英文文献-建模和Micro-Radian精密指向一个更灵活

Modeling and Micro-Radian Precision Pointing of a FlexibleManipulator With the Existence of Static FrictionAbstractControlling flexible structures has been considered as a well-developed topic in the control field. In this topic, vibration suppression and line-of-sight (LOS) poting arein two of the main research issues. Unfortunately, among all works reported, only a few have really achieved high levels of pointing precision. This is mainly because friction, especially static friction, is hard to deal with. In this research, the LOS control of the flexible beam of a standard hub-beam system is studied. The precision required is instead of treating friction as an external noise and trying to compensate for it, the friction phenomenon is represented by a linear spring and is included as part of the system model when the hub sticks. This is, in fact, a good approximation of the hysteretic friction behavior in the presliding phase if the spring stiffness is ap- propriately chosen. This modification greatly enhances the fidelity.Of the system model. What is more important is that it makes ac- tive controls still available in the stick phase to facilitate the fine tuning of the pointing error. Based on this modified model, it is then not hard to synthesize traditional proportionalintegralderivative and linear quadratic Gaussian controls to get LOS pointing with microradian precision.Index TermsDisturbance isolation, flexible structure control (FSC), line-of-sight (LOS) control, precision positioning, static fric- tion, vibration suppression.I. INTRODUCTIONLEXIBLE structure control (FSC) has gained a lot of atten- tion in the space industry as well as in academic research. It has also been an important research issue for robotic manipu- lators. The topics induced by FSC are quite varied. For example, system modeling, large structure identification, model/controller reduction, sensor/actuator location analysis, control theory de- velopment, experimental verification, and so on have all been widely studied since the 1970s. A thorough review on this issue can be found in 1. The main goal of these research works is to get a stable, fast, and precise motion platform which is essentially soft due to its size or due to stringent performance requirements. When it comes to controller synthesis, major efforts have been focused on suppressing the vibration of the system caused by large distance maneuver or external noises. A lot of control algorithms have been put forward, such as the traditional propor- tionalintegralderivative (PID) control, observer-based linear quadratic Gaussian (LQG) control, Lyapunove function-based algorithm, fuzzy control, control, adaptive control, slidingManuscript received April 27, 2006; revised October 19, 2006. Manu- script received in final form February 7, 2007. Recommended by Associate Editor K. Kozlowski. This work was supported in part by the Ministry of Economic Affairs and National Science Council, R.O.C. under Grant92-EC-17-A-05-S10014 and Grant NSC 932218-E-006018.The authors are with the Department of Aeronautics and Astronautics, Na- tional Cheng Kung University, Tainan, Taiwan, R.O.C. (e-mail: .tw; .tw).Color versions of one or more of the figures in this brief are available online at .Digital Object Identifier 10.1109/TCST.2007.903093mode control, pulse width modulation (PWM) control, and so on 212. Smart structures that apply embedded active members have also played an Fimportant role in improving the vibration problem. The application of smart structures can even take care of the microvibrations effectively, which is usually considered impossible using traditional actuators 1215. These strate- gies do provide significant improvement on suppressing system vibration as compared with those cases where nothing is done. However, though one of the main motivations to do these re- search works is to obtain high accuracy pointing which is re- quired by systems, such as space observation equipment, flex- ible robot arms, and other precision instruments, few works have seriously examined how precise is the pointing that they have really achieved. Furthermore, in most of the reports published, the plots that show the closed-loop responses are usually drawn using a scale of degree or radian. This makes it difficult for the reader to see the real steady-state error when it comes to a re- ally high precision pointing problem. Table I lists some research works where the steady-state error can be identified or estimated based on the plots or data shown in the corresponding reports 6, 7, 12. The systems used in these research have similar structures as that used in this study. Obviously, the precision achieved by these works is in the order of milliradian.It is thus the main contribution of this brief to show how to make the line-of-sight (LOS) pointing of a standard flex- ible system achieve microradian precision. There is no active member embedded in the flexible part of the system. In addition, it has significant friction in it. The system discussed in this brief is shown in Fig. 1, which is a standard hub-beam system. It has a brush type permanent magnet (PM) dc torque motor rotating horizontally, which is the only actuator in this system. The bear- ings used in this system are ball bearings with the existence of friction. Two flexible fiberglass beams are clamped to the hub. Each beam is 1000 mm long, 100 mm high, and 1.64 mm thick, and a laser diode is mounted near the tip of one of the beams. A photo sensor is located 10.13 m away from this hub-beam system to measure the position of the laser spot emitted by the diode. The diameter of the effective measuring range of this sensor is 20 mm. In addition to the photo sensor, an encoder with a resolution of 15 s/count is mounted on the motor and a strain gauge is adhered to each beam near the root. The analog-to-dig- ital (A/D) converter