GUIDANCE AND CONTROL OF AUTONOMOUS UNDERWATERVEHICLES

Michael R. C. AndonianDepartment of Mathematics

University of Hawai’i at ManoaHonolulu, HI 96822

Abstract

This paper discusses the design and practical use of efficient trajectories for an autonomousunderwater vehicle. The vehicle model that will be presented is based on the NASA funded DeepPhreatic Thermal Explorer (DEPTHX) and the Omni-Directional Intelligent Navigator (ODIN).The underlying mathematical framework is developed using geometric control theory and differ-ential geometry. Briefly, the control strategies are developed are based on integral curves definedin the configuration space of the rigid body and kinematic reductions of rank one. Presented arethe control schemes for several legitimate mission scenarios and how pre-planned trajectories ex-ploiting the inherent geometry of the vehicle and the environment overcome thruster failure.

Introduction

Autonomous robots not only represent the next great milestone for science, but their practicaluses span a wide range. Autonomous underwater vehicles (AUVs) are such an example. In partic-ular, AUVs are being used to explore hostile environments far to hazardous for humans or mannedvehicles. These environments are of particular interest to researchers studying extremophiles, mi-croorganisms capable of surviving in extraordinary circumstances. Such organisms may poten-tially harbor secrets to the origins of life, both here on Earth and even plausibly extraterrestrial life.For this reason, scientists involved in space exploration and the pursuit of extraterrestrial life haveturned their attention to AUVs. Projects such as the NASA funded first generation Deep PhreaticTHermal eXplorer (DEPTHX) and the second generation, the Environmentally Non-DisturbingUnder-ice Robotic ANtarctiC Explorer (ENDURANCE) are examples of state-of-the-art AUVsdeployed to survey a given underwater environment in preparation for an anticipated opportunityto explore Europa, a moon of Jupiter, which is believed to house oceans beneath its icy crust [6].Other approaches to exploring hostile underwater environments include sending teams of AUVsto survey hydrothermal vents [1], using sonar sensors to create maps, eventually to photographhydrothermal vent sites [9], and by predetermining trajectories for the AUV, while sampling thewater to choose a viable site to study [7]. Still, the fact remains that AUVs have to explore haz-ardous environments, which may compromise the safe return of the vehicle, while still retrievingsatisfactory data such as a high resolution mapping of the surrounding region being studied. Pre-cautionary techniques are thus initially implemented to improve the vehicle’s capability to return tosafely and intact in the event of unexpected damages which may occur during the mission. Basedon the reality of the situations AUVs may face, mission design and preparation are critical whenstudying these dangerous areas and for the vehicle to return with minimal damage. Presented herewill be several strategies for practical missions based on realistic environments such as the under-water volcano off the coast of the island of Hawai’i, Loihi. These missions were developed with

the intent to keep the AUV close to walls for high resolution mapping and show a mission can stillbe accomplished even after thruster actuation failure. The following presents an overview of themodel used for the design and simulation of trajectories which will also in turn be presented. Themodel includes a brief introduction into the mathematics behind the wheel and a description of thevehicle. One should note that the model and the vehicle description are separate entities, meaningthe model presented can be applied to any vehicle; we chose the hybrid DEPTHX-ODIN as thevehicle the model would be applied to in order to illustrate this fact. Following the discussion ofthe model will be two scenarios simulated based on the mathematics and realistic environments.The conclusion will summarize the work done and emphasis will be placed on the versatility of themodel and its benefits.

The AUV Model

To begin the discussion of the motion planning problem for an AUV, we consider a submergedrigid boy in a viscous fluid. The initial question is how will one express the location of the vehicle.Typically, two right-handed, orthogonal, frames of reference are considered, an inertial (earth-fixed) frame and a body-fixed frame. In addition, we chose the vertical axis to be positive inthe direction of gravity. For our purposes, we can naturally express the rigid body as a pointon the configuration space (manifold), Q, known as the special euclidean group, denoted SE(3).

