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Hydrodynamic parameters play a major role in the dynamics and control of Autonomous Underwater Vehicles (AUVs). The performance of an AUV is dependent on the parameter variations and a proper understanding of these parametric influences is essential for the design, modeling, and control of high-performance AUVs. In this paper, the sensitivity of hydrodynamic parameters on the control of a flatfish type AUV is analyzed using robust design techniques such as Taguchi's design method and statistical analysis tools such as Pareto-ANOVA. Since the pitch angle of an AUV is one of the crucial variables in the control applications, the sensitivity analysis of pitch angle variation is studied here. Eight prominent hydrodynamic coefficients are considered in the analysis. The results show that there are two critical hydrodynamic parameters, that is, hydrodynamic force and hydrodynamic pitching moment in the heave direction that influence the performance of a flatfish type AUV. A near-optimal combination of the parameters was identified and the simulation results have shown the effectiveness of the method in reducing the pitch error. These findings are significant for the design modifications as well as controller design of AUVs.

Autonomous Underwater Vehicles provide new alternatives for undersea exploration, relieving human divers from the risks and fatigue of working under constrained underwater environments. These are intelligent robots deployed to carry out predefined underwater tasks without human intervention. Since they are autonomous in nature, dynamics, control, and navigation of these robots are of major concern to the ocean engineering researchers. Developing a highly stable and maneuverable AUV with good payload capability is of major focus in the underwater robotics research today. The dynamic performance and autonomous control of an AUV depend on many factors such as robot shape, weight, buoyancy, propulsion, sensor systems, and control systems. The geometrical shape of the robot determines the hydrodynamic and added mass coefficients and has a very large influence on the robot dynamics [

There are two major types of AUVs presently in use. The most common type is the torpedo-shaped AUV with a cylindrical hull [

Body-fixed frame and earth-fixed reference frame for AUV.

The essential control requirements for any AUV are maneuvering in dive plane, depth control, and station keeping. While all these are critical requirements for an AUV, this paper focuses on the maneuvering in dive plane, especially the pitch angle variation during the forward motion of AUV. The design parameters that influence control requirements are the vehicle buoyancy, vertical distance between centre of gravity (CG) to centre of buoyancy (CB), and the hydrodynamic parameters. Due to safety reasons, most of the AUVs are designed to have some minimum positive buoyancy. Similarly, the vertical distance between CG and CB is always kept constant in order to ensure stability of the vehicle. Therefore, for a given configuration of AUV, the hydrodynamic parameter variation is the most critical and hence the sensitivity of these parameters in the pitch motion of an AUV is analyzed using Taguchi’s design principles in this paper. The paper is organized in the following manner. A short introduction to the modeling and simulation of AUV with open-loop and closed-loop controls is presented in Section

Underwater robots experience a range of forces while moving in the fluid medium and they are generally referred to as hydrodynamic forces. A detailed discussion on hydrodynamic forces on underwater robots can be found in [

Surge Motion:

In order to identify the sensitivity of these parameters, an experimental AUV that is being developed at the Indian Institute of Technology, Madras, India, is considered a test case. The dynamic model of the robot is developed using the Newton-Euler formulation [

AUV positions and orientations for constant forward speed (at

A parametric model of the AUV for the pitch motion is developed to study the pitch angle variation and the parameter sensitivity.

With reference to Figure

Engineering design is increasingly becoming model based, in that its complexity calls for a mathematical model involving multiple quantities, some of which are to be decided by the designer with the purpose of meeting performance specifications, for example, the thrust that an underwater propeller must deliver at a given rpm, under given environment conditions such as ambient temperature, pressure, and so forth. The aim of robust design is to develop products whose performance remains within specifications in the presence of large variations in environment conditions.

It is a procedure to determine the sensitivity of the outcomes of an alternative to changes in its parameters (as opposed to changes in the environment). If a small change in a parameter results in relatively large change in the outcomes, the outcomes are said to be sensitive to that parameter. This may mean that the parameter has to be determined very accurately or the alternative is to redesign such that the sensitivity of the parameter is low [

the model resemblance with the system under study,

the quality of model definition,

factors that mostly contribute to the output variability,

the region in the space of input factors for which the model variation is maximum,

interactions between factors.

