ISSN: 2168-9695
Advances in Robotics & Automation
Make the best use of Scientific Research and information from our 700+ peer reviewed, Open Access Journals that operates with the help of 50,000+ Editorial Board Members and esteemed reviewers and 1000+ Scientific associations in Medical, Clinical, Pharmaceutical, Engineering, Technology and Management Fields.
 
Meet Inspiring Speakers and Experts at our 3000+ Global Conferenceseries Events with over 600+ Conferences, 1200+ Symposiums and 1200+ Workshops on
Medical, Pharma, Engineering, Science, Technology and Business
 

Artificial Neural Network Based Forward Kinematics Solution for Planar Parallel Manipulators Passing through Singular Configuration

Ammar H Elsheikh*, Ezzat A Showaib and Abd Elwahed M Asar
Department of Production Engineering and Mechanical Design, Faculty of Engineering, Tanta University, Tanta, Egypt
Corresponding Author : Ammar H. Elsheikh
Department of Production Engineering and Mechanical Design
Faculty of Engineering, Tanta University, Tanta, Egypt
E-mail: eng_ammar_sheikh@yahoo.com
Received June 30, 2013; Accepted September 18, 2013; Published September 20, 2013
Citation: Elsheikh AH, Showaib EA, Asar AE (2013) Artificial Neural Network Based Forward Kinematics Solution for Planar Parallel Manipulators Passing through Singular Configuration. Adv Robot Autom 2:106. doi:10.4172/2168-9695.1000106
Copyright: © 2013 Elsheikh AH, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Visit for more related articles at Advances in Robotics & Automation

Abstract

It is well known that, the main drawback of parallel manipulators is the existence of singularities within its workspace, an Artificial Neural Network (ANN) based solution is proposed in this paper. The proposed approach can certainly learn the input-output data and discover the non-linear relationships which are inherent in the training data. Additionally, the proposed approach can provide solution of the forward kinematic problem with reasonable errors at and in the vicinity of kinematic singularities. The approach is implemented for the 3-RPR, 3-PRR, and 3-RRR planar parallel manipulators.

