Contour based object-compliant shape control

\(E_{\mathrm{MSN}}\) energy can be used to formulate both shape error \(E_{\mathrm{MSN}}(\gamma(t),\bar{\gamma})\) and deformation cost \(E_{\mathrm{MSN}}(\gamma(t_0),\gamma(t))\). One may suggest combining both in a shape control strategy that seeks reducing an object-compliant (OC) energy: \[ E_{\mathrm{OC}}(\gamma(t),\bar{\gamma})=E_{\mathrm{MSN}}(\gamma(t),\bar{\gamma})+\beta E_{\mathrm{MSN}}(\gamma(t_0),\gamma(t)), \tag{8} \] where parameter \(\beta\geq0\) allows to weigh the deformation cost with respect to the shape error of the control task. Equation (8) constitutes the projection of two optimisation functions onto a single one \(E_{\mathrm{OC}}\). Since both cost functions are defined through the same metric \(E_{\mathrm{MSN}}\), (8) can be also regarded as shape-evolution path (parameterised by \(\beta\)) in the shape space defined by the shape representation (1). This feature means that establishing a specific value of \(\beta\) in (8) is equivalent to defining an new target shape \(\gamma_\beta\) in-between \(\gamma(t_0)\) and \(\bar{\gamma}\), resulting from an interpolation in the shape representation space defined by (1). With these considerations (8) can be expressed as: \[ E_{\mathrm{OC}}(\gamma(t),\bar{\gamma})=E_{\mathrm{MSN}}(\gamma(t),\gamma_\beta). \tag{9} \] The major issue with (9) is that it does not consider per se how the object is deforming, but rather defining a closer reference \(\gamma_\beta\) in shape space. One could try to highly penalise deforming away from \(\gamma(t_0)\) by establishing very high values of \(\beta\) in (8). That would lead to re-defining a target shape \(\gamma_\beta\) in the neighbourhood of \(\gamma(t_0)\) (locally in shape space). Very close-by target shapes \(\gamma_\beta\) might imply a worse performance in shape control tasks. On the other hand, if we relax the deformation cost and establish low values of \(\beta\), the obtained reference shapes \(\gamma_\beta\) still do not guarantee an object-compliant shape evolution path towards \(\bar{\gamma}\).

For this reason, we propose to exploit the multi-scale nature of \(E_{\mathrm{MSN}}\) and define the following object-compliant energy:

\[ E_{\mathrm{OC}}(\gamma(t),\bar{\gamma})=E_{\mathrm{MSN}}(\gamma(t),\bar{\gamma}) \\ +\beta\, \begin{matrix} \\ \text{min} \\ \Pi \end{matrix} \int_{0}^{l(t)}\int_{0}^{r_{\mathrm{OC}}}\|\mathbf{l}(s,r,t)-\mathbf{l}(\Pi(s,t),r,t_0)\|^2\mathrm{d}r\mathrm{d}s, \tag{10} \] where the deformation cost (second term) is defined within a specific scale range \(r\in (0,r_{\mathrm{OC}}]\) (being \(0<r_{\mathrm{OC}}\leq r_{\rm{max}}\)). As the scale value \(r_{\mathrm{OC}}\rightarrow0\), the second term in the right hand side of (10) becomes negligible thus (10) turns equivalent to (3). On the other hand, if \(r_{\mathrm{OC}}=r_{\rm{max}}\), (10) is equivalent to (8). That is, rather than disregarding the deformation cost in the formulation (as (3) does) or defining a specific intermediate shape state between two shapes like (8), the object-compliant energy \(E_{\mathrm{OC}}\) in (10) penalises changes in length and curvature on a specifically targeted range of scales.

This formulation allows to establish high penalisation values to the deformation cost (i.e., establish very high values for \(\beta\)) while not over-constraining the set of low deformation paths. If the target shape \(\bar{\gamma}\) is within (or nearby) any of the shape evolution paths defined by (10), deformations can be highly penalised while still allowing to significantly reduce the shape control error. The choice of \(r_{\mathrm{OC}}\) is closely related to the rigidness of the deformable object. If the object does not present a very high rigidness (e.g., a chewing gum), \(r_{\mathrm{OC}}\) can take very low values whereas objects with higher rigidness (e.g., a cardboard piece) will benefit from higher \(r_{\mathrm{OC}}\) values that penalise large changes in both local and larger-scale curvatures.

A global strain measure could be approximated through the change in the total length of the object \(e(t)={l(t)}/{l(t_0)}\). However, strain \(e(t)\) does not provide a reliable measure when deformations are unevenly distributed over the object. We therefore require a local measure of the strain that allows to detect strain limits being reached in specific object parts.

Under the assumption of isotropic behaviour, we propose characterising the object's local strain through the evolution of the elastic map \(\Pi_\epsilon\) defined between its current state \(\gamma(t)\) and its initial state \(\gamma(t_0)\): \[ \Pi_\epsilon=\begin{matrix} \\ \text{arg min} \\ \Pi_\epsilon \end{matrix} \int_{0}^{l(t)}\int_{0}^{r_{\rm{max}}}\|\mathbf{l}(s,r,t)-\mathbf{l}(\Pi_\epsilon(s,t),r,t_0)\|^2\mathrm{d}r\mathrm{d}s. \tag{11} \] That is, we approximate the local strain \(\epsilon(s,t)\) as: \[ \epsilon(s,t)\approx \left. \frac{\partial\Pi_\epsilon(s,t)}{\partial s}\right |_t . \tag{12} \] Partial derivative \(\partial\Pi_\epsilon(s,t)/\partial s\) provides a measure of local change in the relative density of the object (with respect to its density at \(t_0\)). If \(\epsilon(s,t)=1\) the object mass distribution at point \(s\) is the same as in the initial time instant. On the other hand, if \(\epsilon(s,t)>1\) or \(\epsilon(s,t)<1\) the object has been respectively stretched or compressed at \(s\), respectively.