Adv Syst Sci Appl 2017; 3:1–8
Published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/502

AUV Thrust Allocation with Variable Constraints

V.V. Kostenko1, A.Yu. Tolstonogov1,2∗

1Institute for Marine Technology Problems FEB RAS, Sukhanova st. 5a, Vladivostok, 690091, Russia
2 Far Eastern Federal University, Sukhanova st. 8, Vladivostok, 690091, Russia

Abstract: Modern multipurpose Autonomous Underwater Vehicles (AUVs) represent the next
generation of robotic systems with new technological tasks faced by researchers. One method
of extending the functionality of the vehicle is installing tunnel thrusters in addition to the stern
propulsion system. Thereby the vehicle becomes able to undertake both survey-style missions
and low speed interactions with the environment. But the efficiency of the tunnel thruster depends
strongly on the vehicle’s velocity, due to hydrodynamic aspects. The usual solution to this problem
uses different control models and thrust allocation methods for these types of mission. A unified
approach to thrust allocation is presented in this paper. The approach is based on solving the
allocation problem by quadratic programming with variable constraints, depending on the velocity
of the AUV and the thrusters’ angle of attack. The well-known implementation of active set
method was used to model the proposed allocation method.

Keywords: AUV, propulsion system, thruster hydrodynamics, control allocation problem,
quadratic programming, interior point method, active set method

1. INTRODUCTION

The design of a control algorithm for an underwater vehicle is often divided into several levels
(Fig. 1.1) [1]. First, a high-level motion control algorithm is designed to compute a vector of
virtual unbounded inputs to the vehicle fc (eq. (1.1)) from the target and current vehicle states
and the control type.

Fig. 1.1. Control system structure. fc is the vector of commanded inputs, f is the virtual vector of inputs, τ is
the vector of allocated thrusts, u is the vector of low-level actuator inputs.

∗Corresponding author: tolstonogov.anton@gmail.com

http://ijassa.ipu.ru/ojs/ijassa/article/view/502


2 V.V. KOSTENKO, A.YU. TOLSTONOGOV

ẋ(t) = A(t)x +B(t)f
y(t) = C(t)x (1.1)

Here, x(t) ∈ Rn is the system state vector, t is time, y(t) is the vector of outputs, f ∈ Rm is
the vector of virtual inputs, which should be equal to the command vector fc of the high level
motion algorithm, andA(t), B(t), C(t) are the coefficient matrices of the mechanical system.
The virtual inputs are usually chosen to have a number of forces and torques that equals the
number of degrees of freedom, m, that the motion control system wants to control.

Second, the control allocation algorithm is designed in order to map the vector of virtual
input forces and torques fc to individual thruster forces τ such that the total forces and torques
generated by all thrusters f amount to the commanded virtual input fc.

f = B(x, t)τ (1.2)
Here, B(x, t) is the thruster configuration matrix. It contains the location and orientation of
all thrusters, in the vehicle body-fixed frame. τ ∈ A ⊂ Rp is the vector of actuator thrusts
generated by the vehicle propulsion system, A are constraints from the saturation of the
thrusters or other physical constraints, and p is the number of actuators.

Third, there is a separate high-frequency low-level controller for each actuator, that
controls the desired thrust τi by a low-level control input ui. For each actuator,

τi = hi(x, t, ui) (1.3)
where h is a function, τi is the thrust of actuator i, and ui ∈ U ⊂ R is the low-level control
input of actuator i. Usually, the effector model is linear in u:

τi = hi(x, t, ui) = G(x, t)ui (1.4)
but the relation between the actuator thrust and the low-level control input is not considered
in this article. We assume that each low-level controller maintains the desired thrust τi with
sufficient quality.

This modularity allows the high-level motion control algorithm to be designed without
detailed knowledge about the vehicle propulsion system. In addition to coordinating the
effects of the different thrusters in the system, issues such as thruster/fault tolerance,
redundancy, and control constraints are typically handled within the control allocation
module. In the case of an over-actuated propulsion system, when the number of thrusters
is greater than the number of degrees of freedom (DOF) controlled by the vehicle (p > m),
the control allocation module solves an optimization problem to achieve a minimal power
consumption of the propulsion system.

