Journal of Mechanical Science and Technology 28 (4) (2014) 1519~1527
www.springerlink/content/1738-494x
DOI 10.1007/s12206-014-0139-x
Motion control of industrial robots by considering
serial two-link robot arm model with joint nonlinearities†
Eui-Jin Kim1,2,*, Kenta Seki2 and Makoto Iwasaki2
1Robotics Research Department, Engine& Machinery Research Institute, Hyundai Heavy Industries Co., Ltd., Korea
2Department of Computer Science and Engineering, Nagoya Institute of Technology, Japan
(Manuscript Received April 8, 2013; Revised October 9, 2013; Accepted November 7, 2013)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Abstract
This paper presents a model based motion control approach for industrial robots by considering a serial two-link robot arm model with joint nonlinearities. In order to achieve the desired performance using the model based control approaches, it is important to obtain rele-vant models of both kinematics and dynamics including nonlinear characteristics. Main nonlinear components that lead to trajectory tracking errors of typical multi-axis industrial robot are joint nonlinearities in each axis and dynamic coupling effects between different axes. In this paper, a parametric modeling approach is introduced to reproduce behaviors of a serial two-link robot arm with joint nonlin-earities. Nonlinear stiffness, angular transmission errors, and friction in these two links are directly identified as joint nonlinearities. This approach is applied for the serial two-link arm of a typical multi-axis industrial robot, which has low frequency vibration modes and sig-nificantly affects to the trajectory performance. Effectiveness of the modeling is verified by comparative studies with numerical simula-tions and experiments. Finally, a 2-DOF control scheme with the identified two-link dynamic model and a feedba
ck loop-shaping with a variable notch filter are applied to improve the performance of trajectory tracking and residual vibration suppression.
Keywords: Industrial robot control; Two-link robot arm; Joint nonlinearities; Modeling and identification
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Introduction
Industrial robots are versatile manufacturing machines for industry automation, while the performance requirements become more and more strict for fulfilling various applications.
A wide variety of control strategies have been continuously researched for the fast and precise motion control, which are the main issues in the field of industrial robots. Model-based approaches such as computed torque control, linearization and nonlinear decoupling control by disturbance observer are one of the key concepts for improving the control performance [1, 2]. In order to achieve the desired performance in such control strategies, it is important to obtain relevant models of both kinematics and
dynamics including nonlinear effects, where a kinematic model determines a relation between joint configu-ration and robot end-effector position and orientation, while a dynamic model provides the control inputs to generate a refer-ence motion. It is known that two main nonlinear components deteriorate trajectory tracking errors of typical multi-axis in-dustrial robot: One is the nonlinear characteristic of each axis due to transmission components of joint such as gear reducers, belts, or shafts [3]. The most general nonlinear effects oc-curred in the joints are related to stiffness, friction, and trans-mission errors. The other is a dynamic coupling effect intro-duced between different axes connected in series, which caus-es the deterioration of positioning control during the fast robot motion [4]. Therefore, an accurate modeling of the nonlinear characteristics with consideration of relation between axes is one of the keys to allow a practical design of advanced robot controllers. A variety of literatures on the modeling of robot arm with nonlinear characteristics have been already pub-lished. A common remedy for modeling the nonlinear charac-teristics is to identify the dynamic behavior around certain positions and to estimate the parametric and nonparametric models [5-8]. However, those models are sometimes not in practical use to reproduce the exact trajectory motion behav-iors, because the dynamics of multi-axis industrial robot varies according to its posture. The identification processes for such infinite postures might be huge time consuming tasks.
In this paper, we deal with the problems on high perform-ance model based motion control of serial two-link robot arm, especially focusing on typical multi-axis industrial robots. A parametric modeling approach is introduced to reproduce the behavior of serial two-link robot arm with joint nonlinearities. Nonlinear stiffness, transmission errors, and friction in these
*Corresponding author. Tel.: +82 31 289 5275, Fax.: +82 31 289 5345 E-mail address: kr
† Recommended by Associate Editor Hyungpil Moon
© KSME & Springer 2014
1520 E.-J. Kim et al. / Journal of Mechanical Science and Technology 28 (4) (2014) 1519~1527
links are directly identified as joint nonlinearities. Effective-ness of the modeling approach is verified by comparative studies with numerical simulations and experiments for fre-quency responses. Finally, 2-DOF control scheme with the identified two-link dynamic model and a feedback loop-shaping with a variable notch filter are applied to improve the performance of trajectory tracking and residual vibration
sup-pression. This approach is applied for a serial two-link arm of Hyundai Heavy Industries' robot, which has low frequency vibration modes and significantly affects to the trajectory per-formance.
