Inverse Kinematics Solution for Trajectory Tracking using Artificial Neural Networks for SCORBOT ER-4u

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1 Inverse Kinematics Solution for Trajectory Tracking using Artificial Neural Networks for SCORBOT ER-4u Rahul R Kumar 1, Praneel Chand 2 School of Engineering and Physics The University of the South Pacific Suva, Fiji 1 kumar_ru@usp.ac.fj 2 chand_pc@usp.ac.fj Abstract This paper presents the kinematic analysis of the SCORBOT-ER 4u robot arm using Artificial Neural Networks (ANN). The SCORBOT-ER 4u is a 5-DOF vertical articulated educational robot whose all joints are revolute. The inverse kinematics solution is found using the ANN. This paper uses the forward kinematics to train the ANN. The Denavit-Hartenberg and Geometrical methods act as a feed-forward network (forward kinematic algorithms) to generate data and train the ANN. The algorithm will be tested on a real physical robot (SCORBOT-ER 4u) to evaluate its performance. The modelling and simulations are done using MATLAB 8.0 software. Keywords- Forward Kinematics, Inverse Kinematics, Back- Propagation, Feed-Forward Propagation, Geometrical approach I. INTRODUCTION Mankind has always strived to give a life like quality to its artefacts, and merely intends to find a substitute (robot), that can work in hostile or in-conducive environments. Basically, robots are the specialized, highly automated mechanical manipulators which are controlled by sophisticated electronic control systems and computer systems. Over the past decades, industries are moving towards automation. The reason behind this is the need for accuracy, efficiency in repetitive tasks, high productivity and budgeting for industrial processes. In addition to that, studies in robotics are clearly oriented to eliminate a human operator. By introducing autonomous robotic applications, many challenging tasks can be accomplished with a higher rate of success. The development of the robotic arm for applications such as object sorting needs to undergo the concept of Inverse Kinematics (IK) which governs the notion of motion planning. Motion planning for the robotic arms is essential for real, physical industrial applications. The complexity is enhanced when the Degrees of Freedom (DOF) is increased for an articulated robotic arm. This complexity is in terms of the forward and inverse kinematics control of the robotic arm. In [1, 2] three drawbacks of traditional inverse kinematics (IK) control methods are stated. These are Algebraic, Geometric and Iterative methods to find the solutions for the IK problem. Similarly, agreed by [3] and [4] that for Algebraic method, there s no closed form of solution and specifically not suited for real (physical) robotic applications. As for Geometric method, suggested by [1] and [4] the IK solution is limited up to 3 DOF robots. For the Iterative method, [1] and [4] state that the solution is entirely dependent on the starting point. It has also been pointed out by [3, 5] that Jacobian s method for IK does not provide all the solutions when joint velocities of the configurations become large upon following the desired posture in a Cartesian plane. As a result, state of the art algorithms come into picture now. Most of the methods which have been presented are only designed for a very specific task. However, in general, most of the IK solutions are approached through the concept of ANNs. In [1], to solve the complexity in a multi DOF robotic manipulator, an Adaptive Neuro-Fuzzy Inference System (ANFIS) is used and implemented on real time basis. In terms of the trajectory tracking and optimal control of the robotic manipulator system, [4] and [6] highlight that feed-forward ANNs and Bees Back propagation via ANN is a good means to solve IK problems and optimize the whole system. Enhancing the computation time of the processor for the controlling module is highly recommended by [3] when it comes to the mapping of IK with complex trigonometry. In addition to these methods, [3, 6, 7] suggest that Genetic Algorithm (GA) is also very well suited for the generation of IK for an articulated system. Additionally, presented by [8] is the usage of Bayesian network to solve the problem of IK and robot redundancy resolution. An anthropomorphic configuration of the robotic arm has been mathematically modelled via studying a human arm along with its DOF. Finally, it is also recommended by [1, 3-6, 9]that hybrid systems e.g. combination of ANN and Fuzzy Logic (FL), combination of GA and FL typically provide robust solutions to IK problems. However, integrating, stability space using Lyapuvnov and AFNC algorithm, machine learning (online and offline) is also suggested by [10]. The goal of this paper is to use Denavit-Hartenberg (D-H) [11-13] and Geometrical approach [1, 11] to model the SCORBOT-ER 4u robot arm and derive the Forward Kinematics (FK). The reason why the two approaches have been used for FK is for simplicity and precision as the calculations which revolve around Geometric and D-H approach, are very straight forward and used by [11, 12, 14]

for confirmation of the test results particularly for the arm robots. Using the FK as a Feed Forward network for training the ANN, IK solutions are found.

