The EMG controllers put forward in this paper are compared based on a virtual control task of balancing an inverted pendulum in one degree of freedom, as depicted in Fig. 1. In addition to gravity acting on the pendulum, perturbations, in the form of short torque pulses, tip it over and challenge the user to keep it upright. Via EMG measurements of their lower leg muscles, users can modulate a torque acting on the pendulum to steer its movements. To calculate this control torque, four different controllers are used: A proportional controller, a model-based controller employing a Hill-type muscle model, and two reflex controllers (weak and strong) that augment the model-based approach with simulated stretch reflexes. The following sections elaborate on the torque control architectures and conducted user study design.

Fig. 1

Photo of the experimental setup. A user controls the inverted pendulum via EMG signals from their lower leg muscles. Red and blue arrows symbolize the input signals controlling the pendulum. The red arrow on the screen indicates a perturbation to the right

2.1 EMG-based torque control & virtual control task2.1.1 Hill-type muscle model

The presented torque control architecture contains a Hill-type muscle model, of which two instances are employed for the torque calculation described in Sect. 2.1.3. All parameters and measures listed are general and exist in both instances independently. If a value belongs to muscle 1 or 2 specifically, it is indicated by a 1 or 2 appended to the respective variable’s index.

Based on the review by Caillet et al.  [31], the model used was designed to consist of components that are well-established in the literature, while keeping the number of parameters low. The latter is meant to ease the individualization of the model to participants in future experiments. The model consists of a force-generating element (FG), parallel elastic element (PEE) and series elastic element (SEE). Its structure is shown in Fig. 2. Given the mechanical arrangement of components, the SEE length \(l_\textrm\) and FG length \(l_\textrm\) (which is also the PEE length) add up to the total muscle length

$$\begin l=l_\textrm+l_\textrm, \end$$

(1)

while there is a force equilibrium

$$\begin f_\textrm=f_\textrm+f_\textrm \end$$

(2)

between the relative forces of the SEE, FG and PEE in reference to the maximum isometric force \(F_0\). The total muscle force F is given by

$$\begin F=F_0 f_\textrm. \end$$

(3)

The FG has a force-length and a force-velocity relationship represented by \(f_\textrm\) and \(f_\textrm\). Its relative force

$$\begin f_\textrm=a f_\textrm f_\textrm \end$$

(4)

depends on the activation a. The force-length relation is given by

$$\begin f_\textrm= 1-\left( \dfrac-l_0}\right) ^2 & ,\ \left|\dfrac}-l_0}\right|\le W \\ 0 & ,\ \text \end\right. }. \end$$

(5)

The maximum value of one is reached at the optimal FG length \(l_0\), decreases quadratically across the width W, and becomes zero beyond that  [32, 33].

The force-velocity relation is defined by two rational functions as

$$\begin f_\textrm= \dfrac}}v_\textrm}} & ,\ v\le 0\\ f_\textrm-\left( f_\textrm-1\right) \dfrac}} v_\textrm}} & ,\ v> 0 \end\right. }\ , \end$$

(6)

where v is the time derivative of \(l_\textrm\)  [34]. For \(v=0\), \(f_\textrm\) equals one. It decreases rapidly for concentric contraction (\(v\le 0\)), and increases for eccentric contraction (\(v>0\)) up to the maximum value \(f_\textrm\), present at the maximum relaxation speed \(v_\textrm\). The velocity v is limited to \(\pm v_\textrm\). \(a_\textrm\) and \(a_\textrm\) are shaping parameters for the concentric and eccentric contraction curves, respectively, determining the steepness of \(f_\textrm\) around the static case \(v=0\). Finally, the relative forces of the SEE  [35] and PEE  [32] amount to

$$\begin f_\textrm&= k_\textrm\left( \dfrac-l_\textrm}}\right) ^2 & ,\ l_\textrm\ge l_\textrm\\ 0 & ,\ l_\textrm<l_\textrm \end\right. } \end$$

(7)

$$\begin f_\textrm&= \left( \dfrac-l_0}\right) ^2 & ,\ l_\textrm\ge l_0\\ 0 & ,\ l_\textrm<l_0 \end\right. } \end$$

(8)

with the SEE slack length \(l_\textrm\) and SEE stiffness \(k_\textrm\). Equation 8 is synonymous with setting the PEE slack length and stiffness to \(l_0\) and \(1/W^2\), respectively.

