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Stabilizing Control for a Wheeled Robot Following a Curvilinear Path
10th IFAC Symposium on Robot Control International Federation of Automatic Control September 5-7, 2012. Dubrovnik, Croatia
Stabilizing Control for a Wheeled Robot Following a Curvilinear Path Alexander V. Pesterev ∗ ∗
Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya 65, Moscow, 117997 Russia; [email protected]. Abstract: The problem of finding a stabilizing control for a wheeled robot with constrained control resource following a curvilinear path is studied. The goal of the control is to bring the robot to an assigned path and to stabilize its motion along it. A new change of variable is suggested that reduces the problem of stabilizing robot’s motion to that of stabilizing the zero solution in the form that admits feedback linearization. A control law stabilizing robot’s motion along an arbitrary feasible curvilinear target path is synthesized. The new control is shown to be more efficient than the well-known linearizing feedback obtained from the celebrated chainedform representation of the system equations. For a straight target path, the closed-loop system is shown to be asymptotically stable for any initial conditions except for the case where the initial direction of motion is perpendicular to the target path. Keywords: Wheeled robot, path following, feedback linearization, saturation control 1. INTRODUCTION
ables, one obtains different canonical representations of the original problem and different control laws.
There exist many applications (for example, in agriculture [1, 2] or road construction) where a vehicle is to be driven along some target path with high level of accuracy. Such tasks are performed by automatic vehicles (further referred to as wheeled robots (WR), or simply robots) equipped with navigational and inertial tools and satellite antennas. The problem of bringing the robot to a preassigned curvilinear path and stabilizing its motion along the path is called path stabilization problem. It was discussed in a great number of publications. Various models (e.g., unicycle, simple car, car-like model, tractor with trailers, etc.) and target paths (straight lines, circle arcs, general-form curves) were considered (see, e.g., [3–10] and references therein).
In this paper, a new change of variables is proposed, which reduces equations of robot’s motion to a canonical form that admits feedback linearization. Based on the representation obtained, a new control law stabilizing motion of a WR along an arbitrary curvilinear feasible path is synthesized. Comparison of this control law with two above-mentioned feedbacks demonstrates its unquestionable advantages.
One of the commonly accepted techniques for solving the path following problem is that of feedback linearization. The application of this technique requires that equations of motion of the system under study be converted to a special (canonical) form from which the linearizing feedback can easily be found. In this way, stabilizing feedbacks were synthesized, e.g., for a trailer-like mobile robot following a straight target path [6] and, in the general case of an arbitrary feasible curvilinear target path, for the car-like model [7] and the kinematic model of WR [8]. In a number of works [1–3, 5], the desired representation was derived from the chained-form representation [3, 11]. The canonical equations obtained by means of the change of variables suggested in [7–9] and those derived from the chainedform representation have different forms, and, accordingly, control laws synthesized on the basis of these equations are also different. Thus, applying different changes of vari 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”).
978-3-902823-11-3/12/$20.00 © 2012 IFAC
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2. PROBLEM STATEMENT The WR 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 the target point located at the midpoint of the rear axle and the angle θ between the central line of the platform (which coincides with the direction of the velocity vector) and the x-axis of a fixed reference system x0y. The kinematic equations of such a robot are well known to be (see, for example, [2, 4, 8–10]): x˙ c = v cos θ, y˙ c = v sin θ, θ˙ = vu.
(1)
Here, the dot denotes differentiation with respect to time, v is forward velocity of the target point, and u is the instant curvature of the trajectory described by the target point. Since 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 each other and the path curvature, so that there exists an “efficient mean” angle φ [7, 10]) by the equation u = tan φ/L, (2) 10.3182/20120905-3-HR-2030.00055
IFAC SYROCO 2012 September 5-7, 2012. Dubrovnik, Croatia
where L is the wheelbase distance, it can be taken as the control to simplify the model. A restricted turning angle of the front wheels results in the two-sided phase constraints −¯ u≤u≤u ¯ (−φmax ≤ φ ≤ φmax ),
(3)
where φmax < π/2 and u ¯ = tan φmax /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 trajectory (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 [3] everywhere except for a finite number of points, and maximum curvature k¯ = maxs k(s) of the target path satisfies the constraint k¯ < u ¯. 3. CHANGE OF VARIABLES First, let us turn from variables xc , yc , θ to the so-called path coordinates [3] s, d, and ψ, where s is the value of the path parameter at the closest to the robot point of the target path, d is the deviation of the target point from the target path (distance to the curve with the plus (minus) sign if the target point is on the left (on the right) from the curve when moving in the positive direction of the curve) and ψ is the angular deviation (angle between the velocity vector and the tangent to the target curve at the point closest to the robot). In the path coordinates, equations (1) take the form [1–3, 8] v cos ψ s˙ = 1 − kd d˙ = v sin ψ vk cos ψ , ψ˙ = vu − 1 − kd
(4)
where k ≡ k(s) is the curvature of the target path at the point closest to the robot. The forward velocity v of the robot is assumed to be strictly positive and separated from zero: v(t) ≥ v0 > 0 ∀t ≥ t0 . Since the right-hand side of system (4) is not defined at d = 1/k(s) and k(s) may vary ¯ the change of the Cartesian coordinates from −k¯ to +k, to the path ones is defined in the symmetric neighborhood of the target curve defined by the condition ¯ |d| < 1/k. (5) Then, the right-hand side of system (4) is defined on the Cartesian product of the three one-dimensional sets ¯ 1/k) ¯ × S 1 , where S 1 is the one-dimensional R1 × (−1/k, manifold homeomorphic to a circle, i.e., interval [0, 2π] with the identified ends (angles ψ = 0 and ψ = 2π are assumed to be equal). Let us restrict the range of variation of angle ψ by the interval (−π/2, π/2) and consider the function
ξ˙ = v(t) cos ψ(t) > 0 for any ψ in the considered interval. Hence, we may take ξ for the independent variable. Turning from the differentiation with respect to time to the differentiation with respect to ξ in (4) and denoting the corresponding derivatives by the prime, we obtain 1 s = 1 − kd d = tan ψ (7) k u ψ = − . cos ψ 1 − kd Let us introduce the notation z1 = d, z2 = tan ψ,
(8)
and z = [z1 , z2 ]T . Clearly, mapping (8) is a diffeomor¯ 1/k)×(−π/2, ¯ phism of the open set Dx = (−1/k, π/2) 1 ¯ ¯ of the manifold (−1/k, 1/k)×S and the open set Dz = 1 ¯ 1/k)×R ¯ (−1/k, of the coordinate space R2 . Differentiating both sides of the second equation in (8) by virtue of system (7), we obtain k u z2 = − . (9) 3 2 cos ψ cos ψ(1 − kd) With regard to the inequality cos12 ψ = 1 + tan2 ψ, system (7) in terms of the new variables takes the form 1 s = 1 − kz1 z1 = z2 (10) (1 + z22 )k 2 3/2 z2 = (1 + z2 ) u − . 1 − kz1 4. WHEELED ROBOT WITH UNCONSTRAINED CONTROL RESOURCE Note that the right-hand sides of the last two equations do not explicitly depend on the state variable s. Considering k as a parameter rather than a function of the path coordinate s, we arrive at the system in which two last equations are independent from the first one. Then, the first equation (which describes the rate of motion of the projection of the target point onto the target curve) becomes redundant, and the control law can be synthesized based on only two last equations. Let σ(z) be a linear function, σ(z) = cT z, cT = [c1 , c2 ], c1 , c2 > 0.
