容错机械手设计：有多少自由度？外文文献翻译、中英文翻译.doc
容错机械手设计：有多少自由度？外文文献翻译、中英文翻译
收藏
资源目录
压缩包内文档预览：(预览前20页/共40页)
编号：118693717
类型：共享资源
大小：1.12MB
格式：ZIP
上传时间：2021-03-24
上传人：QQ14****9609
认证信息
个人认证
郭**（实名认证）
陕西
IP属地：陕西
12
积分
- 关 键 词：
-
容错
机械手
设计
多少
自由度
外文
文献
翻译
中英文
- 资源描述：
-
容错机械手设计：有多少自由度？外文文献翻译、中英文翻译,容错,机械手,设计,多少,自由度,外文,文献,翻译,中英文
- 内容简介：
-
Designing Fault Tolerant Manipulators:How ManyDegrees-of-Freedom?Christiaan J. J. Paredis and Pradeep K. KhoslaDepartment of Electrical and Computer Engineering andThe Robotics Institute,Carnegie Mellon University,Pittsburgh, Pennsylvania 15213.AbstractOne of the most important parameters to consider when designing a manipulator is the number of degrees-of-freedom (DOFs). This article focuses on the question: How many DOFs are necessary and sufficient for fault tolerance and how should these DOFs be distributed along the length of the manipulator? A manipulator is fault tolerant if it can complete its task even when one of its joints fail sand is immobilized. The number of degrees-of-freedom needed for fault tolerance strongly depends on the knowledge available about the task. In this article, two approaches are explored. First, for the design of a General Purpose Fault Tolerant manipulator, it is assumed that neither the exact task trajectory, nor the redundancy resolution algorithm are known a priori and that the manipulator has no joint limits. In this case, two redundant DOFs are necessary and sufficient to sustain one joint failure as is demonstrate din two design templates for spatial fault tolerant manipulators. In a second approach, both the Cartesian task path and the redundancy resolution algorithm are assumed to be known. The design of such a Task Specific Fault Tolerant Manipulator requires only one degree-of-redundancy.1 IntroductionAs robots are being used in a growing range of applications, the issue of reliability becomes more and more important. Recently, with the Hubble telescope and the Mars Observer, NASA has experienced firsthand how devastating the consequences can be when a critical component fails during a multi-billion dollar mission. Space applications are particularly vulnerable to failure, because of the adverse environment (cosmic rays, solar particles etc.) and the demand for long term operation. In this context, NASA has started to incorporate fault tolerance in their robot designs (Wu et al. 1993). Reliability is also important in medical robotics, because of the risk of the loss of human life. Although medical staff will probably always be on standby to take over in the case of a manipulator failure, the robot should at least be fail-safe, meaning it should fail into a safe configuration. A third domain of robot applications in which reliability is a major issue is the Environmental Restoration and Waste Management (ER&WM) program of the Department of Energy. Consider, for instance, the use of a manipulator in a nuclear environment where equipment has to be repaired or space has to be searched for radioactive contamination. The manipulator system deployed in these kinds of critical tasks must be reliable, so that the successful completion of the task or the safe removal of the robot system is assured. In this article, we focus on fault tolerance as a technique to achieve reliability in manipulator systems. The traditional approach to reliability has been that of fault intolerance, where the reliability of the system is assured by the use of high quality components. However, increasing system complexity and the necessity for long term operation have proven this approach inadequate. The system reliability can be further improved through redundancy. This design approach was already advocated in the early fifties by von Neumann in connection with the design of reliable computers: “The complete system must be organized in such a manner, that a malfunction of the whole automaton cannot be caused by the malfunctioning of a single component, . , but only by the malfunctioning of a large number of them”(von Neumann 1956, p. 70). This is the basic principle of fault tolerance: add redundancy to compensate for possible failures of components. However, this does not mean that any kind of redundancy added to a system results in fault tolerance. The main goal of this article is therefore to shed some light on the redundancy requirements for fault tolerant manipulators. That is, how much redundancy is needed and how should this redundancy be distributed over the manipulator structure?The redundancy provisions needed for fault tolerance can be incorporated only at a price of increased complexity. This drawback can be overcome by a modular and structured design philosophy, as is advocated in (Schmitz, Khosla, and Kanade 1988; Fukuda et al. 1992; Sreevijayan 1992; Hui et al. 1993;Chen and Burdick 1995). Modularity in hardware and software has the advantage of facilitating testing during the design phase and therefore reducing the chances for unanticipated faults. Modules also constitute natural boundaries to which faults can be confined. By including fault detection and recovery mechanisms in critical modules, the effect of local faults remains internal to the modules, totally transparent to the higher levels of the manipulator system. Such a modular design philosophy is embodied in the Reconfigurable Modular Manipulator System (RMMS) developed in the Advanced Manipulators Laboratory at Carnegie Mellon University (Schmitz, Khosla, and Kanade 1988). The RMMS utilizes a stock of interchangeable link modules of different lengths and shapes, and joint modules of various sizes and performance specifications. By combining these general purpose modules, a wide range of manipulator configurations can be assembled. When one needs a different configurations for a specific task (Paredis, and Khosla 1993), a robot created with the RMMS can be easily taken apart and reconfigured suitably. This reconfigurability can be further exploited to reduce the complexity of fault tolerant manipulators, as is shown in Section 4. Over the past decade, a lot of research has been done in fault tolerance for computer systems (refer to Johnson (1989) for and overview), but only recently has the concept been applied in robotics. Most of the work in fault tolerant robotics is directly based on the results from computer science, and can be classified in three categories:1. Design of fault tolerant robots,2. Fault detection and identification (FDI),3. Fault recovery and