On thruster allocation, fault detection and accommodation ... - CiteSeerX

On thruster allocation, fault detection and accommodation issues for underwater robotic vehicles. Giovanni Indiveri∗ , Gianfranco Parlangeli Dipartimento Ingegneria Innovazione University of Lecce via Monteroni, 73100 Lecce, Italy [email protected]

Abstract— The use of mono-directional thrusters on underwater vehicle poses interesting issues on actuator allocation, fault detection and accommodation. Preliminary results relative to the horizontal motion of an unhabited underwater vehicle (UUV) are presented. Index Terms— Underwater remotely operated vehicles, control allocation.

I. I NTRODUCTION The use of shallow water unhabited underwater vehicle (UUV) systems for environment monitoring is increasingly popular. Typical applications of small size UUVs include hull inspection of marine vessels, monitoring of costal marine protected areas or pipeline inspections. The growth of the application range of these systems can be correlated with the decreasing costs and sizes of UUVs that need also to be always more simple to operate and robust. Indeed often these systems are operated for long lasting missions by non technical staff which can be made possible only thanks to enhanced control systems that may include, besides auto depth and auto heading capabilities, advanced modules as fault detection and accommodation modules. Moreover for the sake of hardware robustness, ease and low cost of maintenance only simple, standard and commercially available components should be used. These requirements induce design constraints that may impact on the control system itself. Take by example the propulsion equipment: most often the adopted DC motors and propellers are adapted from small boat equipment that is designed to deliver thrust in one direction only. Indeed although the motor can reverse direction, the propeller does not have a symmetrical behaviour. This lack of symmetry in the thruster has to be taken into account at the motion control level, in particular when addressing the thruster allocation design problem. It should be noticed that the need to take into account asymmetrical behaviour of UUVs propulsion subsystems may be relevant also when the single thruster, i.e. motor, shaft, propeller and nozzle unit, is specifically designed to be symmetrical. As shown in [1] an asymmetrical behaviour of the overall propulsion system may arise as a consequence of so-called propellerhull interactions or propeller-propeller interactions. The former are present when the water flow in or out of the ∗ Corresponding

author.

thrusters interact with the hull of the vehicle rather than with open waters and the latter are present when the flows relative to two or more thrusters interact with each other rather than with open waters. Both propeller-hull and propeller-propeller interactions are very common and likely to occur in open frame UUVs. An asymmetrical behaviour of the propulsion system is of course also present if a jet technology should be employed [2].

Fig. 1. Horizontal thruster configuration with respect to the body - fixed frame.

II. T HRUSTER ALLOCATION ISSUES Given that restoring forces and moments dominate the roll and pitch dynamics of most open frame UUVs, and assuming low operating speeds, roll and pitch will be assumed to be constantly zero. Moreover heave and the horizontal plane degrees of freedom (DOFs) surge, sway and yaw are assumed to be actuated by different sets of actuators: the former by vertical thrusters and the latter by horizontal ones. As these two sets of actuators belong to orthogonal planes and considering the low operating speeds, horizontal and vertical DOFs are assumed independent. With respect to figure (1), the horizontal forces Fx , Fy and the vertical torque Nz expressed in the depicted body fixed frame < B >, are related to the forces produced by the horizontal plane thrusters τ1 , τ2 , τ3 , τ4 , as follows : 





Fx cθ1  Fy  =  sθ1 Nz J31

cθ2 sθ2 J32

cθ3 sθ3 J33



 τ1 cθ4  τ2   sθ4    τ3  J34 τ4 

(1)

where

15

cψ := cos ψ, sψ := sin ψ J31 := rx sθ1 + ry cθ1

(2)

J32 := rx sθ2 − ry cθ2 J33 := −rx sθ3 − ry cθ3

(4)

J34 := −rx sθ4 + ry cθ4

(6)

10

(3) Thruster force [Kg]

5

(5)

being, as in most UUVs, (rx , −ry )T , (rx , ry )T , (−rx , ry )T , (−rx , −ry )T the symmetrical positions in < B > of thrusters 1, 2, 3 and 4 respectively and θ1 , θ2 , θ3 , θ4 their orientation with respect to the body fixed x axis. Equation (1) will be expressed in compact form as F = J τ , J ∈ R3×4 .