(11)
Consider the feedback of the form u = −σ(z) cos3 ψ + ≡−
k cos ψ 1 − kd k
σ(z) + , (1 + z22 )3/2 1 + z22 (1 − kz1 )
(12)
(6)
Substituting (12) into (10) and discarding the first equation, we obtain the second-order linear system z1 = z2 (13) z2 = −σ(z),
defined on solutions of system (4). By virtue of the assumption of strict positiveness of the forward velocity v(t),
the zero solution of which is globally asymptotically stable. Now, note that linear system (13) is defined in the entire
t ξ(t) =
v(τ ) cos ψ(τ )dτ t0
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IFAC SYROCO 2012 September 5-7, 2012. Dubrovnik, Croatia
coordinate space R2 , whereas system (10) is defined (in what concerns variables z) in set Dz ⊆ R2 . Therefore, a solution to system (13) is also a solution to system (10) closed by feedback (12) only if it lies in Dz . Thus, it can be stated that feedback (12) globally asymptotically stabilizes solution z = 0 of system (10) only in the case of a straight target path, since, only in this case, Dz = R2 (k¯ = 0). In the general case of a curvilinear target path, in order that the entire trajectory of system (13) lie in set Dz , it is sufficient that z(0) belong to an invariant set DzI ⊆ Dz of linear system (13). Now, let us return to the original stabilization problem in terms of the path coordinates (4). Let solution z(ξ) to system (10) closed by feedback (12) belong to Dz . Since the change of variables (8) is a diffeomorphism of the open sets Dx and Dz , the image of solution z(ξ) upon the inverse transform lies completely in the set Dx and is a solution to system (4) closed by feedback (12). Since ξ˙ is strictly positive in Dx and Dx is an open set, there is > 0 such that ξ˙ ≥ > 0 at any point of the considered trajectory. Then, it follows that the study of stability in the time domain can be replaced by the study of stability as ξ → ∞. Thus, we proved the following theorem. Theorem 1. Solution d = 0, ψ = 0 of system (4) closed by feedback (12) is asymptotically stable for any initial conditions belonging to the image of an invariant set DzI ⊆ Dz of linear system (13) upon inverse transform. In the case of a straight target path, DzI = Dz = R2 , and the image of DzI upon the inverse transform coincides with Dx , which, in the given case, has the form Dx = R1 ×(−π/2, π/2). Applying Theorem 1 to the case of a straight target path and introducing the notation Π = {d, ψ : d ∈ R1 , −π/2 < ψ < π/2},
(14)
we obtain the following assertion. Corollary. For a straight target path, solution d = 0, ψ = 0 of system (4) closed by feedback (12), is asymptotically stable for any initial conditions d(0), ψ(0) ∈ Π. Remark. It is easy to see that control (12) stabilizes motion of the WR along a straight target path for π/2 < ψ(0) < 3π/2 as well. In this case, the stable solution is solution d = 0, ψ = π corresponding to the motion in the opposite direction of the target path. Motion of a WR cannot be stabilized by control (12) only if the initial direction of robot motion is perpendicular to the target line: ψ(0) = π/2 or ψ(0) = 3π/2. In this case, u ≡ 0, and the WR endlessly keeps moving in the direction perpendicular to the target path. 5. COMPARISON OF THE SYNTHESIZED CONTROL LAW WITH OTHER LAWS OBTAINED BY FEEDBACK LINEARIZATION
a form that admits feedback linearization by means of the change of variables z1 = d, z2 = sin ψ. In [1–3], the desired representation was derived from the chained-form equations of robot motion [3, 5]. In terms of this paper, it is obtained by taking the independent variable ξ equal to the projection of the path passed by the robot onto the target curve (i.e., ξ = s) and substituting z1 = d, z2 = (1− kd) tan ψ. The disadvantage of the second representation is that the control law synthesized on its basis depends on the derivative of curvature of the target path ks [1–3], whereas, in order that the target path be feasible, it is sufficient that curvature is a piecewise continuous function [3] (this is because the path curvature takes part in the definition of the state variable z2 ). Thus, the control law derived from this representation imposes excessive requirements to the smoothness of the target curve. The disadvantage of the first representation is that it is defined in a smaller domain ¯ 1/k)×(−1, ¯ of the coordinate space R2 (Dz = (−1/k, 1) [8]). As a result, the set of initial values ensuring asymptotic stability of the zero solution of the closed-loop system is generally smaller than that for the closed-loop system obtained from the second representation. In particular, even in the case of a straight target path, the set of initial values ensuring asymptotic stability is bounded. On the other hand, the control synthesized from this representation is determined only by the curvature of the target path and does not depend on its derivative. Control (12) synthesized in this work possesses advantages of both above-mentioned control laws and lacks their drawbacks: it is not restricted to a smaller domain, like the first control law, and does not impose excessive requirements to the smoothness of the target curve, like the second control law. Hence, superiority of the proposed control law over the two others is undisputable. 6. WHEELED ROBOT WITH CONSTRAINED CONTROL RESOURCE Theorem 1 and Corollary have been obtained under the assumption that the control resource is unbounded. Let us show that the latter result is valid also for the robot with a constrained control resource, at least, for functions σ(z) of special form. Namely, let constraints (3) hold. Let us redefine control (12) setting it equal to u ¯ or −¯ u if the right-hand side of (12) is greater than u ¯ or less than −¯ u, respectively, i.e., consider the control given by σ(z) k u = satu¯ − + , (15) (1 + z22 )3/2 1 + z22 (1 − kz1 ) where satu¯ (u) is the saturation function, and restrict our consideration to the functions σ(z) of the form
As mentioned in the Introduction, feedback linearization was used earlier in [1–3, 8] for solving the path stabilization problem for the considered WR model. However, in these works, changes of variables different from that considered in this work were used to convert the equations of motion to a form that admits feedback linearization. In [8], for the t independent variable, the path ξ(t) = 0 v(τ )dτ passed by the robot was used, and the system was reduced to 646
σ(z) = λ2 z1 + 2λz2 , λ > 0.
(16)
For such functions σ(z), the closed-loop system (13) has pole −λ of multiplicity two. Theorem 2. Let the target path be a straight line. Then, system (4) closed by feedback (15), (16) is asymptotically stable for any λ > 0 and arbitrary initial conditions d(0), ψ(0) ∈ Π.
