- Email: [email protected]
Performance Evaluation of Manipulators with Links of Different Cross Sections
Available online at www.sciencedirect.com
ScienceDirect Materials Today: Proceedings 4 (2017) 1778–1787
www.materialstoday.com/proceedings
5th International Conference of Materials Processing and Characterization (ICMPC 2016)
Performance Evaluation of Manipulators with Links of Different Cross Sections B. Naveena, *, B. K. Routb a Student, Dept. of Mechanical Engg., BITS Pilani, Pilani-333031,India Associate. Professor, Dept. of Mechanical Engg., BITS Pilani, Pilani-333031,India
b
Abstract This manuscript identifies the cross section with best positional performance of a arm at the target by using cross section induced dynamic model. The intent here is to obtain high accuracy at the target, and relax the sophistication and effort required to design a sophisticated controller. To illustrate the application, a manipulator with a rotary joint and 1-DOF with different cross sections are compared and here mass as well as same area of cross section are assumed to same. A probabilistic worst-case technique is used to perform the numerical simulations as an alternative to computationally intensive Monte Carlo simulation technique. Finally, different performance measures from arm with different cross sections are used for comparison and results are summarized. ©2017 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of Conference Committee Members of 5th International Conference of Materials Processing and Characterization (ICMPC 2016). Keywords:Link cross section; Dynamic Model; Probabilistic worst-case technique; Most probable distance; Area of spread
1. Introduction Manipulators form a major part of robots to perform high precision tasks like electronic chip assembly, tele operated surgery and high precision manufacturing operations etc. The manipulator’s end effector is expected to operate within safe limits at the target position to avoid accidents due to deviation in output. These are expected to have high accuracy and repeatability in the end effector’s pose [1] which can be obtained by using high-end closedloop control systems which prove to be costly. Instead, choosing a design which relaxes the control requirements is a superior technique. Every engineering system has parameters that are impacted by uncertainties due to manufacturing, loading or end conditions etc. which are inherent. These can be modelled using approaches like probabilistic, fuzzy and interval
* Corresponding author. Tel.: +91-9413891983. E-mail address: [email protected] 2214-7853©2017 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of Conference Committee Members of 5th International Conference of Materials Processing and Characterization (ICMPC 2016).
B. Naveen and B. K. Rout / Materials Today: Proceedings 4 (2017) 1778–1787
1779
analysis [2]. A common approach in modelling uncertainties is probabilistic approach whose theoretical base is the central limit theorem. With the knowledge of parameter mean and its tolerance, it can be modeled as a normally distributed random variable [3]. Due to presence of parameter uncertainties the positional and directional motion performed by the robot are affected. As a result, the actual motion of the manipulator deviates from the desired output intended by the designer. Hence, a careful analysis of the effect of uncertainties on the manipulator’s kinematics and dynamics is necessary. The deviation between actual and planned positions of the endeffector is referred to as end effector error in this paper. Shi and Wuhan [4] developed a probabilistic approach to evaluate manipulator accuracy and reliability by considering the uncertainties in kinematic parameters. Veitschegger, and Wu [5] developed a non-linear error model of end effector’s pose due to error in DH parameters. The influence of link flexibility on the kinematic reliability of the manipulator has also been done [6]. Randomness in link lengths and joint clearances have been considered to evaluate the kinematic reliability [7]. Wu and Rao [2] have considered the problem of allocating optimum tolerance values to the joints by minimizing the positional error at the endeffector while satisfying a set of design constraints. These studies were limited to evaluating the impact of error in kinematic parameters on endeffector error in terms of position only. Further, the impact of a manipulator’s DH parameters on the dynamic performance was studied by evaluating the sensitivity of inertia matrix eigenvalues [8]. Rao and Bhatti [9] developed probabilistic kinematic and dynamic models for a serial manipulator by proposing kinematic and dynamic reliability parameters. The problem of finding the optimal tolerance values and tolerance ranges of kinematic as well as dynamic parameters and significant parameters identification has been performed by Rout & Mittal [3, 10, 11]. All the above studies focused on evaluating or optimizing the tolerances and errors of the kinematic and dynamic parameters. But, Rout and Mittal also obtained [12] values of design parameters of a manipulator which has minimal performance variations and insensitive to the effect of noise factors. The error of the endeffector’s pose is a result of error in the joint torque, joint friction, joint clearances, flexibility of the links, manufacturing tolerances and assembly errors of links and joints and even the environmental conditions. All of these in a way affect the kinematic and dynamic parameters of the manipulator. However, the error due to some of the factors can be systematically modelled and the effect on the performance can be reduced by calibration [13]. In these studies,dynamic models are developed the using a lumped mass system for the links of the manipulator and didn’t consider the cross section. One of the possible reasons for ignoring the cross section is the computational complexity involved with the inertia matrix, which is used in dynamic model of the manipulator. Another direction of research was dynamic performance evaluation and its optimization [14] where uncertainties in operational point at the end effector were considered. A considerable amount of work is done on evaluating the impact on end effector and optimizing parameters and their tolerance values. The focus was to optimize either the pose error or the manufacturing cost or the dynamic performance. All these models considered the link as a lumped mass system and ignored the effect of cross section on performance. Current work focuses on the cross-sectional parameters in evaluating their impact on positional performance at the destination point. 2. Parametric modelling of the manipulator A rotary joint with a link of commonly used cross sections is modelled as 1-DOF robot arm. To investigate the impact of cross section on performance, six commonly used cross sections are considered. Those are solid and hollow circular, rectangular, and trapezoidal cross sections. Thus, suitable kinematic and dynamic models are necessary to introduce the effect of cross section. After the appropriate dynamic models are formulated, effects of uncertainties are incorporated on dynamic parameters and the torque, to simulate the performance. The method for parameter uncertainty modelling is available and discussed in following section. 2.1. Kinematic modelling The mapping between the joint space and task space is performed by kinematics and modelled using the DH convention. The frame assignment is shown in fig. 1, based on which the DH parameters are written.
1780
B. Naveen and B. K. Rout/ Materials Today: Proceedings 4 (2017) 1778–1787
Fig. 1. 1-DOF manipulator with the frames and few inertial parameters.
In this manipulator, payload (mp) is acting at the end effector.The frames are assigned at the centroid of the cross sections at both the ends, so that it helps in computation of simplified inertia tensor as shown below Table 1. Table 1. DH parameters for 1-DOF robot shown in Fig. 1.
ai
Link i
l
1.
