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A mechatronic approach for robotic deburring
Mechatronics 17 (2007) 431–441
A mechatronic approach for robotic deburring G. Ziliani a, A. Visioli b
b,*
, G. Legnani
q
a
a Dipartimento di Ingegneria Meccanica, University of Brescia, Italy Dipartimento di Elettronica per l’Automazione, University of Brescia, via Branze 38, I-25123 Brescia, Italy
Received 30 August 2006; accepted 19 April 2007
Abstract This paper deals with the implementation of a mechatronic methodology for the robotic deburring of planar workpieces with an unknown shape performed by an industrial manipulator. The approach is based on the use of a hybrid force/velocity control law and on a correlated suitable design of the deburring tool. Experimental results, obtained with a two degrees-of-freedom SCARA industrial robot manipulator, show the effectiveness of the method. 2007 Elsevier Ltd. All rights reserved. Keywords: Industrial robot manipulators; Deburring; Hybrid force/velocity control
1. Introduction Despite the recent achievements in the design of control systems for complex robotic tasks, nowadays industrial settings still generally employ robots in fixed and highly structured environments so that reconfiguration efforts are a clear barrier to face the continuous changes required by the market demand. Indeed, robots that are able to autonomously adapt themselves to semi-unstructured tasks would be a key issue to cut re-programming costs and to shorten the lead to production time [1]. This is of main concern in several industrial applications, such as grinding, deburring, chamfering and polishing, where the current standard methodologies require the knowledge of the workpiece shape (and the machining tools are designed based on this assumption) but the capability to cope with a workpiece of an unknown shape would significantly reduce the task programming phase which represents a significant part of the overall cost, especially when frequent q
Partial support for this research has been provided by MIUR scientific research funds. * Corresponding author. Tel.: +39 0303715460; fax: +39 030380014. E-mail addresses: [email protected] (G. Ziliani), antonio. [email protected] (A. Visioli), [email protected] (G. Legnani). 0957-4158/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2007.04.012
changes in production occur. In fact, robot control systems currently adopted in industrial settings for deburring applications typically require the knowledge of the end-effector path. This has to be obtained by means of off-line programming, computer aided design models, teaching by demonstration methods or teaching support devices [2]. The path uncertainties are usually compensated by the adoption of a suitable compliant tool without any feedback action. Various methodologies for robotic deburring have been actually proposed in the literature in the last two decades. They are based for example on impedance control [3], on hybrid control with an internal position loop [4] and on the so-called triangular control [5] (note that in these cases the geometry of the workpiece is known). Soft-computing techniques can be adopted to improve performances and in particular to cope with an unknown geometry of the workpiece (see for example [6,7]). The unknown geometry of the workpiece is also addressed by a geometrical projection method in [8], but therein the burrs are assumed to be small so that the variation in the burr size does not actually influence the cutting force. The idea to design the deburring tool in conjunction with the control algorithm has been pursued in [9,10]. In particular, in [9], under the framework of impedance
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control, a roller bearing mounted on the force sensor is employed for the purpose of tracking. In this case, however, two force sensors are needed. Differently, in [10], an end-effector mounted jig and a proper design of the compliance of the manipulator are used to achieve accurate force guidance. An admittance control law is proposed but no results are given to prove the effectiveness of the method. In any case, when dealing with an unknown object, the robot has to perform a contour tracking task effectively, by imposing an appropriate normal force and an appropriate tangential velocity to the robot end-effector. In this context, the use of classical explicit hybrid force/velocity control [11] is addressed in this paper. It will be shown that it can be effectively adopted, in conjunction with a suitable design of the milling tool, for the planar deburring motion task. It is worth noting that this is an important case in the industrial context, since in a die casting process, burrs often lie on the closure planes of molds. Indeed, by measuring the contact force as well as the machining torque, it is possible to detect the contact angle and to achieve an accurate tracking [12]. Further, the presence of burrs can be detected and consequently the reference tangential velocity can be suitably modified to improve the quality of the deburring. It is remarkable that the dynamic model of the manipulator is not required and therefore, the overall control scheme design is very simple (with respect to the other methods that consider an unknown object shape) and suitable to implement in an industrial context. The paper is organized as follows. First, in Section 2 the experimental setup is presented. Then, in Section 3 the basic hybrid force/velocity controller for contour tracking is described. In Section 4 problems related with the deburring task are analyzed and their solution is proposed in Section 5. Experimental results are shown in Section 6 and conclusions are drawn in Section 7. 2. Experimental setup Although the concepts discussed in this paper can be applied in general, in the following we refer to a two degrees-of-freedom planar SCARA robot, as this is the one adopted in the experiments. The setup available in the Applied Mechanics Laboratory of the University of Brescia consists of an industrial manipulator manufactured by ICOMATIC (Gussago, Italy) with a standard SCARA architecture where the z-axis has been blocked for our planar tasks. For a description of the dynamic model, see [13]. Both links have the same length of 0.33 m. The two joints are actuated by DC motors (driven by conventional PWM amplifiers) through Harmonic Drive speed reducers whose reduction rate is 1/100. Motor rotations are measured by means of two incremental encoders with 2000 pulses/rev. resolution. Velocity is estimated through numerical differentiation whose output is then processed by a low-pass second order Butterworth filter (100 Hz cut-off frequency and 1.0 damping ratio). An ATI 65/5 Force/Torque sensor capable of measuring forces and torques in a ±65 N and
±5 Nm range, respectively, with 0.05 N resolution is mounted at the manipulator wrist. The corresponding signals are processed at 7.8 kHz frequency by an ISA DSP based board and collected by the robot controller at 1 kHz after filtering. The PC-based controller is based on a QNX4 real time operating system and the control algorithms were written in C/C++ language. Acquisition and control are performed at a frequency of 1 kHz. 3. Basic contour tracking algorithm 3.1. Problem formulation With reference to Fig. 1, frame (0) refers to the robot base, while task frame (T) has its origin on the robot end-effector, its n-axis is directed along the normal direction of the piece contour and its t-axis along its tangent; # is the angle between n-axis and x-axis of frame (0). Let Q = [q1,q2]T be the vector of the joint positions and Q_ its first time derivative. Since a suitable belt transmission keeps the end-effector with constant orientation with respect to the absolute frame, force measurements are directly available in frame (0). Let F(0) = [Fx,Fy]T, F(T) = [Ft,Fn]T be the vectors of contact force in frames (0) and (T), respectively. They are related to each other by the equation F(0) = M0T(#)F(T) denoting with Mij the rotation matrix from frame j to frame i. Vector V(T) = [Vt,Vn]T representing the Cartesian velocity in frame (T) can be obtained from the relation _ V ðT Þ ¼ M T 0 ð#ÞV ð0Þ ¼ M T 0 ð#ÞJ ðQÞQ; where J(Q) is the robot Jacobian matrix. Note that, by assuming the absence of tangential friction force, the angle # can be estimated on-line as Fy # ¼ atan2ðF y ; F x Þ ¼ arctan p: ð1Þ Fx However, this assumption is not valid when a deburring task is performed and therefore, another estimation
y (n) (T)
ϑ
(t) q2 (0)
q1
x Fig. 1. Sketch of a SCARA robot following a contour.