IFAC SYROCO 2012 September 5-7, 2012. Dubrovnik, Croatia
The proof of Theorem 2 is given in the Appendix. From Theorem 2, it follows, in particular, that Π is an invariant set of the closed-loop system (angular deviation ψ never exceeds π or becomes less than −π). 8 6
c=20
4
c=15
z2
2
As can be seen, the behavior of a WR closed by feedback (15), (16) in the case of a straight target path depends on ˜ The integral curves of system (17) only one parameter λ. are easily found to be 1 z1 − = c, σ(z) > 0, (18) 1 + z22
0 −2 −4
σ(z)=0
−6 −8 −20
−15
−10
−5
0 z (m)
5
10
15
˜ = λ/¯ ¯, z˜2 = z2 , λ u. ξ˜ = ξ u ¯, z˜1 = z1 u Applying it to (15) and introducing dimensionless control u ˜ = u/¯ u and curvature k˜ = k/¯ u, we find that feedback (15) is also invariant w.r.t. the transformation. Thus, the study of behavior of an arbitrary WR can be reduced to that of the dimensionless WR with u ¯ = 1. Keeping the old notation for the new variables and parameters, we obtain the equations of motion on the saturation segments in the case of a straight target path in the form z1 = z2 (17) z2 = −sign(σ(z))(1 + z22 )3/2 .
and
20
−z1 −
1
Fig. 1. A state-space trajectory of the WR with the constrained resource and contour lines of the function L(z) (trajectories of the system with u = ±¯ u). As an illustration, the dashed line in Fig. 1 shows a statespace trajectory of the WR with u ¯ = 0.05 m−1 (minimum turning radius 20 m) and λ = 1 with the initial point z1 (0) = −15 m, z2 (0) = 4. The solid lines in the figure show trajectories of the WR with the saturated controls u = u ¯ or u = −¯ u (see the proof of Theorem 2 in the Appendix) and the straight line σ(z) = 0. Theorem 2 guarantees asymptotic stability of the system closed by feedback (15) for any λ > 0 but says nothing about how a particular choice of the feedback parameter affects the system behavior. Generally, any system trajectory consists of alternating segments where the system is linear (|u| ≤ u ¯) and nonlinear (control reaches saturation, |u| = u ¯). We will call these segments linearity and saturation segments, respectively. From general considerations, it is clear that, if λ is “small,” then the system is mostly linear, and the control can reach saturation only in a region of very large deviations z1 . However, small λ’s imply slow rate of convergence of the linear system, so that it may take a long time for the system to reach the origin. On the other hand, “large” λ’s mean rapid convergence to zero of the linear system and, simultaneously, frequent switchings to the saturation, so that the major part of the trajectory will consist of many saturation segments alternated with short linearity segments, which also results in slow overall convergence, as well as in great overshooting. The words “small” and “large” above are placed in inverted commas because whether λ is small or large is determined also by the second system parameter u ¯. The fact that the system behavior depends on two parameters greatly complicates the analysis. Therefore, before we proceed, we will get rid of one parameter by converting the equations governing system motion to an equivalent dimensionless form. It is not difficult to check that the linear system (13) is invariant w.r.t the transformation 647
1 1 + z22
= c, σ(z) < 0,
(19)
where c is a constant depending on the initial conditions. To get an idea of how to select the feedback parameter λ, let us recall how the phase portrait of the linear system (13) looks like. This system has one equilibrium point— stable degenerate node z = 0. The phase plane is divided by the asymptote z2 = −λz1 into two half-planes such that any trajectory tends to zero staying in the half-plane to which its initial point belongs and never intersects the asymptote. On any trajectory, both linear and angular deviations z1 and z2 can change sign at most once. In the constrained case, the phase portrait may look quite different. In the general case, a trajectory may wrap around the origin several times, so that the phase portrait looks more like that of a focus (like, e.g., in Fig. 1) rather than of a node. Clearly, such a behavior of the trajectory implies frequent switchings of the control between its extreme values and overshooting, which seems undesirable. From this point of view, the trajectory behavior like that in the linear case looks much more attractive. Let us pose the following problem. To identify the range of λ’s for which trajectories of the nonlinear closed-loop system under study preserve the behavior typical of trajectories of the linear system (13). Selection of the feedback parameter from this range guarantees that any trajectory of the nonlinear system possesses the above-mentioned features of the trajectories of the linear system (13), and the selection of the maximum value from this range ensures the greatest convergence rate in the class of the identified trajectories. The maximum value of λ for which the system under study preserves a node-like phase portrait will be referred to as the “optimal” feedback parameter. Theorem 3. For a straight target path,√ the “optimal” feedback parameter is given by λopt = 3 3/2 ≈ 2.6. For any λ ≤ λopt , any trajectory of the closed-loop system has one saturation segment at most. Proof of Theorem 3. It is easy to see that the “saturation” region, i.e., the region where the inequality
IFAC SYROCO 2012 September 5-7, 2012. Dubrovnik, Croatia
|σ(z)| ≥ (1 + z22 )3/2
holds consists of two nonintersecting sets lying from different sides of the straight line σ(z) = 0, as shown in Fig. 2 for λ = 3. The bold lines depict boundaries of the two one-connected saturation sets; the thin line, the asymptote z2 = −λz1 ; and the dashed line is the straight line σ(z) = 0. In the saturation set lying above (below) the straight line σ(z) = 0, u = −1 (u = 1). 2.5 2 1.5 1 0.5 z2
7. CONCLUSIONS
(20)
0
A new change of variables is proposed by means of which the problem of stabilizing motion of the kinematic model of the wheeled robot following a curvilinear path is represented in the form that admits feedback linearization. On the basis of the representation obtained, a control law is synthesized that ensures stabilization of robot’s motion along a given target path when initial conditions belong to a known invariant set. In the case of a straight target path, the system closed by the synthesized feedback is asymptotically stable for any initial conditions, unless the initial direction is perpendicular to the target line. The asymptotic stability for arbitrary initial conditions is shown to be preserved in the case of a straight target path even if the control resource is bounded. Comparison of the control law synthesized in this work with other similar laws known from the literature demonstrates its obvious advantages.
