In the industrial practice, there exist different norms to evaluate thefinal product quality. For example, ISO 12780 and 12781 define the path straightness, the surface
Link #2 Link #1
Ac
Link #6
1-d.o.f.
spring 1-d.o.f.
spring
F
Ac end-effector
Base Ac
1-d.o.f.
spring
Ac Actuated joint Virtual spring (b)
(a)
Fig. 3 Typical industrial robot and its VJM-based stiffness model.aTypical industrial robot.b VJM model of serial robot
Comparison Study of Industrial Robots for High-Speed Machining 141
flatness and the path roundness that in some cases is also called as thecircularity.
From our experience, the circularity index is the best one for evaluating the machining process quality, since for linear trajectories the force magnitude/direction are constant and their impact can be easily compensated in the control program (Feng et al.2015). In contrast, for the circular path the cutting force direction varies and may lead to irregular path distortion that can be hardly com- pensated. For this reason, the circularity index will be used to evaluate the capacity of industrial robot to perform the machining task.
According to ISO 12181, the circularity evaluation includes two steps: obtaining a reference circle and estimation of the path deviations with respect to this circle.
There are four methods to define the reference circle: Minimum Circumscribed Circle (MCC), Maximum Inscribed Circle (MIC), Minimum Zone Circles (MZC) and Least Squares Circle (LSC). They are illustrated by Fig.4. For all of them, the circularity evaluates the distance between two circles in accordance with the equation
qẳrmaxrmin ð12ị where rmax and rmin are the radii of the circumscribed and inscribed circles, respectively. The principal difference is related to the circle centers that are com- puted using different methods. For example, for MIC the center point is computed for the maximum inscribed circle and it is also used for the minimum circumscribed one. In the MCC method, the center is computed for the minimum circumscribed circle and the inscribed circle is build using the same center point. In the case of LSC, the inscribed and circumscribed circles are found for the center point obtained for the least square circle. In contrast, the MZC method uses a center point for
MZC LSC
MCC
MIC OMIC
OMCC OMZC OLSC
Fig. 4 Circularity evaluation using different industrial norms
142 A. Klimchik et al.
which the distance between the inscribed and circumscribed circles is minimal. In practice, MIC and MCC methods are rarely used if the tool path is essentially distorted and corresponding center points are essentially different (Fig.4). In the latter case, it is preferable to use either MZC or LSC index. Since there is no considerable difference between MZC or LSC, practicing engineers prefer LSC to evaluate the trajectory circularity.
In the frame of the LSC method, the circle center is obtained as a solution of the following optimization problem
Xn
iẳ1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxix0ị2 ðyiy0ị2 q
r
2
! min
x0;y0;r ð13ị
whereðxi;yiịare the measured tool path coordinates,ðx0;y0ịis the circle center and ris its radius. This optimization problem is highly non-linear and cannot be solved analytically. For this reason, a Newton–Raphson method is used.
Using the LSC center point, the circularity is computed in straightforward way in accordance with (12), where
rmaxẳmaxðjpiOLSj;iẳ1;nị; rminẳminðjpiOLSj;iẳ1;nị ð14ị
Herepiẳ ðxi;yiịT is measured tool path points andOLSẳ ðx0;y0ịT is the center of corresponding LS circle.
Assuming that all geometric and elastostatic parameters of the manipulator are known, the resulting tool path for the desired circular trajectory can be computed and evaluated the above presented index. For the set of machining tasks considered in this paper, a benchmark trajectory is a circle of radius 100 mm. For each task and for each path point, there were computed the cutting wrenches Wẳ ẵFTC; MTT which include both the force and the torque components. It is worth mentioning, that the wrench magnitude is constant here, while its direction varies along the path depending on the rotation angleφ. Using this data, the resulting tool path can be computed in the following way
piẳp0ỵR rỵJðhpị Kh1JðhpịTR FCỵJðhpị Kh1 Jðuịh TR M ð15ị
where p0 is the center of the reference circle, ris its radius, rdefines the radius vector of the circle foruẳ0, the matrixRðuiịtakes into account rotation of the target point by the angleui, the superscripts (p) andðuịindicate the position and orientation part of the Jacobian matrix presented asJh ẳ ẵJðhpịT;Jðuịh TT. It should be stressed that dynamics also affects the circularity; however, its influence is much lower comparing to the compliance errors caused by cutting forces.
Based on this approach, it is possible to evaluate robot accuracy with respect to the circularity index for the entire workspace and to determine the region in which machining accuracy is the best (from the circularity point of view).
