Underwater vehicles: a surprising non time-optimal path

Underwater vehicles: a surprising non time-optimal path M. Chyba Abstract his paper deals with the time-optimal problem for a class of underwater vehicles. We prove that if two configurations at rest can be joined by a horizontal or a vertical translation in the body frame coordinates, then there exists a shorter path. Our computations are based on the maximum principle and use that translations in the body frame coordinates are time-equivalent to paths formed by concatenations of 2-singular extremals that are proved to be non time-optimal. ntroduction One of the most important problems in robotics is the motion planning problem. t adresses the question of finding a path between an initial and a final position for a given system such as for instance a wheeled mobile robot, an articulated arm or an underwater vehicle. his problem is a particularly challenging one and has been widely studied during the past few years. Among all the possible paths, the optimal ones with respect to some given criteria are in many ways interesting. We want for instance to minimize the duration of a mission, the fuel comsumption of our vehicle or the energy spent to attain the desired goal. Notice also that when considering the motion planning problem in a cluttered environment, most of the existing planners are based on a steering method coupled with a global geometric obstacle avoidance scheme. Optimal paths satisfy, in general, a particular topological property that makes the algorithm converge and therefore allows one to steer the system even in the presence of obstacles, see [7]. n this paper, we focus on the time-optimal problem for a specific class of underwater vehicles. We consider a simplified model moving in the vertical plane and whose actuation is realized through thrusters with magnitude limits. Mathematically, the evolution of such a vehicle is described by an affine control system Department of Mathematics, 2565 McCarthy Mall, University of Hawaii, Honolulu, H 96822 ẋ = f(x) m i= g i(x)u i and the question we address is that of: given a set of initial and final states, determine among all possible trajectories joining them a time-optimal one. Optimization problems are known as highly difficult problems. A great deal of energy has been spent in the past decades to find a systematic way to characterize optimal trajectories. n particular, differential geometric techniques have been efficiently used to extract information from the maximum principle. However, due to technical and computational difficulties it is still impossible in most cases to find the optimal trajectories for a given criteria, the hard part being to reduce the set of candidate extremals for optimality all the way to a single optimal extremal. n this paper, we are concerned with a more modest goal: given a specific trajectory, we discuss its optimality status. Our motivation for such a study is the incorporation of our methods into a working vessel and is not only based on theoretical aspects. n a long term basis, our main objective is to be part of a general effort to provide enhanced research tools to other scientists. hen, we have to consider the optimal control problem with new eyes and take into account practical issues as for instance the fact that controls with discontinuities cannot be realized by the robot. Using theoretical tools as the maximum principle and differential geometric techniques with the utlimate goal of solving concrete problems is a new direction mathematicians and engineers have been taken recently. o get some insight on the problem, we consider in this paper as a first approach an idealized model. We then have to be careful not to draw to fast any practical conclusion for our underwater vehicles. However, in the conclusion section we include a brief discussion on why the results obtained under our assumptions seem to reflect the reality and hence are a good starting point. n [2], we address the general question of the time-optimality for the important class of controlled mechanical systems. We derive conditions on the Lie brackets of the vector fields describing the system for a trajectory to be optimal. An application on underwater vehicle is considered, and based on the maximum

as follows (see [5] for the details): z x v v 3 θ ẋ ż θ v v 3 Ω = cos θv sin θv 3 cos θv 3 sin θv Ω v 3 Ω m3 v Ω m v v 3 Figure : he model principle we analyze the structure of the singular extremals. Moreover, in [] we include a discussion on the time-optimality status of some specific trajectories and conjecture the time-optimality for the translations in the body frame coordinates. More precisely, we conjecture that if two configurations at rest can be joined by a horizontal or a vertical translation in the body frame coordinates, with one switching for the non singular control, then it is the time-optimal motion. Surprisingly, at least for me in a first time, this conjecture happens to be false. ndeed, in this paper we provide a time-equivalent path to the translations in the body frame coordinates that does not satisfy the necessary conditions of the maximum principle. 