https://orcid.org/0000-0002-4864-4978
Figures
Abstract
For assistive devices such as active orthoses, exoskeletons or other close-to-body robotic-systems, the immediate prediction of biological limb movements based on biosignals in the respective control system can be used to enable intuitive operation also by untrained users e.g. in healthcare, rehabilitation or industrial scenarios. Surface electromyography (sEMG) signals from the muscles that drive the limbs can be measured before the actual movement occurs and, hence, can be used as source for predicting limb movements. The aim of this work was to create a model that can be adapted to a new user or movement scenario with little measurement and computing effort. Therefore, a biomechanical model is presented that predicts limb movements of the human forearm based on easy to measure sEMG signals of the main muscles involved in forearm actuation (lateral and long head of triceps and short and long head of biceps). The model has 42 internal parameters of which 37 were attributed to 8 individually measured physiological measures (location of acromion at the shoulder, medial/lateral epicondyles as well as olecranon at the elbow, and styloid processes of radius/ulna at the wrist; maximum muscle forces of biceps and triceps). The remaining 5 parameters are adapted to specific movement conditions in an optimization process. The model was tested in an experimental study with 31 subjects in which the prediction quality of the model was assessed. The quality of the movement prediction was evaluated by using the normalized mean absolute error (nMAE) for two arm postures (lower, upper), two load conditions (2 kg, 4 kg) and two movement velocities (slow, fast). For the resulting 8 experimental combinations the nMAE varied between nMAE = 0.16 and nMAE = 0.21 (lower numbers better). An additional quality score (QS) was introduced that allows direct comparison between different movements. This score ranged from QS = 0.25 to QS = 0.40 (higher numbers better) for the experimental combinations. The above formulated aim was achieved with good prediction quality by using only 8 individual measurements (easy to collect body dimensions) and the subsequent optimization of only 5 parameters. At the same time, just easily accessible sEMG measurement locations are used to enable simple integration, e.g. in exoskeletons. This biomechanical model does not compete with models that measure all sEMG signals of the muscle heads involved in order to achieve the highest possible prediction quality.
Citation: Grimmelsmann N, Mechtenberg M, Schenck W, Meyer HG, Schneider A (2023) sEMG-based prediction of human forearm movements utilizing a biomechanical model based on individual anatomical/ physiological measures and a reduced set of optimization parameters. PLoS ONE 18(8): e0289549. https://doi.org/10.1371/journal.pone.0289549
Editor: Emiliano Cè, Università degli Studi di Milano: Universita degli Studi di Milano, ITALY
Received: December 6, 2022; Accepted: July 20, 2023; Published: August 3, 2023
Copyright: © 2023 Grimmelsmann et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The data underlying the manuscript are available from: Surface electromyographic recordings of the biceps and triceps brachii for various postures, motion velocities and load conditions: DOI 10.57720/2290.
Funding: This work has been supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, https://www.dfg.de/en/) - ref. no. SCHN 1339/3-1, by the Federal Ministry of Education and Research (BMBF) within the project ITS.ML - ID 01IS18041 A (AS) and by the research training group “Dataninja” (Trustworthy AI for Seamless Problem Solving: Next Generation Intelligence Joins Robust Data Analysis) funded by the German federal state of North Rhine-Westphalia. The submission was funded by the DFG – ref. no. 490988677 and Bielefeld University of Applied Sciences (https://www.fh-bielefeld.de/). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
The model based prediction of limb movements from myoelectric signals as measured in surface electromyography (sEMG) can be used for intuitive human-machine interfaces (HMIs) in close-to-body robotics like in wearables and in actuated orthoses [1–6]. Between the appearance of myoelectric signals (sEMG) and the respective muscle contraction (force generation) as well as between muscle contraction and the actual limb movement, time delays occur. The former delay results from the activation dynamics during static or dynamic contractions of the respective muscle which can be explained by the electrochemical conversion of myoelectric excitation into the release of Ca2+ ions within the muscle sarcomeres followed by sarcomere shortening (concentric contraction) or sarcomere lengthening (excentric contraction) [7, 8]. The latter delay results from the fact that the inherently inert mechanical system of a limb must be accelerated by muscular forces before motion becomes noticeable in terms of a measurable angular velocity of the respective joint or an interaction force e.g. under isometric conditions. The resulting overall time delays lie in the range of e.g. ∼ 106 ms for the human knee extensor [9] and e.g. ∼ 53 ms for muscles of the upper limb [10–12]. The time interval between innervation and movement can potentially be used for a model-based prediction of the limb movement by using sEMG data as input. In order to exploit this time advantage in the control of close-to-body robotic systems, fast model algorithms with sufficient quality for limb motion prediction are needed to reduce the delay between limb and robot motion. Biomechanical limb models at different levels of detail are available as a basis for the above strategy. These contain Hill-type muscle models [7, 13] as subsystems and cover, e.g. for human upper limbs, aspects like joint geometry and its effect on the muscle moment arm [14], the structure of the musculoskeletal system [15], or, length partitioning and properties of muscle and tendon [16]—the latter also with respect to age [17]. As a consequence, each submodel contributes to a growing range of parameters. These parameters must be individually adjusted to the respective user and, if necessary, continuously readjusted during prolonged use of the model, e.g. due to muscle fatigue [18, 19] or due to electrode shifts on the skin and other interface effects [20, 21].
This work proposes a prediction model for limb movements of the forearm, which is based on sEMG measurements of the heads of the main flexor and extensor muscles (biceps and triceps brachii). Particular attention was paid to reducing the parameter space as much as possible I) by reducing the number of parameter using prior knowledge—like the time constants in the activation dynamics, and II) by obtaining hard-to-access physiological parameters from easy-to-access physiological/anatomical measures—like locations and directions of palpable bone processes and the respective intermediate distances and rotations (e.g. acromion of the scapula, epicondyles of the distal humerus and styloid process of ulna)—and to adjust the remaining, free parameters to individuals in an optimization process.
