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Haptic Rendering of Parametric Surfaces Using a Feedback Stabilized ExtremalDistance Tracking Algorithm

Volkan Patoglu R. Brent GillespieDepartment of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA

vpatoglu, [email protected]

Abstract

A new extremal distance tracking algorithm is presentedfor convex parametric curves and surfaces undergoing rigidbody motion. The geometric extremization problem is differ-entiated with respect to time to produce a dynamical systemthat incorporates dependence on both surface shape andrigid body motion. Extremization then takes place by in-tegrating these dynamical equations, but with a feedbackcontroller in place to stabilize the solution. A controllerdesign using feedback linearization is developed that si-multaneously accounts for surface shape and motion whileasymptotically achieving (and maintaining) the extremalpair. Collision detection then takes place in a frameworkfully analogous to that used for multibody simulation. Lo-cal stability results are extended to provide global stabilityfor body shapes composed of pieced-together convex para-metric surface patches using a switching algorithm.

B

A

P

E

minimum distance

minimum distance

virtual coupler

image of end effector

user'shand

end effector

haptic interface

minimum distance

Figure 1. Schematics of haptic rendering.

1. IntroductionHaptic rendering is the process by which virtual objects

are made apparent to be felt and at the same time madeavailable to be manipulated by a human user. Haptic render-ing requires a haptic interface, a computationally mediatedvirtual environment, and a paradigm according to which thetwo are linked. Figure 1 presents a schematic view of a hap-tic interface and the manner in which it is most commonlylinked to a virtual environment. On the left portion of thefigure, mechanical interaction takes place between a humanand the physical haptic interface device, or more specifi-cally, between a fingertip and the device end-effector. In thecomputational domain depicted on the right, an image ofthe device end-effector E is connected to a proxy P throughwhat is called the virtual coupler. The proxy P in turn in-teracts with objects such as A and B in the virtual environ-ment. Proxy P might take on the shape of the fingertip or atool in the user’s grasp.

The virtual coupler is depicted as a spring and damperin parallel, which is a model of its most common computa-tional implementation. Rigid bodies in the virtual environ-ment, including P , have both configuration and shape —they interact with one another according to their dynamicand geometric models. Configuration (including orienta-tion and position) is indicated in Figure 1 using referenceframes (three mutually orthogonal unit vectors) and refer-ence points fixed in each rigid body, while shape is indi-cated by a surface patch. Note that the end-effector image Ehas configuration but no shape. Its interaction with P takesplace through the virtual coupler and requires only the con-figuration of E and P .

Virtual Environment

HUMAN

Haptic Device

Virtual CouplerHap

ticIn

terf

ace

forces

vh

vm

ve

Fh

Fm

Fe

motion

ZOH T

Figure 2. Block diagram of haptic rendering.

The various components in Figure 1, including the hu-man user, haptic device, virtual coupler, and virtual envi-ronment, form a coupled dynamical system whose behav-ior depends on the force/motion relationship of each com-ponent. Figure 2 shows these components interconnected ina block diagram, where the additional indication of causal-ity has been made.

Not apparent in the block diagram is the detail inside thevirtual environment block which must be brought to bear tosimulate systems with changing contact conditions, includ-ing a forward dynamics solver, collision detector, and in-teraction response algorithm. The focus of this paper is acollision detector that treat bodies whose geometry is mod-elled using parameterized functions that may collide, rest,slide, or roll on one another. In particular, we treat objectswhose boundaries are represented using parametric surfacepatches. Patches are joined at boundaries that might cor-respond to discontinuities in curvature or might simply beput in place to effect a decomposition into a set of convexpatches.

Proceedings of the 12th International Symposium on Haptic Interfaces for Virtual Environment and Teleoperator Systems (HAPTICS’04)

0-7695-2112-6/04 $20.00 © 2004 IEEE

The core of this paper contains a new approach for track-ing the pair of extremal points on a pair of convex paramet-ric surfaces. Features of our approach include its guaranteedstability and seamless integration with the forward dynam-ics solver and interaction response algorithms. In section 2,we will undertake a literature review of collision detectors.In section 3, the proposed dynamic model for the trackingalgorithm is layed out in detail. Controller design and anal-ysis takes place in section 4. The switching algorithm thathandles transitions is presented in section 5. In section 6,simulation results are presented. We further discuss our al-gorithm in section 7 and wrap up in section 8.

