Patent Application
20050010326
The present invention relates generally to computer based simulation of the behavior of solid objects when they come into contact and, in particular, to methods and apparatus for synthesizing accurately the virtual interaction between a rigid object and a deformable body, as it is needed to provide for sensory feedback in haptic simulations.
Computers can be used in many instances to predict the behavior of material objects. It is often needed to provide these predictions with a number of computation steps which is known in advance for any of the cases considered by the simulation. This knowledge makes it possible to design computer simulations which are performed in real-time. This is because, with computers, simulation of a material system must be performed from one time step to the next. If it is possible to know a fixed upper bound to the number of computation steps required to compute the state of the simulated system as it changes from one time step to the next, then it is possible to design a computer system able to performing a sufficient number of computation steps per unit of time to provide a sequence of simulation steps rapid enough to approximate closely the behavior of the material objects being simulated.
More specifically, a haptic display enables a user to experience sensory signals by means of a computer. Often it is needed to compute signals related to the behavior of objects coming into contact and to transform these signals into forces or movements to be experienced by a user. To provide the ability to experience these artificially created signals, computations can be performed on the basis of the user's motions which are tracked by a haptic device, or on the basic of forces applied by a user which are measured by the device. The process of computing and generating mechanical signals in response to user interactions is sometimes called haptic rendering. To reduce the possibility of confusion with techniques in computer graphics, a preferred term used in this disclosure is interaction synthesis. Applications and areas that implement interaction synthesis include training simulators, computer aided design and entertainment.
Herein, the term force will mean force and torques and the term position will mean position and orientation. Those of ordinary skill in the art will appreciate that interaction synthesis can be carried out in two possible ways. The most common method employs a haptic device which measures the position of a tool, or a handle, held by the user. When interaction between a virtual tool and a virtual body occurs, forces are computed in response to displacement and returned to the user by the haptic device. A less common method employs a haptic device which measure forces applied by a user to a handle. When interaction between a virtual tool and a virtual body occurs, the haptic device is commanded to move to a position corresponding to the measured force. The former method is often called impedance synthesis, while the latter is termed admittance synthesis. For each case, haptic devices are constructed or controlled in specific ways, but this has no consequence on the present invention.
Accurate synthesis of interaction must be performed in real-time in order to generate virtual interactions which, once reconstructed, will closely resemble actual mechanical interaction. Therefore, to reconstruct details, a computer will have only a brief amount of time in which to compute the virtual interaction after receiving the user's input. For example, if the interaction causes forces to change significantly during a time interval of a 10 millisecond, reconstruction must occur at intervals of about 1 millisecond.
During synthesis, to compute forces (or positions) resulting from a contact between rigid bodies, it acceptable to assume that the bodies contact at one, or few points in close proximity, in order to represent the shape, texture and the position of a body. When one body is rigid and one body is deformable, the virtual contact must simulate the local details of the interaction because the virtual contact depends on the shape of the rigid body, on the elastic properties of the deformable body, on how it is supported, and on its shape. For example, assuming that bodies contact at just one point, or even at a few points, precludes the possibility to design a simulation able to distinguish between a blunt tool and a sharp tool. An accurate representation of these factors has importance for surgical simulators among other applications.
Various methods of simulation for body deformation in computer graphics and haptics have been developed. However, current methods do not simulate contact interaction resulting from local deformation. Specifically, when the deformation of a body is localized around a point of contact, the force of contact as a result of position displacement depends on the shapes of the objects in contact. For example, the shape of a rigid tool used to cause deformation of a deformable object has a critical effect on the force resulting from displacement. A sharp tool will produce a small force given a small indentation, while a blunt tool will produce a large force for the same displacement. Alternatively, a given force applied to a locally deformable body by a rigid tool results in different positions and hence difference deformations, according to the shape of the tool. This is in contrast to an elastic body that is globally deformed, for which a crude representation of contact can suffice to predict the force of contact (or the displacement given a force), since the deformation energy is distributed everywhere in the body. In such cases, only the resultant force is important, but not the force distribution in the zone of contact. A special case is to represent the zone of contact by a single point for computational expediency.
