multiple vessel cooperative dp operations...operations 5.3 multiple vessel cooperative dp operations...

Author’s Name Name of the Paper Session

DYNAMIC POSITIONING CONFERENCE October 10-11, 2017

OPERATIONS SESSION

Multiple Vessel Cooperative DP Operations

By André S. S. Ianagui, Alex S. Huang, Eduardo A. Tannuri

University of São Paulo, Numerical Offshore Tank, TPN-USP

Ianagui,A.S.S., Huang, A.S., Tannuri, E.A.

Operations 5.3 Multiple Vessel Cooperative DP Operations

MTS DP Conference - Houston October 10-11, 2017

Abstract

As offshore operations grow larger and more complex, the requirements for a higher number of agents –

from vessels to equipment – working simultaneously and cooperatively become evident. In this scenario,

the ability to perform such tasks safely coordinating all these elements will eventually reach human limits,

bounding also the complexity that can be achieved. Enhanced levels of autonomy emerge as a response to

these requirements, giving margin to safer, larger and possibly more cost effective operations. In this

approach, DP vessels can be treated as “drone ships”, which can not only perform stationkeeping but also

trajectory tracking tasks collectively. This work intends to propose a guidance and control method for use

in multiple DP vessels to perform motion control of floating, unactuated loads connected through cables.

The idea consists in usage of a cooperative control structure, in which the vessels share its positions and

efforts through a wireless network. A virtual leader is introduced and moved over the cargo required

positions. Coordinated motions of the DP vessels in a rigid formation ensure the load is taken to its set-

point. Simulation results for an illustrative case are presented and discussed.

Abbreviation / Definition

AHTS. Anchor Handling Tug Supply

ASV. Autonomous Surface Vehicles

AUV. Autonomous Underwater Vehicles

DOF. Degree of Freedom

DP. Dynamic Positioning

DPS. Dynamic Positioning System

Protocol. Cooperative Control Law

UAV. Unmanned Aerial Vehicle




Introduction

In a world heading to increasing levels of automation in the daily life, where Unmanned Aerial Vehicles

(UAV’s) will deliver order packages and autonomous cars are expected to take streets soon, it is probably

unknown to most people that DP Vessels were one of the first and most successful applications of

cybernetics to reach industry. Such success can be credited to the DPS’s own autonomous nature, which

permitted the automatic stationkeeping capacity required for years length offshore operations. As such

marine operations grow larger and more complex, the requirements for a higher number of agents – from

vessels to equipment – working simultaneously and cooperatively become evident. In this scenario, the

ability to perform such tasks safely coordinating all these elements will eventually reach human limits,

bounding also the complexity that can be achieved. Enhanced levels of autonomy emerge as a response to

these requirements, giving margin to safer, larger and possibly more cost effective operations. Current

state-of-the-art technologies in robotics and autonomous systems along with great advances in bandwidth

availability and in reliability of wireless communication systems have permitted the development of large

scale multi-robot and multi-autonomous-vehicle architectures, able to solve cooperatively tasks that

wouldn’t be possible with a single agent. In (Murray, 2006) a list of several devised applications for

multi-vehicle operations is presented, including examples for autonomous cars, UAV’s, satellite arrays

and mixed vehicles. Given such examples, it is easy to think also of sets of marine vehicles such as

Autonomous Underwater Vehicles (AUV’s) and Autonomous Surface Vehicles (ASV’s) that could

perform cooperative tasks and eventually enable large and complex offshore and even restricted waters

operations.

DP Vessels, thought of as actual “drone ships” are interesting candidates for the application of marine

multi-agents’ missions. Their flexibility and already highly automated operational capacity might be

enhanced with the introduction of cooperative systems concepts. This represents a step up in the

automation tasks a DP system may perform. Beyond the already mentioned stationkeeping and the

trajectory tracking/path following capacities, already present in most of current industrial systems,

formation keeping tasks with multiple vessels, automated collision avoidance between agents and

distributed load carrying and holding may be introduced. Figure 1 shows an example of an operation that

could take advantage of such system. The installation of an underwater tunnel module uses four tugboats

to position a floating barge. As the tugboats are human-operated, accuracy in such positioning is limited.

To make sure the module keeps its position during the descent and installation, the hoisting barge must be

anchored. This, of course, limits the application of such technique to shallow waters and very mild

environmental conditions. As shall be seen, the simple usage of DP capable tugboats might render

stability issues. The task might be accomplished with less restrictions if the boats were running a

cooperative DP algorithm. Other applications devised for cooperative control systems applied to DP

vessels are subsea installation operations with joint materials hoisting, rescues, multiple tug boats load

manoeuvring and large cargo transportation. When designed as multiple agent tasks, these operations can

be advantageous by reducing overall costs, enhancing operational safety and reliability due to redundancy

and by enabling large configuration flexibility.




Figure 1 - Marmaray Tunnel Section being positioned (source: (Railway Technology, 2017))

It was not until very recently that marine vessels have been discussed as a target for application of a

cooperative control concept. Some works like (Skjetne, Moi, & Fossen, 2002), (Kyrkjebo & Pettersen,

2003), (Ihle, Jouffroy, & Fossen, 2005), and (Arrichiello, Chiaverini, & Fossen, 2006) exploit path

following and trajectory tracking missions of marine vessels working in coordination. Researches in

cooperative control applied to DP systems are even scarcer. In (Queiroz Filho & Tannuri, 2013) an

extensive experimental analysis of a consensus-based control law applied to DP vessels was presented. In

(Peng, Wang, & Wang, 2016) a cooperative DP control system using their so called Dynamic Surface

Control (DSC) combined with an adaptive algorithm that observes and corrects the control law to respond

to oceanic disturbances. Nonlinear methods were presented in (Ianagui & Tannuri, 2015) and (Ianagui,

Queiroz Filho, & Tannuri, 2016), where numerical and experimental results were presented.

