# GEPPETO-OFC: An optimal feedback speech motor control model integrating biomechanical constraints and multisensory goal specification

Ny Tsiky Rakotomalala, Pierre Baraduc, Pascal Perrier

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Ny Tsiky Rakotomalala, Pierre Baraduc, Pascal Perrier. GEPPETO-OFC: An optimal feedback speech motor control model integrating biomechanical constraints and multisensory goal specification. JASA Express Letters, 2025, 5 (9), pp.095202. 10.1121/10.0039197. hal-05247251

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# GEPPETO-OFC: An optimal feedback speech motor control model integrating biomechanical constraints and multisensory goal specification

Ny Tsiky Rakotomalala, ID Pierre Baraduc,ª and Pascal Perrier ID

University of Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 Grenoble, France

ny.tsiky.onintsoa@gmail.com, pierre.baraduc@grenoble-inp.fr, pascal.perrier@grenoble-inp.fr

**Abstract:** We present a speech motor control model that integrates optimal feedback control (OFC) for movement planning and execution with a biomechanical model of the vocal tract. The OFC model was designed to optimize a cost function that combines motor effort and the achievement of multisensory goal zones. We show that the model can account for various aspects of speech production: kinematic properties, coarticulation, and sensorimotor integration. Furthermore, we provide evidence that hearing, proprioception, and tactile feedback may play distinct roles in shaping speech trajectories. © 2025 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

[Editor: Douglas D O'Shaughnessy]

https://doi.org/10.1121/10.0039197

Received: 24 May 2025

Accepted: 18 August 2025

Published Online: 5 September 2025

## 1. Introduction

Several models of speech motor control have been proposed to bridge the gap between high-level linguistic representations and the complex motor commands required for fluent speech production. The DIVA model (Guenther et al., 2006) introduced a hybrid control strategy combining feedforward and feedback mechanisms, with connections to neurophysiology. It effectively explains aspects of speech acquisition and motor learning. However, it relies on a geometric rather than neuro-muscular effector and uses corrective feedback commands through subtraction, which may oversimplify the complexity of sensorimotor integration. The TD model (Saltzman, 1986) incorporates dynamics by representing articulatory movements as coordinated patterns of muscle activations linked to constriction goals. However, it emphasizes neural dynamics over physical dynamics, and neither accounts for auditory feedback nor feedback delays, limiting its ability to address the influence of sensory information or biomechanical constraints. The FACTS model (Parrell et al., 2019) introduces a state feedback control, where commands are adjusted based on the optimally estimated state of the system as derived from feedback signals and their prediction fromfference copy. Although it provides a robust framework for incorporating prediction mechanisms, its ability to account for behaviors that may require auditory targets must be evaluated. Additionally, sensory feedback delays are not incorporated at this stage, and the plant has simplified dynamics.

Among existing models, the GEPPETO model (Patri et al., 2015) is notable for incorporating a realistic biomechanical model of the peripheral system. It accounts for key factors influencing movement, including mass, tissue elasticity, force generation mechanisms, and neurophysiological aspects such as the stretch reflex in the tongue (Ito et al., 2024). GEPPETO employs a purely feedforward control strategy based on the Equilibrium Point Hypothesis (EPH). Despite its simplicity, the model successfully captures several key aspects of speech production, such as articulatory loops (Perrier et al., 2003) or the speed-curvature relationship (Perrier and Fuchs, 2008). This indicates the crucial role of biomechanics in trajectory formation. However, GEPPETO lacks certain essential features, such as a high-level feedback control mechanism during movement execution; thus, GEPPETO cannot react in line to auditory or mechanical perturbations or take into account uncertainty (i.e., noise) in the sensory and motor signals. In addition, it fails to achieve precise time control.