Such a model, in which only a few parameters are determined by an optimization process, should be particularly suitable for use in motion prediction scenarios outlined above. The model does not compete with detailed full-scale dynamic simulations such as OpenSim [22], but is intended to be a reference for such models that aim for rapid adaptability, e.g. in the control of wearable robots.
To approach the question of how such a model can be set up, parameterized and deployed, this paper first describes the experimental setup and the experimental protocol for acquisition of forearm movement data including sEMG recordings of relevant muscles. The resulting dataset based on 31 subjects was published separately [23]. This is followed by a detailed description of the musculoskeletal model of the elbow joint and forearm as a basis for subsequent movement prediction. The section also contains a description of model components as well as of the model parameters. A key point here is the strategy of how to divide the parameters into those that can be determined based on a few anatomical landmarks that can be measured from the outside and those that still need to be adjusted as part of an optimization process. The model was evaluated based on the experiments with 31 subjects described in the data collection section. The methods part of this work concludes with a description of the properties of the chosen error and quality scores in order to compare movement predictions for different experiments and subjects. Finally, the movement prediction results for all subjects and experiments are presented and discussed.
2 Methods
In the process of parameter identification for a musculoskeletal elbow joint and forearm model controlled by surface electromyography (sEMG), the first step was the acquisition of experimental data. For this, sEMG signals of the muscles involved in forearm actuation and the corresponding elbow angle θ were recorded for different subjects and for varying motion sequences (see section 2.1). The experimental data was subsequently used to optimize the parameters of a musculoskeletal model of the elbow joint to predict the movement of the forearm (see section 2.2). The majority of the parameters from the model were set to individual values based on prior knowledge. Finally, the last five parameters were optimized by a global and local optimizing process. Since the entire model is based on two muscle heads for each elbow extensor and flexor muscle group, it was assumed that the total torque of all flexors is caused by the biceps brachii alone, and the total torque of all extensors is caused by the triceps brachii alone (other muscles were not included).
2.1 Acquisition of experimental data
Subjects.
The sEMG data and the corresponding elbow angle θ was acquired for n = 31 subjects (25 male, 3 female and 3 in none of these categories) while the subjects were performing different motion sequences. 29 subjects chose their right arm as the dominant one, 2 chose the left one as their dominant arm. All subjects were healthy and did not have any prior neural diseases when the experiments were performed. Subjects in this study were randomly labeled with an identification number (ID) starting from 20. Smaller numbers came from pre-experiments to adjust the experimental protocol and were not used. Informed consent was obtained from all subjects in accordance with the policies according to the ethical guidelines of the German Society for Psychology (DGPs) and the German Psychologists Association (BdP), and was approved by the Ethics Committee of the University of Bielefeld (EUB 2017–156 02.08.2017). For each subject, the positions of acromion (ac) at the shoulder, and the lateral epicondyle (ec_l) at the elbow were defined as reference points to calculate the length Lac,ec_l of the upper arm. Furthermore, the positions of the medial epicondyle (ec_m)—again at the elbow—and the processus styloideus ulnae (p_su) at the wrist were defined as reference points to calculate the length Lec_m, p_su of the forearm down to the wrist. In addition, also the position of the palm was used to calculate a forearm length Lec_m, palm that includes the location where the strap loop of the force sensor was held (as will be described below). The latter served as a measure of the length of the lever arm for calculating the elbow torque that occurred due to a measured hand force. An overview of the parameters is given in Table 1.
Determination of maximum isometric muscle force.
To determine the maximum isometric muscle force, the maximum hand force Fhand,max had to be measured first. The maximum hand force was converted into a maximum elbow torque via the corresponding moment arm (forearm length Lec_m, palm). Later in the process, a maximum muscle force was determined on the basis of a maximum elbow torque and a lever length of the tendon attachment that will be introduced in Eq (11). The maximum hand force was further differentiated into maximum extension force and maximum flexion force
depending on the direction of the hand force. The former results from the joint action of all muscle (heads) involved in elbow extension (e.g. triceps heads) during maximum voluntary contraction (MVC, see [24, 25]), the latter results from the joint action of all muscle (heads) involved in elbow flexion (e.g. biceps heads) during MVC. Results of the maximum hand force measurements for all subjects are shown in Fig 1(A).
(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.
The system used for the determination of the maximum hand force consisted of an adjustable inelastic strap attached to a piezoelectric force transducer (Typ 9255B, Kistler Instrumente GmbH, Sindelfingen, Germany) which was connected to a charge amplifier (Typ 5011, Kistler Instrumente GmbH, Sindelfingen, Germany). The signal generated by the charge amplifier was recorded with a sample rate of 1kHz using a USB-based A/D-converter card (NI9215, National Instruments corp., Austin, Texas, USA).
To determine the maximum isometric flexion hand force , each subject was instructed to hold the dominant forearm in a horizontal orientation while grabbing the strap via a loop at its end, with the hand oriented such that the thumb pointed upwards. The subjects were then instructed to pull on the strap as strong as possible for a time interval of Δt = 5 s. To ensure that the subject only pulled via the elbow joint, each subject was instructed to minimize shoulder movement while pulling on the strap. The maximum force (
) applied during the time interval Δt was saved and the maximum elbow flexion torque (
) was computed according to the left side of Eq (1). The data for all subjects are shown in Fig 1(B) where data points for the flexion direction are plotted in red.
(1)
To determine the maximum isometric hand force for an elbow extension the subjects were instructed to push as strong as possible with the bottom of the dominant hand placed on the force transducer for a time interval of Δt = 5 s while the maximum force applied was recorded. In this case, subjects were also instructed to hold the dominant forearm in a horizontal orientation and apply force only via the elbow joint and not by e.g. pressing with the upper body. The maximum isometric elbow extension torque
was subsequently computed according to the right side of Eq (1). The data for all subjects are shown in Fig 1(B) where data points for the extension direction are plotted in blue.
Surface electromyography (sEMG).
