基于旋量法的飞机表面清洗机器人运动学分析_英文_.docx

doi: 10 3969 / j issn 10013881 2014 06 011kinematicsanalysisofcleaningrobotforaircraftsurfaces based on screw theory*yu-yang jin1，2 ，ming-lu zhang1 ，qing-ji gao2 ，chang-juan yu11 school of mechanical engineering，hebei university of technology，tianjin 300130，china;2 obotics institute，civil aviation university of china，tianjin 300300，chinaabstract: focusing on 5-dof cleaning robot for aircraft surfaces，the screw theory and exponen- tials formula are used for the forward kinematics solution of robot and the configuration of the end-effector relative to inertial coordinate frame was obtained besides，jacobian matrix of for- ward kinematics is obtained by the twists，which establishes a basis of singularity of mechanism， future real-time control，and manipulabilitykey words: obot，jacobian，kinematics，screw theory，exponential formula1 introductiond-h method and screw theory method are used to solve the kinematics for a robot d-h method is relatively mature and wildly used in analysis of kine- matics for robots the screw theory method only needs to build two coordinate frames，which has a clear geometric meaning literature1 presented a kinematics parameter calibration method for serial ro- bots based on the product of exponential ( poe) for- mula a linearized model describing the relationship between the errors in the end-effector and the errors in the joint twists，and in the zero position twist is obtained by differentiating the kinematics equation literature2gave the result of inverse kinematics by using the combined method of algebraic elimination method and paden-kahan sub-problem methodjacobian matrix relates to static analysis，speed control，and operation3 the domestic research of the robot jacobian matrix in recent years is as fol- lows literature4 used differential transform and vector multiplication method to solve the jacobianeceived: 2013 11 04* project supported by undergraduate innovative training pro- gram ( iecauc13008) yu-yang jin，phd e-mail: yyjin cauc edu cnmatrix literature5 calculated the characteristics of the rehabilitation robots jacobian matrix of the ro- bot by using screw theory this paper built kinemat- ics and jacobian matrix by applying screw theory2 the construction of kinematics coordi- nates for aircraft cleaning robotthrough the analysis of working environment and working process for aircraft surface cleaning robot， mechanism should have large workspace and high stiffness therefore，the robot should have the opti- mal workspace， dexterity and obstacle avoidance space to meet targets during operation according to the functional requirements of cleaning aircraft，the five degree of freedom robot consists of four rotary joint and a prismatic joint，as shown in figure 1figure 1 aircraft cleaning robot in its reference configuration0l33 the solution to kinematics of the clean-q4 = l1 + 3 ，q5= l1 + 3 ing robotl0 l0 + l2 the essence of serial robot is spatial kinematic2) if = 1，e is defined as2chain composed of links with the joint connection d- h method and screw theory method are usually usede= i3 + sin + 2 0 x3x2( 1 cos)( 1)to solve the kinematics for a robot the d-h methodwhere = x30 x1 ，= t -idescribed configuration between each link by link x2x10 frame n-dof serial robot requires a total of n + 1link frame but screw theory does not need to estab-lish the adjacent link frame exponential formula on-e iiusing equation ( 1) ，the exponential of this twistare given by: cos1 sin10 11ly needs two coordinate frames，the base frame s ，and the tool frame t attached to the end-effecter of robot as shown in figure 1，the base frame s ise= sin1cos10 001 attached to a point on the cleaning manipulator which is stationary with respect to link 0 all of vectors ande 22 100 = 0cos2sin2 points are specified relative to the base frame s 0 sin2cos2 all joint angles are defined by right-handed coordi- 100 nate system，the angle about along direction of thee 44 = 0cos4sin4 axis is positive if it represents a clockwise