2. System configuration
All data are measured from a typical six-axis industrial ro-bot as shown in Fig. 1 used for spot welding, sealing, and material handling applications with 165 kg payload. The target object in this paper is the second and third axes of this robot because the accurate control of these two links, which are most significantly coupled than other axes and have vibration modes in the low frequency range, is important to improve the trajectory tracking performance of robot end-effectors. The configuration of serial two-link robot arm with joint flexibility is described in Fig. 2. Each link is composed of AC motor with encoder, RV gear reducer, link inertia, and especially gravity balancing spring for link 1. In this system, motor angu-lar positions are measured every 500μsec by the encoders, and then differentiated to obtain the motor angular velocities. The 3D laser sensor device, which has ±0.018 arc-sec of angular resolution and 3.5μm/m of accuracy, is used to meas-ure link angular positions. Even though only the positions in Cartesian coordinate can be directly measured by this laser
sensor device, the positions can be converted into angular coordinate of joint axis as following Eq. (1)
and shown in Fig. 4. The information, such as motor and link angular position and motor angular velocity, is used to identify stiffness, angu-lar transmission error and friction in the Sec. 4.
2s l arcsin q =èø
(1)
where ,,and x y z are measured positions in Cartesian coor-dinate, and i j are position numbers, and s l is a length between the origin of joint coordinate and the attached point of sensor.
3. Dynamic model
The basic dynamic model of the serial two-link robot arm with joint flexibility [9] can be represented as:
()()[]m m f m g g g m l m
M N K N q t q q q t ++-=&&& (2) ()()(,)()[]l l l c l l g l g g g m l
M N K N q q t q q t q q q ++=-&&& (3)
where 1,2
: Link number
12[,]T m m m q q q = : Motor angular position 12[,]T l l l q q q =
: Link angular position
12(,)m m m M diag J J = : Motor inertia
22l M R ´Î
: Link inertia matrix 21
(,)c l l R t q q ´Î&
: Coriolis and centrifugal force 21()f m R t q ´Î& : Motor friction 21
()g L R t q ´Î
: Gravity torque 21
m R t ´Î
:
Motor input torque 12(,)g g g K diag K K = : Joint stiffness
12(,)g g g N diag N N = : Gear reduction ratio (< 1) .
In this dynamic model, the rigid body dynamic properties of robot arm, such as motor and link inertia, can be known from designed CAD data. Therefore, the link inertia matrix, Corio-lis and centrifugal force, and gravity torque are calculated from the rigid body dynamics as following Eqs. (4)-(6) even though there may be some errors comparing with true values. A considerable literature has been published to handle and identify these errors [10], but it is outside scope of this re-search.
11122122()l l M
M M M M q éù=êúëû
(4) where
222
222
1111212122(2)lc l lc l lc l M m l m l l l l c =+++
2
122122122()lc l lc l M M m l l l c ==+
2
2222lc M m l =
Fig. 1. 6-axis industrial robot from HHI.
1
l q 1
m q 2
l q 2
m q g
x z
1
l l 2
lc l 1
lc l
Fig. 2. Configuration of serial two-link model with joint flexibility.
E.-J. Kim et al. / Journal of Mechanical Science and Technology 28 (4) (2014) 1519~1527
1521
22121222221212(2)(,)c l l lc l l l l l lc l l l m l l s m l l s t q q q q q q éù-+=êúëû
&&&& (5) 1112112122212(())()lc l l l lc l l c l g l g m l c m l c l c gm l c q t ++éù
=ê
úëû
(6)
where m is mass property of link, and l lc l l are kinematic properties of link, and l c , l s , and 12l c are ()l cos q , ()l sin q , and 12()l l cos q q +, respectively. In addition, the identification of joint characteristics, such as stiffness, friction, and angular transmission error, is positively necessary for more accurate dynamic modeling of two-link robot arm with joint nonlineari-ties. Fig. 3 shows a block diagram of serial two-link robot arm with joint nonlinearities. In this figure, ,s t ,f t and ATE q represent the nonlinear components of torsional torque by stiffness, friction, and angular transmission errors, respectively, and the broken line represents the coupling effect between two links, where s M is (11221221M M M M -).
