# Controller for an Asymmetric Underactuated Hovercraft in Terms of Quasi-Velocities

## Abstract

**:**

## 1. Introduction

- (1)
- The proposed algorithm, unlike the scheme in [22] and many other algorithms, is suitable for asymmetric vehicles in the horizontal plane. Thus, it is an extension of the known control concept to a more general class of vehicles.
- (2)
- The theoretical development concerns the inclusion in the control algorithm of components related to the fact that the geometric center does not coincide with the center of mass. These components are then introduced into the stability proof of the system. In addition, the dynamic model of the vehicle is transformed to velocity space and, as a result of this transformation, new variables include the parameters of this model. Consequently, the combination consisting of a velocity transformation and a control algorithm yields a useful tool for estimating the behavior of a vehicle during trajectory tracking.
- (3)
- The simulation verification of the proposed trajectory tracking algorithm for estimating the effect of mutual coupling between velocities and evaluating the reduction in kinetic energy by each quasi-velocity (which includes coupling with other variables).
- (4)
- Pointing out some performance differences between the IQV-based control scheme and two other selected algorithms that use a diagonal inertia matrix model.

## 2. Hovercraft Model

**Assumption 1.**

**Assumption 2.**

## 3. Control Scheme Using Quasi-Velocities

**Theorem 1.**

**Proof.**

**Theorem 2.**

**Proof.**

## 4. Simulation Results

- (1)
- A hovercraft model with parameters similar to those of a vehicle known from the literature was selected;
- (2)
- Two curvilinear trajectories were selected for tracking;
- (3)
- Comparative tests were carried out for the proposed control algorithm based on quasi-velocities, the algorithm using the TSMC method and the original algorithm from [22];
- (4)
- The test results were compared on the basis of the assumed set of criteria and indexes.

#### 4.1. Hovercraft Model and Test Conditions

#### 4.2. Performance Indexes and Evaluation Criteria

- (1)
- The mean integrated absolute error, i.e., $MIA=\frac{1}{{t}_{f}-{t}_{0}}{\int}_{{t}_{0}}^{{t}_{f}}\left|{\chi}_{e}\left(t\right)\right|dt$, where ${\chi}_{e}=\Delta {p}_{x},\Delta {p}_{y},\Delta {\zeta}_{1},\Delta {\zeta}_{2}$;
- (2)
- The mean integrated absolute control, i.e., $MIAC=\frac{1}{{t}_{f}-{t}_{0}}{\int}_{{t}_{0}}^{{t}_{f}}\left|P\left(t\right)\right|dt$, where $P=Tcos\psi ,T,sin\psi $;
- (3)
- The root mean square of the tracking error, i.e., $RMS=\sqrt{\frac{1}{{t}_{f}-{t}_{0}}{\int}_{{t}_{0}}^{{t}_{f}}{\u2225e\left(t\right)\u2225}^{2}dt}$, with $\u2225e\left(t\right)\u2225=\sqrt{\Delta {p}_{x}^{2}+\Delta {p}_{y}^{2}}$ ($\Delta {p}_{x}$, $\Delta {p}_{y}$ mean the position error in the reference frame).

- (1)
- The kinetic energy for the total vehicle and corresponding to each quasi-velocity, i.e., ${K}_{T}=\frac{1}{2}{\nu}^{T}M\nu =\frac{1}{2}{\zeta}^{T}N\zeta =\frac{1}{2}{\sum}_{i=1}^{3}{N}_{i}{\zeta}_{i}^{2}={\sum}_{i=1}^{3}{K}_{i}$.
- (2)
- The mean kinetic energy relating to each quasi-velocity and the total kinetic energy: ${K}_{m}=mean\phantom{\rule{0.166667em}{0ex}}\left(K\right)={\sum}_{i=1}^{3}mean\phantom{\rule{0.166667em}{0ex}}\left({K}_{i}\right)$.
- (3)
- The deformation of each velocity resulting from dynamic coupling, i.e., $\Delta {\zeta}_{i}={\zeta}_{i}-{\nu}_{i}$, where $i=1,2,3$.

