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Trajectory planning for automated robotic deburring on an unknown contour
International Journal of Machine Tools & Manufacture 40 (2000) 957–978
Trajectory planning for automated robotic deburring on an unknown contour Seng-Chi Chen, Pi-Cheng Tung
*
Department of Mechanical Engineering, National Central University, Chung-Li, 32054 Taiwan, ROC Received 20 August 1998; received in revised form 2 September 1999; accepted 26 October 1999
Abstract For a conventionally automated robotic deburring system, a precise model of the mechanism and geometric knowledge of the environment is necessary. Also, the accuracy of the planned trajectory must be high. The trajectory which the robot travels is usually planned with a small depth inside from the constrained surface of the environment. For a workpiece with unknown contour, planning a trajectory may be unfeasible. Therefore, in this study, we present a novel trajectory planning, which allows for arbitrary planning of trajectory with a large distance inside the constrained surface. When the manipulator comes into contact with the environment, the robot controller compensates for the trajectory in real time by applying an innovative geometrical projection method. To demonstrate the feasibility and effectiveness of the proposed method, a Cartesian robot arm on which a grinding tool is rigidly mounted performs precision deburring and chamfering on unknown contours. Experimental results indicate that the manipulator is controlled in terms of automatically deburring the edges of parts with an unknown geometrical configuration. Moreover, its cutting force is maintained at a desired level. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Trajectory planning; Robotic deburring; Geometrical projection method
1. Introduction Burrs, which are nearly always generated when machining is performed on metal parts, can be done so either between a machined surface and a raw surface or at the intersection between two machined surfaces. High labor costs have made manual deburring expensive and inefficient. As a more viable alternative, an industrial robot can remove fine burrs faster and more completely than by hand filing methods, with less dimensional loss.
* Corresponding author. Tel.: +886-3-4267304; fax: +886-3-4254501. E-mail addresses: [email protected] (S.-C. Chen), [email protected] (P.-C. Tung).
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Nomenclature d e E f G I N r0 r S X,Y y z z0 d w0 smax[·] smin[·]
vector of external force on the robot endpoint vector of the input trajectory environment dynamics vector of contact force robot dynamics with a positioning controller an n×n identity matrix the compensator vector of planned trajectory vector of input command disturbance–compliance function Cartesian direction vector of the robot endpoint position vector of the environment deflection vector of the environment position before contact the depth of the planned trajectory inside the constrained surface the bandwidth of the system the maximum singular value the minimum singular value
Subscripts x,y n,t
components in the X and Y directions components in the normal and tangential directions
Robotic deburring can be performed in two ways. If the part is relatively lightweight, the robot can handle the part and hold it against a tool that performs the actual deburring. If the part is heavy, the robot can hold the deburring tool and control its motion around the stationary part. In both ways, the relative motion between the tool and the part is of a continuous-path (CP) type with high repeatability and accuracy. Hence, sensing the contact forces that develop between the end-effector of the manipulator and its interface is very important and provides vital feedback information in the servo control for guiding and controlling the robot in completing its task. Despite the diversity of approaches towards the active force control of deburring, such approaches can be categorized into two classes: hybrid force/position control [1–3] and impedance control [4–7]. The first approach attempts to control the force and position in a non-conflicting manner. Two identical, but separate position and force control loops were designed for the controlled system. The second approach focuses on developing a relationship between the interaction forces and the manipulator position. By controlling the manipulator end-point position and specifying its relationship with the interaction force, the manipulator must maneuver in a constrained environment and maintain appropriate contact forces simultaneously.
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For an impedance control system, the trajectory which the robot arm traverses is usually planned within a depth d from the constrained surface of the environment, as illustrated in Fig. 1(a). To achieve the stability, the depth d is usually set to be small and its value depends on the stiffness of the manipulator. For the unknown contour of an object, planning a trajectory with a certain depth d may be unfeasible. Also, manipulator positional errors can lead to excessively large forces and damage to the manipulator and environment if the commanded positions of the manipulator and the physical constraints conflict with each other. Although the overall positional error is mainly due to robot inaccuracy, errors in machining, parts tolerance, tool wear, and part misalignment in the fixture can also contribute to the inaccuracy [14,15]. To overcome these problems, this study presents an innovative geometrical projection method which allows for trajectory planning with an arbitrary distance inside the constrained surface, as depicted in Fig. 1(b). After geometrical projection, a portion of the planned trajectory which passes through the interior of the environment projects itself onto a new trajectory which is a distance d offset from the constrained surface. When the manipulator comes into contact with the environment, the robot arm moves successively along the new trajectory and maintains a relatively constant contact force on the unknown contour.
2. Cutting force As is generally known, the location and form of burrs are seldom predicted with any real accuracy. In addition, the shape and size of the burr can differ widely between parts since the tools suffer wear and tear during machine-tool fabrication. Herein, the burrs are assumed to be small so that variations in the burr size do not affect deburring and chamfering operations. The burr removal tools selected for our analysis and experiments were grinding wheels, which produce a 45° chamfer on the workpiece edge when the tool is held orthogonally to the part surface. To ensure that a given burr is completely removed, the chamfer width must exceed the burr root width, as illustrated in Fig. 2. The primary characteristics of burrs include burr thickness, burr height and burr length. Notably, burr thickness cannot be accurately measured with height gage indicators. At this time, only optical methods provide precise results. Herein, burr height is used to denote the burr size. Burr length represents the total length of the edge burrs and is critical to the deburring time [16,17]. As Fig. 3 indicates, the resultant cutting force f can be resolved with respect to the part into two vector components: the tangential force ft and normal force fn. Kazerooni et al. [8,9] developed a dynamic model for the deburring process. In their model, the volumetric material removal rate (MRR) of a deburring pass is a function of the velocity of the tool bit along the edge and the cross-sectional areas of both the chamfer and the burr. This relationship can be expressed as MRR⫽(Achamfer⫹Aburr)Vw
(1)
where Vw, Achamfer, and Aburr denote the tool velocity (or feedrate) along the path, the chamfer area, and the burr area, respectively. Those investigators [9] also confirmed that variations in the burr size only slightly influenced the normal force, but produced significant variations in the tangential force for a given tool velocity Vw, when two burrs of different sizes were cut. The tangential force can be linearly related to MRR
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Fig. 1. (a) Planned trajectory and actual trajectory (conventional method). (b) Planned trajectory and actual trajectory (the proposed method).
