Propulsion system optimization for an underwater vehicle.

Dobref, Vasile ; Tarabuta, Octavian ; Badara, Nicolae 等

Abstract: This paper addresses methods of thrust distribution in a propulsion system for an unmanned underwater vehicle. It concentrates on finding an optimal thrust allocation for desired values of forces and moments acting on the vehicle. Special attention is paid to the unconstrained thrust allocation. The proposed methods are developed using a configuration matrix describing the layout of thrusters in the propulsion system.

Keywords: underwater vehicle, propulsion system, thrust allocation

1. INTRODUCTION

The basic modules of the control system for ROVs are depicted in fig. 1. The autopilot computes demanded propulsion forces and moments [[tau].sub.d] by comparing the vehicle's desired position, orientation and velocities with their current estimates.

2. DESCRIPTION OF PROPULSION SYSTEM

The conventional ROVs operate in a crab-wise manner in 4 DOF with small roll and pitch angles that can be neglected during normal operations. Therefore, it is purposeful to regard the vehicle's spatial motion as a superposition of two displacements: the motion in the vertical plane and the motion in the horizontal plane. It allows us to divide the vehicle's propulsion system into two independent subsystems. The most often applied configuration of thrusters in the propulsion system is shown in fig. 2. The first subsystem permits the motion in heave and consists of 1 or 2 thrusters generating a propulsion force Z acting in the vertical axis. The thrust distribution is performed in such a way that the propeller thrust, or the sum of propellers thrusts, is equal to the demanded force [Z.sub.d]. The other subsystem assures the motion in surge, sway and yaw and it is usually composed of 4 thrusters mounted askew in relation to the vehicle's main symmetry axes (see fig. 3). The forces X and Y acting in the longitudinal and transversal axes and the moment N about the vertical axis are a combination of thrusts produced by the propellers of the subsystem. In practical applications the vector of propulsion forces and the moment [tau] acting on the vehicle in the horizontal plane can be described as a function of the thrust vector f by the following expression (2).

[tau] = T([alpha])Pf (2)

where [tau] = [[[tau].sub.1], [[tau].sub.2], [[tau].sub.3]].sup.T]

[[tau].sub.1], [[tau].sub.2], [[tau].sub.3]--force in the longitudinal, transversal and vertical axis. T-thruster configuration matrix,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [[gamma].sub.1] = [[alpha].sub.1] - [[phi].sub.1], [[gamma].sub.2] = [[alpha].sub.2] - [[phi].sub.2], [[gamma].sub.n] = [[alpha].sub.n] - [[phi].sub.n]

[alpha]= [[[alpha].sub.1], [[alpha].sub.2], ..., an]T--vector of thrust angles, [[alpha].sub.i]--angle between the longitudinal axis and direction of the propeller thrust [f.sub.i],

[d.sub.i]--distance of the i-th thruster from the centre of gravity,

[[phi].sub.i]--angle between the longitudinal axis and the line connecting the centre of gravity with the symmetry centre of the ith thruster,

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

f = [[f.sub.1], [f.sub.2], ... , [f.sub.n]]T--thrust vector,

P-diagonal matrix of the readiness of the thrusters:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The computation of f from [tau] is a model-based optimization problem and it is regarded below for two cases, namely, constrained and unconstrained thrust allocations. It will be assumed that the allocation problem is constrained if there are bounds on the thrust vector elements [f.sub.i].

3. UNCONSTRAINED THRUST ALLOCATION

Assume that the vector [[tau].sub.d] is bounded in such a way that the calculated elements of the vector f can never exceed the boundary values [f.sub.min] and [f.sub.max]. Then the unconstrained thrust allocation problem can be formulated as the following leastsquares optimization problem:

[min.sub.f] 1/2 [f.sup.t] Hf (4)

subject to [[tau].sub.d] - Tf = 0 (5)

where H is a positive definite matrix. The solution of the above problem using Lagrange multipliers is shown as

f = T* [[tau].sub.d] (6)

where T* = [H.sup.-1][T.sup.T] [([TH.sup.-1][T.sup.T]).sup.-1] (7)

is recognized as the generalized inverse. For the case H = I, the expression (7) reduce to the Moore-Penrose pseudo inverse:

T* = [T.sup.T]([TT.sup.T]).sup.-1] (8)

The above approach assures a proper solution only for faultless work of the propulsion system and cannot be directly used in the case of thruster damage. To increase its applicability and overcome this difficulty, two alternative algorithms are proposed.

3.1. Solution using singular value decomposition

Singular value decomposition (SVD) is an eigenvalue like decomposition for rectangular matrices. SVD has the following form for the thruster configuration matrix (3):

T = [USV.sup.T] (9)

where

U,V--orthogonal matrices of dimensions 3 x 3 and n x n, respectively,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[S.sub.[tau]]--diagonal matrix of dimensions 3 x 3,

0--null matrix of dimensions 3 x (n - 3).

