POSITION ESTIMATION AND CONTROL OF LOAD SWAY IN QUAY-CRANES

Transcription

1 POSITION ESTIMATION AND CONTROL OF LOAD SWAY IN QUAY-CRANES M.A. LOUDA, M. STEVENS, M.W.M.G. DISSANAYAKE & D.C. RYE Australian Centre for Field Robotics The University of Sydney 26 NSW, Australia. {monica michael dissa Abstract We describe the identification and control of the load position for a quay-crane. Identification and control are realised for the two practically important load motions: namely rotation about the vertical axis, and sway in the direction of the crane long travel axis. The implementation utilises an inertial measurement unit to drive a Kalman filter for state estimation. A camera tracker measures true position in the horizontal plane, and is used to validate filter estimates. Keywords Quay-crane, modelling, estimation, control. 1 INTRODUCTION Quay-cranes transfer containerised cargo between ships and ports. A spreader mounted on a head block grapples the container. The head block is hoisted from a rail-mounted trolley by wire ropes that reeve through sheaves. The crane s reeving arrangement and the inherent rope elasticity cause the container to sway in a harmonic fashion. Quay-crane cycle times are limited by the travel distance required, by the residual load sway following a positioning manoeuvre, and by the degree of alignment between the ship s cell guides and the crane axes. The aim of the work described in this paper is to overcome these problems by increasing the workspace of the crane to six degrees of freedom, by estimating the load position on-line, and by using this position estimate to control load yaw and sway. The work presented here builds on our previous work [1]. Previous approaches to load sway damping have applied forces or imposed velocity through the trolley travel motor. The approach reported here adds damping as a function of the load displacement relative to the trolley, so that the sway is attenuated regardless of the driver inputs to the trolley or the external disturbances present. This approach requires an estimate of the quaycrane load position, which we provide using an observer driven by an inertial measurement unit (IMU) mounted on the load. Bar-Itzhack et al.[2] investigated the possibility of using the model of aircraft dynamics as a means for aiding an IMU. Dissanayake et al.[3] made use of the constraint of a vehicle travelling on a surface to extend the time that a land vehicle could be navigated with an IMU. Note that load sway control applied through trolley motion, as considered by Ridout[4] and Marttinen et al.[5] for example, can also be applied to our system. Key prerequisites to controlling crane load sway are the ability to predict the load s future position in space and to measure independently the load position with an accuracy sufficient to validate the prediction. In this paper, our solution and contribution has been to obtain load position and velocity estimates with an IMU, and from these estimates attain control of the load sway. A linear Kalman filter was designed for load position estimation, and was validated against ground truth measurements obtained using a vision system to directly acquire the load position. The velocity was also estimated and used to provide active damping feedback. This paper is organised as follows. In Section 2 the practical system on which our theories were tested is described. A subsection describes the modelling of this system for position identification and control. Estimation utilising this model is discussed in Section 3, for both the longitudinal (X) andyaw(ψ) directions. Position and velocity estimates are verified both in simulation and on an experimental crane. The purpose of the estimator was to control the crane and this is discussed in Section 4. 2 DESCRIPTION OF ISAAC The reported experiments were conducted on a 1/15 th geometric scale model of a quay-crane, called ISAAC.

