通过优化柔性椭球体对欠驱动冗余度机械臂的自重构外文文献翻译、中英文翻译、外文翻译

附 录 第 38 页 共 62 页 SELF RECONFlGURATlON OF UNDERACTUATED REDUNDANTMANIPULATORS WITH OPTIMIZlNGTHE FLEXIBILITY ELLlPSOlD He Guangping SchooI of MechanicaI and Electrical Engineering， North China University of Technology,Beijing 1 00041， China Lu Zhen SchooI of Automation Science and ElectricaI Engineering Beijing University of Aeronautics and Astronautics， Beijing 1 00083， China Abstract： The multimodes feature， the measure of the manipulating flexibility,and self-reconfiguration contro1 method of the underactuated redun dantman ipulators are investigated based on the optimizing technology The relationship between the configuration of the joint space and the manipulating flexibility of the underactuated redun dan t man ipulator is an alyzed a new measure of man ipulating flexibiltty ellipsoid for the underactuated redundant manipulator with passive oints in Locked mode jsproposed,which can be used to get the optimal configuration for the realization of the self-reconfiguration control Furthermore a timevarying nonlinear contro1 method based on har monic inputs is suggested for fulfilling the self-reconfiguration A simulation example of a threeDofs underactuated manipulator with one passive joint features some aspects of the investigations Key words： Underactuated manipulators Selfreconfiguration Optimization Nonlinear contro1 0 INTRODUCTION Underactuated mechanism and manipulator can be used in some fields such as space technology,cooperation robot and metamorphic mechanism In the space field， for the sake of no losing the useful function or realizing the reconfiguration of the system the fault tolerance based on the underactuated technology is essential when some actuated components reveal some troubles An underacmated manipulator also can be designed as a coopera tion robot that is to say C0B0T The drivers ofthe COBOT are not for acre ating the machine but for providing a ldnematics constraint that is usually nonholonomic The C0B0T needs outside 1otee that is provided by operator and can complete some accurate apphcatlons such as in 华北科技学院毕业设计（论文） 第 39 页 共 62 页 biology engineenng， surgical， and semiconductor manufacture and so on In the field of mechanisms， the metamorphic mechanism has multimodes and can be trans formed from one mode to an other,had been presented recentlyt ,The transform ation between the diferent modes is likely to result in change of the Dofs or constraints of the metamorphic mechanism It is obviously that control the underactuated Redundant actuated an d flexible mechanism cannot be evaded Therefore。 the underacmated system becomes an attractive research field gradually The researches on the underacmated manipulators manifest the system cannot be controlled by kinematics Th e motion of the passive ioint can be moved by the dynamic coupling only L4,3J Jain， etal have shown that the dynamic coupling is second order nonholonomic cons仃 aints 0f the underacmated manipulator Incontrast with the fact that nonholonomic system is st died exten sively about one hundred of years in mechan ics but the motion planning an d control of the nonholonomic system iS no longer than tw o decade an d the st dies limited in the first order non holonomic system (such as wheel moving robott J,hopping robottand space robot WJ)mainlyIn the aspect ofthe research on the un deractuated manipulators Anthoney et alt have investigated the stability ofthe motion Arai et al LII J have proposed a time。 