#### 4.3. Simulation Tests Conducted

#### 4.3.1. Test of Quasi-Velocity Based Controller

**Sine trajectory**. As is evident in Figure 2a, the set trajectory is tracking correctly and the position errors (presented in Figure 2b) in the x and y directions are tending to near zero after about 5 s. As can be observed in Figure 2c, the non-steady state for the control signals disappears after about 15 s of movement. Figure 2d demonstrates that at a similar time, this condition disappears for hovercraft velocities. Additional information about the effect of dynamic parameters on movement is obtained from changes in the value of $\Delta {\zeta}_{1},\Delta {\zeta}_{2}$ as seen in Figure 2e. Because the displacement values of the center of mass position are the same in the x and y directions, $\Delta {\zeta}_{1},\Delta {\zeta}_{2}$ are therefore symmetrical. Figure 2f presents the evolution of kinetic energy when the vehicle is in motion. Using this graph, it is possible to determine which variable causes more or less kinetic energy consumption, assuming dynamic parameters and a desired trajectory. This can be used to design the vehicle and for planning its movement.

**Complex trajectory**. Figure 3 shows the results of the analogous tests conducted for the complex trajectory. It can refer, for example, to a situation in which the vehicle should avoid a fixed obstacle. As can be seen in Figure 3a, the algorithm works properly and the desired trajectory is tracked efficiently. This observation is also confirmed in Figure 3b. However, from Figure 3c, it is apparent that the values of the applied force components do not reach a steady state but are constantly changing and, moreover, are larger than for a sine trajectory. A similar phenomenon can be seen in Figure 3d, which shows the velocities of the vehicle. The changes in the $\Delta {\zeta}_{1},\Delta {\zeta}_{2}$ values are larger than before but also symmetrical (for the same reason as previously), as observed in Figure 3e. The kinetic energy changes, as shown in Figure 3f, have larger values than for the sinus trajectory, and their waveforms are considerably different from the previous ones. So, the selection of the trajectory significantly affects the consumption of this energy.

**Robustness issue**. In addition, the robustness of the proposed algorithm to changes in the hovercraft parameters was examined. The inaccuracy of the model parameters was assumed as W = 0.5 (50%).

#### 4.3.2. Comparison with Terminal Sliding Mode Controller

- (1)
- According to the cited reference, the TSMC method provides robustness against unmodeled dynamics, model uncertainty and external interference from ocean currents and waves. It is also known to be superior to the conventional sliding mode control technique in terms of finite-time convergence and high-precision steady-state tracking. It has been successfully applied in many works.
- (2)
- Despite the fact that the approach based on TSMC is designed for underactuated underwater vehicles, it has been successfully implemented for the hovercraft in [27].
- (3)
- In the cited article, several algorithms based on different methods were investigated through simulation, and the mentioned TSMC method proved successful in tracking different trajectories while two others proved ineffective.

- (1)
- The model with diagonal inertia matrix ${M}_{diag}$ and without friction.
- (2)
- The model with diagonal inertia matrix ${M}_{diag}$ with friction.
- (3)
- The model with non-diagonal inertia matrix ${M}_{sym}$.

**Sine trajectory without friction**and using ${M}_{diag}$. Due to numerical problems, acceptable results could not be obtained as in the case of the algorithm using IQV for the same step time. Therefore, this time was reduced to $\Delta t=0.003$ s (ODE 3 integration method). As can be seen in Figure 5a, the trajectory tracking is correct and the position errors are close to zero after about 10 s, as shown in Figure 5b. However, the time to obtain the error convergence is longer than with an IQV-type algorithm. The values of the force and torque decrease significantly after a short time (Figure 5c), but the torque suddenly increases and decreases after about 8 s, which is not good for the performance of the task. The velocity values are acceptable, as can be seen in Figure 5d.

