Grip forces during fast point-to-point and continuous hand movements

Abstract

Three experiments investigated the grip force exerted by the fingers on an object displaced actively in the near-body space. In one condition (unimanual) the object was held by one hand with the tripod grip and was moved briskly back and forth along one of the three coordinate directions (up–down, left–right, near–far). In the second condition (bimanual) the same point-to-point movements were performed while holding the object with the index and middle fingers of both hands. In the third condition (bimanual) the object was held as in the second condition and moved along a circular path lying in one of the three coordinate planes (horizontal, frontal, sagittal). In all conditions participants were asked to exert a baseline level of grip force largely exceeding the safety margin against slippage. Both grip forces and hand displacements were measured with high accuracy. As reported in previous studies, in the two point-to-point conditions we observed an upsurge of the grip force at the onset and at the end the movements. However, the timing of the transient increases of the grip force relative to hand kinematics did not confirm the hypothesis set forth by several previous studies that grip modulation is a pre-planned action based on an internal model of the expected effects of the movement. In the third condition, the systematic modulation of the grip force also for circular movements was again at variance with the internal model hypothesis because it cannot be construed as a pre-planned action aiming at countering large changes in dynamic load. We argue that a parsimonious account of the covariations of load and grip forces can be offered by taking into account the visco-elastic properties of the neuromuscular system.

Appendix

We describe a simple mechanical model for simulating the experimental results. The central assumption is that the end-point position P is determined by the opposing forces generated by two mass-spring systems, where the springs have constant stiffness K and controllable resting lengths L ₀ (Fig. 10a). Displacements of the end-point position P are generated by modulating appropriately the resting lengths. The scheme described here applies directly to the case where movement and grip forces are aligned. When they are not aligned (e.g., for vertical point-to-point movements) we assume that two such schemes are at work, one responsible for the displacement, the other for holding the manipulandum. Thus, as argued in the Discussion, the grip modulations measured in the holding scheme—within which the frame is not moving—are the indirect reflection of the modulations occurring in the moving scheme. A similar assumption is made in the case of circular motions where the role of the two schemes switches every half-period of the motion.

Fig. 10

Modeling the performance. a A mass under the action of two opposite elastic forces F = K(x − L ₀) can be moved from one equilibrium position P′ to a different equilibrium position P″ by controlling the resting length L ₀ of one or both elastic elements, while keeping constant their stiffness. b A simple mechanical system. The displacement x(t) of a mass M is driven by controlling independently the effective lengths X ₁(t) = L − L ₀₁(t) and X ₂(t) = L − L ₀₂(t) of each spring. Because of the way the springs are attached to the mass, the total compressing force acting on the mass is G _f(t) = K ₁ X ₁(t) + K ₂ X ₂(t) + x(t)(K ₂ − K ₁)

The system is depicted in Fig. 10b. A mass M moves under the joint pull of the springs with stiffness K ₁ and K ₂. The outer ends of the springs are attached at a fixed distance L from the 0 reference and time-varying resting length L ₀₁(t) and L ₀₂(t). Motion is damped by a linear viscous damper with coefficient C. The input to the system are the effective lengths of the springs, defined as X ₁(t) = L − L ₀₁(t) and X ₂(t) = L − L ₀₂(t). The equation of the motion is:

$$M\frac{{{\text{d}}^{2} x(t)}}{{{\text{d}}t^{2} }}\;\; + \;C\frac{{{\text{d}}\,x(t)}}{{{\text{d}}t}}\; + \;(K_{1} \, + \,K_{2} ){\kern 1pt} {\kern 1pt} x(t)\; = \;K_{1} {\kern 1pt} X_{1} (t) - K_{2} \,X_{2} (t)\; = \;F\left( t \right)\quad x(0) = 0,\;\left. {\frac{{{\text{d}}\,x(t)}}{{{\text{d}}t}}} \right|_{t = 0} = 0$$

where x(t) is the signed distance of the mass for the 0 reference. Because of the way the springs are attached, the grip force acting on the mass during the motion is: G _f(t) = K ₁ X ₁(t) + K ₂ X ₂(t) + x(t)·(K ₂ − K ₁).

To simulate the results for point-to-point linear movements, X ₁(t) and X ₂(t) were modeled by two sequences of low-pass-filtered pulses with the same period (T = 2.4 s), amplitude and baseline, each modulating in opposite directions the force exerted by the springs. The leading and trailing edges of the pulses were modeled by generalized sigmoid functions:

$$\begin{aligned} {\text{Leading edge}}{:}\,{\text{LE}}\left( t \right) & = \frac{{1 - \exp \left[ {\frac{1}{{\sigma \left( {\,t - \mu } \right)^{\alpha } }}} \right]}}{{1 + \exp \left[ {\frac{1}{{\sigma \left( {\,t - \mu } \right)^{\alpha } }}} \right]}}\quad [0 \le {\text{LE}}\left( t \right) \le 1; \, t \ge \mu ] \\ {\text{Trailing edge}}{:}\,{\text{TE}}\left( t \right) & = 1 - {\text{LE}}\left( t \right) = \frac{2}{{\exp \left[ {\frac{1}{{\sigma \left( {\,t - \mu } \right)^{\alpha } }}} \right] + 1}} - 1 \quad \left[ {0 \le {\text{TE}}\left( t \right) \le 1;t \ge \mu } \right] \\ \end{aligned}$$

Figure 11 shows a normalized representation of the individual pulses X ₁(t) and X ₂(t) (baseline not shown). We assumed K ₁ = K ₂. Thus, the grip force is proportional to the sum X ₁(t) + X ₂(t), which is also shown in the upper part of the figure. In this scheme grip modulations emerge from the overlap between the leading edge of a pulse and the trailing edge of the previous pulse pulling in the opposite direction. Pulse amplitude and baseline were set to reproduce the average grip force and the prescribed displacement amplitude (40 cm). The grip force time course was simulated by choosing appropriately the slope difference for pulses in opposite directions. The kinematics of the mass was obtained by solving the equation of the motion with the best-fitting parameters M, C and K (see below). Figure 12 compares the simulation with the actual data in the case of transversal U-grip motions (see Fig. 2) where the peaks of the grip force (upper panel) and the acceleration profiles (lower panel) in the two phases of the movement were significantly different.