−0.5 −1
REFERENCES
−1.5
[1]
−2 −2.5 −0.8
−0.6
−0.4
−0.2
0 z1
0.2
0.4
0.6
0.8
Fig. 2. Saturation regions (¯ u = 1, λ = 3). Clearly, in order that the phase portrait of the system look like that of the linear system, the asymptote z2 = −λz1 must not intersect the saturation sets. Indeed, in this case, after leaving, e.g., the lower saturation set, the system cannot enter the upper saturation set, since the latter lies from the other side of the asymptote, which the trajectory of the linear system cannot intersect. Hence, the trajectory will go to the origin staying in the half-plane where it was originated. Now, let us find the condition under which the asymptote does not intersect the saturation sets. In view of symmetry, it will suffice to consider one (say, upper) saturation set. Substituting the right-hand side of (16) for σ(z) into (20) and transposing the term λz2 , we find that, in the saturation region, the inequality λ2 z1 + λz2 ≥ (1 + z22 )3/2 − λz2 holds. The saturation set lies completely above the asymptote if the inequality λ2 z1 + λz2 ≥ 0 holds for any point of the set. The latter inequality holds when (1+z22 )3/2 ≥ λz2 . Setting the equality sign in the inequality, squaring both sides of the equation obtained, and denoting x = z22 , we obtain λ2 x = (1 + x)3 . This equation may be viewed as the equation in the tangency point of the cubic and linear functions of x. Setting derivatives of both functions equal to one another at the point of tangency, λ2 = 3(1+x)2 , and substituting the right-hand side of the last equation for λ2 into the previous equation, we easily find the solution x = 1/2. Substituting it in the last equation, we find that the √ saturation region touches the asymptote when λ = 33/2 and that the ordinate of the tangency point is z2 = 1/2. The theorem is proved. 648
Thuilot, B., Cariou, C., Martinet, P., and Berducat, M (2002). Automatic guidance of a farm tractor relying on a single CP-DGPS. Autonomous Robots, 13, 53–61. [2] Thuilot, B., Lenain, R., Martinet, P., and Cariou, C. (2005). Accurate GPS-based guidance of agricultural vehicles operating on slippery grounds. In J.X. Liu (ed.), Focus on robotics research, Nova Science. [3] De Luca, A., Oriolo, G., and Samson, C. (1998). Feedback control of a nonholonomic car-like robot. In J.-P. Laumond (ed.), Robot motion planning and control, 170–253. [4] LaValle, S.M. (2006). Planning algorirhms. Cambridge University Press. [5] Samson, C. (1995). Control of chained systems. Application to path following and timevarying pointstabilization of mobile robots. IEEE Trans. Automatic Control, 40, 64–77. [6] Sampei, M., Tamura, T., Itoh, T., and Nakamichi, M. (1991). Path tracking control of trailer-like mobile robot. In Proc. IEEE/RSJ Workshop Intelligent Robots Systems, 193–198. [7] Gilimyanov, R.F., Pesterev, A.V., and Rapoport, L.B. (2008). Motion control for a wheeled robot following a curvilinear path. J. Comput. System Sci. Int., 47, 987–994. [8] Pesterev, A.V. and Rapoport, L.B. (2010). Stabilization problem for a wheeled robot following a curvilinear path on uneven terrain. J. Comput. System Sci. Int., 49, 672–680. [9] Pesterev, A.V. (2011). Maximum-volume ellipsoidal approximation of attraction domain in stabilization problem for wheeled robot. In Proc. of the 18th IFAC World Congress, Milan, CD ROM. [10] Rapoport, L.B. (2006). Estimation of attraction domains in wheeled robot control. Automation Remote Control, 67, 1416–1435. [11] Murray, R.M. (1993). Control of nonholonomic systems using chained forms. Fields Inst. Commun., 1, 219–245.
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[12] Andronov, A.A., Leontovich, E.A., Gordon, I.I., and Maier, A.G. (1973). Qualitative theory of second-order dynamic systems. Wiley. Appendix A. PROOF OF THEOREM 2 The proof is based on the well-known fact of the theory of second-order dynamical systems [12], which says that the set of ω-limit points (i.e. the limit points of positive semitrajectories) of bounded trajectories can consist of only equilibrium points, trajectories connecting the equilibrium points, and closed trajectories. Since the system under study has only one equilibrium point (the stable node z = 0), we need to prove that any trajectory (i) is bounded and (ii) cannot be a closed curve. Clearly, no linearity segment can go to infinity, since the linear system has stable node at the origin. This is also true for a saturation segment. Indeed, on the saturation segments, the system moves along the integral curves (18) and (19), which, in the dimensional case, take the form 1 1 z1 − = c and − z1 − = c, (A.1) u ¯ 1 + z22 u ¯ 1 + z22 respectively. Consider, e.g., the first curve in (A.1). When the representing point moves along it in the positive direction, z2 → −∞. However, this curve cannot be a trajectory of the system when σ(z) < 0, so that z2 will stay finite when the system leaves this curve (upon approaching the straight line σ(z) = 0) to become linear. Similarly, z2 will never turn to infinity when moving along the second curve. Taking into account that transition from any (except, perhaps, the first) linearity segment to the saturation mode may occur only in the region of bounded deviations, namely, when |z1 | ≤ γ, where γ = max{|z1∗ |, |z1∗∗ |} and z1∗ and z1∗∗ are abscissas of the points where the asymptote intersects the boundary of either saturation set (see Fig. 2), which take finite values for any finite λ, the trajectory will never leave a bounded set of the phase plane even in the case of an infinite number of switchings. Now, let us prove that no linearity segment, but, perhaps, the first and last one, can lie in the 1st or 3d quadrant of the plane. For definiteness, let us consider the 1st quadrant (z1 > 0, z2 > 0). Suppose that a linearity segment different from the first one lies in it (as can be seen from Fig. 2, this may happen only in a sufficiently small neighborhood of the origin, since the part of the axis z1 for which z1 ≥ u ¯/λ2 lies in the saturation region. Let us prove that, in this case, the control will never reach saturation. Let us denote the coordinate of the point of the intersection of the considered segment and the z1 axis as z10 and the value of function σ(z) at this point as σ0 . Since the system is linear at this point, we have σ0 < u ¯. Considering (z10 , 0) as the initial point, we find that the subsequent motion of the linear system is given by z1 (ξ) = z10 (1 + λξ)e−λξ , z2 (ξ) = −λ2 z10 ξe−λξ . With regard to the relation σ0 = λ2 z10 , we have σ(ξ) = σ0 (1 − λξ)e−λξ and σ (ξ) = −λσ0 (2 − λξ)e−λξ . Then, |σ(ξ)| ≤ σ0 ∀ξ, and the following chain of inequalities holds: |σ(ξ)| max |σ(ξ)| σ0
The last inequality means that control will not reach saturation, and the system will stay linear. Since the linear system is stable, its trajectory tends to the origin, and, hence, the considered segment is the last segment of the trajectory. The proof for the 3d quadrant is similar. Now, consider the function 1 + 1/¯ u. L(z) = |z1 | − u ¯ 1 + z22
(A.2)
It is not difficult to see that it is positive definite in the entire phase plane and is equal to zero at the origin. Its contour lines L(z) = c for 0 < c < 1/¯ u are closed curves consisting of segments of the integral curves (A.1). For c ≥ 1/¯ u, the contour lines are not closed and consist of two unbounded integral curves (A.1). For the sake of illustration, Fig. 1 shows the contour lines corresponding to c = 5, 10, 15, and 20 for the WR with u ¯ = 0.05 m−1 and λ = 1. The function L(z) is differentiable everywhere, except for the points on the line z1 = 0, and its derivative is z2 z2 . L (z) = z2 sign(z1 ) + u ¯(1 + z22 )3/2 Suppose that there exists a closed trajectory. Then, the corresponding curve on the phase plane consists of four segments: two saturation segments and two linearity segments moving along which the system intersects the line σ(z) = 0. As proved above, the latter segments lie entirely in the 2-nd and 4th quadrants, respectively. Let us find the derivative of the function L(z) by virtue of the equations of motion on each of the above-mentioned segments. 1. Saturation segment lying in the 2nd, 1st, and 4th quadrants (σ(z) > 0, u = −¯ u, and z2 = −¯ u(1 + z22 )3/2 ). (a) In the 2nd quadrant (z1 < 0, z2 > 0), we have L (z) = −z2 − z2 = −2z2 < 0; (b) in the 1st and 4th quadrants (z1 > 0), L (z) = z2 − z2 = 0. 2. On the linearity segment in the 4th quadrant (z1 > 0, z2 < 0), we have z2 = −σ(z) and z2 σ(z) σ(z) L (z) = z2 − = z 1 − < 0. 2 u ¯(1 + z22 )3/2 u ¯(1 + z22 )3/2 In the same way, it is proved that the derivative is negative semi-definite on the two other segments. We have proved that the function L(z) monotonically decreases on each segment of the considered trajectory. Hence, having started at some point, the system cannot occur at the same point after passing these four segments. Thus, assuming the existence of a closed trajectory, we arrived at the contradiction, which proves that any trajectory of the system tends to the equilibrium state z = 0.