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Fig. 2.
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Cutting surface area, with 45° chamfer on the workpiece edge.
ft⫽K⫻MRR
(2)
where K depends on material properties while MRR is a geometric quantity. From the above description, we can infer that the ratio of tangential force to resultant force maintains a constant level at a constant tool velocity and chamfer area along the part edge. That is, ft a⫽cos−1 ⫽constant (Vw and Achamfer are constant) |f|
冑
|f|⫽ (f 2n+f 2t)
(3) (4)
Fig. 4(a) summarizes the results for a low carbon steel (AISI 1020) sample part, for a given depth of cut (0.5 mm) on an edge with no burrs. Feedrate along the part edge is set at 6 mm/s. Fig. 4(b) indicates that the ratio of tangential force to the resultant cutting force maintains a relatively constant level. If the cutter remains in contact with the edge and sustains the small cutting force f deemed necessary for edge breaking, compliance can be implemented in both the tangential and the normal directions. In addition, the feedrate can be automatically adjusted according to the burr size by controlling the cutting force. In the robotic deburring process, the tangential directions to the contour can be derived from force measurements. Without previous knowledge of the workpiece’s shape, the manipulator can still comply to the surface of a constrained environment.
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Fig. 3. Illustration of deburring process: Vs=wheel speed; Vw=feedrate; ar=depth of cut; D=wheel diameter; fn=normal force; ft=tangential force; f=resultant force.
3. Dynamic models and stability conditions Consider the system of compliance motion control as represented by the block diagram in Fig. 5. All dynamic models in this system are expressed in terms of input–output mappings. Let G be the mapping from the input actuating signal vector e to the manipulator end-point position y. Let S be the mapping from the external force d to the manipulator end point position y. The mapping S is referred to herein as the disturbance–compliance function which implies that the manipulator end-point moves invariably somewhat in response to the external force, even if the tracking controllers of robot manipulator are typically designed to follow the input actuating signal and to reject disturbances. Similarly, the environment can be considered from the viewpoint of an unstructured model. The function E represents a general mapping from a deflection vector of z to a contact force vector f. Herein, f is assumed to be invariably positive to push the constrained surface, although in some applications, the constrained surface may have a contact force for pulling. Let z0 be the initial position of the constrained surface before contact occurs and y still be the position of the robot manipulator end-point; then z=y⫺z0. An attempt is also made to
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Fig. 4. (a) Tangential and resultant force in deburring process with depth of cut: 0.5 mm, feedrate: 6 mm/s. The initial 2 s and the last 1 s of the data represent the transient part where the grinder comes into contact with and separates from the part, respectively. (b) For a given depth of cut and a constant feedrate, the ratio of tangential force to the resultant force maintains a relatively constant level.
achieve compliant motion control for robots, in which the compensator N is considered to operate upon the contact force. In addition, mappings G, S, E and N are assumed herein to be Lp-stable. Fig. 5 contains two loops: the upper loop is the “natural” feedback loop and the lower one is the “artificial” feedback loop. The latter is referred to herein as a “controlled feedback loop”. If
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Fig. 5. Natural feedback loop and artificial feedback loop.
the robot and environment are in contact with each other (i.e. traveling in the constrained space), then both loops occur. If we consider all the above mapping as linear transfer function matrices, the following equations are obtained: f⫽E[I⫹SE⫹GNE]−1[Gr⫺z0]
(5)
e⫽r⫺N[f]
(6)
where r is the input-command vector and I is an n×n identity matrix. From Eq. (5), knowing G, S, and E and selecting an appropriate N allowed us to shape the contact force as we desired. A large value of N generates a soft robot whereas a small N generates a hard one. Let z0 be equal to zero for convenience. By doing so, Eq. (5) can be rewritten as f⫽[E−1⫹S⫹GN]−1Gr
(7)
in which E⫺1 and S add GN to develop the total compliance in the system. In addition, S represents the natural hardware compliance whereas GN represents the artificial software compliance. Although one can choose a larger value of GN to increase compliant ability, from the viewpoint of stability, an arbitrarily large value of N cannot be chosen. To simplify the block diagram of Fig. 5, we introduce a mapping Q from e to f as f⫽Q[e]
(8)
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Similarly, Q is assumed to be Lp-stable. Fig. 6 depicts the simplified block diagram of Fig. 5. If Q is a linear operator and let z0=0. By using the small gain theorem to analyze system stability [10,11], the following inequality ensures that the closed-loop system is stable. 1 1 ⫽ smax[N]< smax[Q] smax[E[I+SE]−1G]
(9)
where smax[·] is the maximum singular value. As soon as the manipulator encounters the environment, Eq. (6) has a condition of the maximum value |emax|⫽|rmax|⫽d, as |f|⫽0
(10)
where rmax is the maximum depth inside the constrained surface we can plan. The value of rmax depends on the stiffness of the end-effector, where emax is the maximum positional error. Herein, assume that the robot manipulator has a good tracking controller. Hence, the end-point position is approximately equal to the input actuating signal in the unconstrained space. Thus, G⬇I for all w苸[0,w0] with w0 the bandwidth of the system. In this case, the condition for stability can be expressed as
冋 册 冋册
smax[N]
1 d ⫽smin E f
(11)
Inequality (11) clearly reveals that the stiffer environment implies a smaller N must be chosen to stabilize the closed-loop system. In addition, the less sensitive the disturbance–compliance function is implies that a smaller N must be chosen to ensure the stability of the closed-loop system. However, for a planned trajectory with a large depth d, the larger N must be selected to compensate for the motion; however, the condition of stability is violated. A large range of stability can be obtained by installing a passive compliant element such as remote center compliance (RCC) into a robot manipulator [12]. If the RCC has a linear dynamic behavior, then the disturbance–compliance function equals the reciprocal of stiffness (or
Fig. 6.