Elements in SE (3) look like[

R b01×3 1

]where b = (b1,b2,b3)

t ∈ R3 represents a position from

the origin of the inertial frame to the origin of the rigid body frame and R ∈ SO(3) is a 3 by 3rotation matrix representing the orientation of the rigid body. Next, we denote ν = (ν1,ν2,ν3)

t

and Ω = (Ω1,Ω2,Ω3)t as the linear and angular velocities in the body-fixed frame. Using two

different linear transformations (omitted here), one can transform between the linear and angularvelocities of the two reference frames. The resulting kinematic equations for the rigid body are

b = Rν (1)

R = RΩ (2)

where · : R3→ so(3). so(3) is called a Lie algebra and it is, in this case, correlated to SO(3)as the space of skew symmetric matrices in SO(3)

Typically, one constructs the equations of motion for the situation. This construction is lengthyand will not be presented, but for a classical treatment, see [8]. These equations of motion takeinto account external forces and moments of a submerged rigid body in viscous fluid and are givenby

Mv+D(ν)v+Cor (ν)v+g(η) = σ(t) (3)

η = J (η)v (4)

where M is the added mass and inertia matrix, D(ν) is the drag forces, Cor (ν) is the Coriolisand centripetal forces, g(η) corresponds to the restoring forces and moments, J (η) corresponds tothe linear transformations (matrices) mentioned previously to transform linear and angular veloc-ities between reference frames, thus η is a vector representation of the position and orientation ofthe rigid body in the inertial frame, σ(t) represent the external forces and moments acting on the

vehicle, and finally, v = (ν ,Ω)t . In a moment, we will see that σ (t) actually represents the inputcontrols for our system in order to realize a desired motion. However, there exists a geometricrepresentation of these equations that is coordinate invariant. The derivation is very long but canbe found in [11], which also provide a detailed treatment of the mathematics. Here, we will brieflymention an analog of these equations.

The model is further developed by considering the representation of the kinetic energy bya metric, which is a unique Riemannian metric, G, on Q. Furthermore, there is a Levi-Civitaconnection ∇ that is unique and is associated with G. These last two statements are useful theoriesin differential geometry. This affine connection ∇ provides one with an appropriate means fordiscussing accelerations on Q. In fact, the connection ∇ is a projection of elements in the tangentspace of Q onto itself. This means everything we need to discuss exists naturally on Q. So, givena curve γ(t) on Q, which corresponds to configurations (position and orientation) as a function oftime, there are γ ′(t), which represent the linear and angular velocities of the configurations, andthere exists ∇γ ′(t)γ

′(t), which represents accelerations, given as

∇γ ′(t)γ′(t) =

(ν +M−1(Ω×Mν)

Ω+ J−1(Ω× JΩ+ν×Mν)

)(5)

where, M and J are the mass and inertial matrices with added mass and inertia included.To consider the submerged rigid body, gravitational, buoyancy, and viscous dissipative (drag)

forces are considered. The model continues its development as seen in [4], [11], and [12].Byreparametrizing the curves γ(t) so the vehicle will be at rest initially and between kinematic mo-tions, a concatenation of these motions is practical. Ultimately, this leads us to an equation for thenecessary controls,

m

∑i=1

σi(t)I−1i (γ τ (t)) =

(τ′ (t))2

∇VV (γ τ (t))+ τ′′ (t)V (γ τ (t)) (6)

where σi represents the kinematic control vector (corresponding to control signals to be sent tothrusters for desired motions), I−1

i is the ith column of the inverse of the generalized inertia matrixI, γ τ (t) is the reparameterization of the curve γ (t) such that the vehicle begins and stops with

zero velocity, V =m

∑i=1

hiI−1i , hi ∈R, represents a vector field, where the set

I−1

im

i=1 , 1≤m≤ 6, is

a given set of input control vector fields and a subset ofI−1

i6

i=1, and ∇ is the modified connectionof the original connection defined to take into account the fluid dynamics. Here, we will simplynote that decoupling vector fields (DVFs) are a kinematic reduction of rank one. Consider

∇γ ′(t)γ′(t) =

6

∑i=1

σi(t)I−1i (γ (t)) (7)

which is an analogous, coordinate invariant, geometric interpretation of equation 5. Equation 6is the dynamic control system called the affine connection control system (ASSC). Thus, speakingroughly, a kinematic reduction of the dynamic system is when every trajectory of the kinematicsystem, γ ′ (t) , also satisfies the dynamic system. The primary reason for going through this workis because there is well develop theory on the kinematic, 1st order system, bu not so much for the2nd order system. As a result, integral curves of DVFs correspond to kinematic motions which

we concatenate to develop our full trajectories. Thus, using the DVFs to solve for σ allows oneto simulate a desire trajectory by solving for v and η in equations 3 and 4. Frankly, an exampleof DVF would be V1,3 = h1I−1

1 + h3I−13 + h4I−1

4 , which would correspond to a surge-heave-pitchmotion. When the vehicle is fully actuated, every vector field is decoupling, meaning any trajectorycan be realized. However, when the vehicle is under-actuated, it is difficult to know which vectorfields are decoupling. [11] gives a list of the DVFs for any under-actuated situation.