intelligent control.When designing a fault tolerant manipulator, one should decide where to include redundancy so that theoverall reliability is maximum. One should distinguish between hardware, software, analytical, information, and time redundancy (Johnson 1989). Our focus will be on hardware redundancy, which consists of actuation, sensor, communication and computing redundancy. Each of these types of redundancies can still be implemented at different levels. In Sreevijayan (1992), for instance, a four-level subsumptive architecture for actuation redundancy is proposed:Level 1: Dual actuatorsextra actuators per joint,Level 2: Parallel structuresextra joints per DOF,Level 3: Redundant manipulatorsextra DOFs per manipulator arm,Level 4: Multiple armsextra arms per manipulator system.A system can possibly be designed with redundancies at all four levels, resulting in the ability to sustainmultiple simultaneous faults. An example of a fault tolerant design for the space shuttle manipulator is described in Wu et al. (1993). Fault tolerance is here guaranteed by using a differential gear train with dual input actuators for every DOFan implementation of the first level of the four-level subsumptive architecture. In this article, we are more interested in achieving fault tolerance using redundant DOFs (Level 3). We envision the following scenario. A fault detection and identification algorithm monitors the proper functioning of each DOF of a redundant manipulator. As soon as a failure of a subcomponent is detected, one immobilizes the corresponding DOF by activating its fail-safe brake. Automatically, the joint trajectory is adapted to the new manipulator structure and the task is continued without interruption. The strength of this scenario resides in the fact that it can handle a large variety of possible faults, ranging from sensor failures to transmission and actuation failures. All these failures can be treated in the same manner, namely, by eliminating the whole DOF through immobilization. Although fault detection and identification (FDI) is an important part of our scenario for fault tolerance, we will not cover this subject in this article. Instead we refer to the following references (Chow and Willsky 1984; Stengel 1988; Ting, Tosunoglu, and Tesar 1993; Visinsky, Walker, and Cavallaro 1993;Visinsky, Walker, and Cavallaro 1994). In Visinsky, Walker, and Cavallaro (1993), using the concept of analytical redundancy, an FDI algorithm is presented along the lines of Chow and Willsky (1984). The result is a set of four simple equations which test for consistency between the measured position and velocity and the expected acceleration and jerk. This FDI algorithm fits into a three-layer intelligent control framework, consisting of a servo layer, and interface layer and a supervisory layer. The main problem presented to the intelligent controller is to distinguish between failures, disturbances and modeling errors, and to respond to each in the proper way. An overview of intelligent fault tolerant control is given in Stengel (1988). A range of approaches is reported, beginning with robust control, progressing through parallel and analytical redundancy, and ending with rule-based systems and artificial neural networks. The task of the robotics researcher is to apply and modify these approaches to the highly non-linear dynamics of robot manipulators. After a fault has been detected, the failing DOF is immobilized by activating its brake. In Pradeepetal. (1988), the authors analyze the effect of the immobilization of one of the DOFs of three commercial manipulators. They conclude that the robots with decoupled DOFs are more severely “crippled” by the loss of a joint than the ones with strongly coupled DOFs. This can be translated into the guideline that, for the design of fault tolerant manipulators, strong coupling between the DOFs is highly desirable. The results presented in Lewis and Maciejewski (1994a) can be interpreted similarly. A kinematic fault tolerance measure is defined as the minimum kinematic dexterity after joint failure. The maximum kinematic fault tolerance is achieved in a manipulator posture in which each joint contributes equally to the null-space motiona posture with strong coupling between the DOFs. For a manipulator with at least one decoupled DOF, the kinematic fault tolerance measure is always minimal, that is, zero. The same measure can be used as a criterion for the redundancy resolution of the fault-free manipulator. It is shown that the chances for task completion, after immobilization of one joint due to failure, are much better than when traditional pseudo-inverse control is used. However, due to the local nature of the fault tolerance measure, completion of the task cannot be guaranteed on a global scale (Lewis and Maciejewski 1994b). An important conclusion is that the ability to recover from a fault depends strongly on the joint trajectory followed by the fault-free manipulator system.This conclusion led us to explore two approaches to the problem of manipulator fault tolerance. The two approaches differ in the assumptions that are made with regard to the task and with regard to the choice of redundancy resolution algorithm. In a first approach, the goal is to design a general purpose fault tolerant manipulator. We assume that the task is only characterized by the size and position of the task space which is the portion of the Cartesian output space in which the task will take place. No assumptions are made about the path that needs to be followed within the task space or about the redundancy resolution algorithm used to execute the task. Such a general purpose fault tolerant manipulator can fault tolerantly execute any task of which the task space lies inside the fault tolerant workspace of the manipulator. This approach to fault tolerance is further explored in Section 3. In a second approach, the goal is to design a task specific fault tolerant manipulator. In this approach, we assume that the Cartesian path to be followed is known a priori and that the corresponding set of possible joint trajectories can be limited by an appropriate choice of a redundancy resolution algorithm. We show in Section 4 how these additional assumptions allow us to design a fault tolerant manipulator with fewer DOFs than a general purpose fault tolerant manipulator.For both approaches, we will in this article answer the question: How many degrees-of-redundancy (DORs) are necessary and sufficient for fault tolerance?2 Fault Tolerance and ReliabilityThe basic idea presented in this article is to use a manipulators redundant DOFs to compensate for a possible failure of one of the joints. The underlying assumption is that a manipulator that can sustain a joint failure is more reliable than one that cannot. The question is: “Does fault tolerance always result in an increase in reliability?” The answer is given by reliability theory (Johnson 1989). The reliability, R(t), of a component or a system is the conditional probability that the component operates correctly throughout the interval t0, t , given that it was operating correctly at the time t0 . For non-fault-tolerant serial link manipulators, the system fails when any single subsystema DOF or joint module for modular manipulatorsfails. The system reliability can then be computed as the product of the module reliabilities, Ri(t)： Or, in the case that every module is equally reliable with reliability Rmod(t)： If there are n modules and only of those are required for the system to function properlythe system can tolerate (n m) module failuresthen the system reliability is the sum of the reliabilities of all systems with (n m) or fewer faults. Since there are different systems with faults, the system reliability of a fault tolerant system with equal module reliabilities can be written concisely as We can apply this formula to the example of an 8-DOF fault tolerant manipulator, which needs only seven DOFs to function properly. The system reliability of the fault tolerant system is: Compared to for an equivalent 6-DOF non-fault-tolerant system. Both reliabilities are plotted as a function of the module reliability in Figure 1a, while Figure 1b shows the relative system reliability which equals 1 for and . These graphs should be interpreted as follows: when then The system reliability is zero in both cases, i.e.,both systems are guaranteed to fail. when , then That is, both systems are 100% reliable. when then meaning that the fault tolerant system is more reliable than the non-fault-tolerant one. when then The modules are so unreliable that the added complexity of the fault tolerant system reduces the overall performance.Preferably, one would like to operate a system at a reliability close to one, for which the fault tolerant system is the more reliable. To compare both alternatives for modules with a high reliability, it is more instructive to rewrite Equation (4) as an expression for the systems unreliability, When The unreliability for the non-fault-tolerant system is In general, the unreliability of a k-fault tolerant systemone that can sustain k faultsis of the order This means that the reliability of a fault tolerant system increases more significantly when the reliability of the individual modules is high. Best results are thus obtained when fault tolerance is combined with high component reliability, or fault intolerance. 3 General Purpose Fault Tolerant Manipulators1In this section, we discuss the kinematic design of a general purpose fault tolerant manipulator without joint limits. As for non-fault-tolerant manipulators, the kinematic capabilities are mainly characterized by the shape and size of the workspace or rather the fault tolerant workspace (FTWS) in this case. We identify several properties of general purpose fault tolerant manipulators and their workspaces, and propose an 8-DOF design template.To set the stage for our development, we define the following concepts relating to general purpose fault tolerant manipulators: General Purpose Fault Tolerant Manipulator: A manipulator that will still be able to meet the task specifications, even if any one or more of its joints fail and are frozen at any arbitrary joint angles. k-Reduced Order Derivative (k-ROD): When k joints of an n-DOF manipulator fail, the effective number of joints is (n k). The resulting faulty manipulator is called a k-reduced order derivative. Order of Fault Tolerance: A manipulator is fault tolerant of the k-th order, if and only if all possible k-reduced order derivatives can still perform the specified task. We call the manipulator k-fault tolerant. Fault Tolerant Workspace (FTWS): The fault tolerant workspace of a -fault tolerant manipulator is the set of points reachable by all possible -reduced order derivatives.Notice that our definition of a general purpose fault tolerant manipulator reflects the assumption that the redundancy resolution algorithm is not known a priori: a joint failure can occur at an arbitrary angle. n the remainder of this section, if no specific task is mentioned, it is assumed that the task is to reach a onzero volume of points. That is, the task space is an m-dimensional manifold in the m-dimensional utput space of the manipulator. A manipulator with a FTWS of dimension less than m is considered ot to be fault tolerant.3.1 Properties of General Purpose Fault Tolerant Manipulators3.1.1 ExistenceA general purpose manipulator has six DOFs which allow it to position its end effector in an arbitrary sition and orientation anywhere in its workspace. An obvious way to make this manipulator fault tolerant is to esign every joint with a redundant actuator. If one of the actuators of the resulting 2n-DOF fault tolerant manipulator were to fail, the redundant actuator could take over and the manipulator ould still be functional. imilarly, a k-fault tolerant manipulator can be constructed by duplicating very DOF k times, resulting in a (k+1)n-DOF pulator. This argument illustrates that (k+1)nDOFs are sufficient for -th order fault tolerance. In the remainder of this section, we determine the number of DOFs necessary to achieve general purpose fault tolerance.3.1.2 Boundary of the Fault Tolerant WorkspaceIn this section, we show that a boundary point of the FTWS is a critical value. Consider a kfault tolerant planar manipulator, M . A boundary point, , of the FTWS has to be an element of the boundary of the workspace of at least one ROD, M*, obtained by freezing k joints of M. Indeed, if were an interior point of the workspaces of all RODs, then it would by definition be an interior point of the FTWS and not a boundary point. The Jacobian of M* , , can be obtained from the Jacobian of M, by deleting the columns corresponding to the frozen DOFs. Because is a boundary point of the workspace of M*, the Jacobian of M*at Pb is singular. We prove now that JM is singular too.Suppose