Table 1.
Parameters of subjects who participated in the experiments.
Fig 1.
Measured and derived forces and torques of all subjects.
(A) shows measured maximal forces Fhand,max at subject’s dominant hand for elbow flexion (red) and elbow extension (blue) in the upper panel. Lower panels contains box and whisker plot of data of the upper panel where the box represents the interquartile range (IQR), and the whiskers represent data of 1.5 ⋅ IQR. (B) shows maximal elbow torques that result from the maximal hand forces in (A) when eq:T max is used. (C) contains the maximal muscle forces as calculated based on sec:methods model.
Fig 2.
Passive orthosis with sEMG sensors and different experimental postures.
(A) shows the passive orthosis and the sEMG sensors placed on the arm. To fix the orthosis to the arm, flexible straps (black) are used. The mounting points and the overall length of the orthosis is customizable. The sEMG sensors were placed onto the short and long head of the biceps and onto the long and lateral head of the triceps. The wrist rotation is in neutral position with the thumb pointing upwards. (A) and (C) show the lower experimental posture (α approximately 0°). The upper experimental posture (α almost 180°) is shown in (B). The angle between the upper arm and the body (α) is zero when the long axis of the upper arm is pointing towards the ground. In (D) the upper body with the right arm is shown in the coronal plane from an anterior and posterior perspective. Left side shows anterior view with the placement of two biceps sEMG sensors in shades of red. Right side shows posterior view with the placement of two triceps sEMG sensor in shades of blue. Lighter tones of red an blue indicate lateral sensor positions to measure the long head of biceps and the lateral head of tricpes. Darker tones indicate medial sensor position to measure the short head of biceps and the long head of triceps. This color code is consistent throughout the paper. mark the distance from the acromion to the innervation zone.
Fig 3.
Raw values of sEMG signals and elbow joint angle of a single subject.
(A,B,C,D) show sEMG signals for the upper posture. (E,F,G,H) show sEMG signals for the lower posture. Biceps sEMGs are shown in shades of red and triceps sEMGs in shades of blue. Left column (A,C,E,G) show sEMG signals from innermost muscle heads in darker shades and right column (B,D,F,H) show sEMG signals from outermost muscle heads in lighter shades of the respective color. On the right axis the elbow angle (θ) is plotted in green. θ = 0° means that the arm is fully extended. (C,D,G,H) each show a time frame of 0.3 s from the respective signals arranged above.
Fig 4.
Block diagram of musculoskeletal model of elbow joint.
The general signal flow in the diagram is from left to right. Signal sources of the model are shown on the left side and contain sEMG signals and the upper arm angle α (relevant for the arm posture, see Fig 2(B) and 2(C)). sEMG signals of muscle heads involved in elbow flexion, in this work the two heads of the biceps brachii, are depicted in the upper row in shades of red. sEMG signals of muscles heads involved in elbow extension, in this work two of the three heads of the triceps brachii, are depicted in the lower row in shades of blue. Darker shades of a color represent the innermost head of the respective muscle, lighter shades of a color represent the outermost head of the respective muscle. After preprocessing of the sEMG signals, the resulting neural signals were fed into the activation dynamics, added up and served as input to the A-model. The output of the A-model, which is the muscle activation, was fed into the submodel for the contraction dynamics. Its output, which is the respective muscle force, was applied to the joint geometry submodel which calculates the respective torque. Resulting torques were summed up and fed into the equations of motion submodel where the resulting movement of the forearm was calculated. The main output of the overall model is the joint angle θ of the elbow (see Fig 2(A)) which was further used to calculate current muscle velocities and lengths.
Fig 5.
Mechanical model of the arm with dimensions and reference points.