αi 0
di 0
θi θ1
The transformation matrix describing thekinematic modelling is C1 S 0 T1 = 1 0 0
− S1 C1 0 0
0 lC1 0 lS 1 1 0 0 1
(1)
where,C1 is cos(θ1) and S1 is sin(θ1). 2.2. Dynamic modelling The motion of the manipulator under the action of external force and torque is studied. Dynamic model is formulated with the inertia matrix involving the cross sectional parameters which is one of the defining aspects of this study. The inertia matrix for a generic cross section is as follows[17] x 2 dm ∫ xydm I = ∫ ∫ xzdm xdm ∫
∫ xydm ∫ y dm ∫ yzdm ∫ ydm 2
∫ xzdm ∫ xdm ∫ yzdm ∫ ydm ∫ z dm ∫ zdm ∫ zdm ∫ dm + m 2
p
(2)
where, mpis themass of the payload. Payload is assumed to be a point mass which is located at the end effector. So, it effectively contributes only to the last term in the matrix. Since, the frames are at the centroids for all the cross sections,
∫ xydm = ∫ yzdm = ∫ xzdm = ∫ ydm = ∫ zdm = 0
(3)
Since, it is a 1-DOF manipulator, the Coriolis term is zero and, hence, it is ignored in the dynamic model in (4). •• (4) τ 1 = M 11 θ 1 + G1
Where M11 is the inertialterm and G1 is the gravitational term.
B. Naveen and B. K. Rout / Materials Today: Proceedings 4 (2017) 1778–1787
1781
Fig. 2. Different cross sections of single link manipulator (a) Solid cylinder (b) Hollow cylinder (c) Solid rectangle (d) Hollow rectangle (e) Solid trapezoid (f) Hollow trapezoid.
Different cross sections of single link manipulators are shown in fig 2. 2.3. Parameter uncertainty modelling To model the inaccuracies and uncertainties in the parameters of the manipulator, probability distributions are necessary. In this case, each parameter is assumed to follow a Gaussian distribution with known mean and standard deviation [11]. In addition to the above, the dimensional tolerances which are available from the manufacturer and design data book are used to assume the mean and standard deviation as shown below Table 2. Table 2. Dimensions of the cross sectional parameters. Cross section
Dimensions (in m)
Solid cylindrical
r1 – 0.02
Hollow cylindrical
r2 – 0.0150; t 2 – 0.01
Solid rectangular
a3 – 0.04; b4 – 0.03142
Hollow rectangular
a4 – 0.04; b4 – 0.0527; t 4 – 0.006
Solid trapezoidal
a5 – 0.06; b5 – 0.03; h5 – 0.0279
Hollow trapezoidal
a6 – 0.05; b6 – 0.025; h6 – 0.017; t 6 – 0.008
The tolerance in all the parameters is taken to be as 0.1% of their mean dimension. The effect of uncertainties in torque is modelled using the tolerance specified for the motor’s torque. In this case, the worst-case scenario of the torque at a particular instant is used as an input to the model. The tolerance specified on the pull out torque is used for this case and taken to be 0.2% of it. 3. Methodology To simulate the performance and capture worst-case performance, an algorithm of the manipulator is discussed. The joint space trajectory planning, inverse dynamics and forward dynamics approach of manipulator with the proposed algorithm, the end effector’s final position is simulated. Later, the simulated values are consolidated into different performance measures.
B. Naveen and B. K. Rout/ Materials Today: Proceedings 4 (2017) 1778–1787
1782
In the quintic trajectory time law for joints, acceleration is smooth and controllable. Hence, it has been selected for the purpose of investigation. This trajectory is also widely used in the industrial manipulators. The trajectory for position, velocity and acceleration are as follows θ = k1 + k 2t + k3t 2 + k 4t 3 + k5t 4 + k 6t 5 (5) • dθ θ= = k 2 + 2k 3t + 3k 4 t 2 + 4k 5 t 3 + 5k 6 t 4 (6) dt
•
dθ θ= = 2k3 + 6k 4t + 12k5t 2 + 20k6t 3 dt ••
(7)
The constants are obtained by applying the boundary conditions as given in (8). These equations are solved to obtain the values of the constants. •• •• • • (8) θ (t ) = 0;θ (t ) = θ , θ (t ) = 0; θ (t ) = 0 , θ (t ) = 0;θ (t ) = 0 0
f
f
0
f
0
f
3.1. Simulation technique using worst-case approach Implementing the effect of uncertainties in parameters and performing the Monte Carlo simulation on a nonlinear dynamic model involving transcendental functions is a computationally intensive task. Thus, a modification in the form of worst-case technique is implemented for the purpose of numerical simulations [10]. It choses different combinations of worst cases for the parameters as inputs to the model to produce the outputs. Suppose, a parameter ‘x’ is affected by noise whose mean is μaand standard deviation is σa, in Monte Carlo simulation technique, Gaussian function is formulated using μa, σa and points are randomly picked from the distribution as inputs. Whereas, in worst-case technique μa±3σaare selected as the inputs to the dynamic model. In case of multiple inputs, all the combinations of the worst cases are used as inputs. So, in a model with ‘n’ parameters the number of the inputs will be only 2n. In this work, the maximum number of inputs were only 64, which is for hollow trapezoidal cross section, which has six parameters. Whereas,to capture the complete scenario by performing Monte Carlo simulation, generally the number of inputs are in the order of thousands [9]. If the same Monte Carlo simulation is performed with 64 inputs the exact scenario may or may not be captured. But in the worst-case technique, it captures the complete scenario although giving conservative results. It is because higher number of worst outputs will be produced by these worstcase inputs. Whereas, in the Monte Carlo simulation with a few inputs, these many worst outputs might not be captured. The outputs from worst-case inputs are then used to plot the performance ellipse. Further, in the field of state estimation, Unscented Kalman Filter (UKF) is used for non-linear dynamic models to estimate the state vector. Sigma points are selected, based on a few scaling parameters, and the approximated outputs are evaluated based of certain calculated weights [15]. Whereas, in this work we directly choose the worstcase inputs and pass them into the dynamic model.These worst-case dimensional parameters and control inputs are passed into dynamic model and the numerical forward dynamic simulations are performed on MATLAB using the ode45 routine. Thus obtained end effector positions for a particular cross section are used to evaluate its performance based on the proposed performance parameters. 3.2. Methodology The steps used to perform the simulations and obtain the results are explained in this section. Step 1: Develop kinematic model of specified manipulator Step 2: Formulate the Inertia matrix and the dynamic model Step 3: Assume equal cross sectional area, evaluate the cross sectional parameters Step 4: Use trajectory planning in the joint space to obtain properties of motion Step 5: Use inverse dynamics approach to evaluate the joint torque required for the planned trajectory Step 6: Use forward dynamic approach to simulate the performance using worst-case scenario. Step 7: Evaluate the most probable distance and the area of spread Step 8: Evaluate the sum of inertia values and the sensitivity of the eigenvalues of the inertia matrix The procedure used in steps 7-8 are explained in subsequent section.
