Periodic Motions for Estimation of the Attraction Domain in the Wheeled Robot Stabilization Problem

Transcription

1 Periodic Motions for Estimation of the Attraction Domain in the Wheeled Robot Stabilization Problem Lev Rapoport Institute of Control Sciences RAS, Moscow, Russia ( Abstract: Extremum properties of the two-dimensional linear time-varying (LTV) system are used in this paper to estimate boundary of the attraction domain in the problem of the wheeled robot control. The motion is supposed to be planar without a lateral slippage. The control goal is to drive the target point of the robot platform to the specified trajectory and to stabilize the motion along it. The trajectory consists of line segments and circular arcs. The current curvature of the trajectory of the target point is taken as control. The controlmust satisfy two-sided constraints. The attraction domain must be inscribed into certain region of the distance to the trajectory - orientation phase space. Time-varying curvature of the target trajectory is considered as arbitrary varying function which takes values from the specified interval. The feedback linearization scheme is used for synthesis of the control law. The saturation function is then used to take into account control constraints. The closed loop system takes form of the nonlinear system with parametric disturbances. The absolute stability approach is explored for stability analysis. Some nonlinearities take values from the interval. Other nonlinearitiessatisfy sectorconstraints. Along with the nonlinear timevarying system the uncertain linear time varying system is considered. Every solution of the nonlinear system is also solution of the time varying system for certain set of time-varying disturbances. To estimate the attraction domain of the nonlinear closed loop system, the Lyapunov function for the LTV system is constructed. A convexinvariant function is known to exist at the boundary of the absolute stability region. For the second order case, the extremum trajectory, corresponding to the boundary of the absolute stability region, belongs to the level set of the invariant function. The periodic solution has finite number of switches on the period. It circumscribes the boundary of the attraction domain estimate. Two illustrative examples are considered. 1. INTRODUCTION Inthispaperconsideredistheproblemofthecontrolofthe wheeled robot. The robot must be automatically driven along a target trajectory. In general, the control system does not possess global stability. Stability with certain exponent rate is not guaranteed if the system starts from the initial state not belonging to the attraction domain in the state space. The problem of numerical estimation of the attraction domain is addressed in the paper. In many papers, see e.g. Thuilot et al. [00] - Matyukhin [006] and the references therein, a control law is designed which stabilizes the motion along a line segment or stabilizes the motion towards a given point in the plane; the control may be either continuous or discontinuous (Guldner and Utkin [1994], Matyukhin [006]). Here, we consider the case of constrained control. Boundness of control makes it impossible to attain the guaranteed rate of decay of deviation from the target trajectory if motion starts from the arbitrary position. The domain of initial conditions is estimated such that the synthesized control provides the specified rate of exponential convergence. In the paper Rapoport and Morozov [008] the absolute stability approach, quadratic Lyapunov functions, and LMI technique were used to estimate the attraction domain. The trajectory was supposed to consist of line segments and circular arcs of the finite length. To reduce conservatism of estimation, more general class of Lyapunov functions must be considered. In the present paper the stability problem for non-linear time varying system is embedded into a class of LTV twodimensional systems with bounded disturbances. Absolute stability analysis of LTV systems is equivalent to absolute stability analysis of selector-linear differential inclusions (SLDI) analyzed in papers Pyatnitskii and Rapoport [1996], Margaliot and Yfoulis [005], Holcman and Margaliot [00], Zhermolenko [006]. The range, the right hand side of SLDI takes values from, depends on dimensionsoftheattractiondomainoftheinitialnonlineartimevarying system. The maximum dimension of the attracion domain corresponds to the margin of the absolute stability region of SLDI. This work was supported in part by the Integrated Research Program no. 15 of the OEMMPU Division of RAS. Copyright by the International Federation of Automatic Control (IFAC) 5878