There are different approaches to the high-level motion control of a vehicle. The PID
(proportional-integral-derivative) is a widespread control method, but new methods such as
linear quadratic Gaussian (LQG) [2], the H∞ control method [3], or fuzzy logic control, are
being rapidly developed. An interesting method is Model Predictive Control [4], which solves
a suboptimal motion control problem at each iteration, with the ability to combine the first and
second levels of the motion, with the consequent increase in the computational complexity.

In this paper, the problem of high-level control is assumed to be solved and the only the
problem of optimal thrust allocation in the case of an over-actuated vehicle is considered. This
problem is well studied. There is a survey devoted to this problem [1]. Different approaches
to this problem, including linear iterative approach as well as the quadratic programming
approach satisfying optimality criteria, are treated in that survey. There are also papers
published more recently [5–7].

These papers devoted to control allocation are focused on reducing the computational
complexity of the optimal thrust allocation, but there is no mention of the fact that the thruster
constraints can drift dynamically due to thruster hydrodynamics. A new approach, taking into
account the vehicle speed and the thrusters’ angle of attack, is proposed in the present article.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



AUV THRUST ALLOCATION WITH VARIABLE CONSTRAINTS 3

This approach allows controlling both survey-style missions and low speed interactions with
the environment for a vehicle equipped with tunnel thrusters.

2. PROBLEM FORMULATION

Let the vector of virtual inputs computed by the high-level motion control or vehicle operator
be denoted by the generalized force vector f = (fx, fy, fz,mx,my,mz)

T , here fi is the
projection of the force onto the axis i (i equals to the x, y and z axis) and mi is the projection
of the torque onto the axis i in the body-fixed coordinate frame. The body-fixed reference
frame [8] is used in this work. The x axis is directed along the longitudinal vehicle axis
from the vehicle’s stern to the fore, the y axis is directed along the latitudinal vehicle to the
starboard, and the z axis completes the frame to a right-handed coordinate system. Assume
that the system is equipped with p thrusters with control thrust τi(i = 1, . . . , p). This leads to
the following relation between τ = (τ1, τ2, . . . , τp) and the virtual inputs f :

Bτ = f (2.5)

According to [9], the optimization problem can be written as

min
τ,s

1

2
(τTQτ + sTRs)

Bτ = f + s (2.6)
τmin ≤ τ ≤ τmax

where s is the vector of slack variables used to penalize |Bτ− f |, Q and R are the weight
diagonal matrices for the thrusters and DOFs, respectively, τmin and τmax are the vector of
thrust constraints according to Bollard pull tests.

The thrust allocation of an overactuated AUV with tunnel vertical thrusters is a velocity-
dependent problem. For example, the pitch motion of the vehicle on zero velocity is better
created by tunnel thrusters due to their large thrust arm. But the effectiveness of a tunnel
thruster tends to zero at high velocities (Fig. 2.2).

We propose a variable constraints method for the thrust allocation problem. This method
allows reallocating the thrust depending on the vehicle velocity. The method considers thrust
constraints that depend on the vehicle velocity, as well as the thrusters’ angle of attack:

τmin(v, θ) ≤ τ ≤ τmax(v, θ)

where τmin(v, θ) and τmax(v, θ) are the variable thrust constraints depending on the
vehicle velocity and the thrusters’ angle of attack.

3. THE MODEL OF THRUSTER CONSTRAINTS

The thrust and torque generated by a thruster can be described by the following formulas:

τ = Kτ(J0)ρΩ |Ω|D4

m = Km(J0)ρΩ |Ω|D5

where ρ is the density of water, Ω is the rotational speed of the thruster propeller, andD is the
diameter of the propeller.Kτ(J0), Km(J0) are the coefficients of thrust and torque determined

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



4 V.V. KOSTENKO, A.YU. TOLSTONOGOV

(a) v < 0.7ms−1

(b) v > 0.7ms−1

Fig. 2.2. Thrust allocation for pitch motion depending on vehicle velocity.

by the form of the propeller and the characteristics of the motor. J0 = v/ΩD is the advance
ratio (where v is a velocity of thruster input flow, this velocity equals to velocity of the vehicle
in case of the absence of currents).