The diagonal entries si are called the singular values of T. They are positive and ordered so that [[sigma].sub.1][greater than or equal to] [[sigma].sub.2] [greater than or equal to] [[sigma].sub.3] [greater than or equal to]. This decomposition of the matrix T allows us to work out a computationally convenient procedure to calculate the thrust vector f being a minimum-norm solution to (8). The procedure is analyzed for two cases:

1. All thrusters are operational (P = I).

2. One of the thrusters is off due to a fault (P ? I).

3.1.1. Algorithm for all thrusters active

Set [[tau].sub.d] = [[[tau].sub.d1], [[tau].sub.d2], [[tau].sub.3]].sup.T] as the required input vector,

f = [[[f.sub.1], [f.sub.2], ... ,[f.sub.n]].sup.T] = as the thrust vector necessary to generate the vector [[tau].sub.d], and n as the number of thrusters.

A direct substitution of (11) shows that the vector f determined by (8) and (10) can be written in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3.1.2. Algorithm for one non-operational thruster

Assume that the k--the thruster is off. This means that and [p.sub.kk] = 0 The substitution of (11) into (2) leads to the following dependence:

[[tau].sub.d] TPf = [USV.sup.T] Pf (11)

Defining: f' = [[[f.sub.1], ... ,[f.sub.k-1], [f.sub.k+1], ... [f.sub.n]].sup.T]

V* = [V.sup.T] P = [[v.sup.*.sub.1], ... ,[v.sup.*.sub.k-1], 0, [v.sup.*.sub.k+1], ... [v.sup.*.sub.n]]

[V.sup.*.sub.f] = [[V.sup.T] P = [[v.sup.*.sub.1], ... ,[v.sup.*.sub.k-1], [v.sup.*.sub.k+1], ... [v.sup.*.sub.n]]

the expression (11) can be written as

[[tau].sub.d] = [USV.sup.*.sub.f]f' (12)

The matrices U and SV _f have dimensions 3 x 3 and 3 x m, where m = n - 1, so the vector f' can be computed as

f' = [(([SV.sup.*.sub.f]).sup.T] [SV.sup.*.sub.f]).sup.-1] [(SV.sup.*.sub.f]).sup.T][U.sup.T][[tau].sub.d] (13)

Hence, the value of the thrust vector f can be obtained as follows:

f = [[f'.sub.1], ... , [f.sub.k-1], 0, [f'.sub.k],..., [f'.sub.m]].sup.T] (14)

Note that if n = 4 then (15) can be simplified to the form

F' = [([SV.sup.*.sub.f]).sup.-1] [U.sup.T] [[tau].sub.d] (15)

4. SOLUTION USING THE WALSH MATRIX

The solution proposed below is restricted to ROVs having the configuration of thrusters exactly as shown in Fig. 3, i.e., the propulsion system consists of four identical thrusters located symmetrically around the centre of gravity. In such a case [d.sub.j] = [d.sub.k] = d, [[alpha].sub.j] mod ([PI]/2) = [[alpha].sub.k] mod ([PI]/2) = [alpha], [[phi].sub.j] mod ([PI]/2) =[[phi].sub.k] mod ([PI]/2) = [phi] for j, k = 1, . . . , 4 and the thrusters configuration matrix T can be written in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

where [gamma] = [alpha]--[phi]. Then the matrix T has the following properties:

(a) it is a row-orthogonal matrix,

(b) |[t.sub.ij]| = |[t.sub.ik]| for i = 1, 2, 3 and j, k = 1, ... , 4,

(c) it can be written as a product of two matrices: a diagonal matrix Q and a row-orthogonal matrix Wf having values [+ or -]1:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

It allows us to work out a simple and fast procedure to compute the thrust vector f by applying an orthogonal Walsh matrix. It should be emphasized that the use of this method does not require calculations of any additional matrices. The procedure is considered for the case when all thrusters are operational.

4.1. Algorithm for all thrusters active

As in Section 3.1.1, set [[tau].sub.d] = [[[[tau].sub.d1], [[tau].sub.d2], [[tau].sub.d3]].sup.T] as the required input vector and f = [f1, f2, . . . , fn]T as the thrust vector necessary to generate the input vector [[tau].sub.d]. After the thrust vector is substituting, it becomes:

[[tau].sub.d] = [QW.sub.f] Pf (18)

By multiplying both sides of (18) by [Q.sup.-1], the following expression is obtained:

[Q.sup.-1] [[tau].sub.d] = [W.sub.f] Pf (19)

Substituting

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [w.sub.0] = [ 1 1 1 1 ], and assuming that P = I , (21) can be transformed to the form

S = Wf (22)

The matrix W is the Walsh matrix having the following properties: W = [W.sup.T] and [WW.sup.T] - nl, where n = dimW. Hence, the thrust vector f can be expressed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

5. CONCLUSIONS

The paper presents methods of thrust distribution for an unmanned underwater vehicle. To avoid a significant amount of computations, the problem of thrust allocation has been regarded as an unconstrained optimization problem. The described algorithms are based on the decomposition of the thruster configuration matrix. This allows us to obtain minimum Euclidean norm solutions. The main advantage of the approach is its computational simplicity and flexibility with respect to the construction of the propulsion system and the number of thrusters.

6. REFERENCES

Berge S. and Fossen T.I. (1997). Robust control allocation of overactuated ships: Experiments with a model ship Proc. 4-th IFAC Conf. Manoeuvring and Control of Marine Craft, Brijuni, Croatia, pp. 161-171, 1997.

Fossen T.I. (2002). Marine Control Systems, Marine Cybernetics AS, Trondheim, 2002

Garus J. (2004). A method of power distribution in power transmission system of remotely operated vehicle.--J. Th.