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ScienceDirect Procedia IUTAM 13 (2015) 108 – 117

IUTAM Symposium on "Dynamical Analysis of Multibody Systems with Design Uncertainties"

Dynamical Analysis of Autonomous Underwater Glider Formation with Environmental Uncertainties DongyangXuea,b, ZhiliangWua,b,*, ShuxinWanga,b a

Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin 300072, China b School of Mechanical Engineering, Tianjin University, Tianjin, 300072, China

Abstract Application of multiple autonomous underwater gliders (AUGs) is a promising method for large scale, long-term ocean survey. Dynamical behaviors of AUGs are inevitably affected by uncertainties in the marine environment. This paper introduces a statistical method for uncertainty analysis in formation control of a fleet of AUGs. The AUG formation is modeled as a multibody system. Artificial potential fields are constructed between the AUGs and the goal, between the AUGs and the obstacle, and between neighboring AUGs in the formation for motion planning and coordination. Kane’s method is used to describe the dynamics of the formation. Currents are addressed as the environmental uncertainties and criteria are provided for uncertainty analysis. A series of simulations is carried out for quantitative analysis on influence of uncertain factors. Results show that environmental uncertainties may greatly influence the dynamics of AUGs and should be included in the design and control of AUG formations. © 2015 The Authors. Published by Elsevier B.V. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of organizing committee of Institute of Engineering and Computational Mechanics (http://creativecommons.org/licenses/by-nc-nd/4.0/). University Stuttgart. Peer-review of under responsibility of organizing committee of Institute of Engineering and Computational Mechanics University of Stuttgart. Keywords:underwater vehicles; formation; uncertainty; motion planning; multibody system

1. Introduction Persistent observation and feature tracking is a significant support for scientific research, exploration, development and applications in the marine environment. Autonomous mobile underwater vehicles such as autonomous underwater vehicles (AUVs)1,2 and autonomous underwater gliders (AUGs)3,4,5,6, due to their high

*

Corresponding author. Tel.: +86-22-8740 2173; fax: +86-22-8740 2173. Email address: [email protected]

2210-9838 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of organizing committee of Institute of Engineering and Computational Mechanics University of Stuttgart. doi:10.1016/j.piutam.2015.01.007

Dongyang Xue et al. / Procedia IUTAM 13 (2015) 108 – 117

maneuverability and controllability, have advantages over conventional stationary and passive equipments in ocean observation. Coordination and cooperation of multiple mobile underwater vehicles7,8 provide an opportunity to perform tasks much more difficult than one single vehicle can do. Ocean observing and survey can be completed in a more efficient and robust way by an underwater vehicle formation. As applications of multiple coordinated AUGsextended to accurate observation in challenging ocean regions, ideal models which ignore influence of uncertain factors have become impractical in formation design and control for a fleet of AUGs operated in dynamic and constrained environment. Uncertainties that may affect dynamical behavior of AUGs or measurement accuracy include environmental uncertainties, vehicle design uncertainties, and sensor measurement uncertainties. Unexpected uncertainties in the marine environment such as currents may cause a significant deviation from the designed trajectory or a severe distortion in the desired formation structure that may eventually result in loss of local communication within the vehicle network. Manufacturing and assembly uncertainties may affect AUGs in hydrodynamics and maneuverability. Sensor measurement uncertainties directly influence the range and accuracy of marine monitoring. Researchers have devoted their effort to taking uncertain environmental factors into investigation on single AUV and coordination of underwater vehicles. Woolsey (2011)9, Thomasson and Woolsey (2013)10, Fan and Woolsey (2014)11 developed nonlinear dynamic models for underwater vehicles in an unsteady, nonuniform flow. Cui12 defined a system error function and designed an adaptive sliding variable structure control law for multiple AUVs with parameter uncertainty and environmental disturbances. Coordinated and path-following method of a group of vehicles was provided by Ghabcheloo et al. in the presence of communication losses and time delays13. Yang et al.14 developed a robust controller for AUVs with bounded time delays to make sure the AUVs form and keep a desired formation shape and track a desired trajectory. In most of the current effort, environmental uncertainty was either modeled as constant or described by a determined function. This paper investigates the effects of environmental uncertainties on the coordinated operation of a fleet of AUGs. The coordinated underwater gliders are supposed to automatically find an optimal obstacle-free path in a preset formation pattern. The AUG formation is modeled as a multibody system15. Artificial potential field approach is used to describe the interaction between neighboring bodies. Kane’s method is used for dynamic analysis such that the formation configuration can be described by the minimal set of coordinates and complexity of motion planning can be reduced. The statistically random environmental uncertainty that cannot be described by determined functions is described by probabilistic parameters. Criteria are provided to evaluate the effects of the uncertain factors. The uncertain analysis could be used in formation pattern design and adaptive compensation during formation control. The rest of the paper is organized as follows: Section 2 presents multibody modeling of the AUG formation. In Section 3, the main uncertain factors influencing motion and observing abilities of the AUG formation are analyzed. Assessing criteria are also proposed to evaluate the effects of the uncertainties. Simulation results are presented in Section 4. Section 5 summarizes the main contributions and addresses further work on uncertainty analysis in AUG formation control. 2. Multibody Modeling of AUG Formation The AUGs coordinated for ocean survey are usually required to follow an optimal path while maintaining the prescribed formation pattern, as shown in Fig. (1). The multiple AUGs are regarded as a virtual multibody system and for simplicity, the individual agent in the fleet is treated as a particle and is virtually connected with other agents in the multibody system. Interactions between individual vehicles are established to control the formation pattern using artificial potential field approach. When the neighboring vehicles are apart from each other in a distance larger than the desired distance, the interactive potential field will act as an attractive potential field and pull the two vehicles toward each other. Otherwise, when the distance between two vehicles are shorter than expected, they will be pushed away by the interactive potential field to maintain the preset formation pattern. The individual vehicle is also virtually attracted by the goal and repelled by the obstacle in the ocean.

