Myoelectric prosthetic hands can restore or enhance independence for individuals with limb differences, enabling them to perform various activities of daily living [1, 2]. Machine learning-based myocontrol approaches offer intuitive control of advanced prostheses [3] and are currently available in commercial systems [4, 5]. Classification techniques enable control over multiple grasp types by defining an association between muscular activity and the desired grasp [6, 7], while regression methods establish a continuous mapping between the user's muscle activations and motor commands for the degrees of freedom (DoFs) of the prosthesis [8, 9]. These techniques typically learn the myocontrol model in a supervised way, meaning that surface electromyography (sEMG) measurements of the forearm's muscles are associated with prescribed motor commands during a calibration phase.

Supervised myocontrol relies on the assumptions that the distribution of the control signal remains consistent between training and testing conditions, and that training samples are accurately labeled [10]. However, meeting these assumptions in a realistic setting poses methodological challenges. The characteristics of sEMG signals can change over time due to factors like muscle fatigue, limb position, and electrode displacement [11, 12]. Common approaches to reduce this distribution shift involve capturing more of the signal variability in the training data [13, 14] or iteratively recalibrating the system with additional data over time [15, 16]. These methods come therefore at the cost of an increased burden on users by prolonging the data acquisition process. Additionally, accurately labeling samples can be difficult for individuals with limited residual muscle control, such as those with limb differences. Extensive preprosthetic user training is often required to generate muscle signals that are sufficiently distinguishable, stable, and repeatable for myocontrol. This typically includes mental practice, emulation of specific gestures using the phantom limb, and sEMG visualization using biofeedback [17, 18]. However, such training can be demanding and requires supervision from healthcare professionals. The requirement for expert guidance typically confines preprosthetic training to clinical facilities, which increases the associated costs, limits the user's exposure to training, and potentially slows down the adoption of the myocontrol technology [18].

Unsupervised myocontrol is a desirable alternative to supervised myocontrol, as it eliminates the need for hard-to-obtain labeled training data. Existing unsupervised myocontrol approaches derive low-dimensional approximations of the muscular input, corresponding to distinct muscle coactivation patterns, and employ them as control commands for the kinematic or kinetic variables of interest [19–22]. This is based on the neuromotor control principle that the human nervous system efficiently realizes movement by recruiting and coordinating non-redundant muscle synergies [23–25]. In this context, the nervous system treats the activations of each muscle synergy as high-level motor commands that can be combined to generate the muscular activity necessary to accomplish the desired movement. This also entails that information about the synergies' structure and coactivation is encoded into multichannel sEMG measurements of the muscular activity [24].

Nonnegative matrix factorization (NMF) algorithms are commonly utilized to extract muscle synergies from sEMG signals. The advantage of this specific factorization is that it decomposes signals into linear nonnegative combinations of nonnegative components, which mirrors the central nervous system's approach of combining nonnegative antagonistic muscle activations [24–26]. In addition, using these components as control inputs for prosthetic devices enables users to naturally control multiple prosthetic functions at once and adjust their intensity proportionally. However, standard NMF solutions can be ill-posed, and they therefore need particular training procedures or formulations to enforce the identification of minimally overlapping components that could serve as reasonable proxies for muscle synergies [19, 26].

Jiang et al [19] proposed a minimally supervised approach for simultaneous and proportional (SP) myocontrol of a virtual cursor using muscle synergies related to wrist movements. To identify these synergies, they developed a DoF-wise calibration of the myocontrol system, which involved concatenating partial NMF models trained on sEMG data of antagonistic movement pairs, taking advantage of the distinct muscle activation patterns each pair generated. This method reduced supervision compared to traditional approaches, but still required users to perform specific movements in a predefined order, potentially posing challenges for individuals with limb differences.

Building on this work, Lin et al [20] developed an extension that imposed sparsity constraints on NMF to allow for a more flexible calibration procedure. During this calibration, participants were allowed to perform random wrist movements, engaging multiple DoFs of the wrist simultaneously. The sparse NMF formulation encouraged the extraction of minimally overlapping components, which were then manually associated with the control of the desired cursor directions. This manual association was performed to ensure an intuitive correspondence between muscle synergies and cursor directions but required direct supervision during the process. Moreover, their calibration procedure explicitly excluded finger movements, which could hinder the identification of potentially more effective muscle commands and could prove challenging for individuals who struggle to isolate wrist and hand movements.