The computation of muscle activation consists of two steps  [36]. Firstly, the delayed input \(u'\) is calculated by a discrete second-order differential equation

$$ \begin u^ (t) = & \left( + p_ + p_ p_ } \right)u(t - T_ ) \\ & - \left( + p_ } \right)u^ (t - T_ ) - p_ p_ \;u^ (t - 2T_ ), \\ \end $$

(9)

with the discrete poles \(p_\textrm\) and \(p_\textrm\) as well as the delay \(T_\textrm\). \(T_\textrm\) is the sample time. Secondly, there is the following nonlinear relation between the delayed input and the activation a given by

$$\begin a=\dfrac^-1}^A-1}, \end$$

(10)

with the shaping parameter A.

The geometry of muscle insertion and length dependency was adopted from the modeling of the ankle joint by Geyer and Herr  [37]. The muscle length \(l_\textrm\) changes with the joint angle \(\alpha \) as

$$\begin l_\textrm=l_0+l_\textrm-r_0\rho \left( \sin \left( \alpha -\alpha _\textrm\right) -\sin \left( \alpha _0-\alpha _\textrm\right) \right) , \end$$

(11)

where \(r_0\) is the maximum lever arm on the joint, \(\alpha _\textrm\) is the angle at which the maximum lever arm occurs, and \(\alpha _0\) is the angle at which the muscle is just slack with \(l_\textrm=l_0+l_\textrm\). The factor \(\rho \) adjusts the extent of length change and therefore enforces length limits. The lever variable arm r of the muscle on the joint is given by

$$\begin r=r_0\cos \left( \alpha -\alpha _\textrm\right) . \end$$

(12)

The two muscle model instances used are identical, but oppose each other. The relation of their respective parameters regarding the joint mechanics and insertion come down to

$$\begin \frac}}=\frac}} =\frac}}=-1. \end$$

(13)

Therefore, if muscle 1 shortens, muscle 2 lengthens and vice versa.

2.1.2 Simulated stretch reflex

Reflexes are involuntary responses to various stimuli generated in the spinal cord or brain stem  [20]. They can involve multiple sensory organs and muscles and adapt to different motor tasks. Such complex reflex structures are not applicable to the 1-DoF control task discussed here. Therefore, we focus on emulating the simple stretch reflex – which might be the most important muscle reflex  [20] – to be incorporated in the EMG control architecture. A stretch reflex makes a muscle contract when it gets stretched, while at the same time relaxing its antagonists  [20].

In some existing gait simulation  [37] and prosthesis control  [38] studies, positive force feedback was employed in a simulated reflex scheme. This is a similar approach, but cannot be employed in our control experiment. The mentioned papers switch controllers for different gait phases, interrupting the feedback loop during swing phase. In a continuous control loop like it is put forward in this paper, positive feedback from a muscle’s force to its input signal would create an inherently unstable behavior. Because an increase in muscle activation constitutes a further increase in muscle force, positive force feedback would result in permanent maximum contraction of both antagonistic muscles. Therefore, we base the simulated reflex on the speed of the occurring stretch.

In our model, the reflex signal of a muscle is given by

$$\begin u_\textrm= K_\textrm \dot_\textrm\dfrac} & ,\ \dot_\textrm>0 \wedge F>F_\textrm\\ 0 & ,\ \text \end\right. }, \end$$

(14)

where \(K_\textrm\) is a constant gain, \(\dot_\textrm\) is the time derivative of the muscle length and F is the muscle’s force. \(F_\textrm\) is the force of the antagonist muscle (\(F_\textrm=F_2\, ,\ F_\textrm=F_1\)). When \(F_\textrm\) is zero, the reflex is proportional to the muscle lengthening speed with \(u_\textrm=K_\textrm\dot_\textrm\) (for \(\dot_\textrm>0\)). The higher the force of the antagonist is in relation to the respective muscle’s force, the weaker the reflex becomes. This is meant to prevent a reflex to interfere with a quick movement that is driven by the opposing muscle. At all times, the reflex signal \(u_\textrm\) is limited to the interval [0, 1].