Scaling methodand achievedthe position control ofthe s3 stem Lee et al LlZ have presented several kinds of nonlinear control metbo d for un deractuated acrobot These researches on the control of underac mated system have revealed that these methods are nothing but nonlinear,time varying and discrete in nature The fact is mat Brocket have proved that there is no smooth and static state feedback 1aw that stabilizes the system to a given confign ration asymptotically An obvious feature of the nonholonomic system is tllat it is controllable in a configuration space with more dimen sionsthan thatoftheinput space S0matmuch atentions have beenpaid on the control of the nonholonomic system The underactuated mechan ism or manipulator also iS disobedient to the fundamental principle of the mechanism design ingtheory that js the number of the acmated components should equal to that of the DOFs of the mechanism The underactuated manipulator had been suggested firstly is not by reason that it has some merits distinctly， but some researches show that the underactuated mechanism design ed purposely also is aluable Forexample Rivhter, 附 录 第 40 页 共 62 页 havemeasuredmulti dimensions force bya flexible underactuated manipulator,Nakamura etalt have design ed a nonholonomic manipulator based on the wheel rolling contact and controlled a four DOFs planar manipulator by two Inputs,He etal have proposed a collision free motion plan algorithm for the un deractuated redundant manipulator Based on the results that have been discussed above。 we Call conclude that the investigation about the un deractuated manipulator deliberately may result in some new phenom enon to be discovered， techniques to be proposed or theory to be formed therefore could be develop the mechanics potentially In this Pape we explore the static feam re and self-reconfigu ration control method for the underactuated redundant manipulators 1 FLEXIBILITY ELLIPSOID The manipulating stiffness is an important parameter of a manipulator,which can be used in the force or impedance contro1 A manipulator is open chain in mechan ism generally,and the links always are rigid bodies， so the deform ation on end effector results from the ioints mainly The stif mode can be written to the follow equation Approximately: iii KM i=1， 2， ， n （ 1） where: iM 关节 i 的转矩 i 关节 i 的变形量 ik 关节 i 的硬度系数 If the gravitation and the friction in the joints are ignored， supposing there is a force vector mRF on the manipulatorsend efector， the equivalent joint torque can be written as: FJM T （ 2） where: nRM EquiVaIent torque ofthe joint nmRJ JacObianmatrix It is well known that the deformations on the joints and the end effector have a relationship as follows Jx （ 3） where: x Micro displacement ofthe end efector Micro displacement of the joints 华北科技学院毕业设计（论文） 第 41 页 共 62 页 We write Eq(1)as amatrixform and combineitwithEqs (2)and(3)， by some simple calculations， the relationship between X 和 F can bewriten as F)JJk(X T1 （ 4） where: If we define T1JJkC （ 6） Eq (6)is to say the flexibility matrix of the end-efector Where as 1c corresponds to the stifiness matrix in task space The flexibi1ity matrix C can be used to measure the static feature of Manipulator Matrix C is also a function of the Jacobian hence it is changeable in a large rang for it relations to the configuration and the construction parameters The variable features of the manipulator in static can be used to complete some compliant and complex man ipulation such as assemblage， polishing treatment and soon Based on Eqs (5)and(6)，we can see matrix C is symmetric If we define )CCdet( T （ 7） and decompose matrix C by the singular value， from Eq (7)we have m1i i （ 8） where i ， i=1,2,3,m, denotes the singular value ofmatrix C Therefore the expression TCC is a positive definite symmetric matrix， an dan equation canbedefined as: 1x)CC(x TT （ 9） Eq.