Thus, it is apparent that a point contact representation used to simulate virtual interactions between bodies is an idealization which is not sufficient to distinguish between knifes, forceps, or blunt tools interacting with dead or living tissues, or with other materials possibly including soils, or fabrics. Hence, there is a need to provide an improved method of haptic simulation that will provide realistic simulations of contact interaction with a deformable body, whether there is local deformation or global deformation, and that is independent of material homogeneity, isotropy, and linear elasticity of the body, all of which critically influence the behavior of a contact.
In accordance with one aspect, the present invention addresses a method for synthesizing a response to interaction between a simulated tool having a tip and a simulated, deformable body model having its surface modeled as being divided in a plurality of adjacent polygons, preferably triangles, said method comprising the steps of:
By a further feature, the invention comprises a method for synthesizing a response to interaction between a simulated tool having a tip and a simulated deformable body model having its surface modeled as being divided in a plurality of adjacent polygons, said method including the steps of:
According to another aspect of the invention, a method for synthesizing a response of a simulated tool sliding over a simulated deformable body model includes the step of permitting the point of initial contact to move over the surface of the undeformed body such that the magnitude of the tangential component of the deflection projected over the surface of the undeformed body is smaller than a bound that depends on the normal deflection component. This method can be employed wherein the deflection data arising from the point of initial contact, upon moving said point of contact over the surface, is preserved at the value created when the point of contact is about to leave the active polygon until data for a new active polygon is established, and thereafter is determined by the deflection data for such new active polygon.
The above methods may be adapted such that the magnitude of the tangential component of the force projected over the surface of the undeformed body is smaller than a bound that depends on the normal force component. Further, the force data arising from the point of initial contact, upon moving said point of contact over the surface, may be preserved at the value created when the point of contact is about to leave the active polygon until data for a new active polygon is established, and thereafter is determined by the force data for such new active polygon.
For all methods as described which are high rate processes, a process may be provided that is suitable to deliver, at low rate, response data having a lag and associated with an active polygon such that, if the response data represents a conservative force field, the synthesized force signal is dissipative with respect to displacement. This process comprises the detection of establishment of virtual contact and removing from the deflection the offset corresponding to the lag in said detection of establishment of the virtual contact.
The invention also addresses synthesizing a response to an input force acting on a deformable body having a surface modeled as a mesh of polygons, preferably triangles, said method comprising:
This method can be applied wherein receiving data indicative of an input force acting on the deformable body at a point of contact comprises receiving a deflection conveying an input force magnitude and an input force direction with respect to the point of contact. It can also be applied wherein determining a relative position of the point of contact within the currently active polygon comprises determining a weight associated with each of the vertices of the currently active polygon.
According to another aspect of the invention, obtaining force response data associated with the currently active polygon comprises loading the force response data associated with the currently active polygon from a memory. This may be combined with, prior to loading the force response data into the memory, computing the force response data on the basis of physical measurements, on the basis of analytical data, or on the basis of a simulation.
Thus, each polygon in a mesh of polygons may be defined by a plurality of vertices, wherein loading the force response data associated with the currently active polygon comprises loading force response data associated with each vertex of the currently active polygon. Such force response data associated with each vertex of the currently active polygon may convey a magnitude and a direction of the force response at that vertex, optionally in each of at least two basis directions.
Such basis directions may include a first basis direction and a second basis direction, wherein the magnitude component in the first basis direction is a first magnitude component and wherein the magnitude component in the second basis direction is a second magnitude component, and wherein the force response data associated with each vertex of the currently active polygon further conveys the first and second basis directions.
According to a further aspect of the invention, jointly processing the force response data and the relative position to produce data indicative of a force that is reactional to the input force comprises:
Such steps may alternately produce data indicative of a force that is reactional to the input force by inclusion of steps comprising:
This method may be pursued wherein the data indicative of the reactional force conveys a vector that is the sum of the first magnitude component for the reactional force in the first basis direction for the reactional force and the second magnitude component of the reactional force in the second basis direction for the reactional force.
