IEEE TRANSACTIONS ON ROBOTICS, VOL. 26, NO. 1, PP. 160-165, FEBRUARY 2010 1 Inﬂuence of Vibration Modes and Human Operator on the Stability of Haptic Rendering I˜ naki D´ ıaz and Jorge Juan Gil Abstract—Developing stable controllers able to display virtual objects with high stiffness is a persistent challenge in the ﬁeld of haptics. This paper addresses the effect of internal vibration modes and human operator on the maximum achievable virtual stiffness. An 11-parameter mechanical model is used to adequately characterize overall system dynamics. Experiments carried out on LHIfAM and PHANToM haptic interfaces demonstrate the importance of vibration modes to determine the critical stiffness when the user grasps the device. Index Terms—Haptics and haptic interfaces, stability, physical human- robot interaction, vibration modes I. I NTRODUCTION Haptic devices allow users to interact with a certain environment, either remote or virtual, through the sense of touch. With these mechanisms—unlike in conventional robotic systems—both the user and the device inhabit the same workspace. Therefore, stability must be guaranteed to ensure user safety and maintain proper haptic per- formance. Unfortunately, preserving haptic stability usually implies reducing the range of impedances achievable by the system. Hence, rigid virtual objects cannot be perceived as stiff as real ones, and the overall haptic performance is degraded. The critical stiffness of a haptic system depends on many factors, such as inherent interface dynamics, motor saturation, sensor resolu- tion or time delay. Several studies [1]–[3], have analyzed how these phenomena affect the stability and passivity boundaries. However, the mathematical models used to analyze stability rarely consider the existence of internal vibration modes [4], [5]. This paper presents a theoretical and experimental approach that studies the inﬂuence of the ﬁrst vibration mode of the haptic interface on the stability of haptic rendering. The ﬁrst resonant mode of the system is often linked to the ﬂexibility of the mechanical device itself, or to the cable transmission used in these types of devices. The current study also takes into account the inﬂuence of the human operator dynamics. Previous studies on impedance interactions [6], [7], based on rigid haptic models that do not include vibration modes, showed that the worst-case scenario occurs when the operator is not grasping the haptic device. However, this assumption cannot be generalized for the haptic schematic with a vibration mode. In this paper, a model of the human impedance is included and its inﬂuence on stability is evaluated using the LHIfAM [8] and PHANToM haptic interfaces. II. SYSTEM DESCRIPTION AND STABILITY ANALYSIS A. System Model without Human Operator Fig. 1a illustrates a simpliﬁed model commonly used to analyze the stability of haptic systems [3], [7]. It has a mass m and a viscous damping b, and the model assumes that the mechanical device is perfectly rigid. Although the force exerted by the motor Fr and the net force exerted by the user Fu are introduced in different places, a single transfer function is deﬁned for this model, which is G(s)= X Fr + Fu = 1 ms 2 + bs . (1) This work is supported by the Basque Government, project S-PE07TE02. I. D´ ıaz and J. J. Gil are with the Applied Mechanics Department, CEIT, Paseo Manuel Lardiz´ abal 15, E-20018 San Sebasti´ an, Spain (phone: +34 943 212 800; fax: +34 943 213 076; e-mails: {idiaz,jjgil}@ceit.es) and the Control and Electronics Department, TECNUN, University of Navarra, Paseo Manuel Lardiz´ abal 13, E-20018 San Sebasti´ an, Spain. Digital Object Identiﬁer 10.1109/TRO.2009.2037254 E B  B `  K (a) E d >  B `  1 K 1 E 1  d B  K  K d (b) Fig. 1. Mechanical schematic of a perfectly rigid haptic device (a), and a haptic device with a single vibration mode (b). Fig. 1b shows a haptic system with a single vibration mode [9]. In this model, the interface is divided into two masses connected by a link: device mass m1, pushed by the force of the motor, and device mass m2, pushed by the user. The dynamic properties of their connection are characterized by a spring and a damper (kc and bc). Depending on the nature of the vibration mode, the physical interpretation of m1 and m2 changes: mass m1 could be the rotor inertia if the link is modeling the transmission system (usually a steel cable), or an arbitrary mass if the vibration mode is due to the ﬂexibility of the mechanism. This model is a two-input/two-output system and the relationship between output positions and input forces is x =  X2 X1  =  G2(s) Gc(s) Gc(s) G1(s)  Fu Fr  = Gf . (2) Three new transfer functions have been deﬁned: G = 1 p(s)  p1(s) kc + bcs kc + bcs p2(s)  (3) where, p1(s)= m1s 2 +(b1 + bc)s + kc, (4) p2(s)= m2s 2 +(b2 + bc)s + kc, (5) p(s)= p1(s)p2(s) - (kc + bcs) 2 . (6) Introducing an impedance interaction with the virtual environment, the device can be analyzed as a single-input/single-output system, as illustrated in Fig. 2. C(z) is the force model of the virtual contact (which usually includes a spring and a damper) and H(s) is the zero- order-holder. T is the sampling period and t d represents the delay in the loop (e.g., lag caused by ampliﬁer circuitry). The sampled position of the motor is given by X ∗ 1 = Z[Gc(s)Fu(s)] 1+ C(z)Z[H(s)G1(s)e -t d s ] . (7) If the force model only has a virtual spring with stiffness K, sta- bility of the system depends on the following characteristic equation: 1+ KZ[H(s)G1(s)e -t d s ]=0, (8) and the critical stiffness is KCR = Gm{Z[H(s)G1(s)e -t d s ]}, (9) where Gm{.} means gain margin of the transfer function within brackets. From (9) it follows that G1(s) is the relevant transfer function for the stability of the system. B. System Model Including the Human Operator Human impedance models are usually characterized by a second- order linear time-invariant system (mass-spring-damper) [10]–[13]. The model assumes that the human operator has a perfectly rigid grasp on the manipulator’s end-effector. This assumption may not be valid depending on the type of grasp or tension applied to it by