Tracking Control And Trajectory Planning In Layered ... - CiteSeerX

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 46, NO. 2, APRIL 1999

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Tracking Control and Trajectory Planning in Layered Manufacturing Applications Abderrahim Bouhal, Mohsen A. Jafari, Member, IEEE, Wen-Biao Han, and Tong Fang, Student Member, IEEE

Abstract— This paper discusses the improvements acquired by the introduction of a tracking controller and a look-ahead trajectory-planning policy in part fabrication using the layer manufacturing process. The improvements are quantified not only in terms of tracking and contouring errors, but they are also quantified in terms of overfilled and underfilled areas, thus directly relating the quality of parts fabricated through such a process. Index Terms— Fused deposition, layered manufacturing, motion control, software prototyping, tracking, trajectory planning.

I. INTRODUCTION

T

HE layered manufacturing (LM) process significantly reduces part-specific setup manufacturing lead times. It has been primarily used in building prototypes for design conceptualization, verification, and simulation. As the application of LM expands from merely prototyping to functional parts, the fabrication of accurate geometry becomes an important issue. In the case of fused deposition modeling (FDM) the built process, which includes both the positioning and deposition processes, can be a potential source of defects. The errors generated by the built process are essentially due to the accumulation of the deposition errors and the – positioning errors. The current FD technology only supports open-loop positioning and deposition control systems. In the case of rapid prototyping, the open-loop control system does not necessarily impose much of a problem as the tolerance for the various geometrical and dimensional features of the part are relatively not so tight. This is no longer true if the technology is intended to be used for fabrication of functional parts [5]. The focus of this paper will be particularly oriented to tracking control and trajectory profile generation for FD technology. We intend to test our models on the FD machine that is under construction at Rutgers University, Piscataway, NJ. To evaluate the performance of the proposed approach, a simulation model has been developed for the positioning system. Section II presents an overview of the FD technology. The process model is given in Section III. The proposed tracking control strategy and the trajectory profile generation algorithm are respectively discussed in Sections IV and V-A. A program for the evaluation of underfilled and overfilled areas within Manuscript received December 9, 1997; revised September 25, 1998. Abstract published on the Internet January 18, 1999. This work was supported in part by the Office of Naval Research under Grant N00014-96-1-1175. The authors are with the Department of Industrial Engineering, Rutgers University, Piscataway, NJ 08855 USA (e-mail: [email protected]). Publisher Item Identifier S 0278-0046(99)02723-9.

Fig. 1. FD Process.

a layer is discussed in Section V-B. A simulation model and results are given in Section VI for both contour and raster tool path patterns. II. FD TECHNOLOGY A. FD Process Description The FD process is an LM technique for the fabrication of polymer, wax, and ceramic parts using FD technique. The process starts by designing a boundary surface model of the part using a computer-aided design (CAD) package. The CAD model is then sliced [2] into a succession of horizontal layers ordered by their coordinates in the axis. For each layer, the boundaries of the slices are fabricated using a contour tool path, and the interiors are filled with a raster tool path pattern. The density of a raster pattern is controlled by the air gap between raster segments. The tool-path information and the design parameters are then merged into a design file and downloaded to the FDM machine. B. Machine Description An FDM machine [1], [9] consists essentially of a head liquefier (Fig. 1) attached to a carriage moving in the horizontal – plane. The function of the head-liquefier assembly is to heat and pump the filament material (polymer, wax, or ceramic) through the tip onto the modeling surface. The spooled filaments are fed into the liquifier via a set of two feedwheels driven in a counterrotating direction by a small dc servomotor which provides enough torque to the filament to act as a piston during the extrusion phase. The filament softens and melts inside the liquefier, and then it is extruded out of a nozzle tip located at the bottom end of the liquifier.

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Fig. 2. 2-D tracking system.

Fig. 3. x-axis controller structure.

The extruded material is laid down on a platform following the tool path. In general, the contour is laid down first, after which the interior is filled. Once a layer is built, the platform moves down one step in the direction to design the following layer and, hence, up to the top layer of the part. III. MODEL

OF THE

controller structure is similar). In this figure represents the is the analog velocity feedback gain, amplifier gain, is the discrete position feedback gain, and is the leadscrew pitch. The discrete transfer function between the input of the analog velocity loop and the position output is given by