(A) Schematic depiction of upper arm and forearm with elbow joint together with biceps and triceps brachii muscle models. Mechanical support structures representing the bones are plotted as bold lines. Three externally palpable landmarks (acromion, ac as well as medial and lateral epicondyles, ec_m and ec_l) are illustrated in green. The hand/wrist marks the lateral end of the forearm. The musculotendon complexes (MTC) for both muscles were depicted schematically as spring-mass systems with separate tendon elasticities. The MTCs were spanned between the respective insertion points (Abic|tric, Bbic|tric). The angle of the elbow joint θ was defined as 0 for the fully extended arm and positive for counter-clockwise flexion movements. (B) shows a reduced model version of (A) in which most insertion points were placed on the support structures and the tendons were neglected. The graphic also shows the definition of the length specifications. The MTC of the triceps was guided over a pulley wheel with constant radius. (C) Posterior view of upper arm with two heads of biceps brachii in shades of blue and the three externally palpable landmarks in green. Landmarks were used to estimate the depicted length measures (for details see text).
Fig 6.
Comparison between MAE and nMAE.
In each subfigure (A), (B) and (C) there are two exemplary elbow joint movements of different waveform and amplitude. Waveforms in solid green lines have an amplitude of 1, those in solid red line of 0.5. In (A) and (B) the simulated prediction θsim (dashed lines in green|red) is constantly at zero. C) gives an alternative example for a simulated θsim at 80% of the signal amplitude. The resulting error is independent of the range, but dependent on the signal form as shown in Table 2.
Table 2.
Values of MAE, nMAE and QS for different curve shapes and amplitudes as given in Fig 6.
Fig 7.
Selected examples of forearm movements (elbow angle θmeas) with quality scores QS close to [0, 0.25, 0.5, 0.75].
Upper row (A-D) shows four results for the upper posture, the lower row (E-H) for the lower posture. The first column (A) and (E) show two results with QS close to 0; (B) and (F) a QS close to 0.25; (C) and (G) a QS close to 0.5 and (D) and (H) a QS close to 0.75. Different experiments were marked with symbols (cross, square, triangle, circle).
Fig 8.
Optimization results for two exemplary subjects in the same experimental condition (slow, 2 kg) but at different postures.
(A) Lower posture with the movement course having a more sinusoidal character (subj. 48). (B) Upper posture with movement course having a more triangular shape (subj. 24). First row shows neural activation of both biceps muscle heads in shades of red and two triceps muscle heads in shades of blue (triceps data is shown mirrored). Second row shows elbow joint torque components according to the contributing muscles. As shown in Fig 4, both muscle heads are combined in one contraction dynamics (one active torque for each muscle group). Gray signals are torque components originating from passive muscle forces. Third row shows measured (θmeas, green) and simulated (θsim, black) elbow joint angle course. Below the time curves of the free parameters are shown (colour code according to the respective muscles).
Fig 9.
Comparison of all experiments for one subject (subj. id 21).
The selected subject has achieved the highest mean of the QS across all experiments combined. The top part shows the experiments in the upper posture, the bottom part those in the lower posture. For each experiment, curves are depicted with the same color coding as in Fig 8 (torque curves are not shown). Columns represent different experiments which were marked with symbols (cross, square, triangle, circle).
Fig 10.
The nMAE for all subjects in the two different postures.
In (A), results for the lower posture, and in (B) results for the upper posture are shown. Vertical gray lines at 0.25 (triangular) and 0.32 (sinusoidal) indicate the theoretical nMAE for a constant prediction θconst for the respective signal as a reference. The experiments with different movement speeds and different additional weights are shown in four different shades of gray and with different symbols (cross, square, triangle, circle). The diagrams at the top show individual results. The bottom diagrams show aggregated results in individual box and whisker plots for speed and weight combinations. The chronological sequence of the experiments is from top to bottom. Data was published in [23].
Fig 11.
Quality score QS for all subjects in two different postures.
In (A), results for the lower posture, and in (B) results for the upper posture are shown. The general structure of the graphs is the same as in Fig 10. The quality score is calculated as shown in Eq (34).