2. BackgroundTo calculate a global solution in a computationally effi-

cient manner, it is very common to handle the collision de-tection problem in two parts: a far phase which involves acoarse global search for potentially interacting surfaces anda near phase, which is usually based on a fast local opti-mization scheme. Local operations, if they can be used toimprove a proposed extremal distance by some form of gra-dient descent operation, can be made quite fast. Thus theuse of a near phase algorithm contributes to efficiency, es-pecially when the distance problem is called not just oncefor a particular pair of objects, but at each time step in orderto track the evolution of the extremal distance over a num-ber of time-steps. Additionally, restriction to convex objectsor features is often made in the near phase, since in suchcase the distance problem also becomes convex and admitsfast, iterative solution by convex optimization.

2.0.1. Far Phase The far phase is composed of two majorsteps. First, a global proximity test is performed using hier-archies of bounding volumes or spatial decompositions foreach surface patch. The distances between bounding boxesfor each pair of surface patches drawn from all pairs of bod-ies are compared to a threshold distance and patches thatare too distant to be contacting are pruned away. Remain-ing surface patches are declared active. For a full reviewof bounding volume and space decomposition methods, see[8] and references therein.

In the second step of the far phase, approximate interac-tion points on active surfaces are computed. For example,if the geometric models are represented by non-uniform ra-tional B-splines (NURBS), control polygons of the activesurface models can be used to calculate a first order ap-proximation to the closest points. Bounding box centroidsof paired surface patches can be projected onto the polygo-nal control meshes of each other and parameters can be in-terpolated from the mesh node values. These approximateprojections serve as good initialization points for the nearphase of the collision detector.

2.0.2. Near Phase After initialization with the approxi-mations calculated in the far phase, the near phase employsan algorithm to iteratively improve the extremal distance be-tween each active pair of surface patches. Most previous

work in collision detection has concentrated on computingthe distance between convex polyhedral objects. Polyhedralmodels have been extensively studied because their reso-lution is sufficient for many applications, and they are al-most always compatible with 3-D modelling or CAD sys-tems. We first review the near phase algorithms for polyhe-dra, then discuss algorithms that compute the distance be-tween objects bounded by continuous surfaces.

2.1. Polyhedral Models

State of the art algorithms for computing the distancebetween convex polyhedra are based on the algorithm byGilbert, Johnson and Keerthi (GJK) [5] and the algorithmof Lin and Canny [11]. The GJK algorithm makes useof Minkowski difference and (simplex-based) convex opti-mization techniques to calculate the minimum distance. It isan iterative algorithm that generates a sequence of ever im-proving intermediate steps within the polyhedra to convergeto the true solution. The algorithm of Lin and Canny makesuse of Voronoi regions and temporal/spatial coherence be-tween successive queries to navigate along the boundariesof the polyhedra in the direction of decreasing distance. TheV-Clip algorithm by Mirtich [14] is reminiscent of the Linand Canny closest features algorithm but makes several im-provements. H-Collide by Gregory et.al. [6] is a specializedcollision detection framework for haptic interaction whichmakes use of several earlier algorithms to detect collisionsbetween polygonal surfaces at interactive rates.

2.2. Nonpolyhedral Models

Most of the available closest point algorithms for non-polyhedral models address the problem indirectly —by con-verting the problem into a polyhedral one with the use ofadaptively refined meshes. Another indirect approach pro-posed by Adachi [1] and Stewart [20] uses intermediate tan-gent representations.

Although these indirect methods can be successfully im-plemented for some applications, there also exist cases forwhich they are not sufficient. Intermediate representationsfail to approximate surfaces with high curvature, and poly-hedral approximations to complex models can grow verylarge in the number of polygons.

Less literature exists on direct methods for nonpolyhe-dral models. Gilbert et.al. extended their algorithm to gen-eral convex objects in [4]. In a related paper [22], Turnbullmodifies the GJK algorithm to handle convex shapes de-fined using NURBS. Similarly, in [12] Lin and Manochapresent an algorithm for curved models composed of splineor algebraic surfaces by extending their earlier algorithm forpolyhedra.

Also worth noting are the subdivision techniques imple-mented by Von Herzen [23]. Snyder [19] improves thesesubdivision methods by modelling the collision detectionbetween time dependent surfaces as a constrained mini-mization problem. The problem is then solved using inter-val Newton methods.


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In the field of computer graphics, Kriezis [9], and Bajaj[2] propose to model parametric surface intersections usingdifferential equations whose solution may be interpreted asa tracing/marching method. Although the goal of these ap-proaches is only to calculate surface intersections, the man-ner in which the problem is modelled and appropriate pointsare traced is closely related to our extremal pair tracking al-gorithm.