Keywords
Parallel manipulators; Forward kinematics; Singularities; Artificial Neural Network (ANN)
Introduction
Parallel manipulators, due to its closed-loop structure, posses a number of advantages over traditional serial manipulators such as high rigidity, high load-to-weight ratio, high natural frequencies, high speed and high accuracy [1]. However, they also have a few disadvantages such as a relatively small workspace, relatively complex forward kinematics and the most importantly, existence of singularities inside the workspace [2]. Kinematics analysis of parallel manipulators separate in two types, forward kinematics and inverse kinematics. The inverse kinematics, which maps the task space to joint space, is not difficult to solve. On the other hand, the forward kinematics, which maps the joint space to task space, is so hard to solve. Also, the existence of not only multiple inverse kinematic solutions (or working modes) but also multiple forward kinematic solutions (or assembly modes) is another problem in kinematics analysis [3]. The challenging problem is not to find all possible solutions but to directly determine the unique feasible solutions, the actual physical solution, in among all possible solutions starting from a certain initial configuration [4].
Forward kinematics and singularity analysis of planar parallel manipulators have been investigated by many researchers [5-7]. Efforts to solve the forward kinematics of planar parallel manipulators have concentrated on 3-RPR manipulator due to its inherent simplicity. It is established the forward kinematic solution of general 3 DOF planar parallel manipulators can be lead to a polynomial of degree 8 [8]. However, the forward kinematic problem for the manipulator under study leads to a maximum of 6 real solutions. It is worth taking into considerations, the three manipulators under study are kinematically equivalent to each other and, as a result, we derived the forward kinematics equations for 3-RRR and modified it to the two other manipulators. Additionally, the existence of singularities and uncertainties inside the workspace where the manipulator gains some degrees of freedom and become uncountable. In such configurations, the actuated joints forces of the manipulator will become unacceptably large that often reach their allowable limits. To overcome the problem of kinematics singularities a neural network –based approach is developed which has the ability of generalization and can successfully learn relationships that are not present in the training set in an efficient manner.
There have been increasing research interests of Artificial Neural Networks (ANNs) due to their extreme flexibility and the capability of non-linear function approximation. Many efforts have been made on applications of Neural Networks to various types of parallel manipulators [9-13].
In this paper, a supervised neural network approach is developed to control the motion of the 3-RPR, 3-PRR and 3-RRR planar parallel manipulators. Multiple neural networks are used to overcome the problem of the multiple solution branches of either forward or inverse kinematics. This approach also overcomes the problems of singularities and uncertainties’ arising in trajectory planning as it has, like any ANN algorithms, generalization ability. In this approach a network is trained using training data generated from the inverse kinematics. The training is done off-line until reaching acceptable error and a validation test is also done, at each iteration, to avoid model over fitting. It may be noted here that the present work may be considered as an implementation of the artificial neural network approach for serial manipulators passing through singular configuration, as proposed by [14], for planar parallel manipulators.
Kinematics of Parallel Manipulators
Kinematic analysis of parallel manipulators includes solution to forward and inverse kinematic problems. The forward kinematics of a manipulator deals with the computation of the position and orientation of the manipulator end-effector in terms of the active joints variables. Forward kinematic analysis is one of essential parts in control and simulation of parallel manipulators. Contrary to the forward kinematics, the inverse kinematics problem deal with the determination of the joint variables corresponding to any specified position and orientation of the end-effector. The inverse kinematics problem is essential to execute manipulation tasks. Most parallel manipulators can admit not only multiple inverse kinematic solutions, but also multiple forward kinematic solutions. This property produces more complicated kinematic models but allows more flexibility in trajectory planning [15]. In other words, a manipulator configuration can be defined either by actuator coordinates q=[q1, .., qn]T or by Cartesian end-effector coordinates x= [x1, .., xn]T with n the DOF of the manipulator under study. The transformation between actuator coordinates and Cartesian coordinates is an important issue from viewpoint of kinematic control. Computation of the end-effector coordinates from given actuator coordinates (forward kinematics) can be written in the general form
x= ƒFKP(q) (1)
The inverse task which is to establish the actuator coordinates corresponding to a given set of end effector coordinates (inverse kinematics) can be also written in the general form
q= ƒIKP(x) (2)
Then the kinematic constraints imposed by the limbs can be written in the general form
ƒ(x,q)=0 (3)
Differentiating Eq.(3) with respect to time, we obtain a relationship between the input joint rates and the end-effector output velocity
(4)
Where
Inverse kinematic singularity occurs when different inverse kinematic solutions coincide that happens usually at the workspace boundary. Hence the manipulator loses one or more degrees of freedom. Mathematically they can detected by det (Jq)=0
Forward kinematic singularity occurs when different forward kinematic solutions coincide. Hence the manipulator gains one or more degrees of freedom. That happens inside the workspace so it is a great problem. Mathematically they can detected by det (Jx)=0
Manipulators Under Study
The architectures of the planar parallel manipulators under study, 3-RPR, 3-PRR and 3-RRR, are illustrated in Figures 1a-1c, Where R, P, R and P denote revolute, prismatic, actuated revolute and actuated prismatic joints, respectively. For manipulators under study the three fixed pivots A1, A2 and A3 define the geometry of the fixed base, and the three moving pivots C1, C2 and C3 define the geometry of the moving platform, where point O and H are the centroids of the fixed base and moving platform respectively. Three limbs connect the moving platform to the fixed base. Each limb of the 3-RPR is composed of a R, a P, and a R joint in sequence. Each limb of the 3-PRR is composed of a P, a R, and a R joint in sequence. Likewise, each limb of the 3-RRR is composed of three R joints in sequence. The origin of the fixed coordinate frame is located at point A1.The x-axis points along the direction of A1A2 and the y-axis is perpendicular to A1A2. We assume that the manipulators under study are symmetrical, manipulators with equilateral base and moving platform [15]. The moving platform pose, i.e., its position and its orientation, is determined by means of the Cartesian coordinates vector H=[Hx,Hy]T of operation point H and angle φ, namely, the angle between C1C2 and the positive direction of x-axis.
Artificial Neural Networks
