Docking and Undocking a Modular Underactuated Oscillating Swimming Robot
aa r X i v : . [ c s . R O ] F e b Docking and Undocking a Modular Underactuated OscillatingSwimming Robot
Gedaliah Knizhnik and Mark Yim
Abstract — We describe a docking mechanism and strategy toallow modular self-assembly for the Modboat: an inexpensiveunderactuated oscillating swimming robot powered by a singlemotor. Because propulsion is achieved through oscillation, ori-entation can be controlled only in the average; this complicatesdocking, which requires precise position and orientation con-trol. Given these challenges, we present a docking strategy and amotion primitive for controlling orientation, and show that thisstrategy allows successful docking in multiple configurations.Moreover, we demonstrate that the Modboat is also capableof undocking and changing its dock configuration, all withoutany additional actuation. This is unique among similar modularrobotic systems.
I. I
NTRODUCTION
Modular and self-reconfigurable aquatic systems showgreat potential for versatility and utility in modern appli-cations. Individual modules can be made inexpensive, andvarious capabilities can be introduced through cooperationand the modular structure, rather than through the units. Thiscan enable significantly less expensive and more versatilerobots.A single aquatic robot may carry a single flow sensor, forexample, providing little utility. But a system of robotic unitsattached together, each carrying a flow sensor, can provide adetailed picture of the flow field in a particular region witha precise spacial distribution.Very little work, however, has been done on modular andself-reconfigurable robots in aquatic applications. The TEMPproject explored building structures from rectangular roboticmodules that could dock together [1] [2], and the Roboatsproject considered docking multiple surface modules [3] [4],but we are not aware of other docking systems for aquaticsurface operations. Both of these systems involve expensive,holonomic units. Underneath the water’s surface, AMOURhas demonstrated docking capabilities [5], as has ANGELS[6], but both of these systems can reconfigure in only 1D.CoCoRo [7] has also demonstrated docking capabilities [6],but these have been used exclusively for connecting to a fixedpower station, not other modules.In prior work [8] we introduced the Modboat: an inexpen-sive oscillating swimming robot powered by a single actuatorand ready for modular self-assembly; passive flippers convertthe rotation of its two concentric bodies (see Fig. 1) into aforward translation. We showed that pauses in the oscillationcould be used to steer a Modboat towards a target as adiscrete-time single-integrator [9]. Critically, all parts ofthe Modboat oscillate in order to achieve propulsion, and
The authors are with the GRASP Laboratory, University of Pensylvannia,Philadelphia, PA 19104. [email protected] T − π π π d cap φ cap Fig. 1. A simplified diagram of the Modboat, with the docking pointsindicated and labeled and the approximate area of acceptance of the − π/ dock (not to scale) in green. The top body is shown in black, the bottombody in blue, and the flippers in gray. We have approximately measured φ cap ≈ ◦ and d cap ≈ . This is approximately one diameter of theModboat top body. The other dock points have similar areas of acceptanceby symmetry. while swimming its orientation can be controlled only in theaverage . This makes docking — a process involving precisecontrol of both position and orientation — very difficult. Itis not sufficient to plan a path accounting for the propulsion-orientation coupling as with other non-holonomic systems.In this work, we present a mechanism and strategy thatallows a Modboat to dock to/undock from other units withoutadditional actuation. This is unique, as all other similarsystems require additional actuators for docking and latching[1] - [7]. We also describe a novel motion-primitive forcontrolling the instantaneous orientation, which allows achoice of docking configurations. This allows the Modboatto self-assemble and reconfigure in 2D with an affordableand easily scalable system.The rest of this work is organized as follows. We discussthe docking mechanism and the strategy for using it in Sec-tions II and III, respectively. In Section IV we discuss a newmotion primitive for controlling instantaneous orientation,and in Section V we demonstrate how the tail can be usedfor undocking. We experimentally demonstrate docking inSection VI and discuss the results in Section VII.II. D OCKING M ECHANISM
