Tuning and Modeling of Redundant Thrusters for Underwater Robots

Transcription

1 Tuning and Modeling of Redundant Thrusters for Underwater Robots Aaron M. Hanai, Kaikala H. Rosa, Song K. Choi Autonomous Systems Laboratory University of Hawaii Mechanical Engineering Honolulu, HI U.S.A. ABSTRACT This paper covers various aspects of tuning redundant marine thrusters for an autonomous underwater vehicle. Topics include the derivation of the system geometry so as to reveal the nature of the redundancy, an experimental outline for the optimal adjustment of the individual thrusters, thruster modeling, and a feedback control scheme. The mathematical relationships between the various thruster properties are combined with information about the system geometry to produce thruster models. These models may then serve to potentially improve the overall vehicle system robustness. KEY WORDS: Thrusters; AUV; modeling. INTRODUCTION With rapidly evolving technology, the capabilities of undersea vehicles have migrated from remotely operated vehicles (ROVs) to autonomous underwater vehicles (AUVs). As the desired mission tasks become more complex, there is an industry-wide demand for increased autonomy. Applications that include hovering, docking, tracking, or manipulation require the fine motion control of an autonomous system. These precision operations all benefit from an increased understanding of the dynamics and the geometry of both the vehicle and its thrusters (Healey, Rock, Cody, Miles and Brown, 1995; Leonessa and Luo, 2001; Smallwood and Whitcomb, 2002; Whitcomb and Yoerger, 1999). Not that the accurate dynamic modeling of a vehicle's actuators would benefit a relatively small, high-performance AUV. However, a larger, more massive vehicle would not be affected in any significant way by the transient properties of the thruster outputs, in which case the dynamic model would be less critical. In any case, it is vital to have a thorough understanding of the steady state characteristics of the marine thrusters, and to realize which of the variables have the most significant effect on the overall system performance. This paper outlines an optimized process for the tuning of redundant marine thrusters for a large-scale AUV. A vehicle control system generally generates a set of desired body forces, which is related to the individual thruster outputs based on the system geometry. There are an infinite number of possible solutions for a vehicle with redundant thrusters. Hence, there is design freedom to choose particular solutions to satisfy specific goals. A mathematical derivation of this geometrical relationship is presented and will be required for the thruster modeling. Performance gains can also be realized by the proper tuning of the thruster controllers/amplifiers, based on the system configuration and the implemented hardware. It would be ideal if the individual amplifiers were tuned so that the thrusters perform identically relative to each other, and so that the open-loop performance is as expected. In the applied case, system uncertainties necessitate controller feedback and personal engineering experience. Strategies of the empirical implementation are presented, along with experimental procedures and analysis of the data. Improved system performance can be attained by identifying the parameters in the relationship between the input amplifier voltage and output thruster force, then selecting the proper signal for feedback to the control system. It is known that there is a mathematical relationship between thruster shaft velocity and output force. This propeller shaft velocity proves to be a stable feedback signal to the control system. This topic will be explored, explained, and presented through experimental analysis. The groundwork for the steady state thruster model and all experimental procedures were performed on the Semi-Autonomous Underwater Vehicle for Intervention Missions, SAUVIM (Yuh and Choi, 1999), shown in Fig. 1. It is a large-scale vehicle rated for fullocean-depth operation, and is capable of intervention tasks, as it is equipped with an electronic seven degree-of-freedom manipulator. The overall effort of this work serves to develop various experimental relationships for the modeling of the different subsystems of the vehicle, in order to improve the robustness of the entire system.

2 where ν is linear and angular velocities in the body-fixed frame and η is the vehicle position and orientation vector in the earth-fixed frame. M is the inertia matrix, including both rigid body and added mass terms. C is the matrix of Coriolis and centripetal terms, including both rigid body and added mass terms. D is the hydrodynamic damping matrix. The vector g describes the gravitational and buoyant restoring forces. The vector B = F F M M M F (2) x y z x y z T represents the input forces and moments in the body-fixed frame, and is a linear combination of the individual thruster forces [ ] T T1 T n T = (3) such that B = AT (4) The transformation matrix A is entirely a function of the particular vehicle's geometry, and will be referred to as the thruster configuration matrix. This matrix A is generally non-square, and must ultimately be inverted (Nakamura, 1991; Strang, 1998) so that the thrusters can be issued their individual commands as a function of the desired vehiclefixed body forces. In this case, the Moore-Penrose pseudoinverse is defined as: ( ) 1 # T T A A AA = (5) In general, the product T # = A B (6) is a unique least-squares solution for $T$ that minimizes the Euclidian norm Fig. 1 SAUVIM VEHICLE GEOMETRY In general, the control system of an AUV generates a set of desired body forces, which is converted to individual thruster amplifier input voltages, based on the system geometry. For a vehicle with a redundant thruster configuration, the under-determined system would have infinite possible solutions. Hence, there is design freedom to choose particular solutions that satisfy specific goals regarding energy consumption, fault tolerance, or fine motion control. A mathematical derivation of this geometrical relationship follows. Generalized Geometrical Derivation Beginning with the dynamic equation of motion for an AUV, expressed as (Fossen, 1994}: (1) Mν + C( ν ) ν + D( ν ) ν + g( η) = B B AT (7) If there is redundancy in the thruster configuration, then the geometry of the system itself may be exploited to achieve fine motion control. The existence of an infinite number of solutions for a redundant system comes from the addition of homogeneous solutions from the nullspace: In # A A (8) The complete solution is then of the form ( ) T = T + T = A B + I A A z (9) where solution. # # particular homogeneous n z R Vehicle Description n is an arbitrary vector that can be used to scale the The SAUVIM vehicle is under continuing development by the joint effort of Marine Autonomous Systems Engineering, Inc., the

