Trajectory tracking, Path Following and Formation Control of - - PowerPoint PPT Presentation

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Trajectory tracking, Path Following and Formation Control

• f Autonomous Marine Vehicles

Kristin Y. Pettersen Erik Kyrkjebø Even Børhaug

Department of Engineering Cybernetics, NTNU, Norway

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Outline

• I. Trajectory tracking and path following
• Trajectory tracking, path following and manoeuvring

control problems

• Underactuated path following for marine vehicles
• II. Formation control (Multi-object control)
• Different degrees of synchronization
• Formation control of autonomous marine vehicles

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Trajectory tracking, path following and manoeuvring

Fossen 2002

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Tracking approaches

• Path following:
• Trajectory tracking:

Aguiar et al. 2004

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Tracking approaches

• Manoeuvring:

A manoeuvre is a curve in the input and state space that is consistent with the system dynamics

Hauser and Hindman 1995 Skjetne et al. 2004

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Tracking approaches

Advantages and drawbacks

Aguiar et al. 2004: Performance limitations in trajectory tracking due to unstable zero- dynamics can be removed by considering the manoeuvring problem instead Trajectory tracking forces the system to be at a given point on the curve at a given time

– Acceleration and retardation – Formation control and collision avoidance

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Underactuated path following

Underactuated control of mechanical/marine vehicles Acceleration constraint

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Nonholonomic systems

Second-order nonholonomic constraint First-order nonholonomic constraints Holonomic constraints

f

• ,t

f

• ,

,t

Goldstein 1980

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Underactuated control

The underactuated control problem Underactuated vehicles are vehicles with fewer independent control inputs than degrees of freedom. We have an underactuated control problem when we seek to control more degrees of freedom than the number of independent control inputs available. Output feedback state tracking control problem Output feedback output tracking control problem State feedback output tracking control problem State feedback state tracking control problem Lefeber 2000

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Underactuated control

Underactuated control of mechanical/marine vehicles The gravitation and buoyancy vector is important for the stabilizability

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Underactuated control

Brockett’s necessary condition (1983) Coron and Rosier (1994) Pettersen and Egeland 1996

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Underactuated path following

Underactuated ships - the control problem The route of a ship is typically specified by way points The control problem consists of two tasks:

– the geometric task – the dynamic task

Control challenge:

– Surge control is straightforward – Control both sway and yaw without sideway control force.

Way-point manoeuvring

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Underactuated path following

Underactuated ships

• d

t arctan

yk y xk x

• d

t arctan

yk y

LOS methods much used in ship control practice

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Underactuated path following

Underactuated ships Idea: LOS guidance much used in practice Possible to prove stabilization of all 3DOF? Tool: Cascaded systems theory Panteley and Loria 1998

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Underactuated path following

Underactuated ships

• Simplified model (u = U = constant, diagonal matrices):

Pettersen and Lefeber, 2001: A controller was developed that gave global asymptotic stability of the straight-line path.

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Underactuated path following

Underactuated ships

• Full 3DOF nonlinear ship model:

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Underactuated path following

Underactuated ships

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Underactuated path following

Underactuated ships Fredriksen and Pettersen, 2006:

• A coordinate transformation –

moving the body-fixed coordinate system along the mid-ship axis

• Conjecture: A control law designed to make

will stabilize both the sway and yaw dynamics

• d

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Underactuated path following

Underactuated ships The control laws give

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Underactuated path following

Underactuated ships The closed-loop system is – globally asymptotically stable – locally exponentially stable This result yields for any control law that globally exponentially stabilizes where is the LOS angle

• d

u ud

• d

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Underactuated path following

Underactuated ships Experimental results:

• Video

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Extension of the LOS-motivated approach to 3D path following. CASE 1: Straight line path following. 5DOF dynamics model of an AUV:

– Roll motion not considered in the model used for control design purposes. – Three available controls: surge, pitch and yaw. – We account for the effect of pitch/yaw control on sway/heave motion.

(Børhaug and Pettersen, 2005)

Underactuated path following

Underactuated ships

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Underactuated path following

Underactuated Autonomous Underwater Vehicles (AUVs)

Path following control objective: Intermediate LOS control objective: Desired path: LOS angles:

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Underactuated path following

Underactuated Autonomous Underwater Vehicles (AUVs) Steering autopilot AUV dynamics Speed autopilot

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Underactuated path following

Underactuated Autonomous Underwater Vehicles (AUVs)

• Cross-track error dynamics:
• The cross-track error dynamics can be rewritten in terms of the LOS

angles according to:

Errors that can be driven to zero by a suitable controller.

