7 A Survey of Control Allocation Methods for Underwater Vehicles Thor I. Fossen 1,2 , Tor Arne Johansen 1 and Tristan Perez 3 1 Dept. of Eng. Cybernetics, Norwegian Univ. of Science and Techn. 2 Centre for Ships and Ocean Structures, Norwegian Univ. of Science and Techn. 3 Centre of Excellence for Complex Dyn. Syst. and Control, Univ. of Newcastle, 1,2 Norway 3 Australia 1. Introduction A control allocation system implements a function that maps the desired control forces generated by the vehicle motion controller into the commands of the different actuators. In order to achieve high reliability with respect to sensor failure, most underwater vehicles have more force-producing actuators than the necessary number required for nominal operations. Therefore, it is common to consider the motion control problem in terms of generalised forces—independent forces affecting the different degrees of freedom—, and use a control allocation system. Then, for example, in case of an actuator failure the remaining ones can be reconfigured by the control allocation system without having to change the motion controller structure and tuning. The control allocation function hardly ever has a close form solution; instead the values of the actuator commands are obtained by solving a constrained optimization problem at each sampling period of the digital motion control implementation loop. The optimization problem aims at producing the demanded generalized forces while at the same time minimizing the use of control effort (power). Control allocation problems for underwater vehicles can be formulated as optimization problems, where the objective typically is to produce the specified generalized forces while minimizing the use of control effort (or power) subject to actuator rate and position constraints, power constraints as well as other operational constraints. In addition, singularity avoidance for vessels with rotatable thrusters represents a challenging problem since a non-convex nonlinear program must be solved. This is useful to avoid temporarily loss of controllability. In this article, a survey of control allocation methods for over-actuated underwater vehicles is presented. The methods are applicable for both surface vessels and underwater vehicles. Over-actuated control allocation problems are naturally formulated as optimization problems as one usually wants to take advantage of all available degrees of freedom (DOF) in order to minimize power consumption, drag, tear/wear and other costs related to the use of control, subject to constraints such as actuator position limitations, e.g. Enns (1998), Bodson (2002) and Durham (1993). In general, this leads to a constrained optimization Underwater Vehicles 110 problem that is hard to solve using state-of-the-art iterative numerical optimization software at a high sampling rate in a safety-critical real-time system with limiting processing capacity and high demands for software reliability. Still, real-time iterative optimization solutions can be used; see Lindfors (1993), Webster and Sousa (1999), Bodson (2002), Harkegård (2002) and Johansen, Fossen, Berge (2004). Explicit solutions can also be found and implemented efficiently by combining simple matrix computations, logic and filtering; see Sørdalen (1997), Berge and Fossen (1997) and Lindegaard and Fossen (2003). Fig. 1. Block diagram illustrating the control allocation problem. The paper presents a survey of control allocation methods with focus on mathematical representation and solvability of thruster allocation problems. The paper is useful for university students and engineers who want to get an overview of state-of-the art control allocation methods as well as advance methods to solve more complex problems. 