Handling Roll Constraints for Path Following of Marine Surface Vessels using Coordinated Rudder and Propulsion Control

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1 2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 FrB15.5 Handling Roll Constraints for Path Following of Marine Surface Vessels using Coordinated Rudder and Propulsion Control Zhen Li, Jing Sun and Soryeok Oh Abstract The problem of path following for marine surface vessels using coordinated rudder and propeller control is addressed in this paper. The same problem was addressed in [1] using model predictive control (MPC) to deal with roll constraints. The goal of this study is to investigate the benefits as well as the associated cost, in terms of both control performance and computational complexity, of using the propeller as the second control actuator for solving the path following problem. The proposed MPC design is based on a time varying linear model, and the roll limitation is enforced as a hard constraint. The performance evaluation of the MPC using the rudder and propeller is carried out on a nonlinear 4 degree-offreedom surface vessel model. The simulation results verify the effectiveness of the resulting controller and show the advantage of the proposed controller over the one using a rudder as the sole actuator. Sensitivity of the performance over the key design parameters is also investigated to provide design guidelines. I. INTRODUCTION As a representative control problem for marine applications, the path following of marine surface vessels has attracted considerable attention from the control community [2] [10]. The inherent physical limitations in the control inputs, namely the rudder saturation and propeller speed limits, together with the nonlinearities associated with ship kinematics and hydrodynamics, represent some serious challenges in achieving high performance in path following control for marine surface vessels. More recently, enforcing roll constraints while maneuvering in seaways becomes another important design consideration in surface vessel control, given that the roll motion produces the highest acceleration and is considered as the principal villain in sailor seasickness and cargo damage [11]. Typical nonlinear control methodologies, such as those employed in previously mentioned references, do not take these constraints explicitly into account in the design process. Instead, they achieve the constraint enforcement through numerical simulations and trial-and-error tuning of the controller parameters. On the other hand, model predictive control (MPC) approach has been used in several publications to address the rudder saturation. For example, [12] applied MPC for the tracking problem of surface vessels, [13] addressed the roll reduction for heading control, and [1] solved the path following problem with roll constraints using the rudder as the only control input. In this paper, the coordinated rudder and propeller control is proposed to improve the roll response of the This work is supported by the Office of Naval Research under grants N and N Z. Li, J. Sun and S. Oh are with the Department of Naval Architecture and Marine Engineering, University of Michigan, 2600 Draper Road, Ann Arbor, MI 48109, USA lizhen@umich.edu MPC path following controller, motivated by the fact that the dynamic couplings between the vessel surge speed and roll response can be leveraged to mitigate excessive roll with properly designed propeller actions. Figure 1 shows the Bode plots from normalized inputs (rudder angle and propeller speed) of open-loop linearized container ship S175 [14]. The linearization is performed at the equilibriums corresponding to 10 deg rudder angle and 60, 90 and 120 RPM propeller speeds respectively. It can be seen clearly from Figure 1 that the propeller speed has comparable influence with the rudder on the roll response. Therefore, it is expected that properly coordinated propeller speed and rudder control will have substantial benefits in achieving fast maneuvering and mitigating large roll motion, and this work is aimed to investigate the utility of multiple actuators in MPC to enhance path following performance. It should be noted that the propeller speed effects on heading and roll angles are largely dependent on the rudder angle. Thus the Bode plot (shown as Figure 1) will change correspondingly if different equilibrium rudder angle is adopted. Fig. 1. Bode plot from normalized inputs (rudder angle and propeller speed) to roll angle. This paper presents an MPC design of the path following controller for an integrated model of the surface vessel dynamics and 2-DoF path following kinematics. Two inputs, namely the rudder angle and propeller speed, are employed and coordinated to control the vessel. Our study is focused on enforcing roll constraints while achieving satisfactory path following performance. A time varying 3-DoF simplified /10/$ AACC 6010