that JM were non-singular, then at least one of the columns corresponding to a frozen DOF would be outside the column space of the singular matrix, . Physically this means that a small change in the angle of that frozen DOF would cause the end effector of to move in a direction with a component perpendicular to the boundary of the workspace of the ROD, M*, as illustrated in Figure 2.The ROD with this new frozen angle would be unable to reach the point,. As a result, would be outside the FTWS, contradicting the fact that is a boundary point of the FTWS. Thus, is singular and is a critical value.Consequently, the FTWS is bounded by critical value manifolds. For planar positional manipulators, thecritical value manifolds are concentric circles, and the FTWS is an annulus with inner radius and outer radius 3.1.3 Required Degree of RedundancyIn Section 3.1.1, it is shown that, in general,kn redundant DOFsi.e. (k + 1)n DOFs in totalare sufficient to achieve k-th order fault tolerance. For planar positional manipulators, however, we prove that 2k redundant DOFs are necessary and sufficient for k-th order fault tolerance.Necessary: The proof shows that (2k + 1) DOFs (or 2k 1 redundant DOFs) are insufficient, by finding a lower bound for and an upper bound for . First consider the ROD obtained by freezing the first k joints at 0 radians, as illustrated in Figure 3. The maximum reach in the opposite direction is an upper bound for where li is the length of the i-th link. In order for to be positive, we must have that: Making this assumption, we find that is bounded below by the inner radius of the workspace of the ROD obtained by freezing the k last joints at 0 radians, as illustrated in Figure 3: From Equation (8) and Equation (10), it follows that at best resulting in a one-dimensional FTWS. Therefore, a (2k + 1)-DOF planar manipulator cannot be k-thorder fault tolerant.Sufficient: The proof shows that there exists a (2k + 2)-DOF manipulator template that is k-fault tolerant. Consider a manipulator with (2k + 2) links of length l. Because all the links have the same length, it is possible to compensate for a fault in a DOF by choosing a neighboring DOF to be at radians; that is, folded back onto the failing DOF. Even when consecutive DOFs fail, this trace-backmechanism can be used to compensate for failures. The result is that, by sacrificing one DOF to compensate for every fault, the (2k + 2)-DOF manipulator with faults is equivalent to a faultless 2-DOF manipulator. The FTWS of the (2k + 2) DOF manipulator is then the workspace of the equivalent 2-DOF manipulator, that is, 3.1.4 Including OrientationThus far, we have only considered planar positional manipulators. The results can be easily extended to the case in which orientation is considered also, by converting the orientational problem into an equivalent positional problem: An n-DOF manipulator, M, is k-fault tolerant with respect to a set of points, , ifand only if:1. The positional manipulator,M , obtained from by deleting its last link, ln, is k-fault tolerant with respect to the set of points 2.M is (k 1)-fault tolerant while reaching the points in W in any direction.Necessary: The positional manipulator,M , needs at least (2k + 2) DOFs to be k-th order fault olerant with respect to W; therefore, the manipulator M needs at least (2k + 3)DOFs.Sufficient: Again, we show that there exists a (2k + 3)-DOF manipulator template that is k-fault tolerant. Consider a template of which the first (2k + 2) links have length l and the last link has length zero; it is the template described in the previous section with a zero-length link added at the end. For this template, one can again use the trace-back-mechanism to show that it is equivalent to a faultless 3-DOFmanipulator with link lengths l ,l, and 0. the FTWS is thus: This result for planar manipulators with orientation and the result obtained in Section 3.1.3 can besummarized in the following theorem:Theorem:For planar manipulators without joint limits,2k degrees-of-redundancy are necessary and sufficientfor k-th order general purpose fault tolerance.3.2 Spatial Fault Tolerant ManipulatorsFor planar fault tolerant manipulators, we were able to prove that 2k is the required degree of redundancy. The proof was based on geometric workspace analysis. For spatial manipulators, however,the geometric analysis becomes too complex. Therefore, we will demonstrate some properties of spatial fault tolerant manipulators using two examples.As a first example, consider a 5-DOF spatial positional manipulator. Its Denavit-Hartenberg (D-H)parameters are listed in Table 1. This manipulator is first order fault tolerant, and because of its simple kinematic structure, an analytic expression for the boundary of the FTWS can be derived. The FTWS is symmetric with respect to the first axis. A cross-section (the X-Z plane), as shown in Figure 4, can be described by two segments of a circle with radius 2 and center at (x=1,z=0), and a straight line from(x = 2,z = 3) to(x = 2, z = 3) . An important property of this FTWS is that it does not have any holes or a central void, so that the FTWS of the same manipulator scaled by any factor contains the original FTWS. As a result, this fault tolerant manipulator can be used as a design template. Any task space can be enclosed in the FTWS of a scaled version of the design template.In Section 3.1.2, it is shown that the boundary of the FTWS of a planar manipulator coincides with its critical value manifolds. Figure 4 demonstrates that this property also holds for the 5-DOF spatial manipulator considered in this example. The critical value manifolds are computed using the algorithm described in Burdick (1992) and are depicted in a solid line. The bold part of the critical value manifolds is the boundary of the FTWS. The property that the FTWS is bounded by critical value manifolds can be effectively used for the determination of the FTWS. Testing whether a point is an element of the FTWS is a complicated procedure. One has to verify whether that point is reachable for all possible RODs, i.e., for all manipulator structures resulting from a joint failure of every possible joint at every possible joint angle. To find a good approximation of the FTWS, one would have to execute this test for a large number of points. This would be prohibitively slow. However, to improve the efficiency, one can compute the critical value manifolds of the manipulator first. These