Asymmetric Steering Hydrodynamics Identification of a Differential Drive Unmanned Surface Vessel⁎

11th IFAC Conference on Control Applications in 11th IFAC Conference on Control Applications in Marine Systems, Robotics, and Vehicles 11th IFAC Conference on Control Applications in Marine Systems, Robotics, and Vehicles Available online at www.sciencedirect.com Opatija, Croatia, September 10-12, 2018 Marine Systems, Robotics, and Vehicles 11th IFAC Conference on Control Applications in Opatija, Croatia, September 10-12, 2018 Opatija,Systems, Croatia, September 10-12, 2018 Marine Robotics, and Vehicles Opatija, Croatia, September 10-12, 2018

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IFAC PapersOnLine 51-29 (2018) 207–212

Asymmetric Steering Hydrodynamics Asymmetric Asymmetric Steering Steering Hydrodynamics Hydrodynamics Identification of a Differential Drive Asymmetric Steering Hydrodynamics Identification of a Differential Identification of a Differential Drive Drive  Unmanned Vessel Identification of Surface a Differential Drive Unmanned Surface Vessel Unmanned Surface Vessel  Unmanned Surface Vessel ∗∗∗ G. Peeters, ∗∗ R. Boonen, ∗∗∗ ∗∗∗ M. Vanierschot, ∗∗∗

