Ellipsoidal Approximations of the Attraction Domain in the Path ...

ISSN 10642307, Journal of Computer and Systems Sciences International, 2012, Vol. 51, No. 4, pp. 602–615. © Pleiades Publishing, Ltd., 2012. Original Russian Text © A.V. Pesterev, 2012, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2012, No. 4, pp. 131–144.

ROBOTICS

Ellipsoidal Approximations of the Attraction Domain in the Path Following Problem for a Wheeled Robot with Constrained Resource A. V. Pesterev Institute of Control Sciences, Russian Academy of Sciences, ul. Profsoyuznaya 65, Moscow, 117997 Russia Javad GNSS, LLC, The Triumph Palace, Moscow, 125057 Russia Received, December 9, 2011; in final form, February 3, 2012

Abstract—The path following problem for a wheeled robot with constrained resource moving along a given curvilinear path is studied. With the help of an earlier introduced change of variables, the path following problem is reduced to that of stability of the zero solution, and a control law linearizing the system in the case of the unconstrained control resource is synthesized. For the closedloop system, the problem of finding the best ellipsoidal approximation of the attraction domain of the target path is set. To take into account the control constraint, an approach based on absolute stability theory is used. In the framework of this approach, construction of an approximating ellipse reduces to solving a parameterized system of linear matrix inequalities. The LMI system in the considered case can be solved analytically. Owing to this, construction of the best ellipsoidal approximation is reduced to solv ing a standard constrained optimization problem for a function of two variables. The proposed method is further extended to finding the best ellipsoidal approximation with an additional constraint on the maximum deviation from the target path. The discussion is illustrated by numerical examples. DOI: 10.1134/S1064230712040107

INTRODUCTION There are many applications (e.g., in agriculture [1, 2] or road construction) where a vehicle is to be automatically driven along some target curve (target path) with high level of accuracy. Such tasks are per formed by automated vehicles (further referred to as wheeled robots (WRs) or simply robots) equipped with satellite and inertial navigation tools and antennas [1–3]. The problem of bringing the robot from an ini tial state to a preassigned path and stabilizing its motion along the path is called path following problem, and its solution was discussed in a great number of publications (see [1–6] and references therein). In this paper, we consider the wellknown kinematic model of a WR moving without slippage, which is described by three nonlinear differential equations (see, for example, [1, 2, 5, 7, 8]). In view of essential nonlinearity of the considered problem, it is not guaranteed that the synthesized control is capable of bringing the robot to an arbitrary curvilinear path from an arbitrary initial configuration. Therefore, in practice, it is desirable to have a criterion that would allow one to check in the course of motion whether the robot state belongs to the attraction domain of the target path, or, in other words, whether the synthe sized control can stabilize motion of the robot along the given path. For such a criterion, it is suggested to construct ellipsoidal estimates of the attraction domain of the target path or its separate segments [8–11]. The desired ellipsoids are found by applying the approach proposed in [8] for the case where the target path is a straight line or an arc of a circle. The approach is based on the absolute stability theory and reduces construction of the ellipsoid approximating the attraction domain to solving a system of linear matrix inequalities (LMIs) depending on a parameter. In so doing, the stabilization problem reduces to that of stability of the zero solution of a system of equations in deviations. In [9], the abovementioned approach was applied to an arbitrary curvilinear target path with the preliminary use of the change of vari ables suggested in [12]. The problem of constructing the ellipsoidal estimate of maximum volume was considered in [10, 11]. For the control law synthesized in [12, 13], construction of the best ellipsoidal approximation was reduced to a standard constrained optimization problem for a function of two vari ables. The goal of this work is to construct similar estimates for the system closed by the control synthe sized in the recent paper [14]. We will also discuss construction of the best estimates under the additional constraint on the maximum deviation from the target path. 602

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1. PROBLEM STATEMENT The wheeled robot considered in this work is a vehicle moving without lateral slippage with two rear driving wheels and front wheels responsible for steering the platform. In the planar case, the robot position is described by two coordinates (xc, yc) of some point of the platform, the socalled target point, and one angle describing orientation of the platform with respect to a fixed reference system xOy. For the target point, the point located in the middle of the rear axle is taken, and for the angle, the angle θ between the central line of the platform (which coincides with the direction of the velocity vector) and the xaxis. The kinematic equations of robot’s motion are well known to be (see, for example, [1, 2, 5, 7, 8]) and, there fore, are given without derivation: x· c = v cos θ, y· c = v sin θ, · θ = vu.

(1.1)

Here, the dot over symbols denotes differentiation with respect to time, v is a scalar linear velocity of the target point, and u is the instant value of the curvature of the trajectory described by the target point. The curvature u is uniquely related to the turning angle of the front wheels ϕ (although turning angles ϕ1 and ϕ2 of the two front wheels are different, they are related to one another and the path curvature, so that there exists an “efficient mean” angle ϕ [8, 12]) by the equation u = tan ϕ /L, where L is the distance between the front and rear axles. This relationship between the turning angle and the curvature of the trajectory described by the target point simplifies the model and allows us to take curvature u for the control. Bound edness of the turning angle of the front wheels results in the following twosided phase constraints: – u ≤ u ≤ u ( – ϕ max ≤ ϕ ≤ ϕ max ), where 0 < ϕmax < π/2 and u = tan ϕ max /L is the maximal possible curvature of an actual trajectory. It is required to synthesize a control law u that brings the robot to a given target path and stabilizes its motion along the curve. The target curve (path) is given in a parametric form by a pair of functions (X(s), Y(s)), where s is a natural parameter (arc length), and is assumed to be feasible. The latter means that functions X(s) and Y(s) are twice differentiable [4] everywhere except for a finite number of points, and the maxi mum curvature k = maxsk(s) of the target path satisfies the constraint k < u [9–14]. In [14], the following change of variables is suggested: z1 is the deviation of the WR from the target curve (signed distance1 to the point on the target curve that is closest to the robot) and z2 = tan ψ , where ψ is the angular deviation (angle between the velocity vector and the tangent line to the target curve at the curve point closest to the robot). The independent variable ξ is defined on the trajectories of the system in the region –π/2 < ψ < π/2 by the integral t