1.1 Problem formulation Consider an underwater vehicle (Fossen, 2002): =   ηην ν +νν+νν+η=τ J MC D g () () () () (1.1) that is controlled by designing a feedback control law of generalized control forces: n τ= α ∈BuR() (1.2) where p α∈R is a vector azimuth angles and r ∈uRare actuator commands. For marine vehicles, some control forces can be rotated an angle about the z-axis and produce force components in the x- and y-directions, or about the y-axis and produce force components in the x- and z-directions. This gives additional control inputs α which must be computed by the control allocation algorithm. The control law uses feedback from position/attitude T xyz ϕ θψ =η [,,,,, ] and velocity T uvwpqr=ν [,, ,,,] as shown in Figure 1. For marine vessels with controlled motion in n DOF it is necessary to distribute the generalized control forces τ to the actuators in terms of control inputs α and u. Consider (1.2) where nr × α∈BR() is the input matrix. If B has full rank (equal to n) and rn> , you have control forces in all relevant directions, this is an over-actuated control problem. Similarly, the case rn< is referred to as an under-actuated control problem. Computation of α and u from τ is a model-based optimization problem which in its simplest form is unconstrained while physical limitations like input amplitude and rate saturations imply that a constrained optimization problem must be solved. Another complication is actuators that can be rotated at the same time as they produce control forces. This increases the number of available controls from r to r+p. A Survey of Control Allocation Methods for Underwater Vehicles 111 2. Actuator models The control force due to a propeller, a rudder, or a fin can be written Fku= (1.3) where k is the force coefficient and u is the control input depending on the actuator considered; see Table 1. The linear model F=ku can also be used to describe nonlinear monotonic control forces. For instance, if the rudder force F is quadratic in rudder angle δ, that is δ δ = ||,Fk (1.4) the choice ||u δ δ = , which has a unique inverse ()si g nu u δ = , satisfies (1.3). Actuator u α T f Main propeller/longitudinal thrusters pitch/rpm - [,0,0] x F Transverse thrusters pitch/rpm - [0, ,0] y F Rotatable thruster in the horizontal plane pitch/rpm angle α α [,,0]cos sin xx FF Rotatable thruster in the vertical plane pitch/rpm angle α α [sin,0,cos] zz FF Aft rudders angle - [0, ,0] y F Stabilizing fins angle - [0,0, ] z F Table 1. Example of actuators and control variables. For underwater vehicles the most common actuators are: • Main propellers/longitudinal thrusters are mounted aft of the hull usually in conjunction with rudders. They produce the necessary force in the x-direction needed for transit. • Transverse thrusters are sometime going through the hull of the vessel (tunnel thrusters). The propeller unit is then mounted inside a transverse tube and it produces a force in the y -direction. Tunnel thrusters are only effective at low speed which limits their use to low-speed maneuvering and DP. • Rotatable (azimuth) thrusters in the horizontal and vertical planes are thruster units that can be rotated an angle α about the z-axis or y-axis to produce two force components in the horizontal or vertical planes, respectively. Azimuth thrusters are attractive in low-speed maneuvering and DP systems since they can produce forces in different directions leading to an over-actuated control problem that can be optimized with respect to power and possible failure situations. • Aft rudders are the primary steering device for conventional vessels. They are located aft of the vessel and the rudder force y F will be a function of the rudder deflection (the Underwater Vehicles 112 drag force in the x-direction is usually neglected in the control analysis). A rudder force in the y-direction will produce a yaw moment which can be used for steering control. • Stabilizing fins are used for damping of vertical vibrations and roll motions. They produce a force z F in the z-directions which is a function of the fin deflection. For small angles this relationship is linear. Fin stabilizers can be retractable allowing for selective use in bad weather. The lift forces are small at low speed so the most