Parameter sensitivity is usually performed as a series of tests in which the modeler sets different parameter values to see how a change in the parameter causes a change in the dynamic behavior of the system. By showing how the model behavior responds to change in parameter values, sensitivity analysis is a useful tool in model building as well as in model evaluation [

For sensitivity analysis, Taguchi’s robust design is more appropriate because it is a design and data analysis method and is more engineering oriented than science oriented. The distinct idea of Taguchi’s robust design that differs from the conventional experimental design is that of designing for the simultaneous modeling of both mean and variability [

The concept of robust design has many aspects such as

finding a set of conditions for design variables which are robust to noise,

achieving the smallest variation in a product’s function relative to a desired target value,

minimizing the number of experiments using orthogonal arrays and testing for confirmation.

Robust Design is a method, also called the Taguchi Method, pioneered by Byrne and Taguchi [

In setting up a framework for robust design, the classification of the quantities is at play in the design task as given below.

Design variables (DVs) are those quantities to be decided by the designer with the purpose of meeting performance specifications under given conditions.

Design-environment parameters (DEPs) are the quantities over which the designer has no control and define the conditions of the environment under which the designed object will operate.

Performance functions (PFs) are quantities used to represent the performance of the design in terms of design variables and design-environment parameters.

The responses at each setting of parameters are treated as a measure that would be indicative of not only the mean of some quality characteristic but also the variance of the same characteristic. The mean and the variance are combined into a single performance measure known as the signal-to-noise (S/N) ratio [

Taguchi’s method uses an orthogonal array (OA) and analysis of mean to study the effects of parameters based on statistical analysis of experiments. An OA is a fractional factorial matrix which assures a balanced comparison of levels of any factor or interaction of factors. It is a matrix of numbers arranged in rows and columns where each row represents the level of the factors in each run, and each column represents a specific factor that can be changed from each run. The array is called orthogonal because all columns can be evaluated independently of one another. To compare performances of parameters, the statistical test known as the analysis of variance (ANOVA) is used. Further details and technical merits about robust parameter design can be found in the references of [^{3}), ^{7}), ^{4}), ^{6}), and ^{8}) [

Representation of standard orthogonal array.

Once the suitable orthogonal array for experimentation is selected, then the experiments are conducted for the identified conditions and the results are analyzed using statistical methods, such as Pareto-ANOVA, ANOVA, or response curves to identify the optimal parameters for performance enhancement [

As explained in Sections

The first step in the analysis is to choose the orthogonal array for trials. For selecting the orthogonal array, the main criterion is the required number of trials [

Mathematically, this is given as

It was decided to conduct trials using the eight variables at seven levels (more numbers of levels means, more number of experiments, and better accuracy of results), that is, 10%, 25%, 40%, 55%, 70%, 85%, and 100% of the actual values of the variables. So, the number of trials required was found to be 49. The smallest standard orthogonal array that matches with this requirement is ^{8}), which will give a test matrix of 49 different configurations with various combinations of parameters and its physical values (eight parameters at seven levels). Three levels of noise variable (vehicle speed) are considered in the analysis and the test matrix used is shown in Table

Design of Experiments for Pitch Response (^{8}) OA).

Exp. no. | Pitch angle (deg) for | S/N ratio (dB) | ||||||||||

(%) | (%) | (%) | (%) | (%) | (%) | (%) | (%) | 1.13 (m/s) | 1.59 (m/s) | 1.98 (m/s) | ||

