Adaptive, Non-Singular Path-Following Control of Dynamic Wheeled Robots

Transcription

1 Adaptive, Non-Singular Path-ollowing Control of Dynamic Wheeled Robots D. Soetanto, L. Lapierre, A. Pascoal nstitute for Systems and Robotics and Dept. Electrical Engineering and Computers nstituto Superior Técnico Av. Rovisco Pais,, 49- Lisboa, Portugal didik, lpl, Abstract This paper derives a new type of control law to steer the dynamic model of a wheeled robot of unicycle type along a desired spatial path. The methodology adopted for path following control deals explicitly with vehicle dynamics and plant parameter uncertainty. urthermore, it overcomes stringent initial condition constraints that are present in a number of path following control strategies described in the literature. This is done by controlling explicitly the rate of progression of a virtual target to be tracked along the path, thus bypassing the problems that arise when the position of the virtual target is simply defined by the projection of the actual vehicle on that path. The nonlinear adaptive control law proposed yields convergence of the (closed loop system) path following error trajectories to zero. Controller design relies on Lyapunov theory and backstepping techniques. Simulation results illustrate the performance of the control system proposed.. NTRODUCTON Various problems related to motion control of autonomous vehicles (including air, land, and marine robots) have been studied extensively in recent years. The problems addressed in the literature can be roughly classified into three groups: point stabilization - the goal is to stabilize the vehicle at a given target point, with a desired orientation; trajectory tracking - the vehicle is required to track a time parameterized reference, and path following - the vehicle is required to converge to and follow a path, without any temporal specifications. Point stabilization presents a true challenge to control system designers when the vehicle has nonholonomic constraints, since that goal cannot be achieved with smooth (or even continuous) state-feedback control laws, as pointed out in [4]. To overcome this difficulty, two main approaches have been proposed: smooth time-varying control laws [6],[2] and discontinuous and hybrid feedback laws [], [5], [3]. The trajectory tracking problem for fully actuated systems is now well understood and satisfactory solutions can be found in advanced nonlinear control textbooks. However, in the case of underactuated vehicles, that is, when the vehicles have less actuators than state variables to be tracked, the problem is still a very active topic of research. Linearization and feedback linearization methods [], [2] as well as Lyapunov based control laws [6], [] have been proposed. Path following control has received relatively less attention in the literature. See [7], [8] for pioneering work in the area as well as [6], [5] and the references therein. Path following systems for marine vehicles have been reported in [8], [9]. The underlying assumption in path following control is that the vehicle s forward speed tracks a desired speed profile, while the controller acts on the vehicle orientation to drive it to the path. Typically, smoother convergence to a path is achieved (when compared to the behaviour obtained with trajectory tracking control laws) and the control signals are less likely pushed to saturation. This paper addresses the problem of steering the dynamic model of a wheeled robot of unicycle type along a desired path. ts main contribution is twofold: i) it extends the results obtained in [7] for kinematic models of wheeled robots to a more general setting, in order to deal with vehicle dynamics and parameter uncertainty, and ii) it overcomes stringent initial condition constraints that are present in a number of path following control strategies described in the literature. This is done by controlling explicitly the rate of progression of a virtual target to be tracked along the path, thus bypassing the problems that arise when the position of the virtual target is simply defined by the projection of the actual vehicle on that path. This procedure avoids the singularities that occur when the distance to path is not well defined and allows