As input for the muscoskeletal elbow joint and forearm model, sEMG signals of those muscles involved in forearm actuation were used. For the measurement of the respective muscle activity, two wireless sEMG sensors (Delsys Trigno, Delsys, Inc., Boston, MA, USA) were attached to the skin surface above the biceps brachii (bic: short head and long head) and two on the skin surface above the triceps brachii (tric: long head and lateral head [medial head not recorded]). The sEMG sensor is equipped with one sEMG channel with a sampling period of 900 μS with 16 bit resolution at a range of 11mV and a six axis IMU with a sampling period of 6.75ms. Measurements were conducted at a sample rate of 1.111kHz. Before the placement of sensors, the skin was cleaned using isopropanol alcohol. The acromion (ac), the medial epicondyle (ec_m) and the distal insertion of the biceps tendon (di_bic) were palpated by an experienced experimenter. For the biceps heads, the innervation zone lies after 61% of the distance as measured from ac to di_bic [26]. Accordingly, the optimal electrode site is on the muscle belly proximal of the innervation zone (). For the triceps heads, the innervation zone lies after 40% of the distance as measured from ac to ec_m [26]. Again, the optimal electrode site lies proximal of the innervation zone (
). These areas were marked via pen and tape roller accordingly. The individual muscle heads of biceps and triceps were probed by hand and the innervation zones of the respective muscles, which have to be omitted in sensor placement, were marked via pen and tape roller according to [26]. After the preparation of the skin and the determination of positional placement, sEMG sensors with the included IMU sensors were fixed with double-sided adhesive tape on the skin of the subjects. Fig 2(A) shows the placement of the sensors as well as the passive orthesis used for synchronous measurement of the elbow joint angle θ.
(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.
The interface between skin and electrodes was evaluated in a preliminary experiment via instructing the subjects to separately contract the flexors and extensors of the upper arm and subsequent visual inspection of the signal quality.
Orthetic device for measuring the elbow joint angle.
A passive measurement orthosis was used to measure the elbow angle θ synchronously with sEMG recordings while the subjects were performing different motion sequences (Fig 2). The measurement orthosis was custom designed and 3D-printed in-house from PLA plastic. The attachment locations on the body and the overall length of the orthosis can be adjusted to different arm sizes. The elbow angle is determined using a 10-bit magnetic rotary position encoder (AS5043, ams AG, Premstaetten, Austria) which is integrated in the joint of the orthosis. Thus, it is aligned to the rotary axis of the elbow joint of the subject. The analog output of the rotary encoder is fed into a Trigno Analog Adapter (Delsys, Inc., Boston, MA, USA) to allow for synchronous recording of the elbow joint angle and the sEMG signals. Since the limbs of the measurement orthosis were not necessarily aligned and parallel to the long axes of the forearm and upper arm, a calibration was performed by the supervisor during the initial phase of the experiment.
Experimental paradigm.
The subjects were instructed to perform different motion sequences while sEMG signals of the muscles involved in forearm actuation and the elbow joint angle were recorded. The motion sequences consisted of periodic movements of the dominant forearm for two different postures of the respective upper arm (see Fig 2(B) and 2(C)) with three different weights held by the subject. For each posture, subjects were instructed to align the longitudinal axis of the forearm orthogonal to the longitudinal axis of the upper arm resulting in an initial elbow angle of θ0 = 90° (see also Fig 2(A)). In the lower posture subjects were instructed to hold the upper arm vertically pointing downwards and in the upper posture the upper arm was held vertically pointing upwards. In contrast to the lower posture, the upper posture varies among the subjects due to different flexibility in the subjects’ shoulders. To address this inconsistency, the IMU of the Trigno sensor was used to determine the orientation of the upper arm. For each posture a weight held by the subject was varied (w = [2kg, 4kg]) resulting in four different posture-weight combinations that correspondingly led to different loads on the involved muscle groups. After the initial static orientation of the arm for Δt = 5 s, the subjects were instructed to move the forearm back and forth rhythmically about the axis of the elbow joint at as constant angular velocity as possible, resulting in a sine-like modulation of the elbow joint angle. The target range of the movement was ±45° around the initial position. To define the frequency of the sinusoidal elbow-joint rotation, a metronome was used to provide a visual and auditory hint to the subjects. For each posture-weight combination experiments were performed at [0.25Hz (slow) and 0.5Hz (fast)]. After Δt = 30 s of dynamic movement subjects were instructed that the movement should come to an end naturally, implying that the subject could finish the current move and stopped close to the initial position θ0. After each trial the subjects were allowed to rest for at least one minute. After the resting phase, the trials were repeated with a different posture-weight combination. The order was as follows: fast 2 kg, fast 4 kg, slow 2 kg, slow 4 kg—first in the lower posture, than in the upper posture.
Fig 3 gives an exemplary overview of the time behavior for the amplitudes of the data recorded in the experiments described above to serve as input into the predictive musculoskeletal model described in section 2.2. The data in the figure were collected from one single subject. sEMG signals from the two biceps heads are shown in shades of red (light red for the long head [lateral location in upper arm] and dark red for the short head [innermost location on the upper arm]). sEMG signals from two triceps heads are shown in shades of blue (light blue for the lateral head [lateral location on the upper arm] and dark blue for the long head [innermost location on the upper arm]). Upper posture (Fig 3(A)–3(D)) and lower posture (Fig 3(E)–3(H)) are indicated via symbolic figures on the left.
(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.
sEMG signals differ in the variation of the time behavior for the amplitudes between subjects and also characteristics of muscle utilization (co-contraction, reciprocal contraction) are different for the same motion between subjects.
2.2 Musculoskeletal model of the elbow joint
The model, shown in Fig 4, was used to simulate the behavior of components of the musculoskeletal system of elbow and forearm using sEMG signals of the respective muscles to predict the forearm movement. The overall structure of the simulation model is derived from the biochemical and biomechanical target system. Hence, the model used comprises both the electrophysiological properties of muscle activation and the mechanical properties underlying forearm actuation.
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.