ξ(t) =

∫ v ( τ ) cos ψ ( τ ) dτ. t0

By means of this change of variables, the problem of stabilizing motion of the WR along a given path reduces to the stabilization of the zero solution of the system z '1 = z 2 , (1.2)

2

( 1 + z 2 )k 2 3/2 z 2' = ( 1 + z 2 ) u –  , 1 – kz 1

where the prime denotes differentiation with respect to ξ. System (1.2) is defined in the open set Dz = (–1/ k , 1/ k ) × R1 of the coordinate space R2 [14]. 1 The

problem of finding distance from a point to a curve can generally be solved differently for curves of different types and is not discussed here. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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Suppose that the control resource is not constrained. Then, as can easily be seen, by applying the feed back 3 k cos ψ –  σ(z)  k u = – σ ( z ) cos ψ +  ≡ + , 2 3/2 2 1 – kd ( 1 + z2 ) 1 + z 2 ( 1 – kz 1 )

(1.3)

where σ(z) is a linear function, σ(z) = cTz, cT = [c1, c2], c1, c2 > 0, we obtain the linear closedloop system. It is proved [14, Theorem 1] that the zero solution of system (1.2) closed by feedback (1.3) is asymptoti I cally stable for any initial points belonging to an invariant set D z ⊆ Dz of the linear closedloop system. In the case of a constrained control resource, we redefine control (1.3) setting it equal to u or – u if the righthand side of (1.3) is greater than u or less than – u , respectively, i.e., consider the control ⎛ ⎞ σ(z) k u = sat u ⎜ –   +  ⎟ , 3/2 2 ⎝ ( 1 + z 22 ) 1 + z ( 1 – kz )⎠ 2

(1.4)

1

where sat u (u) is the saturation function equal to u when |u| ≤ u and sign(u) u when |u| > u , and restrict our consideration to the functions σ(z) of the form σ(z) = λ2z1 + 2λz2, λ > 0. Under such selection of func tion σ(z) in feedback (1.3), the linear closedloop system has pole –λ of multiplicity two. In [14, Theorem 2], it is proved that, in the case of a straight target path, the zero solution of system (1.2) closed by feedback (1.4) is asymptotically stable for any λ > 0 and any initial conditions (z1(0), z2(0)) ∈ R2. In terms of the original problem statement, this means that the constrained control (1.4) stabilizes motion of the WR along the straight path for any initial conditions unless the initial direction of motion is perpendicular to the target line. Clearly, the last result cannot be extended to the case of an arbitrary target path, at least, because system (1.2) is not defined in the entire space R2. Therefore, construction of an estimate of the attraction domain of the zero solution for a given class of target curves is an important problem: belonging of the sys tem state to this domain guarantees that motion of the WR with a constrained control resource can be sta bilized by means of feedback (1.4) for any target curve from the given class. This work is devoted to constructing ellipsoidal estimates T

Ω = { z : z Pz ≤ 1 },

P > 0,

(1.5)

of the invariant attraction domain of the zero solution of system (1.2) closed by control (1.4). To this end, we apply an approach proposed in [8], which is based on the analysis of absolute stability [15]. In the framework of this approach, construction of the ellipsoid approximating the attraction domain reduces to solving an LMI system depending on a parameter. In the considered twodimensional case, our task is facilitated in that the LMI system can be solved analytically. Such a solution was obtained by the author of this paper in [10]. With its help, finding of the “best” ellipsoidal approximation of the attraction domain in the problem of stabilizing a WR following a curvilinear path was reduced to solving a standard con strained optimization problem for a function of two variables.2 The feedback used in [10] was synthesized based on the change of variables proposed in [12] (see also [7, 9]) and is different from that employed in this study. Therefore, the first goal of this study is an extension of the basic result of work [10]—construc tion of the “best” ellipsoidal estimate of the attraction domain—to the WR with feedback (1.4). We will also consider the problem of finding the best ellipsoidal approximation under an additional constraint on the maximum deviation from the target path. Such a constraint is frequently met in practice, for example, in agricultural applications where room for maneuver is constrained by adjacent beds and/or natural or artificial obstacles, such as roads, fences, rivers, and the like. Therefore, the problem of constructing an estimate of the set of initial conditions for which the synthesized control guarantees asymptotic decrease of the deviation and trajectory of the robot remains in the prescribed strip is of practical significance. 2 It should be noted that analytical solution of the secondorder LMI systems was studied in many publications. The majority of

these works are devoted to finding necessary and sufficient conditions (criteria) for the existence of a common quadratic Lyapunov function for a finite number of secondorder linear systems (see, for example, [16, 17] and references therein). The results reported in [10] are related to a particular system of two LMIs and are obtained independently. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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2. REDUCTION OF THE PROBLEM TO SOLVING AN LMI SYSTEM Let us rewrite system (1.2) closed by feedback (1.4) in the form z '1 = z 2 ,