The subject “cooperative multi-agent systems” is rather broad and involve notions from computer science

to mechatronics. The most studied topics in this area for application in multi-vehicle systems are the

cooperative control methods. These methods involve decision making, cooperative motion planning and

actual control theory required to perform the cooperative tasks. To accomplish such objectives, the

vehicles may be required to collectively adopt several behaviours, such as (but not restricted to)

swarming, flocking, setting a formation and navigate in it, tracking a leader and avoid collisions. These

behaviours can be achieved through the application of collective control laws, referred in literature as

cooperative protocols, which can be either calculated in centralized fashion or distributed among the

agents. While centralized control is easier and can be designed through classical control methods, it lacks

robustness, flexibility and scalability to be applied in larger scale systems. Within this set of

requirements, distributed cooperative methods are usually preferable for application in multi-agent

control. The main features of this architecture are cited as follows:

1) Motion synchronization. A leader tracking algorithm can be used to make all DP vessels move

with controlled relative positioning in all degrees of freedom and following a desired joint

trajectory. Harder tasks, like changing the heading of the whole formation (as opposed to

changing the heading of all individual vessels) are performed in a simple manner. This becomes

even more advantageous with a larger number of vessels;

2) Scalability. The usual cooperative algorithms are distributed and easily scalable, which once

again presents an advantage when many vessels are assigned;

3) Stability. The introduction of restrictions between agents, like cable connections, does not

destabilize the overall system;




4) Load Position Correction. When carrying a load with multiple vessels, the leader position may be

corrected online to relocate the load to a desired position.

While not very different from the control structures that are applied to other vehicles, the cooperative

algorithms needed for DP vessels must address some specific challenges, like the persistent and/or

random punctual disturbances from waves, winds and currents. The communication topologies used

should also be correctly addressed to keep the system safe, reliable and available.

Cooperative Control of DP Vessels

The architecture proposed for the cooperative DP system is presented in Figure 2. The control is

decentralized, so the wireless communication structure illustrated in the figure is not a hardware

implementation. This block represents what will be defined later in terms of a graph communication

topology between the agents. Each vessel sends its own position; relative position references are

broadcast.

Figure 2 General Architecture for the Cooperative DP

Figure 3 shows the internal blocks of each agent depicted in the last figure. The system considers that a

standard, local DP system is running in each single vessel. Wind feedforward compensation is not shown

but is present. The cooperative controller sees the junction DPS + Vessel as a single plant, and sends

position references to it. This controller can be seen, in a way, as a guidance system.




Figure 3 - Control blocks for a single vessel

The cooperative controller may be accompanied by a local Kalman Filter, which estimates the data of all

neighbor agents. This enhances the overall system robustness, as in the event of a network shortage the

neighbor agents’ positions can be integrated (dead-reckoning).

To find a cooperative control rule (hereafter mentioned as protocol), a simple second order model for the

motion equations of the joint block Vessel + DPS is considered. Equation (1) presents such model.

[�̇�𝟏�̇�𝟐] = [

0 𝐼3×3𝐴0 𝐴1

] [𝒙𝟏𝒙𝟐] + [

0𝐵1]𝒖

𝒚 = [1 0] [𝒙𝟏𝒙𝟐]

(1)

In which 𝒙𝟏 and 𝒙𝟐 are the state vectors corresponding to the global reference frame positions (easting,

northing and heading, represented here by 𝑥,𝑦 and 𝜓) and velocities (�̇�,�̇� and �̇�), respectively, 𝒚 is the

observation vector and 𝒖 is the desired position vector. The matrixes 𝐴0, 𝐴1 and 𝐵1 are given by

𝐴1: = [

−2𝜁𝑥𝜔𝑛𝑥2 0 0

0 −2𝜁𝑦𝜔𝑛𝑦2 0

0 0 −2𝜁𝜓𝜔𝑛𝜓2

] ; 𝐴0: = [

−𝜔𝑛𝑥2 0 0

0 −𝜔𝑛𝑦2 0

0 0 −𝜔𝑛𝜓2

] ;

𝐵1 ∶= −𝐴0

(2)

The variables 𝜔𝑛𝑖 and 𝜁𝑖, 𝑖 = 𝑥, 𝑦, 𝜓 are the natural frequency and the relative damping factor of the

system, respectively. Their values depend on the DP control system tuning, with 𝜁𝑖 usually between 0.6

and 0.9 (Fossen, 2011) and 𝜔𝑛𝑖 with small values, depending on the vessel size. Both can be easily

recovered from step responses of the DP vessel by expressions (3) and (4).

𝜔𝑛 ≔ −2𝜋

Δ𝑡√1 − 𝜁2

(3)

𝜁 ≔ −

ln(𝑀𝑝)

√𝜋2 + (ln(𝑀𝑝))2

(4)




Here, 𝑀𝑝 is the overshoot as a fraction of the step input and Δ𝑡 is the time interval between two

consecutive peaks on the plot of the step response.