B. Naveen and B. K. Rout / Materials Today: Proceedings 4 (2017) 1778–1787
1783
4. Performance measures This investigation is intended to identify the cross section which provides performance close to the target having least spread (variations) of the end effector at the destination. For this purpose, measures like most probable distance and area of spread are utilized and the focus here is to capture the variability in performance. 4.1. Most probable distance and area of spread The performance measures chosen should accurately depict the scenario. Researchers used various measures like positional error [11], S/N ratio [3], kinematic and dynamic reliabilities [8], Euclidean Norm of the inertia matrix eigenvalue derivatives etc. [7].Mean positional error considers only the distance of the point, and the actual position is immaterial to it. Even, Signal to Noise ratio (S/N) is a function of mean positional error. So both of them have the same drawback. Whereas, in this problem the actual position of the end effector changes with cross section. Even if the mean positional error is same for two different cross sections, their performance is actually different because the end effector’s finalpositions are different. Hence, performance is evaluated using the error in positional coordinates at the target point which is termed as Most Probable Distance (MPD). The mean of all the points reached by the end effector is called as Most Probable Point (MPP). So, with the given tolerances in the parameters and joint torque, the manipulator is most probable to reach the MPP instead of the intended target point. The distance of MPP from the target point is the MPD. For the ith experiment, the error in the positional coordinates is given by ∆xi = xi − xt ; ∆yi = yi − yt ; ∆zi = zi − zt (9) where the subscript ‘t’ denotes target. The mean error in the positional coordinates, is given by
1 N 1 N 1 N ∆xi ; ∆y = ∑ ∆y i ; ∆z = ∑ ∆z i ∑ N i =1 N i =1 N i =1 A positive and negative values indicate overshoot and undershoot in the target coordinates respectively. The MPP and MPD (μ) at the end effector is given by MPP = xt + ∆x, yt + ∆y, zt + ∆z ∆x =
(
)
2
2
(10)
(11)
2
(12) MPD( µ ) = ∆x + ∆y + ∆z The spread of the data around the MPP is found using variance and covariance of the positional coordinates.
σ xx ( x, x) σ xy ( x, y ) σ xz ( x, z ) Cov( x, y, z ) = σ yx ( y, x) σ yy ( y, y ) σ yz ( y, z ) σ zx ( z , x) σ zy ( z , y ) σ zz ( z , z )
a = sλ1 ;
b = sλ2 ; c = sλ3
(13)
(14)
Where, a, b, c are the length of the axes of the performance ellipsoid, λ1, λ2, λ3 (λ1> λ2 > λ3)are the Eigenvalues of the covariance matrix.For a value of s = 3, the performance ellipse just enclosed all the worst outputs for all the cross sections. To consolidate the error-spread at the end effector into a single parameter, the volume of performance ellipsoid (V) has been used. In case of 2-D the area of ellipse (A) will be appropriate. 4 V = πabc ; A = πab (15) 3 In general, for a matrix the determinant is equal to the product of its eigenvalues. So the above area of spread can also be interpreted as a scaled determinant of the covariance matrix. Also, for optimal design, the determinant of the covariance matrix is minimized, which is termed as d-optimality criterion, to reduce the effect of noise and extract as much information as possible from the experiments [16]. In this case also, the covariance matrix with lower determinant is the one with better performance in terms of area of spread.The pink dots in Fig. 3 represents individual performance of worst-case simulation. The minor axis of the performance ellipse is very small compared to the major axis because the tolerance along the minor axis direction is negligible compared to its perpendicular
1784
B. Naveen and B. K. Rout/ Materials Today: Proceedings 4 (2017) 1778–1787
direction. Along the major axis all the geometric parameters as well as torque contribute whereas in the minor axis direction only the tolerance due to the length is the contributor and the effect of other parameters is negligible. The ellipse is expected to be broader along the minor axis direction in case of manipulator with multiple links, since all the parameters will effect both the directions.
Fig. 3. The performance ellipse and the end effector positions.
4.2. Sensitivity analysis To get an insight into the performance results, a sensitivity analysis of the inertia matrix was performed. Each inertia matrix can be uniquely characterized by their Eigenvalues and Eigenvectors. Hence, the sum of Eigenvalues is computed and their sensitivity analysis w.r.t. the dimensional parameters is performed.Tr is the trace of the inertia tensor matrix which is also the sum of the eigenvalues. The numerical value of I11and I44 are same for all the cross sections. Hence, the sum of I22 and I33values give an understanding on the mass distribution. TI = I 22 + I 33 (16) It is intuitive to understand that for higher value of TI higher value of torque is required; consequently, higher will be the error in supplied torque. Furthermore, the effect of dimensional tolerances of the inertia matrix is due to all the terms of the inertia matrix. Let the sum of eigenvalues(K) in parametric form beK = f (a1,a2….an), where ai is the dimensional parameter [16]. n ∂f ∆ai ∂f = ∑ ai (17) ∂ai ai i =1 As the coefficient of tolerance to parameter ratio varies, its effect on the functional error also varies. The functional error is expected to be as close to zero as possible. Higher the magnitude of these coefficients higher is the effect on the output due to the tolerance to parameter ratio. Hence the vector length (L) consisting of the coefficients will determine the effect of parameter tolerance on the output error. ∂f L = ∑ ai ∂ai i =1 n
2
(18)
The interaction between the torque and dimensional parameters is not integrated in the eigenvalue analysis. Hence, it is not used for primary analysis and only used to get a physical interpretation.
B. Naveen and B. K. Rout / Materials Today: Proceedings 4 (2017) 1778–1787
1785
5. Results and discussion The selected arm is used to perform tasks with gravity loading and payload. However, to identify exclusively the effect of inertial parameters on the performance, the payload and the gravity were removed.Hence, to get a complete understanding, for the same initial and final positions the performance parameters are evaluated for fast and slow motions with payload, gravity, and without payload& gravity.The following observations are obtained from the simulation results of cases in Table 3 available in fig. 4. Important findings are (a) Solid rectangular and solid cylindrical have the worst MPD and worst area of spread respectively in all the cases; (b) Hollow cylindrical has the least MPD 3 out of 4 times and on the other occasion it is close to minimum; (c) Solid trapezoidal has the least area of spread 3 out of 4 times and very close to minimum in the remaining case Table 3. The four different cases used for the analysis of the manipulator (With effect of gravity and without it).
θ i (rad)
θf
1.
0
2.
Total time of
Payload mass
Gravity
motion (sec)
(kg)
(m/s2)
π/3
0.5
5
9.81
0
π/3
2
5
9.81
3.
0
π/3
0.5
0
-
4.
0
π/3
2
0
-
Case No.
