23 -18 Robotics and Automation Handbook start yes no no yes stop interaction calculator collision detector simulation over? forward dynamics solver dynamic model geometric model interaction model FIGURE 23.7 Flowchart showing interconnection of the dynamics solver, collision detector, and interaction calculator. 23.5.1 Collision Detector To calculate a global solution in a computationally efficient manner, it is very common to handle the collision detection problem in two parts: a broad phase, which involves a coarse global search for potentially interacting surfaces, and a narrow phase, which is usually based on a fast local optimization scheme on convex surface patches. To handle nonconvex shapes, preprocessing is performed to decompose them into sets of convex surface patches. However, there exist shapes for which such a decomposition is not possible, for example, a torus. 23.5.1.1 Broad Phase Thebroad phaseiscomposedoftwomajorsteps. First,aglobalproximitytestisperformedusinghierarchies of bounding volumes or spatial decompositions for each surface patch. Among the most widely used bounding volumes and space decompositions are the octrees [36], k-d trees [37], BSP-trees [38], axis- aligned bounding boxes (AABBs) [39], and oriented bounding boxes (OBBs) [40]. During the global proximity test, the distances between bounding boxes for each pair of surface patches drawn from all pairs of bodies are compared with a threshold distance and surfaces that are too distant to be contacting are pruned away. Remaining surfaces are set to be active. In the second step of the broad phase, approximate interaction points on active surfaces are calculated. For example, if the geometric models are represented by non-uniform rational B-splines (NURBS), control polygons of the active surface models can be used to calculate a first order approximation to the closest points on these surfaces. Specifically, bounding box centroids of each interacting surface patch can be projected onto the polygonal control mesh of the other patch. Using the strong convex hull property of Copyright © 2005 by CRC Press LLC Haptic Interface to Virtual Environments 23 -19 NURBS surfaces, one can determine the candidate span and interpolate the surface parameters from the node values. These approximate projections serve as good initialization points for the narrow phase of the collision detector. 23.5.1.2 Narrow Phase After initialization with the approximations calculated in the broad phase, the narrow phase employs an algorithm to iteratively update the minimum distance between each active pair of surfaces. Previous work in narrow phase collision detection has concentrated on computing the minimum dis- tance between convex polyhedral objects. State-of-the-art algorithms for computing the distance between convex polyhedra are basedonthealgorithmby Gilbert, Johnson, and Keerthi (GJK) [41,42],thealgorithm of Lin and Canny [43], and the algorithm by Mirtich [44]. The GJK algorithm makes use of Minkowski difference and (simplex-based) convex optimization tech- niques to calculate the minimum distance. The iterative algorithm generates a sequence of ever im- proving intermediate steps within the polyhedra to converge to the true solution. The algorithm of Lin and Canny makes use of Voronoi regions and temporal/spatial coherence between successive queries to navigate along the boundaries of the polyhedra in the direction of decreasing distance. The V-Clip al- gorithm by Mirtich is reminiscent of the Lin and Canny closest features algorithm but makes several improvements. Less literature exists on direct collision detection for nonpolygonal models. Gilbert et al. extended their algorithm to general convex objects in [45]. In a related paper [46], Turnbull and Cameron modify the widely used GJK algorithm to handle convex shapes defined using NURBS. Similarly, in [47, 48] Lin and Manocha present an algorithm for curved models composed of spline or algebraic surfaces by extending their earlier algorithm for polyhedra. Finally, a new class of algorithms, namely, minimum distance tracking algorithms for parametric sur- faces, is presented in [49–52]. These algorithms are designed to maintain the minimum distance between two parametric surfaces by integrating the differential kinematics of the surfaces as the surfaces undergo rigid body motion. 23.5.2 Interaction Calculator The interaction calculator, triggered into action by the collision detector, computes inputs (forces and moments) and new initial conditions (in the case of impulse response) for the forward dynamics solver. In so doing, it realizes interaction between bodies, whether such interaction occurs during a finite or an infinitesimal period of time. As indicated in Figure 23.7, the interaction calculator calls upon an interaction model and the geometric model to compute its response. If impulse response is also used, the interaction calculator additionally calls upon portions of the dynamic model. The most widely used interaction model in both haptic rendering and multibody dynamics is called the penalty method: it assumes compliant contact characterized by a spring-damper pair. The force response is then proportional to the amount and rate of interpenetration. The interaction