For multiple vessels, the dynamic state equation (1) is to be indexed. Consider the each DP vessel

dynamics of the form

�̇�𝒊 = 𝐴𝒙𝒊 + 𝐵𝒖𝒊, 𝑖 = 1,2,… ,𝑁 (5)

Here 𝑛 is the number of vessels to be cooperatively controlled. Our objective is to define a decentralized

set point generation law 𝒖𝒊 = 𝑓(𝒙𝒊,𝒋,…,𝑵) that fulfills the required tasks. The strategies presented are based

on the definition of a consensus problem, where multiple agents’ outputs are required to reach a

consensus value in a coordination variable. By defining this basic problem, standard cooperative tasks

such as formation keeping and leader-following behavior can be described. A thorough discussion on

linear consensus theory can be found in (REN, BEARD, & ATKINS, 2007).

Suppose that the vessels’ controllers can communicate wirelessly sending each its own position and

velocity and receiving such data from neighbor vessels. Here the concept of neighbor does not mean

necessarily physical proximity, but depends on how the communication topology is defined. In the

cooperative control area, the usual way to represent such topology is by means of communication graphs.

A graph is defined as a pair 𝒢 = (𝑉, 𝐸), where 𝑉 is a set {𝑣1, 𝑣2, … , 𝑣𝑛} called vertices or nodes set and 𝐸

is a set of pairs {𝑣𝑖, 𝑣𝑗} called edges. In a multi-agent communication structure, each vertex represents a

single agent and each edge represents the availability of information between edges.

(a) (b)

Figure 4 – (a) A weighted digraph. (b) A spanning tree

In a standard or undirected edge, (𝑣𝑖, 𝑣𝑗) = (𝑣𝑗, 𝑣𝑖). This means, from the multi-agent network design

point of view, that information is shared bi-directionally, i.e. the agent nodes connected through this edge

have access to each other’s states. In an oriented edge, (𝑣𝑖, 𝑣𝑗) ≠ (𝑣𝑗 , 𝑣𝑖), which means that information

is shared in only one direction and only one agent has access to the other’s states. Graphically, an oriented

edge is represented through an arrow in which the point enters 𝑣𝑗 and the tail leaves 𝑣𝑖. An oriented or

directed graph is also called a digraph.

The in-degree of a node 𝑣𝑖 is the number of edges pointing to 𝑣𝑖. The neighbors of node are the set 𝑁𝑖 =

{𝑣𝑗: (𝑣𝑗 , 𝑣𝑖) ∈ 𝐸}, i.e., it is the set of nodes with edges pointing to 𝑣𝑖. The neighbor set number of

elements |𝑁𝑖| is equal to the (in) degree of 𝑣𝑖. Each edge (𝑣𝑗, 𝑣𝑖) can be associated with a weighting

factor 𝑎𝑖𝑗 > 0 (Figure 4a). When this is considered, the neighbor set can be defined as 𝑁𝑖 = {𝑣𝑗: 𝑎𝑖𝑗 > 0}.

A directed path is a sequence of edges 𝑣0, 𝑣1, … , 𝑣𝑟: (𝑣𝑖, 𝑣𝑖+1) ∈ 𝐸. If there is a directed path from 𝑣𝑖 to 𝑣𝑗, it is said that 𝑣𝑖 is connected to 𝑣𝑗. A graph is said to be strongly connected if any pair (𝑣𝑖 , 𝑣𝑗) ∈

𝑉, 𝑖 ≠ 𝑗 is connected.




A (directed) tree (Figure 4b) is a digraph in which every node, except one called root node, has in-degree

equal to one. A graph has a spanning tree if a subset of its edges forms a spanning tree. This is equivalent

to say that all the vertices of this graph are reachable if starting from a single vertex and following the

edges arrows. A graph can have multiple spanning trees. A set of root nodes or leader set is the set of all

roots of all spanning trees of a graph. If a graph is strongly connected, it has at least one spanning tree and

all vertices are roots.

For a graph with edges weighted through 𝑎𝑖𝑗, the adjacency matrix 𝐴 is defined as

𝐴 = [𝑎𝑖𝑗] {𝑎𝑖𝑗 > 0 𝑖𝑓 (𝑣𝑗, 𝑣𝑖) ∈ 𝐸

𝑎𝑖𝑗 = 0 𝑖𝑓 (𝑣𝑗 , 𝑣𝑖) ∉ 𝐸 (6)

The weighted in-degree (𝑑𝑖) of a node 𝑣𝑖 is the sum of the elements of the 𝑖 − 𝑡ℎ row of the adjacency

matrix:

𝑑𝑖 =∑ 𝑎𝑖𝑗𝑁

𝑗=1 (7)

The in-degree matrix 𝐷 is defined as

𝐷 = 𝑑𝑖𝑎𝑔(𝑑𝑖) (8)

With the in-degree definition, the Laplacian matrix can also be defined as

𝐿 = 𝐷 − 𝐴 (9)

Now, return to the problem of finding 𝒖𝒊 = 𝑓(𝒙𝒊,𝒋,…,𝒏) for equation (5). A linear consensus control law (or

protocol) for this system is will be given by (Lewis, Zhang, Hengster-Movric, & Das, 2014):

𝒖𝒊 = 𝑐𝐾∑ 𝑎𝑖𝑗(𝒙𝒋 − 𝒙𝒊)𝑗∈𝑁𝑖

(10)