(rad)
In cases 3-4, the MPD for cylindrical cross section is least and solid rectangular is highest with others following the same pattern in both the cases. Apart from the solid cylindrical section which has the highest area of spread, other cross section have not much difference in both the cases.This leads to a result that hollow cylindrical cross section has a better performance compared to other cross section in terms of both closeness to the target and area of spread around the target without the payload and gravity. Further, without gravity and payload, when the task time was increased by four times, the MPD and the area of spread of different cross sections have not changed much and their ratios lie close to one. Whereas, under the action of gravity and with payload, for different speeds, by the same increase in the task time, the MPD and area of spread have increased in the ranges of hundreds and seventy thousand respectively. It clearly shows that the performance is dominantly effected by the gravity and payload as well as their coupling with the velocity of the end effector. *The cross section index mapping to cross section is: 1 – Solid cylindrical, 2 – Hollow cylindrical, 3 – Solid rectangular, 4 – Hollow rectangular, 5 – Solid trapezoidal, 6 – Hollow trapezoidal Practically, manipulators are expected to work under the gravity and with payload. It is evident from the results of case 2 that the MPD and area of spread is very high as compared to that of case 1. Under gravity, slow motion of the manipulator leads to a worst performance where the MPD and area of spread have greatly increased compared to fast motion. But the pattern of MPD is almost the same in both the cases with cylindrical sections having least and solid rectangular section the highest. But the same pattern is not followed in case of without gravity and payload. This shows that the effect of gravity and payload is different on different cross sections. It is because different dimensional tolerances impact differently to the error in the mass of the link and hence the gravitational force effect changes leading to different performances. These observations are obtained from the results in fig. 5.
1786
B. Naveen and B. K. Rout/ Materials Today: Proceedings 4 (2017) 1778–1787
Fig. 4. MPD vs cross section (on left) and area of error ellipse vs cross section (on right) for four cases starting from case 1*.
Fig. 5. Sum of inertias vs cross section (on left) and vector length vs cross section (on right)*.
B. Naveen and B. K. Rout / Materials Today: Proceedings 4 (2017) 1778–1787
1787
The important findings are: (a) solid cross sections have lower inertia values compared to their hollow counterparts which is an expected; (b) hollow rectangular and solid cylindrical have the highest and least sum of inertia values respectively; (c) solid trapezoidal has the smallest vector length, while it is highest for solid cylindrical.From the figs. 4-5, it can be deduced that solid trapezoidal cross section with least vector length has least area of spread in most of the times and the hollow cylindrical cross section with medium range inertia value has the least MPD. It can also be observed that the vector length for hollow cylindrical section is close to minimum and the area of spread also is close to the minimum most of the times. 6. Conclusion This papers uses commonly available cross sections for the links of a 1-DOF with rotary joint.Uncertainties in the dimensional parameters and input torque was introduced in the deterministic dynamic model. Since the dynamic model is non-linear, a worst-case technique has been implemented instead of Monte Carlo simulations.Two parameters which consolidates the performance were suggested, which are mean positional distance (MPD) and the area of spread. The performance wasevaluated and analyzed under different working conditions for different speeds and with gravity and payload and without them. It can be concluded that the hollow cylindrical section has the least MPD with area of spread close to the minimum which is attributed to medium range inertia value and a low vector length. The solid trapezoid sectionwith minimum vector length has the least area of spread. Based on the requirements of the application, appropriate cross section for the link can be chosenby user which gives the least MPD or area of spread. References [1] B. K. Rout & R. K. Mittal, “Tolerance design of manipulator parameters using design of experiment approach.” Struct Multidisc Optim (2007), pp. 445–462. [2] Weidong Wu, S.S. Rao, “Uncertainty analysis and allocation of joint tolerances in robot manipulators based on interval analysis.” Reliability Engineering and System Safety 2007, 92, pp. 54–64. [3] B. K. Rout, R.K. Mittal, “Tolerance design of robot parameters using Taguchi method.” Mechanical Systems and Signal Processing 20 (2006) pp. 1832–1852. [4] Zhongxiu Shi, Technology Wuhan, “Reliability Analysis & Synthesis of Robot Manipulators.” 1994 Proceedings Annual Reliability and Maintainability Symposium, pp. 201 – 205. [5] W.K. Veitschegger, Chi-Haur Wu, “Robot Accuracy Analysis Based on Kinematics.” IEEE Journal of Robotics and Automation, Vol. RA-2, No. 3, September 1986, pp. 171 -179. [6] Tong Li, Qingxuan Jia, Gang Chen, Hanxu Sun, Jian Zhang, “Kinematics Reliability Analysis for Manipulator Considering Elasticity.” IEEE 9th Conference on Industrial Electronics and Applications (ICIEA), 2014. [7] Mahesh D. Pandey, Xufang Zhang, “System reliability analysis of the robotic manipulator with random joint clearances.” Mechanism and Machine Theory (2012), Vol.58, pp.137–152. [8] Ibrahim, M. Y., Cook, C. D. & Tieu, A. K., “Effect of a robot's geometrical parameters on its optimal dynamic performance.” Proceedings IEEE International Conference on Intelligent Control and Instrumentation, Singapore, 1992, pp. 820-825. [9] S.S. Rao, P.K. Bhatti, “Probabilistic approach to manipulator kinematics and dynamics.” Reliability Engineering and System Safety (2001), Vol. 72, pp. 47-58. [10] B.K. Rout, R.K. Mittal, “Optimal manipulator parameter tolerance selection using evolutionary optimization technique.” Engineering Applications of Artificial Intelligence 2008, Vol.21, 509–524. [11] B.K. Rout, R.K.Mittal, “Screening of factors influencing the performance of manipulator using combined array design of experiment approach.” Robotics and Computer-Integrated Manufacturing 25 (2009) pp. 651–666. [12] B.K. Rout, R.K. Mittal, “Optimal design of manipulator parameter using evolutionary optimization techniques.” Robotica (2010) Vol. 28, pp. 381–395. [13] C. Mavroidis, S. Dubowsky, P. Drouet, J. Hintersteiner and J. Flanz, “A Systematic Error Analysis of Robotic Manipulators: Application to a High Performance Medical Robot.” Proceedings of the 1997 International Conference in Robotics and Automation, Albuquerque, NM. [14] Alan P. Bowling, John E. Renaud, Jeremy T. Newkirk, Neal M. Patel, Harish Agarwal, “Reliability-Based Design Optimization of Robotic System Dynamic Performance.” Journal of Mechanical Design, APRIL 2007, Vol. 129, pp. 449 – 454. [15] Eric A Wan, Rudolph van der Merve, “The Unscented Kalman Filter,” in Kalman Filtering and Neural Networks, Simon Haykin, John Wiley & Sons, Ltd., Publication, 2001. [16] Peter Goos, Bradley Jones, “An optimal screening experiment” in Optimal Design of Experiments: A Case Study Approach, John Wiley & Sons, Ltd., Publication, 2011. [17] R.K. Mittal, I.J. Nagrath, Robotics and Control, Tata McGraw Hill Publication, New Delhi, 2003.