(9)describes the equation of a generalized ellipsoid This ellipsoid is to say the generalized flexibility ellipsoid(GFE) The principal axes of the ellipsoid are equal to the singular values of matrix C respectively For some intuitionistic sake， a planar two links manipulatorthat the length oflinks 2,1i,m0.1L i ， is regarded as an example(Fig.1)， and some GFEs are shown in Fig.2 and Fig.3 附 录 第 42 页 共 62 页 Fig.1 Planar 2R manipulator Fig.2 GFE of a full actuated planar 2R manipulator Fig.3 GFE ofa full-actuated plan ar 2R man ipulator These figures show that the measure depends on the configuration and the structure param eters W hereas a full actuated manipulator 1s unable to be changed to the structure par am eters generally Therefore， the GFE can be changed by diferent configuration(Fig.2)but not the structure parameters(change from Fig.2 to Fig.3) When some passive ioints are introduced into the fu11 actuated manipulator,for some convenience supposing that the passive joints are equipped with brakes and position sensor so matthe brak es can switch the passive ioints betw een the free swing mode and the locked mode thus the underactuated red undant manipulator reveals some redundant DOFs in kinemat ics but cannot be shown in“self-motion”for that the dimension of the inputs space is not more than that of the task space typically on the other hand， switching the mode of the passive ioins can 华北科技学院毕业设计（论文） 第 43 页 共 62 页 reconfiRalre the un deractuated manipulator， and the system reveals some dexterousness in adapting to diferent works 2 FLEXIBILITY M ATIUX Supposed that there are s passive joints in an underactuated redundant manipulator,and the passive joints are equipped with brakes When the passive joints are freeswing mode， the micromotion equations ofthe manipulator can be written as: ppaa JJx （ 10） where mRX Micro displacement of the end efector nmRJ Sub matrix of the Jacobian of the manipulator corresponding to the actuated joints 3pn R,R Micro displacement in actuated joints and the passive joints respectively When the passive joints are locked， the micromotion equation can be described as: qJx i （ 11） where mRX Microdisplacement of the end efector nmi RJ Jacobian of the manipulator as passive joints locked nRq Microdisplacement in the actuated joints It is obviously that Eqs (11)and(3)have the same forms Eqs (10)and(11)indicate that an underactuated manipulator has two diferent model in kinematics In other words， the system has the feature of multimodes in kinematics A planar 3R manipulator shown in Fig.4 can be regarded as an example of this The second joint of the manipulator is passive， and the others are actuated When the passive joint is free， 3R can be chosen as the generalized coordinates If the passive joint is locked， the DOFs of the manipulator chan ges to tw o， and the generalized coordinates can be selected as2Rq For the reason of q obviously thus the Jacobian matrix satisfies the