IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 1, JANUARY 2002 3
Control Law Design for Haptic Interfaces to VirtualReality
Richard J. Adams and Blake Hannaford
AbstractThe goal of control law design for haptic displays is toprovide a safe and stable user interface while maximizing the op-erators sense of kinesthetic immersion in a virtual environment.This paper outlines a control design approach which stabilizes ahaptic interface when coupled to a broad class of human opera-tors and virtual environments. Two-port absolute stability criteriaare used to develop explicit control law design bounds for two dif-ferent haptic display implementations: impedance display and ad-mittance display. The strengths and weaknesses of each approachare illustrated through numerical and experimental results for athree degree-of-freedom device. The example highlights the abilityof the proposed design procedure to handle some of the more diffi-cult problems in control law synthesis for haptics, including struc-tural flexibility and noncollocation of sensors and actuators.
Index TermsControl system human factors, force feedback,haptic display, stability criteria, virtual reality.
T HE WORD haptic means of or relating to the sense oftouch. Haptic feedback is a new and relatively unexploredway of conveying information between a human and a com-puter. A video monitor provides visual information, speakersprovide audio information, in an analogous manner, a haptic dis-play conveys kinesthetic information to the operator through thesense of touch. The haptic display generates force feedback cueswhich may represent the resistance of a virtual wall, the rough-ness of a virtual texture, or the weight of a virtual mass. Hapticdevices come in all shapes and sizes, ranging from large indus-trial manipulators to motorized desktop mice.
The field of haptics has been led by research applications.Some of the most exciting work is in surgery simulation. Thegoal is to permit student surgeons to safely practice proceduresusing haptic and graphical interfaces which accurately reflectreal surgical conditions , . Another burgeoning area isforce feedback in computer-aided design (CAD). By allowingthe designer to actually touch the objects under design, hapticsmay greatly increase efficiency and creativity. It is not surprisingthat the most rapidly evolving side of haptics is consumer prod-ucts. The first consumer haptic devices were force feedback joy-sticks for computer gaming. With the acceptance of standardapplication programming interfaces (APIs), the number of forcefeedback enabled computer games has skyrocketed, and new in-terfaces, such as force feedback steering wheels, have appeared.
Manuscript received September 13, 1999; revised September 20, 2000. Man-uscript received in final form August 7, 2001. Recommended by Associate Ed-itor D. W. Repperger.
The authors are with the Department of Electrical Engineering, University ofWashington, Seattle, WA 98195-2500 USA.
Publisher Item Identifier S 1063-6536(02)00327-5.
Fig. 1. Excalibur and the virtual building block simulation.
The advent of the force feedback mouse  and force feedbackenabled graphical user interfaces (GUIs) may eventually bringhaptic technology to the majority of computer users desktops.
The virtual building block (VBB) system is an example ofhaptic simulation, developed at the University of WashingtonBioRobotics Laboratory to demonstrate the utility of haptictechnology in CAD and virtual prototyping applications. Thesystem currently uses the Excalibur three-axis force displaybuilt by Haptic Technologies Inc. of Seattle, WA , . Fig. 1illustrates the simulation. The user can select and manipulateindividual blocks or groups of blocks. Haptic feedback rep-resents each objects inertia, prevents objects from impedingon each other, and renders interaction forces when blocks aresnapped together.
In one sense, haptic interaction is very different from othermodes of human-computer interface. In haptic display, there isa bidirection flow of information, both from the device to thecomputer and from the computer to the device. The haptic dis-play in not just a force player, rendering preset force effects. Itis a force feedback device which creates forces in response tothe users input. As with any device where feedback is present,the potential for instability exists. Instabilities will at best de-grade the quality of the force feedback information to the humanoperator. In the case of devices with significant inertia and forceoutput, the consequences of instability may be more severe.
