通過優(yōu)化柔性橢球體對欠驅(qū)動冗余度機(jī)械臂的自重構(gòu)外文文獻(xiàn)翻譯、中英文翻譯、外文翻譯

附 錄 第 38 頁 共 62 頁 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 華北科技學(xué)院畢業(yè)設(shè)計(jì)（論文） 第 39 頁 共 62 頁 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 錄 第 40 頁 共 62 頁 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 關(guān)節(jié) i 的轉(zhuǎn)矩 i 關(guān)節(jié) i 的變形量 ik 關(guān)節(jié) i 的硬度系數(shù) 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 華北科技學(xué)院畢業(yè)設(shè)計(jì)（論文） 第 41 頁 共 62 頁 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 錄 第 42 頁 共 62 頁 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 華北科技學(xué)院畢業(yè)設(shè)計(jì)（論文） 第 43 頁 共 62 頁 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 錄 第 44 頁 共 62 頁 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 華北科技學(xué)院畢業(yè)設(shè)計(jì)（論文） 第 45 頁 共 62 頁 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 錄 第 46 頁 共 62 頁 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 華北科技學(xué)院畢業(yè)設(shè)計(jì)（論文） 第 47 頁 共 62 頁 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 錄 第 48 頁 共 62 頁 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 華北科技學(xué)院畢業(yè)設(shè)計(jì)（論文） 第 49 頁 共 62 頁 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 錄 第 50 頁 共 62 頁 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 華北科技學(xué)院畢業(yè)設(shè)計(jì)（論文） 第 51 頁 共 62 頁 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 錄 第 52 頁 共 62 頁 通過優(yōu)化柔性橢球體對欠驅(qū)動冗余度機(jī)械臂的自重構(gòu) 何廣平 北方工業(yè)大學(xué)機(jī)電工程學(xué)院 北 京 100041 陸震 北京航空航天大學(xué)自動化學(xué)院 北京 100083 摘要 ：根據(jù)優(yōu)化技術(shù)，欠驅(qū)動冗余度機(jī)械臂的多模型特征、柔性操作的測量、自重構(gòu)的控制方法已被調(diào)查研究。分析了空間關(guān)節(jié)的結(jié)構(gòu)變形和欠驅(qū)動冗余度機(jī)械臂柔性操作之間的關(guān)系，處于鎖定模式下欠驅(qū)動冗余度機(jī)械臂的一種新型柔性橢球體操作的測量被提出，能應(yīng)用于獲得自重構(gòu)控制的最理想結(jié)構(gòu)。