2 A convex invariant function is known to exist at the boundary of the absolute stability of SLDI. This invariant function is used as the exact Lyapunov function for attraction domain estimation. The extremum trajectory, corresponding to the boundary of the absolute stability in the second order case belongs to the level set of the invariant function and is exactly the periodic orbit, see Pyatnitskii and Rapoport [1996]. The periodic solution has finite number of switches on the period and describes boundary of the estimate of the attraction domain of the initial nonlinear time-varying system. Numerical examples illustrating the proposed technique conclude the paper. Results are compared with the estimate earlier obtained using quadratic Lyapunov function.. KINEMATIC SCHEME The motion of a wheeled robot is assumed to be twodimensional. Its orientation is defined by a single angle, see Fig. 1. Let X =(x,y) T be the point of the plane, the symbol T denotes the matrix transpose. The target point is located at the middle of the rear axle and is denoted by X c =(x c,y c ) T, the orientation is defined by an angle θ between the centre line of the platform and the x-axis. y L X c V r H X 0 l Target path Fig. 1. The kinematics scheme of the wheeled robot. The value u reciprocal to the radius of turn of the target point is the instantaneous curvature, u =1/ X c X 0, where X 0 denotes the instantaneous curvature center. Let L and H be dimensions of the platform as shown in Fig. 1. Then relationships ul 1 uh/ =tanα l, x ul 1+uH/ =tanα r (1) relate the curvature u of the target point to the steering angle of the left and right front wheels, α l,α r.the counterclockwise direction is taken as positive, a left turn associates with a positive value of u. This value has opposite sign for right turn. The relation (1) allows to simplify the model, and the value u is further taken as control. Denoting v c = V c inthe forwardmotionandv c = V c while in the reverse motion, arrive at the following well known model: ẋ c = v c cosθ, ẏ c = v c sinθ, () θ = v c u. The limitations on the steering angle impose two-sided constraints on the value of curvature: ū u ū. (3) The expression for the quantity ū is easily derived from the value of the maximum steering angle. Taking (3) into account, the equations () take the form ẋ c = v c cosθ, ẏ c = v c sinθ, θ = v c sū(u), where sū(u) is the saturation function. In what follows, the target trajectory parametrization and the change of variables described in Rapoport and Morozov [008] is used. It is shortly described below for convenience. Thetargettrajectoryconsistsofline segmentsandcircular arc segments. For brevity they will be referred to as trajectory segments and denoted by s i, i =1,...,n.The number of segments can be as large as pleased. Every segment has its curvature c i ; the line segments have zero curvature. The following feasibility condition is supposed to be satisfied (4) c i < ū. (5) Let ξ be the length parameter and l i be the length of the segment s i. In the courseof motion in the neighborhoodof the trajectory, one of the segments is considered current. The motionequations (4)arefurther rewrittenin the form where the parameter ξ is taken as an independent variable. The motion along the current segment s i is considered started if the parameters ξ exceeds the value b i b 1 =0, b i = b i 1 +l i 1 for i =,,n, (6) and it is considered terminated as soon as ξ exceeds the value b i+1. After the segment s i is over, the segment s i+1 becomes current. The following assumption is supposed to be satisfied Assumption 1. Adjacent segments have a common tangent at the connection point The passage to the successive segment is accompanied by abrupt change of the desired curvature c i,which,in turn, necessitates the abrupt change of the control u or, according to formulas (1), that of the steering angle of the front wheels. In the situations where the dynamics of the front wheel drive cannot be neglected (in contrast to what was assumed further), the passage from segment to segment is accompanied by transition processes. Then the change of variables is applied which was initially proposedincordessesetal.