Artificial neural network (ANN) is an algorithm that model brain performs a particular task, and is usually implemented using electronic components or simulated in software on digital computers. It has the ability of imitating of the mechanisms of learning and problem solving functions of the human brain which are flexible, powerful, and robust. In artificial neural networks implementation, knowledge is represented as numeric weights, which are used to gather the relationships between data that are difficult to realize analytically, and this iteratively adjusts the network parameters to minimize the sum of the squared approximation errors using a gradient descent method [14]. One category of the artificial neural networks is the multilayer perceptron (MLP) which be considered a supervised back propagation learning algorithm. It consists of an input layer, some hidden layers and an output layer as shown in Figure 2. MLP is trained by back propagation of errors between desired values and outputs of the network using some effective algorithms such as gradient descent algorithm. The network starts training after the weight factors are initialized randomly. Weight adjusting takes place until, we get reasonable errors or no more weight changes occur. There is no available theoretical procedures to choose the appreciate network architecture, i.e. number of hidden layers and number of neurons of each layer. This depends on the problem under investigation and user’s experience.
Results of Numerical Simulations
Simulations have been conducted for the 3-RPR, 3-PRR, and 3-RRR planar parallel manipulators to demonstrate the performance of the developed approach. First point H (the centroid of the end-effector) is moved along a given trajectory which passing through singular locus then the correct active prismatic joint or joint angle variables to track this trajectory are calculated using the inverse kinematic model of the simulated manipulator which give a unique solution for a given working mode. Then, those active prismatic joint or joint angle variables are fed to the MIP to track the trajectory and the tracking errors are calculated. The simulated manipulators are assumed to be ideal mechanisms with no flexibility and no joint clearance that affect the accuracy of the manipulator. Also, the prismatic joints are assumed to have unlimited length. A two-hidden layer MLP with back propagation learning is considered here. The input layer has as many nodes as the number of inputs to the map, namely three actuator variables. Similarly the output layer will have three nodes which represent the pose of the end-effector. The number of neurons in the hidden layers and its configuration are used as a design parameter. Sigmoid and linear activation functions are used for all hidden and output layer nodes respectively. Supervised learning scheme is used in which the network is taught to learn the map by observing the inputs and outputs. The network is trained by 10,000 training input-output patterns generated, randomly within the workspace of the manipulator, from the inverse kinematic model. Random initialization is used for the weights. For each manipulator, different configurations of the MLP network were tested to get the optimal configuration used for solve the problem. About 36 multi-layer feed forward networks with two hidden layers are trained. All these networks were trained over 1,000 training epochs to ensure the success of the training process and to avoid over fitting the model. Simulation results showed that 40×60 multilayer perceptron neural network with two hidden layers had the best performance when the minimum tracking error is used as performance index. All manipulators under study are symmetric with three identical limbs. Each side of the moving end-effector equilateral triangle is 100 mm, while that of the base is 300 mm. The lengths of the proximal links and the distal links are 120 mm and 80 mm, respectively.
3 RPR planar parallel manipulator
Three end-effector trajectories are specified as straight lines which cross over singularity loci at H1(265, 58.499) mm and H2(265,114.706) mm as shown in Figure 3. The first trajectory is a vertical straight line starting at Hi(265,40) mm with orientation angle φ=15° and ending at point Hf(265,140) mm with the same orientation it is obvious the selected trajectory passes through singular points H1 and H2. The tracking errors in x- and y-directions are depicted in Figure 4. The maximum tracking error along the trajectory points is 0.0027 mm which happens in the vicinity of kinematic singularities. We also note that there is a significant increasing in the tracking error near the singularity points. Anyway, the developed approach can provide solution for the problem with reasonable errors.
The second trajectory is a horizontal straight lines starting at Hi(245,58.499) mm with orientation angle φ=15° and ending at point Hf(275,58.499) mm with the same orientation it is obvious the selected trajectory passes through singular point H1. The tracking errors in xand y-directions are depicted in Figure 5. The maximum tracking error along the trajectory points is 0.0023 mm which also happens in the vicinity of kinematic singularities.
Finally, the third trajectory is a horizontal straight lines starting at Hi(245,114.706) mm with orientation angle φ=15° and ending at point Hf(275,114.706) mm with the same orientation. The selected trajectory passes through singular point H2. The tracking errors in xand y-directions are depicted in Figure 6. The maximum tracking error along the trajectory points is 0.0027 mm which also happens in the vicinity of kinematic singularities.
3 PRR planar parallel manipulator
Two end-effector trajectories are specified as straight lines which cross over singularity loci at H1(200,115.470) mm as shown in Figure 7. The first trajectory is a vertical straight line starting at Hi (200,102) mm with orientation angle φ=0° and ending at point Hf(200,122) mm with the same orientation it is obvious the selected trajectory passes through singular point H1. The tracking errors in x- and y-directions are depicted in Figure 8. The maximum tracking error along the trajectory points is 0.0015 mm which happens in the vicinity of kinematic singularities.
The second trajectory is a horizontal straight lines starting at Hi(187,115.470) mm with orientation angle φ=0° and ending at point Hf(207,115.470) mm with the same orientation it is obvious the selected trajectory passes through singular point H1. The tracking errors in xand y-directions are depicted in Figure 9. The maximum tracking error along the trajectory points is 0.0015 mm which also happens in the vicinity of kinematic singularities.
3 RRR planar parallel manipulator
In the same way, two end-effector trajectories are specified as straight lines which cross over singularity loci at H1(155, 4.845) mm as shown in Figure 10. The first trajectory is a vertical straight line starting at Hi(155,4) mm with orientation angle φ=0° and ending at point Hf(155,24) mm with the same orientation it is obvious the selected trajectory passes through singular point H1. The tracking errors in xand y-directions are depicted in Figure 11. The maximum tracking error along the trajectory points is 0.005 mm which happens in the vicinity of kinematic singularities.
The second trajectory is a horizontal straight lines starting at Hi(142,4.845) mm with orientation angle φ=0° and ending at point Hf(162,4.845) mm with the same orientation it is obvious the selected trajectory passes through singular point H1. The tracking errors in xand y-directions are depicted in Figure 12. The maximum tracking error along the trajectory points is 0.0053 mm which also happens in the vicinity of kinematic singularities.
Conclusion
In this paper, we proposed to use neural networks for forward kinematic solution of three different architectures of planar parallel manipulators, which can be elaborated to generate the best estimation of forward kinematics of the manipulators under study. Even though the manipulators passing through the kinemaic singularities, the proposed approach can provide solution for the problem with reasonable errors. The results of this paper can be used to find the forward kinematics solutions at critical points (singularity points) which can be then avoided, as long as we specify them, in dynamic control stage.
References