relationship of 附 录 第 44 页 共 62 页 Fig.4 Planar 3R manipulator By reason that the underactuated manipulator have diferent kinematics modes， one can select an optimal configuration and reconfigure the manipulator for adapting to a diferent task An essential problem is how to predict the perform ance of the manipulator for fu11一actuated model based on the underactuated mode of it Unlike a fu11 actuated redundant manipulator,the underactuated redundan t manipulator cannot improve the perform ance by itself and implement a ma nipulation task simultaneously to the reason of fewer dimensions in input space than the task space A feasible approach is to decompose these tasks to indiferent time for actualizing For example when the manipulator is working in the un deractuated mode， one can reconfigure the mechanism for an appropriate configuration Whereas when the manipulator isworking in the fu11.actuated mode.one can control it to manipulation Substantively the manipulator working in underactuated mode can realize some manil：、 ulation such as position control or discrete pointtopoint motion But this is not the central of this Paper We pay atention to the static feature an d self-reconfiguration control method of the underactuated redundant manipulator The kinematical equations of the tw o modes of the underactuated manipulator can be established by many methods(such as denavitHartenberg method) but there is a dificulty in the structure parameter to be defined for a multiDOF manipulator that is complicated in mechan ism To resolve this problem next we analyze the relationship betw een the tw o modes of the underactuatedredundant manipulator 华北科技学院毕业设计（论文） 第 45 页 共 62 页 Given a special configuration of the manipulator,and supposing that has mn the deform ations of the end efector under the tw o modes of the mechan ism will be the same This can be expressed as:ppaai JJqJ （ 12） Let: 0JJaaaa （ 13） That means the micromotion occurred in joint space does not change the position of the end efector From Eq (1 3)， an expression canbewriten as: aapp JJ （ 14） where () denotes the MoorePenrose pseudoinverse Substituting Eq(14)into Eq(12)，we obtain aappi J)JJI(qJ （ 15） Eq(1 5)describes the same configuration of the two modes of the un deractuated manipulator Therefore the micromotion denoted by the tw o kinds of generalized coordinates will be same Let q， aform ulationis obtained as: appi J)JJI(J （ 16） Eq(16)shows the relationship of the Jacobian in the two modes， which can be used to predict the perform ance of the fullactuated mode Substituting Eq(16)into Eq(5)， a new flexi bility matrix of underactuated manipulator in fullactuated mode can bewriten as: Ti1i JkJC （ 17） The GFE for the underactuated manipulator can also be de fined according as Eq(7).Eq(17) indicates the static feature of the system after the mechanism reconfigured.One more， we show it by the planar 3R manipulator(Fig4),For a nonredundant manipulator if we give a point in the task space there is only one flexibility ellipsoid coresponding to it.By contrary， there are many ones coresponding to a point in task space for a redundant mechanism Supposing the links length of the 3R manipulator are120 .5L L m m and 3 1.0L mm， the initial configuration is 30,60,60321， some of the GFE scoresponding to the initial position ofthe endeffector are given in Fig.5 It is obviously there are a lot of configurations in ioint space coresponding to one