2 linear vessel model is adopted in the controller design and a corresponding 4-DoF nonlinear container model is used in simulations in order to study interactions between the path following maneuvering control and roll dynamics. The path following performance and roll response are analyzed by numerical simulations, and compared with the MPC using the rudder as the sole actuator. The results show that the controller using both rudder and propeller achieves improved roll response without compromising the path following performance in terms of the convergence speed. Furthermore, the moderate computational demand associated with additional propulsion control is affordable for real-time MPC implementation. The paper is organized as follows: in Section II, the 4-DoF container model S175 and the corresponding time varying 3- DoF linear model are presented along with the Serret-Frenet formulation to facilitate the path following control design. In Section III, the MPC algorithm is developed to address the path following problem with roll constraints using rudder and propeller. The simulation results are presented in Section IV together with discussions on the tuning of key controller parameters, followed by the conclusions in Section V. II. PATH FOLLOWING ERROR DYNAMICS AND MARINE SURFACE VESSEL MODEL A. Path Following Error Dynamics In the open literature, the path following problem is often simplified into a regulation problem by adopting proper path following error dynamics ( [9], [15] [18]). For this approach, the Serret-Frenet frame ( [19], [20]) is often adopted to derive the error dynamics. Fig. 2. Illustration of the coordinations in the earth frame (inertial frame) {E}, the ship body-fixed frame {B} and the Serret-Frenet frame {SF}. Fig. 2 shows the definitions of the errors used for path following control. The origin of the frame {SF} is located at the closest point on the path curve C from the origin of the frame {B}. The error dynamics based on the Serret-Frenet equations are introduced in [18], given as: ψ = ψ ψ SF = κ (usin ψ vcos ψ) + r, (1) 1 eκ ė = usin ψ + vcos ψ, (2) where e, defined as the distance between the origins of {SF} and {B}, and ψ := ψ ψ SF, are referred to as the crosstrack error and heading error respectively, u, v, r are the surge, sway and yaw velocity respectively. ψ is the heading angle of the vessel and ψ SF is the path tangential direction as shown in Fig. 2 [18], κ is the curvature of the given path. The control objective of the path following problem is to drive e and ψ to zero. The path for surface vessels to follow in the open sea is often a straight line or a way-point path, which consists of piecewise straight lines. In these cases, the curvature κ is zero, therefore the heading error dynamics (1) could be simplified as: B. Marine Surface Vessel Model ψ = r. (3) Marine surface vessels have 6 degrees of freedom. For maneuvering of surface vessels, normally 3 DoFs are of interest, namely the surge, sway and yaw. In some cases, the surge motion is treated as a decoupled dynamics, leaving 2 DoF (sway, yaw) in the maneuvering model [11]. In this paper, in order to address the path following problem with roll constraints, a 4-DoF model is needed, including 3-DoF discussed in traditional maneuvering [14] and the roll p as the additional DoF to represent seakeeping characteristics. A mathematical model for a single-screw high-speed container ship (often referred to as S175 in the marine engineering community) in surge, sway, roll and yaw has been presented in [14]. This 4-DoF dynamical ship model is highly nonlinear with 10 states and 2 control inputs: X = [u,v,r, p,x,y,ψ,φ,n,δ] T and U = [n c,δ c ] T. u, v, r and p are the surge velocity, sway velocity, yaw rate and roll rate with respect to the ship-fixed frame respectively, the corresponding displacements with respect to the inertial frame are denoted as x, y. ψ and φ are heading and roll angles. Other two states are the propeller shaft speed n and the rudder angle δ. The inputs to the