and Michio Sugeno Abstract This paper deals a tracking trajectory controller design of a tricycle robot as a non-holonomic system with a piecewise multi-linear PM model. The approximated model is fully parametric. Input-output I/O dynamic feedback linearization is applied to stabilize PM control system. We also apply a method for a tracking control based on PM models to the tricycle robot. Although the controller is simpler than the conventional I/O feedback linearization controller, the control performance based on PM model is the same as the conventional one. Examples are shown to confirm the feasibility of our proposals by computer simulations. Index Terms piecewise model, tracking trajectory conrol, dynamic feedback linearization. I. INTRODUCTION WE propose the tracking trajectory control of a tricycle robot using dynamic feedback linearization based on piecewise multi-linear PM models. Wheeled mobile robots are completely controllable. However they cannot be stabilized to a desired position using time invariant continuous feedback control []. The wheeled mobile robot control systems have a non-holonomic constraint. Non-holonomic systems are much more difficult to control than holonomic ones. Many methods have been studied for the tracking control of wheeled robots. The backstepping control methods are proposed in e.g. [], []. The sliding mode control methods are proposed in e.g., [4], [], and also the dynamic feedback linearization methods are in e.g., [6], [7], [8]. For non-holonomic robots, it is never possible to achieve exact linearization via static state feedback [9]. It is shown that the dynamic feedback linearization is an efficient design tool to solve the trajectory tracking and the setpoint regulation problem in [6], [7]. In this paper, we consider PM model as a piecewise approximation model of the tricycle robot dynamics. The model is built on hyper cubes partitioned in state space and is found to be bilinear bi-affine [], so the model has simple nonlinearity. The model has the following features: The PM model is derived from fuzzy if-then rules with singleton consequents. It has a general approximation capability for nonlinear systems. It is a piecewise nonlinear model and second simplest after the piecewise linear P model. 4 It is continuous and fully parametric. The stabilizing conditions are represented by bilinear matrix inequalities Manuscript received December, 6; revised January, 7. This work was supported by Grant-in-Aid for Scientific Research C:68 of Japan Society for the Promotion of Science. T. Taniguchi is with IT Education Center, Tokai University, Hiratsuka, Kanagawa, 99 Japan taniguchi@tokai-u.jp M. Sugeno is with Tokyo Institute of Technology. BMIs [], therefore, it takes long computing time to obtain a stabilizing controller. To overcome these difficulties, we have derived stabilizing conditions [], [], [4] based on feedback linearization, where [] and [4] apply inputoutput linearization and [] applies full-state linearization. We propose a dynamic feedback linearization for PM control system and apply the tracking control [] to a tricycle robot system. The control system has the following features: Only partial knowledge of vertices in piecewise regions is necessary, not overall knowledge of an objective plant. These control systems are applicable to a wider class of nonlinear systems than conventional I/O linearization. Although the controller is simpler than the conventional I/O feedback linearization controller, the tracking performance based on PM model is the same as the conventional one. Wheeled robot dynamics has some trigonometric functions. The trigonometric functions are smooth functions and of class C. The PM models are not of class of C. In the tricycle robot control, we have to calculate the third derivatives of the output. Therefore the derivative PM models lose