Inter-Ing 2007 INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 15-16 November 2007. UNCERTAINTY REGION SIMULATION FOR A SERIAL ROBOT STRUCTURE MARIUS SEBASTIAN RUSU, LUCIAN GRAMA PETRU MAIOR UNIVERSITY, TG-MUREŞ Keywords: vbratons composton, uncertanty regon, smulaton Abstract: As the undesred vbratons n mechatronc and robotc systems lead to the alteraton of ther behavor and fnally of ther performances, t s recommed that n the begnnng of the desgn phase, vbratons predcton and removal should be consdered as an mportant requrement. The knowledge of the nfluence of vbratons over the desgned system provdes the bass for elmnatng ther undesred effects as much as possble. Ths paper presents a smulaton of the uncertanty based vbratons effect model appled to a seral robot. 1. Introducton Buldng a robot that meets the proect requrements s the goal of the product desgn and development. The requrements mpose performance crtera to the new system. These performance crtera are usually constraned by the undesred phenomena, such as vbratons. The vbraton reecton should be an mportant goal of the robot proect, as ts avodance can lead to an neffcent and useless system n the. Postonng s one maor feature of the robot and ths s strongly nfluenced by mechancal, electrcal and even algorthm faults. Undesred vbratons can alter the poston of the fnal effector and to make the robot less precse. Ths s why, knowng the behavor of these vbratons can offer a soluton for performance mprovements. In the next pages we shall present a method of modelng the effect of vbratons over a smple 1-degree-of-freedom robot and then a more complex n-degrees-of-freedom robot. 2. About the uncertanty regon model of the postonng alteraton of the effector We shall buld our smulatons on the uncertanty regon model of the postonng alteraton of the robot s effector. Ths model states that the effector s poston can be placed n an uncertanty regon, usually around ts deal poston. The smplest model presents an uncertanty surface around the deal pont where the effector s characterstc pont can be found. A more complex approach makes the dfference between longtudnal and transversal waves n the robot sold body and proposes a dsc shaped uncertanty surface surroundng the central deal poston. Stll these two models are lmted to surface areas, whle the volume of the obect s not taken n consderaton. Therefore, a more realstc representaton arses where the uncertanty regon s shaped as a volumetrc sphere around the deal poston. The effector can be found anywhere nsde the sphere s volume. Usually, the hghest probablty to fnd the effector s closer to the deal poston, therefore, nsde the sphere, there s a normal probablty dstrbuton feld. III-4-1
III-4-2 To enhance ths new perspectve, one shall consder the volumetrc dsc shaped normal dstrbuton to model the uncertanty regon. These two volumetrc models are presented n Fg. 1. a) sphercal volume uncertanty regon b) dsc shaped volume uncertanty regon Fg. 1 Volumetrc uncertanty regon models Startng from a smple, yet general, rotaton onts seral robot structure n Fg. 2, a forward knematcs model can be deduced n the form of the equatons n (1). Fg. 2 Seral planar robot model ± ± ± n n f n f l arctg l y l x 1 ' 1 1 ' 1 1 ' sn cos ε ε ε (1) Ths, along wth a trval 1-degree-of-freedon robot model wll be the bass for the followng smulatons.
3. Smulaton of vbratons nduced alteraton for 1-DOF robot We shall consder a trval 1 DOF robot havng a rotaton movement wth a 10 to 170 openng and a 0.9 step. The arm length s 30cm and the devaton coeffcent s 0.6. We shall buld the deal workspace and then some altered workspaces. The smulaton program s wrtten n Matlab 6.5 and ts source code s detaled n the next pages. The graphcal results are shown n Fg. 3 Fg. 6. Fg. 3 1 DOF robot deal workspace Fg. 4 Uncertanty regon borders Fg. 4 shows the borders of the uncertanty regon around the desred poston of the effector. The followng two mages show the nfluence of the Gaussan dstrbuted perturbaton and the unform dstrbuted perturbaton of the workspace of the trval robot. Fg. 5 Normal dstrbuted workspace alteraton Fg. 6 Unform dstrbuted workspace alteraton III-4-3