manifolds partition the Cartesian output space of the manipulator in sectors that are either entirely inside the FTWS or entirely outside the FTWS. Thus, by checking whether one point of a sector is an element of the FTWS, one can check whether the whole sector is in the FTWS. The number of FTWS-membership tests is so reduced to the number of sectors in the partition of the output space.As a second example of a spatial general purpose fault tolerant manipulator, consider an 8-DOFmanipulator, with D-H parameters listed in Table 2. It is the same manipulator as in the previous example, with a zero-length 3-roll-wrist added at the end. Using a Monte Carlo method, it has been determined that this manipulator is first order fault tolerant while reaching all the points, in the FTWS of example one, in any direction. This property can be demonstrated with the following arguments. When one of the first five DOFs fails, the manipulator can still reach any position in the FTWS (because the 5-DOF positional manipulator is FT) and can take any orientation at this position using the intact 3-rollwrist.When one of the DOFs in the wrist fails, we are left with a 7-DOF manipulator which has enough orientational capabilities to reach any point in the FTWS in any orientation. Consequently, one could call this the dextrous FTWS. Since there are again no holes or voids in the FTWS, this manipulator can also be used as a design template.Finally, one should notice that both examples have only two redundant DOFs, which indicates that two degrees-of-redundancy are also sufficient for 1-fault tolerance of spatial manipulators. Whether the theorem in Section 3.1.4 also holds for higher orders of fault tolerance of spatial manipulators requires further research.4 Task Specific Fault Tolerant ManipulatorsIn the previous section, we considered the design of fault tolerant manipulators for general use. Weproved that two redundant DOFs are necessary and sufficient for first order fault tolerance. However, as we will show in this section, a simpler kinematic structure is often sufficient when one specific task is considered. The disadvantage of this approach is that a task specific fault tolerant manipulator is only suited for a very limited set of similar trajectories. Unlike general purpose fault tolerance, task specific fault tolerance might require a different manipulator structure for every task. However, by using a modular manipulator system such as RMMS, it is possible to quickly reconfigure the manipulator to custom-tailor it for a specific task.4.1 Local versus Global Fault ToleranceConsider the task of reaching all the points in an -neighborhood, of the point .Suppose that p can be reached by an n-DOF manipulator in a posture,. If the posture, is nonsingular,then there exists an , such that the manipulator can reach any point in . However,for k-fault-tolerance, any point in needs to be reachable even when k of the joints of the manipulator are frozen. This is possible if and only if the Jacobians of all k-RODs in the posture are non-singular, i.e., have at least rank m. We call such a posture, locally fault tolerant. The Jacobian of a k-ROD, JROD, can be obtained by deleting the columns of the fault-free Jacobian that correspond to the frozen DOFs; the dimensions of JROD are m *(n k). A necessary condition for JROD to be of rank m is that n has to be larger than or equal to (m + k). Indeed, the manipulator needs to have at least n functional DOFs, even after a failure of k of them. That means that k DORs are necessary for local fault tolerance.Are k DORs also sufficient for local fault tolerance? Consider a manipulator with n = m + k DOFs; the Jacobian of a -ROD,JROD, is then a square matrix. A posture, is locally fault tolerant if the Jacobians of all m RODs are full rank. When the rank of any is less than m, the robot is in an internal singularity. The difference between singular, locally fault tolerant and internally singular postures is illustrated in Figure 5. The locus of internal singularities is a set of (m + k 1)-dimensional surfaces in ; or (n 1)-dimensional surfaces, when . Thus, nearly all postures of a manipulator with k DORs are locally k-fault tolerant. This can be summarized in the following theorem.Theorem:k degrees-of-redundancy are necessary and sufficient for k-th order local fault tolerance.The fact that a posture is locally k-fault tolerant guarantees that the manipulator can reach every point in a neighborhood of the end effector position, even after failure of k DOFs. However, this neighborhood can be small. To extend this result to larger trajectories, we have to formulate a global fault tolerance condition. This can best be illustrated with an example.4.2 Example of Task Specific Fault ToleranceBefore we present the global fault tolerance condition, we modify the definition of fault tolerance to include task specificity: Task Specific Fault Tolerant Manipulator: A manipulator is 1-fault tolerant with respect to the task of following the Cartesian path, if and only if there exists a fault tolerant joint trajectory,. Fault tolerant joint trajectory: A trajectory, is 1-fault tolerant with respect to the task of following the Cartesian path , if for every DOF j=1nand for every instant,ft, there exists an alternate trajectory, for which:1) maps ontounder the forward kinematics2) 3) (i.e., the -th DOF is frozen at the time ft)There are two main difference between this definition and the one for general purpose fault tolerance. The first difference is that we assume that the task is to follow a specific Cartesian path, rather than an unspecified path inside the task space. The second difference is that we no longer require that every point along the Cartesian path be reachable when a joint fails at an arbitrary angle, but only at an angle that occurred previously along the fault tolerant joint trajectory. Under these assumptions, k-fault tolerance can be achieved with only k redundant DOFs.As an example, consider a 3-DOF planar manipulator with normalized link lengths of 1. We want to determine whether this manipulator is 1-fault tolerant with respect to the task of following the path, shown in Figure 6, which is parametrized as with 0 a 1. For every point, along the path, one can compute a preimage.3 Because the manipulator in this example has one DOR, the preimage of every point is a one-dimensional subset of Tn, and can be parametrized in this case as with (In general, the