Simplified version of Fig. 5.
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impedance) of the RCC. In addition, the passive compliant element can be used to stabilize the system of the robot in a compliant motion control operation. Although these elements are capable of quick responses and are inexpensive, they are only useful for a small class of tasks, in particular the peg-in-the hole problem, and are sensible to the direction of insertion, which must be the vertical axis. They can correct a lateral misalignment of at most about 2 mm. 4. Trajectory planning Most industrial robots generally rely on teach programming, a process that is tedious and time consuming. An alternative method uses computer-aided design (CAD) data and a representation of the robot’s environment to compute manipulator coordinates off-line. Unfortunately, robots usually have less accuracy so that off-line programming is ineffectual without modification. For a workpiece with an unknown contour, planning a trajectory may be unfeasible. An alternative approach can be used, e.g. a roller bearing [9] or a properly designed jig [13], which is mounted to the robot end-effector, thereby ensuring accurate trajectory tracking of the actual geometrical edge of the workpiece. However, these machines frequently cost too much to justify their use in a deburring system. Also, the mechanism is more complex. In contrast, the projection method proposed herein is more flexible and more reliable in terms of deburring tasks. The geometrical projection methods can be classified into three general cases:
Fig. 7. The projection method for tracking a straight edge of a part.
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(1) in which the robot arm tracks along a straight edge; (2) which has a curved edge; and (3) which is around the contour of an object. The dashed line in Fig. 7 denotes the planned trajectory, it is a circle in the plane. Basically, the planned trajectory can also be a simple closed curve which does not cross itself. Herein, O is the starting point, the arc trajectory AA⬘ inside the environment is divided into small straightline segments which are denoted as AB, BD, DF, …, and so on. The number of segments is critical to the operating time and the smoothness of the motion. For convenience in explanation, AB, BD, DF, … are enlarged, as shown in Fig. 7, where O⬘ is the first contact point between the manipulator and the environment. The distance O⬘A is the deflection of the robot manipulator end-point, which depends on the stiffness of the end-effector. When the manipulator moves along the planned trajectory and reaches the point A, the next point C relative to point A can be obtained from the given slope of AB, and that of the tangential direction of the contact force at point A by the following equation, AC⫽AB cos q1 where q1⫽p⫺tan−1
冉
(12)
冊
m2−m1 1+m1m2
m1, m2 are the slopes of AC and AB, respectively. Notably, the slope of AC can be measured
Fig. 8.
The projection method for tracking a curved edge of a part.
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indirectly using the force sensor. By repeating this process, the point D, F, … on the planned trajectory will project to E, G, …, respectively. Initially, the manipulator approaches and contacts the workpiece at O⬘ from the starting point O, then tracks along the points A, C, E, G, …, A⬘. Finally, it departs the workpiece from point O⬙, then returns to the original point O. Fig. 8 indicates that when the manipulator moves along the planned trajectory and reaches the point A, the slope of AC, i.e. the slope of the tangential direction of the contact force at that point, can be obtained. Herein, we set A as the center and AB as the radius (coordinate at point B is known), and plot an arc C1 which intersects AC at C. The point C is the next position which the manipulator moves to relative to the point A. When the manipulator arrives at the point C, the orientation of CE is the tangential direction of the contact force at C. Again, we set A as the center and AD as the radius (coordinate at point D is known), and plot an arc C2 which intersects AB and AC at points B⬘ and C⬘, respectively. According to trigonometry, the two triangles are similar. That is
Fig. 9. The projection method for tracking a closed surface.
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왕BB⬘D⬵왕CC⬘E BB⬘⫽BD cos q3⫽CC⬘⫽CE cos q2 Hence cos q3 CE⫽BD cos q2
(13)
According to Eq. (13), the coordinate of E relative to point C, i.e. the next position of the manipulator maneuvers, can be obtained. By repeating this process, points B, D, F, … on the planned trajectory project themselves on C, E, G, …, respectively. The curvature of a curve, once approaching infinity, becomes a straight line. Therefore, the case of moving along a straight edge is basically a unique case of moving along a curved edge. Thus, the method applied to a curved edge can also be applied to a straight edge. According to Fig. 9, the manipulator contacts the environment at the point C and maintains a desired contact force. Let CD be parallel to AB (the magnitude and orientation of AB are given). From trigonometry CD⫽AB
L1 L2
(14)
Fig. 10. Block diagram of the control system.
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The point A, B, E, F, … on the planned trajectory projects itself onto C, D, G, H, …. 5. Experimental results The block diagram of the control system is illustrated in Fig. 10, and the main experimental set up is displayed in Fig. 11. The maximum travel distances of the X, Y, and Z axes for the Cartesian manipulator are 300, 200, and 100 mm, respectively. The manipulator is driven by three AC servo motors. The positions of the three axes are then measured by optical linear scales with a resolution of 0.001 mm. Signals from optical linear scales go through decoders to become up/down pulse signals. Next, synchronous counters are used to counter the number of pulse signals and the position of the three axes can be obtained. A force sensor with a resolution of 0.1 N is installed on the manipulator. For the sensor itself the maximum sampling rate is about 1 kHz,
Fig. 11. The main experimental equipment.
