五自由度MOTOMAN焊接机器人运动学分析与仿真设计含5张CAD图
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XXX附录一:平面冗余机械手的运动学控制扩展运动分配方案机械工程系,浦项市科学技术研究所125年邮政信箱,浦项市790 - 600年(朝鲜)(最终形式收到:1991年4月28日)摘要数度的平面机械手的运动控制的冗余是一个困难的问题,因为重计算,维负担或缺乏适当的技术。扩展运动分配方案,基于平面冗余机械手分解为一系列n冗余当地武器和分发效应器在关节的运动速度水平,本文提出了。配置指数被定义为未成年人的产品对应雅可比矩阵,用于指导全球冗余机械手。提高方案的性能,自动控制,负责内部联合运动,不运动导致效应器,可以使用可选保证全局最优操纵。的重复性问题讨论了冗余机械手使用方案。计算机模拟的结果显示,详细分析了平面特性和9-DOF操纵者,为例。关键词:运动控制;平面机械手;分配方案,配置指数。1. 简介卓越的灵活性和多功能性,人类手臂展品在执行各种任务,主要归因于手臂的运动学冗余。机器人机械手叫做(动)冗余如果它具有更多的自由度(自由度)比是必要的来执行指定的任务。额外的自由度的冗余机械手可以用来实现一些如避免奇点,!避障、2或关节限制回避。3虽然一个或两个冗余度可以用来满足上述，变得非常有吸引力,因为它的使用灵活性和敏捷运动在一定的任务在一个复杂的环境。然而,研究冗余机械手的运动控制冗余度大的尚未执行广泛因为缺乏适当的技术和/或计算负担。大部分的冗余机械手的运动控制方法专注于解决冗余运用广义逆或伪逆机械手基于性能函数可操纵性等措施,1条件数,兼容性索引,7的机械手的雅可比矩阵。然而,大多数这样的瞬时优化方案是基于本地决策可能不能保证全局最优。9 - 11最近几个全局优化方案被提出,但需要大量的计算,使实时应用程序不现实的。这些伪逆的方法可用于控制平面冗余机械手不管偶数和奇数的自由度。然而,当机械手等大型冗余自由度平面特性或9-DOF某个任务需要在复杂的环境中,伪逆的方法有很大的困难在制定性能函数,发现其梯度向量甚至借助象征MACSYMA等计算。 另一个不同的概念,提出了由李等。12平面四自由度机械手,分解成两个冗余机械手当地武器,称为和前臂,在一个中间的手臂位置或任务点称为肘部。即冗余机械手转换为连续的双臂系统合作,与合作由肘部。这个方案被称为任务分配方案使用一个名为面向任务的可操纵性的新措施措施(汤米),代表任务之间的差异需求和机械手的可操纵性,全球指导任务执行。然而,任务分配方案有以下限制: 适它只适用于连冗余机械手的自由度,因为机械手应该分解为。因此该计划变得很难被应用到一个平面机械手单自由度,如平面五自由度或7自由度机械手。 它并不倾向于保留重复性如果所需的奇异值向量对汤米不正确的选择。 汤米需要先验知识所需的奇异值和奇异向量的任务执行,这在实践中是相当困难的。本文提出一种新方案,可以应用于任何平面冗余机械手通过扩展和推广上述任务分配方案。冗余机械手是把这里当作系统多个冗余手臂串行连接在一起。这个计划叫做扩展运动分配方案摘要,因为它包括运动分配方法和解决运动方法。3虽然基本结构类似于李等人工智能。2,控制方案和方法指导机器人运动的新方案在很大程度上不同于。使用冗余机械手的固有特性,可以可选地包括在自动控制方案来提高可操纵性或获得更理想的配置。灵巧指数,称为配置指数,也是开发作为判据来确定执行自动控制和/或性能函数的最大化解决运动方法。第二节提出了方案的结构,和第三节描述了控制由配置指数。在第四节中,数值模拟了平面8-00F和9-00F机械手来验证该方法的有效性与讨论。结论在第五节。2。扩展运动分配方案 方案的基本结构分解为一系列冗余机械手。基本上是2-link机械手模块,但最后可以2自由度或机械手偶数和奇数自由度机械手,分别。甚至当平面冗余机械手的自由度,所有模块和效应器的运动是分发给每个按照下列运动分布的方法。一个节点被定义为相邻之间的联合。图1说明了平面冗余机械手表示为连续多个辅助臂合作制度。机械手的关节速度,可以表示为一个组合的联合辅助臂速度：图1多个辅助臂系统在平面的情况下。, （1）辅助臂的数量。效应器速度, 辅助臂生成的参照基础构架可以描述的，: 一致的笛卡尔速度的第i个节点，方程(2)表明,效应器运动可以显式地表示辅助臂的共同运动。在这个方程式中,，为奇异值,可以表示分解为： 正交矩阵为奇异值，和等级一组标准正交基向量z范围是由 和，在，节点速度,可以用标准正交基向量的线性组合, 和作为： 中，因此可以被写作 现在由运动分布的问题，找到一个加权K的最低标准的解决方案满足方程(5)和最小化以下运动分布函数。拟议的运动分布函数G是： 运动控制图2可操纵性的第i个辅助臂椭球。运动分布函数的物理意义是,所有的可操纵性椭圆体辅助臂应该尽可能轮通过函数的最小化。更具体地说,一个节点的分解速度的权重， ，成为的轴， 和在第i个辅助臂选择轴的长度的倒数第i个辅助臂的可操纵性椭球。这种分解是实现更高的可操纵性以上各向同性配置的可操纵性椭圆体辅助臂更圆。如图2所示,可操纵性的轴或更多的各向同性配置的可操纵性椭圆体辅助臂更圆。如图2所示,可操纵性的轴椭球配合 和主轴的长度等于奇异值,，与一致，j=1,2.除了在辅助臂里K的权重,权重的辅助臂对整个机械手是由运动分布标准为:标准的分布效应器运动子武器是由权重因素,被称为“圆度的因素,圆度因素应基于:(1)与成反比尔(术语(2)有一个物理意义,它正比于规范化的雅可比行列式的绝对值,也就是说,另一方面,这一术语的第i个辅助臂体积成正比的可操纵性椭圆体。大量的椭球意味着它的圆度。因此,可以说,较大的型号更圆的第i个辅助臂的可操纵性椭球。上述标准的目的是一个,大圆度因子应辅助臂细长上可操纵性椭球椭球更圆。如果我们仔细观察权重的方式,它可以很容易地发现,有一个一致性增加可操纵性。 (7)在l以2 X 1拉格朗日乘子向量。最小的必要条件的G(K),结果是 （8） （9）假设我们可以从选择2线性独立的行,方程(9)可分为 (10) (11)可以由2线性独立的行,剩下的（n-2）,行分别，类似，可以通过选择制定2 线性独立的行对应的行，确定从方程(10)和方程（11），得到 （现在,可以唯一确定K方程(8)和(12)。K,获得上面.S可以从方程（4)中获得，可以从方程（4）和方程（13）中获得单个的辅助臂面冗余机械手使用方程可以有效地指导，(14)当所有辅助臂模块。另外,因为最小化的解决方案K方程(6)方程(5)的主题是W最低标准方程(5)的解决方案,它可以获得的方程（15） （15）当机械手的自由度,最后一个(s-th)辅助臂 3联系模块。在整个环节中,理由选择3-link模块作为最后的子部门是辅助臂接近效应器容易分配给执行某一任务,同时优化性能的功能。最后3-link模块的基本控制方法是解决运动方法已广泛用于动控制冗余机械手。然而,剩下的(s-1)辅助臂是由上面提到的运动分布法。这两种方法将使用专门为了方便。扩展运动分配方案平面机械手与奇怪的自由度可以概括为:运动分布来解决运动的转换方法如下执行,反之亦然。然后解决运动方法应用,否则运动分布方法的应用。上述标准的物理意义是解决运动方法将用于“当最后辅助臂比任何其他更有可能将容易在执行一个给定的任务。当运动解决方案应用到最后辅助臂，效应器的运动只分配到最后也的辅助臂就是说, （16） 一般解决方法，在方程（17）中可以得到 （在，和相关迹象的最大化和最小化性能函数,分别;I是一个33单位矩阵。方程的系数K(18)是一个积极的标量常数和,梯度向量的,被描述为: (19)一个合适的选择K可能基于手臂配置,硬件限制关节速度和试探法。扩展运动分配方案包括运动分配方法和解决运动方法适用于一般冗余机械手，扩展运动分配方案的流程图偶数和奇数的自由度机械手是图3和图4中所示,分别。尽管上述方案可以指导操纵者圆满,自动控制的内在属性冗余机械手,另外也可以用于提高性能和将在下一节中讨论。3.自动控制扩展运动在第二节分布方案已经制定一个有效的技术来控制平面冗余机械手。然而,提供更理想的配置或执行另一个任务,如一个避障一个自动控制也可以使用可选。