ScienceDirect Materials Today: Proceedings 4 (2017) 1778–1787
www.materialstoday.com/proceedings
5th International Conference of Materials Processing and Characterization (ICMPC 2016)
Performance Evaluation of Manipulators with Links of Different Cross Sections B. Naveena, *, B. K. Routb a Student, Dept. of Mechanical Engg., BITS Pilani, Pilani-333031,India Associate. Professor, Dept. of Mechanical Engg., BITS Pilani, Pilani-333031,India
b
Abstract This manuscript identifies the cross section with best positional performance of a arm at the target by using cross section induced dynamic model. The intent here is to obtain high accuracy at the target, and relax the sophistication and effort required to design a sophisticated controller. To illustrate the application, a manipulator with a rotary joint and 1-DOF with different cross sections are compared and here mass as well as same area of cross section are assumed to same. A probabilistic worst-case technique is used to perform the numerical simulations as an alternative to computationally intensive Monte Carlo simulation technique. Finally, different performance measures from arm with different cross sections are used for comparison and results are summarized. ©2017 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of Conference Committee Members of 5th International Conference of Materials Processing and Characterization (ICMPC 2016). Keywords:Link cross section; Dynamic Model; Probabilistic worst-case technique; Most probable distance; Area of spread
1. Introduction Manipulators form a major part of robots to perform high precision tasks like electronic chip assembly, tele operated surgery and high precision manufacturing operations etc. The manipulator’s end effector is expected to operate within safe limits at the target position to avoid accidents due to deviation in output. These are expected to have high accuracy and repeatability in the end effector’s pose [1] which can be obtained by using high-end closedloop control systems which prove to be costly. Instead, choosing a design which relaxes the control requirements is a superior technique. Every engineering system has parameters that are impacted by uncertainties due to manufacturing, loading or end conditions etc. which are inherent. These can be modelled using approaches like probabilistic, fuzzy and interval
* Corresponding author. Tel.: +91-9413891983. E-mail address: [email protected] 2214-7853©2017 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of Conference Committee Members of 5th International Conference of Materials Processing and Characterization (ICMPC 2016).
B. Naveen and B. K. Rout / Materials Today: Proceedings 4 (2017) 1778–1787
1779
analysis [2]. A common approach in modelling uncertainties is probabilistic approach whose theoretical base is the central limit theorem. With the knowledge of parameter mean and its tolerance, it can be modeled as a normally distributed random variable [3]. Due to presence of parameter uncertainties the positional and directional motion performed by the robot are affected. As a result, the actual motion of the manipulator deviates from the desired output intended by the designer. Hence, a careful analysis of the effect of uncertainties on the manipulator’s kinematics and dynamics is necessary. The deviation between actual and planned positions of the endeffector is referred to as end effector error in this paper. Shi and Wuhan [4] developed a probabilistic approach to evaluate manipulator accuracy and reliability by considering the uncertainties in kinematic parameters. Veitschegger, and Wu [5] developed a non-linear error model of end effector’s pose due to error in DH parameters. The influence of link flexibility on the kinematic reliability of the manipulator has also been done [6]. Randomness in link lengths and joint clearances have been considered to evaluate the kinematic reliability [7]. Wu and Rao [2] have considered the problem of allocating optimum tolerance values to the joints by minimizing the positional error at the endeffector while satisfying a set of design constraints. These studies were limited to evaluating the impact of error in kinematic parameters on endeffector error in terms of position only. Further, the impact of a manipulator’s DH parameters on the dynamic performance was studied by evaluating the sensitivity of inertia matrix eigenvalues [8]. Rao and Bhatti [9] developed probabilistic kinematic and dynamic models for a serial manipulator by proposing kinematic and dynamic reliability parameters. The problem of finding the optimal tolerance values and tolerance ranges of kinematic as well as dynamic parameters and significant parameters identification has been performed by Rout & Mittal [3, 10, 11]. All the above studies focused on evaluating or optimizing the tolerances and errors of the kinematic and dynamic parameters. But, Rout and Mittal also obtained [12] values of design parameters of a manipulator which has minimal performance variations and insensitive to the effect of noise factors. The error of the endeffector’s pose is a result of error in the joint torque, joint friction, joint clearances, flexibility of the links, manufacturing tolerances and assembly errors of links and joints and even the environmental conditions. All of these in a way affect the kinematic and dynamic parameters of the manipulator. However, the error due to some of the factors can be systematically modelled and the effect on the performance can be reduced by calibration [13]. In these studies,dynamic models are developed the using a lumped mass system for the links of the manipulator and didn’t consider the cross section. One of the possible reasons for ignoring the cross section is the computational complexity involved with the inertia matrix, which is used in dynamic model of the manipulator. Another direction of research was dynamic performance evaluation and its optimization [14] where uncertainties in operational point at the end effector were considered. A considerable amount of work is done on evaluating the impact on end effector and optimizing parameters and their tolerance values. The focus was to optimize either the pose error or the manufacturing cost or the dynamic performance. All these models considered the link as a lumped mass system and ignored the effect of cross section on performance. Current work focuses on the cross-sectional parameters in evaluating their impact on positional performance at the destination point. 2. Parametric modelling of the manipulator A rotary joint with a link of commonly used cross sections is modelled as 1-DOF robot arm. To investigate the impact of cross section on performance, six commonly used cross sections are considered. Those are solid and hollow circular, rectangular, and trapezoidal cross sections. Thus, suitable kinematic and dynamic models are necessary to introduce the effect of cross section. After the appropriate dynamic models are formulated, effects of uncertainties are incorporated on dynamic parameters and the torque, to simulate the performance. The method for parameter uncertainty modelling is available and discussed in following section. 2.1. Kinematic modelling The mapping between the joint space and task space is performed by kinematics and modelled using the DH convention. The frame assignment is shown in fig. 1, based on which the DH parameters are written.
1780
B. Naveen and B. K. Rout/ Materials Today: Proceedings 4 (2017) 1778–1787
Fig. 1. 1-DOF manipulator with the frames and few inertial parameters.
In this manipulator, payload (mp) is acting at the end effector.The frames are assigned at the centroid of the cross sections at both the ends, so that it helps in computation of simplified inertia tensor as shown below Table 1. Table 1. DH parameters for 1-DOF robot shown in Fig. 1.
ai
Link i
l
1.