The method may also be pursued wherein the data indicative of the reactional force conveys a vector having:
According to a further aspect, the methods of the invention may comprise:
Then, if the new currently active polygon is different from the currently active polygon,
Alternately, a further aspect, the methods of the invention may comprise:
This method can be repeatedly performed and can be continued until an Nth execution of step h) yields that it is time to update the currently active polygon, N being greater than one.
The method of the invention can further comprise synthesizing the force that is reactional to the input force and this can be done by applying the force that is reactional to the input force to a physical tool. This can be obtained by applying the input force on the deformable body using the physical tool.
According to a further aspect of the invention, an apparatus for synthesizing a response to an input force acting on a deformable body having a surface modeled as a mesh of polygons, comprises:
This apparatus may further comprise a memory unit for storing the force response data associated with each of the polygons. The processing unit is then operable to obtaining force response data associated with the currently active polygon by loading the force response data associated with the currently active polygon from the memory.
The invention further includes computer-readable media tangibly embodying a program of instructions executable by a computer to perform a method of synthesizing a response to an input force acting on a deformable body having a surface modeled as a mesh of polygons, the method comprising:
These and other aspects and features of the present invention will now become apparent to those of ordinary skill in the art upon review of the following description of the invention in conjunction with the accompanying drawings.
In the accompanying drawings:
a shows a virtual deformable body in its undeformed state represented with its boundary modeled in accordance with an embodiment of the present invention;
b shows a virtual rigid object initially coming into contact with a virtual deformable body at point c common to the boundary of the rigid object and the deformable body.
c shows a virtual rigid object interfering with a virtual deformable body. Point c, the point of initial contact is on the boundary of the undeformed deformable body. Point x is the point on the boundary of the virtual rigid object which initially coincided with c. An actual material body would be deformed by a real tool so that no actual interference would occur.
a is a sub-flowchart of flowchart of
a and 6b are sub-flowcharts of flowchart of
With reference to
As shown on
Referring to
Referring now to
As illustrated by
The points x(t) and c(t) define a deflection dc(t)=c(t)−x(t) (the time dependence quantities like those will be omitted from now on). Each such deflection dc is vectorial in nature and thus is associated with a magnitude and a direction (where bold underline notation signifies a vector quantity) Once the system is operating and providing a reaction. For each point c of initial contact with the body, a force fc is associated to a deflection dc. In some embodiments, the signal x is provided as input data 140 to a computer system 100 equipped with a processing unit 120 and a memory 110. In some other embodiments, the signal fc is provided as input data 140 of a computer system 100.
For each vertex 210 of the mesh made of patches 220 that represents the virtual deformable body 200, the memory 110 stores a set of basis vectors and associated response information as shown by
It is noted that the deflection-force response curves or the force-deflection curves may be obtained during the course of a preprocessing stage using various approaches, such as finite element code, analytical solutions from first principles, by performing measurements made on physical samples, or by manually tuning parameters in order to reach a desired virtual interaction as experienced by the user. To make clear how such responses may be obtained, consider the example of an elastic block of homogeneous rubber bonded to a hard surface so that the deformation of the bonded surface is zero. Next consider a spherical tool indenting this block in a manner similar to tests which are performed to measure the hardness of materials, that is, in a direction perpendicular to the exposed surface. If the block if homogenous, to each deflection there will correspond a force in a direction orthogonal to the surface. In such special case, the vector uzl,m will be preferably chosen to coincide with the geometric normal vector to the surface of the undeformed body because, in this case, only one curve will have to be recorded. Consider next that the body is not homogeneous, that there is, for example, a steel ball embedded in the rubber in the vicinity of the point of initial contact. The same test performed by driving the spherical tool orthogonally into the surface will yield a response force which will be at an angle. Although the deflection is forced to have only one component, the force response may have three components. In this example, it is more economical to perform the test with the deflection aligned with response force and to select the vector uzl,m to be aligned in the same direction, since only one response curve will have to be recorded. However, during synthesis as will shortly be described, the result will be identical as-that obtained using an orthogonal deflection and three response curves.