The four sEMG signals recorded from each of the two different heads of the biceps and triceps served as an input to the model and were preprocessed separately resulting in a muscle-specific neural activation. In the model, corresponding neural activations of each muscle head are summed up and the sum is used as input for a non-linear submodel (so called A-model, [8]). For each antagonistic muscle activation provided, here the biceps and triceps, two independent contraction dynamics and joint geometries are used to simulate muscular force and muscular torque generation. In the context of this work, the biceps acted as the only flexor and the triceps as the only extensor in the simulation of musculoskeletal mechanics. The equation of motion submodel contains the musculoskeletal mechanics and was used to calculated the elbow joint angle and is driven by the torques generated by the involved muscles. The corresponding muscle velocities and muscle lengths obtained from the muscular mechanics are fed back to drive the contraction dynamics submodel. The described signal flow is shown in Fig 4. Details of the submodels will be introduced below.
sEMG prepossessing.
The sEMG signals measured at the two heads of the biceps brachii and the triceps brachii, respectively, served as inputs for the musculoskeletal model of the elbow joint. Each of the sEMG signals was preprocessed, i.e. amplified, filtered, and rectified, to eliminate interfering signals and to raise the input signal to a level that was comparable for different subjects.
At first, the input was amplified by a factor k. This factor was chosen in the optimization process such that the range between no contraction and a maximum voluntary contraction leads to a signal interval between minus one and one, or rather between zero and one for the following activation dynamics (after filtering and rectification). The value of k is also related to the resistance of the electrode-skin-interface and may drift over time. Signal filtering was achieved with a 4th order Butterworth bandpass with cutoff frequencies of flow = 4Hz and fhigh = 400 Hz [27] to reduce noise. Rectifying the signal, as introduced in [7], resulted in the neural activation e which ranges mostly between zero and one. This range can be interpreted as the activation level of the neurons. All parameters of this paragraph are summarized in S1 Table.
Activation dynamics.
The activation of a muscle as a result of neural excitation (recorded as sEMG signals) was modelled in the activation dynamics submodel. This submodel represents the electrochemical conversion of neural signals (spike trains on the muscle fibre) into the release of Ca2+ ions in the muscle sarcomeres, ultimately enabling mechanical contraction of the sarcomeres and—as a consequence—of the muscle as described e.g. by Zajac or Buchanan and co-workers [7, 8]. In terms of signal flow, the activation dynamics submodel converts the neural signal e into the neural activation u. It was formulated as a non-linear, first order differential equation [7] as shown in Eq (2).
(2)
Here, τact is the time constant and β is a dimensionless parameter which essentially sets the time response when the activation is suspended as compared to the beginning of the activation. τact (17.3 ms) and β (0.35) were set, such that the step response matches a reference step response as formulated in [28]. Neural activation of one muscle were summed up and fed into the A-model [8]. The A-model represents a nonlinear behavior when the neural activation u(t) is translated into the muscle activation a(t) (which results from a nonlinear relationship between stimulation frequency and force [8]). The respective formulation for the A-model is shown in Eq (3).
(3)
Eq (3) contains only one parameter A to shape the nonlinearity which lies in the interval −0.001 ≤ A ≤ −3 [8]. A coarse search was initially performed on different subjects and resulted in a value of A = −0.25. All parameters of this paragraph are summarized in S2 Table.
Contraction dynamics.
The contraction dynamics submodel models the relationship between muscle activation a(t) and muscle force FM(t). Besides activation, the muscle force also depends on the current muscle length LM(t), and the current muscle contraction or muscle relaxation speed vM(t) [7]. The length dependence of the muscle force is divided into an activation-dependent (active) component FL(ΔLM(t)) and an activation-independent (passive) component Fp(ΔLM(t)), where the length dependence is represented as a deviation ΔLM(t) from the resting (optimal) length LM,0 of the muscle:
(4)
Based on activation, muscle length deviation from resting length, and muscle contraction (relaxation) velocity, the muscle force can be formulated as
(5)
With Fmax being the maximum isometric force of the respective muscle (biceps or triceps) as later described in Eq (28).
The length-dependent part FL(ΔLM(t)) (force-length-function) of the active muscle force reflects the amount of overlap of actin and myosin filaments in the muscle sarcomeres to potentially form cross-bridges. Deviations from the (optimal) resting length of the muscle lead to smaller forces. According e.g. to Geyer and co-workers [29], the force-length-function can be formulated as
(6)
where w is a parameter to control the width of the bell-shaped force-length-function. According to [7, 8, 16, 30] this parameter was set to a standard value of w = 0.5 to reflect a maximum length increase/decrease of the muscle length by 50% w.r.t. the resting length. The shape factor c was set to c = ln0.05 to fullfill FL(ΔLM(t) = ±w ⋅ LM,0) = 0.05 (i.e. a force decline down to 5% of the maximum force at a length deviation of 50% from the resting length [29]), as well es the value range condition given in Eq (6).
The velocity-dependent part Fv(vM(t)) (force-velocity-function) of the active muscle force was formulated in a two-part equation as proposed e.g. by Geyer et al. [29] and shown in Eq (7).
(7)
The shortening part of Eq (7) describes the effect formulated by Hill [13] for active muscle shortening. The lengthening part follows a formulation by Aubert [31] with N as a dimensionless maximum (force) value reached at vM(t) = vM,max. Thelen has shown that N varies with age [17] from N = 1.4 for young adults to N = 1.8 for old adults. Since the mean age of the subjects in this study was 25.3 years (Table 1), N was set to N = 1.4. The shape factor Kv changes the curvature of the lengthening part and was set to Kv = 5 [29–31]. The maximum velocity was set to vM,max = 10 ⋅ LM,0 ⋅ s−1 [17, 32].
The passive part of the muscle force is generated by an non-linear elasticity (attributed to the elastic protein titin) which acts in parallel to the contractile elements (actin myosin complex) in the sarcomere and which accounts for the return of the unactivated muscle to its resting state [7]. Zajac modelled this as shown in Eq (8) [7].
(8)
The factor Kp sets the slope of the quadratic function. Values for Kp vary between Kp = 3 [32–34] to Kp = 4 [35]. Thelen has shown that values for Kp vary with age between Kp = 2.78 for young adults to Kp = 4 for older adults [17] (Because Thelen uses a different formulation for passive elasticity, the values were converted accordingly.). Since the mean age of the subjects in this study was 25.3 years (Table 1), Kp was set to the lower boundary Kp = 2.78. All parameters of this paragraph are summarized in S3 Table.