calculator need only query the geometric model using these surface parameters to determine the interpenetration vector. To compute a replacement (a resultant applied at the center of mass and a couple applied to the body) for the set of all contact forces and moments acting on a body for use as input to the dynamic model, the interaction calculator queries the geometric model using the surface parameters and then carries out the appropriate dot and cross products and vector sums. Using replacements for the set of interaction forces allows the dynamic model to be fully divorced from the geometric model. The penalty contact model allows interpenetration between the objects within the virtual environment, which in a certain sense is nonphysical. However, the penalty method also eliminates the need for impulse response models, since all interactions take finite time. Important alternatives to the penalty method in- clude the direct computation of constraint forces using the solution of a linear complementarity problem espoused by Baraff [53, 54] and widely adopted within the computer graphics community. The formula- tion by Mirtich and Canny [55] uses only impulses to account for interaction, with many tiny repeated Copyright © 2005 by CRC Press LLC 23 -20 Robotics and Automation Handbook impulses for the case of sustained contact. Note that computation of impulses requires that the interaction calculator query the dynamic model with the current extremal point parameters and system configuration to determine the effective mass and inertia at each contact point. Itisimportant tocarefully distinguishbetween theroleof thepenaltymethodand thevirtual coupler. The virtual coupler serves as a filter between the virtual environment and haptic device whose design mitigates instability and depends on the energetic properties of the closed-loop system components. The penalty method, on the other hand, though it may also be modelled by a spring-damper coupler, is an empirical law chosen to produce reasonable behavior in multibody system simulation. Parameter values used in the penalty method also have implications for stability, though in this case it is not haptic rendering system but rather numerical stability that is at issue. Differential equation stiffness is, of course, also determined by penalty method parameter values. Associated with the penalty method is another subtle issue, that of requiring a unique solution to the maximum distance problem when the interpenetrated portions of two bodies are not both strictly convex or there exists a medial axis [56] within the intersection. This is the issue which the god-object or proxy methods address [57, 59]. 23.5.3 Forward Dynamics Solver The final component comprising the haptics-equipped simulator is a differential equation solver that advances the solution of the equations of motion in real time. The equations of motion are a set of differential equations constructed by applying a principal of mechanics to kinematic, inertial, and/or energy expressions derived from a description of the virtual environment in terms of configuration and motionvariables(generalizedcoordinatesandtheir derivatives)and mass distribution properties.Typically the virtual environment is described as a set of rigid bodies and multibody systems interconnected by springs, dampers, and joints. In such case the equations of motion become ordinary differential equations (ODEs), expressing time-derivatives of generalized coordinates as functions of contact or distance forces and moments acting between bodies of the virtual environment. In Figure 23.4, representative bodies A, B, and P comprise the virtual environment, while the forces and moments in the springs and dampers that make up the virtual coupler between bodies P and E are inputs or arguments to the equations of motion. The configuration (expressed by the generalized coordinates) and motion (expressed by the generalized coordinate derivatives) of bodies A, B, and P are then computed by solving the equations of motion (see also Figure 23.5). One may also say that the state (configuration and motion) of the virtual environment is advanced in time by the ODE solver. Meanwhile, the collision detector runs in parallel with the solution of the equations of motion, monitor- ing the motion of bodies A, B, and P and occasionally triggering the interaction calculator. The interaction calculator runs between time-steps and passes its results (impulses and forces) to the equations of motion for continued simulation. Quite often the virtual environment is modeled as a constrained multibody system, and expressed as a set of differential equations accompanied by algebraic constraint equations. For example, constraint appending using the method of Lagrange Multipliers produces differential-algebraic equations (DAEs). In such case, a DAE solver is needed for simulation. Note that DAE solvers are not generally engineered for use in real-time or with constant step-size and usually require stabilization. Alternatively, constrained multibody systems may be formulated as ODEs and then simulated using standard ODE solvers using constraint-embedding techniques. Constraint embedding can take place symbolically (usually undertaken prior to