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Modeling and Control Approach for a Complex-Shaped Underwater Vehicle

G.A.C.T. Bandara*, D.A.A.C. Ratnaweera, D.H.S. Maithripala

Department of Mechanical Engineering, University of Peradeniya, KY 20400, Sri Lanka

Abstract: This paper presents the trajectory tracking and the path planning algorithm based on an adaptive control law to operate a complex-shaped low speed autonomous underwater vehicle (AUV) in a challenging environment of non-linearity, time variance and unpredictable external disturbances. Firstly, computational fluid dynamic (CFD) simulations are used to compute the added mass matrix and the damping matrix. Secondly, the adaptive controller is implemented to track the desired trajectory. This desired state-dependent regressor matrix-based controller provides consistent results even under hydrodynamic parametric uncertainties. The stability of the developed controller is verified using Lyapunov’s direct approach. Moreover, the proposed control law adopts quaternions to represent the attitude errors and thus avoids the representation of singularities that occur when using the Euler angle description of the orientation. Thirdly, an efficient underwater path planning algorithm is developed based on vehicle-fixed-frame error variables. The simulations are done to compute the optimal path of the AUV which minimizes the travelling time. Finally, an optimal thrust allocation for the desired values of forces and moments acting on the vehicle is found via a model-based unconstrained thrust allocation. The results show that the AUV asymptotically converges on the desired trajectory and the path with a minimum time. At this moment, the propulsion forces approach zero, which further assures the accuracy of the controller. Hence, the effectiveness and the robustness of the developed algorithm are acceptable to design the AUV.

Keywords: autonomous underwater vehicle (AUV), CFD modelling, adaptive control, trajectory, path planning, optimal thrust allocation

Received August 19, 2019; Revised September 26, 2019; Accepted October 16, 2019

Citation: G.A.C.T. Bandara, D.A.A.C. Ratnaweera, D.H.S. Maithripala, Modeling and Control Approach for a Complex-Shaped Underwater Vehicle. American Journal of Mechanical Engineering, 7 (4). 158-171. January 2019. doi:10.12691/ajme-7-4-2

1. Introduction

Over last two decades, autonomous underwater vehicles (AUVs) have played major roles in underwater applications, and have a large variety of types and shapes. Especially small size, complex-shaped AUVs have more flexibility to reach the areas where remote operating vehicles (ROVs) and human occupied vehicles (HOVs) cannot be deployed. AUVs can also be operated in risky and hazardous environments. Meanwhile, complex-shaped AUVs have more DOFs than conventional torpedo-shaped AUVs to increase the maneuverability in complex underwater spaces.

There are three key factors to be considered in the motion control of an AUV: First, an accurate hydrodynamic model, second, an advanced control system, and third, an optimum thrust allocation. Yamamoto revealed in [1] that a model-based control system is more effective in case of dynamics of the AUV is known to some extent. Furthermore, an empirical model generally fails to represent the dynamics of the AUV over a wide operating region that was noticed by Ferreira [2]. Thus, to get an accurate hydrodynamic model of the complex-shaped AUVs is vital to design a controller. In addition to that, it is an arduous task to build a suitable controller for an AUV due to the complexity of hydrodynamic parameters containing highly non-linear and coupled terms. Moreover, the literature in the field of the optimum thrust allocation is little to be known [3]. Particularly, the importance of the optimization of the thrust forces of AUVs has been disregarded so far.

Scaled and full-scaled experiment, empirical formula, and computational approach are the existing methods to model the underwater vehicles. The scaled and full-scaled experiments are the most expensive methods because these ones require costly devices like towing tanks, but provide accurate hydrodynamic parameters. References [4,5] provide the methods to do the experiments without the towing tank. Furthermore, the empirical formulas usually give the acceptable results on the underwater vehicles that have slender bodies such as torpedo-shaped AUVs as noticed in [6]. Lastly, computation approaches are the best way to find the hydrodynamic parameters under low cost. Nevertheless, vast knowledge and experience are the prime factors to model the complex-shaped AUVs. Potential and finite element theory-based software such as ANSYS FLUENTTM, ANSYS AQWATM, ANSYS CFXTM, and WAMITTM is used in this method. WAMITTM overcomes the other software to compute the added mass matrix [7]. Similarly, ANSYS CFXTM leads the case of finding damping matrix [8].