3 Autonomous Systems Laboratory of the University of Hawai'i, and the Naval Undersea Warfare Center, Rhode Island. It is equipped with an electronic seven degree-of-freedom robotic manipulator and will be capable of untethered autonomous operation at the generally considered full ocean depth of 6000 meters. The vehicle is currently in its shallow-water form of development, which provides for easier testing and fabrication at relatively low cost. In its current manifestation, the vehicle is equipped with aluminum pressure vessels to house the electronics, shallow-water foam, and a fiberglass fairing, and is rated for operation at 1300 meter depth. The subsequent stage of development will be to pressure harden the vehicle for full ocean depth by upgrading the pressure housings to titanium, and adopting syntactic foam for flotation. Overall, SAUVIM is a relatively large vehicle of about 6 by 2 by 2 meters in size, and a mass of about 1800 kg in shallow water form, and 3600 kg in deep water form. The geometry of the thruster setup is shown in Fig. 2. Fig. 2 SAUVIM Thruster Geometry direction. Also, none of the thruster outputs are symmetrically bidirectional by design, due to the hull and propeller shapes. Furthermore, the two longitudinal thrusters are of a different model that provides more thrust than the other six. In application, maximum measured outputs have been on the order of 245 N in the forward direction, and 115 N in reverse for the larger, high-output thrusters. The smaller thrusters have displayed at most 110 N in the forward direction, and 40 N in reverse. These thrusters are shown in Fig. 3. Furthermore, SAUVIM's center of mass is not precisely at the centroid of the vehicle, and will in fact change over time due to several factors. These include the deployment and motion of the robotic manipulator, the movement of ballast for motion compensation and vehicle trim, or various degrees of payload increase and reduction from respective object retrieval and placement (10) The thruster configuration matrix for SAUVIM is shown in Eq. 10. The numbers expose the lack of symmetry in the placement of the thrusters on the vehicle's frame. Also, the last three rows of the first four columns reveal the coupling between the longitudinal thrusters and pitch, as well as the coupling between the lateral thrusters and roll motion. In the study of the geometry, these off-diagonal coupling terms in the matrix are not negligible. However, a study of the vehicle's dynamics would also reveal that SAUVIM is quite stable in roll and pitch, such that these terms are reduced. A combination of the dynamics with the geometry is important, and is a topic of ongoing efforts. Part of the dynamics may eventually be incorporated into the parameters of the matrix, such as the center of mass, which may change over time in certain conditions. TUNING Based on the system configuration and the implemented hardware, the proper tuning of the thruster amplifiers can produce significant performance gains. Factors to consider include the thruster dead zone, saturation level, offset bias, and current limit. It would be best if the individual amplifiers were tuned so that all of the thrusters perform in an identical manner. In the ideal case, the open-loop performance would behave exactly as desired. In the applied case, factors such as measurement uncertainties, electronic signal noise, non-conservative forces, and outside disturbances result in system errors. This compels the need for controller feedback, as well as personal engineering experience. Fig. 3 SAUVIM Thrusters Numerical Representation of Vehicle Geometry There are several intrinsic asymmetries in SAUVIM's hardware configuration. To begin with, the vehicle geometry and the thruster arrangement is intended for motion primarily in the forward or surge Experiments were necessary for all of the SAUVIM thrusters, but the data presented henceforth is from just one of the large longitudinal thrusters. The numbers are representative of the behavior of the other seven thrusters and are sufficient for the description of the concepts and tuning procedures presented in the scope of this paper. For the experimental setup of these thruster tests, certain arrangements needed to be made. As the SAUVIM vehicle is relatively large, the thrusters, batteries, and controllers were removed from the frame, and