(2.1)

z '2 = – Φ ( σ, z ), where ⎞ k ( 1 + z 22 ) 2 3/2 ⎛ k σ Φ ( σ, z ) = ( 1 + z 2 ) s u ⎜   –   ⎟ + . 2 1 – kz 1 ⎝ ( 1 + z 22 ) 3/2 1 + z 2 ( 1 – kz 1 )⎠

It is obvious that, for any admissible path, Φ(0, 0) = 0. Then, it follows that z = 0 is solution of system (2.1) and, on the strength of the strict inequality k < u , condition Φ(σ, z) = σ holds in a neighborhood of the origin; i.e., the system is linear at least in some neighborhood of the origin. Let us introduce the notation σ 0 = max σ ( z ) , z∈Ω

U 0 = min U ( z ), z∈Ω

2 3/2

U ( z ) = u ( 1 + z2 )

2

k ( 1 + z2 ) –  . 1 – k z1

Let, for some 0 < β0 ≤ 1, the inequality β0 σ0 ≤ U0

(2.2)

holds in Ω. Then, it is not difficult to show (see also [9, 10]) that the plot of the righthand side of the sec ond equation in (2.1) considered as a function of σ lies in the sector formed by the straight lines Φ = σ and Φ = β0σ for any z ∈ Ω. Maximum of the linear function σ(z) = cTz on the ellipsoid is found analytically: as shown in [8], σ0 =

T

–1

c P c,

(2.3)

where P –1 is the inverse of matrix P determining the ellipsoid. Unlike σ0, the exact minimum U0 of the nonlinear function U(z) on the ellipsoid cannot be found analytically. Therefore, instead of minimum U0 of U(z) on the ellipsoid, we will use its lower bound k , ˜ 0 = u –  U 1 – kα 1

(2.4)

˜ 0 ≤ U in D , the fulfillment of the inequality β σ ≤ U ˜ 0 guarantees the ful where α1 = maxΩ|z1|. Since U 0 z 0 0 fillment of inequality (2.2). Along with the problem of stability of the zero solution of the nonlinear system (2.1), we consider the problem of absolute stability of the zero solution of the linear nonstationary system z '1 = z 2 ,

(2.5)

z '2 = φ ( ξ, σ ),

where φ(ξ, σ) = β(ξ)σ and the parametric perturbation β(ξ) satisfies the conditions of existence of an absolutely continuous solution of system (2.5). The range of perturbation variation is defined to be β 0 ≤ β ( ξ ) ≤ 1.

(2.6)

By the definition of system (2.5), the plot of function φ(ξ, σ) lies in the same sector where the plot of function –Φ(σ, z) lies. Evidently, any solution of the nonlinear system is also a solution of the linear non stationary system (2.5) under an appropriate choice of the perturbation. Let us denote the matrices of the JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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linear systems with constant coefficients corresponding to the boundaries of the sector [β0, 1] as A β0 and A, respectively, ⎛ 0 1 ⎞ A = ⎜ ⎟. ⎝ – λ 2 – 2λ ⎠

⎛ 0 1 ⎞ A β0 = ⎜ ⎟, ⎝ – λ 2 β 0 – 2λβ 0 ⎠

It is known [15, 18, 19] that stability of the zero solution of system (2.5) for all possible β(ξ) of form (2.6) implies stability of the zero solution of system (2.1) (and, hence, that of system (1.2)) closed by feedback (1.4) for initial conditions z1(0), z2(0) ∈ Ω. In turn, in order that the zero solution of system (2.5) be stable for all possible φ(ξ, σ) of the considered form, it is sufficient that the linear matrix inequalities T

PA + A P ≤ 0,

(2.7)

T

PA β0 + A β0 P ≤ 0

(2.8)

hold, i.e., there exists a common quadratic Lyapunov function for the linear systems with the matrices A β0 and A. Thus, construction of an ellipsoidal estimate of the attraction domain of the zero solution of system (2.1) reduces to finding a positive definite matrix P and parameter β0 satisfying the LMI system (2.7), (2.8) for which, in set (2.5), the scalar inequality (2.2) holds.3 Indeed, the fulfillment of the scalar inequality (2.2) guarantees that the righthand side of system (2.1) for ∀z ∈ Ω belongs to the sector [β0, 1] and, hence, the study of stability of the nonlinear system can be replaced by the study of absolute stability of the linear nonstationary system (2.5). At the same time, the fulfillment of the matrix inequalities (2.7) and (2.8) guarantees that the ellipsoid Ω defined by formula (1.5) is an invariant attraction domain of the zero solu tion of system (2.5). Solution of the LMI system is easily found numerically (for example, by means of the solvers available in the Robust Control Toolbox in Matlab). Since, in the general case, the fulfillment of inequality (2.2) can be verified only after the ellipsoid has been constructed, the estimate is to be sought iteratively. For this reason, solution of optimization problems (when it is required to construct, in a sense, the “best” ellipsoid) seems problematic in view of nontrivial implicit dependence of the set of feasible matrices P (defined by inequalities (2.2), (2.7), and (2.8)) on parameter β0. In the twodimensional case, however, efficient solution of optimization tasks is possible through the use of an analytical solution of the LMI system (2.7), (2.8) reported in [10]. Some results from [10] needed for the subsequent discussion and adapted with regard to the specific features of the system under consid eration are presented in the next section. 3. ANALYTICAL SOLUTION OF THE LMI SYSTEM It was proved in [10, Theorem 1] that the LMI system (2.7), (2.8) has solutions for any λ and β0 satis fying the condition 1/9 ≤ β0 ≤ 1 and that any solution can be represented in the form ˜ = ⎛⎜ λp˜ 1 p˜ 2 /2 ⎞⎟ , P ⎝ p˜ 2 /2 p˜ 3 /λ ⎠