The Modboat is composed of two concentric bodies, asseen in Fig. 1. The bottom body provides a mounting pointfor the flippers, which rotate freely, and for the tail. The topody is larger and contains the control hardware; it is the primary body of the Modboat. The robot weighs . , andthe top body has a diameter of . . A full descriptionof the design can be found in our prior work [8].Note that the flippers do not protrude from the top bodyfootprint, which prevents them from mechanically interferingwith neighboring docked boats. The tip of the tail, however,does protrude. The purpose for this will be discussed inSection V.Permanent magnets are advantageous for low-cost dockingapplications, as they can passively align and guide the mod-ules when docking. The resulting dock consumes no energywhen active, which is also advantageous due to limitedonboard energy capacity. Magnetic docks also provide areasonably large area of acceptance.Four permanent magnets are therefore placed at the fourcardinal points inside the top body, as shown in Fig. 1.This allows the circular Modboat to form a square latticewhen docked. Each Modboat is assigned either a magnetic N or S designation, and each magnet is placed so that thedesignated pole faces outward; this gives the boats polarityrather than the docks. Nongendered/hermaphroditic magneticdocking setups are possible, but they show lower areas ofacceptance [10]. A valid square lattice can easily be formedfrom such polarized boats (consider a chess-board), and it isreasonable in large self-assembly applications to assume aninfinite number of modules of either polarity.Each magnet is rated for a pull strength of ; two layersof / acrylic and the air gap due to a flat magnet ina cylindrical shell results in an effective docking strengthof . . Each Modboat weighs . (or . ), and thisdocking strength has been experimentally verified to besufficient to hold two boats together when swimming.The area of acceptance of a magnetic dock depends greatlyon the fluid and robot velocities, as well as on magneticinteractions between the four magnets on each boat involved.This makes it difficult to explicitly quantify. We roughlyapproximate the area of acceptance, however, as a ◦ wideregion extending to ≈ from the boat’s edge, whichis shown in Fig. 1; another magnet entering this regionwill be captured. This assumes the fluid and the boatsare approximately at rest, and the boats are approximatelyaligned in orientation .We define a dock by its orientation ψ ∈ {− π/ , , π/ , π } relative to the ”front” of the Modboat (at ) as shown in Fig.1, and by a subscript identifying each boat. The orientationof the Modboat top body in the world-frame is given by θ .Since the dock directions are given relevance by theModboat’s preferred front direction, we want to be able tochoose which pair of docks are used on each side. Themethod for doing so is discussed in Sections III-A and IV, In fact, our experiments show that even a small velocity pointed awaydrastically reduces the area of acceptance. In our previous work [8] [9] we use θ to refer to the orientation ofthe bottom body and θ t to refer to the top body orientation. We use θ here for the top body to avoid unnecessary subscripts, as only the top bodyorientation is relevant in this work. T C R R R R d app d dock φ max Fig. 2. A diagram of the docking strategy regions used by a mobile boat(not shown) to dock to the target boat T . The strategy is shown in detail inFig. 3. Region dimensions are not to scale. and we define: Definition 1. A front-dock involves the ψ b = 0 dock. Definition 2. A side-dock involves the ψ = ± π/ docks. Definition 3. A rear-dock involves the ψ = π dock. III. D
OCKING S TRATEGY
Conventional techniques for self-assembly generally addsingle units to the assembled whole rather than combininglarge sub-assemblies [11]. This is often sensible from adynamics perspective, since individual modules are often farmore mobile than a conglomerate. Moreover, the dynamicsof multiple Modboats swimming together have not beenexplored to date, so we consider docking as defined inDefinition 4, with nomenclature as in Definitions 5 and 6.
Definition 4.
The docking problem is guiding a singleswimmer from initial location ( x , y ) , orientation θ , anddesired dock ψ b to dock with a target at location ( x t , y t ) ,orientation θ t , and desired dock ψ t . The target cannot swimor adjust its orientation. Definition 5.
The mobile boat is the active swimmer beingguided in the docking problem.
Definition 6.
The target boat is the boat to which the mobileboat is docking. The target boat may not be fixed but isassumed to be stationary.
A single targeted Modboat may be able to adjust itsorientation to assist in the dock, simplifying the trajectoryfor the mobile boat. A target that is docked to other units,however, cannot do so. Thus the mobile boat is tasked withachieving both the position and orientation necessary fordocking.