G. Ziliani et al. / Mechatronics 17 (2007) 431–441 θ,Q
Vt
Vt , d (t )
Vn
Vn , d (t ) = 0
Fn , d (t )
MT 0J PID
u PID ,Vt
fˆ1
.
kV,ff
-
Kv,fb
u P ,Vn
J T M 0T
k F,ff
-
PI Fn
u PI , F
θ,Q
Q
τ1 τ2
fˆ2
Robot
F
MT0 θ,Q
Fig. 2. Basic hybrid force/velocity control scheme.
method has to be employed, as it will be described in Section 5. The aim of a contour tracking task is to control the normal force and the tangential velocity of the robot probe along the directions n and t of the task frame (T), respectively. 3.2. Control law The following hybrid force/velocity control law has been considered (see the control scheme in Fig. 2): s1 s¼ ð2Þ ¼ J T ðQÞM 0T ð#ÞðU ðT Þ þ K R RÞ þ f^ ; s2 where R = [Vt,d, Fn,d]T is the feedforward vector based on force and velocity references, KR = diag[kV,ff,kF,ff] the corresponding diagonal matrix of gains, f^ ¼ ½f^ 1 ðq_ 1 Þ; f^ 2 ðq_ 2 ÞT is a term that compensates for the joint friction torques [14] and uPID;V U ðT Þ ¼ ; ð3Þ uPI;F þ K v;fb ðV n;d ðtÞ V n ðtÞÞ where uPID,V is the tangential velocity PID output, uPI,F is the normal force PI output, Vn,d(t) = 0, Vn(t) is the velocity of the end-effector in the normal direction and Kv,fb is a proportional gain. Note that the use of a normal force derivative term has been avoided in (3) (indeed, only the proportional and the integral actions have been employed) as the derivation of such a signal is ill-conditioned [15,16]. Conversely, the adoption of a normal force velocity feedback loop has been proven to be effective to compensate for the possible large force oscillations due to the kinetostatic behavior of the manipulator [17] (note that the compliance of the robot is mainly concentrated in the third link).
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1. the estimation of the contour direction to allow the contour tracking without the loss of contact in the presence of disturbances due to the cutting force; 2. the choice and the modulation of the correct contact force set-point to provide a satisfactory surface quality and a complete burr removal. When a robot manipulator accomplishes a material removing task such as deburring and chamfering, the cutting force can introduce disturbances in the force control loop that can yield to the loss of contact and can cause an incorrect measure of the normal direction. Depending on the task, many parameters have to be considered such as the tool tangential velocity, the tool typology and sharpening, the direction of the angular velocity with respect to the path feedrate, the workpiece stiffness, the cutting depth and the height and thickness of burrs. For example, if the material is homogeneous, then the cutting force increases proportionally to the amount of material removed. If the manipulator end-effector is moving with constant speed along the contour, then the force will vary proportionally to the depth of cut. The cutting process therefore generates also a reaction force that is not directed tangentially to the surface. Indeed, the normal component of this contact force can give some problem to the force control introducing disturbances in the control loop and it may cause the manipulator to penetrate in the workpiece material or to detach depending on the verse of rotation of the mill with respect to the tool feedrate. In Fig. 4 the forces acting on a workpiece during a milling process are reported. For simplicity it is assumed that only one tool tooth at a time is in contact with the surface and that all the forces act in one single point. For each tool tooth in contact with the surface, there is a cutting force, F~c , acting in the direction of the cutter tangential speed and a normal force, F~r , acting in the radial direction. The radial force is usually assumed to be proportional to the cutting force, i.e., F r ¼ F c tan ðb cÞ; where b is the friction angle (which is in general not easy to estimate) and c is the tool front bevel angle (see Fig. 3). The cutting force on the workpiece Fc can be expressed as [18]
Tool tooth
γ
Workpiece surface
4. Robotic deburring For the robotic deburring of workpieces of unknown geometry, two main problems have to be actually solved:
Fig. 3. The tool front bevel angle c for a single tool tooth in contact with the surface.
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Fc ¼
G. Ziliani et al. / Mechatronics 17 (2007) 431–441
bdvu ; x
where b is the tool wheel width, d is the depth of cut, v is the feedrate (i.e., the tangential velocity of the end-effector), u is the specific energy of the material and x is the tool wheel speed. It appears that also in this case the involved parameters are very difficult to estimate. F~n and F~t are the components of the total force ~ F in the task frame. If the tool path feedrate direction coincides with the direction of the peripherical velocity of the tool tooth in contact with the surface (Fig. 4, case ‘‘a’’), the machining force acting on the workpiece has a component F~t directed as the tracking tangential velocity, while the normal component F~n is directed from the surface to the tool. The reaction forces generated on the tool (and measured by the force sensor) have therefore, opposite directions. Note that in this case the normal force tends to pull the tool inside the workpiece surface and eventually the tool and the robot may stall while the tangential force acts against the tool feedrate. Conversely, if the tool path feedrate is opposite to the peripherical velocity of the tool tooth in contact with the surface (Fig. 4, case ‘‘b’’), the normal force generated on the tool tends to push it away from the surface while the tangential force tends to increase the tool feedrate. Further, for a correct deburring task the cut depth should vary proportionally to the burr height. The cut depth depends on the normal force (and on the value of b + c) and so the normal force set-point should vary proportionally to the burr height. Unfortunately burrs size and thickness can be very different and discontinuous, especially in casting workpieces. For these reasons, an accurate model-based estimation of the cutting force is very difficult to obtain.