G. Peeters, Boonen, ∗ R. ∗∗ ∗∗∗ M. Vanierschot, ∗∗∗ ∗∗ ∗∗∗ G. Peeters, R. ∗∗ Boonen, M. Vanierschot, M. DeFilippo, P. Robinette, ∗∗ P. Slaets ∗∗∗ M. DeFilippo, P. Robinette, ∗ ∗∗∗ ∗∗ P. Slaets ∗∗∗ G. R. ∗∗ Boonen, M. Vanierschot, M.Peeters, DeFilippo, P. Robinette, P. Slaets ∗∗∗ ∗ M. DeFilippo, ∗∗ P. Robinette, ∗∗ P. Slaets ∗∗∗ ∗ SB PhD fellow at FWO, Faculty of Engineering Technology, KU ∗ SB PhD fellow at FWO, Faculty of Engineering Technology, KU SB PhD fellow at FWO, Faculty of Engineering Technology, KU Leuven, 3000 Leuven, Belgium(e-mail: [email protected]). Leuven, 3000 Leuven, Belgium(e-mail: [email protected]). ∗ ∗∗ SB PhD fellow at FWO, Faculty of Engineering Technology, KU Leuven, 3000 Leuven, Belgium(e-mail: [email protected]). AUV Lab, MIT Sea Grant College, Cambridge, MA 02139, USA ∗∗ MIT Sea Grant Cambridge, MA ∗∗ AUV 3000 Leuven, Leuven, Belgium(e-mail: [email protected]). AUV Lab, Lab, MIT Sea Grant College, College, [email protected]) Cambridge, MA 02139, 02139, USA USA (e-mail: [email protected], (e-mail: [email protected], ∗∗ ∗∗∗ AUV Lab, Sea GrantTechnology, College, [email protected]) Cambridge, MA 02139, USA (e-mail: [email protected], [email protected]) Faculty ofMIT Engineering KU Leuven, 3000 Leuven, ∗∗∗ of Engineering Technology, KU ∗∗∗ Faculty(e-mail: [email protected], [email protected]) Faculty ofBelgium(e-mail: Engineering Technology, KU Leuven, Leuven, 3000 3000 Leuven, Leuven, peter.slaets@kuleuven). peter.slaets@kuleuven). ∗∗∗ Faculty ofBelgium(e-mail: Engineering Technology, KU Leuven, 3000 Leuven, Belgium(e-mail: peter.slaets@kuleuven). Belgium(e-mail: peter.slaets@kuleuven). Abstract: This paper identifies the asymmetric steering characteristics of a Wave Adaptive Abstract: This paper the asymmetric steering characteristics of Abstract: This(WAM-V) paper identifies identifies steering of aa Wave Wave Adaptive Adaptive Modular Vessel deployedthe as asymmetric an Unmanned Surfacecharacteristics Vessel (USV). Differentially steered Modular Vessel (WAM-V) deployed as an Unmanned Surface Vessel (USV). Differentially steered Abstract: This paper identifies the asymmetric steering characteristics of a Wave Adaptive Modular as an Unmanned Vessel (USV).lateral Differentially steered propellersVessel create(WAM-V) a virtualdeployed rudder movement withoutSurface explicitly inducing rudder forces. propellers create(WAM-V) a rudder movement withoutSurface explicitly inducing lateral rudder steered forces. Modular deployed as an Unmanned Vessel (USV). Differentially propellers a virtual virtual rudder explicitly inducing lateral forces. However,Vessel acreate rotating propeller willmovement generate awithout small lateral force, depending on rudder its rotational However, a rotating propeller will generate aawithout small lateral force, depending on its propellers a virtual rudder explicitly inducing lateral forces. However, acreate rotating propeller will generate walk. small force, depending on rudder its rotational rotational direction and speed, also known as movement propeller Thelateral WAM-V USV uses two similar propellers direction and speed, also known as propeller walk. The WAM-V USV uses two similar propellers However, a rotating propeller will generate a small force, depending onthe its vessel rotational direction and speed, as propeller walk. Thelateral WAM-V USV uses two similar propellers to manoeuvre, hencealso theknown propeller walk effects are doubled. Consequentially, has to manoeuvre, hence the propeller walk are doubled. Consequentially, the vessel has direction andturning speed, as propeller walk.in The WAM-V USV uses two similar propellers to manoeuvre, hencealso theknown propeller walk effects effects are doubled. Consequentially, the vessel has asymmetric characteristics which result different steering behaviours when turning asymmetric turning characteristics which result in steering behaviours when turning to manoeuvre, hence the propeller walk effects aredifferent doubled. Consequentially, vessel has asymmetric turning characteristics which result in different behaviours when turning port or starboard. Heading measurements and virtual ruddersteering movements suffice the for identifying port or starboard. Heading measurements and virtual rudder movements suffice for identifying asymmetric turning characteristics which result indodifferent steering behaviours when turning port or starboard. Heading measurements and virtual rudder movements suffice for identifying these turning characteristics at a certain speed. To so, a first order Nomoto model was chosen theseor turning characteristics at aa certain certain speed. To do do so, so, firstmovements order Nomoto Nomoto model model was chosen port starboard. Heading measurements andmodel virtual rudder forport, identifying these turning characteristics at To aaidentified: first order was chosen as identification model. Three varieties ofspeed. this were one forsuffice turning one for as identification model. Three varieties of this model were identified: one for turning port, one for these turning characteristics atvarieties a averages certainofspeed. To aforementioned dowere so, aidentified: first order Nomoto model was chosen as identification model. this one for turning port, one for turning starboard, and Three one that the model two cases. These offline identified turning starboard, and Three one that averages the two aforementioned cases. These offline identified as identification varieties of this were identified: one for turning port, one for turning starboard, onemultiple that averages the model two aforementioned cases. These offline identified Nomoto models model. canand serve objectives. They can be used for simulation purposes, which Nomoto models can serve multiple objectives. They can be used for simulation purposes, which turning starboard, and one that averages the two aforementioned cases. These offline identified Nomoto models can serve multiple objectives. They can be used for simulation purposes, which themselves can be used to test control algorithms offline. Moreover, the coefficients of the themselves can can be used used to test control control algorithms offline. Moreover, the coefficients coefficients of the Nomoto serve multiple objectives. They can beIntegral used forDerivative simulation purposes, which themselves can be test offline. Moreover, the of the Nomoto models model itself can to be used to tunealgorithms a Proportional (PID) controller. Nomoto model itself can be used to tune a Proportional Integral Derivative (PID) controller. themselves can be used test control algorithms Moreover, the coefficients of the Nomoto model itself can to be can usedalso to tune a Proportional Integral Derivative (PID) controller. Finally, the Nomoto models be used as a feedoffline. forward term in control algorithms. Finally, Nomoto be as term in algorithms. Nomoto model itself models can be can usedalso to tune a Proportional Integral Derivative controller. Finally, the the Nomoto models can also be used used as aa feed feed forward forward term in control control(PID) algorithms. © 2018, IFAC (International Federation Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Finally, theidentification, Nomoto models can alsoofbe used ashydrodynamics, a feed forward term in control algorithms. Keywords: asymmetric, steering, differential drive Keywords: identification, asymmetric, steering, hydrodynamics, differential drive Keywords: identification, asymmetric, steering, hydrodynamics, differential drive Keywords: identification, asymmetric, steering, hydrodynamics, differential drive 1. INTRODUCTION controller then changes the heading based on its desired 1. INTRODUCTION INTRODUCTION controller then changes the based on 1. controller thenlevels changes the heading heading basedcomplexity, on its its desired desired value. Several of accuracy, and thus can value. Several levels of accuracy, and thus complexity, can 1. INTRODUCTION controller then changes the heading based on its desired Several levels of accuracy, and thus complexity, can be used to model the hydrodynamical model of a USV. The rising maturity levels of test platforms, sensors, ac- value. be used to model the hydrodynamical model of a USV. The rising maturity levels of test platforms, sensors, acvalue. Several levels of accuracy, and thus complexity, be used to model the hydrodynamical model of a USV. A truncated Taylor-series expansion using odd terms as The rising maturity levels of test platforms, sensors, actuators, communication systems, and computing power A truncated Taylor-series expansion using odd termscan as tuators, communication systems, and computing computing power used to model the Abkowitz hydrodynamical model ofterms a but USV. A truncated Taylor-series expansion using odd as The rising maturity levels of test platforms, sensors, ac- be proposed by Abkowitz (1964) can be used, it tuators, communication and power unlock a wide spectrum ofsystems, purposes for Unmanned Surface proposed by Abkowitz Abkowitz (1964) can be used, but it unlock a wide spectrum of purposes