Simple Adaptive Dynamical Control of Vehicles Driven ... - IEEE Xplore

ICCC2009 • IEEE 7th International Conference on Computational Cybernetics • November 26-29, 2009 • Palma de Mallorca, Spain

Simple Adaptive Dynamical Control of Vehicles Driven by Omnidirectional Wheels 1

J.K. Tar1 , I.J. Rudas2 , I. Nagy3

Telematics and Communications Informatics Knowledge Centre, 2 John von Neumann Faculty of Informatics, 3 Bánki Donát Faculty of Mechanical & Safety Engineering Budapest Tech, H-1034 Budapest, Bécsi u´ t 96/B, Hungary [email protected],[email protected],[email protected] 4

K.R. Kozłowski4

Chair of Control and Systems Engineering, Computing Science & Management, Poznań University of Technology, Piotrowo 3a, 60-965 Poznań, Poland [email protected] 5

J.A. Tenreiro Machado5

Department of Electrotechnical Engineering, Institute of Engineering of Porto, Rua Dr. Antonio Bernardino de Almeida, 4200-072 Porto, Portugal [email protected] localization measurements with the precision predefined by the user. In the so obtained useful navigation territory via skeletonisation “optimal trajectories” can be planned that are as far from the critical edges of the E-zone as possible. These “skeletons” can be refined by the usual spline functions to obtain smooth trajectories exempt of corners at which the vehicle should stop due to its dynamical inertia (e.g. [5]). The viable spline curves can replace the vertices of the graphs the edges of which can be weighted by the usual techniques (e.g. [3]) to measure the appropriateness of different possible trajectories that can connect the initial and ending points between which the robot has to move within the workspace. This weighting technique has been refined in [6] in which besides the edges of the graph the vertices are also weighted according to the appropriate turns the curvature of which can limit the maximum allowed velocity of the robot motion on the basis of dynamical considerations. The aim of the present paper is to make a step forward by filling in the formal framework offered by this improved weighting technique. For this the particular dynamic properties of a given mechanical system has to be investigated in details. For instance, the maximum allowed speed in a curve depends on the actual height of the mass center point of the vehicle that can turn over if the velocity exceeds this limit. In Section II the dynamic model and a PID type control of a vehicle of triangular shape driven by three omnidirectional wheels is detailed. In Section III the principles of a simple adaptive control using local basins of attraction instead of some more complicated Lyapunov function technique is briefed.

Abstract— Precise control of Automatic Guided Vehicles (AGVs) navigating between the aisles of manufacturing systems by the use of local markers is an important task. On the basis of the geometric model of the workspace and the vehicles and that of the sensor uncertainties precise trajectories were recently generated along which the vehicle safely can move. In order to achieve precise trajectory tracking the effects of the system’s dynamical uncertainties (modeling errors and possible external perturbations) have to be compensated. In the present paper a simple, fixed point transformations based adaptive control is proposed for this purpose. The proposed method is tested via simulation for a vehicle of triangular shape, driven by three omnidirectional wheels. The method is also able to monitor and evade the conditions that may lead to turning over the vehicle.

I. I NTRODUCTION The traditional algorithms developed for planning trajectories for mobile robots (e.g. the BUG –a wall tracing algorithm used by insects– and its variant TBA –Tangent Bug Algorithm– [1] share the common feature that each of them plans the trajectory in the close vicinity of the obstacles. Application of potential fields for navigation purposes also is a popular approach (e.g.[2]). To evade collisions that may happen due to the reduced precision of the localization of the robot these methods normally are combined with the use of a “safety zone” that corresponds to certain augmentation of the configured obstacles. The extent of increasing the size of the obstacles normally is not based on detailed calculations. Instead of that rough approximations are applied that may exclude useful but very narrow passages that would be taken into account in the case of more precise estimation of the localization errors or uncertainties. To improve the efficiency of navigation the proper extent of augmentation of the obstacles was calculated in [4] by taking into account the properties of the given marker- and measuring system by the use of which the so called “E-zone” was introduced. Within this zone the robot cannot carry out