此外,如果我们注意到运动控制的基础上扩展运动的局部最优分配方案在某个瞬间的时候,我们需要自动控制来保证全局最优操纵。 为此,需要使用一些指数量化机器人线圈形状来确定是否需要自动控制。一个名为配置的新灵巧指数,指数,负责的转换方面的定义是: （20） 配置指数可以提供标准确定机械手是在全球范围内的配置合适的或不是因为这个指数间接表明可操纵性。当配置指数恶化低于某一阈值,执行特定的子任务被称为自动控制通过改变手臂配置到配置的绝对值指数高于阈值增加。特别是自动控制应用于通过第S第一辅助臂和第(s - 1) 的偶数和奇数自由度机械手。于自动内部关节机械手的运动,不运动导致效应器,它应该满足零空间，约束的零空间机械手雅可比行列式,J.这意味着引导时自动配置指数低于某一阈值,所谓的参考联合配置是作为参考的理想配置。特别是用参考联合配置重新配置机手械,以避免障碍物或某一物体的周围形成一个包装结构。的使机械手是通过控制节点位置的方式来驱动辅助臂对参考联合配置， （21） （22）这个优化问题是一样的,前面讨论的运动分布方案。因此,K的解决方案,在这里可以得到使用方程描述的类似的程序(7)-()2)。自动的关节速度同样由方程(14)。除了它的使用作为全球标准的指导在自动控制、配置索引也被用作性能功能最大化在过去的三自由度suba辅助臂在前面的扩展运动分配方案。例如,配置指数平面五自由度机械手,性能函数H方程(18)的解决运动方法,成为 （23）5.数值例子在这个仿真、特性和9-DOF平面作为一个选择带有转动关节机械手示例显示如何扩展运动分布方案可以应用于控制平面冗余运动方法,控制平面冗余操纵者。1平面特性机械手的平面特性操纵者8联系(单位米)被选中的整个长度是3.9米。给定的任务就是从最初的位置沿着x坐标的正方向与速度常数效应器最后的位置的初始位置1日2日和3日节点为(0.8,0.5),(1.2,0.3),(1.4,1.4)。辅助臂的数量是4,也就是说。s = 4。圆度因子分别为0.05,0.05,0.1,和0.65根据第二节使用的标准。在配置指数阈值触发自动控制被选为0.002。为了方便起见,我们定义效应器的向前运动的运动从一个内部外部的位置。同样,我们也向后运动定义为从外部的运动效应器内部位置。最初的关节角度向后运动将同最后联合角度向前运动。当然,效应器向后运动速度是相反的方向,而其重要性是一样向前运动。图5和图6的仿真结果说明向前和向后运动使用扩展运动分配方案没有自动控制,特别是，值得注意的是,没有自我运动控制的方案倾向于保留重复性这些数据所示,预计。相同的任务图5和图6可以做更好的定性当我们使用配置索引自动。向前运动的仿真结果使用扩展运动分配方案与自动控制如图7所示的联合运动平滑比图5。这个事实可以看到在图8配置指数的值显示在运动。大配置高可操纵性指数表示,这是一个好迹象光滑的行为。向后运动的仿真结果与自动省略了,因为它的行为类似于向前运动。从这些结果,可以说,自动提高可操纵性的运动学控制配置指数恶化低于阈值时,虽然扩展运动分配方案可以独自引导冗余机械手。图5仿真结果的平面特性操纵(前进运动没有自动控制)。图6仿真结果的平面特性操纵者(向后运动没有自动控制)。图7仿真结果的平面特性操纵(前进运动没有自动控制)。没有自我运动控制的方案倾向于保留重复性这些数据所示,预计。相同的任务图5和图6可以做更好的定性当我们使用配置索引自动。向前运动的仿真结果使用扩展运动分配方案与自动控制如图7所示的联合运动平滑比图5。这个事实可以看到在图8配置指数的值显示在运动。大配置高可操纵性指数表示,这是一个好迹象光滑的行为。向后运动的仿真结果与自动省略了,因为它的行为类似于向前运动。从这些结果,可以说,自动提高可操纵性的运动学控制配置指数恶化低于阈值时,虽然扩展运动分配方案可以独自引导冗余机械手。平面9-DOF机械手的平面9-DOF操纵者9联系(整个长度是5.0米)选为扩展运动分配方案的另一个例子。图8配置指数的平面特性机械手(移动)。图9仿真结果的平面9-DOF机械手(前进运动没有自动控制)。在配置指数阈值触发自动控制是110 - 6。图9说明了前进运动的仿真结果使用扩展的动态分配方案自动控制向前运动的仿真结果使用扩展运动分配方案与自动控制图10所示。的配置前进运动指数并没有自动控制的方案是如图11所示。同样的评价的作用自动控制,如平面特性的操纵者。图11配置指数的平面9-DOF机械手应用,本文提出的方法可以(移动)。6.结论提出了扩展的运动分布方案平面机械手的运动控制与几个程度的冗余,无论其景深是奇数还是偶数。该方法被证明有效地通过运动实现本地机械手的运动任务分配方法和运动方法解决。这个方案的显著优势甚至自由度机械手的方案倾向于保留，这里提出延长运动分配方案是概念上很简单有效,适用于任何平面冗余机械手。灵巧指数,称为配置指数,被用作性能函数解决运动控制和标准指导自动控制。全球任务执行指导有效地通过控制自动通过渐进的关节运动时参考联合配置配置指数恶化低于某个阈值。模拟演示的,配置指数,隋,表保存方面反映,是一个很好的性能指标在全球指导自动控制。自动的运动控制也显示增加可操纵性。另一点要提到的是,这种自动控制还可以方便地应用于避障问题,正确使用引用联合配置。尽管该方法有一个缺点,它不能直接应用于空间机械手使用目前的结构,方案可以推广到覆盖空间机械手与修改,在这个问题上和工作正在进行。对于实际的应用程序,本文提出的方法可用于实时控制平面机器人与数度冗余,机器人在复杂环境中工作等灵巧操作和控制POSTECH 7-00F直接驱动机器人。16进一步的研究将集中在逆运动学问题的共同立场水平在关节速度级别(不)机械手有大量冗余。参考文献1. T.Yoshikawa ,“分析和控制冗余机器人的“机器人技术研究:第一个国际研讨会(M. Brady and R. Pauleds)(麻省理工学院出版社、剑桥、质量)(1984)439 - 439。2. A.A. Maciejewski and CA. Klein,“冗余机械手避障在动态变化的环境中”。机器人研究1、3号109 - 117(1985)。3. A. Liegeois, “自动监控的配置和多体的行为机制”IEEE反式。在系统中,人,Cybern。SMC-7,12号,868 - 871(1977)。4. G.S. Chirikjian and J.W. Burdick, “避障算法操纵者，机器人与自动化,辛辛那提(1990年5月)第625 - 631页。5. J.K. Salisbury and J.J. Craig, “力控制和运动学问题”。机器人技术研第一4-17(1982)。6. R. Dubey and J.Y.S. Luh,“更高的灵活性冗余机器人控制，机器人与自动化,罗利(1987年3月),页1066 - 1072。7. K.W. Jeong, W.K. Chung, and Y. Youm, “发展POSTECH 7自由度直接驱动机器人”3日ISRAM Conf . .温哥华(7月.1990)页577 582。附录二Kinematic control of planar redundant manipulators by extended motion distribution scheme W.l. Chung, W.K. Chung and Y. Youm MechanicaL Engineering Department, Pohang Institute of Science and TechnoLogy, P.O. Box 125, Pohang 790-6()() (Korea) (Received: in Final Form April 28 , 1991) SUMMARY The kinematic control of a planar manipulator with several-degrees of redundancy has been a difficult problem because of the heavy computat,ional burden and/or lack of appropriate techniques. The extended motion distribution scheme, which is