CLF-based Tracking Control for UAV Kinematic Models with Saturation Constraints

Transcription

1 CDC3-IEEE45 CLF-based Tracking Control for UAV Kinematic Models with Saturation Constraints Wei Ren Randal W. Beard Department of Electrical and Computer Engineering Brigham Young University Provo, Utah 846, Abstract This paper considers the trajectory tracking problem for unmanned air vehicles UAVs). We assume that the UAV is equipped with an autopilot which reduces the twelve degree-of-freedom DOF) model to a six DOF model with altitude, heading, and velocity command inputs. In this paper we restrict our attention to planar motion. One of the novel features of our approach is that we explicitly account for heading and velocity input constraints. For a UAV, the velocity is constrained to lie between two positive constants, and therefore presents particular challenges for the control design. We propose a control Lyapunov function CLF) approach. We first introduce a CLF for the input constrained case, and then construct the set of all constrained inputs that render the CLF negative. The control input is then selected from this feasible set. The proposed approach is then applied to a simulation scenario, a UAV is assigned to transition through several targets in the presence of multiple dynamic threats. Introduction The stabilization and tracking of dynamical systems with nonholonomic constraints has received recent attention in the literature. Reference [] provides a nice overview of the developments in control of nonholonomic systems. An inherent challenge, identified by Brockett s well-known necessary condition for feedback stabilization [], is that nonholonomic systems cannot be stabilized via smooth time-invariant state feedback. A simple but classical example of a nonholonomic system is a mobile robot which serves as an interesting topic for stabilization [3] and tracking [3, 4, 5, 6]. Unmanned aerial vehicles UAVs) equipped with low-level altitude-hold, velocity-hold, and heading-hold autopilots can be modeled by kinematic equations of motion that are similar to the kinematic equations of motion of mobile robots. The input constraints, however are Corresponding author. quite different. While mobile robots and UAVs have similar angular velocity constraints, the linear velocity constraints are quite different. In particular, UAVs have a minimum velocity constraint that is greater than zero, due to the stall conditions of the aircraft. On the other hand, mobile robots may have negative linear velocity, greatly simplifying the tracking problem. This paper deals with the issue of tracking control for UAV kinematic models with physically motivated heading rate and velocity constraints. We approach the problem using control Lyapunov functions CLFs) [7, 8, 9]. While our approach is designed for UAVs in particular, it is also valid for mobile robot kinematic models with input constraints. Current approaches to tracking control of mobile robots includes linear model approaches [, ], the sliding mode approach [5], backstepping [, 4, 6, 3], and passivity based approaches [4], to name a few. Reference [5] proposes a robust tracking strategy for nonholonomic wheeled mobile robots using sliding modes. The approach asymptotically stabilizes the mobile robot to a desired trajectory and is robust to bounded external disturbances. In [4], a tracking control methodology using time-varying state feedback based on the backstepping technique is proposed for both a kinematic and simplified dynamic model of a two-degree-of-freedom mobile robot, local and global tracking problems are solved under certain conditions. Using the backstepping technique and the LaSalle s invariance principle, [3] proposed a controller with saturation constraints which can simultaneously solve both the tracking and regulation problems of a unicycle-modeled mobile robot. With their approach, mobile robots can globally follow any path specified by a straight line, a circle, or a path approaching the origin using a single controller. Reference [4] developed a model-based control design strategy via passivity and normalization approaches to deal with the problem of global stabilization and global tracking control for the kinematic model of a wheeled mo-