因此，基于簡諧振動隨時間變化非線性控制方法認(rèn)為能完成其自重構(gòu)。被動關(guān)節(jié)三連桿欠驅(qū)動機(jī)械臂等仿真例子在一些調(diào)查方面起重要作用。 關(guān)鍵詞 ：欠驅(qū)動機(jī)械臂；自重構(gòu)； 優(yōu)化；非線性控制 0.前 言 欠驅(qū)動裝置和機(jī)械臂能應(yīng)用于許多領(lǐng)域，例如太空技術(shù)、合作機(jī)械人、變形裝置。在太空領(lǐng)域里，由于沒有失去有用功能或了解系統(tǒng)的自重構(gòu)。當(dāng)驅(qū)動構(gòu)件出現(xiàn)一些問題時，基于欠驅(qū)動技術(shù)的誤差出現(xiàn)是不可避免的。欠驅(qū)動機(jī)械臂也能被設(shè)計(jì)為合作機(jī)器人，也就是說 COBOT。 COBOT 的驅(qū)動不是作驅(qū)動裝置而是提供動力學(xué)非函數(shù)約束。 COBOT 需要操作人員提供外力才能完成準(zhǔn)確的應(yīng)用，例如在生物工程學(xué)上外科手術(shù)和半導(dǎo)體制造等等。在機(jī)械領(lǐng)域機(jī)械變形有多種模態(tài)，并能從一種模態(tài)向另一種模態(tài)轉(zhuǎn)變。引 用不同模態(tài)之間的改變可能導(dǎo)致連桿數(shù)目的變化或機(jī)械變形的約束限制。很顯然，欠驅(qū)動控制、冗余度驅(qū)動和柔性裝置是不可避免的。因此，欠驅(qū)動系統(tǒng)逐漸的成為研究領(lǐng)域一個具有吸引力的話題。 從力學(xué)角度看，研究欠驅(qū)動機(jī)械臂系統(tǒng)是不可能控制的。被動關(guān)節(jié)的運(yùn)動是必須靠與動力裝置連接。 Jain等表明動力裝置是欠驅(qū)動機(jī)械臂的非完整性約束是二階的。在機(jī)械實(shí)際上，與非完整性約束廣泛被研究比較也有 100多年歷史，然而，關(guān)于這種系統(tǒng)的運(yùn)動規(guī)劃和控制技術(shù)的研究只是近 10的事情，研究多針對輪式移動機(jī)器人、跳躍機(jī)器人、航空航天機(jī)器人等一階非完整 性約束系統(tǒng)。關(guān)于欠驅(qū)動機(jī)械臂的研究觀點(diǎn)， Anthoney等研究運(yùn)動的穩(wěn)定性， Arai 等提出隨時間變化方法完成系統(tǒng)的位置控制。 Lee 等為欠驅(qū)動機(jī)器人提供了多種非線性控制方法。欠驅(qū)動研究的這些方法已從本質(zhì)上揭示了它是非線性的，并且是隨時間變化的、抽象的。事實(shí)上， Brockett 已證實(shí)這并沒有消除阻礙和穩(wěn)定給定結(jié)構(gòu)系統(tǒng)的靜電狀況反饋。很顯然，非線性系統(tǒng)的特征在組合空間多自由度華北科技學(xué)院畢業(yè)設(shè)計(jì)（論文） 第 53 頁 共 62 頁 是可以控制的。所以，非線性系統(tǒng)的控制研究受到更多的關(guān)注。 欠驅(qū)動機(jī)構(gòu)和機(jī)械臂是對傳統(tǒng)機(jī)械設(shè)計(jì)基本原理相違背的，傳動機(jī)械設(shè)計(jì)基本原理認(rèn)為 ，原動件的數(shù)目要與自由度的數(shù)目相等時，機(jī)構(gòu)才具有確定的運(yùn)動。欠驅(qū)動機(jī)械臂首先被提出并不是由于它的價值優(yōu)點(diǎn)，但一些研究表明，欠驅(qū)動機(jī)構(gòu)的故意設(shè)計(jì)也是很有價值的。例如， Rivhter 等獲得由柔性欠驅(qū)動機(jī)械臂多維受力的測量。 Nakamura 等設(shè)計(jì)出了輪式滾動接觸的非完整機(jī)器人和平面四連桿二驅(qū)動機(jī)械臂的控制。 