Fig. 11

Lower panel: schematic representation of the driving input to the mechanical system depicted in Fig. 10. The modulations of the effective lengths X ₁(t) = L − L ₀₁(t) and X ₂(t) = L − L ₀₂(t) (colored lines) are modeled by a sequence of low-pass-filtered rectangular pulses. The net force acting on the mass is F(t) = K ₁ X ₁(t) − K ₂ X ₂(t). Upper panel the sum K ₁ X ₁(t) + K ₂ X ₂(t) (black line). When K ₁ = K ₂, this sum is equal to the grip force. The scheme shows how grip forces arise because of the overlap between the trailing edge of an impulse and the leading edge of the following impulse. The example illustrates a situation where the overlap for left-to-right movements is larger than the one in the opposite direction giving rise to a larger grip pulse (see Fig. 2). Scales for X ₁(t) and X ₂(t) and for the grip force are in arbitrary units

Fig. 12

Point-to-point movements. U-grip with holding frame, movements in the transversal (X) direction. Comparison of real (black dots) and simulated (red lines) performance. The data points are an undersampled version of the experimental results already shown in Fig. 2a. a Grip force. The indicated average is relative to the simulation. b Displacement and acceleration

The model behavior was also compared with the results for the two main components of circular movements. We assumed that movement trajectories are generated by a combination of two mechanical systems as the one in Fig. 10b acting along orthogonal axes. For one system the effective lengths vary as X ₁(t) = A _x sin(ωt) + B _x and X ₂(t) = A _x sin(ωt + θ _x ) + B _x . For the orthogonal system they vary as Y ₁(t) = A _y cos(ωt) + B _y and Y ₂(t) = A _y cos(ωt + θ _y ) + B _y (ω = 2π/T). Thus, for the X-axis, the driving force K ₁ X ₁ − K ₂ X ₂ is an harmonic function F sin(ωt + ψ) where

$$F = A_{x} \sqrt {K_{1}^{2} - 2K_{1} K_{2} \cos \left( {\theta_{x} } \right) + K_{2}^{2} } \quad tg\left( \psi \right) = \frac{{K_{2} \sin \left( {\theta_{x} } \right)}}{{K_{2} \cos \left( {\theta_{x} } \right) - K_{1} }}$$

As in the case of rectilinear movements, grip forces emerge because the phase difference θ produces a partial overlap between the components K ₁ X ₁ and K ₂ K ₂ of the driving force. The experimental results were simulated by making again for each axis separately the simplifying assumption K ₁ = K ₂ = K, so that the grip force is GF(t) = K(X ₁ + X ₂) = Gsin(ωt + φ) where

$$G = A{\kern 1pt} K\,\sqrt {2 + 2\,\cos \left( {\theta_{x} } \right)} \quad tg\left( \varphi \right) = \frac{{\sin \left( {\theta_{x} } \right)}}{{1 + \cos \left( {\theta_{x} } \right)}}$$

Because trajectories were not perfectly circular, the amplitude parameter A was estimated independently for each axis from the data. Then, we determined the model parameters affording the best fit to both the actual grip force profile and to the kinematics of the movement. Figure 13 compares experimental and simulated data in the case of the X-axis for movements in the frontal (X–Y) plane. The approximation was equally good for the Y axis in the frontal plane and for both X- and Z-axis for movements in the horizontal plane. Our scheme assumes that movements were strictly planar. Therefore, it cannot account for grip modulations measured in the sagittal plane (Figs. 8c, 9c), which may in part reflect the significant deviations of the movement from planarity.

Fig. 13

Circular movements. B-grip with holding frame, Transversal (X) component of the movement in the frontal (XY) plane. Comparison of real (black dots) and simulated (red lines) performance. The data points are an undersampled version of the experimental results already shown in Fig. 9c. a Grip force. The indicated average is relative to the simulation. b Displacement and acceleration

The driving force is proportional to the stiffness K. Thus, in fitting the simulation to the data only the ratios C/M and K/M can be specified independently. However, we verified that the stiffness values required to mimic grip forces in linear and circular movements are at least realistic. From the average body mass for individuals in the age range of the participants (M _b  = 78.24 kg, Ogden et al. 2004), and the average ratio between arm and body mass (M _a /M _b  = 0.062, Martin and Chaffin 1972), one obtains: M _a  = 4.85 kg. For linear U-grip movements, the fitting shown in Fig. 12 required a ratio K/M = 138.9, yielding an estimated stiffness K = 6736 N/m. The required ratio C/M = 9.69 yielded the estimate C = 470 N s/m. For circular movements (Fig. 13) the fitting required K/M = 55.56 and C/M = 8.33. Because both arms were involved, we assumed that the moving mass was twice as large as in single arm movements. This yielded an estimated equivalent stiffness K = 5384 N/m and an estimated equivalent viscosity C = 808 N s/m. Though approximate, stiffness estimates are well in keeping with those reported by Hu et al. (2012) for maximally stiffened arms in the horizontal plane.