Guidance and Navigation of Autonomous Underwater Vehicles

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Departments of Mathematics, University of Hawai i at

1 Guidance and Navigation of Autonomous Underwater Vehicles Monique Chyba - November 9, 2015 Departments of Mathematics, University of Hawai i at Mānoa Elective in Robotics 2015/ Control of Unmanned Vehicles

2 Autonomous Underwater vehicles Oceans cover three-fourths of the earth and have a large influence on the global climate. They have a great impact on the lives of human beings, plants and animals. They are a huge source of mineral resources. However, many of these resources still remain untapped and potentially unknown. With the advancement of technology, man has stepped on moon. Several unmanned missions have been carried out successfully to the outer reaches of space. Scientists have been successful in sending robots as far as Mars. However, planning a successful voyage to the abyssal world still remains elusive for researchers. Executing a manned voyage to the deep sea is extremely risky because of the unknown environment. Moreover, the oceanic environment is not ideal for humans, as the ambient pressure is unbearable at depths as shallow as 200m. Thus unmanned underwater robots prove to be an ideal platform to perform deep sea research.

1886. The term torpedo comes from the Torpedo fish, which is

3 Whitehead Automobile Fish Torpedo The Whitehead torpedo was the first self-propelled or locomotive torpedo ever developed The term torpedo comes from the Torpedo fish, which is a type of ray that delivers an electric shock to stun its prey

4 Sample AUVs Figure: Bluefin Robotics autonomous underwater vehicle (AUV) Figure: Semi-AUV for Intervention Missions, UH Manoa Figure: DEep Phreatic THermal Figure: Omni-Directional

5 Typical control problems As the missions increase in complexity, there is an increased demand in terms of autonomy of the vehicle. Given a prescribed mission, a critical component to the autonomy of the underwater vehicle is its ability to design its own motion. 1 Stabilization: Design the control u (as a function of state, or time, or possibly both) so that a desired state is a stable equilibrium. 2 Output tracking: Obtain the control u so that the output follows a specified trajectory. 3 Motion planning: Steer the state from an initial state to a final one. 4 Optimal control: Do any of the above while minimizing some cost function, e.g., time, or control energy.

6 Motion Planning for AUVs Highly complex: nonlinear system Optimization: time and/or energy Under-actuated: design of the vehicle and/or actuator failure Focus on the structure of the system: Geometric control AUVs have proven to be an asset in many areas of oceanographic research. With this ever increasing role, research has turned its focus to examining the control of such vehicles. This research interest can be addressed via the implementation of control strategies developed using a differential geometric approach. This architecture permits to exploit the inherent nonlinear structure of AUVs and other mechanical systems.

7 AUVs Mechanical System AUVs fall into the class of simple mechanical systems; their Lagrangians are of the form kinetic energy minus potential energy. Geometric control theory provides a useful framework for the study of simple mechanical systems. The equations of motion for a rigid body submerged in a real fluid subject to external potential and dissipative forces can be written as a Forced Affine Connection Control System (FACCS) on the differentiable configuration manifold, Q = R 3 SO(3), as seen in F. Bullo and A. D. Lewis. Geometric Control of Mechanical Systems. Springer, 2005.

8 Rigid Body in a Real Fluid We consider water to be a real fluid (versus an ideal fluid) in order to emphasize the inclusion of the dissipative terms in the equations of motion. Real fluid: Fluid which is viscous and incompressible with rotational flow Figure: ODIN in its pool environment

9 Second-order Forced Affine Connection Control System (FACCS) An FACCS is a 5-tuple (Q,, D, Y = {Y 1,, Y m }, U), where Q is the configuration manifold for the system, is an affine connection defined on Q D is a regular linear velocity constraint such that restricts to D, Y is a set of input vector fields, usually taken to be the external control of the system U R m is the control set in which the input controls take their values, We refer to the set Y as the set of input control vector fields.