Stabilizing Control for a Wheeled Robot Following a Curvilinear Path Alexander V. Pesterev ∗ ∗
Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya 65, Moscow, 117997 Russia; [email protected]. Abstract: The problem of finding a stabilizing control for a wheeled robot with constrained control resource following a curvilinear path is studied. The goal of the control is to bring the robot to an assigned path and to stabilize its motion along it. A new change of variable is suggested that reduces the problem of stabilizing robot’s motion to that of stabilizing the zero solution in the form that admits feedback linearization. A control law stabilizing robot’s motion along an arbitrary feasible curvilinear target path is synthesized. The new control is shown to be more efficient than the well-known linearizing feedback obtained from the celebrated chainedform representation of the system equations. For a straight target path, the closed-loop system is shown to be asymptotically stable for any initial conditions except for the case where the initial direction of motion is perpendicular to the target path. Keywords: Wheeled robot, path following, feedback linearization, saturation control 1. INTRODUCTION
ables, one obtains different canonical representations of the original problem and different control laws.
There exist many applications (for example, in agriculture [1, 2] or road construction) where a vehicle is to be driven along some target path with high level of accuracy. Such tasks are performed by automatic vehicles (further referred to as wheeled robots (WR), or simply robots) equipped with navigational and inertial tools and satellite antennas. The problem of bringing the robot to a preassigned curvilinear path and stabilizing its motion along the path is called path stabilization problem. It was discussed in a great number of publications. Various models (e.g., unicycle, simple car, car-like model, tractor with trailers, etc.) and target paths (straight lines, circle arcs, general-form curves) were considered (see, e.g., [3–10] and references therein).
In this paper, a new change of variables is proposed, which reduces equations of robot’s motion to a canonical form that admits feedback linearization. Based on the representation obtained, a new control law stabilizing motion of a WR along an arbitrary curvilinear feasible path is synthesized. Comparison of this control law with two above-mentioned feedbacks demonstrates its unquestionable advantages.
One of the commonly accepted techniques for solving the path following problem is that of feedback linearization. The application of this technique requires that equations of motion of the system under study be converted to a special (canonical) form from which the linearizing feedback can easily be found. In this way, stabilizing feedbacks were synthesized, e.g., for a trailer-like mobile robot following a straight target path [6] and, in the general case of an arbitrary feasible curvilinear target path, for the car-like model [7] and the kinematic model of WR [8]. In a number of works [1–3, 5], the desired representation was derived from the chained-form representation [3, 11]. The canonical equations obtained by means of the change of variables suggested in [7–9] and those derived from the chainedform representation have different forms, and, accordingly, control laws synthesized on the basis of these equations are also different. Thus, applying different changes of vari 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”).
978-3-902823-11-3/12/$20.00 © 2012 IFAC
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2. PROBLEM STATEMENT The WR 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 the target point located at the midpoint of the rear axle and the angle θ between the central line of the platform (which coincides with the direction of the velocity vector) and the x-axis of a fixed reference system x0y. The kinematic equations of such a robot are well known to be (see, for example, [2, 4, 8–10]): x˙ c = v cos θ, y˙ c = v sin θ, θ˙ = vu.
(1)
Here, the dot denotes differentiation with respect to time, v is forward velocity of the target point, and u is the instant curvature of the trajectory described by the target point. Since 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 each other and the path curvature, so that there exists an “efficient mean” angle φ [7, 10]) by the equation u = tan φ/L, (2) 10.3182/20120905-3-HR-2030.00055
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where L is the wheelbase distance, it can be taken as the control to simplify the model. A restricted turning angle of the front wheels results in the two-sided phase constraints −¯ u≤u≤u ¯ (−φmax ≤ φ ≤ φmax ),
(3)
where φmax < π/2 and u ¯ = tan φmax /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 trajectory (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 [3] everywhere except for a finite number of points, and maximum curvature k¯ = maxs k(s) of the target path satisfies the constraint k¯ < u ¯. 3. CHANGE OF VARIABLES First, let us turn from variables xc , yc , θ to the so-called path coordinates [3] s, d, and ψ, where s is the value of the path parameter at the closest to the robot point of the target path, d is the deviation of the target point from the target path (distance to the curve with the plus (minus) sign if the target point is on the left (on the right) from the curve when moving in the positive direction of the curve) and ψ is the angular deviation (angle between the velocity vector and the tangent to the target curve at the point closest to the robot). In the path coordinates, equations (1) take the form [1–3, 8] v cos ψ s˙ = 1 − kd d˙ = v sin ψ vk cos ψ , ψ˙ = vu − 1 − kd
(4)
where k ≡ k(s) is the curvature of the target path at the point closest to the robot. The forward velocity v of the robot is assumed to be strictly positive and separated from zero: v(t) ≥ v0 > 0 ∀t ≥ t0 . Since the right-hand side of system (4) is not defined at d = 1/k(s) and k(s) may vary ¯ the change of the Cartesian coordinates from −k¯ to +k, to the path ones is defined in the symmetric neighborhood of the target curve defined by the condition ¯ |d| < 1/k. (5) Then, the right-hand side of system (4) is defined on the Cartesian product of the three one-dimensional sets ¯ 1/k) ¯ × S 1 , where S 1 is the one-dimensional R1 × (−1/k, manifold homeomorphic to a circle, i.e., interval [0, 2π] with the identified ends (angles ψ = 0 and ψ = 2π are assumed to be equal). Let us restrict the range of variation of angle ψ by the interval (−π/2, π/2) and consider the function
ξ˙ = v(t) cos ψ(t) > 0 for any ψ in the considered interval. Hence, we may take ξ for the independent variable. Turning from the differentiation with respect to time to the differentiation with respect to ξ in (4) and denoting the corresponding derivatives by the prime, we obtain 1 s = 1 − kd d = tan ψ (7) k u ψ = − . cos ψ 1 − kd Let us introduce the notation z1 = d, z2 = tan ψ,
(8)
and z = [z1 , z2 ]T . Clearly, mapping (8) is a diffeomor¯ 1/k)×(−π/2, ¯ phism of the open set Dx = (−1/k, π/2) 1 ¯ ¯ of the manifold (−1/k, 1/k)×S and the open set Dz = 1 ¯ 1/k)×R ¯ (−1/k, of the coordinate space R2 . Differentiating both sides of the second equation in (8) by virtue of system (7), we obtain k u z2 = − . (9) 3 2 cos ψ cos ψ(1 − kd) With regard to the inequality cos12 ψ = 1 + tan2 ψ, system (7) in terms of the new variables takes the form 1 s = 1 − kz1 z1 = z2 (10) (1 + z22 )k 2 3/2 z2 = (1 + z2 ) u − . 1 − kz1 4. WHEELED ROBOT WITH UNCONSTRAINED CONTROL RESOURCE Note that the right-hand sides of the last two equations do not explicitly depend on the state variable s. Considering k as a parameter rather than a function of the path coordinate s, we arrive at the system in which two last equations are independent from the first one. Then, the first equation (which describes the rate of motion of the projection of the target point onto the target curve) becomes redundant, and the control law can be synthesized based on only two last equations. Let σ(z) be a linear function, σ(z) = cT z, cT = [c1 , c2 ], c1 , c2 > 0.