Here, 𝑁𝑖 is the set of neighbors of the 𝑖 − 𝑡ℎ agent and 𝑎𝑖𝑗 are the weight entrances of the graph edges, 𝑐

is a positive synchronization variable (usually set to 1) and 𝐾 is a gain matrix to be designed. The stability

properties of the closed loop system with protocol (10) is equivalent of the stability properties of the 𝑁

systems

�̇̃�𝒊 = (𝐴 − 𝑐𝜆𝑖𝐵𝐾)�̃�𝒊, 𝑓𝑜𝑟 𝑖 = 1,2,… ,𝑁 (11)

In which 𝜆𝑖 are the eigenvalues of the Laplacian matrix and �̃�𝒊 is the relative error vector. Calculation of

the gains 𝑐 and 𝐾 required to reach stability are an extensive topic; their values depend on the graph

topology. Stability attendance for the gains can be found through standard linear theory applied in each

block of equation (11), for instance, the Routh Stability Criteria (Ogata, 1970). Throughout the text it will

be assumed that the graphs have fixed structure and at least one spanning tree. Optimal definition of gains

for consensus problems are given in (Lewis, Zhang, Hengster-Movric, & Das, 2014) and in (LI & Duan,

2015).




Even though the standard consensus theory can provide control over the relative states of multiple agents,

it is rather limited (in the way it was presented) to enforce some desired behaviors on the agents that are

of operational practice. The protocol and graph topology are now discussed in order to provided

Formation control and Leader Following Behavior, which are more applicable in the operations of marine

vehicles. Formation control can be achieved through the introduction of reference relative position signals

𝛿𝑖𝑗 between the agents in the control inputs. For the most general case,

𝒖𝒊 = 𝑐𝐾∑ 𝑎𝑖𝑗(𝒙𝒋 − 𝒙𝒊 − 𝜹𝒋𝒊)𝑗∈𝑁𝑖

(12)

The reference signals do not need to be constant along time, which means that a trajectory generation law

can be input to enforce a dynamically varying formation. When thinking about a DP system mass-spring-

damper analogy (Figure 5-a), the introduction of such law corresponds to adding extra spring-damper sets

between the vessels, as can be seen in Figure 5-b.

(a) (b) Figure 5 – (a) Mass-Spring-Damper DP Analogy (b) Mass-Spring DP+Cooperative Protocol Analogy

The consensus algorithm does not provide means to control the general location of the agents, and the

formation function from equation (12) controls only relative position. Full controllability of the

cooperative positioning behavior can be achieved through the introduction of a leader agent. This is

consensus leader following is referred in literature as consensus tracking.

Consensus tracking can only be introduced in digraphs, by the usage of one root node of a spanning tree

that has in-degree equal to zero. In this case, the row of the adjacency, the in-degree and the Laplacian

Matrixes corresponding to this node has all elements equal to zero. The overall result is that the

consensus value reached for every agent, except the leader, is the initial leader state (Lewis, Zhang,

Hengster-Movric, & Das, 2014). Note that the leader node is not required to be real; it can be defined as

only a sequence of signals representing a desired dynamics. In fact, this behavior is even desirable as the

eventual failure of an agent defined as the leader system could handle the cooperative system

uncontrollable. In Figure 5-b such method is exemplified with L, the central leader node reference. In a

load carrying situation, as the one to be studied next, the leader reference position may be modulated to

online correct the overall formation position in order to take and keep the load in a desired position,

eliminating offset errors caused by environmental loads. This may be done through a modified PID

controller applied to the leader virtual dynamics input:




𝒖𝑳 = 𝒙𝑫 + 𝐾𝑃(𝒙𝑫 − 𝒙𝒍𝒐𝒂𝒅) + 𝐾𝐷(�̇�𝑫 − �̇�𝒍𝒐𝒂𝒅) + 𝐾𝐼∫(𝒙𝑫 − 𝒙𝒍𝒐𝒂𝒅)𝑑𝑡 (13)

In which 𝒖𝑳 is the virtual leader input vector, 𝒙𝑫 is the virtual leader desired position vector and 𝒙𝒍𝒐𝒂𝒅 is

the load measured position vector. 𝑲𝑷, 𝑲𝑫 and 𝑲𝑰 are diagonal gain matrices.

Case Study: Load Hold Back and Manoeuvring

As an example application, a case study using four DP AHTS and an unactuated Drilling Rig will be

described and simulated. In a larger scale, this setup is similar to what would be required to position the

tunnel construction module mentioned in the introduction. The system considered is an expansion of the

situation studied in (Huang, et al., 2017), where a hold back vessel is used to help a Drilling Rig to keep

its position during a partial failure in its energy supply system, degrading its stationkeeping capacity. The

overall results showed that the cable connection of two DP vessels running separate auto positioning

might cause control instability, reinforcing results seen in (JENSSEN, 2008) and (IMCA, 2000). As shall

be seen next, the introduction of a cooperative algorithm will enhance the stability limits of the system,

regardless the cable flexibility.

Linear Analysis

In this section, the stability of the simplified system composed of three bodies (i = 0 - Drilling Rig, 1 -

AHTS, 2 - AHTS) with total inertia 𝑚𝑖 (considering the hydrodynamic added mass) connected by cables

of horizontal restoring coefficient 𝑘. Both AHTS have DP controllers and keep their positions while

connected to the central load (drilling rig). This simplified analysis will consider the vessels' motion only

in the 𝑥 axis (surge direction) as shown in the figure below.