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Underactuated path following

Underactuated Autonomous Underwater Vehicles (AUVs)

We propose controllers based on sliding mode with eigenvalue decomposition to regulate ζ(t) and ξ(t) to zero (see e.g. Fossen 2002): 1) Surge and pitch control: 2) Yaw control: The controls render ζ=0 and ξ=0 UGES.

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• Closed loop system is a cascade:

Underactuated path following

Underactuated Autonomous Underwater Vehicles (AUVs)

UGES UGAS + ULES Globally bounded

Cascaded system is UGAS + ULES.

(Application of Panteley et. al. 1998)

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Underactuated path following

Underactuated Autonomous Underwater Vehicles (AUVs)

Control strategy: summary

• Proposed controller guarantees tracking of the LOS angles,

θLOS and ψLOS, and desired surge speed ud(t).

• The controller error dynamics is UGES.
• The nominal cross-track error dynamics is UGAS + ULES.
• The cascade of the controller error dynamics and cross-track

error dynamics is UGAS + ULES.

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Underactuated path following

Underactuated Autonomous Underwater Vehicles (AUVs)

• Video

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Underactuated path following

Underactuated Autonomous Underwater Vehicles (AUVs)

CASE 2: Curves paths in 3D. LOS angles are given relative to the path-fixed Serret-Frenet coordinate Frame, not the earth-fixed inertial frame.

(Børhaug and Pettersen, 2006)

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Underactuated path following

Underactuated Autonomous Underwater Vehicles (AUVs)

• Video

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Multi-object control

What happens when we want to control a group of objects?

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Multi-object?

• Two or more objects with a common objective

– Not necessarily all are identical objects – Not necessarily all are controlled by us – Not necessarily all communicate with everyone

• Control objective determines control strategy

– Multiple sensor control – Data acquisition – Surveillance

Formation control

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Multi-object strategies

• Apply a single-object strategy such as path-following,

trajectory tracking or manoeuvring to multiple objects and introduce some type of synchronization between the objects

• Apply a multi-object strategy that inherently

incorporate synchronization between the objects

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Synchronization?

Synchronization Cooperation Coordination Nominal behaviour Error situation Nominal behaviour Error situation

Cooperation Coordination

degree of synchronization

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Path-following LOS application

• A LOS based approach is a single-object approach

that has proven effective for path following of underactuated as well as fully actuated vehicles.

• A LOS based approach is intuitive, effective and

widely used. Also, it is applicable to a wide set of different vehicles with different dynamic capabilities.

• How can we adapt the LOS framework to multi-

vehicle synchronized control?

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When is the synchronized LOS approach applicable?

• All vehicles follow a predefined path.
• The paths must be parameterized in terms of a

suitable synchronization variable.

x1 x2

x1 x2

x1 x2 x1 x2

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Desired speed

From single-object to multi-

• bject LOS control

Steering autopilot LOS guidance

• Speed

autopilot

• Synchronization

controller

Desired path

Synchronization variables

(Borhaug et. al 2006)

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Towards Synchronized LOS Control

• We can use the single-object LOS control scheme to control

multiple vehicles to individual paths.

• We can then use the synchronization controller to synchronize

the motion of the vehicles along the paths.

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Synchronization and Consensus

• Synchronization requires consensus.
• Consensus requires information sharing.
• Information can be shared directly through communication or

indirectly through sensing.

• The objective of the synchronization controller is to achieve

consensus among the vehicle on the overall group motion, i.e. inter-vehicle spacing and path speed.

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Information Sharing

• A directed graph (or digraph) is a mathematical object that is

suitable for modeling the information flow among the vehicles.

• Nodes represents vehicles. Information flows along the arcs,

either by means of direct communication or sensing.

• The graph can change (discretely) with time.

1 2 3 4

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Graph connectedness: Important Definitions

• A node is called a center node, if it is reachable from every other

node in the graph.

• A digraph G(t) is said to be quasi strongly connected (QSC) if it

has a center node.

• A union digraph G(t)[t1,t2] is a graph whose arcs are obtained

from the union of the arcs in G(t) over the time interval [t1,t2].