POSITIONING SYSTEM

In our proposed framework, the positioning system is a linear open-frame – table that uses high-precision ground ball screws and permanent magnet ac servomotors. Two linear incremental encoders attached to the axis and axis sense the position of the head-liquefier assembly. Since the stator ac motor currents are essentially sinusoidal and are regulated within fixed bands, the dynamic equations of the permanent magnet ac motor can be reduced to (1) [6], which is similar to a dc motor equation (1) (mN) is the inertia of the load for axis (rad/s) where is the motor rotational speed, (mN s) is the viscous friction (mN/A) is the torque constant, (A) is the motor factor, (mN) is the motor torque (see [4] and input current, and [7]). IV. CONTROLLER STRUCTURE Fig. 2 shows the basic schematic of a two-dimensional (2-D) tracking control structure. A tracking feedforward compensator is added in order to compensate the regulation dynamic and to impose an appropriate tracking dynamic between the new reference (or and the actual output . Fig. 3 shows desired) input the detailed structure of the -axis controller (the -axis

(2) where

The proportional and derivative gains were chosen near critically damped closed-loop behavior. The closed-loop transfer and the actual function between the reference input is given by position output

(3) has two complex conjugate poles and one negative real pole close to the origin of the plan. The feedforward compensates the closed-loop transfer compensator and imposes a faster tracking dynamic. function is However, because the zero of negative, it should not be canceled, otherwise, it will result in

BOUHAL et al.: TRACKING CONTROL AND TRAJECTORY PLANNING IN LM APPLICATIONS

a lightly damped oscillatory output. To ensure good tracking performance without canceling this closed-loop zero, a zero phase error tracking controller (ZPETC) was proposed in [8] in the context of welding applications. In the ZPETC [3], cancels the regulation dynamics the compensator and any stable zeros in . It also defined by applies a feedforward dynamic scale factor to guarantee zero phase errors between the reference input and the actual position . In this case, the position tracking compensator will be given by (4), shown at the bottom of the page.The compensation is realizable, because the desired and, in particular, the two-step lookinput sequence ahead (LA) value, are available well in advance. Hence, at , the feedforward control action will depend on the instant , shown by (5), at the bottom of the two-step LA value of page.The transfer function (6) between the desired trajectory and the plant output expresses that the plant input output is a moving average of the desired trajectory with a unite steady-state gain (6) V. TRAJECTORY PLANNING Unlike the computer numerical control (CNC) technology, which according to the geometrical information provided to it, uses linear, circular, or spline interpolators, in FD technology only linear interpolation is used. Linear interpolation as such contributes to tracking and contour errors, which are partially responsible for the voids and defects in the built process. Therefore, a given layer, including the contours and the interior rasters, can be looked upon as a set of line segments. To keep the deposited road width constant along each line segment, it is necessary to maintain a constant ratio between the output flow rate and the positioning speed during the built process. Under certain assumptions it is possible to write (see Fig. 4)

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Fig. 4. Feeding flow and positioning speed.

feedback is on the controlled torque provided by the dc motor. It is, therefore, essential to keep the speed of the liquefier head constant, as long as possible, in order to avoid the difficulty of changing the deposition flow rate in such an open-loop system. This constitutes the first constraint for the design of the trajectory-planning model. Another constraint on the trajectory-planning model is the fact that the liquefier-head speed is not continuous. At the connecting point between two segments, an infinite acceleration or torque is sometimes needed to exactly follow the reference geometry (at sharp corners). However, since the torque is finite, the linear system model (1) can represent a “good” approximation of the real system only when there is no saturation (8) At sharp corners, the contour errors cannot be avoided unless the deposition head stops at the turning points. However, it is not suitable to stop and start at each corner, especially when the segments are very small, because this will produce a very slow average speed, besides the difficulty of controlling the deposition flow rate at stop/start points. Instead of a complete stop, it is more appropriate to reduce the speed within certain tolerances with respect to (7).

(7) A. Modeling and Solution Methodology (m /s) is the material flow rate, (m/s) where (m ) is the liquefier-head positioning speed, and (m ) are, respectively, the cross-section areas of and the filament and the deposited road. With the current state of technology, the material flow rate can only be driven in an open loop without any feedback on the flow. The only

We can formalize the trajectory-planning model as an optimization problem where one wants to minimize the voids by properly setting the liquefier-head speed at various locations, subject to the two constraints that were discussed above. Unlike many processes where time is of essential importance, we are not going to consider the built time in our calculations.

