International Journal on Mechanical Engineering and Robotics (IJMER)
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Design of a Heading Autopilot for Mariner Class Ship with Wave Filtering Based on Passive Observer
1Mridul Pande, 2K K Mangrulkar
1,2Aerospace Engg Dept DIAT (DU), Pune Email: 1[email protected]
Abstract : In this paper, the heading control of a large Mariner class platform is improved in order to counteract the undesirable effect of waves on the actuator system.
Wave filter used here is a linear passive observer and includes features like estimation of both the low frequency heading and heading rate from the measurement devices which give values post addition of disturbances due to sea waves. Matlab/Simulink tool is used to show the proposed solution in the control loop. The experimental results confirm the suitable filtering, the estimating properties of the observer and navigation response expected, reducing control action and thus the vibrations of the rudder.
I. INTRODUCTION
This paper is based on theory, methods and problems described by Professor Thor Ivar Fossen in “Marine Control Systems”, 2002. A part of this paper is to identify the so called Nomoto approximation model for the Mariner Class Vessel, based on identification methods in Matlab. The obtained model has been used to design a heading autopilot for the tanker using a simple PD/PID control routine. To ensure robustness of the autopilot, a Leuenberger wave filter has been implemented. In section 3, the model has been tested through an open-loop analysis, a so called “pullout manoeuvre”. Identification and verification of the Nomoto approximation is carried out in section 4 using the “turning circle test”. Section 5 is devoted to the design of a PID controller for the autopilot. Further section 6 introduces the model for wave force and wave induced motion in addition to a description of the Leuenberger wave filter. The wave model that uses Linear Models and an observer is quite advantageous and has been tested by other researchers in various Marine applications.
II. RIGID BODY SHIP DYNAMICS
Ship response in waves is considered as 6 degree of freedom motion in space. For manoeuvring study 3 DOF motion namely surge ,sway, yaw is considered. Fossen (1994) derives the 3 DOF non linear model for a marine vessel written in vectorial setting as:
MRB v +CRB v v = τRB
Where v = [u v r]T is a generalised velocity vector with body fixed reference frame and τRB = [X Y N]T is a generalised vector of external forces and moments (includes added mass, damping and control input terms), while MRB is system inertia matrix and CRB is the rigid body coriolis and centripetal matrix.. The Hydro and Aerodynamic laboratory in Lyngby, Denmark has performed both planar motion mechanism tests and full scale manoeuvring predictions for a Mariner class vessel, the main data and dimensions of which are
Length overall 171.8 m
Length between perpendiculars LPP 160.93 m
Maximum Beam 23.17 m
Design Draft 8.23 m
Design Displacement 18451 m3
Design Speed 15 knots
For this vessel the dynamic equations of motion in surge, sway and yaw are:
m′− Xu ′ 0 0
0 m′− Yv ′ m′x′G− Yr ′ 0 m′x′G− Nv ′ Iz′− Nr ′ ∆u ′
∆v ′
∆r ′ = ∆X′
∆Y′
∆N′ where the non linear forces and moments ∆X′, ∆Y′ and
∆N′ are defined as (Prime system I with LPP and U as normalisation variables. Here Yv indicates the hydrodynamic derivative of the sway force Y to the sway speed v evaluated at the reference condition; m is the mass of the ship; Iz is the moment of inertia about the z-axis; v is the sway speed; u is the surge speed and r is the yaw rate; ψ is the heading angle defined by ψ = r.
The non-dimensional co-efficient in this model are m′ = 798 ∗ 10−5, Iz′= 39.2 ∗ 10−5
x′G = -0.023
Figure 1 Sway-yaw-roll motion coordinate system.
“Mariner Model” is a full Non Linear Dynamics MIMO model of a generic ship. The dynamic aspects of the model have been referenced from Marine GNC Toolbox which is available on “TTK 4190 Guidance and Control” at NTNU and are not further discussed in this paper.