A. Strategy
When swimming, the Modboat is controlled by a headingfrom it to a target waypoint. We create a docking strategyby using a discrete and non-holonomic approximation of theattractive well used by Saldana et. al [12]. This strategy canbe summarized in three stages: wim towards C Swim towards T Orient to dockDoneEnter R Enter R and (3)Dock Enter R Enter R Fig. 3. An illustration of the docking procedure followed by the mobileboat. The procedure moves linearly unless an abort is triggered, at whichpoint it resets. Distancing:
Achieve distance from the target so thata perpendicular approach towards the desired dock isavailable.2)
Homing:
Approach the target along the perpendicularto establish a drift velocity toward the desired dock.3)
Orienting:
Control the orientation to present the de-sired dock on the mobile boat to the desired dock onthe target boat.Fig. 2 demonstrates the principle elements of this dockingstrategy, which are implemented as a finite state machine asshown in Fig. 3. From an initial state in R the mobile boat isdriven towards a virtual waypoint at C defined by a distance d app , the orientation of the target θ t , and the desired targetdock ψ t . This serves to establish an (almost) perpendicularapproach to the target dock.Once it is within R (i.e. within the cone and further than d app from the target), the mobile boat is directed towards thetarget at T . This waypoint is maintained within both R and R , serving to establish a drift velocity towards the target.Finally, within R the mobile boat orients itself to presentthe desired dock ψ b using the method described in SectionIV. The Modboat cannot simultaneously control its orienta-tion and the direction of its translation, so it relies on thepreviously established velocity to drift the rest of the way tothe target while the orientation is being set. The transitionto R is based both on a distance criterion (being closer tothe target than d dock ) and an angular velocity criterion thatwill be discussed in Section IV.Because the Modboat is underactuated, we insist thattransitions occur in order without skipping. Skipping R ,for example, could create a scenario where the drift velocitywas insufficiently directed at the target for a successful dock.However, if at any point the Modboat returns to R the dockattempt is aborted and the procedure resets.IV. O RIENTATION C ONTROL
In prior work [9] we focused on controlling the heading ofthe Modboat, defined at discrete time intervals as the averageorientation over a cycle . Self-assembly, however, requiresthat the instantaneous orientation of the boat matches that of the target. We therefore need to bridge the transition fromswimming — treated discretely — to docking, which mustbe treated continuously.An additional motion primitive to control the instantaneousorientation provides this bridge. We can use the bottom bodyas a reaction wheel to control the orientation of the topbody. Because angular acceleration activates the flippers andinduces translation, this motion primitive is of limited use,but we theorize that it can be useful when:1) The orientation can be controlled slowly , allowing slowmotion that does not activate the flippers OR2) The orientation needs to be controlled only briefly , sothat induced translation is unlikely to matter.Item 1 may be relevant in docking scenarios where thetarget boat is free to rotate. It can spend a relatively long timeachieving its orientation while the mobile boat approaches.For the docking problem as defined in Definition 4, however,we rely instead on item 2; the orientation control maneuverfor the mobile boat will occur in a short period before thedock, minimizing the effect of any induced translation.Orientation control is therefore achieved by defining a PIDcontroller as in (1) and (2), where θ is the orientation of thetop body and α is commanded as angular acceleration to themotor. To minimize flipper activation, we limit the allowableangular velocity of the motor during this mode to / s . e = θ d − θ (1) α = K p e + K d dedt + K i Z edt (2)It is important to note that — because oscillations inducetranslations in the Modboat system — it is desirable that theorientation controller be overdamped, but this induces a slowrise time. A long drift period could compensate for this, butit would require higher precision in the approach heading,which we cannot guarantee. TABLE IO