10 Assumptions for our AUV 1 The origin of the body-fixed frame to be C G. 2 The body to have three planes of symmetry with body axes that coincide with the principal axes of inertia. The kinetic energy metric is actually a Riemannian metric given on Q = SE(3): (b, R): ( ) M 0 G = 0 J with M = mi 3 + M f and J = J b + J f where J b, M f and J f are diagonal.

11 Equations of motion for a rigid body submerged in a real fluid The equations of motion for a rigid body submerged in a real fluid subjected to external forces can be written as γ γ = G # (P(γ (t))) + G # (F (γ (t))) + 6 i=1 I 1 i (γ(t))σ i (t), where G # (P(γ (t))) represents the potential force from gravity and the potential force from the vehicle s buoyancy, G # (F (γ (t))) represents the dissipative drag force, I 1 i = G # π i = G ij X j, which may be( represented ) as the i th column of the matrix I 1 = M 1 0, and σ 0 J 1 i (t) are the controls.

12 The Modified Connection The magnitude of the drag force acting on a rigid body is proportional to the square of the velocity of the body (i.e. D = C D ρaν ν ). Thus, we are able to define the new connection in the following way Xi X j = { Di G ii X i Xi X j i = j i j, since we know Xi X i = 0. The equations of motion for the forced affine connection control system become γ γ = G # (P(γ (t))) + 6 i=1 I 1 i (γ(t))σ i (t). The above system is a second order affine connection control system (ACCS) on SE(3).

13 Underwater Mission As long as the vehicle has the ability to directly control each of the six DOF, the motion planning problem is easily addressed, and trajectories can even be optimized with respect to a cost function such as time or energy consumption. However, AUVs are mechanical systems which inevitably malfunction for one reason or another; batteries fail, actuators quit and electronics short out. Considering the loss of actuator(s) or specific vehicle design, if the vehicle does not have direct control on one or more DOF, we consider it to be under-actuated. In this scenario, the motion planning problem is more difficult, as some final configurations may not be realizable.

14 Optimization As we give a vehicle more responsibilities, assuming all else is constant, it will require an increase in efficiency. From a practical point of view, efficiency is measured time or energy consumption. Emergency avoidance places a heavier weight on time minimization, while long duration observation missions will require more energy efficiency.

15 Geometric Optimal Control 1 Theoretical analysis: Maximum principle ; Bang-bang arcs; Singular arcs Optimal control structure: Combination of bang-bang arcs and singular arcs. Question: Finite number of switchings? No. 2 Numerical algorithm: Direct methods (rewriting of the optimal control problem as a finite dimensional optimization problem); Indirect methods (based on the application of the maximum principle and are usually called single or multiple shooting methods)

16 Bang-bang ans singular arcs From a practical point of view, bang-bang controls correspond to thrusters operating at their extreme values (maximum acceleration or deceleration of the AUV in a prescribed direction), while singular controls correspond to low thrust application for small corrections (to maintain a prescribed orientation for instance). It is then clear that both types of control will be necessary. Optimal Control Structure Combination of bang-bang arcs and singular arcs. Question: Finite number of switchings? Answer: No, Chattering

19 Comparison 1 Optimal strategy: time optimal, singular arcs, high number of switchings! 2 Pure motion strategy: time consuming, piecewise constant controls!

20 Implementation The geometric control theory generates continuous controls as a function of time, whereas ODINs input requires a piece-wise constant control structure over discretized time intervals. The reason for this type of input is based on the combination of the refresh rate of the controller, the voltage to thrust relation used for the thrusters and an effort to keep the thrusters operating in a steady state to reduce their transient output response. In addition to the control structure, we must also link the piece-wise constant thrusts via a linear junction since it is impossible for a physical actuator to change outputs instantaneously.