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Fig.1Formation control structure with APFs

Fig.2Virtual multibody system

2.1. Kinemics of the multibody system The schematic of the simplified multibody system with N agents is shown in Fig.(2). The agents are regarded as particles with full actuation. Bk in the figure represents the kth agent. The three-dimensional system configurations are denoted by the Cartesian coordinates. It is assumed that all the bodies are fixed in the Cartesian reference frame which is denoted with the unit vector [N1, N2, N3]. In the multibody system, each agent has three degrees of freedom. Thus the system with N bodies has 3N degrees of freedom. The position coordinates of the agents can be chosen as the generalized coordinate, which is given by

ql

[ x11 , x12 , x13 ,..., xk 1 , xk 2 , xk 3 ...]

(1)

where xkn (k=1..N,n=1,2,3) represents the position coordinates of the kth body with respect to the inertial frame and n denotes the three axes of reference frame.The generalized speed can be expressed by

ql

[ x11 , x12 , x13 ,..., xk 1 , xk 2 , xk 3 ...]

(2)

where x kn is the time derivative of xkn. The position, velocity, and acceleration of Bk can therefore be expressed by 3

rk

¦x

kn

Nn

n 1

(3)

3

vk

¦ x n 1

kn

Nn

(4)

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Dongyang Xue et al. / Procedia IUTAM 13 (2015) 108 – 117 3

ak

¦ x

kn

Nn

(5)

n 1

The partial velocity array can be obtained by 3

vk l m

wv k wql

w ¦ vkn N n n 1

wql

(6)

where l=1…3N represents the degree of freedom, n,m=1,2,3 represent the three axes of the coordinate system, and vkn xkn is the velocity component of the agents. 2.2. Dynamics of the multibody system Three type of artificial potential fields are applied in coordination of the AUG formation, as shown in Fig.(1), where the green dash-dot line denotes the attractive force exerted by the goal, the black dashed line denotes the repulsive force by the obstacle, and the red solid line denotes the interactive force between agents in the formation. The dissipative force is also contained in the control architecture to ensure convergence from the initial condition. Equation (7)~(9) describe the attractive potential field between the goal and the individual agents in the formation, the repulsive potential field between the obstacle and the agents, and the interactive potential field between agents in the formation, respectively.