The following figure outlines the process carried out for the proposed technique to demonstrate the IK solutions via ANN. Figure 1: Conceptual Framework of the Proposed Technique II.

Using the D-H convention, the orthonormal coordinate system is attached to each link of the robot arm (Fig.2). Table I lists the D-H parameters for the SCORBOT-ER 4u. A.

2 for confirmation of the test results particularly for the arm robots. Using the FK as a Feed Forward network for training the ANN, IK solutions are found. The data is generated using the FK for different sets of joint angles. A suitable training algorithm for IK is also determined in this paper. The following figure outlines the process carried out for the proposed technique to demonstrate the IK solutions via ANN. Figure 1: Conceptual Framework of the Proposed Technique II. FORWARD KINEMATICS USING DENAVIT-HERTENBURG MODEL D-H parameters work with the following:,,, (quadruple) which are link twist angle, link length, link offset and joint angle respectively. Using the D-H convention, the orthonormal coordinate system is attached to each link of the robot arm (Fig.2). Table I lists the D-H parameters for the SCORBOT-ER 4u. A. Frame Assignment to SCORBOT-ER 4u is the distance between the point of intersection of axis with the axis to the origin of the frame. This distance is taken along the axis. is the distance measured between the point of intersection of axis with axis and the origin of the frame 1. - is the angle between and axes about axis in the Right Hand sense. B. D-H Convention The final transformation matrix from joint to 1 derived in [11-13]is due to the following: i. Rotation about by ii. Translation along by distance iii. Translation by distance along axis iv. Rotation by angle about axis (1),,, ) (2) Hence, writing in matrix form gives:,,, cos sin cos sin cos cos sin cos sin cos sin sin (3) 0 sin cos The above matrix is the standard D-H Parameter matrix. For simplicity let: Figure 2: Frame Assignment to SCORBOT-ER 4u TABLE I D-H Parameters for SCORBOT-ER 4u to 155 * to 130 * to 130 * to 130 * to 570 * The following explains how the quadruple for each joint (in the table above) is determined: *Indicates that the angles are varied. is the twist angle between and axes about in the Right Hand sense. sin (4) cos (5) sin (6) cos (7) sin (8) sin (9) The simplified D-H Parameter Matrix is: Now, (10) 0 5, 1 5, which gives rise to (11): Therefore: (11) (12)

The first three notations of the last column represent the,, coordinates in a Cartesian plane. Also note that varies as per the robotic manipulator s given range [15].

(13) Since the finding of IK is complicated, this paper will not cover IK analytically.

3 The first three notations of the last column represent the,, coordinates in a Cartesian plane. Also note that varies as per the robotic manipulator s given range [15]. The following matrix (13) represents the end-effector position and orientation which is compared and is equivalent to the matrix in (12). This step is done to find the IK analytically. (13) Since the finding of IK is complicated, this paper will not cover IK analytically. However, using a feed-forward ANN (which simply trains the forward kinematics) will be used to find the IK solutions. TABLE II Joint Details Mechanical Structure Vertical Articulated Number of Axes 5 axes plus servo gripper Axis 1: Base rotation 310 Axis 2: Shoulder rotation +130 / 35 Axis 3: Elbow rotation ±130 Axis 4: Wrist pitch ±130 Axis 5: Wrist roll Unlimited (mechanically) ±570 (electrically) The joints of SCORBOT-ER 4u are all revolute. The base is associated with the yaw angle, the shoulder, elbow and wrist are associated with the pitch angle and the wrist is also associated with the roll angle. III. FORWARD KINEMATICS USING GEOMETRICAL APPROACH V. SCORBOT-ER 4U WORKING ENVELOP Figure 3: SCORBOT-ER 4u s Elevation Fig.3 describes the position and orientation of the end effector relative to the fixed frame attached to the base of the robot. The Geometrical approach is used to find the forward kinematics in the 2D plane for the SCORBOT-ER 4u. From the figure, the following equations are derived: (14) cos cos (15) sin sin (16) is the yaw angle to navigate in the x-z axis. The end effector s direction is related to the actual joint displacements (pitch as shown in figure 3). For the computation of the IK, a feed-forward neural network is trained. Random joint angles in compliance with the ranges specified previously are generated for training data. IV. SCORBOT-ER 4U JOINT SPECIFICATIONS The SCORBOT-ER 4u is a 5DOF educational robotic arm. Its extents are represented by TABLE II of all the 5 joints. Figure 4: Operating Range (Side View) Figure 5: Operating Range (Side View) Based on the Joint details provided in TABLE II, Fig.4 and Fig.5 characterize the working envelope of SCORBOT-ER 4u [15]. The maximum reach of the robot is 610mm in terms of yaw angle and the maximum height (pitch length with respect to the base) is 1040mm.