There are so many control systems proposed to the track trajectories and to plan the desired path for AUVs. The horizontal tracking control for the AUVs based on a non-linear sliding mode incremental feedback model was introduced in [9] to track the desired trajectories. The error dynamics on the horizontal plane were proposed in [10] to plan and track the trajectories using the closed loop tracking controller and the backstepping method was used to stabilize the system. To track the given trajectories in the presence of ocean currents, a feedback controller was introduced in [11] using ling of sight (LOS), and a sliding mode controller based on both LOS and cross track error approach was presented in [12]. The trajectory planning for the AUVs was addressed in [13] to provide ocean processes of the real-time ocean model. The wheel robot was controlled to track the desired trajectories in [14] using a linearized fuzzy adaptive controller with the backstepping feedback. The trajectory tracking of the AUVs was implemented using the adaptive tracking controller based on a radial basis function neural network (RBF-NN) [15]. A dynamic surface control (DSC) and minimal learning parameters (MLP) based on a robust adaptive neural network tracking control for the underwater vehicle have been used in [16]. The learning method of a neural network has been used to model the stability and robustness adaptive controller for a nonholonomic robot [17]. In [18,19], they have described the path planning method based on a velocity field using the starting point and the desired ending point within a minimum time period. The coordinate path followed by underactuated AUVs was outlined, based on the convergence of geometric errors with respect to the origin of the vehicle that coincides with the center of gravity [20]. The path tracking based on the marching algorithm has been used in [21] for the path planning on a fixed depth. The combined path following and trajectory tracking for the AUVs have been discussed in [22] using the backstepping method, where Lyapunov’s direct approach was used to develop the kinematic of the AUVs. The optimal path planning for the AUVs in fast flowing, complex fluid flow was proposed using the methods of cost function, parametrization, and principle of minimum energy [19].

Recommended techniques of finding the hydrodynamic, trajectory tracking and path planning methods for AUVs have been noted in literature. However, an uncertainty of their hydrodynamic parameters, including non-linear hydrodynamic effects, parameter variations, and ocean current disturbance, causes difficulties when designing the suitable controllers. To overcome these problems, there should be a controller that could be operated under the well-defined hydrodynamic model.

In our work, the complex-shaped AUV that was used for the simulations is shown in Figure 1. The placement of 6 thrusters is only considered in Section 5. Furthermore, the configuration details of the AUV are listed in Table 1.

Figure 1. 3D CAD images of the AUV [23]

Table 1. Characteristics of the complex-shaped AUV

This paper presents a modelling and adaptive controlling approach for the complex-shaped AUV. In Section 2, the standard notions for marine vehicles are introduced. Section 3 is focused to develop the model of the complex-shaped AUV to find the hydrodynamic parameters using the computational approach, ANSYS FLUENTTM. In Section 4, the vehicle-fixed frame adaptive controller is introduced to track the desired trajectories under the presence of hydrodynamic uncertainties, and uses the quaternion-based attitude error. Finally, the path planning method is developed based on the trajectory tracking controller, and the approach to find the minimum thrust allocation is proposed to increase the effectiveness of the propulsion system in Section 5. To the best of our knowledge, this is the first time that the representation of the quaternion-based attitude error is used for the trajectory tracking and path planning with a complete hydrodynamic analysis.

2. AUV Modeling

This section is committed to represent the AUV kinematics and dynamics containing highly non-linear and coupled terms which make the mathematical model challengeable. NED-frame (North East Down) and B-frame (Body fixed frame) are the two coordinate frames for marine systems introduced for convenience by Fossen in [24] and shown in Figure 2.

Figure 2. NED frame and B-frame of underwater vehicles

The AUV dynamics are based on the marine vehicle formulation by Fossen in [24] and [25], and by the Society of Naval Architects and Marine Engineers (SNAME) [26]. Positions, angles, linear and angular velocities, force and moment are shown in Table 2. The position vector (η), velocity vector (υ), and force vector (τ) are defined as follows.

[picture] | (1)

[picture] | (2)

[picture] | (3)

Table 2. The notation of SNAME for marine vessels

The dynamic equation of motion is given by

[picture] | (4)

Rigid body mass-inertia matrix defined in (5). is the mass of the AUV and is the position vector from the origin of the body frame to the Center of Gravity (COG). In this case, the origin of the body-fixed frame is coincided with the COG of the AUV. Then, the mass-inertia matrix is extremely reduced. It is assumed that the AUV is symmetric about and planes. Therefore, the inertia components become roughly diagonal form

[picture] | (5)

Let be the added mass and inertia.