for global convergence of the actual path of the vehicle to the desired path. This is in striking contrast with the results described in [7] for example, where only local convergence has been proven. To the best of the authors knowledge, the idea of exploring the extra degree of freedom that comes from controlling the motion of a virtual target along a path appeared for the first time in [2] for the control of wheeled robots. This idea was later extended to the control of marine craft in [3]. However, none of these references addresses the issues of vehicle dynamics and parameter uncertainty. urthermore, the methodologies adopted in [2], [3] for control system design build on an entirely different technique that requires the introduction of a nonsingular transformation in the original error space. nterestingly enough, a very recent publication explores the same concept of a virtual target for path following of wheeled robots [7]. n this paper, controller design builds on the work reported in [7] on path following control and relies heavily on Laypunov based backstepping techniques [6] in order to extend kinematic control laws to a dynamic setting. Parameter uncertainties are dealt with in an adaptive framework by

2 y O } ig.. x q p y 2R 2L r P Q m, t θ c θ m s } Unicycle s parameters and frame definitions augmenting Lyapunov candidate functions with terms that are quadratic in the parameter errors. See [], where identical techniques were used in the design of an adaptive control law to steer the dynamic model of a wheeled robot to a point, with a desired orientation, in the presence of parameter uncertainty. Due to space limitations, only the key ideas involved in the development of a nonsingular, adaptive path following control law will be presented. The reader is referred to [9] for complete details and proofs of all statements. The paper is organized as follows. Section introduces the problem of path following control for a wheeled robot of unicyle type. Section develops a nonlinear, adaptive, path following control law to deal with vehicle dynamics and parameter uncertainty. The performance of the control system proposed is illustrated in simulation in Section V. inally, Section V contains the conclusions and describes some problems that warrant further research.. PATH OLLOWNG CONTROL. PROBLEM ORMULATON This section reviews the dynamic model of a wheeled robot and provides a rigorous formulation of the problem of steering it along a desired path. The reader is referred to [] and [7] for background material. The following assumptions are made regarding the robot depicted in igure. The vehicle has two identical parallel, nondeformable rear wheels which are controlled by two independent motors and a steering front wheel. t is assumed that the plane of each wheel is perpendicular to the ground and that the contact between the wheels and the ground is pure rolling and nonslipping, i.e., the velocity of the center of mass of the robot is orthogonal to the rear wheels axis. t is further assumed that the masses and inertias of the wheels are negligible and that the center of mass of the mobile robot is located in the middle of the axis connecting the rear wheels. Each rear wheel is powered by a motor which generates a control torque τ i ; i =,2. v A. Kinematic Equations of Motion The solution to the problem of path following derived in [7] admits an intuitive explanation: a path following controller should look at i) the distance from the vehicle to the path and ii) the angle between the vehicle velocity vector and the tangent to the path, and reduce both to zero. This motivates the development of the kinematic model of the vehicle in terms of a Serret-renet frame } that moves along the path; } plays the role of the body axis of a virtual target vehicle that should be tracked by the real vehicle. Using this set-up, the abovementioned distance and angle become the coordinates of the error space where the control problem is formulated and solved. n this paper, motivated by the work in [7], a renet frame } that moves along the path to be followed is used with a significant difference: the renet frame is not attached to the point on the path that is closest to the vehicle. nstead, the origin of } along the