model are the commanded propeller speed n c and rudder angle δ c respectively. The actuator input saturation and rate limits are also incorporated in this model. The 4-DoF nonlinear model [14] is one of the most comprehensive ship models available in open literature. It captures the fundamental characteristics of the ship dynamics and covers a wide range of operating conditions. However, due to the complexity associated with the nonlinear constrained optimization, adopting the 10th order ship model for MPC implementation is computationally prohibitive. In this work, the 10th order nonlinear model is used as a virtual ship for simulation and performance evaluation, while a reduced order linear model is used for control design. Suppose the nonlinear system (3 DoF: sway, yaw and roll), together with the path following error dynamics, has the following form: x = f ( x,ū), (4) 6011

3 where x = v r p φ e ψ [ δ, ū = n ]. (5) Given the current state x 0 and control ū 0, the overall nonlinear system (4) can be linearized into: (d x) = Ā( x 0,ū 0 )d x + B( x 0,ū 0 )dū + C( x 0,ū 0 ), (6) where d x and dū are the deviations from linearization point x 0 and ū 0 respectively. Furthermore, Ā( x 0,ū 0 ) = f ( x 0,ū 0 ) x a 11 a 12 a 13 a a 21 a 22 a 23 a = a 31 a 32 a 33 a , (7) a a b 11 b 12 B( x 0,ū 0 ) = f ( x b 21 b 22 0,ū 0 ) = b 31 b 32 ū 0 0, (8) C( x 0,ū 0 ) = f ( x 0,ū 0 ). (9) Notice that all the values of model parameters a (.) and b (.) depend on the linearization point ( x 0,ū 0 ). It should also be noted that the nominal model employed in MPC path following control using the rudder only [1], linearized around the equilibrium with δ 0 = 0 deg and n 0 = 66.9 RPM, is used commonly in linear controller design [11]. For this application, however, this model does not capture the dynamic influence of propeller speed on the state, because in this case b 12 = b 22 = b 32 = 0. Indeed, as the ship is close to the desired path (small e, ψ and δ), the propeller speed has little influence on the path following performance. However, the propeller influence is more pronounced when the ship is away from the desired path and large maneuvering is used to steer the ship. Therefore, a single linear model can not be used to support the control design objective pursued in this paper. To incorporate the propeller effects on the system dynamics, the time varying linear model is employed in the MPC formulation, which is described in the next section. III. MPC FORMULATION FOR PATH FOLLOWING CONTROL USING RUDDER AND PROPELLER Given a sampling time T s and current state x k and previous control input ū k 1, the plant (6) can be discretized as: d x k+1 = A( x k,ū k 1 )d x k + B( x k,ū k 1 )dū k +C( x k,ū k 1 ). (10) Notice that the time varying linear model (10) can capture the dynamic relation between the propeller speed and system states except few cases such as δ k 1 = 0. Then the MPC online optimization problem can be formulated as follows: at each time k, find the optimal control sequence {dū k,dū k+1,,dū k+n p 1 } to minimize the following cost function (11): J( x k ) = N p j=1 subject to equation (10) and [(ū k+ j 1 ū d ) T R(ū k+ j 1 ū d ) + x T k+ j Q x k+ j], (11) ū min ū k+ j 1 ū max, j = 1,2,,N p, (12) dū max dū k+ j 1 dū max, j = 1,2,,N p, (13) x max x k+ j x max, j = 1,2,,N p, (14) where ū d is the desired steady state control action (for straight-line pathes, we have ū d = [0,n d ] T with n d being the desired propeller speed). The inequalities (12), (13) and (14) stand for the control input saturation, input rate limit and state limit respectively (the inequalities (12), (13) and (14) for vector have to be satisfied element-by-element). Notice that x k+ j = d x k+ j + x k and ū k+ j 1 = dū k+ j 1 + ū k 1. Q and R are the corresponding weighting matrices for states and inputs and N p is the predictive horizon. The control law is given by dū k = dū k. Since the cost function (11) is quadratic in x and ū and all the constraints are linear, we can use quadratic programming (QP) to solve the optimization problem. In this study, the optimization and simulation are performed in MATLAB. For successful MPC implementation, the controller parameters, such as the sampling time T s, prediction horizon N p and weighting matrices Q and R, should be properly selected. In general, small sampling time provides more timely feedback but requires more frequent optimization. Likewise, the length of the predictive horizon N p is a basic tuning parameter