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II. T HE DYNAMIC M ODEL OF THE V EHICLE AND THE PID- TYPE C ONTROL As is well known the conventional vehicles with Ackerman steering system suffer from significant limitation: due to geometric reasons it is impossible to simultaneously prescribe

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J. K. Tar et al. • Simple Adaptive Dynamical Control of Vehicles Driven by Omnidirectional Wheels

their precise location and orientation while tracking a given trajectory. For planning trajectories for such devices various practical techniques are available (e.g. [7]). In contrast to the Ackerman devices AGVs using omnidirectional wheels (e.g. [8]) can precisely track arbitrary trajectories at least from kinematic point of view. On this reason for our purposes a triangular structure similar to that in [8] was chosen as a paradigm. The AGV was supposed to have canonical triangular shape of side lengths L = 2 (m). The orientation of the active forces were supposed to be described by the orthogonal unit vectors eA , fA , eB , fB and eC , fC at wheels A,B, and C in the (x, y) plane in which the direction of the appropriate e vectors is identical to that of the straight line connecting the geometric center of the triangle with the appropriate vertices. These vectors were assumed to rigidly rotate around the vertical axis with angle q3 . Each wheel has the common constant vertical vector component ez along which the contact constraint forces originating from the ground act. At the vertices of the triangle three heavy wheels and drive systems were located of mass M = 30 (kg). It was assumed that further 5 · M (kg) mass was located over the geometric center of the triangle at the height of hD = 0.5 (m). The vehicle was assumed to move on a horizontal plane with prescribed nominal location of the N horizontal projection of its hypothetical mass center point S N (m) and nominal rotational pose q3 (rad) around the vertical axis. Utilizing the well known fact that the acceleration of the mass center point of a rigid body multiplied by its full mass is equal to the sum of the external forces acting on that system, and that the time-derivative of momentum of the system computed with respect to the actual mass center point is equal to the momentum of the external forces (torque) with respect to this point the required active driving force components FAeA , FAfA , FBeB , FBfB , and FCeC , FCfC , as well as the hypothetical vertical constraint force components FAz , FBz , and FCz can be calculated. The trajectory tracking rule was prescribed on purely kinematical basis using a PIDtype control, i.e. the “desired accelerations” t wereNdetermined D N N N ˙ ˙ ˙ ¨ as ξ = ξ +P (ξ −ξ)+D(ξ − ξ)+I t0 [ξ(τ ) −ξ(τ )]dτ with the same feedback parameters for ξ = S1Real , ξ = S2Real , Real and q Real denote the realized and ξ = q3Real in which S 3 position of the hypothetical mass center point and the realized rotational angle of the AGV. The appropriate equation is

¨ + g ) 5M (m) (S P˙ (m)

=

A D

B C E F

the hypothetical mass center point with the wheels in the points A,B, and C. Since in the commonly available omnidirectional wheels the small ones can only rotate without being able to exert contact forces in the direction of their rotation, the FAeA = 0, FBeB = 0, and FCeC = 0 conditions normally hold, and the big matrix in (1) can be reduced to a smaller one having unique inverse. By calculating it the FAfA , FBfB , and FCfC components can be obtained by the dynamic model and can be exerted by the drives, while the FAz , FBz , and FCz computed vertical force components can simply be dropped. These components will be determined by the actual motion of the system. Supposing that the AGV cannot turn over the equations of the realized motion can be summarized in (2) as ⎡ ⎤ −q˙32 Θ23 ⎢ ⎢ q˙32 Θ13 ⎥ ⎥ + −FABCef = H ⎢ ⎢ ⎣ ⎦ ⎣ 0 5M g 0 ⎡

