Combined trajectory tracking and path following for ...

PROCEEDINGS OF MED'01 - 9T H MEDITERRANEAN CONFERENCE ON CONTROL AND AUTOMATION, DUBROVNIK, CROATIA, JUNE 2001.

Combined trajectory tracking and path following for marine craft P. Encarnac~ao A. Pascoal Institute for Systems and Robotics and Dept. Electrical Eng. Instituto Superior Tecnico Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal E-mail:fpme,[email protected]

Abstract |The paper presents a solution to the problem of combined trajectory tracking and path following for marine craft. This problem is motivated by the practical need to develop control systems for marine vehicles that can yield good trajectory tracking performance while keeping some of the desired properties associated with path following. The solution described builds on and extends previous work by Hindman and Hauser on so-called maneuver modi ed trajectory tracking. The theoretical tools used borrow from Lyapunov stability theory and backstepping techniques. Simulations with a nonlinear model of an underactuated marine craft illustrate the performance of the combined trajectory tracking and path following controller derived. Keywords | Trajectory Tracking, Path Following Systems, Nonlinear Control, Backstepping Techniques, Autonomous Marine Crafts.

tracked, the problem is still a very active topic of research. Linearization and feedback linearization methods [9], [10], as well as Lyapunov based control laws [2], [11], [12] have been proposed. Applications to underactuated surface vessels can be found in [13], [14], [15]. Path following control has received relatively less attention than the other two problems. See for example [2], [16] and the references therein. Path following systems for marine vehicles have been reported in [17], [18], [19], [20]. While on a path following mode, the vehicle forward speed need not be controlled accurately since it is sucient to act on the vehicle orientation to drive it to the path. Typically, smoother convergence to a path is achieved in this case when compared to the performance obtained with trajectory tracking controllers, and the control signals are less likely pushed to saturation [21]. These three classes of problems have traditionally been addressed separately. In fact, Brockett's condition implies that the point stabilization problem cannot be simply solved by designing a trajectory tracking control law and applying it to a trajectory that degenerates into a single point. Conversely, it is not clear how to extend stabilization methodologies now available to trajectory tracking problems. However, a paper by Hindman and Hauser [21] that has not received much attention in the literature shows clearly how to go from a trajectory tracking to a path following controller. In their work, the authors assume that a trajectory tracking controller is available and that a Lyapunov function is known that yields asymptotic stability of the resulting control system about a desired trajectory. The Lyapunov function captures the distance between the vehicle's posture (position and orientation) and the desired posture along the trajectory as measured in some P -metric, that is, V = k (t)  (t)k2P ;

I. Introduction

Over the last few years, there has been considerable interest in the development of powerful methods for motion control of autonomous vehicles. The problems of motion control addressed in the literature can be roughly classi ed in three groups:  point stabilization, where the goal is to stabilize a vehicle at a given point, with a desired orientation;  trajectory tracking, where the vehicle is required to track a time parameterized reference, and  path following, where the vehicle is required to converge to and follow a desired path, without any temporal speci cations. Point stabilization presents a true challenge to control system designers when the vehicle has nonholonomic constraints since, as pointed out in the celebrated result of Brockett [1], there is no smooth (or even continuous) constant state-feedback control law that will do the job. To overcome this diculty two main approaches have been proposed: smooth time-varying control laws [2], [3] and discontinuous feedback laws [4], [5], [6], [7]. The trajectory tracking problem for fully actuated systems is now well understood and satisfactory solutions can be found in standard nonlinear control textbooks [8]. However, when it comes to underactuated vehicles, this is, when the vehicle has less actuators then state variables to be

where  (t) is the vehicle's posture,  (t) is the desired posture, and kxkP = xT P x with P > 0. The key idea in the work of Hindman and Hauser can now be simply explained as follows. To execute a path following maneuver, the vehicle should look at the "closest point on the path" and adopt the corresponding hypothetical vehicle's posture as a reference to which it should converge. Thus, instead of feeding a desired posture  (t) to the tracking controller, the posture corresponding to the path point that is clos-

This work was supported in part by the Commission of the European Communities under contract no. MAS3-CT97-0092 (ASIMOV) and the Portuguese PRAXIS and PDCTM programs under projects CARAVELA and MAROV. The rst author bene ted from a PRAXIS XXI Graduate Student Fellowship.