2 X acceleration(m 2 /s) Figure 1: Idealised Modified Stewart Platform Reeving Geometry The reeving of ISAAC differs from that of a conventional quay-crane, being based on a modified Stewart platform. This novel reeving arrangement utilises six ropes instead of the conventional four [6]. It confers the ability to microposition the load. That is, to manipulate the load in six degrees of freedom, without moving the crane s trolley or the gantry. By constraining all six spatial degrees of freedom of the load, the six rope reeving increases the stiffness and workspace of the crane. An IMU was fitted inside the model container proximal to the load centre of mass, and was used to determine the linear acceleration in the X, Y and Z directions in a reference frame fixed to the load, as well as the angular rates of rotation about these axes. Position is not observed by the IMU. A camera was therefore positioned on the trolley such that it was looking directly down on top of the container. With the knowledge of the container dimensions, the ground truth positions were determined for X and ψ. 2.1 MODELLING Load sway control depends on accurate measurement of the position and orientation of the spreader and its load. The Jacobian and its inverse is the relationship between the six Cartesian co-ordinates and the displacement expressed in the change in rope lengths, or joint space. Knowledge of the geometrical arrangement of the ropes confers the ability to determine the Jacobian and its inverse, in the following manner. The notation used is shown in Figure 1, where an idealised representation of the reeving between our trolley and load is given. Figure 2: ISAAC X Excitation X Acceleration Output from the IMU Consider a small displacement dp of the spreader from the unloaded 1 position. In the world coordinates F : {t; X, Y, Z, φ, θ, ψ}, this displacement can be expressed as: dp =[dx,dy,dz,dφ,dθ,dψ] T. It is assumed that the angular displacements in roll φ, pitch θ, and yaw ψ are small so that dp is a vector quantity, and the order of rotations is unimportant. The inverse kinematic equations may be written as a trigonometric function of the spreader position: l = f(x, Y, Z, φ, θ, ψ) =f(p), where the vector of rope segment lengths is l = [l 1,l 2,l 3,l 4,l 5,l 6 ] T. The perturbation in rope segment lengths required to effect dp is therefore dl = J.dp, (1) where J = l/ p is the Jacobian which transforms from world co-ordinates to rope co-ordinates Dagalakis et al.[7] show how the Jacobian is derived, for a similar situation. As the IMU in conjunction with the Kalman filter estimates displacements in world space, and our control is achieved by varying the rope lengths, this transformation is essential. A generic system model is required within the Kalman filter. The representation chosen is a second order damped harmonic system, described by q(t)+2ζω n q(t)+ωnq(t) 2 =. (2) Figures 2 and 3 show the natural response of ISAAC to excitation in X and ψ, for a height of 2.6 metres below the trolley. 1 that is, the geometric position defined by the unstrained rope segment lengths l 1,l 2,..., l 6

3 ψ rate(rad/s) Figure 3: ISAAC ψ Excitation Yaw Rate Output from the IMU 2.2 Scaling Considerations The results obtained from ISAAC do not scale linearly to a full-size crane. A simulation was therefore performed to determine the expected difference. A natural frequency ratio of 1 : 15, consistent with the scale of lengths, was assumed. As regards the scaling of estimates, the standard deviation in position estimates from the simulation was three times that for ISAAC. The standard deviations for the velocity and bias state estimates were approximately the same as for ISAAC. 3 ESTIMATION 3.1 Design ISAAC allowed the development and testing of a variety of control strategies. Our approach has been to use discrete-time stochastic estimation techniques to provide on-line estimates of system states. We use Kalman filters to estimate the velocities in the two axes critical to load sway control. Position estimates from these filters are used for direct control of the container position. It is assumed that, given real-time state estimates of sufficient accuracy, load-sway control can be treated as a subsidiary problem. Solutions to this problem abound, and we present a candidate controller in the next section. In this section we describe the characteristics of the stochastic estimation problem and present results from Kalman filters using the harmonic system model augmented with a bias state. The load mounted strap-down IMU provides noise-corrupted inertial observations. In this configuration, a full dynamic estimator must maintain platform orientation estimates that match the true orientation of the system. In a system that is experiencing relatively small accelerations, perhaps less than.5g for load-sway, the effect of gravitational acceleration can dominate inertial acceleration measurements if the platform alignment is incorrect [8]. Maintaining strap-down inertial platform alignment is notoriously difficult [9] for a body in free motion. The effect of gyro drift and other sensor non-linearities results in a growing axis orientation error that must be corrected externally. The container hoisted by a crane, however, is not in free motion. By noting that the system acts as a harmonic resonator [1] we can build a constrained prediction model that allows us to estimate the dynamic behaviour of the container and cancel the effects of gyro drift. Critical to simplifying the estimation task is an axis decoupling assumption. It is assumed that each inertial axis of the system acts as a separate harmonic resonator, and that coupling between modes is insignificant for estimation purposes. In practice, non-linearities in the ropes result in some axis coupling. We have found that the rate of energy transfer between modes (equivalent to system non-linearity) is sufficiently low that it does not effect estimation. Given this assumption we can use the one degree of freedom model from Equation 2. X motion results in an IMU linear acceleration, and ψ gives rise to a rotation rate measurement. We use two very similar models for these cases. In the case of the X axis the acceleration observations are the sum of the inertial and gravitational components. Since the model must cope with changes to the gravitational axis direction, we augment the system model with a bias state. This lumped bias state serves to estimate both the gravitational bias and any accelerometer zero offset bias. For the ψ model we found that an additional bias term is also useful, and in this case it estimates the rotational rate bias of the gyro. The ψ Kalman filter discrete state space equations are ψ k+1 ψ k+1 ψ bk+1 where = 1 dt ωndt 2 1 2ζω n dt 1 + dt 2 2 v k (dt ζω n dt 2 )v k (dt)b k ψ k ψ k ψ bk (3) ψ bk = the bias state at time step k b k = discrete white Gaussian bias noise. The observation equation is ψ k = [ 1 1 ] ψ k ψ k + w k (4) ψ bk.