He 等針對欠驅(qū)動冗余度機(jī)械臂提出一種自由碰撞運(yùn)動規(guī)劃演算法。從以上討論的結(jié)果來看，我們可推斷出在研究欠驅(qū)動時，可能遇到一些未被發(fā)現(xiàn)的新問題，如所提到的技術(shù)和理論的形成。因此，我們改善這裝置具有很大的潛能性 。 這篇論文中，我們對欠驅(qū)動機(jī)械臂的靜態(tài)特征和自重構(gòu)控制方法進(jìn)行探索與研究。 2.柔性橢球體模型 機(jī)械硬度是機(jī)械臂的一個重要要素，它是用來抵抗受力和阻礙力的能力。對于開式鏈接機(jī)械臂而言，鏈接部分是非常重要的部分。所以末端位姿的變形將會對連桿帶來不良影響。轉(zhuǎn)矩可以近似滿足如下方程： iii KM i=1， 2， n （ 1） 式中 iM 關(guān)節(jié) i 的轉(zhuǎn)矩 i 關(guān)節(jié) i 的變形量 ik 關(guān)節(jié) i 的硬度系數(shù) 如果忽略關(guān)節(jié) i 的重力和摩擦力不計(jì)，假設(shè)機(jī)械臂末端位姿力矢 mRF ，則轉(zhuǎn)矩方程又可以寫成： FJM T （ 2） 式中 : nRM 關(guān)節(jié)的轉(zhuǎn)矩 nmRJ 雅可比矩陣 眾所周知，關(guān)節(jié)有會有變形，機(jī)械臂末端位姿有如下關(guān)系式： Jx （ 3） 式中 : x 機(jī)械臂末端位姿矢量 關(guān)節(jié)的位姿矢量 將（ 1）式寫成矩陣的形式，結(jié)合（ 2）、（ 3）式，經(jīng)簡單的計(jì)算， X 和 F 之間的關(guān)系如下： 附 錄 第 54 頁 共 62 頁 F)JJk(X T1 （ 4） 式中 : 如果定義 T1 JJkC （ 6） （ 6）式是末端位姿的柔性矩陣。然而，在太空工作 1c 強(qiáng)度矩陣一致。柔性矩陣 C可以用來測量機(jī)械臂的靜態(tài)特征。矩陣 C 也有雅可比函數(shù)功能。因此，它在組合和構(gòu)造要素較大范圍內(nèi)是可改變的，在穩(wěn)定條件下機(jī)械臂的可變特征能用于完成一些 應(yīng)該的復(fù)雜的操作。如裝配、拋光、維修等等。由（ 5）、（ 6）式可知矩陣 C 是對稱性矩陣。 如果定義 )CCdet( T （ 7） 對矩陣 C進(jìn)行微分，方程式（ 7）我們又可以得到 m1i i （ 8） 式中 i ， i=1,2,3,， m應(yīng)用了矩陣 C 的單一性。因此， TCC 是其對 稱矩陣，有如下關(guān)系： 1x)CC(x TT （ 9） 式（ 9）被描述為橢球體曲線方程，當(dāng)橢球體的主要曲線與矩陣 C的單一值相等時，這橢球體也被認(rèn)為是一般柔性橢球體 GFE。由于直觀原因，圖一中平面 2連桿機(jī)械臂的的連桿長 2,1i,m0.1L i ， GFE 如圖（ 2）和（ 3）所示。 圖一 平面 2R 桿機(jī)械臂 圖 2 平面 2R 桿全驅(qū)動機(jī)械臂的 GFE 模型華北科技學(xué)院畢業(yè)設(shè)計(jì)（論文） 第 55 頁 共 62 頁 圖 3 平面 2R 桿 全驅(qū)動機(jī)械臂的 GFE 模型 這些圖示表明測量是需要依賴組合和機(jī)構(gòu)要素。然而全驅(qū)動機(jī)械臂并不能改變其機(jī)構(gòu)要素。因此，由于不同的構(gòu)件（圖 2），而不是結(jié)構(gòu)要素（從圖 2 改變到圖 3）， GFE模型是可以改變的。當(dāng)被動關(guān)節(jié)被引進(jìn)作為全驅(qū)動機(jī)械臂時，為了方便使用，假設(shè)這些被動關(guān)節(jié)具有制動裝置和位置控制，以便被動關(guān)節(jié)能在自由模式和鎖定模式下進(jìn)行制動。然而在運(yùn)動學(xué)上，欠驅(qū)動機(jī)械臂揭示了一些冗余度連桿問題，并沒有表明在輸入方式下的自運(yùn)動不如工作狀態(tài)下的自運(yùn)動。