RIENTATION CONTROL
PID
COEFFICIENTS
Mode K p K i K d Aggressive 30 0 30Non-Aggressive 10 0 30
Since magnetic docking can compensate for angular er-rors, we therefore choose a gain-scheduled controller model.The initial orientation controller uses an aggressive set ofPID coefficients to quickly rise most of the way to the desiredorientation. Once the orientation is within . of thetarget value and the angular velocity is below / s weshift to an overdamped non-aggressive controller to preventoscillations, should the drift period last long enough. ThePID coefficients are given in Table I.Figure 4 shows an example of the performance of thegain-scheduled control approach as compared to using onlythe non-aggressive controller. The non-aggressive controllertakes to reach the . threshold, while the hybridcontroller takes only without a significant increase in Fig. 4. Sample step response of the non-aggressive (dashed) and gain-scheduled (solid) orientation controllers to a step input of ≈ π (dotted).Gain-scheduling speeds up the rise-time without an appreciable increasein oscillations. This results in more drift overall but is comparable in theinitial few seconds, which is sufficient for docking. The vertical lines markreaching within . of the desired orientation. oscillation. We do observe more translation when using thegain-scheduled control approach, but this is offset by thesignificantly faster correction time when used for dockingas per item 2. Thus, even though the hybrid controllereventually drifts nearly further, it drifts less in thetime needed to rise to the desired orientation.In order to facilitate docking using orientation control, weadd an angular velocity criterion to the strategy in SectionIII-A. The distance transition criterion to R is supplementedwith (3), where θ b is the orientation of the mobile boat, θ des is the desired orientation that will align the mobile and targetdocks, and the error term is wrapped to ( − π, π ] .This guarantees that the transition occurs when the propul-sive oscillations are already turning the Modboat in thedesired direction, easing the presentation of the desired dock ψ b . The minimum angular velocity ω trans ensures this willhappen quickly, and the orientation controller then brings thisrotation to a halt at the desired angle. | ˙ θ b | ≥ ω trans and ˙ θ b ( θ b − θ des ) < (3)V. U NDOCKING
Although we have presented a strategy for docking, modu-lar self-assembly and reconfiguration also requires the abilityfor modules to undock from each other. In the case of theModboat, this is achieved without additional actuation byusing the tail.Fig. 1 shows a simplified diagram of the Modboat, inwhich the tail is shown as an extension of the bottom body.The flippers are designed to sit within the footprint of the topbody to prevent mechanical interference with neighboringmodules [8]. The tail, on the other hand, is designed toprotrude at its peak, with its curve parametrized so thatdistance to the center of rotation increased linearly with theangle. This means that two neighboring Modboats (such asshown in Fig. 5) can rotate their tails into each other tomechanically force separation of the magnetic docks.The motors used have sufficient torque to separate twodocked boats, but the separation provided by two tails is in-sufficient to completely disable the magnetic attraction. Thusa re-dock occurs immediately after separation if insufficientforce is applied by the tail.Considering Fig. 5, we must examine two cases. Assumingwe focus on boat A , either A B (a)
A B (b)Fig. 5. (a) Two Modboats A and B docked together, with ψ A = − π/ and ψ B = π/ . (b) To undock, A would rotate its tail by π/ and B would rotate its by − π/ , forcing the boats apart.