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est to the vehicle is provided as a reference. The reference where  2 Rn is the state and  2 Rm is the control input. The functions f : D ! Rn and g : D ! Rn+m are smooth posture is formally computed as  ( ()), where in a domain D  Rn that contains  = 0 and f (0) = 0.  () =arg min k  ( )k2P Let  (t) be a reference trajectory for  and let  (;  (t))  2R a trajectory tracking control law for (1) such that, with is a projection mapping that computes the 2trajectory time  =  (;  (t)),  ! . Assume that local asymptotic  that minimizes the distance k  ( )kP . Under some stability of the trajectory tracking system can be proven technical conditions on path shape and path parametriza- using the Lyapunov function tion, it is shown that this strategy actually drives the vehiV1 (; t) = k  (t)k2P ; (2) cle to the path. Interestingly enough, the work of Hindman and Hauser goes even one step further. In fact, it shows where P is a positive de nite weighting matrix (notice that how to combine in a single control law trajectory tracking V1 can be simply viewed as the tracking error measured in and path following behaviors, thus achieving smooth spa- the P metric). Assume also that tial convergence to the trajectory as well as time converV_ 1 (; t) = 2 h  (t) ; _ _ (t)iP gence. This is accomplished by modifying the projection  k1 V1 (; t) ; function  () through the addition of a time dependent T penalty term to obtain where hx; yiP = x P y is the inner product relative to P h and k1 is a positive scalar. The condition V_1  k1 V1  (; t) = arg min (1 )2 k  ( )k2P + can be weakened if one is able to prove stability with V_1  2R i 2 negative semide nite by resorting to LaSalle-Yoshizawa or 2 (t  ) : Barbalat's lemmas [8]. To convert the trajectory tracking control law into a comThe constant  > 0 weighs the relative importance of convergence in time over spatial convergence to the path. If bined trajectory tracking and path following control law,  = 0 is chosen, pure path following is achieved (0 (; t) = let the square of the distance from the path point ( ( ) ;  )  ()). If  = 1, 1 (; t) = t since  = t is always a mini- to the point (; t) be given by the function mizer of (t  )2 , and the original trajectory tracking conU (; t;  ) = (1 )2 V1 (;  ) + 2 (t  )2 troller is recovered. When 0 <  < 1, the system displays = (1 )2 k  ( )k2P + 2 (t  )2 : attributes of both schemes. Since the Lyapunov function that is used to prove stabil- Given U (; t;  ), de ne the mapping ity of the trajectory tracking control system is also used to de ne the distance to the path, the methods proposed by  (; t) =arg min U (; t;  ) =  2R Hindman and Hauser are intrinsically suitable for vehicles h i arg min (1 )2 k  ( )k2P + 2 (t  )2 ; (3) with negligible dynamics. This follows from the fact that  2R in practical applications the distance to the path should only take into account kinematic variables such as vehicle where  is a positive scalar that allows balancing the position and possibly orientation, thus avoiding the need relative importance of time convergence to the trajecto feedforward dynamic variables. With the objective of tory against spatial convergence to the path. If  = 0, applying similar methods to the control of marine craft,  (; t) =arg min k  ( )k2P gives the trajectory time   2R this paper extends the key results of Hindman and Hauser that minimizes the distance k  ( )k2P and a pure path to encompass systems that capture the motion of vehicles with non negligible dynamics. This is done by resorting to following controller is obtained. If  = 1,  (2; t) = t backstepping techniques after modi cation of a standard (since  = t is always the minimizer of (t  ) ), yieldtracking control law for marine craft. Using the methodol- ing the original trajectory tracking controller. When 0 < ogy developed, the distance to the path is evaluated by tak-  < 1, the  mapping gives the trajectory time  of the ing into consideration the kinematic variables only, namely path point ( ( ) ;  ) that is closest to (; t) in the metric induced by U (; t;  ), and a mixed trajectory tracking the vehicle position. The paper is organized as follows: Section II derives and path following behavior is achieved. Under some mild a control law that enables balancing combined require- technical conditions on the reference path shape and paments of trajectory tracking and path following for vehicles rameterization (see [21] for details),  (
) is unique and with non negligible dynamics. The application to an au- continuous. Note that a necessary condition for  =  (; t) to be a tonomous marine craft is described in Section III. Finally, minimizer of U (; t;  ) is that Section IV presents the main conclusions. II. Combined trajectory tracking and path following control