Yeung et al [21] designed an adaptive version of the paradigm by Lin et al [20], in which the factorization model was automatically updated during operation to account for changes in muscle synergies caused by the nonstationarity of sEMG and the user's adaptation to the myocontrol system. The same quasi-unsupervised calibration procedure was followed to build a myocontrol model for a prosthetic wrist, which involved performing specific actions in an unstructured manner and manually defining a biomimetic motor mapping between muscle synergies and wrist actions. The myocontrol system automatically updated the factorization model when it detected model degradation, characterized by increased coactivation of antagonistic muscle synergies. Model updates were made possible by adopting an incremental NMF approach with sparsity constraints and a forgetting mechanism to gradually reduce the influence of older data. Even though this incremental NMF formulation allowed for fully unsupervised model updates, the paradigm still relied on a partially supervised and constrained calibration procedure to create a biomimetic motor mapping for myocontrol in the first place.

Other approaches have also attempted to reduce the amount of supervision necessary for defining biomimetic motor mappings. This includes methods that identify relationships between muscular activity and kinematic variables in a shared latent space [27, 28], or those leveraging musculoskeletal models to estimate forearm muscle forces directly from electromyographic recordings [29]. However, these strategies still require a loosely supervised calibration phase, involving synchronized acquisition of sEMG and ground truth data for the estimated kinematic variable.

As an alternative to biomimetic mappings, abstract motor mappings can be adopted to implement fully unsupervised myocontrol. This type of mapping, commonly used in supervised myocontrol approaches, links muscle activations to hand gestures without requiring a direct physiological relationship between them [30, 31]. Research shows that humans can learn such arbitrary mappings, including muscle synergy-based ones, through closed-loop interaction with a myocontrol system, making them a viable approach for prosthetic control [24, 32]. Abstract motor mappings based on muscle synergies provide flexibility and robustness, enabling users to control complex hand actions with comfortable, reliable, and stable muscle activations [22, 33], while also being more resistant to variations in myoelectric signals due to the muscle synergies' focus on underlying muscle coactivation structures [24].

Gigli et al [22] used abstract motor mappings to devise a fully unsupervised coadaptive simultaneous and proportional myocontrol paradigm for hand and wrist actions. Similarly to the method from Yeung et al [21], this also originated as an adaptive extension of the work of Lin et al [20]. However, this approach completely eliminated the need for initial model calibration and allowed users to identify viable muscle inputs autonomously. This was achieved through a combination of adaptive NMF, an abstract motor mapping, and a straightforward interaction strategy. An adaptive sparse NMF formulation was devised to extract muscle synergies from the user's sEMG in realtime. An abstract motor mapping was established by arbitrarily associating the extracted muscle synergies with predefined hand actions of the myocontrolled hand. As users interacted with the system and discovered action-triggering muscle patterns, the synergies were continuously refined for enhanced control. This approach provided an adaptive and low-dimensional visualization of the muscle space, enabling users to discover complex muscle coactivation patterns, including those difficult to discern with standard biofeedback methods. Moreover, the approach demonstrated performance comparable to state-of-the-art supervised adaptive myocontrol approaches.

A limitation of all existing unsupervised myocontrol paradigms that rely on NMF, is that the number of components for sEMG factorization must be set to match the preexisting number of independent muscle synergies that the user can generate. Specifically, allowing too many NMF components might lead the factorization model to identify components unrelated to physiological muscle synergies, potentially resulting in unintended activations of the myocontrolled hand. Determining how many independent and stable muscle synergies the user can elicit is challenging. First, the amount of sensors used by the sEMG measurement system limits the number of detectable synergies [34]. Second, the individual's preexisting motor capacities can significantly impact the number of synergies elicited [25, 35]. Lastly, the number of distinct synergies may increase over time as the individual progressively familiarizes themselves with more motor tasks [25, 36]. In practice, determining the number of independent muscle synergies often requires extensive collaboration between the user and a clinician, making more autonomous methods for identifying and refining available synergies desirable.