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the transmission of the data to the computer uses a direct memory access (DMA) mode to obtain the data with a high transfer rate. Finally, the position and force signals are fed back into a personal computer and the control signals are sent to a servo motor from the personal computer through a D/A converter, as shown in Fig. 10. In the following figures, the unit of the position coordinate is in a counter number. Following calibration, one counter is equivalent to 0.932 µm. Experiments in this study largely focused on compliant motion control rather than position control. It is satisfactory to use conventional PID control plus feedforward control to derive the positioning controller of the manipulator. The material of the specimen is made of low carbon steel, an effectively infinitely stiff environment. While considering disturbance–compliance functions Sx and Sy in the X and Y directions as the reciprocal of the manipulator stiffness, we had Sx=Sy=37 unit distance (unit distance=0.932 µm) per Newton. Therefore, the stability inequality can be rewritten as |Nx|<|Sx|⫽37
(15a)
|Ny|<|Sy|⫽37
(15b)
If |Nx| and |Ny| were less than 37, the system was considered stable. However, if |Nx| and |Ny| exceeded this value, no conclusion was made. Fig. 12 illustrates the closed-loop system of compliant motion control of the manipulator in our experiments, where r0 is the planned trajectory and r is the actual trajectory after geometrical projection. In the first experiment, the workpiece has a straight sharp edge. Burr height ranges from zero to 0.93 mm, and the root thickness ranges from zero to 0.17 mm. Fig. 13(a) depicts the planned trajectory and an actual path which the end-effector traverses along a straight edge, respectively. The manipulator end-point closely follows an input command vector when traveling in a free
Fig. 12. Closed-loop system of a robot’s compliant motion.
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Fig. 13. (a) Experimental measurement of the trajectories for tracking a straight edge. (b) Experimental measurement of the contact force. (c) Before deburring (magnified 20 times). (d) Surface finish after deburring (magnified 20 times).
space, and tracks along the new trajectory when traveling in the constrained space. Fig. 13(b) displays the contact force. Fig. 13(c) and (d) display the specimen before and after deburring, respectively. In the second experiment, the workpiece is a circular disk. Burr height ranges from zero to 0.81 mm, and the root thickness ranges from zero to 0.19 mm. Fig. 14(a) depicts the result of
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Fig. 13. (continued)
the manipulator position for tracking along a curved edge. Fig. 14(b) displays the contact force. Once the end-effector encounters the workpiece, the contact force increases slightly due to impact and then decreases instantly and stays at a relatively constant level. Fig. 14(c) and (d) display the specimen before and after deburring, respectively. Fig. 14(d) displays the smoothness of the edge after deburring. In the third experiment, the specimen is a circular disk of diameter 20 mm. Burr height ranges from zero to 0.84 mm, and the root thickness ranges from zero to 0.15 mm. Fig. 15(a) and (b) present the results for tracking around the contour of the circular object. Notably, a circular trajectory is planned inside the circular object, and two circles have distinct centers, as shown in Fig. 15(a). For this case, the diameter of the planned trajectory is markedly smaller than that of
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Fig. 14. (a) Experimental measurement of the trajectories for tracking a curved edge. (b) Experimental measurement of the contact force. (c) Before deburring (magnified 20 times). (d) Surface finish after deburring (magnified 20 times).
the circular object. In addition, conventional controllers for compliant motions fail because the end-effector may produce an excessively large force or even crash into the supporting surface of the object. The manipulator maintains a fairly regular contact force of about 1.5 N during edge following, as displayed in Fig. 15(b).
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Fig. 14. (continued)
Experimental results confirm that the burrs are successfully removed even though a secondary burr is generated. The size of the secondary burr is very small so that it still has a high quality finish on the edge of the part. 6. Conclusions As mentioned earlier, conventional controllers for compliant motion are constrained. Both desired trajectories and desired forces must be known. Off-line trajectory planning must be carefully performed. The trajectory frequently offsets a small distance inside from the constrained
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Fig. 15. (a) Experimental measurement of the trajectories for tracking a circular disk. (b) Experimental measurement of the contact force.
surface. In addition, positional error as a consequence of robot inaccuracy can lead to an excessively large contact force. For the case of unknown contours, most controllers for compliant motions fail since the desired trajectory cannot be obtained. In comparison with the conventional method of robotic deburring, the proposed method in this study has the following features:
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1. the method provides an effective strategy for automated robotic deburring and chamfering running on an unknown contour in real-time; 2. knowledge of the coordinate data and location of a workpiece is unnecessary; 3. the system stability is unaffected, particularly in terms of the projection algorithm installed; 4. the proposed method allows a trajectory in which the robot arm moves to be arbitrarily planned with a large distance inside the constrained surface; and 5. The path planning can be achieved by implementing the simple algorithms on a microprocessor and is easily carried out in practical applications.
Acknowledgements The authors would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC 82-0422-E-008-138. Mr Hong kuochih of Delta Electronics Co., is also appreciated for providing in-house engineering and technical support.
References [1] R.P. Paul, B.E. Shimano, Compliance and control, in: Proceedings of the Joint Automatic Control Conference, Purdue University, 1976, pp. 694–699. [2] M.H. Raibert, J.J. Craig, Hybrid position/force control of manipulators, Transactions of ASME Journal of Dynamic Systems, Measurement, and Control 102 (1981) 126–133. [3] M.T. Mason, Compliance and force control for computer controlled manipulators, IEEE Transactions on Systems, Man, and Cybernetics 11 (1981) 418–432. [4] N. Hogan, Impedance control: an approach to manipulation, Part 1: theory, Part 2: implementation, Part 3: application, Transactions of ASME Journal of Dynamic Systems, Measurement, and Control 107 (1985) 1–24. [5] H. Kazerooni, Automated robotic deburring using impedance control, IEEE Control System Magazine 8 (1988) 21–25. [6] J.K. Salisbury, Active stiffness control of a manipulator in Cartesian coordinates, in: Proceedings of 19th IEEE Conference on Decision and Control, 1980, pp. 95–100. [7] N. Hogan, Stable execution of contact tasks using impedance control, in: Proceedings IEEE International Conference on Robotics and Automation, 1987, pp. 1047–1054. [8] H. Kazerooni et al., An approach to automated deburring by robot manipulators, Transactions of ASME Journal of Dynamic Systems, Measurement, and Control 108 (1986) 354–359. [9] M.G. Her, H. Kazerooni, Automated robotic deburring of parts using compliance control, Transactions of ASME Journal of Dynamic Systems, Measurement, and Control 113 (1991) 60–66. [10] H. Kazerooni, T.I. Tsay, Stability criteria for robot compliant maneuvers, in: Proceedings IEEE International Conference on Robotics and Automation, 1988, pp. 1166–1172. [11] M. Vidyasagar, C.A. Desoer, Feedback System: Input–Output Properties, Academic Press, New York, 1975. [12] J.L. Navins, D.E. Whitney, Computer-controlled assembly, Scientific American 238 (1978) 62–74. [13] J.M. Schimmels, Design of an admittance control law for precision edge tracking in robotic deburring, Advances in Robotics, Mechatronics, and Haptic Interfaces, the ASME Winter Annual Meeting 49 (1993) 281–286. [14] Y. Koren, Robotics for Engineers, McGraw-Hill, New York, 1985. [15] F.M. Proctor, K.N. Murphy, Keynote address: advanced deburring system technology, ASME Winter Annual Meeting 45 (1989) 1–12.