U

k att

k U rep

UI

0 ° ®1 2 ° ka Rgk ¯2

0 Rgk d d goal Rgk ! d goal

1 1 2 1 ) ° kr ( R d 2 ® ok obs °0 ¯ 1 2 2 °° k I ( 2 rij d 0 ln( rij )) ® ° k ( 1 d 2 d 2 ln( d )) 0 1 °¯ I 2 1

(7)

0 Rok d d obs Rok ! d obs

(8)

0 rij d1 rij t d1

where i, j,k =1,2,…,N (N is the number of agents in the formation); ka -- the scalar attractive control gain; dgoal -- the equivalent radius of the attractive area; Rgk -- the distance between the kth agent and the goal; kr -- the scalar repulsive control gain; dobs -- the distance of influence by the obstacles; Rok -- the distance between the kth agent and the effective obstacle; rij -- the distance between the neighboring ith and jth agents; kI -- the scaling interactive control gain; d0 -- a constant denoting the critical point between attraction and repulsion; d1 -- the limited distance of interaction.

(9)

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The Kane's equation is used to study the dynamic characteristics of the multibody system, which is Fl Fl

0, l 1, 2, , 3N

(10)

where Fl is the generalized active force and Fl is the generalized inertia force. The generalized force is calculated by the force associated with the generalized coordinate ql (l=1,2,…, 3N) and can be written as: N

Fl

¦F

k

k 1

wv k wql

N

3

¦¦

Fkm vklm N m

k 1m 1

(11)

where Fk is the resultant force on the kth body, Fkm is the component of the Fk with respect to the inertial frame, vklm is the partial velocity array, and Nm represents the component of the unit coordinate system vector. Fl is contributed by the generalized attractive force, the generalized repulsive force, the generalized interactive force, and the generalized dissipative force as:

Fl

Fal Frl FIl Fdisl , l 1,2,,3N

(12)

3. Uncertainty analysis

In this section, a statistical method is presented to analyze the influence of random uncertain factors on formation control of the AUGs. The main uncertainties are discussed according to observation conditions and requirements of ocean survey. The assessing criteria are also provided to evaluate effects of the uncertainties on the motion and the ocean observation abilities of the AUGs. Fleets of multiple underwater gliders can be deployed in sensitive and remote regions. Flows and currents sometimes might enormously influence the fleet motion, especially in shallow and coastal areas. Uncertain ocean currents are featured by random speed and direction. The effect of ocean currents on formation design and control of multiple AUGs can be investigated by combining the current velocity with the vehicle velocity11. The current velocity with respect to the inertial frame can be given by:

v current

vc1 N1 vc 2 N 2 vc 3 N 3

(13)

where vcn (n=1, 2, 3) represents the components with respect to the unit vector. The velocity and the acceleration of the kth agent in the multibody system can then be obtained as:

vk

( xk1 vc1 )ȃ1 ( xk 2 vc 2 )N 2 ( xk 3 vc 3 )N 3

(14)

ak

( xk1 vc1 )N1 ( xk 2 vc 2 )N 2 ( xk 3 vc 3 )N 3

(15)

The generalized velocity is expressed as:

ql

[ x11 vc1 , x12 vc 2 , x13 vc 3 ,..., xk1 vc1 , xk 2 vc 2 , xk 3 vc 3 ]

(16)

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Fig.3Schematic of trajectory error

Fig.4Schematic of shape error

The effects of currents on the fleet motion can be separated into two aspects. One is that the trajectory of the AUG formation would deviate from the planned path, resulting in a trajectory error (TE). The other one is that the formation shape might be altered, causing a formation shape error (SE). The two errors are defined as follows. x Trajectory error (TE): the difference in the trajectory of the centroid of the AUG formation, as shown in Fig.(3). The centroid can be obtained based on the geometry shape of the formation. The trajectory error is used to identify influence of the ocean currents on the overall motion of the AUG formation16. x Shape error (SE): the difference between the perimeters of the actual formation shape and the desired geometry. As shown in Fig.(4), formation shape error can be used to estimate the distortion of the AUG formation. 4. Numerical Simulation

In this section, the impact of ocean currents on the coordinated control of an AUG formation is investigated using numerical simulation. The formation consists of three AUGs. The AUGs are designed to move from an initial position to the goal in an equilateral formation and avoid the obstacle in the environment. In the simulation, it is assumed that the AUG formation moves at a depth of less than 250 meters under the ocean surface in the presence of random currents. Ocean currents can usually be characterized by current velocity and direction, varying greatly in space and time. Due to the various existing patterns and complicated distributions, it is almost impossible to use an accurate model or generate a precise probability distribution to describe the ocean currents in the global scope. But for a specific ocean area, ocean currents can be modeled by statistically random variables, representing current velocity and direction. By analyzing the speed and occurring probability of the currents in the South China Sea, two assumptions are made in the simulation: (1) Currents are simplified as single direction currents with constant velocities parallel to the horizontal plane, as shown in Fig.(5). Directions of the currents in the horizontal plane are assumed to be evenly distributed between 0Û to 360Û with a step of 45Û, as shown in Fig.(6); (2) Current velocity is assumed to obey a Gaussian distribution with a mean of 0.5 m/s and a standard deviation of 0.19 m/s, as shown in Fig. (7).