preimage of a point might consist of disjoint manifolds, in which case each manifold should be parametrized separately). The continuous function, describes a 2- dimensional surface in , as is shown in Figure 7. Any joint trajectory that maps onto the specified Cartesian path, can be formulated as , or . According to the definition of task specific fault tolerance, the manipulator is fault tolerant if and only if a fault tolerant trajectory, can be found. It is clear that every posture of a fault tolerant trajectory, has to be locally fault tolerant. However, this requirement is not sufficientbecause a fault at a point, might cause another point, to become unreachable, even if the posture were locally fault tolerant.Therefore, one should exclude as possible postures for a fault tolerant trajectory not only internally singular postures, but also postures that, in the case of failure, would cause an internal singularity further along the path. This is the condition for global fault tolerance. For our example, the set of acceptable postures is shown in Figure 8. The same set of postures can be represented in the (a, b)-planethe white regions in Figure 9. A fault tolerant trajectory exists when a continuous function,with 0 a 1, can be found for which all postures, , are acceptable, i.e., satisfy the global fault tolerance condition. One such trajectory is shown in dashed line in Figure 9.The dotted lines correspond to internally singular postures, while the solid lines are contours of constant joint angles , and , respectively. These contours represent the joint trajectories that are followed once a failing DOF has been immobilized. The gray areas are postures for which the global fault tolerance condition is violated. Notice that the contour lines in these areas do not reach the end of the Cartesian path, but instead get stuck at an internal singularity. It is also important to notice that the acceptable areas (in white) can be described by fault tolerant joint intervals for each of the DOFs. For example, the area in which the fault tolerant joint trajectory is drawn in Figure 9, can be characterized by the joint intervals depicted in Figure 13. As long as the joint angles stay within these fault tolerant joint intervals at each point along the path, fault tolerant execution of the path is guaranteed.4.3 Redundancy Resolution Algorithms for Task Specific Fault ToleranceWe have shown that, to fault tolerantly follow a Cartesian path, it is important to follow a joint space trajectory that consists only of postures satisfying the global fault tolerance condition. How can we achieve this practically?An obvious implementation is to compute a fault tolerant joint trajectory off-line and store it as a set of via points in joint space. At execution time, a joint space controller can be used to follow this trajectory until a joint failure is detected. At that point, the failing joint is immobilized and an inverse kinematics algorithm is used on-line to compute the appropriate joint angles for the remaining DOFs. Although simple and computationally inexpensive, this scheme has several disadvantages. First, the fault tolerant joint trajectory cannot be adjusted at run time to satisfy secondary task requirements, such as obstacle avoidance. Also, when a failure occurs, one has to change the joint trajectory generation algorithm instantaneously, making it difficult to achieve a smooth transition from operation with DOFs to operation with DOFs.To avoid these drawbacks, we suggest that one uses a kinematic controller of the form:both before and after a failure has occurred, as is also suggested in Lewis and Maciejewski (1994b).The transition to operation with one fewer DOF can simply be achieved by zeroing out the column ofthe Jacobian that corresponds to the immobilized DOF. The gradient projection method(Ligeois1977)can be used to assure that the manipulator stays within the fault tolerant joint intervals, indicated in Figure13. By choosing an appropriate potential function, , one can even combine the requirement for fault tolerance with a secondary task requirement. An additional advantage is that this scheme can accommodate for small changes in the Cartesian path, while still maintaining the property of fault tolerance. As long as the joint trajectory stays within the fault tolerant joint intervals, it is guaranteed that every point further along the Cartesian path will be reachable even after joint failure. This is particularly important in sensor based systems in which small trajectory corrections are made based on sensor feedback.5 SummaryIn this article, we have shown that making a manipulator fault tolerant by adding redundant DOFs is an effective way to increase the reliability of a manipulator. However, not every redundant manipulator is fault tolerant. Thus, an important problem for the design of fault tolerant manipulators is: How many DOFs are necessary and sufficient for fault tolerance and how should these DOFs be distributed along the length of the manipulator? We have shown that, depending on the assumptions that are made about the task, the answer to this question varies.For general purpose fault tolerant manipulators without joint limits, two degrees-of-redundancy are necessary and sufficient to sustain one fault. This conclusion was illustrated with two spatial general purpose fault tolerant manipulator designs: a 5-DOF positional manipulator and an 8-DOF positional and orientational manipulator. Both manipulators have a fault tolerant workspace without any holes or voids so that one can scale the designs to fit any task. For task specific fault tolerant manipulators, only one degree-of-redundancy is necessary and sufficient for 1-fault tolerance. However, one might have to use a different manipulator and recompute the fault tolerant joint intervals, for every task. This drawback can be partially overcome by using a modular manipulator system that can be quickly reconfigured to suit a particular task.AcknowledgmentsThis research was funded in part by the Department of Energy under grant DOE-F902-89ER14042, by Sandia National Laboratories under contract AC-3752-A, by the Department of Electrical and Computer Engineering, and by The Robotics Institute at Carnegie Mellon University.The authors would also like to thank the reviewers for pointing out some flaws in the original manuscript. Thanks to their comments and suggestions the quality of this article has been improved considerably.References1 Burdick, J. W. 1988. Kinematic analysis and design of redundant robot manipulators. STAN-CS-88-1207. Stanford, Calif.: Stanford University Computer Science. Burdick, J. W. 1992 (May 1214, Nice, France). A recursive method for finding revolute-jointed manipulator singularities. Proc. 1992 IEEE Int. Conf. Robot. Automat. Los Alamitos, Calif.: IEEE, pp. 448453.2 Chen, I.