Comparison Study of Industrial Robots for High-Speed Machining 143
4 Experimental Result
The technique developed in this paper has been applied to the comparison study of four industrial robots of Kuka family. There were compared with respect to the circular machining task of 100 mm radius that was placed in different workspace points. Some details concerning the examined robots and their principal parameters are given in Table1. These robots have similar kinematics and provide comparable repeatability/accuracy without loading. However, their payload capacities and workspace size are different. Elastostatic parameters of the examined robots have been identified using dedicated experimental study using methodology developed in our previous works (Klimchik et al.2013b,2014b). Corresponding results are presented in Table2.
In the experimental study, it was used the cutting tool of the radiusRẳ5 mm with three teeth ðzẳ3ị. Its machining parameters are the following:
jẳ90;c0 ẳ7;ksẳ45;fzẳ0:08 mm=rev;apẳ5 mm;Knẳ750 N=mm2. It is assumed that cutting tool is completely engagedðuẳ180ịalong the trajectory.
This corresponds the cutting forceFCẳ ẵ440 N;1370 N;635 Nand cutting torque Mẳ ẵ0 Nm;3 Nm;10:5 Nm. These vectors are rotating while the tool is moving along the circular trajectory.
In our study it was assumed that the reference circular trajectory was located in the plane of joint q2and q3movements (XOZ if q1= 0), which is the most critical one for all articulated robots in machining application. For these conditions, the circularity maps have been computed for all examined manipulators. Relevant
Table 1 Principal characteristics of examined robots Robot Repeatability
(mm)
Workspace volume (m3)
Working radius (m)
Maximum payload (kg) KR 100
HA
0.05 46 2.6 100
KR 270 0.06 55 2.7 270
KR 360 0.08 118 3.3 360
KR 500 0.08 68 2.8 500
Table 2 Stiffness parameters of examined robots Robot Equivalent joint compliances,μm/N
k1 k2 k3 k4 k5 k6
KR 100 HA 1.92 0.34 0.56 3.31 3.83 5.42
KR 270 0.54 0.29 0.42 2.79 3.48 2.07
KR 360 0.86 0.17 0.25 2.17 1.47 2.96
KR 500 0.47 0.14 0.19 0.72 0.95 1.44
144 A. Klimchik et al.
results are presented in Figs.5,6,7 and 8. In addition, thesefigures contain the optimal regions for locating the machining tasks of size 100×100, 200×200 and 500×500 mm. Summary of circularity indices for different machining task is given in Table3. It should be stressed that in the case of machining task different from the considered one, the results should be scaled according to cutting force magnitude.
As follows from the presented results, robot Kuka KR 500 ensures the best performance for the considered technological task. This advantage is achieved due to less complaint actuators, which obviously affect the robot price. In general, all examined robots ensure circularity level about 3 mm within entire workspace (without compensation). So, if this accuracy is sufficient, the robot can be chosen taking into account the workspace and payload properties. It should be stressed that robot KR 100 cannot be used for machining of hard materials because of the
0.5
0.5
0.5
0.5
0.5 0.5 0.5
0.5 0.5
1
1
1
1
1 1 1
1 1
1.5
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3 3 3 3
3
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-2 -1 0 1 2 3
-0.5 0 0.5 1 1.5 2 2.5 Fig. 5 Circularity maps for 3
robot KR 100 HA
0.5
0.5 0.5
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3 3
-2 -1 0 1 2 3
-0.5 0 0.5 1 1.5 2 2.5 Fig. 6 Circularity maps for 3
robot KR 270
Comparison Study of Industrial Robots for High-Speed Machining 145
payload limitation. However, because of its lower price comparing to other examined robots, it is competitive for machining if the cutting force magnitude is less than 1 kN and the desired circularity is about 1 mm. On the other hand, robot KR 360 is competitive for large-dimensional tasks only and for milling with forces higher than 2.5 kN. Otherwise, KR 270 is preferable that may ensure better per- formance within its workspace. The obtained results are also summarized in Table4that presents robots suitable for machining with desired accuracy (in terms of the circularity) for different force magnitudes. The latter allows practicing engineers to justify robot selection for given machining task.
0.5
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-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Fig. 7 Circularity maps for 4
robot KR 360
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-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
-1 -0.5 0 0.5 1 1.5 2 2.5 3 Fig. 8 Circularity maps for 3.5
robot KR 500
146 A. Klimchik et al.
5 Conclusions
The paper presents an industry oriented technique for evaluation of robot capacity to perform machining operations. It proposes methodology for evaluation of the circularity widely used in industrial practice. Obtained theoretical results have been applied to the experimental study that provides comparison analysis of four industrial robots with respect to their accuracy in machining. In future, this methodology will be applied to a wider set of industrial robots, technological tools and materials.