1) Boat B is itself docked to an unspecified number ofother modules (not shown in the figure).2) Boats A and B are the only boats in the structure.In case 1, the presence of additional modules preventsboat B from rotating and increases its effective inertia duringthe separation from A . We have experimentally determinedthat an angular velocity of / s is sufficient to guaranteeseparation in this case, when both A and B rotate their tailsthrough at least π/ . Thus, considering Fig. 5 withoutloss of generality, A rotates its tail by π/ and B rotatesits by − π/ , each at / s , which separates the boatsand imparts enough force to cause them to drift apart.In case 2, however, additional energy is spent rotatingand moving boat B , which lowers the imparted separationdistance and prevents successful undocking. In this case, asufficient strategy is instead for A to initiate a swim with φ (the centerline of the motor rotation [9]) given by thedock connected to B (in Fig. 5 this is φ = π/ , so thatthe tail faces − π/ ), while B rotates its tail to face A .This swimming motion creates thrust away from B usingthe flippers, which adds to the tail-based separation andensures successful undocking. This method can also be usedsuccessfully in case 1.It is significant to note that in both cases we observedthat, while a tail velocity of / s resulted in an immediatere-dock in the original configuration, a velocity of / s consistently resulted in a re-dock in a new configuration. Incase 1 boat A re-docked rotated by − π/ in of tests while B maintained its orientation, while in case 2 boat B would also rotate by π/ .VI. E XPERIMENTS
Experiments were performed in a . × . × . tankof water equipped with an OptiTrack motion capture systemthat provided planar position, orientation, and velocities at . A MATLAB interface recorded the data, calculatedthe heading and control parameters, and sent commands tothe Modboat for course correction. A set of base parameters (cid:2) T T A φ (cid:3) = (cid:2) . . (cid:3) was used for all ex-periments, with pause control applied over these parameters(see [9]). Testing was done in calm water with no externalcurrents applied.To simulate docking to a larger assembly of boats thetarget boat was fixed to the bottom of the tank. This targetwas used for the majority of testing, but we also evaluateddocking with a single free-floating non-actuated target. This ABLE IIR
ESULTS OF DOCKING EXPERIMENTS
Succ./TriesTarget ψ b,des ψ t ψ b d dock [m]Static π/ − π/ π π/ − π/ OCKING TIMES FROM . AWAY , PRESENTED AS µ ± σ Approach [s] Drift [s] Latch [s] Total [s]Modboats . ± . . ± . ± TEMP [1] was intended to evaluate if significant differences existbetween the two cases, and if the docking strategy wouldfunction for a target that could be pushed away by the watermoving with the incoming mobile boat.The results of the docking tests are presented in TableII; front-docking and side-docking were performed reliably,while rear-docking was largely unsuccessful and so notattempted for the free-floating target. The time taken to dockfrom . is shown in Table III. Without loss of generalitythe desired target dock was set to ψ t = 0 for all experiments,and the desired mobile dock ψ b was varied. The orientationtransition velocity was set to ω tran = 0 . / s and theapproach distance to d app = 1 . . The approach cone wasset to φ max = 40 ◦ wide.Two measures of success are presented in Table II. Thefirst records whether the mobile boat hit the desired dock onthe target ψ t with any dock ψ b , which measures the successof the approach method. The second records if the correctdock ψ b was used, which evaluates the orientation controlmethod. If the mobile boat hits the wrong target dock ψ t the ψ b criterion is marked as a failure, regardless of whichmobile dock was involved.The mobile boat was started at a random location inthe experimental tank and allowed to run for 3 minutes oruntil a dock occurred, whichever came first. The desiredangle θ des maintained by orientation control was dynamicallydetermined by the line from the mobile boat to the targetdock point. This allowed it to compensate for misalignmentin the approach vector.VII. D ISCUSSION
In our prior work [9], the Modboat required waypoints in radius for reasonable performance. In this work,however, the target waypoint (another boat) was only . in radius. Nevertheless, our overall docking strategy and ◦ wide conical approach region allowed the Modboat toconsistently approach and hit the target. This can be seenin Table II, where the success rate for hitting the target -1 -0.5 0 0.5 1 1.511.52 Fig. 6. Sample trajectory (blue) for docking with a free-floating target.Dashed regions correspond to those in Fig. 2, but the target (black) isenlarged by the radius of the mobile boat. accurately (as indicated by docking with the correct ψ t ) is when rear-docks are excluded (and otherwise). Anexample trajectory is shown in Fig. 6.This performance is due to a combination of factors aimedto combat the poor precision in the Modboat’s low-costand non-holonomic design. Discrete steering control providespoor position tracking, because the heading is calculatedbased on the current position but achieved at a differentone; a wide conical approach provides plenty of room forthe heading to converge. An observed downside of thisapproach is that heading towards the target does not drivethe mobile boat away from the edges of the cone, whichleads to occasional aborts