(2)

in which the appropriate matrix elements of the actual inertia matrix Θ occur, and H = [ez ×xA , ez ×xB , ez ×xC ; 1, 1, 1, 0] i.e. it is a matrix of size 4 × 4, and FABCef := (eA × xA )FAeA + (eB × xB )FBeB + (eC × xC )FCeC + (fA × xA )FAfA + (fB × xB )FBfB + (fC × xC )FCfC . In the simulations the inverse of H was calculated to obtain the realized angular acceleration q¨3Real and the realized vertical constraint , FBReal , and FCReal . force components at the wheels as FAReal z z z If one of these components becomes negative the vehicle becomes apt to turn over. Besides this, the horizontal driving forces cannot be exerted in the lack of appropriate vertical pressing forces at the appropriate wheel. For safety reasons these components must be over a safety limit to evade turning up of the vehicle. On the basis of the approximate and the exact model proper simulation investigations can be carried out for the proposed PID control for various nominal trajectories to be tracked. As it can generally be expected proper increase in the feedback gains can reduce the tracking error even in the case of a very rough available dynamic model. Alternative, more intelligent approach is the application of adaptive control at reduced PID feedback gains that may introduce into the system less drastic transients in the initial phase of the control than the initial transients of a simple PID controller. Adaptive improvement of the simple PID-type controllers has a huge literature we cannot properly survey here being in lack of satisfactory room. The here proposed method corresponds to an “iterative” approach and from certain points of view is akin to that applied by Preitl & Precup in [9] for tuning Mamdani-type two-degreeof freedom PI-fuzzy controllers. However, in our case a far simpler structure and procedure is applied for the purposes of reaching adaptivity. It seems to be especially advantageous when the identification of the model parameters (e.g. that of the friction as in [10]) is a hard, difficult task. In the next section this adaptive controller is outlined.

⎡

⎤ Fe ⎣ Ff ⎦ Fz

⎤ FAReal z ⎥ FBReal ⎥ z Real ⎦ FC z q¨3Real

(1)

in which P (m) denotes the model momentum vector, g is the gravitational acceleration, and the block matrices of size 3 × 3 of the big matrix are defined as A = [eA , eB , eC ], B = [fA , fB , fC ], C = [0, 0, 0; 0, 0, 0; 1, 1, 1], D = [eA × (m) (m) (m) (m) (m) xA , eB ×xB , eC ×xC ], E = [fA ×xA , fB ×xB , fC × (m) (m) (m) (m) xC ], and F = [ez × xA , ez × xB , ez × xC ], Fe = T T [FAeA , FBeB , FCfC ] , Ff = [FAfA , FBfB , FCfC ] , and Fz = (m) (m) (m) [FAz , FBz , FCz ]T . The xA , xB , and xC vectors connect

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III. A DAPTIVE C ONTROL BASED ON THE E XCITATION R ESPONSE S CHEME Certain control task can be formulated by using the concepts of the appropriate ”excitation” Q of the controlled system to which it is expected to respond by some prescribed or ”desired response” rd . The appropriate excitation can be computed by the use of some inverse dynamic model Q = ϕ(rd ). Since normally this inverse model is neither complete nor exact, the actual response determined by the system’s dynamics, ψ, results in a realized response rr that differs from the desired one: rr ≡ ψ(ϕ(rd )) ≡ f (rd ) = rd . It is worth noting that the functions ϕ() and ψ() may contain various hidden parameters that partly correspond to the dynamic model of the system, and partly pertain to unknown external dynamic forces acting on it. The controller normally can manipulate or “deform” the input value from rd so that rd ≡ ψ(rd ). Such a situation can be maintained by the use of some local deformation that can properly “drag” the system’s state in time meandering along some trajectory. To realize this idea a fixed point transformation was introduced in [13] that is quite “robust” as far as the dependence of the resulting function on the behavior of f (•) is concerned. This robustness can approximately be investigated by the use of an affine approximation of f (x) in the vicinity of x and it is the consequence of the strong nonlinear saturation of the sigmoid function tanh(x): G(x|xd ) := := (x + K) 1 + B tanh(A[f (x) − xd ]) − K G(x |xd ) = x , if f (x ) = xd , then G(−K|xd ) = −K, (x+K)ABf (x) d + 1+ G(x|x ) = cosh(A[f (x)−xd ])2 d +B tanh(A[f (x) − x ]), G(x |xd ) = (x + K)ABf (x ) + 1.

|G(x ) − x | ≤ |G(x ) − xn | + |xn − x | = = |G(x ) − G(xn−1 )| + |xn − x | ≤ ≤ H|x − xn−1 | + |xn − x | → 0, xn → x .