Our contribution is a near-phase algorithm designedspecifically for parametric surface patches. It is a track-ing algorithm like the near-phase algorithms designed forpolyhedral models and thus depends on convexity and ac-quires efficiency. Unlike those near-phase models, however,our algorithm in its basic form is designed to operate on fea-tures rather than bodies. (The extremal distance problemon the previously identified extremal features is quite triv-ial for polyhedra, but not for nonpolyhedral bodies.) Toextend our algorithm to handle bodies (composed of sur-face patches), we employ a switching algorithm that han-dles transitions when tracked pairs encounter surfaceboundaries. That switching algorithm also uses Voronoi re-gions and thus is closely related to the Lin-Canny algo-rithm. Thus the switching algorithm extends a local solutionto become global for convex bodies. Finally, the inclu-sion of a coarse global search (far phase) may be used toextend our algorithm to cover nonconvex bodies and fur-ther to determine whether an extremum is a maximum orminimum solution.

Thompson et.al. also contribute a tracking type closestpoint algorithm for non-polyhedral models. Their near-phase algorithm is based on Newton’s method. Extensionsto this work include [21], which handles a moving surfaceand [7], which makes use of higher order derivatives andtangent plane projections at singularities. Finally in [15]this approach is generalized to surface-to-surface interac-tions and combined with the “velocity formulation”, whichkeeps track of the exact extremal distance during contactand penetration as surfaces move, given exact initial condi-tions.

Recently, Duindam [3] et.al. and Kry et.al. [10] have for-mulated kinematic models for contact simulation of smoothsurfaces which are similar to the velocity formulation in[15] and dynamic formulation of the motion of the extremalpoints in our previous work [16].

All of these models account for the dependence of ex-tremal point motion on both surface motion and surfaceshape. However, our approach is unique in that it uses feed-back. In our approach, to continually solve the relationshipbetween point motion and surface shape and motion, a feed-back control problem is formulated and solved with the de-sign of a stabilizing controller. The controller output is pre-cisely the motion of each of the extremal points, and may beused to update the parameter values that locate the pointsthemselves. The speeds along the tangent curves are pro-duced by the controller as functions of the surface motions

and surface shapes. These speeds may be integrated to ar-rive at the closest points, where integration is the essentialprocess of “maintenance”.

In our algorithm, we differentiate the essentially geomet-ric problem to form the “differential kinematics” of the clos-est points, and then we integrate these differential kinemat-ics, but with a feedback loop in place to ensure that the in-tegration is robust and stable. As a result, collision detec-tion takes place in a framework fully analogous to that usedfor the simulation of the dynamical response. In fact, colli-sion detection and forward dynamics solution can take placethrough the solution of a single set of coupled differentialequations. Similar feedback stabilization techniques will befamiliar to roboticists. In particular, to solve for the inversekinematics of robot manipulators, it is customary to inte-grate the differential kinematics in a feedback loop to avoiddrift and numerical disturbances [17].

u

v

r

s

∆R

fu

fv

hr

hs

A

B

N

f(u, v) h(r, s)

Figure 3. Two parametric surfaces with a positionvector ∆R between two arbitrary points

3. ModelingLet there exist parametric representations for the sur-

faces shown in Figure 3. We use f to denote a position vec-tor to a point on the left surface. And we use f(u, v) to re-fer to the mapping from �2 to �3 that generates the Carte-sian coordinates [x(u, v) y(u, v) z(u, v)]T from the inde-pendent parameters u and v.

The following development relies on the existence ofsurface continuity through at least two differentiations. Wealso require that both surfaces be strictly convex. However,the method can be generalized to piecewise continuous sur-faces and non-convex surfaces using an appropriate switch-ing method, as mentioned above.

Let fu(u, v) and fv(u, v) denote the first partial deriva-tives with respect to u and v of the parametric surface atthe point f(u, v). Similarly, let fuu(u, v) denote the partialderivative with respect to u of fu and so on. Note that thefirst partials are tangent to the isoparametric curves of u andv respectively.

Two parametric surfaces are plotted in Figure 3 with theircorresponding isoparametric curves. On these surfaces, twoarbitrary points f(u, v) and h(r, s) and surface tangentsevaluated at these points are shown. ∆R(u, v, r, s) is a vec-tor between these arbitrary points.


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M =

fu ·fu+∆R·fuu fv ·fu+∆R·fuv −hr ·fu −hs ·fufu ·fv+∆R·fuv fv ·fv+∆R·fvv −hr ·fv −hs ·fv

fu ·hr fv ·hr −hr ·hr+∆R·hrr −hs ·hr+∆R·hrs

fu ·hs fv ·hs −hr ·hs+∆R·hrs −hs ·hs+∆R·hss

, b =

−(AωωωωωωωωωωωωωB × h) · fu−(AωωωωωωωωωωωωωB × h) · fv

(BωωωωωωωωωωωωωA × f) · hr

(BωωωωωωωωωωωωωA × f) · hs

Note that when the difference vector ∆R is normal toboth surfaces, the requirements of the extremal distance aresatisfied. In such case the values, denoted u�, v�, r� ands�, of the parameters u, v, r, and s locate the extremal pairf(u�, v�) and h(r�, s�). The extremal distance is then equalto the Euclidian norm of ∆R.