The deflection-force or force-deflection response curves can be stored in the memory 110 as look-up tables, regression models, or function fit, for example. Generally speaking, the memory requirements for storing the deflection-force response curves (and the basis vectors) for individual nodes (vertices) are low. To illustrate, 240 megabytes of memory will suffice to represent the nonlinear response of an entire human body of 2 m2 of surface at 1 mm of resolution using 6 million nonlinear deflection-force response curves, if it is assumed that 10 parameters, each occupying four bytes (a floating point number), are sufficient to specify-these curves and that each parameter is represented by a single precision floating point number. The 240 megabyte requirement for memory is easily satisfied by most conventionally available computers.
As will be described in greater detail later on, the processing unit 120 runs a program which processes the basis vectors and the deflection-force response curves or the force-deflection response curves in the memory 110 in accordance with the virtual deflection dc (or the interaction force fc) at the contact point c conveyed by the signal 140 to synthesize a virtual interaction. For impedance synthesis, the response force fc is provided in the form of a signal 150 to an input of the haptic device 130. Upon receipt of the signal 150 from the computer system 100, the haptic device 130 applies the response force conveyed by the signal 150 to the tool 230. For admittance synthesis, the deflection d; is used to provide a position x=c−dc in the form of a signal 150 to an input of the haptic device used to displace the held handle. In other instances, the interaction conveyed by the signal 150 can be displayed visually on a display or be processed by any other interface capable of transforming the output data signal 150 into forces or positions that can be sensed by the user, for example locally or at a remote location, or by other users, as in a classroom teaching situation.
For the case of impedance synthesis, upon sensing the response force, the user of the tool 230 will be under the impression that the tool 230 has come into contact with the virtual deformable body 200. For admittance synthesis, the user of the tool 230 will experience the tool being moved to the location it should occupy given force she or he applied to the deformable body 200 and will be under the impression that the tool 230 has come into contact with the virtual deformable body 200. Depending on the accuracy and speed with which the interaction is computed by the computer system 100 (and the accuracy and speed with which it is transformed by the haptics device 130 into actual forces or displacements), the force or the displacement felt by the user of the tool 230 approximates that which would have been felt, had the virtual deformable body 200 actually been present.
One possible algorithm 500 that can be executed by the processing unit 120 with a view to computing the signals needed to simulate a virtual interaction is now described with reference to the flowchart in
At step 510, data indicative of an input position of the haptic device causing the virtual tool to come into contact at an initial point of contact c with the virtual deformable body 200 is received via the signal 140. After suitable coordinate transformations which are well known from those skilled in the art, the process is in a position to notify to another part of the processor the position of the held handle. The processor reserves a small amount of memory called a buffer the store all information relative to a patch m. The information relative to patch m is termed is active element. If no patch is active, the buffer contain a special element termed the null element. At step 520 the existence of point c is determined. It is a simple operation since it performed with reference to patch m only. There can be two outcomes to the test 521.
When there is no contact, the algorithm 500 proceeds with step 540 where possible interference with any other possible element is determined. As it is well known by those skilled in the art, this determination may in general consume an unpredictable number of steps since it typically involves search. Many methods are available to reduce the number of these steps typically relying on recursive data structures such as “octrees” or “oriented bounding box” (OBB) trees, else or in addition to methods which take advantage of coherence from one time step to the next, that is, to seed the search from the last element that was found to be in contact, including methods known in computer graphics as “picking” methods. However these improvements have no consequence on the present invention. In any case, given unpredictability, it is preferable to carry out interference detection in a separate process which will be detailed later on. For now, we consider that step 540 is performed in a small and predictable number of steps.
If an interference between the rigid and the deformable objects is detected, the algorithm branches at step 541 and the element which contains point c is identified at step 550 and corresponding the response data retrieved from memory and put in the buffer reserved for the active element at step 555. If no interference is detected, the active element is set to null at step 560. The device data is set to zero at step 570 before proceeding to the next iteration.