(7)

In its most general formulation the problem of thruster allocation consists in determining the horizontal thrusters positions and orientations such that equation (7) admits an inverse. As J ∈ R3×4 , if J should have full rank, there would be ∞1 inverse solutions. Thus one may try to pick, among these infinite number of solutions, the optimal one according to a suitable criteria. In practice thruster positions are fixed at the corners of the UUV frame in a common horizontal plane and only their orientations θ1 , θ2 , θ3 , θ4 need to be found. The optimal criteria to be satisfied is commonly chosen to be the minimum of a quadratic cost function as follows: ) minτ 21 τ T W τ such that (8) W = W T , W > 0 and F d = J τ being W a weight matrix. If J has full rank, the solution to equation (8) is given by: τ∗ † JW

=

† JW Fd

:=

−1

W

J

T

¡

JW

−1

J

¢ T −1

(9) .

(10)

The general non-optimal solution to equation F d = J τ for a full rank J matrix and a symmetrical positive definite W matrix would be given by τ

† = JW Fd + k

(11)

k

∈

(12)

ker J.

It should be noticed that in the given hypothesis for the weighting matrix W , the only necessary and sufficient condition for solution τ ∗ (9) to exist is that J must be full rank. In this respect the thruster orientation problem would be reduced to finding the orientations θ1 , θ2 , θ3 , θ4 such that J has full rank. A. Nonnegative solution issues As briefly discussed in the introduction of this paper, it may be necessary to solve the thruster allocation problem with the additional constraint that all the components of τ be nonnegative, namely τ ≥ 0. This additional requirement may occur, by example, in the following situations:

0

−5

−10

−15 −10

−8

−6

−4

2 0 −2 Applied voltage [V]

4

6

8

10

Fig. 2. Cavitation tunnel data of generated thrust versus applied DC motor voltage of a single symmetrical thruster of the CNR-ISSIA (former IAN) ROMEO UUV vehicle [3]. Notice that being the DC motor electrical time constant much smaller than the mechanical one, applied voltage may be approximately assumed proportional to the propeller revolutions per minute n.

mono-directional jet thrusters are employed; propeller based thrusters are employed, but the propellers are designed for forward propulsion only and their efficiency in reverse motion is critically low; • DC motors are used with mono-directional power amplifiers (no H-bridge) to obtain higher electrical efficiency and simpler power amplifier design; • high precision hovering maneuvers are required: in this case given that the output thrust of a propeller based thruster scales proportionally to n|n| (see figure (2)) being n the propeller revolutions per minute, one may desire to avoid working in the low efficiency region centered on n = 0. An interesting allocation problem to be considered is then the following: ) minτ ≥ 0 21 τ T W τ such that (13) W = W T , W > 0 and F d = J τ • •

that differs from (8) only for the additional constraint that τ should have nonnegative components, i.e. τ ≥ 0. Notice that problem (13) has interesting analogies in other robotic applications as space vehicles using jet based thrusters or tendon driven robotic devices. As far as the existence of solutions to problem (13), the following important result [4] [5] holds: Lemma 1 Given a matrix J ∈ Rm×n , the following conditions are equivalent: 1) for all F ∈ Rm×1 , there exists τ ∈ Rn×1 , τ ≥ 0 such that F = J τ . 2) J is full rank and its kernel has sign definite vectors, i.e. ∃ k > 0 such that J k = 0. For Lemma 1 to be applicable it is necessary that m < n, i.e. the system needs to be overactuated. Suppose that all conditions of Lemma 1 are satisfied and that k r > 0 is

any positive vector belonging to the kernel of J. Then, given any symmetrical positive definite W ∈ Rn×n matrix, denoting with (x)k the k-th component of vector x , a nonnegative solution to F d = J τ would be given by: † τ nn = JW F d + kr σ

being I

=

½

³ h, q, . . . :

† JW

´

Fd

³ h

< 0,

† JW

(14) ¾

´

Fd

q

< 0, . . . (15)

the set of indexes corresponding to negative components † F d, of JW   0 if I is empty † (J F ) σ ≥ σl := (16)  maxj∈I W d j otherwise (kr )j † and JW given by equation (10).

