Kurfess T.R. Robotics and Automation Handbook

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start

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

Thebroadphase iscomposedof two major steps.First, a global proximitytestis 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 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

Haptic Interface to Virtual Environments 23

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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 polyhedraare based on the algorithmby Gilbert, Johnson, and Keerthi (GJK) [41, 42], the algorithm

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

-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.

Itis important tocarefullydistinguishbetweenthe role of thepenalty methodand the virtualcoupler. 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(generalizedcoordinatesand their 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.

Haptic Interface to Virtual Environments 23

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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.

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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 predicting and minimizing the adverse consequences of ﬂexibilitywillbe 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

-2 Robotics and Automation Handbook

very susceptible to ﬂexibility problems as arm mass and compliance increase with length. The weight of a

device that must be carried on a mobile platform or launched into space is directly penalized, leading to

severe tradeoffs between ﬂexibility and mobility. When access through narrow openings is desired, as for

servicing a nuclear facility, the cross section of an arm is limited with consequences for ﬂexibility. In order

to reduce cycle time, high accelerations produce loads that excite ﬂexible behavior. The time for vibrations

to settle to the required precision can extend the very cycle time that required the offending accelerations.

Static deﬂections as well as vibrations may impact the arm’s credentials in a given task if tip positioning

is based on joint measurement. In summary, long, lightweight, thin, quick, and accurate arms are likely

to have problems arising from ﬂexibility. For a more complete discussion of the nature of ﬂexible devices,

the reader can refer to Book [1].

24.1.3 Description of Contents

Discussed in the following pages is the modeling of elastic behavior of material and of the structures

composed of that material. A brief introduction to statics is useful for modeling the modes of failure that

can occur, including excessive ﬂexibility, fracture, and buckling. We focus on ﬂexibility, but that may not be

the active constraint. Kinematics of a ﬂexible arm, discussed next, describes material deformation as well as

joint motion. The dynamic equations are ﬁrst derived for lumped parameter models consisting of springs

and masses, and then controlled joints are added between the springs. Distributed effects are examined in

two types of models, frequency domain linear models and discretized time domain models. The discussion

of control begins with the simplest feedback controls, and considers their limitations. It progresses to more

complex schemes intended to overcome those limitations. Open loop and feed forward techniques are

equally important and provide the trajectory that the feedback control will try to follow. Improving the

performance may also require rethinking the operational strategies used to deploy the robot arm.

24.2 Modeling

Whenmodeling ﬂexible behavior,weareacknowledgingthe deformableproperties ofengineeringmaterials

that introduce several practical considerations in addition to ﬂexibility. What are the constraints on the

structural geometry and selection of the arm materials? Flexibility must be limited to achieve acceptable

behavior (accuracy, natural frequency, etc.) but other constraints on geometry should be examined as well.

One or more of these constraints will be the active constraint that limits one in creating a light, long, or

fast arm. The arm may yield, it may fail in fatigue, or it may buckle before the ﬂexibility constraint comes

into effect. It is not within the scope of this article to comprehensively treat all of these issues, but it is

essential that they be identiﬁed to the reader.

24.2.1 Statics

24.2.1.1 Mechanics of Deformable Bodies

The classical treatment of deformable bodies is appropriate to the analysis of most manipulator struc-

tures. Exceptions are components made of anisotropic materials, that is, materials having a pronounced

directionality in their properties. Truly, these are some of the most exciting materials for combating the

limitations of ﬂexibility. The material itself, typically a composite of two or more elementary materials, is

engineered to contend with the loading conditions expected. The general nature of some arms, however,

prevents a highly specialized design. The quantitative aspects of our analysis will only apply to isotropic

materials.

24.2.1.1.1 Stress vs. Strain

Hooke’s law describes a material where stress (force per unit area) is proportional to strain (change in

dimension per unit dimension). The constant of proportionality between normal stress and extension for

a material in plane strain is the elastic or Young’s modulus, given the symbol E . Another hypothetical

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

, σ

, or the normal and shear

stresses at any surface. The Von Mises stress is found as





+ σ

− σ

(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

-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

) 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

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.

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,

constructionusing this ﬁbertoadvantage is aseriousdifﬁcultyand warrantedonly 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