Lastly, the vehicle model we used to showcase the above theory was a hybrid of the ODIN andthe DEPTHX. This is to show that the theory is flexible to any vehicle. Here are the parameters ofthe model vehicle:

m : 1360 kg jΩ1b : 1008 kgm2

mν1f : 3750 kg jΩ2

b : 1008 kgm2

mν2f : 3750 kg jΩ3

b :1114 kgm2

mν3f : 6222 kg j

Ω1,2,3f : 0 kgm2

Diameter (m) : (1.9,1.9,1.9)

Center of Gravity (CG): (0,0,−7) mm Center of Buoyancy (CB): (0,0,−7) mm

B = ρgV : 1215.8 N W = mg :1214.5 N

drag 1 (D1): 500.0 kg/m2 drag 2 (D2): 500.0 kg/m2

drag 3 (D3): 777.8 kg/m2 drag 4 (D4): 280.0 kg/m2

drag 5 (D5): 280.0 kg/m2 drag 6 (D6): 318.2 kg/m2

Table 1: AUV DEPTHX-ODIN model parameters

Moreover, we assumed the vehicle to have a thruster configuration similar to ODIN’s, as seenin the following figure. Since the rigid body is expressed in SE (3), which is 3 dimensional, it hassix degrees of freedom (DOF). To have motion in all six DOF from thruster actuation is consideredfully actuated and allows the rigid body to realize any kinematic motion. In addition, this model isover-actuated, since there is redundancy with the thruster controls. A simple linear transformationis performed on the control signals from the eight thrusters to yield the six dimensional controlsignal, σ .

Figure 1: Thruster configuration for and DEPTHX-ODIN vehicle. Such a configuration gives therigid body motion in the six DOF and is thus fully actuated.

Missions

Once everything desired is known, masses, inertias, the DVFs for a desired trajectory, etc.,simulations were made in Mathematica (and MATLAB) script. For a user defined DVF, the aboveparameters, and the correct function τ (t) that has the vehicle start and stop with zero velocity, anydesired trajectory was realized through a computer simulation by computing the necessary controls,σ , and solving numerically equations 3 and 4 for v = (ν ,Ω)t and η . To show the application ofmathematics presented and the versatility of the code and model, we simulated various missions.These mission designs were based on legitimate situations an AUV may face when exploringunderwater caves and the Loihi seamount, an active underwater volcano off the coast of the Islandof Hawai’i. In addition, minimization of energy and the event of under-actuation were also kept inmind during the design process to emphasize the practicality of these trajectories.

Mission 1: Cave Exploration

This mission saw many changes throughout the semester. The following figure shows theevolution of the mission.

a) b) c)

Figure 2: Trajectories for cave exploration. Left most image (a) is a spiral motion down with aconstant radius, (b) is a spiral motion down with an expanding radius, and (c) is a concatenation ofmotions.

The first mission, seen in Figure 2(a), was developed to explore the walls of an underwatercave (or basin) with cylindrical features. Here, the dimensions of the cave are 40 m in radius and150 m in depth. If the intent of the mission is to successfully map the cave, staying close to thewalls to get a high resolution image is necessary. This choice of trajectory allows the vehicle toremain close and closely inspect the wall surface. The trajectory depends on solely on motionscorresponding to sway, heave, and yaw (the kinematic motion of such is generated by the linearcombination of the vector fields I−1

2 , I−13 , and I−1

6 , respectively). In other words, the vehicle maylose the ability of motion in the other DOF, but it must have the necessary thruster configuration to

perform a sway, heave, and yaw. Since the DEPTHX-ODIN model has the thruster configurationcorresponding to full actuation every vector field is decoupling and as such, it can perform thistrajectory, even in certain under-actuated scenarios. For example, the vehicle may lose actuationin the thrusters which produce the sway motion, but with a simple rotation, the thrusters originallyused to produce a surge can be used for a sway. This is a direct exploitation of the symmetry of thevehicle. If the vehicle were to be fixed with an exceptional array of sensors and a SLAM algorithm,the detailed 3-D map of the basin could be acquired in a few attempts.

The second mission seen in Figure 2(b) is very similar to the previous mission. The goals ofminimizing energy and staying close to the wall of the basin were kept as well. In this scenario, thebasin is similar to a cone, with the top, initial radius of 20 m and expands to 60 m at 100 m deep.Like the previous mission, the vehicle must have actuation in sway, heave, and yaw to successfullyaccomplish this motion.

