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Designing Fault Tolerant Manipulators:
How Many Degrees-of-Freedom?
Christiaan J. J. Paredis and Pradeep K. Khosla
Department of Electrical and Computer Engineering and
The Robotics Institute,Carnegie Mellon University,Pittsburgh, Pennsylvania 15213.
Abstract
One 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 Introduction
As 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 the
overall 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 actuators—extra actuators per joint,
Level 2: Parallel structures—extra joints per DOF,
Level 3: Redundant manipulators—extra DOFs per manipulator arm,
Level 4: Multiple arms—extra arms per manipulator system.
A system can possibly be designed with redundancies at all four levels, resulting in the ability to sustain
multiple 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 DOF—an 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 motion—a 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 Reliability
The basic idea presented in this article is to use a manipulator’s 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 subsystem—a DOF or joint module for modular manipulators—fails. 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 properly—the system can tolerate (n – m) module failures—then 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 system’s unreliability,
When The unreliability for the non-fault-tolerant system is
In general, the unreliability of a k-fault tolerant system—one that can sustain k faults—is 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 Manipulators
In 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 Manipulators
3.1.1 Existence
A 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 Workspace
In this section, we show that a boundary point of the FTWS is a critical value. Consider a k–fault 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 va