2 bile robot in the presence of input saturations. Saturated, Lipschitz continuous, and time-varying feedback laws are obtained in their approach. However, in all of these approaches, negative velocities are allowed with exclude them from being used on UAVs. The CLF approach introduced by Artstein [7] and Sontag [8] can be used to find asymptotically stabilizing controllers for nonlinear systems. Artstein s theorem [7] states that the existence of a differentiable CLF implies and is also implied by) the existence of a continuous feedback stabilizer [5]. Based on a known CLF, universal formulas are found to generate explicit feedback stabilizing control laws for cases of unbounded controls [9], bounded controls [6], and positive controls [7]. In Ref. [8], a class of pointwise min-norm control laws are defined based on robust CLF, which minimize the instantaneous control effort while maintaining some desired negativity of the worst-case Lyapunov derivative. In [9], the satisficing approach is developed to design and analyze state-feedback control laws based on CLFs. The satisficing framework parameterizes a set of asymptotically stabilizing control laws that obey an instantaneous cost-benefit inequality. Under certain definitions and parameterizations, these control laws guarantee inverse-optimality and desirable stability margins. This paper proposes a different approach to the input constrained tracking control than those papers mentioned above. Instead of introducing a specific control law and then proving its stability, we propose a valid CLF for tracking control of the kinematic model with input constraints, and derive a state-dependent, timevarying set of feasible control values from which different controllers can be instantiated. In this sense, the approach is similar to the satisficing approach proposed in [9]. To our knowledge the CLF proposed in this paper has not appeared previously in the literature. Selection from the feasible control set, guarantees accurate tracking as well as satisfaction of the saturation constraints. Different control strategies can be derived by selection from the feasible control set according to some auxiliary performance index. This approach introduces a great deal of flexibility to the tracking control problem. It is worthwhile to mention that the universal formulas introduced in [6, 7] are not feasible in the UAV case due to the velocity constraints. The salient features of our approach are as follows. First, under the proposed tracking CLF framework with input constraints, we allow the reference velocity and angular velocity to be piecewise continuous while other papers for tracking control e.g. [4, 4, 3]) constrain them to be uniformly continuous in order to apply Barbalat s lemma. This in turn relaxes the potential restrictions for the other approaches. Second, we apply the tracking control approach with input constraints to a UAV scenario, the UAV is assigned to transition through several targets in the presence of dynamic threats. Instead of following a simple path like straight lines and circles e.g. [4, 4, 3]), the UAV tracks a real-time trajectory generated dynamically from a trajectory generator according to the current situation. The remainder of the paper is organized as follows. In Section, we formally state the tracking control problems that will be addressed in the paper. In Section 3, we propose a valid CLF for tracking control with input constraints. A nonlinear tracking control is then instantiated from the feasible control set and is presented in Section 4. In Section 5, simulation results are given. Finally Section 6 contains our conclusion. Problem Statement Following [], we assume that each UAV is equipped with standard autopilots for heading hold and Mach hold. In order to focus on the essential issues, we will assume that altitude is held constant. Let x, y), ψ, and v denote the inertial position, heading angle, and velocity for the UAV respectively. Then the resulting kinematic equations of motion are ẋ = v cosψ) ẏ = v sinψ) ) ψ = α ψ ψ c ψ) v = α v v c v) ψ c and v c are the commanded heading angle and velocity to the autopilots, and α ψ and α v are positive constants []. In addition, we assume that each UAV has the constraints that v min v v max and ω max ψ ω max, v min > and ω max > is the saturated heading rate. Assuming that α v is large compared to α ψ, Eq. ) reduces to ẋ = v c cosψ) ẏ = v c sinψ) ) ψ = α ψ ψ c ψ). Letting ψ c = ψ + α ψ ω and v c v, Eq. ) becomes ẋ = v cosψ) ẏ = v sinψ) 3) ψ = ω with input constraints that v min v v max and ω max ω ω max. Note that if v min = v max, then Eq. 3) is the kinematic model for a mobile robot with input constraints.

