Contact geometry and mechanics predict friction forces during tactile surface exploration

When we touch an object, complex frictional forces are produced, aiding us in perceiving surface features that help to identify the object at hand, and also facilitating grasping and manipulation. However, even during controlled tactile exploration, sliding friction forces fluctuate greatly, and it is unclear how they relate to the surface topography or mechanics of contact with the finger. We investigated the sliding contact between the finger and different relief surfaces, using high-speed video and force measurements. Informed by these experiments, we developed a friction force model that accounts for surface shape and contact mechanical effects, and is able to predict sliding friction forces for different surfaces and exploration speeds. We also observed that local regions of disconnection between the finger and surface develop near high relief features, due to the stiffness of the finger tissues. Every tested surface had regions that were never contacted by the finger; we refer to these as “tactile blind spots”. The results elucidate friction force production during tactile exploration, may aid efforts to connect sensory and motor function of the hand to properties of touched objects, and provide crucial knowledge to inform the rendering of realistic experiences of touch contact in virtual reality.

. Lengths L i of finger-surface contact regions, determined from the video capture and analysis, as a function of the fingertip position. Mean of 15 trials L 1 in green, Mean of 15 trials L 2 in red and Mean of 15 trials L 1 in blue. Shaded regions: 1 s.d. Step Step Step Step Step

Supplementary data 2: Nonlinear force components
We investigated whether the model predictions would improve if we allowed for a nonlinear polynomial dependence of the deformation component on the surface slope, dh(x)/dx. To this end, we replaced p 1 dh(x)/dx in Equation (4) by polynomial functions, with terms p n (dh(x)/dx) n of successively higher powers, n up to N, where N is the model order. Fit quality is assessed via the normalized mean square error ε(F T ,F T ) (NMSE) between the measured force F T and the model estimateF T , where Higher quality fits were achieved with increasing polynomial order, but the best results, as measured by normalized root mean square error ε between the model output and measurements, were nearly achieved by a linear model. The errors for models of quadratic order or higher in dh(x)/dx were negligibly better (Fig. S2). A model of "zeroth" order, which omitted the deformation contribution to sliding friction forces, yielded errors that were 150% higher (ε = 1.27 vs. 0.5). Supplementary Information continues on the next page.

2/5 Supplementary data 3: Force Model decomposition
The forces as predicted by the model were the sum of two components F int and F de f accounting for interfacial shear frictional force and force caused by deformation respectively (Fig. S3). Figure S3. Total force computed by the model and the partial components F int and F de f . The force caused by fingertip deformation F de f is one order of magnitude larger in the bump and step up surfaces. The shear force estimated F int has comparable level to that of F de f only on the step down surfaces. Force estimatesF T , F int , and F de f in blue and force measurements F T in black. Mean in solid lines, 1 standard deviation in shaded colors.

3/5 Supplementary data 4: Model parameter distributions
The force components F int and F de f were associated with two parameters p 0 , a pressure term, and p 1 , which weighted the importance of deformation. Their values were estimated from the data in each trial, accounting for trial to trial variations in contact pressure and force (Fig. S4). The model also depended on a friction coefficient, µ, which was estimated once for each participant, yielding µ = 0.55 for subject 1, and µ = 0.6 for subject 2. By cleaning the surfaces and applying talc, we controlled friction forces during the experiment.

Supplementary data 5: Fingertip average sliding speed
Subjects were instructed to slide their fingertips on the surfaces with specified sliding speed (40 mm/s, 80 mm/s, or 120 mm/s). The speed varied due to motor behavior and contact with relief features on the surface. We assessed the speeds by measuring the time, t s , that it took the fingertip to slide across the middle 24 mm of the surface during each trial, and computed the average sliding speed as v = 24/t s . The results (Fig. S5) show show that the average speeds varied around the specified speed, but preserved a general rank ordering. Superimposing forces captured at different sliding speeds on each surface reveals that if there is an effect of speed on forces, it is small and inconsistent ( Figure S6; same data as shown in Figure 4). Figure S6. Measured forces F T (x) grouped by surface and subject (15 trials per case). Mean of 5 trials at 40 mm/s in red, mean of 5 slides at 80 mm/s in blue and mean of 5 trials at 120 mm/s in orange. Shaded regions: 1 standard deviation.

Supplementary data 7: Analytical form of the surface shapes
The surfaces were fabricated using electrical discharge machining, yielding a smooth finish with specified geometry. The height h(x) varied along the length of the surface according to the following expression.