for Unmanned Surface A truncated Taylor-series expansion using odd terms proposed by Abkowitz Abkowitz (1964) can be used, but as it tuators, communication computing power needs the identification of plenty of hydrodynamic parameunlock a wide spectrum ofsystems, purposesand for Unmanned Surface Vehicles (USVs). In the maritime sector, most applicaneeds the identification of plenty of hydrodynamic parameVehicles (USVs). In the maritime sector, most applicaproposed Abkowitz Abkowitz (1964) can be used, but it the by identification of plenty of hydrodynamic parameunlock a wide spectrum purposes for Unmanned Surface needs ters, needing different tests for different parameters Caccia Vehicles (USVs). In either theof maritime sector, most applications for USVs are militarily or environmentally ters, needing different tests for parameters Caccia tions for for USVs USVs are are either militarilysector, or environmentally environmentally the identification of plenty of hydrodynamic parameters, needing different tests for different different parameters Caccia Vehicles Inineither the most applicaet al. (2008). A robot-like vectorial model suits better for tions militarily oriented. (USVs). The USV thismaritime paper fulfilsora surveillance and needs et al. (2008). A robot-like vectorial model suits better for oriented. The USV in this paper fulfils a surveillance and different testsvectorial for different parameters Caccia et al.needing (2008). A robot-like model suits Here, better for tions for The USVs areineither militarily ora surveillance environmentally computer implementation and control design. the oriented. USV this for paper fulfils measurement-taking role both inland, harbour, and ters, computer implementation and control design. Here, the measurement-taking role for both inland, harbour, and et al. (2008). A robot-like vectorial model suits better for computer implementation and control design. Here, the oriented. The USV inrole this for paper fulfils aAdaptive surveillance and hydrostatics can be modelled under measurement-taking both inland, harbour, and hydrodynamics coastal waters. This USV is a Wave Moduhydrodynamics and hydrostatics can be modelled under coastal waters. This USV is a Wave Adaptive Moducomputer implementation and control design. Here, the hydrodynamics and hydrostatics can be modelled under measurement-taking role for both inland, harbour, and and the control can benefit from the coastal waters. This Marine USV is Advanced a Wave Adaptive lar Vessel (WAM-V) Research Modu(2013) different assumptions assumptions and control from the lar Vessel Vessel (WAM-V) Marine Advanced Research Modu(2013) different and hydrostatics can can beofbenefit modelled under different assumptions and the theproperties control can benefit from the coastal waters. USV is a Wave understanding of the physical these modelling lar (WAM-V) Marine Advanced Research (2013) hence making it This a WAM-V USV. In theAdaptive unmanned sail- hydrodynamics understanding of the physical properties of these modelling hence making it a WAM-V USV. In the unmanned sailassumptions and theproperties control canof benefit frominthe understanding of the physical these modelling lar (WAM-V) Marine Advanced Research assumptions. Different implementations are discussed I. hence making it avessel, WAM-V USV. In theto unmanned ing Vessel mode of the software needs control (2013) it,sailfor different Different implementations are discussed in I. ing mode mode of the the vessel, software needs tounmanned control it, it,sailfor assumptions. understanding of the physical properties of modelling assumptions. Different implementations arethese discussed inthe I. hence making it avessel, WAM-V USV. In theto Fossen (2011). A similar vectorial approach consists of ing of software needs control for example: the Mission Oriented Operating Suite (MOOS) Fossen (2011). A similar vectorial approach consists of the example: the Mission Oriented Operating Suite (MOOS) assumptions. are consists discussed inthe I. (2011).Different A similarimplementations vectorial approach of ing mode (2013) of the software needs to(IvP) control it, for Fossen mathematical modelling group (MMG) model, which modexample: the Mission Oriented Operating Suite (MOOS) Newman -vessel, Interval Programming Benjamin mathematical modelling group (MMG) model, which modNewman (2013) (2013) - Interval Interval Programming (IvP) Benjamin Fossen (2011). A similar vectorial approach consists of the mathematical modelling group (MMG) model, which modexample: the Mission Oriented Operating Suite (MOOS) els the effects of propeller, hull and rudder modularly. This Newman Programming (IvP) Benjamin et al. (2010) software. This MOOS-IvP software lets a els the effects of propeller, hull and rudder modularly. This et al. al. (2010) (2010) software. This MOOS-IvP (IvP) software lets aa mathematical modelling group (MMG) which models the approach effects of propeller, hull and modularly. This Newman (2013) - Interval Programming Benjamin MMG for a WAM-V USVrudder wasmodel, shown in Pandey et software. This MOOS-IvP lets behaviour based-controller steer the vessel software Brooks (1985). approach for USV was shown in behaviour based-controller steer the vessel Brooks (1985). els the effects of(2016), propeller, hull and rudder modularly. This MMG approach for aa WAM-V WAM-V USV was shown in Pandey Pandey et al.MOOS-IvP (2010) software. This MOOS-IvP software lets a MMG and Hasegawa however they did not take lateral behaviour based-controller steer the vessel Brooks (1985). The software is designed based on three archiand Hasegawa (2016), however they did not take lateral The MOOS-IvP software is designed based on three archiMMG approach for a WAM-V USV was shown in Pandey and Hasegawa (2016), however they did not take lateral behaviour based-controller steer the vessel Brooks (1985). into account. It is also possible to derive The MOOS-IvP software designeddriver basedparadigm, on three architecture philosophies: (i) a is back-seat (ii) a propeller forces forces into It is possible to derive tecture philosophies: (i) aa is back-seat driver paradigm, (ii) aa propeller and Hasegawa (2016), however did not and take lateral propeller forcesfunctions into account. account. It they is also also possible toheading derive The MOOS-IvP software designedand based on archiSISO transfer (e.g. between rudder tecture philosophies: (i) back-seat driver (ii) publish and subscribe middleware, (iii)paradigm, thethree aforemenSISO transfer functions (e.g. between rudder and heading publish and subscribe middleware, and (iii) the aforemenpropeller forces into account. It is also possible to derive transferwhich functions (e.g. between rudderofand heading tecture philosophies: a back-seatBenjamin driver (ii) a SISO or yaw-rate) diminishes the number parameters publish and subscribe(i)middleware, and (iii)paradigm, the aforementioned behaviour-based autonomy et al. (2010). or yaw-rate) which diminishes the number of parameters tioned behaviour-based autonomy Benjamin et al. (2010). SISO transfer functions (e.g. between rudder and heading or yaw-rate) which diminishes the number of parameters publish andissubscribe middleware, and (iii) the aforemento identify aa vessel. A discussion of the most comtioned behaviour-based autonomy Benjamin etwhich al. (2010). The latter implemented as an IvP-Helm, is an needed to vessel. A discussion of the most comThe latter latter is implemented implemented as an an IvP-Helm, IvP-Helm, which is an an needed or yaw-rate) which thesecond number of parameters needed to identify identify a diminishes vessel. Aand discussion oforder, the most comtioned behaviour-based autonomy Benjamin al. (2010). mon models: Nomoto’s first Norrbin’s The is as is additional MOOS application that decides etwhich the desired mon models: Nomoto’s first and second order, Norrbin’s additional MOOS application that decides the desired needed to identify a vessel. A discussion of the most common models: Nomoto’s first and second order, Norrbin’s The latter is implemented as an IvP-Helm, which is an non-linear first order and Bech’s non-linear second order, additional MOOS application that decides the desired speed and heading for the USV. A conventional PID non-linear first order and Bech’s non-linear second order, speed and and MOOS headingapplication for the the USV. USV. Adecides conventional PID mon models: Nomoto’s and(1982). secondThe order, Norrbin’s non-linear first order andfirst Bech’s non-linear second order, additional that A the desired can be found in Van Amerongen second order speed heading for conventional PID can in Van The second order  This research non-linear first and Bech’s (1982). non-linear was funded Flanders Foundation PID can be be found found in order Van Amerongen Amerongen (1982). Thesecond second order, order speed and heading for bythe USV.Research A conventional  This research funded by Flanders Research Foundation  This research was was funded by Flanders Research Foundation can be found in Van Amerongen (1982). The second order