based on decomposing a planar redundant manipulator into a series of nonredundant/redundant local arms (referred to as subarms) and distributing the motion of an end-effector to subarms at the joint velocity level, is proposed in this paper. The configuration index , which is defined as the product of minors corresponding to subarms in the Jacobian matrix, is used to globally guide the redundant manipulators. To enhance the performance of the proposed scheme, a self-motion control, which handles the internal joint motion that does not contribute to the end-effector motion, can be used optionally to guarantee globally optimal manipulation . The repeatability problem for the redundant manipulators is discussed using the proposed scheme. The results of computer simulations are shown and analyzed in detail for planar 8-DOF and 9-DOF manipulators, as examples. KEYWORDS: Kinematic control ; Planar manipulators; Distribution scheme; Configuration index. 1. INTRODUCTION The remarkable dexterity and versatility that the human arm exhibits in performing various tasks can be attributed largely to the kinematic redundancy of the arm . A robotic manipulator is called (kinematically) redundant if it possesses more degrees of freedom (DOF) than is necessary for performing a specified task. The extra degrees of freedom of a redundant manipulator can be used to achieve some subgoaJs such as singularity avoidance ,! obstacle avoidance,2 or joint limit avoidance. 3 Although one or two degrees of redundancy can be used to satisfy the above subgoaJs, the use of hyper-redundancy4 becomes very attractive because of its flexibility and dexterity in motion for a certain task in a complex environment. However, the studies on the kinematic control of the redundant manipulators with large degrees of redundancy has not been performed extensively because of the lack of appropriate techniques and/or computational burdens. Most of the approaches to the kinematic control of a redundant manipulator focus on resolving redundancy by applying the generalized inverse or pseudo-inverse to manipulator lacobians based on performance function such as the manipulability measure condition number,5 manipulator-velocity-ratio,6 compatibility index ,7 the minors of the manipulator Jacobian matrix. Most of such instantaneous optimization schemes, however, are based on local decision which may not guarantee global optimality. Several global optimization schemes have been proposed recently ,9- 11 but require a large amount of computation which makes real-time applications unrealistic. These pseudo-inverse approaches can be used to control planar redundant manipulators regardless of even and odd degrees of freedom . However, when a manipulator with large redundant degrees of freedom such as a planar 8-DOF or 9-DOF is required for a certain task in complex environment , the pseudo-inverse approaches have great difficulty in formulating a performance function and finding its gradient vector even with the aid of symbolic calculations such as MACSYMA. Another different concept was proposed by Lee et al. 12 for a planar 4-DOF manipulator where the redundant manipulator is decomposed into two nonredundant local arms, referred to as the basearm and the forearm, at an intermediate arm location or task point called the elbow. That is, a redundant manipulator is transformed into a serially cooperating dual-arm system, with the cooperation between the subarms being carried