Grasping kinematics from the perspective of the individual digits: a modelling study

Abstract

Grasping is a prototype of human motor coordination. Nevertheless, it is not known what determines the typical movement patterns of grasping. One way to approach this issue is by building models. We developed a model based on the movements of the individual digits. In our model the following objectives were taken into account for each digit: move smoothly to the preselected goal position on the object without hitting other surfaces, arrive at about the same time as the other digit and never move too far from the other digit. These objectives were implemented by regarding the tips of the digits as point masses with a spring between them, each attracted to its goal position and repelled from objects' surfaces. Their movements were damped. Using a single set of parameters, our model can reproduce a wider variety of experimental findings than any previous model of grasping. Apart from reproducing known effects (even the angles under which digits approach trapezoidal objects' surfaces, which no other model can explain), our model predicted that the increase in maximum grip aperture with object size should be greater for blocks than for cylinders. A survey of the literature shows that this is indeed how humans behave. The model can also adequately predict how single digit pointing movements are made. This supports the idea that grasping kinematics follow from the movements of the individual digits.

Figure 1. Example set of trajectories generated by the model.

A: Paths of the tips in the horizontal plane. B: A perspective plot of the tips' paths and of the rod on the table. C: Mean velocity profile of both tips. D: Time-course of grip aperture.

Figure 2. Simulating grasping movements to target objects of various sizes.

A: Paths of the tips when grasping a cylinder at an angle of 30 degrees. B: Paths of the tips when grasping a cylinder at an angle of 0 degrees. C: Paths of the tips when grasping a block. D: Mean velocity profiles of both tips when grasping a cylinder at an angle of 30 degrees. E: Mean velocity profiles of both tips when grasping a cylinder at an angle of 0 degrees. F: Mean velocity profiles of both tips when grasping a block. G: Aperture profiles when grasping a cylinder at an angle of 30 degrees. H: Aperture profiles when grasping a cylinder at an angle of 0 degrees. I: Aperture profiles when grasping a block.

Figure 3. The dependence of kinematic parameters of grasping on target size and shape.

A: Movement time (MT), time to maximum grip aperture (tMGA), time to maximum height (tMH) and time to maximum velocity (tMV). B: Maximum grip aperture (MGA) and maximum height (MH). C: Maximum velocity (MV). D: The slope of the relationship between MGA and target distance (Figure 5B) decreases with target size.

Figure 4. Distribution of the slopes of linear fits to the relationship between MGA and target size.

The slopes are taken from the data of 38 published studies (see text for selection). The red lines represent the values of the slopes predicted for the two object shapes by our model and the blue lines represent the single value predicted by the model of Smeets and Brenner.

Figure 5. The dependence of kinematic parameters of grasping on target distance.

The target was a cylinder or a block (width 4 cm). A: Movement time (MT), time to maximum grip aperture (tMGA), time to maximum height (tMH) and time to maximum velocity (tMV) all increase with target distance. B: Maximum grip aperture (MGA) and maximum height (MH) increase with target distance. C: Maximum velocity (MV) increases with target distance.

Figure 6. Simulating a grasping movement in the proximity of obstacles.

Solid red lines represent the condition with no obstacles. Dashed red lines represent the condition with obstacles at ‘b’ and ‘c’. A: Paths of the tips in the horizontal plane. Circles indicate possible obstacle positions. B: Velocity profiles, averaged over both tips. C: Time-course of grip aperture.

Figure 7. Grip aperture profiles during trials with constant and changing target diameters.

The black lines indicate representative trials measured by Paulignan et al. (Fig. 8 of [71]). To remove the effect of marker placement in the experimental data, we shifted the emperical curves downwards so that the minimum aperture is 0 cm. The red lines represent the grip aperture profiles generated by our model and the blue lines represent the grip aperture profiles generated by the model of Smeets and Brenner (Fig. 3 of [70]). ‘S’ and ‘L’ indicate the conditions in which the diameter was constant, 1.5 cm or 6 cm respectively. ‘S-L’ indicates the condition in which the diameter changed from 1.5 cm to 6 cm and ‘L-S’ indicates the condition in which the diameter changed from 6 to 1.5 cm.

Figure 8. Results of simulating a grasping movement with a large initial aperture.

The target was a 5.5 cm high cylinder with a diameter of 4 cm placed at a distance of 30 cm on a horizontal surface, as in the experiment by Hesse and Deubel . The simulation started with the tips 8 cm apart. A: The position profiles of the tips when simulating grasping. B: The development of aperture in time. The black line represents a single trial of a representative subject measured by Hesse and Deubel (Fig. 3d of [38]). To remove the effect of marker placement in the experimental data, we shifted the emperical curve downwards so that the initial aperture is 8 cm. The red line represents the grip aperture profile given by our model.

Figure 9. Approach angles of index finger and thumb when grasping trapezoid-shaped objects.

Black solid squares indicate the experimentally found angles (α in inset, with the standard error across subjects), reproduced from Fig. 4 of . The observed angles are more similar to the angles predicted by our model (red lines) than to the angles predicted by the model of Smeets and Brenner (blue lines) or to simple grip closure (grey open circles).

Figure 10. Paths of the tips in the horizontal plane for grasping and pointing.

In both the simulation and the experimental study (Fig. 2 of [76]) the target was a cube (sides of 5 cm) placed at a distance of 30 cm from the digits. To simulate reaching to push on the side of the cube with the index finger, a realistic position was chosen as the goal position of the thumb (1 cm to the side of and 3 cm closer than the goal position of the index finger; indicated with a grey cross).

Figure 11. Distances used in the equations.

The distances are indicated with respect to one of the tips. The distances to only one point on the surface of the target and obstacle (both indicated with d_o) are shown; the model takes into account all relevant points and their respective distances (Fig. 12).

Figure 12. The repulsive force.

Three examples to show the direction of the repulsive force and which surface areas exert a repulsive force on the tip.