Arm Dynamics Simulation

The ability to mathematically model the movement of a robot manipulator is a prerequisite to the undcrstanding of the key factors that influence a manipulator’s performance. This paper presents a manipulator modcl which has been uscd to simulate and control a real robot arm. A method of describing the arm by its rotational characteristics, a sct of equations called the Inverse Arm, and an algorithm called Forward Arm are prcscnted. Thc Forward Arm simulatcs the movcment of an arm, and the Inverse Arm provides a means of computing thc corrcct voltages to apply to an arm to achieve a desired movement. A rm I):,: nani ics Si niulation 1 23 Novcmbcr 1982 Introduction [\ mathematical model of a physical sysrcm, such as a robot msnipulator(arm), is one of thc most uscful tools ;r\ailablc for studying thc system's behavior. 'fhc model. usually in the form of a compiiter program, can bc used to study the system in several ways. l h e dcvclopmcnt of an accurate modcl lcads to a full understailding of all of the key elcmcnts of the system. Thc model provides a means of tcsting the system under conditions that woald be dmgcrous o r impossible for the rcal system. 1,argcr sysrcms that contain the modclcd systcm can be tcstcd with thc modcl instcad of the rcal system. As part of the Carnegic-Mcllon University Direct-Drive Manipulator Project CCMU 1111 Arm Project) we have dc\dopcd R niathcmatical inodd of the manipulator. The madicrnatical cquations are based on a Ncwton-Eider analysis of frce-body dynamics developed for robotic manipulators. [SI [ll] This paper describes the structure of the modcl that simulates h e dynamic motions of our manipulator. The model is divided into three parts. 0 A de/nilcd dcicriplion of /he slnrcturr o,f/he a m . The description of the stnicture is contained in a Manipulator Database which consists of two parts: the kinematic and the dynamic. 'The kinematic dcscriptjon specifics the re!ative positions between the links of thc a r m and givcs the axes of rotxion h r each of thc joints. This description is easily detcrinincd from the nicchanical drawings of the aim. The d~naniic description contains thc momcnt of inertia, tlic center of Inass, and the mass for cach of thc links. A computer program was written to calculate thew values from a (Imbw fik(thc Parts I)atab;tsc) that contains a description of evcry piece of the arm. e 77ie Invme A17n. This is a set of equations which, hhen evaluated, yield the motor voltages rcquircd to produce certain accelerations. This is the inverse of a real arm which produces accclcrations given the voltages. The Inverse A m part of the model is necdcd for tlic third part which is the Forward Arm. 0 The Forward Arm. This part of the model contains an algorithm which can compute valucs for the acceleration of the joiiits in thc ann when the motor voltages zrc specified. When the accclcrations are integrated over a period of time, the new positions and velocities can bc de tetmined. Arm Description 'Ibc CMU DD Arm consists of seven links, numbcred 0 to 6, going from thc base (link 0) down to the hand (link 6 ) . Therc are six joints numbered 1 to 6 . ?'he odd numbercd joints are rofafrcjrial joints which rotate in thc sainc manner as the turning of a screw. 'I'l-~e evcn numbcred joints are pivofol joints, which move in a manner similar to that of a pcrson's elbow. Each link has a coordinate frame associatcd with it. The Denavit-Hartenburg convention [5] for assigning coordinate framcs to nianipulator links is used to specify the coordinate frames of the manipulator because it simplifies the evaluation of the cquations used in the Iiiverse Ann and Forward Ami parts of thc model. Arm Ilynamics Simulation 2 23 Novcmbcr 1982 ‘Hie I)cn,ii4-J laltCfiblJrg convention specifics that link i+ I rotatcs around thc Z axis of link i, dcnotcd Zi, \~,hcn joint i t 1 t u i m . L i n k 1, thcrcfore. rotatcj around Zo at joint 1 and link 2 around at joint 2, ctc. ‘I’he X axis o f cncli link’s coordinatc systcm points along the common normal of the link’s Z axis and the Z axis of thc prckious link. I f thcrc is no conimon normal, such as whcn thc two Z axcs intcrscct, the direction of the X axis is arbitrary. so long as it is pcrpcndicular to the Z axis. The Y-axis is perpendicular to both the X and I , axes and c o ~ t i i ) l ~ t c ~ 3 right-handcd coordin;ite fr:ime. The coordinatc frames spccificd by the IlcnavitHi1;.tcnburg convcrition for the Ch4Ll DIl Arm arc shown in figure 1. [4] I( inzma t ic Description Thc dcscription is a datnbasc which contains three pieces of information, dcnotcd a, r, and a. for cach joint. For any joint i, a spccifics thc nnglc of rotation ncccssary to bring Zi-l parallel to or collinear with Zi. ‘l’hc paramctcr r spccifics dx distanci? aloiig thc Z l axis from the link i-I coordinate system to the link i coordinate systcm. l’hc a unramctclspccifks Lhe distance from link i-I to link i along the Xi-1 axis. Once the coordinarc Frame origins :!rc dctcrinincd in thc Dena\it-llartenburg conventicn thc a, r, and a parameters can be obtaincd from the ~i~rcclianical drawings. ‘l’lic kincmatic dcscription spccifics the gcncral organization of a manipulator. D y !i am i c D e s c rip t i on Obtaining a dynamic description of the arm requires a greater effort than the kinematic dcscription. l’he d;, n,!mic dcscription consists of the moment of inertia tensor. center of mass vector, and the mass scalar of cach link. To dctcrininc this data we have dcvelopcd a Parts I)ntabasel for the Ch2U-IlIl Ann which has iiifoimXiori o n cach of thz six links. Within thc databasc, each iink is separatcd intci several p~7rt .s which are nu;?iljcrcd in the mcchanical drawings. h c h par/ is broken down into several secfions. F ~ c h section is dc:ciibcd as a cylinder. semi-cylindcr, rectangle, sphere, or prism. The characteristics of each of the parts in c u r inaniplrlator can be approximated by combinations of these shapes. The databasc contains ditncnsions, dcnsiries, and locations of each section relative to the pari that it bclongs to. With this infoinlation we can dctcrmine.thc moinent of inertia, center of mass, aiid mass of each section. Once the infomiation is calculated for cach section, it can then be determined for cach par1 using the formulas for transforming moments of inertia. [7] [lo]’ ‘I’hc dynamic dcscription of cach link can be computed in a similar manncr from the parf information. The resultant dcscription is saved as part of the Nlanipulator Database and is rcferrcd to by the rcsL of the arm model. I t must be rccalcularcd if the const~uction of the arm changes. Thew are other pieces of data related to the dynnniic description of the ann which can only be expcrirnently dctcnnincd. The most important of tliesc arc the motor constants. Each motor has a resistance, R. arid a torqw constant, Kt. ’l’hc resistance of the motor relates the current in each motor’s armature to the \. oltagc applicd across cach motor’s terminals. The torque constant relates thc torque produccd by the motor tu t5c currcrit in the armature and it relates the spccd of the motor to the back EMF(E1ectro-Motive Force). ’ Ilie link coordinatc slstcins used in the parts database are ;;signed for convenience. The software which gencratcs Uie dynamic dcscriprion changes thcsc coordinatc syaems to those of the Dcnavit-IIarteiiburg convention whcn the klanipulnlor Ilatabase is produced. LChnpie: 10.5 The Inertia Tensor