[picture] | (6)

where the conversation of SNAME is used; etc.

The rigid body induced Coriolis-centripetal matrix, and the added mass induced Coriolis-centripetal matrix are given by and respectively. In our case, matrix is neglected due to the slow speed of the AUV.

[picture] | (7)

[picture] | (8)

where and also here the SNAME notion is used.

denotes the total hydrodynamic damping matrix. The hydrodynamic damping of the underwater vehicles normally contains four types of damping forces as below:

[picture] | (9)

where and are the linear, quadratic, potential, and wave drift term respectively.

If the vehicle is symmetric about all the planes, then becomes a diagonal matrix describing in [27] as follows:

[picture] | (10)

The axial quadratic drag force of the AUV can be model with

[picture] | (11)

where is the fluid density; is the cross-sectional area and is the damping coefficient;

The notion of (11) gives the quadratic drag matrix as shown below:

[picture] | (12)

It is considered that the weight of the AUV is approximately equal to the buoyance force is the gravitational and buoyance matrix. It can be simplified as in (13) as pointed out by [24]. is the position vector for the COG to the Center of Buoyancy (COB).

[picture] | (13)

is the forces and moments vector of the propulsion input, and the environmental forces and moments are defined as

The kinematic equation of the AUV can be expressed as

[picture] | (14)

where is the velocity transformation matrix between the body-fixed frame of the AUV and the earth-fixed frame and is the rotation matrix expressing the transformation from the earth-fixed frame to the body-fixed frame. This velocity transformation matrix is further represented as below.

[picture] | (15)

where

and and are the short notations for and respectively.

To overcome the possible occurrence of representation singularities, it might be convenient to resort to the quaternion representation The relationship between and the time derivative of the quaternion is given by the quaternion propagation equations. (See Appendix A)

3. Computational Solutions for Dynamic and Hydrodynamic Parameters

This section represents how the numerical methods are used to find the rigid body mass-inertia matrix, added mass matrix, and damping matrix. First, the SOLIDWORKS software is used to compute the mass-inertia matrix Next, ANSYS FLUENTTM is used to find both added mass and damping matrix

3.1. Rigid Body Mass Inertia Matrix

The COG and the inertia parameters of the AUV are very difficult to calculate analytically as shown in (16) in [24] because they have many different density components. The mass and inertia of a small particle can be defined as follow.

[picture] | (16)

where is the density of an element of volume is the total volume of the AUV, is the distance between the element of volume and the COG.

The best way to calculate the mass-inertia matrix of the AUV is to use the CAD software, SOLIDWORKS. The SOLIDWORKS’ model of the AUV is shown in Figure 1 and is listed in (17). Floating and payloads are not examined here. It is acceptable to neglect the off-diagonal inertia elements compared with the diagonal terms, expressed as follows. The units are measured in and

[picture] | (17)

3.2. Added Mass Matrix

The added mass of the AUV really depends on the geometry of the underwater vehicle as mentioned in [28]. Hence, the use of empirical formulas predicting and a hull approximation by elementary shapes are inaccurate for the complex-shaped AUV. For the case of a complex-shaped AUV, matrix cannot pursue the empirical formulas noted in [5] and [6]. ANSYS FLUENTTM is advanced to compute the hydrodynamic characteristics of marine objects, offering the function to figure out the added mass matrix, and it requires a closed geometry of the AUV for the calculation. In this case, the speed of the AUV is low enough to neglect the off-diagonal elements. Thus, the diagonalized matrix is considered for the MATLAB simulations done in Section 4 and Section 5. The units are measured in and

[picture] | (18)

3.3. Damping Matrix

In this case, can be neglected comparing to other terms, and is also ignored since the AUV operates under certain depth, shown in [24].

These formulas are impractical to calculate the damping matrix. Hence, ANSYS FLUENTTM based on the finite element theory is used to compute the relationship between the damping forces and velocity of the AUV. The AUV is fixed in a rectangular tank in which the velocity of the fluid varies from 0 to 0.6 (approximation of general AUV operating speed), with the speed interval of

The velocity stream-line views of ANSYS FLUENTTM results in different directions are shown in Figure 3. The configuration details used for the simulation are listed in Table 3, and the computed damping forces and moments are listed in Table 4 and Table 5. A second order polynomial relationship is noticed between damping forces/moments and velocities, see Figure 4 and Figure 5.