(4)

(5)

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The reason is that LM in its entirety is much faster and much more economical than its closest competitor (i.e., injection molding). Ideally speaking, such an optimization model requires the dynamic models of both positioning and deposition systems. Unfortunately, the dynamic model of the deposition system is unknown to a large extent, thus leaving us with only the dynamic model of the positioning system. Therefore, in our analysis, we will assume that the deposition system is providing a constant material flow, according to (7), with no disturbances or errors allowed. From the two constraints described above, one could conclude that the liquefier head must move with a constant speed as much as possible, except at the sharp corners. Indeed, if the liquefier-head speed were sufficiently low, such that turns can be made satisfying the above constraints, one would then prescribe constant speed. However, for higher reference speed requirements and sharp corners or very small line segments, the idea of constant speed may not be appropriate. To “optimize” the speed profile within a given line segment, it is necessary to take into account in advance the length of the next segment and the sharpness of the next corner, so that enough time is given for acceleration or deceleration. This is the main trust of the LA policy that we propose for LM applications. As a comparative evaluation, two algorithms have been considered, constant speed (CS) and LA. The CS algorithm keeps the speed of the subsystem liquefier head constant, except at the starting and ending phases of the contours and the raster. The LA algorithm determines the starting and the ending speeds for the current segment, according to the lengths of the current and the next segment, and according to the angle between these two segments. At the sharp corners, the liquefier-head speed is reduced according to the sharpness of the corner. The above two algorithms only use linear interpolators. The flow chart in Fig. 5 describes the different computational steps required in each of these two interpolation algorithms. Different speed profiles can be chosen (such as constant, linear, or exponential profile). In this application, a constant profile has been used in order to reduce the complexity of the real-time calculation and communication load. To evaluate the performance of these two algorithms, a program has been developed in order to calculate the defects (underfills and overfills) within each layer. The amount of defects gives a good evaluation of the tracking and contouring errors in terms of underfills and overfills.

Fig. 5. General structure of the proposed interpolation algorithms.

simulated tool path, along with the tracking and contour errors, will be used for the selection of the appropriate interpolation algorithm. VI. SIMULATION MODEL A simulation model of the two-axes positioning system described above was developed in Matlab in order to evaluate the performance of the tracking controller and the interpolation algorithms presented in Section V. Fig. 6 illustrates the simulation model for the axis. The model of the positioning system, the regulation feedback, and the feedforward tracking compensator are described in Section IV. The simulation model takes into account torque and controller saturation. A. Simulation Results

B. Evaluation of Underfilled and Overfilled Areas The computation of overfilled and underfilled areas is done by mapping the contours, raster, and background into a binary image. The background is represented by “0” pixel entries and the area covered by contours and raster are represented by “1” pixel entries. The underfilled areas are then equal to the number of “0” pixels within the contour multiplied by the pixel size and the overfilled areas are equal to the number of overlapped “1” pixels multiplied by the pixel size. The results for underfills and overfills of the reference tool path and the

To evaluate the performance of the feedforward tracking controller, a “high”-frequency sinusoidal reference input position was applied to the simulation model, with and without the presence of a tracking compensator. Fig. 7(a) and (b) represents the reference input position (by solid line) and the system output position (by dotted line) obtained, respectively, with and without tracking compensator. These results show the improvement, in terms of tracking performance, acquired by the introduction of the feedforward compensator. As expected, with this compensator the tracking dynamic of the closed-loop

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Fig. 6. Simulation model (x axis).

(a) (a)

(b) Fig. 7. (a) With tracking compensator. (b) Without tracking compensator. (b)

system is faster and can follow high-speed reference input without introducing any lag. We also performed simulation with sharp corner references. We noted that, for low-speed references, the position follows closely the reference with very small contouring errors at the corner. The errors obtained with high-speed references were not acceptable for this kind of application. To this end, the proposed LA tracking algorithm detects the points along the path where the curvature is sharp and used this information to slow down to a speed which will allow the corner to be traversed without saturation and within acceptable error. To compute defects, a typical raster tool path is used as an input reference for the simulation model of the positioning system. Fig. 8(a) and (b) shows the system response to the reference inputs obtained from the 2-D raster information, by

Fig. 8. (a) Constant speed (Vmax (Vmax = 12 cm/s).

=

12 cm/s). (b) LA algorithm

using, respectively, a CS interpolation algorithm and an LA interpolation algorithm. The amount of underfilled and overfilled areas and the average contouring errors are summarized in in the real path, the contour error Table I. For a point is defined as the perpendicular distance from that point to the reference path. It is given by where is the equation of the reference input segment. The utilization of the LA algorithm has reduced the average contouring errors by about 40% compared to the CS algorithm. Fig. 9(a) and (b) corresponds to the contouring errors obtained, respectively, from Fig. 8(a) and (b). As expected, Fig. 9(b)

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CS

AND

TABLE I LA PERFORMANCE FOR TYPICAL RASTER TOOL PATH.