4 .4 X filter X video.5 X Acceleration(m 2 /s) Figure 4: ISAAC X + ψ Excitation X Filter Acceleration and Bias States Figure 6: ISAAC X + ψ Excitation X Filter Output and Camera Yaw Rate(rad/s) Figure 5: ISAAC X + ψ Excitation Yaw Filter Velocity and Bias States For the X motion, the equations of motion are similar to the yaw equations, with X replacing ψ, and with X acceleration measurements in place of yaw rates. The observation matrix for the X acceleration Kalman filter was determined to be Ẋ k = [ ωn 2 + g/l 2ζω n 1 ] X k Ẋ k X bk + w k, (5) where the (g/l)x k term compensates for gravity.the estimator optimally apportions observations depending on how they match the harmonic or bias predictions of the model. After a period at rest an external impulse is applied to the container to cause X and ψ motion. Results for filter velocity and bias estimates are shown in Figures 4 and 5. Biases are close to zero. Analysis of the filter innovations (observation error from prediction) show that the filter is able to follow the impulsive motion rapidly and tracks well within a fraction of a cycle. In addition the filter is able to estimate its bias state with only a small deviation caused by the impulse. Similar results are obtain when micropositioning is used to alter the container s spatial orientation. 3.2 Position Tracking An external system is required to obtain a ground truth to determine if the position estimates closely match the true positions of the container. For this purpose a camera tracker was implemented to capture the motion whilst inertial data was recorded. White paper squares were placed on the corners of the model container. These features were extracted and bounding boxes estimated. From these boxes the pose of the container was determine using its geometry. The capture, processing and boxing algorithms used were developed at the ACFR. The camera ground truth position and the filter positions are shown for X and ψ in Figures 6 and 7 respectively. In these figures, ISAAC was excited in both the X and ψ motions, so as to assess the validity of the decoupled model assumption. 3.3 Robustness The fidelity of the harmonic resonator model is predicated on knowledge of the natural frequency and damping coefficient. In principle it is possible to estimate these states [1]. In practice, robust estimates are only achievable when there is a close match between the model and the physical system. As we expect to encounter many external influences including the (modelled) bias, such an extension of the estimator

5 5 ψ filter ψ video base K 5 b M.5 q.5 Figure 8: One Degree of Freedom Base-excited Springmass System Figure 7: ISAAC X + ψ Excitation Yaw Filter Output and ψ Camera is not advisable. In ISAAC we directly calibrated nominal model parameters based on rope length. The on-line estimator simply extrapolates from this calibrated information. Our estimator must be robust to the expected mismatch between the model and the system. Robustness was validated by deliberately introducing an incorrect model calibration. It was shown that the single axes filters were robust to disturbances in other axes as well as to variation in the actual and filter model natural frequencies and damping ratios. The variation in natural frequency had a greater propensity for affecting the response. Good dynamic response is still obtained for a 2% mismatch in natural frequency which is much greater than expected in practice. 3.4 Technical Difficulties There is no optimal tuning for the estimator parameters. Observation noise data were obtained by direct measurement and were essentially Gaussian. We modelled harmonic and bias prediction errors as Brownian processes. The prediction parameters were chosen to give good innovation consistency and also to exhibit good dynamic behaviour for unmodelled interaction. It is assumed that these unmodelled interactions will be common and the design criterion was that the estimator must respond within in fraction of a cycle to them. This required de-tuning the estimator to slow its response to bias change, and increasing its sensitivity to observation noise so as to quickly track external interaction. X Acceleration(m 2 /s) Figure 9: Damped ISAAC X + ψ Excitation X Filter Acceleration and Bias States 4 CONTROL The main purpose of the estimator was to control ISAAC. Since the IMU readings encode the absolute (or inertial ) values, the one degree of freedom system was represented as in Figure 8. The equation of motion is M q + C q + K(q b) =, (6) where q is the absolute displacement of the load mass. Rearranging the above equation and introducing the natural frequency and damping ratio gives q +2ζω n q + ω 2 nq = ω 2 nb. (7) It was decided to drive the base motion, b, soasto increase the damping of the system: b = 2 ζ c q = K gi q, (8) ω n where K gi is a damping constant that can be adjusted by the driver. Since the filter estimates absolute motion of the load, commanded trolley motion and micropositioning must be subtracted from load motion before the rope