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models also account for the overshoot, caused by the coupling between yaw rate and sway speed Technology (2015). The first order Nomoto model Nomoto (1957) adequately replaces the second order model as for the latter, one of the poles of the transfer function is nearly cancelled out by the zero Van Amerongen (1982). Moreover, due to the fast response time of the WAM-V USV, the overshoot angles were small. The virtual rudder movements during the experiments were generated by differentially steering both propellers. Doing so, no lateral rudder forces are explicitly induced on the vessel by the virtual rudder, as would be the case with a classical rudder. Hence the coupling between yaw and sway (i.e. second order Nomoto model) is less dominant. Nevertheless, the turning direction and rotational speed of the fixed pitch propellers generates a small lateral force, called propeller walk, and thus sway speed Pocora et al. (2015). However the effect of the torque of the propeller wake, due to its longitudinal distance from the centre of gravity of the vessel, has a bigger influence on the turning characteristics and will be implicitly included by the first order model. One must also notice that additional lateral forces will occur due to Corioliscentripetal forces, both being rather small and most likely accompanied by wind and current disturbances. A nonlinear least squares solver is used to identify the model. If accurate yaw-rate measurements are present together with a higher coupling between yaw-rate and sway (e.g. with real rudders), a least squares support vector mechanism could be used as in Moreno-salinas and Chaos (2013). This paper experimentally identifies the asymmetric first order turning characteristics of the WAM-V USV i.e. the asymmetric first order transfer function between applied rudder as input and yaw-rate as output, fitted to the first order Nomoto model. A starboard, port side and average Nomoto model are identified to demonstrate the asymmetric steering behaviour. This paper continues as follows, section II correlates the hydrodynamics of the USV to the first order model of Nomoto. Section III explains how a PID controller can be developed, modified and augmented. Section IV shows the equipment and section V clarifies the methodology for the actual implementation. The results are discussed in section VI, followed by their conclusion in section VII. 2. SHIP HYDRODYNAMICS 2.1 Vectorial model Several assumptions or methods can be used to model the hydrodynamics and hydrostatics, for a thorough listing look at I. Fossen (2011). Three degrees of freedom (3 DOF) suffice to describe the planar motion (i.e. surge u, sway v and yaw-rate r) of the WAM-V USV, where the hydrostatic forces are assumed to be constant since heave, pitch or roll are assumed to be zero. Moreover, as this paper studies the steering hydrodynamics for a cruising speed, i.e. constant surge, two degrees of freedom, sway and yaw, suffice i.e. ν = [v, r]T as shown in (1), where matrix M accumulates all the terms concerning acceleration induced forces and matrix N the speed dependent forces. τv and τr respectively depict the transversal and rotational actuation forces. 208