B. Optimal nonnegative solutions Equations (14 - 16) define a generic nonnegative solution to the F d = J τ problem when the conditions of Lemma 1 are satisfied and W is symmetric and positive definite. The solution to problem (13), i.e. an optimal solution to F d = J τ with respect to the quadratic cost 1 T τ Wτ , (17) 2 can be found replacing equation (14) in (17), namely ¢−1 1 T ¡ c( τ nn ) = F d JW −1 J T Fd 2 1 2 T + σ kr W kr . (18) 2 The first term on the right hand side of equation (18) is c( τ ∗ ) with τ ∗ given by (9), namely it would be the least possible value of the quadratic cost function c given in (17) if the unconstrained optimization problem (8) would be considered and τ would be left free to take any value. By direct inspection of equation (18), it then follows that the solution to the constrained problem (13) will be given by:

c( τ ) =

τ ∗nn

=

k ∗r , σ ∗

:

† JW F d + k ∗r σ ∗ 1 ∗ 2 ∗T σ k r W k ∗r = 2 1 2 T = min σ kr W kr σ≥σl ,kr 2

(19)

(20)

being σl defined in equation (16). The explicit determination of the optimal nonnegative solution given by (19) is easier in the case that J satisfies the hypothesis of Lemma 1 with J ∈ Rm×(m+1) : indeed in this case dim ker J = 1 and the problem of selecting a sign definite vector according to (20) vanishes. This result is summarized in the following: Lemma 2 Given J ∈ Rm×(m+1) satisfying Lemma 1, denote with ku : k ku k = 1 , ku > 0 , J ku = 0

(21)

the unique positive unit vector in the kernel of J, a solution to problem (13) with an arbitrary symmetrical positive definite W ∈ R(m+1)×(m+1) matrix is given by: † τ ∗nn = JW F + σ∗ ku (22)  d  0 if I is empty † (JW F d )j σ∗ = (23)  maxj∈I otherwise k ( u )j † being I and JW given by equations (15) and (10) respectively.

The existence of a positive vector in the kernel of J implies that if all the actuators are concurrently active with a due distribution, these do not produce any force / torque on the system. Besides on overactuated UUVs and spacecrafts, this situation generally occurs in tendon driven robotic systems as robotic hands where the σ gain in equation (14) can be exploited to regulate the tension of tendons. III. FAULT DETECTION AND ACCOMMODATION ISSUES In the hypothesis of having an overactuated system without nonnegative constraints, the allocation problem is commonly formulated as in equation (8) that is solved by equations (9-10). In this case, as shown, by example, in [6] [8], the additional degrees of freedom related to the choice of the W matrix may be exploited to compensate, when possible, for thruster failures and, eventually, for saturation [7]. Control reallocation to compensate for thruster faults in the presence of nonnegative control constraints may not always be possible, even in the overactuated case. A possible approach relative to the planar motion of a UUV in presence of nonnegative thrusts is discussed in the following sections. A. Fault detection and isolation. A basic step for an effective fault tolerant control design is a FDI unit which detects an abnormal behavior of the dynamics and isolate the faulty component. In this work only shutdown faults are considered, these being the most usual within this context. Whenever a fault happens, the dynamic behavior of the controlled system differs from the nominal behavior which in turn is as close as possible to a desired reference. A usual way to set a detection algorithm is to compute the difference between the actual behavior of the system with the the best prevision possible of the nominal system [9](that is the model of the system without faults). In the present setting, the positivity of the thrusts gives a significant help for deriving a simple isolation rule. Indeed a shutdown fault on the i-th actuator can be thought of as having an external force F e balancing the i-th actuator thrust τ i being equal in norm and opposite in direction. This will result in an unmodeled drift motion in direction of −τ i that allows to uniquely detected the fault at navigation level.

B. Fault accommodation. When a fault acts on an actuator, three actuators remain available for control, the configuration matrix reduces to J ∈ R3×3 and, when singularities are avoided, it represents a bijective endomorphism in R3 . This in turn destroys the feasibility of the positive solution for a given vector of force distribution making the problem of fault management impossible to be solved in the given hypothesis. Thus either new hardware and/or a more flexible structure of the control architecture must be considered to manage the presence of faults. Possible choices are to equip the vehicle with some reversible and/or actively reorientable actuators. One may wonder whether a subgroup of the actuators must be equipped with a certain advanced technology to have a feasible solution for each force vector, and then the minimum number of actuators should be sought. In case that reversible thrusters should be considered, each actuator should be reversible in order to have a feasible solution for each force vector. By direct computation it can be found that, in order to get the opposite of the i-th column of the J matrix as a desired force vector, the opposite of the i-th canonical vector should be given to the actuators; so the choice of using reversible actuators implies the necessity of the reversibility of each actuator. In case that actively reorientable thrusters should be considered, the minimum number of reorientable actuators necessary to achieve tolerance to a single thruster (shutdown) fault is two. Suppose that the first actuator is faulty. A useful way to exploit the different degrees of freedom is to decompose the thrust of the rotatable actuator into the two components; this way configuration matrix J contains no time varying terms. Let actuators 1 and 4 be on orientable supports and assume actuator 1 to be faulty: the available actuators are 2, 3 (with fixed angle θ) and 4 (with variable angle φ or, equivalently, with the two components τ4x and τ4y that can be chosen arbitrarily).       τ2 Fx −cθ −cθ 1 0  τ3    Fy  =  −sθ sθ 0 1   τ4x  (24) z1 (θ) z2 (θ) ry rx Nz τ4y where z1 = ry cθ + rx sθ and z2 = −ry cθ − rx sθ. In order to suitably choose the set where to search the feasible solution, notice that components τ4x and τ4y can assume real values, both positive and negative, since φ can take all values in (0, 2π), while τ2 and τ3 can take real nonnegative values. The issue is then to verify the existence of a solution the feasibility region © (possibly optimal) within ª F1 = x ∈ R4 |x1 ≥ 0, x2 ≥ 0 . The following result ensures the existence of a feasible solution for an arbitrary given desired reference force vector. ¢ ¡ ¯z T it F¯x F¯y N Proposition Given a vector is compute a solution of (24) ¡ always possible ¢to T τ¯2 τ¯3 τ¯4x τ¯4y belonging to F1 . The proposed solution is optimal in the sense of (13).

