Figure 1.
Representation of the double-pendulum model of the arm.
The centers of the inertial ellipsoids represented in the figure are located at the centers of mass of the body segments. The length of the upper arm is , and the center of mass is at
from the shoulder center of rotation. Hand and forearm are considered as a unit of length
with no joint at the wrist. The resulting center of mass for the segment is obtained by the combination of those of the hand and forearm and is located at
from the elbow. The size of each ellipsoid depends on both mass and inertial tensor of the segment. The dimensions of each ellipsoid along the major and minor axes (eigenvectors) are computed as
, where
are the principal moments of inertia of the tensor
, and
is the mass of the segment. During simulated movements, the hand's center of mass follows the trajectory shown as the dashed brown line. In the figure the hand center of mass is at position (0.4,0)m, which is the configuration used for the postural tests.
Figure 2.
Mechanical models used in the simulations.
A) Time-variant second-order viscoelastic linear system (Kelvin-Voigt). B) third-order viscoelastic linear system (Poynting-Thomson). C) Time-invariant second-order cubic viscoelastic system (Duffing). The schematics highlight the different force fields of the D'Alembert equation (2) when the internal forces generated by the dynamics are negligible. In the figure, each force field is dependent to the mechanical elements that generate it.
Table 1.
Inertial and geometrical parameters used in the simulations.
Figure 3.
Representation of the imposed reaching trajectory and the multipliers for the stiffness time profiles.
In the left panel, the reaching profile for the x (solid) and y (dashed dotted) components of movement are represented using the convention of Figure 1. The co-ordinates shown in light blue refer to the position of the hand's center of mass used in the static (postural) condition. For the first part of the trajectory, a constant stiffness and damping are imposed at the beginning of the movement (right panel). Subsequently, after the application of a force impulse perturbation, the joint stiffness is modulated by means of the gain profiles depicted on the right panel. We imposed a constant (green), slow sigmoidal (red), a combination of linear and sinusoidal (blue), and sharp sigmoidal gain (black), respectively. The same time-varying profiles are also imposed to stiffness and damping during the simulated static condition.
Figure 4.
Orthogonal projection of the reassigned spectrogram for each separate joint.
Orthogonal projection of the reassigned spectrogram for the variables (A,B) and
(C,D) calculated with the maximum noise level (SNR = 10 dB). Due to the different orientation of the eigenvector matrix
, the second frequency of
(A,B) has a lower power compared to that calculated for
(C,D); hence, the oscillation is still present but it is just above the noise level. The estimation of instantaneous frequency
and instantaneous amplitude
are however very clear when analyzing the spectrogram of
(C,D).
Figure 5.
Short Time Fourier Transform (STFT) Spectrogram on the left and Reassigned Spectrogram (RS) on the right for a simulated arm reaching movement with sigmoidal joint stiffness. Based on the classical spectrogram, the partial derivatives of the STFT phase with respect to time and frequency were calculated. This process identifies the location of the stationary phase with respect to the location of the window in the time and frequency domain. The time delay and frequency shift obtained with this process are then used to “reassign’ the position of maximum energy. Savitzky-Golay polynomial filtering allows for easy calculation of the RS peaks envelope. The envelope is depicted in both the classical and reassigned spectrogram in black. Note that it would be difficult to estimate accurately the peaks' envelope in the classical spectrogram due to the lower frequency accuracy.
Figure 6.
A) Reassigned spectrogram of a perturbed movement. This panel illustrates the effect of an impulsive perturbation on the spectrogram of the elbow angular rotation, the frequency of the oscillations excited by the impulse are clearly identifiable. B) Time signal of the elbow rotation corresponding to the reassigned spectrogram in A).
Figure 7.
Stiffness estimation comparisons.
