Diff-MSM: Differentiable Musculoskeletal Models

• Diff-MSM is a family of models that implement differentiable biomechanics, enabling gradient-based optimization across muscle activation, joint torques, and kinematics.
• They integrate Hill-type muscle formulations, ODE-based forward dynamics, and physics-informed losses from sEMG data for robust subject-specific parameter estimation.
• These frameworks support simulation and control in complex systems—from human movement to robotic actuation—enhancing both physiological plausibility and computational efficiency.

Differentiable MusculoSkeletal Model (Diff-MSM) denotes a family of musculoskeletal modeling frameworks in which biomechanical components are implemented as differentiable mappings, allowing gradient-based optimization across muscle activation and contraction dynamics, muscle-to-joint torque mappings, skeletal kinematics and dynamics, and, in some systems, perception and control. In the recent literature, Diff-MSM appears both as a physics-informed learning paradigm for estimating muscle forces and joint kinematics from surface electromyography (sEMG), and as an end-to-end differentiable simulation and identification framework for subject-specific muscle and bone parameters, full-body human motion, and musculoskeletal robot control (Zhang et al., 2022, Zhou et al., 18 Aug 2025, Heng et al., 5 Jun 2026).

1. Terminological scope and defining properties

The core distinction of Diff-MSM is not a single anatomical model or software stack, but the requirement that the relevant physiological and mechanical transformations remain differentiable with respect to learnable parameters, latent states, or control variables. In physics-informed EMG-driven models, this means that neural predictors and biomechanical constraints are optimized jointly through a composite loss, with the physics entering as soft constraints rather than post hoc checks (Zhang et al., 2022). In parameter-identification settings, differentiability is used to propagate gradients from observable motion back through joint torques to Hill-type muscle and bone parameters, eliminating the need to measure internal joint torques directly (Zhou et al., 18 Aug 2025). In simulation-oriented work, differentiability extends to musculotendon paths, contact, rigid-soft coupling, and ODE-based forward dynamics (Wang et al., 2022, Qiao et al., 2022).

This usage also spans several neighboring formulations. Some papers emphasize differentiable optimization rather than neural prediction, as in the identification of bone and superficial muscle parameters without reconstructing deep muscle EMGs (Sanderink et al., 26 Sep 2025). Others embed an MSK model directly into a neural architecture as an ODE loss and identify physiological parameters during training (Shi et al., 2023). Full-body systems broaden the term further by coupling muscle-driven mechanics with hierarchical reinforcement learning, multisensory observations, or monocular vision pipelines (Zuo et al., 2023, Zuo et al., 29 May 2025, Koleini et al., 24 Nov 2025).

A recurring motivation is the gap between two older paradigms. Physics-based neuromusculoskeletal models can interpret the dynamic interaction between neural drive, muscle dynamics, body motion, and joint kinetics, but they are described as slow, especially for complex models. Purely data-driven models are fast and simple to implement, but they may not reflect the underlying neuromechanical processes and may generalize poorly across subjects or motions (Zhang et al., 2022, Zhang et al., 2022). Diff-MSM therefore emerges as a hybrid response: preserve biomechanical structure while retaining end-to-end trainability.

2. Mathematical structure and differentiable mechanics

Most Diff-MSM formulations are organized around standard multibody dynamics coupled to muscle models. A representative full-body equation is

$$M(q)\ddot{q} + c(q,\dot{q}) = J_{m}^\top f_m(act) + J_c^\top f_c + \tau_{\text{ext}},$$

where generalized motion is driven by muscle forces, constraint forces, and external loads (Zuo et al., 2023). In arm- and joint-level models, the muscle-to-torque mapping is often written as

$\tau = J_m^\top F_m$

or, in scalar moment-arm form,

$\tau_t = \sum_{n=1}^N r_n F_t^n,$

with $r_n$ exported from OpenSim in one physics-informed EMG framework (Zhou et al., 18 Aug 2025, Zhang et al., 2022).

The muscle side is typically based on Hill-type formulations. In the simultaneous identification framework, muscle force is expressed as

$$F_{m} = F_{active}(\tilde{l}_{m}, \tilde{v}_{m}, \pi_{m}, a) + F_{passive}(\tilde{l}_{m}, \pi_{m}),$$

while the forward dynamics are parameterized by bone parameters $\pi_b$ through the Articulated Body Algorithm (ABA),

$\ddot{q} = f_{ABA}(q, \dot{q}, \tau, \pi_b).$

The loss is the normalized MSE between predicted and ground-truth joint accelerations (Zhou et al., 18 Aug 2025). In EMG-embedded PINN formulations, activation dynamics and contraction dynamics are themselves part of the computation graph, including the nonlinear activation map

$a_n = \frac{e^{A e_n} - 1}{e^A - 1},$

with $A$ treated as a trainable physiological parameter (Shi et al., 2023).