αi 0
di 0
θi θ1
The transformation matrix describing thekinematic modelling is C1 S 0 T1 = 1 0 0
− S1 C1 0 0
0 lC1 0 lS 1 1 0 0 1
(1)
where,C1 is cos(θ1) and S1 is sin(θ1). 2.2. Dynamic modelling The motion of the manipulator under the action of external force and torque is studied. Dynamic model is formulated with the inertia matrix involving the cross sectional parameters which is one of the defining aspects of this study. The inertia matrix for a generic cross section is as follows[17] x 2 dm ∫ xydm I = ∫ ∫ xzdm xdm ∫
∫ xydm ∫ y dm ∫ yzdm ∫ ydm 2
∫ xzdm ∫ xdm ∫ yzdm ∫ ydm ∫ z dm ∫ zdm ∫ zdm ∫ dm + m 2
p
(2)
where, mpis themass of the payload. Payload is assumed to be a point mass which is located at the end effector. So, it effectively contributes only to the last term in the matrix. Since, the frames are at the centroids for all the cross sections,
∫ xydm = ∫ yzdm = ∫ xzdm = ∫ ydm = ∫ zdm = 0
(3)
Since, it is a 1-DOF manipulator, the Coriolis term is zero and, hence, it is ignored in the dynamic model in (4). •• (4) τ 1 = M 11 θ 1 + G1
Where M11 is the inertialterm and G1 is the gravitational term.
B. Naveen and B. K. Rout / Materials Today: Proceedings 4 (2017) 1778–1787
1781
Fig. 2. Different cross sections of single link manipulator (a) Solid cylinder (b) Hollow cylinder (c) Solid rectangle (d) Hollow rectangle (e) Solid trapezoid (f) Hollow trapezoid.
Different cross sections of single link manipulators are shown in fig 2. 2.3. Parameter uncertainty modelling To model the inaccuracies and uncertainties in the parameters of the manipulator, probability distributions are necessary. In this case, each parameter is assumed to follow a Gaussian distribution with known mean and standard deviation [11]. In addition to the above, the dimensional tolerances which are available from the manufacturer and design data book are used to assume the mean and standard deviation as shown below Table 2. Table 2. Dimensions of the cross sectional parameters. Cross section
Dimensions (in m)
Solid cylindrical
r1 – 0.02
Hollow cylindrical
r2 – 0.0150; t 2 – 0.01
Solid rectangular
a3 – 0.04; b4 – 0.03142
Hollow rectangular
a4 – 0.04; b4 – 0.0527; t 4 – 0.006
Solid trapezoidal
a5 – 0.06; b5 – 0.03; h5 – 0.0279
Hollow trapezoidal
a6 – 0.05; b6 – 0.025; h6 – 0.017; t 6 – 0.008
The tolerance in all the parameters is taken to be as 0.1% of their mean dimension. The effect of uncertainties in torque is modelled using the tolerance specified for the motor’s torque. In this case, the worst-case scenario of the torque at a particular instant is used as an input to the model. The tolerance specified on the pull out torque is used for this case and taken to be 0.2% of it. 3. Methodology To simulate the performance and capture worst-case performance, an algorithm of the manipulator is discussed. The joint space trajectory planning, inverse dynamics and forward dynamics approach of manipulator with the proposed algorithm, the end effector’s final position is simulated. Later, the simulated values are consolidated into different performance measures.
B. Naveen and B. K. Rout/ Materials Today: Proceedings 4 (2017) 1778–1787
1782
In the quintic trajectory time law for joints, acceleration is smooth and controllable. Hence, it has been selected for the purpose of investigation. This trajectory is also widely used in the industrial manipulators. The trajectory for position, velocity and acceleration are as follows θ = k1 + k 2t + k3t 2 + k 4t 3 + k5t 4 + k 6t 5 (5) • dθ θ= = k 2 + 2k 3t + 3k 4 t 2 + 4k 5 t 3 + 5k 6 t 4 (6) dt
•
dθ θ= = 2k3 + 6k 4t + 12k5t 2 + 20k6t 3 dt ••
(7)
The constants are obtained by applying the boundary conditions as given in (8). These equations are solved to obtain the values of the constants. •• •• • • (8) θ (t ) = 0;θ (t ) = θ , θ (t ) = 0; θ (t ) = 0 , θ (t ) = 0;θ (t ) = 0 0
f
f
0
f
0
f
3.1. Simulation technique using worst-case approach Implementing the effect of uncertainties in parameters and performing the Monte Carlo simulation on a nonlinear dynamic model involving transcendental functions is a computationally intensive task. Thus, a modification in the form of worst-case technique is implemented for the purpose of numerical simulations [10]. It choses different combinations of worst cases for the parameters as inputs to the model to produce the outputs. Suppose, a parameter ‘x’ is affected by noise whose mean is μaand standard deviation is σa, in Monte Carlo simulation technique, Gaussian function is formulated using μa, σa and points are randomly picked from the distribution as inputs. Whereas, in worst-case technique μa±3σaare selected as the inputs to the dynamic model. In case of multiple inputs, all the combinations of the worst cases are used as inputs. So, in a model with ‘n’ parameters the number of the inputs will be only 2n. In this work, the maximum number of inputs were only 64, which is for hollow trapezoidal cross section, which has six parameters. Whereas,to capture the complete scenario by performing Monte Carlo simulation, generally the number of inputs are in the order of thousands [9]. If the same Monte Carlo simulation is performed with 64 inputs the exact scenario may or may not be captured. But in the worst-case technique, it captures the complete scenario although giving conservative results. It is because higher number of worst outputs will be produced by these worstcase inputs. Whereas, in the Monte Carlo simulation with a few inputs, these many worst outputs might not be captured. The outputs from worst-case inputs are then used to plot the performance ellipse. Further, in the field of state estimation, Unscented Kalman Filter (UKF) is used for non-linear dynamic models to estimate the state vector. Sigma points are selected, based on a few scaling parameters, and the approximated outputs are evaluated based of certain calculated weights [15]. Whereas, in this work we directly choose the worstcase inputs and pass them into the dynamic model.These worst-case dimensional parameters and control inputs are passed into dynamic model and the numerical forward dynamic simulations are performed on MATLAB using the ode45 routine. Thus obtained end effector positions for a particular cross section are used to evaluate its performance based on the proposed performance parameters. 3.2. Methodology The steps used to perform the simulations and obtain the results are explained in this section. Step 1: Develop kinematic model of specified manipulator Step 2: Formulate the Inertia matrix and the dynamic model Step 3: Assume equal cross sectional area, evaluate the cross sectional parameters Step 4: Use trajectory planning in the joint space to obtain properties of motion Step 5: Use inverse dynamics approach to evaluate the joint torque required for the planned trajectory Step 6: Use forward dynamic approach to simulate the performance using worst-case scenario. Step 7: Evaluate the most probable distance and the area of spread Step 8: Evaluate the sum of inertia values and the sensitivity of the eigenvalues of the inertia matrix The procedure used in steps 7-8 are explained in subsequent section.