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[16] X. Chen, W.B. Rowe, Analysis and simulation of the grinding process—Part I, generation of the grinding wheel surface, International Journal of Machine Tools and Manufacture 36 (8) (1996) 871–882. [17] X. Chen, W.B. Rowe, Analysis and simulation of the grinding process—Part II, mechanics of grinding, International Journal of Machine Tools and Manufacture 36 (8) (1996) 883–896.
Trajectory planning for automated robotic deburring on an unknown contour Seng-Chi Chen, Pi-Cheng Tung
*
Department of Mechanical Engineering, National Central University, Chung-Li, 32054 Taiwan, ROC Received 20 August 1998; received in revised form 2 September 1999; accepted 26 October 1999
Abstract For a conventionally automated robotic deburring system, a precise model of the mechanism and geometric knowledge of the environment is necessary. Also, the accuracy of the planned trajectory must be high. The trajectory which the robot travels is usually planned with a small depth inside from the constrained surface of the environment. For a workpiece with unknown contour, planning a trajectory may be unfeasible. Therefore, in this study, we present a novel trajectory planning, which allows for arbitrary planning of trajectory with a large distance inside the constrained surface. When the manipulator comes into contact with the environment, the robot controller compensates for the trajectory in real time by applying an innovative geometrical projection method. To demonstrate the feasibility and effectiveness of the proposed method, a Cartesian robot arm on which a grinding tool is rigidly mounted performs precision deburring and chamfering on unknown contours. Experimental results indicate that the manipulator is controlled in terms of automatically deburring the edges of parts with an unknown geometrical configuration. Moreover, its cutting force is maintained at a desired level. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Trajectory planning; Robotic deburring; Geometrical projection method
1. Introduction Burrs, which are nearly always generated when machining is performed on metal parts, can be done so either between a machined surface and a raw surface or at the intersection between two machined surfaces. High labor costs have made manual deburring expensive and inefficient. As a more viable alternative, an industrial robot can remove fine burrs faster and more completely than by hand filing methods, with less dimensional loss.
* Corresponding author. Tel.: +886-3-4267304; fax: +886-3-4254501. E-mail addresses: [email protected] (S.-C. Chen), [email protected] (P.-C. Tung).
0890-6955/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 9 ) 0 0 0 9 9 - 1
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Nomenclature d e E f G I N r0 r S X,Y y z z0 d w0 smax[·] smin[·]
vector of external force on the robot endpoint vector of the input trajectory environment dynamics vector of contact force robot dynamics with a positioning controller an n×n identity matrix the compensator vector of planned trajectory vector of input command disturbance–compliance function Cartesian direction vector of the robot endpoint position vector of the environment deflection vector of the environment position before contact the depth of the planned trajectory inside the constrained surface the bandwidth of the system the maximum singular value the minimum singular value
Subscripts x,y n,t
components in the X and Y directions components in the normal and tangential directions
Robotic deburring can be performed in two ways. If the part is relatively lightweight, the robot can handle the part and hold it against a tool that performs the actual deburring. If the part is heavy, the robot can hold the deburring tool and control its motion around the stationary part. In both ways, the relative motion between the tool and the part is of a continuous-path (CP) type with high repeatability and accuracy. Hence, sensing the contact forces that develop between the end-effector of the manipulator and its interface is very important and provides vital feedback information in the servo control for guiding and controlling the robot in completing its task. Despite the diversity of approaches towards the active force control of deburring, such approaches can be categorized into two classes: hybrid force/position control [1–3] and impedance control [4–7]. The first approach attempts to control the force and position in a non-conflicting manner. Two identical, but separate position and force control loops were designed for the controlled system. The second approach focuses on developing a relationship between the interaction forces and the manipulator position. By controlling the manipulator end-point position and specifying its relationship with the interaction force, the manipulator must maneuver in a constrained environment and maintain appropriate contact forces simultaneously.
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For an impedance control system, the trajectory which the robot arm traverses is usually planned within a depth d from the constrained surface of the environment, as illustrated in Fig. 1(a). To achieve the stability, the depth d is usually set to be small and its value depends on the stiffness of the manipulator. For the unknown contour of an object, planning a trajectory with a certain depth d may be unfeasible. Also, manipulator positional errors can lead to excessively large forces and damage to the manipulator and environment if the commanded positions of the manipulator and the physical constraints conflict with each other. Although the overall positional error is mainly due to robot inaccuracy, errors in machining, parts tolerance, tool wear, and part misalignment in the fixture can also contribute to the inaccuracy [14,15]. To overcome these problems, this study presents an innovative geometrical projection method which allows for trajectory planning with an arbitrary distance inside the constrained surface, as depicted in Fig. 1(b). After geometrical projection, a portion of the planned trajectory which passes through the interior of the environment projects itself onto a new trajectory which is a distance d offset from the constrained surface. When the manipulator comes into contact with the environment, the robot arm moves successively along the new trajectory and maintains a relatively constant contact force on the unknown contour.
