**6.1. Data processing**

As for humans the cerebral cortex constructs an image of touched objects from the fragmented information provided by skin receptors, in a similar way tactile data processing is needed to enable the robot response to external stimuli. Starting from sensor data, the development of methods for touch interpretation is not an easy task.

Towards this scope, this chapter presents the basic ideas of a *force reconstruction* algorithm. The tactile stimuli applied on the outer surface of the elastomer and filtered by the soft layer, are assumed to be received by the sensors as stress components. Any tactile recognition task is haunted by the following problem: given the output of the sensor array, derive the force distribution on the outer surface. The problem is fatally an ill-posed one: in fact the set of the data is discrete (the sensor number is finite) and the set of the unknowns is a continuum (a force distribution is a vector field).

The proposed model exploits a discretization of the original continuous problem by constructing an array of concentrated forces at discrete points on the skin surface, starting from the information provided by the sensor array (normal stress measurements).

**Figure 11.** (a) A distribution of forces (not shown in the figure) acts on the cover layer. The pure normal stress component is recorded by the sensor array which is located beneath the elastomer layer. (b) From the sensor electrical outputs - through the developed algorithm – the force is reconstructed at discrete points over the outer layer.

If the force distribution is discretized into an array of point forces, a superposition of the Boussinesq equations (Section 3.1) for the single point forces can be used as a particularly simple and attractive approximation in the case of linear elasticity.

Mathematically it reduces to a linear vector equation *b = Cx*, where *C* is in general a rectangular matrix. The discretized problem consists in solving this equation for the force vector *x*.

628 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

recorded (in accordance with *d33* results reported in Section 3.1).

development of methods for touch interpretation is not an easy task.

**6. Towards system integration** 

(a force distribution is a vector field).

points over the outer layer.

**6.1. Data processing** 

The tests have been conducted by varying the amplitude of the input force for different frequency values. Figure 10 shows the measured charge at 12 Hz, 162 Hz and 362 Hz compared with the value obtained using the Boussinesq's equation (see Section 3.2). As it can be seen a good correspondence can be found. A good linearity is achieved over the whole explored range, and the same voltage-to-force behavior for different frequencies is

As for humans the cerebral cortex constructs an image of touched objects from the fragmented information provided by skin receptors, in a similar way tactile data processing is needed to enable the robot response to external stimuli. Starting from sensor data, the

Towards this scope, this chapter presents the basic ideas of a *force reconstruction* algorithm. The tactile stimuli applied on the outer surface of the elastomer and filtered by the soft layer, are assumed to be received by the sensors as stress components. Any tactile recognition task is haunted by the following problem: given the output of the sensor array, derive the force distribution on the outer surface. The problem is fatally an ill-posed one: in fact the set of the data is discrete (the sensor number is finite) and the set of the unknowns is a continuum

The proposed model exploits a discretization of the original continuous problem by constructing an array of concentrated forces at discrete points on the skin surface, starting

**Figure 11.** (a) A distribution of forces (not shown in the figure) acts on the cover layer. The pure normal stress component is recorded by the sensor array which is located beneath the elastomer layer. (b) From the sensor electrical outputs - through the developed algorithm – the force is reconstructed at discrete

If the force distribution is discretized into an array of point forces, a superposition of the Boussinesq equations (Section 3.1) for the single point forces can be used as a particularly

simple and attractive approximation in the case of linear elasticity.

from the information provided by the sensor array (normal stress measurements).

The inverse Boussinesq's problem has a direct and unique solution only if *C* is a square fullrank matrix. This would imply that the number of sensor outputs is the same as the number of force components. If we have one output per sensor (typically, the normal stress) and want to explore 3 -component surface forces, the number of points in the force array should be one third of that in the sensor array, to meet the condition. Such a solution would be rather poor in definition.

We examined a solution where the number of point forces (applied to the nodes of a predetermined grid) is equal to the number of sensors. Initial assumptions are made (some can be relaxed in further developments) in that the problem is considered to be static, the surface plane, deflections are small and the elastomer layer Poisson ratio close to 0.5. The method consists in creating a series of *x* solutions – written as a function of two scalar parameters - to the ill-posed problem. Those are built by using the Moore-Penrose pseudo inverse matrix [29] and an orthogonal projector acting on a not completely defined *w* vector which, however, fulfills the essential physical constraints (for instance, in a contact problem compressive normal forces and horizontal forces in the friction cone are expected). The eligible solution is the one which minimizes a cost functional corresponding to the norm of the difference between the generic *x* solution and the *w* vector.

**Figure 12.** 3D representation of the test case, with only the normal traction field evidenced.

A check of the algorithm was made in MATLAB on a modeled skin structure consisting in a rectangular patch 40×20 mm2 in size with a 10×10 point-like sensor grid at the bottom and 3mm thick elastomer layer (Figure 12). An hertzian force distribution with normal and tangential components (directed at 45° to the longitudinal axis of the patch) was applied to an elliptical area on the outer surface. The model was previously evaluated by a FEM computation and the normal stress at the center of each sensor was calculated.

The algorithm described above has been applied to retrieve the 3 traction components on the loading area, having the sensor stress values as inputs (*b* vector). The result concerning the sole normal component is shown in Figure 13.

**Figure 13.** Algorithm output: normal traction on the outer surface.

The shape of the distribution as well as its peak value appears to be in very good accord with the assigned data. The tangential components, which are not shown in the present context, can be also retrieved within a certain approximation.

A crucial feature of the present research is that the proposed algorithm should be efficiently implemented on digital hardware. This in turn allows for real-time implementation of tactile data processing into embedded electronics systems.

In a wider perspective, this work is intended as a first step towards the integration of different techniques for tactile information processing (e.g. computational intelligence), possibly implemented at different levels of the transmission line towards the robot central processor. In its difference from statistical approaches such as Machine Learning, the advantage of the present algorithm is that it does not require time-consuming training and data set analysis.