1 ˜ P =  P , ˜ detP

(3.1)

where p˜ 1 , p˜ 2 , and p˜ 3 are coordinates of an arbitrary point of the threedimensional space belonging to the intersection of the solution cones of inequalities (2.7) and (2.8). The solution cone of inequality (2.7) is given by either of the following two equivalent inequalities: 2 2 2 ( p˜ 3 – p˜ 2 – p˜ 1 ) + ( p˜ 2 – 2p˜ 1 ) ≤ 4p˜ 1 ,

(3.2)

or 2

2

2

( p˜ 1 – p˜ 2 – p˜ 3 ) + ( p˜ 2 – 2p˜ 3 ) ≤ 4p˜ 3 .

(3.3)

3 This result was obtained in [10]; a similar result for the threedimensional case was presented in [9, Theorem 1].

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The solution cone of inequality (2.8) is determined by the condition 2

2

2

β 0 ( p˜ 3 – p˜ 2 – p˜ 1 /β 0 ) + ( p˜ 2 – 2p˜ 1 ) ≤ 4p˜ 1

(3.4)

or, equivalently, 2 2 2 1  ( p˜ 1 /β 0 – p˜ 2 – p˜ 3 ) + ( p˜ 2 /β 0 – 2p˜ 3 ) ≤ 4p˜ 3 . β0

(3.5)

As can be seen from (3.2)–(3.5), the solution cones of both inequalities written in terms of the coordi nates p˜ 1 , p˜ 2 , and p˜ 3 do not depend on λ. Since solutions of an LMI are determined up to a scalar factor, the number of variables can be reduced ˜ reduces to finding a point belonging to the intersection of two to two, and finding of a feasible matrix P ellipses obtained by crosssectioning the cones by a plane. In [10], for the cut plane, the plane p˜ 1 = const was taken, and the equations of the elliptic sections were obtained from (3.2) and (3.4) by dividing both 2 sides of the inequalities by p˜ 1 (clearly, p˜ 1 ≠ 0 everywhere except for the origin), with the equations of the elliptic sections being written in terms of the variables q 1 = p˜ 2 /p˜ 1 ,

q 2 = p˜ 3 /p˜ 1 .

(3.6)

It is shown in [10] that the maximum values α1 and α2 of the coordinates z1 and z2 on any ellipse Ω con structed by means of the matrix P of form (3.1) depend only on p˜ 3 and p˜ 1 , respectively, α 1 ≡ max z 1 = Ω

p˜ 3 /λ,

α 2 ≡ max z 2 = Ω

λp˜ 1 .

(3.7)

Thus, for p˜ 1 = const, all ellipses are inscribed in the strip |z2| ≤ α2, where α2 = λp˜ 1 , and, for p˜ 3 = const, in the strip |z1| ≤ α1, where α1 = p˜ 3 /λ . In Section 4, we consider the problem of finding the “best” ellipsoidal approximation in the strip |z1| ≤ α1. Therefore, it is more convenient to deal with the elliptic sections in the plane p˜ 3 = const. Equations of these sections are obtained in the same way as it was done in [10]. Dividing both sides of inequalities (3.3) 2 and (3.5) by p˜ 3 and introducing the notation q˜ 1 = p˜ 2 /p˜ 3 ,

q˜ 2 = p˜ 1 /p˜ 3 ,

(3.8)

we obtain the following equations of the desired sections: 2 2 ( q˜ 2 – q˜ 1 – 1 ) + ( q˜ 1 – 2 ) ≤ 4

(3.9)

and 2 2 q˜ q˜ 1 ⎛ 2 – q˜ 1 – 1⎞ + ⎛ 1 – 2⎞ ≤ 4. ⎠ ⎝ β0 ⎠ β0 ⎝ β0

(3.10)

For brevity, we will refer to the ellipse obtained by sectioning the solution cone of the LMI (2.8) by some plane as the β0ellipse in the corresponding section. Comparing (3.6) and (3.8), we easily find for mulas for transforming the “old” variables q into the “new” ones: q1 q˜ 1 = , q2

1 q˜ 2 = . q2

(3.11)

Note also that equation (3.9) of the section of the solution cone of the LMI (2.7) is invariant with respect to the abovespecified transformation of the coordinates q (cf. [10, formula (2.6) and Fig. 2]). Figure 1 shows ellipse (3.9) (bold line) and the three ellipses (3.10) corresponding to β0 = 0.8, 0.4, and 1/9. The last ellipse (in the left lower corner of the figure) has only one common point with ellipse (3.9). This is the point q˜ 1 = 2/5, q˜ 2 = 1/5 (the point q1 = 2, q2 = 5 in terms of the “old” coordinates [10, The orem 1]). If β0 < 1/9, the ellipses do not intersect. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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4 3 2

β0 = 0.4

1

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0 q1

Fig. 1. Crosssections of the solution cones of the LMIs (2.7) (bold line) and (2.8) by the plane p˜ 3 = const for β0 = 0.8; 0.4; and 1/9.