If the result of step 521 is in the case of contact between the rigid object and the active element, then at step 530, the processing unit 120 finds the data pertaining to the active polygon in the reserved buffer (the intervening step 525 will be described later and may be skipped for now). This data includes basis vectors uxl,m, uyl,m, and uzl,m for and deflection-force response curves for each of the vertices I=1,2,3 of the active polygon as illustrated by
At step 530, the processing unit 120 proceeds to determine the relative position of the initial point of contact c within the active polygon m. This position can be used to define an interpolation having weights nl,m(c), l=1,2,3. One suitable choice for the interpolation function nl,m(c) is based on natural coordinates as it will be known from those skilled in the art:
nl,m(c)=Al,m(c)/Am
where Am is the area of the currently active polygon m and, as illustrated in
uxc=Σl=1,3nl,m(c)uxl,m
uyc=Σl=1,3nl,m(c)uyl,m
uzc=Σl=1,3nl,m(c)uzl,m
At step 535, for impedance synthesis, the virtual contact force is, in principle, computed according to
fc=fxc(dxc,dyc,dzc)uxc+fyc(dxc,dyc,dzc)uyc+fzc(dxc,dyc,dzc)uzc,
where dxc, dyc, and dzc are the components of dc in the basis Uc as illustrated by
However, in actual practice it often sufficient given a proper selection of the bases Ul,m to only consider the deflection components that have the greatest influence on corresponding force response component and to neglect the others:
fc=fxc(dxc)uxc+fyc(dyc)uyc+fzc(dzc)uzc,
where dxc, dyc, dzc are the components of the deflection d with respect to the basis vectors at c. At step 535, the components of the virtual contact force, namely fxc(.), fyc(.), and fzc(.) are obtained from the stored displacement-force response curves and the interpolation weights in the following manner
fxc(dxc)=Σl=1,3nl,m(c)fxl,m(dxc)
fyc(dyc)=Σl=1,3nl,m(c)fyl,m(dyc)
fzc(dzc)=Σl=1,3nl,m(c)fzl,m(dzc)
The resulting data representing the response force is output at step 580. It should be noted that in many cases, one could be satisfied with even more simplified representations of a contact. For example, the above set of equation may be replace by two: one for the response in the normal direction z and one in a radial direction r that provides the best fit (yielding cylindrical coordinates), or even just one in the normal direction.
For admittance synthesis, at step 535 the virtual contact position is computed applying the same simplifications according to
dcdxc(fxc)uxc+dyc(fyc)uyc+dzc(fzc)uzc,
where fzc and frc are the components of the force fc exerted by the user on the handle along the basis vectors at c. The individual components of the virtual deflection, namely dxc( ), dyc( ), and dzc( ) are obtained from the stored displacement-force response curves and the interpolation weights in the following manner:
dxc(fxc)=Σl=1,3nl,m(c)dxl,m(fxc)
dyc(fyc)=Σl=1,3nl,m(c)dyl,m(fyc)
dzc(fzc)=Σl=1,3nl,m(c)dzl,m(fzc)
Thus, the point of initial contact c displaced by dc indicates the position of the virtual deflected-contact which-respect to c and hence the position x where the haptic device must be commanded to move to in order to simulate the interaction.
So far the we have described virtual interaction synthesis which did not involved sliding. In other words the simulated contact is sticking only. But before we describe the more general case of sliding virtual interaction, an improved algorithm is preferably needed to realize virtual interaction synthesis requiring a predictable number of computation steps to be accomplished within a set time period.
With reference to
The flowchart in
Instead of one single buffer to store the data pertaining to the active element, two are now provided, one for the high rate process and one for the low rate process, such that the data stored in the high rate buffer can be accessed frequently and predictably.
In the flowchart of
One method hereby disclosed which provides a passive virtual interaction despite the combined actions of the low and high rate processes. This property will be further discussed later, but at step 630, it can be shown that it is sufficient that an additional variable be computed to compensate for the time tag. It suffices to compute a deflection offset d0 which is such that the initial potential energy stored in the virtual interaction when a new element is activated is equal or smaller than the potential energy stored in the virtual interaction with the previous element at the moment it was replaced. Since any force, including virtual interaction forces, derive from a potential, we can call Ei−1(x) the potential energy of virtual interaction at the time when the active element is replaced and Ei(x) the initial potential energy of the virtual interaction of the new active element for the same position x of the virtual rigid body. It is well known that a necessary condition for passivity is:
Ei(x)≦Ei−1(x)
Among the several methods to enforce this inequality even though the new active element was determined at an earlier time than when it was activated, one of them can readily-be illustrated. Step 630 computes a deflection offset dc0i which is substracted from dc to ensure that Ei(x)≦Ei−1(x) is verified. By accounting for this offset, the force is computed from dc−dc0i according to
fc=fxc(dxc−dxc0i)uxc+fyc(dyc−dyc0i)uyc+fzc(dzc−dzc0i)uzc.