moved through the desired path smoothly and without actuator saturation. This algorithm also has an open structure that can include the coordination between the deposition and positioning control subsystems. REFERENCES [1] J. W. Comb, W. R. Priedeman, and P. W. Turley, “Layered manufacturing control parameters and material selection criteria,” Manuf. Sci. Eng., vol. 68, no. 2, pp. 547–556, 1994. [2] A. Dolenc and I. Makela, “Slicing procedure for layered manufacturing techniques,” Comput. Aided Des., vol. 26, no. 2, pp. 119–126, Feb. 1994. [3] S. Endo, M. Tomizuka, and Y. Hori, “Robust digital tracking controller design for high-speed positioning systems,” in Proc. American Control Conf., 1993, pp. 2494–2498. [4] H. K. Joon, H. Y. Chung, and II. K. Dong, “Robust position control of AC servo motors,” in Proc. IECON’95, vol. 1, pp. 621–626. [5] K. A. Mukesh, A. Bandyopadhyay, R. V. Weeren, A. Safari, S. Danforth, N. Langrana, V. R. Jamalabad, and P. J. Whalen, “FDC, rapid fabrication of structural components,” Amer. Ceramic Soc. Bull., vol. 75, no. 11, pp. 60–65, Nov. 1996. [6] P. Pillay and R. Krishnan, “Modeling of permanent magnet motor drives,” IEEE Trans. Ind. Electron., vol. 35, pp. 537–541, Nov. 1988. [7] L. Werner, “Microcomputer control of high performance ac drives—A survey,” Automatica, vol. 22, no. 1, pp. 1–18, 1986. [8] M. Tomizuka, “Design of digital tracking controllers for manufacturing applications,” Manuf. Rev., vol. 2, no. 2, pp. 82–90, June 1989. [9] M. J. Wozny, “Systems issues in solid free form fabrication,” in Proc. Solid Free Form Fabrication Symp., Univ. of Texas, Austin, 1992, pp. 1–15.

(a)

Abderrahim Bouhal received the M.S. and Ph.D. degrees in electrical engineering from the National Institute of Applied Science, Lyon, France, in 1990 and 1994, respectively. He is currently a Research Associate and PartTime Lecturer in the Department of Industrial and Systems Engineering, Rutgers University, Piscataway, NJ. His research interests are adaptive control theory, nonlinear systems, layered manufacturing, and manufacturing automation.

(b) Fig. 9. (a) Contour error with constant speed (Vmax error with LA policy (Vmax = 12 cm/s).

= 12 cm/s). (b) Contour

displays a good reduction of the average value of the contouring errors along the raster tool path. The corresponding underfilled area has also been reduced to an acceptable level. The reduction of underfilled areas is not in the same proportion as the reduction of the contouring errors, because the reference input obtained from the tool path file contains certain amounts of underfills and overfills. In the ideal case, the improvement of the tracking accuracy will only reach the amount of underfilled and overfilled areas embedded within the reference input during the generation of the tool path file. The only way to reduce the underfilled areas, within the reference input, is to design a tool path file with negative offset. This, however, will create more overfills because of the overlapping roads in the raster. An acceptable tradeoff will essentially depend on the material characteristics and the degree of accuracy required.

Mohsen A. Jafari (M’73) received the M.S. and Ph.D. degrees in industrial engineering and operations research and the M.S. degree in computer science from Syracuse University, Syracuse, NY. He is currently an Associate Professor in the Department of Industrial and Systems Engineering, Rutgers University, Piscataway, NJ. His research interests are design of operational control of manufacturing systems, computer-aided manufacturing, and machine vision.

VII. CONCLUSION The control of an LM system requires essentially a highprecision positioning system in coordination with an accurate material extrusion and deposition subsystem. The synchronization between both subsystems should be perfect and reliable in order to eliminate the deposition errors at start/stop points and at sharp corners. The control structure and the trajectory-generation algorithm described above provide an accurate positioning subsystem, which allows the liquefier-head subsystem to be

Wen-Biao Han received the B.S. degree in electrical and computer engineering in 1991 from Harbin Institute of Technology, Harbin, China, and the M.S. degree in pattern recognition and intelligent control in 1994 from the Chinese Academy of Sciences, Shenyang, China. He is currently working towards the Ph.D. degree in the Department of Industrial and Systems Engineering, Rutgers University, Piscataway, NJ. His research interests are manufacturing process modeling, intelligent control, and layered manufacturing.

BOUHAL et al.: TRACKING CONTROL AND TRAJECTORY PLANNING IN LM APPLICATIONS

Tong Fang (S’98) received the B.S. degree in electrical engineering from Hefei Polytechnic University, Hefei, China, in 1988 and the M.S. degree in systems engineering from the University of Science and Technology of China, Hefei, China, in 1992. He is currently working towards the Ph.D. degree in the Department of Industrial and Systems Engineering, Rutgers University, Piscataway, NJ. His research interests are image processing, intelligent control, and layered manufacturing.

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