       m11 m12 v˙ n n v τ + 11 12 = v m21 m22 r˙ n21 n22 r τr       M

(1)

N

2.2 Nomoto’s first and second order model

Nomoto’s models are transfer functions relating the rudder input, δ, to the yaw-rate output, r. The second order transfer function (2) has three time constants T1 , T2 , and T3 , and one gain constant K. Equation (3) expresses their implicit connection to their hydrodynamic parameters of (1). Additionally, one can also express a similar transfer function between rudder angle and sway speed, where two constants will be different: K = Kv and T3 = Tv . Finally, (2) can be simplified to (4) where T = T1 + T2 − T3 . Which is an adequate assumption as the overshoot of the WAM-V USV is low and because in practice the zero (1 + T3 s) and pole (1 + T2 s) almost cancel each other out Technology (2015). K(1 + T3 s) r (s) = (2) δ (1 + T1 s)(1 + T2 s) det(M ) n11 m22 − n12 m21 − n21 m12 T1 T2 = , T1 + T2 = , det(N ) det(N ) m21 b1 − m11 b2 n21 b1 − n11 b2 , KT3 = K= det(N ) det(N ) (3) K r (s) = (4) δ 1 + Ts 2.3 Asymmetric turning characteristics If a propeller turns clock-wise to generate a forward thrust, an additional lateral force (W in Fig. 1) arises. The WAMV USV vessel has two clock-wise turning propellers placed at a longitudinal distance d from the centre of gravity of the vessel, shown in Fig. 1. Hence, together the propellers generate an additional turning moment Mw = 2 ∗ (W ∗ d). To demonstrate this asymmetric turning behaviour of the WAM-V USV, three Nomoto first order functions are identified: (i) turning left (port), (ii) turning right (starboard), and (iii) the average of turning left and right. Naming the torque generated by the virtual rudder Th , discussed in section 4.2, the total actuation forces become, τr = Th + Mw and τv = 2W .

Cg

d

T

T W

Fig. 1. Propeller walk

W

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209

3. PID HEADING CONTROL Equation (5) implicitly couples the vessels hydrodynamics with the first order Nomoto transfer function. Take m = m22 and d = n22 so that the time constant T = m/d, consequentially (5) can be simplified to (6) m22 r˙ + n22 r = τr (5) 1 1 (6) r˙ + r = τr T m To control the heading of the WAM-V USV, a PID controller can be designed and tuned. The total actuation force τr can be split in the following components in the time domain (7). Where the error, ψ˜ = ψ − ψd , represents the difference between the current heading and the desired heading.  t ˜ ˜ )dτ − KD ψ˜˙ (7) τr = τF F − KP ψ − KI ψ(τ 0

Pole placement (9) enables to design the PID gains by specifying the control bandwidth (8), I. Fossen (2011). Choose ωb and ξ arbitrarily, and calculate ωn via equation (8). For example, choosing a critically damped vessel i.e. ξ = 1, will make ωb equal 0.64ωn . After choosing a bandwidth, the controller gains can be calculated. Where KI is often chosen as KI = T1i KP = ω10n KP making the integrator 10 times slower as the natural frequency.   ωb = ωn 1 − 2ξ 2 + 4ξ 4 − 4ξ 2 + 2 (8) ωn2 T

ωn3

2ξωn T − 1 (9) , KD = 10K K   1 (10) τF F = m r˙d + rd T The feedforward term can be model based on the Nomoto first order model (10), making the PID controller an augmented model based controller. KP =

K

, KI =

4. EQUIPMENT 4.1 The WAM-V USV test platform Fig. 2 shows a picture of the used WAM-V USV vessel and a drawing of the back view, its specifications are listed in Table 1. The rear of the vessel, approximately one fifth its length, is able to hinge, guaranteeing the inflow of water for the propellers. The sensor platform has a spring-damped mounting on both the cylindrical hulls. In total this generates high flexibility, making the vessel wave adaptive. A Hemisphere Vector v102 GPS Compass series measured the heading at 5 Hz, with a 0.75 degree accuracy. Table 1. WAM-V USV specifications Parameter Length Beam Hull diameter Weight Max payload Cruise speed u0 Max. propulsive power Propelor diameter