21 SWITCHING TIME PARAMETRIZATION ALGORITHM Remark: a translational displacement can always be achieved by a thrust strategy with a single switching time. We extend this idea by imposing the structure of the control strategy. We fix the number of switching times along the trajectory, preferably to a small number, and we numerically determine the optimal trajectory from these candidates.

26 Experiments

27 Experiments

28 Experiments

29 Experiments

30 Introduction Equations of Motion Experiments Motion Planning Optimal Control Kinematic Motion Pau

31 Experiments

32 Experiments MOVIE

33 Kinematic Reduction Solutions to the motion planning problem can be designed using a geometric reduction procedure. The kinematic reduction is a first order control system on the configuration space Q whose controlled trajectories are also controlled trajectories of the second order dynamic system, possibly up to reparameterization. Of particular interest for the under-actuated scenario, we will consider kinematic reductions of rank one which are called decoupling vector fields. Characterization of a kinematic reduction for the FACCS which includes the external dissipative and restoring forces experienced by a real vehicle does not exist.

34 Kinematic Reduction Let Σ dyn = (Q,, Y, R 6 ) be a C affine connection control system with Y having locally constant rank. A driftless system Σ kin = (Q, X, R m ) ( m < 6) is a kinematic reduction of Σ dyn if 1 X is a locally constant rank subbundle of TQ and if, 2 for every controlled trajectory (γ, u kin ) for Σ kin, there exists u dyn such that (γ, u dyn ) is a controlled trajectory of Σ dyn. The rank of the kinematic reduction Σ kin at q is the rank of X at q.

35 Decoupling Vector Fields Definition A decoupling vector field for an affine-connection control system is a vector field V on M having the property that every reparametrized integral curve for V is a trajectory for the affine-connection control system. More precisely, let γ : [0, S] M be a solution for γ (s) = V (γ(s)) and let s : [0, T ] [0, S] satisfy s(0) = s (0) = s (T ) = 0, s(t ) = S, s (t) > 0 for t (0, T ), and (γ s) : [0, T ] TM is absolutely continuous. Then γ s : [0, T ] M is a trajectory for the affine-connection control system. Additionally, an integral curve of V is called a kinematic motion for the affine-connection control system.

36 Characterization of Decoupling vector Fields A necessary and sufficient condition for V to be a decoupling vector field for an affine-connection control system is that both V and V V are sections of Y. Notice that if Y = TM (i.e. the ACCS is fully-actuated) then every vector field is a decoupling vector field, and if Y has rank k = 1 (i.e. the ACCS is single-input) then V is a decoupling vector field if and only if both V and V V are multiples of Y 1. In the under-actuated setting, decoupling vector fields are found by solving a system of homogeneous quadratic polynomials in several variables.

37 Kinematically Controllable An interesting question is the minimal number of inputs vector fields which we need in order to reach any configuration from the origin using exclusively kinematic motions. Definition A submerged rigid body in an real fluid is said to be kinematically controllable if every point in the configuration space SE(3) is reachable from the origin via a sequence of kinematic motions. In order to determine whether our system is kinematically controllable we need to determine the involutive closure of the set of decoupling vector fields.

38 Results Theorem If the set of decoupling vector fields contains at least one translational control vector field and two distinct rotational control vector fields, then the submerged rigid body in a real fluid is kinematically controllable. Corollary A three input rigid body submerged in a real fluid with one translational and two rotational input is kinematically controllable. A four input rigid body submerged in an real fluid with at least two rotational inputs vector fields is kinematically controllable. A m input rigid body submerged in an real fluid with m {5, 6} is kinematically controllable.