III. OPEN LOOP ANALYSIS
To investigate the ship’s stability a “pullout manoeuvre”
is done under ideal conditions, i.e no environmental disturbances. The test is performed by turning the rudder to a certain prescribed angle (corresponding to a time limited step input before turning it back to neutral after the ship has attained a steady turning rate). The simulation was performed with both positive and negative rudder angles of 10 degrees with step time of 1000 seconds. The yaw rate was plotted versus time in the figure below:
Figure 2 Yaw Rate plotted against time for pullout manoeuvres
According to Fossen, a ship is said to be straight line stable if both curves from the pullout manoeuvre converge to the same value, which is the case for this particular model. Ideally the ship should sail with zero yaw rate after the rudder has been set to neutral, the static offset from zero is regarded as imperfection in the ship dynamics and would require a rudder offset to be corrected. However, we may conclude that the ship is straight line stable due to mutual convergence of the two independent tests.
IV. IDENTIFICATION AND EVALUATION
4.1 The Nomoto Approximation
As mentioned in section 2, the model is considered as a full non linear model presented as
MRB v +CRB v v = τRB + w
where “w” is the environmental disturbance. To make the identification and control of the model feasible we use the Nomoto second order approximation which describe the heading dynamics of the ship through a simple transfer function as follows:-
r
δ = 1+TK(1+T3s)
1s (1+T2s)
where, the zero (1+ T3s) and the pole (1 + T2s) are due to the coupling effect from the sway mode on the yaw dynamics and where K is the static yaw rate gain, and T1, T2 and T3 are time constants, In practice, because the pole term (1 + T2s) and the zero term (1 + T3S) nearly cancel each other, a further simplification of can be done to give the first order Nomoto model:
ψ δ = K
s(1+Ts )
This linear model is a transfer function from the commanded rudder angle δ to the ship’s heading angle ψ, and has only two parameters to be identified, the Nomoto gain K and the time constant T . These both will be identified before designing a controller for the ship. Nomoto’s approximations are valid under the assumptions of small angle perturbations, low ship velocity and movement in vertical plane. Thus the Nomoto model may be considered as a linear approximation for the full model ship.
4.2 Identification of First Order Nomoto Parameters The transfer function needed for observer design is the response yaw angle versus the rudder angle. To determine the Nomoto parameters and thus obtain an identified model of the ship a so called turning circle manoeuvre is performed by using an open loop scheme i.e some step variation in rudder position are introduced (in form of step inputs in simulation) and rate of yaw output is registered. The input signal δ and output yaw rate r are processed in System Identification Toolbox of Matlab obtaining the following parameters for Nomoto second order Nomoto approximation.
r
δ= 0.0049807 (s+0.009087 ) s2+ 0.02533 s+0.0001623
The non linear response is plotted along with reduced second order model to ascertain the level of approximation and the following results were achieved:
Figure 3 Non Linear response together with Nomoto’s second order approximation
Expressing the transfer function in terms of Time constant and further reducing it to Nomoto’s First order model where (T=T1+T2-T3) as per section 4.1 we come to the following approximation:-
ψ
δ = s(1+47.44s)0.2803
Thus obtaining K= 0.2803 and T=47.44. The non linear response is plotted along with reduced first order model
Figure 4 Non Linear response together with Nomoto’s First order approximation
V. CONTROL DESIGN
5.1 Choice of Controller
We have chosen to apply a PID/PD controller for the autopilot. To ensure robustness and to reduce the wear and tear of the actuators, the controller is implemented with a wave filter which is described in section 6.1.
Considering a control law of PD type, in the form δ = KP(ψd− ψ)- Kdψ
Here KP and Kd are the controller design parameters.