In contrast, the stochastic optimal feedback control theory (Todorov and Jordan, 2002), a popular modeling approach in computational motor control, integrates sensory feedback and noise in real-time while also allowing for specifying goals at adequate time points (chosen with respect to prosody, pragmatics, interaction context, etc.). In addition, it introduces the concept of motor effort and provides a principled account of motor variability. While we and others (Houde and Nagarajan, 2011; Simko and Cummins, 2011; Parrell and Lammert, 2019) have already used either the concept of command optimization or the concept of state estimation for speech motor control, this letter introduces the first model that incorporates a biomechanical effector within the integrated framework of stochastic optimal feedback control.

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<footnote>ªAuthor to whom correspondence should be addressed.</footnote>
## 2. Model

The GEPPETO model is based on the premise that speech consists of goal-directed movements, with these goals being sensory in nature and related to phonemes. Our modeling work adopts this approach.

### 2.1 Planning

To generate speech movements, phonemic goals were concurrently defined in the auditory, proprioceptive, and tactile modalities. Each auditory goal was represented as an ellipsoid in the space of the first three formants, based on production data (Ménard, 2002), which were adjusted to fit the acoustic vowel space of the model (see the supplementary material, S1). Any position inside an ellipsoid was a valid production of the corresponding phoneme. Somatosensory goals were defined as ellipsoids in the three-dimensional (3D) reduced-dimensional space of tongue positions, designed to align as closely as possible with the auditory goals. When applicable, a tactile goal specified the valid tongue-palate contact area on the palate; these were inferred from cineradiographic data (Bothorel et al., 1986).

Each multisensory goal was set at two specified times, for reaching and maintaining the phoneme. This ensured the phoneme's clear perceptibility. Timings were derived from acoustic and articulatory data [vowel-consonant-vowel (VCV) and vowel-consonant-vowel-consonant-vowel (VCVCV) sequences] recorded from the same individual whose anatomy was mimicked by the biomechanical model (Perrier and Fuchs, 2008). The data were VCV and VCVCV sequences (logatomes and meaningful words) uttered at normal speaking rate within carrier sentences, used to study anticipatory coarticulation in French.

### 2.2 The plant to be controlled

GEPPETO controls a two-dimensional (2D) finite element (FE) biomechanical model of the tongue (192 quadratic elements and 221 nodes). The model includes seven muscles responsible for the main deformations and displacements of the tongue in the mid-sagittal plane of the head [Fig. 1(B)]. In the figure, the bold lines represent macro-fibers, over which the muscle force is distributed. The colored quadrilaterals are elements within the FE structure, whose mechanical stiffness increases with muscle activation. The resting configuration was extracted from an X-ray picture of a male speaker (Badin et al., 1995).

To reduce computation time, a reduced model [referred to as the *model of tongue biomechanics* in Fig. 1(B)] was developed to approximate the dynamics of the FE model. This involved representing the sixteen 2D nodes of the tongue contour—both necessary and sufficient for computing sensory features—with a five-dimensional (5D) vector belonging to the latent space of an autoencoder (AE). Then, a Long Short Term Memory (LSTM) network was trained on 21 361 tongue trajectories generated with GEPPETO to learn the plant dynamics (i.e., predict tongue velocity at the next 2 ms time step; see the supplementary material, S2 for details).

The jaw being fixed in the position of an $l/E$, and assuming a lip aperture of 1.8 cm$^2$, the anatomical reference data of Badin et al. (1995) allowed us to deduce the area function of the complete vocal tract from the external contour of the tongue (decoded from the 5D representation). The first three formants (spectral maxima) were computed from the transfer function of the vocal tract (Badin and Fant, 1984). At rest, the acoustics were close to a schwa (neutral configuration).

In our study of the impact of sensory and motor noise, the noises were assumed to be additive Gaussian, and their standard deviation was set so that their impact on articulatory variability corresponded to experimental data (Mooshammer et al., 2024).

