A haptic system is a robotic computer interface which aims to provide tactile feedback for human operators when they manipulate virtual environments (VEs) or remote environments (REs). The tactile feedback is emulated by applying forces, vibrations, or motions to the human users through a haptic device/interface, e.g. a robot arm. Transparency and stability are two important criteria for designing a haptic system. Transparency is related to the realism of user's touch sensation and stability guarantees the safety of the user while interacting with VEs/REs. Because of the nature of the human tactile sensory system, a transparent haptic system demands an update rate greater than 500 Hz, i.e. most commercial haptic devices work at 1 KHz. On the other hand, many haptic applications are multirate systems. The multirate property of a haptic system is due to either the slow update rate of the VE or the impairments of computer networks such as limited transmission bandwidth or packet loss.

Wave transformation is wildly used in teleoperation to cope with both constant and varying time delays. This work aims to use wave transformation to tackle the challenges imposed by multirate property of a haptic system. First, passive multirate wave variables control (PMWVC) is introduced. PMWVC guarantees the passivity of the communication channels through which the fast haptic device is connected to the slow VE/RE. It is shown that to maintain the passivity of the system, aliasing should be avoided in the communication channels, i.e. by using anti-aliasing filters.

Next, PMWVC strategy is applied to two different applications:​ i) multiuser cooperative haptics and ii) haptic interaction with an unknown VE. In the first application, two users at two different locations manipulate a common virtual object simulated on a central server. The users are connected to the central server through a LAN network. The second application is a single user application in which PMWVC is used to connect the haptic device to an unknown slowly updated VE. Since in this application the VE is unknown, the computational delay of the VE significantly affects the stability of the overall system. To tackle this problem, a nonlinear algorithm based on passivity analysis is proposed. In both examples, numerical and experimental results validating the analytical results are provided. The results show that by using PMWVC, it is possible to significantly improve the performance of a multirate haptic system in terms of transparency and stability.

The second half of this work is devoted to improving the performance of PMWVC in all frequency ranges. In order to study the performance of PMWVC, lifting is used to convert the multirate haptic system to a unirate system. By using this technique, it is shown that velocity estimation plays a critical role in a haptic application with PMWVC, especially in high frequencies. Considering this fact, a method for designing a passive velocity filter in wave domain is proposed.

Finally, a filter bank structure is introduced which enables utilizing a local model in conjunction with PMWVC. In this structure, the outgoing signal sent to the VE is split into two frequency ranges. Low frequency content of the signal is fed to the original VE and high frequency content of the signal is sent to the local model. By using lifting the performance of the proposed structure is studied. The results show that the proposed method improves the transparency of the system in all frequency ranges and unlike utilizing a local model in power domain, it does not impose any restriction on the stability of the system.