out by the elbow. This scheme which is called the task distribution scheme used a new measure called Task Oriented Manipulability Measure (TOMM), which represents the discrepancy between the task requirements and the manipulators manipulability, to globally guide the task execution. The task distribution scheme, however, has the following restrictions. It can be applied to only a redundant manipulator with even degrees of freedom because the manipulator should be decomposed into nonredundant subarms only. Thus the scheme becomes very difficult to be applied to a planar manipulator with odd degrees of freedom , such as planar 5-DOF or 7-DOF manipulators. It does not tend to preserve repeatability if the desired singular values and vectors for TOMM are not properly chosen . TOMM requires a priori knowledge of the desired singular values and singular vectors for task execution, which is rather difficult in practice. This paper presents a new scheme which can be applied to any planar redundant manipulator by extending and generalizing the above task distribution scheme. A redundant manipulator is treated here as a multi-subarm system where multiple nonredundant/redundant arms are serially linked together. The scheme is called the extended motion distribution scheme in this paper because it includes both the motion distribution method and the resolved motion method.3 Although the basic structure is similar to that of Lee et aI. , 2 both the control scheme and the way to guide the robot motion in the new scheme are largely different from those of Lee et al. To use the intrinsic property of redundant manipulators, a self-motion control can be optionally included in the proposed scheme to enhance manipulability or to obtain a more desirable configuration. A dexterity index, which is called the configuration index, is also developed to be used as a criterion to determine the execution of self-motion control and/or a performance function to be maximized in the resolved motion method. Section 2 presents the structure of the proposed scheme, and Section 3 describes the seU-motion control guided by the configuration index. In Section 4, the numerical simulations are made for planar 8-00F and 9-00F manipulators to verify the effectiveness of the proposed method with discussions. Concluding remarks are made in Section 5.2. EXTENDED MOTION DISTRIBUTION SCHEME The basic structure of the scheme is to decompose the redundant manipulator into a series of subarms. The subarm is basically a 2-link manipulator module, but the last subarm can be either a 2-link or 3-link manipulator for even and odd degrees of freedom manipulators, respectively. When a planar redundant manipulator has even degrees of freedom, all of the subarms are 2-link modules and the motion of an end-effector is distributed to each subarm according to the following motion distribution method. A node is defined as a joint between adjacent subarms. Figure 1 illustrates a planar redundant Fig1. Multi-subarm system in planar case manipulator represented as a serially cooperative multi-subarm system. The joint velocity of the manipulator, iJ Em, can be expressed as a combination of the joint velocities of subarms: , （1）where s is the number of subarms. The end-effector velocity, xn generated by the subarms with reference to the base frame can be described in terms of iJa; for i=1,2, . ,s: ，: (2) where J; E m2X2 and x e; are the submatrix of the Jacobian 2XfI matrix, J E