On the Synthesis of Haptic Textures

Abstract

Advanced, synthetic haptic virtual environments require textured virtual surfaces. We found that texturing smooth surfaces often reduces the system passivity margin of a haptic simulation. As a result, a smooth virtual surface that can be rendered in a passive manner may loose this property once textured. We propose that any texture algorithm is associated with a characteristic number that expresses the relative change in loop gain. We further found that a passive virtual interaction can have severe unwanted artifacts if the synthesized force field is not conservative. The energy characteristics of seven algorithms are analyzed. Finally a new texture synthesis algorithm, which operates by modulating a friction force during scanning, is shown to have several advantages over previous ones.

Reprinted from Gianni Campion and Vincent Hayward, “On the Synthesis of Haptic Textures.”IEEE Transactions on Robotics, Volume 24, Number 3, 527–536, 2008.

Appendices

Appendix 1: Characteristic Number of AlgorithmF

An upper bound for$q_{\mbox{\scriptsize{\textsf{\textbf{F}}}}}$ $h(p^{x})=A\sin (2\pi p^{x}/l)$ is

(5.31)

Appendix 2: Jacobian Matrix of AlgorithmD

For the god-object method, the assumptions are:

• the boundary curveh(p ^x) is smooth and differentiable.

• there is just one point$(p^{x}_{h},p^{z}_{h})$ onh that minimizes the distance between the (p ^x,p ^z) and the boundary.

• probe is ‘inside’ the texture.

We know that:

$$ (p^z-p^z_h)=-(p^x-p^x_h)/h'(p^x_h)$$

(5.32)

while the linearization around (p ^x,p ^z) gives:

(5.33)

(5.34)

The energy function can be described by

$$ E_{\mbox{\scriptsize{\textsf{\textbf{D}}}}} = - \kappa_0[(p^x-p^x_h)^2+(p^z-p^z_h)^2]/2 .$$

(5.35)

Differentiating (5.35) gives

(5.36)

(5.37)

(5.38)

where we used Eq. (5.32), (5.33), and (5.34). The derivation of the force is complete by defining$\boldsymbol{f}_{\!\mbox{\scriptsize{\textsf{\textbf{D}}}}}(\boldsymbol{s})_{x} = 0 \mbox{ if } h'(p^{x}_{h})=0$.

The same reasoning can be used to derive

$$ \boldsymbol{f}_{\!\mbox{\scriptsize{\textsf{\textbf{D}}}}}(\boldsymbol {s})_z = -\kappa_0(p^z-p^z_h)$$

(5.39)

and$\boldsymbol{f}_{\!\mbox{\scriptsize{\textsf{\textbf{D}}}}}(\boldsymbol {s})_{z} = 0 \mbox{ if }1/h'(p^{x}_{h}) =0$. The Jacobian can be easily computed:

$$ {\mathbf{J}_{\!\boldsymbol{f}}}_{\!\mbox{\scriptsize{\textsf{\textbf{D}}}}}(\boldsymbol{s}) = -\kappa_0 \left[\begin{array}{c@{\quad}c}1-\frac{\partial p^x_h}{\partial p^x} & -\frac{\partial p^x_h}{\partial p^z}\\[4pt]-\frac{\partial p^z_h}{\partial p^x} & 1-\frac{\partial p^z_h}{\partial p^z}\end{array}\right]$$

(5.40)

hence

$${\mathbf{J}_{\!\boldsymbol{f}}}_{\!\mbox{\scriptsize{\textsf{\textbf{D}}}}}(\boldsymbol{s}) = -\kappa_0 \left[\begin{array}{c@{\quad}c}1-\frac{\partial p^x_h}{\partial p^x} & -\frac{\partial p^z_h}{\partial p^z}/h'(p^x_h)\\[4pt]-h'(p^x_h)\frac{\partial p^x_h}{\partial p^x} & 1-\frac{\partial p^z_h}{\partial p^z}\end{array}\right] .$$

(5.41)

Knowing that:

$$ \frac{\partial p^x_h}{\partial p^x}= \cos(\mathrm{atan}(h'(p^x_h)))^2=1/(1+h'(p^x_h)^2)$$

(5.42)

and

$$ \frac{\partial p^z_h}{\partial p^z}= \sin(\mathrm{atan}(h'(p^x_h)))^2=h'(p^x_h)^2/(1+h'(p^x_h)^2)$$

(5.43)

we can write

$$ {\mathbf{J}_{\!\boldsymbol{f}}}_{\!\mbox{\scriptsize{\textsf{\textbf{D}}}}}(\boldsymbol{s}) = -\frac{\kappa _0}{1+h'(p^x_h)^2} \left[\begin{array}{c@{\quad}c}h'(p^x_h)^2 & -h'(p^x_h) \\[4pt]-h'(p^x_h) & 1\end{array}\right] .$$

(5.44)

Appendix 3: Erratum to “On the Synthesis of Haptic Textures”

Reprinted from Gianni Campion and Vincent Hayward, Erratum to “On the Synthesis of Haptic Textures.”IEEE Transactions on Robotics, Volume 25, Issue 2, 475, 2008.

In Sect. IV.D.3 of reference [4] (5.5.4 of this book) it is stated that the characteristic number of the first variant of the ‘god-object’ method applied to texture synthesis is 1. However, the assumptions used to simplify the Jacobian matrix neglect the effect of curvature of the height function, hence this statement is not always true. The ‘god-object’ method generates a force field based on minimizing the distance between the interaction point,p=(p ^x,p ^z), and a point$p_{h}=(p_{h}^{x},p_{h}^{z})$ on the texture,z=h(x). In other words, there exists a functionu(x,y) such that$p_{h}^{u}, p_{h}^{h(u)}$ minimizes ‖p−p _h ‖ and such that the vectorp _h −p is normal to the curveh at pointu. In a reference frame at located atu (tangent and normal to the curve, coordinatesξ,ν, see Fig. 5.1), the complete expression of the Jacobian matrix of the force field is (the original expression was given in global coordinates)

$$\kappa_0 \left(\begin{array}{c@{\quad}c}h''(0) \nu/ (1-h''(0)\nu) & 0 \\0 & -1\end{array}\right).$$

(5.45)

Upon evaluation of the norm of this matrix, it can be found that the characteristic number is indeed 1 except when the interaction point approaches the center of curvature of the curve describing the height function. Specifically, it is

(5.46)

(5.47)

(5.48)

which has a singularity forν=1/h″(0) and tends to 1 asν→∞. In practice not all centers of curvature represent a problem. For a sine wave$A \sin(2\pi p^{x} / l)$ there is one singular point per period at (l,1−(l/2π)²). Notice that as the spatial frequency increases the singular points are closer to the surface, becoming less likely to be encountered. Moreover, the increase of stiffness happens only in the convex regions where limit cycles are less likely to occur.