4 VI. THE ARTIFICIAL NEURAL NETWORK A. Data Generation Using the Geometrical and D-H modelling based forward kinematics (section II and section III), position and orientation data of the robot arm is generated for different sets of joint angles. The input data is the localization vector,, and the target data is the corresponding joint angles for the Geometrical approach ANN model. Note the Geometrical approach takes into account the cylindrical coordinate system, hence, the base angle (yaw) and the roll angle (of the end effector) does not need to be trained as the output for the base and roll angle would be similar to the input. Therefore, in this case, only the elevation (side view) of the robot is considered. (Fig.5) For the D-H ANN model, the input data is the localization vector,, and the target data is the corresponding joint angles,,,,, where is the roll angle of the end effector. B. Data Division The actual workspace shapes up with the accumulating 1000 sets of generated data using the forward kinematics, of which 70% was used for training, 15% for validation and 15% for testing. C. Training Algorithms Used Five different training algorithms were used and tested to identify the most suitable model for IK solutions. These methods were: Gradient Descent with momentum (GDM) Gradient Descent (GD) Scaled conjugate gradient back-propagation (SCG) Resilient back-propagation (RBP) Random order incremental training with learning functions (RI) All the models consisted of 100 neurons in the hidden layer. Fig.6 and Fig.7 show the network architecture for Geometrical and DH methods respectively: VII. RESULTS AND DISCUSSION The TABLE III below portrays the performance, time taken, number of neurons and accuracy of the Geometrical and D-H methods using the five different training algorithms. The best training algorithm is the Resilent back-propagation algorithm in both approaches (Geometrical and D-H). TABLE III Performance and Test Accuracies for Geometrical and D-H Approach Geometrical Approach Network Type Performance Training Function Time No. % Function (MSE) Neurons Accuracy GDM 5.60E-04 49s GD 6.39E-03 48s SCG 3.55E-03 42s RBP 1.34E-06 23s RI 1.20E-01 32s DH Method Feed-Forward Back- Propagation Network Type Feed-Forward Back-Propagation Performance Training Function Time No. % Function (MSE) Neurons Accuracy GDM 1.70E-01 49s GD 1.80E-01 48s SCG 1.00E-03 31s RBP 1.20E-06 11s RI 2.26E s Figure 8: Performance of Resilent Back-Propagation using MSE for Geometrical Method Figure 6: ANN Structure for Geometrical Approach Figure 7: ANN Structure for D-H Method Figure 9: Regression plot of Test Data using Resilent Back-Propagation for Geometrical Method

Fig. 8 Fig. 11 represent the training performance and the regression plot of the test data for the best model which was trained using the Resilient Back-propagation (RBP) algorithm.