Figure 6: Representation of the system.

Ships 1 and 2 are hold back vessels based on the Maersk Handler AHTS (IMO: 9246724) and the central

load (vessel 0) is a drilling rig based on the West Eminence (IMO: 8768438). The hold back vessels

controller gains are presented in Table 1 and the main properties of the vessels are presented in Table 2.

The stability of the system will be verified for two cases:

• Case 1 - Ships 1 and 2 each have their independent DP controller.

• Case 2 – There is a cooperative controller in addition to the individual controllers of ships 1 and

2.

x

𝑘 𝑘

𝑚1

𝑚0

𝑚2




Table 1: DP control gains

DP Gain AHTS

Proportional Surge (𝑘𝑁/𝑚) 2.05 × 101

Proportional Sway (𝑘𝑁/𝑚) 0.87 × 101

Proportional Yaw (𝑘𝑁.𝑚/𝑟𝑎𝑑)

0.66 × 104

Derivative Surge (𝑘𝑁. 𝑠/𝑚) 8.36 × 102

Derivative Sway (𝑘𝑁. 𝑠/𝑚) 5.51 × 102

Derivative Yaw (𝑘𝑁.𝑚. 𝑠/𝑟𝑎𝑑)

3.62 × 105

Integral Surge (𝑘𝑁/𝑚. 𝑠) 1.82 × 10−1

Integral Sway (𝑘𝑁/𝑚. 𝑠) 0.39 × 10−1

Integral Yaw (𝑘𝑁.𝑚/𝑟𝑎𝑑. 𝑠) 2.58 × 101

Table 2: Main properties of the vessels

Vessel Drilling Rig AHTS

LOA 118.6m 80.0m

Beam 72.7m 18.0m

Draft 23.5m 6.5m

Displacement 56,629ton 7,240ton

The dynamics of the complete system (Case 1) is represented by the block diagram in Figure 7, the

vessels' reference positions (𝑅1, 𝑅2) are the inputs, and the vessels' actual positions (𝑋1, 𝑋2) are the

outputs.

Figure 7: Block diagram of the system (Case 1).

The hold back vessels DP system is composed by a notch wave filter and a PID controller, of dynamics

𝐺𝑜1/𝐺𝑜2 and 𝐺𝑐1/𝐺𝑐2, respectively. Results are held if a Kalman Filter is used. The vessels actuators are

considered to have a first order dynamic (𝐺𝑎1 and 𝐺𝑎2) of time constant 0.5 seconds, the simplified

unidirectional dynamic of the vessels is represented by blocks 𝐺𝑝0, 𝐺𝑝1 and 𝐺𝑝2. The transfer functions

for each block are:

𝐺𝑝𝑖 =1

𝑚𝑖 . 𝑠2, 𝑚1 = 𝑚2 = 7.9 × 10

6𝑘𝑔; 𝑚0 = 80 × 106𝑘𝑔 (14)

+

-

PID Controller Actuators

+

+

+

- PID Controller Actuators

-

+

-

+

-

+

-

+

𝐺𝑝0 𝑋0

Notch Wave Filter

Notch Wave Filter

Vessel 1

Vessel 2

Vessel 0

Cable

Cable

𝐺𝑜1

𝐺𝑜2

𝐺𝑐1

𝐺𝑐2

𝐺𝑎1

𝐺𝑎2

𝑘

𝐺𝑝1

𝐺𝑝2

𝑋1

𝑋2

𝑅1

𝑅2

𝑘




Gci = KPi + KDis +KIis, control gains obtained from Table 1 (15)

Ga1 = Ga2 =1

0.5 s + 1 (16)

Go2 =∏s2 + 2ζωis + ωi

2

(s + ωi)2

3

i=1

, with ζ = 0.1; ωi = 0.5; 0.73; 1.1 rad/s (17)

When adding the cooperative controller (equation (12)) the inputs (𝑅1, 𝑅2) will be calculated as:

{

𝑅1 = [𝑘11 𝑘12] ([𝑥0𝑥0̇] − [

𝑥1�̇�1] + [

𝑥2�̇�2] − [

𝑥1�̇�1] − 𝛿01 − 𝛿21) + 𝑥1

𝑅2 = [𝑘21 𝑘22] ([𝑥0𝑥0̇] − [

𝑥2�̇�2] + [

𝑥1�̇�1] − [

𝑥2�̇�2] − 𝛿02 − 𝛿12) + 𝑥2

(18)

Where 𝑘𝑖𝑗 are the cooperative controller gains, 𝑥𝑖 are the measured states, and the dot superscript

indicates the time derivative of the signal. Now, the new inputs to the system will be: the virtual leader

desired position and velocity (𝑥0 , �̇�0) and the desired relative states vectors 𝛿𝑖𝑗 = −𝛿𝑗𝑖 = ([𝑥𝑖 �̇�𝑖] −

[𝑥𝑗 �̇�𝑗])𝑡 . Cooperative controller gains used were 𝑘11 = 𝑘21 = 1.2361 and 𝑘12 = 𝑘22 = 34.7205.

With the systems defined, we can calculate their poles as a function of the cables horizontal restoring

coefficient (𝑘). The system will be stable if its poles are always in the left half complex plane, and

unstable otherwise. The root locus for both cases are depicted below, with the poles changing from green

to black as 𝑘 increases:

(a) (b)

Figure 8 - (a) Poles of the system without cooperative control – Case 1. (b) Poles of the system with

cooperative control – Case 2.