and retries due to drifting out ofit. Additionally, the magnetic dock points are able to com-pensate for misalignment of the approach vector, and thedynamic choice of θ des based on the mobile boat’s locationplaces the desired mobile dock in the most favorable orien-tation for capture. This allows the mobile boat to achieveinitial contact at an offset point on the target, yet still becaptured by the correct target dock ψ t . This produces a success rate for front and side docking, which our methodperforms reliably.In evaluating docking performance with a free-floatingtarget, our chief concern was that the target might be pushedaway by the oncoming mobile boat. Since we do not yethave a strategy for station-keeping with Modboats, thiswould mean that an alternative strategy would be neededto seed structures from single boats. In reality however,this effect is negligible; magnetic attraction is far moreprominent, and the free-floating target is able to rotate andtranslate to compensate for misalignment when a front-dockis attempted.Free-floating targets do present an issue for side-docking,however. The orientation-control phase of the docking ma-neuver brings the mobile side-docks ψ b = ± π/ close to thetarget, and if the approach vector is misaligned they may endup closer to the target side-docks than to the front-dock. Fora fixed target this is not a problem, as orientation control willrotate through that configuration before magnetic attractiondominates. But if the target is able to rotate — as the free-oating target is — then it will do so and a side-to-side dockresults ( ψ t = ± π/ and ψ b = ∓ π/ ).We compensate for this by increasing the dock transitiondistance d dock , as shown in the second half of Table II. Theextra space allows the orientation controller to rise to thedesired orientation before magnetic forces are significant,and the dock occurs as before.Rear-docking, however, has proven largely unsuccessfulusing our approach (Table II). This occurs because theorientation-controlled maneuver to perform a ◦ rotationcompletely arrests the drift towards the target, and theoscillations that occur serve to then push the mobile boataway. This results in a lower ψ t success rate and a ψ b successrate below . Lowering d dock can increase the chance ofcapture but leaves insufficient time to complete the maneuver,resulting in erroneous side-docks.We can, however, still accomplish docking to all four dockpoints. As mentioned in Section V, the tail can be used torotate the dock point on the mobile boat by ± π/ when thetarget is docked to at least one other boat. This techniquesucceeded in 15 out of 16 attempts, so it can be considereda successful primitive for reorienting the mobile dock. Thusrear-docking can — in all cases except seeding a structure —be replaced by a front or side-dock followed by a tail-basedreorientation.Docking times following our strategy compare favorablyto those displayed by the TEMP project [1], as shown inTable III. Although we introduce an additional drift phasefor orientation control, time is saved due to the lack of activelatching. This allows the Modboat to dock in a comparabletime frame to other modular systems.VIII. C ONCLUSIONS
In this work we have presented a design and a strategyfor docking the Modboat — a single-motor oscillating swim-ming robot — to other Modboats. We have also demonstratedthat the Modboat design is capable of undocking from adocked position by using its tail to break the magneticconnections to its neighbors. Together, these results allowModboats to self-assemble and reconfigure into potentiallylarge structures following a 4-connected lattice. This dockingand undocking is accomplished with only one actuator permodule — without the need for additional actuators for(un)docking — which the authors believe is the first instancefor any modular self-reconfigurable robot system.We have developed an additional motion-primitive forthe Modboat, which allows control of the instantaneousorientation of the top body. In combination with our dockingstrategy, this enables the Modboat to dock in either a front-facing or a side-facing configuration despite being able tocontrol its orientation only in the average while swimming.This method cannot reliably achieve successful rear-facingdocks, but we can achieve rear-docks by side-docking andthen reorienting using the tail. The combination of thesemethods allows Modboats to achieve a full set of poten-tial configurations, with applications for orientation-sensitivepayloads such as flow sensors. In this work we have considered docking to/undockingfrom only a single target. In future work we will examinethe full range of docking and undocking scenarios andassemble a large structure. We also plan to explore steeringfunctionality to drive the Modboat towards the center of theapproach cone and decrease the number of retries needed,and to consider the effects of disturbances such as fromexternal flows. We have also verified experimentally that thedocking mechanism described in this work is sufficient toallow multiple Modboats to swim together, but this typeof motion has not been explored. In future work we willconsider the various types of motion allowed by two ormore Modboats swimming while linked together in variousorientations. A
CKNOWLEDGMENT
We thank Dr. M. Ani Hsieh for the use of her instrumentedwater basin in obtaining all of the testing data.R
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