path is made to evolve according to a conveniently defined function of time, effectively yielding an extra controller design parameter. This simple strategy plays a key role in lifting the stringent initial condition constraints that arise with the path following controller described in [7]. The notation that follows is by now standard. See for example [7], [8]. Consider igure, where P is an arbitrary point on the path to be followed and Q is the center of mass of the moving vehicle. Associated with P, consider the corresponding Serret-renet frame }. The signed curvilinear abscissa of P along the path is denoted s. Clearly, Q can either be expressed as q = (X,Y,) in a selected inertial reference frame } or as (s,y,) in }. Stated equivalently, Q can be given in (X,Y ) or (s,y ) coordinates (see igure ). Let the position of point P in } be vector p. urther let R = R(θ c ) be the rotation matrix from } to }, parameterized locally by the angle θ c. Define ω c = θ c. Then, ωc = θ c = c c (s)ṡ () ċ c (s) = g c (s)ṡ where c c (s) and g c (s) = dcc(s) ds denote the path curvature and its derivative, respectively. The velocity of P in } can be expressed in } to yield ( ) dp = [ṡ ] T dt t is also straightforward to compute the velocity of Q in } as ( ) ( ) ( ) dq dp = + R dr + R (ω c r) dt dt dt where r is the vector from P to Q. Multiplying the above equation on the left by R gives the velocity of Q in } expressed in } as R ( dq dt ) = ( ) dp + dt ( ) dr + ω c r (2) dt

3 Using the relations ( ) ( ) dq dr = [Ẋ dt Ẏ ]T, = [ṡ ẏ ] T dt and ω c r = = θ = c c (s)ṡ c c (s)ṡy c c (s)ṡs s y equation (2) can be rewritten as Ẋ ṡ( c c (s)y ) + ṡ R Ẏ = ẏ + c c (s)ṡs Solving for ṡ and ẏ yields ṡ = [ cos θ c ẏ = [ sin θ c ] [ ] Ẋ sin θ c Ẏ ] [ Ẋ cos θ c Ẏ. ṡ ( c c y ) ] c c ṡs. At this point it is important to notice that in [7] s = for all t, since the location of point P is defined by the projection of Q on the path, assuming the projection is well defined. One is then forced to solve for ṡ in the equation above. However, by doing so c c y appears in the denominator, thus creating a singularity at y = c c. As a result, the control law derived in [7] requires that the initial position of Q be restricted to a tube around the path, the radius of which must be less than c c,max, where c c,max denotes the maximum curvature of the path. Clearly, this constraint is very conservative since the occurrence of a large c c,max in even a very small section of the path only will impose a strict constraint on the initial vehicle s position, no matter where it starts with respect to that path. By making s not necessarily equal to zero, a virtual target that is not coincident with the projection of the vehicle on the path is created, thus introducing an extra degree of freedom for controller design. By specifying how fast the newly defined target moves, the occurrent of a singularity at y = c c is removed. The velocity of the unicycle in the } frame satisfies the equation [ ] [ ] Ẋ cos θm = v (4) Ẏ sin θ m where θ m and v denote the vehicle s yaw angle and its body-axis speed, respectively. Substituting (4) in (3) and introducing the variable θ = θ m θ c gives the kinematic model of the unicycle in the (s,y ) coordinates as ṡ = ṡ ( c c y ) + v cos θ ẏ = c c ṡs + v sin θ θ = ω m c c ṡ (3) (5) where ω m = θ m. B. Dynamics. Problem ormulation The complete dynamical model of the system is obtained by augmenting (5) with the equations v = m ω = ω m c c s g c ṡ 2 (6) where ω m = N, m and are the mass and moment of inertia of the unicycle, respectively, and and Γ are the external force and moment applied to the vehicle. inally, and N can be rewritten in terms of the control inputs τ and τ 2 as = (τ+τ2) R N = L(τ τ2) (7) R where R is the radius of the rear wheels and 2L is the distance between them. With the above notation, the problem under study can be formulated as follows: Given a desired speed profile v d (t) > v min > for the vehicle speed v, derive a feedback control law for τ and τ 2 to drive y, θ, and v v d asymptotically to zero in the presence of plant parameter uncertainties in m,,r, and L.. NONLNEAR CONTROLLER DESGN This section introduces a nonlinear closed loop control law to steer the dynamic model of a wheeled robot described by (5)-(6) along a desired path, in the presence of parametric uncertainties. Controller design builds on previous work by [7] on path following control and relies heavily on backstepping techniques. The reader will find in [6] a lucid exposition of interesting theoretical and practical issues involved in backstepping. See also [] for related work on wheeled robot control in the face of parameter uncertainty. A. Nonlinear Controller Design Using a Kinematic Model The analysis that follows in inspired by the work in [7], [8] on path following control for kinematic models of wheeled robots. Recall from the problem definition in Section 2 that the main objective of the path following control law is to drive y, s and θ to zero. Starting at the kinematic level, these objectives can be embodied in the Lyapunov function candidate (see [7]) V = 2 (s2 + y 2 ) + 2γ (θ δ(y,v)) 2 (8) where it is assumed that lim t v(t) δ(,v) = y v sinδ(y,v), y v. The choice of the function δ is instrumental in shaping the transient maneuvers during the path approach phase [7]. The derivative of V can be easily computed to give

4 V = s (v cos θ ṡ) + y v sinδ + ( γ θ δ)( θ δ + γy v sin θ sin δ θ δ Let the kinematic control laws for s and θ be defined as ṡ = v cos θ + k s ; θ = δ sin θ sin δ γy v θ δ (9) k 2 (θ δ). Then V. B. Backstepping the Dynamics The above feedback control law applies to the kinematic model of the wheeled robot only. n what follows, using backstepping techniques, that control law is extended to deal with the vehicle dynamics. Notice how in the kinematic design the speed of the robot v(t) was assumed to follow a desired speed profile, say v d (t). n the dynamic design this assumption is dropped, and a feedback control law must be designed so that the tracking error v(t) v d (t) approach zero. Notice also that the robot s angular speed ω m (or equivalently the variable θ) was assumed to be a control input. This assumption will be lifted by taking into account the vehicle dynamics. ollowing [6] define the virtual control law for θ (desired behaviour of θ in (9)) as ). ζ = δ sinθ sin δ γy v k 2 (θ δ) () θ δ and let ɛ = θ ζ be the difference between actual and desired values of θ. Replacing θ by ɛ + ζ in the computaton of V gives V = k s 2 + y v sinδ (θ δ)2 γ + (θ δ) ɛ γ Augment now the candidate Lyapunov function V with the terms ɛ 2 /2 and (v v d ) 2 to obtain V 2 = V + [ ɛ 2 + (v v d ) 2]. 2 Simple computations show that if ɛ = k3 γ (θ δ) ɛ v = v d k4 m (v v d), () then, V2. t is now straightforward to compute the control inputs and N (and thus τ and τ 2 ) by solving the dynamics equation (6) to obtain N = f ( ) k 3 ε (2) = mf 2 ( ) k 4 (v v d ) where f ( ) = ζ γ (θ δ) + c c s + g c ṡ 2 f 2 ( ) = v d The above control law makes V 2 negative semidefinite. This fact plays an important role in the proof of convergence of the robot to the path. C. Parameter Adaptation We now consider the case where the unicycle s physical constants m,, L, R are not known accurately. Still, the controller should be able to achieve accurate path following in spite of parameter uncertainty. t is convenient to redefine the control inputs as u = τ τ 2 and u 2 = τ + τ 2. This yields the dynamic equations (see (6)-(7)) ωm = u c (3) v = u2 c 2 where c = R L and c 2 = mr are positive unknown parameters. Define the parameter errors c i = c i c i ; i =,2 where c i, i =,2 are assumed values of the parameters. Notice that ċ i = c i ċ i = c i ; i =,2, since the derivative of the real value of the parameters is zero. n what follows we assume the controller does not have direct access to acceleration measurements. Thus, acceleration must be estimated by relying on the dynamical model of the robot. This in turn complicates the error convergence analysis, due to parameter uncertainty. n order to deal with this problem augment V 3 with the parameter error terms to obtain the final candidate Lyapunov function V 3 = V 2 + k 5 2 c 2 + k 6 c 2 2. (4) c 2 c 2 Let the parameter update law be c = f( )ɛ k 5 c 2 = k 6 (f 2 ( )(v v d ) + u2 c 2 [ δ y (sin θ c c s cos θ) ] sin θ sin δ +c c cos θ γy θ δ, t is now straightforward to show that V3. D. Complete System. Convergence Analysis (5) rom the presentation above, the complete adaptive path following control