(4)

A possible way of applying this simple idea elaborated for SISO to Multiple Input – Multiple Output (MIMO) systems is its separate application for each component of a vector valued r, rd . To mathematically substantiate this statement consider the fact that any contractive map acting in linear, normed, complete metric spaces (i.e. in Banach spaces) generally yields Cauchy sequences that must be convergent due to the completeness of the space. Therefore, all the above considerations can be applied for MIMO systems if instead n of the absolute n values the following norm is used for x ∈ : ||x|| := i=1 |xi |. If a multiple dimensional sigmoid function σ : n → n defined as yi = σ (i) (xi ) {i = 1, 2, ..., n}

(5)

in its each component is contractive, i.e. for ∀i ∃0 ≤ Mi < 1 so that |σ (i) (a) − σ (i) (b)| ≤Mi |a − b| then it can be n (i) (i) stated that ||σ (a) − σ (b)|| := i=1 |σ (ai ) − σ (bi )| ≤ n n [maxi=1 {Mi }] j=1 |aj − bj | ≡ M ||a − b||, 0 ≤ M < 1, i.e. it is contractive in the normed n space, too. A alternative possibility for applying the same idea outlined in (3) of adaptivity is the application of a sigmoid function projected to the direction of the response-error defined in the ˜ = nth control cycle as h := f(xn ) − xd , e := h/||h||, B σ(A||h||), so that

(3)

˜ xn + BK ˜ e. xn+1 = (1 + B)

(6)

(If ||h|| is very small, instead of normalizing with it the approximation xn+1 = xn can be applied since then the system already is in the very close vicinity of the fixed point.) In the adaptive control of the vehicle driven by three omnidirectional wheels both variants of this simple control idea was applied with the sigmoid function σ(x) := x/(1 + |x|). This approach is far simpler than the conventional, Lyapunov function based techniques e.g. the Slotine & Li adaptive controller (a comparison was given e.g. in [12]).

It is evident that the transformation defined in (3) has a proper (x ) and a false (−K) fixed point, but by properly manipulating the control parameters A, B, and K the good fixed point can be located within its basin of attraction, and the requirement of |G (x |xd )| < 1 can be guaranteed. This means that the iteration can have considerable speed of convergence even nearby x , and the strongly saturated tanh function can make it more robust in its vicinity, that is the properties of f (x) have less influence on the behavior of G. (Of course, instead of the function tanh any sigmoid function, i.e. any bounded, monotone increasing, smooth function can be used.) It is not difficult to show that in the case of Single Input – Single Output (SISO) systems the G(x|xd ) (or in a simpler notation the G(x)) functions can realize contractive mapping around x , i.e. the conditions of |G | ≤ H < 1 [0 ≤ H < 1] can be maintained by properly choosing the parameters of this function and if f is finite and its sign and absolute value can be estimated in the vicinity of x . Then the sequence of points {x0 , x1 = G(x0 ), ..., xn+1 = G(xn ), ...} obtained via iteration form a Cauchy Sequence that is convergent (xn → x ) in the real numbers and converge to the solution of the Fixed Point Problem x = G(x ):

IV. C OMPUTATIONAL R ESULTS To represent unknown external perturbations in the simulations a weighty “point-like” burden of mass Mcoupl = 15 kg was assumed to be connected to one of the wheels by a spring of stiffness k = 1000 N/m and L0 = 1 m zero force length as a dynamically coupled sub-system unknown by the controller. It was assumed to move over the plain by a viscous damping constant μ = 5 N s/m. In the 1st set of computations non-adaptive control was simulated with the PID parameters P = 25/s2 , D = 10/s, and I = 5/s3 for a nominal trajectory containing many smaller turns. Representative results are given in Fig. 1. It can be seen that the vehicle should roll over in several stages of the motion in the non-adaptive case.