The system without cooperative control (Figure 3) is stable only if the cables’ horizontal restoring

coefficients are below 164 kN/m. While the system with cooperative control (Figure 4) is always stable.

For every value of 𝑘, its poles are always to the left of the imaginary axis. Assuming a steel wire and a

polyester cable with the properties given in Table 3, we can plot the horizontal restoring coefficient as a

function of the force transmitted by the cable (Figure 9).

-0.2 -0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Lugar das Raízes

Im

Re

↑k𝑘𝑙𝑖𝑚 = 164 𝑘𝑁/𝑚

Root Locus

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5

-1

-0.5

0

0.5

1

Root Locus

Im

Re




Table 3: Cable properties

Gain Steel Polyester

Diameter (𝑚𝑚) 85.7 143

Weight in air (𝑘𝑔/𝑚) 30.7 14.3

Weight in water (𝑘𝑔/𝑚) 25.5 13.3

Axial Rigidity EA (𝑘𝑁) 2.38 × 105 8.23 × 104

Figure 9: Cables restoring coefficient as a function of the horizontal force.

The 164 kN/m limit (without cooperative control) can be easily reached for 400m cables with forces as

low as 50 tonf, and would be even smaller for shorter cables, therefore, in these cases, the hold back

vessels can’t keep their positions. However, if the cooperative control was implemented, the system

would always be stable and the operation could be safely executed. The benefits of using cooperative

control becomes evident when simulating the step responses of both cases for 𝑘 = 200 𝑘𝑁/𝑚, as shown

by the graphs below, (Out(1) = 𝑋1 and Out(2) = 𝑋2):

(a) (b) Figure 10 - (a) Step response of the system w/o Cooperative algorithm (unstable). (b) Step response of the

system with Cooperative Algorithm (stable).

The cooperative control logic introduces a relative position damping effect compensating the additional

stiffness introduced by the cables, and effectively stabilizing an otherwise unstable system.

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

Force (tonf)

Horizonta

l R

esto

ring C

oeff

. (k

N/m

)

Steel Wire Length 950m

Steel Wire Length 400m

Polyester 950m

Polyester 400m

-2

0

2

4

6

To:

Out(

1)

0 2000 4000 6000 8000 10000 12000-2

0

2

4

6

To:

Out(

2)

Linear Simulation Results

Time (seconds)

Am

plit

ude




Full Scale Simulation

In this section the application of the cooperative algorithm in a load maneuvering situation is presented.

The results are obtained through numeric simulation, using the Numerical Offshore Tank’ (TPN-USP)

numeric simulator ( (TPN, 2016) and (Tannuri, et al., 2014)). Results from this simulator for DP offshore

operations have been validated in several works, such as (Huang, et al., 2017) and (Orsolini, Castelpoggi,

Yuba, Machado, & Malafaia, 2016). The vessel’s dynamic behavior and its environmental responses in

the simulator have been calibrated through extensive experimental study (TPN , 2011). Remaining

relevant data on the AHTS are presented in Table 4 and Table 5.

Each vessel runs an DP system with an extended Kalman Filter, with design matrixes given by Table 6.

For simplicity, the rudder positions are kept constant in zero and the azimuth thruster is kept in 90°,

emulating an extra bow tunnel thruster. The vessels are connected through a communication graph

represented in Figure 11, with a virtual leader (node L) and design matrices given in Table 7. Optimal

gain was calculated through LQR.

Table 4: AHTS Data

Parameter Value

𝑚 (𝑘𝑔) 7.24000 × 106

𝐼𝑧 (𝑘𝑔.𝑚2) 2.75000 × 109

𝑋�̇� (𝑘𝑔) 6.40970 × 105

𝑌�̇� (𝑘𝑔) 6.40380 × 106

𝑁�̇� (𝑘𝑔.𝑚2) 1.56150 × 109

𝑁�̇� = 𝑌�̇� (𝑘𝑔.𝑚) 7.97240 × 106 𝑥𝐺 0.0

𝐿𝑂𝐴 (𝑚) 80.0

𝐿𝑃𝑃 (𝑚) 69.3

𝐵𝐸𝐴𝑀 (𝑚) 18.0

𝐷𝑟𝑎𝑓𝑡 (𝑚) 6.6

Table 5: Propeller Data

Propeller Power (kW)

Max.

Thrust

(kN)

Tunnel Bow 883 117.6

Azimuth Bow 883 147.0

Tunnel Stern 883 117.6

Main 1 6440 965.3

Table 6 - Extended Kalman Filter Design Matrices

Matrix Value

Process

Covarianc

e

𝐐𝟏

=

[ 𝟎. 𝟎𝟏 𝟎 𝟎 𝟎 𝟎 𝟎𝟎 𝟎. 𝟎𝟏 𝟎 𝟎 𝟎 𝟎𝟎 𝟎 𝟒 × 𝟏𝟎−𝟖 𝟎 𝟎 𝟎𝟎 𝟎 𝟎 𝟏𝟎𝟎 𝟎 𝟎𝟎 𝟎 𝟎 𝟎 𝟏𝟎𝟎 𝟎𝟎 𝟎 𝟎 𝟎 𝟎 𝟒 × 𝟏𝟎−𝟒]

;

