Haptic Interface to Virtual Environments 23-2123.6 Concluding Remarks In this chapter, we have presented a framework for rendering virtual objects for the sense of touch.. Spatial motion
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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 intosets 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
The broad phase is composed of two major steps First, a global proximity test is performed using hierarchies
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 patchesdrawn 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), controlpolygons of the active surface models can be used to calculate a first order approximation to the closestpoints on these surfaces Specifically, bounding box centroids of each interacting surface patch can beprojected onto the polygonal control mesh of the other patch Using the strong convex hull property of
Trang 2Haptic Interface to Virtual Environments 23-19
NURBS surfaces, one can determine the candidate span and interpolate the surface parameters from thenode values These approximate projections serve as good initialization points for the narrow phase of thecollision detector
dis-of Lin and Canny [43], and the algorithm by Mirtich [44]
The GJK algorithm makes use of Minkowski difference and (simplex-based) convex optimization 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 Linand Canny makes use of Voronoi regions and temporal/spatial coherence between successive queries tonavigate 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 severalimprovements
tech-Less literature exists on direct collision detection for nonpolygonal models Gilbert et al extended theiralgorithm to general convex objects in [45] In a related paper [46], Turnbull and Cameron modify thewidely used GJK algorithm to handle convex shapes defined using NURBS Similarly, in [47, 48] Lin andManocha present an algorithm for curved models composed of spline or algebraic surfaces by extendingtheir earlier algorithm for polyhedra
Finally, a new class of algorithms, namely, minimum distance tracking algorithms for parametric faces, is presented in [49–52] These algorithms are designed to maintain the minimum distance betweentwo parametric surfaces by integrating the differential kinematics of the surfaces as the surfaces undergorigid body motion
The most widely used interaction model in both haptic rendering and multibody dynamics is called thepenalty method: it assumes compliant contact characterized by a spring-damper pair The force response isthen proportional to the amount and rate of interpenetration The interaction calculator need only querythe 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 ofall contact forces and moments acting on a body for use as input to the dynamic model, the interactioncalculator queries the geometric model using the surface parameters and then carries out the appropriatedot and cross products and vector sums Using replacements for the set of interaction forces allows thedynamic 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 impulseresponse 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 problemespoused 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
Trang 323-20 Robotics and Automation Handbook
impulses for the case of sustained contact Note that computation of impulses requires that the interactioncalculator query the dynamic model with the current extremal point parameters and system configuration
to determine the effective mass and inertia at each contact point
It is important to carefully distinguish between the role of the penalty method and the virtual coupler Thevirtual coupler serves as a filter between the virtual environment and haptic device whose design mitigatesinstability and depends on the energetic properties of the closed-loop system components The penaltymethod, on the other hand, though it may also be modelled by a spring-damper coupler, is an empiricallaw chosen to produce reasonable behavior in multibody system simulation Parameter values used in thepenalty method also have implications for stability, though in this case it is not haptic rendering systembut 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 themaximum 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 proxymethods address [57, 59]
23.5.3 Forward Dynamics Solver
The final component comprising the haptics-equipped simulator is a differential equation solver thatadvances the solution of the equations of motion in real time The equations of motion are a set ofdifferential equations constructed by applying a principal of mechanics to kinematic, inertial, and/orenergy expressions derived from a description of the virtual environment in terms of configuration andmotion variables (generalized coordinates and their derivatives) and mass distribution properties Typicallythe virtual environment is described as a set of rigid bodies and multibody systems interconnected bysprings, 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 forcesand moments acting between bodies of the virtual environment InFigure 23.4,representative bodies A,
B, and P comprise the virtual environment, while the forces and moments in the springs and dampers thatmake 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 generalizedcoordinate derivatives) of bodies A, B, and P are then computed by solving the equations of motion (seealsoFigure 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, ing the motion of bodies A, B, and P and occasionally triggering the interaction calculator The interactioncalculator runs between time-steps and passes its results (impulses and forces) to the equations of motionfor continued simulation