(11)
Consider the feedback of the form u = −σ(z) cos3 ψ + ≡−
k cos ψ 1 − kd k
σ(z) + , (1 + z22 )3/2 1 + z22 (1 − kz1 )
(12)
(6)
Substituting (12) into (10) and discarding the first equation, we obtain the second-order linear system z1 = z2 (13) z2 = −σ(z),
defined on solutions of system (4). By virtue of the assumption of strict positiveness of the forward velocity v(t),
the zero solution of which is globally asymptotically stable. Now, note that linear system (13) is defined in the entire
t ξ(t) =
v(τ ) cos ψ(τ )dτ t0
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coordinate space R2 , whereas system (10) is defined (in what concerns variables z) in set Dz ⊆ R2 . Therefore, a solution to system (13) is also a solution to system (10) closed by feedback (12) only if it lies in Dz . Thus, it can be stated that feedback (12) globally asymptotically stabilizes solution z = 0 of system (10) only in the case of a straight target path, since, only in this case, Dz = R2 (k¯ = 0). In the general case of a curvilinear target path, in order that the entire trajectory of system (13) lie in set Dz , it is sufficient that z(0) belong to an invariant set DzI ⊆ Dz of linear system (13). Now, let us return to the original stabilization problem in terms of the path coordinates (4). Let solution z(ξ) to system (10) closed by feedback (12) belong to Dz . Since the change of variables (8) is a diffeomorphism of the open sets Dx and Dz , the image of solution z(ξ) upon the inverse transform lies completely in the set Dx and is a solution to system (4) closed by feedback (12). Since ξ˙ is strictly positive in Dx and Dx is an open set, there is > 0 such that ξ˙ ≥ > 0 at any point of the considered trajectory. Then, it follows that the study of stability in the time domain can be replaced by the study of stability as ξ → ∞. Thus, we proved the following theorem. Theorem 1. Solution d = 0, ψ = 0 of system (4) closed by feedback (12) is asymptotically stable for any initial conditions belonging to the image of an invariant set DzI ⊆ Dz of linear system (13) upon inverse transform. In the case of a straight target path, DzI = Dz = R2 , and the image of DzI upon the inverse transform coincides with Dx , which, in the given case, has the form Dx = R1 ×(−π/2, π/2). Applying Theorem 1 to the case of a straight target path and introducing the notation Π = {d, ψ : d ∈ R1 , −π/2 < ψ < π/2},
(14)
we obtain the following assertion. Corollary. For a straight target path, solution d = 0, ψ = 0 of system (4) closed by feedback (12), is asymptotically stable for any initial conditions d(0), ψ(0) ∈ Π. Remark. It is easy to see that control (12) stabilizes motion of the WR along a straight target path for π/2 < ψ(0) < 3π/2 as well. In this case, the stable solution is solution d = 0, ψ = π corresponding to the motion in the opposite direction of the target path. Motion of a WR cannot be stabilized by control (12) only if the initial direction of robot motion is perpendicular to the target line: ψ(0) = π/2 or ψ(0) = 3π/2. In this case, u ≡ 0, and the WR endlessly keeps moving in the direction perpendicular to the target path. 5. COMPARISON OF THE SYNTHESIZED CONTROL LAW WITH OTHER LAWS OBTAINED BY FEEDBACK LINEARIZATION
a form that admits feedback linearization by means of the change of variables z1 = d, z2 = sin ψ. In [1–3], the desired representation was derived from the chained-form equations of robot motion [3, 5]. In terms of this paper, it is obtained by taking the independent variable ξ equal to the projection of the path passed by the robot onto the target curve (i.e., ξ = s) and substituting z1 = d, z2 = (1− kd) tan ψ. The disadvantage of the second representation is that the control law synthesized on its basis depends on the derivative of curvature of the target path ks [1–3], whereas, in order that the target path be feasible, it is sufficient that curvature is a piecewise continuous function [3] (this is because the path curvature takes part in the definition of the state variable z2 ). Thus, the control law derived from this representation imposes excessive requirements to the smoothness of the target curve. The disadvantage of the first representation is that it is defined in a smaller domain ¯ 1/k)×(−1, ¯ of the coordinate space R2 (Dz = (−1/k, 1) [8]). As a result, the set of initial values ensuring asymptotic stability of the zero solution of the closed-loop system is generally smaller than that for the closed-loop system obtained from the second representation. In particular, even in the case of a straight target path, the set of initial values ensuring asymptotic stability is bounded. On the other hand, the control synthesized from this representation is determined only by the curvature of the target path and does not depend on its derivative. Control (12) synthesized in this work possesses advantages of both above-mentioned control laws and lacks their drawbacks: it is not restricted to a smaller domain, like the first control law, and does not impose excessive requirements to the smoothness of the target curve, like the second control law. Hence, superiority of the proposed control law over the two others is undisputable. 6. WHEELED ROBOT WITH CONSTRAINED CONTROL RESOURCE Theorem 1 and Corollary have been obtained under the assumption that the control resource is unbounded. Let us show that the latter result is valid also for the robot with a constrained control resource, at least, for functions σ(z) of special form. Namely, let constraints (3) hold. Let us redefine control (12) setting it equal to u ¯ or −¯ u if the right-hand side of (12) is greater than u ¯ or less than −¯ u, respectively, i.e., consider the control given by σ(z) k u = satu¯ − + , (15) (1 + z22 )3/2 1 + z22 (1 − kz1 ) where satu¯ (u) is the saturation function, and restrict our consideration to the functions σ(z) of the form
As mentioned in the Introduction, feedback linearization was used earlier in [1–3, 8] for solving the path stabilization problem for the considered WR model. However, in these works, changes of variables different from that considered in this work were used to convert the equations of motion to a form that admits feedback linearization. In [8], for the t independent variable, the path ξ(t) = 0 v(τ )dτ passed by the robot was used, and the system was reduced to 646
σ(z) = λ2 z1 + 2λz2 , λ > 0.
(16)
For such functions σ(z), the closed-loop system (13) has pole −λ of multiplicity two. Theorem 2. Let the target path be a straight line. Then, system (4) closed by feedback (15), (16) is asymptotically stable for any λ > 0 and arbitrary initial conditions d(0), ψ(0) ∈ Π.