4. Identification
The nonlinear behaviors of robot arm result from the super-position of coupled phenomena, such as joint stiffness, trans-mission error, friction, and gravity. Therefore, the experimen-tal procedure is designed such that it isolates and decouples these effects according to the purpose of identification as far as possible.
4.1 Stiffness
The stiffness of typical industrial robots is complexly af-fected by the compliance of their actuators and transmission components, geometric and material properties of the links, and the active stiffness provided by position control system.
For such reason, the stiffness characteristics provided by manufacturers are different from those of actual system. Therefore, the modeling, analysis, synthesis and control of stiffness have attracted the attention of many researchers [11, 12]. In this paper, a direct measurement approach using grav-ity effect is adopted to the identification of joint stiffness for target industrial robot arm. The gravity torque applied to each joint at every pose of system can be calculated from Eq. (6). Since other terms except the gravity torque in Eq. (3) can be neglected at a static state, the gravity torque is regarded as the torsional torque. The identification procedure of stiffness for each joint is as follows. Firstly, link 1 move
s from pose 1 to pose 2, which are the minimum and maximum gravity poses respectively, by 10° as shown in the left figure of Fig. 4, while next disk loads of 10kg are attached and detached one by one till maximum payload for applying additional torsional torque. Finally, the link goes back to pose 1 by 10°. The motor and link positions are measured at every step for calculating torsional angles. Although the same procedure can be applied for identifying torsional angles of link 2 as shown in the right figure of Fig. 4, the coupled torsional angle of link 1 should be considered and excluded from the measured torsional angle. Therefore, the torsional angle of each link at every step can be identified by
111
221221s m l s m l g s g N N q q q q q q q =-=-- (7)
where 1s q and 2s q are torsional angles of each link, and m q and l q are angular positions of motor and load measured by the encoder and the laser sensor respectively. Fig. 5 shows experimental results for the relationship between torsional torque and torsional angle of each link, which represent nonlinear characteristics of joint stiffness. The experimental results can be identified by a 3rd order polynomial function as follows and the corresponding parameters are listed in Table 1.
331()s s s s A A q q q t =+ (8)
where s t is torsional torque and 31and A A are polynomial coefficients.
1
m t 1
m q 1
l
m t 2
m 2
l
Fig. 3. Block diagram of serial two-link robot arm with joint nonlin-earities.
L i n k 1
L i n k 2
Link2
Link1
pose 1
pose 2
pose 1
pose 2
Sensor attached point Load
added point
Link1
Link2 L i n k 2
(,,)i i i x y z (,,)
j j j x y z l
q s
l
(a) (b)
Fig. 4. Experimental setup for stiffness identification of (a) link 1; (b)link 2.
1522 E.-J. Kim et al. / Journal of Mechanical Science and Technology 28 (4) (2014) 1519~1527
4.2 Angular transmission errors
The angular transmission errors ATE q are basically caused by kinematic and assembling errors, which are synchronous to the relative rotation of motor and gear. According to the con-ventional studies, the ATE q in the gearing system can be cate-gorized into two main components, which are th
e angular transmission errors to motor angle TEM q and the angular transmission errors to load angle TEL q . In this research, how-ever, only TEM q is modeled as ATE q , since TEL q can be neg-ligible in the amount comparing to TEM q and the load angle cannot be used for positioning control [13]. The angular transmission errors, therefore, are mathematically formulated as the periodic function in Eq. (9) for motor angular position m q .
()()(())m m n
m A i
m TE A i cos i i q q q f =+å
(9)
where i is pulsational order for motor angle, m A is ampli-tude of pulsation, and m f is phase of pulsation. Even though the angular transmission errors are generally defined by the difference between motor angular position m q and load an-gular position l q , the torsional angle s q modeled in the Sec. 4.1 should be considered as shown in Eq. (10) since the differ-ences between m q and l q include the torsional angle s q caused by stiffness and gravity effect.
l ATE g m s N q q q q =-+ (10)
Fig. 6 shows comparative waveforms of angular transmis-sion errors in the experiment and the model corresponded to parameters in Table 2. In this figure, the model can precisely reproduce the actual angular transmission errors.