@ U (; t;  ) = @

  2 (1 )2   ( (; t)) ; d (  (; t)) d

Consider the following general nonlinear control system _ = f () + g ()  (1)

22 (t  (; t)) = 0:

2

(4)

Consider now the candidate Lyapunov function Y (; t) =min U (; t;  ) = U (; t;  (; t)) 2

R

= (1 )2 k  ( (; t))k2P + 2 (t  (; t))2 :

(1) and a corresponding Lyapunov function V1 (; t) have been found. The vehicle dynamics can be included in the (5) control design through a backstepping step [22] as follows. Consider the change of variables z =  :

Dierentiating Y with respect to time gives

The variable z can be interpreted as the dierence between the virtual control variable  and the virtual control law . Thus, if z tends to zero, the system approaches the manifold corresponding to the "kinematic system". This motivates the choice of 1 V2 = Y (; t) + z T z 2 as a candidate Lyapunov function. Dierentiating V2 with respect to time yields V_2 = @Y @ () [f () + g () ] +  @Y () g () z + z T  _ @  (1 )2 k1 V1 (;  (; t)) + @Y () g () z + z T  _ : @ Setting @Y T  = _ gT () () Kdynz; @

D

Y_ (; t) = 2 (1 )2   ( (; t)) ; E d _ d d ( (; t)) dt (; t) P +  22 (t D(; t)) 1 ddt (; t) = 2 (1 )2   (E (; t)) ; _ d ( (; t)) P +  d h D d 2 1 dt (;Et) (1 )2  i ( (; t)) ; d ( (; t)) + 2 (t  (; t)) : d

From (4), the second term of this expression is zero, hence Y_ (; t) = (1 )2 V_1 (;  (; t))  (1 )2 k1 V1 (;  (; t))  0: To conclude asymptotic time convergence to the path one must resort to Barbalat's Lemma [8] since Y_ is not negative de nite in t  . Assuming that the rst and second derivatives of  ( ) (the velocity and the acceleration along the reference path, respectively) are bounded, it is easy to prove that Y is bounded. Thus, Y_ is uniformly continuous. Since Y_  0 and Y is bounded below by zero, Y (; t) ! Y1 as t ! 1, where Y1 is positive and nite. Now, Barbalat's Lemma allows for the conclusion that Y_ (; t) ! 0 as t ! 1. This implies that  !  ( (; t)) as t ! 1. Using (4), it follows that  (; t) ! t as t ! 1. Therefore,  !  (t) as t ! 1, that is, the system tracks the reference path asymptotically. In order to apply the method described to motion control of autonomous vehicle, some modi cations are required. This stems from the fact that the equations of motion of these vehicles exhibit a clear split between kinematics and dynamics and the speci cation of a trajectory to be tracked includes the kinematic variables only. For example, in the case of a marine craft a reference trajectory may simply include the desired evolution of the center of mass (and possibly the orientation) of that craft. In this case, however, the methodology proposed by Hindman and Hauser still provides a powerful tool to obtain a path following controller for the craft assuming a trajectory tracking control law exists and its stability has been proved using Lyapunov theory. In fact, the work of Hindman and Hauser provides a clear recipe to suitably modify the Lyapunov function for trajectory tracking so as to yield a new Lyapunov function for path following stability. Suppose for example that (1) captures the kinematics of a vehicle, and that (after a suitable coordinate transformation) the dynamics can be cast in the form _ = ; (6) where  is the actual input vector. Suppose also that a stabilizing trajectory tracking controller  =  (;  (t)) for

where Kdyn is a positive de nite matrix, it follows that V_2  (1 )2 k1 V1 (;  (; t)) z T Kdynz  0:

Using the same arguments as before, convergence to the path can be concluded. In practice, the modi cations required to deal with more realistic cases may warrant further complexity of the resulting control laws, as the next section will show. However, the key recipe provided by Hindman and Hauser still applies. III. Application to a surface craft

The methodology described in Section II is now applied to the design of a combined trajectory tracking and path following controller for the DELFIM autonomous surface craft developed at the Instituto Superior Tecnico of Lisbon, Portugal. See [23], [24] for details on the construction, modeling, and control of the DELFIM vehicle. A. Vehicle model Following standard practice, the equations of motion of a marine craft can be developed using a global coordinate frame fU g and a body- xed coordinate frame fB g that moves with the vehicle. The following notation is required [25], [26]: U pBorg := (x; y)T - position of the origin of fB g measured in fU g; B vBorg := (u; v)T - velocity of the origin of fB g relative to fU g, expressed in fB g (i.e., body- xed linear velocity); B !B := r - angular velocity of fB g relative to fU g, expressed in fB g (i.e., body- xed angular velocity). The symbol UB R ( ) denotes the rotation matrix from

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fB g to fU g, parameterized by the yaw angle . With this where X(
) and Y(
) denote hydrodynamic force coenotation, the dynamics and kinematics of the marine craft cients. The
D (
) matrix can
be further decomposed as D B vBorg ; r = D1 B vBorg + D2 r, where can be cast in the form   Dynamics:
Xu Xuu u Xuuu u2 Xvv v B B vB
B vB org + CRB org org = RB Iz r_ = N

MRB B v_ B

D1

(7) (8)

= UB R ( ) B vBorg _ = r;

where RB denotes the vector of external forces and N is the external yawing torque. The symbols MRB and Iz denote the rigid body mass matrix and the moment of inertia about the zB axis respectively, and CRB represents the matrix of Coriolis and centripetal terms. The vector RB can further be decomposed as RB = add (; _  ) + body ( ) + ;

(11) T



where add is the added mass term and  = B vBT org ; r . The term body consists of the hydrodynamic forces acting on the vehicle's body. The vector  = (Tc ; 0)T denotes the forces applied to the vehicle by the propulsion system, where Tc is the force exerted on the vehicle's body by the two back propellers working in common mode. A simpli ed model for the yawing torque N is [24], [27] N = Nv uv + Nr ur + Td;

(12)

where Td represents the torque generated by the two back propellers in dierential mode. The symbols Nv and Nr denote hydrodynamic torque coecients. Combining equations (7)-(8) and (11)-(12), the body xed dynamic model of the AUV can be written as [24], [27] M B v_ Borg + rJM B vBorg +


D B vBorg ; r B vBorg =



Tc



(13)

0

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(14)

where M includes the rigid body mass matrix and added mass terms, and J is the skew symmetric matrix 



J = 01 01 :

The term rJM B vBorg captures Coriolis and
centripetal, as well as added mass terms, and D B vBorg ; r includes the hydrodynamic damping terms. For the DELFIM vehicle, a simpli ed model of the hydrodynaminc damping terms is 