-M., and Burdick, J. W. 1995 (May 21-27, Nagoya). Determining task optimal modular robot assembly configurations. Proc. 1995 IEEE Int. Conf. Robot. Autom. Los Alamitos, Calif.: IEEE.Chow, E. Y., and Willsky, A. S. 1984. Analytical redundancy and the design of robust failure detection systems. IEEE Trans. Automat. Contr. 29(7):603614.3 Fukuda, T., et al. 1992. Concept of cellular robotic system (CEBOT) and basic strategies for its realization. Comput. Electr. Eng. 18(1):1139.4 Hui, R., et al. 1993 (May 26, Atlanta). Design of the IRIS facilitya modular, reconfigurable andexpandable robot test bed. Proc. 1993 IEEE Int. Conf. Robot. Autom. Los Alamitos, Calif.: IEEE, pp. 155160.5 Johnson, B.W. 1989. Design and analysis of fault-tolerant digital systems. Reading, Mass.: Addison-Wesley.6 Lewis, C. L., and Maciejewski, A. A. 1994a. Dexterity optimization of kinematically redundant manipulators in the presence of joint failures. Comput. Electr. Eng. 20(3):273288.7 Lewis, C. L., and Maciejewski, A. A. 1994b (May 813, San Diego). An example of failure tolerant operation of a kinematically redundant manipulator. Proc. 1994 IEEE Int. Conf. Robot. Autom. Los Alamitos, Calif.: IEEE, pp. 13801387.8 Ligeois, A. 1977. Automatic supervisory control of the configuration and behavior of multibody mechanisms. IEEE Trans. Syst. Man Cybern. 7:868871.9 Nakamura, Y. 1991. Advanced robotics: redundancy and optimization. Reading, Mass.: Addison-Wesley.10 Paredis, C. J. J., and Khosla, P. K. 1993. Kinematic design of serial link manipulators from task specifications. Int. J. Robotics Res. 12(3):274287.11 Paredis, C. J. J., and Khosla, P. K. 1994 (May 813, San Diego). Mapping tasks into fault tolerant manipulators. Proc. 1994 IEEE Int. Conf. Robot. Autom. Los Alamitos, Calif.: IEEE, 696703. Pradeep, A. K., et al. 1988. Crippled motion in robots. IEEE Trans. Aerosp. Electron. Syst. 24(1):213.12 Schmitz, D. E., Khosla, P. K., and Kanade, T. 1988 (Nov. 610, Sydney). The CMU reconfigurable modular manipulator system. Proc. 19th Int. Symp. Exp. Rob. (ISIR). New York: Springer-Verlag, pp. 473488.13 Sreevijayan, D. 1992. On the design of fault-tolerant robotic manipulator systems. M.S. thesis. University of Texas at Austin Department of Mechanical Engineering.14 Stengel, R. F. 1988. Intelligent failure-tolerant control. IEEE Control Syst. Mag. 11(4):213. Ting, Y., Tosunoglu, S., and Tesar, D. 1993 (May 26, Atlanta). A control structure for fault-tolerant operation of robotic manipulators. Proc. 1993 IEEE Int. Conf. Robot. Autom. Loa Alamitos, Calif.: IEEE, pp. 684690.15 Visinsky, M. L., Walker, I. D., and Cavallaro, J. R. 1993 (May 26, Atlanta). Layered dynamic fault detection and tolerance for robots. Proc. 1993 IEEE Int. Conf. Robot. Autom. Los Alamitos, Calif: IEEE, pp. 180187.16 Visinsky, M. L., Walker, I. D., and Cavallaro, J. R. 1994 (May 813, San Diego). New dynamic modelbased fault detection thresholds for robot manipulators. Proc. 1994 IEEE Int. Conf. Robot. Autom. Los Alamitos, Calif.: IEEE, pp. 13881395.17 Von Neumann, J. 1956. Probabilistic logics and the synthesis of reliable organisms from unreliable components. Automata studies (Annals of mathematics studies, no. 34), eds. C. E. Shannon and J. McCarthy. Princeton: Princeton University Press.18 Wu, E. C., Hwang, J. C., and Chladek, J. T. 1993. Fault-tolerant joint development for the Space Shuttle remote manipulator system: analysis and experiment. IEEE Trans. Robot. Autom. 9(5):675684.容错机械手设计：有多少自由度？Christiaan J. J. Paredis and Pradeep K. Khosla电气与计算机工程系和机器人研究所联合部， 卡内基梅隆大学， 美国宾夕法尼亚州匹兹堡市15213 。摘要：设计一个机械手需要考虑的最重要的参数是其自由度的数量。本文的重点：为满足其容错功能有多少自由度才是必要和足够的，沿机械手的长度应如何分配这些自由度？当一个机械手的某个关节因磨花而固定，在这种情况下仍能完成它的任务，我们就说这个机械手具有容错功能。为满足其容错功能，其自由度数紧密依赖于对任务所获得的知识。在这篇文章中，对以下两种方法进行了探索。第一种方法，为设计一个通用容错机械臂，假定没有确切的任务轨迹，也不是被称为先验的冗余第算法，机器手没有关节限制。在这种情况下，两个多余的自由度以弥补一个关节失败被证明是必要和足够的，以此做为设计空间容错机械臂模板的标准。在第二个方法中，笛卡儿任务路径和冗余第算法假定是已知的。对一项任务的具体容错机械臂的设计只需要一个自由度的冗余。1简介由于机器人的应用越来越广泛，可靠性问题变得越来越重要。最近，美国宇航局经历了在通过哈勃望远镜对火星观察这一价值数十亿美元的任务中，当一个关键组成部分失败所产生的灾难性后果。太空应用因环境恶劣（宇宙射线，太阳粒子等）并要求长期运行而特别容易失败。在此背景下，美国航天局已经开始把容错功能加入到机器人设计中（吴等人。1993年）。对于医疗机器人，可靠性也是重要的，因为存在丧失人生命的风险。尽管医务人员可能会长久的精力集中，在情况下依然会出现操作失误，机器人至少应万无一失，这意味着它应该成为一个失败的安全配置。在机器人应用的第三领域-环境恢复和废物管理计划的能源部中，可靠性仍然是一个重要的问题。例如，机械手被使用在核环境下修复设备或在被放射性污染的空间中进行搜索。部署在这类关键任务中的机器人系统必须是可靠的，因此，成功地完成任务或安全拆除机器人系统是有保证的。在本文中，我们侧重于容错功能做为一种实现机械手系统可靠性的技术。传统保证可靠性的方法的障碍是容错，在这种情况下系统的可靠性是靠使用高品质的元件来保证的。然而，为保证长期运行，通过增加系统的复杂性和必要性已经证明这种做法不够。系统的可靠性还可以进一步通过冗余来改善。在五十年代初，这种设计方法已经被冯诺依曼采用于设计可靠的计算机：“完整的系统必须是以这种方式来组织，即一个单一的组成部分发生故障不会引起整个自动机的故障， . ，但是，只因它的大部分出现故障“（冯诺依曼1956年，第70页）。容错的基本原则是：添加冗余以弥补部件可能发生的故障。然而，这并不意味着任何形式的冗余添加到系统中都能实现容错功能。因此本文的主要目的是阐明容错机械手所需的冗余。也就是，需要有多少冗余和冗余在机械手结构上应如何分布？容错所需的冗余规定只有以增加复杂性为代价可以组织在一起。这个缺陷可以被一种模块化和结构化得设计理念克服，这一理念在(Schmitz, Khosla, and Kanade 1988; Fukuda et al. 1992; Sreevijayan 1992; Hui et al. 1993;Chen and Burdick 1995)被提倡。硬件和软件模块化的优点是在设计阶段便于测试，因此减少了突发故障的机会。模块化同时也构成了对那些可被限制的障碍的自然边界。由于在关键模块中包含故障检测和回收机制，局部故障的影响依然存在于内部模块，完全透明的更高层次的机械手系统。这种模块化的设计理念体现在由卡内基梅隆大学先进的机械手实验室开发的可重构模块化的机械臂系统(RMMS) (Schmitz,Khosla,Kanade 1988年)。可重构模块化的机械臂系统(RMMS)利用股票有多种不同长度和形状的互换链接模块和有各种尺寸和性能规格的联合模块。结合这些通用模块，宽范围的机械手结构可组装。当为了一个特定的任务需要不同的配置,（Khosla Paredis1993),一个可重构模块化的机械臂系统的机器人可以轻易的拆解和进行适当的改装。这种重构可以进一步减少容错机械臂复杂性，见第4节。在过去的十年中,很多对计算机系统容错功能的研究已经完成(参阅约翰逊(1989)和概述),但直到最近才被应用在机器人的概念。关于容错机器人的大部分的作直接地以计算机科学的成果为基础,这些工作可分为三大类：1、 容错机器人的设计，2、 故障检测和鉴定(FDI)，3、 故障恢复和智能控制。 当设计一个容错机械手时, 选择应该包括冗余,以便总体可靠性是最大的。冗余应该分为硬件、软件、分析、信息和时间冗余(约翰逊1989年12月初版)。我们的焦点将集中在硬件冗余,其由驱动、传感器、通信和计算冗余。每一种类型的冗余仍可以应用在不同级别。在Sreevijayan(1992),例如,一个驱动冗余四级结构被提出： 1级：双驱动器-每个关节多驱动器， 2级：并行结构-每个自由度多关节， 3级：冗余机械手-每个机械臂多自由度， 4级：多臂-每个机械手系统多臂。 一个以四个级别冗余设计的系统，有承受多个同步故障的能力。 在Wu et al. (1993)中描述了一个关于航天飞机机械手容错设计的例子。这里通过使用差动轮系与双输入驱动器的每个自由度来保证容错的，实施四个级别归属知识架构的第一级别。在这篇文章中,我们更有兴趣采用冗余自由度的方法实现容错 (等级3)。我们想象的情况是这样的。一个故障检测与识别算法监视冗余机械手各自由度是否运转正常。一检测到某个关节失败，它就通过激活其故障安全制动来固定相应的自由度。关节的轨迹是持续而自动地适应新的机械手结构和任务。这个场景的力量存在于这样一个事实,即它能处理各种可能的故障,从传感器传输故障和驱动故障。所有这些故障可以用同样的方式处理,即通过固定消除全自由度的方式。既然故障检测与识别是我们对容错推测的重要组成部分,在这篇文章中我们不会覆盖这个主题的。相反，我们参考以下参考文献(Chow and Willsky 1984; Stengel 1988; Ting, Tosunoglu, and Tesar 1993; Visinsky, Walker, and Cavallaro 1993;Visinsky, Walker, and Cavallaro 1994)。