Data processing and analysis

For each trial, the measurement included the response (correct, incorrect, or no-response) and the maximum radial displacement

where dx and dz are the displacements of the handle in the X and Z directions with respect to its initial position at t = 0.

Human arm stiffness was also estimated in the experiment by considering the applied robot interaction force and the resulting hand displacement. While direct measurement of the interaction force was not available, the commanded robot interaction force was used as an approximation of the interaction force value, from which the 2-dimensional endpoint stiffness of the arm was estimated through the following procedure^36,45,60:

The quasi-static stiffness of the arm is related to the interaction forces and the hand displacement such that

where F_x and F_z are the robot interaction force in the X and Z-direction; dx (t) and dz (t) are the displacements in X and Z, respectively, K_xx, K_xz, K_zx, and K_zz are the elements of the 2-dimensional stiffness matrix. To avoid dynamic effects, measurements at 3 s were used. Then, the stiffness elements K_xx, K_xz, K_zx, and K_zz were determined for each participant for each of the four conditions (RH*MH, RH*ML, RL*MH, and RL*ML) by using the linear least square regression method. However, the magnitude of the velocities of the hand (=robot handle) at ~3 s for all 1960 trials was small (V_x: 1.39 ± 1.23 mm/sec and V_z: 1.49 ± 1.21 mm/s). Therefore, we can assume that at ~3 s, the hand is quasi-static and only the stiffness component of the arm dynamics is present at that time.

A two-way measure of analysis of variance (ANOVA) was used to find the effect of the robot interaction force and muscle contractions on the measurement of the sensitivity of the interaction force direction. A generalized linear mixed-effects model was also used to analyze the data in a trial-by-trial manner, where the binomial outcome measure was whether participants responded correctly on that particular trial. For this, no-response was considered as an incorrect response. This analysis included fixed effects of robot interaction force, muscle contraction, motion direction (X/Z and +/−), and the logarithm of the maximum radial displacement on that particular trial, with a random intercept for participant and by-participant random slopes for muscle contraction. The maximum radial displacement was transformed to its logarithmic value due to the skewness and kurtosis of the raw data sets.