3 In this paper we will assume the existence of a reference trajectory x r, y r, ψ r, v r, ω r ) which satisfies ẋ r = v r cosψ r ) ẏ r = v r sinψ r ) 4) ψ r = ω r under the constraints that v r and ω r are piecewise continuous and satisfy the constraints v min + ɛ v r v max ɛ 5) ω max + ɛ ω r ω max ɛ, ɛ and ɛ are positive constants. The inclusion of ɛ i in the constraints of the reference trajectory generator, guarantees that there is sufficient control authority to track the trajectory. We will see that as ɛ i approach zero, the feasible control set will vanish. The control objective is to find feasible control inputs v and ω such that x r x + y r y + ψ r ψ as t. Transforming the tracking errors expressed in the inertial frame to the robot frame, the error coordinates [] can be denoted as x e y e ψ e = cosψ) sinψ) sinψ) cosψ) x r x y r y ψ r ψ 6) Accordingly, the tracking error model can be represented as ẋ e = ωy e v + v r cosψ e ) ẏ e = ωx e + v r sinψ e ) 7) ψ = ω r ω. Following [3], equation 7) can be simplified as and ẋ = u ẋ = ω r u )x + v r sinx ) 8) ẋ = ω r u )x + u, [ u u x x x ] [ = = ψ e y e x e ω r ω v v r cosx ) 9) The input constraints under the transformation become ɛ u ɛ ]. v min v r cosx ) u v max v r cosx ). ) It is also easy to see that v min v max + ɛ v min v r cosx ) v min + v max ɛ ɛ v max v r cosx ) v max ɛ. Obviously, Equations 6) and 9) are invertible transformations, which means x, x, x ) =,, ) is equivalent to x e, y e, ψ e ) =,, ) and x r, y r, ψ r ) = x, y, ψ). Therefore, the original tracking control objective is converted to a stabilization objective, that is, our goal is to find feasible control inputs u and u to stabilize x, x, and x. Note from Equation 8) that when both x and x go to zero, that x becomes uncontrollable. To avoid this situation we introduce another change of variables. Let x = mx + x π, ) m > and π = x + x +. Accordingly, x = x m x mπ. Obviously, x, x, x ) =,, ) is equivalent to x, x, x ) =,, ). Therefore it is sufficient to find control inputs u and u to stabilize x, x, and x. With the same input constraints ), Equation 8) can be rewritten as x = m ax bx ))u + ax bx )ω r + av r sin x + bu ẋ = ω r u )x + v r sinx ) ) ẋ = ω r u )x + u, a = +x and b = xx, and it is obvious π 3 π 3 that < a, < b <, and < ax bx <. 3 CLF for Tracking Control with Saturation Constraints In this section, we find a valid CLF for tracking control with input constraints. Consider the following class of affine nonlinear time-varying systems ẋ = fx, t) + gx, t)u, 3) x IR n, u IR m, and f : IR n IR + IR n and g : IR n IR + IR n m are locally Lipschitz in x and piecewise continuous in t. Definition A continuously differentiable function V : IR n IR + IR is a control Lyapunov function CLF) for the system??) with input constraints u U IR m if it is positive definite, decrescent, radially unbounded in x, and satisfies inf u U { V t + V fx, t) + gx, t)u) x } W 3 x), 4)

4 x and t W 3 x) is a continuous positive definite function. In order to find a CLF for systems with bounded input constraints, we prefer the partial derivative of V to be bounded. Accordingly, we have the following lemma. Lemma If W x) = x T x +, then W x) is continuously differentiable, radially unbounded, positive definite, and W x. Proof: Trivial. Lemma will be used to construct a CLF for system. The following lemma defines a valid CLF for tracking control with input constraints. Lemma 3 Consider the function ) x V = k W + W x ), k > the function x and W x) is defined in Lemma, and W 3 x) = α + α + α, 5) α = γ x ) ) x x α = k γ v min + ɛ ) sin π mπ α = γ k x ) ) x x π π ) ) x v min + ɛ ) cos v min, mπ and = + x, and γ, γ, γ, ). If m > max d = { d ɛ +, d ɛ +, k + 3 ) + ω max + v max + } v acos min v min+ɛ ), 4, π γ + ) k + γ + γ ) k 3 ) ɛ ɛ + γ, d = k M + k + ) M + v max + ω max ) v min ɛ ɛ + γ, 