1 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 30.03 | |||

2 | 10 | 25 | 25 | 40 | 55 | 70 | 85 | 100 | 19.30 | |||

3 | 10 | 40 | 40 | 55 | 70 | 85 | 100 | 25 | 19.57 | |||

4 | 10 | 55 | 55 | 70 | 85 | 100 | 25 | 40 | 31.99 | |||

5 | 10 | 70 | 70 | 85 | 100 | 25 | 40 | 55 | 29.28 | |||

6 | 10 | 85 | 85 | 100 | 25 | 40 | 55 | 70 | 18.57 | |||

7 | 10 | 100 | 100 | 25 | 40 | 55 | 70 | 85 | 18.96 | |||

8 | 25 | 10 | 25 | 25 | 25 | 25 | 25 | 25 | 24.70 | |||

9 | 25 | 25 | 55 | 85 | 40 | 10 | 100 | 70 | 34.86 | |||

10 | 25 | 40 | 85 | 40 | 10 | 100 | 70 | 55 | 14.79 | |||

11 | 25 | 55 | 40 | 10 | 100 | 70 | 55 | 85 | 26.66 | |||

12 | 25 | 70 | 10 | 100 | 70 | 55 | 85 | 40 | 20.73 | |||

13 | 25 | 85 | 100 | 70 | 55 | 85 | 40 | 10 | 25.05 | |||

14 | 25 | 100 | 70 | 55 | 85 | 40 | 10 | 100 | 39.61 | |||

15 | 40 | 10 | 40 | 40 | 40 | 40 | 40 | 40 | 23.15 | |||

16 | 40 | 25 | 85 | 70 | 100 | 55 | 10 | 25 | 40.72 | |||

17 | 40 | 40 | 70 | 100 | 55 | 10 | 25 | 85 | 28.95 | |||

18 | 40 | 55 | 100 | 55 | 10 | 25 | 85 | 70 | 13.98 | |||

19 | 40 | 70 | 55 | 10 | 25 | 85 | 70 | 100 | 16.99 | |||

20 | 40 | 85 | 10 | 25 | 85 | 70 | 100 | 55 | 20.80 | |||

21 | 40 | 100 | 25 | 85 | 70 | 100 | 55 | 10 | 24.08 | |||

22 | 55 | 10 | 55 | 55 | 55 | 55 | 55 | 55 | 22.51 | |||

23 | 55 | 25 | 40 | 100 | 85 | 25 | 70 | 10 | 10.92 | |||

24 | 55 | 40 | 100 | 85 | 25 | 70 | 10 | 40 | 32.62 | |||

25 | 55 | 55 | 85 | 25 | 70 | 10 | 40 | 100 | 26.69 | |||

26 | 55 | 70 | 25 | 70 | 10 | 40 | 100 | 85 | 13.46 | |||

27 | 55 | 85 | 70 | 10 | 40 | 100 | 85 | 25 | 17.68 | |||

28 | 55 | 100 | 10 | 40 | 100 | 85 | 25 | 70 | 33.20 | |||

29 | 70 | 10 | 70 | 70 | 70 | 70 | 70 | 70 | 22.19 | |||

30 | 70 | 25 | 10 | 55 | 25 | 100 | 40 | 85 | 20.91 | |||

31 | 70 | 40 | 55 | 25 | 100 | 40 | 85 | 10 | 10.92 | |||

32 | 70 | 55 | 25 | 100 | 40 | 85 | 10 | 55 | 34.86 | |||

33 | 70 | 70 | 100 | 40 | 85 | 10 | 55 | 25 | 25.45 | |||

34 | 70 | 85 | 40 | 85 | 10 | 55 | 25 | 100 | 21.99 | |||

35 | 70 | 100 | 85 | 10 | 55 | 25 | 100 | 40 | 18.30 | |||

36 | 85 | 10 | 85 | 85 | 85 | 85 | 85 | 85 | 22.01 | |||

37 | 85 | 25 | 100 | 10 | 70 | 40 | 25 | 55 | 30.59 | |||

38 | 85 | 40 | 10 | 70 | 40 | 25 | 55 | 100 | 20.69 | |||

39 | 85 | 55 | 70 | 40 | 25 | 55 | 100 | 10 | 10.92 | |||

40 | 85 | 70 | 40 | 25 | 55 | 100 | 10 | 70 | 36.72 | |||

41 | 85 | 85 | 25 | 55 | 100 | 10 | 70 | 40 | 24.69 | |||

42 | 85 | 100 | 55 | 100 | 10 | 70 | 40 | 25 | 18.31 | |||

43 | 100 | 10 | 100 | 100 | 100 | 100 | 100 | 100 | 21.90 | |||

44 | 100 | 25 | 70 | 25 | 10 | 85 | 55 | 40 | 16.15 | |||

45 | 100 | 40 | 25 | 10 | 85 | 55 | 40 | 70 | 28.07 | |||

46 | 100 | 55 | 10 | 85 | 55 | 40 | 70 | 25 | 20.68 | |||

47 | 100 | 70 | 85 | 55 | 40 | 70 | 25 | 10 | 27.01 | |||

48 | 100 | 85 | 55 | 40 | 70 | 25 | 10 | 85 | 38.27 | |||

49 | 100 | 100 | 40 | 70 | 25 | 10 | 85 | 55 | 16.09 |

The simulation trials were conducted for the states mentioned in the OA. For each experiment, the pitch angle was recorded (refer to Table

The plot of SNR for various parameters is shown in Figure

Response curves.