Musculoskeletal mechanics.
As the biceps and triceps muscles contract, their forces are propagated to the bones of the upper and forearm via the insertion points of the respective tendons, causing torques in the elbow joint. The tendon is generally assumed to be a non-linear spring with a high stiffness in the linear region [36]. At the maximal force, the elongation is about 3% [7]. As the experiment conducted here is far from maximum force, the lengthening of the tendon was neglected. With this assumption, Eq (9) is valid.
(9)
With an inelastic tendon, the velocity of the tendon (vT) and the velocity of the muscle (vM) are equal. The simplified model of the mechanical configuration of the elbow joint in Fig 5(B) was the basis for the following considerations. For the sake of simplicity, single insertion points Abic and Atric were assumed for the upper tendons of the biceps and triceps. Each of the two designators also stands for the distance of the respective insertion point to the medial epicondyle (ec_m). Also for the lower tendons of biceps and triceps, single insertion points Bbic and Btric were assumed at the forearm. Again, each of the designators represents the distance of the respective insertion point to the medial epicondyle. As shown Fig 5(B), the insertion point of the lower triceps tendon is located at the osseous structure extension of ulna called olecranon.
(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).
In addition to the elbow angle θ(t), for the biceps the auxiliary angle ϕbic(θ(t)) was introduced which is the angle between the biceps muscle and the forearm. With changing elbow angle θ(t), also the biceps moment arm and the triceps moment arm
change. As the muscle forces are always positive, the signs of the moment arms were chosen such that the positive sign of the biceps moment arm
leads to a positive torque and thus to a counterclockwise rotation of the elbow joint and the negative sign of the triceps moment arm
to a negative torque and thus to a clockwise rotation. With the distance from upper insertion point of the upper biceps tendon Abic to medial epicondyle (which represents the elbow axis) and the distance from medial epicondyle to the insertion point of the lower biceps tendon Bbic, the current length of the biceps muscular tendon complex
and the respective moment arm
can be calculated. According to [37], the angle ϕbic(θ(t)) for the calculation of the muscle moment arms can be calculated as follows:
(10)
The moment arm of the biceps was calculated according to:
(11)
And finally, the length of the muscle tendon complex resulted in:
(12)
As can be seen in Fig 5(B), the length Bbic is equal to the maximum length of the moment arm . Murray et al. conducted a study on the courses of elbow muscles’ moment arms [38] from which Bbic = 0.0472m was adopted. As the length Abic is not directly accessible, it was derived from the length of the upper arm Lac,ec_l by using a fixed ratio
derived from [39] (Abic = 0.293m and Lforearm = 0.416m) and [40] (Lac,ec_l = 0.618 ⋅ Lforearm). Abic can then be calculated for each subject individually according to Eq (13).
(13)
The individual length Abic was further used to derive the individual resting length
of the biceps muscle tendon complex for each subject in Eq (14). For this, an average ratio
has to be found. This requires a mean value for the upper biceps insertion point which is given by Winters and Stark with
[41]. Furthermore, the mean length of the muscle tendon complex of the biceps at optimal muscle length is also given by Winter and Stark with
. Using these two values results in mean ratio
. With this,
can be calculated individually as follows:
(14)
The resting length of the muscle tendon complex from Eq (14) contains the resting length of the muscle
as well as the length of the tendon
. This condition is also applied to the triceps. To calculate the resting length of the muscle
from the resting length of the muscle tendon complex
for each subject, again a fixed mean ratio
was used. It was calculated based on average values as given in Winters and Stark [41]. These values are the average length of the muscle tendon complex
, and
, as well as the average length of the resting biceps and triceps muscle
, and
. This results in a
and
.
(15)
(16)
The resulting individual resting muscle length was used in the contraction dynamics (Eqs (6) and (8)).
The origin of the triceps long head is at the scapular and the origin of the respective lateral head is at the humerus. The goal was to calculate a combined Atric for the two heads for each individual subject based on the measured individual length Lac,ec_l and a mean ratio as shown in Eq (18). For the latter, an average
had to be determined. For that, it was assumed that the origin of the triceps Atric can be found halfway between the acromion and the axillary fold. A mean distance between the acromion and the axillary fold is given by Gordon and coworkers [42, p. 346] (based on the ANSUR I dataset, [43]) with
for male subjects and with
for female subjects. The corresponding mean distance between acromion and lateral epicondyle is given with
for male subjects and
for female subjects [42, p. 80]. The resulting
was calculated as follows:
(17)
This resulted in a mean distance between elbow rotation axis and triceps origin for male subjects of
and for female subjects of
. This mean distance together with the mean value of
was further used to calculate the mean ratio
. For male subjects this resulted in
and for female in
.
(18)
The lower tendon of the tricpes ends at the olecranon. Here, it was assumed that the olecranon and the the medial epicondyle are at the same height in the sagittal plane if the elbow is fully extended [44]. Therefore, it can be assumed that
applies which leads to:
(19)
In the movement range of the elbow [0°…120°], the length of the triceps varies from to
[16]. For the case of a fully extended elbow, Eq (20) can be used to calculate the resting length of the triceps muscle:
(20)
With Eq (16) rearranged to
(21)
(22)
inserted into Eq (19), and using Eq (20) LM 54 allows to calculate the resting length of the triceps muscle according to:
(23)
(24)
As the moment arm
is nearly constant over the elbow angle [14], it was modeled as a pulley with constant radius. Before running the simulation, the moment arm
can be calculated based on the maximum shortening/ lengthening range of the muscle according to:
(25)
Note that the maximum shortening/ lengthening range in [16] is specified for a third of a revolution.
The calculated values for the the resting length (, Eq (24)) and the moment arm (
, Eq (25)) were used to calculate
during simulation:
(26)
The only unknown parameter
compensates inaccuracies and is subject to the optimization process.
The length of the muscle and the velocity were fed back to the contraction dynamics (cf. Fig 4). The force from the contraction dynamics () was multiplied by the moment arm (
) to get the elbow torque as generated by the muscle.