6) and M > and M > will be specified in the proof, then W 3 x) is a continuous positive definite function and V is a constrained CLF for system ) with input constraints ). Proof: Obviously V is positive definite, decrescent, and radially unbounded, therefore it remains to show that Equation 4) is satisfied. Using straightforward calculations, it is easy to show that under the hypothesis, W 3 x) is continuous and positive definite. Differentiating V we obtain the following expression after some algebraic manipulation: V + W 3 x) = σ + σ + σ 3 u + σ 4 u + σ 5, 7) σ = k x π v r sinx ) + α σ = x ax bx ) ω r + av r sin x )) + α σ 3 = x m ax bx )) π σ 4 = k x ) x x π π σ 5 = α. The goal is to show that we can always find a control satisfying ) to drive V + W 3 x). Three cases will be considered with respect to x. Case : x. From 7), we know that σ < k v max ɛ + γ v min + ɛ )), σ < ω max ɛ ) + v max ɛ ) + γ, σ 4 < k +, and σ 5 < k γ ɛ. When x <, we can choose u = v max v r cosx ), which is always positive. Thus σ 4 u <. When x, we can choose u = v min v r cosx ). Thus σ 4 u < k + )v min+v max ɛ ). Combining these two cases, we know that σ 4 u < k + )v min +v max ɛ ). We can also choose u = ɛ sgn x ). As a result, V + W 3 x) < x m ax bx ))ɛ + d, d is defined in 6). Therefore, after some manipulation, we know that if m > d ɛ +, V W3 x). Case : < x <. In this case, we know that x m < π 4, x mπ < π 4, and x < since m > 4 π. We also know that σ β v r, x β = k π sinx ) + γ sin x mπ )). When x, obviously β / x has a finite positive upper bound. When x, it can be easily verified that β. Accordingly, we know that β / x ) when x. In fact, in the case of x, β x when

5 x. In addition, β / x is continuous when < x <, which means that it has a finite positive upper σ bound. Thus x has a finite positive upper bound k M when < x <, M can be easily specified by numerical methods. Also, when x <, we can choose u = v max v r cosx ), which is greater than or equal to ɛ from section. In this case, it is easy to see that σ 4 u +σ 5. When x, we can choose u = v min v r cosx ). In this case, since cosx ) >, we know that σ 4 u + σ 5 ) β, β = k ) x π v min v min + ɛ ) cosx )) + σ 5. When x, it is obvious that β / x has a finite positive upper x x bound. When x, it can be verified that β since m > Accordingly, we know that. v acos min v min +ɛ ) β / x when x. In fact, in the case of x, β / x when x. In addition, β / x is continuous when < x <, which means that it has a finite positive upper bound. Thus σ 4 u + σ 5 )/ x has a finite positive upper bound k + )M, M can also be easily specified by numerical methods. We can also choose u = ɛ sgn x ). It is also easy to see that σ ω max ɛ ) + v max ɛ ) + γ ) x. As a ) result, V + W3 x) d ɛ m ax bx )) x, d is defined in 6). Therefore, if m > d ɛ +, V W3 x). Case 3: x =. In this case, 7) can be written as V + W 3 x) = σ + σ 4 u + σ 5. Similarly, we know that x mπ < π 4 since m > 4 π. Since x =, it is easy to see that σ. When x, we can choose u = v min v r cosx ). When x <, we can choose u = v max v r cosx ). Thus it can be verified that σ 4 u + σ 5 since m >. As a result, we know that V v acos min W v min +ɛ ) 3 x). Therefore, V is a valid CLF for system ) under saturation constraints ). Note that a very conservative upper bound is found for m in each case for simplicity of the proof. In reality, m can be much smaller than the upper bound specified above. 