In order to identify the optimal parameter combination and the contribution ratio of each parameter, statistical analysis of the results was carried out using Pareto-ANOVA analysis.

For Pareto-ANOVA analysis, the sum of S/N ratios, sum of squares of differences, and contribution ratio were calculated as follows:

For example, the calculation of above values for

Pareto-ANOVA analysis.

Parameters | Sum of S/N ratio, | Sum of squares of differences | Contribution ratio in % | ||||||

level 1 | level 2 | level 3 | level 4 | level 5 | level 6 | level 7 | |||

167.70 | 168.68 | 157.08 | 154.62 | 163.92 | 168.18 | 4489.094 | 3.30 | ||

166.50 | 155.62 | 165.76 | 169.65 | 167.06 | 168.55 | 1270.777 | 0.93 | ||

167.05 | 169.15 | 155.09 | 164.79 | 168.10 | 168.55 | 1406.652 | 1.03 | ||

168.32 | 154.93 | 165.08 | 168.29 | 170.19 | 154.24 | 4677.602 | 3.44 | ||

128.71 | 140.81 | 177.19 | 171.52 | 182.14 | 178.84 | 21419.23 | |||

156.14 | 156.99 | 163.89 | 166.88 | 167.84 | 168.07 | 4335.005 | 3.19 | ||

198.43 | 171.46 | 154.11 | 129.22 | 120.70 | 139.82 | 89949.91 | |||

138.93 | 167.11 | 167.65 | 168.93 | 169.23 | 167.16 | 8539.692 | 6.28 |

From Table

In order to show the effectiveness of this analysis in identifying optimum parameters to reduce the pitch angle variations of the AUV, simulations were carried out using the optimal values of the hydrodynamic parameters. The simulation results are summarized in Table

Comparative results.

Condition | Pitch angle (deg) for | ||||||||||

(%) | (%) | (%) | (%) | (%) | (%) | (%) | (%) | 1.13 (m/s) | 1.59 (m/s) | ||

Original values | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | |||

Optimized values | 25 | 25 | 55 | 85 | 100 | 10 | 10 | 70 |

Comparative results of AUV positions and orientations for open loop control.

Comparative results of positions and orientations for closed loop control.

The significance of the analysis is in the redesign of the AUV based on the sensitivity analysis. Once the most sensitive parameters were identified, designer can focus his attention on redesigning the physical features of the AUV to achieve the desired values for these parameters. For example, additional control planes in the aft or increased control plane areas will reduce the hydrodynamic moment (

A systematic study on the sensitivity of various hydrodynamic parameters on the performance of an underwater robot was presented. Using statistical design techniques, critical parameters affecting the diving plane motion of the robot were identified and optimal combination of parameters to reduce the pitch angle error was determined. Simulation results have shown the effectiveness of the analysis in improving the dynamic performance. Further studies may include the robot motion in a three-dimensional plane and the sensitivity of environmental disturbance parameters as noise variables. Experimental validation of above findings will be taken up as soon as the prototype AUV is ready for trials.

The authors wish to thank the editor and the anonymous referees for their valuable comments and suggestions that improved the paper.