A) Each graph represents the temporal variation of a specific component of the stiffness matrix as depicted in Figure 3. The hand is in a static posture at position (0.4,0) as represented in Figure 1. A “sinlin” damping profile is imposed and four profiles of stiffness are presented: constant (green), sigmoidal (red), sinlin (blue), sharp (black). Dashed stiffness profiles are those imposed in the simulation, while the solid-line profiles are the estimations obtained with the proposed spectrographic method. “X” represents the estimations of stiffness using a “full regression” from an imposed displacement. Each point represents the average stiffness within a 200 ms window.”◊” refers to the “steady state” estimations, notice that since the estimation is done at the end of the perturbation plateau, there is a time shift between “X” and “◊” of 75 ms. “O” represents the estimations using a full regression with an imposed force. Eight perturbations were applied to obtain each point of the stiffness with a regression. Only one impulsive perturbation was applied to obtain each full stiffness profile with the spectrogram technique. The different subpanels represent estimations of each element of the stiffness matrix with four different levels of noise. B) Equivalent estimations to those presented in A) but obtained during the movement condition, during which the hand's center of mass moves along the trajectory represented in Figure 3. Same nomenclature.
Figure 8.
Damping estimations comparisons.
Estimations equivalent to those in Figure 7 for the damping parameters, when a “sigmoidal” stiffness profile is imposed. The nomenclature is the same as in Figure 7.
Table 2.
Repeated measures ANOVA among estimation methods with stiffness and damping time-profiles as random factors along the interval 2.5–5 s.
Table 3.
Repeated measures ANOVA among estimation methods with stiffness and damping time-profiles as random factors along the interval 2.6–3.175 s.
Table 4.
Pairwise repeated measures ANOVA between estimation methods with stiffness and damping time-profiles as random factors along the interval 2.6–3.175 s.
Figure 9.
Effect of neglecting damping on stiffness estimation.
Dashed lines represent the imposed stiffness (red), and damping (black) time-profiles, in accordance with the color-code of Figure 3. The solid lines represent the estimated values of stiffness coefficients when the natural frequencies of the system are assumed equal to the resonant frequencies
therefore neglecting the damping contributions
in equation (46).
Table 5.
Effect of neglecting damping on stiffness estimation with stiffness varying sigmoidally.
Table 6.
Repeated measures ANOVA for the percentage RMS error using spectrogram technique among different inertial methods with directions of perturbation, stiffness and damping time-profiles as random factors along the interval 2.5–5 s.
Table 7.
Repeated measures ANOVA for the percentage RMS error using force full regression among different inertial methods with stiffness and damping time-profiles as random factors along the interval 2.5–5 s.
Table 8.
Repeated measures ANOVA for the percentage RMS error using displacement full regression among different inertial methods with stiffness and damping time-profiles as random factors along the interval 2.5–5 s.
Table 9.
Repeated measures ANOVA for the percentage RMS error using displacement steady state regression among different inertial methods with stiffness and damping time-profiles as random factors along the interval 2.5–5 s.
Figure 10.
Representation of the eigenvectors and relative errors during simulations.
A) FIRST ROW: Representation of the time-invariant coefficients of the eigenvector matrix (
) for the static simulations. Dashed lines represent the coefficient of the imposed matrix
and solid lines represent the estimated
for the different stiffness time profiles: constant (green), sigmoidal (red), sinlin (blue), sharp (black). SECOND ROW: The coefficients of matrix
and their estimations for the dynamic case, where the variation of hand position makes the coefficients time-varying. B) Effect of misestimating the orientation of
on the stiffness coefficients. The estimations presented in this work are within the shaded blue area. C) RIGHT: Representation of the eigenvectors at the beginning of the estimation (time = 2.5 s) for the postural case. Coefficients of the imposed matrix
are in magenta. LEFT: Error in the eigenvector orientation throughout the estimation time window. The reference eigenvectors are shown in magenta. D) Same as panel B for the dynamic condition.
Figure 11.
Mode Synchronization and parameter estimation with complex modes.
A) LEFT: unsynchronized first mode. RIGHT: imposed (Magenta) vs. estimated (Black) eigenvectors using unsynchronized modes. The error is equal to half the rotation imposed on the damping matrix to simulate non-classical damping. B) Synchronized modes and eigenvector respectively. C) Estimation of stiffness and damping for a sigmoid-sinlin Stiffness-Damping time-profile, when signals are corrupted with signal dependent noise.
Figure 12.
Estimation of normalized force for non-linear and higher-order systems.
A) Estimation of normalized force for the Duffing model for 2 DOF. Solid line represents estimation, while dashed line depicts the imposed value. B) Estimation of normalized force for the Poynting-Thomson model.