state of the task space These configurations are coresponding to diferent generalized flexibility ellipsoids respectively Thus an underactuated redundant manipulator has the capability of 附 录 第 46 页 共 62 页 reconfiguration Generally， we expect the generalized flexibility ellipsoid has a similar perform an ce in diferent direction of the principle axes In other words， the ellipsoid is more similar to a bal1 So the first configuration has the best perform ance among the three configurations that is given in Fig.5 3 NONLINEAR CONTROL For finding an approach that can control the underactuated manipulator efficaciously,we analyze the dynamic system The dynamic equations for the underactuated manipulator can be written as: McIIapapaaa （ 18） 0cII ppppaTap （ 19） Where denotes me mass inenia matrix is the vector of Coriolis， centrifugal， gravitational and frictional torque，is torque vector of the actuated ioints, is the generalized coordinates vector corresponding to the actuated joints，pwhereas coresponding to the passive joints Eq(19)is second order nonholonomic constraints generally that have been proved by Jain,etalt An underactuated redundant manipulator has the capability of improving the performan ce of the mechanism for a given position in task space by reconfiguration For the reason of less dimension of input 华北科技学院毕业设计（论文） 第 47 页 共 62 页 space than that of the ioint space， the position control of the passive ioint can only be realized by the dynamiccoupling Based on the Brocket “s theoryt” there is no smooth static state feedback law that asymptotically stabilizes the system to a given configuration Therefore， the results that have been proposed for controlling of the nonholonomic system are nonlinear ,time varying and discrete n nature A nonlinear control method that has a manner of harm onic function for actuated joint is proposed in Ref 17 The basis of this method is that the passive joints is will deviate their equilibrium position when the actuated joint is moving in a periodic manner(Fig.6) The harmonic motion is input to the actuated joint， such as tcosAa （ 20） tsinA （ 21） tcosA 2a （ 22） Where: AAmplitLlde ofthe harmonic function WAngular frequency ofthe harmonic function If we approximate Eq(22)by the first item of its exponential progression， and substitute it to Eq(19)， we obtain 2Tapp1ppp AIcI （ 23） Generally， the angular frequency 09 is a large number, therefore the period T=2of the harmonic function is a small one， and the items such as TPPPP ICI ,1， and can be treated as con stant in the time of a period The integral of Eq (23)can be approximately written as 附 录 第 48 页 共 62 页 22Tapp1pp TAIcI21p （ 24） Eq(24)indicates an approximate deviation after a periodic time It is obviously that the value of the integral depends on the amplitude and angular frequency of the harmonic inputs This is the reason of that the harmonic inputs in actuated joints can control the motion of the passive one 4 SELF-RECONFIGURATION CONTROL The self-reconfiguration needs a stable control technology Based on the harmonic input nonlinear control method that given in section 3 briefly,next we will design a new control scheme to implement the self-reconfiguration motion This method will be used to optimize the generalized flexibility ellipsoid for a given position in the task space Given denotes an expected configu ration that is derived from some optimizing method， 0 is the actual position of the