The last mission involving cave exploration can be seen in figure 2(c). In this scenario, wecontinue to make use of the inherent mathematical framework provided and consider a missionwhere we lose actuation in surge an pitch. The vehicle begins its descent into a basin using thehelical motion previously described. Upon reaching the bottom of the 110 deep, 15 m in radiussinkhole, it continues the exploration by entering a tunnel 5 m in diameter and 15 m in length witha slight arc. Exploration of the tunnel is accomplished by following a parabolic-like curve usinga surge-heave-pitch motion (the kinematic motion of such is generated by the linear combinationof the vector fields I−1

1 , I−13 , and I−1

5 , respectively). However, once inside the tunnel, the vehicleloses actuation of four thrusters resulting in the loss of motion corresponding to surge and pitch.The thruster loss can be seen in Figure 3.

Figure 3: ODIN-like vehicle with loss of four thrusters (grey). An AUV’s health monitor candetect loss of thrusters.

With the loss of surge and pitch, every vector field is no longer decoupling. Thus, our set ofinput vector fields becomes I−1

1 = I−12 , I−1

2 = I−13 , I−1

3 = I−14 , and I−1

4 = I−16 , i.e., we can only

sway, heave, roll and yaw. From [10] and [11], the DVFs are V = h2I−12 + h4I−1

4 (heave and yaw)and V = h1I−1

1 + h2I−12 + h3I−1

3 (sway-heave-roll).As a result, the vehicle exists by use of a roll-heave motion. The vehicle’s thrusters roll the

vehicle to the desired angle and then apply thruster required to heave. Geometrically, the pathexiting the tunnel is same as the path entering. Now, to make even more use of the mathematics,upon exiting the tunnel, instead of using a pure heave motion to ascend, we designed a morecomplex concatenation of motions. The kinematic motions concatenated were sway-heave-roll

motions seen in red in Figure 2(c). Here, the descent took one hour and required 58 kJ of energy,the tunnel exploration fully actuated took 3 minutes and required 2616 J, the exit of the tunnelunder-actuated also took three minutes but 2254 J were needed, and finally the ascent took one hourand required 10.6 kJ. Of course, lengthening the mission time would also decrease the amount ofenergy required. This mission and a more detailed discussion of the mathematics can be found in[3].

Mission 2: Loihi Seamount

Most recently, we have begun the development of a simulation of a mission to map the summitof Loihi, an active underwater volcano off the southeast coast of the Island of Hawai’i. Naturally,this mission would take a long time to accomplish with the current parameters of the vehiclemodel we have used for the exploration of the basin presented above. The vehicle must be largerand faster in order to successfully accomplish the mission within a reasonable time frame andalso be unaffected by ocean currents. However, the reader should be reminded that what is beingpresented here is a model, meaning things such as vehicle size, weight, thruster configuration,etc, can easily be modified and accounted for. That being said, our current simulations do usethe same DEPTHX-ODIN hybrid. However, due to external perturbations that may compromisethe vehicle’s planned trajectory, ocean currents for instance, we have opted to add an additionalfeedback controller into the simulations. Details about the feedback controller implemented canbe found in [5]. This mission involves a survey of a region of the summit referred to as Pele’sPit. The full expedition concatenates various trajectories into a single mission. Unfortunately, thesimulations for this mission have yet to be complete, so visuals of the trajectories used are not yetavailable, but the mission is as follows.

Initially, the vehicle is fully actuated as it descends 1000 m just below the summit of Loihi. thevehicle then arcs over the ridge and heads straight for the crater labelled (a) in the following Figure4. Using the helical trajectory, this time a surge-heave-pitch motion, the vehicle successfully canmap the walls of this region. The exit of this crater is a pure heave motion. Next, the vehicle movestowards the ridge labelled (b) below. Upon reaching this ridge, it experiences thruster failure andloses two of its horizontal thrusters, those corresponding to the sway motion. The vehicle stillmanages to arc over the ridge (not using pure motions) and heads towards the crater labelled (c)in Figure 4. At this point, the surge-heave-pitch helical motion cannot be realized. However, dueto the symmetry of the vehicle, a simple rotation can cause the dead thrusters to now correspondto sway, meaning we revive the surge-heave-pitch helical motion. To remain in the “spirit of theproblem”, we continue to consider the surge motion as “lost”, but instead note that we are stillactuated in sway and roll. Thus, we use the original helical motion described previously. Finally,the vehicle simply leaves the seamount. Currently, we are developing this mission further to thepoint where the vehicle becomes so under-actuated, only pure motions are possible.