Select your language of interest to view the total content in your interested language
 
Share This Article
   
 
   
 
Relevant Topics
Disc ACS Nano
Disc AI Algorithm
Disc Advanced Materials
Disc Applied Theories on Machines
Disc Artificial Intelligence and Philosophy
Disc Automated Mining
Disc Automated Reasoning and Inference
Disc Automation
Disc Automation Devices
Disc Automobile Engineer
Disc Automotive Engineering
Disc Automotive Industry
Disc Capacitor
Disc Case-based reasoning
Disc Circuits
Disc Coal Consumption
Disc Cognitive Aspects of AI
Disc Commonsense Reasoning
Disc Computational Neuroscience
Disc Computer Simulation
Disc Constraint Processing
Disc Contingency Analysis
Disc Control System
Disc Current
Disc Diesel Engine
Disc Digital Image Processing
Disc Directional Microphone
Disc Dynamic Information
Disc Dynamic Stabilizer
Disc Electrical Engineering
Disc Electricity
Disc Electromagnetism
Disc Engine
Disc Engine Performance
Disc Engineering Design
Disc External Force
Disc Fluid Bodies
Disc Flying Wheel
Disc Fuel Economy
Disc Fuzzy Logic
Disc Helmet
Disc Heuristic Search
Disc High-Level Computer Vision
Disc Human Centered
Disc Human-Machine-Interfaces
Disc Human-Robot Interaction
Disc Hydraulic Engineering
Disc Ignititon System
Disc Industrial Robotics
Disc Intelligent Interfaces
Disc Intelligent Robotics
Disc Machine
Disc Mathematical Model
Disc Mechanical Properties
Disc Mechanical Systems
Disc Mechanism
Disc Mechatronics
Disc Medical Device
Disc Medical Robotics
Disc Microphone Array
Disc Mobile Device
Disc Mobile Repots
Disc Mobile Robot System
Disc Modular Architecture
Disc Nano/Micro Robotics
Disc Neural Networks
Disc Neurorobotics
Disc Omnidirectional Microphone
Disc Oscillator
Disc Power Analysis
Disc Power System Stabilizers
Disc Product Quality
Disc Quaternary Logics
Disc Robot Audition
Disc Robotic Rehabilitation
Disc Robotics
Disc Robotics In Medical
Disc Robotics Methods
Disc Robotics for Application
Disc Robotics for Mechanism
Disc Semiconductor
Disc Sensing and Perception
Disc Sensor
Disc Simulation
Disc Social Robots
Disc Spark Ignition
Disc Spectrum
Disc Splitting Method
Disc Telerobotics
Disc Thermodynamics Methods
 