Several papers extend differentiability to geometric and inertial effects often omitted in conventional musculoskeletal simulation. “Differentiable Simulation of Inertial Musculotendons” models muscle inertia by distributing muscle mass along the musculotendon centerline and expressing the motion of musculotendon mass points through chained Jacobians,

$$\mathbf{J}_{a r} = \mathbf{J}_{a m} \mathbf{J}_{m r},$$

so that reduced skeletal coordinates drive both bones and musculotendons (Wang et al., 2022). “Empirically-based Multibody Dynamics for Modeling the Human Body Musculoskeletal System” derives ODEs rather than DAEs by embedding holonomic constraints as smooth functions of reduced generalized coordinates, and introduces a closed-form formula for the moment arm matrix,

$\tau = J_m^\top F_m$0

for general musculotendon models (Ehsani, 2023).

These formulations are not mathematically identical, but they share a common computational pattern: latent biomechanical quantities are not treated as isolated intermediate calculations. They remain inside a differentiable graph through which gradients can flow to neural weights, physical parameters, or control inputs.

3. Physics-informed learning from sEMG and partial observations

A major branch of Diff-MSM research uses sEMG as input and predicts muscle forces, joint kinematics, or both. In “Physics-informed Deep Learning for Musculoskeletal Modelling,” the architecture is explicitly hybrid: pre-processed EMG and time indices are passed through a CNN, and the equation of motion is injected as a soft constraint into the total loss,

$\tau = J_m^\top F_m$1

The physics-based penalty is

$\tau = J_m^\top F_m$2

and all terms are differentiable with respect to network outputs (Zhang et al., 2022). The reported CNN uses 1 convolutional layer with kernel size 3, 128 kernels, stride 1, padding 3, followed by ReLU, batch normalization, dropout at rate 0.3, two fully connected blocks with 128 hidden nodes, and a regression block. Training used stochastic gradient descent with momentum, batch size 1, maximum 1200 iterations, initial learning rate 0.01, and rapid convergence in fewer than 600 iterations (Zhang et al., 2022).

The transfer-learning extension in “Boosting Personalised Musculoskeletal Modelling with Physics-informed Knowledge Transfer” separates a generic network from a personalized network. The generic model is trained on multi-subject data with the same structure of data loss plus physics-based regularization, while the personalized model freezes feature extraction layers and fine-tunes only subject-specific layers with limited labeled data from a new subject (Zhang et al., 2022). This formulation directly addresses the stated problem that a pre-trained data-driven model may degrade seriously on a new subject.

“The bionic neural network for external simulation of human locomotor system” takes a related but distinct route by embedding the MSK model into the neural network as an ODE loss function. The total loss is

$\tau = J_m^\top F_m$3

where one residual enforces forward dynamics and another enforces muscle-force consistency under the embedded contraction model, while physiological parameters are automatically estimated during training (Shi et al., 2023). This is notable because the parameters $\tau = J_m^\top F_m$4 are treated as trainable weights rather than fixed calibrations.

The most explicit treatment of partial observability appears in “Physiologically Constrained Musculoskeletal Neural Network for Multi-DoF Joint Kinematics Estimation from Partially Observed sEMG.” MSK-NN combines a CNN-based muscle activation estimator with an embedded MSK forward dynamics module, and it is trained without direct supervision of internal biomechanical variables such as muscle-tendon forces or joint torques (Heng et al., 5 Jun 2026). Its composite loss integrates a joint kinematics loss, a data-driven muscle synergy loss, and an anatomy-guided trend loss. The system estimates activations for both measured and unmeasured muscles, and the paper reports that the optimized MSK parameters remain within physiological limits while the estimated activation of an input-excluded muscle exhibits strong temporal agreement with its recorded sEMG envelope (Heng et al., 5 Jun 2026).