𝐐𝟐 = [𝟏𝟎 𝟎 𝟎𝟎 𝟐𝟎 𝟎𝟎 𝟎 10

] ;

𝐐𝟑 = 𝟏 × 𝟏𝟎19 ∙ [

1 0 00 80 00 0 5 × 1011

] ;

𝐐 = [

𝐐𝟏 𝟎 𝟎𝟎 𝐐𝟐 𝟎𝟎 𝟎 𝐐𝟑

]

Noise

Covarianc

e

𝐑 = [𝟏 × 𝟏𝟎−𝟐 𝟎 𝟎

𝟎 𝟏 × 𝟏𝟎−𝟐 𝟎𝟎 𝟎 𝟒 × 𝟏𝟎−𝟖

]

Table 7 – Cooperative Control Matrices

Matrix Value

Adjacenc

y A =

[ 0 0 0 0 01 0 1 0 11 1 0 1 01 0 1 0 11 1 0 1 0]

In-Degree D =

[ 0 0 0 0 00 3 0 0 00 0 3 0 00 0 0 3 00 0 0 0 3]

Laplacian L =

[ 0 0 0 0 0−1 3 −1 0 −1−1 −1 3 −1 0−1 0 −1 3 −1−1 −1 0 −1 3 ]

Gain

K

= [1.24 0 0 34.7 0 00 1.24 0 0 70.5 00 0 1.24 0 0 46.9

]

Throughout the simulation mild environmental conditions were considered, with general intensities given

by Figure 12.




Figure 11 - Communication Structure

Figure 12 - Environmental Conditions

A set of four AHTS’s are required to maintain formation and move the Drilling Rig connected to them

(Figure 13). The system starts with all DP vessels running standalone and simply holding the Rig, after an

initial stabilization, the cooperative control starts to run and corrects the Drilling Rig position to a desired

one. A set of maneuvers then occur as follows, as a modified version of a 4 corner test (VÆRNØ,

BRODTKORB, SKJETNE, & CALABRÒ, 2017), where surge, sway, yaw and coupled motions are

verified.

Figure 13 – AHTS’s holding the Drilling Rig.

i. The DP vessels start in formation running standalone;

ii. After an initial stabilizing period required for the EKF to estimate and compensate the DP

current, the cooperative control is turned on and the vessels are commanded to enter

formation. The formation pattern is illustrated in Figure 14. The formation set points are

given by

𝜹𝟏𝑳 = [−394−237−149

] ; 𝜹𝟐𝑳 = [394−237−31

] ; 𝜹𝟑𝑳 = [39423731

] ; 𝜹𝟒𝑳 = [−394237149

]

𝜹𝟏𝟐 = −𝜹𝟐𝟏 = [−7880

−118] ; 𝜹𝟐𝟑 = −𝜹𝟑𝟐 = [

0−474−62

] ; 𝜹𝟑𝟒 = −𝜹𝟒𝟑 = [7880

−118] ; 𝜹𝟏𝟒 = −𝜹𝟒𝟏 = [

047462

]

iii. After formation is assembled, the leader vessel is commanded to a sequence of set-points in

𝑥, 𝑦 and 𝜓 (Figure 15), waiting for stabilization after each command:

[

𝑥𝑑𝑦𝑑𝜓𝑑

] = [000] → [

10000] → [

1001000] → [

10010045

] → [010045

] → [000]




The virtual leader vessel dynamics considered was very slow, corresponding to a second order system

with all degrees of freedom having natural frequency of 0.005 rad/s and damping ration of 0.9. The virtual

leader required positions are online corrected to accurately position the load, with the exception of the

heading, as the formation used does not require control in this DOF.

Figure 14 - Stabilized Formation

Figure 15 - Drilling Rig Set Point Sequence

Simulations were run for 12000s, with a step size of 0.5s. The total motion footprint is shown in Figure

16, with each motion step being depicted in Figure 18.

Figure 16 - Complete Motion Footprint

Figure 17 – Drilling Rig Trajectory

The drilling rig positions in Easting, Northing and Heading for the initial phase, when the cooperative

controller is not active are plotted in Figure 19. The steady state error in Easting is of about 14m, while in

Northing is of 45m. Heading error is kept under 4°. Given the environmental conditions coming mainly

from northeast), such errors are expected, as the cable flexibility permits a large offset from the Rig

(despite the DP Vessels set point errors are low). When the cooperative controller is started (Figure 20), the error drops to zero and follows the desired set-points with dynamics close to the one designed

for the virtual leader, reaching the step values in around 1000s.

-50 0 50 100 150

-50

0

50

100

No

rth

ing

(m

)

Easting (m)

Drilling Rig Center Trajectory

Before Consensus




(1)

(2)

(3)

(4)

(5)

(6)

Figure 18 – Motion Sequence

Figure 19 – Drilling Rig positions time series and error series (non-cooperative)

0 500 1000 1500 2000 2500-20

-15

-10

-5

0

5

Ea

stin

g (

m)

Drilling Rig Position(No Cooperative)

0 500 1000 1500 2000 2500-5

0

5

10

15

20

Ea

stin

g E

rro

r (m

)

Drilling Rig Position Error (No Cooperative)

0 500 1000 1500 2000 2500-60

-40

-20

0

20

No

rth

ing

(m

)

0 500 1000 1500 2000 2500-20

0

20

40

60

No

rth

ing

Err

or

(m)

0 500 1000 1500 2000 2500-1

0

1

2

3

4

Time(s)

He

ad

ing

(°)

0 500 1000 1500 2000 2500-4

-3

-2

-1

0

1

He

ad

ing

Err

or

(°)

Time(s)

Reference

Actual




Figure 20 - Drilling Rig positions time series and error series (cooperative)

The total relative positioning errors, a measuring of the vessels’ deviation from the required formation is

plotted next for each degree of freedom. Note that for all DOF’s the relative positioning errors are

dropped when the cooperative controller is turned on. This is translated as the synchronization

characteristic of controller, permitting then maneuvers tight formation control.