simulation time) or numerically (possibly undertaken during simulation and in response to run- time events). Alternatives to the forward dynamics/virtual coupler formulation described above have been developed for tying together the dynamic model and the haptic interface. For example, the configuration and motion of the haptic device image (body E in Figure 23.5) might be driven by sensors on the haptic interface. A constraint equation can then be used to impose that configuration and motion on the dynamic model of the virtual environment. Copyright © 2005 by CRC Press LLC Haptic Interface to Virtual Environments 23 -21 23.6 Concluding Remarks In this chapter, we have presented a framework for rendering virtual objects for the sense of touch. Using haptic interface technology, virtual objects can not only be seen, but felt and manipulated. The haptic interface is a robotic device that intervenes between a human user and a simulation engine, to create, under control of the simulation engine, an appropriate mechanical response to the mechanical excitation imposed by the human user. ‘Appropriate’ here is judged according to the closeness of the haptic interface mechanical response to that of the target virtual environment. If the excitation from the user is considered to be motion, the response is force and vice-versa. Fundamentally, the haptic interface is a multi-input multi-output system. From a control engineering perspective, the haptic interface is a plant simultaneously actuated and sensed by two controllers: the human user and the simulation engine. Thus the behavior of the haptic interface is subject to the influence of the human user, the virtual environment simulator, and finally its own mechanics. As such, its ultimate performance requires careful consideration of the capabilities and requirements of all three entities. While robotics technology strives continually to instill intelligence into robots so that they may act autonomously, haptic interface technology strives to strike all intelligence out of the haptic interface—to get out of the way of the human user. After all, the human user is interested not in interacting with the haptic interface, but with the virtual environment. The user wants to retain all intelligence and authority. As it turns out, to get out of the way presents some of the most challenging design and analysis problems yet posed in the greater field of robotics. References [1] Goertz, R.C., Fundamentals of general-purpose remote manipulators, Nucleonics, vol. 10, no. 11, pp. 36–45, 1952. [2] Baarspul, M., Review of flight simulation techniques, Progress in Aerospace Sciences, vol. 27, no. 1, pp. 1–20, 1990. [3] Collinson,R., Fly-by-wireflight control,ComputingandControlEngineering Journal,vol.10,pp.141– 152, August 1999. [4] Burdea, G.C., Force and Touch Feedback for Virtual Reality, Wiley Interscience, New York, NY, 1996. [5] Marayong, P., Li, M., Okamura, A.M., and Hager, G.D. Spatial motion constraints: theory and demonstrations forrobotguidanceusing virtualfixtures,in IEEEInternational ConferenceonRobotics and Automation, pp. 1954–1959, 2003. [6] Griffiths, P. and Gillespie, R.B., Shared control between human and machine: Haptic display of automation during manual control of vehicle heading, in 12th International Symposium of Haptic Interfaces for Virtual Environment and Teleoperator Systems (HAPTICS’04), Chicago, IL, pp.358–366, March 27–28, 2004. [7] Bettini, A., Marayong, P., Lang, S., Okamura, A.M., and Hager, G. Vision assisted control for ma- nipulation using virtual fixtures, IEEE Transactions on Robotics and Automation, in press. [8] Reinkensmeyer, D., Standard Handbook of Biomedical Engineering & Design, ch. Rehabilitators, pp. 35.1–35.17, McGraw-Hill, New York, 2003. [9] Gillespie, R.B., Hoffman, M., and Freudenberg, J., Haptic interface for hands-on instruction in system dynamics and embedded control, in IEEE Virtual Reality Conference, (Los Angeles, CA), pp. 410–415, March 22–23, 2003. [10] Okamura,A.M.,Richard, C.,andCutkosky, M.R.,Feelingisbelieving:usinga force-feedbackjoystick to teach dynamic systems, ASEE J. Engineering Education, vol. 92, no. 3, pp. 345–349, 2002. [11] Lawrence, D.A., Stability and transparency in bilateral teleoperation, IEEE Transactions on Robotics & Automation, vol. 9, pp. 624–637, October 1993. [12] Seguna, C.M., The design, construction and testing of a dexterous robotic end effector, Proc. IEEE R8 Eurocon 2001 Int. Conf. Trends Commun., vol. 1, pp. XXXVI–XLI, July 2001. [13] Adams, R.J., Klowden, D., and Hannaford, B., Stable haptic interaction using the excalibur force dis- play, in Proceedings of International ConferenceonRobotics and Automation, vol. 1, pp. 770–775, 2000. Copyright © 2005 by CRC Press LLC 23 -22 Robotics and Automation Handbook [14] Miller, B.E., Colgate, J.E., and Freeman, R.A., Guaranteed stability of haptic systems with nonlinear virtual environments, IEEE Transactions on Robotics and Automation, vol. 16, no. 6, pp. 712–719, 2000. [15] Klatzky, R.L., Lederman, S.J., and Reed, C., Haptic integration of object properties: texture, hard- ness, and planar contour, Journal of Experimental Psychology-Human Perception and Performance, vol. 15, no. 1, pp. 45–57, 