Acknowledgments The work presented in this paper was partially funded by the project FEDER ROBOTEX№38444, France.
Table 3 Circularity indices for examined robots, mm
Robot Min Max Middle
KR 100 HA 1.03 3.36 1.91
KR 270 0.84 3.13 1.64
KR 360 1.02 2.81 1.80
KR 500 0.41 1.42 0.76
Table 4 Suitability of examined robots for different machining tasks (without compensation) Circularity (mm) Force magnitude
500 N 1000 N 2000 N 3000 N
0.2 KR 500
0.5 KR 100 KR 270 KR 500
KR 270 KR 500
KR 360 KR 500
1.0 KR 100 KR 100 KR 270 KR 500
KR 270 KR 270 KR 360
KR 360 KR 360 KR 500
KR 500 KR 500
1.5 KR 100 KR 100 KR 270 KR 500
KR 270 KR 270 KR 360
KR 360 KR 360 KR 500
KR 500 KR 500
2.0 KR 100 KR 100 KR 270 KR 360
KR 270 KR 270 KR 360 KR 500
KR 360 KR 360 KR 500
KR 500 KR 500
Comparison Study of Industrial Robots for High-Speed Machining 147
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Comparison Study of Industrial Robots for High-Speed Machining 149
Adaptive Robust Control and Fuzzy-Based Optimization for Flexible Serial Robot
Fangfang Dong, Jiang Han and Lian Xia
Abstract An adaptive robust control design for flexible joint serial robot is con- sidered. The system contains uncertainty which is assumed to lie in fuzzy set. Since the overall system does not meet the matching condition, a virtual control is implanted and the system is transformed into fuzzy dynamical system. Then an adaptive robust controller can be designed to guarantee the uniform boundedness and uniform ultimate boundedness of the transformed system. No other knowledge of uncertainty is required other than the existence of its bound. The control is deterministic and is not IF-THEN heuristic rules-based. The optimization of control parameters is also considered by solving two quartic equations. The extreme minimum value of the equation is proven to be the optimal solution.
Keywords Fuzzy controlAdaptive robust controlUncertain systemFlexible
joint robot Uniform ultimate boundedness
1 Introduction
Flexible joint serial robot is widely used in industry applications. It provides less mass, faster motion and stronger payload capacity. All these advantages make it become incomparable. However, a flexible joint serial robot is a highly coupled, nonlinear, complicated system, and often surfers from matched or mismatched uncertainty. Therefore, scholars devoted themselves to find an effective control strategy for achieving high performance in the past decades. Although significant developments have been made, the simplicity, efficiency, and reliability of controls forflexible robots are still unsolved.
Once the exact information of uncertainty is known a priori, some deterministic control schemes can be applied, such as PD control (Piltan and Sulaiman2011),H1
F. DongJ. HanL. Xia (&)
School of Mechanical Engineering, Hefei University of Technology, 193 Tunxi Road, Hefei 230009, Anhui, China
e-mail: xia_nian@126.com
©Springer International Publishing Switzerland 2017
D. Zhang and B. Wei (eds.),Mechatronics and Robotics Engineering for Advanced and Intelligent Manufacturing, Lecture Notes
in Mechanical Engineering, DOI 10.1007/978-3-319-33581-0_12
151
control (Shen and Tamura1995), sliding mode control (Huang and Chen2004) and so forth. To deal with the uncertainty, Alonge et al. (2004) and Yeon et al. (2008) adopted robust control and the performance would be guaranteed as long as uncertainty lies in some (typically compact) set. By using adaptive methodology, the unknown parameter of controller can approach to the real value for achieving better tracking trajectory (Chien and Huang2007; Pradhan and Subudhi2012). The fuzzy-neural control (Sun et al.2003) and other feasible strategies are well sum- marized in, e.g., Kiang et al. (2015) and its bibliographies.
If the uncertainty cannot be clearly identified, which means the known portion cannot be separated from the unknown portion, a fuzzy perspective via the degree of occurrence of certain applications comes into account as an alternative. Fuzzy theory was originally used to describe the object, which is in lack of a sharp boundary (Zadeh1965), in a human linguistic manner. Thus, fuzzy theory is mostly applied to fuzzy reasoning, estimation, decision-making, etc. The road of fuzzy theory incorporate with system theory (i.e., fuzzy dynamical system) is insuffi- ciently investigated. Past efforts on this domain can be found in Hanss (2005) and Klir and Yuan (1995). Our work endeavors to adopt a new fuzzy dynamical system approach to complement the previous efforts and to broaden the scope.
