Optimal Design of Robust PID Controller for Robot Arm Manipulators Using Quantitative Feedback Theory (QFT) Method Keivan Torabi Z., Amir Ali Amiri Moghadam*-p
Abstract In this paper, application of Quantitative Feedback Theory (QFT) to design optimal robust PID controllers for robot arm manipulators is proposed. In reality robots have uncertain mathematical models. Uncertainties in the models are caused by lack of knowledge about the dynamics of the robot, external disturbance, pay load changes, and friction, etc. Thus application of robust control methods for high precise control of robots is inevitable. As robot arm manipulators have multivariable nonlinear coupled transfer functions, therefore, using QFT technique at first converts the nonlinear plant into a family of linear and uncertain plants. This is achieved using fixed-point theorem and then for illumination of cross-coupling effect between degrees of freedom suitable disturbance rejection bounds will be designed. An optimal robust PID controller will be designed for the linear processes. In order to illustrate the algorithm the authors applied a two degree of freedom robot and the optimal robust PID controller is designed for tracking problem. Finally, the robustness and optimality of the design will be checked by means of a nonlinear simulation in tracking different trajectories. Key Words: QFT, Robot arm manipulators, Robust PID, Uncertainty, Nonlinear simulation
* Corresponding author K.Torabi Z. is with Department of Mechanical Engineering, University of Kashan, Kashan, Iran (email:
[email protected]). Amir Ali Amiri Moghadam is a PhD student in Department of Mechanical Engineering, University of Kashan, Kashan, Iran (
[email protected])
1
1. Introduction The purpose of a robot is to control the movement of its gripper to perform various industrial jobs, such as, assembly, material handling, painting, and welding [1]. Robot manipulators have complex nonlinear dynamics that might make accurate and robust control difficult. Fortunately, robots have several extremely nice physical properties that make their control straight forward as they are in the class of Lagrangian dynamical systems [2]. There are several methods for controlling of a robot such as: classical joint control, digital control, adaptive control, robust control, learning control, force control, and teleoperation. There are two basic methodologies for dealing with the effect of uncertainty in a system namely adaptive control and robust control. In adaptive control design approach, the controller will estimate the system’s parameter online and then will tune itself based on these estimates. In the robust control design approach, the controller has a fixed structure which will satisfy the system specifications over whole range of plant uncertainty. Although adaptive control can be applied to a wider class of problems, the application of robust control will lead to a simpler controller as the structure of controller is fixed requiring no time for tuning [3]. As mentioned above the uncertainty in the dynamics of robot is inevitable, therefore, in this paper the robust control QFT has been selected to control the robot. The application of robust control methods for control problems is always a difficult job for engineers, thus in this article the design steps are described in Fig.1 below which illiterates the algorithm used to model and control an arm manipulator.
Fig. 1 Controlling flow chart of robot arm manipulator As the chart indicates in the first step the robot will be modeled through the Solid Works and this model will be transferred to the Matlab by means of Simmechanics
2
then the nonlinear dynamic of robot will be simulated with Matlab and will be tasted by using of an analytical soft ware such as Vissual Nastran in the next step the nonlinear system will converted to the uncertain linear plant sets by application of globe soft ware (it is a numerical computational program based on the fixedpoint theorem) , and then by applying the QTF technique for the uncertain linear plant sets the suitable controllers and prefilters will be designed . In the last phase of design simulation for nonlinear system will be done. There are many practical systems that have high uncertainty in open-loop transfer functions which makes it very difficult to have suitable stability margins and good performance in command following problems for the closed-loop system. Therefore a single fixed controller in such systems is found among of "robust control" family. Quantitative Feedback Theory (QFT) is a robust feedback control-system design technique initially introduced by Horowitz (1963, 1979), which allows direct design to closed-loop robust performance and stability specifications. Since then this technique has been developed by him and others [4], [5], [6], [7], and [8]. In many techniques from "robust control" family such as H design is based on magnitude of transfer function in frequency domain, but QFT is not only concerned with aforementioned subject, but also able to take into account phase information in the design process. The unique feature of QFT is that the performance specifications are expressed as bounds on frequency-response loop shapes in such a way that satisfaction of these bounds imply a corresponding approximate closed-loop satisfaction of some time-domain response bounds for given classes of inputs and for all uncertainty in a given compact set. Consider the feedback system shown in diagram Fig.2. This system has the twodegrees of freedom structure. In this diagram p(s) is uncertain plant belong to a set p (s ) p (s , ); p where is the vector of uncertain parameters for uncertainty structured of p(s) which to take valves in p and also G(s) is the fixed structure feedback controller and F(s) is the prefilters, and D(s) is the disturbance at the plant output.