In the subsequent discussions, we need the following lemma. L e m m a 1. Let q˜ 1 , q˜ 2 be coordinates of an arbitrary point belonging to ellipse (3.9). The least value of β0 for which inequality (3.10) is also satisfied is given by the formula ⎧ 2q˜ 1 q˜ 2 + 2q˜ 2 – q˜ 21 – q˜ 1 ( q˜ 1 – 4q˜ 2 ) ( q˜ 21 – 4q˜ 2 ) ⎪ , 2 2 ( 1 – q˜ 1 ) ˜β ( q˜ , q˜ ) = ⎪ 0min 1 ⎨ 2 ⎪ q˜ 2 2 ⎪  , q˜ 1 = 1. ⎩ 4q˜ 2 – 1

q˜ 1 ≠ 1, (3.12)

P r o o f. The lemma follows directly from [10, Lemma 3]. Let us consider an arbitrary point q˜ 1 , q˜ 2 that simultaneously belongs to ellipse (3.9) and to the boundary of ellipse (3.10). Having performed substitu tion (3.11) in the latter equation, after simple transformations, we obtain the equation 2

2

β 0 ( q 2 – q 1 – 1/β 0 ) – ( q 2 – q 1 – 1 ) = 4, which is the equation of the boundary of the section obtained by cutting the solution cone of LMI (2.8) by the plane p˜ 1 = const [10, formula (31)]. Thus, the problem of finding the desired minimal β0 reduces to a similar problem in the plane p˜ 1 = const, which was solved in [10, Lemma 3]. Substituting q˜ 1 , q˜ 2 for q1, q2 into the righthand side of formula (34) in [10], we obtain formula (3.12). Lemma 1 gives us the least value of β0 for which a given feasible matrix P satisfies the LMI system (2.7), (2.8). Finally, let us present the formula for σ0 in terms of q˜ 1 and q˜ 2 . It is easy to check (see also formula (41) in [10]) that the substitution of the inverse matrix P –1, where P is defined by (3.1), into formula (2.3) yields 3 σ0 = λ ( p˜ 3 – 2p˜ 2 + 4p˜ 1 ) . From this formula, using definition (3.8) of variables q˜ 1 and q˜ 2 , we obtain

σ0 = α1 λ

2

2 1 – 2q˜ 1 + 4q˜ 2 ≡ α 1 λ σ 0 ( q˜ 1, q˜ 2 ).

(3.13)

4. FINDING THE “BEST” ELLIPSOIDAL ESTIMATE OF THE ATTRACTION DOMAIN Let us apply the approach proposed in [10] to finding the maximumarea ellipsoidal approximation of the attraction domain. Since the area of an ellipse is inversely proportional to the square root of the deter minant of matrix P determining the ellipse [20], of interest is the matrix P with the least determinant, or, JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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˜ with the greatest determinant. Such a matrix P ˜ will be referred to as the by virtue of (3.1), the matrix P matrix of the “best” ellipsoidal approximation of the attraction domain of the zero solution of system (2.1). Consider a ray in the space of three variables p˜ 1 , p˜ 2 , and p˜ 3 belonging to the solution cone of the LMI (2.7). Such a ray is uniquely determined, for example, by the coordinates q˜ 1 and q˜ 2 of the point of intersection of the ray by the plane p˜ 3 = 1. Ellipses Ω corresponding to different points belonging to the ray have iden ˜ differ by tical shapes and differ from one another only in terms of size, and the corresponding matrices P scalar factors and can be represented in the form ⎛ 2 ˜ 2 λq˜ 1 /2 2⎜ λ q ˜ ˜ ˜ P ( α 1, q 1, q 2 ) = α 1 ⎜ ˜ ⎝ λq 1 /2 1

⎞ ⎟, ⎟ ⎠

(4.1)

˜ grows monotonously as α where point q˜ 1 , q˜ 2 belongs to ellipse (3.9) (the bold line in Fig. 1). Since det P 1 increases, the maximum area of the ellipse is achieved at the greatest value of α1 for which inequality (2.2) still holds. Then, with regard to Lemma 1 and formula (3.13), we arrive at the following equation in the desired greatest value of α1: ˜ 0 ( α )/α = λ 2 β˜ 0min ( q˜ , q˜ )σ 0 ( q˜ , q˜ ). U 1 1 1 2 1 2

(4.2)

Substituting (2.4) into (4.2), we obtain the linear u = α1Q( q˜ 1 , q˜ 2 ) (for k = 0) or quadratic 2 k Q( q˜ 1 , q˜ 2 ) α 1 – (Q( q˜ 1 , q˜ 2 ) + uk )α1 + u – k = 0 (for k ≠ 0) equation, where Q( q˜ 1 , q˜ 2 ) = λ2 β˜ ( q˜ , q˜ ) σ ( q˜ , q˜ ). From these equations, the desired α is easily found to be4 0min

1

2

0

1

1

2

⎧ u , k = 0, ⎪  ˜ ˜ ⎪ Q ( q 1, q 2 ) α 1 ( q˜ 1, q˜ 2 ) = ⎨ 2 2 ⎪ Q ( q˜ 1, q˜ 2 ) + uk – ( Q ( q˜ 1, q˜ 2 ) – uk ) + 4k Q ( q˜ 1, q˜ 2 ) ⎪ , 2kQ ( q˜ 1, q˜ 2 ) ⎩

(4.3) k ≠ 0.