m, coresponding to iJa;, and the Cartesian velocity of the i-th node, respectively. The equation (2) indicates that the end-effector motion can be explicitly represented in terms of the joint motions of subarms. In this equation, J; for i = 1, 2, . , s can be expressed by the singular value decomposition13 as:with the singular values a, 02 and rank (J;) = 2. A set of orthonormal basis vectors of rangeof J;, r!Il(J;) is formed by u, and U2 where U= U,U2. The node velocity, Xe;, can be represented by a linear combination of orthonormal basis vectors, ;u, and ;U2, as: ， Hence x can be written as The problem of motion distribution now consists of finding a weighted minimum norm solution of K satisfying equation (5) and minimizing the following motion distribution function. The proposed motion distribution function, Gis: （6）Fig. 2. Manipulability ellipsoid of the i-th subarm.The physical meaning of the motion distribution function is that all of the manipulability ellipsoids for subarms should be as round as possible through the minimization of the function. To be more specific, the weightings for the decomposition of a node velocity, Xi into the principal axes of Ji, i.e., iU1 and iU2 , within the i-th subarm are chosen to be the reciprocals of the lengths of the principal axes of the i-th subarms manipulability ellipsoid.1 This decomposition is to achieve higher manipulability or more isotropic configuration by making the manipulability ellipsoids of subarms more round. As shown in Figure 2, the principal axes of the manipulability ellipsoid coincide with iU1 and iU2 and the length of a principal axis is equal to the singular value, iOj , corresponding to iUj for j = 1, 2. In addition to this weighting of K within subarms, the weighting of each subarm for the whole manipulator is determined by the motion distribution criterion as:Criterion The distribution of an end-effector motion to sub arms is determined by weighting factors, a/s for i =1, 2, . , s, called the roundness factors . The roundness factors should be given based on: (1) E= l (li = 1. (2) (li is inversely proportional to Isin 0 2il. The Isin 0Zil term in (2) has a physical meaning that it is proportional to the absolute value of the normalized determinant of the Jacobian Ji, that is, On the other hand, the term Idet (Ji)1 is proportional to the volume of the i-th subarms manipulability ellipsoidl. The large volume of the ellipsoid means its roundness. Thus, it can be said that the larger Isin 02il the more round the i-th subarms manipulability ellipsoid. The purpose of the above criterion is that a ,large roundness factor should be given to the subarm with a slender manipulability ellipsoid to make its ellipsoid more round. If we observe carefully the way of weighting, it can be easily found that there is a consistency to increase manipulability. To obtain K minimizing G(K) subject to xe = UK, let us define the Lagrangian function L(K) as follows.where l is a 2 X 1 Lagrangian multiplier vector. The necessary conditions for the minimum of G(K), aL/al=O and aL/aK=O, result in (8) （9）respectively. Assuming that we can select 2 linearly independent rows from UT, equation (9) can be divided into where U E m2x 2 and iffE m(n-2) X2 can be formulated by 2 linearly independent rows of UT and the remaining respectively. Similarly, can be formulated by selecting 2 rows of W corresponding to 2 linearly independent rows of UT , and selecting the remaining (n -2) rows of