We define scalars Ψu,Ψv as the projections of the differ-ence vector ∆R onto the tangents fu and fv of surface f ;and similarly we define Ψr, and Ψs as the projections of thedifference vector ∆R onto the tangents hs and hr of sur-face h as follows.

Ψu �= ∆R · fu (1)

Ψv �= ∆R · fv (2)

Ψr �= ∆R · hr (3)

Ψs �= ∆R · hs (4)

When the projection errors are all zero, the conditions forthe extremal pair are met: the difference vector ∆R is per-pendicular to both surfaces at f and h.

Note that it is possible to define the extremal distancecondition by an alternate set of equations as presented in[15]. This alternative formulation makes use of surface tan-gents of one surface and normals of both surfaces. Althoughwe use the set (1) - (4) in our further derivation in this pa-per, similar results can be achieved using the alternate set.

Given the set of equations (1) - (4), one way to find theextremal pair is to search for the solution u�,v�,r�,s� thatminimizes the projection errors using a gradient descent al-gorithm. This procedure would require the computation ofa Jacobian for use in Newton Iteration. This approach is un-dertaken in [15].

In the present work, rather than taking the Jacobian of thesystem of equations (1) through (4) with respect to the inde-pendent parameters u, v, r and s, we differentiate them withrespect to time. The differentiation operation causes the mo-tion of the surfaces and the time rates of change of the pa-rameters du/dt, dv/dt, dr/dt and ds/dt, called the para-metric velocities, to show up in the expression for the pro-jection error derivatives.

Note that one must effectively freeze time (and conse-quently the motion of the bodies) while using a gradient de-scent algorithm to find the extremal pair. In contrast, takingthe time derivative of equations (1) through (4) produces adynamic expression where the time rates of change of theprojection errors are expressed in terms of the motion andshape of the surfaces.

It is worth mentioning that although we use vector ex-pressions throughout the paper, one needs to express eachvector consistently in a single reference frame before in-terpreting the operations as matrix operations. Where dot

products and cross products appear, we use boldface no-tation to indicate operations which may be performed in abasis-independent fashion. Once suitably expressed in a ref-erence frame, standard matrix operations may be used, andwe indicate this using normal typeface. Note also that sincethe right hand sides of equations (1) through (4) are basis-independent vector expressions, it is important to specifya frame in which differentiation is to be performed. Wechoose to express the vectors in the first two equations, (1)and (2), in the body frame A and the vectors in last twoequations, (3) and (4), in the body frame B (see Figure 3).This choice results in simpler matrix expressions.

Consider the case where each surface is attached to arigid body in motion. In Figure 3 these bodies are namedA and B. Assume that the configuration of bodies A and Bwith respect to a reference frame N is known. Then motionof these bodies with respect to the reference frame N willbe specified by the vectors NωωωωωωωωωωωωωA, NωωωωωωωωωωωωωB, NvAo and NvBo .

Using the notationAddt

(.) to indicate differentia-tion in reference frame A, and noting that, for any vectorβββββββββββββ, A d

dt(βββββββββββββ) = B d

dt(βββββββββββββ) + AωωωωωωωωωωωωωB × βββββββββββββ, we take time deriva-

tives of Equations (1)-(4) as follows

Ψu = A d

dt[(f − h) · fu] (5)

Ψv = A d

dt[(f − h) · fv] (6)

Ψr = B d

dt[(f − h) · hr] (7)

Ψs = B d

dt[(f − h) · hs] (8)

and rearrange into matrix form to produce the following dif-ferential equations for the projection errors

Ψ = M U + b (9)x = U (10)y = x (11)

where Ψ =

Ψu

Ψv

Ψr

Ψs

, U =

dudtdvdtdrdtdsdt

and M and b are shown at the top of the page.In this state space realization the state variables Ψ are

taken to be the projection errors. The inputs U are timederivatives of surface parameters whereas the system out-puts are denoted by y. The desired outputs from the algo-rithm are the estimates x = [u, v, r, s]T of the parametricvalues of extremal points on each surface, u∗, v∗, r∗ and s∗,at every instant of time. These estimates can be calculatedas a by-product of the control effort that regulates the pro-


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jection errors to zero. Details of this procedure are shown inthe next section.

4. ControlEquations (9)-(11) define a nonlinear dynamic model to

maintain the extremal pair on two surfaces undergoing rigidbody motion. It characterizes the projection error deriva-tives in state space form and formulates the extremal dis-tance problem as a standard nonlinear control problem.