The second solution of the quadratic equation (with the plus sign before the square root) exceeds 1/ k , i.e., lies in the region where system (1.2) is not defined, and, thus, is not admissible. Substituting α1( q˜ 1 , q˜ 2 ) ˜ is uniquely determined by the found for α1 into (4.1), we find that the determinant of the desired matrix P ˜ ≡ D( q˜ , q˜ ). selected ray and is a function of variables q˜ and q˜ : det P 1

2

1

2

Now, in order to construct the best ellipsoidal approximation of the attraction domain, we need to find maximum of the function of two variables the domain of which is constrained by ellipse (3.9). Thus, we have proved the following theorem (which is similar to Theorem 2 in [10]): T h e o r e m 1. Let q˜ 1* , q˜ 2* be solution of the problem of maximization of the function of two variables given by 2 q˜ 1⎞ 2 4 ⎛ ˜ ˜ ˜ ˜ ˜ D ( q 1, q 2 ) = λ α 1 ( q 1, q 2 ) q 2 –  ⎝ 4⎠

(4.4)

in domain (3.9), where α1( q˜ 1 , q˜ 2 ) is defined by formula (4.3), and let α *1 = α1( q˜ *1 , q˜ *2 ). Then, matrix ˜ ( α * , q˜ * , q˜ * ) given by formula (4.1) is the matrix of the best ellipsoidal approximation of the attraction P 1

1

2

domain of the zero solution of system (2.1). This theorem reduces the problem of construction of the best ellipsoidal approximation of the attrac tion domain to the standard constrained optimization problem for a function of two variables defined on 4 Note that, for the control law synthesized in [10], the similar scalar equation in α can be solved only numerically. 2

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a convex set (ellipse). The maximum of this function can be found either by applying graphical visualiza tion methods or numerically (for example, by means of the fmincon program available in Matlab). As has already been noted, in practice, of great interest is the problem of finding a set of initial values such that any trajectory of the WR beginning in this set does not go beyond the boundaries of a prescribed strip. Let us denote the maximum allowable deviation of the robot from the target path as α 1 and set the problem of finding the best (in the sense of area) invariant ellipsoidal approximation of the attraction domain of the zero solution of system (2.1) inscribed in the strip |z1| ≤ α 1 . ˜ of form (4.1) completely belongs to the strip |z | ≤ α 1 if By virtue of (3.7), any ellipse with matrix P 1

α1 < α 1 . Let us apply the saturator function sat α1 to the function α1( q˜ 1 , q˜ 2 ) defined by formula (4.3) and ˜ 1 ( q˜ 1 , q˜ 2 ): denote the function obtained as α ˜ 1 ( q˜ 1, q˜ 2 ) = sat [ α 1 ( q˜ 1, q˜ 2 ) ]. α α1

(4.5)

˜ of form (4.1) with α = ˜ 1 ( q˜ 1 , q˜ 2 ) ≤ α 1 any ellipse constructed by means of the matrix P Since α 1 ˜ 1 ( q˜ 1 , q˜ 2 ) belongs to the considered strip. This brings us to the following evident extension of Theorem α 1 to the case under consideration. C o r o l l a r y f r o m T h e o r e m 1. Let q˜ *1 , q˜ *2 be solution of the problem of maximization of the function of two variables given by ˜2 2 4 ˜ 1 ( q˜ 1, q˜ 2 ) ⎛ q˜ 2 – q1⎞ D ( q˜ 1, q˜ 2 ) = λ α ⎝ 4⎠ ˜ 1 ( q˜ 1 , q˜ 2 ) is defined by formula (4.5), and let α ˜ 1* = α ˜ 1 ( q˜ 1* , q˜ 2* ). Then, matrix in domain (3.9), where α ˜ (α ˜ * , q˜ * , q˜ * ) given by formula (4.1) is the matrix of the best ellipsoidal approximation of the attraction P 1

1

2

domain of the zero solution of system (2.1) in the strip |z1| ≤ α 1 . 5. NUMERICAL EXAMPLES To illustrate the above discussed, we constructed several invariant ellipsoidal estimates of the attraction domains for a typical wheeled robot built on the basis of a car with u = 0.2 m–1 (minimal turning radius Rmin = 5 m). As an example, we considered motion of the WR closed by feedback (1.4) with the exponent λ = 0.3 along an admissible target path with maximum curvature k = 0.15 m–1. Figures 2 and 3 show the plot of the maximized function D( q˜ 1 , q˜ 2 ) and its contour lines, respectively. Maximum of D( q˜ , q˜ ) is achieved at q˜ = 0.35 and q˜ = 0.25 (for this point, β˜ ( q˜ , q˜ ) = 0.1251). 1

2

1

2

0min

1

2

The corresponding ellipsoidal set is shown in Fig. 4. The hatching in the figure shows the sets where con trol reaches saturation. The bold lines depict boundaries of the saturation sets, and the thin line is the straight line σ(z) = 0. In the upper saturation set, u = – u on target curves with curvature k = – k . The boundary of this set is found from the equation σ(z)  k –  –  = – u. 2 3/2 2 ( 1 + z2 ) 1 + z 2 ( 1 + kz 1 ) In the lower saturation set, u = u on target curves with positive maximal curvature k = k . The boundary of this set is found from the equation k σ(z) –   +  = u. 2 3/2 2 ( 1 + z2 ) 1 + z 2 ( 1 – kz 1 ) JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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det P 0.05 0.04 0.03 0.02 0.01 0 6

4 4 q2

2 0

0

3

2 q1

1

Fig. 2. Function D( q˜ 1 , q˜ 2 ).

q2 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0 q1

Fig. 3. Contour lines of function D( q˜ 1, q˜ 2 ).