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Fig.5Singledriction currents

Fig.6Current directions

The initial locations of the three agents in the formation are B1 (200,200,200), B2 (200,182.32, 182.32), and B3 (200,206,175.5). The initial velocities are 2 m/s, 1.5 m/s and 1 m/s respectively. The effective radius of the goal area is defined as 50 meters. The central point of the sphere of influence is located at the coordinate origin. The obstacle is a sphere with a radius of 20 meters located at (150,140,140) with an influence distance of 70 meters. Each agent is designed to maintain a desired distance of 20 meters to the other agents. The simulation results are show in Fig.(8~12). Figure (8) illustrates the centroid trajectories of the AUG formation with and without currents. The black line represents the ideal centroid trajectory and the other colored lines show the motion of the AUG formation when currents with different velocities come from different directions. It can be observed in the figure that although the ocean currents more or less push the AUG formation away from the desired trajectory, the formation would finally reach the goal under the influence of the artificial potential fields. It is also indicated that as the agents approach the goal area or get far away from the obstacle, the impact of currents become predominant. The current whose direction is close to the ideal moving direction has less influence on formation coordination, and vice versa. The simulation result reveals that the artificial potential fields must be adjusted when environmental uncertainties are taken into account in the formation control. 0.03 0.025

PD

0.02 0.015 0.01 0.005 0 -0.4

-0.2

0

0.2 0.4 0.6 0.8 current speed /m/s

1

1.2

Fig.7Probability density distribution of current velocity

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0 - degree current 45 - degree current 90 - degree current 135 - degree current 180 - degree current 225 - degree current 270 - degree current 315 - degree current no current

200 150 z/m

100 50 0 200 100 0 y/m

-100 -100

100

0

200

x/m

Fig.8Centroid trajecties (CT) of AUG formation

The formation trajectory error (TE) and shape error (SE) defined in Section 3 directly indicate the influence of ocean currents and are used as the evaluation criteria. A large TE or SE represents a big impact on the AUG formation. For any randomly generated current, the averages of TE and SE at each time step are calculated as the overall TE and SE of the formation. A set of 100 current samples are used for the error estimation. The probability density distributions of TE and SE are shown in Fig. (9) and Fig. (10) respectively. As shown in these figures, the influence of uncertain currents is much bigger on the formation trajectory than on the formation shape. Formation TE ranges from about 4 meters to 90 meters while SE ranges from 0.1 meters to 0.9 meters. It is shown in Fig. (10) that SE also follows a Gaussian distribution, indicating that currents affect little on formation shape. However, TE varies nonlinearly with current velocity. The fitted curve in Fig. (11) shows an exponential relationship between TE and current velocity. The relationship can be given by:

H TE =67.1e0.1947v 63.65e5.19v c

c

(17) 0.04

PD

0.03 0.02 0.01 0 0

20

40

TE /m

60

80

100

Fig.9Probability density distribution of TE 0.03 0.025

PD

0.02 0.015 0.01 0.005 0 -0.9

-0.8

-0.7

-0.6

-0.5 -0.4 SE /m

-0.3

-0.2

Fig.10Probability density distribution of SE

-0.1

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Dongyang Xue et al. / Procedia IUTAM 13 (2015) 108 – 117 80 Original data Fitted curve

70

TE /m

60 50 40 30 20

0.1

0.2

0.3

0.4 0.5 0.6 current velocity /m/s

0.7

0.8

0.9

Fig.11Fitted curve of TE

When ocean currents are strong, the AUG formation may not be able to move toward the goal or even be pushed backward by the currents. Such extreme situation occurs when the currents have a velocity larger than or equivalent to the vehicle’s velocity. Figure (12) and Fig. (13) illustrate the trajectory of the AUG formation when the current comes with a velocity of1.5m/sand a direction of 0ewith respect to the x axis. The other initial conditions are the same as previously mentioned. The trajectories of the three agents are shown in Fig. (12). Comparison between the actual centroid trajectory and the ideal trajectory under no currents is given in Fig. (13). It is apparent that the current has a great impact on the motion of the AUG formation. The agents do not move to the expected goal. Such result is unacceptable in practice, indicating that current influence should be considered in AUG design and formation design. Development of hybrid-driven underwater glider with a propeller could provide a larger velocity to overcome strong currents.