The control input vector U is composed of time deriva-tives of surface parameters, i.e. the elements of U are speedsalong the tangent curves. The objective of the controller isto continually update these speeds to regulate the projectionerrors to zero, i.e. to maintain the extremal pair on the sur-faces.

With the model (9)-(11) in hand, the extremal pair on thesurfaces can be dynamically tracked making use of a controlloop with exact feedback linearization. Exact feedback lin-earization is feasible since the plant is implemented in thecomputer and at any instant of time the specific values of Mand b are exactly known. Note that feedback linearization isfundamentally different than Jacobian linearization in thatfeedback linearization is achieved by exact state transfor-mation and feedback, rather than by linear approximationsof the dynamics for a small range of operation [18].

First, in order to feedback linearize the model, an innerfeedback loop is designed. Assuming the matrix M is notsingular in the range of operation, we define the control in-put vector U in terms of a new input vector µ as

U = M−1 (µ − b) (12)

and apply this control input to (9). Then the nonlinear modelis algebraically transformed into an equivalent linear model

Ψ = µ (13)

Second, an outer loop linear controller is used to imposethe desired linear dynamics to equation (13). In this paper,a full state linear feedback

µ = −K Ψ (14)

is utilized to stabilize the closed loop dynamics andto achieve desired performance: to keep projection er-rors small. However, it is possible to synthesize differentouter loop controllers to satisfy various design objec-tives.

Exponential stability of the overall closed loop systemis guaranteed since there are no internal dynamics associ-ated with this input-output linearization. This observationfollows from the fact that the relative degree of the systemis the same as its order and input-output linearization leadsto input-state linearization [18].

Figure 4 shows the block diagram of the completed con-troller design. Recognizable here is the inner loop that ren-ders the system linear from input µ to output x. The outer

U = M−1(µ − b)

µ = −K Ψ

∫ [Ψ = M U + bx = U

]dt

motionmotion

Ψ

x

xµ U

Figure 4. Block diagram showing inner feedbacklinearization loop and outer linear control loop

loop achieves desired dynamics of the linear system via fullstate feedback with diagonal matrix gain K.

Furthermore, the desired outputs, i.e. the parametric val-ues of the extremal pair, are continually maintained usingthe control input vector U . This is simply achieved by inte-grating the input vector U with initial conditions extractedfrom the starting points. In practice, the state vector Ψ isaugmented with the input vector U to perform all integra-tions in a single operation.

U = M−1(µ − b)

µ = −K Ψ

Ψ = Ψ(x)∫

[x = U ] dt

motion

Ψ

x

x

x

µ U

Figure 5. Block diagram of the alternative imple-mentation

In fact, even a simpler implementation is possible. Fig-ure 5 demonstrates this equivalent case. Since the projec-tion errors Ψ can be directly calculated through equations(1) to (4), the derived dynamic model can be replaced bythis set of nonlinear equations. Note that although the de-rived dynamic model is replaced, the controller design re-mains unchanged.

Finally, it is important to mention that the algorithm neednot be initialized with the exact extremal points. Any initialpoint within the region of attraction of the designed nonlin-ear controller (which is the entire convex surface) will con-verge to the extremal pair since the controller is exponen-tially stable. Moreover the convergence rate can be adjustedby tuning the controller gain K.

5. Hybrid SystemsWe have adopted the mathematical language of hybrid

dynamical systems to describe collision detection betweenbodies composed of parametric surface patches. Hybrid sys-tems contain both discrete and continuous state variablesand exhibit both discrete and continuous state dynamics.In certain hybrid systems, continuous and discrete dynam-ics not only coexist, but interact such that changes in thecontinuous and discrete dynamics occur in response to con-tinuous and/or discrete state variables. In the case of col-lision detection between parametric bodies, however, thecoupling between the continuous and discrete dynamics issomewhat limited. When tracking the extremal distance be-tween a body and a point, the continuous state the describes


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the motion of the extremal point on the body is decoupledfrom the discrete state that specifies the relevant feature onthe body. In particular, the switching times do not dependon the continuous states. On the other hand, when trackingthe extremal distance between two bodies (either in a planeor in space) the switching times do in fact depend on cer-tain portions of the continuous states.

In the following, let us develop a hybrid mathematicalmodel to describe the most general case: extremal distancetracking between two bodies. Thereafter, we will show howcertain arguments may be removed from the argument listsfor the various functions to describe the case of extremaldistance tracking between a body and a point.

