Mapping Tasks into Fault Tolerant Manipulators

Transcription

1 Proceedings of the 994 IEEE International Conference on Robotics and Automation, San Diego, CA, Ma 9 3, 994. Mapping Tasks into Fault Tolerant Manipulators Christiaan J.J. Paredis and Pradeep K. Khosla Department of Electrical and Computer Engineering and The Robotics Institute, Carnegie Mellon Universit, Pittsburgh, PA 53 Abstract The application of robots in critical missions in hazardous environments requires the development of reliable or fault tolerant manipulators. In this paper, we define fault tolerance as the abilit to continue the performance of a task after immobilization of a joint due to failure. Initiall, no joint limits are considered, in which case we prove the eistence of fault tolerant manipulators and develop an analsis tool to determine the fault tolerant work space. We also derive design templates for spatial fault tolerant manipulators. When joint limits are introduced, analtic solutions become infeasible but instead a numerical design procedure can be used, as is illustrated through an eample. : Introduction Consider the use of manipulators in a nuclear environment where equipment has to be repaired or space has to be searched for radioactive contamination. A manipulator that is deploed for this kind of critical missions must have man desirable attributes. It must be versatile. That is, the manipulator sstem must be capable of accomplishing a wide variet of tasks and possibl be capable of reconfiguring itself. It must be easil maintainable and cost effective. Ease of maintenance would significantl reduce the downtime and therefore increase the utilit of the sstem. It must also be reliable or fault tolerant. That is, if one of the subsstems were to fail, the reduced sstem should still be capable of accomplishing its mission objectives. The propert of fault tolerance is a must in critical missions that ma involve high costs if the mission were to fail due to a failure in the manipulator sstem. Recentl, fault tolerant (or failure tolerant) robotics has been the subject of several publications [,, 5, 6], in which different aspects of the problem are addressed. Visinsk et al. [6] propose a framework to include failure detection in fault tolerant robot sstems. Lewis and Maciejewski [], on the other hand, discuss the importance of the controller and the redundanc resolution algorithm. In this paper, the focus is on kinematic design of fault tolerant manipulators. : Definition of fault tolerance To set the stage for our development, we define the following properties of fault tolerant manipulators []: Fault Tolerant (FT) Manipulator: An n-dof manipulator that will still be able to meet the task specifications, even if an one or more of its joints fail and are frozen at an arbitrar 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 fault manipulator is called a k-reduced order derivative. Order of Fault Tolerance: An n-dof manipulator is FT of the k-th order, if and onl if all possible k-reduced order derivatives can still perform the specified task. We call the manipulator k-fault tolerant. Fault Tolerant Work Space (): The fault tolerant work space of a k-ft manipulator is the set of points reachable b all possible k-reduced order derivatives. These definitions differ from the concept of fault tolerance as proposed b Maciejewski []. Instead of attributing the propert of fault tolerance to a manipulator, he quantifies a measure of fault tolerance for a manipulator posture and describes a technique to determine the optimal FT posture, based on the singular value decomposition of the Jacobian matri. If a joint fails in this optimal posture, the resulting reduced order derivative will have maimum possible deterit. However, a failure at a different angle ma make the eecution of the task impossible.

