Modelling and identification of the Charlie2005 ASC

Transcription

1 Modelling and identification of the Charlie ASC Massimo Caccia, Gabriele Bruzzone and Riccardo Bono Abstract This paper discusses a practical model, with respect to the performance of basic sensors available on-board the vessel, and the corresponding identification procedure for guidance and control of the Charlie ASC. The work is supported by extended at sea trials. Index Terms marine robotics, modelling, identification, autonomous vehicles. I. INTRODUCTION The main goal of the research presented in this paper is the definition of a practical model for autonomous surface craft (ASC) and of the corresponding identification procedure, where practical basically stands for consistent, from the point of view of the degree of accuracy, with the quality in terms of noise and sampling rate of the measurements provided by the sensors commonly available on-board a small and relatively low-cost vehicle. On-board sensor-based identification is well known in the field of marine robotics and, in the case of ROVs, provides low cost accurate estimates of the coefficients of simplified models usually employed for control system design 1]]3]. As discussed in 4], in the case of poor sensor measurements, special attention has to be paid to the design of the experiments, i.e. of the manoeuvres performed by the vehicle, in order to guarantee the observability of the hydrodynamic model parameters: in particular, steady-state maneouvres are very useful for identifying drag and actuator interaction effects. The general methodology developed for ROV modelling and identification has been applied to the Charlie ASC, a small autonomous catamaran prototype for harbour and coastal survey and sampling, assuming as reference model of the vessel dynamics the nonlinear model of Blanke for speed and steering equations ] integrated with the rudder model reported in 6]. Steady-state manoeuvres have been designed for the identification of hydrodynamic drag on the basis of on-board compass and GPS data, while conventional zig-zag manoeuvres have been executed for a rough estimate of the yaw moment of inertia, including its added component. The Charlie prototype vehicle, see Figure 1, is the evolution of the Charlie ASC, originally designed for calm water and very low speed sea surface microlayer sampling 7], enhanced with the integration of a rudder-based steering system and a new software control system fully based on This work has been partially funded by PRAI-FESR Regione Liguria prot. Coastal and harbour underwater anti-intrusion system. The authors would like to thank Giorgio Bruzzone and Edoardo Spirandelli for their fundamental support in hardware development and operating at sea. M. Caccia, G. Bruzzone and R. Bono are with CNR-ISSIA, Via De Marini 6, Genova, Italy, {max,gabry,riccardo.bono}@ge.issia.cnr.it standard GNU/Linux 8]. Experiments revealed that, for a Fig. 1. Charlie. small catamaran equipped with compass and GPS, the sway velocity is negligible, and the coupled effects of the vehicle speed with the propeller revolution rate and rudder angle are not observable, leading to a simplified uncoupled model consisting of speed equation and yaw steering equation. The paper is organised as it follows. Section II presents basic nomenclature and modelling of vehicle kinematics, thruster and rudder force, and vessel dynamics, discussing the approximations of the theoretical models, and focusing on steady-state models, which are fundamental for on-board sensor-based drag identification. Experiments are reported in Section III, where after a brief description of the experimental set-up, the performance of the available sensors is discussed, and attention is focused on steady-state experiments for the identification of speed and steering equations and zig-zag manoeuvres for the identification of the yaw timeconstant. II. MODELLING AND NOMENCLATURE A. Kinematics and water flows Assuming that the vessel motion is restricted to the horizontal plane, i.e. neglecting pitch and roll, two reference frames are considered: an inertial, earth-fixed frame <e>, where position and orientation xyψ] of the vessel are usually expressed, and a body-fixed frame <b>, where surge and sway velocities (with respect to the water) and yaw rate uvr], and force and moments X Y N], are represented. The vehicle speed in the earth-fixed and body-fixed frames

