Designing Equally Fault-Tolerant Conﬁgurations for Kinematically Redundant Manipulators Rodney G. Roberts † , Salman A. Siddiqui † , and Anthony A. Maciejewski ‡ Abstract— In this article, the authors examine the problem of designing nominal manipulator Jacobians that are optimally fault-tolerant to multiple joint failures. In this work, optimality is deﬁned in terms of the worst case relative manipulability index. Building on previous work, it is shown that for a robot manipulator working in three-dimensional workspace to be equally fault-tolerant to any two simultaneous joint failures, the manipulator must have precisely six degrees of freedom. A corresponding family of Jacobians with this property is identiﬁed. It is also shown that the two-dimensional workspace problem has no such solution. I. INTRODUCTION Fault-tolerant design of serial or parallel manipulators is critical for tasks requiring robots to operate in remote and hazardous environments where repair and maintenance tasks are extremely difﬁcult [1]-[10]. In such cases, operational reliability is of prime importance. By adding kinematic redundancy to the robotic system, the robot may still be able to perform its task even if one or more joint actuators fail [11]. However, simply adding kinematic redundancy to the system does not guarantee fault tolerance [12]. One must strategically plan how the kinematic redundancy should be added to the system to ensure that fault tolerance is optimized [13]. One approach to the problem of designing fault-tolerant robots is to optimize some measure of fault tolerance. This measure can be either global, i.e., over a speciﬁed region of the workspace, or local, i.e., at a speciﬁc conﬁguration. Global measures, such as those in [14], [15], are more appropriate for tasks that require large motions throughout the workspace, whereas local measures [11], [16] are more appropriate for dexterous operations in a relatively small location, e.g., laser pointing [6] and manipulation of nuclear material [9]. In this article we focus on a local measure called the relative manipulability index, which was ﬁrst introduced in [12] to quantify the fault tolerance of kinematically re- dundant serial manipulators. Relative manipulability indices have also been used to study the fault tolerance of redundant Gough-Stewart platforms [17]. In the next section, we describe the relative manipulability index and its relationship to the null space of the manipulator Jacobian. In Section III, bounds are derived for the minimum This work was supported by the National Science Foundation under Contract IIS-0812437. † R. G. Roberts and S. A. Siddiqui are with the Department of Electrical and Computer Engineering, Florida State University, Tallahassee, FL 32310- 6046, USA (e-mail: rroberts@eng.fsu.edu; siddiqui@eng.fsu.edu). ‡ A. A. Maciejewski is with the Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523-1373, USA (e-mail: aam@colostate.edu). relative manipulability index. These bounds motivate the goal of ﬁnding optimally fault-tolerant manipulator conﬁgurations in Section IV. Examples of optimally fault-tolerant conﬁgu- rations are presented. Lastly, conclusions appear in Section V. II. THE RELATIVE MANIPULABILITY INDEX For a serial manipulator, the relative manipulability index is deﬁned in terms of the manipulator Jacobian J , which relates the manipulator’s joint velocity ˙ θ to its end-effector velocity v by the equation v = J ˙ θ. (1) In this work, we will assume that the manipulator is not operating at a kinematic singularity so that J has full rank. A joint failure signiﬁcantly affects the kinematics of the robot. Two types of joint failures for serial manipulators have been examined in the literature. One type is a free-swinging failure. In this case, the failed joint becomes passive. The other type, which we will study in this work, is a locked- joint singularity. When a locked-joint failure occurs, say in joint i, that component of the joint velocity is zero. Consequently, the end-effector motion is characterized by i J , i.e., the Jacobian J with its i-th column removed. Multiple locked-joint failures are handled in the same way, i.e., the corresponding columns of the Jacobian are removed. The relative manipulability index corresponding to locked- joint failures in joints i 1 ,...,i f is deﬁned to be ρ i1,··· ,i f = w( i1···i f J ) w(J ) , (2) where i1···i f J denotes the manipulator Jacobian after the columns i 1 ,...,i f corresponding to the failed joints are removed and where w(J )=  det(JJ T ) is the manipu- lability index of J [18]. This quantity is a local measure of the amount of dexterity that is retained when a manipulator suffers one or more locked-joint failures. The value of a relative manipulability index ranges from zero to one. A zero value would indicate a loss of full end-effector motion at that conﬁguration after the failed joints are locked. In other words, a zero relative manipulability index means that the reduced manipulator Jacobian i1···i f J does not have full rank. A relative manipulability index of one would indicate that no dexterity is lost at that conﬁguration. In this case the joints in question do not contribute to end-effector motion at the operating conﬁguration prior to their failure, i.e., those joints only produce self-motion [12], [19]. 41st Southeastern Symposium on System Theory University of Tennessee Space Institute Tullahoma, TN, USA, March 15-17, 2009 T2B.2 978-1-4244-3325-4/09/$25.00 ©2009 IEEE 335 Authorized licensed use limited to: COLORADO STATE UNIVERSITY. Downloaded on April 9, 2009 at 19:22 from IEEE Xplore. Restrictions apply.