2 In the rest of this paper, if no specific task is mentioned, it is assumed that the task consists of reaching a nonzero volume of points in the task space, i.e., an m-di- mensional manifold in the m-dimensional task space. A manipulator that satisfies the fault tolerance condition onl while reaching a manifold of dimension less than m, is considered not to be fault tolerant. 3: Properties of fault tolerant manipulators 3.: Eistence frozen θ ROD M * p b work space boundar of M * Suppose that we have an n-dof manipulator,, that satisfies all the kinematic requirements of a given task. It has been shown that such a manipulator can be found b using the task based design approach described in [8] or [3]. An obvious wa to make this manipulator FT is to design ever joint with a redundant actuator. If one of the actuators of the resulting n-dof FT manipulator were to fail, the redundant actuator could take over and the manipulator would still be functional. Similarl, a k-ft manipulator can be constructed b duplicating ever DOF k times, resulting in a ( k )n-dof manipulator. This same manipulator structure can also be interpreted according to the four level subsumptive architecture for FT manipulators, proposed in [5]. The four levels are: I) Dual Actuators: etra actuators per joint. II) Parallel Structures: etra joints per DOF. III) Redundant Manipulators: etra DOFs per arm. IV) Multiple Arms: etra arms per sstem. According to this classification, the ( k )n-dof manipulator is an n-dof manipulator with ( k ) actuators for ever DOF redundanc of level I. In the rest of this paper, we focus our attention on fault tolerant manipulators with redundancies of level III. 3.: Boundar of the fault tolerant work space M n In this section, we show that a boundar point of the is a critical value (A critical value is an end-effector position that can be reached in a singular configuration, i.e., that is the image of a critical point []). Consider a k-ft planar manipulator, M. A boundar point, p b, of the has to be an element of the boundar of the work space of at least one ROD, M *, obtained b freezing k joints of M. Indeed, if p b were an interior point of the work spaces of all RODs, then it would b definition be an interior point of the and not a boundar point. The Jacobian of M *, J M*, can be obtained from the Jacobian of M, J M, b deleting the columns corresponding to the frozen DOFs. Because p b is a boundar point of the work space of, the Jacobian of M * frozen θ d θ Figure : A ROD unable to reach a point outside the. M * at p b is singular. We prove now that J M is singular too. Suppose that J M were non-singular, then at least one of the columns corresponding to a frozen DOF would be outside the column space of the singular matri, J M*. Phsicall this means that a small change in the angle of that frozen DOF would cause the end effector of M to move in a direction with a component perpendicular to the boundar of the work space of the ROD, M *, as illustrated in Figure. The ROD with this new frozen angle would be unable to reach the point, p b. As a result, p b would be outside the, contradicting the fact that p b is a boundar point of the. Thus, J M is singular and p b is a critical value. Consequentl, the is bounded b critical value manifolds. For planar positional manipulators, the critical value manifolds are concentric circles, and the is an annulus with inner radius R min and outer radius. R ma 3.3: Required degree of redundanc In Section 3., it is shown that, in general, kn redundant DOFs i.e. ( k )n DOFs in total are sufficient to achieve k-th order fault tolerance. For planar positional manipulators, we prove now that k DOFs are also necessar for k-th order fault tolerance. p b

3 (k) joints R ma = R min, (4) resulting in a one-dimensional. Therefore, a ( k ) -DOF manipulator cannot be FT 3.4: Including orientation R ma k frozen joints Thus far, we have onl considered planar positional manipulators. The results for positional manipulators can be easil etended to the case in which orientation is considered also, b converting the orientational problem into an equivalent positional problem: R min k frozen joints (k) joints R ma Figure : An upper bound for and a lower bound for. R min The proof shows that ( k ) DOFs (or k redundant DOFs) are insufficient, b finding a lower bound for R min and an upper bound for R ma. First consider the ROD obtained b freezing the first k joints at radians, as illustrated in Figure. The maimum reach in the opposite direction is an upper bound for : R ma k l k i = k R ma i = k () where is the length of the i-th link. In order for to be positive, we must have that: R ma k l k i = k i = k. () Making this assumption, we find that is bounded below b the inner radius of the work space of the ROD obtained b freezing the k last joints at radians, as illustrated in Figure : R min k i = k. (3) From Equation () and Equation (3), it follows that at best l k k i = R min An n-dof manipulator, M, is k-ft with respect to a set of points, W = { ( i, i, ϕ i )}, if and onl if:. the positional manipulator, M ', obtained from M b deleting its last link, l n, is k-th order fault tolerant with respect to the set of points W' = { ( i l n cosϕ i, i l n sinϕ i )},. M ' is ( k ) -FT while reaching the points in W' in an direction. The positional manipulator, M ', needs at least ( k ) DOFs to be k-ft with respect to W' ; therefore, the manipulator M needs at least ( k 3) DOFs. Now, consider a ( k 3) -DOF manipulator with the first ( k ) links having length, l, and the last link having length zero. It is eas to verif that this manipulator s k-th order is: W = { (,, ϕ) l and ϕ [ π, ) } Thus, ( k 3) DOFs are necessar and sufficient for k-th order fault tolerance of planar manipulators when orientation is included This result and the result obtained in Section 3.3 can be summarized in the following theorem: Theorem: For planar manipulators, k redundant DOFs are necessar and sufficient for k-th order fault tolerance. 4: Spatial fault tolerant manipulators For planar FT manipulators, we were able to prove that k is the required degree of redundanc. The proof was based on geometric work space analsis. However, geometric analsis becomes too comple for spatial manipulators, especiall since we are dealing with redundant manipulators. Therefore, we will demonstrate some properties of spatial FT manipulators using two eamples. As a first eample, consider a 5-DOF spatial positional manipulator. Its Denavit-Hartenberg (D-H) parameters are listed in Table. This manipulator is first order FT, (5) 3