### 2.3 Control

To control the biomechanical model of the tongue, the *optimal controller* minimizes a cost function to determine the time course of the commands ($\lambda$) required to achieve the desired goals. Based on the EPH (Feldman, 1986), $\lambda$ defines the threshold muscle length that determines the activation onset of $\alpha$-motoneurons, responsible for generating active forces (Perrier et al., 2003). For a command $\lambda$, a muscle having a length $l$ has an activation $A = [l - \lambda]^+$. The generated muscle force is $F = \rho [e^A - 1]$, in which $\rho$ specifies the muscle force generation capacity and $c$ is a form parameter accounting for the gain of feedback from the muscle to the motoneurons. Consequently, for a given length $l$, *decreasing* $\lambda$ *increases* muscle activation and force. Then, reaching a target requires specifying the $\lambda$-values that define the mechanical equilibrium point at the target position, potentially considering interactions with external loads. In the original GEPPETO model, commands vary at a constant velocity from the initial to the final $\lambda$-value, leading to a ramp-like variation up to a holding phase. In the GEPPETO-optimal feedback control (OFC) model, the $\lambda$ commands were free to adopt any time course.

The cost function balances two key objectives: minimizing motor effort (which reduces higher-order derivatives and mitigates the effects of motor noise); accurately reaching speech targets. The motor effort term is represented by the $L_2$-norm of the difference between successive control commands, where only a decrease in $\lambda$ values is penalized, as indicated by the positive part. The accuracy term measures distance between actual sensory feedback and target values, considering for each phoneme three components, normalized to have comparable magnitude: (1) the error between the estimated somatosensory feedback and the somatosensory target, (2) the error between the estimated auditory feedback and the
<figure><img src="image_4.png" /><figcaption>Fig. 1. (A) Model architecture. Each phoneme *p* is associated with two goals (*2p* − 1, *2p*) sharing a multisensory target zone *Z*<sub>2*p*−1</sub> = *Z*<sub>2*p*</sub> to be reached at onset time *T*<sub>2*p*−1</sub> and final time *T*<sub>2*p*</sub>; *N*<sub>*g*</sub>, number of goals; λ, motor commands; *s* and *ŝ*, actual and estimated state (position, velocity, and acceleration); *F*, *P*, and *ϑ*, auditory, proprioceptive and tactile feedback, combined in the vector *j*; *ŝ* and *ŷ*, predicted state and sensory feedback. *ε*<sub>*z*</sub> (*z* ∈ {λ, *F*, *P*, *ϑ*}), additive motor and sensory noise; *τ*<sub>*F*</sub> and *τ*<sub>*s*,*i*</sub>, sensory delays; *K*, feedback gain matrix of the EKF; *C*, cost function; *N*, total number of discrete time steps; *t*<sub>*n*</sub>, *n*-th time step. (B) Three steps (bullets) in the development of the *model of tongue biomechanics*. LSTM, LSTM network of 20 nodes. See text and the *supplementary material*.</figcaption></figure>

auditory target, and (3) the error between the closest node on the estimated tongue contour and the tactile target on the
palate (with a fixed penalty if there is no tongue-palate contact). This is done with the function dist that measures the dis-
tance between the estimated sensory outcome ŷ(T_g) and the intended multisensory target zone Z_g. The parameter α sets
the effort-error trade-off. Additional terms ensure stability at goals by minimizing velocity and acceleration at T_g, with w_v
and w_a acting as weighting parameters. Details on these calculations can be found in the *supplementary material*, S3 and
S4. This formulation follows the approach of Todorov and Jordan (2002), using a composite value function, which integra-
tes neuromuscular effort, accuracy in sensory spaces, and stability.

The optimal estimator computes an estimate of the system's state, which is essential for evaluating the cost function, using an extended Kalman filter (EKF). Specifically, the *internal model*, a surrogate LSTM network, predicts the system state from an efference copy of the control commands. From this estimate, the EKF computes the expected sensory feedback, and the state estimate is finally corrected by comparing these expectations to the actual, delayed, sensory feedback (see the *supplementary material*, S5). From this state estimate, optimal commands to the current targets can be computed. In practice, a reoptimization of the commands was performed every 20 ms. Optimization was performed using the MATLAB `fminsearch` function.

