**7. Applications**

186 Haptics Rendering and Applications

1 1 <sup>2</sup> *n n nn t u u uu*

<sup>1</sup> 2 *<sup>n</sup> n nn*

Zhuang's mass concentration method is employed to approximate the distributed mass in real-time (Zhuang & Canny, 2000). To avoid inverting a large sparse matrix at each time step, Zhuang's method approximates the force matrix M with a diagonal matrix. Diagonalization results in approximating the mass continuum as concentrated masses at each nodal point of the finite element model. This simplifies the nonlinear system of equations used to form a set of independent algebraic equations that can be solved with

A prototype system has been implemented. Although the FE model of the spine is a simplified beam model, the updating rate of FEM evaluation is still far below the 1000 Hz required for effective vibration-free haptic rendering. The FE model has 472 elements and 473 nodes and the time step of the explicit integration is 0.02 second. The updating rate is approximately 200 Hz using a PC graphic workstation. To avoid vibration due to the updating rate discrepancy, a force interpolation method (Zhuang & Canny, 2000) was adopted to smoothen the force feedback. Haptic feedback between two simulated states is linearly interpolated at a suitably high frequency. However, at time tn, we do not have the information of tn+1. Zhuang's solution uses an intentional delay of an FEM deformation time step, up to 1/100s. User experimentation found that such a small lag in time is within the

Different from the finite element spine model whose locomotion is directly calculated in the haptic simulation, the multi-body spine model in the compliant stage utilize the preinterpolated displacement-force functions of all vertebrae to conveniently compute the movement of the spine. Based on the magnitude of the haptic external forces applied by the users, dynamic properties of all vertebrae can be easily calculated via these functions and

The purpose of interpolating displacement-force functions of all vertebrae is to facilitate computational process of dynamic properties of the spine model during haptic simulation. To do that, some constraints are imposed on the spine model. The pelvis is fixed in 3D space. Then, constant forces are applied on each specific vertebra in the thoracic region in each axisaligned direction during simulation. The force magnitude is gradually increased with an equal increment in subsequent simulations. Corresponding to each value of force, dynamic characteristics of all vertebrae (e.g. translation, rotation) can be automatically obtained using the plots in LifeMOD as a reference. The dynamic properties are recorded after the spine model is stabilized. Based on these recorded dynamic properties, displacement-force

tolerance of human perception for virtual interaction with soft objects.

the locomotion of the whole spine model can be rapidly observed.

(8)

*<sup>u</sup> u u ut t* (9)

11 1 1 ( ) *Mu F R u Du nn n n* (10)

2

The other 4 terms of the strain are defined similarly. The algorithm equations are:

relatively high updating rate.

**6.3.2 Multi-body spine model** 