effective operating condition is in transit. • Control surfaces can be mounted at different locations to produce lift and drag forces. For underwater vehicles these could be fins for diving, rolling, and pitching, rudders for steering, etc. Table 1 implies that the forces and moments in 6 DOF due to the force vector [,,] xz T y FFFf = can be written x y z zy yz xz zx y xxy F F F Fl Fl Fl Fl Fl Fl ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡⎤ ⎢ ⎥ == ⎢⎥ ⎢ ⎥ × ⎣⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ − − − f rf τ (1.5) where [,,] xz T y lllr = are the moment arms. For azimuth thrusters in the horizontal plane the control force F will be a function of the rotation angle. Consequently, an azimuth thruster will have two force components cos x FF α = and sin , y FF α = while the main propeller aft of the vehicle only produces a longitudinal force , x FF = see Table 1. 2.1 Thrust configuration matrix for non-rotatable actuators The control forces and moments for the fixed thruster case (no rotatable thrusters) can be written τ =Tf (1.6) where nr× ∈T R is the thrust configuration matrix. The control forces satisfies, ,=fKu (1.7) with control inputs 1 , , ][. T r uu=u The force coefficient matrix rr × ∈K R is diagonal, 1 { , , }. r diag k k = K (1.8) The actuator configuration matrix is defined in terms of a set of column vectors n i ∈t R according to 1 ( ) [ , , ]. r α =Ttt (1.9) If we consider 6 DOF motions, the columns vectors can be derived from (1.5) and (1.9) according to A Survey of Control Allocation Methods for Underwater Vehicles 113 N N N ⎡⎤ ⎡⎤ ⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ == = − ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ − ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ ⎣⎦ tunnel thruster stabilizin g fin main propeller and aft rudder 0 0 1 0 1 0 0 0 0 1 0 0 i i i i i ii i z y z x y x l l l l l l tt t (1.10) 2.2 Thrust configuration matrix for rotatable actuators A more general representation of (1.6) is, () () , τ= α α Tf =T Ku (1.11) where the thrust configuration matrix )( nr × ∈T Rα varies with the azimuth angles 1 [ , , ] . T p αα α= (1.12) The azimuth thruster in the horizontal plane are defined in terms of the column vector α α α α α α α α α αα α ⎡⎤ ⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥⎢⎥ ⎢⎥⎢⎥ ⎢⎥ ⎢⎥ == − ⎢⎥ ⎢⎥ ⎢⎥⎢⎥ − ⎢⎥⎢⎥ ⎢⎥ ⎢⎥ − ⎣⎦ ⎣⎦ −      azimuth thruster in az the horizontal plane sin cos 0 sin cos 0 , cos sin sin sin cos sin cos sin i i i ii ii i i i i i ii yi zi zi zixi xiy i yi l ll l l ll l tt imuth thruster in the vertical plane (1.13) where the coordinates (,,) iii x y z lll denotes the location of the actuator with respect the body fixed coordinate system. Similar expressions can be derived for thrusters that are rotatable about the x- and y-axes. 2.3 Extended thrust configuration matrix for rotatable actuators When solving the control allocation optimization problem an alternative representation to (1.10) is attractive to use. Equation (1.11) is nonlinear in the controls α and u. This implies that a nonlinear optimization problem must be solved. In order to avoid this, the rotatable thrusters can be treated as two forces. Consider a rotatable thruster in the horizontal plane (the same methodology can be used for thrusters that can be rotated in the vertical plane), =cos cos , FF xi i ku ii i i α α = (1.14) Underwater Vehicles 114 =sin sin . FF y ii ku ii i i α α = (1.15) Next, we define an extended force vector according to eee = fKu (1.16) such that eee τ = TK u (1.17) where e T and e K are the extended thrust configuration and thrust coefficient matrices, respectively and e u is a vector of extended control inputs where the azimuth controls are modelled as cos sin ix i i iy i i uu uu α α = = (1.18) The following examples show how this model can be established for an underwater vehicle equipped with two main propellers and two azimuth thrusters in the horizontal plane. Example 1: Thrust configuration matrices for an ROV/AUV with rotatable thrusters The horizontal plane forces X and Y in surge and sway, respectively and the yaw moment N satisfy (see Figure 2), () 8 TKuτ= α (1.19) 11 22 34 11 22 33 1122 44 000 1011 000 0100 . 