Figure 21 – Total relative Position Errors

Power usage is presented in Figure 22. Due to the constantly changing set-points provided by the

cooperative controller, some larger actuation activity is expected. All tunnel stern thrusters are largely

required, as there are the only ones providing lateral force in the stern of the vessel. Note that vessels 3

and 4 are more required overall. Once again, this is due to the environmental loads, coming from

Northeas. Vessel 3 in speciall is highly demanded, as for the formation used, this is the one aligned to the

environmental vectors. Cable tension series (Figure 23) also indicate this effect, as the tensions in cable 3

2000 4000 6000 8000 10000 12000-50

0

50

100

150Drilling Rig Position(Cooperative)

Ea

stin

g (

m)

2000 4000 6000 8000 10000 12000-150

-100

-50

0

50

100

Ea

stin

g E

rro

r (m

)

Drilling Rig Position Error (Cooperative)

2000 4000 6000 8000 10000 12000-50

0

50

100

150

No

rth

ing

(m

)

2000 4000 6000 8000 10000 12000-200

-100

0

100

200

No

rth

ing

Err

or

(m)

2000 4000 6000 8000 10000 12000-20

0

20

40

60

He

ad

ing

(°)

Time(s)2000 4000 6000 8000 10000 12000

-50

0

50

Time(s)

He

ad

ing

Err

or

(°)

Reference

Actual

0 1000 2000 3000 4000 5000 6000 7000 8000-40

-20

0

20

40

60

80

Total Easting Relative Position Error

Time(s)

Ea

stin

g E

rro

r(m

)

Consensus Start

0 1000 2000 3000 4000 5000 6000 7000 8000-150

-100

-50

0

50

100

Total Northing Relative Position Error

Time(s)

No

rth

ing

Err

or(

m)

Consensus Start

0 1000 2000 3000 4000 5000 6000 7000 8000-40

-30

-20

-10

0

10

20

30

40

50

60

Total Heading Relative Position Error

Time(s)

He

ad

ing

Err

or(

m)

Consensus Start




and 4 reach much higher levels, specially during the rig repositioning in both northing and easting

positive directions.

Figure 22 - AHTS's power usage

Figure 23 - Cable Tensions

Conclusions

This paper has shown an application of a high-level guidance controller to DP systems using robotics

concepts to automate a relative positioning and a towing operation. Despite our case study is set in a

situation with a relatively low vessel quantity, this method is easily scalable, permitting a larger number

of vessels to present the described behaviours. By its own nature, the cooperative controller enforces

synchronization, as long as the vessel agents are able to perform the required motion.

Linear analysis showed that this method can be used in hold back operations stabilizing the overall system

with (at least in theory) a large cable elasticity range. The simulated study case showed that the multiple

towing vessels can not only stabilize a load position, but also move it to a desired location and heading

with low error.

TUNNEL BOW AZIMUTE BOW TUNNEL STERN MAIN MAIN20

20

40

60

80

100

120

140

160

180

200

Tim

e (

min

)

Maersk Handler1: Commands

Stop

Dead Slow

Slow

Half

Full


50

100

150

200

250

Tim

e (

min

)


Stop

Dead Slow

Slow

Half

Full


50

100

150

200

250

Tim

e (

min

)


Stop

Dead Slow

Slow

Half

Full


50

100

150

200

250

Tim

e (

min

)


Stop

Dead Slow

Slow

Half

Full

0 2000 4000 6000 8000 10000 120006

7

8

9

10

11Cable 1

F (

ton

)

Time(s)0 2000 4000 6000 8000 10000 12000

5

10

15

20

25Cable 2

F (

ton)

Time(s)

0 2000 4000 6000 8000 10000 120000

50

100

150

200

250

300

350

400Cable 3

F (

ton

)

Time(s)0 2000 4000 6000 8000 10000 12000

0

50

100

150

200

250

300Cable 4

F (

ton)

Time(s)




In this study case, enhancements can be achieved by usage of smarter rules for formation settling. In

specific, grasping allocation rules, similar to thrust allocation of standalone DPS’s can be used. Moreover,

auto weathervaning methods may also be applied to reduce the required loads in individual vessels. In

actual applications, where communication hardware limitations may arise, Local Kalman Filters should

be used to perform dead-reckoning of neighbour positions.

From the software point-of-view, most aspects of cooperative control methods are already solved in a

robust manner. To achieve applicability in actual operations, most remaining issues rely on hardware and

wireless communication standards definitions, besides adequate human interfaces to command and

control the required inputs.

Acknowledgements

The first author acknowledges the Higher Education Personnel Improvement Coordination (Capes) for the

scholarship. The third author acknowledges the Brazilian National Council for Scientific and

Technological Development (CNPq) for the research grant process 308645/2013-8.

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multiple vessel cooperative dp operations...operations 5.3 multiple vessel cooperative dp operations...

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