Fig. 2 Two degree of freedom feedback system Simply QFT controller design method can be summaries as fallows: In parametric uncertain systems, we must first generate plant templates prior to the QFT design (at a fixed frequency, the plant’s frequency response set is called a template). Given the plant templates, QFT converts closed loop magnitude specifications into magnitude constraints on a nominal open- loop function (these are called QFT bounds). A nominal open loop function is then designed to simultaneously satisfy its constraints as well as to achieve nominal closed loop stability. In a two degree-of-freedom design, a pre-filter will be designed after the loop is closed (i.e., after the controller has been designed) [9]. In this article the QFT design problem is finding (if possible) suitable G(s) and F(s) such that the output signal y(s) tracks accurately the reference signal R(s) , despite the presence of uncertainty in p(s) .
3
2. Design of QFT controller for nonlinear systems In QFT method, the nonlinear plant is converted to family of linear and uncertain processes. For this purpose, literature on QFT offers two different techniques [10], namely Linear Time Invariant Equivalent (LTIE) of nonlinear plant, and Non Linear Equivalent Disturbance Attenuation (NLEDA) techniques. In both methods, limited accepted output is the main tool to translate nonlinearities of the plant into templates for the first technique, or disturbance bounds for the second technique. In this paper the first method is used. For this conversion at the first acceptable out put set is introduced and then the input set produced. Dividing of output laplasian to input laplasian, introduces transfer function between each input and output. This transfer function will be linear and uncertain. Response condition is provided with use of fixed-point theorem [11].In this paper the Golubev method [12] is applied for each input-output set, in order to reach directly to a linear time-invariant transfer function, relating acceptable plant input-output data set. 3. Application of QFT technique for MIMO systems Application of QFT to MIMO uncertain system is still remains to be one of the most difficult controlling problems for engineers. The first application of QFT to MIMO systems was introduced by Horowitz [13]. After that this technique has been extended by [14], [15], and [16]. One of the simplest MIMO techniques that can be applied to robot arm manipulators is the one which was introduced by Ching cheng c. et al [17]. In this method the basic idea is to convert the closed loop transfer function to an off-diagonal matrix which can be described as below. t ij ( j ) j i (1) ij ( ) 1, for t jj ( j ) Where t ij ( j ) denotes the relation between jth input to the ith output. By means of fixed point theorem it is proved that we could substitute the MIMO system by its equivalent SISO systems provided that suitable disturbance rejection bounds are designed. Suitable disturbance rejection model would be disturbance at the plant output: 1 Y ( j ) (2) TD TD ( j ) 1 L D( j ) In reference [17] it is shown that in order to achieve an off-diagonal closed loop transfer function the below inequality must be held ij ( ) 1 (3) min for i j , j 1,2,, n, h 1 li ( j ) qii ( j ) q ( j ) max ij 1 Where l i is the open loop transfer function, P 1 , and ij is a small positive qij function which can bound the closed loop transfer function. Therefore based on above inequality and the iteration algorithm which is described in reference [17] one can design suitable disturbance rejection bounds. 4
4. Optimal design of robust PID controller for robot arm manipulators In this part we will apply our algorithm for a two degree of freedom robot. The objective of this part is to synthesize suitable controller and prefilter such that, first the closed loop system is stable, second it can track desired inputs, and third illumination of cross-coupling effects by using suitable robust disturbance rejection bounds. 1) Stability margin P( j )G( j ) 1.2 1 P( j )G( j )
2) The tracking specification is overshoot (=5%) and the settling time (=0.08 s) for all plant uncertainty which can be describe with second order system.