An alternative approach is to use a progressive learning strategy for myocontrol functions, where users begin with a single function and gradually 'unlock' additional functions as they master existing ones. This method mirrors the progressive nature of human motor development, which involves the ongoing expansion and refinement of motor functions [25, 37, 38]. Throughout an individual's life, innate reflexes are integrated with newly acquired rudimentary motor skills, which are then refined and combined to form more advanced and specialized skills. This progression is connected to the development of muscle synergies, as new motor skills are achieved by adapting preexisting muscle synergies to meet the demands of tasks and efficiency [25, 38]. Moreover, the challenge point framework theory suggests that a progressive motor learning approach would support the acquisition of new motor functions. In fact, adapting the task difficulty to an individual's current skill level has proven helpful to regulate the learning workload and ultimately accelerate motor learning [39–41].

A sequential NMF formulation could be employed to implement a progressive motor learning procedure [42, 43]. This algorithm learns the factorization model by adding one component at a time, ensuring the stability of existing components when new ones are introduced. However, existing sequential NMF methods are not suitable for incremental settings as they are based on iterative warm reinitialization of progressively larger models and retraining on historical data to preserve the continuity of the existing components. Simply discarding the historical data when attempting online sequential NMF is unlikely to be successful, as the lack of context may lead to a loss of continuity in the existing components. Therefore, there is a need for an online factorization method that maintains component continuity without requiring the storage of historical data.

In this work, we introduce progressive unsupervised myocontrol (PUM), a fully unsupervised and coadaptive paradigm inspired by the progressive nature of human motor learning. PUM enables users to autonomously learn to control the functions of a myocontrolled hand one at a time. These functions are implemented through an abstract mapping between the users' muscle synergies and the desired actions of the hand and wrist. Users refine muscle synergies for myocontrol autonomously while familiarizing themselves with the system and request to unlock new functions as they become proficient with the existing ones. The result is a coevolving and coadaptive interaction dynamics between the user and the system. To achieve this, we extend the adaptive NMF from [22] with an algorithmic procedure to increase the number of components while preserving the existing ones without explicitly storing historical data. Moreover, we adjust the loss function to reduce the overlap between the identified components and to improve their stability over time.

In a multi-session user study, we evaluate how effectively PUM enables users and the myocontrol system to synergistically learn a control model in a completely unsupervised manner. We specifically investigate the performance of individuals with limb differences (LD), who stand to benefit the most from an unsupervised myocontrol paradigm, in comparison to non-disabled (ND) participants, who represent the best-case scenario for myocontrol due to their more extensive motor skills. Moreover, we examine how PUM compares to a non-progressive unsupervised myocontrol (UM) paradigm, based on that of [22], which serves as a baseline for identifying potential advantages and limitations of our approach. Our assessments and comparisons are based on the workload associated with the progressive learning of motor skills, as well as the evolution and retention of myocontrol performance in a series of target achievement control (TAC) tests involving a virtual hand.

The paper is structured as follows. In section 2, we detail the methods employed for the PUM paradigm. Section 3 presents the study's findings, followed by a discussion of their implications in section 4. The appendix includes mathematical derivations of the factorization algorithm utilized in the myocontrol paradigms.

In this section, we introduce the PUM paradigm and discuss its relation to the non-progressive UM paradigm adapted from Gigli et al [22]. We then outline a multi-session study where we assess the effectiveness of PUM in enabling participants to progressively learn, refine, and retain control of a virtual hand's functions, and compare its performance to that of UM.

2.1. PUM

The PUM paradigm is inspired by the way humans progressively develop their motor skills when learning new tasks. The system factorizes muscular inputs into muscle synergies and arbitrarily maps them to functions of the myocontrolled virtual hand, while users learn the motor mapping by interacting with the system. Upon user request, the system adapts the number of synergies to accommodate an increasing number of functions, while aiming for minimal disruption to previously learned synergies. A schematic overview of the control paradigm is presented in figure 1.