2 he model We make the following assumptions on the vehicle: it is of ellipsoidal form, neutrally buoyant and uniformly distributed. he motion of the vehicle is restricted in the vertical plane. he absolute position of the vehicle is described by its horizontal position x and its vertical position z, while the orientation is represented by the angle θ, see Figure. hen, the configuration variable corresponding to our vehicle is the triple q = (x, z, θ). Let us introduce the horizontal and vertical velocities v, v 3 of the vehicle in the body frame coordinates (see Figure ) and Ω the scalar angular rate in the plane. hen, neglecting the viscosity effects, the equations of motion of such a vehicle under the assumptions that it is submerged in an infinitely large volume of incompressible, irrotational and inviscid fluid at rest at infinity are described as a conservative mechanical system where, ( ) are the body-fluid mass terms in the body horizontal and vertical directions and is the body-fluid moment of inertia in the plane. he fundamental reason why we consider the velocities in the body frame coordinates instead of having a mechanical system defined on the state variable (q, q), as it is done usually, is that this formulation makes our result much easier to interpret from a pratical point of view. Let us now describe the inputs we use to control our vehicle. As mentioned previsouly, we consider a vehicle actuated by thrusters. We assume the thrusters to be such that one control denoted u is a force in the body -axis, one denoted u 2 is a force in the body 2-axis and the third one u 3 is a pure torque in the plane. t follows that the equations of motion of our fully actuated underwater vehicle are described by an 3 affine control system: ẇ = f(w) g i (w)u i, where i= the drift f is given by the equations of motion of the conservative system described above, the g i are the following constant vector fields g =, g 2 =, g 3 = and the inputs u i are measurable bounded functions. o reflect the fact that the thrusters have limited power, we assume the following constraints on the inputs: { u i }, i =, 2, 3 (our results can be easily generalized to domains of control of the form {α i u i β i ; α i, β i R}, α i < < β i, i =, 2, 3). n a forthcoming article [3], we prove that fully actuated controlled mechanical systems are controllable even if we assume bounds on the controls. hus, it follows that our system is controllable.

3 Adjoint equations Our analysis is based on the maximum principle [6]. t provides necessary conditions for a trajectory to be time-optimal. For our problem, the maximum principle states that along a time-optimal trajectory w defined on the interval [, ] and with u being the corresponding optimal control, there exists an absolutely continuous function λ = (λ,, λ 6 ) defined on [, ] that never vanishes and such that it maximizes the Hamiltonian H(w, λ, u) = λ t( 3 ) f (w) g i (w)u i i= over the set of all possible controls. Moreover, λ is a solution of the following equations: λ 3 = λ (v sin θ v 3 cos θ) λ = () λ 2 = (2) λ 2 (v cos θ v 3 sin θ) (3) λ 4 = λ cos θ λ 2 sin θ λ 5 Ω λ 6 v 3 α (4) λ 5 = λ sin θ λ 2 cos θ λ 4 Ω λ 6 v α (5) λ 6 = λ 3 λ 4 v 3 λ 5 v (6) where α = m3 m is a nonzero constant. he function λ is called the adjoint vector and the equations ()-(6) are called the adjoint equations. A triple (w, λ, u) that solves the maximum principle is called an extremal. Since the proof of our result is based on the necessary conditions of the maximum principle let us introduce some definitions and notations. Remark first that the Hamiltonian takes the form H(w, λ, u) = λ t f (w) λ 4 u λ 5 u 2 λ 6 u 3. hen, since along an optimal path w with control u we have the maximization condition: max v H(w(t), λ(t), v) = H(w(t), λ(t), u(t)), it follows that: u i 3 (t) = sign(λ i (t)) if λ i (t), i = 4, 5, 6. As a consequence, the structure of the optimal paths is governed by the zeroes of the functions λ 4, λ 5, λ 6. ndeed, when the sign of one of these functions changes, we have to switch the corresponding component of the control from one extreme to the other. We call these functions the switching functions. Notice that these functions are given by λ i3 = λ t g i (w). f there exists a nonempty interval such that a given switching function is identically zero, no information is provided directly by the maximum principle on the component u i. We then say that the extremal is u i -singular on that interval. A time t s such that u i is not almost everywhere constant on any interval of the form ]t s ε, t s ε[, ε > is called a switching time for u i. 4 Results As mentioned previously, our main result concerns the time-optimality status of some specific trajectories, namely horizontal and vertical translations in the body frame coordinates between two configurations at rest. heore A horizontal or vertical translation in the body frame coordinates between two configurations at rest is not time-optimal. o prove our result, several steps are necessary. First, let