some dynamics. Thus we propose the derivative PM models of the trigonometric functions. This paper is organized as follows. Section II introduces the canonical form of PM models. Section III presents a dynamic feedback linearization of the car-like robot. Section IV proposes a tracking controller design using dynamic feedback linearization based on PM model of the tricycle robot. Section V shows examples demonstrating the feasibility of the proposed methods. Finally, section VI summarizes conclusions. II. CANONICA FORMS OF PIECEWISE BIINEAR A. Open-oop Systems MODES In this section, we introduce PM models suggested in []. We deal with the two-dimensional case without loss of generality. Define vector dσ, τ and rectangle R στ in twodimensional space as dσ, τ d σ, d τ T, R στ [d σ, d σ ] [d τ, d τ ]. σ and τ are integers: < σ, τ < where d σ < d σ, d τ < d τ and d, d, d T. Superscript T denotes a transpose operation. ISBN: ISSN: Print; ISSN: Online IMECS 7

2 Proceedings of the International MultiConference of Engineers and Computer Scientists 7 Vol I, IMECS 7, March - 7, 7, Hong Kong For x R στ, the PM system is expressed as σ τ ẋ ω x i ω j x f o i, j, iσ jτ σ τ x ωx i ω j x di, j, iσ jτ where f o i, j is the vertex of nonlinear system ẋ f o x, ω σ x d σ x /d σ d σ, ω σ x x d σ/d σ d σ, ω τ x d τ x /d τ d τ, ω τ x x d τ/d τ d τ, and ωx i, ω j x [, ]. In the above, we assume f, and d, to guarantee ẋ for x. A key point in the system is that state variable x is also expressed by a convex combination of di, j for ωx i and ω j x, just as in the case of ẋ. As seen in equation, x is located inside R στ which is a rectangle: a hypercube in general. That is, the expression of x is polytopic with four vertices di, j. The model of ẋ fx is built on a rectangle including x in state space, it is also polytopic with four vertices fi, j. We call this form of the canonical model parametric expression. B. Closed-oop Systems We consider a two-dimensional nonlinear control system. ẋ fo x g o xux, 4 y h o x. The PM model is constructed from a nonlinear system 4. ẋ fx gxux, y hx, where σ τ fx ωx i ω j x f o i, j, iσ jτ σ τ gx ω x i ω j x g o i, j, iσ jτ σ τ hx ωx i ω j x h o i, j, iσ jτ σ τ x ωx i ω j x di, j, iσ jτ and f o i, j, g o i, j, h o i, j and di, j are vertices of the nonlinear system 4. The modeling procedure in region R στ is as follows: Assign vertices di, j for x d σ, d σ, x d τ, d τ of state vector x, then partition state space into piecewise regions see Fig.. Compute vertices f o i, j, g o i, j and h o i, j in equation 6 by substituting values of x d σ, d σ and x d τ, d τ into original nonlinear 6 d τ f σ, τ d τ ω σ ω τ f σ, τ d σ f x f σ, τ ω σ ω τ f σ, τ d σ σ τ Fig.. Piecewise region f x iσ jτ ωi ωj f i, j, x R στ functions f o x, g o x and h o x in the system 4. Fig. shows the expression of fx and x R στ. The overall PM model is obtained automatically when all vertices are assigned. Note that fx, gx and hx in the PM model coincide with those in the original system at vertices of all regions. III. DYNAMIC FEEDBACK INEARIZATION OF TRICYCE ROBOT We consider a tricycle robot model. ẋ cos θ ẏ θ sin θ tan ψ u u, 7 ψ where x and y are the position coordinates of the center of the rear wheel axis, θ is the angle between the center line of the car and the x axis, ψ is the steering angle with respect to the car. The control inputs are represented as u v s cos ψ u ψ, where v s is the driving speed. Fig shows the kinematic model of tricycle robot. The steering angle ψ is constrained by ψ M, < M < π/. The constraint [8] is represented as ψ M tanh w, where w is an auxiliary variable. Thus we get ψ Msech wµ u, ẇ µ We substitute the equations of ψ and w into the tricycle robot model. The model is obtained as ẋ cos θ ẏ θ sin θ tanm tanh w u µ 8 ẇ ISBN: ISSN: Print; ISSN: Online IMECS 7