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The proof of Theorem 2 is given in the Appendix. From Theorem 2, it follows, in particular, that Π is an invariant set of the closed-loop system (angular deviation ψ never exceeds π or becomes less than −π). 8 6
c=20
4
c=15
z2
2
As can be seen, the behavior of a WR closed by feedback (15), (16) in the case of a straight target path depends on ˜ The integral curves of system (17) only one parameter λ. are easily found to be 1 z1 − = c, σ(z) > 0, (18) 1 + z22
0 −2 −4
σ(z)=0
−6 −8 −20
−15
−10
−5
0 z (m)
5
10
15
˜ = λ/¯ ¯, z˜2 = z2 , λ u. ξ˜ = ξ u ¯, z˜1 = z1 u Applying it to (15) and introducing dimensionless control u ˜ = u/¯ u and curvature k˜ = k/¯ u, we find that feedback (15) is also invariant w.r.t. the transformation. Thus, the study of behavior of an arbitrary WR can be reduced to that of the dimensionless WR with u ¯ = 1. Keeping the old notation for the new variables and parameters, we obtain the equations of motion on the saturation segments in the case of a straight target path in the form z1 = z2 (17) z2 = −sign(σ(z))(1 + z22 )3/2 .
and
20
−z1 −
1
Fig. 1. A state-space trajectory of the WR with the constrained resource and contour lines of the function L(z) (trajectories of the system with u = ±¯ u). As an illustration, the dashed line in Fig. 1 shows a statespace trajectory of the WR with u ¯ = 0.05 m−1 (minimum turning radius 20 m) and λ = 1 with the initial point z1 (0) = −15 m, z2 (0) = 4. The solid lines in the figure show trajectories of the WR with the saturated controls u = u ¯ or u = −¯ u (see the proof of Theorem 2 in the Appendix) and the straight line σ(z) = 0. Theorem 2 guarantees asymptotic stability of the system closed by feedback (15) for any λ > 0 but says nothing about how a particular choice of the feedback parameter affects the system behavior. Generally, any system trajectory consists of alternating segments where the system is linear (|u| ≤ u ¯) and nonlinear (control reaches saturation, |u| = u ¯). We will call these segments linearity and saturation segments, respectively. From general considerations, it is clear that, if λ is “small,” then the system is mostly linear, and the control can reach saturation only in a region of very large deviations z1 . However, small λ’s imply slow rate of convergence of the linear system, so that it may take a long time for the system to reach the origin. On the other hand, “large” λ’s mean rapid convergence to zero of the linear system and, simultaneously, frequent switchings to the saturation, so that the major part of the trajectory will consist of many saturation segments alternated with short linearity segments, which also results in slow overall convergence, as well as in great overshooting. The words “small” and “large” above are placed in inverted commas because whether λ is small or large is determined also by the second system parameter u ¯. The fact that the system behavior depends on two parameters greatly complicates the analysis. Therefore, before we proceed, we will get rid of one parameter by converting the equations governing system motion to an equivalent dimensionless form. It is not difficult to check that the linear system (13) is invariant w.r.t the transformation 647
1 1 + z22
= c, σ(z) < 0,
(19)
where c is a constant depending on the initial conditions. To get an idea of how to select the feedback parameter λ, let us recall how the phase portrait of the linear system (13) looks like. This system has one equilibrium point— stable degenerate node z = 0. The phase plane is divided by the asymptote z2 = −λz1 into two half-planes such that any trajectory tends to zero staying in the half-plane to which its initial point belongs and never intersects the asymptote. On any trajectory, both linear and angular deviations z1 and z2 can change sign at most once. In the constrained case, the phase portrait may look quite different. In the general case, a trajectory may wrap around the origin several times, so that the phase portrait looks more like that of a focus (like, e.g., in Fig. 1) rather than of a node. Clearly, such a behavior of the trajectory implies frequent switchings of the control between its extreme values and overshooting, which seems undesirable. From this point of view, the trajectory behavior like that in the linear case looks much more attractive. Let us pose the following problem. To identify the range of λ’s for which trajectories of the nonlinear closed-loop system under study preserve the behavior typical of trajectories of the linear system (13). Selection of the feedback parameter from this range guarantees that any trajectory of the nonlinear system possesses the above-mentioned features of the trajectories of the linear system (13), and the selection of the maximum value from this range ensures the greatest convergence rate in the class of the identified trajectories. The maximum value of λ for which the system under study preserves a node-like phase portrait will be referred to as the “optimal” feedback parameter. Theorem 3. For a straight target path,√ the “optimal” feedback parameter is given by λopt = 3 3/2 ≈ 2.6. For any λ ≤ λopt , any trajectory of the closed-loop system has one saturation segment at most. Proof of Theorem 3. It is easy to see that the “saturation” region, i.e., the region where the inequality
IFAC SYROCO 2012 September 5-7, 2012. Dubrovnik, Croatia
|σ(z)| ≥ (1 + z22 )3/2
holds consists of two nonintersecting sets lying from different sides of the straight line σ(z) = 0, as shown in Fig. 2 for λ = 3. The bold lines depict boundaries of the two one-connected saturation sets; the thin line, the asymptote z2 = −λz1 ; and the dashed line is the straight line σ(z) = 0. In the saturation set lying above (below) the straight line σ(z) = 0, u = −1 (u = 1). 2.5 2 1.5 1 0.5 z2
7. CONCLUSIONS
(20)
0
A new change of variables is proposed by means of which the problem of stabilizing motion of the kinematic model of the wheeled robot following a curvilinear path is represented in the form that admits feedback linearization. On the basis of the representation obtained, a control law is synthesized that ensures stabilization of robot’s motion along a given target path when initial conditions belong to a known invariant set. In the case of a straight target path, the system closed by the synthesized feedback is asymptotically stable for any initial conditions, unless the initial direction is perpendicular to the target line. The asymptotic stability for arbitrary initial conditions is shown to be preserved in the case of a straight target path even if the control resource is bounded. Comparison of the control law synthesized in this work with other similar laws known from the literature demonstrates its obvious advantages.
−0.5 −1
REFERENCES
−1.5
[1]
−2 −2.5 −0.8
−0.6
−0.4
−0.2
0 z1
0.2
0.4
0.6
0.8
Fig. 2. Saturation regions (¯ u = 1, λ = 3). Clearly, in order that the phase portrait of the system look like that of the linear system, the asymptote z2 = −λz1 must not intersect the saturation sets. Indeed, in this case, after leaving, e.g., the lower saturation set, the system cannot enter the upper saturation set, since the latter lies from the other side of the asymptote, which the trajectory of the linear system cannot intersect. Hence, the trajectory will go to the origin staying in the half-plane where it was originated. Now, let us find the condition under which the asymptote does not intersect the saturation sets. In view of symmetry, it will suffice to consider one (say, upper) saturation set. Substituting the right-hand side of (16) for σ(z) into (20) and transposing the term λz2 , we find that, in the saturation region, the inequality λ2 z1 + λz2 ≥ (1 + z22 )3/2 − λz2 holds. The saturation set lies completely above the asymptote if the inequality λ2 z1 + λz2 ≥ 0 holds for any point of the set. The latter inequality holds when (1+z22 )3/2 ≥ λz2 . Setting the equality sign in the inequality, squaring both sides of the equation obtained, and denoting x = z22 , we obtain λ2 x = (1 + x)3 . This equation may be viewed as the equation in the tangency point of the cubic and linear functions of x. Setting derivatives of both functions equal to one another at the point of tangency, λ2 = 3(1+x)2 , and substituting the right-hand side of the last equation for λ2 into the previous equation, we easily find the solution x = 1/2. Substituting it in the last equation, we find that the √ saturation region touches the asymptote when λ = 33/2 and that the ordinate of the tangency point is z2 = 1/2. The theorem is proved. 648
Thuilot, B., Cariou, C., Martinet, P., and Berducat, M (2002). Automatic guidance of a farm tractor relying on a single CP-DGPS. Autonomous Robots, 13, 53–61. [2] Thuilot, B., Lenain, R., Martinet, P., and Cariou, C. (2005). Accurate GPS-based guidance of agricultural vehicles operating on slippery grounds. In J.X. Liu (ed.), Focus on robotics research, Nova Science. [3] De Luca, A., Oriolo, G., and Samson, C. (1998). Feedback control of a nonholonomic car-like robot. In J.-P. Laumond (ed.), Robot motion planning and control, 170–253. [4] LaValle, S.M. (2006). Planning algorirhms. Cambridge University Press. [5] Samson, C. (1995). Control of chained systems. Application to path following and timevarying pointstabilization of mobile robots. IEEE Trans. Automatic Control, 40, 64–77. [6] Sampei, M., Tamura, T., Itoh, T., and Nakamichi, M. (1991). Path tracking control of trailer-like mobile robot. In Proc. IEEE/RSJ Workshop Intelligent Robots Systems, 193–198. [7] Gilimyanov, R.F., Pesterev, A.V., and Rapoport, L.B. (2008). Motion control for a wheeled robot following a curvilinear path. J. Comput. System Sci. Int., 47, 987–994. [8] Pesterev, A.V. and Rapoport, L.B. (2010). Stabilization problem for a wheeled robot following a curvilinear path on uneven terrain. J. Comput. System Sci. Int., 49, 672–680. [9] Pesterev, A.V. (2011). Maximum-volume ellipsoidal approximation of attraction domain in stabilization problem for wheeled robot. In Proc. of the 18th IFAC World Congress, Milan, CD ROM. [10] Rapoport, L.B. (2006). Estimation of attraction domains in wheeled robot control. Automation Remote Control, 67, 1416–1435. [11] Murray, R.M. (1993). Control of nonholonomic systems using chained forms. Fields Inst. Commun., 1, 219–245.