4. Subject-specific parameter identification and personalization

A second major branch of Diff-MSM focuses on system identification. “Diff-MSM: Differentiable MusculoSkeletal Model for Simultaneous Identification of Human Muscle and Bone Parameters” introduces a single-stage, end-to-end differentiable pipeline that simultaneously identifies muscle parameters of a Hill-type model and bone dynamic parameters for a human arm with 3 DoF at the shoulder, 2 DoF at the elbow, 8 major muscles, and 3 movable bones (Zhou et al., 18 Aug 2025). The pipeline differentiates from measurable muscle activation, through joint torque, to observable motion, and does so without the need to measure internal joint torques. In the reported simulations, 50 runs were initialized with parameters sampled within 10% standard deviation of ground truth; the proposed method achieved the lowest loss and parameter error among compared methods, and muscle parameters could end up with an average of the percentage errors of the estimated values as low as 0.05%. The best run on unseen data achieved a normalized mean squared error of $\tau = J_m^\top F_m$5 (Zhou et al., 18 Aug 2025).

“Parameter Identification of a Differentiable Human Arm Musculoskeletal Model without Deep Muscle EMG Reconstruction” addresses a different observability bottleneck: deep muscle EMGs. The model includes 30 bone dynamic parameters for humerus, ulna, and radius, and 50 superficial muscle parameters, with muscle activations computed via EMG-to-excitation dynamics, activation dynamics, and Hill-type contraction dynamics (Sanderink et al., 26 Sep 2025). Deep muscle forces are not modeled directly; instead, for each time instant they are assigned the least-squares solution

$\tau = J_m^\top F_m$6

and the loss is the normalized mean squared error of the unachievable component of the dynamics. The reported result is that estimation accuracy is comparable to a similar method with all muscle EMGs available (Sanderink et al., 26 Sep 2025). This directly counters the assumption that deep-muscle reconstruction is a prerequisite for usable subject-specific identification.

Personalization also appears in exercise analysis. “Muscle Activation Estimation by Optimizing the Musculoskeletal Model for Personalized Strength and Conditioning Training” builds a whole-body OpenSim model from the Rajagopal model and the Bilateral Upper Extremity Trunk Model, simplifies the Hill-type muscle force equation to $\tau = J_m^\top F_m$7 for optimization, and calibrates maximum isometric force with an EMG-based optimization method (Wu et al., 19 Feb 2025). The optimization minimizes joint-torque mismatch under EMG-derived activation distribution constraints, and subsequent activation estimation is evaluated on bench press and deadlift. The paper reports that the “proposed” method consistently yields the lowest RMSE relative to measured activation envelopes (Wu et al., 19 Feb 2025).

A plausible implication is that Diff-MSM personalization is increasingly shifting from sequential calibration procedures toward joint optimization over muscle, skeletal, and latent-force variables. That trend is explicit in the single-stage formulation of (Zhou et al., 18 Aug 2025) and implicit in the differentiable least-squares integration of (Sanderink et al., 26 Sep 2025).

5. Differentiable simulation, perception, and control systems

Diff-MSM is also used as a simulation substrate for control, reinforcement learning, and perception-driven inference.

System	Scale	Differentiable feature
Diff-MSM (Zhou et al., 18 Aug 2025)	Human arm: 3 DoF shoulder, 2 DoF elbow, 8 muscles	End-to-end automatic differentiation for simultaneous muscle and bone parameter identification
MS-Human-700 (Zuo et al., 2023)	90 body segments, 206 joints, 700 muscle-tendon units	Full-body muscle-actuated simulation with hierarchical low-dimensional control
SMS-Human (Zuo et al., 29 May 2025)	175 rigid segments, 278 joints, 1,266 muscle-tendon units	Multisensory human model with muscle-level control in MuJoCo
MonoMSK (Koleini et al., 24 Nov 2025)	OpenSim-style 24-segment skeleton	Inverse dynamics transformers with differentiable FK and FD ODE layers

On the simulation side, differentiability has been extended to musculotendon inertia, soft articulated bodies, and hybrid rigid-soft robots. “Differentiable Simulation of Inertial Musculotendons” supports higher-order time integrators, inverse dynamics, Hill-type muscle models, and differentiability, while allowing inertial and non-inertial musculotendons to be mixed and matched (Wang et al., 2022). “Differentiable Simulation of Soft Multi-body Systems” develops a Projective Dynamics framework with a top-down matrix assembly algorithm and a generalized dry friction model; it reports more than an order of magnitude acceleration for system identification and supports muscles, joint torques, or pneumatic tubes as actuation (Qiao et al., 2022). “EquiMus” models linear elastic actuators with a 3-2-1 energy-equivalent lumped-mass construction, reports static RMSE below 0.02 rad, dynamic RMSE below 0.05 rad, and simulation rates above 140× real time in MuJoCo (Zhu et al., 11 Nov 2025).