The nascent field of controls for haptic displays offers inter-esting challenges. As with most control problems, there are two
10636536/02$17.00 2002 IEEE
4 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 1, JANUARY 2002
sometimes conflicting goals, performance and stability. Goodperformance implies forces and velocities experienced by thehuman operator are nearly identical to those generated by theapplication software. This is the transparency of the hapticinterface . The performance of any haptic display is limitedat one extreme by how well it can simulate free motion and atthe other extreme by how well it can generate rigid constraints.The minimum and maximum achievable impedance of a hapticinterface delineates its impedance range , another usefulmeasure of haptic performance. The inability to render arbi-trarily small or large impedance to the human operator is due,in part, to the onset of instability. The stability of a haptic in-terface is strongly affected by both the application software andthe nature of human-device contact. The application software,which we will also refer to as the virtual environment, deter-mines what forces are rendered through the haptic display to thehuman operator. If the application calls for an unachievable levelof impedance, for example by creating an excessively rigid vir-tual wall, the haptic display may oscillate violently. The mannerin which the human operator contacts the haptic display alsoaffects the stability of the overall system. With improperly de-signed controls, instability may ensue when the operator graspsthe device very tightly or, conversely, oscillations may occurwhen the device is released.
In early applications of haptic display, no distinction wasmade between the virtual environment and the control law forthe haptic device. In effect, the virtual environment was the con-trol system. The stiffness and damping of a virtual wall becameproportional-plus-derivative feedback gains on device position.Some examples are found in . This approach has twomajor drawbacks. First, it is very difficult to ensure that a com-plex dynamic virtual environment translates into a stabilizingcontrol law for a haptic device. The application software mustbe tuned and tested extensively, and even then, stability is not as-sured. Second, since the virtual environment must be tuned fora specific device, the application software must be redesignedif it is to be used with a different haptic display. In other words,a haptic enabled CAD package that works properly with oneforce feedback mouse may induce instabilities when linked toa different mouse which has slightly less mechanical damping.An improved approach to control law design for haptic displayswould separate the problem of device control from the designof application software.
The use of an artificial coupling between the haptic displayand the virtual environment was first proposed by Colgateet al.. Zilles and Salisbury  suggested a similar godobjectapproach which couples a haptic device to a virtual environmentthrough a virtual spring-damper. Adams and Hannaford  putthe problem of stable haptic simulation into a two-port frame-work. The two-port approach allows for rigorous stability andperformance analysis for a very general class of haptic displays.Virtual couplings may be designed for devices with structuralflexibility, force sensing, noncollocated sensors and actuators,and measurement delay.
This paper builds upon the theoretical work presented in. A review of the two-port framework for analysis anddesign of haptic interfaces precedes the introduction of newdevelopments. Conservatism in previous results is reduced
Fig. 2. Network model of haptic simulation.
using a new technique which restricts the assumed human op-erator impedance to realistic levels. A model for the Excaliburforce display is developed, and numerical data provided for acritical design point. The work of  is extended to permitregulator and virtual coupling design for a much broader classof haptic displays. Two haptic display implementations areexplored, numerically and experimentally, for Excalibur. Thestrong agreement between theoretical and experimental resultshighlights the utility of the proposed two-port framework forhaptic interface analysis and design.
II. TWO-PORT FRAMEWORK FORSTABILITY AND CONTROL
Two-port methods are rooted in linear circuit theory, wherethey are used to characterize the effects of different loading con-ditions on two terminal electrical networks. We can consider amechanical analog to this electrical two-port, the haptic inter-face, which is subject to variable loading conditions both at thepoint of interaction with the human operator and at the pointof interaction with the virtual environment. In using this me-chanical analog, we substitute velocities for currents in repre-senting flow and forces for voltages in representing effort. Thetwo-port haptic interface model characterizes the exchange ofenergy between the human operator and the virtual environment.It is useful in studying the stability of the overall system and indescribing the performance of the haptic interface. Fig. 2 showsthe network model of haptic simulation. The human operatoraffects the velocity, , and force, , at the physical point ofcontact with the haptic display. The virtual environment modu-lates the velocity, , and force, , at the point of informationexchange with the haptic interface. The virtual environment isa digital system. The star superscript indicates that a variable isdiscrete, defined only at the time of sampling.