Figure 3. ANSYS FLUENTTM velocity streamline views of the AUV at 0.5 m/s: (a) in surge direction; (b) in sway direction; (c) in heave direction

Table 3. Configuration details of ANSYS FLUENTTM

Table 4. Damping forces at different linear velocities

Table 5. Damping torques at different angular velocities

Figure 4. Estimated damping force with linear velocity

Figure 5. Estimated damping torque with angular velocity

4. Trajectory Tracking Control Law

The dynamics of the AUV denoted in (4) include the uncertainty of the parameters in the damping matrix. It is designed to recompense the uncertainties; therefore, the vehicle-fixed-frame adaptive controller is proposed to achieve a consistent AUV performance by estimating uncertain parameters. The parameters in the control law are adjusted by an adaption mechanism. The proposed adaptive controller is designed to force the AUV to follow the desired trajectory with the existence of a parameter uncertainty. For this, a regression matrix should be defined.

Let us consider the vehicle-fixed variables

[picture] | (19)

[picture] | (20)

where being the desired position, and the quaternion-based attitude error is given by is the desired velocity of the AUV in the body-fixed frame.

The error vector is given by:

[picture] | (21)

with positive definite matrix.

The vehicle regressor matrix

[picture] | (22)

The adaptive control law can be defined in the following form.

[picture] | (23)

Such that,

[picture] | (24)

where is a positive definite matrix. The parameter estimate is updated by

[picture] | (25)

where is a suitable positive definite matrix of an appropriate dimension, and is selected in such a way that the tuning law provides convergent characteristics.

The adaptive law proposed in the equation can drive the AUV in a desired manner, and guarantee the AUV’s stability. The structure of the proposed control is given in the block diagram below.

Figure 6. MATLAB SIMULINK layout of the proposed adaptive control law

4.1. Stability of the Proposed Control Law

The following Lyapunov candidate function is used to examine the stability of the system.

[picture] | (26)

where and

The selected Lyapunov candidate function satisfies the conditions given in (27).

[picture] | (27)

Indeed, the system is asymptotically stable in the sense of Lyapunov because satisfies the above conditions.

The time derivative of (26) is given by

[picture] | (28)

Being and as follows,

[picture] | (29)

[picture] | (30)

where as is a constant definite vector

Substituting (21) into (29), as shown in [29].

[picture] | (31)

where

Substituting the value of from (4) into (31) and simplifying for

[picture] | (32)

[picture] | (33)

[picture] | (34)

is a skew-symmetric matrix for the AUVs. Substituting the system dynamics given in (4) into above equation.

[picture] | (35)

Therefore, equation (28) becomes,

[picture] | (36)

Substituting the control input into (36)

[picture] | (37)

[picture] | (38)

[picture] | (39)

Substituting from (25) into (39)

[picture] | (40)

[picture] | (41)

Equation (41) satisfies the Lyapunov stability criterion of the AUV dynamics with a stable controller. Therefore, the adaptive control law proposed in (23) gives a stable closed system.

4.2. Validation of the Adaptive Controller

The derivation of the regressor matrix of a high-DOF is very cumbersome. Therefore, for the sake of simplicity, only five DOFs are considered.

[picture] | (42)

where

and

When the regression matrix is used, the AUV model given in (42) can be written down in the form of linear parameters. For a real-time implementation, it is clear that the regression matrix, a state dependent matrix, is needed to calculate in each control cycle, seen in [30].

The regression matrix defined in (22) can be written as follows:

[picture] | (43)

The number of simulations is performed with the desired trajectories to check the validity of the proposed control. The parameters of the AUV used for simulations are listed in Table 6.

The desired trajectory in the form of a circular path.

[picture] | (44)

Note: The selected trajectory functions should be continuous to compute the second derivative with respect to time.

Figure 7. Circular trajectory comparison between the desired path and actual path projected onto: (a) xy plane; (b) xz plane and plotted; (c) in space

Table 6. Parameters of the AUV used for the simulation

Figure 8. Actual position and attitude of the AUV compared with the desired position and attitude: (a) after 120 ; (b) after 630 s

Figure 9. Earth-fixed linear and angular velocities: (a) after 120 s; (b) after 630 s

Figure 8 elaborates that the actual position and orientation of the AUV accurately track the desired position and orientation after

Figure 10. Position and attitude errors of the AUV after 200 s

Figure 9 illustrates that the velocity errors tended to zero within 85 s from the beginning. And also, Figure 10 and Figure 11 asymptotically converge to zero. Thus, the proposed controller accurately commands the AUV to follow the desired path.