Consider a system described by a state space S =⋃n

i=1 Si, where the state x within each mode Si evolves ac-cording to the differential equation

x(t) = Fi(x(t), θ(t)), (15)

where θ(t) is an exogenous input representing the speci-fied motion of the bodies. Note that the state vector x hasthe same dimension in all modes, a property that can beaccommodated by setting unused elements to zero. Associ-ated with each mode Si is a set of transitions to other modes.And associated with each transition in that set is a switchingfunction f and a resetting function φ. Let us index the mem-bers of the set of transitions (and likewise the set of f ’s andset of φ’s) with a superscript j and indicate the size of eachset with mi, which in general is different for each mode. Asubscript i indicates association with the ith mode. Mem-bers f j

i of the set of switching functions Ji trigger transi-tions out of mode Si :

Ji ={f j

i (x(t), θ(t)) = 0, (j = 1, ...,mi)}

(16)

The time t∗ that, together with the state x− = x(t∗) andspecified motion θ(t∗), produces a zero of transition func-tion f j

i is called a switching instant; it triggers the associ-ated transition. Once a transition is triggered, the associatedmember φj

i of the set of resetting functions Φi is executedto relate the initial state values x+ in the new mode to the fi-nal state values x− in the last mode.

Φi ={x+ = φj

i (x−, t∗), (j = 1, ...,mi)

}(17)

A special case of the resetting functions is the set of initialconditions for the initial mode S1.

To particularize the description of a hybrid system con-tained in Eqs. (15) through (17) for the case of collision de-tection between a body and a point, one may remove the ar-gument x from each switching function in (16). The switch-ing functions depend only on the specified motion of thebody and point, not on the motion of the extremal point onthe body. In fact, for collision detection between two bod-ies, the switching between features on one body dependsonly on the motion of the extremal point on the other bodyand the motion, not the motion of the extremal point on itsown surface.

Evaluation of the hybrid system can be viewed as a se-quence of subproblems, each characterized by a continuousevolution in a mode terminated by an event (i.e. zero cross-ing of a switching function), and then evaluation of the re-setting functions to initialize the new mode.

A transition from one discrete state to another one is trig-gered when a discrete state (a feature) encounters a transi-tion condition (i.e. an active feature goes out of the Voronoiregion of the other active feature). These discrete transitionsare handled in a manner similar to the Lin and Canny al-gorithm [11]. Note that whenever a transition occurs, notonly the discrete state changes, but also the continuous sys-tem model changes and the continuous states are updatedaccording to the reset relations Φ. It is convenient to rep-resent the discrete dynamics of the hybrid system using anautomaton as in Figure 7.

To analyze convergence to the extremal pair, first con-sider the case when both objects are stationary. In section 4it was shown that, once initialized with the correct pair offeatures (but not necessarily with the correct extremal pairwithin the features), our algorithm accounts for the initial-ization errors and disturbances, exponentially converging tothe extremal pair. This exponential convergence within thefeature is guaranteed since the basin of attraction of the con-trolled system spans all the feature itself. Therefore (for thestationary case) in order to achieve global asymptotic sta-bility, it would be sufficient to introduce a far phase that lo-cates the correct pair of features to initialize the near algo-rithm.

Now, consider the case when objects are in motion. Intro-duction of motion converts the problem into a tracking prob-lem. In our method, within each feature, the motion of theobjects are exactly accounted for by the feedforward term ofthe controller. However, some extra effort is required to han-dle transitions between features that are due to the motion.In order to be able to track the extremal distance throughfeatures, the switching algorithm is put in place. The switch-ing algorithm decides when and how to handle transitionsbetween features according to Voronoi criteria. Therefore,with the installment of the switching algorithm, our methodsimultaneously tracks the extremal pair and the correct setof features containing them.

Even under motion, convergence to the extremal pair isguaranteed since the motion is handled in a feedforwardmanner and the transitions are handled in such a way thateach time a transition is triggered the extremal distance isguaranteed not to increase.

6. Simulation ResultsWe developed computer simulations to check the dy-

namic formulation and the control algorithm discussed inthe previous sections. Our simulations are implemented inMatlab/Simulink and sample results are presented below.

Figure 6 shows a convex planar body A formed by join-ing three convex curves C1, C2, and C3 at vertices Va, Vb,and Vc. Also shown in Figure 6 are the six lines that bound


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AC1

C2

C3

Va

Vb

Vc

Figure 6. A convex planar body A composed ofcurves C1, C2, C3 joined at vertices Va, Vb, Vc. Linesindicate boundaries between interior and exteriorVoronoi regions.

C1

C2 C3

Va

Vb

Vc

t=

34.4

t = 48.7

t=

86.5

t = 91.6t = 137.9

t=

143.