2 are related by ẋ = ucosψ vsinψ+ẋ C ẏ = usinψ+ vcosψ+ẏ C (1) ψ = r where ẋ C ẏ C ] T is the sea current. The vessel s absolute speed, denoted by U, is expressed by U = u + v () Assuming a vessel equipped with stern propellers and Fig.. Charlie nomenclature. rudders mounted behind the propellers, as in the case of the considered catamaran (see Figure ), the average flow velocity over a stern propeller v a, called speed of advance, is affected by the forward motion in the direction of the motion of the hull acquired by the water around the stern, which induces the so-called wake speed w such that v a =(1 w) U (3) In addition, the flow passing the rudder v av is influenced by the propeller too. A possible expression of the average flow is 6] vav = va + C T T (4) where T is the propeller generated thrust. B. Thruster model For vessels equipped with stern thrusters, the thrust available for the propulsion T a is usually modelled as T a =(1 t) T where the thrust deduction number t (typical values are.-.) takes into account the interaction between propeller and hull. This available thrust may be considered as independent of the vessel speed 6], and, neglecting the term given by the speed of advance, the following equations holds: T a =(1 t) T n n n n () C. Rudder model As far as the rudder is concerned, the model by Lewis presented in 6], where the total resultant force is nearly normal to the center plane of the rudder, is assumed. Denoting with δ the rudder angle, δ a the relative angle between the rudder and the flow, and δ f the angle of the flow in the vehicle-fixed reference frame, the magnitude of the rudder force is expressed as { ( ) c F = F vavsin π δ a δs if δ a <δ s c F vavsign (δ a ) if δ a δ s where δ s is the rudder stall angle and the angle of attack δ a is computed as ( ) v + Lr δ a = δ δ f = δ atan u where L is the range from the rudder to the vehicle center of mass in the longitudinal direction. In the case the sway speed v is negligible and the yaw rate r is small, the angle of attack can be approximated by the rudder angle δ. Thus, in the case δ a δ <δ s, the rudder force can be written as ( ) π F = c F vavsin δ c F v π av δ = k F v δ s δ avδ s where k F = πcf δ s and the approximation sinα α has been assumed. D. Dynamics The vessel speed and steering equations are given by the nonlinear model by M. Blanke (1981) ], which is detailed in the following assuming to steer the vessel only using the rudder, i.e. applying the same revolution rate n to both the propellers. 1) Speed Equation: (m X u ) u = X h (u)+x p (n)+x R (δ, n)+ + (m + X vr ) vr +(mx G + X rr ) r where: the term X u can be neglected, since it is typically less than % of the vessel mass m; X h (u) is the surge hydrodynamic drag constituted, generally speaking, by a linear and a quadratic term: X h (u) =k u u + k u u u u X p (n, ) is the thrust exerted by the thrusters, i.e. in the case of the vehicle configuration shown in Figure, from eqn. () X p (n, ) =(1 t) T n n n n X R (δ, ) is the resistance due to the rudder, that, after some computation and assuming the approximation sinδ δ, reduces to X R (δ, ) = k F (1 w) U δ + k F C T T n n n n δ

3 the term (m + X vr ) vr could be neglected if the sway speed is negligible; the term (mx G + X rr ) r is usually negligible as well as the factor r, which is less than.3 for a turning rate of 1 deg/s. Thus, assuming the propeller revolution rate n positive and normalizing in order to have unitary coefficients for the control action n, the resulting speed equation for a small catamaran is ˆm u u = ˆk u u + ˆk u u + ˆk u δ U δ + ˆk n δ n δ + n (6) For details about the equations coefficients, the reader can m refer to 9]. Here, it is worth noting that ˆm u = (1 t)t n n, where m is the vehicle mass, which can easily determined, and T n n 1, since the thruster and propellers are the same of the Romeo ROV working in free water 4]. Thus, given the tipycal values of t, the normalised mass can be assumed ˆm u m 1.8 ) Steering Equations: Neglecting the added mass derivatives Yṙ and N v ], the steering equations assume the form: (m Y v ) v = Y h (v)+y R (δ, ) (m + Y ur ) ur (I zz Nṙ)ṙ = N h (r)+n R (δ, )+N p (n, ) where: Y h (v) and N h (r) are the sway and yaw hydrodynamic drags constituted, generally speaking, by a linear and a quadratic term: Y h (v) = k v v + k v v v v N h (r) = k r r + k r r r r while the coupled drag terms are neglected because v and r are tipycally small; Y R (δ, ) is the sway action of the rudder which, assuming the approximation cosδ 1 and applying the values of the average flow, gets: Y R (δ, ) = k F (1 w) U δ + k F C T