2.1.3 EMG-based torque controllers

In the experiment, three different control architectures are used, one of which comes in two variants, creating a total of four controllers. Firstly, the model-based controller calculated the control torque based on the presented hill-type muscle model. Secondly, two reflex controllers extend the model-based one by incorporating the reflex model with two different intensities. Lastly, a simple proportional controller calculates a control torque without the muscle model.

Model-based controller

The model-based controller employs two instances of the hill-type muscle model. They are driven by the EMG input \(u_\textrm\) as

$$\begin u=u_\textrm. \end$$

(15)

The control torque \(\tau \) is computed from the forces of the two muscle models as

$$\begin \tau =F_1 r_1+F_2 r_2. \end$$

(16)

While all muscle forces are always positive, the muscles act on the pendulum in opposite directions, as depicted in Fig. 2. To represent this geometry, the lever arms have opposite signs in the model (see sect. 2.1.5 for details on model parameters).

Fig. 2

Schematic representation of the experimental setup. Participants wore an orthopedic ankle-foot brace that locks the ankle, while two surface electromyography sensors (marked red and blue) measure activity of lower leg muscles (solid lines). The control torque is calculated by one of three controller types: proportional, model-based and reflex. Together with the perturbations, the control torque influences the pendulum dynamics. Depending on the pendulum position, muscle length is fed back into the model-based and reflex controllers (dashed lines). In addition, muscle lengthening velocity is fed back into the reflex controllers (dotted lines). Through the pendulum display on the screen, visual feedback enables the user to balance the pendulum

Reflex controllers

The reflex controllers are based on the same muscle model (including identical parameters), but the EMG amplitude gets combined with the reflex signal to compute the muscle model input u, given by

$$\begin u=u_\textrm\left( 1-u_\textrm\right) +u_\textrm, \end$$

(17)

where \(u_\textrm\) is the antagonist’s reflex. Therefore, a reflex signal increases the same muscle’s input by addition, while impeding the antagonist. This mimics the stretch reflex impacts mentioned in sect. 2.1.2. It is to be noted that because of the opposing orientation of the muscles, \(\dot_\textrm\) can only be positive in one muscle at a time. Consequently, there can never be a reflex in both muscles at the same time. Like the reflex signal, the muscle input signal u is limited to the interval [0, 1].

To investigate the influence of the reflex strength, a weak and a strong reflex variant are used. Besides the strong reflex controller having an increased gain \(K_\textrm\) compared to the weak one, the two are identical.

Proportional controller

In the simple proportional controller, the torque was computed directly from the EMG inputs by

$$\begin \tau =\left( u_\textrm-u_\textrm\right) K_\textrm , \end$$

(18)

with the constant gain \(K_\textrm\). The EMG measurements are subtracted from each other, causing the EMG channels to have opposite effects on the control torque.

2.1.4 Inverted pendulum dynamics

The virtual inverted pendulum consists of a massless rod with a point mass at its top. It has the angular position \(\alpha \) and its dynamics are given by

$$\begin ml^2\ddot=ml\textrm\sin \left( \alpha \right) +\tau +\tau _\textrm, \end$$

(19)

where m, l, and \(\textrm\) are the mass, pendulum length, and gravitational constant, respectively. The null position (\(\alpha =0\)) is the upright position, while positive angles represent a tilt to the right. \(\tau \) is the torque exerted by the EMG controller, \(\tau _\textrm\) is the perturbation torque introduced in the experimental trials. The perturbations occur as rectangle pulses of \(0.5\,\textrm\) duration. All perturbations have the same amplitude \(}_\textrm\), but vary in their sign.

The pendulum can only move freely up to \(\alpha =\pm 45^\circ \), where it comes to a hard stop. These limits are indicated on the screen by solid gray triangular blocks.

2.1.5 Model parameters

The presented equations describing the behavior of the controllers contain a multitude of parameters. Their values shape the model behavior and are therefore an important design element. In general, the pendulum control task is not meant to resemble the behavior of a human leg or body. The focus is rather on providing an intuitive control task with an adequate difficulty that allows to compare the discussed control approaches.