5 Fig. 8 Fig. 11 represent the training performance and the regression plot of the test data for the best model which was trained using the Resilient Back-propagation (RBP) algorithm. The RBP required very less computation time and yielded higher accuracy upon testing the Geometrical and D-H network models. The SCG algorithm is also a good approach for finding the IK solution, however in comparison with RBP, it requires more iteration (i.e. more computation time to achieve similar accuracy). Figure 10: Performance of Resilent Back-Propagation using MSE for D-H Method Figure 11: Regression plot of Test Data using Resilient Back-Propagation for D-H Method Figure 12: End Effector Trajectory tracking Test results for Geometrical Method Figure 13: End Effector Trajectory tracking Test results for D-H Method Fig.12 and Fig.13 show the end-effector trajectory tracking for both methods i.e. ANN trained Geometrical method (in 2D plot) and ANN trained D-H method (3D plot). The test data coincides with the end-effector plots as the points for RBP algorithm (red dots on the graph in Fig. 13&14) mostly lie along the desired points (blue dots) in both 2D and 3D graphs. VIII. CONCLUSION In this paper two methods of forward kinematics were presented. The two methods (Geometrical & DH methods) were manipulated as a feed forward ANNs. The ANN trained Geometrical approach provided the IK solutions for a 2D model and the ANN trained D-H method yielded 3D IK solutions of the SCORBT-ER 4u. According to the results, the best training algorithm was determined to be Resilient Back- Propagation Algorithm having test accuracy as 99.92% for the Geometrical method and 99.96% for the D-H method. The performance of the Resilient Back-Propagation algorithm is said to be the most reliable and accurate in terms of computation time, MSE and the percentage test accuracy. This will be an advantageous when testing the algorithm on real (physical) robot (SCORBOT-ER 4u). Hardware results will be included in the final version of the paper. IX. RECOMMENDATIONS Having trained and simulated the models (i.e. ANN trained Geometrical and D-H methods), both the models give reasonable accuracy. We are currently performing hardware testing and a comparison of simulated and hardware results are expected for the final version of this paper. Furthermore, two additional training algorithms can also be tested. These are Levenberg-Marquardt [16] and Bayesian regulation backpropagation algorithms [17]. REFERENCES [1] H. Chaudhary and R. Prasad, "Intelligent inverse kinematic control of scorbot - er v plus robot manipulator " International Journal of Advances in Engineering & Technology, vol. 1, [2] A.-V. Duka, "Neural network based inverse kinematics solution for trajectory tracking of a robotic arm," Procedia Technology, vol. 12, pp , [3] M. Tarokh and M. Kim, "Inverse kinematics of 7-DOF robots and limbs by decomposition and approximation," IEEE transactions on robotics, vol. 23, pp , [4] H. Chaudhary, R. Prasad, and N. Sukavanum, "Position analysis based approach for trajectory tracking control of scorbot -er - v plus robot manipulator," International Journal of Advances in Engineering & Technology, vol. 3, [5] E. Mattar, "A Practical Neuro-fuzzy Mapping and Control for a 2 DOF Robotic Arm System," Int. J. Com. Dig. Sys, vol. 2, pp , 2013.

6 [6] V. Banga, R. Kumar, and Y. Singh, "Fuzzy-genetic optimal control for robotic systems," International Journal of the Physical Sciences, vol. 6, [7] A. Pajaziti and H. Cana, "Robotic Arm Control with Neural Networks Using Genetic Algorithm Optimization Approach." [8] P. K. Artemiadis, P. T. Katsiaris, and K. J. Kyriakopoulos, "A biomimetic approach to inverse kinematics for a redundant robot arm," Autonomous Robots, vol. 29, pp , [9] H. Chaudhary, V. Panwar, R. Prasad, and N. Sukavanam, "Adaptive neuro fuzzy based hybrid force/position control for an industrial robot manipulator," Journal of Intelligent Manufacturing, pp. 1-10, [10] M. J. Er and Y. Gao, "Robust adaptive control of robot manipulators using generalized fuzzy neural networks," Industrial Electronics, IEEE Transactions on, vol. 50, pp , [11] M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot modeling and control vol. 3: Wiley New York, [12] V. A. Deshpande and P. George, "Analytical Solution for Inverse Kinematics of SCORBOT-ER-Vplus Robot." [13] J. Iqbal, R. ul Islam, and H. Khan, "Modeling and Analysis of a 6 DOF Robotic Arm Manipulator," Canadian Journal on Electrical and Electronics Engineering, vol. 3, pp , [14] P. V. Patil and S. S. Ohol, "Performance Analysis of SCORBOT ER 4u Robot Arm," [15] R. SCORBOT-ER, "4U User Manual Catalog# Rev," ed: B. [16] G. Lera and M. Pinzolas, "Neighborhood based Levenberg- Marquardt algorithm for neural network training," Neural Networks, IEEE Transactions on, vol. 13, pp , [17] J. Liao and R. Kirsch, "Predicting the initiation of minimum-jerk submovements in three-dimensional target-oriented human arm trajectories," in Engineering in Medicine and Biology Society (EMBC), 2012 Annual International Conference of the IEEE, 2012, pp

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