3. Results

We first examined the trajectories produced by the GEPPETO-OFC model, using a VCV sequence as a benchmark
(/aki/, Fig. 2). For simplicity, this initial assessment did not include sensory or motor noise. On some panels, the results
of GEPPETO, based on the EPH, are presented for comparison [black lines; for clarity, they are not shown in Figs. 2(B),
2(C), and 2(D)].

We observe that for the palatal sounds /k/ and /i/ GEPPETO-OFC successfully reaches the ellipsoid targets at the
specified times and during the specified durations across different sensory domains: somatosensory [Fig. 2(B)], auditory
<figure><img src="image_5.png" /><figcaption>Fig. 2. (A) Time evolution of tangential velocity for two points on the tongue surface: in the palatal region (top panel) and in the pharyngeal region (bottom panel). Colored curves represent trajectories generated by GEPPETO-OFC; black curves correspond to the EPH-based GEPPETO model. White areas indicate transition periods between phonemes, colored regions represent phoneme holding phases. (B) Trajectories in somatosensory space, where gray dots indicate the initial positions and colored dots mark the positions reached at the phoneme initial and final times. Colored ellipses are central slices of the ellipsoid targets. (C) Trajectories in auditory space [same conventions as in (B)]. (D) Kinematic trajectories in the mid-sagittal plane. (E) Time evolution of the seven muscle commands ($\lambda$). Note that muscle activation corresponds to a *decrease in* $\lambda$. GGP, posterior genioglossus; GGA, anterior gen; HYO, hyoglossus; STYLO, styloglossus; VERT, verticalis; SL, superior longitudinal; IL, inferior longitudinal. Their effect on muscle activations is illustrated for the hyoglossus on the bottom right sub-panel. Graphical conventions as in (A).</figcaption></figure>

[Fig. 2(C)], and tactile (for stop consonants, not shown). The articulation of the posterior vowel /a/, located between the rest position and consonant /k/, requires the achievement of large, rapid movements, which, with our choice of effort-error trade-off generates some hypometry [Figs. 2(B) and 2(C)]. Globally, accuracy is achieved thanks to an efficient control of the velocity, which is significantly reduced during the phoneme holding phases [Fig. 2(A)]. The temporal evolutions of the muscle commands highlight distinct strategic differences between GEPPETO-OFC and the EPH-based GEPPETO model (Perrier et al., 2003). While the EPH-based model has simpler command profiles [Fig. 2(D)], it does not meet the precise timing requirements of the sequence [Fig. 2(A)]; in particular, troughs of velocity occur in the transition periods instead of during phonemic goals, as in GEPPETO-OFC. In fact, because it takes into account the inertia of the physical system, GEPPETO-OFC is often “one step ahead”: for example, the hyoglossus muscle [Fig. 2(E), bottom right] is deactivated during the holding phase of /a/ to prepare the raising of the tongue necessary to reach /k/. Last, as the complete sequence is optimized, the reaching of the intermediate goals depends on their phonemic context: for example, the /k/ glides inside the allowed target zone to smoothly connect the /a/ to the /i/ [Fig. 2(B), in somatosensory space; Fig. 2(C), in auditory space; Fig. 2(D), in physical space]. As in GEPPETO, coarticulation is a natural consequence of effort minimization constrained by sensory requirements.

To look more precisely into these coarticulation features, we compared the production of /aki/ vs /aka/. Consistent with experimental data, we observed that the realization of /k/ strongly depends on the later phonemic target; the tongue is more forward for an upcoming /i/ than for an upcoming /a/ [Fig. 3(A)], both tongue shapes being valid sensory realizations for the consonant. This dissimilarity of posture in the two contexts originates in differing motor commands when reaching /k/: notably, the $\lambda$ commands of the posterior genioglossus and the hyoglossus diverge early on after the initial /a/ [Fig. 3(B)].