2. Cutting force As is generally known, the location and form of burrs are seldom predicted with any real accuracy. In addition, the shape and size of the burr can differ widely between parts since the tools suffer wear and tear during machine-tool fabrication. Herein, the burrs are assumed to be small so that variations in the burr size do not affect deburring and chamfering operations. The burr removal tools selected for our analysis and experiments were grinding wheels, which produce a 45° chamfer on the workpiece edge when the tool is held orthogonally to the part surface. To ensure that a given burr is completely removed, the chamfer width must exceed the burr root width, as illustrated in Fig. 2. The primary characteristics of burrs include burr thickness, burr height and burr length. Notably, burr thickness cannot be accurately measured with height gage indicators. At this time, only optical methods provide precise results. Herein, burr height is used to denote the burr size. Burr length represents the total length of the edge burrs and is critical to the deburring time [16,17]. As Fig. 3 indicates, the resultant cutting force f can be resolved with respect to the part into two vector components: the tangential force ft and normal force fn. Kazerooni et al. [8,9] developed a dynamic model for the deburring process. In their model, the volumetric material removal rate (MRR) of a deburring pass is a function of the velocity of the tool bit along the edge and the cross-sectional areas of both the chamfer and the burr. This relationship can be expressed as MRR⫽(Achamfer⫹Aburr)Vw
(1)
where Vw, Achamfer, and Aburr denote the tool velocity (or feedrate) along the path, the chamfer area, and the burr area, respectively. Those investigators [9] also confirmed that variations in the burr size only slightly influenced the normal force, but produced significant variations in the tangential force for a given tool velocity Vw, when two burrs of different sizes were cut. The tangential force can be linearly related to MRR
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Fig. 1. (a) Planned trajectory and actual trajectory (conventional method). (b) Planned trajectory and actual trajectory (the proposed method).
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Fig. 2.
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Cutting surface area, with 45° chamfer on the workpiece edge.
ft⫽K⫻MRR
(2)
where K depends on material properties while MRR is a geometric quantity. From the above description, we can infer that the ratio of tangential force to resultant force maintains a constant level at a constant tool velocity and chamfer area along the part edge. That is, ft a⫽cos−1 ⫽constant (Vw and Achamfer are constant) |f|
冑
|f|⫽ (f 2n+f 2t)
(3) (4)
Fig. 4(a) summarizes the results for a low carbon steel (AISI 1020) sample part, for a given depth of cut (0.5 mm) on an edge with no burrs. Feedrate along the part edge is set at 6 mm/s. Fig. 4(b) indicates that the ratio of tangential force to the resultant cutting force maintains a relatively constant level. If the cutter remains in contact with the edge and sustains the small cutting force f deemed necessary for edge breaking, compliance can be implemented in both the tangential and the normal directions. In addition, the feedrate can be automatically adjusted according to the burr size by controlling the cutting force. In the robotic deburring process, the tangential directions to the contour can be derived from force measurements. Without previous knowledge of the workpiece’s shape, the manipulator can still comply to the surface of a constrained environment.
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Fig. 3. Illustration of deburring process: Vs=wheel speed; Vw=feedrate; ar=depth of cut; D=wheel diameter; fn=normal force; ft=tangential force; f=resultant force.
3. Dynamic models and stability conditions Consider the system of compliance motion control as represented by the block diagram in Fig. 5. All dynamic models in this system are expressed in terms of input–output mappings. Let G be the mapping from the input actuating signal vector e to the manipulator end-point position y. Let S be the mapping from the external force d to the manipulator end point position y. The mapping S is referred to herein as the disturbance–compliance function which implies that the manipulator end-point moves invariably somewhat in response to the external force, even if the tracking controllers of robot manipulator are typically designed to follow the input actuating signal and to reject disturbances. Similarly, the environment can be considered from the viewpoint of an unstructured model. The function E represents a general mapping from a deflection vector of z to a contact force vector f. Herein, f is assumed to be invariably positive to push the constrained surface, although in some applications, the constrained surface may have a contact force for pulling. Let z0 be the initial position of the constrained surface before contact occurs and y still be the position of the robot manipulator end-point; then z=y⫺z0. An attempt is also made to
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Fig. 4. (a) Tangential and resultant force in deburring process with depth of cut: 0.5 mm, feedrate: 6 mm/s. The initial 2 s and the last 1 s of the data represent the transient part where the grinder comes into contact with and separates from the part, respectively. (b) For a given depth of cut and a constant feedrate, the ratio of tangential force to the resultant force maintains a relatively constant level.
achieve compliant motion control for robots, in which the compensator N is considered to operate upon the contact force. In addition, mappings G, S, E and N are assumed herein to be Lp-stable. Fig. 5 contains two loops: the upper loop is the “natural” feedback loop and the lower one is the “artificial” feedback loop. The latter is referred to herein as a “controlled feedback loop”. If
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Fig. 5. Natural feedback loop and artificial feedback loop.