monitor-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, constraintappending 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 engineeredfor use in real-time or with constant step-size and usually require stabilization Alternatively, constrainedmultibody systems may be formulated as ODEs and then simulated using standard ODE solvers usingconstraint-embedding techniques Constraint embedding can take place symbolically (usually undertakenprior 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 developedfor 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 Aconstraint equation can then be used to impose that configuration and motion on the dynamic model ofthe virtual environment
Trang 4Haptic 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 Usinghaptic interface technology, virtual objects can not only be seen, but felt and manipulated The hapticinterface 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 excitationimposed by the human user ‘Appropriate’ here is judged according to the closeness of the haptic interfacemechanical 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-inputmulti-output system From a control engineering perspective, the haptic interface is a plant simultaneouslyactuated 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 thecapabilities and requirements of all three entities
While robotics technology strives continually to instill intelligence into robots so that they may actautonomously, haptic interface technology strives to strike all intelligence out of the haptic interface—toget out of the way of the human user After all, the human user is interested not in interacting with thehaptic 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 problemsyet posed in the greater field of robotics
[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 for robot guidance using virtual fixtures, in IEEE International Conference on Robotics 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
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[8] Reinkensmeyer, D., Standard Handbook of Biomedical Engineering & Design, ch Rehabilitators,
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[9] Gillespie, R.B., Hoffman, M., and Freudenberg, J., Haptic interface for hands-on instruction in
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[10] Okamura, A.M., Richard, C., and Cutkosky, M.R., Feeling is believing: using a force-feedback joystick
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
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[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 Conference on Robotics and Automation, vol 1, pp 770–775, 2000.
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[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: From Neural Networks to Artificial Intelligence, vol 4,
[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.
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[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,
[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,
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[51] Nelson, D.D., Johnson, D.E., and Cohen, E., Haptic rendering of surface-to-surface sculpted model
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[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
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Copyright © 2005 by CRC Press LLC
Trang 724 Flexible Robot Arms
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 theirapplication These loads may result from gravity, acceleration, applied contact forces, or interaction withprocess 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, methodsfor predicting and minimizing the adverse consequences of flexibility 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 sounddesign 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 ofmore demanding requirements, further steps must be taken It is these further steps that are the focus ofthis 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 andvibration natural frequency, for the dynamic case, are numerical indicators of this problem Long arms are
Trang 8Flexible 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 angleγ 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 shearstress 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 theyare applied as well as environmental conditions The purposes of this article are served by special cases ofloading that can be used to compute a reasonable prediction of the stress state at the critical location of thegeometry
The nature of the material (e.g., brittle or ductile) and the local geometry (which leads to stress centration factors) are also required to produce a reasonable prediction of structural failure For repeatedloading of structural or drive components, one must be concerned about fatigue failure as a result of crackdevelopment and propagation Although various criteria for failure prediction have been developed, theVon Mises effective stress [2] is appropriate and convenient for multi-axial stress It can be calculated fromthe principal normal stresses (found on planes with no shear stress)σ1, σ2, σ3, or the normal and shear
con-stresses at any surface The Von Mises stress is found as
σ=
σ2+ σ2+ σ2− σ1σ2 − σ2σ3 − σ1σ3 (24.1)
Static failure in ductile materials is predicted based on failure in shear when the Von Mises stressσexceeds the shear strength, normally taken to be one-half the tensile yield strength Fatigue failure ofductile materials is predicted whenσexceeds the stress value that varies with the number of cycles ofloading, and with the mean and alternating stress levels The Modified Goodman diagram is one acceptedway 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 thenumber of cycles of loading is large enough The purpose here is not to elaborate on these methods ofmachine design, but to provide the engineer with an indication of when these stress-based failure modesdominate 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 theycan 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 maximumstress in tension and compression do not generally fail in shear, but in tension Alternative methods ofpredicting failure will not be covered here because the components that we analyze for flexibility are seldomconstructed from these materials; however, the extreme conditions of temperature might occur for specialapplications
