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Research Papers

# Energy-Consistent Force Feedback Laws for Virtual Environments

[+] Author and Article Information
Arash Mohtat

e-mail: amohtat@cim.mcgill.ca

József Kövecses

e-mail: jozsef.kovecses@mcgill.ca
Department of Mechanical Engineering and Centre for Intelligent Machines,
McGill University

A system is called BIBO stable if it has bounded gain. Refer to [26] for more details.

In the absence of any physical dissipation such as viscous damping or friction, the closed loop will be only critically stable. The undamped physical mass is a simple example of such a worst-case scenario that will be used later.

In this paper we will only deal with such VEs, i.e., VEs whose continuous-time counterparts are passive.

The sign convention is consistent: Passivity violation is implied by a positive leak reflected into energy expressions through a negative sign.

The “impulsive” behavior (sudden spiky reactions) of the basic POPC is not a result of singularity. Even when the singularity is avoided in the aforementioned way, the basic POPC will still exhibit spiky reactions (see Fig. 2). This behavior originates from the very wait-then-react policy of the basic POPC, i.e., waiting for accumulation and then attempting to dissipated all the accumulated violation in a single step.

The POPC-REF, as a nonlinear operation, does not seem to lend itself to such a simple symbolic analysis. The approximate limit $κ=0.96$ is obtained via simulation and contains overestimation due to artificial dissipation of numerical solvers.

This refers to no net energy generation over the event, and is different from exact (interval-wise) passivity in its strict sense.

This convention has been satisfied in all previous derivations, e.g., the spring force is considered $Kx$ (and not $-Kx$). Figure 5(b) clearly obeys this convention.

A pure force source can be considered a worst-case scenario for impedance devices [31].

The leaks are about 400 times smaller at this rate.

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Computing and Information Science in Engineering. Manuscript received February 6, 2013; final manuscript received February 25, 2013; published online May 14, 2013. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 13(3), 031003 (May 14, 2013) (13 pages) Paper No: JCISE-13-1018; doi: 10.1115/1.4023918 History: Received February 06, 2013; Revised February 25, 2013

## Abstract

When digitally realized, virtual environments (VEs) do not perfectly match the physical environments they are supposed to emulate. This paper deals with energy aspects of such a mismatch, i.e., artificial energy leaks. A methodology is developed that employs smooth correction (SC) and leak dissipation (LD) to achieve a stable interconnection of the VE with the haptic device. The SC-LD naturally blends with the original laws for rendering the VE and gives rise to modified force feedback laws. These laws can be regarded as energy-consistent discretizations of their continuous-time counterparts. For some fundamental examples including virtual springs and masses, these laws are analytically reduced to simple closed-form equations. The methodology is then generalized to the multivariable case. Several experiments are conducted including a 2-DOF coupled nonlinear VE example, and a scenario leading to a sequence of contacts with a virtual object. Besides the conceptual advantage, simulation and experimental results demonstrate some other advantages of the SC-LD over well-known time-domain passivity methods. These advantages include improved fidelity, simpler implementation, and less susceptibility to produce impulsive/chattering response.

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## Figures

Fig. 1

The closed-loop feedback interconnection of the haptic device HHD and the target environment HT

Fig. 2

The VS corrected by different POPC methods

Fig. 3

The VS corrected by the basic SC method

Fig. 4

The VS corrected by different SC methods

Fig. 5

The VS-VM example: (a) schematic and (b) block-diagram interconnection

Fig. 6

Simulation of physical mass colliding to a VO

Fig. 7

The SC implemented in EP coordinates

Fig. 8

(a) The 2-DOF haptic device. (b) Notation.

Fig. 9

Experiment 1a results: K=250N/m at 50Hz

Fig. 10

Experiment 1b results: K = 300 N/m at 500 Hz

Fig. 11

Experiment 1b results: K = 3000 N/m at 500 Hz

Fig. 12

The contact experiment

Fig. 13

Experimental results: K = 100 N/m at 50 Hz

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