W, respectively. Determining l from equation (10) and substituting into equation (11), we have Now, K can be uniquely determined by equations (8) and (12). With K, K = KiK . KY obtained above, Xei for i = 1, 2, . . . , s can be obtained from equation (4) as: The individual subarm of the planar redundant manipulator can be effectively guided by using equation (14) when all of the subarms are 2-link modules. Alternatively, since the solution K which minimizes equation (6) subject to equation (5) is the W-weighted minimum norm solution of equation (5), it can be obtained byWhen the manipulator has odd degrees of freedom , the last (s-th) subarm is a 3-link module. Among the whole links, the reason to choose the 3-link module as a last sub arm is that the subarm close to an end-effector C3n be easily assigned to perform a certain task while optimizing a performance function. The basic control method for the last 3-link module is the resolved motion method which has been widely used to kinematically control redundant manipulators. However, the remaining (s -1) subarms are guided by the motion distribution method mentioned above. The two methods will be used exclusively each other for the sake of convenience. The extended motion distribution scheme for planar manipulators with odd degrees of freedom can be summarized as: The switching from the motion distribution to the resolved motion method and vice versa are performed as follows. If 1(lsin 82s 1+ Isin 82+ 11) is greater than Isin 82i l for i = 1, 2, . , s -1, then the resolved motion method is applied, otherwise the motion distribution method is applied. The physical meaning of the above criterion is that the resolved motion method will be used only when the last subarm has more possibility to move easily than any other subarm in executing a given task. When the resolved motion scheme is applied to the last subarm, the motion of an end-effector is assigned to only the last subarm, that is, The general solution to the inverse problem of finding Oas in equation (17) is given by where Jt is the Moore-Penrose inverse of Js ; + and signs are related to the maximization and minimization of a performance function H( 0.,), respectively; f is a 3 x 3 identity matrix. The coefficient K in equation (18) is a positive scalar constant and ilH(Oas ), the gradient vector of H(Ous ), is described as: A suitable selection of K may be based on arm configurations, hardware limits on joint velocities, and heuristics. The extended motion distribution scheme includes the motion distribution method and the resolved motion method to be applied to general redundant manipulators. The flowcharts of the extended motion distribution scheme for even and odd degrees of freedom manipulators are illustrated in Figures 3 and 4, respectively. Although the above scheme can guide the manipulator satisfactorily, the self-motion control which is the intrinsic property of the redundant manipulator, can also be additionally used to improve the performance and will be discussed in the next section. 