Recommended Journals
Disc Electrical Engineering Journals
Disc Computer Engineering Journals
Disc Mechanical Engineering Journals
Disc Electrical & Electronic Journals
Disc Lovotics Journals
Disc Automobile Engineering Journals
  View More»
 
Recommended Conferences
Disc 2nd Automation and Robotics Congress
Jun 13-15, 2016 Philadelphia, USA
Disc Design and  Production Engineering Conference    
Jul 25-26, 2016 Berlin, Germany
Disc  Neurorehabilitation Conference
July 21-22, 2016 Brisbane, Australia
View More»
 
Article Tools
Disc Export citation
Disc Share/Blog this article
 
Article usage
  Total views: 11184
  [From(publication date):
November-2013 - May 24, 2016]
  Breakdown by view type
  HTML page views : 7439
  PDF downloads :3745
 
 

Post your comment

captcha   Reload  Can't read the image? click here to refresh

 
OMICS International Journals
 
Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals
 
 
OMICS International Conferences 2016-17
 
Meet Inspiring Speakers and Experts at our 3000+ Global Annual Meetings
 
 

Contact Us

Agri, Food, Aqua and Veterinary Science Journals

Dr. Krish

agrifoodaquavet@omicsinc.com

1-702-714-7001 Extn: 9040

Clinical and Biochemistry Journals

Datta A

clinical_biochem@omicsinc.com

1-702-714-7001Extn: 9037

Business & Management Journals

Ronald

business@omicsinc.com

1-702-714-7001Extn: 9042

Chemical Engineering and Chemistry Journals

Gabriel Shaw

chemicaleng_chemistry@omicsinc.com

1-702-714-7001 Extn: 9040

Earth & Environmental Sciences

Katie Wilson

environmentalsci@omicsinc.com

1-702-714-7001Extn: 9042

Engineering Journals

James Franklin

engineering@omicsinc.com

1-702-714-7001Extn: 9042

General Science and Health care Journals

Andrea Jason

generalsci_healthcare@omicsinc.com

1-702-714-7001Extn: 9043

Genetics and Molecular Biology Journals

Anna Melissa

genetics_molbio@omicsinc.com

1-702-714-7001 Extn: 9006

Immunology & Microbiology Journals

David Gorantl

immuno_microbio@omicsinc.com

1-702-714-7001Extn: 9014

Informatics Journals

Stephanie Skinner

omics@omicsinc.com

1-702-714-7001Extn: 9039

Materials Science Journals

Rachle Green

materialsci@omicsinc.com

1-702-714-7001Extn: 9039

Mathematics and Physics Journals

Jim Willison

mathematics_physics@omicsinc.com

1-702-714-7001 Extn: 9042

Medical Journals

Nimmi Anna

medical@omicsinc.com

1-702-714-7001 Extn: 9038

Neuroscience & Psychology Journals

Nathan T

neuro_psychology@omicsinc.com

1-702-714-7001Extn: 9041

Pharmaceutical Sciences Journals

John Behannon

pharma@omicsinc.com

1-702-714-7001Extn: 9007

Social & Political Science Journals

Steve Harry

social_politicalsci@omicsinc.com

1-702-714-7001 Extn: 9042

 
© 2008-2016 OMICS International - Open Access Publisher. Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version