**Sine trajectory with friction**and using ${M}_{diag}$. The same step time $\Delta t=0.003$ s and ODE 3 integration method were applied. From Figure 6a,b, it can be observed that the sine trajectory tracking is not as accurate when frictional forces appear in the dynamics model. However, in this case, the force and torque have smaller values and the suddenly jumping value of the torque is clearly smaller, as shown in Figure 6c. The velocity values are now also slightly lower, as can be seen in Figure 6d.

**Complex trajectory without friction**and using ${M}_{diag}$. For this desired trajectory, the numerical problems were even greater than for the previous one. The ODE 3 method was adopted, but the time step was reduced 10 times, i.e., $\Delta t=0.0003$ s. As can be noted in Figure 7a, and which is also confirmed by the results in Figure 7b, the algorithm works properly, ensuring trajectory tracking. The time to reach steady state is about 10 s, so the controller works quickly. Figure 7c shows that the force and moment values are acceptable. However, after about 15 s, there is a sharp change in the value of the moment due to a change in the shape of the desired trajectory. This also indicates the sensitivity of the algorithm when this situation occurs.

**Complex trajectory without friction**and using ${M}_{diag}$. The algorithm did not provide any acceptable results even when reducing the integration step.

**Sine and complex trajectory without friction**and using ${M}_{sym}$. The algorithm proved to be completely useless and the task of trajectory tracking remained unsolved.

#### 4.3.3. Comparison with Original Controller

#### 4.3.4. Discussion of Results

- It provides information on the asymmetric movement of the hovercraft (with a displacement in the center of mass caused, for example, by an additional load);
- It makes it possible by simulating the effect of a set trajectory on the vehicle motion, velocity deformation and kinetic energy consumption for the vehicle and for the individual quasi-velocity corresponding to motion in each direction;
- A small effect of the inaccuracy of the model parameters (with a 50% increase in their values) on the tracking accuracy was noticed, except for the kinetic energy values;
- If possible for the realization of the set task of trajectory tracking, it is better to choose trajectories described by less complicated mathematical functions.

**Benefits of IQV controller compared to TSMC algorithm**.

- The TSMC method can give satisfactory results but for a diagonal inertia matrix. For a symmetric matrix, the problem of tracking the desired trajectory, as shown by the tests performed, was not solved at all. This means that if the model errors are significant, the algorithm will be ineffective.
- For the proposed IQV algorithm, no such significant numerical problems arise as in the TSMC method.
- The applicability of the proposed IQV algorithm is greater than that of the TSMC scheme, because the former is suitable for both a model with a diagonal inertia matrix and a symmetric matrix, and it also takes into account functions representing friction.
- The selection of the control parameters in the case of the offered algorithm is intuitive, while the controller parameters in the TSMC method must be selected with a lot of time unless another selection method is added. Even then, when a set of parameters is selected correctly, a small change in them can lead to a loss of performance of the algorithm. Such an inconvenience is not present in the control scheme proposed in this paper.

**Benefits of IQV controller compared to original algorithm**.

- An IQV controller is suitable for asymmetric vehicles in the longitudinal and lateral directions, so the geometric center can be different from the center of mass.
- During the realization of the tracking task of the desired trajectory, information on changes in the behavior of the vehicle due to the displacement of the center of mass is obtained simultaneously. Thus, it is a combination of a control algorithm and a scheme for analyzing vehicle dynamics.
- It is possible to estimate the effect of dynamic parameters on changes in vehicle velocity and kinetic energy reduced by each quasi-velocity (taking into account couplings). In the classical algorithm, the effect of couplings is ignored.

**Advantages, drawbacks and limitations of the IQV controller**

- Because only two controller parameters and three design parameters are changed, it may therefore be difficult to achieve very accurate tracking (however, this may be because according to the mathematical proof only uniform stability is guaranteed).
- The control scheme is suitable for studying hovercraft with rudders, while so far there is no way to directly extrapolate to marine vehicle models with rudders.