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
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TABLE 24.1 Representative Properties (Various Sources)
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 andequal in compression and tension It is almost completely independent of alloying modifications of metalcomposition 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 dynamicflexibility limitations We, therefore, see that the lighter weight, lower stiffness material like aluminum iscomparable to the heavier but stiffer steel On the other hand, if a large payload is accommodated, steelgains an advantage because the payload adds less mass on a percentage basis to the steel arm Materials likeKevlar®are dramatically better in this figure of merit (five times better than most metals) Unfortunately,construction using this fiber to advantage is a serious difficulty and warranted only in exceptional situations.The cost is also high
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 weldedconstruction looses less energy, thus has less damping than a bolted construction Any relative motionbetween two arm components can remove vibrational energy, and when the damping is very low a smallimprovement 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 Thiscan create a dilemma for the designer who wants high accuracy in joint positioning and, thus, chooses
Trang 10Flexible Robot Arms 24-5
high feedback control gains to control the joint In terms of material damping, a wide range of dampingvalues 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 presentpurposes it will be more valuable to compose the robot from idealized structures tractable to analyticaldetermination 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 dynamicdeflection the mass properties are also needed Geometry will contribute to both mass and elastic properties
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 deflectionδ 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 ais
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 naturalfrequency 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 formost flexible robot analysis; its mass can be considered lumped, if considered at all, and does not warrantconsideration 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 experiencetorsion in conjunction with bending or compression, and to a reasonable approximation the effects can beadded 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 ofθ, where
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Y
FIGURE 24.1 Geometry of bending deformation.
When the cross section is not circular an approximate analysis is often used that is accurate as long as thecross section is not prone to extreme warping with torsion In this analysis
24.2.1.1.3.3 Beams and Bending
The simple theory of beams is remarkably accurate for slender beams with cross sections that are not
extreme in their departure from a rectangle If the beam has a deflection w such that at the position x,
the second derivative∂2w /∂x2is approximately equal to 1/r , where r is the radius of curvature of the
neutral axis As illustrated in Figure 24.1, the further assumption that planes perpendicular to the neutralaxis remain planes dictates the amount of elongationδ that takes place on the outer edge of the element
of length dx With any finite curvature, an angle θ subtends the distance dx = r θ at a radius r from the
center of the local circular arc, that is, along the neutral axis The strain along the neutral axis is zero by
definition Moving to the outermost fiber of the beam, a distance c from the neutral axis, means the angle
θ subtends a greater distance of dx + δ = (c + r )θ The combination of these relationships means the
strain at the outermost fiber is
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
Trang 12Flexible Robot Arms 24-7
distance from the neutral axis times the differential area
this beam a force F y and a moment M z , and the resulting deflection w (l ) and rotation θ(l) at the end will
24.2.1.1.3.4 Combinations of Loading
In many situations it is acceptable to superimpose torsion, bending, and compression from the abovesimple cases to calculate deformation and stress The worst case stress will be where maximum axialnormal stress from bending is increased by the normal stress from axial forces At this same point the shearfrom torsion occurs and is maximum The plane stress approximation is valid here and the von Mises
stress is found (assuming a circular cross section of outer radius R) to be
TR J
2
Extreme cases require a more complex analysis beyond the scope of this section The boundary betweenacceptable and unacceptable is fuzzy and case dependent, but if any component of loading is modified by
a significant fraction of the factor of safety, a more detailed analysis should be undertaken For example,
if axial torsion T on a beam is realigned by 10◦, the bending moment on the beam is increased by about0.18 T An extremely slender beam could undergo 10◦of deformation without failure but the addition of
an additional bending moment of 0.18 T should be further considered An exception to this proportionaleffect is buckling behavior which is treated next
24.2.1.1.4 Buckling
Bending becomes the most pronounced mode of stress and deflection in most beam type elements,including arm links The na¨ıve designer is, therefore, tempted to place the maximum material at themaximum distance from the neutral axis Because bending may take place about any axis perpendicular tothe neutral axis, circular cross sections are attractive as they also resist torsion about the beam axis well Whynot carry this to the extreme of very thin walls located at extreme distances from the neutral axis so that thearm cross section is similar to an aluminum beverage can? The limitation may come from one or more of
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