system is described by the equations (5) and (6), the control law u = c f ( ) k 2 ε (6) u 2 = c 2 f 2 ( ) k 4 (v v d ), the parameter adaptation scheme (5) and the virtual target dynamics for ṡ in (9) ; in the above functions all parameters are replaced by their estimated values. The main result of the paper is stated next. See [9] for complete details. Theorem : Consider the closed loop adaptive control system given by the dynamic model of the unicycle (5)- (6), the control law (6), the adaptation scheme (5), and the virtual target dynamics for ṡ in (9). Let v d be a desired velocity profile. Assume that v d and its first and second order

5 derivatives are bounded. urther assume that v d does not tend to zero as t tends to infinity. Then y, s, θ, and v v d tend asymptotically to zero in the presence of bounded plant parameter uncertainties in m,, R, and L. Stated differently, the robot converges to the prescribed path asymptotically, in the presence of parametric modelling uncertainties. ndication of Proof. The proof is inspired by the work of [7]. The key steps in the proof can be briefly described as follows: since V 3 is negative semidefinite and bounded below, V 3 is bounded and has a limit. Therefore, s, y, θ, ɛ, and v are bounded since δ and v d are assumed to be bounded. rom the equations above it then follows that ṡ, ẏ, ɛ, and v are bounded as well. t is now easy to compute the second derivative of V 3 and prove it is bounded. Therefore, V 3 is uniformly continuous and Barbalat s Lemma implies that V3 as t. The theorem now follows using standard arguments in stability theory. V. SMULATON RESULTS This section illustrates the performance of the path following control law derived in simulation, using a dynamic model of a wheeled robot. The reference and actual robot paths are shown in igure 2. The desired speed v d (t) was set to ms. The function δ(y,v) used in the simulation was δ(y,v) = sign(v)θ a tanh(y ) with θ a = π 4. The values of the design parameters were γ = and K i = ; i =,...,4. The following initial conditions were adopted in the simulation (units are omitted): x = [s () y () θ() v() θ() s() c () c 2 ()] T = [ π/4 5 5] T The actual values of parameters c and c 2 are 4 and 6, respectively. igures 2 through 5 summarize the results of the simulation. igure 2 shows the envelope defined by the initial condition constraint imposed in [7]. Notice how the lateral distance to the path is driven to zero. t is important to remark that the errors in the parameter estimates do not go to zero. The objective of following the desired path is achieved nonetheless. V. CONCLUSONS The paper derived a nonlinear, adaptive control law for accurate path following of a dynamic wheeled robot in the presence of parameter uncertainty. The key idea behind the new control law developed was to control explicitly the rate of progression of a virtual target to be tracked along the path, thus bypassing the singularity problems that arise when the position of the virtual target is simply defined by the projection of the actual vehicle on that path. Controller design relied on backstepping techniques. The paper offered a formal proof of convergence of the robot to the path. Simulation results illustrated the performance of the control system proposed. The paper did not address the problems y(t) [m] s [m] y [m] c cmax x(t) [m] ig. 2. Desired and actual robot paths ig. 3. Time history of s (t) and y (t) caused by sensor noise and actuator saturation. These issues warrant further research. V. ACKNOWLEDGEMENTS This work was supported by the Portuguese CT POS Programme under framework QCA. The first author was partially supported by a post doctoral grant from the CT. The second author benefited from a grant of the EC, under project REESUB. V. REERENCES [] A. Aguiar, A. Atassi, and A. Pascoal, Regulation of a Nonholonomic Dynamic Wheeled Mobile Robot with Parametric Modeling Uncertainty Using Lyapunov unction, Proc. 39 th EEE Conference on Decision and Control, Sidney, Australia, December 2. [2] M. Aicardi, G. Casalino, A. Bicchi, and A. Balestrino, Closed Loop Steering of Unicyle-Like Vehicles via Lyapunov Techniques, EEE Robotics and Automation Magazine, pp , March 995.

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