4.3 Friction
Friction is typically characterized by its relationship with velocity. To determine the friction-velocity relationship, the robot was commanded to move at a constant velocity and the mean torque required for maintaining the velocity was taken to be the friction torque for that value of velocity. Furthermore, the effect of gravity can be negated by bidirectional drive ex-periments. Each axis is driven back and forth at constant ve-locities of ±1, ±5, ±10,..., ±100 in vpu (velocity per unit) for multiple bidirectional drive experiments. After col-lecting the data for each velocity, the data was fit to the fric-tion model in Eq. (11).
()()||c v s f s sgn F F F exp v q q q q t æöæö=++-ç÷ç
÷ç÷ç÷èøèø
&&&& (11)
where c F ,v F ,s F , and s v are coefficients for coulomb, vis-cous, and stribeck friction respectively. The identified results
Table 1. Parameters of stiffness model.
Torsional angle [ppu]
T o r s i o n a l t o r q u e [t p u ]
(a)
Torsional angle [ppu]
T o r s i o n a l t o r q u e [t p u ]
(b)
Fig. 5. Comparative waveforms of nonlinear stiffness curve between experiment and model of (a) link 1; (b) link 2.
Table 2. Parameters of angular transmission errors model.
Motor angle(rad)
T r a n s m i s s i o n e r r o r (p p u )
(a)
Motor angle(rad)
T r a n s m i s s i o n e r r o r (p p u )
(b)
Fig. 6. Comparative waveforms of angular transmission errors between experiment and model of (a) link 1; (b) link 2.
E.-J. Kim et al. / Journal of Mechanical Science and Technology 28 (4) (2014) 1519~1527
1523
for the nonlinear friction are shown in Fig. 7 and the corre-sponding parameters are listed in Table 3.
4.4 Model verification
The identified model is verified through experiments per-formed using the second and third axes of a t
ypical six-axis industrial robot. The behavior of two-link robot arm changes according to poses of the robot. Therefore, sufficiently distin-guishable poses are selected depending on the inertia and gravity for the model validation. For these poses, frequency characteristics can be obtained by an FFT analysis using input and output data of robot arm in the close loop system as shown in Fig. 8, where ID u is exciting signals, u is measured input data for motor torque, and y is measured output data for motor angular position. The comparative results of frequency response between the experiment and simulation are shown in Figs. 9 and 10. The left figures are the frequency responses of nonparametric modal models using experimental data, while the right figures are the frequency responses of identified two-link robot arm model with joint nonlinearities in simulation.
The first two vibration modes in the low frequency ranges are expressed by the identified two-link robot arm model, even though over third vibration modes are not appeared. Further-more, the gain characteristics in the low frequency range relate to friction and the resonant/anti-resonant frequency relate to stiffness and angular transmission error. The comparison re-sults show that the identified model in the Sec. 4.1−4.3 repre-sents real two-link robot arm system well.
5. Motion control scheme
The PD position controller including PI velocity control with state feedback and dynamic feedforward compensator with two-inertia model of single link [14, 15] is used as a con-ventional control scheme for the motion control of industrial robots. This conventional scheme will be denoted by Control-ler1 hereafter. In this paper, two kinds of motion control scheme based on the identified flexible two-link model in the Sec. 4 are introduced to cope with joint nonlinearities and dynamic coupling effects of industrial robots while the robot is moving. One is a feedback loop shaping approach using a
Table 3. Parameters of friction model.
Motor velocity(vpu)
T o r q u e (t p u )
(a)
Motor velocity(vpu)
T o r q u e (t p u )
(b)
Fig. 7. Comparative waveforms of friction curve between experiment and model of (a) link 1; (b) link 2.
Controller
Robot
u
M
q y
ID
u R
q
Fig. 8. Block diagram for measurement of frequency response.
G a i n [d B /u n i t ]
Frequency [fpu]
P h a s e [d e g ]
G a i n [d B /u n i t ]
Frequency [fpu]
P h a s e [d e g ]
(a) (b)
Fig. 9. Comparative waveforms of frequency response on some poses of link 1: (a) nonparametric modal model; (b) proposed model.
G a i n [d B /u n i t ]
Frequency [fpu]
P h a s e [d e g ]
G a i n [d B /u n i t ]
Frequency [fpu]
P h a
s e [d e g ]
(a) (b)
Fig. 10. Comparative waveforms of frequency response on some poses of link 2: (a) nonparametric modal model; (b) proposed model.
Fig. 11. Block diagram of proposed motion control scheme.
发布评论