4 tests were conducted apart from the vehicle for convenience. There was a need for a sufficiently large volume of water for the high-output thrusters, so the experiments were conducted at a swimming pool. The test setup is shown in Fig. 4, and includes the controllers/amplifiers, the lever system with force sensor, the thrusters, and a 144 Volt power supply. Fig. 5 Output Thrust vs. Input Voltage The initial experiment collected the output thruster force as a function of the input command voltage and is shown for a large longitudinal thruster in Fig. 5. Note that there is no thrust data between zero and 18N due to hardware limitations in the force sensor. The first data set, represented by the triangles, was taken with the gain setting at a low value. It is apparent that the input signal is bound by +/-10V. The shape of the curve already reveals the bidirectional asymmetry, as the slope in the forward direction is about double that of the slope in the reverse direction. The next step of the process is to increase the gain to a large value, represented by the circles of the second data set in Fig. 5. Here, the thrusters run into the current limit before the input voltages can reach their maximum values. These two data runs have then effectively bounded the horizontal and vertical axes of the plot. Extrapolating the data to the horizontal-axis reveals the dead-zone of the thruster. Note that the increase in gain actually reduces the size of the dead-zone. From the plot, it is also obvious that the curves are not exactly centered about the origin. This can be tuned by the offset bias adjustment. So, for this particular thruster, the maximum thrust is known to be about 250 N forward, and 110 N in reverse, at +/-10 V respectively. The final step is to tune accordingly, so that both maximum output thrust and input resolution are attained, as is shown by the squares of Fig. 5. Fig. 4 Thruster Test Setup For the input signal, a measured voltage is sent to the controllers. There are potentiometers on the controllers to adjust offset bias, current limit, and loop gain. The output voltage and current were monitored, and the signals from the Hall-effect sensors were recorded and converted to the angular velocity of the propeller shaft, given the known planetary gear ratio from the motor shaft. The output forces of the thrusters were recorded via a force sensor attached to the lever arm of the test rig. FEEDBACK SIGNAL AND MODELING Identifying the parameters in the relationship between the input amplifier voltage and output thruster force, then selecting the proper signal for feedback to the control system can serve to improve the robustness of the system. The direct transfer function between command voltage and thrust is affected by various factors, including the amplifier gain and input power. For an applied AUV system, there would be controller tuning mismatches due to human imprecision and error, as well as an inevitable drop in battery voltage, both of which would affect performance. It has been established through theory, that there is a parabolic relationship between thruster shaft velocity and

5 output force that is primarily based on the physical parameters of the propeller. The equation is therefore independent of the input voltage and any variance therein. These claims are verified through the experimental data plotted in Fig. 6. The plot demonstrates that the gain settings do not affect the curve, deeming the velocity output (from the Hall-effect signal) of the controllers to be a reliable measure of the actual output force of the thrusters. The propeller shaft velocity then proves to be a stable feedback signal to the control system. Fig. 7 Block Diagram CONCLUSIONS The experience gained through this experimentation process should provide a practical guideline to optimize the performance of an autonomous underwater vehicle designed for fine motion control applications. Fig. 6 Output Thrust vs. Shaft Velocity Squared In addition to being a reliable feedback measure, for modeling purposes, the relationship between the square of the propeller shaft velocity and output thrust is simply linear. This relationship is only a function of mechanical hardware specifications, such as the propeller blade angle. Being independent from electronic specifications, adjustments can be made to the controllers without affecting the thruster model. Fig. 7 is a block diagram that outlines the uses of the different models that were developed. Following the forward path in red, the thruster configuration matrix (TCM) must be inverted, as discussed above, to produce the desired individual thruster forces. The thrust to voltage relationship is simply modeled as a pair of linear equations (forward and reverse) per thruster. Following the green path of Fig. 7, the thrusters generate forces that move the vehicle. Various sensors feed back the position, velocity, and acceleration information to the motion controller. This can also be passed to a vehicle model to approximate the body forces. Now, following the feedback path traced in blue, the shaft velocities as measured from the thruster controllers are readily converted to thruster forces through the linear (thrust versus shaft velocity squared) thruster models. These approximated thruster forces are then converted into vehicle body forces through the TCM. There now exists redundant body force data to be filtered. Important topics include the thorough understanding of the geometric configuration of the vehicle, and the effect of the thruster controller settings. The experiments revealed the empirical relationship between voltage, propeller shaft velocity, and thrust, which can be modeled by a pair of linear approximations per thruster. The thruster configuration matrix, combined with the different thruster models can provide redundant vehicle body force information that can be filtered with other sensor data. Future work includes a comparison between the body force approximations from the geometrical thruster models and that of the dynamic vehicle model. Along these lines, integration of the vehicle dynamics with the geometry would provide a more complete and welltuned vehicle model. Also, since SAUVIM is equipped with redundant thrusters, the system can be further modified with a fault-tolerant scheme. ACKNOWLEDGMENTS This research was sponsored by ONR grants N and N through the Autonomous Systems Laboratory, and ONR grants N , N , & N through MASE, Inc.

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