D B vBorg ; r =



Xu

Xuu u Xuuu u2 Yr r

Xvv v Yv u





0

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 0 0 : 0

Yv u

Observe that the DELFIM model was written in such a way as to separate the surge and sway dynamics from the yaw dynamics. This is because control design will resort to two backstepping-like steps, the rst to include the surge and sway dynamics, and the second to include the yaw dynamics. B. Trajectory tracking controller As explained in Section II, the starting point for the derivation of a combined trajectory tracking and path following controller for the surface craft is the availability of a trajectory tracking controller for the same vehicle. In what follows, the general structure of the trajectory tracking controller for underactuated vehicles described in [14] is adopted. In his work, Godhavn presents both a nonadaptive and an adaptive controller where the latter aims at estimating and rejecting constant ocean currents. Both schemes are applicable to surface vehicles that are only actuated in surge and yaw. In this paper, only the nonadaptive version will be used since, to derive the adaption laws for the adaptive case, the estimation errors are included in the Lyapunov functions, thus preventing the use of the Hauser and Hindman methods. The controllers developed by Godhavn are applicable to low speed slender ships for which decoupling the surge from the steering (sway and yaw) dynamics is natural. Since that is not the case for the DELFIM vehicle, a slight modi cation in the design method is necessary. The presentation that follows borrows considerably from the work in [14]. Let  (t) = (xref (t) ; yref (t))T be a reference for the position U pBorg and let p~ = U pBorg  (t) be the corresponding position error. Notice how in this case the reference trajectory only includes the desired position of the vehicle, thus avoiding specifying its orientation explicitly. Consider the Lyapunov function 1 V1 = p~T p~: (15) 2 Dierentiating V1 with respect to time yields
V_1 = p~T U RB B vBorg _ (t) : Let U RB B vB = _ (t) k1 p~; org where k1 is a positive scalar. Then V_1 = k1 p~T p~ is negative de nite in p~. Following backstepping techniques, de ne z1 = U RB B vBorg (_ (t) k1 p~)

(9) (10)

org

=

D2 r =

Kinematics: U p_ B

vBorg

;

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as the dierence between the virtual control variable Finally, consider the Lyapunov function U RB B vB and the virtual control law _ (t) k1 p~, and org 1 let V2 be the Lyapunov function V3 = V2 + z22 2 1 (16) with time derivative V2 = V1 + z1T z1 : 2 V_3 = k1 p~T p~ k2 z1Tz1  The time derivative of V2 is given by 0 T U 1   z1 RB M 0 1 a + V_ 2 = k1 p~T p~ k2 z1T z1 +    z2 hIz0 (Td + Nv uv + Nr ur) + h2 Tc   a ; z1T U RB M 1 0 = k1 p~T p~ k2 z1T z1 + z1T U RB M 1 z0 + 2   where k2 > 0, and h 0 z2 Iz (Td + Nv uv + Nr ur) + h2 : a = h0 r + h1 with The control signal B h0 = (JM MJ
+ D2 ) vBorg Td = Nv uv Nr ur    h1 = D1 B vBorg B vBorg k1 M B vBorg Iz 0 T U 1 U T h2 + z1 RB M M RB (~p  (t) k1 _ (t) + k2 z1 ) : 1 + k3 z2 ; (18) h0 Choosing where k3 is a positive scalar, renders   Tc = 1 0 a (17) V_3 = k1 p~T p~ k2 z1T z1 k3 z22 yields negative semide nite. In [14] it is shown, using LaSalle V_2 = k1 p~T p~ k2 z1Tz1  arguments, that the control law (17)-(18) provides exponential tracking of z1 as long as saturation of the inputs is z1T U RB M 1  0 01  a : avoided, and the vehicle surge velocity u is kept nonzero. It is now straight forward to derive a combined trajectory To include the yaw dynamics, consider now tracking and path following controller for the surface craft. To do this, one starts by identifying the kinematic equation 1  0 1  a; z2 = (9) with the general model (1) by making  = (x; y)T and   as the vehicle inertial velocity, that is,  = U RB B vBorg . with  > 0. The time derivative of a can be computed to Hence, f () = 0 and g () = I , where I denotes the idengive tity matrix. The Lyapunov function (15) should then be a_ = h0 r_ + h2 ; modi ed according to (5). The last step consists of rede ning the Lyapunov function (16) as where 1 h2 = (JM MJ
+ D2 ) B v_ Borg r+ V2 = Y (; t) + z1T z1 ; 2 _D1 B vBorg
B vBorg + B B B D1 vBorg v_ Borg k1 M v_ Borg where Y is de ned in (5). The details are omitted. U R_ T (~ B p  (t) k1  (t) + k2 z1 )
C. Simulation results M U RBT p~_ (3) (t) k1  (t) + k2 z_1 : Simulations with the model of the autonomous surface Therefore, craft DELFIM were performed choosing k1 = 1, k2 = k3 =   0:1, and  = 0:017, as in [14]. In the simulations, the initial
z_2 = 1 0 1 hIz0 (Td + Nv uv + Nr ur) posture of the vehicle was set to (x; y; )jt=0 = 35 m; 0; 2 1  0 1 h2 : and the desired velocity along the reference path was chosen as 1 ms 1 . The performance achieved with pure trajecIn order to simplify the expressions, de ne tory tracking ( = 1), pure path following ( = 0), and for   1 combined trajectory tracking and path following ( = 0:2) h0 =   0 1  h0 1 was assessed. Figure 1 shows the vehicle following a path h1 =   0 1  h1 consisting of two straight line segments joined together by h2 = 1 0 1 h2 : a cubic spiral, for the case when  = 1. Figure 2 depicts the actuator signals, that is, the force Tc and the torque Td With these de nitions, generated by the common and dierential thruster modes, respectively, and Figure 3 shows how the tracking errors h0 z_2 = (Td + Nv uv + Nr ur) + h2 : x (t) xref (t) and y (t) yref (t) converge to zero. The I z