manipulator Let de （ 25） Where eVector ofjoint position errorsdecomposing Eq(24)to following form padadapaee （ 26） a sliding model is given by a1aa ekeS （ 27） and the convergence law is selected as: a3a2a Sk)Ss g n (kS （ 28） where，0,0,0 321 KKK， and sgn()indicates a sigmoid function， which has the form of: If vector has a manner of ： Tn1 SSS ， following equation may be obtained Eq(27)manifests that the motion of the actuated joint will be stable for satislying the Lyapunov stability theorem Let the inputs of the actuated joints accord with Eqs(20),(2 1)，and select the parameters 0K4K,0Kp2dd ， so that following relations can be obtained 华北科技学院毕业设计（论文） 第 49 页 共 62 页 Substituting the twice time derivation of the second row of Eq(26) into Eq(30)， a result can be given by Let the inputs ofthe actuatedjoints be substituting Eqs(31)and(32)into Eq(19)， the following relations hold And the amplitude of the harmonic can be givenby Thus if the passive joint is not in the expected position， the control input of the actuated joint is defined by Eqs(32)and(34) On the other hand， if the passive joint is in the expected position， the control input will change to the following equation From Eq(27)， the time derivation is Combining Eqs(28)and(35)， the control law is given by It is obviously that the method suggested here is nonlinear and time varying， which obeys the BrockeRs theory Rearranging the algorithm above， the controllaw can be written as When ep=0 is satisfied When ep0 is satisfied 5 SlM ULATION STUDY In this section the planar 3R manipulator is selected as a simulation mode1 which is shown in Fig4， Supposing that the second ioint of the manipulator is passive， and the other ioints are actuated Ifthe initial configuration 30,60,60321 ， for improving the perform ance of the fiexibility ellipsoid 。 A better configuration is 附 录 第 50 页 共 62 页 15.81,89.17,85.24 321 ,which is given in section 3 We regard the later one as the expected configuration In accordance with the method that is suggested in section 4 the simulation result is shown in Fig7,Fig7 a indicates Fig.7 Self-reconfiguration of the 3R underactuated manipulator 1 Joint1 2 Joint2 3 Joint3 the joints position errors with respect to time； Fig.7b is the joints.trajectory respect to time；Fig.7c shows the transformation of the manipulators configuration in the self-reconfiguring control； Fig.7d shows the phase relations between the speed and position of the ioints Obviously,the manipulator is reconfigured to the expected configurationbyitsel 6. concluding remark The underactuated technology is a crucial problem not only for fault toleran ce of space robot systems but also for cooperation robot and metamorphic mechanisms The 华北科技学院毕业设计（论文） 第 51 页 共 62 页 underactuated redundant manipulator has the capability of realizing the mechan is mrecon figuration by it The new generalized flexibility ellipsoid measure of the underactuated redundant manipulator with passive joints braked mode is suggested， and the measure can be used to optimize the static performance of the system A novel nonlinear control algorithm based on the harmonic function can implement the motion ofthe self-rcconfigurafion The simulation results by a three DOFs un deractuated mallipulator prove the measure and the control algorithm is efectua1 附 录 第 52 页 共 62 页 通过优化柔性椭球体对欠驱动冗余度机械臂的自重构 何广平 北方工业大学机电工程学院 北 京 100041 陆震 北京航空航天大学自动化学院 北京 100083 摘要 ：根据优化技术，欠驱动冗余度机械臂的多模型特征、柔性操作的测量、自重构的控制方法已被调查研究。