00 0 sin cos sin cos 000 xy x y yy ku X ku Y ku Nl l l l ll ku αααα ⎡ ⎤⎡ ⎤ ⎡⎤ ⎡⎤ ⎢ ⎥⎢ ⎥ ⎢⎥ ⎢⎥ ⎢ ⎥⎢ ⎥ = ⎢⎥ ⎢⎥ ⎢ ⎥⎢ ⎥ ⎢⎥ ⎢⎥ −− ⎢ ⎥⎢ ⎥ ⎣⎦ ⎣⎦ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ (1.20) Fig. 2. ROV/AUV equipped with two azimuth thrusters (forces F 1 and F 2 ) and two main propellers (forces F 3 and F 4 ). The azimuth forces are decomposed along the x- and y-axis. A Survey of Control Allocation Methods for Underwater Vehicles 115 By using the extended thrust vector, (1.19) can be rewritten as, eee 8 TK uτ= (1.21) 1234 1 1 1 1 22 2 2 3 3 4 4 00000 00000 101 0 1 1 00 000 010 1 0 0 . 000 00 00 0000 0 00000 x y x y xxyy u k u k X uk Y u k Nl lll k u k u ⎡ ⎤ ⎡⎤ ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎡⎤ ⎡⎤ ⎢ ⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢ ⎥ = ⎢⎥ ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢⎥ ⎢⎥ − ⎢ ⎥ ⎣⎦ ⎢⎥ ⎣⎦ ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎣⎦ ⎣ ⎦ (1.22) Notice that e T is constant while ()T α depends on . α This means that the extended control input vector e u can be solved directly from (1.21) by using a pseudo-inverse. This is not the case for (1.20) which represents a nonlinear optimization problem. The azimuth controls can then be derived from the extended control vector e u by mapping the pairs 11 (,) x y uu and 22 (,) x y uu using the relations, 22 111 11 2 1 2 222 222 ,atan2(,), ,atan2(,). xy yx xy yx uuu uu uuu uu α α =+ = =+ = (1.23) The last two controls u 3 and u 4 are elements in u e . □ 3. Linear quadratic unconstrained control allocation The simplest allocation problem is the one where all control forces are produced by thrusters in fixed directions alone or in combination with rudders and control surfaces such that constant, ( ) constant. = ==TT α α Assume that the allocation problem is unconstrained-i.e., there are no bounds on the vector elements , ii f α and i u and their time derivatives. Saturating control and constrained control allocation are discussed in Sections 4-5. For marine craft where the configuration matrix T is square or non-square ()rn≥ , that is there are equal or more control inputs than controllable DOF, it is possible to find an optimal distribution of control forces f, for each DOF by using an explicit method. Consider the unconstrained least-squares (LS) optimization problem (Fossen & Sagatun, 1991), { } subject to: min T J = − = f fWf Tf 0.τ (1.24) Here W is a positive definite matrix, usually diagonal, weighting the control forces. For marine craft which have both control surfaces and propellers, the elements in W should be Underwater Vehicles 116 selected such that using the control surfaces is much more inexpensive than using the propellers. 3.1 Explicit solution for α = constant using lagrange multipliers Define the Lagrangian (Fossen, 2002), (, ) ( ), TT L =+−ffWf Tfλλτ (1.25) where r ∈Rλ is a vector of Lagrange multipliers. Consequently, differentiating the Lagrangian L with respect to ,f yields 1 1 2 2 TT L − ∂ =−= = ∂ Wf T 0 f W T f λ ⇒λ. (1.26) Next, assume that 1 T− TW T is non-singular such that 111 1 2( ) 2 TT−−− == = τ λ⇒λ τ.Tf TW T TW T (1.27) This gives −− = 11 2( ) , T λ τTW T (1.28) Substituting (1.28) into (1.27) yields, 111†† ,(), TT ww −−− ==τfT T WTTWT (1.29) where † w T is recognized as the generalized inverse. For the case W=I, that is equally weighted control forces, (1.29) reduces to the Moore-Penrose pseudo inverse, 1† ). TT − =TT(TT (1.30) Since † , w =fTτ the control input vector u can be computed from (1.7) as, 1† w − = τ.uKT (1.31) Notice that this solution is valid for all α but not optimal with respect to a time-varying α. 3.2 Explicit solution for varying α using Lagrange multipliers In the unconstraint case a time-varying α can be handled by using an extended thrust representation similar to Sørdalen (1997). Consider the ROV/AUV