the robot and environment are in contact with each other (i.e. traveling in the constrained space), then both loops occur. If we consider all the above mapping as linear transfer function matrices, the following equations are obtained: f⫽E[I⫹SE⫹GNE]−1[Gr⫺z0]
(5)
e⫽r⫺N[f]
(6)
where r is the input-command vector and I is an n×n identity matrix. From Eq. (5), knowing G, S, and E and selecting an appropriate N allowed us to shape the contact force as we desired. A large value of N generates a soft robot whereas a small N generates a hard one. Let z0 be equal to zero for convenience. By doing so, Eq. (5) can be rewritten as f⫽[E−1⫹S⫹GN]−1Gr
(7)
in which E⫺1 and S add GN to develop the total compliance in the system. In addition, S represents the natural hardware compliance whereas GN represents the artificial software compliance. Although one can choose a larger value of GN to increase compliant ability, from the viewpoint of stability, an arbitrarily large value of N cannot be chosen. To simplify the block diagram of Fig. 5, we introduce a mapping Q from e to f as f⫽Q[e]
(8)
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Similarly, Q is assumed to be Lp-stable. Fig. 6 depicts the simplified block diagram of Fig. 5. If Q is a linear operator and let z0=0. By using the small gain theorem to analyze system stability [10,11], the following inequality ensures that the closed-loop system is stable. 1 1 ⫽ smax[N]< smax[Q] smax[E[I+SE]−1G]
(9)
where smax[·] is the maximum singular value. As soon as the manipulator encounters the environment, Eq. (6) has a condition of the maximum value |emax|⫽|rmax|⫽d, as |f|⫽0
(10)
where rmax is the maximum depth inside the constrained surface we can plan. The value of rmax depends on the stiffness of the end-effector, where emax is the maximum positional error. Herein, assume that the robot manipulator has a good tracking controller. Hence, the end-point position is approximately equal to the input actuating signal in the unconstrained space. Thus, G⬇I for all w苸[0,w0] with w0 the bandwidth of the system. In this case, the condition for stability can be expressed as
冋 册 冋册
smax[N]
1 d ⫽smin E f
(11)
Inequality (11) clearly reveals that the stiffer environment implies a smaller N must be chosen to stabilize the closed-loop system. In addition, the less sensitive the disturbance–compliance function is implies that a smaller N must be chosen to ensure the stability of the closed-loop system. However, for a planned trajectory with a large depth d, the larger N must be selected to compensate for the motion; however, the condition of stability is violated. A large range of stability can be obtained by installing a passive compliant element such as remote center compliance (RCC) into a robot manipulator [12]. If the RCC has a linear dynamic behavior, then the disturbance–compliance function equals the reciprocal of stiffness (or
Fig. 6.
Simplified version of Fig. 5.
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impedance) of the RCC. In addition, the passive compliant element can be used to stabilize the system of the robot in a compliant motion control operation. Although these elements are capable of quick responses and are inexpensive, they are only useful for a small class of tasks, in particular the peg-in-the hole problem, and are sensible to the direction of insertion, which must be the vertical axis. They can correct a lateral misalignment of at most about 2 mm. 4. Trajectory planning Most industrial robots generally rely on teach programming, a process that is tedious and time consuming. An alternative method uses computer-aided design (CAD) data and a representation of the robot’s environment to compute manipulator coordinates off-line. Unfortunately, robots usually have less accuracy so that off-line programming is ineffectual without modification. For a workpiece with an unknown contour, planning a trajectory may be unfeasible. An alternative approach can be used, e.g. a roller bearing [9] or a properly designed jig [13], which is mounted to the robot end-effector, thereby ensuring accurate trajectory tracking of the actual geometrical edge of the workpiece. However, these machines frequently cost too much to justify their use in a deburring system. Also, the mechanism is more complex. In contrast, the projection method proposed herein is more flexible and more reliable in terms of deburring tasks. The geometrical projection methods can be classified into three general cases:
Fig. 7. The projection method for tracking a straight edge of a part.
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(1) in which the robot arm tracks along a straight edge; (2) which has a curved edge; and (3) which is around the contour of an object. The dashed line in Fig. 7 denotes the planned trajectory, it is a circle in the plane. Basically, the planned trajectory can also be a simple closed curve which does not cross itself. Herein, O is the starting point, the arc trajectory AA⬘ inside the environment is divided into small straightline segments which are denoted as AB, BD, DF, …, and so on. The number of segments is critical to the operating time and the smoothness of the motion. For convenience in explanation, AB, BD, DF, … are enlarged, as shown in Fig. 7, where O⬘ is the first contact point between the manipulator and the environment. The distance O⬘A is the deflection of the robot manipulator end-point, which depends on the stiffness of the end-effector. When the manipulator moves along the planned trajectory and reaches the point A, the next point C relative to point A can be obtained from the given slope of AB, and that of the tangential direction of the contact force at point A by the following equation, AC⫽AB cos q1 where q1⫽p⫺tan−1
冉
(12)
冊
m2−m1 1+m1m2
m1, m2 are the slopes of AC and AB, respectively. Notably, the slope of AC can be measured
Fig. 8.
The projection method for tracking a curved edge of a part.
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indirectly using the force sensor. By repeating this process, the point D, F, … on the planned trajectory will project to E, G, …, respectively. Initially, the manipulator approaches and contacts the workpiece at O⬘ from the starting point O, then tracks along the points A, C, E, G, …, A⬘. Finally, it departs the workpiece from point O⬙, then returns to the original point O. Fig. 8 indicates that when the manipulator moves along the planned trajectory and reaches the point A, the slope of AC, i.e. the slope of the tangential direction of the contact force at that point, can be obtained. Herein, we set A as the center and AB as the radius (coordinate at point B is known), and plot an arc C1 which intersects AC at C. The point C is the next position which the manipulator moves to relative to the point A. When the manipulator arrives at the point C, the orientation of CE is the tangential direction of the contact force at C. Again, we set A as the center and AD as the radius (coordinate at point D is known), and plot an arc C2 which intersects AB and AC at points B⬘ and C⬘, respectively. According to trigonometry, the two triangles are similar. That is
Fig. 9. The projection method for tracking a closed surface.
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왕BB⬘D⬵왕CC⬘E BB⬘⫽BD cos q3⫽CC⬘⫽CE cos q2 Hence cos q3 CE⫽BD cos q2
(13)
According to Eq. (13), the coordinate of E relative to point C, i.e. the next position of the manipulator maneuvers, can be obtained. By repeating this process, points B, D, F, … on the planned trajectory project themselves on C, E, G, …, respectively. The curvature of a curve, once approaching infinity, becomes a straight line. Therefore, the case of moving along a straight edge is basically a unique case of moving along a curved edge. Thus, the method applied to a curved edge can also be applied to a straight edge. According to Fig. 9, the manipulator contacts the environment at the point C and maintains a desired contact force. Let CD be parallel to AB (the magnitude and orientation of AB are given). From trigonometry CD⫽AB
L1 L2
(14)
Fig. 10. Block diagram of the control system.