分析了空间关节的结构变形和欠驱动冗余度机械臂柔性操作之间的关系，处于锁定模式下欠驱动冗余度机械臂的一种新型柔性椭球体操作的测量被提出，能应用于获得自重构控制的最理想结构。因此，基于简谐振动随时间变化非线性控制方法认为能完成其自重构。被动关节三连杆欠驱动机械臂等仿真例子在一些调查方面起重要作用。 关键词 ：欠驱动机械臂；自重构； 优化；非线性控制 0.前 言 欠驱动装置和机械臂能应用于许多领域，例如太空技术、合作机械人、变形装置。在太空领域里，由于没有失去有用功能或了解系统的自重构。当驱动构件出现一些问题时，基于欠驱动技术的误差出现是不可避免的。欠驱动机械臂也能被设计为合作机器人，也就是说 COBOT。 COBOT 的驱动不是作驱动装置而是提供动力学非函数约束。 COBOT 需要操作人员提供外力才能完成准确的应用，例如在生物工程学上外科手术和半导体制造等等。在机械领域机械变形有多种模态，并能从一种模态向另一种模态转变。引 用不同模态之间的改变可能导致连杆数目的变化或机械变形的约束限制。很显然，欠驱动控制、冗余度驱动和柔性装置是不可避免的。因此，欠驱动系统逐渐的成为研究领域一个具有吸引力的话题。 从力学角度看，研究欠驱动机械臂系统是不可能控制的。被动关节的运动是必须靠与动力装置连接。 Jain等表明动力装置是欠驱动机械臂的非完整性约束是二阶的。在机械实际上，与非完整性约束广泛被研究比较也有 100多年历史，然而，关于这种系统的运动规划和控制技术的研究只是近 10的事情，研究多针对轮式移动机器人、跳跃机器人、航空航天机器人等一阶非完整 性约束系统。关于欠驱动机械臂的研究观点， Anthoney等研究运动的稳定性， Arai 等提出随时间变化方法完成系统的位置控制。 Lee 等为欠驱动机器人提供了多种非线性控制方法。欠驱动研究的这些方法已从本质上揭示了它是非线性的，并且是随时间变化的、抽象的。事实上， Brockett 已证实这并没有消除阻碍和稳定给定结构系统的静电状况反馈。很显然，非线性系统的特征在组合空间多自由度华北科技学院毕业设计（论文） 第 53 页 共 62 页 是可以控制的。所以，非线性系统的控制研究受到更多的关注。 欠驱动机构和机械臂是对传统机械设计基本原理相违背的，传动机械设计基本原理认为 ，原动件的数目要与自由度的数目相等时，机构才具有确定的运动。欠驱动机械臂首先被提出并不是由于它的价值优点，但一些研究表明，欠驱动机构的故意设计也是很有价值的。例如， Rivhter 等获得由柔性欠驱动机械臂多维受力的测量。 Nakamura 等设计出了轮式滚动接触的非完整机器人和平面四连杆二驱动机械臂的控制。 He 等针对欠驱动冗余度机械臂提出一种自由碰撞运动规划演算法。从以上讨论的结果来看，我们可推断出在研究欠驱动时，可能遇到一些未被发现的新问题，如所提到的技术和理论的形成。因此，我们改善这装置具有很大的潜能性 。 这篇论文中，我们对欠驱动机械臂的静态特征和自重构控制方法进行探索与研究。 2.柔性椭球体模型 机械硬度是机械臂的一个重要要素，它是用来抵抗受力和阻碍力的能力。对于开式链接机械臂而言，链接部分是非常重要的部分。所以末端位姿的变形将会对连杆带来不良影响。转矩可以近似满足如下方程： iii KM i=1， 2， n （ 1） 式中 iM 关节 i 的转矩 i 关节 i 的变形量 ik 关节 i 的硬度系数 如果忽略关节 i 的重力和摩擦力不计，假设机械臂末端位姿力矢 mRF ，则转矩方程又可以写成： FJM T （ 2） 式中 : nRM 关节的转矩 nmRJ 雅可比矩阵 众所周知，关节有会有变形，机械臂末端位姿有如下关系式： Jx （ 3） 式中 : x 机械臂末端位姿矢量 关节的位姿矢量 将（ 1）式写成矩阵的形式，结合（ 2）、（ 3）式，经简单的计算， X 和 F 之间的关系如下： 附 录 第 54 页 共 62 页 F)JJk(X T1 （ 4） 式中 : 如果定义 T1 JJkC （ 6） （ 6）式是末端位姿的柔性矩阵。然而，在太空工作 1c 强度矩阵一致。柔性矩阵 C可以用来测量机械臂的静态特征。矩阵 C 也有雅可比函数功能。因此，它在组合和构造要素较大范围内是可改变的，在稳定条件下机械臂的可变特征能用于完成一些 应该的复杂的操作。如装配、抛光、维修等等。由（ 5）、（ 6）式可知矩阵 C 是对称性矩阵。 如果定义 )CCdet( T （ 7） 对矩阵 C进行微分，方程式（ 7）我们又可以得到 m1i i （ 8） 式中 i ， i=1,2,3,， m应用了矩阵 C 的单一性。因此， TCC 是其对 称矩阵，有如下关系： 1x)CC(x TT （ 9） 式（ 9）被描述为椭球体曲线方程，当椭球体的主要曲线与矩阵 C的单一值相等时，这椭球体也被认为是一般柔性椭球体 GFE。由于直观原因，图一中平面 2连杆机械臂的的连杆长 2,1i,m0.1L i ， GFE 如图（ 2）和（ 3）所示。 图一 平面 2R 杆机械臂 图 2 平面 2R 杆全驱动机械臂的 GFE 模型华北科技学院毕业设计（论文） 第 55 页 共 62 页 图 3 平面 2R 杆 全驱动机械臂的 GFE 模型 这些图示表明测量是需要依赖组合和机构要素。然而全驱动机械臂并不能改变其机构要素。因此，由于不同的构件（图 2），而不是结构要素（从图 2 改变到图 3）， GFE模型是可以改变的。当被动关节被引进作为全驱动机械臂时，为了方便使用，假设这些被动关节具有制动装置和位置控制，以便被动关节能在自由模式和锁定模式下进行制动。然而在运动学上，欠驱动机械臂揭示了一些冗余度连杆问题，并没有表明在输入方式下的自运动不如工作状态下的自运动。另一方面，被动关节的制动模式能使欠驱动机械臂具有重构能力 ,系统具有敏 捷性而使其能适合不同的工作。 2. 柔性矩阵 假设在欠驱动冗余度机械臂中 s 连杆为被动关节 ，被动关节装有制动装置，当被动关节处于自由状态时，其速度运动方程可以写成为：ppaa JJx （ 10） 式中 mRX 机械臂末端位姿矢量 nmRJ 驱动机械臂的雅可比矩阵 3pn R,R 分别为驱动和被动机械臂的广义坐标矢量 当机械臂中被动关节处于锁定状态时，系统运动方程可变为 qJx i （ 11） 式中 mRX 机械臂末端位姿矢量 nmi RJ 锁定状态下被动关节机械臂的雅可比矩阵 附 录 第 56 页 共 62 页 nRq 驱动关节的机械臂广义坐标 很显然，方程（ 11）和（ 3）是同一形式，方程（ 10）和（ 11）表明欠驱动机械臂在运动学上具有不同的模式。换句话说，在运动学上系统具有多中模式特征。图（ 4）平面 3R 连杆机械臂就是很好的例子。机械臂的第二关节是被动关节，其他的都是驱动关节。当被动关节处于自由状态时， 3R 被选做为广义坐标变量。如果被动关节处于自锁状态，机械臂的维数将变为 2 维，这广义坐标变量为 2Rq ，显然由于 0q ，但雅可比矩阵有如下关系： 图 4 平面 3R 杆机械臂 由于欠驱动机械 臂存在不同的运动模式，一种可以用来优化和机械臂的机构组合及自重构以使用不同的工作。预测如何完成基于欠驱动下的全驱动机械臂操作是不可避免的问题。不象全驱动冗余度机械臂那样，欠驱动冗余度机械臂并不能改善其操作工作，执行机械臂任务类似于输入空间的体积比工作空间少的缘故。有一条可行的途径就是在不同的时间分解机构的工作。例如，当机械臂工作处于驱动模式下，机构组合能进行机构自重构。然而当机械臂工作在全驱动模式下，其功能之一就是能控制机构的运动。事实上，处于欠驱动工作模式下的机械臂能辩别机构的运动，如位置控制或间断点对应 点运动。但是这并不是此论文所讨论的重点。我们应关注的是欠驱动冗余度机械臂的静态特征和机构自重构控制方法。 欠驱动机械臂两中模式的运动方程可以被多种方法描述。但是在复杂的机械装置中多连杆机械臂的机构要素定义还存在一定的困难。为了解决这些问题，我们将进行分析华北科技学院毕业设计（论文） 第 57 页 共 62 页 欠驱动冗余度机械臂的两种模式间的关系。 假定一种特殊的机械臂组合机构，假设有 mn ，处于装置的两种模式下的末端位姿表达式是一致