39 Dynamical Control F V i t = flow along the integral curve of the vector field V i for a duration [0, t]. Initial configuration=η init, final configuration=η final. Solve the kinematic motion planning problem by concatenating the flows of the decoupling vector fields from η init to η final : F ɛ kv ak t k F ɛ 1V a1 t 1 (η init ) = η final. For each j {1,..., k}, choose a C 2 -reparameterization τ : [0, t j ] [0, t j ] such that τ (0) = τ ( t j ) = 0 (so that the kinematic motion begins and ends at rest, zero velocity). Define γ : [0, t j ] Q as the integral curve t F ɛ j V aj t. Then the dynamic control σ : [0, t j ] R m is defined by m σ α (t)i 1 α (γ τ(t)) = (τ (t)) 2 Vaj V aj (γ τ(t))+τ (t)v aj (γ τ(t)). α=1 If we let u : [0, T ] R m be the control formed by the concatenation of σ 1,..., σ k, then (γ, u) is a controlled trajectory for Σ dyn.

40 Summary 1 Compute decoupling vector fields for an AUV submerged in an ideal fluid subject to no restoring forces. 2 Show that the dissipative force of viscous drag can be absorbed into the affine connection. This allows to partially extend the notions of kinematic reduction and decoupling vector field to a real fluid and express this mechanical system as an ACCS using the new affine connection. 3 With this extension, calculate the decoupling vector fields for an AUV submerged in a real fluid subject to no restoring forces. 4 Determine the dynamic control strategy from the kinematic motion. 5 Compensate for the restoring forces in an ad-hoc way.

41 Implementation 1 Calculate the continuous control strategy σ(t) for the dynamic system by use of the geometric theory. 2 Discretize σ(t) to obtain an implementable piece-wise constant control structure σ 0 (t). 3 Upload σ 0 (t) to ODIN s CPU. ODIN autonomously implements σ 0 (t) in full open-loop. On-board, ODIN converts σ 0 (t) from a 6-dimensional control to an 8-dimensional control. The 8-dimensional control is converted from force (N) to voltage (V). The voltage controls are send to the eight independent thrusters. Position and orientation data are collected. 4 Collected data are post-processed and analyzed.

42 Autonomous Underwater Vehicle for Basin Exploration NASA funded Deep Phreatic Thermal Explorer (DEPTHX) and Environmentally Non-Disturbing Under-ice Robotic Antarctic Explorer (ENDURANCE). Both projects involve deploying an AUV to survey, integrating simultaneous localization and mapping technology, the underwater environment of Lake Bonney in Antarctica (ENDURANCE), and a group of five sinkholes of Sistema Zacatn (DEPTHX) in preparation and anticipation for and opportunity to explore the moon of Jupiter, Europa, for life in its icy oceans. A major focus of our work will be the ability of the AUV to map the environment as efficiently as possible during the descent to gain as much knowledge as possible for the environment for future descents as well as to help the AUV while ascending after the exploration took place. Notice that a precise mapping is also critical when deciding which locations are best for sampling.

43 Sample Basin Mission Figure: map of the Church Sink (Leon County, FL, USA - vertical section) Figure: Six thrusters - Five DOF Figure: Mission 1 Figure: Mission 2

44 Sample Basin Mission Figure: Horizontally motion in a tunnel with obstacles Figure: Six DOF Figure: Five DOF Figure: Controls

Actuators failure We assume two failures: first suppose the loss of two thrusters, for instance due to falling rocks or water infiltrations, so only a control reallocation is needed.

45 Actuators failure We assume two failures: first suppose the loss of two thrusters, for instance due to falling rocks or water infiltrations, so only a control reallocation is needed. Then we suppose the vehicle to suffer the failure of two more thrusters, therefore only half of its thrusters is still available. In particular, the AUV will lose its actuation in surge and pitch. Figure: Tunnel using roll and heave Figure: The calculated trusters signals Decoupling vector fields: axial motions(heave and yaw) and

vehicles motion planning particularly efficient also in case of

46 Map and sample the summit of the Loihi submarine volcano Figure: The control redundancy due to the presence of eight thrusters keeps the vehicles motion planning particularly efficient also in case of thruster failures. Figure: Graphic three-dimensional model of the main shape of Loihi