Size 4.88 2.44 0.40 270 80 1.4 2 x 2240 0.25

Fig. 2. Side and back view of the WAM-V USV Marine Advanced Research (2013) 4.2 Virtual rudder The WAM-V USV has no physical rudder, instead it is differentially steered by the two propellers (Torqueedo Cruise 4.0) at the back. In order to identify the first order Nomoto model, a virtual rudder needs to be constructed. By giving both propellers a different amount of thrust Thlef t and Thright , a turning moment Th arises: Th = b ∗ (Thright − Thlef t ) = b ∗ Th (11) on the vessel, where b denotes the lateral distance between both propellers. This generates a turning movement as a traditional rudder would also provide. Still, a traditional rudder would also explicitly generate a lateral force, as the water is deflected on the rudder. This is not the case with a differentially trusted vessels as both thrust forces are parallel to the longitudinal axis of the bodyfixed reference frame. However, the propeller walk will generate a lateral force and additional turning moment as discussed in section 2.3. The thrust of each propeller is expressed in percentages and is bounded by [-100%, 100%], representing maximum speed backwards or forwards as both the propellers’ shafts can turn either way. The virtual rudder has similar artificial boundaries i.e. [-100%, 100%]. For example, to induce a differentially steered rudder of 20%, each propeller either gets half of the rudder i.e. 10% added to or subtracted from its thrust, depending on the desired turning direction. 5. METHODOLOGY

Dimension m m m kg kg m/s W m

5.1 Zigzag MOOSApp A fully autonomous Zigzag manoeuvre was used to identify the first order transfer function where the alternating steps in rudder generated changes in heading and thus yaw-rates. Therefore an open-loop MOOS Application (MOOSApp) ran with the IvP-helm disabled, to guarantee 209

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that the movements of the vessels were only due to the Zigzag MOOSApp. When the vessel was deployed, it sped up, using 50% of the maximal thrust, meaning Thlef t = Thright = 50%. and after 10 seconds (ensuring a steady speed regime) the vessels started its configured number of 5 zigzags with rudder changes from 20% to 20%. These alternating rudder changes resulted in thrust combinations of Thlef t = 40% and Thright = 60%, to turn left, and the other way around to turn right. A completion threshold of 1.00 degrees was programmed equally divided around the desired heading change which triggered a new rudder movement. This threshold guaranteed logging of the heading change, which had a accuracy of 0.75 degrees.

behaviour. This was of interest to show the propeller walk effect. Additionally there was a fin-like casting device mounted at the front right of the vessel, approximately half a meter from the centreline. This mounted sensor could make the vessel turn easier to the right than to the left, due to its location. The positions of the vessel during the three tests are visible in top plot of Fig. 3. The first heading change due to a rudder movement δr of every zigzag test has the biggest transient behaviour, as the total speed of the vessels needs to settle, visible in Fig. 3 bottom. Consequentially the first rudder movement and its heading response are ignored during the following identification procedures. 6. RESULTS

5.2 Model-fitting

transient

(13)

step

A non-linear least-squares solver minimised the square sum of the difference between the model and the measurements as shown in (14) and (15), by altering the T and K parameters. The best fit for these parameters was found by using the Levenberg-Marquardt algorithm. Whenever the rudder, δr , changed direction, the relative time of the exponentials were reset to zero and the initial yaw-rate, r0 , got set to its previous value simulating a new transient and step response. min||f (x)||22 = min(f1 (x)2 + f2 (x)2 + ... + fn (x)2 ) (14) f (x) = r(k) − yaw rate(k)

= r0 .e−t(k)/T − K.δr .(1 − e−t(k)/T ) − yaw rate(k)

(15)

To further demonstrate this asymmetric discrepancy, summary plots for zigzag test 1 and 3 (as they ran in opposite spatial direction they diminished the effects of environmental disturbances) are shown in Fig. 4 and Fig. 5 re-100

5.3 Zigzag experiments

-200 -300 zigzag1

-400 -100

210

zigzag2

0

zigzag3

50

100

150

Distance [m] 1.5

Speed [m/s]

In total, three zigzag tests of each five zigzags were conducted on the Charles river in Boston. The Charles has locks on both sides resulting in a lake-like behaviour of the water i.e. very slow and local currents. The number of zigzags, five every run, was arbitrarily kept low in order not to interfere with other water users during the autonomous testing. The wind strength was around 1 Beaufort, with no real constant direction, though mainly coming from the North. The three runs were conducted perpendicular to each other, making run one and three running in the opposite direction, in order to diminish disturbances as much as possible to uncover a potential asymmetrical

-50

20

1 0.5

0 u v

0

r

-20

Degrees [°]

r(k) = r0 .e−t(k)/T + K.δr .(1 − e−t(k)/T )      