Fig.12Three agent trajectoriesunder 1.5m/s current

Fig.13Centroid trajectorycomparison

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5. Conclusion

In this paper a statistical method has been used to analyze the influence of environmental uncertainties on the formation motion of AUGs. Multibody dynamic modelling has been adopted to analyze the dynamical behaviors of the AUG formation. Artificial potential field method is used for motion planning and coordination. Currents are discussed as the environmental uncertain factor. Simulations have been conducted and formation trajectory error and shape error have been proposed to evaluate current influence. Results show that influence of environmental uncertainties may be significant and uncertainty analysis should be included in the design of underwater glider formations. Future work will be focus on uncertainty analysis under more complicated situations. Development of adaptive compensation methods will also be conducted for improvement on formation motion planning. Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant No. 5125277, 50835006, 51005161, 51105268), National Key Technology R&D Program (Grant No. 2011BAC12B01-02), and Tianjin University Elite Scholar Program. References 1. Yuh J. Design and control of autonomous underwater robots: A survey. Auton Robot 2000;8:7-24. 2. Wernli RL. AUVs - A technology whose time has come. In: Proceedings of the 2002 international symposium on underwater technology. Tokyo, Japan,2012; 3. Eriksen CC, Osse TJ, Light RD, Wen TW, Sabin PL, Ballard JW, Chiodi AM. Seaglider: a long-range autonomous underwater vehicle for oceanographic research. IEEE J OceanicEng2001;26:424-436. 4. Sherman J, Davis RE, Owens WB, Valdes J. The autonomous underwater glider Spray. IEEE J OceanicEng2001;26:437-446. 5. Webb DC, Simonetti PJ, Jones CP. SLOCUM: an underwater glider propelled by environmental energy. IEEE J OceanicEng2001;26:447-452. 6. Wang SX, Sun XJ, Wang YH, Wu JG, Wang XM. Dynamic Modeling and Motion Simulation for a Winged Hybrid-Driven Underwater Glider. China Ocean Eng2011;25:97-112. 7. Fiorelli E, Leonard NE, Bhatta P, Paley DA, Bachmayer R, Fratantoni DM. Multi-AUV control and adaptive sampling in Monterey Bay. IEEE J OceanicEng 2006;31:935-948. 8. Leonard NE, Paley D, Lekien F, Sepulchre R, Fratantoni D, Davis R. Collective motion, sensor networks, and ocean sampling. P IEEE2007;95:48-74. 9. Woolsey CA. Vehicle Dynamics in Currents. Technical Report VaCAS-2011-01, Virginia Center for Autonomous Systems, VirginiaTech, Blacksburg, VA. Available from: URL: http://www.unmanned.vt.edu/discovery/reports.html. 10. Thomasson P, Woolsey C. Vehicle motion in currents. OceanEng2013;38:226-242. 11. Fan SS, Woolsey CA. Dynamics of underwater gliders in currents. OceanEng 2014;84:249-258. 12. Cui RX, Xu DM, Yan WS. Formation control of Autonomous Underwater Vehicles under fixed topology. In: IEEE International Conference on Control and Automation. Guangzhou, China,2007. 13. Ghabcheloo R, Aguiar AP, Pascoal A. Coordinated path-following in the presence of communication losses and time delays. SIAM J Control Optim2009;48:234-265. 14. Yang HZ, Wang CF, Zhang FM. Robust Geometric Formation Control of Multiple Autonomous Underwater Vehicles with Time Delays. In: American Control Conference (ACC).Washington DC 2013. 15. Yang Y, Wang SX, Wu ZL, Wang YH. Motion planning for multi-HUG formation in an environment with obstacles. OceanEng 2011;38: 2262-2269. 16. Chen BZ, Dario P. Team formation and steering algorithms for underwater gliders using acoustic communications. ComputCommun 2012;35:1017-1028.

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