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Grip-force modulation in multi-finger prehension during wrist flexion and extension

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Abstract

Extrinsic digit muscles contribute to both fingertip forces and wrist movements (FDP and FPL–flexion, EDC–extension). Hence, it is expected that finger forces depend on the wrist movement and position. We investigated the relation between grip force and wrist kinematics to examine whether and how the force (1) scales with wrist flexion–extension (FE) angle and (2) can be predicted from accelerations induced during FE movement. In one experiment, subjects naturally held an instrumented handle using a prismatic grasp and performed very slow FE movements. In another experiment, the same movement was performed cyclically at three prescribed frequencies. In quasistatic conditions, the grip force remained constant over the majority of the wrist range of motion. During the cyclic movements, the grip force changed. The changes were described with a linear regression model that represents the thumb and virtual finger (VF = four fingers combined) normal forces as the sum of the effects of the object’s tangential and radial accelerations and an object-weight-dependent constant term. The model explained 99 % of the variability in the data. The independence of the grip force from wrist position agrees with the theory that the thumb and VF forces are controlled with two neural variables that encode referent coordinates for each digit while accounting for changes in the position dependence of muscle forces, rather than a single neural variable like referent aperture. The results of the cyclical movement study extend the principle of superposition (some complex actions can be decomposed into independently controlled elemental actions) for a motor task involving simultaneous grip-force exertion and wrist motion with significant length changes of the grip-force-producing muscles.

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Acknowledgments

The study was supported in part by NIH grants AG-018751, NS-035032, and AR-048563. We thank the reviewers for their insightful comments.

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Correspondence to Satyajit S. Ambike.

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Appendices

Appendix 1

Estimation of unmeasured kinematic variables using Newton’s second law

Figure 2 depicts three coordinate frames located at the handle’s CG. The first frame \(\left\{ {\hat{e}_{x} - \hat{e}_{y} - \hat{e}_{z} } \right\}\) is fixed to the handle. The second frame \(\left\{ {\hat{e}_{r} - \hat{e}_{t} - \hat{e}_{{x^{\prime } }} } \right\}\) is attached to the lever of the support mechanism. Along the lever is the vector \(\hat{e}_{r} ,\hat{e}_{{x^{\prime } }}\) points vertically upwards, and \(\hat{e}_{t}\) is perpendicular to both \(\hat{e}_{t}\) and \(\hat{e}_{{x^{\prime } }} .\) The third frame \(\left\{ {\hat{e}_{{x^{\prime } }} - \hat{e}_{{y^{\prime } }} - \hat{e}_{{z^{\prime } }} } \right\}\) is attached to the palm. The unit vector \(\hat{e}_{{x^{\prime } }}\) is common to the palm-fixed and lever-fixed frames. Any tilt of the handle is assumed to be due to variable finger configuration, and the palm maintains its vertical orientation. Also, the angle between the lever and vector r is assumed to be constant, so that the angular velocity and acceleration of the lever and the hand are equal.

The handle tilt was fairly constant during majority of the movement away from extreme wrist flexion and extension positions. Therefore, the portions of a half-cycle wherein the angular velocity was below 10 % of the peak angular velocity during that half-cycle were excluded from this analysis. To confirm these observations, we repeated the cyclic movement test at all weight and frequency levels and measured the handle kinematics for one subject using a vision-tracking system (Qualisys Motion Capture Systems, sampling frequency 100 Hz). During these trials, the maximum variations in the parameters θ, β, and r (the magnitude of vector r) were 10.6°, 5.2°, and 7.2 mm, respectively, and the vertical displacement of the CG was 8.5 ± 2.1 mm. The parameters r, θ, and β for each movement half-cycle are assumed as constant, and the vertical movement of the handle was considered insignificant.