The closed loop dynamics resulting from the ship dynamics and the PD controller are
Tψ +(1+KKd) ψ +KKP ψ = KKPψd
This expression corresponds to a 2nd order system of the form
ψ +2ζωnψ +ωn2 ψ = ωn2ψd
with natural frequency ωn (rad/s) and relative damping ratio ζ. The relative damping is typically chosen in the interval 0.8 < ζ < 1.0, whereas the choice of ωn will be
limited by resulting bandwidth of the rudder ωδ
(rad/sec) and the ship dynamics 1/T (rad/sec) according to
𝟏 𝐓 < 𝛚𝐧 (𝟏 − 𝟐𝛇𝟐 𝟐+ (𝟒𝛇𝟒− 𝟒𝛇𝟐+ 𝟐)) < 𝛚𝛅
For a critically damped ship (ζ = 1), the closed loop bandwidth ωb is related to the natural frequency ωn of the closed loop system by a factor of 0.64ωn. The values of KP and Kd are dependent on ωn and ζ (which are treated as design parameters), by following relations KP= TωKn2 , Kd= 2TζωKn−1
5. 2 Bode Plot Analysis
Recalling the Nomoto transfer function G(s) and the PD controller C(s), the corresponding open loop transfer function C(s) G(s) is given by
ψ
ψref = KKP+KKds
Ts2+ KKd+1 s+KKP
Below is the Bode plot of the OLTF for damping ratio (ζ = 1) and (ωn= 0.15), K= 0.2803 and T=47.44 resulting in KP and Kd =3.80 and 47.206 respectively.
Figure 5 Bode Plot for the OLTF with PM= 78.6 deg (at 0.289 rad/s)
To test the stability of the non linear plant with the proposed PD controller with above gains, we give a transport delay of (4.5 second) corresponding to the phase margin of 78.6 degrees achieved on the reduced model. Now we increase the transport delay till we
achieve sustained oscillations indicating marginal stability and thus corresponding phase margin for the controller with non linear plant. The sustained oscillations for the plant are achieved on introducing a transport lag of 9seconds, which corresponds to 81 degrees phase margin. Thus, the proposed controller gains are viewed to be suitable for autopilot control of the non linear model.
Figure 6 Sustained oscillations achieved on inducing 9 seconds transport delay
VI. MODEL FOR WAVE FORCE AND WAVE INDUCED MOTION
The main environmental disturbances affecting marine vehicles during navigation are waves generated by winds and ocean currents. Taking into account the effect of waves, the initial equation may be modified as:
τRB = τ + τWaves
and Here τWaves represents the moment and forces caused by waves. Wave forces are modelled as sum of linear and Non- linear components as:
τWaves = τWlin + τWnlin
The second term on the right hand side of the expression corresponds to low frequency component and is commonly treated as input disturbance and is modelled by a Bias term, hence it is not discussed here. On the other hand the Linear high frequency component is usually modelled by wave elevation transfer function approximation of the wave spectrum selected. In this paper a 2nd order wave transfer function approximation is considered of the following form:
ψH ={s2+2ζωKws
ns+ωn2}w(s) Where Kw =2ζωnσ
Here ωn is the dominating frequency, ζ is the relative damping ratio, ψH is the yaw angle induced by the waves, w is a zero mean Gaussian white noise and σ is a
parameter related to wave intensity that is adjusted on the level of affectation of waves or the wave intensity. In addition, the wave frequency response of the ship is generated by using the principle of linear superposition.
The total motion can be separated in LF and HF components as shown in fig 7 below thus obtaining the expression:-
Figure 7 Total motion of the vessel as sum of LF and HF components
ψ = ψL+ ψH
Taking this into consideration we can write the LF yaw dynamics and HF motion as
ψL = rL
rL = − 1 T rL +K T δ ξH = ψH
ψH = - 2ζωnψH - ωn2ξH +Kw w
Where ψL and rL are LF states , ψH is the HF yaw and ξH is HF state form introduced to represent in state space form the HF wave induced motion.
6.1 Observer Structure
The main purpose of the observer or the state estimator is to reconstruct the unmeasured LF components from disturbed measurements since the measurement consists of both LF and a HF component. This is crucial in heading control because the oscillatory motion due to 1st order wave induced disturbance will, if it enters the feedback back loop, cause wear and tear of the actuators and increase control energy consumption. Therefore the filtering must be accomplished before the signals are used in feedback control.