The length of the pendulum is chosen to \(l=10\,\textrm\). With the gravitational torque being proportional to ml and the inertia being proportional to \(ml^2\), the length l determines their ratio and therefore how quickly the pendulum tips over. In preliminary tests, these values yielded a manageable pendulum behavior. The pendulum mass was chosen arbitrarily to \(m=1\textrm\). This determines the amount of gravitational torque occurring at every angle. However, how much user input is needed to overcome gravity can be adjusted independently by assigning \(F_0\), later on. The gravitational constant is set to \(\textrm=9.81\,\mathrm \).

For the proportional controller, the gain \(K_\textrm\) was chosen so that 30 % normalized EMG amplitude is necessary to just overcome gravity at the \(\pm 45^\circ \) limit stop. This yields \(K_\textrm=23.12\,\textrm\).

For the muscle model, the most influential parameters are the maximum isometric force \(F_0\) and the length parameters \(l_0\) and \(l_\textrm\)  [31]. Since they determine the relation of joint angle and muscle length, the geometrical parameters \(r_0\), \(\rho \), \(\alpha _\textrm\), and \(\alpha _0\) also play a major role in designing the control behavior. The two muscle model instances were decided to be identical. This symmetry does not reproduce the physiological strength relation of plantar- and dorsiflexor muscles, but yields a more intuitive control experiment. The parameters were adopted from the soleus model by Geyer and Herr [37] and are shown in Table 1.

Table 1 Muscle model parameters adopted from Geyer and Herr [37]Table 2 Model parameters taken from literature

By taking these parameters from a gait simulation study  [37], we expect a reasonably natural behavior and resulting length range for the muscle, at least within the range of motion that would be usual for the soleus (interval \([-20^\circ ,40^\circ ]\) for muscle 1, interval \([-40^\circ ,20^\circ ]\) for muscle 2  [37]). To keep the control task challenging and comparable to the proportional controller, it is also important to consider the passive elastic behavior of the muscles. It would be possible to parameterize them in a way so that the nonlinear elastic elements keep the pendulum upright without user input. With the given values for \(a_0\), both muscles are slack and therefore forceless for \(u=0\) within \(\pm 10^\circ \). Therefore, no passive assistance is provided in that angular range. To further facilitate an unbiased comparison between the controllers, the maximum isometric force \(F_0\) was chosen so that the same input of 30 % is needed to cancel gravity at \(\pm 45^\circ \) without changing the length of the FG (\(v=0\)). The remaining muscle model parameters were taken from literature and are listed in Table 2.

The reflex model only has the single parameter \(K_\textrm\) and the influence of its value on the controller performance is part of the scope of this study. At very low values the reflex has negligible influence on pendulum motion, whereas excessively high values over-damp the dynamics, producing unnatural and less intuitive behavior in the control experiment. Some studies suppose a gait-phase dependent stretch reflex contribution of about 50 % to soleus activation  [39]. This relates to the total activation including additional neural signals stimulating the muscle. In the pendulum model, the reflex contribution needs to be defined relative to the maximum input of \(u=1\). Since our reflex model also does not incorporate modulation based on gait phase or other system states, a different approach for the parameterization is required. We based \(K_\textrm\) around the maximum speed reached by the pendulum tipping over freely. Qualitatively assessing the effects of different gains in preliminary exploratory tests, we chose levels of 5 % and 10 % for \(u_\textrm\) for that speed, covering the estimated appropriate range for a noticeable reflex effect without inadequately high damping. This yields \(K_\textrm=5.84\) for the weak and \(K_\textrm=11.68\) for the strong reflex controller. Because small variations in \(K_\textrm\) seemed barely noticeable in the preliminary tests, only these two levels were included in the study.

Finally, the perturbation amplitude was set to \(}_\textrm=5.78\,\textrm\), which is a fourth of the proportional controller gain \(K_\textrm\). Therefore, each perturbation equals an EMG input of \(25\,\%\) normalized amplitude for \(0.5\,\textrm\). This also was determined in preliminary tests to be a substantial but manageable perturbation strength.

2.2 Experimental setup & user interface

As shown in Fig. 1, users sit in front of a screen that displays the pendulum. They are wearing an orthopedic ankle-foot brace that locks the ankle in a neutral angular position, which serves two purposes. First, it allows users to quickly produce meaningful muscle tensions and EMG amplitudes, because independently of the leg position, the foot can always push against a resistance. Second, it prevents any perception of a connection between the angular positions of the ankle and the pendulum, since the ankle does not move.

