Dynamics of a general multi axis robot with analytical optimal torque analysis

The robot equations of motion are obtained from the implemented program and verified against those obtained using only Lagrange equation. The output of program for the 3 DOF robot was used to find the optimal torque using analytical optimization analysis for a given set of parameters. This procedure analysis can be used as a benchmark analysis for any optimization technique.
Nội dung trích xuất từ tài liệu:
Dynamics of a general multi axis robot with analytical optimal torque analysisJournal of Automation and Control Engineering, Vol. 1, No. 2, June 2013Dynamics of a General Multi-axis Robot withAnalytical Optimal Torque AnalysisAtef A. Ata, Mohamed A. Ghazy, and Mohamed A. GadouDepartment of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria 21544,EgyptEmail: atefa@alexu.edu.eg, {mohghazy, eng.mohamed.aref}@gmail.comAbstract—Robot dynamics is considered one of the mostimportant issues in robot design and control. Manytechniques were developed to find equations of motion. Oneof these techniques is Lagrange-Euler method which issuitable for numerical simulation. In this paper animplementation of Lagrange-Euler to find equations ofmotion for any general multi-axis robot giving only robotconfigurations is introduced. The program is verified for a 3Degree-of-Freedom robot. The robot equations of motionare obtained from the implemented program and verifiedagainst those obtained using only Lagrange equation. Theoutput of program for the 3 DOF robot was used to find theoptimal torque using analytical optimization analysis for agiven set of parameters. This procedure analysis can be usedas a benchmark analysis for any optimization technique.generalized coordinates. In the comparison betweenNewton-Euler and Lagrange-Euler Silver [4] showed thatthe computational complexities of the two techniques arethe same.In this paper, an implementation based of LagrangeEuler technique to determine the equations of motion foran n-axis robot is presented. An example for a 3 DOFrobot is illustrated to verify the proposed algorithm. Ananalytical optimization approach is investigated as abenchmark for minimum energy using any optimizationtechnique. The paper is organized as follows: section IIcontains the equations of motion in compact form anddetails of the algorithm to get each term. Section IIIpresents the case study for a 3 DOF robot while sectionIV is devoted to analytical optimization analysis followedby discussion and conclusions in section V and references.Index Terms—dynamics, lagrange-euler, genetic algorithm,trajectory planning, optimal.II.I.INTRODUCTIONDYNAMICAL ANALYSISThe objective of Lagrange-Euler method is to getequations of motion for a robot provided that robotconfigurations and the kinematic equation in terms ofDenavit-Hartenberg are given. Lagrange-Euler techniquedepends on finding kinetic energy of a body which ischanged with its spatial and angular velocity in generalmotion and finding its potential energy. In the case ofrigid body dynamics, the only source for potential energyis gravity. It is suitable for rigid robot but there weresome research activities on using this technique forflexible robotic manipulators and this is mentioned byLin and Yuan [5].The general form of robot equation of motion can begiven using Lagrange-Euler technique in the followingform:Optimal trajectory planning is regarded as a veryimportant area for research where some constraints andobjectives are required to be optimized. Some examplesof objective functions are minimum path for amanipulator travel to achieve its target, minimum time intravel, and minimum applied torques on manipulatorjoints. Also there may be constraints on the maximumtorque that can be applied on any joint. Robot dynamicmodel is important as it provides relationship between theapplied forces/torques and the motion of robotmanipulator.Many techniques have been developed to find theequations of motion of a multi-degree-of-freedom robotsuch as Newton-Euler and Lagrange-Euler. A shortreview in the field of this research including fundamentalwork and present techniques can be found in Featherstoneand Orin [1]. Hollerbach [2] proved that Newton-Eulerapproach can be formulated as a recursive structurewhich can be faster than treating the manipulator as awhole; he also showed that Lagrange-Euler can be usedin a recursive manner. Lagrange-Euler technique is sosuitable for numerical solution. Khalil [3] provided moredetails about these techniques and presented sometechniques on conversion between Cartesian andnnj 1j 1 k 1(1)is thewhere is the actuator torque of joint i,actuator inertia of joint i, is the generalized coordinaterepresented by p if the joint is prismatic or represented byif the joint is revolute, n is the total number of links andterms can begotten as follows:Dij nTrace U pj J pU pi T p  max ( i , j )Manuscript received September 11, 2012; revised December 22,2012.©2013 Engineering and Technology Publishingdoi: 10.12720/joace.1.2.144-148nTi  Dij q j  I i act  qi  Dijk q j qk  Di144(2)Journal of Automation and Control Engineering, Vol. 1, No. 2, June 2013Dijk nTrace U pjk J pU piT The developed program is able to find equations ofmotion for any general multi-axis robot giving robotconfiguration and its A matrices. Table I describesprogram inputs in details.Program main units are:(3)p  max ( i , j , k )nDi    m p g TU pi r p(4)p 1A. ModelThe model is the main procedure of the program. It isused to do summations and to calculate D terms whichare needed to be computed.where is the pseudo inertia matrix for link i and isdefined by:  I xx  I yy  I zz2I yxJi  I zxmi xiI xyI xzI xx  I yy  I zzI yz2I zyI xx  I yy  I zz2mi zimi yimi xi mi yi mi zi mi B. GetD2This module is used to calculate termsC. GetD3This module is used to calculate terms(5)Ajq j* Ai.D. GetD1This module is used to calculate terms .The following is a flowchart for the main componentof the implemented programwhereis the mass of link i andarecoordinates of center of link i relative to the linkcoordinate frame. On th ...