1989. [16] Klatzky, R.L., Lederman, S.J., Pellegrino, J., Doherty, S., and McCloskey, B., Procedures for haptic object exploration vs. manipulation, Vision and Action: The Control of Grasping, pp. 110–127, 1990. [17] Lederman, S.J. and Klatzky, R.L., An introduction to human haptic exploration and recognition of objects for neuroscience and AI, Neuroscience:FromNeuralNetworkstoArtificial Intelligence, vol. 4, pp. 171–188, 1993. [18] Katz, D., The World of Touch, 1925, edited and translated by Lester E. Krueger, Lawrence Erlbaum, Hillsdale, NJ, 1989. [19] Gibson, J.J., The Senses Considered as Perceptual Systems, Boston: Houghton Mifflin, vol. 2, pp. 1453–1458, 1966. [20] Schaal, S. and Atkeson, C.G., Robot juggling: an implementation of memory-based learning, Control Systems Magazine, vol. 14, no. 1, pp. 57–71, 1994. [21] Schaal, S., Sternad, D., and Atkeson, C.G., One-handed juggling: a dynamical approach to a rhythmic movement task, Journal of Motor Behavior, vol. 28, no. 2, pp. 165–183, 1996. [22] Klatzky, R.L. and Lederman, S.J., Identifying objects from a haptic glance, Perception and Psychophysics, vol. 57, no. 8, pp. 1111–1123, 1995. [23] Lederman, S.J. and Klatzky, R.L., Extracting object properties by haptic exploration, Acta Psychologica, vol. 84, pp. 29–40, 1993. [24] Klatzky, J.M., Loomis, R.L., Lederman, S.J., Wake, H., and Fujita, N., Haptic identification of objects and their depictions, Perception and Psychophysics, vol. 54, no. 2, pp. 170–178, 1993. [25] Aristotle, De Anima, translated by Hugh Lawson-Tancred, Penguin Books, New York, NY, 1986. [26] Katz, D., The World of Touch, 1925, edited and translated by Lester E. Krueger, Lawrence Erlbaum, Hillsdale, NJ, 1989. [27] Heller, M.A. and Schiff, W., eds., The Psychology of Touch, Lawrence Erlbaum, Hillsdale, NJ, 1991. [28] Ellis, R.E., Sarkar, N., and Jenkins, M.A., Numerical methods for the force reflection of contact, Journal of Dynamic Systems, Measurement, and Control, vol. 119, pp. 768–774, Decemeber 1997. [29] Gillespie, R.B. and Cutkosky, M.R., Stable user-specific haptic rendering of the virtual wall, in Proceedings of the ASME Dynamic Systems and Control Division, vol. 58, pp. 397–406, November 1996. [30] Hajian, A. and Howe, R.D., Identification of the mechanical impedance at the human finger tip, Journal of Biomechanical Engineering, Transactions of the ASME, vol. 119, pp. 109–114, February 1997. [31] Colgate, J.E. and Schenkel, G.G., Passivity of a class of sampled-data systems: application to haptic interfaces, Journal of Robotic Systems, vol. 14, pp. 37–47, January 1997. [32] Colgate, J.E., Coordinate transformations and logical operations for minimizing conservativeness in coupled stability criteria, Journal of Dynamic Systems, Measurement, and Control, vol. 116, pp. 643–649, December 1994. [33] Colgate, J.E., Stanley, M.C., and Brown, J.M., Issues in the haptic display of tool use, in Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Control, pp. 140–145, 1995. [34] Miller, B.E., Colgate, J.E., and Freeman, R.A., Guaranteed stability of haptic systems with nonlinear virtual environments, IEEE Transactions on Robotics & Automation, vol. 16, pp. 712–719, December 2000. [35] Hannaford, B. and Ryu, J H., Time-domain passivity control of haptic interfaces, IEEE Transactions on Robotics & Automation, vol. 18, pp. 1–10, February 2002. [36] Moore, M. and Wilhelms, J. Collision detection and response for computer animation, Computer Graphics SIGGRAPH 1988, vol. 22, pp. 289–298, 1988. Copyright © 2005 by CRC Press LLC Haptic Interface to Virtual Environments 23 -23 [37] Held, M., Klosowski, J.T., and Mitchell, J.S.B., Evaluation of collision detection methods for virtual reality fly-throughs, Proceedings of Seventh Canadian Conference Computer Geometry, pp. 205–210, 1995. [38] Naylor, B., Amatodes, J.A., and Thibault, W., Merging BSP trees yields polyhedral set operations, Computer Graphics SIGGRAPH 1990, vol. 24, pp. 115–124, 1990. [39] Beckmann, N., Kriegel, H.P., Schneider, R., and Seeger, B., The r ∗ -tree: an efficient and robust access method for points and rectangles, Proceedings of ACM SIGMOD International Conference on Management of Data, pp. 322–331, 1990. [40] Barequet, G., Chazelle, B., Guibas, L.J., Mitchell, J.S.B., and Tal, A., Boxtree: a hierarchical representation for surfaces in 3D, EuroGraphics’ 96, vol. 15, no. 3, pp. 387–484, 1996. [41] Gilbert, E.G., Johnson, D.W., and Keerthi, S.S., Fast procedure for computing the distance between convex objects in three-dimensional space, IEEE Journal of Robotics and Automation, vol. 4, no. 2, pp. 193–203, 1988. [42] Ong, C.J. and Gilbert, E.G., Fast versions of the Gilbert-Johnson-Keerthi distance algorithm: Additional results and comparisons, IEEE Transactions on Robotics and Automation, vol. 17, no. 4, pp. 531–539, 2001. [43] Lin, M.C. and Canny, J.F. A fast algorithm for incremental distance calculation, in IEEE International Conference on Robotics and Automation, vol. 2, pp. 1008–1014, 1991. [44] Mirtich, B., V-Clip: fast and robust polyhedral collision detection, ACM Transactions on Graphics, vol. 17, no. 3, pp. 177–208, 1998. [45] Gilbert, E.G. and Foo, C.P., Computing the distance between general convex objects in three- dimensional space, IEEE Transactions on Robotics and