(27)
The direction of the angle and the geometric dimensions of the arm are shown in Fig 5(B).
Equation of motion.
For the simulation of the mechanical system, the masses and centers of gravity (cog) of the arm segments were required. The masses of forearm and hand were based on findings by Bernstein [45]. For the forearm mass mforearm a portion of 1.82% of the body mass for both women and men is given. The hand mass mhand accounts for 0.7% of the body mass for men and for 0.55% of the body mass for women. The center of gravity (cog) of the forearm Lforearm,cog is at 42% of arm segment length as measured from the elbow joint. As the fingers of the hand are closed to hold an additional weight (dumb-bell), the cog was assumed to be at Lhand.
The last missing parameter in the simulation setup of the mechanical system is the damping coefficient d. This parameter varies between up to
[46]. The damping coefficient d of the elbow joint in this model is set to
. This value was determined by a coarse search.
By using the moment arms calculated in Eqs (11) and (25) for θ = 90°, the maximum forces of the two muscles for the contraction dynamics were calculated as follows:
(28)
is the maximum torque as calculated in Eq (1).
At this point, all but five parameters of the overall model were set. Only these five free parameters were used in the optimization process. To calculate the angular acceleration of the elbow joint based on the described masses and their distribution, torques which result from the flexor (biceps long head + biceps short head) and extensor (triceps lateral head + triceps long head) muscles were summed up. The elbow joint was assumed to be a simple revolute joint (see [37, 41]). Elbow joint angle (θ) and angular velocity (ω) were computed by numeric integration. Initial values were calculated based on measured elbow angle (θmeas). Initial angular acceleration () was determined using the measured angular velocity (ωmeas). Since the shoulder movement is not in the focus of this investigation—the shoulder joint in the simulation was oriented according to the shoulder angles measured in the IMU units which are integrated in the biceps sEMG sensors. As described in the experimental paradigm, the subject holds an additional weight madd in the hand. This is in addition to the weight of the forearm and the hand. The simulation was conducted in Matlab 2019a using the Simulink and Simscape toolboxes (The Mathworks Inc., Natick,MA, USA). The simulation outputs the course of the elbow joint angle θsim as generated by the model. During the optimization process of the five free parameters, θsim was compared to the measured angle θmeas.
Signal preparation and model optimization.
In the optimization process, the goal was to find those values of the remaining five free parameters that minimize the error between simulated elbow angle θsim and measured elbow angle θmeas. Only the four gain values k for the raw sEMG signals of the biceps and triceps muscle heads (see paragraph on sEMG preprocessing and Fig 4, left side) and the offset of the musculoskeletal mechanics (see Eq (26)) were not set to subject dependent values and had to be optimized. To get a robust optimization, the start values were estimated before the optimization. The measured signals of the conducted experiments were split into separate intervals according to the respective phase of the movement. At the beginning of each experiment, there was a nearly motionless time interval which is called static phase in the following. This was followed by the first period of the cyclic movements which started at the initial value of θmeas = 90°, contained one upwards and one downwards movement of the forearm and ended at the initial value of θmeas = 90°. The first three full periods of each experiment were used for the optimization.
Before the start of the optimization, the upper and lower limits of the free parameters had to be estimated to constrain the optimization algorithm during its search. During the static phase, the values of the gains were estimated. In the static phase, the elbow torque Tstatic as generated by the muscles was mainly caused by the arm weight marm, the hand weight mhand and the additional weight madd due to the dumbbell. The estimates of the gains
were calculated based on Eqs (29) and (30).
(29)
(30)
In Eq (29), the angle α indicates the posture of the upper arm (α ≈ 0° → upper arm pointing downwards, α ≈ 180°→ upper arm pointing upwards, measured by the IMU in the EMG-sensors). In Eq (30),
is the mean muscle activation during the static phase. The estimated gain values
can only be determined in the lower posture (α ≈ 0°) for the biceps muscle and in the upper posture (α ≈ 180°) for the triceps muscle. Based on the estimated value
, upper and lower limits of the optimization process for the gains
were set according to
(31)
to enforce realistic values. The limits of the offset
were set to ±0.05m.
The five free parameters (parameter set) were optimized in two steps. In the first step, a random walk search was used to scan the overall parameter space in the previously defined limits. This helped to escape a local minimum that was otherwise found for small parameter values when the forearm is mainly driven by gravity. For the random walk, 1044 parameter sets were randomly taken from the interval between zero and one of a uniform distribution (rand(n,m) function, Matlab V2019b, The Mathworks Inc., Natick, MA, USA). Each parameter of the parameter sets was mapped to the interval between the lower and upper limits. For each parameter set, the model was simulated and the sum of squared errors for the elbow angle trajectories was stored. Afterwards, the parameter set with the smallest squared error was used in the next step. In the second optimization step, a gradient descent algorithm (lsqnonlin, Matlab V2019b) was initialized based on the parameter set with the lowest error from the first step. The lower limits of gain values () (Eq (31)) were set to zero in this step to allow the optimizer to fade out an EMG. A maximum of 20 iterations and a minimum step size of 1 ⋅ 10−3 were chosen as termination criteria. Trust-region-reflective was used as optimization algorithm. For each of the subjects, the eight experiments were optimized by tuning the five free parameters while the previously fixed 37 parameters remain unchanged throughout all experiments.
2.3 Characteristics of chosen error and quality scores
To evaluate the predictive performance of the musculoskeletal model, EMG-data as recorded in the experiments (see section 2.1) was fed into the model and the model was used to simulate the course of the elbow angle θsim. Based on the difference between simulated course θsim and real course θmeas of the elbow angle the free parameters of the model were optimized and evaluated. For the comparison between measured angle and simulated angle in section 3, the mean absolute error (MAE) was calculated as shown in Eq (32).
(32)
With n being the number of measured values during one experiment. The mean absolute error (MAE) is an error measure which is independent of the length n of the measurements and allows comparison of results across experiments with different recording times. In addition, the normalization of the MAE to the range of the θmeas as shown in Eq (33) allows for the comparison across different experiments/subjects with different movement behavior.