Proof The proof is constructive and a vector satisfying ¢ belonging to F1 for a given vector ¡ both (24) and ¯z T will be computed. F¯x F¯y N Directly from the definition it is possible to derive the structure of the kernel of J.   rx cθ       r sθ + r cθ y x  ker[J] = span   sθ(2rx cθ + ry sθ)       −ry cθsθ If θ (fixed) is chosen between (0, π2 ), the first two components are positive and, invoking Lemma 1, this ensures the existence of a feasible solution for every choice of the desired reference force vector. Moreover, in view of Lemma 2, a feasible input minimizing (17) is given by (22), where σ ∗ is chosen according to (23) with respect to the first two components: ³ ´ ³ ´   † †  JW Fd Fd  JW 1 2 σ ∗ = max 0, − ,−  rx cθ ry sθ + rx cθ  ¥ IV. C ONCLUSIONS Based on the results in [4] and [5] a control allocation solution for the horizontal motion of an UUV with 4 monodirectional thrusters is presented. It is shown that having two reorientable mono-directional thrusters the shutdown failure of a single actuator could be perfectly accommodated if reorientation was instantaneous. Future work will address, among the rest, the role of actuator reorientation dynamics on the overall system performance. R EFERENCES [1] M. Caccia, G. Indiveri and G. Veruggio, ”Modelling and identification of open-frame variable configuration unmanned underwater vehicles” IEEE Journal of Oceanic Engineering, vol. 25, no. 2, pp. 227 - 240, April 2000. [2] AJO-1 project co-financed by the Italian Ministry of Research MURST (PRIN 1999-2000, Navigazione, guida e controllo di veicoli subacquei), http://www.diee.unica.it/%7Episano/AUV.html [3] G. Indiveri, ”Modelling and Identification of Underwater Robotic Systems” Ph.D. Thesis, DIST - University of Genova, Italy and IAN - National Research Council (CNR), Italy, 1998. [4] P. A. Servidia and R. S. Pe˜na, ”Spacecraft Thruster Control Allocation Problems”, IEEE Trans. on Automatic Control, Vol. 50, N. 2, pp. 245 - 249, Feb. 2005. [5] R. S. Pe˜na, R. Alonso and P. A. Anigstein, ”Robust Optimal Solution to the Attidute/Force Control Problem”, IEEE Trans. on Aerospace and Electronic Systems Vol. 36, N. 3, pp. 784 - 792, Jul. 2000. [6] E. Omerdic and G. Roberts, ”Thruster Fault Diagnosis and Accommodation for Open-frame Underwater Vehicles”, Control Engineering Practice, N. 12, pp. 1575-1598, 2004. [7] N. Sarkar, T. Podder, and G. Antonelli, ”Fault-Accommodating Thruster Force Allocation of an AUV Considering Thruster Redundancy and Saturation”, IEEE Tran. on Robotics and Automation, Vol. 18, N. 2, pp. 223-233, 2002. [8] T. K. Podder and N. Sarkar, ”Fault-tolerant control of an autonomous underwater vehicle under thruster redundancy”, Robotics and Autonomous Systems, vol. 34, pp. 39 - 52, 2001. [9] A. Alessandri, M. Caccia and G. Veruggio ”Fault detection of actuator faults in unmanned underwater vehicles”, Control Engineering Practice, vol. 7, pp. 357 - 368, 1999.