4 Nonlinear Tracking Control based on CLF With the CLF given in Lemma 3, our goal in this section is to find a family of feasible tracking control laws based on this CLF. Motivated by equation 7), we consider the following expression λ u + λ u = λ, 8) λ = σ 3, λ = σ 4, and λ = σ σ σ 5. In the u u plane, Equation 8) denotes a line which separates the control space into two halves: λ u + λ u λ and λ u + λ u > λ. The half plane λ u + λ u λ represents unconstrained control values that ensure that V W 3 x). The input constraints ) produce a time-varying rectangle in the u u plane. One interpretation of Lemma 3, is that it guarantees that there is always a nonempty intersection between the areas described by λ u + λ u λ and the input constraints ). We call this nonempty intersection the feasible control set and denote it by Ft, x). We have the following lemma. Lemma 4 If the time-varying feedback control law kt, x) satisfies. kt, ) =,. kt, x) Ft, x), x, 3. kt, x) is locally Lipschitz in x and piecewise continuous in t, x and t, then this control solves the tracking problem with input constraints, that is, x r x + y r y + ψ r ψ as t. Proof: Straightforward using standard Lyapunov stability theory for time-varying systems. As a preliminary result, the following lemma gives a simple control law chosen from the feasible control set. Lemma 5 If sat u x) = sat u x) = k sat t, x) = ) satu η x ), sat u η x ) { x, x ɛ 9) sgnx)ɛ, x > ɛ v min v r cosx ), x < v min v r cosx ) v max v r cosx ), x > v max v r cosx ) x, otherwise ) then there exist positive constants η and η such that k sat t, x) satisfies the conditions of Lemma 4. Proof: Obviously k sat t, x) satisfies the first and third conditions in Lemma 4. We will show that they also stay in the feasible control set Ft, x). In the proof for Lemma 3, we use discontinuous signum like functions for u and u. Here we use two saturation functions instead. When x and x are large, these two sets of,

6 functions are the same, which means the proposed control law stays in the feasible control set Ft, x). When x and x are close to zero, we can see that u = η x and u = η x. By properly tuning η and η, which may depend on other parameters in Equation 7), the proposed control law can be made to guarantee that V + W 3 x) based on the property of W 3 x) in Equation 5). We have second order items x ) and x π ) instead of first order items x and x π in α and α respectively in W 3 x) in order that besides discontinuous sigum like functions the proposed saturation functions for u and u can also make σ 3 u + σ 4 u dominate the other terms in Equation 7) to guarantee V + W 3 x) when x and x are small. Since the proof follows a similar line as that in Lemma 3, a detailed proof is omitted here. Waypoint Path Planner Dynamic Trajectory Smoother Trajectory Tracker Low-Level Autopilot UAV In Lemma 5 we used a simple control law that stays in the feasible control set. Other continuous saturation functions like atan, tanh are also possible as long as they stay in the feasible control set. In the case of v r and ω r being uniformly continuous, it is also possible to use geometrical strategies to find feasible control laws e.g. choose the geometrical center of the feasible control set Ft, x) as feasible controls). One advantage of the CLF-based approach used in this paper is that it only requires v r and ω r to be piecewise continuous instead of being uniformly continuous, which results in wider potential applications than other approaches which requires uniform continuity. Note that if we go back to the original system defined by 3), we can see that v = sat u η x ) + v r cosx ) and ω = ω r + sat u η x ), which is piecewise continuous in t since v r and ω r are piecewise continuous in t. The other advantage is that it provides the possibility to use other advanced strategies to choose feasible controls from Ft, x). For example, at each time t, a feasible control may be generated from Ft, x) while optimizing some performance index function or minimizing some cost function at the same time, which introduces more flexibility and benefits to the tracking control problem than specifying a fixed control law in advance. In addition, it is also possible to propose a suboptimal controller from Ft, x) based on the combination of model predictive techniques and the tracking CLF see e.g. []). 