• A digraph G(t) is said to be uniformly quasi strongly connected

(UQSC) if there exists T > 0 such that for all t ≥ 0, the union digraph G([t,t+T]) is QSC. (Lin et. al. 2005)

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Synchronization Control

• We consider along-path synchronization.
• The objective is to synchronize the motion of the vehicles, i.e. to

form a desired formation pattern, and move synchronously with desired speed.

?

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Synchronization Control

• The along-path dynamics can be written as

– xj: along-path position/arc length. – uref,j: reference speed to be tracked by the speed autopilot. – εLOS,j(t) : converging error signal due to path following and speed tracking

• If the path is straight, uref,j corresponds to the desired surge
• speed. If the path is curved, uref,j corresponds to the desired total

speed.

• We can now use the speed reference uref,j as a

virtual control to synchronize the vehicles.

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Synchronization Control Law

• We propose the following synchronization control law

– g(x): saturation-like function. – ud(t): positive desired group speed.

– γji ≥ 0: linkage parameters.

– If vehicle j has access to xi, then γji > 0. Otherwise, γji = 0. The control law uses

• nly locally available

information!

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Properties of the Synchronization Control Law

• The synchronization control law

guarantees asymptotic consensus of all vehicles if the communication graph is Uniformly Quasi Strongly Connected (UQSC).

• In other words, all synchronization variables converge

asymptotically to each other and

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Synchronization control LOS path following and speed control

Control System: Overview

• Hierarchical structure:
• Synchronization control at higher level.
• Speed tracking and LOS PF at lower level.
• LOS PF system remains unchanged from the

single-object case.

• The synchronization controller uses extra

freedom in choosing the speed of the vehicle.

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Simulation illustration

• Video

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Leader-follower synchronization

• Inherently a multi-object control strategy
• Resembles the tracking problem of being at a specific

point at a specific time

• This is a coordinated approach to synchronization

– Example: Underway replenishment (UNREP)

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Underway ship replenishment

• Can we apply this operation to civilian ships?

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Available measurement systems

• The ships obtain their position through GPS and ship

sensors

• The replenished – leader – ship transmits position

and heading through the AIS standard

• Automatic Identification System (AIS)

– Digital VHF radio communication

• Dynamic updates :

< 14 knots: every 4 s > 14 knots: every 2 s

• Mandatory broadcast for all

ships > 65 feet

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Control problem

• Stabilize the follower to the leader using only position

measurements – output feedback control problem

• No paths or trajectories are present – the leader is

manoeuvring freely

• No model is available for the leader – can be an any

(unknown) ship

• The problem suggests a leader-follower coordinated

solution with limited feedback

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Control approach

• Design a tracking control law
• Design a state observer for the follower
• Design an error observer
• Derive the leader states
• Semi-globally bounded

closed-loop errors

Leader

• bserver

Follower Leader Follower

• bserver

Error

• bserver

Control (Kyrkjebø et. al 2005)

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Control overview

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Experimental illustration

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Conclusions

• Choose your tool depending on the application!

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References

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The 5th IFAC/EURON Symposium on Intelligent Autonomous Vehicles, Lisbon, Portugal, July 2004.

• BROCKETT, R., Asymptotic stability and feedback stabilization, Differential Geometric Control Theory, Eds. R.W. Brockett,

R.S. Millman and H.J. Sussmann, Birkhäuser, pp. 181-191, 1983.

• BØRHAUG E., PAVLOV A., PETTERSEN K. Y., Cross-track formation control of underactuated autonomous underwater

vehicles, Group Coordination and Cooperative Control, Springer Verlag, 2006.

• BØRHAUG E., PETTERSEN K. Y., Cross-track control for underactuated autonomous vehicles, Proc. 44th IEEE Conference
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Cybernetics, Trondheim, Norway, 2002.

• FREDRIKSEN, E., PETTERSEN, K., Global K-exponential way-point manoeuvring of ships: Theory and Experiments,

Automatica, Vol. 42, No. 4, pp. 677-687, 2006.

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Symposium on Nonlinear Control Systems Design, Lake Tahoe, CA, USA, IFAC, pp. 595 - 600, 1995.

• KYRKJEBØ, E., PETTERSEN, K. Y., Output synchronization control of Euler-Lagrange systems with nonlinear damping

terms, Proc. 44th IEEE Conference on Decision and Control, Seville, Spain, 2005

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Journal on Control and Optimization, March 2005.

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Orlando, Florida, Dec. 2001, pp. 940-945.

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