rotation asviewed along the direction of the axis 0 sin4cos4 1) for each joint，construct a twist i ，i is the 55 100 twist of the ith joint relative to the base coordinateframe s for a revolute joint，the pure rotation a-bout an axis l = q + by an amount ，the twiste= 0cos5sin5 0 sin5cos5 3) a twist has the form: i = ( i qi ，i )，where i isi is associated with a screw of jointt3a unit vector in the direction of the twist axis and3i，if = 0，e = i ，where = q01qi is any point on the axis for a prismatic，ejoint，which is described by a linear displacement i along a directed axis，where positive displace-if 0e = e is defined as( i e ) ( )+ t ( 2)ment is taken along the direction for axis when aspecial cases is pure translation along an axis，we01the individual exponentials eii are given by:have that i =vi where vi is a unit vector311 cos1 sin100 sin1cos100 0pointing in the direction of translation the corre- sponding twist is:e= 0010 0001 0 1 0 1 = 0 ，2 = 0，3 = 1 ， 1000 1 0 0 e22 = 0cos2sin2 l0 sin2 1 1 0 sin2cos2l0 ( 1 cos2 ) 4 = 0，5 = 0 0001 0 0 1000 0 0 q1 = 0，q2 = 0 ，e33 0103 = 0010 l0 l0 0001 1000the velocity of the individual jointse44 = 0cos4sin4 ( l1 + 3 ) ( 1 cos4 ) l0 sin4 vss 0 sin4 cos4 ( l1 + 3 ) sin4 + l0 ( 1 cos4 ) s6 nst = jst ( ) ( 4) 0001 1000 where jst ( ) bianjsis the spatial manipulator jaco-st ( ) = 12n ( 5)e55 = 0cos5sin5e24 where = ad ( e 11 e i 1i 1 ) ，it shows thati iwhere 0 sin5cos5e34 0001 js ( ) 6 n is a configuration-dependent matrix，stand the contribution of the ith joint velocity to theend-effector velocity is independent of the configura-e24 = ( l1 + 3 ) ( 1 cos5 ) sin5 ( l0 + l2 )e34 = ( l1 + 3 ) sin5 + ( l0 + l2 ) ( 1 cos5 )4) if we define gli 1 li ( i ) as the transformationtion of later joints in the chain according to twist co-ordinates，the twist coordinates related to rotary joint is given:between the adjacent link frames，then combining the = ( q ， ) tiii iindividual joint motions，the forward kinematics mapwhere iis the unit vector of the direction of the ax-of manipulator has the form:is of rotary joint in current configuration:gst ( ) = e 11e 22e nngst ( 0)( 3)i= e珚11 e珚22 e珚i 1i1 i( 6)gst ( 0) is the rigid body transformation between t1) solving equation ( 6) for i :and s when the clean manipulator is in its refer-ence configuration ( i = 0) :1= 1= 0; 0; 1 tcos sin011 1 il1 gst ( 0) = l3 2z1= e 2= sin1cos10 0=0l0 + l2 cos001 011 sin1 this completes the derivation of the forward ki-nematics of robot00 -sin1 cos2 4 jacobian of the manipulatorv = ez1e x 21 = cos1 cos2 based on the relationship between the joint an-34 = ez1e x 204 = sin2 gles and the end-effector configuration，we need to cos1 sin1 0 10 0 1figure out the linear mapping relationship between the sin1cos10 0cos2sin2 0 =end-effector velocity and the velocity of the individualjoints traditional，we describes the jacobian of a manipulator by differentiating the forward kinematics， 001 0 sin2cos2 0 cos1 sin but usually it is not easy to obtain the solving process and results，so we describe the jacobian of the for-ward kinematics map by terms of twists because the1 0z x x cos1 exponentials formula can obtain a very natural and5 = e 1 e2 e4 5 = sin1 explicit description of jacobian，it avoids destruction of geometric structure of rigid body motionlet gst : q se ( 3 ) be the forward kinematics 0 2) qi is a unit vector of any point on the axis in current configurationmap for a robot if the joints move along the path( qi ) = e珋11 e珋22 e珋i 1i1 ( ri ( 0) )( 7)( t) q，the instantaneous spatial velocity of theend-effector related to the base coordinate frame ineither configuration is given:1solving equation ( 7) for qi : 0 1 0 vs1q1 = 0 ，q2 = 0 st = g st ( ) gst ( ) so the end-effector velocity is linearly related to l0 l0 0 0 ( l1 + 3 )sin1 cos2 st = 0 024 8; 0; 0 110 4; 0，0 090 7; 0 104 7 tq4 = 0 + ez1e x2 l1 + 3 = ( l1 + 3 )cos1 cos2 figure 2 shows 3d model of aircraft cleaning ro-l l0 0 00 0 ( l1+ 3 )sin2 0 bot in adams we assembled the parts and added constraint，performanced