1 Trajectory tracking, Path Following and Formation Control of Autonomous Marine Vehicles Kristin Y. Pettersen Erik Kyrkjebø Even Børhaug Department of Engineering Cybernetics, NTNU, Norway
2 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
3 Trajectory tracking, path following and manoeuvring Fossen 2002
4 Tracking approaches • Path following: Aguiar et al. 2004 • Trajectory tracking:
5 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
6 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
7 Underactuated path following Underactuated control of mechanical/marine vehicles Acceleration constraint
8 Nonholonomic systems Second-order nonholonomic constraint First-order nonholonomic constraints f � � , � , t � � 0 Holonomic constraints f � � , t � � 0 Goldstein 1980
9 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
10 Underactuated control Underactuated control of mechanical/marine vehicles The gravitation and buoyancy vector is important for the stabilizability
11 Underactuated control Brockett’s necessary condition (1983) Coron and Rosier (1994) Pettersen and Egeland 1996
12 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 Way-point – the dynamic task manoeuvring Control challenge: – Surge control is straightforward – Control both sway and yaw without sideway control force.
13 Underactuated path following Underactuated ships LOS methods much used in ship control practice y k � y � d � t � � arctan � x � x k � y k � y � d � t � � arctan � � �
14 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
15 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.
16 Underactuated path following Underactuated ships •Full 3DOF nonlinear ship model:
17 Underactuated path following Underactuated ships
18 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 d dynamics
19 Underactuated path following Underactuated ships The control laws give
20 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 � � � d u � u d � where is the LOS angle d
21 Underactuated path following Underactuated ships Experimental results: • Video
22 Underactuated path following Underactuated ships 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 )
23 Underactuated path following Underactuated Autonomous Underwater Vehicles (AUVs) Desired path: LOS angles: Path following control objective: Intermediate LOS control objective:
24 Underactuated path following Underactuated Autonomous Underwater Vehicles (AUVs) Speed autopilot AUV dynamics Steering autopilot
25 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.
26 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.
27 Underactuated path following Underactuated Autonomous Underwater Vehicles (AUVs) • Closed loop system is a cascade: Globally bounded UGES UGAS + ULES Cascaded system is UGAS + ULES. (Application of Panteley et. al. 1998 )
28 Underactuated path following Underactuated Autonomous Underwater Vehicles (AUVs) Control strategy: summary • Proposed controller guarantees tracking of the LOS angles, θ LOS and ψ LOS , and desired surge speed u d (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.
29 Underactuated path following Underactuated Autonomous Underwater Vehicles (AUVs) • Video
30 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 )
31 Underactuated path following Underactuated Autonomous Underwater Vehicles (AUVs) • Video
32 Multi-object control What happens when we want to control a group of objects?
33 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 Formation control – Surveillance
34 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
35 Synchronization? Synchronization Cooperation Coordination Nominal behaviour Error situation Nominal behaviour Error situation Cooperation Coordination degree of synchronization
36 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?
37 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. x 1 x 2 x 1 x 2 x 2 x 1 x 1 x 2
38 From single-object to multi- object LOS control - Desired Speed Synchronization Synchronization speed variables controller autopilot Desired LOS Steering path autopilot guidance - - ( Borhaug et. al 2006 )
39 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.
40 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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