3 Proceedings of the International MultiConference of Engineers and Computer Scientists 7 Vol I, IMECS 7, March - 7, 7, Hong Kong Fig.. y x, y Kinematic model of tricycle robot In this case, we consider η x, y T as the output, the time derivative of η is calculated as ẋ cos θ u η. ẏ sin θ µ The linearized system of 8 at any points x, y, θ, w is clearly not controllable and the only u affects η. To proceed, we need to add some integrators of the input u. Using dynamic compensators as ψ u ν, ν µ, the tricycle robot model 8 can be dynamic feedback linearizable. The extended model is obtained as ẋ u cos θ ẏ u sin θ θ ẇ u tanm tanh w µ µ 9 u ν ν The time derivative of η is calculated as η f h ν cos θ u f h tanm tanh w sin θ ν sin θ u, tanm tanh w cos θ where h, h x, y. Since the controller µ, µ doesn t appear in the equation η, we continue to calculate the time derivative of η. Then we get η f h g f hµ f h g f h f h g f h µ g f h g f h. µ Equation shows clearly that the system is input-output linearizable because state feedback control µ g f h f h g f h v reduces the input-output map to y v. The matrix g f h multiplying the modified input µ, µ is non-singular if u. Since the modified input is obtained as µ, µ, the integrator with respect to the input v is added to the original input u, u. Finally, the stabilizing controller of the tricycle robot system 7 is presented as a dynamic feedback controller: u ν, ν µ, u Msech wµ θ x IV. PM MODEING AND TRACKING CONTROER DESIGN OF THE TRICYCE ROBOT MODE A. PM Model of the Tricycle Robot Model We construct PM model of the tricycle robot system 9. The state spaces of θ and w in the tricycle robot model 9 are divided by the vertices x π, π/6,..., π} and the vertices x 4.,.,...,.}. The state variable is x x, x, x, x 4, x, x 6 T x, y, θ, w, u, v T. σ σ ẋ σ 4 i σ i σ i 4 σ 4 x f d i x x f d i x µ µ. 4 x 4f d 4i 4x x 6 We can construct PM models with respect to f x, f x and f x. The PM model structures are independent of the vertex positions x and x 6 since x and x 6 are the linear terms. This paper constructs the PM models with respect to the nonlinear terms of x and x 4. Note that trigonometric functions of the tricycle robot 9 are smooth functions and are of class C. The PM models are not of class C. In the tricycle robot control, we have to calculate the third derivatives of the output y. Therefore the derivative PM models lose some dynamics. In this paper we propose the derivative PM models of the trigonometric functions. B. Tracking Controller Design Using Dynamic Feedback inearization Based on PM Model We define the output as η x, x T in the same manner as the previous section, the time derivative of η is calculated as σ fp h η ẋ f d fp h ẋ x i x f d i x where the vertices are f d i cos d i and f d i sin d i. The time derivative of η doesn t contain the control inputs µ, µ. We calculate the time derivative of η. We get η h x f d i x 6 x f d i η h 4 x f d i x 6 x f d i 4 x 4 f d 4 i 4 x, 4 4 x 4 f d 4 i 4 x, ISBN: ISSN: Print; ISSN: Online IMECS 7

4 Proceedings of the International MultiConference of Engineers and Computer Scientists 7 Vol I, IMECS 7, March - 7, 7, Hong Kong where f d 4 i 4 tanm tanh d 4 i 4 /. We continue to calculate the time derivative of η. We get η h g h µ g h µ σ4 η x x f x x 6 x x d i x f d i x f d i µ x f d i x f d i 4 x 4 f d 4 i h g h µ g h µ σ4 x x 6 x x f d i x f d i µ x f d i 4 x 4 f d 4 i 4 4 x 4 f d 4 i 4 µ, 4 x 4 f d 4 i x 4 f d 4 i 4 4 x 4 f d 4 i 4 µ. The vertices f d i, f d i and The controller of is designed as µ, µ T g h h g h v g f p h g h g h fp h g h h g f p h g h g h g v h where v is the linear controller of the linear system. ż Az Bu, y Cz, where z h, fp h, h, h, fp h, h T R 6, A, B, C If x, there exists a controller µ, µ T of the tricycle robot model since det g h. In this case, the state space of the tricycle robot model is divided into vertices. Therefore the system has local