( ji ) T ( ji ) ( ji ) Where ( ji ) and ( ji ) are lower bound and upper bound respectively. T ( ji ) is the input-output relation from the input R(s) to the output Y(s). 1 3) Suitable robust disturbance rejection bounds ( ( ) ) for reducing 1 li ( j ) the cross-coupling effects between joints. Table.1 shows ( ) for each link over the design frequencies.
i Link 1 Link 2
0.01
0.7
2
10
100
1000
10000
0.0001 _______
0.0050 0.0005
0.0286 0.0016
0.2044 0.0084
0.9555 0.1701
0.9999 1.0000 0.9023 0.9992
Table.1 Robust disturbance rejection bounds 4.1. Dynamics of two arm manipulators A two-link robotic manipulator is shown in Fig. 3 where m1 and m2 are the masses of links 1 and 2, respectively; l1 and l2 are the length of the links 1 and 2, respectively. The dynamic equation of the robotic manipulator is: (4) M (q)q C(q, q)q G(q) Where q q1 q2 is an 2 1 vector of joint position, q q1 q2 T
T
vector of joint velocity, q q1 q2
T
is an 2 1
is an 2 1 vector of joint acceleration,
1 2 is an 2 1 vector of control input torque; M (q) is an 2 2 inertial T
matrix that can be described as :
5
Fig. 3 two degree of freedom robot arm manipulator 1 1 2 2 ( 3 m1 m2 )l1 3 m2l2 m2l1l2 cos q2 M (q) 1 1 m2l22 m2l1l2 cos q2 3 2
1 1 m2l22 m2l1l2 cos q2 3 2 1 2 m2l2 3
(5)
C (q, q) is a 2 2 matrix of coriolis and centrifugal forces that can be described as:
1 2 m2l1l2 (2q2 ) C ( q, q ) 1 m l l q sin q 21 2 1 2 2
1 m2l1l2 q2 sin q2 2 0
(6)
And G(q) is a 2 1 gravity vector that can be represented as: 1 1 ( 2 m1 m2 ) gl1 cos q1 2 m2 gl2 cos(q1 q2 ) G (q) 1 m2 gl2 cos(q1 q2 ) 2
(7)
Where g represents a gravity acceleration constant. Assuming following quantities for mass and length of each link (m1=2kg, m2=3kg, L1=0.4m and L2=0.6m) the dynamic equation of the robot can be derived as below. 1 0.9467 0.72 cos q2 q1 0.36 0.36 cos q2 q2 0.72q1q2 2 0.36q2 sin q2 15.68cos q1 8.82 cos q2 q1 0.36 0.36 cos q q 0.36q 0.36q 2 sin q 8.82 cos q q 2 1 2 1 2 2 1 2
(8)
4.2. Linearization In the first step the nonlinear multivariable dynamic of robot will be simulated in Matlab.
6
Fig. 4 Simulation of robot dynamic in Matlab Next step involves linearization by using Golubev method. As we discussed in part 2, application of this method involves producing acceptable output set, and then based on nonlinear dynamic obtaining input set. Dividing of output laplasian to input laplasian, introduces transfer function between each input and output. This transfer function will be linear and uncertain. By generating different trajectory for robot arm suitable output set will be produced then according to nonlinear dynamic of robot associated input set (required torque in joints) will be obtained. In order to find the linear uncertain plant transfer function, numerical methods can be used. Therefore by means of numerical software which applies in frequency domain (with minimization of square error between input and output of nonlinear transfer function) the uncertain linear family of processes was achieved. Nonlinear dynamics of robot is modeled with an uncertain linear two by two matrix transfer function Fig. 5.