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The point A, B, E, F, … on the planned trajectory projects itself onto C, D, G, H, …. 5. Experimental results The block diagram of the control system is illustrated in Fig. 10, and the main experimental set up is displayed in Fig. 11. The maximum travel distances of the X, Y, and Z axes for the Cartesian manipulator are 300, 200, and 100 mm, respectively. The manipulator is driven by three AC servo motors. The positions of the three axes are then measured by optical linear scales with a resolution of 0.001 mm. Signals from optical linear scales go through decoders to become up/down pulse signals. Next, synchronous counters are used to counter the number of pulse signals and the position of the three axes can be obtained. A force sensor with a resolution of 0.1 N is installed on the manipulator. For the sensor itself the maximum sampling rate is about 1 kHz,
Fig. 11. The main experimental equipment.
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the transmission of the data to the computer uses a direct memory access (DMA) mode to obtain the data with a high transfer rate. Finally, the position and force signals are fed back into a personal computer and the control signals are sent to a servo motor from the personal computer through a D/A converter, as shown in Fig. 10. In the following figures, the unit of the position coordinate is in a counter number. Following calibration, one counter is equivalent to 0.932 µm. Experiments in this study largely focused on compliant motion control rather than position control. It is satisfactory to use conventional PID control plus feedforward control to derive the positioning controller of the manipulator. The material of the specimen is made of low carbon steel, an effectively infinitely stiff environment. While considering disturbance–compliance functions Sx and Sy in the X and Y directions as the reciprocal of the manipulator stiffness, we had Sx=Sy=37 unit distance (unit distance=0.932 µm) per Newton. Therefore, the stability inequality can be rewritten as |Nx|<|Sx|⫽37
(15a)
|Ny|<|Sy|⫽37
(15b)
If |Nx| and |Ny| were less than 37, the system was considered stable. However, if |Nx| and |Ny| exceeded this value, no conclusion was made. Fig. 12 illustrates the closed-loop system of compliant motion control of the manipulator in our experiments, where r0 is the planned trajectory and r is the actual trajectory after geometrical projection. In the first experiment, the workpiece has a straight sharp edge. Burr height ranges from zero to 0.93 mm, and the root thickness ranges from zero to 0.17 mm. Fig. 13(a) depicts the planned trajectory and an actual path which the end-effector traverses along a straight edge, respectively. The manipulator end-point closely follows an input command vector when traveling in a free
Fig. 12. Closed-loop system of a robot’s compliant motion.
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Fig. 13. (a) Experimental measurement of the trajectories for tracking a straight edge. (b) Experimental measurement of the contact force. (c) Before deburring (magnified 20 times). (d) Surface finish after deburring (magnified 20 times).
space, and tracks along the new trajectory when traveling in the constrained space. Fig. 13(b) displays the contact force. Fig. 13(c) and (d) display the specimen before and after deburring, respectively. In the second experiment, the workpiece is a circular disk. Burr height ranges from zero to 0.81 mm, and the root thickness ranges from zero to 0.19 mm. Fig. 14(a) depicts the result of
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Fig. 13. (continued)
the manipulator position for tracking along a curved edge. Fig. 14(b) displays the contact force. Once the end-effector encounters the workpiece, the contact force increases slightly due to impact and then decreases instantly and stays at a relatively constant level. Fig. 14(c) and (d) display the specimen before and after deburring, respectively. Fig. 14(d) displays the smoothness of the edge after deburring. In the third experiment, the specimen is a circular disk of diameter 20 mm. Burr height ranges from zero to 0.84 mm, and the root thickness ranges from zero to 0.15 mm. Fig. 15(a) and (b) present the results for tracking around the contour of the circular object. Notably, a circular trajectory is planned inside the circular object, and two circles have distinct centers, as shown in Fig. 15(a). For this case, the diameter of the planned trajectory is markedly smaller than that of
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Fig. 14. (a) Experimental measurement of the trajectories for tracking a curved edge. (b) Experimental measurement of the contact force. (c) Before deburring (magnified 20 times). (d) Surface finish after deburring (magnified 20 times).
the circular object. In addition, conventional controllers for compliant motions fail because the end-effector may produce an excessively large force or even crash into the supporting surface of the object. The manipulator maintains a fairly regular contact force of about 1.5 N during edge following, as displayed in Fig. 15(b).
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Fig. 14. (continued)
Experimental results confirm that the burrs are successfully removed even though a secondary burr is generated. The size of the secondary burr is very small so that it still has a high quality finish on the edge of the part. 6. Conclusions As mentioned earlier, conventional controllers for compliant motion are constrained. Both desired trajectories and desired forces must be known. Off-line trajectory planning must be carefully performed. The trajectory frequently offsets a small distance inside from the constrained
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Fig. 15. (a) Experimental measurement of the trajectories for tracking a circular disk. (b) Experimental measurement of the contact force.
surface. In addition, positional error as a consequence of robot inaccuracy can lead to an excessively large contact force. For the case of unknown contours, most controllers for compliant motions fail since the desired trajectory cannot be obtained. In comparison with the conventional method of robotic deburring, the proposed method in this study has the following features:
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1. the method provides an effective strategy for automated robotic deburring and chamfering running on an unknown contour in real-time; 2. knowledge of the coordinate data and location of a workpiece is unnecessary; 3. the system stability is unaffected, particularly in terms of the projection algorithm installed; 4. the proposed method allows a trajectory in which the robot arm moves to be arbitrarily planned with a large distance inside the constrained surface; and 5. The path planning can be achieved by implementing the simple algorithms on a microprocessor and is easily carried out in practical applications.
Acknowledgements The authors would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC 82-0422-E-008-138. Mr Hong kuochih of Delta Electronics Co., is also appreciated for providing in-house engineering and technical support.
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