4 z Figure 3: A cross-section of the boundar of the of a 5-DOF spatial manipulator (bold) as part of its critical value mani- and because of its simple kinematic structure, an analtic epression for the boundar of the can be derived. The is smmetric with respect to the first ais. A cross-section (the X-Z plane), as shown in Figure 3, can be described b two segments of a circle with radius and center at ( =, z = ), and a straight line from ( =, z = 3) to ( =, z = 3). An important propert of this is that it does not have an holes or a central void, so that the of the same manipulator scaled b an factor, λ >, contains the original. As a result, this FT manipulator can be used as a design template. An specified set of points can be reached in a first order FT wa b a scaled version of the template. In Section 3., it is shown that the boundar of the of a planar manipulator coincides with its critical value manifolds. Figure 3 demonstrates that this propert also holds for the 5-DOF spatial manipulator considered in this eample. The critical value manifolds are comput- DOF i d i a i α i 9 a 3 -a 9 4 b 5 -b Table : D-H parameters of a 5-DOF first order FT spatial manipulator, without orientation. ed using the algorithm described in [] and are depicted in a solid line. The bold part of the critical value manifolds is the boundar of the. As a second eample, consider an 8-DOF manipulator, with D-H parameters listed in Table. It is the same manipulator as in eample one, with a zero-length 3-rollwrist added at the end. Using a Monte Carlo method, it has been determined that this manipulator is first order FT while reaching all the points in the of eample one, in an direction. This propert can be demonstrated with the following arguments. When one of the first five DOFs fails, the manipulator can still reach an position in the (because the 5-DOF positional manipulator is FT) and can take an orientation at this position using the intact 3-roll-wrist. When one of the DOFs in the wrist fails, we are left with a 7-DOF manipulator which has enough orientational capabilities to reach an point in the in an orientation. Consequentl, one could call this the detrous. Since there are again no holes or voids in the, this manipulator can also be used as a design template. Finall, one should notice that both eamples have onl two redundant DOFs, which seems to indicate that the theorem in Section 3.4 is etendable to spatial manipulators. 5: Joint limits In Section, the definition of a FT manipulator included the specification that joints could fail at an arbitrar angle. Thus far, we did not consider joint limits and therefore at an arbitrar joint angle meant at an angle between and π. From now on, however, an arbitrar joint angle is restricted to be within the joint limits. Using two eamples, we discuss now how the introduction of joint limits changes the properties of FT manipulators. First, consider a 4-DOF positional planar manipulator with links of length one. Figure 4 illustrates how the DOF i d i a i α i 9 a 3 -a 9 4 b 5 -b Table : D-H parameters of an 8-DOF first order FT spatial manipulator. 4

5 [-8,8] [-5,5] [-,] [-9,9] section of fewer RODs has to be taken to obtain the. As a second eample, we revisit the 5-DOF spatial manipulator of the previous section. As depicted in Figure 5, the of this 5-DOF manipulator shrinks rapidl when the joint angle intervals are reduced. Moreover, a central hole appears and the splits into two disconnected parts. When the joint angle interval becomes smaller than [ 5, 5 ], the vanishes. From the above eamples, it is clear that the analsis, and therefore also the design of FT manipulators is ver comple when joint limits are introduced. Whereas we were able to generate design templates for the case of no joint limits (or joint limits of ±8 ), finding a general design template is infeasible due to the infinite number of possible joint limits. Therefore, in the net section, we propose a numerical procedure to find the kinematic structure of a FT manipulator when joint limits are considered. 6: Task based design of fault tolerant manipulators 6.: Mathematical formulation In this section, we formulate a method to determine the kinematic structure of a manipulator which is FT with respect to the task of reaching a set of positions/orienta-.5 [-6,6] [-3,3] Figure 4: The of a 4-DOF planar manipulator with joint limits. work space changes when the joint limits (the same limits for all joints) var from ± 8 through ± 3. As one would epect the shrinks, but it also changes position. A point that is outside the of a manipulator without joint limits can become inside the when joint limits are introduced. This can be understood b thinking of the as the intersection of the work spaces of all RODs. When joint limits are introduced, the work space of ever ROD reduces in size, but the inter- z Figure 5: A cross-section of the of a 5-DOF spatial manipulator with joint limits. 5