In the analysis and design of haptic interfaces, we are in-terested in a particular representation of the two-port system,known as animmittance matrix. A matrix , which maps theinput to output , is an immittance matrix characterization ofthe haptic interface if . There are fourways to form such a matrix. These are the impedance matrix,,the admittance matrix, , the hybrid matrix, , and the alter-nate hybrid matrix, . Any one of these forms can be formedas a algebraic combination of the elements of any of the others,provide that the matrices exist. By definition, if any one of thesematrices is positive real, then all of the others will be as well.The converse is also true. More details on two-port representa-tion can be found in , .
We will say that the overall haptic simulation isstableif, forgiven human operator and virtual environment impedance, the
ADAMS AND HANNAFORD: CONTROL LAW DESIGN FOR HAPTIC INTERFACES TO VIRTUAL REALITY 5
resulting characteristic equation has no roots in the right half-plane and only simple roots on the imaginary axis. Unfortu-
nately, we do not havea priori knowledge of the level of humanor virtual environment impedance. We would like the hapticsimulation to remain stable under any foreseeable variation. Auseful notion for handling this issue of robustness is absolutestability. A linear two-port is said to be absolutely stable if thereexists no set of passive terminating one-port impedances forwhich the system is unstable. Thus, if the haptic interface can bemade absolutely stable, the haptic simulation will remain stableas long as the human operator and virtual environment are pas-sive.
Llewellyns stability criteria provide both necessary and suf-ficient conditions for absolute stability of linear two-ports ,
has no poles in the right half-plane, only simple
poles on the imaginary axis,
Together, the two inequalities imply . The satis-faction of the absolute stability criteria for any one of the immit-tance matrix forms ( , , , or ) is necessary and sufficientfor them to hold for the other three . By demonstrating thatthe haptic interface two-port satisfies these criteria, the stabilityof the system is assured for any level of passive human operatorand virtual environment impedance.
Treating energetic interaction between the human arm anda mechanical device as passive appears to be a reasonable as-sumption. In experimental studies, Hogan found that despiteneural feedback within the arm and a high degree of adapt-ability in the neuromuscular system, the impedance exerted bya human is passive . Requiring that the virtual environmentact as a passive operator can be challenging. It is intuitive thatthe simulation of physically motivated effects (masses, springs,dampers) should obey conservation laws of physics, and thusbe passive. However, formulating numerical integration routineswhich achieve strict adherence to these laws can be difficult.Brown  showed that explicit discrete-time passive integra-tion of the equations of motion is impossible. Fortunately, expe-rience has demonstrated that absolutely stable haptic interfacesare very robust when coupled to virtual environments which arealmost passive , .
For some haptic interfaces, representing the human operatoras an arbitrary passive impedance can be overly conservative.Arbitrary passivity allows for the extreme cases of zero humanimpedance (no contact) and a perfectly rigid (infinitely stiff)human grasp. If a haptic display has very little mechanicaldamping, it becomes difficult to achieve any level of perfor-mance while maintaining absolute stability. In this case wemay want to assume some minimum level of impedance, ,which will be provided by the human operators contact withthe device. This strategy should always be coupled with someform of dead-mans switch since the system may well beunstable if the operator breaks contact with the haptic device.
Fig. 3. Human operator impedance model.
If the haptic display implementation includes force sensingat the human-device interface, the stability analysis may bedominated by the unrealistic scenario of infinitely high humanstiffness. This conservatism can be abated by postulating amaximum level of impedance for the human operator.Fig. 3 shows how the analysis model can be modified to includeminimum and maximum levels of human impedance. Wecharacterize the human operator in terms of three impedances,
, , and . is an arbitrary passive impedance function,, and . Notice that when
is zero (short circuit) the resulting human impedance is. When is infinite (open circuit), the resulting human
impedance is .The stability conditions in this paper are contingent upon
the linearity of the system and t...