6 5 modified to add stiffness as well as damping. This, along with the application of other crane control theories to the trolley, is the subject of future work. Yaw Rate(rad/s) Figure 1: Damped ISAAC X + ψ Excitation Yaw Filter Velocity and Bias States length changes required to cause active damping can be determined. Figure 9 and 1 show the X and ψ responses, when damping is active. The damped X response settles to within 2% of the final value in 15 seconds, seven times faster than the undamped system response. The damped ψ response settles in 7.7 seconds, twelve times faster than the undamped system. The control gain was limited by saturation of the actuators, which will vary with the particular crane design. Possibly, a different controller would have to be designed for hydraulic actuators on a full-scale crane. Alternatively, the specification of the crane s hydraulic actuators could include these factors in their design. 5 CONCLUSION AND FUTURE WORK The estimator was tuned and robust. The camera ground truth measurement showed a small, average 15% mismatch in X amplitude and 3% mismatch in ψ for the undamped system. Active control damped the X motion sevenfold and the ψ motion twelvefold. The demanded horizontal-plane control could occasionally excite unwanted roll and pitch motions, which were assumed negligible in this work. To improve the control, modelling of the low and high frequency roll and pitch dynamics and their inclusion in the horizontal plane controller would further enhance ISAAC s operation and improve accuracy above that already achieved. Pitch gyro observations show that the low frequency mode coincides with X motion and the high frequency mode coincides with load oscillation in pitch about its own axis. This second mode is excited much more readily when unexpected disturbances are applied. Currently, the control law is being 6 ACKNOWLEDGEMENTS Many thanks must go to Keith Willis and Rob Dawkins for helping with the feature extraction from the camera image. References [1] M.A. Louda, D.C. Rye, M.W.M.G. Dissanayake and H.F.Durrant-Whyte INS-based identification of quay-crane spreader yaw In: Proc. of the IEEE 1998 Int. Conf. on Robotics and Automation, 1998, , Leuven, Belgium. [2] I.Y. Bar-Itzhack and M.Koifman, Inertial navigation system aided by aircraft dynamics, IEEE Trans. on Control Systems Technology, 7(4), 1999, [3] G. Dissanayake, S. Sukkarieh, E. Nebot and H.F. Durrant-Whyte, A new algorithm for the alignment of inertial measurement units without external observation for land vehicle applications, In: Proc. of the 1999 IEEE Int. Conf. on Robotics and Automation, 1999, , Michigan, USA. [4] A.J. Ridout, Feedback control of an overhead crane, ME thesis, (Dept. Electrical Engineering, University of Technology, Sydney, 1989). [5] A. Marttinen, J. Virkkunen and R. Salminen, Control study with a pilot crane, IEEE Trans. on Education, 33(3), 199, [6] M.W.M.G Dissanayake, J.W.R Coates, D.C. Rye, H.F. Durrant-Whyte and M.A. Louda, Control of load sway in enhanced container handling cranes, In: Proc. 5 t h Int. Symp. on Experimental Robotics, 1997, , Barcelona, Catalonia. [7] N.G. Dagalakis, J.S. Albus, B-L. Wang, J. Unger and J.D. Lee, Stiffness study of a parallel link robot crane for shipbuilding applications, J. Offshore Mechanics and Arctic Engng., 111, 1989, [8] E.M. Nebot and H.F. Durrant-Whyte, Initial calibration and alignment of an inertial navigation unit, In: Proc. of M2VIP, 1997, , Toowoomba, Australia. [9] P.S. Maybeck, Stochastic Models, Estimation and Control: Volume 1, (London: Academic Press, 1979). [1] M.S. Grewal and A.P. Andrews, Kalman Filtering: Theory and Practice (New Jersey: Prentice- Hall, 1993).