Finally, we investigated the impact of motor and sensory noise on the features of articulatory trajectories. Our goal was twofold: first, to assess the robustness of the model to imprecision in the sensory and motor signals, and second, to determine whether an imbalance in sensory noise—when one modality is more reliable than the other—leads to noticeable differences in behavior. To this end, we chose a simple CV syllable (/te/) as the goal, imposed a significant level of motor noise, and varied the levels of sensory noise in the following three ways: (1) significant noise across all sensory
<figure><img src="image_6.png" /><figcaption>Fig. 3. (A) /a/ and /k/ tongue postures reached in /aki/ and /aka/ with the OFC model in the mid-sagittal plane. (B) Time course of the muscle commands ($\lambda$) for three primary movement-driving muscles; as described previously, decreasing $\lambda$ corresponds to muscle activation (GGP, genioglossus; HYO, hyoglossus; STYLO, styloglossus). Light gray areas, phoneme holding phases.</figcaption></figure>

modalities (the three sensory inputs are weakly reliable), (2) significant somatosensory and tactile noise and no auditory noise (hearing is significantly more reliable), and (3) significant auditory noise and no somatosensory noise (somatosensory and tactile inputs are significantly more reliable). Regarding our first aim (explored with situation 1), the model demonstrated a strong resistance to motor and sensory noise, maintaining stability. Interestingly, despite the introduction of noise, the variability across movement repetitions of the motor, articulatory, and acoustic signals was moderate [Fig. 4(A)]. The average behavior of the model was close to what is observed without noise (see the supplementary material, S7). Now, when a sensory modality was more reliable than another, trajectories were different from each other: for example, discrepancies begin at the end of consonant /t/ with a later onset of the backward movement toward the following vowel /e/ when somatosensation is more reliable than audition [condition (3)] [Fig. 4(B), top subplots]. Acoustic consequences of this divergence can logically be noticed, mainly on F3 [Fig. 4(C)]. In terms of motor commands, differences are mostly confined to the posterior genioglossus muscle, which plays a major role in tongue protrusion and is the major mover of the target consonant /t/ [Fig. 4(D)].

## 4. Discussion

GEPPETO-OFC is the first—to our knowledge—optimal feedback control model of speech production that enables in a same framework (1) to control muscle activations in a biomechanical model of the tongue, (2) to monitor sensory feedback on-line, and (3) to use sensory feedback to constantly (re)-optimize motor planning before and during the execution of movements. This model allowed us to explore the interactions between the biomechanical constraints of the articulators and the putative objectives of the central nervous system: achieving the sensory goals while minimizing effort. We first demonstrated the effectiveness of the system: it can reach the specified phonemic targets at specified (realistic) times and for specified (realistic) durations. Second, the model adequately replicated articulatory and acoustic context-dependent trajectories to the same phonemic target, showing natural coarticulation patterns. Third, it could withstand realistic levels of motor and sensory noise, and its behavior was influenced by a difference in sensory reliability between sensory channels.

While showing the proper reaching of sensory goals by such a system may look trivial, it is not entirely. The tested consonants /t/ and /k/ are associated with nonlinear biomechanical and acoustic dynamics, which makes control more complex than the direct or linearly indirect control of vocal tract geometry postulated in many speech production models. In parallel, the optimality principle ensures the coordination of several partially agonist or antagonist tongue muscles; while the timing of the speech goals was fixed, the time course of each muscle activation emerged from the biomechanically constrained optimization. Note that while we did not explore this possibility, our framework could allow us to adjust the timing of the upcoming goals online, as a function of the sensory feedback, e.g., using the principle Tilsen (2022) proposed.