used to measure the output voltages of the strain gauge and photo sensor has a resolution of 12 bits. The goal then of this research is to design a control system that can maneuver the laser beam from its arbitrary initial position to the center of the photo sensor with an error of less than 50 m. This accuracy is equivalent to saying that the error of the hub angle plus the deflection angle at the tip of the beam should be less than 5rad. Obviously, the whole control process will be divided into two sets. The first set involves the movement of the laser beam into the effective range of the photo sensor so that fine positioning can be started. This requires the error of the hub angle to be less than 0.057 if the beam is completely straight, or the deflection at the tip of the beam to be smaller than 0.72mm if the hub is pointing exactly at the target. In this set, both encoder and strain gauges are available for feedback. The second set involves the fine tuning of the hub position and the suppression of the residue vibration of the beam so that the error of the hub angle is less than 5 rad (0.00028), and the tip deflection is less than 3.6m. Only the photo sensor is available for feedback in this set.As mentioned earlier, similar systems have been widely studied in the literature, but a few of them have achieved this accuracy. For example, none of the systems listed in Table I have achieved the accuracy that is good enough to move the laser beam into the effective range of the photo sensor to start the fine tuning. At first glance, the main difficulties in achieving the stringent LOS accuracy are coping with the vibration of the beam, the inducement of the nonlinear behavior of the large deflection of the beams, and the possible spillover problem. As a matter offact, these issues have been studied quite thoroughly. However, it turns out that it is the friction embedded in the roller bearings and motor that should be overcome as it comes to precision. This is, in fact, the critical factor that influences the steady-state error. In a real situation, the hub usually sticks at some position other than the designated angle as it gets close to it. If this happens, according to the AmontonCoulomb friction model, all control efforts will be counterbalanced by the static friction force and from then on, the beam just behaves like a cantilevered beam. No control is valid anymore. The LOS of the beam is hence sticking at a wrong angle, and this is why people usually get a steady-state error even if the beam vibration is completely set- tled. To sort out this problem, people use the Coulomb model, the LuGre model 16, or other friction models to estimate the friction force and then compensate for it using feedforward con- trols 79, 12. However, due to the varying properties of friction, models with constant friction parameters can never pre- dict the exact friction force. The deviations from the true fric- tion force causes detrimental disturbance to systems where high in-position stability is demanded, let alone the fact that those models used are usually only an approximation for some lim- ited cases.In this research, friction is not considered as an external noise to be compensated at all. Instead, it is considered as part of the system. More precisely, it is treated as a linear spring and in- cluded in the model as the hub sticks. This simple modification is, in fact, an appropriate approximation of the hysteretic fric- tion behavior as far as precision positioning is concerned. This gives a much more accurate representation of the system. Furthermore, it allows control to be available in the stick phase to keep suppressing the residue vibration of the beam and im- proving the LOS of the whole system.In Section II, the static friction behavior is briefly reviewed, and both the traditional and modified hub-beam models are studied. The experimental verification of the new model is also examined. Sections III and IV describe the control strategy used in this research together with the experimental verification. Section V is the conclusion. II. MODELING OF THE HUB-BEAM SYSTEMThis section begins with an examination of the problem of the hub-beam model developed using the conventional method. The standard way to derive the system model in most literatures is to first find the mode shapes of the beams according to the boundary conditions of a cantilevered beam. Then these mode shapes in the form of space functions are normalized and treated as a base to represent the deflection of each beam. More pre- cisely, the deflection of each beam is a linear combination of these mode shapes. The amount of contribution of each mode shape to the total deflection is then a time function. The gov- erning equations are derived using Lagrangian dynamics with these time functions as part of the general coordinates. Another general coordinate is the rotational angle of the hub. The ex- ternal forces applied to this system are the motor torque, the friction force, the air damping force, and the damping in the bearings and motor brushes. The air damping force is usually ig- nored for simplicity. The friction is usually represented by the AmontonCoulomb friction model. The system is finally lin- earized for controller design based on the assumptions of slow rotation and small vibration. Though these assumptions are usu- ally not theoretically valid, they are still close approximations of the real case. The details of the derivation are omitted to make the presentation concise. Interested readers can find these derivations in 18. The derived model is summarized in the following.A. Traditional Model of the Hub-Beam SystemConsider the system geometry shown in Fig. 2, where are the Newtonian and body-fixed frame, respectively, and is chosen so that the laser beam will hit at the center of the photo sensor if beam A coincides with . Let and be the th mode shape