us restrict ourself to horizontal translations in the body frame coordinates. n a previous paper [], we completely characterize the 2-singular trajectories, i.e. the extremals such that 2 components of the control are singular at the same time. We show that for such extremals there exists a bound on the number of switchings for the nonsingular component of the control and that this bound is in fact. Moreover, we also show that if along a trajectory we have Ω and v 3 (i.e. along a horizontal translation in the body frame coordinates), the components u 2, u 3 of the control must be identically zero and hence are singular. We can then conclude that if a horizontal translation in the body frame coordinates between two configurations at rest is optimal, it a 2-singular extremal and, due to the form of the constraints on the domain of control, there is exactly one u -swicthing to be performed at the half-time of the trajectory. n other words, we accelerate as much as possible for the first half and then decelerate at maximum speed for the second half. As this trajectory satisfies the necessary conditions of the maximum, it is a candidate for time-optimality. Surprisingly, as stated in heore, it happen not to be. Let us start by considering the simple situation when the initial and final configurations of the underwater vehicle are such that θ = and differ only by their horizontal component. his means we assume the initial and final states of the vehicle to be (x, z,,,, ) and (x 2, z,,,, ) where x 2 > x. he horizontal translation between these two configurations with exactly one u -swithing at half-time

is defined on the time interval = 8 (x 2 x ) and corresponds to the inputs u (t) = if t [, 2 ], u (t) = if t [ 2, ] and u 2, u 3 identically zero, see Figure 2. Figure 2: Horizontal translation n coordinates, we have that w(t) = ( t2 2 t x, z,,,, ) if t [, t2 2 ] and w(t) = ( 2 t (4x 3x 2 ), z,, t,, ) if t [ 2, ]. As mentioned previously, this is a 2-singular extremal, the two singular components of the control being u 2, u 3. For this simple situation we can explicitely compute an adjoint vector. ndeed, using the fact that along this trajectory we must have λ 5 = λ 6, the adjoint equations provide the information that λ has to be of the form (λ,,, λ t λ 2,, ) where λ is any constant greater than. heore contradicts the conjecture that it is a time-optimal path. n other words, we claim that there exists another trajectory defined on a time interval ˆ linking these two configurations at rest and such that ˆ < 8 (x 2 x ). o prove our result, we will show that there exists a time-equivalent path to the horizontal translation that does not satisfy the necessary conditions of the maximum principle and hence that cannot be time-optimal. he key idea is the following remark. By adding a vertical motion to the horizontal translation previously studied we do not alter the time of the trajectory, see Figure 3. Figure 3: ime- equivalent trajectory More precisely, the trajectory defined on = 8m (x 2 x ) and given by w(t) = ( t2 2 t x, 2 t t 2,,,, ) if t [, t2 2 ] and w(t) = ( 2 t (4x 3x 2 ), t2 2 2 t z,, t, t, ) if t [ 2, ] also links the initial and final states: (x, z,,,, ) and (x 2, z,,,, ). he corresponding control is u (t) = u 2 (t) = if t [, 2 ] and u (t) = u 2 (t) = if t [ 2, ] while u 3 is deduced from the fact that along this trajectory we have Ω, which gives u 3 (t) = v (t)v 3 (t) m3 m. Depending on the positions x, x 2 and on the structural constants of the vehicle,,, the component u 3 of the control might not be admissible. n other words, we might not have u 3 (t) along the whole trajectory.o make this trajectory admissible, we should introduce a time reparametrization but then we would loose timeequivalence between these two paths and comparaison would not make any sense. So instead of reparametrizing this trajectory, notice that our remark can be generalized to any vertical motion with a finite number of switchings on the component u 2 of the control. he reason is that as long as we keep the angular velocity Ω identically zero along the trajectory (which is equivalent to saying that the angle θ is constant) the variables x and z are decoupled, x depending only on u and z on u 2. Along all these motions we have max(v ) = 2, hence we can choose a trajectory with enough switchings on u 2 such that max(v 3 ) will satisfy max(v ) max(v 3 ) m3 m < and make the control an admissible one. More precisely, if 2n is the number of switchings on u 2 (it has to be an odd number in order to satisfy both the initial and final velocities v 3 to be zero), since the maximum of the velocity v 3 is given by: max(v 3 ) = 2(n), to get an admissible time-equivalent trajectory we have to choose n such that 2 2 2 (n) <. o summarize, we found that to the horizontal translation in the body frame coordinates with θ and one u -switching there exists a time-equivalent trajectory with no angular velocity, with at least 3 switchings on the component u 2 of the control (if the control is admissible with only one switching, then it will be admissible with any number of switchings) and such that u 3 (t) < along the trajectory. We now claim that this time-equivalent trajectory is not timeoptimal. ndeed, if it was an optimal trajectory it has to be a solution of the maximum principle. n other words, there should exist a nonzero adjoint vector such that the adjoint equations are satisfied and such that it maximizes the Hamiltonian. As the componnent of the control u 3 belongs strictly to the inside of the do-