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J. K. Tar et al. • Simple Adaptive Dynamical Control of Vehicles Driven by Omnidirectional Wheels

Nominal and Computed Trajectories on the Plane (x,y)

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Fig. 1. The results of the non-adaptive control: position and orientation tracking (1st row); position and orientation error (2nd row); acceleration and rotational acceleration (3rd row); vertical constraint forces and power consumption of the system (4th row) vs. time

10

20

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Fig. 2. The results of the adaptive orientation and position control according to the realization (5): position and orientation tracking (1st row); position and orientation error (2nd row); acceleration and rotational acceleration (3rd row); vertical constraint forces and power consumption (in 104 W ) of the system (4th row) vs. time

Fig. 2 is the counterpart of Fig. 1 in the sense that in its case adaptive control parameters were introduced for tracking of the orientation as Kq3 = −4000, Bq3 = 1, and Aq3 = 10−4 , and the position as Kq3 = −4000, Bq3 = 1, and Aq3 = 10−6 in the multi-dimensional sigmoid defined in (5). Figure 2 reveals improvement in both the position and orientation control, and in the control of the appropriate accelerations. The conditions of rolling over are met only in the beginning of the motion in which the connected subsystem is considerably excited. (Of course it is easy to find example for stable motion by the reduction of the speed of the cart along the same nominal trajectory.) The more quiet motion of the controlled cart does not cause serious excitation of the connected system the motion of which is also curbed by friction effects. Practically the tracking errors and acceleration errors of the non-adaptive motion can serve as unified measures of the significance of the errors in the dynamic model and the external perturbations. They reduction by the adaptive control can also serve as the measure of adaptivity. In this sense turning on the adaptive control seriously improves the behavior of the controlled system. The second implementation-variant of the same adaptive control idea defined by (6) yields simultaneously very good results for both the orientation and the position tracking [Fig. 3]. Its superiority becomes evident by comparing Figs. 2

and 3. The precision of the position became drastically better without corrupting the precision of the orientation. It is also worth noting that the adaptive method works well for quite extreme nominal trajectories in which the vehicle’s assumed mass center point actually stops while the orientation very sharply varies [Fig. 4]. This trajectory is similar to the horn of a ram. The center point in Fig. 4 corresponds to the most irregular behavior. At this point the absolute value of the tracking and orientation error have maximums that quickly relax as the vehicle leaves the “critical point”. In this point drastic variation happens in the horizontal constraint forces and in the horizontal acceleration of the assumed mass center point. V. C ONCLUSIONS In this paper a dynamic simulation was carried out for the control of a triangle-shaped vehicle driven by three omnidirectional wheels. This system can be controlled in principle by the use of some GPS. It was shown that a special, simple, fixed point transformations based adaptive control investigated in two variants can cause quite precise trajectory and orientation tracking. The proposed approach has limited number of control parameters and far more easily can be constructed than e.g. a traditional Lyapunov function based solution.

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The proposed method was investigated via numerical simulations. The conditions needed for the evasion of turning over the vehicle can be checked by monitoring the minimal vertical constraint force component that must be positive. The simulation program here reported can be used for qualifying the graphs of the possible trajectory in a special weighting technique that can take into account kinematic and dynamic details for mobile robots navigating within a safe zone. Furthermore, the allowable maximum velocity at which turning up the vehicle is just evaded can be found by this program.