It should be emphasized that there exist admissible target paths (k ≤ k ) for which the closedloop sys tem is nonlinear in the constructed ellipsoid. Nevertheless, if the point representing the system belongs to such an ellipsoid, then exponential stabilizability of the system is guaranteed for any admissible paths. If, when finding an estimate of the attraction domain, we confined ourselves to the region where the closedloop system is linear (as is often the case), we would obtain a much worse estimate, since such an ellipsoid lies in the nonhatched region. Thus, the use of the absolute stability methods for analysis of sys tems with saturated control allows one to improve considerably the estimate of the attraction domain compared to that obtained from the analysis of system linearity. Next, for the system under consideration, the problem of finding the best invariant ellipses in the strips |z1| ≤ 1 and |z1| ≤ 0.75 m was posed. Maximum of the determinant is achieved at q˜ 1 = 0.5, q˜ 2 = 0.45 ( β˜ 0min ( q˜ 1 , q˜ 2 ) = 0.1925) for the strip |z1| ≤ 1 m and at q˜ 1 = 0.65, q˜ 2 = 0.75 ( β˜ 0min ( q˜ 1 , q˜ 2 ) = 0.2787) for the strip |z1| ≤ 0.75 m. Both ellipses are depicted by the bold lines in Fig. 5. For the sake of comparison, the earlier constructed maximumarea ellipse (see Fig. 4) is shown in Fig. 5 by the dashed line. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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−1.0

−0.5

0

0.5

1.0

1.5 z1, m

Fig. 4. The best ellipsoidal approximation of the attraction domain of the zero solution for target paths with maximum curvature k = 0.15 m–1 for λ = 0.3.

z2 0.25 0.20 0.15 0.10 0.0 0 −0.05 −0.10 −0.15 −0.20 −0.25 −1.5

−1.0

−0.5

0

0.5

1.0

1.5 z1, m

Fig. 5. The best invariant ellipses inscribed in the strip |z1| ≤ 1 and |z1| ≤ 0.75 m (the dashed line depicts the maximum area ellipse without constraints on the deviation).

6. COMPARISON OF TWO CONTROL LAWS Finally, we compare the “best” ellipsoidal estimate of the attraction domain discussed above with the similar estimate obtained in [10] for the same wheeled robot closed by the control law synthesized in [10, 11]. Advantages of the control law employed in this work compared to two other control laws consid ered in [1, 2] and [10, 11], which are also based on the feedback linearization, are briefly discussed in [14]. It is noted, in particular, that the stability domain of the closedloop system considered in this work is gen erally greater than that in the case of control from [10, 11]. This can be explained by the following consid erations. Since the state variables z2 in the two systems are different5 (in [10–12], z2 = sinψ, whereas, in [14] and in the current paper, z2 = tan ψ ), the ranges of the corresponding transformations to zcoor 5 Variables z in both systems are identical, z = d, where d is the deviation from the target path. 1 1

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z2 3 2 1 0 −1 −2 −3 −10

−5

0

5

10 z1, m

Fig. 6. Comparison of the estimates for two control laws ( k = 0.05, λ = 0.3).

z2 3 2 1 0 −1 −2 −3

−20 −15 −10

−5

0

5

10

15

20 z1, m

Fig. 7. Comparison of the estimates for two control laws ( k = 0, λ = 0.3).

dinates are also different: (–1/ k , 1/ k ) × (–1, 1) and (–1/ k , 1/ k ) × R1, respectively. Since the second range is greater than the first one, it can adopt a greater invariant set. From comparison of the two considered changes of variables, it is not difficult to conclude that the ellipsoidal estimates of stability domains for the two systems should almost coincide when the stability margin is small. One encounters such a situation in the case of large curvature of the target path ( k close to the maximum possible value u ) or “aggressive” control law (large exponent λ). In this case, the angular deviation in the domain of stability is not large (ψ  π/4); hence, sinψ ≈ tan ψ , and the images of any system state belonging to the stability domain upon the two transformations practically coincide. However, in the case of a large stability margin (small curvature of the target path k  u and/or con servative control law with small λ), when the domain of stability is great, the estimates for the two systems can differ considerably. Indeed, for the system with control from [10, 11], any invariant ellipsoid belongs JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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to the strip |z2| ≤ 1, whereas, for the system closed by control (1.4), there are no a priori constraints on the variables z1 and z2. It was interesting to verify the above reasoning by numerical calculations. Since the variables z2 in the two considered systems are different, in order to compare the two sets, it is necessary to represent equa tions of both ellipses in the same variables (one of them will cease to be an ellipse, of course). We applied feedback (1.4) and that synthesized in [10, 11] with the same exponent λ = 0.3 to one and the same WR from the previous examples and constructed the best ellipsoidal estimates for target curves of different cur vature. As expected, for large values of curvature (0.1 ≤ k < u ), the estimates almost coincided (and, for this reason, are not presented here). For small values of maximum curvature, however, the estimate of the stability domain for the control law (1.4) was considerably bigger than the estimate in the case of the con trol law from [10, 11]. Figures 6 and 7 show the best ellipsoidal estimates of the attraction domains for both control laws and target paths of maximum curvature k = 0.05 and 0 (straight target path), respectively. The bigger set (ellipse) in both figures is the best ellipsoidal estimate of the attraction domain for the closedloop system considered in this paper. The second (nonellipsoidal) set is the image of the best ellip soidal estimate of the attraction domain for the WR closed by the control law from [10, 11] upon the trans formation (z1 = d, z2 = sinψ) → (z1 = d, z2 = tan ψ ). As was expected, the difference in the areas of the two estimates for the two control laws is significant and grows as the path approaches the straight line. CONCLUSIONS The problem of stabilization of motion of a wheeled robot with constrained resource along a given tar get path is considered. By means of the control law synthesized in [14], the problem has been reduced to the study of stability of the zero solution of a system of differential equations. For the closedloop system obtained, the problem of construction of the best (in the sense of area) ellipsoidal approximation of the attraction domain and a similar problem with the additional constraint on the maximum deviation from the target path have been posed and solved. It has been shown that these optimization problems reduce to the standard constrained optimization problem for a function of two variables. The discussion is illustrated by numerical examples. It should be noted that, in various applications related to construction of estimates of the attraction domain, there may arise other optimization problems as well, which differ from that considered in the paper by the performance index and/or by the presence of additional constraints on the desired ellipsoidal approximation. For example, the performance index may be the trace of the matrix or some nonlinear function of its elements. Along with the constraint on the maximum deviation, one may consider a con straint on the maximum angular deviation or some linear combination of the linear and angular devia tions, and so on. Clearly, any optimization problems of this kind can easily be solved in the framework of the approach proposed in the paper. ACKNOWLEDGMENTS This work was supported by the Department of Power Engineering, Mechanical Engineering, Mechanics, and Control Process of Russian Academy of Sciences (Program no. 1 “Scientific foundations of robotics and mechatronics”). REFERENCES 1. Cordesses, L., Cariou, C., and Berducat, M., Combine Harvester Control Using Real Time Kinematic GPS, Precision Agriculture, 2000, no. 2, pp. 147–161. 2. Thuilot, B., Cariou, C., Martinet, P., and Berducat, M., Automatic Guidance of a Farm Tractor Relying on a Single CPDGPS, Autonomous Robots, 2002, no. 13, pp. 53–61. 3. Rapoport, L., Gribkov, M., Khvalkov, A., Matrosov, I., Pesterev, A., and Tkachenko, M., Control of Wheeled Robots Using GNSS and Inertial Navigation: Control Law Synthesis and Experimental Results, Proc. of 19th Int. Technical Meeting of the Satellite Division of The Institute of Navigation, Fort Worth, USA, Sept. 2006. 4. De Luca, A., Oriolo, G., and Samson, C., Feedback Control of a Nonholonomic CarLike Robot, Robot Motion Planning and Control, Laumond, J.P., Ed., Springer, 1998, pp. 170–253. 5. LaValle, S.M., Planning Algorithms, Cambridge University Press, 2006. 6. Samson, C., Control of Chained Systems. Application to Path Following and Timevarying PointStabilization of Mobile Robots, IEEE Trans. Automatic Control, 1995, vol. 40, no. 1, pp. 64–77. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