5 Simulation Results In this section, we simulate a scenario a UAV is assigned to transition through several known targets in the presence of dynamic threats. The overall system architecture is shown in Figure. The design of the Waypoint Path Planner WPP) and Dynamic Trajectory Smoother DTS) are described in [3, 4] and [5, 6], respectively. The WPP produces waypoint paths that change in accordance with the dynamic threat envi- Figure : Path Planning Architecture. ronment. The DTS smoothes through these waypoints and produces a feasible time-parameterized trajectory that satisfies Equations 4) and 5). The parameters used in this paper are given in Table 5. Parameter Value v min. m/s) v max. m/s) ω max.7 rad/s) ɛ. m/s) ɛ. rad/s) v r [.,.8] m/s) ω r [.5,.5] rad/s) α ψ 5 α v 5 m k γ.5 γ.5 γ.5 η η Table : Parameter values used in simulation. In Figure shows the problem scenario. The dots are threat locations to be avoided. The waypoint path planner described in [3, 4] generates the waypoint path which is shown in green. The dynamic trajectory smoother described in [5, 6] generates the reference trajectory, which is shown in red. The actual trajectory is shown in blue. The trajectory tracking errors for position and head-

7 .8.65 v r m/s) time s).5.75 ω r rad/s).75.5 time s) Figure : The simulation scenario: waypoint path green), smoothed reference trajectory red), and actual trajectory blue). Figure 4: The reference control inputs v r and ω r..75 x r x m) v m/s).5.5 time s) time s) y r y m) time s).5 ω rad/s).85 ψ r ψ rad).5.7 time s) time s) Figure 5: The control inputs v and ω. Figure 3: The trajectory tracking errors expressed in the inertial frame. ing angle are plotted in Figure 3. We can see that the tracking error for heading angle converges faster than that for x and y coordinate, which is due to the weighting factor m in the definition of x. The reference control inputs v r and ω r are plotted in Figure 4. Obviously, v r and ω r are only piecewise continuous instead of uniformly continuous. The reference control inputs generated by the trajectory generator satisfy their constraints respectively, that is, v r [.,.8] m/s and ω r [.5,.5] rad/s. The control inputs v and ω are plotted in Figure 5. We can see that v and ω are within the range [, ] m/s and [.7,.7] rad/s respectively, which satisfies the input constraints. The original heading rate plot ω is somewhat chattering due to the abrupt change of the reference heading rate ω r, discrete implementation of the system with a sample rate of Hz, and the high gain m in the definition of x. The chattering phe- nomenon is overcome by adding a low pass filter after the control signal ω. 6 Conclusion A tracking CLF for a UAV kinematic models with input constraints is derived. Based on this CLF, a feasible control set is formed. This feasible control set facilitates the generation of a variety of feasible control strategies that not only guarantee accurate tracking but also optimize auxiliary performance functions. A simple saturation control strategy generated from the feasible control set was used and applied to a nontrivial simulation scenario. Acknowledgments This work was funded by AFOSR grants F and F496--C-94, and by DARPA grant NBCH3. References [] I. Kolmanovsky and N. H. McClamroch, Developments in nonholonomic control problems, IEEE