5. The mechatronic design 5.1. Contact direction estimation with machining tangential force As already mentioned, the estimation of the angle # between the n-axis and the x-axis of frame (0) is essential for the control algorithm. The knowledge of the cutting force, which can be derived by considering a model of the forces involved in a milling task, could be used to estimate the angle. However, in a deburring task many of the parameters may vary and may not be accurately known (for example, the geometry of the casting burr) and therefore, it is almost impossible to have a sufficiently accurate model-based estimation of #. Thus, we propose a method that is based on the use of z-axis torque measure obtained from the six axes force/torque sensor and on the knowledge of the radius of the cutting mill, which can be easily a priori measured. Specifically, with reference to Fig. 4 we have that
Fig. 4. Notation of forces generated by the mill on the workpiece surface: (a) the tool path feedrate direction coincides with the direction of the peripherical velocity of the tool tooth in contact with the surface; (b) the tool path feedrate is opposite to the peripherical velocity of the tool tooth in contact with the surface.
• Fx and Fy are the forces along the axes x and y, respectively, of the force sensor frame; • Ftot is the total force; • Fn and Ft are the normal and tangential forces, respectively with respect to the workpiece surface; • # is the actual contact angle; • #* is the ‘‘apparent’’ contact angle calculated as atan2 (Fy,Fx) (see (1)); • D# is the angle between the actual and the measured contact directions; • sz is the z-axis measured torque and • r is the radius of the cutting mill. In the hypothesis that the z-axis of the mill frame coincides with the sensor frame axis, the tangential force due to the machining operation is sz Ft ¼ : ð4Þ r With trivial trigonometric relations the angle D# can easily be written as Ft D# ¼ arcsin ; ð5Þ F tot where F tot ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 2x þ F 2y ¼ F 2n þ F 2t :
ð6Þ
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Thus, the actual contact angle can be simply calculated from the measured angle with a correction depending on the z-axis measured torque as follows: # ¼ # D# or, substituting Eq. (5) in (7) it can be written as Ft # ¼ atan2ðF y ; F x Þ arcsin : F tot
ð7Þ
ð8Þ
However, if the vertical axis of the mill does not coincide with the vertical axis of the force sensor, then Eq. (4) cannot be used. This situation is depicted in Fig. 5, where the center O 0 of the force sensor frame is not coaxial with the center of the mill frame O due to the offset (Dx, Dy). In this case, the measured torque can be expressed as s0z ¼ sz þ F y Dx F x Dy;
ð9Þ
where sz = Ftr is the machining torque and the term Fy Æ Dx Fx Æ Dy represents the torsional moment with respect to the point O generated by the force Ftot applied in O 0 . Note that because of the terms Fy Æ Dx and Fx Æ Dy the force sensor measures a torque even if the tangential force Ft is null. The machining torque can be written therefore as sz ¼
s0z
F y Dx þ F x Dy
ð10Þ
and substituting Eq. (10) in (4) and (5) the value of Ft can be estimated as Ft ¼
s0z F y Dx þ F x Dy r
ð11Þ
and the actual contact angle can still be obtained with (8).
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5.2. Mill center calibration The position of the mill frame center O 0 can be estimated with a simple least squares based calibration procedure. This calibration is necessary, especially if the radius of the mill is comparable to the offset of the mill from the z-axis of the force sensor. The procedure is based on the contour tracking of a planar object with the mill mounted on the end-effector switched off, i.e., rolling on the contour so that the machining force is null. During the tracking the values of Fx, Fy and s0z are collected. Since the machining force is null, the measured torque is caused only by the offset of the z-axis of the mill with respect to the sensor frame, i.e., s0z ¼ F x Dy þ F y Dx. By considering N samples, the following linear systems result: 8 0 ¼ F x1 Dy þ F y 1 Dx s z1 > > > 0 > > s ¼ F x2 Dy þ F y 2 Dx > z > < 2 .. .. . . > > > 0 > > > szN 1 ¼ F xN1 Dy þ F y N1 Dx > : 0 s zN ¼ F xN Dy þ F y N Dx which can also be written in matrix form as s0z ¼ F X ; where 2
3 s0z1 6 s0 7 6 z2 7 7 6 7 sz ¼ 6 6 ::: 7; 6 s0 7 4 zN 1 5 s0zN
2
F x1 6 F 6 x2 6 ::: F ¼6 6 6 4 F xn1 F xn
3 F y1 F y2 7 7 7 ::: 7 7 7 F y n1 5 F yn
and X ¼ ½ Dy
Dx T :
Let F+ be the pseudoinverse matrix of F. Then, Dx and Dy can be calculated, with the least square criterion, as X ¼ F þ s0z :
ð12Þ
Remark 1. It is worth stressing that the calibration procedure has to be performed just once on a workpiece whose shape guarantees a small condition number of F, in order to provide a good estimation of X, disregarding the kind of workpieces that have to be subsequently deburred. For example, a circular object tracked on all its contours guarantees this condition.
5.3. The deburring tool Fig. 5. Forces acting on the cutting tool and measured by the force sensor during a machining task without offset between the sensor and the mill.
A major problem in the hybrid force/velocity control based deburring of workpieces of unknown geometry is
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the selection of the normal force set-point value. Actually, in order to measure the orientation of the surface, the contact force must be non-null. Further, the amount of material removed by the mill obviously depends on the contact force. Thus, it appears that this is a crucial issue for the performance of the deburring task. In fact, if no information is given to detect the shape of the burrs or to decide whether the surface must be worked or not, the mill will remove the same amount of material on all the contours, even when the surface is already finished. In other words, the size and the consistency of the burrs can be very different and the contact force should vary, in principle, to ensure a complete burrs removal and to prevent workpiece damages. For tender material as aluminum or plastic, the problem is actually more significant because the tool can penetrate inside the workpiece even for very low contact forces. Moreover, as shown in Fig. 4a, if the tool path feedrate direction coincides with the direction of the peripherical velocity of the tool tooth in contact with the surface, the reaction force Fn will tend to pull the mill inside the workpiece. This force is added to the normal force applied by the force control, but an increased normal force generates a higher cutting force and so a higher force Fn and so on. Eventually, the tool may get stuck in the workpiece. In any case, the use of the alternative approach depicted in Fig. 4b cannot be considered as a stable contact between the tool and the surface could not be maintained. In order to effectively address these problems, a novel special deburring tool has been designed. The tool has two ball bearings mounted on the sides of the cutting cylindrical surface. The external diameter of the ball bearings is the same as that of the tool (see Fig. 6). The two ball bear-
ings (or one of them if the burrs are placed on the edges of the workpiece) have to be placed in contact with the surface where no burrs are present, while the cutting part has to be placed along the plane where the burrs lie (see Fig. 7). In this way, the angular velocity of the tool has the same direction of the feedrate velocity, but the ball bearings avoid the penetration of the tool in the workpiece and also solve the problem of the choice of the set-point contact force value, which can be selected as a reasonably high value, in order to ensure that the thickest burrs are removed. Indeed, if no burrs are found on the tool path, the force applied to the workpiece will be only the ball bearings contact force and no material is removed from the workpiece. Further, the use of this kind of tool also makes the detection of the presence of burrs possible. Indeed, the measure of the z-axis torque increases when a burr is encountered (proportionally to the thickness of the burr) and this can be adopted to appropriately adapt the reference tangential velocity and therefore to improve the overall task time (see Section 6).