IFAC SYROCO 2012 September 5-7, 2012. Dubrovnik, Croatia
[12] Andronov, A.A., Leontovich, E.A., Gordon, I.I., and Maier, A.G. (1973). Qualitative theory of second-order dynamic systems. Wiley. Appendix A. PROOF OF THEOREM 2 The proof is based on the well-known fact of the theory of second-order dynamical systems [12], which says that the set of ω-limit points (i.e. the limit points of positive semitrajectories) of bounded trajectories can consist of only equilibrium points, trajectories connecting the equilibrium points, and closed trajectories. Since the system under study has only one equilibrium point (the stable node z = 0), we need to prove that any trajectory (i) is bounded and (ii) cannot be a closed curve. Clearly, no linearity segment can go to infinity, since the linear system has stable node at the origin. This is also true for a saturation segment. Indeed, on the saturation segments, the system moves along the integral curves (18) and (19), which, in the dimensional case, take the form 1 1 z1 − = c and − z1 − = c, (A.1) u ¯ 1 + z22 u ¯ 1 + z22 respectively. Consider, e.g., the first curve in (A.1). When the representing point moves along it in the positive direction, z2 → −∞. However, this curve cannot be a trajectory of the system when σ(z) < 0, so that z2 will stay finite when the system leaves this curve (upon approaching the straight line σ(z) = 0) to become linear. Similarly, z2 will never turn to infinity when moving along the second curve. Taking into account that transition from any (except, perhaps, the first) linearity segment to the saturation mode may occur only in the region of bounded deviations, namely, when |z1 | ≤ γ, where γ = max{|z1∗ |, |z1∗∗ |} and z1∗ and z1∗∗ are abscissas of the points where the asymptote intersects the boundary of either saturation set (see Fig. 2), which take finite values for any finite λ, the trajectory will never leave a bounded set of the phase plane even in the case of an infinite number of switchings. Now, let us prove that no linearity segment, but, perhaps, the first and last one, can lie in the 1st or 3d quadrant of the plane. For definiteness, let us consider the 1st quadrant (z1 > 0, z2 > 0). Suppose that a linearity segment different from the first one lies in it (as can be seen from Fig. 2, this may happen only in a sufficiently small neighborhood of the origin, since the part of the axis z1 for which z1 ≥ u ¯/λ2 lies in the saturation region. Let us prove that, in this case, the control will never reach saturation. Let us denote the coordinate of the point of the intersection of the considered segment and the z1 axis as z10 and the value of function σ(z) at this point as σ0 . Since the system is linear at this point, we have σ0 < u ¯. Considering (z10 , 0) as the initial point, we find that the subsequent motion of the linear system is given by z1 (ξ) = z10 (1 + λξ)e−λξ , z2 (ξ) = −λ2 z10 ξe−λξ . With regard to the relation σ0 = λ2 z10 , we have σ(ξ) = σ0 (1 − λξ)e−λξ and σ (ξ) = −λσ0 (2 − λξ)e−λξ . Then, |σ(ξ)| ≤ σ0 ∀ξ, and the following chain of inequalities holds: |σ(ξ)| max |σ(ξ)| σ0
The last inequality means that control will not reach saturation, and the system will stay linear. Since the linear system is stable, its trajectory tends to the origin, and, hence, the considered segment is the last segment of the trajectory. The proof for the 3d quadrant is similar. Now, consider the function 1 + 1/¯ u. L(z) = |z1 | − u ¯ 1 + z22
(A.2)
It is not difficult to see that it is positive definite in the entire phase plane and is equal to zero at the origin. Its contour lines L(z) = c for 0 < c < 1/¯ u are closed curves consisting of segments of the integral curves (A.1). For c ≥ 1/¯ u, the contour lines are not closed and consist of two unbounded integral curves (A.1). For the sake of illustration, Fig. 1 shows the contour lines corresponding to c = 5, 10, 15, and 20 for the WR with u ¯ = 0.05 m−1 and λ = 1. The function L(z) is differentiable everywhere, except for the points on the line z1 = 0, and its derivative is z2 z2 . L (z) = z2 sign(z1 ) + u ¯(1 + z22 )3/2 Suppose that there exists a closed trajectory. Then, the corresponding curve on the phase plane consists of four segments: two saturation segments and two linearity segments moving along which the system intersects the line σ(z) = 0. As proved above, the latter segments lie entirely in the 2-nd and 4th quadrants, respectively. Let us find the derivative of the function L(z) by virtue of the equations of motion on each of the above-mentioned segments. 1. Saturation segment lying in the 2nd, 1st, and 4th quadrants (σ(z) > 0, u = −¯ u, and z2 = −¯ u(1 + z22 )3/2 ). (a) In the 2nd quadrant (z1 < 0, z2 > 0), we have L (z) = −z2 − z2 = −2z2 < 0; (b) in the 1st and 4th quadrants (z1 > 0), L (z) = z2 − z2 = 0. 2. On the linearity segment in the 4th quadrant (z1 > 0, z2 < 0), we have z2 = −σ(z) and z2 σ(z) σ(z) L (z) = z2 − = z 1 − < 0. 2 u ¯(1 + z22 )3/2 u ¯(1 + z22 )3/2 In the same way, it is proved that the derivative is negative semi-definite on the two other segments. We have proved that the function L(z) monotonically decreases on each segment of the considered trajectory. Hence, having started at some point, the system cannot occur at the same point after passing these four segments. Thus, assuming the existence of a closed trajectory, we arrived at the contradiction, which proves that any trajectory of the system tends to the equilibrium state z = 0.