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y vehicle trajectory for  = 0 (pure path following

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case) is presented in Figure 4. Figure 5 shows the actuator signals. The trajectory tracking errors x (t) xref (t) and y (t) yref (t), and the modi ed errors x (t) xref ( (; t)) and y (t) yref ( (; t)) are plotted in Figures 6 and 7, respectively. From Figure 7 it is possible to see that the vehicle actually converges to the path. However, from Figure 6 it is clear that it is always about 1:1 m ahead of the "reference" vehicle. Finally, Figures 8 to 11 refer to the combined trajectory tracking and path following controller that is obtained by setting  = 0:2. Figure 8 shows the x y vehicle trajectory, while Figure 9 displays the actuator signals. From the tracking errors in Figure 10 and from the modi ed errors in Figure 11, the combined behavior can be interpreted as follows: the vehicle converges quickly to the desired path and only then will it react to achieve zero trajectory tracking error. Notice that in this particular example the overall spatial approach to the desired path does not show drastic dierences in the three dierent simulations. However, the control activity is considerably smoother in the cases where  6= 1, thus showing the bene ts of path following.

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modi ed trajectory tracking and allows for its application to the control of vehicles with non negligible dynamics. The methodology described has been used to develop a combined trajectory tracking and path following controller for a fully actuated underwater vehicle and also applied to the coordinated control of autonomous marine craft [28], [29]. Further work will address the problem of controlling surface crafts in the presence of parameter uncertainty and external disturbances, as well as reduced motion sensor information. References

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[1] R. W. Brockett, \Asymptotic stability and feedback stabilization," in Dierential Geometric Control Theory, R. W. Brock-

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[5]

ett, R. S. Millman, and H. J. Sussman, Eds., Boston, 1983, pp. 181{191, Birkhauser. C. Canudas de Wit, H. Khennouf, C. Samson, and O. J. Srdalen, \Nonlinear control design for mobile robots," in Recent Trends in Mobile Robotics, Yuan F. Zheng, Ed. 1993, vol. 11, pp. 121{156, World Scienti c Series in Robotics and Automated Systems. J. M. Godhavn and O. Egeland, \A Lyapunov approach to exponential stabilization of nonholonomic systems in power form," IEEE Transactions on Automatic Control, vol. 42, no. 7, pp. 1028{1032, July 1997. A. Aguiar and A. Pascoal, \Stabilization of the extended nonholonomic double integrator via athlogic-based hybrid control," in Proceedings of SYROCO'2000, 6 IFAC Symposium on Robot Control, Vienna, Austria, Sept. 2000. A. Astol , \Exponential stabilization of a wheeled mobile ro-

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2

λ

x−x (π ) [m]

0

ref

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−4

−6

0

50

100

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200 t [s]

250

300

350

0

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λ

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Fig. 11. Modi ed error  (t)  ( (; t)) for  = 0:2.

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