Fig. 7. The deburring tool.
Fig. 6. Forces acting on the cutting tool and measured by the force sensor during a machining task with offset between the sensor and the mill.
G. Ziliani et al. / Mechatronics 17 (2007) 431–441
Fig. 8. The deburring tool for the removal of central and edging burrs.
Remark 2. It is worth noting that the devised methodology can be easily extended to chamfering by using a conical cutter instead of a cylindrical one (see Fig. 8).
6. Experimental results The original tool chosen for the deburring task is a high speed 20,000 rpm pneumatic mill. Irregularities in the rotation of the tool are transmitted to the force sensor introducing high frequency noise in the force control loop. To
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Fig. 9. The chamfering tool.
face this problem a low-pass filter with a cut-off frequency of 100 Hz has been introduced in the force/torque acquisition. As a first experiment, in order to verify the effectiveness of the contact angle estimation method (see Section 5.1) a deburring task has been performed on an artificial burr created with an iron sheet fixed between two straight aluminum pieces. The iron sheet was 3 mm high and 1 mm thick. The modified deburring tool has been used and the tool path feedrate and tool rotation have opposite directions. The reference normal force has been fixed at 30 N, while the reference tangential velocity has been fixed at
Fig. 10. The artificial burr after the attempt of removal without the contact angle compensation.
Fig. 11. The artificial burr after the removal with the contact angle compensation.
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5 mm/s. In Fig. 9 the artificial burr is shown after the deburring task executed without the compensation of the contact angle (see (1)). The burr has not been removed completely and the tool has contoured the burr because of the wrong contact angle estimation. In Fig. 10 the artificial burr is shown after the deburring task has been executed with the compensation of the contact angle presented in Section 5.1 (see (8)). It appears that in this case the burr has been completely removed. To give a better insight in the results, in Fig. 11 the measured con-
tact angle without and with the compensation procedure is reported for comparison. It can be noted that the angle remains almost constant with the use of the compensation technique, while without compensation a larger deviation from the actual value is observed (about 75). In Fig. 12 the grinder torque during the deburring task in the two cases is reported. The torque measured is non-null only when the tool is in contact with the burr. Note that the contact time without the angle compensation is shorter. Since the tangential velocity set-point and the burr length are the
Fig. 12. The contact angle estimation without (top) and with (bottom) the compensation during the deburring of an artificial burr.
Fig. 13. The torque along the z-axis measured by the sensor during the deburring of the artificial burr without (top) and with (bottom) the compensation.
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same, this means that the tangential velocity incorrectly increased in the first case (the path is executed in about 2 s instead of 4 s). This is explained by the fact that an incorrectly estimated normal angle makes the direction in which the force is controlled to have a component directed in an unconstrained direction (i.e., the tangential direction in which the velocity is controlled). As a second experiment, the system has been tested on a real aluminum die casting workpiece. In this case, the z-axis torque is used to detect the presence of the burrs and to consequently reduce the reference tangential velocity (see Section 5.3). Thus, the reference tangential velocity is fixed at 6 mm/s but if the z-axis torque value exceeds a fixed threshold of 0.05 Nm, the reference velocity is (smoothly) reduced to 0.8 mm/s. Then, when the burr is completely removed and the measured torque decreases again, the set-point returns (smoothly) to the previous value. The original workpiece is shown in Fig. 13. Note that the burrs lay in different positions along the die closure plane and on the edging of the surface as indicated by the arrows. The height of some burrs exceeded 15 mm and the thickness was about 1 mm. A picture taken during the deburring task is shown in Fig. 14 and different pictures of the workpiece after the
robotic deburring are shown in Figs. 15–17, demonstrating the high performances achieved by the system. For a thorough analysis of the results, the estimation of the contact angle with and without the compensation technique is
Fig. 14. An aluminum die casting workpiece before the deburring task.
Fig. 17. Part of the aluminum die casting workpiece after deburring.
Fig. 15. The new milling tool during the deburring task.
Fig. 18. Zoom of the aluminum die casting workpiece after the deburring.
Fig. 16. The aluminum die workpiece casting after the deburring task.
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reported in Fig. 18, while the measured z-axis torque and the tangential velocity set-point are, respectively, shown in Figs. 19 and 20, showing that the velocity set-point reduces when the measured torque exceeds the selected threshold of 0.05 Nm. Note that until the tangential velocity set-point is reached at the beginning of the task (after slightly more than 1 s in this case), this method is not applied (see Fig. 21). Videos of the experiments are available at http:// www.ing.unibs.it/~visioli/deburring.html. 7. Conclusions Fig. 19. Comparison between the estimated contact angle with and without the compensation method in a deburring task of an aluminum workpiece.
In this paper, we have presented a mechatronic methodology for the robotic deburring of workpieces of unknown shape. The combined design of the milling tool and of a hybrid force/velocity controller, which is based on measurements provided by a force/torque sensor, allows to obtain high performances as demonstrated by the obtained results. Due also to its simplicity, the overall methodology appears suitable to implement in industrial settings. Indeed, the task is performed by the robot almost fully autonomously, since the knowledge of the geometry of the workpiece is not required, thus yielding to a significant reduction of the programming time (especially for complex paths). References
Fig. 20. Measured z-axis torque in a deburring task of an aluminum workpiece.
