Bachmayer Et Al.: an Accurate Four-quadrant Nonlinear Dynamical Model for Marine Thrusters

This paper reports two speci c improvements in the nite-dimensional nonlinear dynamical modeling of marine thrusters. Previously reported four-quadrant models have employed airfoil theory considering only axial uid ow and using sinusoidal lift/drag curves. First, we present a thruster model incorporating the e ects of rotational uid velocity and inertia on thruster response. Second, we report a novel method for experimentally determining nonsinusoidal lift/drag curves. The model parameters are identi ed using experimental thruster data (force, torque, uid velocity). The models are evaluated by comparing experimental performance data with numerical model simulations. The data indicates that thruster models incorporating both reported enhancements provide superior accuracy in both transient and steady-state response. Recent advances in underwater position and velocity sensing enable real-time centimeter-precision position measurements of underwater vehicles [1, 2, 3, 4, 5, 6]. With these advances in position sensing, our ability to precisely control the hovering and low-speed trajectory of an underwater vehicle is limited principally by our understanding of (i) the vehicle's dynamics and (ii) the dynamics of the bladed thrusters commonly used to actuate dynamically-positioned marine vehicles. This paper addresses the latter problem. Recent results indicate that the transient (unsteady) dynamics of marine thrusters can be approximated by a simple nonlinear nite-dimensional lumped-parameter dynamical system [7, 8, 9, 10, 11, 12, 13, 14]. Healey et al. [10] present a nonlinear model that is based on the motor electro-mechanical dynamics and thin-foil propeller hydrodynamics using sinusoidal lift and drag functions. Propeller and uid dynamics are approximated by a two-dimensional second order nonlinear dynamical system with state variables of axial uid velocity and propeller rotational velocity. We will refer to this model as the axial ow model . In [13] the authors report experiments that corroborate the utility of the axial ow model, but also identify discrepancies between the thruster's transient response and the model predictions. This paper examines two possible sources for the reported discrepancies rotational uid ow and lift/drag curve pro les. Models employing experimentally derived (non sinusoidal) lift/drag curves are shown to more accurately agree with experimental performance than models employing sinusoidal lift/drag curves. We describe the experimental setup in section 2. In section 3, we report on the results of an improved hydrodynamic model and compare its results to experimental data and the axial ow model. In section 4, we report on a novel procedure to generate lift and drag curves that is based on experimental data and our new model. Conclusions are given in Section 5. Previously reported studies of unsteady thruster dynamics have measured propeller position, electrical motor current, and axial thrust [8, 15, 9, 13, 7]. One of these studies also made measurements of uid ow velocity [13]. No previously reported study has made measurements of actual thruster torque. To enable a more precise characterization and experimental validation of thruster dynamics, we have constructed a new thruster test facility, that measures directly three-axis force and torque, three-dimensional ow velocity, motor current, and propeller position (Table 2). Bachmayer and Whitcomb are with the Department of Mechanical Engineering, Johns Hopkins University, Baltimore, Maryland, 21218, USA, email: [email protected], [email protected]. Grosenbaugh is with the Deep Submergence Lab at Woods Hole Oceanographic Institution, Woods Hole, MA, 02543, USA, email: [email protected]. The authors gratefully acknowledge the support of the O ce of Naval Research and the National Science Foundation under Grants #N00014-97-1-0487 and #BES-9625143 (Whitcomb), and #N00014-96-1-5014 (Grosenbaugh). WHOI contribution #9883. 2 Preprint of IEEE Journal of Oceanic Engineering (25)1:146-159, January 2000 Name Unit Description im(t) A Motor Current ~ V (t) m s Flow Velocity Vector vz(t) m s Axial Flow Velocity ! (t) rad s Rotational Flow Vel. ! u (t) rad s Rotational Flow Vel. Upstream ! d (t) rad s Rotational Flow Vel. Downstream p(t) N m2 Pressure Òprop(t) rad s Prop. Rotational Vel. L(t) N Lift Force D(t) N Drag Force ~ F (t) Nm Volumetric Body Force Vector F (t) N Hydrodynamic Force Q(t) Nm Hydrodynamic Torque (t) rad Angle of Attack (t) rad Angle of Incidence l m Duct Length, Axial Flow Length lw m Rotational Flow Length R m Propeller Radius A0 m 2 Duct Cross Section Area Ð rad Pitch Angle % Kg m Density kt Nm A Motor Torque Constant kf1 Nm s rad Linear Friction Coe cient kf0 Nm Stick Sriction Imech Kg m rad Motor,Shaft and Prop. Inertia Kz Axial Flow Form Factor KÊ Rotational Flow Form Factor CLmax Max. Lift Coe cient CDmax Max. Drag Coe cient Table 1: Nomenclature Name Accuracy Max. Sample Rate Description im(t) 13bit 2000Hz Motor Current Òprop(t) 4096 counts/rev 2000Hz Prop. Rotational Vel. ~ Fxyz(t) 16bit 2% (FS) 8000Hz Force Vector ~ Qxyz(t) 16bit 2% (FS) 8000Hz Torque Vector ~ Vxyz(t) 1% of measured Velocity 25Hz Flow Velocity Vector Table 2: New Thruster Test Facility Instrumentation II-A Structure The new thruster test stand, shown in Figure 1 was designed to provide a rigid mount for a force sensor and thruster while minimizing disruption of the ow. The