47 Creating Bathymetric Maps Using AUVs in the Magdalena River Figure: Thruster s Configuration Figure: Simulated Magdelena Environment

48 Underwater Mission Approximately 90% of the goods traded throughout the world are carried by the international shipping industry. Currently, there are more than 50,000 merchant ships trading internationally. This fleet belongs to more than 150 nations, and employs over one million seafarers. With a high volume of ships arriving from worldwide destinations, it is of utmost importance to monitor and protect the ports that facilitate each country s trading market. To this end, it has become an interest of border police and port authorities to examine the hulls of ships for potentially dangerous attachments, for instance explosives, before they enter the harbor.

49 Bulbous Bow Region of a Ship Figure: USS George H.W. Bush (CVN 77). The bulb is a protrusion from the front of the hull, positioned to sit just below the design water line. Hydrodynamically, the bulb serves the purpose of reducing the height of the bow wake of the vessel, thus decreasing hull drag and achieving better efficiency.

50 Automating Ship Hull Surveys Currently, these tasks are performed by highly-skilled human divers. Such labor intensive work introduces fatigue and poses multiple potential risks to the divers. In particular, in the presence of hazardous elements these risks can be life-threatening. In an effort to provide a more comprehensive and cost-effective solution to this problem, engineers have been working on automating this process by employing Autonomous Underwater Vehicles (AUVs).

51 Current techniques Use of a Doppler Velocity Logger (DVL) to allow the vehicle to lock onto the ships hull and perform fixed-distance, hull-relative motions to complete a survey. Highly-effective in inspecting the sides of the hull. If the DVL loses lock on the ships hull, the AUV loses localization, and thus is unable to complete the mission without intervention.

52 Survey the uniquely-shaped bulbous bow Figure: Front view of a bulbous bow. Also depicted is the trajectory to survey the sides of the Figure: Overall path for complete bulb. inspection of a bulbous bow.

53 Piece-wise constant control strategy for the concatenation of the motion Time Applied Thrust (6-dim.) (N) Time Applied Thrust (6-dim.) (N) 0 (0,0,0,0,0,0) 42.3 (30.99, 0, 30.04, 0, 0, 0) 0.9 (0.45, 0, 1.2, 0, -2.91, 0) 43.5 (30.99, 0, 30.04, 0, 0, 0) 5.9 (0.45, 0, 1.2, 0, -2.91, 0) 44.4 (32.7, 0, 30.04, 0, 0, 0) 6.8 (-10.7, 0, 7.6, 0, -2.91, 0) 45.1 (32.7, 0, 30.04, 0, 0, 0) 31.9 (-10.7, 0, 7.6, 0, -2.91, 0) 46 (32.7, 0, , 0, 0, 0) 32.8 (4.56, 0, 5.02, 0, -2.91, 0) 46.3 (32.7, 0, , 0, 0, 0) 37.7 (4.56, 0, 5.02, 0, -2.91, 0) 47.2 (-30.99, 0, , 0, 0, 0) 38.6 (-32.7, 0, 30.04, 0, 0, 0) 48.4 (-30.99, 0, , 0, 0, 0) 41.4 (-32.7, 0, 30.04, 0, 0, 0) 49.3 (0,0,0,0,0,0)

54 Figure: Strategy Three: Concatenation of the semi-circle trajectory and the horizontal survey. Solid (blue) line represents actual evolution, dash-dot (red) line represents the theoretical evolution. φ θ ψ Experiments X (N) 0 x (m) Z (N) 0 y (m) M (N m) 0 z (m) Time (s) Time (s) Time (s)

55 Pau Mahalo nui loa

Autonomous Underwater Vehicles: Equations of Motion

Autonomous Underwater Vehicles: Equations of Motion Monique Chyba - November 18, 2015 Departments of Mathematics, University of Hawai i at Mānoa Elective in Robotics 2015/2016 - Control of Unmanned Vehicles