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7. Pesterev, A.V. and Rapoport, L.B., Stabilization Problem for a Wheeled Robot Following a Curvilinear Path on Uneven Terrain, J. Comput. Systems Sci. Int., 2010, vol. 49, no. 4, pp. 672–680. 8. Rapoport, L.B., Estimation of Attraction Domain in a Wheeled Robot Control Problem, Automation Remote Control, 2006, vol. 67, no. 9, pp. 1416–1435. 9. Pesterev, A.V. and Rapoport, L.B., Construction of Invariant Ellipsoids in the Stabilization Problem for a Wheeled Robot Following a Curvilinear Path, Automation Remote Control, 2009, vol. 70, no. 2, pp. 219–232. 10. Pesterev, A.V., An Algorithm for Constructing Invariant Ellipsoids in a Stabilization Problem for a Wheeled Robot, Automation Remote Control, 2009, no. 9, pp. 1528–1539. 11. Pesterev, A.V., MaximumVolume Ellipsoidal Approximation of Attraction Domain in Stabilization Problem for Wheeled Robot, Proc. of the 18th IFAC World Congr., Milan, 2011, CD ROM. 12. Gilimyanov, R.F., Pesterev, A.V., and Rapoport, L.B., Motion Control for a Wheeled Robot Following a Curvi linear Path, J. Comput. Systems Sci. Int., 2008, vol. 47, no. 6, pp. 987–994. 13. Pesterev, A.V., Rapoport, L.B., and Gilimyanov, R.F., Control of a Wheeled Robot Following a Curvilinear Path, Sixth EUROMECH Nonlinear Dynamics Conference, St. Petersburg, 2008, CD ROM. 14. Pesterev, A.V., Synthesis of a Stabilizing Control for a Wheeled Robot Following a Curvilinear Path, Automation Remote Control, 2012, no. 7. 15. Pyatnitskii, E.S., Absolute Stability of Nonstationary Nonlinear Systems, Avtom. Telemekh., 1970, no. 1, pp. 5–15. 16. Shorten, R.N. and Narendra, K.S., Necessary and Sufficient Conditions for the Existence of a Common Qua dratic Lyapunov Function for a Finite Number of Stable Second Order Linear TimeInvariant Systems, Int. J. Adaptive Control Signal Processing, 2002, vol. 16, pp. 709–728. 17. Pakshin, P.V. and Pozdyaev, V.V., A Criterion of Existence of a Common Quadratic Lyapunov Function for a Set of Linear SecondOrder Systems, J. Comput. Systems Sci. Int., 2005, vol. 44, no. 4, pp. 519–524. 18. Gelig, A.Kh., Leonov, G.A., and Yakubovich, V.A., Ustoichivost’ nelineinykh sistem s needinstvennym sostoya niem ravnovesiya (Stability of Nonlinear Systems with a Nonunique Equilibrium State), Moscow: Nauka, 1978. 19. Formal’skii, A.M., Upravlyaemost’ i ustoichivost’ sistem s ogranichennymi resursami (Controllability and Stability of Systems with Constrained Resources), Moscow: Nauka, 1974. 20. Boyd, S., Ghaoui, L.E., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control The ory, 1994, Philadelphia: SIAM.

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