Figure 11. Earth-fixed frame velocity errors of the AUV after 200 s

Figure 12. Forces and torques applied to the AUV: (a) after 50 s; (b) after 630 s

In Figure 12, it is true that initially, both forces and torques have some definite values showing that the AUV accomplished the required acceleration, but after 30s, the torques in case of pitch and yaw converged to zero, which means the angular accelerations became to zero. The forces in case of surge, sway and heave indicate the simple harmonic variation around zero to catch the sinusoidal position inputs, which include the high amplitude in these directions.

5. AUV Path Planning with Optimum Thrust Allocation

5.1. Path Planning Algorithm Using the Adaptive Control Law

In this case, according to the thrust placement given in Figure 1, there are 4 DOFs: surge, roll, pitch, and yaw, in which the AUV is able to maneuver with the controller inputs for the surge roll pitch and yaw DOFs, but the roll DOF behaves like a self-stabilizer. This AUV can move from its current position, called set-point A, to desired potion, called set-point B as shown in Figure 13.

Figure 13. Visualization of the coordinate parameters

The desired point is always considered as the origin of the frame that is parallel to the earth fixed frame, while the origin of the body-fixed frame which coincides with the COG is the current position The error vector describes the position error between the desired point and the current location of the AUV, as shown in Figure 13. The control inputs are provided to steer the AUV along the instantaneously desired path as the optimum path between and to minimize the The AUV is also guided asymptotically towards the desired point Note that this control inputs are only valid for and

[picture] | (45)

[picture] | (46)

[picture] | (47)

[picture] | (48)

[picture] | (49)

[picture] | (50)

where k is a positive constant.

This control model has the position controller, attitude controller, speed controller and plant (AUV). The desired and current position given in the earth fixed frame are the inputs for the position controller which gives the desired attitude and as the outputs. The controller discussed in Section 4 is based on the body-fixed frame. Thus, all the inputs for the adaptive controller should be converted to the body-fixed frame. The outputs of the attitude controller are the desired propulsion torques in the roll, pitch, and yaw directions Furthermore, and are the inputs for the speed controller giving the propulsion force in the surge direction as the output. The inputs of the plant are and see Figure 14.

The MATLAB SIMULINK block diagram of this controller is shown in below.

Figure 14. MATLAB SIMULINK layout of the controller

5.2. Optimum Thrust Allocation

In the thrust allocation, optimizing methods are used to minimize the thrust forces to save the power consumption and increase the efficiency of the AUVs. This complex-shaped underwater vehicle contains 6 thrusters and they allow to control the four DOFs: surge, roll, pitch, and yaw. The thruster configuration, thruster forces, thruster numbering and distances are shown with respect to the body-fixed frame in Figure 15.

Figure 15. Location of the thrusters on the AUV, definition of the distances, and direction of the positive thrust

The force and torque vector can be defined as follows, seen in [25,31]:

[picture] | (51)

where is the thruster configuration matrix, indicating the direction of the force in the surge DOF, and the direction of the toques in the roll, pitch, and heave DOFs,

[picture] | (52)

here, the distances are and and also is the thrust vector.

[picture] | (53)

The optimum thrust allocation from the required can be considered as a model-based optimization problem and it can be categorized under two major categories: constrained thrust allocation and unconstrained thrust allocation. For this AUV, the unconstrained thrust allocation is used because it does not pay an attention about the bounded limitations (minimum and maximum) for the thrust vector. This approach has several advantages than the constrained thrust allocation; it consumes minimum computational time, and requires an on-board computer with low computational power.

Assumption:

The vector of propulsion forces and torques is bounded in such a way that the computed elements of the thrust vector never exceed the boundary values of and

Under above assumption, this thrust allocation can be expressed as a problem of the least-squares optimization as follows:

[picture] | (54)

where H is a positive definite matrix.

The solution of (54) using Lagrange multipliers shown in [25] is given by

[picture] | (55)

[picture] | (56)

This approach provides a suitable solution for the faultless work of the propulsion system, and cannot be used directly in case of damaged thrusters.

5.2.1. Singular Value Decomposition (SVD) for the Solution

SVD is an eigenvalue-like decomposition for rectangular matrices [32]. After applying the SVD for the thrust configuration matrix, it becomes as follows:

[picture] | (57)

Where and are orthogonal matrices of dimensions and respectively. is the number of thrusters in the AUV.