3. SELF-MOTION CONTROL The extended motion distribution scheme has been formulated in Section 2 as an effective technique to control planar redundant manipulators. However, to provide more desirable configuration or to perform another task such as an obstacle avoidance, a self-motion control can also be used optionally. Additionally, if we note that the kinematic control based on the extended motion distribution scheme is locally optimal at a certain instant of time, we need self-motion control to guarantee globally optimal manipulation. To this end, it is required to use some index, which quantifies robot COil figurations to determine whether the self-motion control is needed or not. A new dexterity index, called the configuration index, which is responsible for the switching of aspects/s is defined as: When the configuration index is deteriorated below a certain threshold value, a certain subtask called the self-motion control is performed by changing ann configurations until the absolute value of configuration index increases above the threshold value. Especially the self-motion control is applied to the first through s-th subarms and the (s - 1)-th subarms for an even and odd degrees of freedom manipulator, respectively. Since the self-motion is the internal joint motion of a manipulator that does not contribute to the end-effector motion,it should satisfy the null space constraint in the null space of a manipulator Jacobian, I. This implies that ， 5. NUMERICAL EXAMPLES In this simulation, 8-DOF and a 9-DOF planar manipulators with revolute joints are selected as an example to show how the extended motion distribution scheme can be applied to control the planar redundant manipulators. 1 Planar 8-DOF Manipulator The planar 8-DOF manipulator which has 8 links with /1 = 1.0, 12 = 0.8, 13 = 0.6, 14 = 0.5, Is = 0.4, 16 = 0.3, 17 = 0.2, and 18 = 0.1 (unit in meters) is selected whose entire length is 3.9 meters. The given task is to move from the initial location Xc = ,l.5 O.Or along the positive direction of x-coordinate with the constant end-effector velocity, xe = 0.05 Or mls to the final position Xc = 3.50.0r. The initial locations of the 1st, 2nd, and 3rd nodes were given as (0.8,0.5), (1.2,0.3), and (1.4,0.2). The number of subarms is 4, i.e., s = 4. The roundness factors were 0.05, 0.1, 0.2, and 0.65 according to the criterion in Section 2 using Isin 82i l for i = 1, 2, 3, 4. The weighting factors Wi for the kinematic control of self-motion were chosen as 0.7, 0.15, 0.1, and 0.05 proportional to II Or; - Oaill for i = ,2,3,4. The k;s in equation (22) were chosen as k I = k2 = k3 = k4 = 0.05 sec-I. The reference joint configuration was selected as (Jr = 90, -45V, (Jr2 = -45, 45r, (Jr3 = -90, 45r, alld (Jr4 = -45, -45r. The threshold value in the configuration index for triggering the self-motion control was selected as 0.002. For convenience, we define forward motion as the motion in which the end-effector moves from an inner position to an outer position. Similarly, we also define backward motion as the motion in which the end-effector moves from an outer position to an inner position. The initial joint angles of backward motion are set to be the same as the final joint angles of forward motion. Of course, the end-effector velocity of backward motion is in the opposite direction while its magnitude is the same as that of forward motion. Figures 5 and 6 illustrate the simulation results for forward and backward motion using the extended motion distribution scheme without the self-motion control, respectively. It is noticeable that Fig. 5. Simulation result of the planar 8-DOF manipulator (forward motion without the self-motion control).Fig. 6. Simulation result of the planar 8-DOF manipulator (backward motion without the self-motion control).Fig. 7. Simulation result of the planar 8-DOF manipulator (forward motion without