Fig. 21. Tangential velocity set-point in a deburring task of an aluminum workpiece.
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A mechatronic approach for robotic deburring G. Ziliani a, A. Visioli b
b,*
, G. Legnani
q
a
a Dipartimento di Ingegneria Meccanica, University of Brescia, Italy Dipartimento di Elettronica per l’Automazione, University of Brescia, via Branze 38, I-25123 Brescia, Italy
Received 30 August 2006; accepted 19 April 2007
Abstract This paper deals with the implementation of a mechatronic methodology for the robotic deburring of planar workpieces with an unknown shape performed by an industrial manipulator. The approach is based on the use of a hybrid force/velocity control law and on a correlated suitable design of the deburring tool. Experimental results, obtained with a two degrees-of-freedom SCARA industrial robot manipulator, show the effectiveness of the method. 2007 Elsevier Ltd. All rights reserved. Keywords: Industrial robot manipulators; Deburring; Hybrid force/velocity control
1. Introduction Despite the recent achievements in the design of control systems for complex robotic tasks, nowadays industrial settings still generally employ robots in fixed and highly structured environments so that reconfiguration efforts are a clear barrier to face the continuous changes required by the market demand. Indeed, robots that are able to autonomously adapt themselves to semi-unstructured tasks would be a key issue to cut re-programming costs and to shorten the lead to production time [1]. This is of main concern in several industrial applications, such as grinding, deburring, chamfering and polishing, where the current standard methodologies require the knowledge of the workpiece shape (and the machining tools are designed based on this assumption) but the capability to cope with a workpiece of an unknown shape would significantly reduce the task programming phase which represents a significant part of the overall cost, especially when frequent q
Partial support for this research has been provided by MIUR scientific research funds. * Corresponding author. Tel.: +39 0303715460; fax: +39 030380014. E-mail addresses: [email protected] (G. Ziliani), antonio. [email protected] (A. Visioli), [email protected] (G. Legnani). 0957-4158/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2007.04.012
changes in production occur. In fact, robot control systems currently adopted in industrial settings for deburring applications typically require the knowledge of the end-effector path. This has to be obtained by means of off-line programming, computer aided design models, teaching by demonstration methods or teaching support devices [2]. The path uncertainties are usually compensated by the adoption of a suitable compliant tool without any feedback action. Various methodologies for robotic deburring have been actually proposed in the literature in the last two decades. They are based for example on impedance control [3], on hybrid control with an internal position loop [4] and on the so-called triangular control [5] (note that in these cases the geometry of the workpiece is known). Soft-computing techniques can be adopted to improve performances and in particular to cope with an unknown geometry of the workpiece (see for example [6,7]). The unknown geometry of the workpiece is also addressed by a geometrical projection method in [8], but therein the burrs are assumed to be small so that the variation in the burr size does not actually influence the cutting force. The idea to design the deburring tool in conjunction with the control algorithm has been pursued in [9,10]. In particular, in [9], under the framework of impedance
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control, a roller bearing mounted on the force sensor is employed for the purpose of tracking. In this case, however, two force sensors are needed. Differently, in [10], an end-effector mounted jig and a proper design of the compliance of the manipulator are used to achieve accurate force guidance. An admittance control law is proposed but no results are given to prove the effectiveness of the method. In any case, when dealing with an unknown object, the robot has to perform a contour tracking task effectively, by imposing an appropriate normal force and an appropriate tangential velocity to the robot end-effector. In this context, the use of classical explicit hybrid force/velocity control [11] is addressed in this paper. It will be shown that it can be effectively adopted, in conjunction with a suitable design of the milling tool, for the planar deburring motion task. It is worth noting that this is an important case in the industrial context, since in a die casting process, burrs often lie on the closure planes of molds. Indeed, by measuring the contact force as well as the machining torque, it is possible to detect the contact angle and to achieve an accurate tracking [12]. Further, the presence of burrs can be detected and consequently the reference tangential velocity can be suitably modified to improve the quality of the deburring. It is remarkable that the dynamic model of the manipulator is not required and therefore, the overall control scheme design is very simple (with respect to the other methods that consider an unknown object shape) and suitable to implement in an industrial context. The paper is organized as follows. First, in Section 2 the experimental setup is presented. Then, in Section 3 the basic hybrid force/velocity controller for contour tracking is described. In Section 4 problems related with the deburring task are analyzed and their solution is proposed in Section 5. Experimental results are shown in Section 6 and conclusions are drawn in Section 7. 2. Experimental setup Although the concepts discussed in this paper can be applied in general, in the following we refer to a two degrees-of-freedom planar SCARA robot, as this is the one adopted in the experiments. The setup available in the Applied Mechanics Laboratory of the University of Brescia consists of an industrial manipulator manufactured by ICOMATIC (Gussago, Italy) with a standard SCARA architecture where the z-axis has been blocked for our planar tasks. For a description of the dynamic model, see [13]. Both links have the same length of 0.33 m. The two joints are actuated by DC motors (driven by conventional PWM amplifiers) through Harmonic Drive speed reducers whose reduction rate is 1/100. Motor rotations are measured by means of two incremental encoders with 2000 pulses/rev. resolution. Velocity is estimated through numerical differentiation whose output is then processed by a low-pass second order Butterworth filter (100 Hz cut-off frequency and 1.0 damping ratio). An ATI 65/5 Force/Torque sensor capable of measuring forces and torques in a ±65 N and
±5 Nm range, respectively, with 0.05 N resolution is mounted at the manipulator wrist. The corresponding signals are processed at 7.8 kHz frequency by an ISA DSP based board and collected by the robot controller at 1 kHz after filtering. The PC-based controller is based on a QNX4 real time operating system and the control algorithms were written in C/C++ language. Acquisition and control are performed at a frequency of 1 kHz. 3. Basic contour tracking algorithm 3.1. Problem formulation With reference to Fig. 1, frame (0) refers to the robot base, while task frame (T) has its origin on the robot end-effector, its n-axis is directed along the normal direction of the piece contour and its t-axis along its tangent; # is the angle between n-axis and x-axis of frame (0). Let Q = [q1,q2]T be the vector of the joint positions and Q_ its first time derivative. Since a suitable belt transmission keeps the end-effector with constant orientation with respect to the absolute frame, force measurements are directly available in frame (0). Let F(0) = [Fx,Fy]T, F(T) = [Ft,Fn]T be the vectors of contact force in frames (0) and (T), respectively. They are related to each other by the equation F(0) = M0T(#)F(T) denoting with Mij the rotation matrix from frame j to frame i. Vector V(T) = [Vt,Vn]T representing the Cartesian velocity in frame (T) can be obtained from the relation _ V ðT Þ ¼ M T 0 ð#ÞV ð0Þ ¼ M T 0 ð#ÞJ ðQÞQ; where J(Q) is the robot Jacobian matrix. Note that, by assuming the absence of tangential friction force, the angle # can be estimated on-line as Fy # ¼ atan2ðF y ; F x Þ ¼ arctan p: ð1Þ Fx However, this assumption is not valid when a deburring task is performed and therefore, another estimation
y (n) (T)
ϑ
(t) q2 (0)
q1
x Fig. 1. Sketch of a SCARA robot following a contour.