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Symbol | Explanation |
---|---|

DOF | degree of freedom |

IQV | inertial quasi-velocities |

LFC | Lyapunov function candidate |

$R\left(\varphi \right)$ | transformation matrix between the body frame and the inertial frame |

M | vehicle inertia matrix |

$C\left(\nu \right)$ | Coriolis and centripetal forces vector |

$D\left(\nu \right)$ | drag vector |

$\tau $ | vector of control inputs (the total force and torque) |

${b}_{T}$ | input scaling coefficient |

a | length of the arm from the measured center of mass to the surface of the rudder |

N | diagonal vehicle inertia matrix in terms of IQV |

$\mathsf{{\rm Y}}$ | velocity transformation matrix |

f | vector of total forces |

${\tau}_{r}$ | total torque |

$\zeta $ | vector of IQV including the vehicle velocities and the inertial parameters |

${z}_{1}$ | tracking error in the inertial frame |

${z}_{2}={\dot{z}}_{1}$ | time derivative of the tracking error in the inertial frame |

${\dot{z}}_{2}={\ddot{z}}_{1}$ | second time derivative of the tracking error in the inertial frame |

$\delta $ | arbitrary vector for the point to be controlled in the body frame |

B | control signal transformation matrix |

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**Figure 2.**Simulation results for sine trajectory: (

**a**) desired (${p}_{d}$) and realized (p) trajectory; (

**b**) position errors $\Delta {p}_{x},\Delta {p}_{y}$ (in inertial frame); (

**c**) input signals components (applied force) $Tcos\psi ,Tsin\psi $; (

**d**) actual velocities $u,v,r$; (

**e**) changes in quasi-velocity ($\Delta {\zeta}_{1},\Delta {\zeta}_{2}$); (

**f**) changes in kinetic energy (${K}_{1},{K}_{2},{K}_{3},{K}_{T}$) during motion.

**Figure 3.**Simulation results for complex trajectory: (

**a**) desired (${p}_{d}$) and realized (p) trajectory; (

**b**) position errors $\Delta {p}_{x},\Delta {p}_{y}$ (in inertial frame); (

**c**) input signals components (applied force) $Tcos\psi ,Tsin\psi $; (

**d**) actual velocities $u,v,r$; (

**e**) changes in quasi-velocity ($\Delta {\zeta}_{1},\Delta {\zeta}_{2}$); (

**f**) changes in kinetic energy (${K}_{1},{K}_{2},{K}_{3},{K}_{T}$) during motion.

**Figure 4.**Changes in kinetic energy (${K}_{1},{K}_{2},{K}_{3},{K}_{T}$) during motion—results with 50% parameter increase in parameter values: (