The function Kτ(J0) can be fitted as a linear function of J0 at J0 > 0 and v > 0 [10]:

K+
τ (J0) = K0

τ (0)− a1J0 (3.7)

where K+
τ (J0) is the thrust coefficient Kτ(J0) at J0 > 0 and equi-directional state of ambient

flow and axial thruster flow. K0
τ is the thrust coefficient at J0 = 0 and a1 is a coefficient

determined by the shape of the propeller. The equation may be rewritten as a function of
velocity by fixing the value of Ω:

Kτ(v) = K0
τ − b1v (3.8)

where b1 = a1/DΩ.
The thrusters’ efficiency also depends on the thrusters’ angle of attack θ. At small angle

of attack (θ < θ∗1, where θ∗1 is the first Critical Incoming Angle [10]), Kτ(v, θ) can be written
as

Kθ
τ (v, θ) = K0

τ +
(
K0

τ −K+
τ (v)

) [
sin

(
θ

θ∗1

π

2

)
− 1

]
(3.9)

where Kθ
τ (v, θ) is the thrust coefficient at vehicle velocity equals to v and the thrusters’ angle

of attack equals to θ,K+
τ (v) is determined by equation 3.8, and θ∗1 is the first Critical Incoming

Angle: θ∗1 = π/2− a1v.
During Bollard pull tests (at v = 0), the thrust of the actuators can be obtained as a

function of the input code or the current. The maximum thrust of the thruster can be written
as a function of Kτ:

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



AUV THRUST ALLOCATION WITH VARIABLE CONSTRAINTS 5

τMax = K0
τρΩMax

∣∣ΩMax
∣∣D4 (3.10)

where τMax is the maximum thrust generated by the thruster during the Bollard pull test
and ΩMax is the number of revolution per second of the propeller at maximum thrust.

Hence, the maximum thrust τMax for a stern thruster at vehicle velocity v and the
thrusters’ angle of attack θ can be obtained from equations (3.7), (3.9) and (3.10):

τMax(v, θ) = τMax − Cv
[
1− sin

(
θ

θ∗1

π

2

)]
(3.11)

where τMax(v, θ) is the maximum thrust generated by the thruster at vehicle velocity v and
the thrusters’ angle of attack θ, C = b1/ρΩMax

∣∣ΩMax
∣∣D4. More complicated cases, when

J < 0 or v < 0, can be obtained from the appropriate equations [10].
The steady state performance of a tunnel thruster can be fitted as an exponential function

of the velocity ratio v/vjet [11]:

τ(v) = τ(0)exp

[
−c
(

v

vjet

)2
]

where τ(v) is the the resulting force of the tunnel thruster for a vehicle velocity of v, τ(0)
is the thruster’s performance in the Bollard pull test, and vjet is the axial velocity of flow
generated by the thruster:

vjet =

√
τ(0)

ρS
. (3.12)

Here, ρ is the density of water and S is the cross sectional area of the tunnel thruster.
Hence, the tunnel thruster constraints can be written as

τ(v)max = τ(0)maxexp

[
−c
(

v

vjet,lim

)2
]

(3.13)

where τ(v)max is the maximum thrust when the vehicle’s motion has velocity v and τ(0)max

is the thruster constraint provided by the Bollard pull tests of the thruster.

4. PROPULSION SYSTEM SETUP

The model of the AUV “MT-2012” [12] propulsion system was used for algorithm simulation.
The propulsion system consists of five thrusters: four thrusters located at the stern of the
vehicle at an angle of α to the longitudinal axis (x axis), and a vertical tunnel thruster located
at the forward part of the vehicle (Fig. 4.3).