G. Ziliani et al. / Mechatronics 17 (2007) 431–441 θ,Q
Vt
Vt , d (t )
Vn
Vn , d (t ) = 0
Fn , d (t )
MT 0J PID
u PID ,Vt
fˆ1
.
kV,ff
-
Kv,fb
u P ,Vn
J T M 0T
k F,ff
-
PI Fn
u PI , F
θ,Q
Q
τ1 τ2
fˆ2
Robot
F
MT0 θ,Q
Fig. 2. Basic hybrid force/velocity control scheme.
method has to be employed, as it will be described in Section 5. The aim of a contour tracking task is to control the normal force and the tangential velocity of the robot probe along the directions n and t of the task frame (T), respectively. 3.2. Control law The following hybrid force/velocity control law has been considered (see the control scheme in Fig. 2): s1 s¼ ð2Þ ¼ J T ðQÞM 0T ð#ÞðU ðT Þ þ K R RÞ þ f^ ; s2 where R = [Vt,d, Fn,d]T is the feedforward vector based on force and velocity references, KR = diag[kV,ff,kF,ff] the corresponding diagonal matrix of gains, f^ ¼ ½f^ 1 ðq_ 1 Þ; f^ 2 ðq_ 2 ÞT is a term that compensates for the joint friction torques [14] and uPID;V U ðT Þ ¼ ; ð3Þ uPI;F þ K v;fb ðV n;d ðtÞ V n ðtÞÞ where uPID,V is the tangential velocity PID output, uPI,F is the normal force PI output, Vn,d(t) = 0, Vn(t) is the velocity of the end-effector in the normal direction and Kv,fb is a proportional gain. Note that the use of a normal force derivative term has been avoided in (3) (indeed, only the proportional and the integral actions have been employed) as the derivation of such a signal is ill-conditioned [15,16]. Conversely, the adoption of a normal force velocity feedback loop has been proven to be effective to compensate for the possible large force oscillations due to the kinetostatic behavior of the manipulator [17] (note that the compliance of the robot is mainly concentrated in the third link).
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1. the estimation of the contour direction to allow the contour tracking without the loss of contact in the presence of disturbances due to the cutting force; 2. the choice and the modulation of the correct contact force set-point to provide a satisfactory surface quality and a complete burr removal. When a robot manipulator accomplishes a material removing task such as deburring and chamfering, the cutting force can introduce disturbances in the force control loop that can yield to the loss of contact and can cause an incorrect measure of the normal direction. Depending on the task, many parameters have to be considered such as the tool tangential velocity, the tool typology and sharpening, the direction of the angular velocity with respect to the path feedrate, the workpiece stiffness, the cutting depth and the height and thickness of burrs. For example, if the material is homogeneous, then the cutting force increases proportionally to the amount of material removed. If the manipulator end-effector is moving with constant speed along the contour, then the force will vary proportionally to the depth of cut. The cutting process therefore generates also a reaction force that is not directed tangentially to the surface. Indeed, the normal component of this contact force can give some problem to the force control introducing disturbances in the control loop and it may cause the manipulator to penetrate in the workpiece material or to detach depending on the verse of rotation of the mill with respect to the tool feedrate. In Fig. 4 the forces acting on a workpiece during a milling process are reported. For simplicity it is assumed that only one tool tooth at a time is in contact with the surface and that all the forces act in one single point. For each tool tooth in contact with the surface, there is a cutting force, F~c , acting in the direction of the cutter tangential speed and a normal force, F~r , acting in the radial direction. The radial force is usually assumed to be proportional to the cutting force, i.e., F r ¼ F c tan ðb cÞ; where b is the friction angle (which is in general not easy to estimate) and c is the tool front bevel angle (see Fig. 3). The cutting force on the workpiece Fc can be expressed as [18]
Tool tooth
γ
Workpiece surface
4. Robotic deburring For the robotic deburring of workpieces of unknown geometry, two main problems have to be actually solved:
Fig. 3. The tool front bevel angle c for a single tool tooth in contact with the surface.
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Fc ¼
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bdvu ; x
where b is the tool wheel width, d is the depth of cut, v is the feedrate (i.e., the tangential velocity of the end-effector), u is the specific energy of the material and x is the tool wheel speed. It appears that also in this case the involved parameters are very difficult to estimate. F~n and F~t are the components of the total force ~ F in the task frame. If the tool path feedrate direction coincides with the direction of the peripherical velocity of the tool tooth in contact with the surface (Fig. 4, case ‘‘a’’), the machining force acting on the workpiece has a component F~t directed as the tracking tangential velocity, while the normal component F~n is directed from the surface to the tool. The reaction forces generated on the tool (and measured by the force sensor) have therefore, opposite directions. Note that in this case the normal force tends to pull the tool inside the workpiece surface and eventually the tool and the robot may stall while the tangential force acts against the tool feedrate. Conversely, if the tool path feedrate is opposite to the peripherical velocity of the tool tooth in contact with the surface (Fig. 4, case ‘‘b’’), the normal force generated on the tool tends to push it away from the surface while the tangential force tends to increase the tool feedrate. Further, for a correct deburring task the cut depth should vary proportionally to the burr height. The cut depth depends on the normal force (and on the value of b + c) and so the normal force set-point should vary proportionally to the burr height. Unfortunately burrs size and thickness can be very different and discontinuous, especially in casting workpieces. For these reasons, an accurate model-based estimation of the cutting force is very difficult to obtain.