[000],thuilotet al.[00]for the case of straight lines and extended to the composite trajectories consisting of line segments and circular arcs 5879

3 in Rapoport and Morozov [008] and papers cited there. Let η be distance of the target point from the current segment. In other words, it is distance between the target point and the point of the segment nearest to the target point. It is well defined for both lines and arcs. Let also ψ be the angle between the center line of the platform and the tangent to the current segment at the point nearest to the target point. In the new variables, the control goal is to ensure η 0. The following assumptions are supposed to be satisfied. Assumption. The linear velocity of the platform v c (t)is positive, separated from zero v c (t) v 0 > 0 (7) and satisfies the existence conditions for absolutely continuous solutions of the system of differential equations (4). Assumption 3. Along the trajectories of the controlled system (4), the following relation holds: cosψ(t) ε>0. (8) Assumption 3 will be further removed. As will be shown, if this assumption is satisfied at the initial state it will hold along the whole trajectory. Further, let z 0 = ξ, z 1 = η, z =tanψ. Letalso substitute the time derivative with the derivative in the variable ξ. Omitting details described in Rapoport and Morozov [008] arrive at the equations describing motion of the robot along the composite path: z 1 =(c(ξ)z 1 +1)z, z = u(c(ξ)z 1 +1)(1+z) 3 +c(ξ)(1+z (9) ), where the quantity c(ξ) takes values c i when the current segment becomes s i according to the condition b i ξ b i+1. The value ξ is easily calculated for the current segment usingmeasurementsofthe position.themeasurementand control system consisting of GNSS and inertial measurement sensors is described for example in Rapoport et al. [006]. Solution of the system of differental equations (9) with a non-smooth in z =(z 1,z ) T and discontinuous in ξ right hand side is considered in the A.F. Filippov sense. 3. CONTROL LAW DESIGN Ignoring two-sided constraints on control and using feedback linearization technique (see Khalil [00]) leads to the choice of control u in the form u = σ +c(ξ)(1+z ) (10) (c(ξ)z 1 +1)(1+z )3 for some λ>0and σ =λz +λ z 1, (11) where λ is desired rate of exponential decrease. Then the closed loop system (9) takes the form z 1 =(c(ξ)z 1 +1)z, z = σ (1) which is asymptotically stable in the neighbourhood of the origin. However, in general, control (10) does not satisfy the two-sided constraints (3). On the other hand, taking control in the form ( ) σ+c(ξ)(1+z u = ) sū, (13) (c(ξ)z 1 +1)(1+z )3 may not guarantee that z 1 and z decrease exponentially with given rate of exponential stability. Moreover, undesirable overshoot in variations of the variables is possible. Especially dangerous is overshoot in the variable z 1.This means that even starting sufficiently close to the target trajectory, the robot may perform large oscillations going far enough even if the closed loop system is stable. In what follows, considered is the problem of description of the attraction domain inscribed into the band of certain width around the axis z and guaranteeing prescribed exponential convergence rate. The symbol ξ in the expression c(ξ) will be omitted if that does not lead to misunderstanding. Rewrite the last equation in (9) taking the control u as (13) ( ) σ +c(1+z z = s ) (cz1 ū +1 ) (1+z (cz 1 +1)(1+z )3 )3 Then + c(1+z ). = Φ(z,σ). ( σ +c(1+z ) ) Φ(z,σ)=sū(cz1+1)(1+z )3 (14) c(1+z ). The system (9) takes the form z 1 =(c(ξ)z 1 +1)z, z = Φ(z,σ). 