Automation, vol. 6, no. 1, pp. 53–61, 1990. [46] Turnbull, C. and Cameron, S., Computing distances between NURBS-defined convex objects, in Proceedings IEEE International Conference on Robotics and Automation, vol. 4, pp. 3685–3690, 1998. [47] Lin, M.C. and Manocha, D., Interference detection between curved objects for computer animation, in Models and Techniques in Computer Animation, Thalmann, N.M. and Thalmann, D. (eds.), Springer-Verlag, Tokyo, pp. 43–57, 1993. [48] Lin, M.C. and Manocha, D., Fast interference detection between geometric models, Visual Computer, vol. 11, no. 10, pp. 542–561, 1995. [49] Thompson II, T.V., Johnson, D.E., and Cohen, E., Direct haptic rendering of sculptured models, in Proceedings Symposium on Interactive 3D Graphics, pp. 167–176, 1997. [50] Johnson, D.E. and Cohen, E. An improved method for haptic tracing of a sculptured surface, in Proceedings of ASME International Mechanical Engineering Congress and Exposition, vol. 64, pp. 243–248, 1998. [51] Nelson, D.D., Johnson, D.E., and Cohen, E., Haptic rendering of surface-to-surface sculpted model interaction, in Proceedings ASME Dynamic Systems and Control Division, vol. 67, pp. 101–108, 1999. [52] Patoglu, V. and Gillespie, R.B., Extremal distance maintenance for parametric curves and surfaces, Internationl Conference on Robotics & Automation 2002, pp. 2817–2823, 2002. [53] Baraff, D., Analytical methods for dynamic simulation of nonpenetrating rigid bodies, Computer Graphics, vol. 23, no. 3, pp. 223–232, 1989. [54] Baraff, D., Fast contact force computation for nonpenetrating rigid bodies, in Computer Graphics Proceedings. Annual Conference Series SIGGRAPH 94, vol. 1, pp. 23–34, 1994. [55] Mirtich, B. and Canny, J., Impulse-based simulation of rigid bodies, in Proceedings of the Symposium on Interactive 3D Graphics, pp. 181–188, 1995. [56] Ramamurthy, R. and Farouki, R.T., Voronoi diagram and medial axis algorithm for planar domains with curved boundaries-II. Detailed algorithm description, Journal of Computational and Applied Mathematics, vol. 102, pp. 253–277, 1999. [57] Zilles, C. and Salisbury, J., A constraint based god-object method for haptic display, in IEE/RSJ International Conference on Intelligent Robotsand Systems, Human Robot Interaction, and Cooperative Robots, vol. 3, pp. 146–151, 1995. [58] Ruspini, D.C., Kolarov, K., and Khatib, O., The haptic display of complex graphical environments, in Computer Graphics Proceedings. Annual Conference Series, vol. 1, pp. 345–352, 1997. Copyright © 2005 by CRC Press LLC 24 Flexible Robot Arms Wayne J. Book Georgia Institute of Technology 24.1 Introduction Motivation Based on Higher Performance • Nature of Impacted Systems • Description of Contents 24.2 Modeling Statics • Kinematics • Dynamics • Distributed Models • Frequency Domain Solutions • Discretization of the Spatial Domain • Simulation Form of the Equations • Inverse Dynamics Form of the Equations • System Characteristic Behavior • Reduction of Computational Complexity 24.3 Control Independent Proportional Plus Derivative Joint Control • Advanced Feedback Control Schemes • Open Loop and Feedforward Control • Design and Operational Strategies 24.4 Summary 24.1 Introduction Robots, as mechanical devices, are subject to mechanical strain as a result of loads experienced in their application. These loads may result from gravity, acceleration, applied contact forces, or interaction with process dynamics. An effect of strain is deflection and consequent flexibility. No robot escapes this conse- quence, although the effects may be justifiably ignored in some instances. In the following pages, methods for predictingandminimizingtheadverseconsequences offlexibility will be presented.Primarily, arm-like, serial link devices will be examined, although lessons may often be extended to parallel link (nonserial) manipulators and mobile robots. 24.1.1 Motivation Based on Higher Performance Flexibility can be reduced by reducing strain in critical locations. This should first be attempted by sound design practices for structural and drive components. Wise choices for cross sections and materials ulti- mately reach their limits and some flexibility will remain. If the resulting design is inadequate because of more demanding requirements, further steps must be taken. It is these further steps that are the focus of this section. Higher performance may be required in terms of speed, accuracy, payload mass, arm weight, or arm bulk. Typically, several or all of these measures have an impact on the utility of the design. 