[picture] | (58)

- diagonal matrix of dimensions

- null matrix of dimensions

The singular values of called and are positive and the order of them should be like To calculate the optimum thrust vector being a minimum-norm solution to (54), the decomposition form of is necessary to compute. Finally, can be written in the form of

[picture] | (59)

5.3. Simulation Results and Discussion

In this section, we present the simulation results of the path planning algorithm and the optimized thrust allocation defined in Section 5.1 and Section 5.2 respectively, including the uncertainty of the hydrodynamic parameters. To simulate the proposed path following algorithm, the AUV requires the earth-fixed frame co-ordinates of the set-point A and B for the path planning. It starts form the set-point A (-8, -8, -2) and is driven through the co-ordinates of the set-point B, see Table 7. All of these points were randomly selected for the simulation. The AUV automatically selects the next set point B in the sequence given in Table 7. Figure 16 a) and Figure 16 b) show the projection of the path that the AUV followed onto and plane, respectively. Figure 16 c) presents the path of the AUV in space.

Table 7. Coordinates of set-point B denoted in the earth-fixed frame

Figure 16. Path of the AUV drawn by its COG: (a) projected onto xy plane; (b) projected onto xz plane; (c) in space

Figure 17. Desired and actual position of the path in the earth-fixed frame

Figure 18. Position error of the path in the earth-fixed frame

The comparison between the desired and actual position is given in Figure 17. The Position error converges to zero at each set-point B, and just after that it increases to match with the new error, as shown in Figure 18.

Figure 19. Desired and actual attitude of the path in the earth-fixed frame

Figure 20. Desired and actual attitude error of the path in the earth-fixed frame

Figure 19 illustrates the desired and actual attitude. In the attitude error given in Figure 20, it is observed that the roll angle always tries to maintain its value around zero like a self-stabilizer. It is important to note that the pitch and yaw angles always vary to match with the optimum path between the current position A and the set-point B.

Figure 21. Force and torques applied to the AUV

The Propulsion force in the surge direction and torques in the roll, pitch and yaw directions are indicated in Figure 21. The force and the torques suddenly increase just after reaching each set-point B since the position and attitude errors instantly grow due to the new target, but they asymptotically converge to zero within indicating that linear and angular velocities become constant. Thus, we can assume that the AUV runs towards the next set-point B smoothly.

and are calculated as below using to compute the optimum thrust allocation for the above discussed AUV;

[picture] | (60)

[picture] | (61)

[picture] | (62)

[picture] | (63)

[picture] | (64)

Figure 22. Optimum thrust forces acting on 6 thrusters

The optimum thrust forces derived in (59) are shown in Figure 22. This figure indicates the variation of the thrust vector under the assumption of the unconstrained thrust allocation as discussed in Section 5.2. Generally, and are much higher than the others since they provide the propulsion force in the surge direction. Particularly, at the beginning of each new set-point B, all the thrusters show their peak thrust forces. And also, each pair of the thrusters acts in the opposite direction to create the moments for at least one attitude.

6. Conclusion

By avoiding the expensive devices, this paper presents the numerical solution to find the hydrodynamic parameters and it showed that the damping is varied in the quadratic form leading the non-linearity in the AUV model, which should be carefully considered when any linear based controller is designed. Furthermore, an adaptive control law is proposed to track the trajectories, including the uncertainty of the hydrodynamic parameters and using quaternion-based attitude error. Lyapunov’s direct approach is used to verify the stability of the proposed control law. It is also developed to plan the optimum path through the given co-ordinates using the concept of the convergence of geometric errors. Finally, an algorithm based on the decomposition of the thrust configuration matrix is used to optimize the unconstrained thrust allocation for the given six thrusters' configuration. The accuracy of the developed control law is verified by the results of the numerical simulation.

Appendix A

By defining the mutual orientation between two frames of common origin of the rotation matrix.

[picture] | (65)

where is the angle and is the unit vector of the axis expressing the rotation needed to align the two frames. is the matrix operator performing the cross product between two vectors.

[picture] | (66)

The unit quaternion is

[picture] | (67)

with

[picture] | (68)

where for

The unit quaternion satisfies the condition

[picture] | (69)

Let’s define the vector as follows:

[picture] | (70)

The relationship between and is given as follows:

[picture] | (71)

where

[picture] | (72)

[picture] | (73)

and

[picture] | (74)

References

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