The propulsion system of the vehicle can be described by the thrusters’ configuration
matrix B [13]:

B = [bb, bl, bu, br, bf ]

where bd, bl, bu, br are column vectors for (down, left, up, right) the stern thrusters and bf is
the column vector for the forward tunnel thruster. The column vectors in four DOFs (surge,
heave, pitch and yaw) take the form described below.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



6 V.V. KOSTENKO, A.YU. TOLSTONOGOV

Fig. 4.3. Propulsion system of “MT-2012”.

For the up and down stern thrusters:

bu =

 cosα
− sinα

0
luz cosα− lux sinα

 , bd =

 cosα
sinα

0
ldz cosα− ldx sinα

 .

For the left and right stern thrusters:

bl =

 cosα
0

lly cosα− llx sinα
0

 , br =

 cosα
0

lry cosα− lrx sinα
0

 .

For the vertical tunnel thruster:

bf =

 0
1

−ltunnelx
0


where ri = [lix, l

i
y, l

i
z] is the arm vector of thruster i (where i = u, d, l, r, f ). For this

propulsion system, for reasons of symmetry, luz = lry = −ldz = −lly = Lsternzy and lux = ldx =

llx = lrx = Lsternx .
The parameters of the propulsion system are shown in Table 4.1.

Table 4.1. Parameters of the AUV “MT-2012” propulsion system

Lsternx , m 1.88 α, grad 22.5◦
Lsternzy , m 0.23 f sternlim , N 120.1/-68.2
ltunnelx , m 1.20 f tunnellim , N 122.1/-122.0

5. SIMULATION SETUP

The proposed allocation control method with variable constraints was tested on depth
maneuvering at different velocities, 0.3, . . . , 1.8 ms−1 (corresponding to forward forces 3.6,
. . . , 129.6 N, and 50 Nm of pitch torque.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



AUV THRUST ALLOCATION WITH VARIABLE CONSTRAINTS 7

The optimal thrust allocation problem for different velocities was solved by MATLAB
Optimization Toolbox. Since Q > 0 and R > 0 (equation (2.6)) this is a convex quadratic
problem in τ, so the interior point method or active set method for convex optimization
problem can be used.

The results of the thrust allocation for different vehicle velocities are shown in Fig. 5.4.

Fig. 5.4. Thrust allocation for different velocities.

There are some implementations of quadratic programming solvers in the C/C++
languages. These implementations are faster than the MATLAB version, and are easy to
integrate in vehicle software. There are different types of quadratic programming solvers:
qpOASES implements the active set method [14], which allows using a hot start to reduce
the calculation time for each new iteration using the data of the previous one, and there are
several libraries with implementations of the interior point method: [15] and [16].

6. CONCLUSION

To create a multi-purpose AUV capable of undertaking both survey-style missions and low
speed interactions requires an over-actuated propulsion system and a sophisticated thrust
allocation algorithm. One method of extending the functionality of the vehicle is installing
tunnel thrusters to the vehicle propulsion system.

A new approach for thrust allocation algorithm is proposed in the present article. The
method based on the quadratic cost optimal problem with constraints. The main feature of
the method is that thrust constraints depend on the vehicle velocity, as well as the thrusters’
angle of attack. This method takes into account the decrease in the thrust of tunnel and stern
thrusters at high velocities, and reallocates the thrust accordingly.

The proposed algorithm was tested with a simulation model of the real AUV “MT-2012”
propulsion system. The propulsion system consists of five thrusters: four thrusters located
at the stern of the vehicle and the vertical tunnel thruster located at the forward part of the
vehicle. The result shows the adaptive thrust reallocation depending on vehicle velocity in
case of depth maneuvering.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



8 V.V. KOSTENKO, A.YU. TOLSTONOGOV

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Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)

https://dx.doi.org/10.1109/ICSMC.2012.6378129
https://dx.doi.org/10.1109/ICSMC.2012.6378129
https://dx.doi.org/10.1109/ARSO.2015.7428209
https://dx.doi.org/10.1109/OCEANS.2016.7761251
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https://dx.doi.org/10.5772/6705
https://github.com/YimingYAN/cppipm

	Introduction
	Problem Formulation
	The Model of Thruster Constraints
	Propulsion System Setup
	Simulation Setup
	Conclusion