1 2
Nonlinear System
1 2
q1 q2
P ( s, )
q1 q2
Fig. 5 Substitution of nonlinear system with its associated linear system Seven different paths were used in order to achieve suitable linear model. Therefore the uncertain linear matrix transfer function is as below: P ( s, ) P12 ( s, ) P( s, ) 11 P21 ( s, ) P22 ( s, ) Where q a11 ; a11 [0.62,1.1] and b11 [0.01,30] P11 1 1 s( s b11 ) q a12 ; a12 [0.24,0.75] and b12 [2.2, 2.37] P12 1 2 s( s b12 ) q a21 ; a21 [0.3, 0.001] and b21 [1.81, 2.3] P21 2 1 s( s b21 ) q a22 ; a22 [2.2, 2.5] and b22 [0.01,30] P22 2 2 s( s b22 )
7
4.3. Optimal design of PID controller A realistic definition of optimum in LTI systems is the minimization of the highfrequency loop gain k while satisfying the performance bounds. This gain affect the high-frequency response since lim[ L( j )] K ( j ) where is the excess of
poles over zeros assigned to L( j ) . Thus, only the gain K has a significant effect on the high-frequency response, and the effect of the other parameter uncertainty is negligible .It has been shown that if the optimum L ( j ) exists then it lies on the performance bounds at all i , and it is unique [14]. In this part we will introduce a simple algorithm for designing optimal PID controller. A PID controller has a transfer function; k (9) G pid ( s) k p i k d s s And is therefore completely defined by three terms k p (proportional gain),
k i (integral gain) and k d (derivative gain). Our method for designing optimal PID controller is based on designing a specific lag-lead compensator which transforms into PID controller under special conditions. Consider closed-loop system in Fig. 2 that G(s) is a below lag-lead compensator: 1 1 s s T1 T2 (10) G ( s) Ka a 1 s s T1 aT2 In order to achieve a PID controller let us move a towards infinity value, so we will have: KT1 1 1 lim G ( s) ( s ) ( s ) a s T1 T2 1 1 K 1 (11) ) ( ) KT1 s T1 T2 T2 s So we have a PID controller which is defined by: T T2 K kp K( 1 ) , k i , and k d KT1 T2 T2 Up to now, two real pole of lag-lead compensator have been specified, one located in infinity and the other in the origin. In order to define the PID controller in the next step of this algorithm, we must specify the gain K and situation of two zeros of G(s). But as we mentioned before, the optimum L ( j ) must exactly lies on its performance bounds at all frequency values ( i ). Therefore in the last phase of this algorithm we can find the suitable location of these two zeros by trial and error procedure using The Interactive Design Environment (IDE) of QFT [18]. As another alternative we can also use Genetic Algorithms in order to automate the placement of zeros therefore practically we can automate the design of robust optimal PID controller. Application of G.A. for automation of loop shaping is described in reference [19]. Under special circumstances using only one zero in the loop shaping phase will result in PI controller (consider the lag- compensator) and the PD controller will be resulted by the omission of the pole in the origin. G( s) KT1 (
8
4.4. Design of controllers using QFT At the first step we must define the plant uncertainty (template), thus the boundary of the plant templates have been computed and are shown in Fig. 6. Next having plant templates and required performance specifications we can compute robust performance bounds, which are shown in Fig. 7. Then by having robust performance bounds in the loop-shaping phase of design by applying the algorithm which was developed in part 4.3 we can design suitable PID controller .Figure 8 depicts the loop shaping of open loop transfer function. In both design you can observe that the nominal plants lie on their performance bounds which confirm the optimal design of robust controllers. According to the loop shaping phase optimal robust PID controllers are as below: 1.377 10 5 4 G 1 . 