5. The mechatronic design 5.1. Contact direction estimation with machining tangential force As already mentioned, the estimation of the angle # between the n-axis and the x-axis of frame (0) is essential for the control algorithm. The knowledge of the cutting force, which can be derived by considering a model of the forces involved in a milling task, could be used to estimate the angle. However, in a deburring task many of the parameters may vary and may not be accurately known (for example, the geometry of the casting burr) and therefore, it is almost impossible to have a sufficiently accurate model-based estimation of #. Thus, we propose a method that is based on the use of z-axis torque measure obtained from the six axes force/torque sensor and on the knowledge of the radius of the cutting mill, which can be easily a priori measured. Specifically, with reference to Fig. 4 we have that
Fig. 4. Notation of forces generated by the mill on the workpiece surface: (a) the tool path feedrate direction coincides with the direction of the peripherical velocity of the tool tooth in contact with the surface; (b) the tool path feedrate is opposite to the peripherical velocity of the tool tooth in contact with the surface.
• Fx and Fy are the forces along the axes x and y, respectively, of the force sensor frame; • Ftot is the total force; • Fn and Ft are the normal and tangential forces, respectively with respect to the workpiece surface; • # is the actual contact angle; • #* is the ‘‘apparent’’ contact angle calculated as atan2 (Fy,Fx) (see (1)); • D# is the angle between the actual and the measured contact directions; • sz is the z-axis measured torque and • r is the radius of the cutting mill. In the hypothesis that the z-axis of the mill frame coincides with the sensor frame axis, the tangential force due to the machining operation is sz Ft ¼ : ð4Þ r With trivial trigonometric relations the angle D# can easily be written as Ft D# ¼ arcsin ; ð5Þ F tot where F tot ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 2x þ F 2y ¼ F 2n þ F 2t :
ð6Þ
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Thus, the actual contact angle can be simply calculated from the measured angle with a correction depending on the z-axis measured torque as follows: # ¼ # D# or, substituting Eq. (5) in (7) it can be written as Ft # ¼ atan2ðF y ; F x Þ arcsin : F tot
ð7Þ
ð8Þ
However, if the vertical axis of the mill does not coincide with the vertical axis of the force sensor, then Eq. (4) cannot be used. This situation is depicted in Fig. 5, where the center O 0 of the force sensor frame is not coaxial with the center of the mill frame O due to the offset (Dx, Dy). In this case, the measured torque can be expressed as s0z ¼ sz þ F y Dx F x Dy;
ð9Þ
where sz = Ftr is the machining torque and the term Fy Æ Dx Fx Æ Dy represents the torsional moment with respect to the point O generated by the force Ftot applied in O 0 . Note that because of the terms Fy Æ Dx and Fx Æ Dy the force sensor measures a torque even if the tangential force Ft is null. The machining torque can be written therefore as sz ¼
s0z
F y Dx þ F x Dy
ð10Þ
and substituting Eq. (10) in (4) and (5) the value of Ft can be estimated as Ft ¼
s0z F y Dx þ F x Dy r
ð11Þ
and the actual contact angle can still be obtained with (8).
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5.2. Mill center calibration The position of the mill frame center O 0 can be estimated with a simple least squares based calibration procedure. This calibration is necessary, especially if the radius of the mill is comparable to the offset of the mill from the z-axis of the force sensor. The procedure is based on the contour tracking of a planar object with the mill mounted on the end-effector switched off, i.e., rolling on the contour so that the machining force is null. During the tracking the values of Fx, Fy and s0z are collected. Since the machining force is null, the measured torque is caused only by the offset of the z-axis of the mill with respect to the sensor frame, i.e., s0z ¼ F x Dy þ F y Dx. By considering N samples, the following linear systems result: 8 0 ¼ F x1 Dy þ F y 1 Dx s z1 > > > 0 > > s ¼ F x2 Dy þ F y 2 Dx > z > < 2 .. .. . . > > > 0 > > > szN 1 ¼ F xN1 Dy þ F y N1 Dx > : 0 s zN ¼ F xN Dy þ F y N Dx which can also be written in matrix form as s0z ¼ F X ; where 2
3 s0z1 6 s0 7 6 z2 7 7 6 7 sz ¼ 6 6 ::: 7; 6 s0 7 4 zN 1 5 s0zN
2
F x1 6 F 6 x2 6 ::: F ¼6 6 6 4 F xn1 F xn
3 F y1 F y2 7 7 7 ::: 7 7 7 F y n1 5 F yn
and X ¼ ½ Dy
Dx T :
Let F+ be the pseudoinverse matrix of F. Then, Dx and Dy can be calculated, with the least square criterion, as X ¼ F þ s0z :
ð12Þ
Remark 1. It is worth stressing that the calibration procedure has to be performed just once on a workpiece whose shape guarantees a small condition number of F, in order to provide a good estimation of X, disregarding the kind of workpieces that have to be subsequently deburred. For example, a circular object tracked on all its contours guarantees this condition.
5.3. The deburring tool Fig. 5. Forces acting on the cutting tool and measured by the force sensor during a machining task without offset between the sensor and the mill.
A major problem in the hybrid force/velocity control based deburring of workpieces of unknown geometry is
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the selection of the normal force set-point value. Actually, in order to measure the orientation of the surface, the contact force must be non-null. Further, the amount of material removed by the mill obviously depends on the contact force. Thus, it appears that this is a crucial issue for the performance of the deburring task. In fact, if no information is given to detect the shape of the burrs or to decide whether the surface must be worked or not, the mill will remove the same amount of material on all the contours, even when the surface is already finished. In other words, the size and the consistency of the burrs can be very different and the contact force should vary, in principle, to ensure a complete burrs removal and to prevent workpiece damages. For tender material as aluminum or plastic, the problem is actually more significant because the tool can penetrate inside the workpiece even for very low contact forces. Moreover, as shown in Fig. 4a, if the tool path feedrate direction coincides with the direction of the peripherical velocity of the tool tooth in contact with the surface, the reaction force Fn will tend to pull the mill inside the workpiece. This force is added to the normal force applied by the force control, but an increased normal force generates a higher cutting force and so a higher force Fn and so on. Eventually, the tool may get stuck in the workpiece. In any case, the use of the alternative approach depicted in Fig. 4b cannot be considered as a stable contact between the tool and the surface could not be maintained. In order to effectively address these problems, a novel special deburring tool has been designed. The tool has two ball bearings mounted on the sides of the cutting cylindrical surface. The external diameter of the ball bearings is the same as that of the tool (see Fig. 6). The two ball bear-
ings (or one of them if the burrs are placed on the edges of the workpiece) have to be placed in contact with the surface where no burrs are present, while the cutting part has to be placed along the plane where the burrs lie (see Fig. 7). In this way, the angular velocity of the tool has the same direction of the feedrate velocity, but the ball bearings avoid the penetration of the tool in the workpiece and also solve the problem of the choice of the set-point contact force value, which can be selected as a reasonably high value, in order to ensure that the thickest burrs are removed. Indeed, if no burrs are found on the tool path, the force applied to the workpiece will be only the ball bearings contact force and no material is removed from the workpiece. Further, the use of this kind of tool also makes the detection of the presence of burrs possible. Indeed, the measure of the z-axis torque increases when a burr is encountered (proportionally to the thickness of the burr) and this can be adopted to appropriately adapt the reference tangential velocity and therefore to improve the overall task time (see Section 6).