6 tions, W = { p j = ( j, j, z j, ϑ j, ϕ j, ψ j )}, and for which the joint limits for each of the DOFs are given. We present a technique based on the design method proposed in [4]. In this approach, the Denavit-Hartenberg parameters are determined through the optimization of a penalt, which is positive when an design specification is violated and is zero when all the specifications are met. This is epressed mathematicall as: min min [ pen_f ( DH, θ post, task, cnstr )], (6) DH task points postures where pen_f( ) determines the penalt for the violation of the constraints b a manipulator design, DH, in a posture, θ post, reaching a specific task point. A manipulator is first order FT with respect to a task if all possible first order RODs can fulfill the task. Equation (5) can then be adapted to include first order fault tolerance as follows []: min ma [ min pen_f ( )] DH task points θ j = j θ i ( i j) The maimization ields zero when all the RODs, resulting from a failure of joint j, are able to reach the task point. As a result, the minimization over the D-H parameters equals zero if and onl if all possible RODs are able to reach ever specified task point. 6.: Implementation n Equation (6) contains three nested optimizations. For each ROD, the innermost minimization finds the posture, for which the penalt function is minimal. The task that we consider is to reach a certain point without violating the joint limit constraints. The penalt function therefore consists of two parts. A first part penalizes the failure to reach the point (without considering joint limits). This penalt is equal to the norm squared of the residue, r, namel, the distance between the actual position/orientation of the end effector and its desired position/orientation. A second part, penalizes the joint limit violations. This penalt is equal to n i = (7) ma ma(, θ i θ i ) min { ma(, θ i θ i ) }. (8) The resulting minimization of the full penalt function, is a nonlinear least squares (NLLS) problem, and is commonl solved in robotics applications using a Newton- Raphson scheme [6] (cfr. numericanverse kinematics literature). This scheme converges ver quickl when the penalt at the optimum is zero, i.e., for a zero residual NLLS problem. However, it is possible that a task point measure of isotrop penalt frozen angle θ j frozen angle θ j Figure 6: The penalt function and the measure of isotrop. is outside the reach of the manipulator, in which case the minimization becomes a large residual problem. For this class of problems, the Newton-Raphson scheme (or Gauss-Newton algorithm) converges slowl and is outperformed b the BFGS algorithm (Broden-Fletcher-Goldfarb-Shanno [5]). As shown b Fletcher and Xu [4], the best overall performance is obtained b a hbrid algorithm, which combines the advantages of both the Gauss- Newton and BFGS algorithms. The second optimization in Equation (6) is the maimization over the frozen joint angle, θ j. A tpical eample of the objective function is shown in Figure 6. Notice that the function and its gradient are zero possibl over a large part of the domain, namel, where the task point is reachable. To solve the maimization, we need to incorporate additionanformation. The ke observation is that when the desired point is outside the reach of the manipulator, the innermost minimization algorithm alwas gets stuck at the boundar of the work spaces of the manipulator, where the Jacobian matri of the manipulator loses rank []. The measure of isotrop,, is defined as [7]: m det ( JJ T ) =, (9) trace ( JJ T )/ m with m the number of Cartesian coordinates. For a singular Jacobian, is zero; otherwise it is positive. Consequentl, a minimization over leads to regions where the penalt is possibl positive. Thus, the derivative of with respect to θ j, can be used to compensate for the lack of gradient information in the norm of the residue. As one can see in Figure 6, however, the measure of isotrop is a multimodal function of θ j. As there is onl one optimization variable, an efficient global optimization algorithm, such as the P * algorithm developed b Zilinskas [7], can be used. P * is a combination of a 6

7 In this paper, we defined fault tolerance as a propert of a small class of redundant manipulators. Based on the definition, we were able to constructivel prove the eistence of FT manipulators b duplicating the joints. When no joint limits are considered, we proved analticall that, k redundant DOFs are necessar and sufficient for fault tolerance of planar manipulators. We also proved that the boundar of the consists of critical values. For spatial manipulators, design templates were introduced. A manipulator able to reach a specified set of points can be designed b scaling the template appropriatel. For manipulators with joint limits, analtical solutions be- 5 DOF solution p = (.5,,.5) DOF i d i a i α i p = (,.5,.5) ()..7 9 p 3 = (.75,,.5).. Notice that we do not consider end effector orientation at this time. The joint limits for all five DOFs are and 5. From the eample in Section 5, we know that five degrees of freedom are sufficient for first order fault toler Table 3: D-H parameters of a 5-DOF first order FT spatial solution with joint limits. ance. Our numerical results confirm this. The D-H parameters of a possible design are listed in Table 3. Figure 7 shows a cross-section along the XZ-plane of the boundar of the. one-stage Baesian algorithm and Brent s local minimization resulting in smart global coverage and accurate local refinement. The third and outermost optimization minimizes the total penalt of a kinematic configuration, defined b the ( 3n ) D-H parameters. Unfortunatel, this function also has multiple minima. Therefore, we need to emplo a global optimization method, such as simulated annealing. Simulated annealing was first proposed as a combinatorial optimization algorithm [9], but can be easil adapted to continuousl varing optimization variables. The method is basicall a random iterative improvement algorithm with the modification that, under certain conditions, an increase in the objective function is accepted. A new trial configuration is generated randoml in the neighborhood of the current configuration. The condition for acceptance of this trial configuration is: 8: Summar F obj ep ( F obj T) random [, ) () which depends on a control variable, T, the temperature. The algorithm is started at a high temperature, for which all new configurations are accepted. After each iteration the temperature is decreased until no new acceptable configuration can be found. The search is then frozen. We adapted this basic algorithm to include the special properties of our objective function, in particular, the algorithm is stopped when a new trial configuration has an objective function value equal to zero, even when the search is not et frozen. We know that a kinematic configuration with a penalt of zero satisfies all the design requirements. 7: Numerical results accept > accept To illustrate the approach developed in the previous section, we give an eample of a first order FT spatial manipulator design: a five DOF spatial manipulator that is required to be first order FT while accomplishing the task of reaching three points: z Figure 7: A cross-section of the of the 5-DOF solution 7