847 10 291.468s 1 s 3 G 2 1.490 10 4 2.466 10 208.242s s And the suitable pre-filters are: 1 f1 s s ( 1)( 1) 58.86 241.9 1 f 2 s s ( 1)( 1) 51.78 123.1 5. Analysis of design As we described before application of QFT to nonlinear system involves transformation of nonlinear system to its equivalent linear set of transfer functions thus in analysis phase of design we must run both linear and nonlinear simulation. 5.1. Linear simulation In this part we will investigate robust stability of closed-loop system and also tracking specifications in both time domain and frequency domain for all considered uncertainty of the robot dynamics. Frequency domain stability is shown in Fig.9. The frequency-domain closed-loop response is shown in Fig.10.And consequently the time-domain closed-loop response is shown in Fig.11. Hence according to linear simulation the robot has robust stability and also can satisfy tracking specifications. 5.2. Nonlinear simulation In the previous part we run linear simulation and get satisfactory result, but the main objective in applying QFT to nonlinear systems is to get satisfactory result through nonlinear simulation, thus in this section we will run nonlinear simulation. Block diagram of controlling system has been shown in Fig. 12. In this block diagram we used nonlinear dynamics of robot for nonlinear simulation. By moving the robot in different trajectories we will test the robustness of our design in tracking problem. For this purpose we will use 2 different paths while as a modeling error the robot
9
parameters will be increased by 20 % and then 50% respectively. The result of tracking problem for an elliptical path is shown in Fig .13 and the associated tracking error is also shown in Fig. 14. Figure .15 shows tracking problem of an arbitrary path while the tracking error is depicted in Fig. 16.According to this simulation the robust PID controller demonstrates excellent robustness properties and tracking ability.
Sum
teta1_d qd1
q1 plant of robot
qd2 teta2_d
PreFIlter
Sum1
Plant
Controller
teta1 q2 teta2
tt time
Clock
Fig.12 block diagram of controlling system As it is mentioned previously in the QFT design literature, an optimal design is a design which can satisfy all of its robust performance bounds while at the same time has the lowest possible high-frequency loop gain. Thus based on loop-shaping phase of design one can recognize the optimality of our design, but as another alternative for checking the optimality of design we change the PID gain and find out that the increase of gain will not improve the robot tracking ability while decreasing of gain will increase the tracking error Fig.17 and Fig. 18. Control efforts for tracking of elliptical and arbitrary paths also are shown in Fig.19. 6. Conclusion Due to the presence of uncertainty in the dynamics of robot arm manipulators the application of robust control methods for achieving high accuracy in tracking is inevitable. PID controllers have a wide range of applications in industrial systems, thus in this paper the robust control QFT was successfully implemented to design a robust PID controller for the robot arm manipulators. Since the application of robust control QFT for controlling robot arm manipulators is complicated, the design procedure introduced a new algorithm for designing an optimal robust PID controller for robot arm