(33)
The behavior of the two error measures nMAE and MAE for different signal shapes and amplitudes is illustrated by the imaginary movement data θ for an elbow joint in Fig 6. In Fig 6(A) two triangular curve shapes with different amplitude (
as solid line in red|green) are shown. In this example, it was assumed that the simulation model for both cases makes a constantly incorrect prediction of θsim = 0 (dashed line in red|green) which would be the mean posture of the given imaginary elbow movement. In Fig 6(B) two sinusoidal curve shapes again with different amplitude (
as solid line in red|green) are shown together with the again incorrect simulation model prediction of θsim = 0 (dashed line in red|green).
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.
The respective MAE and nMAE are given in the first two rows of Table 2 for Fig 6(A) and in row three and four for Fig 6(B). It can be seen that the nMAE is independent of the signal amplitude (green vs. red) but sensitive to signal shape ((A) vs. (b)). Fig 6(C) exemplarily shows simulation model predictions (dashed lines in red|green) that are at 80% of the amplitude of the respective measurements (solid lines in red|green) again resulting in the same nMAE value. An nMAE of 0.1 at a maximum angle range of 90° corresponds to a mean deviation of 0.1 ⋅ 90° = 9° between θmeas and θsim. The minimal value of the nMAE is zero.
Note that the nMAE is not independent of the waveform. Therefore, it is only conditionally suitable for the comparison between different movements of subjects.
To improve the comparability between subjects, a quality score QS was introduced which normalizes the nMAE by using the nMAE of θconst (a constant mean elbow joint angle) per subject and experiment. This results in a ratio that should range between slightly below zero and one. This value is calculated as shown in Eq (34).
(34)
The quality score QS is zero if θsim is equal to θconst, i.e. if no or only a very small movement is apparent. This is the case in the examples in Fig 6(A) and (6B) for which the QS values are shown in the first four rows of Table 2. The quality score can become smaller than zero if θsim is not equal to θconst and has a constant curve. A quality score of greater than zero means a simulation in which θsim is not constant, i.e. the forearm is moving. In the model described in section 2.2, this movement can only occur through an active force due to the sEMG signals. With a quality score of one, θsim corresponds to the course of θmeas. Since the quality score is normalized to the respective nMAE at θconst, the quality score is independent of the signal form.
3 Results
Subjects performed forearm movements according to the experimental paradigm as described in section 2.1 (2 upper arm postures, 2 weights (dumbbell, [2kg, 4kg]), 2 movement speeds ([0.25Hz (slow) and 0.5Hz (fast)]) for the forearm movement). A metronome was used as acoustic reference. The subjects were asked to move the forearm periodically in a continuous manner reaching the reversal points of the movement in time with the rhythmic acoustic signal. This requirement was expected to lead to a rather sinusoidal movement behaviour of the elbow angle. Fig 7 shows selected examples for different courses of θmeas (green) for the upper (A-D) and lower posture (E-H). It can be seen that subjects sometimes tended to accelerate strongly at the extreme positions leading to a more triangular rather than sinusoidal waveform.
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).
As described, 36 parameters of the simulation model were set based on measurements of anatomical features and values from literature and were therefore valid for a subject independently of the experimental conditions. The remaining 5 free parameters were optimized per experimental condition. Fig 7 depicts the simulation results θsim in black. The subjects and experiments were chosen exemplary to give an impression of the quality score QS in approximately 25% steps. Fig 8 course shows data from two subjects in the same experimental condition (slow, 2 kg) but at different postures ((A) lower posture, (B) upper posture). The subject in Fig 8(A) shows a more sinusoidal course of θmeas as compared to the more triangular course in Fig 8(B).
(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).
The fourth and fifth rows show the values of the optimised 5 free parameters. In particular, the values for the sEMG amplification factors k vary with the EMG signal strength/quality (location of the applied electrodes and quality of the electrode-skin interface). A k of 0 means that this EMG was faded out by the optimiser. As described, 36 model parameters were fixed for one subject and used across all experiments. Among the subjects these parameters differ but were not optimized. The 5 free model parameters per subject are used for the adaptation to postures and experimental conditions. A comparison of the four experiments per posture for one subject is shown in Fig 9. The values for the k and Loffset are given in a normalized form. The boundaries for the optimization process (see section 2.2) are used for the normalization.
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).
In total, experiments were conducted with 31 subjects (Data was published in [23]) and the simulation models were parameterized (36 parameter for each subject) and optimized (5 free parameters per experimental condition). The quality of the sEMG-based model predictions for the movement trajectories θsim of the forearm with respect to the measured trajectories θmeas is depicted in Fig 10 as nMAE for all subjects. Fig 10(A) contains the results for the lower posture (biceps dominant), (B) for the upper posture (triceps dominant).
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].
In the upper panel of Fig 10(A), the nMAE values for each subject are plotted in one row. The nMAE result for each experimental conditions is marked with a symbol (cross, square, triangle, circle) and thus represents the optimization of the five free parameters for the respective condition. The lower panel of Fig 10(A) shows summarized data of the upper panel as box plots, one for each experimental condition (speed-weight-combination). Values for the respective quartiles are given in the top part of S5 Table. The median values (Q2) of the nMAE for the lower posture correspond—according to the arrangement in the lower panel of Fig 10(A)—from top to bottom (fast-2 kg ≙ 0.20, fast-4 kg ≙ 0.18, slow-2 kg ≙ 0.17, slow-4 kg ≙ 0.16).
The upper panel of Fig 10(B) follows the same structure as the upper panel in Fig 10(A) but for the upper posture. The lower panel of Fig 10(B) also shows summarized data as box plots. The median values (Q2) of the nMAE for the upper posture correspond—according to the arrangement in the lower panel of Fig 10(B)—from top to bottom (fast-2 kg ≙ 0.21, fast-4 kg ≙ 0.19, slow-2 kg ≙ 0.17, slow-4 kg ≙ 0.18).