T n n (n L n L + n R n R ) δ N R (δ, ) and N p (n, ) are the steering action of the rudder and propellers, respectively, modelled taking into account a possible small displacement of the center of mass with respect to the longitudinal axis of the vessel. After some calculations, assuming the approximations sinδ = δ and cosδ =1and neglecting the term due to the small displacement, the rudder steering action can be written as N R (δ, ) = k F L (1 w) U δ + + k F LC T T n n n n δ As far as the propeller steering action is concerned N p (n, ) =(1 t) T n n n n Thus, assuming n positive and normalizing in order to have unitary coefficients for the control action n δ, the resulting steering equations for the catamaran are: ˆm v v = ˆk v v + ˆk v v v v ˆk u δu δ + ˆk ur ur n δ(7) Î r ṙ = ˆk r r + ˆk r r r r + ˆk u δu δ + ˆk n n + n δ (8) Detailed expressions of the steering equation coefficients can be found in 9]. E. Negligible sway speed case In the case the vessel sway speed is negligible, from eqn. (), the absolute speed U is equal to the surge speed u, and the speed of advance v a assumes the form v a =(1 w) u The resulting speed and steering equations reduce to: ˆm u u = ˆk u u + ˆk u u + ˆk u δ u δ + ˆk n δ n δ + n (9) Î r ṙ = ˆkr r + ˆk r r r r + ˆk u δu δ + ˆk n n + n δ (1) F. Steady-state model When working at steady-state, i.e. when constant control input n and δ are applied for a sufficiently long time interval, the vessel moves at constant surge and sway velocities and yaw rate with respect to the water. By integrating kinematics (1) given constant values of u vr] T and assuming constant sea current ẋ C ẏ C ] T, the vehicle motion in the earth-fixed frame, for r, gets x(t) = u r sin (r t + ψ ) sinψ ] + v r cos (r t + ψ ) cosψ ]+ẋ C t + x y(t) = u r cos (r t + ψ ) cosψ ]+ (11) ψ(t) = r t + ψ + v r sin (r t + ψ ) sinψ ]+ẏ C t + y where t = t t, ψ = ψ (t ), x = x (t ) and y = y (t ). On the other hand, at steady-state, vessel dynamics, given by equations (6), (7) and (8), get: ˆk u u ˆk u u ˆk u δ U δ ˆk n δ n δ = n ˆk v v + ˆk v v v v ˆk u δu δ + ˆk ur ur = n δ ˆk r r ˆk r r r r ˆk u δu δ ˆk n n = n δ 1) Steady-state model with negligible sway: in the case the vessel sway velocity is negligible, i.e. v, from eqn. (11) its steady-state kinematics, for r, is x(t) = u r sin (r t + ψ ) sinψ ]+ẋ C t + x y(t) = u r cos (r t + ψ ) cosψ ]+ẏ C t + y (1) ψ(t) = r t + ψ and, from eqns. (9) and (1), the vessel steady-state speed and steering equations are: ˆk u u ˆk u u ˆk u δ u δ ˆk n δ n δ = n (13) ˆk r r ˆk r r r r ˆk u δu δ ˆk n n = n δ(14)

4 III. EXPERIMENTS 184 GPS raw data A. Experimental set-up Experiments have been carried out with Charlie, a small autonomous catamaran prototype which is.4 m long, 1.7 m wide and weighs about 3 Kg in air. The vessel is equipped with a GPS Ashtech GG4C integrated with a KVH Azimuth Gyrotrac, able to compute the True North given the measured Magnetic North and the GPS-supplied geographic coordinates. This sensor system configuration and the goal of identifying the vessel hydrodynamics require trials be executed in very calm water with no sea currents and wind in order to minimise the wind and wave effects and vehicle drift in the earth-fixed reference frame. On this basis, preliminary trials have been performed in the Genova-Voltri harbour with the catamaran executing steadystate maneouvres at constant thruster revolution rate and rudder angle. Experimental conditions were optimal with near total absence of wind and sea currents. It is worth noting that, since the vessel is equipped with servo-amplifiers closing a hardware thruster revolution rate control loop with time constant negligible with respect to the system, in the following the propeller revolution rate n is often substituted by the reference voltage V. B. Sensor performance The KVH Azimuth Gyrotrac provides True North and pitch and roll measurements at Hz. The standard deviation of the residuals between the measured and off-line smoothed heading is about 1. degrees. The GPS Ashtech GG4C provides vessel position and speed measurements at 1 Hz. The true course and speed over ground is given as well as vessel latitude and longitude. Although the artificial degradation of the signal through the process of selective availability was removed in, GPS accuracy, when used in instantaneous stand-alone mode, remains of the order of 1- m 1]. In particular, the Ashtech GG4C latitude and longitude signals are piecewise continuous, presenting steps of the type shown in Figure 