In addition, fc should take the value zero when dzc<dzc0i.
The energy inequality is easily solved for dc0i so that Ei(x)=Ei−1(x)=0. This deflection adjustment is applied immediately at the next iteration.
It is noted that because of the preprocessing performed in order to compute the deflection-force (or the force-deflection) response curves, there is no difficulty in computing the basis vectors at step 530 and the force (or the position) at step 535 in real-time. It is noted that the chosen interpolation function ensures continuity of the response force over the surface of the virtual deformable body 200, but-that any other interpolation function providing continuity could be used. It is also noted that realism in the simulation of the interaction results from two interpolation steps: one to determine the basis vectors at the point of initial contact c, and one to determine the response of the interaction, be it via the impedance or via the admittance method. This is because the synthesized signals result from the products and sums of individually continuous quantities.
Refering again to
It is apparent from the foregoing that the approach used to model the haptic interaction is applicable to almost arbitrary shapes of bodies in contact and can be employed whether the bodies are homogeneous or not, isotropic or not, linear-elastic or not. Moreover, the method is not limited to the case where the simulated deformable body has a fixed shape and fixed material properties. If it is desired to simulate virtual interaction with bodies that vary with time, for example to simulate the interaction with a beating heart or a heaving diaphragm in a surgical simulation application, the determination of the point c of initial and sliding contact will be carried out with a time-varying body boundary rather than a fixed one as it was described so-far. To make this point obvious, consider that the position of the virtual object driven by the hand of the user was considered to vary with time. So-far, the determination of the deflection at each step in time was carried out using the formula dc=c−x. Exactly the same method will apply if the boundary of the virtual body varies under the action of agents such as muscle contraction or change in blood pressure as in the simulation of living tissues which are independent from the movements of the user. In such cases, x would represents the relative position to a point of initial contact on a varying-boundary. The boundary of the simulated body can change, for example cyclically, but the interpolation equations remain the same.
Similarly, it could be desired that the virtual interaction simulation be made dependent on changes in the materials properties. An example of such need is found in the simulation of surgical procedures where changes in material properties could be result of variations due to physiological factors such as haemostatic or haemodynamic phenomena, slow diffusion of fluids, muscle contraction and others, then exactly the same simulation method described in the foregoing would apply but now making the force deflection or the deflection force curves depend on additional continuous states based on the time derivatives of dc (d′c, d″c) or of discrete states-(s1, s2, . . . , sn). The only modification then would be to provide for different responses corresponding to these states. These different responses are noted with left superscripts
This will then yield for impedance synthesis
fc=fxc(dxc,d′xc,d″xc)uxc+fyc(dyc,d′yc,d″yc)uyc+fzc(dzc,d′zc,d″zc)uzc
but now, using additional interpolation coefficients wi(v1,v2, . . . ,vp) where the variables vi represent additional physical factors that have influence on the resulting force, and where the force components Sifxc, Sifyc, and Sifyxc along each coordinates are in turn interpolated according to the position of point c (or projections of c):
fxc(dxc,d′xc,d″xc,v1,v2, . . . ,vp)=wi(v1,v2, . . . ,vp)Sifxc(dxc,d′xc,d″xc)
fyc(dyc,d′yc,d″yc,v1,v2, . . . ,vp)=wi(v1,v2, . . . ,vp)Sifyc(dyc,d′yc,d″yc)
fzc(dzc,d′zc,d″zc,v1,v2, . . . ,vp)=wi(v1,v2, . . . ,vp)Sifzc(dzc,d′zc,d″zc)
For admittance synthesis, a similar method applies but to deflection-force responses.