main of control: u 3 (t) <, it is a consequence of the maximization condition that this component of the control has to be a singular one. t follows that along our time-equivalent trajectory we have λ 6 since we pointed out in Section 3 that λ 6 is the switching function corresponding to the control u 3. As a consequence, from the adjoint equations (4), (5) we have that λ 4 = λ and λ 5 = λ 2 (recall that there is no angular velocity along our path and that θ ). Since λ, λ 2 are cyclic coordinates, or in other words are constant along an extremal, we deduce that the switching functions λ 4, λ 5 are linear functions with respect to t: λ 4 = λ t λ 4 (), λ 5 = λ 2 t λ 5 (). hen, as the switchings of the controls u and u 2 happen when λ 4 and λ 5 respectively vanish we deduce that both nonsingular controls u, u 2 have at most one swicthing. t contradicts the fact that along our time-equivalent trajectory u 2 has at least three switchings. o generalize our result to any horizontal translation in the body frame( coordinates, simply ) remark cos θ sin θ that the rotation matrix applied to sin θ cos θ the coordinates x, z transforms the horizontal-vertical translations with θ = to horizontal-vertical translations in the body frame coordinates whith θ as the motion angle of direction. his close the proof that horizontal translations are not time-optimal trajectories. Finally, notice that due to the form of the equations of motions all the computations we made can be duplicated for vertical translations in the body frame coordinates. n that case, we add horizontal motions keeping the scalar rate Ω identically zero. 5 Conclusion Since we can prove that a car-like robot moving in the plane cannot go faster than along the straight line with its orientation in the direction of motion, the results presented here were at first surprising. However, there is a crucial difference between a car-like robot and an underwater vehicle. ndeed, the first one is a nonholonomic system with 2 controls while the second one is a fully controlled mechanical system with 3 controls. t follows that for the underwater vehicle a faster strategy to the one presented here would be to use the full power of 2 controls instead of only one (for instance u, u 2 in the case of a horizontal motion) by rotating first the vehicle to an optimal angle. his result is presented in [4] where we emphasize that our nonoptimality result is in fact very natural. Notice that the problem of finding a time-optimal trajectory between any two configurations at rest, including the ones on Figure 2, is still an open problem. First, remark that for this last situation if we consider the horizontal translation coupled with some rotation but no vertical motion, in other words we consider v 3 along the trajectory, then the trajectory is not optimal neither. ndeed, along such a trajectory the velocity v is not affected with respect to the horizontal translation and moreover, using the equations of motions for our fully actuated underwater vehicle, we get that ẋ(t) = cos θv (t) which is less or equal to the horizontal speed v (t) of the horizontal translation. We can deduce that to be optimal the path as to leave the horizontal line and needs to have a nonzero angular velocity. n other words, none of the velocities v, v 3, Ω can be identically zero along the time optimal path. Going further into this study, a first important result would be to give a bound on the number of switchings for each component of the control for the optimal path. ndeed, in case there is an accumulation point of zeroes for one of the swicthing functions it is not realistic from a practical point of view and going along the straight line might still have to be considered. his join our remark made in the introduction that practical issues may interfere with the optimal strategy provided by the theory. References [] M. Chyba, N.E. Leonard, E.D. Sontag, Optimality for underwater vehicles, Proceedings of the 4th Conference on Decision on Control, pp., 2 [2] M. Chyba, N.E. Leonard, E.D. Sontag, Singular trajectories in multi-input time-optimal problems. Application to controlled mechanical systems, Journal on Dynamical and Control Systems, Vol. 9 (), pp. 73 88, 23 [3] M. Chyba, E.D. Sontag, Controllability of dynamic extensions with bounded controls: Application to controlled mechanical systems, in preparation [4] M.Chyba, R.N. Smith, G.R. Wilkens, S.K. Choi, Underwater vehicles: how to go faster?, in preparation [5] N.E. Leonard, Stability of a bottom-heavy underwater vehicle, Automatica, Vol 33, pp. 33 346, 997.

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