Fig. 7. The deburring tool.
Fig. 6. Forces acting on the cutting tool and measured by the force sensor during a machining task with offset between the sensor and the mill.
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Fig. 8. The deburring tool for the removal of central and edging burrs.
Remark 2. It is worth noting that the devised methodology can be easily extended to chamfering by using a conical cutter instead of a cylindrical one (see Fig. 8).
6. Experimental results The original tool chosen for the deburring task is a high speed 20,000 rpm pneumatic mill. Irregularities in the rotation of the tool are transmitted to the force sensor introducing high frequency noise in the force control loop. To
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Fig. 9. The chamfering tool.
face this problem a low-pass filter with a cut-off frequency of 100 Hz has been introduced in the force/torque acquisition. As a first experiment, in order to verify the effectiveness of the contact angle estimation method (see Section 5.1) a deburring task has been performed on an artificial burr created with an iron sheet fixed between two straight aluminum pieces. The iron sheet was 3 mm high and 1 mm thick. The modified deburring tool has been used and the tool path feedrate and tool rotation have opposite directions. The reference normal force has been fixed at 30 N, while the reference tangential velocity has been fixed at
Fig. 10. The artificial burr after the attempt of removal without the contact angle compensation.
Fig. 11. The artificial burr after the removal with the contact angle compensation.
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5 mm/s. In Fig. 9 the artificial burr is shown after the deburring task executed without the compensation of the contact angle (see (1)). The burr has not been removed completely and the tool has contoured the burr because of the wrong contact angle estimation. In Fig. 10 the artificial burr is shown after the deburring task has been executed with the compensation of the contact angle presented in Section 5.1 (see (8)). It appears that in this case the burr has been completely removed. To give a better insight in the results, in Fig. 11 the measured con-
tact angle without and with the compensation procedure is reported for comparison. It can be noted that the angle remains almost constant with the use of the compensation technique, while without compensation a larger deviation from the actual value is observed (about 75). In Fig. 12 the grinder torque during the deburring task in the two cases is reported. The torque measured is non-null only when the tool is in contact with the burr. Note that the contact time without the angle compensation is shorter. Since the tangential velocity set-point and the burr length are the
Fig. 12. The contact angle estimation without (top) and with (bottom) the compensation during the deburring of an artificial burr.
Fig. 13. The torque along the z-axis measured by the sensor during the deburring of the artificial burr without (top) and with (bottom) the compensation.
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same, this means that the tangential velocity incorrectly increased in the first case (the path is executed in about 2 s instead of 4 s). This is explained by the fact that an incorrectly estimated normal angle makes the direction in which the force is controlled to have a component directed in an unconstrained direction (i.e., the tangential direction in which the velocity is controlled). As a second experiment, the system has been tested on a real aluminum die casting workpiece. In this case, the z-axis torque is used to detect the presence of the burrs and to consequently reduce the reference tangential velocity (see Section 5.3). Thus, the reference tangential velocity is fixed at 6 mm/s but if the z-axis torque value exceeds a fixed threshold of 0.05 Nm, the reference velocity is (smoothly) reduced to 0.8 mm/s. Then, when the burr is completely removed and the measured torque decreases again, the set-point returns (smoothly) to the previous value. The original workpiece is shown in Fig. 13. Note that the burrs lay in different positions along the die closure plane and on the edging of the surface as indicated by the arrows. The height of some burrs exceeded 15 mm and the thickness was about 1 mm. A picture taken during the deburring task is shown in Fig. 14 and different pictures of the workpiece after the
robotic deburring are shown in Figs. 15–17, demonstrating the high performances achieved by the system. For a thorough analysis of the results, the estimation of the contact angle with and without the compensation technique is
Fig. 14. An aluminum die casting workpiece before the deburring task.
Fig. 17. Part of the aluminum die casting workpiece after deburring.
Fig. 15. The new milling tool during the deburring task.
Fig. 18. Zoom of the aluminum die casting workpiece after the deburring.
Fig. 16. The aluminum die workpiece casting after the deburring task.
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reported in Fig. 18, while the measured z-axis torque and the tangential velocity set-point are, respectively, shown in Figs. 19 and 20, showing that the velocity set-point reduces when the measured torque exceeds the selected threshold of 0.05 Nm. Note that until the tangential velocity set-point is reached at the beginning of the task (after slightly more than 1 s in this case), this method is not applied (see Fig. 21). Videos of the experiments are available at http:// www.ing.unibs.it/~visioli/deburring.html. 7. Conclusions Fig. 19. Comparison between the estimated contact angle with and without the compensation method in a deburring task of an aluminum workpiece.
In this paper, we have presented a mechatronic methodology for the robotic deburring of workpieces of unknown shape. The combined design of the milling tool and of a hybrid force/velocity controller, which is based on measurements provided by a force/torque sensor, allows to obtain high performances as demonstrated by the obtained results. Due also to its simplicity, the overall methodology appears suitable to implement in industrial settings. Indeed, the task is performed by the robot almost fully autonomously, since the knowledge of the geometry of the workpiece is not required, thus yielding to a significant reduction of the programming time (especially for complex paths). References
Fig. 20. Measured z-axis torque in a deburring task of an aluminum workpiece.
Fig. 21. Tangential velocity set-point in a deburring task of an aluminum workpiece.
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