manipulators subject to QFT constraints. The basic design steps can be summarized as linearization of robot dynamics, design of suitable robust disturbance rejection bounds by minimization of Sensitivity Function, application of a specific lag-lead compensator which transforms into a PID controller under special conditions, linear simulation, and nonlinear simulation. Our research indicates that an increase of accuracy in tracking problem has a direct relationship with reduction of the cross-coupling effect between joints by designing suitable disturbance rejection bounds, reduction of settling time in tracking bounds for associated linear system, and improvement of associated linear uncertain system modeling. In the last part of this paper, robustness and optimality of our design is examined through nonlinear simulation. As the results indicate the robust PID controller has a good tracking ability while at the same time is robust against variation of robot dynamic and moving through different paths. It is also shown that 10
an increase of gain will not improve the robot’s tracking ability while decreasing the gain will increase the tracking error, thus the design is optimal. Although this algorithm has been applied to a 2 DOF robot, this method is applicable to robot arm manipulators in general. 7. References [1] J. Y. S. Luh,”An anatomy of industrial robots and their controls,” IEEE Trans.Automat. Conter, vol. AC-28, no. 2, pp.133-153, Feb.1983. [2] Lewis, F.L.; et.al. ”robotics” Mechanical Engineering Handbook .Ed. Frank Kreith .Boca Raton: CRC press LLC, 1999. [3] S. S. Ge, T. H. Lee and C. J. Harris,” Adaptive Neural Networks Control of Robotic Manipulators”, Vol. 19, World Scientific Publishing, 1998 [4] I. M. Horowitz. Synthesis of Feedback Systems. Academic Press, 1963. [5] I. M. Horowitz and M. Sidi. Synthesis of feedback systems with large plant ignorance for prescribed time domain tolerances. Int. J. Control, 16:287-309, 1972 [6] I. M. Horowitz and M. Sidi. Optimum synthesis of nonminimum phase feedback system with plant uncertainty. Int. J. Control, 27:361-386, 1978. [7] I. M. Horowitz. Quantitative Feedback Design Theory (QFT), volume 1. QFT Publications, 4470 Grinnel Ave., Boulder, Colorado 80303, USA, 1992. [8] C.H. Houpis. Quantitative Feedback Theory (QFT) For the Engineer: A Paradigm for the Design of Control Systems for Uncertain Nonlinear Plants. Wright Laboratory, 1995. [9] Yaniv O., Quantitative Feedback Design of linear and non-linear control systems, Kluwer Academic Publishers, Norwell, Massachussets, 1998. [10] N. Niksefat and N. Sepehri, Robust Force Controller Design for an Electrohydraulic Actuator Based on Nonlinear Model, in Proc.IEEE Conf. Robotics and Automation, Detroit, MI, pp.200-206,1999. [11] Horowitz, l, QUANTITATIVE FEEDBACK THEORY. QFT publication, 1993. [12] B. Golubev and I. M. Horowitz, Plant Rational Transfer function Approximation from Input-Output Data, International Journal of Control , Vol. 36,No. 4, pp. 711-723,1982. [13] Horowitz, I.: ‘Survey of quantitative feedback theory (QFT)’, Int. J. Control, 1991, 53, pp. 255–291 [14] Houpis, C.H., Rasmussen, S.J., and Garcia-Sanz, M.: ‘Quantitative feedback theory, fundamentals and applications’ (Marcel Dekker, New York, 2005, 2nd edn.)
11
[15] D’Azzo, J.J., and Houpis, C.H.: ‘Linear control system’ (McGraw-Hill, New York, 1995), pp. 580–737 [16] Horowitz, I.: ‘Quantitative feedback theory (QFT)’ (QFT Publications, Boulder CO, 1993), Vol. 1 [17] Ching cheng c.et.al," Quantitative Feedback Design of uncertain Multivariable Control systems, INT.J.CONTROL, VOL. 65, NO. 3, 537-553, 1996. [18] Borghesani, C., Chait, Y., and Yaniv, O., 1998, MatlabTM Quantitative Feedback Theory Toolbox, Mathworks Inc. [19] Min-Soo Kim, Chan-Soo Chung, Automatic Loop-Shaping of QFT Controllers Using GAs and Evolutionary Computation, Springer Berlin / Heidelberg, Volume 3809/2005