To identify the asymmetric turning behaviour of the vessel, separate Nomoto coefficients for turning left or right were calculated. Table 2 summarises these asymmetric Nomoto coefficients i.e. time constants T [s] and gains K[s−1 ] for each zigzag test. To identify the global averaged behaviour, i.e. supposing the vessel behaves symmetrically, the average Nomoto gains and time constants were calculated for each zigzag test separately. And, afterwards, the total average of these separate averages was taken, in order to identify the global symmetric Nomoto model, resulting in T = 1.27s, K = 0.12s−1 . Note that this is not simply the average of the left and right turning coefficients as the average uses the consecutive turns as identification data instead of each turn separately. From these results, and table 4, it is clear that the vessel has a bigger gain coefficient when turning left compared to turning right, the actual difference is 36% on average for the asymmetrical gain coefficients. Additionally, table 3 shows the time spans of every rudder movement δx (−20%, 20%, −20%, 20%), where it is also visible that turning to the left generally takes less time, 15.87s on average, than turning to the right, 21.82s on average. Hence turning right, on average, takes 38% longer than turning left.

Distance [m]

The Nomoto model needs yaw-rate measurements as an output to the rudder input. In order to derive these signals, the time derivative of the heading was calculated as follows (12). heading(k + 1) − heading(k) yaw rate(k) = (12) time(k + 1) − time(k) Note that this increased the√ standard deviation of the dataset with a magnitude of 2. However, this was partly countered by smoothing the data with a moving mean smoothing filter over a timespan of 23 measurements. This filter was empirically selected, as it gave the best results in terms of model fit. Next, the calculated yaw-rates were compared to the first order model in the time domain (13), where r(k) expresses the yaw-rates in the time domain at the time step k.

-0.5 0

10

20

30

40

50

60

70

80

90

Time [s]

Fig. 3. Top: North East Down position zigzag tests — Bottom: decomposed surge (u) and sway (v) speed (left y-axis) and rudder movements (right y-axis).

IFAC CAMS 2018 Opatija, Croatia, September 10-12, 2018

G. Peeters et al. / IFAC PapersOnLine 51-29 (2018) 207–212

Angle [°]

50

rudder

measured heading

0

Yaw rate [°/s]

-50 0

10

20

0

10

20

30

40

50

30

40

50

60

70

80

90

60

70

80

90

80

90

5 0 -5

measurements

Heading [°]

spectively. These plots show: (top) the rudder step and measured heading changes, (middle) the measured yawrates and simulated yaw-rates based on the symmetric model, and (bottom) the measured heading and simulated headings based on the symmetric model. In both cases, the predicted headings diverge to the right compared to the measurements (positive angles denote turning to the right), due to the fact that the headings were predicted using a symmetric model. To show the strength of the asymmetric model, Fig. 6 illustrates a similar plot for zigzag test 3 whilst using the asymmetric Nomoto model, which results in a better alignment between measured and simulated headings and yaw-rates compared to Fig. 5. To further clarify the difference between the symmetric and asymmetric methods, Fig. 7 and Fig. 8 respectively compare the simulated headings and yaw-rates for both methods for zigzag test 3. Figure 8 also implicitly shows the difference in gain coefficients K and thus size of the yaw-rates.

211

symmetric model

50 0 -50 0

10

20

30

40

50

60

70

Time [s]

Fig. 5. Summary zigzag 3, with symmetric Nomoto model

Table 2. Summary asymmetric Nomoto coefficients zigzag3

50

combined

left

right

left

right

left

right

symmetry

1.36 0.13

1.32 0.11

1.30 0.16

1.45 0.11

1.40 0.16

1.19 0.11

1.27 0.12

δ4

δ5

16.20s 17.27s 14.75s

23.85s 19.62s 19.43s

15.79s 15.80s 14.96s

24.57s 21.10s 22.35s

Table 4. Nomoto coefficients and PID gains

T K KP KI KD

left

right

symmetric

1.35 0.15 16.16 1.60 15.04

1.32 0.11 21.54 2.19 19.85

1.27 0.12 19.00 2.00 17.20

0

10

20

0

10

20

30

40

50

30

40

50

Angle [°]

rudder

70

80

90

60

70

80

90

80

90

0 -5

assymetric model

50 0 -50 0

10

20

30

40

50

60

70

Time [s]

Fig. 6. Summary zigzag test 3, asymmetric Nomoto model

measurements

50

60

5

measurements

Heading [°]

zigzag1 zigzag2 zigzag3

δ3

measured heading

-50

Table 3. Time between rudder movements δ2

rudder

0

Yaw rate [°/s]

T K

zigzag2

Angle [°]

zigzag1

assymetric model

symmetric model

60

measured heading

50

0

40 -50 20

40

60

80

100

30

5

Heading [°]

Yaw rate [°/s]

0

0

20 10 0

-5 0

20

40

60

80

100

-10

Heading [°]

measurements

symmetric model

-20

50

-30

0

0

-50 0

20

40

60

80

10

20

30

40

50

60

70

80

90

Time [s]