在Visinsky, Walker, and Cavallaro (1993)中，使用分析冗余度的算法这一概念, 故障检测与识别算法依照Chow and Willsky (1984)曲线提出。其结果是一组四个测试测量位置、速度、预期加速度和冲击之间一致性的简单方程。故障检测与识别算法适合于一个三层智能控制结构，这一结构包括伺服层，接口层和监控层。主要的问题是提出了智能控制器能区分故障、干扰和建模误差,并对每一个误差给以恰当方式的回应。斯腾格尔(1988)给出了只能容错控制的概述。一系列被报道的方法是，以鲁棒控制开始，通过平行的和分析的冗余进展，以规则系统和人工神经网络结束。研究机器人的任务是运用机器人高度非线性动力学的方法,并且修改这些方法。当检测到某个故障后，通过激活失效自由度对应的制动器来使其固定。在Pradeep et al.(1988)中，作者分析了影响三个机械手某个自由度固定的方法。他们得出这样的结论: 具有解耦自由度的机器人比那些拥有强耦合自由度的机器人因某个关节的损坏而具有更严重的“跛脚”。，自由度之间强耦合的方法是非常可取的，并且是可以成为容错机械手的设计的指导。这些在Lewis and Maciejewski (1994a)中提及的结果可被理解成同样的。运动容错测量定义为:关节出现故障后最小的运动灵敏度,。最大的运动容错在各个关节同样有助于零空间运动的具有强自由度耦合的姿势中实现。对于一个至少有一个解耦的自由度的机械手,运动学容错能力的测量,总是最小，并且为零。同样这个测量量可以作为无故障机械手冗余分辨率的一个标准。结果表明，当某个关节因出现故障而固定时，机械手完成任务的机会明显多于采用传统的伪倒置控制。然而,由于局部容错措施的性质, 就无法保证全面完成这项任务。一个重要的结论是：从故障中恢复的能力很大程度上取决于遵循无故障机械手系统的关节轨迹。这个结论带我们去探索两种方法来解决这个机械手容错性问题。这两种方法分别是关于任务和冗余算法的不同假设。在第一种方法中，目的是设计一个具有通用容错功能的机械手。我们假设任务只是以大小和位置来描述的任务区间，任务即将发生在笛卡尔输出区间，并且任务区间是此区间的一部分。没有任何假设取决于遵循任务区间的轨迹或取决于被用来执行任务的冗余分辨率算法。这种通用容错机械手可以执行任务区间在此机械手容错能力范围内的任何任务。该方法对容错机制进行了深入探究，见第3节。第二种方法中,目的是设计出能完成特定任务的容错机械手。在此方法中，我们假设遵循已知的笛卡尔轨迹和相应的关节轨迹设置受限于适当的冗余分辨率算法。我们在第4节中展示我们怎样用这些额外假设去设计比通用容错机械手有更少自由度的容错机械手。对以上两种方法，我们将在文章中回答这个问题：为实现容错功能，多少冗余自由度才是必要和足够的？2 容错和可靠性本文提出的最基本的理念是用机械手冗余自由度的方法去弥补某个关节可能出现的故障。这个根本假设就是一个可以承受某个关节失效的机械手更加可靠。问题就是：“容错总是可以增加可靠度吗？”可靠性理论给出了这个问题的答案。某个部件或系统的可靠性R(t)，是此部件或系统在整个区间t0, t中正常运行的条件概率，已知部件在时刻t0时运行正常。对无容错串行连接机械手，当模块化机械手的任何一个子系统的一个自由度或关节模块失效，则整个系统就会失效。于是，通过计算产品的模块可靠性，可以计算出整个系统的可靠性R(t)：或者,在此情况下,每个模块的可靠性用Rmod(t)来表示： 如果这里有n个模块，但只有m个模块被用于实现系统相应的功能，这个系统可以承受(n m)个模块失效，则系统的可靠性就是所有子系统可靠性的总和，该系统有(n m)或更少的故障。因为此系统有个不同的子系统，个故障，且该容错系统各模块的可靠性相同，则整个系统的可靠性可以简明的表示为： 我们可以把上面的公式用于下面的例子：该机械手有八个自由度，但只有七个自由度用于实现相应的功能。该容错系统的系统可靠性为：相比于对等效为一个六自由度的无容错系统。这两个系统的可靠性以模块可靠性的函数画在图1a中，同时图1b展示相关系统的可靠性 以上等式等于1，当和时。这两个图的解释如下： 当则。在这两种情况下，系统可靠性为零，这两个系统肯定会失效。 当，则。在这种情况下，系统是百分百可靠的。 当则。这意味着容错系统比无容错系统更可靠。 当则在此情况下，所有的模块是非常不可靠的，容错系统增加的复杂性会降低系统的整体性能。最好使系统可靠性接近1，这样容错系统更加可靠。比较两个可供选择的具有高可靠度的模块，改写方程（4）去表达系统的不可靠性会更有意义， 当时。无容错系统的不可靠性是： 一般来说，k-容错系统，即可以承受k个故障的系统，其不可靠性为。这就意味着当独立模块的可靠性高时，容错系统的可靠性的增加会更加显著。最好的结果是包括容错与部件的高可靠性或与故障不耐性的结合。相关系统可靠性系统可靠性 模块可靠性 模块可靠性 图1a 8自由度容错机械手的系统可靠性 图1b 8自由度容错机械手的系统可靠性和6自由度无容错机械手的系统可靠性 与6自由度无容错机械手的系统可靠性的对比3 通用容错机械手在本节中,我们讨论了一种无关节极限的通用容错机械手的运动学设计。对于无容错机械手，在此情况下，工作空间和大小或更确切地说容错工作空间是其主要特征。我们确认通用容错机械手的若干性质和工作空间,并提出了一个8自由度的设计模板。 为了设定我们开发阶段，我们定义了以下与通用容错机械手有关联的概念： 通用容错机械手：即使一个或更多的关节失效和在任意关节角被固定时，仍能完成任务要求的机械手。 k-降阶导数：当n自由度机械手的k个关节失效，有效关节的数目是(n k)个。这个最终故障机械手被称作k-降阶导数。 容错顺序：当且仅当其所有可能的k阶降阶导数在指定的任务中运行时，这个机械手是有k阶容错顺序的。我们就说这个机械手有k阶容错 容错工作区：k阶容错机械手的容错工作区是一系列以所有可能的k阶降阶导数表示的关节可靠性。请注意，我们对通用容错机械手的定义反映了，关节失效发生在任意角度这一解决冗余算法的假设，不是已知条件。在这节剩余部分，如果没有提及任何指定任务，则假设这个任务是要实现非零关节。那就是说，此任务空间是一个在机械手m输出空间中的m面空间。如果一个机械手的容错工作区少于m空间，则此机械手被认为是不具有容错功能的。3.1 通用容错机械手的性能3.1.1 存在性一个通用机械手的末端受动器若可以到达工作区的任意位置和在任意方向运动，则此机械手要有6个自由度。一个使机械手具有容错功能的明显的方法是设计所有的关节都有多余的驱动器。如果有2n自由度的容错机械手的一个驱动器失效了，多余的驱动器能承担未完成的工作，则这个机械手仍然是功能齐全的。相似地，通过重复自由度k次产生一个(k + 1)n自由度的机械手，最终构造一个k阶容错机械手。这个论点说明(k + 1)n自由度足以满足k阶容错。在本节的剩余部分，我们将确定为实现通用容错功能所需要的自由度的数目。3.1.2 容错工作区的边界在本节中，我们说明容错工作区的边界点是一个重要的值。设想一个机械手k阶容错曲面，M。容错工作区的边界点，必须至少是一降阶导数M*工作区间边界的一个元素，它可以由k个曲面M的临界点获得。的确，如果是所有降阶导数工作区间内的一点，则根据定义它就是容错工作区内的一点，而不是边界点。导数M*的函数行列式可以通过据临界自由度删除列数这一方法，从导数M的函数行列式得到。因为是导数M*工作区间的边界点，所以导数M*在点处是唯一的。我们现在证明也是唯一的。假设不是唯一的，则至少有一个与临界自由度相关的数会在这个唯一矩阵的区间的外面。这意味着在临界自由度下，角度很小的变化就会使降阶导数M最后的效应以垂直于降阶导数M*工作区间边界的方向上移动，见图2的图解。这个降阶导数在新的临界角时达不到点。结果，点将会在容错区间的外面，这与点是容错区间的边界点这一事实相矛盾。因此，函数是唯一的，并且点是关键值。因此，容错区间是关键值的有限集合。对曲面位置机械手来讲，这些关键值的集合是一组同心圆，并且容错区间是外径为、内径为的圆环区域。图2 降阶导数达不到容错区间外的一点图3 外径的上限和内径的下限3.1.3冗余所需自由度在3.1.3节中，讲的是，一般情况下，kn个冗余自由度，也就是说总共(k + 1)n个自由度，可以实现k阶容错。尽管如此，我们需要证明对曲面位置机械手来讲，要实现k阶容错，2k冗余自由度是必须和足够的。必要性：可以找到内径的一个下限和外径的一个上限，这表明(2k + 1)个自由度（或（2k 1）个冗余自由度）是不够的。首先，假设此降阶导数在0度时获得k个最前关节点，见图3.在相反方向上所能得到的最大值就是外径的上限：上式中li 是第i个连杆的长度。为了使为正值，我们必须使：做这样的假设：我们发现是在0度得到k个最后关节的降阶导数的区间内径的下界，见图3：根据等式（8）和等式（10），得：产生一个一维容错区间。因此，一个有(2k + 1)自由度的曲面机械手没有k阶容错功能。充分性：证据表明存在一个有(2k + 2)个自由度的机械手模板，其有k阶容错。设想一个机械手有(2k + 2)个长度l的连杆。因为所有的连杆长度相等，这可以弥补因选择一个邻近自由度到角所引起的失效。即使当连续的自由度失效时，跟踪备份机制就会被用来弥补所有的实效。结果就是，通过牺牲一个自由度去弥补每一个实效，此有k个失效的(2k + 2)自由度机械手等效于一个完美的2自由度机械手。(2k + 2)自由度机械手的容错区间就是其等效的完美2自由度机械手的区间,也就是：3.1.4包含的取向到目前为止，我们只考了虑曲面位置机械手。结果同样很容易延伸到考虑定向的例子中，可以把定向问题转换位等效位置问题：n自由度机械手M，若有k阶容错能力且遵循一组关节的集合：，当且仅当满足以下条件时：1、通过删除机械手M最后连杆得到的位置机械手，有k阶容错且遵循一组关节的集合。 2、当在任何方向上达到关节集合时，机械手是(k 1)阶容错的。必要性：为了有遵循关节集合的k阶顺序容错，位置机械手，至少要有(2k + 2)个自由度；因此，机械手M至少要有(2k + 3)个自由度。充分性：我们再次以存在一个阶容错的(2k + 3)自由度的机械手模板来讲解。假设模板的前(2k + 2)个连杆长度为，最后的连杆长度为0；也就是前一节所描述的在尾部加一个长度为0的连杆的机械手。对次模板，它可以再次用跟踪备份机制来表明它可以等效为一个连杆长度分别为的无故障三自由度机械手。它的容错区间为：定理：对无关节限制的曲面机械手来讲，为满足阶顺序通用容错性，2冗余自由度是必要和足够的。3.2 空间容错臂我们可以证明，对曲面容错机械手来讲，2冗余自由度是必要的。这个证据是基于几何空间分析的。尽管如此，对空间机械手进行几何分析太复杂了。因此，我们将使用两个例子来证明一些空间机械手的性质。第一个例子，假设一个五自由度空间位置机械手。它的迪纳维特-哈坦伯格参数列于表1。这个机械手是一阶容错的，并且因为它的动态结构简单，我们可以导出它的容错区间边界的解析表达式。它的容错区间相对于第一个轴对称。图4中表示的截面（X-Z面），可以由圆心在(x=1,z=0)，半径为2的圆和从点(x = 2, z = 3)到点(x = 2, z = 3)的直线这两段曲线来描述。容错区间的一个重要性质是它没有任何的空穴和中央无效，因此相同机械手的按的元素缩放的容错区间包含原始容错区间。结果，这个容错机械手可以被用作设计模板。任何工作区间都会被封闭于一个设计模板规模化版本的容错区间内。在节中显示，曲面机械手容错区间的边界符合其临界值集合。图4证明了，在这个例子中此个性质同样适用于五自由度空间机械手。该临界值集合用在伯迪克（1992）中描述的算法来计算和用实线来描述。该临界值集合的粗体部分是该容错区间的边界。性质如下：容错区间受限于临界值集合，因此临界值集合可以有效地被用来检测容错区间。检测一个点是不是容错区间的元素，这是一个复杂的程序。我们必要去验证所有可能的降阶导数是否都能到达那个点，也就是说所有机械手的结构是否会导致每个关节在每个关节角度的关节失效。为了找到与容错区间的近似区间，必要对大量的关节进行这样的检测。这过程缓慢的让人望而却步。尽管如此，为了提高效率，首先计算机械手的临界值集合。这些临界值集合将机械手的笛卡尔输出空间隔成一些扇形区域，这些扇形区域全部在容错区间内，或全部在容错区间外表1 无定向的五自由度一阶容错空间机械手的D-H参数自由度 i1019002a103-a19004b105-b1-表2 八自由度一阶容错空间机械手的D-H参数自由度 i1019002a103-a19004b105-b0900610900700900800-因此，通过检查某个领域的一个点是不是容错区间的元素，可以检查整个区域是不是属于容错区间。容错区间成员检测的次数是可以减少到在隔断的输出空间区域的数目。图4 作为五自由度空间机械手临界值集合的一部分的容错区间边界的截面取一个八自由度机械手，作为空间通用容错机械手的第二个例子，其D-H参数见表2。此机械手的尾部加了一个长度为0的三维转动腕，与前面的例子相同。利用蒙特卡罗方法，当该机械手在任何方向都能到达例1中容错区间的所有点，我们可以确定其是一阶容错的。该性质可以用以下的参数证明。当前五个自由度中的一个失效，该机械手依然可以达到容错区间的任何位置(因为五自由度机械手是容错的)，并且使用这个三维转动腕可以达到此位置的任何方向。当转动腕中的自由度的一个失效，我们只剩下一个七自由度机械手，其有足够的定向能力在任何方向都能达到容错区间的任何点。因此我们称其为灵巧容错区间。既然在其容错区间没有空穴和空隙，该机械手同样可以被用作设计模板。最后，我们应该注意到这两个例子都只有两个冗余自由度，这表示两个冗余自由度可以满足位置机械手的一阶容错性。我们还要做进一步的研究：3.1.4节中的这个定理是否同样适用于位置机械手的高阶容错性。4 特定任务机械手在前节中，我们考虑了通用容
- 温馨提示:
1: 本站所有资源如无特殊说明，都需要本地电脑安装OFFICE2007和PDF阅读器。图纸软件为CAD,CAXA,PROE,UG,SolidWorks等.压缩文件请下载最新的WinRAR软件解压。
2: 本站的文档不包含任何第三方提供的附件图纸等，如果需要附件，请联系上传者。文件的所有权益归上传用户所有。
3.本站RAR压缩包中若带图纸，网页内容里面会有图纸预览，若没有图纸预览就没有图纸。
4. 未经权益所有人同意不得将文件中的内容挪作商业或盈利用途。
5. 人人文库网仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对用户上传分享的文档内容本身不做任何修改或编辑，并不能对任何下载内容负责。
6. 下载文件中如有侵权或不适当内容，请与我们联系，我们立即纠正。
7. 本站不保证下载资源的准确性、安全性和完整性, 同时也不承担用户因使用这些下载资源对自己和他人造成任何形式的伤害或损失。
人人文库网所有资源均是用户自行上传分享，仅供网友学习交流，未经上传用户书面授权，请勿作他用。
|
2:不支持迅雷下载,请使用浏览器下载
3:不支持QQ浏览器下载,请用其他浏览器
4:下载后的文档和图纸-无水印
5:文档经过压缩，下载后原文更清晰
|
川公网安备: 51019002004831号