另一方面，被動關(guān)節(jié)的制動模式能使欠驅(qū)動機(jī)械臂具有重構(gòu)能力 ,系統(tǒng)具有敏 捷性而使其能適合不同的工作。 2. 柔性矩陣 假設(shè)在欠驅(qū)動冗余度機(jī)械臂中 s 連桿為被動關(guān)節(jié) ，被動關(guān)節(jié)裝有制動裝置，當(dāng)被動關(guān)節(jié)處于自由狀態(tài)時，其速度運(yùn)動方程可以寫成為：ppaa JJx （ 10） 式中 mRX 機(jī)械臂末端位姿矢量 nmRJ 驅(qū)動機(jī)械臂的雅可比矩陣 3pn R,R 分別為驅(qū)動和被動機(jī)械臂的廣義坐標(biāo)矢量 當(dāng)機(jī)械臂中被動關(guān)節(jié)處于鎖定狀態(tài)時，系統(tǒng)運(yùn)動方程可變?yōu)?qJx i （ 11） 式中 mRX 機(jī)械臂末端位姿矢量 nmi RJ 鎖定狀態(tài)下被動關(guān)節(jié)機(jī)械臂的雅可比矩陣 附 錄 第 56 頁 共 62 頁 nRq 驅(qū)動關(guān)節(jié)的機(jī)械臂廣義坐標(biāo) 很顯然，方程（ 11）和（ 3）是同一形式，方程（ 10）和（ 11）表明欠驅(qū)動機(jī)械臂在運(yùn)動學(xué)上具有不同的模式。換句話說，在運(yùn)動學(xué)上系統(tǒng)具有多中模式特征。圖（ 4）平面 3R 連桿機(jī)械臂就是很好的例子。機(jī)械臂的第二關(guān)節(jié)是被動關(guān)節(jié)，其他的都是驅(qū)動關(guān)節(jié)。當(dāng)被動關(guān)節(jié)處于自由狀態(tài)時， 3R 被選做為廣義坐標(biāo)變量。如果被動關(guān)節(jié)處于自鎖狀態(tài)，機(jī)械臂的維數(shù)將變?yōu)?2 維，這廣義坐標(biāo)變量為 2Rq ，顯然由于 0q ，但雅可比矩陣有如下關(guān)系： 圖 4 平面 3R 桿機(jī)械臂 由于欠驅(qū)動機(jī)械 臂存在不同的運(yùn)動模式，一種可以用來優(yōu)化和機(jī)械臂的機(jī)構(gòu)組合及自重構(gòu)以使用不同的工作。預(yù)測如何完成基于欠驅(qū)動下的全驅(qū)動機(jī)械臂操作是不可避免的問題。不象全驅(qū)動冗余度機(jī)械臂那樣，欠驅(qū)動冗余度機(jī)械臂并不能改善其操作工作，執(zhí)行機(jī)械臂任務(wù)類似于輸入空間的體積比工作空間少的緣故。有一條可行的途徑就是在不同的時間分解機(jī)構(gòu)的工作。例如，當(dāng)機(jī)械臂工作處于驅(qū)動模式下，機(jī)構(gòu)組合能進(jìn)行機(jī)構(gòu)自重構(gòu)。然而當(dāng)機(jī)械臂工作在全驅(qū)動模式下，其功能之一就是能控制機(jī)構(gòu)的運(yùn)動。事實(shí)上，處于欠驅(qū)動工作模式下的機(jī)械臂能辯別機(jī)構(gòu)的運(yùn)動，如位置控制或間斷點(diǎn)對應(yīng) 點(diǎn)運(yùn)動。但是這并不是此論文所討論的重點(diǎn)。我們應(yīng)關(guān)注的是欠驅(qū)動冗余度機(jī)械臂的靜態(tài)特征和機(jī)構(gòu)自重構(gòu)控制方法。 欠驅(qū)動機(jī)械臂兩中模式的運(yùn)動方程可以被多種方法描述。但是在復(fù)雜的機(jī)械裝置中多連桿機(jī)械臂的機(jī)構(gòu)要素定義還存在一定的困難。為了解決這些問題，我們將進(jìn)行分析華北科技學(xué)院畢業(yè)設(shè)計(jì)（論文） 第 57 頁 共 62 頁 欠驅(qū)動冗余度機(jī)械臂的兩種模式間的關(guān)系。 假定一種特殊的機(jī)械臂組合機(jī)構(gòu)，假設(shè)有 mn ，處于裝置的兩種模式下的末端位姿表達(dá)式是一致