4. ESTIMATION OF THE ATTRACTION DOMAIN (15) Let z 0 be the initial data vector. Let us describe the set of z 0 having the property that along the trajectories of the system (9) starting at z 0 the solution z(ξ) decreases exponentially at the rate µ, where0<µ λ. To estimate this domain the Lyapunov function v(z) will be used. The exponential stability domain will be estimated as Ω v 0 = {z : v(z) v 0 } (16) for appropriate choice of the constant v 0 > 0. Given positive value α 1 consider the band-like region and denote Π(α 1 )={z : z 1 α 1 } (17) c =max i=1,, c i, (18) u 1 =ū(1 cα 1 ) c. (19) The following auxiliary statement holds: Lemma 4. Assume that z Π(α 1 ) and the number α 1 satisfies the inequality u 1 > 0. (0) 5880

4 Then the following inequalities hold and s u1 (σ) Φ(z,σ) σ for σ>0 (1) σ Φ(z,σ) s u1 (σ) forσ 0. () Proof. The proof is very close (but a bit different) to the proof of the similar Lemma in Rapoport and Morozov [008] and is given here for convenience. It follows from (14) that { σ1 for σ σ 1, Φ(z,σ)= σ for σ 1 <σ<σ, (3) σ for σ σ, where σ 1 =ū(1+c(ξ)z 1 )(1+z )3 +c(ξ)(1+z ), (4) σ =ū(1+c(ξ)z 1 )(1+z )3 c(ξ)(1+z ). (5) It follows from (17) that σ 1 u 1,σ u 1. (6) Combination (3) and (6) gives (1) and (). Proof of Lemma 4 is completed. We will be looking for the attraction domain of the system (9) inscribed into the set Π(α 1,α )=Π(α 1 ) {z : σ α } (7) for some α > 0. Along with the function Φ(z,σ)inthe formulation of system (15), introduce the function where β(ξ) satisfies the conditions φ(ξ,σ)=β(ξ)σ, (8) k β(ξ) 1, k=min{ u 1 α,1}. (9) According to the absolute stability approach the graph of the function Φ(z,σ), satisfying the conditions (1), (), is inscribedintoa sector ontheplaneσ Φ for σ satisfying condition σ α. Conditions (9) define the size of the sector. Further, introduce the function γ(ξ) satisfyingthe conditions (1 cα 1 ) γ(ξ) (1+ cα 1 ). (30) We next expand the class of systems (15) considering systems Along with (31) consider the system z 1 = γ(ξ)z, z = β(ξ)σ. (31) z 1 = +γ(ξ)z, z = µz β(ξ)σ (3) where 0 <µ<λ. Stability of the zero point of the system (3) implies exponential stability of the zero point of the system (31) with exponential rate µ. Let us denote δ(ξ)=(β(ξ),γ(ξ)). Let (k)betheclass of measurable functions δ(ξ) defined on the real axis and satisfying conditions (9) and (30). Given initial data z 0 let z δ (z 0,ξ) be solution of the system (3) corresponding to the function δ( ) (k) and satisfying initial conditions z δ (z 0,0) = z 0. Our goal is to maximize the dimension of the attraction domain having α as large as possible. In other words, according to (9) we have to minimize the lower sector bound k preserving absolute stability of the system (3) in the class (k). Absolute stability is understood as asymptotic stability of the zero solution uniform over the class (k). Let k be the least upper bound of such values k that system(31)isabsolutelystableforβ(ξ)andγ(ξ)satisfying constraints (9) and (30) respectively. As is proven in Pyatnitskii and Rapoport [1996] and papers cited there, if 0 <k <, the positive convex invariant function v(z) exists such that the following extremal property holds: max v(z δ(z 0,ξ)) = v(z 0 ) (33) δ( ) (k ) for any ξ 0. Conditionµ<λimplies the inequality k < 1. (34) Let v 0 be the largest value that the set Ω v 0 defined in (16) belongs to the set Π(α 1, u1 k ): v 0 =max{w :Ω w Π(α 1, u 1 k )}. (35) It follows from (33) that the set Ω v 0 satisfying condition (35) is invariant for the system (3). Moreover, it follows from (3) that it is invariant for the system (31) and every solution started in it is exponentially convergent to zero with the exponential rate µ. Accordingto construction of the system (3) and construction of the set (k )every solution of the system (15) starting in Ω v 0 remains there and converges exponentially to zero. Thus we have proved the following statement. Theorem 5. LetthesetΩ v 0 defined in (16) satisfies condition (35). Then it is the invariant set of the system (15). Moreover, every solution starting in this set converges to zero exponentially with the exponential rate µ. Note now that the function v(z) is positive for all z 