Figure 1.  Schematics for the progressive unsupervised myocontrol (PUM) paradigm. While the user interacts with the system to learn myocontrol functions, the factorization model refines the identified muscle synergies by conducting periodic unsupervised model updates. Myoelectric control is achieved by factorizing muscle activity into muscle synergies and arbitrarily mapping the encoding coefficients to predefined myocontrol functions, obtaining motor commands for the orange myocontrolled hand. Users progressively unlock more myocontrol functions on demand. This scheme evolves to accommodate user skill evolution and allows the user and system to coadaptively refine control over new motor functions. A gray virtual hand serves as a reference during performance evaluation, presenting the users with functions to replicate in realtime with the myocontrolled hand.

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Standard image High-resolution image 2.1.1. Progressive incremental sEMG factorization

We introduce progressive incremental sparse nonnegative matrix factorization (P-ISNMF), an algorithm that adaptively computes an NMF model and enables online identification of additional components while preserving existing ones without model retraining. It builds upon incremental sparse nonnegative matrix factorization (ISNMF) [22], an adaptive NMF variant with sparsity constraints and a forgetting mechanism to discount outdated information. In addition to incorporating a progressive mechanism, our proposed approach features an improved objective function that results in sparser and more stable components.

NMF approximates a nonnegative matrix V of size n × s as the product of nonnegative factors W of size n × r and H of size r × s, that is, [44]. When the columns of V represent a series of s n-dimensional data samples, the columns of W represent a set of r basis vectors and those of H a series of s r-dimensional encoding coefficients that indicate the relative contribution of the bases to each data sample. In the context of myoelectric control, where data samples correspond to positive envelopes of the myoelectric signal, the bases and encoding coefficients can be loosely interpreted as muscle synergies and their activations.

Algorithm 1. ISNMF. © 2022 IEEE. Reprinted, with permission, from [22]. Input stream of n-dim nonnegative samples Parameter r, β, γ, µ, ε, tmax 1    2    3    4 while true do 5      6   n × k matrix with k new samples from    7   If m = 1 then 8     positive random matrix   9   else 10     11   end 12   positive random matrix   13      14      15   repeat 16        17        18        19        20        21        22   until    23      24      25 end

ISNMF [22] is an incremental solution to the NMF problem that updates the factorization model with new data while discounting the contribution of previous data, without the need for storing it. We present a refined version of this algorithm that features an improved objective function for a sparser factorization and increased stability, and we provide a complete derivation in the appendix. In the following, we employ block notation for matrices, with subscripts identifying specific matrix blocks and superscripts representing the matrix status at particular updates. For instance, V j denotes the data samples received during the jth update, W m indicates the bases values at the mth update, and corresponds to the encoding coefficients computed during the mth update for the block of data samples collected at the jth incremental update (with for obvious reasons). Furthermore, all product, division, or power operators applied to matrices in the update rules are understood to be elementwise. At the mth update, the algorithm refines the factorization model by minimizing the following objective function that incorporates new data and discounts past contributions

The forgetting factor diminishes the influence of old data exponentially via , ensuring the model adapts without excessive reliance on historical data. and denote the Frobenius and the elementwise L0.5 norms respectively. The scalars and determine the regularization strength for the bases and encoding matrices.

The new objective function improves the one from the original method [22] by also scaling the regularizer of the bases with the exponential forgetting factor. The motivation for this change is to ensure that all three terms are balanced identically, regardless of the number of block updates 5 . Furthermore, the new objective function replaces L1 regularization for the encoding coefficients with a sparser L0.5 regularizer. A preliminary empirical validation confirmed that both modifications had the desired effect on stability and sparsity.

An incremental solution to this problem based on multiplicative updates is given in algorithm 1. The derivation of the incremental algorithm, found in the appendix, relies on the assumption that the model undergoes small changes in each update, meaning that old encodings remain practically unchanged when new data arrives. As a result, they no longer have to be optimized and past data samples and encoding coefficients can be aggregated into fixed-size history matrices rather than being stored explicitly. This, in turn, leads to a significant reduction in the computational and memory complexities of an incremental update, which are now constant in the number of updates [45, 46]. The hyperparameter r specifies the number of NMF components and is chosen to be lower than the data dimension. The tolerance ε > 0 and the maximum number of iterations establish the stopping condition for the iterative minimization of the objective function within each model update. The elements of W