B. Naveen and B. K. Rout / Materials Today: Proceedings 4 (2017) 1778–1787
1783
4. Performance measures This investigation is intended to identify the cross section which provides performance close to the target having least spread (variations) of the end effector at the destination. For this purpose, measures like most probable distance and area of spread are utilized and the focus here is to capture the variability in performance. 4.1. Most probable distance and area of spread The performance measures chosen should accurately depict the scenario. Researchers used various measures like positional error [11], S/N ratio [3], kinematic and dynamic reliabilities [8], Euclidean Norm of the inertia matrix eigenvalue derivatives etc. [7].Mean positional error considers only the distance of the point, and the actual position is immaterial to it. Even, Signal to Noise ratio (S/N) is a function of mean positional error. So both of them have the same drawback. Whereas, in this problem the actual position of the end effector changes with cross section. Even if the mean positional error is same for two different cross sections, their performance is actually different because the end effector’s finalpositions are different. Hence, performance is evaluated using the error in positional coordinates at the target point which is termed as Most Probable Distance (MPD). The mean of all the points reached by the end effector is called as Most Probable Point (MPP). So, with the given tolerances in the parameters and joint torque, the manipulator is most probable to reach the MPP instead of the intended target point. The distance of MPP from the target point is the MPD. For the ith experiment, the error in the positional coordinates is given by ∆xi = xi − xt ; ∆yi = yi − yt ; ∆zi = zi − zt (9) where the subscript ‘t’ denotes target. The mean error in the positional coordinates, is given by
1 N 1 N 1 N ∆xi ; ∆y = ∑ ∆y i ; ∆z = ∑ ∆z i ∑ N i =1 N i =1 N i =1 A positive and negative values indicate overshoot and undershoot in the target coordinates respectively. The MPP and MPD (μ) at the end effector is given by MPP = xt + ∆x, yt + ∆y, zt + ∆z ∆x =
(
)
2
2
(10)
(11)
2
(12) MPD( µ ) = ∆x + ∆y + ∆z The spread of the data around the MPP is found using variance and covariance of the positional coordinates.
σ xx ( x, x) σ xy ( x, y ) σ xz ( x, z ) Cov( x, y, z ) = σ yx ( y, x) σ yy ( y, y ) σ yz ( y, z ) σ zx ( z , x) σ zy ( z , y ) σ zz ( z , z )
a = sλ1 ;
b = sλ2 ; c = sλ3
(13)
(14)
Where, a, b, c are the length of the axes of the performance ellipsoid, λ1, λ2, λ3 (λ1> λ2 > λ3)are the Eigenvalues of the covariance matrix.For a value of s = 3, the performance ellipse just enclosed all the worst outputs for all the cross sections. To consolidate the error-spread at the end effector into a single parameter, the volume of performance ellipsoid (V) has been used. In case of 2-D the area of ellipse (A) will be appropriate. 4 V = πabc ; A = πab (15) 3 In general, for a matrix the determinant is equal to the product of its eigenvalues. So the above area of spread can also be interpreted as a scaled determinant of the covariance matrix. Also, for optimal design, the determinant of the covariance matrix is minimized, which is termed as d-optimality criterion, to reduce the effect of noise and extract as much information as possible from the experiments [16]. In this case also, the covariance matrix with lower determinant is the one with better performance in terms of area of spread.The pink dots in Fig. 3 represents individual performance of worst-case simulation. The minor axis of the performance ellipse is very small compared to the major axis because the tolerance along the minor axis direction is negligible compared to its perpendicular
1784
B. Naveen and B. K. Rout/ Materials Today: Proceedings 4 (2017) 1778–1787
direction. Along the major axis all the geometric parameters as well as torque contribute whereas in the minor axis direction only the tolerance due to the length is the contributor and the effect of other parameters is negligible. The ellipse is expected to be broader along the minor axis direction in case of manipulator with multiple links, since all the parameters will effect both the directions.
Fig. 3. The performance ellipse and the end effector positions.
4.2. Sensitivity analysis To get an insight into the performance results, a sensitivity analysis of the inertia matrix was performed. Each inertia matrix can be uniquely characterized by their Eigenvalues and Eigenvectors. Hence, the sum of Eigenvalues is computed and their sensitivity analysis w.r.t. the dimensional parameters is performed.Tr is the trace of the inertia tensor matrix which is also the sum of the eigenvalues. The numerical value of I11and I44 are same for all the cross sections. Hence, the sum of I22 and I33values give an understanding on the mass distribution. TI = I 22 + I 33 (16) It is intuitive to understand that for higher value of TI higher value of torque is required; consequently, higher will be the error in supplied torque. Furthermore, the effect of dimensional tolerances of the inertia matrix is due to all the terms of the inertia matrix. Let the sum of eigenvalues(K) in parametric form beK = f (a1,a2….an), where ai is the dimensional parameter [16]. n ∂f ∆ai ∂f = ∑ ai (17) ∂ai ai i =1 As the coefficient of tolerance to parameter ratio varies, its effect on the functional error also varies. The functional error is expected to be as close to zero as possible. Higher the magnitude of these coefficients higher is the effect on the output due to the tolerance to parameter ratio. Hence the vector length (L) consisting of the coefficients will determine the effect of parameter tolerance on the output error. ∂f L = ∑ ai ∂ai i =1 n
2
(18)
The interaction between the torque and dimensional parameters is not integrated in the eigenvalue analysis. Hence, it is not used for primary analysis and only used to get a physical interpretation.
B. Naveen and B. K. Rout / Materials Today: Proceedings 4 (2017) 1778–1787
1785
5. Results and discussion The selected arm is used to perform tasks with gravity loading and payload. However, to identify exclusively the effect of inertial parameters on the performance, the payload and the gravity were removed.Hence, to get a complete understanding, for the same initial and final positions the performance parameters are evaluated for fast and slow motions with payload, gravity, and without payload& gravity.The following observations are obtained from the simulation results of cases in Table 3 available in fig. 4. Important findings are (a) Solid rectangular and solid cylindrical have the worst MPD and worst area of spread respectively in all the cases; (b) Hollow cylindrical has the least MPD 3 out of 4 times and on the other occasion it is close to minimum; (c) Solid trapezoidal has the least area of spread 3 out of 4 times and very close to minimum in the remaining case Table 3. The four different cases used for the analysis of the manipulator (With effect of gravity and without it).
θ i (rad)
θf
1.
0
2.
Total time of
Payload mass
Gravity
motion (sec)
(kg)
(m/s2)
π/3
0.5
5
9.81
0
π/3
2
5
9.81
3.
0
π/3
0.5
0
-
4.
0
π/3
2
0
-
Case No.