for MPC controllers. Generally speaking, performance of the controlled system improves as N p increases, at the expense of additional computation effort [21]. Furthermore, the weighting matrices Q and R are used as the main tuning parameters to shape the closed-loop response for desired performance [22]. The simulation based tuning process of these key controller parameters is presented in the next section. IV. SIMULATION RESULTS The MPC using rudder and propeller is implemented and simulated on the 4 DoF nonlinear S175 container model. The propeller speed limit (10 n 160 RPM), together with the rudder saturation and its rate limits ( δ 35 deg and δ 5 deg/sec), are incorporated in simulations. No rate limit is imposed on the change of the propeller speed. Since only the relative penalty on x and ū will influence the performance, we choose the diagonal elements of matrices Q and R to have the form of {0,0,0,0,1/100 2,c 1 }, {c 2,c 3 /70 2 }, namely, the 6012

4 cost function is J = N p j=1 [(e k+ j/100) 2 +c 1 ψ 2 k+ j +c 2δ 2 k+ j 1 + c 3 ((n k+ j )/70) 2 ], with c 1, c 2 and c 3 being positive constants. Note the difference in the order of magnitude of the states and controls: the cross-track error e is in hundreds, heading error ψ and rudder angle δ are in π and the deviation of propeller speed is around 70, we use the normalized errors and controls to properly scale the variables in the cost function. A. Sampling time and Prediction Horizon Choices The proper sampling rate for discrete dynamical systems should be about 4-10 samples per rise time [23], which is about 18 seconds for the roll dynamics of the container ship S175. Therefore, the sampling time between 1 to 4 seconds is a rational choice in this application. The simulations for the prediction period of 120 seconds (with different sampling time T s and prediction horizon N p ) are performed to evaluate the effect of the sampling time. Simulations show that the system responses with 1 second and 2 second sampling period are almost indifferentiable, while the responses with 3 or 4 second sampling interval start to deviate. Considering the high computational demand for short sampling times, we conclude that T s = 2 sec is a good trade-off for the implementation of MPC controller for the container ship under consideration. For the selection of prediction horizon, we performed simulations of different prediction horizons with the same sampling time, which are shown in Figure 3. The gains c 1 = 6, c 2 = 1 and c 3 = 0 are employed in this simulation and the sampling time is 2 seconds. Figure 3 shows that a longer predictive horizon leads to a better path following performance, in terms of less overshoot, but the benefits of extending the prediction horizon diminish beyond N p = 40. Given the heavy computational cost associated with long prediction horizon (in our simulations, the computational time for each optimization with N p = 160 and N p = 80 are about 16 and 4 times of the one for N p = 40, respectively), it can be concluded that a value of achieves a good trade-off for the predict horizon N p, given 2 seconds as the sampling period. The same conclusion can be drawn from simulations performed for many other gain sets. B. Effects of Weighting Matrices Q and R The guidelines given in [1] for tuning matrices Q and R of the MPC path following controller with rudder are given as follows: 1) Set c 2 = 1, and vary c 1 to achieve desired path following performance; 2) Fix c 1 as selected in 1), vary c 2 to tune for different rudder and roll responses. These guidelines are also useful for coordinated rudder and propeller controller tuning. In this section, we focus on investigating the performance sensitivity to the gain c 3. With gains c 1 = 6 and c 2 = 1, three values of c 3, namely 0, 1 and 100, are used in simulations, which are summarized in Figure 4. The simulation results for different values of c 3 are illustrated in Figure 4. From Figure 4, one can see that the propeller action is dictated by the value of c 3 : the large Fig. 3. Simulation results of the ship response with different prediction horizon. value of c 3 will prevent the change of propeller speed. More specifically, if c 3 is extremely large (c 3 = 100), the propeller speed will be kept almost constant, leading to the results as if only the rudder were used. It is also shown from Figure 4 that the smaller the value of c 3, the faster path following convergence speed and the smaller maximum roll