The main contributions of this work are threefold. First, the uncertainty bound is described by using fuzzy set. This renders the crisp bound description a special case and provides an alternative to probability description. Second, by creatively implanting afictitious control, the system is transformed into a new representation.
An adaptive robust control, which is deterministic and not if-then rules-based, is designed to guarantee the uniform boundedness and uniform ultimate boundedness of the transformed system. Third, a fuzzy-based system performance index based on the fuzzy-based uncertainty bound description is proposed. The optimal choice of the control parameter is formulated as a (semi-infinite) constrained optimization problem. The extreme minimum value to this optimization problem can be obtained by solving two scalar quartic algebraic equations.
2 Fuzzy Dynamical Model of the System
Consider aflexible joint serial robot described as follows
Dðq1;r1ị€q1 ỵCðq1;q_1;r1ịq_1 ỵGðq1;r1ị ỵKðr1ịðq1q2ị ẳ0 Jðr2ị€q2 ỵFðq2;q_2;r2ịq_2Kðr2ịðq1q2ị ẳu;
ð1ị
whereq1ẳ ẵq2q4. . .q2nT is link position vector and q2ẳ ẵq1q3. . .q2n1T is joint position vector, here, the superscripts 1;2;. . .;2n denote thenth coordinate of the robot.r12R1Rn1andr22R2Rn2are unfictitious control implanted into the certain parameter vectors with R1 and R2 prescribed and compact. Let qẳ
qT1 qT2
T
be a 2n-dimension vector representing the generalized coordinate for
152 F. Dong et al.
the system. The joint flexibility is denoted by a linear torsional spring whose elasticity coefficient is represented by a diagonal positive matrixKðr1;2ị.Dðq1;r1ị is the link inertia matrix andJðr2ịis a diagonal matrix representing the inertia of actuator;Cðq1;q_1;r1ịq_1andFðq2;q_2;r2ịq_2represent the Coriolis and centrifugal forces of links and actuators, respectively; Gðq1;r1ịis the gravitation force, and u denotes the input force from the actuators. We note that certain entries of the uncertain parameter vectorsr1andr2may represent the same physical parameters because these two subsystems are connected.
Assumption 1 The inertia matrix Dðq1;r1ịis uniformly positive definite (Chen et al.1998). That is, there exists a constant k[0;k1[0;k2 0;k3 0, such that
k1 ỵk2k k ỵq1 k3k kq1 2 kDðq1;r1ị k k; 8q12Rn ð2ị We state this as an assumption other than a fact.
Let us rewrite thefirst part of (1) as
Dðq1;r1ị€q1 ỵCðq1;q_1;r1ịq_1 ỵGðq1;r1ị ỵKðr1ịq1Kðr1ịðq2u1ị
ẳKðr1ịu1 ð3ị
where u1 is a fictitious control implanted into the system. It is only used to for- mulate the real controluwithout changing the dynamics of original system. Withu1 introduced, the system could be divided into link position subsystem and joint position subsystem, which are controlled byu1 andu, respectively.
Let x1ẳq1;x2ẳq_1;x3ẳq2u1;x4ẳq_2u_1, then X1ẳ ẵx1 x2T; X2ẳ ẵx3 x4T;Xẳ ẵX1 X2T. Thus, by multiplying K1ðr1ịon both sides of (3) the dynamics offlexible joint robot can be expressed as follows by using new state variable:
D^ðx1;r1ịx_2ẳ C^ðx1;x_1;r1ịx_1G^ðx1;r1ị x1ỵx3ỵu1; ð4ị Jðr2ịx_4ẳ Jðr2ị€u1Fðx1;x_1;x3;x_3;r2ịx_3Fðx1;x_1;x3;x_3;r2ịu_1
Kðr2ịx3ỵKðr2ịx1Kðr2ịu1ỵu; ð5ị whereD^ ẳK1D;C^ ẳK1C;G^ ẳK1G.
Assumption 2
1. Suppose the initial state isXðt0ị ẳX0, wheret0is the initial time. For each entry ofX0, namelyX0i; iẳ1;2;. . .;4n, there exists a fuzzy setM0iin a universe of discourse HiR, characterized by a membership function lHi:Hi! ẵ0;1 (Chen2011). That is,
Adaptive Robust Control… 153