Figure 8 Model Used by the observer
Observer Equations
To obtain the observer equations, injection terms have to be added to the dynamic equations
ψ L = rL +K1ψ r L = − 1 T rL +K
T δ + K2ψ
ξH
= ψH + K3ψ
ψ H = - 2ζωnψH - ωn2ξH +Kw w + K4ψ
Where ψ = ψ − ψ - ψL H, is the estimation error, ψ is the yaw angle measurement, ψL and rL is the LF yaw and yaw rate estimates respectively, ψH is HF yaw estimate and K1, K2, K3, K4 are the estimator gains to be found.
Using the notation, Δx = x − x , the estimation error dynamics are written in the state space form as:
Δψ L ΔrL ΔξH
ΔψH =
−K1 1 0 −K1
−K2 −1T 0 −K2
−K3 0 0 1 − K3
−K4 0 −ωn2 −2ζωn− K4 Δψl
Δrl ΔξH
ΔψH
The characteristic equation equation of the error dynamic can be found by det [sI-M] and takes the form Π s = s4+ a3s3+a2s2+a1s +a0
To obtain the gains for the observer design, the error dynamics must be satisfied:
Πi=14 (s-pi) =π(s)
Where pi (i=1..4) are the four designed parameters specifying the values of desired location of error dynamics poles. The gains of the observer are calculated by K=P−1Q, where
P=
ωn2/T ωn2 0 0
ωn2+ 2ζωn/T 2ζωn −ωn2/T 0 2ζωn+ 1/T 1 −ωn2 1/T
1 0 0 1
And
Q =
P1234
−(P124+123+234+134+ ωn2/T) P12+24+23+14+13+34− (ωn2+ 2ζωn/T)
−(P1+2+3+4+ 2ζωn+ 1/T) 6.2 TUNING PROCEDURE
To minimize the estimation errors and to satisfy faster convergence of the error dynamics that correspond to the LF states , the real components of the two poles associated with these states of the observer (P1, P2) are chosen slightly left to the open loop poles of the 1st order Nomoto model. The other two poles are chosen to the left of these two poles to guarantee that the HF estimation error corresponding to the 1st order wave disturbance should converge much faster than the LF
states. Taking this into account the fixed gain observer is simulated as per Figure 9:
Figure 9 Simulation scheme
VII. RESULTS AND DISCUSSION
The aim of the observer is to smooth the control action, with reduction of HF components and therefore minimizing the positioning error and reducing vibrations of the actuator. Initially, the proposed controller along with the observer acting as a wave filter is simulated with the Linear model. The simulation is performed with controller parameters KP and Kd =3.80 and 47.206 respectively as discussed in Section 5.2. Parameters of wave disturbance correspond to a moderate weather condition. The wave model damping ratio is fixed at ζ = 0.1, ωn = 0.5 rad/sec and Kw = 0.03 [Fossen].
The desired yaw angle was chosen to be 10 degrees. The state estimates were computed by choosing the poles as according to:-
p1=-1.1/T; p2= -10−4 ; p3=p4= -15 ζωn; which yields the estimator gain vector as:
0.0109 −0.0006 −1.2786 1.3946 Using these estimator gains, the following result were obtained with the Linear Model.
Figure 10 Disturbed and LF output of yaw along with estimation error
Figure 11 Disturbance and Disturbance Estimation
Figure 12 Control requirement smoothening when controller was used with the wave filter
VIII. RESULTS ON NON LINEAR PLANT
The same controller gains were applied on the 3 DOF non linear plant for achieving heading control. However, the estimator gains need to be recalculated to compensate for the non linear plants pole location. The satisfactory estimator gains yielded were
24.3819 6.0733 9.4289 −19.3956 The high gains result in slightly higher estimation errors, however the control effort reduction is still considerably high. The results of simulation with non linear plant are as follows :
Figure 13 Disturbed and LF output of yaw along with estimation error
Figure 14 Control requirement reduction when controller was used with the wave filter
IX. CONCLUSION
The application of a passive wave filter along with design and application of a PD controller on a 3 DOF non linear marine model is presented. A better performance of the heading control loop is evident. The HF motion caused by the ocean waves is reduced and only desirable control action is received by the actuator.
Simulation results are given to support the results.
Further the observer algorithm is fairly simple and easy to implement.
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