8 come infeasible. Therefore, we introduced a numerical design method based on an earlier developed task based design approach. All the different steps of our development were illustrated with design eamples. 9: Acknowledgment This research was funded in part b DOE under grant DE-F9-89ER44, b Sandia under contract AC- 375-A, b the Department of Electrical and Computer Engineering, and b The Robotics Institute, Carnegie Mellon Universit. : References [] W.K.F. Au, C.J.J. Paredis, P.K. Khosla, Kinematic Design of Fault Tolerant Manipulators, in Proceedings of the Allerton Conference, October, 99, Urbana-Champagne, Illinois. [] J.W. Burdick, Kinematic Analsis and Design of Redundant Robot Manipulators, Stanford Computer Science Report no. STAN-CS-88-7, 988. [3] J. Denavit, R.S. Hartenberg, A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices, Journal of Applied Mechanics, Vol., No., pp. 5, June 955. [4] R. Fletcher, C. Xu, Hbrid Methods for Nonlinear Least Squares, IMA Journal of Numerical Analsis, Vol. 7, No. 3, pp , Jul 987. [5] R. Fletcher, Practical Methods of Optimization, Second Edition, John Wile & Sons, New York, 987. [6] P.K. Khosla, C.P. Neuman, F.B. Prinz, An Algorithm for Seam Tracking Applications, The International Journal of Robotics Research, Vol. 4, No., pp. 7 4, Spring 985. [7] J.-O. Kim, P.K. Khosla, Deterit Measures for Design and Control of Manipulators, in Proceedings of IROS 9: International Workshop on Intelligent Robots and Sstems, pp , Osaka, Japan, Nov. 3 5, 99. [8] J.-O. Kim, Task Based Kinematic Design of Robot Manipulators, Ph.D. thesis, Robotics Ph.D. Program, Carnegie Mellon Universit, Pittsburgh, PA, August 99. [9] S. Kirkpatrick, C.D. Gelatt Jr., M.P. Vecchi, Optimization b Simulated Annealing, Science, Vol., No. 4598, pp , Ma 983. [] D. Kohli, M.-S. Hsu, The Jacobian Analsis of Workspaces of Mechanical Manipulators, Mechanism and Machine Theor, Vol., No. 3, pp , 987. [] C.L. Lewis, A.A. Maciejewski, Deterit Optimization of Kinematicall Redundant Manipulators in the Presence of Joint Failures, to appear in Computers and Electrical Engineering, 993. [] A.A. Maciejewski, Fault Tolerant Properties of Kinematicall Redundant Manipulators, in Proceedings of the 99 IEEE International Conference on Robotics and Automation, Cincinnati, Ohio, pp , Ma 99. [3] C.J.J. Paredis, An Approach for Mapping Kinematic Task Specifications into a Manipulator Design, M.S. Thesis, Electrical and Computer Engineering Department, Carnegie Mellon Universit, September 99. [4] C.J.J. Paredis, P.K. Khosla, Kinematic Design of Serial Link Manipulators from Task Specifications, The International Journal of Robotics Research, Vol., No. 3, pp , June 993. [5] D. Sreevijaan, On the Design of Fault-Tolerant Robotic Manipulator Sstems, M.S. Thesis, Mechanical Engineering Department, The Universit of Teas at Austin, June 99. [6] M.L. Visinsk, I.D. Walker, and J.R. Cavallaro, Laered Dnamic Fault Detection and Tolerance for Robots, in Proceedings of the 993 IEEE International Conference on Robotics and Automation, Atlanta, Georgia, pp. 8-87, Ma 993. [7] A. Zilinskas, Optimization of One-Dimensional Multimodal Functions, Algorithm AS 33, Applied Statistics, Vol. 3, pp ,