As supplement to Fig 10, the quality of the sEMG-based model predictions for the movement trajectories is depicted in Fig 11(A) and 11(B) as quality score QS for all subjects also represented as box plots in the respective lower panels.
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).
The values for the respective quartiles of the QS box plots are given in S6 Table. The median values (Q2) of the QS for the lower posture are (fast-2 kg ≙ 0.26, fast-4 kg ≙ 0.30, slow-2 kg ≙ 0.32, slow-4 kg ≙ 0.40) for the lower posture and (fast-2 kg ≙ 0.25, fast-4 kg ≙ 0.33, slow-2 kg ≙ 0.34, slow-4 kg ≙ 0.31) for the upper posture. The prediction qualities for both postures show medians of the QS above 0.25 which corresponds to a fit of the simulated data to the measured data which is visibly similar.
4 Discussion
In this work, a model has been presented that predicts limb movements of the human forearm based on sEMG signals from the main muscles involved in the biological actuation of the elbow joint. The prediction is based on a biomechanical model of the humerus, forearm and elbow joint including the associated muscles. Flexion of the elbow mainly relies on three muscles, the two-headed biceps brachii, the single-headed brachioradialis and the single-headed brachialis. The latter is located below the biceps. Some weaker flexor muscles located in the forearm were omitted. Elbow extension is also based on two muscles, the three-headed triceps brachii and the single-headed anconeus (the latter also tightens the joint capsule). Although in principle all these muscles are involved in the joint movement of the elbow, only the sEMGs of those muscles—respectively muscle heads—were recorded that can be easily measured from the outside (lateral and long head of triceps and short and long head of biceps). This was done on the assumption that this restriction is important for applications in exoskeletal controls which require a simple electrode placement that is as unaffected as possible by positional shifts [20, 21]. At the same time, this reduced sEMG electrode setup might be associated with a reduction in the quality of movement prediction (see quality score). Biomechanical prediction models use a variety of parameters such as muscle lengths, locations and orientation of points of the skeletal subsystem under consideration, or physiological parameters such as time constants of muscle activation dynamics, to name a few. For the prediction model in this study, the total number of parameters that can be set in principle is 42 (see section 2.2). On the one hand, setting such a number of parameters individually, e.g. for modelling motion prediction within exoskeletal controls, is not practical. For an online optimization of all parameters, on the other hand, plausible physiological constraints would have to be met. The strategy chosen here is to measure easily accessible and palpable body points of the shoulder (acromion), elbow (medial/lateral epicondyles and olecranon) and wrist (styloid processes of radius/ulna) to estimate desired internal, not directly observable lengths via known correlations (respective references are given in section 2.2). Together with a maximum force measurement of the forearm, the number of parameters still to be set could be reduced from 42 to 5 (see section 2.2).
Assuming that these 5 parameters are sufficient to individualise the prediction model, movement experiments were carried out with 31 subjects in which cyclic forearm movements were each measured at two different speeds and two different weights in two different postures. The data set in this study is therefore quite extensive and uses more subjects than comparable work which fits movement models based on a smaller number of subjects and conditions (e.g. 1 to 5 subjects in [32, 47–50]). The data set of this study has been made available in a separate publication [23] for further scientific use. The corresponding results of the described modelling task were presented in detail in section 3. Two aspects in particular should be emphasised. Firstly, the results show individualised per-subject models in which 37 out of 42 parameters were determined by a single adjustment based on anatomical measurements, i.e. independent of experimental conditions. The downstream optimization of 5 parameters in relation to the concrete experimental conditions (posture, weight, speed) can therefore be carried out more quickly and can be used in later work as a basis for online adaptations, e.g. during the ongoing operation of an exoskeleton. Secondly, the quality of the prediction or fit of the forearm movement was expressed in different error or quality measures. In the representation as quality score (QS), the median of the QS varies over all experimental conditions and over all subjects between approximately 0.25 and about 0.4. Visual inspection of the results suggests that the median and the interquartile range (IQR) of the nMAE for experiments with 4 kg additional weight are lower than for the experiments with 2 kg additional weight. Also, the median and IQR of the nMAE for experiments with slow movements are lower than for those with fast movements. The median and the IQR of the nMAE for experiments with 4 kg are almost equal for both speeds. However, the slow 4 kg experiments show higher nMAE values than in the same experiments in the lower posture. This might be related to a stronger/faster fatigue of the subjects during movements with the heavier weight in the unfamiliar upper posture. The depiction as QS values shows the same tendencies as already described for the nMAE values in Fig 10.
Fig 7 shows exemplarily to which movement curves these QS values lead to. In particular, the phase (correct direction of the respective motion) of the low-frequency components of the movement curves seem to be mainly correct. Deviations between the actual and predicted course of the forearm movements seem to be more strongly reflected in the amplitude differences of the movements than in the phase of the movement. This is most likely explained by the fact that four of the freely adjustable parameters are the gain factors k of the sEMG signals and therefore influence the active torque/active movement.
Future work includes the further development of the domain model shown here towards a hybrid model that integrates additional, data-driven (ML-based) sub-models. Such a hybrid approach could, on the one hand, further increase the prediction quality and, on the other hand, identify and address over-simplifications in sub-models of the existing domain model. Such a hybrid approach should also combine the explainability of a pure domain model with the adaptability of a purely data-driven (black box) approach [51]. The quality of the movement prediction of the domain model in this work will serve as a reference for the described next steps.
Supporting information
S2 Table. Parameters for activation dynamics submodel.
https://doi.org/10.1371/journal.pone.0289549.s002
(PDF)
S3 Table. Parameters for contraction dynamics submodel.
https://doi.org/10.1371/journal.pone.0289549.s003
(PDF)
S4 Table. Parameters for musculoskeletal submodel.
https://doi.org/10.1371/journal.pone.0289549.s004
(PDF)
S5 Table. Values of quartiles for nMAE in different postures as shown in Fig 10.
https://doi.org/10.1371/journal.pone.0289549.s005
(PDF)
S6 Table. Values of quartiles for quality score QS in different postures as shown in Fig 11.
https://doi.org/10.1371/journal.pone.0289549.s006
(PDF)
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