24.1.2 Nature of Impacted Systems There are early indications that a robot design will be challenged in terms of flexibility. Static deflection and vibration natural frequency, for the dynamic case, are numerical indicators of this problem. Long arms are Copyright © 2005 by CRC Press LLC Flexible Robot Arms 24 -3 deformation is pure shear, in which a rectangular specimen distorts into a parallelogram with a change in the corner right angles by a shear angle γ . The proportionality between shear stress and shear angle is the shear modulus commonly denoted G. Mechanics of materials instructs us that the faces of a cubic differential element in a specimen under stress will have different predictable levels of normal and shear stress depending on the orientation of the face. Mohr’s circle is often used to display and compute the values at an arbitrary angle. Failure of the material correlates with these stresses and the manner in which they are applied as well as environmental conditions. The purposes of this article are served by special cases of loading that can be used to compute a reasonable prediction of the stress state at the critical location of the geometry. The nature of the material (e.g., brittle or ductile) and the local geometry (which leads to stress con- centration factors) are also required to produce a reasonable prediction of structural failure. For repeated loading of structural or drive components, one must be concerned about fatigue failure as a result of crack development and propagation. Although various criteria for failure prediction have been developed, the Von Mises effective stress [2] is appropriate and convenient for multi-axial stress. It can be calculated from the principal normal stresses (found on planes with no shear stress) σ 1 , σ 2 , σ 3 , or the normal and shear stresses at any surface. The Von Mises stress is found as σ = σ 2 1 + σ 2 2 + σ 2 3 − σ 1 σ 2 − σ 2 σ 3 − σ 1 σ 3 (24.1) Static failure in ductile materials is predicted based on failure in shear when the Von Mises stress σ exceeds the shear strength, normally taken to be one-half the tensile yield strength. Fatigue failure of ductile materials is predicted when σ exceeds the stress value that varies with the number of cycles of loading, and with the mean and alternating stress levels. The Modified Goodman diagram is one accepted way to determine this value [2]. For ferrous materials, a fatigue limit exists. This is a stress at which there is no limit to the number of cycles, i.e., the material will never fail. Aluminum and other nonferrous materials do not exhibit such a limit and will eventually fail (according to this theory) for low levels of stress if the number of cycles of loading is large enough. The purpose here is not to elaborate on these methods of machine design, but to provide the engineer with an indication of when these stress-based failure modes dominate the deflection based failure modes. Techniques accounting for local factors of stress concentration due to sharp notches and corners will be found in machine design texts also. Multiplying factors may be applied to the stress or the strength to account for these factors. Coefficients are based on a combination of empirical and theoretical results. Again, the purpose here is not served by diversion into these topics. The reader should be aware that they can be applied and do not change the fundamental results of the following discussion. Brittle materials (e.g., gray cast iron or extremely cold materials) or materials with uneven maximum stress in tension and compression do not generally fail in shear, but in tension. Alternative methods of predicting failure will not be covered here because the components that we analyze for flexibility are seldom constructed from these materials; however, the extreme conditions of temperature might occur for special applications. 24.2.1.1.2 Material Properties The discussion of stress and strain above incorporates a discussion of material properties without elabora- tion. It serves us well to examine these properties in more detail as they relate to arm flexibility. In particular we will examine the elastic and shear moduli, the density, damping, and the strength of these materials. Recognize that bulk materials such as steel or aluminum are not directly comparable in all respects to glass or Kevlar ® fibers; however, the numbers give one a useful comparison of the potential of the materials for use in robot construction. 24.2.1.1.2.1 Elastic and Shear Modulus The elastic modulus E and the shear modulus G are related through Poisson’s ratio µ as G = E /(2 + 2µ). For a discussion of flexibility these are the two most important material characteristics, closely Copyright © 2005 by CRC Press LLC 24 -4 Robotics and Automation Handbook TABLE 24.1 Representative Properties (Various Sources) Elastic Modulus E Shear Modulus G Density ρ Yield (2%) Material (GPa) (GPa) (Mg/m 3 ) E /ρ Strength (MPa) Steel-carbon 1040 cold rolled 206.8 80.8 7.8 26.5 490 Stainless steel 301 cold rolled 206.8 80.8 7.8 26.5 1138 Aluminum 6061 heat treated 71.7 26.8 2.8 25.6 276 Kevlar 49 ® 120 X 1.45 82.8 2300 a S Glass fibers 85.5 X 2.49 34.3 X Magnesium AZ91A die cast 44.8 16.8 1.8 24.9 152 Titanium T1-13 V-11 Cr-3 Al 113.8 42.4 4.4 25.9 1172 Alloy heat treated X = not available or not relevant. a Fails at 1.9% strain. followed by density. For bulk materials the elastic modulus is independent of orientation of the stress and equal in compression and tension. It is almost completely independent of alloying modifications of metal composition that may have a dramatic effect on strength. 