the self-motion control). the scheme without the self motion control tends to preserve repeatability as shown in these figures, as was expected. The same task as Figures 5 and 6 can be done much better qualitatively when we use the configuration index to do the self-motion. The simulation result for the forward motion using the extended motion distribution scheme with the self-motion control is shown in Figure 7 where the joint motion is smoother than that of Figure 5. This fact can be seen in Figure 8 where the values of configuration index during the motion are shown. The large configuration index denotes high manipulability, which is a good indication for smooth behavior. The simulation result for the backward motion with the self-motion is omitted here because its behavior is similar to that of forward motion. From these results, it can be said that the kinematic control of self-motion improves manipulability when the configuration index deteriorated below the threshold value, although the extended motion distribution scheme alone can guide the redundant manipulator. 2 Planar 9-DOF manipulator The planar 9-DOF manipulator which has 9 links with 11 =1.5, 12=0.8, 13=0.7, 14=0.7, 15 =0.4, 16=0.3, 17 = 0.3, 18 = 0.2, and 19 = 0.1 (entire length is 5.0 meters) is selected as another example for the extended motion distribution scheme. The given task is to move from theinitial location Xc = 1.5 O.OTalong the positive direction of x-coordinate with the constant end-effector velocity, xe0.05 ov mls to the final position Xe = 4.5 O.ov. The initial locations of the 1st, 2nd, and 3rd nodes were given as (0.5,0.8), (1.0,0.5), and (1.3,0.2). The number of subarms is 4, i.e., s =4. The roundness factors were 0.1, 0.2, and 0.7 according to the criterion in Section 2 usingFig. 8. Configuration index of the planar 8-DOF manipulator (forward motion).Fig. 9. Simulation result of the planar 9-DOF manipulator (forward motion without the self-motion control).Isin 82;1 for i = 1, 2,3. The weighting factors w;s for the kinematic control of self-motion were 0.7, 0.2, and 0.1 proportional to 116r; -6;11 for i = 1,2,3. The k;s in0 equation (22) were chosen as kl = k2 = k3 = 0.05 sec-I. The reference joint connguration was selected as Or! = 30, -30f, Or2= 30 -60f, and Or3 =30, -30f. The threshold value in the configuration index for triggering the self-motion control was 1 x 10-6 . Figure 9 illustrates the simulation results of forward motion using the extended motion distribution scheme without self-motion control. The simulation result for the forward motion using the extended motion distribution scheme with the self-motion control is shown in Figure 10. The configuration index of forward motion for the scheme with and without the self-motion control is illustrated in Figure 11 . The same remarks on the role of the self-motion control , as in the case of the planar 8-DOF manipulator, can be made.Fig. 11. Configuration index of the planar 9-DOF manipulator applications, the method proposed in this paper can be (forward motion).6. CONCLUSIONS This paper presented the extended motion distribution scheme for the kinematic control of a planar manipulator with several degrees of redundancy, whatever its DOF is even or odd. The proposed method was shown to efficiently achieve local motions of the manipulator task through the motion distribution method and the resolved motion method. A no
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