We also provided a first look into the coarticulation properties of this model. While an in-depth analysis is outside the scope of this article, we show for a limited set of phoneme sequences that the behavior of GEPPETO-OFC is consistent with experimental observations. The sliding movement observed during the holding phase of /k/ in /aki/ (Fig. 2) aligns with empirical data (Mooshammer et al., 1995). Similarly, the differential placement of the tongue when anticipating a next /i/ or /a/ (Fig. 3) replicates the coarticulation patterns observed in human speech with cineradiography (Gay, 1977; Bothorel et al., 1986): the tongue shape for /k/ is more anterior in the anterior post-consonant vocalic context /aki/ than in the posterior context /aka/. Additionally, anticipatory effects are evident in motor commands to key muscles. The posterior genioglossus, crucial for producing /i/, shows higher activation in /aki/ than in /aka/. The hyoglossus, whose activation is necessary for /a/ but not for /k/, is activated during the production of /k/ in /aka/ but not in /aki/. Both observations correspond to anticipation in muscle activation during the consonant to achieve the post-consonantal vowel. We also observe carry-over coarticulation in the activation of the genioglossus, which is crucial for /k/ but not for /a/: the small $\lambda$-value necessary for the /k/ remains low in /a/. While further analysis is needed for a more precise evaluation, these trends support the hypothesis that the OFC model realistically accounts for coarticulation
<figure><img src="image_7.png" /><figcaption>Fig. 4. Time evolution of the median trajectories along with the first and third quartiles from 20 simulations during the production of the syllable /te/, in the presence of motor and sensory noise. Green, with motor, somatosensory, and auditory noise; red, with motor and somatosensory noise; blue, with motor and auditory noise. Light gray areas indicate phoneme holding phases. (A) Kinematic space, tangential velocities of two nodes on the external contour of the tongue. (B) Kinematic space, horizontal (top) and vertical (bottom) displacements of two nodes on the external contour of the tongue relative to the resting position. (C) Acoustic space, first three formants. (D) Motor space, variations of the motor commands to the seven muscles.</figcaption></figure>

phenomena. At present, our study focuses on tongue posture and does not capture coarticulatory effects in the jaw or other articulators, a limitation arising from the current biomechanical model. Future work will extend the model to include additional articulators.

Last, we have shown that this optimal feedback control of a biomechanical model of the tongue has satisfying properties in the presence of sensory and motor noise. First, the tongue reaches satisfactorily the sensory targets even in the presence of motor noise and significantly noisy and delayed sensory signals. This was expected from a system using an internal (near-)optimal estimation of the current state, but we suspect that the tongue properties, a shape-from-contraction system, have a stabilizing effect. Second, we observe that different levels of reliability across sensory modalities lead to different trajectories, which is a non-trivial finding. Indeed, here, trajectory differences emerge from differences in state estimation (the more reliable modality being weighted more). We believe that the differences between conditions (2) and (3) might be due to the fact that, in the absence of reliable auditory information (condition 3), the strength of the tactile consonantal goal becomes crucial. In fact, the tactile goal forces the tongue to remain in contact with the palate in the anterior part of the alveolar region. When hearing is absent, the controller does not allow any compromise on the tactile constraint and strictly maintains tongue/palate contacts in the specified tactile target area up to the end of the consonant. In order to validate this hypothesis, we have run the same simulations in the absence of tactile feedback, and, as expected, in conditions (1), (2), and (3), the simulated trajectories are significantly closer to each other (see the **supplementary material**, S6). This hypothesis is also consistent with the fact that in condition (1), for which we believe that the state estimation gives a strong weight to the internal sensory predictions, trajectories are close to those simulated in condition (2). These latter results suggest that experimentally observed idiosyncrasies in speech production could also stem from personal reliance on a preferred sensory modality, as evidenced in sensorimotor adaptation experiments (Lametti et al., 2012).

**Supplementary Material**

Refer to the **supplementary material** for methodological details and for additional simulation results.

**Acknowledgments**

This research was funded by the Multidisciplinary Institute in Artificial Intelligence of the Université Grenoble Alpes (MIAI@Grenoble Alpes, ANR-19-P3IA-0003). During the preparation of this work, ChatGPT was occasionally used in order to improve readability.
## Author Declarations

### Conflict of Interest

The authors have no conflicts of interest to disclose.

## Data Availability

The data that support the findings of this study, including simulation code, are openly available at https://github.com/Ny-Tsiky/GEPPETO-OFC.

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