PM models. Note that all the linearized systems of these PM models are the same as the linear system. T. In the same manner of, the dynamic feedback linearizing controller of the PM system is designed as ü µ, u Msech x 4 µ, 4 µ µ h g f hv. The stabilizing linear controller v F z of the linearized system is designed so that the transfer function CsI A B is Hurwitz. Note that the dynamic controller 4 based on PM model is simpler than the conventional one. Since the nonlinear terms of controller 4 contain not the original nonlinear terms e.g., sin x, cos x, tanm tanh x 4 but the piecewise approximation models. C. Tracking Control for PM System We apply a tracking control [] to the tricycle robot model 7. Consider the following reference signal model ẋr f r, η r h r. The controller is designed to make the error signal e e, e T η η r as t. The time derivative of e is obtained as fp h ė η η r p fp h p fr h r. fr h r Furthermore the time derivative of ė is calculated as ë η η r fp h p fr h r h p f r h r Since the controller µ doesn t appear in the equation ë, we calculate the time derivative of ë. e η η r fp h p h g h p µ µ fr h r f r h r The tracking controller is designed as ü µ, u Msech x 4 µ, µ µ h f r h r g hv. The linearized system and controller v F z are obtained in the same manners as the previous subsection. The coordinate transformation vector is z e, ė, ë, e, ė, ë T. Note that the dynamic controller based on PM model is simpler than the conventional one on the same reason of the previous subsection. V. SIMUATION RESUTS We apply the previous tracking control method to the tricycle robot model 7. Although the controller is simpler than the conventional I/O feedback linearization controller, the tracking performance based on PM model is the same as the conventional one. In addition, the controller is capable to use a nonlinear system with chaotic behavior as the reference model. In the following simulations, the tricycle length is. [m] and the angle constrain M is π/ [rad.]. ISBN: ISSN: Print; ISSN: Online IMECS 7

5 Proceedings of the International MultiConference of Engineers and Computer Scientists 7 Vol I, IMECS 7, March - 7, 7, Hong Kong A. Ellipse-shaped reference trajectory Consider an ellipse model as the reference trajectory. R cos θ x r, x r R sin θ x r where R and R are the semiminor axes and x r, x r is the center of the ellipse. Fig. shows the simulation result. The dotted line is the reference signal and the solid line is the tricycle tracking trajectory. The semiminor parameters R and R are and. The initial positions are set at x, y, and x r, y r,. Fig. 4 shows the control inputs u and ν of the tricycle. Fig. shows the error signals of the tricycle position x, y. y [m],, x [m] B. Trajectory tracking control using ellipse-shaped reference models Fig.. Ellipse-shaped reference signal and the tricycle tracking trajectory Arbitrary tracking trajectory control can be realized using the ellipse-shaped tracking trajectory method. The controller design procedure is as follows: Assign passing points p x i, p y i, i,..., n. We consider the passing points:,,,, 6,, 8, and, 7 Construct some ellipses trajectories to connect the passing points smoothly. From, to,, the trajectory : x r cos θ, 6 sin θ where π/ θ π. From, to 6,, the trajectory : x r 6 cos θ, 7 sin θ ν u where π/ θ. From 6, to 8,, the trajectory : x r 8 cos θ 8, 8 sin θ Fig. 4. Control inputs u and ν of the tricycle where θ π/. From 8, to, 7, the trajectory 4: x r 6 cos θ 8, 9 sin θ 6 where π/ θ π/. Design the controllers for the ellipse tracking trajectories 6-9. We show a tracking trajectory control example for the tricycle robot system. Fig. 6 shows the reference signals 6-9 and the tricycle tracking trajectory. The dotted line is the reference signal and the solid line is the tricycle tracking trajectory. Fig. 7 shows the control inputs u and ν of the tricycle. Fig. 8 shows the error signals with respect to the tricycle position x, y. error of x error of y t Fig.. Error signals of the tricycle position x, y ISBN: ISSN: Print; ISSN: Online IMECS 7