0. It means, that the set Ω v 0 is bounded. It follows from the Theorem 5 that z 1,z take bounded values. In particular, it means that z =tanψ is bounded and so, ψ does not cross the value π/. It means in turn that the condition (8) will be satisfied along the whole trajectory if it is satisfied at the initial state. To find the boundary of the set Ω v 0, the one parameter family of periodic motions of the two dimensional system (3) corresponding to the critical value k,seepyatnitskii and Rapoport [1996], is first found. Then find the largest value of the parameter among those values that the periodic solution is inscribed into the set Π(α 1, u1 k ). Let y(δ) be the right hand side of the system (3) corresponding to the value δ (k). It follows from (33) that for any z v(z) max =0, (36) δ (k ) y(δ) where v(z)/ y(δ) is the directional derivative of the convex function v(z) along the vector y(δ)atthepoint 5881

5 z. Letδ (z) =(β (z),γ (z)) be the value of δ giving the maximum value in (36) for certain z. Inthetwodimensional case the extremum Lyapunov function is differentiable. Then v(z) v(z) = y(δ)t y(δ) z where v(z)/ z is the gradient of the function v(z). Let g (z) be the vector, orthogonal to the vector y(δ (z)) and directed outside of the set Ω v 0 at its boundary: g (z)=[ (µz β (z)σ), ( +γ (z)z )] T. Taking last two relationships into account, rewrite (36) as max (β,γ) (k ) [ γ(β (z)σ µ)z β(γ (z)z + )σ + β (z)µσz 1 +γ (z)µz ]=0 (37) reaching the extremum value 0 at γ = γ (z)andβ = β (z). Let µ be small enough. The plane z 1,z can be split into six sectors as shown in Fig.. Sectors are defined by inequalities S 1 = {z :(σ µz k )z 0}, S = {z :(σ µz )(σ µz k ) 0}, S 3 = {z : σ(σ µz ) 0}, S 4 = {z :(z + )σ 0}, (1 cα 1 ) S 5 = {z :(z + (1+ cα 1 ) )(z + S 6 = {z : z (z + (1+ cα 1 ) ) 0}. S 5 S 6 S 3 S z S 4 S 1 Fig.. Sectors on the plane z 1 z. (1 cα 1 ) ) 0}, Analysis of (37) in sectors S 1 S 6 leads to the following result Theorem 6. β (z)andγ (z) take the following values: (a)β (z)=k,γ (z)=(1+ cα 1 )forz S 1 ; (b)β (z)=k,γ (z)=(1 cα 1 )forz S S 3 S 5 S 6 ; (c)β (z)=1,γ (z)=(1 cα 1 )forz S 4. The critical value k is adjusted such that the extremal trajectory becomes periodic. The periodic trajectory is then inscribed into the set Π(α 1, u1 k ). 5. EXAMPLES To illustrate the proposed method two examples have been consideredbelow.first, attraction domain has been z 1 constructed for ū =0., λ =, c =0.1, µ =0.001, α 1 =0.35. In Fig. 3 the attraction domain is presented together with trajectories of the system (15) corresponding to the switching values of c(ξ) = ± c. Periodic solution corresponds to the extremal trajectory of the system (3) as described in the previous section with k = Fig. 3. Attraction domain for ū =0., λ =, c =0.1, µ =0.001 Magenta lines limit the region Π(α 1 ). Black ellips denotes the attraction domain estimate obtained with the quadratic Lyapunov function using LMI technique as described in Rapoport and Morozov [008], Rapoport [006]. That is easily seen that using the Lyapunov function v(x) allows to estimate the attraction domain less conservative than using quadratic Lyapunov functions. Note that the convex Lyapunov function v(x) is not constructed explicitly. Only boundary of its level set is constructed, which is sufficient for estimation of the boundary of the attraction domain. Inanotherexample,showninFig.4,theattractiondomain is obtained for the case of the straight line ( c =0)and ū =0., λ =0.3, µ =0.01, α 1 = 7. ACKNOWLEDGEMENTS This work was supported in part by the Integrated Research Program no. 15 of the OEMMPU Division of RAS. REFERENCES Cordesses L., Cariou C., and Berducat M. Combine Harvester Control Using Real Time Kinematic GPS. Precision Agriculture : , 000. Thuilot B., Cariou C., Martinet P., and Berducat M. Automatic Guidance of a Farm Tractor Relying on a Single CP-DGPS. Autonomous Robots 15:53-61,00. Rapoport L.B. et al. Control of Wheeled Robots Using GNSS and Inertial Navigation: Control Law Synthesis and Experimental Results. In Proc. ION GNSS 006, The 19th International Technical Meeting, pages

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