(rad)
In cases 3-4, the MPD for cylindrical cross section is least and solid rectangular is highest with others following the same pattern in both the cases. Apart from the solid cylindrical section which has the highest area of spread, other cross section have not much difference in both the cases.This leads to a result that hollow cylindrical cross section has a better performance compared to other cross section in terms of both closeness to the target and area of spread around the target without the payload and gravity. Further, without gravity and payload, when the task time was increased by four times, the MPD and the area of spread of different cross sections have not changed much and their ratios lie close to one. Whereas, under the action of gravity and with payload, for different speeds, by the same increase in the task time, the MPD and area of spread have increased in the ranges of hundreds and seventy thousand respectively. It clearly shows that the performance is dominantly effected by the gravity and payload as well as their coupling with the velocity of the end effector. *The cross section index mapping to cross section is: 1 – Solid cylindrical, 2 – Hollow cylindrical, 3 – Solid rectangular, 4 – Hollow rectangular, 5 – Solid trapezoidal, 6 – Hollow trapezoidal Practically, manipulators are expected to work under the gravity and with payload. It is evident from the results of case 2 that the MPD and area of spread is very high as compared to that of case 1. Under gravity, slow motion of the manipulator leads to a worst performance where the MPD and area of spread have greatly increased compared to fast motion. But the pattern of MPD is almost the same in both the cases with cylindrical sections having least and solid rectangular section the highest. But the same pattern is not followed in case of without gravity and payload. This shows that the effect of gravity and payload is different on different cross sections. It is because different dimensional tolerances impact differently to the error in the mass of the link and hence the gravitational force effect changes leading to different performances. These observations are obtained from the results in fig. 5.
1786
B. Naveen and B. K. Rout/ Materials Today: Proceedings 4 (2017) 1778–1787
Fig. 4. MPD vs cross section (on left) and area of error ellipse vs cross section (on right) for four cases starting from case 1*.
Fig. 5. Sum of inertias vs cross section (on left) and vector length vs cross section (on right)*.
B. Naveen and B. K. Rout / Materials Today: Proceedings 4 (2017) 1778–1787
1787
The important findings are: (a) solid cross sections have lower inertia values compared to their hollow counterparts which is an expected; (b) hollow rectangular and solid cylindrical have the highest and least sum of inertia values respectively; (c) solid trapezoidal has the smallest vector length, while it is highest for solid cylindrical.From the figs. 4-5, it can be deduced that solid trapezoidal cross section with least vector length has least area of spread in most of the times and the hollow cylindrical cross section with medium range inertia value has the least MPD. It can also be observed that the vector length for hollow cylindrical section is close to minimum and the area of spread also is close to the minimum most of the times. 6. Conclusion This papers uses commonly available cross sections for the links of a 1-DOF with rotary joint.Uncertainties in the dimensional parameters and input torque was introduced in the deterministic dynamic model. Since the dynamic model is non-linear, a worst-case technique has been implemented instead of Monte Carlo simulations.Two parameters which consolidates the performance were suggested, which are mean positional distance (MPD) and the area of spread. The performance wasevaluated and analyzed under different working conditions for different speeds and with gravity and payload and without them. It can be concluded that the hollow cylindrical section has the least MPD with area of spread close to the minimum which is attributed to medium range inertia value and a low vector length. The solid trapezoid sectionwith minimum vector length has the least area of spread. Based on the requirements of the application, appropriate cross section for the link can be chosenby user which gives the least MPD or area of spread. References [1] B. K. Rout & R. K. Mittal, “Tolerance design of manipulator parameters using design of experiment approach.” Struct Multidisc Optim (2007), pp. 445–462. [2] Weidong Wu, S.S. Rao, “Uncertainty analysis and allocation of joint tolerances in robot manipulators based on interval analysis.” Reliability Engineering and System Safety 2007, 92, pp. 54–64. [3] B. K. Rout, R.K. Mittal, “Tolerance design of robot parameters using Taguchi method.” Mechanical Systems and Signal Processing 20 (2006) pp. 1832–1852. [4] Zhongxiu Shi, Technology Wuhan, “Reliability Analysis & Synthesis of Robot Manipulators.” 1994 Proceedings Annual Reliability and Maintainability Symposium, pp. 201 – 205. [5] W.K. Veitschegger, Chi-Haur Wu, “Robot Accuracy Analysis Based on Kinematics.” IEEE Journal of Robotics and Automation, Vol. RA-2, No. 3, September 1986, pp. 171 -179. [6] Tong Li, Qingxuan Jia, Gang Chen, Hanxu Sun, Jian Zhang, “Kinematics Reliability Analysis for Manipulator Considering Elasticity.” IEEE 9th Conference on Industrial Electronics and Applications (ICIEA), 2014. [7] Mahesh D. Pandey, Xufang Zhang, “System reliability analysis of the robotic manipulator with random joint clearances.” Mechanism and Machine Theory (2012), Vol.58, pp.137–152. [8] Ibrahim, M. Y., Cook, C. D. & Tieu, A. K., “Effect of a robot's geometrical parameters on its optimal dynamic performance.” Proceedings IEEE International Conference on Intelligent Control and Instrumentation, Singapore, 1992, pp. 820-825. [9] S.S. Rao, P.K. Bhatti, “Probabilistic approach to manipulator kinematics and dynamics.” Reliability Engineering and System Safety (2001), Vol. 72, pp. 47-58. [10] B.K. Rout, R.K. Mittal, “Optimal manipulator parameter tolerance selection using evolutionary optimization technique.” Engineering Applications of Artificial Intelligence 2008, Vol.21, 509–524. [11] B.K. Rout, R.K.Mittal, “Screening of factors influencing the performance of manipulator using combined array design of experiment approach.” Robotics and Computer-Integrated Manufacturing 25 (2009) pp. 651–666. [12] B.K. Rout, R.K. Mittal, “Optimal design of manipulator parameter using evolutionary optimization techniques.” Robotica (2010) Vol. 28, pp. 381–395. [13] C. Mavroidis, S. Dubowsky, P. Drouet, J. Hintersteiner and J. Flanz, “A Systematic Error Analysis of Robotic Manipulators: Application to a High Performance Medical Robot.” Proceedings of the 1997 International Conference in Robotics and Automation, Albuquerque, NM. [14] Alan P. Bowling, John E. Renaud, Jeremy T. Newkirk, Neal M. Patel, Harish Agarwal, “Reliability-Based Design Optimization of Robotic System Dynamic Performance.” Journal of Mechanical Design, APRIL 2007, Vol. 129, pp. 449 – 454. [15] Eric A Wan, Rudolph van der Merve, “The Unscented Kalman Filter,” in Kalman Filtering and Neural Networks, Simon Haykin, John Wiley & Sons, Ltd., Publication, 2001. [16] Peter Goos, Bradley Jones, “An optimal screening experiment” in Optimal Design of Experiments: A Case Study Approach, John Wiley & Sons, Ltd., Publication, 2011. [17] R.K. Mittal, I.J. Nagrath, Robotics and Control, Tata McGraw Hill Publication, New Delhi, 2003.