Fg. 7 Normal dstrbuted uncertanty regon Fg. 8 Unform dstrbuted uncertanty regon Images n Fg. 7 and Fg. 8 present a more detaled smulaton of the uncertanty regon around the desred workspace. 4. Generalzaton of the workspace alteraton for an n-dof seral robot We shall consder a seral rotaton n-dof robot as descrbed n Fg. 2. Ths case wll assume that each element of the chan can alter ts angle ndepently and the cumulatve effect wll be found n the placement of the effector. The followng fgures are made for a 2-DOF seral robot, but the smulaton s not lmted to ths. The robot s a rough structure wth the followng elements: 1: angles from 0 to π rad wth a 0.05 rad step angle, length 30 cm, devaton 0.8 cm; 2: angles from 0 to 2π rad wth a 0.05 rad step angle, length 9 cm, devaton 0.7 cm. Fg. 9 to Fg.12 presents the 2-DOF robot s workspace wth and wthout alteraton. Fg. 9 2-DOF robot deal workspace Fg. 10 2-DOF snus alteraton III-4-4
Fg. 11 2-DOF unform alteraton Fg. 12 2-DOF normal alteraton 5. The smulatons Matlab source code Ths chapter ncludes some of the Matlab source code for the smulatons above. Source code for the 1-DOF smulatons % ROBOT DEFINITION % angles (mn:step:max) angles p*(10/180) : p*(0.9/180) : p*(170/180); % arm length length 30; % uncertanty parameters defnton devaton 0.6; fgure (1); % IDEAL WORKSPACE x length * cos ( angles ); y length * sn ( angles ); subplot ( 221 ); grd on; hold on; ttle ('1 DOF robot deal workspace'); axs ( [ -length-3 length+3 0 length+3 ] ); plot ( x,y ); % ALTERED WORKSPACE BORDERS xb1 x + devaton * cos(angles); yb1 y + devaton * sn(angles); xb2 x - devaton * cos(angles); yb2 y - devaton * sn(angles); subplot ( 222 ); ttle ( '1 DOF Altered workspace borders' ); grd on; hold on; axs ( [ -length-3 length+3 0 length+3 ] ); plot ( x,y, 'b' ); plot ( xb1,yb1, 'r' ); plot ( xb2,yb2, 'r' ); leg ( 'deal workspace', 'uncertanty regon border' ); % NORMAL ALTERATION EXAMPLE gaussan_nose devaton * randn ( sze(angles) ); xna x + gaussan_nose.* cos(angles); yna y + gaussan_nose.* sn(angles); subplot ( 223 ); ttle ( '1 DOF Gaussan alteraton example' ); III-4-5
grd on; hold on; axs ( [ -length-3 length+3 0 length+3 ] ); plot ( x,y, 'b' ); plot ( xna,yna, 'r' ); leg ( 'deal workspace', 'gaussan dstrbuted alteraton' ); % UNIFORM ALTERATION EXAMPLE whte_nose devaton * ( rand ( sze(angles) ) - 0.5) * 5; xua x + whte_nose.* cos(angles); yua y + whte_nose.* sn(angles); subplot ( 224 ); grd on; hold on; ttle ( '1 DOF Unform alteraton example' ); axs ( [ -length-3 length+3 0 length+3 ] ); plot ( x,y, 'b' ); plot ( xua,yua, 'r' ); leg ( 'deal workspace', 'unform dstrbuted alteraton' ); fgure (2); Workspace calculaton recursve functon % functon: workspace % descrpton: computes recursvely the vector % of ponts that descrbe the robot's % workspace % nputs: onts: the lst ont obects % : current ont % ponts: the current ponts vector % output: the new vector of ponts functon [p] workspace ( onts,, ponts ) % stops teratng when fndng the effector f ( length(onts) ) % calculates every posble poston for the current % knematc chan confguraton for ( 1:length(onts().angles) ) onts().ndex ; % enlarge the ponts vector n length(ponts) + 1; ponts ( n ).x 0; ponts ( n ).y 0; % calculate the new posble poston for ( k1:length(onts) ) ponts ( n ).x ponts ( n ).x + onts(k).length * cos(onts(k).angles(onts(k).ndex)) + onts().nose(onts(k).ndex) * cos(onts(k).angles(onts(k).ndex)); ponts ( n ).y ponts ( n ).y + onts(k).length * sn(onts(k).angles(onts(k).ndex)) + onts().nose(onts(k).ndex) * sn(onts(k).angles(onts(k).ndex)); else % f the current terator s an ntermedate element % store the new angle ndex and recurse toward the % effector for ( 1:length(onts().angles) ) onts().ndex ; ponts workspace ( onts, +1, ponts ); p ponts; III-4-6
return The recursve functon above terates through the knematcs chan and determnes the resultng workspace. For readablty, packed obects have been used. 6. Conclusons Gatherng the desgn data and smulatng the vbraton nduced alteraton of the effector, whch drectly deterorates the robot s workspace, can gve a detaled vew of the problems to avod, as well as provdes the bass for reecton measures that should be taken. The uncertanty regon model ncludes the most common devatons and can be regarded as the goal of a perturbatons flterng n the desgn process. 7. Bblography 1. Merovtch, L, Prncples and technques of vbratons, Prentce-Hall Internatonal, New Jersey, 1997 2. Lupea, I., Robot s vbrat, Ed. Daca, Clu Napoca, 1996 3. Ispas, C., and others, Vbratons des systemes technologques, Ed. Agr, Bucurest, 1999 4. Moldovan, L., Vbrat mecance, Ed. UPM, Tg-Mures, 1996 5. Vstran, M, Roboţ ndustral, Ed. UT Clu, Clu-Napoca, 1994 6. Vorel, H, ş.a., Roboţ. Structură, cnematcă ş caracterstc III-4-7