4

t = 174.9 t = 179.1

Figure 7. The automaton used to govern the mo-tion of the extremal point on body A of Figure 6.Switching times for the simulation shown in Fig-ure 8 are also indicated.

the external Voronoi regions of A and the three lines, calledmedial axes that bound the internal Voronoi regions of A.The medial axes are each the locus of points equidistant totwo curves of A. The automaton that describes the hybriddynamics of an extremal point lying on the boundary of Ais shown in Figure 7. The three large ellipses in Figure 7each describe a mode in which the extremal point lies ona particular curve. The continuous dynamics within a par-ticular mode govern the motion of the extremal point solong as it lies on the correspondingly labelled curve. Thethree smaller circles each describe the (trivial) dynamics ofthe extremal point while it lies on the correspondingly la-belled vertex. The arrows indicate the transition functionsthat switch between modes. The two outer circular loopsindicate transitions from one external Voronoi region to an-other, while the two inner triangular loops (not involvingvertices) indicate transitions across medial axes from oneinternal Voronoi region to another.

Figure 8 shows a simulation that produces the trackingbehavior of an extremal point on A as a point P (the othermember of the extremal pair) traces out a pre-specified spi-ral path that begins inside A and ends outside A. In all, 100snapshots of the simulation are shown, or 100 extremal dis-tances connecting the extremal pair. To begin, the extremalpoint is initialized with the wrong value and the algorithmquickly converges to the correct answer as indicated by thedotted lines and sweeping arrow. Thereafter, one end the ex-tremal distance is either perpendicular to a curve of A or an-chored on a vertex of A. The times at which transitions be-tween modes occur are shown both in Figure 8 and on theautomaton in Figure 7. For example, at t = 34.4 seconds, Pcrosses the medial axis and tracking on C1 jumps to track-

A

t = 34.4

t = 48.7 t = 86.5

t = 91.6

t = 137.9

t = 143.4

t = 174.9

t = 179.1

t = 66.9

Figure 8. The extremal distance is shown be-tween A and a point tracing out a spiral that be-gins inside and ends outside A. Simulated transi-tion times are indicated.

0. 2

0

0.2

0.4

0.6

0.8

1

0 20034.4 48.7 86.5 91.6 137.9 143.4 174.9 179.1

u1 u2

u3

u1 u2

u3

Simulation Time

Para

met

ers

Figure 9. The trajectory and sequence of param-eters u1, u2, u3 on curves C1, C2, C3 for the simula-tion shown in Figure 8. Transition times are alsoindicated.

ing on C2. At t = 66.9 seconds, P crosses the boundaryof A and the definition of extremal changes from maximumpenetration depth to minimum distance, but tracking contin-ues on C3. At t = 86.5 seconds, the extremal point anchorson Vc and at t = 91.6 seconds it continues on C1 again. Thetransition times are also indicated for this particular simula-tion in Figure 9, which is a plot of the curve parameters ui

versus time. All three curve parameters ui, (i = 1, 2, 3) canbe varied from 0 to 1 to trace out body A while the particu-lar sequence and trajectory of values shown in Figure 9 per-tains to the evolution of the extremal point shown in Figure8. Also visible in Figure 9 is the fast convergence of startuperror at the very beginning of the simulation.

Figure 10 shows twelve irregularly spaced, labelledsnapshots from the simulation of the extremal pair track-ing problem for two convex planar bodies. During thesimulation body A remains fixed and B undergoes a mo-tion in which it spins around its own center while itscenter traces out an inward spiral centered on A. The ac-tive curves or vertices, which correspond to similarlylabelled modes in the automata not shown, are also indi-cated in Figure 10. Figure 11 shows the curve parame-


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A AA

A A AA

A A AA

B

B

B

BB

BB

B

B

B

BB

1 2 3 4

5 6 87

9 10 11 12

At = 0 t = 1.30 t = 3.25 t = 4.64

t = 8.30 t = 9.30 t = 10.10 t = 11.42

t = 12.30 t = 12.76 t = 13.15 t = 14.00

C1

C1C1 C2

C3

C3C3C3

CbCb

Cb

Cc

CcCc

Cc

CcCcV2

V2

V3

Va

Vb

VcVc

Figure 10. The extremal distance between planarbodies A and B as A remains fixed and B spinsand traces a circle around A. Twelve snapshotsare shown, taken at irregular intervals. The activecurves or vertices of A and B are indicated.