3. Thus, the x and y position measurements have been modelled as x (t) = x (t)+ y (t) = ỹ (t)+ k x k S(t t k ) i=1 k y k S(t t k ) i=1 where S(τ) is the step function, i.e. S(τ) =,τ < and S(τ) = 1,τ 1, and x k and y k represent the disturbance step amplitudes. After that discontinuities have been detected, by testing the smoothness of the first and second derivative of the position signals, and compensated, the x and y have been smoothed off-line as in the case of the heading signal, showing a zero mean measurement noise, with a standard deviation of.17 m and a maximum error lower than.6 m along each component. x m] Fig y m] GPS Ashtech GG4C: raw measured x-y path. C. Steady-state manoeuvres Steady-state manoeuvres at constant propeller revolution rate and rudder angle have been executed in order to estimate hydrodynamic drag parameters. As shown by eqn. (11), at steady-state the vessel with respect to the water moves along a circle which is shifted in the x and y directions by the effects of sea current and wind. For a set of steady-state manoeuvres, the catamaran surge and sway and sea current speed have been estimated on the basis of eqns. (11). Results showed that the vessel sway speed is not observable given the available sensors. Indeed, in many cases, the sign of the estimated sway and yaw rate was not opposite. Thus, the catamarn surge has been estimated through Least Squares on the basis of Equation (1) which neglects sway. On the other hand, the yaw rate has been estimated on the basis of heading measurements. The dependency of the vessel yaw rate from the rudder angle is shown in Figure 4. Table I reports results in identifying through Least Squares the parameters of model (14) and their percentile error computed as ˆσ θ θ, where the standard deviation ˆσ θ of the estimated parameter θ is computed as ( ) ˆσ θ = diag (H T H) 1 σɛ being σɛ the gaussian zero mean measurement noise variance. As suggested in 11], if such variance is unknown it can be estimated by σɛ = (y H ˆθ) T (y H ˆθ) dim(y) dim(θ). As shown by the first three lines of Table I, including explicitely in the model a term relating the normalized applied torque n δ to the additional torque induced by the vessel speed, i.e. u δ, involves the computation of values for the coefficients ˆk r, ˆk r r and ˆk u δ, which are physically inconsistent. This is basically due to the fact that, as discussed in the following, the vessel speed u is about proportional to the propeller revolution rate n, and this makes the coefficient ˆk u δ not observable with respect to the available sensors. Then the observability of the linear and quadratic drag has been evaluated. As shown in the fourth line of Table I the combined estimation of the linear and quadratic yaw drag is not reliable given the available data: in particular, the uncertainty on ˆk r is higher than %. The coefficients of

5 V s m TABLE I YAW DRAG COEFFICIENTS. THE PARAMETER ERRORS ARE EXPRESSED IN PERCENTILE. ˆk r ˆkr r ˆku δ V s ] ] V s V s m ] ˆk n σˆkr ˆk r σˆkr r ˆk r r σˆku δ ˆk u δ σˆkn ˆk n rad] σɛ V 4 rad ] Comments rad ˆkr unconsistent ˆkr unconsistent ˆkr r unconsistent ˆkr unreliable linear drag quadratic drag TABLE II SURGE DRAG COEFFICIENTS. THE PARAMETER ERRORS ARE EXPRESSED IN PERCENTILE. σˆku ˆk u ˆku u ˆku δ ˆkn δ ˆk ] ] ] ] u V s V s 1 m m rad rad σˆku u ˆk u u σûu δ ˆk u δ σˆkn δ ˆk n δ σɛ V 4 ] Comments ˆku and ˆk u δ unconsistent quadratic drag ˆku δ unreliable high measurement noise pure linear or quadratic yaw drag models are given in the last two lines of Table I respectively. The measured and estimated yaw rates according to the models considering linear, quadratic and linear-quadratic drags are plotted in Figure 4, which agrees with the results of Table I indicating the quadratic drag model as the one that best fits the vessel steady-state behaviour observed by the available on-board sensors. Thus, the yaw steady-state L: r deg/s] 1 1 Linear Quadratic yaw drag models: r vs. δ given n (measured and estimated) V =9. V = drag induced only by the propeller revolution rate or by the surge speed are very uncertain (in particular, the latter), while a linear expression of the surge drag does not match accurately the data, i.e. σɛ is quite high (see the last row of Table II). Thus, the model k u u k n δ n δ = n is selected, and the corresponding measured and estimated surge speed is shown in Figure. u m/s] Quadratic surge drag model: u vs. n given δ (measured and estimated) δ=. δ=. δ=7. δ=1. δ=1. δ= Q: r deg/s] V =. V = L Q: r