24.2.1.1.2.2 Density Density of the structural material is critical to natural frequency and to the torque requirement of actuators to lift or accelerate an arm. This is especially true when the arm itself is the main load to be carried, as in many cases. If geometry is fixed, the natural frequency is proportional to √ E /ρ in typical bending modes. Thus, the ratio of modulus to density is a strong indicator of the material’s resistance to dynamic flexibility limitations. We, therefore, see that the lighter weight, lower stiffness material like aluminum is comparable to the heavier but stiffer steel. On the other hand, if a large payload is accommodated, steel gains an advantage because the payload adds less mass on a percentage basis to the steel arm. Materials like Kevlar ® are dramatically better in this figure of merit (five times better than most metals). Unfortunately, constructionusingthis fibertoadvantage isa serious difficulty andwarrantedonly in exceptionalsituations. The cost is also high. 24.2.1.1.2.3 Strength Strength refers in general to the load bearing capacity of the material before failure, and is usually cited in terms of the stress occurring at the point of failure. Since failure may occur in many modes, the reference to strength by itself is insufficient. At low values of stress, structural materials deform elastically and hence reversibly. As stress rises above the elastic range a permanent deformation occurs. Yield strength is the top stress of the elastic range of the material, often defined at a point when 0.2% of strain will not reversibly disappear on removal of the load. Ultimate strength (tensile, compressive, or shear) refers to stress at the point of fracture, or separation of two regions of the material. These failure modes could be the result of a single loading event, whereas fatigue failure is the result of repetitive loading and is more complicated to describe or predict. For steels and some titanium materials a predictable fatigue limit seems to exist beyond which an indefinite number of cycles of loading can be sustained. The fatigue limit is the corresponding stress and is typically 40–50% of the ultimate tensile strength. 24.2.1.1.2.4 Damping Increased damping reduces the settling time for any vibrations that occur in the arm. Damping inherent in an arm is due to many things, particularly the material and the manner of construction. A welded construction looses less energy, thus has less damping than a bolted construction. Any relative motion between two arm components can remove vibrational energy, and when the damping is very low a small improvement can result in substantial reductions in settling time. Thus, arms that are back drivable tend to absorb vibrational energy better than arms which are not back drivable due to high ratio gearboxes, hydraulic actuators blocked by valves with positive overlap, or even a high position feedback gain. This can create a dilemma for the designer who wants high accuracy in joint positioning and, thus, chooses Copyright © 2005 by CRC Press LLC Flexible Robot Arms 24 -5 high feedback control gains to control the joint. In terms of material damping, a wide range of damping values is observed and composites tend to have more damping than metals. 24.2.1.1.3 Idealized Structures and Loading Exact evaluation of the stress in a robot component may require finite element analysis. For present purposes it will be more valuable to compose the robot from idealized structures tractable to analytical determination of stress and deflection. In particular we will use bars in compression, shafts in torsion, and beams in bending. For static deflection the load and the elastic properties are needed. For dynamic deflectionthemass propertiesare alsoneeded. Geometrywillcontributetobothmass andelasticproperties. 24.2.1.1.3.1 Bars and Compression A bar element is considered for axial loading. Almost pure compression or tension will result for a strut or similar element pinned at each end. In this case the static deflection δ is also axial and predicted simply and accurately by the assumption of plane strain. If the cross-sectional area A is constant over the length L,deflection δ under applied axial force F a is δ = F a L AE = α XF F a (24.2) If A is not constant, the deflection is obtained by integrating a differential deflection (axial strain) over the length with the area A(x) variable. Normal stress is approximately σ = F a /A The dynamic behavior of a bar alone is rarely of concern with robots of conventional size if the bar only experiences compression. This is because there are usually other components exhibiting more severe deflection and because buckling constrains the ratio L/A to high values. The bar element can be used to represent “tension only” components including cables and bands. In these cases buckling does not limit L/A and the deflection can be substantial. Dynamic behavior incorporates mass as well. The mass of the bar in tension is reduced as the cross-sectional area is reduced with the resulting effect that the natural frequency is related to the speed of sound in the bar, which is invariant with the geometry and high relative to other dynamic phenomena that impact robot performance. Thus, the bar plays the role of a spring for most flexible robot analysis; its mass can be considered lumped, if considered at all, and does not warrant consideration as a distributed parameter effect. 