0. 2

0

0.2

0.4

0.6

0.8

1

ur

1 122 3 4 5 6 7 8 9 10 11

0 153.25 4.39 4.70 8.24 8.32 9.23 12.76 13.18

u1 u2 u3

u1

rb

rc

Simulation Time

Para

met

ers

Figure 11. The trajectory and sequence of param-eters u1, u2, u3 on curves C1, C2, C3 and ra, rb, rc oncurves Ca, Cb, Cc for the simulation shown in Fig-ure 10. Transition times are indicated.

ters ui of A and ri of B. The sequencing and evolutionof both curve parameters ui and ri that locate the ex-tremal points on A and B can be seen. The snapshottimes are also indicated in Figure 11 as are the transi-tion times. The start up error is demonstrated in Figure10-1 and Figure 10-2 and the fast convergence of startup er-ror at the very beginning of the simulation is noticeable inFigure 11.

C

Vα

Vβ

S1

S2Cb

Cc

Figure 12. The extremal distance is shown be-tween a spatial body C and a point tracing out ahelix.

Figure 12 shows simulation that produces the trackingbehavior between a spatial body C and a point that tracesout a helix. Body C is formed by joining three parametricsurfaces at their intersecting curves and vertices. The sur-faces as labelled S1, S2, and S3, the curves Ca, Cb, and Cc

and the vertices Vα and Vβ . Extremal distances at each timestep are shown.

7. DiscussionThe control theoretic approach to extremal point tracking

that we have described in this paper possesses several sig-nificant advantages. First of all, it is very general —in thesense that various feedback control designs and discretiza-tion choices exercised within our approach yield various al-gorithms, some of which have previously appeared in theliterature.

A second significant advantage of our approach is thatlocal exponential stability can be guaranteed within eachfeature. This stability guarantee holds no matter how fast thebodies might be moving or how sharp their curvature mightbe, since these effects are accounted for in the controller de-sign. Specifically, the motion (known exactly through thesolution of the forward dynamics) is essentially fed for-ward using the term b and the dependence on curvature isaccounted for with the use of M−1 (see Eq. (12)) in theproduction of the linearized system dynamics. The stabil-ity properties of closed loop system then follow from lin-ear controls theory. Furthermore, the convergence rate canbe chosen arbitrarily fast for the tracking problem as posedin continuous time. During haptic rendering, inexact veloc-ity measurements from the haptic interface would pose noproblem for our algorithm since a virtual coupler covers sta-bility concerns and the feedforward term in our controllerrequires only the computed velocity.

Since the controller and its associated dynamics will beimplemented in discrete time, the important question is thepreservation of the stability properties through discretiza-tion. Since our system equations are in fact linear, sta-bility would be preserved through discretization using thethe trapezoidal rule or Backward Euler integration scheme,even for convergence rates set arbitrarily fast using aggres-sive gain values K. But since such implicit methods arenot an option for real-time haptic rendering, we must con-sider preservation of stability through discretization usingexplicit methods. As it turns out, there exist standard tech-niques whereby the convergence rate (determined by gainK) and step size may be traded off against one anotherwhile maintaining stability. In [13] standard discrete timecontroller design techniques are utilized to calculate propercontroller gains given an explicit integration method andfixed integration step size. With these techniques, stabilityof the algorithm is guaranteed even after discretization.

Moreover, with proper design of the ‘far’ phase and theswitching algorithm for the collision detector as discussedin section 5, global stability of the extremal tracking con-


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troller can be achieved, even for geometric models made ofseveral surface patches pieced together.

Finally, whenever the far phase and switching in place,our algorithm is capable of tracking both the minimum dis-tance when the objects are disjoint and the maximum pen-etration depth when they are interpenetrated. The switch-ing algorithm involves consideration of the Voronoi regionsfor each of the knitted shapes, which can be pre-computed.When bodies are disjoint determination of switching definesa convex problem and the switching is restricted to one ofthe neighboring features. However, the interpenetration casedefines a non-convex problem and in this case the switch-ing algorithm requires a global search (the far phase) to as-sure convergence.

8. ConclusionsWe are interested in pursuing a combined simula-

tion/collision detection approach since it results in analgorithm that is easy to implement and that makes max-imum use of all the data available to track the extremalpoints. The proposed algorithm is suitable for both dy-namic simulation and haptic rendering due to the con-tinuous availability of surface normals and penetrationdistances that are necessary to calculate the collision re-sponse and/or the haptic feedback.

Our algorithm treats the extremal point problem for ob-jects modelled using parametric curves and surfaces in adirect manner, without resorting to polyhedral approxima-tions. Thus it serves the needs of CAD/CAM and virtualenvironment systems that require smoothness independentof rendering. Additionally, our algorithm is suited to real-time implementation. Finally, our algorithm features con-venient tuning of convergence properties through design ofthe feedback gain K and enjoys immunity to start-up er-rors.

AcknowledgmentsThe authors gratefully acknowledge the support of

the National Science Foundation under award num-ber IIS:PECASE #0093290, and fruitful discussions withElmer Gilbert.

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