Full-body and task-level control papers emphasize high-dimensional actuation. MS-Human-700 introduces a model with 90 body segments, 206 joints, and 700 muscle-tendon units, and combines physiological muscle synergies with an autoencoder-based latent representation for Soft Actor-Critic control (Zuo et al., 2023). SMS-Human extends the anatomical scale further to 206 labeled bones grouped into 175 rigid segments, 278 explicitly parametrized joints, and 1,266 muscle-tendon units, together with visual, vestibular, proprioceptive, and tactile inputs, and uses a stage-wise hierarchical deep reinforcement learning framework (Zuo et al., 29 May 2025). In both cases, the central problem is not only mechanical realism but tractable control over hundreds to more than a thousand muscle actuators.

Vision-based inference now also enters the Diff-MSM domain. MonoMSK couples transformer-based inverse kinematics and inverse dynamics with differentiable forward kinematics and a forward-dynamics ODE solver, forming a physics-regulated inverse-forward loop for monocular recovery of both kinematics and kinetics (Koleini et al., 24 Nov 2025). On BML-MoVi, it reports an MAE_angle of 1.93 compared with 2.84 for BioPose, alongside MAE_lambda of 0.0139 and MAE_tau of 0.0498 (Koleini et al., 24 Nov 2025).

Animal and robotic systems broaden the scope further. A parametric musculoskeletal canine model built from accurate 3D muscle meshes introduces a sigmoid-smoothed activation dynamics to replace a hard activation/deactivation switch and thereby improve convergence in differentiable control frameworks (Barbera et al., 30 Jun 2025). In musculoskeletal robotics, Diff-Muscle reformulates learning from a high-dimensional muscle-activation space into a lower-dimensional joint space by exploiting differential flatness, and reports a hit rate of 95.1%, success rate of 80.3%, aggregate muscle activation of 0.0278, and up to 14 consecutive dual-robot exchanges in table tennis (Zhao et al., 9 Mar 2026).

6. Misconceptions, limitations, and open directions

A common misconception is that differentiable musculoskeletal modeling is synonymous with purely neural surrogates. The literature shows a broader landscape: soft-constrained CNNs and FNNs are one part of the field, but equally central are differentiable multibody formalisms, differentiable least-squares estimators, ODE-based forward simulators, and reinforcement-learning environments with analytic muscle mappings (Zhang et al., 2022, Ehsani, 2023, Sanderink et al., 26 Sep 2025, Zhao et al., 9 Mar 2026).

Another misconception is that internal biomechanical variables must be directly measured to obtain usable subject-specific models. Several papers explicitly work around this limitation: simultaneous identification without joint torque measurements (Zhou et al., 18 Aug 2025), parameter identification without deep muscle EMG reconstruction (Sanderink et al., 26 Sep 2025), and kinematics estimation from partially observed sEMG without direct supervision of internal forces or torques (Heng et al., 5 Jun 2026).

The main unresolved difficulties described across the papers are practical rather than conceptual. Internal biomechanical variables remain difficult to measure in vivo; deep muscle EMGs are invasive or impractical; generic models may degrade on unseen subjects; and high-dimensional actuation spaces make control inefficient (Zhou et al., 18 Aug 2025, Sanderink et al., 26 Sep 2025, Zhang et al., 2022, Zhao et al., 9 Mar 2026). Some works also identify structural sources of non-differentiability, such as hard switching in activation dynamics or path discontinuities at wrapping surfaces, and replace them with smooth interpolations or neural approximators (Barbera et al., 30 Jun 2025, Wang et al., 2022).

This suggests that the current research frontier is less about establishing whether musculoskeletal models can be differentiable and more about deciding which components should be made differentiable, at what anatomical scale, and under which sensing assumptions. The field already includes wrist-scale EMG estimators, human arm identification systems, whole-body sensory-musculoskeletal simulators, monocular inverse-forward pipelines, canine locomotion models, and musculoskeletal robots (Zhang et al., 2022, Zhou et al., 18 Aug 2025, Zuo et al., 29 May 2025, Koleini et al., 24 Nov 2025, Barbera et al., 30 Jun 2025, Zhao et al., 9 Mar 2026). What remains open is the consolidation of these strands into models that retain physiological plausibility, computational tractability, and robustness under partial observation and out-of-distribution motion.