In such more general cases, the same assurances of continuity of the simulated interaction, of accuracy, and of low computational demand will be provided, given the two step interpolation process, if the variations of the boundary and of material properties as smooth functions of time. Similarly, the same method applies for even more general cases needed, for example, to represent plasticity, visco-elastic relaxation, fluid shift, or damage of materials where now the response curves can be multi-valued and branch at reversal points. In these even more general cases, the same assurances of continuity of the simulated interaction, of accuracy, and of low computational demand will be provided, given the two step interpolation process described, provided that the response curves are continuous. An example of simulation of varying material properties due to the cutting action of a sharp tool causing the responses to vary according to the depth of cutting crack with two discrete interaction-states was provided in (Mahvash and Hayward 2001). The same method applies equally well to this more general case.
Another important case in which the same method applies is when interaction depends on the location of x with respect to the rigid object. In such case, three variables v1, v2, and v3 may be used to represent the position of x with respect to the rigid object (such as when rolling is synthesized). If even more realism is required to account for this dependency, several states s1, s2, . . . , sn, may be used to account for location of x with respect to a mesh used to approximate the rigid object geometry. Force-deflection (or deflection-force) responses are predetermined im each of these states and the above interpotation equations are employed. Similarly, the interpolation weights may be computed from area coordinates to define appropriate weigthing functions wi.
When artificially created friction is of importance, its effect can be conveniently be included in the simulation of virtual interactions between bodies. From the work of Dahl (1969) it is 1 wt that the effect of friction may be viewed as the result of tangential elastic deformation of surfaces in contact. When the tangential load remains under a threshold, by definition of elasticity, loading and then unloading tangentially a contact will cause it to return to its initial state. When the loading exceeds the said threshold, a permanent change is made, the surfaces slip with respect to one another, something which is analogous to plasticity (Hayward and Armstrong 2000, Dupont et al. 2002. In these references a defect of the original description by Dahl which can-be shown to lack the sticking phase is corrected). While this was intended to describe the behavior of friction between hard solids, that is, at microscopic scales, it applies equally well to deformable bodies but with the difference that the relative displacement between the bodies is allowed to occur at macroscopic scales. Thus, this difference is only quantitative but not qualitative. Consider for example the case of an object in contact with the skin at the back of the hand. Movements of this object over distances of many millimeters cause elastic deformation of the skin before slip occurs, but past a certain distance, slip takes pace. Upon reversal, slip is removed. At microscopic scales, these movements are known to those skilled in art as pre-sliding movements and hence, this term applies equally to the tangential deformations of deformable bodies. During presliding force and displacement are related by an elastic behavior. During sliding, the tangential force component is rather independent from the contact area but greatly dependent on the normal force component, as stated by Amontons' laws
It is therefore apparent that the synthesis of virtual interactions between deformable bodies described thus far is only a special case of the more general case of interaction with friction. It is the special case when the bodies in contact are in the stuck state, and therefore, the quantities needed to simulate friction with finite presliding have already been introduced.
Recalling that given the boundary of an undeformed body, an initial point of contact c is defined on it. When the body deforms after contact due to the movement of the rigid body relatively to it, the local deformation is represented by a second point x which tracks the position of initial point of contact on the surface of the rigid object inside the boundary of the undeformed object. Hence, a deflection dc=x−c is readily defined at all times during a virtual contact. For the case of impedance synthesis, a method was disclosed whereby all the components of the contact force fc could be computed given x for any initial contact point with assurances of accuracy and continuity, using a two step interpolation process, when in fact, the exact value of this force was known only at a finite number of points on the boundary of the undeformed body. For admittance synthesis, the same method could be used to compute x given fc for any initial contact point.
To allow the possibility of simulated sliding friction, all what is now needed is to ensure that the point c moves on the surface of the undeformed body when the magnitude of the projection dt of d over the surface of the undeformed body (and correspondingly ftc) is above a threshold. It is needed to provide for the additional specification of a coefficient μ, or more generally a univalued function μ( ), to relate the magnitude of the tangential force component to the normal force component:
ftc=μ(fzc)=(fxc2+fyc2)1/2
The point c should occupy a new location on the surface of the undeformed body only when ftc is found to exceed μ(fzc). The new location is such that ftc=μ(fzc). An efficient solution to this equation for the incremental movement of c location is already provided in (Hayward and Armstrong, 2000) for a variety of cases, including small stick-slip oscillations.