Plant Templates (parametric part w/o hardware)
Plant Templates (parametric part w/o hardware)
0.01
0.01 50
50
0
Magnitude (dB)
Magnitude (dB)
0.7
0.7
0
2 10
-50
2 10
-50 100
100 -100
-100 1000
1000 -150 -350
-300
-250
10000 -200 -150 Phase (degrees)
-150
-100
-50
-350
0
-300
-250
10000 -200 -150 Phase (degrees)
-100
-50
0
A) Joint one B) Joint two Fig. 6 The boundary of the plant templates Intersection of Bounds
Intersection of Bounds 120
=0.01
200
=0.7
100 80
150
=2
=0.7
100
Magnitude (dB)
Magnitude (dB)
60
=2 50
=10 0
=10
40 20
=100 0 -20
>=10000
=100
=1000
=>10000
-40
-50
=1000 -350
-300
-250
-200 -150 Phase (degrees)
-100
-50
-60
0
-350
12
-300
-250
-200 -150 Phase (degrees)
-100
-50
0
A) Joint one B) Joint two Fig. 7 Robust performance bounds
A) Joint one
B) Joint two Fig. 8 Loop shaping
Weight: --
Weight: -0
0
-50
-50
-100 Magnitude (dB)
Magnitude (dB)
-100
-150
-150
-200
-200 -250
-250 -300 -1
10 -2
10
0
10
2
10 Frequency (rad/sec)
4
0
10
1
10
6
10
10
A) Joint one
2
10
3
4
5
10 10 Frequency (rad/sec)
6
10
7
10
10
B) Joint two Fig. 9 Robust stability Weight: --
0
0
-50
-50
-100
-100 Magnitude (dB)
Magnitude (dB)
Weight: --
-150
-150
-200
-200
-250
-250
-300
-300 -2
10
0
10
2
10 Frequency (rad/sec)
4
10
6
-1
10
10
13
0
10
1
10
2
10
3
4
10 10 Frequency (rad/sec)
5
10
6
10
7
10
A) Joint one B) Joint two Fig. 10 Frequency domain response
1.2
1.2
1
1
0.8
0.6
Y(t)
Acceptable Output Bounds
Controller Output
0.4
Controller Output
0.2
0.02
0.04
0.06
0.08
0.1 time (s)
0.12
0.14
0.16
0.18
0
0.2
0
0.02
0.04
0.06
A) Joint one
0.08
0.1 time (s)
0.12
B) Joint two Fig. 11 Time domain response
Elliptical path with zero condition 0.1
0.05
Y : meter
0
0
-0.05
-0.1
Tracking path Desired path
-0.15 0.4
0.5
0.6
0.7 X : meter
0.8
0.9
1
Fig. 13 Tracking problem for an elliptical path -4
15
Error (rad) in joint ne
0
Acceptable Output Bounds
0.6
0.4
0.2
x 10
Original 20 % variation 50 % variation
10 5 0 -5
0
1
2
3 Time (s)
4
5
6
-4
15
Error (rad) in jint two
Y(t)
0.8
x 10
Original 20 % variation 50 % variation
10 5 0 -5
0
1
2
3 Time (s)
14
4
5
6
0.14
0.16
0.18
0.2
Fig. 14 Joint Error for variation of system parameters Arbitrary path with zero conition 0.4 Tracking path Desired path
0.35 0.3 0.25
Y : meter
0.2 0.15 0.1 0.05 0 -0.05 -0.1
0.4
0.5
0.6
0.7
0.8
0.9
1
X : meter
Fig. 15 Tracking problem for an arbitrary path -3
x 10
Error (rad) in joint one
4 2 0
Original 20 % variation 50 % variation
-2 -4
0
1
2
3 Time (s)
4
5
6
-3
x 10
Error (rad) in joint two
2 1 0
Original 20 % variation 50 % variation
-1 -2
0
1
2
3 Time (s)
4
5
6
Fig. 16 Joint Error for variation of system parameters Elliptical path with zero condition 0.15 Tracking path Desired path 0.1
Y : meter
0.05
0
-0.05
-0.1
-0.15
-0.2
0.4
0.5
0.6
0.7
0.8
0.9
1
X : meter
Fig. 17 Decreasing of gain will increase the tracking error (elliptical path)
15
Arbitrary path with zero conition 0.4 Desired path Tracking path
0.35 0.3 0.25
Y : meter
0.2 0.15 0.1 0.05 0 -0.05 -0.1
0.4
0.5
0.6
0.7
0.8
0.9
1
X : meter
Actuator force (Nm) in oint two
Actuator force (Nm) in Joint one
Fig. 18 Decreasing of gain will increase the tracking error (arbitrary path) 100 Elliptical path Arbitrary path
50 0 -50 -100
0
1
2
3 Time (s)
4
5
6
30 Elliptical path Arbitrary path
20 10 0 -10 -20
0
1
2
3 Time (s)
4
5
Fig. 19 Actuator force for tracking problem
16
6