angle. Because in this case, the propeller is actively controlled to slow down the vessel speed to allow faster turn with large rudder command without inducing excessive roll motion. To further confirm it, we performed the linear analysis of the closed-loop system with LQR controller, and it can be shown that the roll response is less sensitive to the rudder input for the coordinated rudder and propeller control. C. Quantitative Comparisons of One-input (Rudder) and Two-input (Rudder and Propeller) MPC The MPC designed using time varying linear models with rudder and propeller as inputs is implemented and simulated with the full order original nonlinear model S175 and compared with simulations of one-input case, where the propeller speed is maintained at 66.9 RPM, which corresponds to 7 m/s ship service speed. 6013

5 Fig. 4. Simulation results of the ship response with different penalties on the propeller speed. Fig. 5. Comparisons of one-input and two-input MPC performance with roll constraints. First we compare the performance of these two controllers when no roll constraint is imposed. For the comparison, see the case of c 3 = 0 in Figure 4 as the coordinated rudder and propeller control and the case of c 3 = 100 as the rudder only control. Figure 4 shows that the introduction of additional propeller control reduces the roll response. Moreover, this improvement is achieved with faster path following convergence speed. When the vessel makes large turns, the coordinated rudder and propeller controller predicts that the large roll motion will happen, thus decides to slow down the propeller speed to reduce the vessel forward speed. As a result of forward speed reduction, the vessel has the capability to make an easier turn while keeping the roll motion small. Furthermore, we compare these two controllers implemented on the original nonlinear system with roll constraints. In simulations, the allowed maximum roll angle is set to 4 deg. The controller gains are chosen as c 1 = 6, c 2 = 1 and c 3 = 0 for the two input case and c 1 = 6 and c 2 = 1 for the one input case. The corresponding results are summarized in Figure 5. As shown in Figure 5, these two controllers both achieve path following while satisfying the roll constraints. Although they have the same maximum roll angle and similar RMS roll angle, which is due to the constraint enforcement capability of MPC, the coordinated rudder and propeller controller has much faster path following convergence speed. Finally, the simulations are performed with tightening rudder saturation (without roll constraints) to further compare the performance of one-input and two-input MPC controller, which are shown in Figure 6 (with the same controller gains as previous simulations). In the simulations, the maximum rudder angle allowed is 20 deg, reflecting a reduced rudder control authority. Figure 6 shows that the coordinated rudder and propeller controller can effectively reduce the roll actions compared with the controller using rudder only and lead to significantly faster path following convergence speed. To quantitatively evaluate the controller performance, three performance indices are introduced and evaluated, namely the maximum roll angle φ max, Root Mean Square (RMS) roll angle φ RMS and path convergence time t con (t con is defined as the time the vessel finally approaches the path with cross-track error less than 10 m). The summary of performance of different controllers is given in Table I, which shows that the roll response is largely reduced with the twoinput MPC control, both in maximum and RMS values. Furthermore, using propeller also reduces the convergence time. Table I also shows that the advantage of introducing the propeller control is more pronounced when the rudder is limited in its control authority. Specifically, more roll reduction and relatively faster path following convergence speed are observed. To sum up all the comparisons, we can conclude that adding propeller as an actuator helps to reduce the roll response and improve path following convergence speed. Using a desktop computer with P4 2.4 CPU and 2G RAM, the optimization problem of the coordinated rudder and 6014