A

BC

Figure 4: Loihi Seamount. Modifications made to original image. The red dot is the starting pointof the mission. Numeric values are depth in fathoms. Image from [10]

Conclusion

The missions designed here can certainly be implemented on vehicles, to an extent, such asthe ODIN or the DEPTHX. The idea is to place these pre-planned trajectories into the softwarearchitecture of an AUV as mission plans or contingency plans. Since the DEPTHX is consideredunder-actuated (roll and pitch have been neutralized), the vehicle cannot perform some of themore complex motions such as the spiral. On the other hand, the DEPTHX also does not fullyexploit its capabilities. By use of a method termed iterative exploration, the vehicle must descendand ascend repeatedly in order to successfully map an environment. In their defense, the schemeworks remarkably well if there is no prior information of the environment. However, if a team hasa rough idea of the geography of the environment prior to the mission, an adequate implementationof the presented control strategy would be more useful in two crucial ways. The first is the factthat a geometric pre-planned trajectory can allow a vehicle such as the DEPTHX to remain closeto the walls. In return, any array of sensors (in particular sonars) would yield high quality data.Second, and as a consequence of the previous fact, the number of attempts to obtain a satisfactorymap would be minimized to an extent, thus resulting in energy conservation. But the DEPTHXis a special vehicle, and it is thus important to remark again that the paper presents a model. Inother words, this type of geometric control planning can be applied to and used for any vehicle.Moreover, any vehicle design can use this theory to discuss the event of under-actuation on theirvehicle, rather than implementing an abortion system which has the vehicle surface immediately.In fact, these schemes become necessary when deciding if the vehicle can even return safely.As seen in the mission designs and simulations above, even in the event of under-actuation, thevehicle was capable of performing tasks and, if conditions are “nice”, complete a given mission.This is a critical benefit of the theory presented here. In addition, since these trajectories areglobal path planners, they will not be deceived by the geometry of a region. For example, ifan AUV were to enter an enclosed dome relying on sensors, it is highly plausible the vehiclemay get trapped. However, there are some drawbacks to pre-planning these trajectories. Thesetrajectories (and indirectly the control scheme in its entirety) are feed-forward controllers anddo not account for uncertainties. Thus, some means of tracking is necessary, which is why theLoihi mission included the feedback controller. In spite of this, the benefits of the techniques

presented do continue to accumulate. When coupled with a feedback controller, the system overallbecomes very robust. As stated previously, the AUV can implement the techniques presented inthis paper to analyze its health and environment to decide what trajectories are can be realizeddue to thruster actuation or the geometry/geography of the environment. If the vehicle’s sensorsbecome useless, due to common scenarios such as murky water or electrical failures, the reliabilityof the feedback controller falls but the trajectories developed by this control scheme can still behighly useful for dead reckoning. In addition, the AUV can diagnose an unfavorable situationand determine if a mission can be completed or if (and how) it can return to the surface safely.Finally, these simulations and the our associated geometric control strategy approach can allowone to determine the pros and cons of a certain thruster configuration. By using our code in theprocess of developing an AUV, one can visually see what the vehicle can and cannot do, i.e., onehas the knowledge of permissible motions for the AUV. This is very beneficial for designing anAUV without sophisticated and expensive sensors.

Acknowledgements

I’d like to thank NASA and the Hawai’i Space Grant Consortium for allowing me this opportu-nity to extend my education far beyond the classroom. Because of this project, I have seen and feltfirst-hand what research is like, not to mention the acquiring of valuable public speaking skills. I’dlike to thank Dr. Ed Scott and Dr. Luke Flynn for their genuine words of encouragement, even ifit was only in passing. A special thank you to Marcia Rei Sistoso, who was always exceptionallyhelpful and understanding, and the rest of the staff in POST 501, Joana Choy, Amber Imai, BiancaSoriano, and Lauren Kamei for their additional help. A tremendous thanks to Dario Cazzaro, Ser-gio Grammatico and Luca Invernizzi; none of these simulations would have been possibly withouttheir high quality knowledge as engineers and computer programmers. Thanks to Dr. GeorgeWilkens for supplemental help regarding the mathematics involved with this entire project. Mostof all, my gratitude and thanks goes to my mentor Dr. Monique Chyba, especially for offering methis opportunity in the first place. Her guidance and concern as a professor and educator has dra-matically raised my level as a student and undergraduate mathematician. Thanks again to everyonefor their enthusiasm, support, and this wonderful opportunity.

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