The inertial forces acting on the handle in the lever-fixed frame are,

$$\bar{F}_{rad} = mr\omega^{2} \hat{e}_{r} , $$
(7)
$$\bar{F}_{tang} = mr\alpha \hat{e}_{t} , $$
(8)

where m is mass of the handle. These forces are transformed into the palm-fixed frame. The finger forces measured by the sensors in their respective coordinate frames are also transformed into the palm-fixed frame. Then, the equation of motion along the \(\hat{e}_{{z^{\prime } }}\) direction is written as

$$\left( {F_{Z}^{TH} + F_{Z}^{VF} } \right)\cos \beta - \left( {F_{X}^{TH} + F_{X}^{VF} } \right)\sin \beta = F_{tang} \sin \theta + F_{rad} \cos \theta ,$$
$$\left( {F_{Z}^{TH} + F_{Z}^{VF} } \right)\cos \beta - \left( {F_{X}^{TH} + F_{X}^{VF} } \right)\sin \beta = mr(\alpha \sin \theta + \omega^{2} \cos \theta ), $$
(9)

where \(F_{i}^{TH}\) is the force along direction i for the thumb, and \(F_{i}^{VF}\) is the sum of the forces along direction i measured by the other four finger sensors, both expressed in the handle frame.

Next, a nonlinear, constrained optimization routine (MATLAB function fmincon) was utilized to minimize the squared error between the left- and the right-hand sides of Eq. 9 summed over all time instants t:

$$\hbox{min}\,J:= \mathop \sum \limits_{t} \left[ {\left( {F_{Zt}^{TH} + F_{Zt}^{VF} } \right)a + mb - \left( {m\alpha_{t} } \right)r\sin \theta - \left( {m\omega_{t}^{2} } \right)r\cos \theta } \right]^{2} ,$$

subject to

$$\begin{gathered} 0.86 \le a \le 1, \hfill \\ - \infty \le b \le \infty , \hfill \\ r_{0} - 0.03 \le r \le r_{0} + 0.03, \hfill \\ 0 \le \theta \le \pi , \hfill \\ \end{gathered}$$

where r 0 is the radial distance (meters) from the wrist center to the handle CG measured in a static pose before the start of the cyclic movement trials for each subject. The constraints on r take into account the physically possible change in finger configuration that would change its value. The parameter a represents the term cos β in Eq. 9, and it is constrained such that β lies between zero and 30 degrees. The parameter b represents the contribution of the net finger force along \(\hat{e}_{x}\) to the acceleration of the handle along \(\hat{e}_{{z^{\prime } }} .\)

The optimization yielded values for the parameters a, b, r and θ, which were then used to compute the left- and the right-hand sides of Eq. 9 for each movement half-cycle. These two time series were used to compute the RMS error normalized by the mean of the absolute value of the net finger force along \(\hat{e}_{z}\) and the correlation between the two series. The movement cycles with correlation greater than 0.9 and normalized RMS error less than 0.25 were selected for further analysis. This procedure effectively rejects the movement half-cycles that have excessive handle tilt.

Appendix 2

Additional validation for the regression model

We followed a two-step analysis for further validation of the regression model (see Eqs. 5, 6). First, 67 % of the movement half-cycles available for each task condition (frequency and weight) were chosen at random, and the model parameters were obtained via regression. Next, the predictive ability of the model was tested on the remaining 33 % of the half-cycles. The accelerations \(a_{y}^{\prime }\) and \(A_{z}^{\prime }\) for these tests were inputs to the model, and the forces output by the model, \(F_{{z^{\prime } }}^{TH}\) and \(F_{{z^{\prime } }}^{VF}\), were compared with the measured forces by computing the normalized RMS error, the correlation coefficient and the VAF for the time series.

Figure 8 shows the histograms of the three metrics used to compare the predicted and the measured force time series. The figure combines data for all movement half-cycles for all subjects (114 movement half-cycles in all). The normalized RMS error is less than 10 % for 68 half-cycles; the correlation coefficient is greater than 0.98 for all half-cycles; and VAF is greater than 85 %, for 110 half-cycles. The model successfully predicts the TH and VF normal forces from task kinematics.

Fig. 8
figure 8

Histograms of the various metrics used to quantify the predictive power of the regression model are shown. The metrics are computed from the time series of the measured forces and those predicted using the regression model. Data is for all subjects

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Ambike, S.S., Paclet, F., Latash, M.L. et al. Grip-force modulation in multi-finger prehension during wrist flexion and extension. Exp Brain Res 227, 509–522 (2013). https://doi.org/10.1007/s00221-013-3527-z

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