fundamental vibration frequency of the stand predicted by Finite-Element Analysis and veri ed experimentally was 48Hz. The fundamental natural frequency of the entire structure would have been 135Hz if not for the compliance of the load cell. II-B Force and Torque Sensing The 6-axis force sensor used in the experiment was a JR3 Model 67M25S-I40. It is constructed from 15-5 stainless steel and equipped with on-board bridge excitation, ampli cation, 16-bit Analog-to-Digital (A/D) conversion, and a digital telemetry output providing full 6-axis digital readings at 8000Hz. The on-board electronics and the digital output of the load cell provide an unusually high immunity from ambient electrical interference. The top of the load cell was bolted to the base of the mast while the bottom was attached to the thruster interface, as shown in Figure 2. The sensor housing was lled with mineral-oil to isolate the sensor from the water. Bachmayer et al.: An Accurate Four-Quadrant Nonlinear Dynamical Model for Marine Thrusters 3 Figure 1: New Thrust Test Stand Setup with Thruster and Acoustic Doppler Velocimeter installed . II-C Flow Sensing For ow measurement we employed a Sontek 10MHz Acoustic Doppler Velocimeter (ADV) which gives the threedimensional velocity averaged over a sampling volume of about 0:3cm at a user selectable sampling rate of up to 25Hz. The sampling volume is 5cm or 10cm away from the ADV sensor depending on the model, as shown in Figure 3. The position of the ow sensor with respect to the thruster is pictured in Figure 1. II-D Thruster Con guration The test stand was designed to accommodate a wide variety of thruster types. At present, six di erent types of thrusters have been tested. The data presented here was generated using a THL404 thruster, Figure 2, manufactured by Deep Sea Systems International. The thruster has a 3 phase DC-brush-less motor in an oil-compensated housing. It is equipped with a resolver for position feedback. The propeller is a 3 bladed nylon propeller, Vetus model BP121, with symmetric, constant pitch blades surrounded by a 15cm aluminum duct. The motor is driven by a modi ed PWM-ampli er with resolver feedback. The ampli er is con gure as a current ampli er. It is equipped with a resolver-to-digital converter providing an encoder signal with a resolution of 4096 counts per revolution. II-E Datalogging All data is logged onto one central PC. The PC, with a 200MHz Pentium processor running MS-DOS, hosts the JR3 Input Output (I/O) card and a separate I/O-card with 8-channels of 13-bit analog I/O, 32bit digital I/O, 8 channels of quadrature encoder input, and a real-time clock with a 139 nanosecond resolution. The ow sensor is interfaced using the standard serial ports of the PC. Delays and cycle times where benchmarked and proved to be easily capable of logging a total of 72 double precision variables at at sampling rate of 2kHz. Data is logged onto RAM during an experiment, and saved to the hard drive at the conclusion of the run. II-F Sampled Data A sample of output data is given in Figure 4. It shows a thruster and ow response to a given command. The upper plot shows a +2:15Nm commanded torque step at t= 0:5s, with a 2:15Nm step at t= 5:5s. The second plot shows the rotational velocity of the propeller. The third and fourth plots show the thrust and the reaction torque as measured by the JR3 sensor. The force and torque data was post processed with a low-pass 5 order zero-phase acausal lter with cuto frequency of 25Hz to suppress artifacts of the test stand's rst resonant mode. 4 Preprint of IEEE Journal of Oceanic Engineering (25)1:146-159, January 2000 Figure 2: Six-Axis Force Sensor Assembly. The bottom three graphs of Figure 4 show the downstream axial, rotational, and radial ow velocities in m=s. In the rst part of the experiment, t= 0:5s to t= 5:5s, the ow sensor is downstream of the propeller. We observe that this ow is dominated by turbulent axial ow with a rotational component and no radial ow. During the second part of the experiment, t= 5:5s to t= 10s, the ow sensor is upstream of the propeller. Here the ow data is dominated by laminar axial and radial ow components, with no detectable rotational ow. The following section describes our approach for obtaining a thruster model from experimental data. In the rst part, we introduce a motor dynamical-model and discuss how to obtain model parameter values for the di erent parameters. The second part of the section deals with the hydrodynamic modeling of the system. Two possible ow models are introduced and compared to experimental data. III-A Motor Model Our Thruster employs a direct drive 3-phase DC-brushless Motor driven by a pulse width modulated (PWM) power ampli er operating in current control mode. A DC-brushless motor in current command is commonly modeled as: Imech Òprop = kt im friction(Òprop) Qhyd (1) where Qhyd denotes the motor shaft load. The plant (1) consists of four terms, an inertia term, a torque input term, a friction term, and a load term. The input into the motor is electric current scaled by the torque constant kt. The constant is usually provided by the manufacturer of the motor but its value depends on the commutation scheme used for the calibration. In our case the value was provided for sinusoidal commutation, which is obtained using the RMS current value and the resulting torque. Since our Bachmayer et al.: An Accurate Four-Quadrant Nonlinear Dynamical Model for Marine Thrusters 5 x y Transmitting Transducer