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ACKNOWLEDGMENT


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The authors gratefully acknowledge the support by the National Office for Research and Technology (NKTH) using the resources of the Research and Technology Innovation Fund within the projects No. OTKA-K 063405, OTKA-CNK 78168 and No. RET-10/2006. We also express our thanks for the support our research obtained from the Bilateral Science & Technology Research Projects No. PT-12/2007 and No. PL14/2008.

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R EFERENCES

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[1] T. Lozano-Perez: Spatial planning: a configuration space approach. IEEE Trans. on Computers Vol. 32, 1983. [2] J. Vaˇscˇ a´ k: Navigation of Mobile Robots Using Potential Fields and Computational Intelligence Means, Acta Polytechnica Hungarica Vol. 4, No. 1, 2007. [3] G. Tan, D. Mamady: Real-Time Global Optimal Path Planning of Mobile Robots Based on Modified Ant System Algorithm. ICNC 2006, Part II, LNCS 4222, Springer-Verlag Berlin Heidelberg 2006, pp. 204–214 [4] I. Nagy: Path Planning Algorithm Based on User Defined Maximal Localization Error. Periodica Polytechnica, BUTE, Hungary, 2005, pp.43– 57. [5] U. Reif: Best bounds on the approximation of polynomials and splines by their control structure. Computer Aided Geometric Design, Vol.17 No.6, 2000, pp. 579589. [6] I. Nagy, L. Vajta: Local Trajectory Optimization Based on Dynamical Properties of Mobile Platform. In the Proc. of the 2001 IEEE International Conference on Intelligent Engineering Systems (INES 2001), Helsinki, Finland, September 16-18, 2001, pp. 285–290 [7] R. M. Murray, S. Sastry: Nonholonomic motion planning: Steering using sinusoids. IEEE Transactions on Automatic Control, Vol. 38 No. 5, pp. 700–716, 1993. [8] J.M. Holland: Basic Robotics Concepts. Howard W. Sams, Macmillan, Inc., Indianapolis, IN., 1983. [9] S. Preitl, R.-E. Precup: Iterative Feedback Tuning in Fuzzy Control Systems. Theory and Applications, Acta Polytechnica Hungarica Vol. 3, No. 3, 2006. [10] L. Márton, B. Lantos: Identification and Model-based Compensation of Striebeck Friction, Acta Polytechnica Hungarica Vol. 3, No. 3, 2006. [11] J.K. Tar, I.J. Rudas and K.R. Kozłowski: Fixed Point TransformationsBased Approach in Adaptive Control of Smooth Systems. Lecture Notes in Control and Information Sciences 360 (Eds.: M. Thoma and M. Morari), Robot Motion and Control 2007 (Ed.: Krzysztof R. Kozłowski), Springer Verlag London Ltd., pp. 157–166, [12] J.K. Tar, I.J. Rudas, Gy. Hermann, J.F. Bit, J.A. Tenreiro Machado: On the Robustness of the Slotine-Li and the FPT/SVD-based Adaptive Controllers. WSEAS Transactions on Systems and Control, Issue 9, Volume 3, September 2008, pp. 686–700 [13] J.K. Tar, J.F. Bitó, I.J. Rudas, K.R. Kozłowski, J.A. Tenreiro Machado: Possible Adaptive Control by Tangent Hyperbolic Fixed Point Transformations Used for Controlling the Φ6 -Type Van der Pol Oscillator. In: Proc. of the 6th IEEE International Conference on Computational Cybernetics (ICCC 2008), November 27–29, 2008, Stará Lesná, Slovakia, pp. 15–20.

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Fig. 3. The results of the adaptive orientation and position control according to the realization (6): position and orientation tracking (1st row); position and orientation error (2nd row); acceleration and rotational acceleration (3rd row); vertical constraint forces and power consumption of the system (4th row) vs. time



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Fig. 4. Fully adaptive control according to the realization (5) for very “sharp” trajectory: the position and orientation tracking of the trajectory (1st row); the vertical constraint forces and the acceleration of mass center point of the approximate model (2nd row); the error in the position and pose (3rd row) vs. time

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