8 Control Systems Magazine, vol. 5, pp. 36, December 995. [] R. W. Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory R. S. Millman and H. J. Sussmann, eds.), pp. 8 9, Birkhaüser, 983. [3] C. C. de Wit, B. Siciliano, and G. Bastin, Theory of Robot Control. London: Spriner-Verlag, 996. [4] Z.-P. Jiang and H. Nijmeijer, Tracking control of mobile robots: A case study in backstepping, Automatica, vol. 33, pp , 997. [5] J.-M. Yang and J.-H. Kim, Sliding mode motion control of nonholonomic mobile robots, Control Systems Magazine, vol. 9, pp. 5 3, April 999. [6] T. Fukao, H. Nakagawa, and N. Adachi, Adaptive tracking control of a nonholonomic mobile robot, IEEE Transactions on Robotics and Automation, vol. 6, pp , October. [7] Z. Artstein, Stabilization with relaxed controls, Nonlinear Analysis, Theory, Methods, and Applications, vol. 7, no., pp , 983. [8] E. D. Sontag, A Lyapunov-like characterization of asymptotic controllability, SIAM Journal on Control and Optimization, vol., pp , May 983. [9] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control, vol. 34, pp , April 989. [] A. M. Bloch, N. H. McClamroch, and M. Reyhanoglu, Controllability and stabilizability properties of a nonholonomic control system, in Proceedings of the 9th Conference on Decision and Control, Honolulu, HI), pp. 3 34, December 99. [] A. W. Divelbiss and J. T. Wen, Trajectory tracking control of a car-trailer system, IEEE Transactions on Control Systems Technology, vol. 5, pp , May 997. [] R. Fierro and F. Lewis, Control of a nonholonomic mobile robot: backstepping kinematics into dynamics, in Proceedings of the 34th Conference on Decision and Control, New Orleans, LA), pp , December 995. [3] T.-C. Lee, K.-T. Song, C.-H. Lee, and C.-C. Teng, Tracking control of unicycle-modeled mobile robots using a saturation feedback controller, IEEE Transactions on Robotics and Automation, vol. 9, pp , March. [4] Z.-P. Jiang, E. Lefeber, and H. Nijmeijer, Saturated stabilization and track control of a nonholonomic mobile robot, Systems and Control Letters, vol. 4, pp ,. [5] E. D. Sontag, Control-lyapunov functions, in Open Problems in Mathematical Systems and Control Theory V. D. Blondel, E. D. Sontag, M. Vidyasagar, and J. C. Willems, eds.), Communications and Control Engineering, ch. 4, pp. 6, Springer, 999. [6] Y. Lin and E. D. Sontag, Universal formula for stabilization with bounded controls, Systems and Control Letters, vol. 6, pp , June 99. [7] Y. Lin and E. Sontag, Control-lyapunov universal formulas for restricted inputs, Control-Theory and Advanced Technology, vol., pp. 98 4, December 995. [8] R. A. Freeman and P. V. Kokotovic, Robust Nonlinear Control Design: State-Space and Lyapunov Techniques. Systems & Control: Foundations & Applications, Birkhauser, 996. [9] J. W. Curtis and R. W. Beard, Satisficing: A new approach to constructive nonlinear control, IEEE Transactions on Automatic Control, in review). Available at [] A. W. Proud, M. Pachter, and J. J. D Azzo, Close formation flight control, in Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, Portland, OR), pp. 3 46, American Institute of Aeronautics and Astronautics, August 999. Paper no. AIAA [] Y. J. Kanayama, Y. Kimura, F. Miyazaki, and T. Noguchi, A stable tracking control scheme for an autonomous mobile robot, in Proceedings of the IEEE International Conference on Robotics and Automation, pp , 99. [] B. Kouvaritakis and M. Cannon, Nonlinear Predictive Control: Theory and Practice. London, United Kingdom: The Institution of Electrical Engineers,. [3] R. W. Beard, T. W. McLain, M. Goodrich, and E. P. Anderson, Coordinated target assignment and intercept for unmanned air vehicles, IEEE Transactions on Robotics and Automation, vol. 8, pp. 9 9, December. [4] T. McLain and R. Beard, Cooperative rendezvous of multiple unmanned air vehicles, in Proceedings of the AIAA Guidance, Navigation and Control Conference, Denver, CO), August. Paper no. AIAA [5] E. P. Anderson and R. W. Beard, An algorithmic implementation of constrained extremal control for UAVs, Monterey, CA), August. AIAA Paper No [6] E. P. Anderson, R. W. Beard, and T. W. McLain, Real time dynamic trajectory smoothing for uninhabited aerial vehicles, IEEE Transactions on Control Systems Technology, in review).