The univalued function μ( ) 340 may be recorded at the nodes of the virtual deformable body as in
Referring again to the flowchart in
Referring now to
It is known to those skilled in the art that self-sustaining oscillations can develop while interacting with a force reflecting viral environment. It can however be demonstrated that these oscillations do not occur if all the interconnected components of a system in a virtual environment possess the property of passivity. Passivity for a component means that this component does not generate energy. It can be shown that passivity is related to the existence of conservative fields (force or velocity fields as the case may be), a property that can be demonstrated for the fields generated by the double interpolation process. It is also the case that these fundamental properties will be nearly maintained even in the case when the simulated deformable bodies have time varying boundaries and time varying material properties, provided that these properties vary sufficiently slowly.
A correspondingly passive force reflecting the virtual environment not only more closely simulates the response of a nominal environment, but as mentioned above, also guarantees the stability of the interactions during which the user does not intend to destabilize the interaction. A discrete-time simulation of a nominally passive continuous-time environment will generally become active. However, with a sufficiently high update rate, the discrete-time simulation of a passive continuous-time environment will result in passive force responses. This is because any material system such as the mechanical components used to manufacture a haptic device always yield residual dissipation no matter-how well it is manufactured.
Thus, to ensure a passive response, high update rates of the force response are desired win synthesizing a response to an input force acting on a deformable body. That is, evaluation of interaction forces through interpolation of stored responses are advantageously performed at a high rate, something that the algorithm represented by the flow chart 600 provides because the identification of the currently active polygon as the rigid virtual object 230 moves along the surface of the virtual deformable body 200 can be performed at a lower rate. The resulting lag will only have a marginal effect on the accuracy of the synthesis.
In fact, with the approach disclosed here, the fidelity of the sensations experienced by a user will be entirely function of the ability of the haptic devices employed to transform the signals resulting from synthesis, namely, the ability of the device to create forces without deleterious effects resulting from transmission friction, damping and structural dynamics for the case of impedance synthesis, or the ability of the device to track position commands without limitations resulting from tracking rate, tracking accuracy and other imprecision resulting from its construction and its control for admittance synthesis.
In some situations, it is needed to couple a virtual interaction synthesis experienced by users via mechanical signals as described herein with concomitant a graphical representation using a computer graphic screen or with other methods such as stereo video projection goggles and the like. With reference to
The processing unit 120 may be implemented as one or several arithmetic and logic unit (ALU) having access to a code memory (not shown) which stored program instructions for the operation of the ALU. The program instructions could be stored on a medium which is fixed, tangible and readable directly by the processor, (e.g., removable diskette, CD-ROM, ROM, or fixed disk), or the program instructions could be stored remotely but transmittable to the processing unit 120 via a modem or other interface device (e.g., a communications adapter) connected to a network over a transmission-medium. The transmission medium may be either a tangible medium (e g, optical or analog communications lines) or a medium implemented using wireless techniques (e.g., microwave, infrared or other transmission schemes).
Those skilled in the art should also appreciate that the program instructions stored in the ode memory can be compiled from a high level program written in a number of programming languages for use with many computer architectures or operating systems. For example, the high level program may be written in assembly language, while other versions may be written in a procedural programming language (e.g., “C”) or an object oriented programming language (e.g., “C++” or “JAVA”)
Those skilled in the art should further appreciate that in some embodiments of the invention, the functionality of the processing unit 120 may be implemented as pre-programmed hardware or firmware elements (e.g., application specific integrated circuits (ASICs), electrically erasable programmable read-only memories (EEPROMs), etc.), or other related components.
Also, it will be understood that the model of the virtual deformable body 200, along with the basis vectors and the associated force responses, can be downloaded into the memory 110 via the internet or they can be provided on a medium such as a CD-ROM
While specific embodiments of the present invention have been described and illustrated, it will be apparent to those skilled in the art that numerous modifications and variations can be made without departing from the scope of the invention as defined in the appended claims.
This application claims the benefit of prior provisional application Ser. No. 60/473,445 filed May 28, 2003.
| Number | Date | Country | |
|---|---|---|---|
| 60473445 | May 2003 | US |