function of beams A and B, respectively, where beam A has the laser diode near the tip. The corresponding general coordinates are, separately, and . In this research, the first two modes for each beam are included in the system model. The contributions of higher modes are negligible. Let is the resistance, and is the back constant of the motor. The beam strain at the position where the strain gauge ad- heres is obtained from the relation. is one half of the thickness of the beam. The location of the light spot hit by the laser beam on the photo sensor is approximated by the expression where is the inclination angle of the photo sensor with respect to , and is the location of the laser diode which is 0.9 m in this re The natural frequencies of the first two mode shapes are analytically calculated and experimentally verified. Their damping ratios are experimentally obtained by observing the decay speed of each mode. These data are shown in Table II. The motor parameters and dynamic friction are identified using the modified ARMA model introduced in 19 with both beams being removed. The motor parameters are contained in Table III. Sub- stituting the mode shape parameters into (1), we can get a tenth- order state space equation for the mechanical part of the whole hub-beam system. The damped natural frequencies of the beams can then be obtained from the poles of the state matrix and are shown in Table IV. It can clearly be seen that each anti-symmetric mode has a frequency higher than that of the corresponding symmetric mode. This is because the hub is free to rotate for the anti-symmetric mode while the symmetric mode is equivalent to the cantilevered vibration with the hub being fixed. Furthermore, the frequencies of those symmetric modes are a little smaller than the corresponding mode frequency shown in Table II. This is due to the extra tip mass of the laser diode. It is also noted that the system is not a completely state controllable system, since the symmetric mode cannot be controlled by the hub. For comparison, the simulation of the model subjected to a pulse input is shown in Fig. 3 together with the experimental result. It is obvious that the phases of these two responses grow apart as time passes. The frequency spectrum of the experi- mental result is then made and shown in Fig. 4. It shows that the first mode is a nonminimum phase mode, and the second mode keeps being a minimum phase mode, with the anti-resonance frequency immediately following the resonance frequency . This coincides with the real situation. What is interesting is that the frequency of the first mode shifts from 1.43 to 1.0 Hz, and that of the second mode shifts from 7.53 to 6.78 Hz. This ex- plains what happens in Fig. 3, since the vibration frequencies in the numerical simulation are fixed and will not shift from one value to another. As mentioned earlier, the modes excited by the motor should be anti-symmetric modes only, and this is exactly what happens in the beginning. However, it shifts to the response of a can- tilevered beam upon comparing these frequencies with those shown in Table IV. This implies that the hub sticks due to fric- tion. To take this factor into consideration, we have to set a small velocity value below which the friction force is transferred from dynamic friction to static friction. When this happens, according to the AmontonCoulomb model, the torque produced by the static friction becomes equal but opposite to the torque applied on the hub by the beams. Hence, when it sticks, the system would be the same as the case where there are two cantilevered beams that are clamped to a fixed pole. A model derived this way does deliver a response that is very close to the experimental re- sults as far as the beam strain is concerned. However, it does not coincide with what really happens to the hub; small hub motion kept being observed as time passes. It also eliminates the pos- sibility of designing a control system to improve the LOS error that was left the moment it stuck, since all control efforts will be counterbalanced by the static friction too. For these reasons, a more accurate model is needed.B. Presliding Dynamics of the HubIt has been found in 1899 21 that there exists a small amount of elastic deformation in the static friction or the so-called pres- liding phase. This observation contradicts what was said by the AmontonCoulomb model. Many new friction models have been introduced since the 1960s, and among them the Dahl model 22 and the LuGre model 16 might be two of the most popular models used in the control field. Many research used these two models to predict the friction force and then compen- sate for it using feedforward control. However, these models are only an approximation of the real case, especially in the presliding phase. Moreover, the friction parameters are usually varying with time and position. Trying to evaluate friction force using any existing model hardly helps in getting the real value. A varying error always exists. This varying error might be ac- ceptable for cases when positioning precision is not very strin- gent. However, it is vital to an ultra-high precision positioning system especially when high in-position stability and high re- peatability are required. Therefore, we need a more appropriate way to cope with friction.The details of the presliding dynamics have further been ex- amined in 23 and are briefly summarized as follows.1) Though the friction related parameters vary with time and position, they are, in fact, temporarily time invariant and locally position invariant. To be temporarily time invariant means that the friction properties remain constant during short intervals of interest. This short interval is usually long enough for control purposes. Meanwhile, to be locally po- sition invariant means that the friction properties remain unchanged if the system stays in the presliding rang