Adjusting Output-Limiter for Stable Haptic Interaction with Deformable Objects Kyungno Lee * and Doo Yong Lee * , Senior Member, IEEE * Department of Mechanical Engineering, KAIST, Daejeon, Republic of Korea {Junis, lee.dooyong}@kaist.ac.kr Abstract—This paper presents a control method using an adjusting output-limiter for stable haptic rendering in a virtual environment. In a simulation of force-reflecting interaction with deformable objects in a virtual environment, a quick computation of the accurate impedance of deformable objects is rare. This is particularly true when physics-based models, such as tensor-mass models or mass-spring models, are used. The problem is aggravated if the simulation involves changes in the geometry and/or impedance of the deformation model, such as cutting or suturing. The proposed control method guarantees stable haptic interactions with deformable objects of unknown and/or varying impedance. The method is based on the time-domain passivity theorem and the two- port network model. The controller adjusts the maximum permissible force to guarantee the passivity of the haptic system at every sampling instant. The controller notes only the magnitude of the reflective force, and does not depend on properties of the employed force model. This allows the proposed control method applicable to haptic systems involving deformable objects with unknown, nonlinear, and/or time-varying impedance. Designs of the controllers are presented for impedance-type and admittance-type haptic systems. The method is also extended for multiple degrees-of-freedom. Keywords: Haptic control; Haptic rendering; Passivity I. INTRODUCTION Multimodal interaction involving haptic and visual feedback enhances the fidelity and experience of the simulation in a virtual environment. It is important in haptic rendering to generate a realistic reflective force in real time. Medical simulation [1], [2] often requires physics-based deformation models such as finite-element models and mass-spring models to display realistic deformation and reflective force. Estimation methods [3], [4], [5] exploiting previous positions and forces are used for real-time computation of a reflective force, due to the low update rate that results from the computational complexity of the deformation model. Estimation error can jeopardize the stability of the haptic system. Moreover, the impedance of the deformation model is comparatively small, nonlinear, time-varying and often unknown. It is difficult to compute accurate impedance values for deformable objects in real time. This problem is magnified if the simulation involves changes in the geometry and/or impedance of the deformation model, such as cutting or suturing. Hence, it is difficult to guarantee the stable haptic interactions with the deformation model. The controller is necessary to maintain the fidelity of the reflective force while guaranteeing the stability. Barbagli et al. [6] described an adaptive local haptic model that allowed users to interact haptically with a deformable object simulation featuring computational delays and low servo rates. Asymptotic stability is guaranteed when the stiffness of the local haptic model is smaller than that of the deformable object at a contact point; however, it is necessary for the real stiffness of deformable objects to be estimated. Mahvash and Hayward [7] propose a passivity condition considering the combined effects of zero-order-hold and computational delay. The passive-force response is synthesized from the local force fields and the passivity condition. It can be applied when the impedance of the deformable object is known beforehand. Control methods based on a virtual coupling [8], [9] can guarantee the stable haptic interaction with a rigid body, regardless of the impedance of a rigid object. However, the virtual coupling cannot decouple an operator from a virtual environment when the impedance of the virtual environment is small. It hinders the operator from feeling realistic impedance from the deformation model, which degrades the fidelity of the reflective force. Hannaford and Ryu [10] employ a time-domain passivity strategy that has the potential to generate a large impulse. Time-domain passivity control with reference energy following [11] is proposed to make the output of the passivity controller smooth. It is difficult to compute the correct reference energy because the update rate of the physics-based deformation model is low and the correct reflective force is unknown during the intersample period. Stramigioli et al. [12] propose a port-Hamiltonian approach to track and dissipate energy excess, but it is not clear how it can be applied to nonlinear multidimensional virtual environments. The energy bounding algorithm [13] makes the system stable, regardless of the range of the impedance and the sampling frequency; however, it deteriorates the fidelity of the reflective force, because it controls the reflective force computed from the physics- based deformation model at every sampling instant. An adjusting output-limiter (AOL) is presented in this paper for stable force-reflection in virtual environments. The AOL is developed based on the passivity theorem and on a two-port network model. It monitors the maximum permissible force to guarantee the passivity of the haptic system at every sampling instant. In addition, the AOL adjusts the reflective force for the passivity while sacrificing the fidelity of the reflective force only if the force computed from the deformation model is larger than the maximum permissible force. Otherwise, the high fidelity of the reflective force is maintained. The AOL can maintain the stability of the haptic system regardless of 3URFHHGLQJV RI WKH WK 0HGLWHUUDQHDQ &RQIHUHQFH RQ &RQWURO  $XWRPDWLRQ -XO\     $WKHQV  *UHHFH 7