model in Example 1 where, ee eee τ=Tf =TK u (1.32) Application of (1.29) now gives, A Survey of Control Allocation Methods for Underwater Vehicles 117 † 1 , ew eee − = τ = fT uKf (1.33) where 112234 [,,,,,] T exyxy uuuuuu=u and 123456 [,,,,,]. T e ffffff=f The optimal azimuth angles and thrust commands are then found as 22 22 111 12 11 1 22 22 222 34 22 2 5 3 3 6 4 4 1 2 1 ,atan2(,), 1 ,atan2(,), , . xy yx xy yx uuu ff uu k uuu ff uu k f u k f u k α α =+= + = =+= + = = = (1.34) The main problem is that the optimal solution for 1 α and 2 α can jump at each sample which requires proper filtering. In the next sections, we propose other solutions to this problem. 4. Linear quadratic constrained control allocation In practical systems it is important to minimize the power consumption by taking advantage of the additional control forces in an over-actuated control problem. It is also important to take into account actuator limitations like saturation, tear and wear as well as other constraints such as forbidden sectors, and overload of the power system. In general this leads to a constrained optimization problem. 4.1 Explicit solution for α = constant using piecewise linear functions (non-rotatable actuators) An explicit solution approach for parametric quadratic programming has been developed by Tøndel et al. (2003) while applications to marine vessels are presented by Johansen et al. (2005). In this work the constrained optimization problem is formulated as { } ,, min max 12 min , , subject to: TT f r Jf fffff β = =+ ≤ −≤ ≤ fs fWf sQs Tf s fff ++ τ ≤ (1.35) where n s ∈ R is a vector of slack variables and forces 12 [, , , ] R Tr r ff f=∈f (1.36) Underwater Vehicles 118 The first term of the criterion corresponds to the LS criterion (1.25), while the third term is introduced to minimize the largest force max | | ii ff= among the actuators. The constant 0 β ≥ controls the relative weighting of the two criteria. This formulation ensures that the constraints min max iii fff≤ ≤ ( 1, , )ir = are satisfied, if necessary by allowing the resulting generalized force Tf to deviate from its specification τ . To achieve accurate generalized force, the slack variable should be close to zero. This is obtained by choosing the weighting matrix 0. >QW Moreover, saturation and other constraints are handled in an optimal manner by minimizing the combined criterion (1.35). Let 21 min max [, , ,] R , TT T T nr β + + ∈pff=τ (1.37) denote the parameter vector and, 1 [,,] R . TT T rn f + + ∈zfs= (1.38) Hence, it is straightforward to see that the optimization problem (1.35) can be reformulated as a QP problem: { } 11 22 subject to: min TT J = = Φ+ ≤ z zzzRp Az Cp Az Cp (1.39) where: 1 ()1 1(1)(2) 11 :,: 1 0 rn r rn nr n rn n r rn ×× +× ×× ++×+ ×× ⎡⎤ ⎡ ⎡⎤⎤ ⎢⎥ == ⎢ ⎢⎥⎥ ⎢⎥ ⎢ ⎢⎥⎥ ⎣ ⎣⎦⎦ ⎢⎥ ⎣⎦ Φ W0 0 0 0Q0 R0 00 1 1 112 ,, : 1 1 1 1 1 1 rr rn r rr rn r rr rn nn n rr rn ×× × ×× × ×× ×× ×× − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡⎤ ⎢ ⎥ =− = ⎣⎦ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎣⎦⎦ # # I0 0 I0 0 I0 ATI0 A I0 (1.40) 1 1 1(21)2 1 1 ,, : rn rr rr r rn rr rr r nn n r rn rr rr r rn rr rr r ×××× ×××× ××+ ×××× ×××× − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎡⎤ == ⎣⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0I00 00I0 CI0 C 0000 0000 [...]... 393 -41 2 Harkegård, O (2002) Efficient Active Set Algorithms for Solving Constraint Least Squares Problems in Aircraft Control Allocation Proc of the 41 st IEEE Conference on Decision and Control (CDC’02), 2002 Johansen, T A (20 04) Optimizing Nonlinear Control Allocation Proc of the IEEE Conf Decision and Control (CDC’ 04) , pp 343 5- 344 0, Nassau, Bahamas Johansen, T A., T I Fossen and S P Berge (20 04) Constraint... algorithms to bind the position error drift Particle Filter cloud after 9 pings Point Mass PDF Contours after 9 pings 120 100 100 80 60 60 North offset[m] North offset[m] 80 40 40 20 20 0 0 -20 -20 -40 -20 -10 0 10 20 30 East offset[m] 40 50 60 70 -40 -20 0 20 East offset[m] 40 60 Fig 8 The contour lines of the PMF’s probability density function (left) and the particle filter cloud (right) after processing... with ⎡ wb ⎢ 0 W = ⎢ ⎢0 ⎢ ⎢0 ⎣ 0 wb 0 0 0 0 ws 0 0⎤ ⎥ 0⎥ , 0⎥ ⎥ ws ⎥ ⎦ ⎡ I 4 4 ⎤ M = ⎢ ⎥, ⎢ −I 4 4 ⎥ ⎣ ⎦ ⎡Δ⎤ N = ⎢ ⎥, ⎢Δ⎥ ⎣ ⎦ b ⎡ δ max ⎤ ⎢ b ⎥ δ min Δ = ⎢ s ⎥, ⎢δ ⎥ ⎢ max ⎥ s ⎢ δ min ⎥ ⎣ ⎦ (1. 