6 Proceedings of the International MultiConference of Engineers and Computer Scientists 7 Vol I, IMECS 7, March - 7, 7, Hong Kong y [m] Fig. 6. ν [m] u [m] Fig ,7 Trajectory 4 Trajectory Trajectory, Trajectory 8, 6,, 4 6 x [m] Reference signals 6-9 and the tricycle tracking trajectory x [m] error of x [m] error of y [m] Control inputs u and ν of the tricycle VI. CONCUSIONS We have proposed a trajectory tracking controller design of a tricycle robot as a non-holonomic system with PM models. The approximated model is fully parametric. I/O dynamic feedback linearization is applied to stabilize PM control system. PM modeling with feedback linearization is a very powerful tool for analyzing and synthesizing nonlinear control systems. We also have applied a method for tracking controller to the tricycle robot. Although the controller is simpler than the conventional I/O feedback linearization controller, the tracking performance based on PM model is the same as the conventional one. Examples have been shown to confirm the feasibility of our proposals by computer simulations. REFERENCES [] B. d Andréa-Novel, G. Bastin, and G. Campion, Modeling and control of non holonomic wheeled mobile robots, in the 99 IEEE International Conference on Robotics and Automation, 99, pp.. [] R. Fierro and F.. ewis, Control of a nonholonomic mobile robot: backstepping kinematics into dynamics, in the 4th Conference on Decision and Control, 99, pp [] T.-C. ee, K.-T. Song, C.-H. ee, and C.-C. Teng, Tracking control of unicycle-modeled mobile robots using a saturation feedback controller, IEEE Transactions on Control Systems Technology, vol. 9, no., pp. 8,. [4] J. Guldner and V. I. Utkin, Stabilization of non-holonomic mobile robots using lyapunov functions for navigation and sliding mode control, in the rd Conference on Decision and Control, 994, pp [] J. Yang and J. Kim, Sliding mode control for trajectory tracking of nonholonomic wheeled mobile robots, IEEE Transactions on Robotics and Automation, pp , 999. [6] B. d Andréa-Novel, G. Bastin, and G. Campion, Dynamic feedback linearization of nonholonomic wheeled mobile robot, in the 99 IEEE International Conference on Robotics and Automation, 99, pp. 7. [7] G. Oriolo, A. D. uca, and M. Vendittelli, WMR control via dynamic feedback linearization: Design, implementation, and experimental validation, IEEE Transaction on Control System Technology, vol., no. 6, pp. 8 8,. [8] E. Yang, D. Gu, T. Mita, and H. Hu, Nonlinear tracking control of a car-like mobile robot via dynamic feedback linearization, in University of Bath, UK, no. ID-8, 4. [9] A. D. uca, G. Oriolo, and M. Vendittelli, Stabilization of the unicycle via dynamic feedback linearization, in the 6th IFAC Symposium on Robot Control,. [] M. Sugeno, On stability of fuzzy systems expressed by fuzzy rules with singleton consequents, IEEE Trans. Fuzzy Syst., vol. 7, no., pp. 4, 999. [] K.-C. Goh, M. G. Safonov, and G. P. Papavassilopoulos, A global optimization approach for the BMI problem, in Proc. the rd IEEE CDC, 994, pp [] T. Taniguchi and M. Sugeno, Piecewise bilinear system control based on full-state feedback linearization, in SCIS & ISIS,, pp [], Stabilization of nonlinear systems with piecewise bilinear models derived from fuzzy if-then rules with singletons, in FUZZ- IEEE,, pp [4], Design of UT-controllers for nonlinear systems with PB models based on I/O linearization, in FUZZ-IEEE,, pp [] T. Taniguchi,. Eciolaza, and M. Sugeno, ook-up-table controller design for nonlinear servo systems with piecewise bilinear models, in FUZZ-IEEE,. time Fig. 8. Error signals of the tricycle position x, y ISBN: ISSN: Print; ISSN: Online IMECS 7