Stability Analysis of Trilateral Haptic CollaborationJian Li∗Mahdi Tavakoli†Qi Huang‡∗ ‡School of Energy Science and Engineering, University of Electronic Science and Technology of China,Chengdu, Sichuan, China 611731∗ †Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, CanadaA BSTRACTThis paper presents a criterion for absolute stability of a generalclass of three-port networks. Trilateral haptic systems, which haverecently found many interesting applications, can be modeled asthree-port networks. Traditionally, existing criteria (Llewellyn’scriterion) have facilitated the stability analysis of bilateral hapticsystems modeled as two-port networks. If the same criteria wereto be used for stability analysis of a three-port network, its thirdport would need to be assumed known for it to reduce to a two-portnetwork. However, this is restrictive because, according to the def-inition of absolute stability, all three terminations of the three-portnetwork must be allowed to be arbitrary (while passive).In this paper, extending Llewellyn’s criterion, we present closed-form necessary and sufficient conditions for absolute stability of ageneral class of three-port networks – the three terminations needto be passive but are otherwise arbitrary. To this end, we firstfind a symmetrization condition under which a general asymmetricimpedance (or admittance) matrix Z 3×3 has an equivalent symmet-ric counterpart Z eq ; this Z eq models a reciprocal three-port networkwith the same stability characterization as the general nonreciprocalthree-port network modeled by Z. Then, based on the equivalenceof passivity and absolute stability for the equivalent reciprocal net-work, an absolute stability condition for the original nonreciprocalnetwork is derived. To show how the resulting absolute stability cri-terion can be utilized at the system design stage, we have applied itto the problem of designing controllers for triple-user collaborativehaptic virtual environment systems. The validity of the resultingabsolute stability conditions have been verified via simulations.Index Terms: H.5.2 [Information Interfaces and Presentation]:User Interfaces—Haptic I/O; I.2.9 [Artificial Intelligence]: Prob-lem Solving, Control Methods, and Search—Control theory1 I NTRODUCTIONFor coupled stability analysis of bilateral teleoperation system, thehuman operator’s and the environment’s dynamics and the teleop-erator immitance (z, y, h, and g) parameters are needed. Here, theteleoperator comprises the master, the slave, their controllers, andthe communication channel. In practice, the models of the humanoperator and the environment can be unknown, uncertain, and/ortime-varying. Thus, absoluteorunconditionalstabilityofabilateralteleoperator assuming that the human operator and the environmentdemonstrate passive behaviors is analyzed via Llewellyn’s stabilitycriterion for two-port networks [1, 2, 3]. For brevity, absolute orunconditional stability is simply referred to as “stability” in the restof the paper. “Coupled stability” will refer to BIBO stability of anetwork when it is coupled to terminations at all if its ports.∗ e-mail: jian1@ualberta.ca† e-mail:mahdi.tavakoli@ualberta.ca‡ e-mail:huangqi@uestc.edu.cnRecently, new application scenarios have emerged that involvethe collaboration of multiple users in teleoperation of a robot or inperforming a haptic virtual task. Examples of these new applica-tions are tele-rehabilitation [4], surgical training, [5], and cooper-ative multi-robot systems[6]. Specifically, dual-user teleoperationof a robot and triple-user collaborative haptic virtual environmentshave given rise to trilateral haptic systems. A difference between atrilateral and a bilateral haptic system is that they are modeled as athree-port and a two-port network, respectively. Thus, conventionaltheories for stability analysis of bilateral haptic systems will not beadequate for trilateral haptic systems.In contrast to the stability criteria for two-port networks, whichhave only involved conditions on the immitance parameters of thetwo-port network and are independent of the port terminations, pastresearch has been struggling to find a similar stability condition forthree-port networks independent of the port terminations. Instead,inpastresearch[7,8,9], thethirdportwasassumedtobecoupledtoa known termination such that the three-port network reduced to atwo-port network, paving the way for the application of Llewellyn’scriterion. The limiting factor of this approach is that the resultingstability condition will inevitably depend on the immitance of thethird port’s termination. This is restrictive because not allowing allthree terminations of the three-port network to be arbitrary (whilepassive) contradicts the very definition of stability (again, through-out this paper, all references are to absolute or unconditional stabil-ity).Using the aforementioned approach, namely, reducing a giventhree-port network to a two-port network by assuming a known ter-mination for the third port, Boehm et al. in [10] established nineconditions for determining the stability of a three-port network de-scribed by its scattering (S) parameters. The approach in [8] re-duced a three-port network to three two-port networks by terminat-ing each of the three ports, and managed to reduce the number ofconditions from nine to three. Also, Kuo et al. [7] reduced a three-port network to a two-port network by coupling the third port toa known termination and then required the input reflection coeffi-cients at the first and the second ports to be less than unity. Unfor-tunately, in the above approaches, a degree of freedom is lost whenthe third port is coupled to a known termination. Thus, there is aneed for a tool that can directly analyze the stability of trilateralhaptic systems modeled as three-port networks without reducingthem to two-port networks. Such a tool, which will guarantee thecoupled stability of the system under all passive but otherwise ar-bitrary terminations for all three ports, is developed in this paper.Unlike past work, we would like to have a stability condition di-rectly in the immittance (e.g., impedance Z) domain and not in thescattering (S) domain. While the S-parameters are most accuratelymeasured for higher-frequency systems such as microwave circuits,Z-parameters can be accurately measured in lower-frequency sys-tems including robotic systems. In fact, the measurement of Z-parameters approaches zero in microwave circuits where the fre-quencies are very high (over 1 GHz), making the use of reflec-tion coefficients and scattering parameters justifiable for the stabil-ity analysis. This explains the abundance of scattering parametersbased stability conditions in the microwave systems literature (see,611IEEE World Haptics Conference 201314-18 April, Daejeon, Korea978-1-4799-0088-6/13/$31.00 ©2013 IEEE