6 following performance and roll response were analyzed by numerical simulations and compared with the MPC using rudder as the only actuator. The simulations showed that the coordinated controller has the advantage over the controller using rudder only with improved roll response and path following performance. Fig. 6. Comparisons of one-input and two-input MPC performance with tightened rudder saturation. TABLE I COMPARISONS OF PERFORMANCE INDICES. φ max φ RMS t con [deg] [deg] [sec] without One-input roll Two-input constraints Change (%) deg One-input roll Two-input constraints Change (%) deg One-input rudder Two-input saturations Change (%) propeller MPC with 2 second sampling interval and 60 step predictive horizon can be solved in about 0.9 second in simulations, compared to around 0.6 second for using rudder as the only actuator. Based on our experience, real-time implementation should not be a problem given this moderate computational demand. V. CONCLUSION The coordinated rudder and propeller MPC design of the path following controller with roll constraints for an integrated model of the surface vessel dynamics and 2-DoF path following kinematics was presented. Two actuators, namely the rudder and propeller, were employed and coordinated to achieve path following. Time varying 3-DoF simplified linear vessel model was adopted in the controller design and a corresponding 4-DoF nonlinear container model was used in simulations to evaluate the vessel response. The path REFERENCES [1] Z. Li, J. Sun, and S. Oh, Path following for marine surface vessels with rudder and roll constraints: an mpc approach, Proceedings of the IEEE American Control Conference, [2] M. Breivik and T. Fossen, Path following of straight lines and circles for marine surface vessels, Proceedings of the 6th IFAC CAMS, [3] K. Do and J. Pan, Underactuated ships follow smooth paths with integral actions and without velocity measurements for feedback: theory and experiments, IEEE Transactions on Control Systems Technology, vol. 14, no. 2, pp , [4] P. Encarnacao and A. Pascoal, Combined trajectory tracking and path following for marine vehicles, Prceeding 9th Mediterranean Conference on Control and Automation, [5] T. Fossen, M. Breivik, and R. Skjetne, Line-of-sight path following of underactuated marine craft, Proceeding of the Sixth IFAC Conference Maneuvering and Control of Marine Crafts, pp , [6] Z. Jiang, Global tracking control of underactuated ships by lyapunov s direct method, Automatica, vol. 38, pp , [7] E. Lefeber, K. Pettersen, and H. Nijmeijer, Tracking control of an underactuated ship, IEEE Transactions on Control Systems Technology, vol. 11, pp , [8] Z. Li, J. Sun, and S. Oh, A robust nonlinear control design for path following of a marine surface vessel, Proceedings of the IFAC Conference on Control Applications in Marine Systems, [9] K. Pettersen and E. Lefeber, Way-point tracking control of ships, Prceedings of 40th IEEE Conference Decisions and Contorl, [10] R. Skjetne, T. I. Fossen, and P. Kokotovic, Robust output maneuvering for a class of nonlinear systems, Automatica, vol. 40, pp , [11] T. Fossen, Marine Control Systems. Marine Cybernetics, [12] A. Wahl and E. Gilles, Track-keeping on waterways using model predictive control, Proceedings of the IFAC Conference on Control Applications in Marine Systems, [13] T. Perez, C. Tzeng, and G. Goodwin, Model predictive rudder roll stabilization control for ships, Proceedings of Maneuvering and Control of Marine Craft, [14] T. Fossen, Guidance and Control of Ocean Vehicles. John Wiley and Sons, Inc., [15] M. Breivik and T. Fossen, Principles of guidance-based path following in 2d and 3d, Proceedings of 44th IEEE Conference on Decision and Control, pp , [16] L. Lapierre, D. Soetanto, and A. Pascoal, Nonlinear path following with applications to the control of autonomous underwater vehicles, Proceedings of 42nd IEEE Conference on Decision and Control, vol. 2, pp , [17] R. Rysdyk, Uav path following for constant line-of-sight, Proceeding of the 2nd AIAA, [18] R. Skjetne and T. I. Fossen, Nonlinear maneuvering and control of ships, Proceeding of the MTS/IEEE OCEANS, [19] C. Samson, Trajectory tracking for unicycle type and two steering wheels mobile robots, Proceedings of ICARV, pp. RO 13.1, [20] A. Micaelli and C. Samson, Path following and time varying feedback stabilization of a wheeled mobile robot, Technical Report. INRIA., [21] S. Qin and T. Badgwell, A survey of industrial model predictive control technology, Control Engineering Practice, vol. 11, pp , [22] M. Morari and J. Lee, Model predictive control: past, present and future, Computers and Chemical Engineering, vol. 23, pp , [23] K. Astrom and B. Wittenmark, Computer-Controller Systems: Theory and Design. Prentice Hall,