EMG activity of their soleus and tibialis anterior muscles is measured as the control input and the following data processing is carried out. The data is measured at 2148 Hz using Delsys Trigno Avanti sensors (Delsys, Natick, MA, USA). The data is first filtered with a second-order \(20\,\textrm\) Butterworth highpass, then rectified and filtered with a second-order \(2\,\textrm\) Butterworth lowpass  [10]. The resulting envelope gets normalized by a maximum value determined before the experiment from maximum voluntary contraction. Finally, the resulting EMG envelope is downsampled to the model sample rate of \(T_\textrm=500\,\textrm\) and fed into the controller as the EMG input signals \(u_\textrm\) (soleus) and \(u_\textrm\) (tibialis anterior).

The calculation of control torque and pendulum dynamics is implemented in Simulink, based on Matlab R2024b (Mathworks, Natick, MA, USA). The model runs at 500 Hz using the Simulink Desktop Real-Time Toolbox.

A two-dimensional animation of the pendulum is displayed in a Matlab App as shown in Fig. 1. Two colored lines (red for muscle 1, blue for muscle 2) illustrate muscles pulling on the pendulum. They increase in width based on the EMG input amplitude, providing feedback about the input to the user. The displayed attachment and corresponding length change of the lines do not match the simulated lever arms and muscle model lengths. The lines rather serve the purpose of confirming an intended control input to the user, helping to move the pendulum in the intended direction. The same is true for the displayed proportions of the pendulum itself. During perturbation occurrences, a red arrow above the pendulum indicates their presence and direction. The desired upright position is illustrated by a dashed silhouette of the upright pendulum.

2.3 Experimental protocol

To compare the controllers, we recruited 16 healthy participants (9 male, 7 female; age: \(33.2\pm 12.1\); body weight: \(74.3\pm 14\,\textrm\)) for a repeated-measures design experimental study. All participants gave informed consent. The experimentation was conducted in accordance with the Declaration of Helsinki and a vote from the Ethics Commission at FAU (vote 22-275-S).

The participants were seated in a chair and the EMG electrodes were placed on the soleus and tibialis anterior of their dominant leg (15 right, 1 left). The electrode positions were located by muscle palpation and observation of the EMG signals during flexion and extension of the ankle. The participants then donned the ankle-foot brace. They were free to rest or hold their braced foot in any comfortable position. The participants were introduced to the experimental setup and the control task of balancing the pendulum and keeping it as upright as possible. In a familiarization trial of 1–2 min using the model-based controller, they tried out the setup without perturbations. The model-based controller was used here because in line with the research questions, it acts as a baseline for the performance comparisons in Sect. 3. Afterwards, every participant performed four 2.5-minute trials, one with each controller. The order of controllers presented was randomized to eliminate effects of practice or fatigue on the results. The first 30 s of every trial were not evaluated and contained no perturbations. In the remaining 2 min, 20 perturbations with random signs occurred at randomized times with 3–7 s in between. After each trial, participants reported on the perceived task load by filling out a NASA TLX questionnaire  [40].

2.3.1 Evaluation metrics

In accordance with the research questions stated in Sect. 1, the independent variable in our experiment is the choice of EMG controller, and the dependent variables are control performance and user load. To answer the two questions, we mainly consider the comparison of the model-based controller with the proportional controller and the reflex controllers. Differences between the proportional and reflex controllers are not the focus.

To assess control performance, we take the angular root mean square error (RMSE) in relation to the upright position (\(\alpha =0\)) as the main indicator. Considering an angular range of \(\pm 5^\circ \) as successfully balanced, we evaluate the percentage of time spent outside this tolerance as another performance metric. Finally, hitting the \(\pm 45^\circ \) limits can be considered a failure in handling the pendulum. Therefore, we also take the number of \(\pm 45^\circ \)-limit hits as well as the percentage of time in contact with these limits into account.

User load is evaluated based on two metrics. The raw TLX scores  [41] of the NASA TLX questionnaires reflect on the perceived task load of every trial. In addition to this subjective measure, to also consider objective physical effort, we gauge the mean normalized EMG amplitude across both muscles for every trial.