100

Time [s]

Fig. 4. Summary zigzag 1, with symmetric Nomoto model 211

Fig. 7. Comparison symmetric and asymmetric model for heading

IFAC CAMS 2018 212 Opatija, Croatia, September 10-12, 2018

G. Peeters et al. / IFAC PapersOnLine 51-29 (2018) 207–212

7. CONCLUSION measurements

assymetric model

symmetric model

4 3 2

Yaw rate [°/s]

1 0 -1 -2 -3 -4 -5 0

10

20

30

40

50

60

70

80

90

Time [s]

Fig. 8. Comparison symmetric and asymmetric model for yaw-rates

A PID controller can be developed according to section III above, using the average symmetric Nomoto model for simulation simplicity. Taking an arbitrary bandwidth of rad ωb = 1 rad s and a damping of ξ = 0.9, gives ωn = 1.34 s . Generating: KP = 19.00, KI = 2.00, KD = 17.20. Fig. 9 demonstrates how well the closed loop system, i.e. the PID controlled heading, works if the vessel behaved completely symmetrical. As system input, 3 arbitrary sine-waves of amplitude 0.25m and ω of 0.1, 1.0 and 10 rad/s are respectively benchmarked as desired heading. As visible, the PID controller works best below frequencies of 1rad/s, which is normal as the bandwidth was chosen to be 1 rad/s, here the amplitude of the desired value is reached with a small lag. Around ω = 1, the lag builds up and the amplitude is not reached any more. At higher frequencies, the PID control can not follow the desired heading and starts to lag, with a maximum lag of 90 degrees. Based on the asymmetric Nomoto coefficients, table 4 summaries similar PID gains for the asymmetric vessel. These coefficients differ in the order of 32 to 37 % from each other, i.e. comparing left to right.

Angle [°]

0.1 rad/s 0.2 0 d

-0.2

controlled plant

0

2

4

6

8

10

6

8

10

6

8

10

Angle [°]

1 rad/s 0.2 0 -0.2 0

2

4

Angle [°]

10 rad/s 0.2 0 -0.2 0

2

4

Time [s]

Fig. 9. Simulation PID controlled heading changes 212

Virtual rudder and heading measurements sufficed to identify an asymmetrical turning behaviour of a differentiallysteered WAM-V USV which uses two right-handed propellers. Further research could elaborate more on the speed dependency of these small lateral propeller forces which induce the asymmetrical turning behaviour. Using a lefthand and a right-hand propeller would solve the asymmetrical behaviour, as then the lateral propeller effects would cancel each other out during straight sailing. During turning manoeuvres there would be small effects which would be mirrored when comparing left and right turns, hence these effects will not be noticed. When using two left-handed or two right-handed propellers, one can change the PID gains as discussed in this paper, after identifying the asymmetrical Nomoto models. Another option would be, to modify the size of the virtual rudder depending on the turning direction. ACKNOWLEDGEMENT We thank the Marine Autonomy Lab from CSAIL, MIT for their cooperation and help during the experiments. REFERENCES Abkowitz, M. (1964). Lectures on Ship Hydrodynamics Steering and Maneuverability. Technical report, Hydroand Aerodynamic’s Laboratory, Lyngby. Benjamin, M.R., Schmidt, H., Newman, P.M., and Leonard, J.J. (2010). Nested autonomy for unmanned marine vehicles with MOOS-IvP. Journal of Field Robotics, 27(6), 834–875. doi:10.1002/rob.20370. URL http://doi.wiley.com/10.1002/rob.20370. Brooks, R.A. (1985). A robust layered control system for a mobile robot. Massachusetts Institute of technology. Artificial intelligence laboratory. Memo 864. Massachusetts institute of technology, Cambridge (Mass.). Caccia, M., Bruzzone, G., and Bono, R. (2008). A Practical Approach to Modeling and Identification of Small Autonomous Surface Craft. 33(2), 133–145. I. Fossen, T. (2011). Handbook of marine craft hydrodynamics and motion control. R Marine Advanced Research, I. (2013). 14 ’ WAM -V  USVe. Technical report. URL http://www.wam-v.com/ 14-wam-v-usv. Moreno-salinas, D. and Chaos, D. (2013). Identification of a Surface Marine Vessel Using LS-SVM. Journal of Applied Mathematics, 2013, 11. Newman, P. (2013). A MOOS-V10 Tutorial. Technical report. Nomoto, K. (1957). On the Steering Quality of Ships. International Shipbuilding Progress, 4, 354–370. Pandey, J. and Hasegawa, K. (2016). Study on turning manoeuvre of catamaran surface vessel with a combined experimental and simulation method. 49, 446–451. Pocora, A., Lupu, S., and Cosmin, K. (2015). Simulated propeller walk on a 13.300 teu container ship. XVIII, 55–59. Technology, A. (2015). Ships Steering Autopilot Design by Nomoto Model. (June). Van Amerongen, J.O.B. (1982). Adaptive Steering of Ships. Ph.D. thesis, Delft University of Technology.