**a**) sine trajectory; (

**b**) complex trajectory.

**Figure 5.**Simulation results for TSMC and sine trajectory (without friction): (

**a**) desired (${p}_{d}$) and realized (p) trajectory; (

**b**) position errors $\Delta {p}_{x},\Delta {p}_{y}$ (in inertial frame); (

**c**) input signals components (applied force and torque) ${\tau}_{u},{\tau}_{r}$; (

**d**) actual velocities $u,v,r$.

**Figure 6.**Simulation results for TSMC and sine trajectory (with friction): (

**a**) desired (${p}_{d}$) and realized (p) trajectory; (

**b**) position errors $\Delta {p}_{x},\Delta {p}_{y}$ (in inertial frame); (

**c**) input signals components (applied force and torque) ${\tau}_{u},{\tau}_{r}$; (

**d**) actual velocities $u,v,r$.

**Figure 7.**Simulation results for TSMC and complex trajectory (without friction): (

**a**) desired (${p}_{d}$) and realized (p) trajectory; (

**b**) position errors $\Delta {p}_{x},\Delta {p}_{y}$ (in inertial frame); (

**c**) input signals components (applied force and torque) ${\tau}_{u},{\tau}_{r}$; (

**d**) actual velocities $u,v,r$.

Symbol | Value | Unit |
---|---|---|

m | 12 | kg |

J | 1.65 | kgm${}^{2}$ |

${d}_{u0}$ | 1.0 | kg/s |

${d}_{u}$ | 4.5 | kg/s |

${d}_{v0}$ | 1.0 | kg/s |

${d}_{v}$ | 4.5 | kg/s |

${d}_{r0}$ | 0.5 | kg/s |

${d}_{r}$ | 0.8 | kg/s |

${d}_{ur}$ | 1.0 | kg/s |

${d}_{vr}$ | 0.5 | kg/s |

Sine | Complex | ||
---|---|---|---|

Index | Trajectory | Trajectory | |

MIA | $\Delta {p}_{x}$ | 0.0045 | 0.0072 |

$\Delta {p}_{y}$ | 0.0398 | 0.0371 | |

$\Delta {\zeta}_{1}$ | 0.0244 | 0.0327 | |

$\Delta {\zeta}_{2}$ | 0.0244 | 0.0327 | |

MIAC | $Tcos\psi $ | 5.1426 | 5.9542 |

$Tsin\psi $ | 2.4981 | 2.9009 | |

RMS | $\left|\right|e\left|\right|$ | 0.1365 | 0.1307 |

KE | ${K}_{m}$ | 0.7055 | 1.1793 |

Sine Trajectory | Sine Trajectory | Complex Trajectory | ||
---|---|---|---|---|

Index | without Friction | with Friction | without Friction | |

MIA | $\Delta {p}_{x}$ | 0.0713 | 0.1209 | 0.0260 |

$\Delta {p}_{y}$ | 0.0955 | 0.1071 | 0.0487 | |

MIAC | ${\tau}_{u}$ | 1.5790 | 2.4533 | 2.0831 |

${\tau}_{r}$ | 0.3539 | 0.4137 | 0.3547 | |

RMS | $\left|\right|e\left|\right|$ | 0.2567 | 0.2657 | 0.1593 |

KE | ${K}_{m}$ | 0.6293 | 0.5866 | 0.9914 |

Sine | Complex | ||
---|---|---|---|

Index | Trajectory | Trajectory | |

MIA | $\Delta {p}_{x}$ | 0.0043 | 0.0047 |

$\Delta {p}_{y}$ | 0.0375 | 0.0391 | |

MIAC | $Tcos\psi $ | 4.7606 | 5.5258 |

$Tsin\psi $ | 1.8247 | 2.7876 | |

RMS | $\left|\right|e\left|\right|$ | 0.1392 | 0.1316 |

KE | ${K}_{m}$ | 0.6813 | 1.0801 |

Sine Trajectory | Sine Trajectory | Complex Trajectory | ||
---|---|---|---|---|

Index | without Friction | with Friction | without Friction | |

MIA | $\Delta {p}_{x}$ | 93.7% | 96.3% | 72.3% |

$\Delta {p}_{y}$ | 58.3% | 62.8% | 23.8% | |

RMS | $\left|\right|e\left|\right|$ | 46.8% | 48.6% | 18.0% |

**Table 6.**IQV tracking performance vs. original algorithm [22].

Sine Trajectory | Complex Trajectory | ||
---|---|---|---|

Index | with Friction | with Friction | |

MIA | $\Delta {p}_{x}$ | −4.7% | −53.2% |

$\Delta {p}_{y}$ | −6.1% | 5.1% | |

RMS | $\left|\right|e\left|\right|$ | 1.9% | 0.7% |

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**MDPI and ACS Style**

Herman, P.
Controller for an Asymmetric Underactuated Hovercraft in Terms of Quasi-Velocities. *Appl. Sci.* **2023**, *13*, 4965.
https://doi.org/10.3390/app13084965

**AMA Style**

Herman P.
Controller for an Asymmetric Underactuated Hovercraft in Terms of Quasi-Velocities. *Applied Sciences*. 2023; 13(8):4965.
https://doi.org/10.3390/app13084965

**Chicago/Turabian Style**

Herman, Przemyslaw.
2023. "Controller for an Asymmetric Underactuated Hovercraft in Terms of Quasi-Velocities" *Applied Sciences* 13, no. 8: 4965.
https://doi.org/10.3390/app13084965