the kinematics simulationq5 = 0 + ez1e x 2 l1 + 3 + ez1e x 2e x 4 0 =substituting angle variable and velocity value of each， l0 0 l2 joint into 3d modelwe measured the angular veloci- sin1 ( l2 sin( 2 + 4 ) + ( l1 + 3 ) cos2 ) ty and linear velocity the simulation result showsthat the jacobian matrix solution is correct cos1 ( l2 sin( 2 + 4 ) + ( l1 + 3 ) cos2 ) l2 cos( 2 + 4 ) ( l1 + 3 ) sin2 + l0according to equation ( 4 ) ，as long as the ve- locity of joint angle is given，the jacobian matrix of the end effector relative to the inertial coordinate frame will be obtainedjswherest = 1 ，2 ，3 ，4 ，5 t1 =0; 0; 0; 0; 0; 1t2 =l0 s1 ; l0 c; 0; c1 ; s1 ; 0t3 = c2 s1 ; c1 c2 ; s2 ; 0; 0; 0t4 =41 ; 42 ; 43 ; c1 ; s1 ; 0where41 = s1 ( l0 s2 ( l1 + d3 ) )，42 = c1 ( l0 s2 ( l1 + d3 ) )， = c ( l + d ) c2 + c ( l + d ) s2432131213155152531 1 = ; ; ; c ; s ; 0t ，where51 = s1 ( l0 + l2 c2 + 4 s2 ( l1 + d3 ) )，s253 = c12132 2 + 452 = c1 ( l0 + l2 c2 + 4 s2 ( l1 + d3 ) )， 2 ( c ( l + d ) + l s) 1 ( c2 ( l1 + d3 ) + l2 s2 + 4 ) )at the end of this procedure，the complete ma-nipulator jacobian is determined，where the various quantities are defined above5 the numerical solution of jacobian ma- trixfor example，we can solve the jacobian matrix of manipulator given 1 ，2 ，d3 ，4 as follow:figure 2 kinematics simulation of manipulator6 conclusion1) the kinematics of manipulator in this paper was completely described by twist coordinates of each joints we represent the forward kinematics using the product of exponentials formula to obtain the configu- ration of end-effector in se( 3 ) ，and the configura- tion of tool coordinate frame is independent of actual sequence of rotation and translation for each joint the twist of each joints are specified with respect to only base frame，simplifying the mechanism analysis2 ) jacobian matrix describes the relationship the velocity of end-effector and the velocity of each joint， which provides a basis for the trajectory planning of robot and real time control3) the velocity relation between the joint and end-effector was derived，and to study the relation- ship between the force spiral of the end-effector and joint torquejsst ( ) = 00 200 0 0 500 02 798 1 2 798 10 0 000 00 000 0 0 000 00 000 0 000 866 01 500 0 0 500 00 0 000 00 0 000 00 000 0 0 1 000 00 1 000 01 000 0 eferences1 gao wenbin，wang hongguang，jiang yong a cal-vs 1 000 00000when 1 = 0 1047 rad / s，2 = 0 078 5 rad / s，d3 = 0. 05 m / s，4 = 0 061 0 rad / s，5 = 0 048 8 rad / saccording to equation ( 4 ) ，the instantaneous spatial velocity of the end-effector is:ibration method for serial robots based on poe formulaj obot，2013，35( 2) : 156 1612 qian donghai，wang xinfeng，zhao wei，et al al- gorithm for the inverse kinematics calculation of 6 dof robots based on screw theory and paden-kahan sub-prob- lemsj journal of mechanical engineering，2009，45( 9) : 72 763 ni shoudong，wen jufeng，yan jingping the estab- lishment of the jacobian for the 4 dof redundant robotj chinese journal of scientific instrument，2001，22( 4) : 381 3824 song mengjun，zhang jianhua，zhang minglu，jia wei jacobian matrix analysis of leg for a new kind de- formable mobile robotj journal of machine design，2013，30( 3) : 21 265 murray m，li zexiang，sastry s s a mathematical introduction to robotic manipulation m cc press，19946 lan zhi，li zhenliang，li ya calculation of jacobin matrix of a 5 dof upper limb re-habilitation robot based on screw theoryj journal of machine design， 2011，28( 5) : 51 53基于旋量法的飞机表面清洗机器人运动学分析*金玉阳1，2 ，张明路1 ，高庆吉2 ，于常娟11 河北工业大学 机械工程学院，天津 300130;2 中国民航大学 机器人研究所，天津 300300摘要: 针对 5 自由度飞机表面清洗机器人，通过旋量和指数积公式求解了运动学正解，得出末端执行 器相对惯性坐标系的位形。采用运动螺旋求解了运动学正解的雅可比矩阵，为机构的奇异性、实时 控制、可操作性的研究提供了理论基础。关键词: 机器人; 雅可比矩阵; 运动学; 旋量理论; 指数积公式中图分类号: tp242introduction of the fluid control engineering institute of kunming university of science and technologythe fluid control engineering institute of kunming university of science and technology was set up in 1996 the researches of institute concentrate on electro-hydraulic( pn