24.2.1.1.3.2 Shafts and Torsion Pure torsion is a fair representation of drive shaft loading. Arm structural elements also may experience torsion in conjunction with bending or compression, and to a reasonable approximation the effects can be added or superimposed. The simple case we will build on is a circular cross section, either solid or hollow. The torque T acting on a circular tube of length l with cross-section polar moment of inertia J produces a rotation of one end with respect to the other of θ,where θ = Tl JG = α θ T T (24.3) The pure shear τ resulting on the plane perpendicular to the axis of the torsion is approximately linear with the radial distance r, giving the maximum at the outer radius R of τ max = TR J (24.4) Copyright © 2005 by CRC Press LLC [...]... j Mi j (24 .55) j =i where Copyright © 20 05 by CRC Press LLC 1 0 Hi = 0 0 0 0 1i 1 0 0 0 1 0 0 0 1 (24 .56) 2 4 -2 2 Robotics and Automation Handbook and −θzi j −θxi j θxi j 0 zi j 0 θ zi j Mi j = −θ yi j θ yi j 0 0 0 0 0 xi j yi j dh i ˙ ˙ j ˙ = h i = Wi h i + Wii h j dt ˙ ˙ ˙ W j = W j−1 A j + W j−1 A j (24 . 57) (24 .58) (24 .59) and ¨ ˙ ¨ ˙ ¨ W j = W j −1 A j + 2W j −1... imposed and the resulting two equations employed to great benefit −W −W = R1 B2 R3 A4 C 5 B6 R7 M M V V 0 (24 . 42) 7 In the example of Figure 24 .3, the left end is pinned and the right end is free, yielding −W u11 u 21 z0 = = M u31 V 0 u41 u11 u 12 u31 u 32 u 32 u33 −W u24 u34 M u 42 u43 u44 u 12 u13 u 22 u23 u14 V 6 (24 .43)... shorter period and in turn a higher critical load The critical compressive load is Pcritical = π 2 EI π 2 EAk2 = , 2 2 l eff l eff π 2 Ek2 Pcritical = 2 A l eff (24 . 12) where k = radius of gyration corresponding to the cross section shape l eff = the effective column length, dependent on end-constraint conditions = l for pinned-pinned ends = 2l for fixed (clamped)-free ends = 0 .70 7l for fixed-pinned ends... ε=c ∂ 2w ∂ 2w d x =c 2 2 dx ∂x ∂x (24 .7) By the simple beam theory the normal stress σ in the outermost fiber would be Eε or σ = Ec ∂ 2w dx ∂ x2 (24 .8) and would vary linearly along y, the distance from the neutral axis The moment M(x) developed at the cross section by normal stress is found by integrating over the area of the cross section the stress times the Copyright © 20 05 by CRC Press LLC 2 4 -7 Flexible... Other terms include mi mi mi ˙ δi j C i j + B2i = ˙ δi k δi j C i k j j =1 (24 .71 ) k=1 j =1 where Ci j = 1i 1 2 µ[η 0 0 1]T [xi j mi (24 . 72 ) (24 .73 ) zi j δi j C i k + C iT + k j =1 1i 1 2 mi mi B3i = C i + Ci = 0]dη δi k δi j C i kl yi j 0 k=1 j =1 µ[η 0 0 1]T [η 0 0 1]dη (24 .74 ) 0 m1 Di k = C i k + δil C il k (24 .75 ) 1=1 mi G i = Ci + δi k C i k + C iT k (24 .76 ) k=1 Lagrange’s equations will require... by CRC Press LLC 2 4-1 8 Robotics and Automation Handbook station 7 R1 B2 R3 R7 B6 station O A4, C5 FIGURE 24 .3 Transfer matrix representation of an arm For a controlled joint with the controller transfer function (joint torque/joint angle) = k( j ω) 1 0 C = 0 0 0 0 0 0 1 0 0 0 1 1 1/k( j ω) 0 (24 .41) When combined together to represent the planar arm with two joints shown in Figure 24 .3,... 1 (24 .18) Copyright © 20 05 by CRC Press LLC 2 4-1 2 Robotics and Automation Handbook Dissipation effects also warrant modeling It is typically possible to include friction or viscous damping into either the actuator forcing or to incorporate these losses into the equations for either lumped or distributed models using the variables that are dictated by the mass and compliance representations 24 .2. 3.1.1... vectors T Z 0 = d0 T φ0 T , T Z n = dn T φn T (24 .26 ) where d0 , dn are the position vectors from the origin of −1 (inertial frame) to the origins of 0 and n, respectively, and 0 , n are the vectors of small rotations of mass 0 and n, respectively, from an initial undeformed orientation Copyright © 20 05 by CRC Press LLC 2 4-1 4 Robotics and Automation Handbook In the case of joint angle rotations, the... j −1 A j + 2W j −1 A j + W j −1 A j (24 .60) ˙ ˙ Aj = Ujq j ¨ ˙j ¨ A j = U2 j q 2 + U j q j (24 .61) (24 . 62) U j = ∂ A j /∂q j (24 .63) where U2 j = ∂ 2 A j /∂q 2 j (24 .64) q j is the joint variable of joint j W j = Wj E j ˙ ˙ ˙ W j = Wj E j + Wj E j ¨ ¨ ˙ ˙ ¨ W j = W j E j + 2W j E j + W j E j (24 .65) mj ˙ δ jk Mjk ˙ Ej = k=1 and mj ¨ δ jk Mjk ¨ Ej = k=1 With expressions for the velocity of any point... 1i K = dk i i =l Copyright © 20 05 by CRC Press LLC 0 (24 .68) 2 4 -2 3 Flexible Robot Arms Careful examination of the expanded form of this equation shows that the following terms appear m1 m1 ˙ ˙ δi j δi k C i k j Bli = (24 .69) j =1 k=1 where 11 1 2 Ci k j = µ[xi k yi k z i k 0]T [xi j yi j zi j (24 .70 ) 0]dη 0 is a 4 × 4 matrix, it is nonzero only in the 3 × 3 upper-left-hand corner It can also be shown . ConferenceonRobotics and Automation, vol. 1, pp. 77 0 77 5, 20 00. Copyright © 20 05 by CRC Press LLC 23 -2 2 Robotics and Automation Handbook [14] Miller, B.E., Colgate, J.E., and Freeman, R.A., Guaranteed stability. (MPa) Steel-carbon 1040 cold rolled 20 6.8 80.8 7. 8 26 .5 490 Stainless steel 301 cold rolled 20 6.8 80.8 7. 8 26 .5 1138 Aluminum 6061 heat treated 71 .7 26 .8 2. 8 25 .6 27 6 Kevlar 49 ® 120 X 1.45 82. 8 23 00 a S. stations i and i +1), and h is the distance to the center of mass from station i. Copyright © 20 05 by CRC Press LLC 24 -1 8 Robotics and Automation Handbook station O station 7 R 1 R 3 R 7 B 2 B 6 A 4 ,C 5 FIGURE