54) where wb and ws represent the weighting for the bow and stern fins—note that only their relative value is of importance 126 Underwater Vehicles Step 2: Rudder Allocation In nominal operational conditions, we... is the preferred method for obtaining maximum position accuracy The survey vessel is 1 34 Underwater Vehicles Position relative to start of navigation (0,0) 4 2 x 10 HUGIN INS position Start point GPS surface fix 1.8 1.6 1 .4 North (m) End of mission 1.2 1 0.8 0.6 0 .4 0.2 0 Start of mission -1 -0.5 0 East (m) 0.5 1 4 x 10 Fig 3 Mission example with GPS surface fix using a HUGIN AUV, September 2003 equipped... 98-108, Reston, VA Fossen, T I (19 94) Guidance and Control of Ocean Vehicles John Wiley and Sons Ltd., ISBN 0 47 1- 941 13-1 Fossen, T I (2002) Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles, Marine Cybernetics AS, ISBN 82-92356-00-2 Fossen, T I and S I Sagatun (1991) Adaptive Control of Nonlinear Systems: A Case Study of Underwater Robotic Systems Journal... with UTP as the only source for position updates A 30 kHz UTP system with approximately 1 .4 km range was used The AUV ran two straight lines of 7.5 km with two transponders 4 km apart With NavLab navigation post-processing, the navigation data was compared to independent USBL data stored on the 136 Underwater Vehicles survey vessel The difference between the UTP post-processed navigation solution and... be changed slowly only This suggests that the following criterion should be minimized (Johansen et al., 20 04) , 121 A Survey of Control Allocation Methods for Underwater Vehicles r ⎧ 3/2 ⎪ min ⎨ J = ∑Pi fi + sT Qs f , α, s ⎪ i =1 ⎩ + (α − α 0 )T Ω(α − α 0 ) + ρ ⎫ ⎬ ε + det(T(α)W −1T?(α)) ⎭ (1 .44 ) subject to T(α)f = τ + s fmin ≤ f ≤ fmax α min ≤ α ≤ α max Δα min ≤ α − α 0 ≤ Δα max where • • ∑r=1Pi fi... behind the propellers Fig 4 INFANTE-AUV Picture courtesy of Dynamic Systems and Ocean Robotics Laboratory (DSOR), Instituto Superior Tecnico de Lisboa, Portugal Copyright (c) 2001 DSOR-ISR Standard hydrofoil theory, see for example Marchaj (2000), establishes that the lift force produced by the hydrofoils is directed perpendicular to the incoming flow while the drag 1 24 Underwater Vehicles force is directed... y-directions, weighted against the magnitudes A Survey of Control Allocation Methods for Underwater Vehicles | (fi cos αi )2 + (fi sin αi )2 | 123 (1 .47 ) representing azimuth thrust Hence, singularities and azimuth rate limitations are not weighted in the cost function If these are important, the QP formulation should be used 5 .4 Explicit solution using the singular value decomposition and filtering techniques... and A Bemporad (2003a) An Algorithm for Multi-parametric Quadratic Programming and explicit MPC solutions Automatica, vol 39, pp 48 949 7 Tøndel, P., T A Johansen and A Bemporad (2003b) Evaluation of Piecewise Affine Control via Binary Searh Tree Automatica, vol 39, pp 743 - 749 Webster, W C and J Sousa (1999) Optimum Allocation for Multiple Thrusters Proc of the Int Society of Offshore and Polar Engineers . (20 04) . Optimizing Nonlinear Control Allocation. Proc. of the IEEE Conf. Decision and Control (CDC’ 04) , pp. 343 5- 344 0, Nassau, Bahamas. Johansen, T. A., T. I. Fossen and S. P. Berge (20 04) with max 44 min 44 max min 000 000 ,,,, 00 0 000 b b b b s s s s w w w w δ δ δ δ × × ⎡ ⎤ ⎡⎤ ⎢ ⎥ ⎢⎥ Δ⎡⎤ ⎡⎤ ⎢ ⎥ ⎢⎥ ===Δ= ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ −Δ ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎣⎦ ⎣ ⎦ I WMN I (1. 54) where. Survey of Control Allocation Methods for Underwater Vehicles 117 † 1 , ew eee − = τ = fT uKf (1.33) where 1122 34 [,,,,,] T exyxy uuuuuu=u and 12 345 6 [,,,,,]. T e ffffff=f The optimal