deg/s] V =49. V = δ deg] Fig. 4. Measured and estimated catamaran yaw rate vs. rudder angle and propeller revolution rate. from top to bottom: linear yaw drag, quadratic yaw drag, linear and quadratic yaw drag are assumed model ˆk r r r r ˆk n n = n δ is selected. The relationship between the catamaran surge and the propeller revolution rate given the rudder angle is shown in Figure. When identifying model (13), the temptative of distinguishing between the influence on the rudder drag induced by the propeller revolution rate and surge speed leads again to physically unconsistent estimated values, i.e. positive values of ˆk u and ˆk u δ (see Table II, first row). As shown in the second and third row of Table II, the estimate of the rudder u m/s] V Volt ] n rps] δ=. δ=. δ= 7. δ= 1. δ= 1. δ= 1. Fig.. Measured and estimated catamaran surge vs. propeller revolution rate given rudder angle. D. Yaw inertia identification: zig-zag manoeuvres On the basis of steady-state modelling and identification, yaw drag dynamics can be modelled by ṙ = ˆk r r Î r r r + 1 Î r n δ where, neglecting pitch and roll, the yaw rate r is assumed equal to the heading first derivative ψ, and the term ˆk n n, originated by small asymmetries in the vessel mass distribution, is considered as an external disturbance.

6 When executing at-field experiments the presence of environmental external disturbance is often not negligible. Anyway, during zig-zag maneouvres the main effect of disturbances is an offset in the steady-state yaw rates with respect to the nominal ones. Thus, assuming a linear drag model, the yaw rate has the form: r(t) =A +(r A) e ( ) (t t ) τ (1) where τ = Ir is the yaw dynamics time-constant. ˆk r The parameters of eqn. (1) can be identified through a set of suitable zig-zag manoeuvres at low yaw rate due to the adoption of a linear approximation of the drag torque. Given a set of N manoeuvres and the length L (in samples) of the corresponding transients, the steady-state values A i, i = 1...N are computed as the mean of a set of yaw rate measurements at steady-state, and the following cost function to be minimized with respect to τ is defined: c(τ) = N i=1 k=1 L { ]} r(t i + k t) A i +(r(t i ) A i ) e k t τ (16) where t i,i=1...n denotes the manoeuvre starting time and t is the sampling interval. In order to identify the yaw dynamics time constant according to the above discussed approach, experiments have been carried out executing zig-zag maneouvres, varying the rudder angle between ±deg (see the top plot of Figure 6). The sampling interval t was.1s, and the yaw rate was computed in post-processing after smoothing the measured heading signal. The time constant computed by minimising eqn. (16) was 3.7s, which, assuming the value of ˆk r for the pure linear drag model given in Table I, corresponds to a normalised moment of inertia equal to V s.the measured and predicted (according to the identified linear drag model) yaw rate during the identification tests are shown in the bottom plot of Figure 6. A validation of δ deg] r deg/s] 1 1 Yaw inertia identification (n = V ): rudder angle Yaw inertia identification (n = V ): measured and predicted yaw rate measured predicted time s] Fig. 6. Identification of the yaw moment of inertia: zig-zag manoeuvres. From top to bottom: commanded rudder angle; measured (blue) and predicted (red) yaw rate. the estimated model of yaw dynamics is shown in Figure 7, where data collected during auto-heading manoeuvres at different surge thrusts are shown. The normalised thrust and commanded rudder angle are plotted in the top graph in red and green lines respectively, while the online measured yaw rate, computed by differentiating the heading signal, is drawn in the second picture from the top. In particular, the behaviour of yaw dynamics assuming pure linear (third graph from the top) and quadratic drag (bottom graph) has been evaluated. As expected, the linear drag model performs better than the quadratic drag model at low angular rates, while it predicts too big yaw rates for high rudder angles. n V ]; δ deg] r deg/s] r LD deg/s] r QD deg/s] Yaw dynamics validation: auto heading trials (7 October ) 6 4 n V ] 4 δ deg] time s] Fig. 7. Validation of yaw dynamics model: auto-heading manoeuvres. From top to bottom: commanded thrust (red) and rudder angle (green); online measured yaw rate; linear drag model predicted yaw rate; quadratic drag model predicted yaw rate REFERENCES 1] A. Morrison and D. 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