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Stochastic Optimal Feedforward-Feedback Control for Partially Observable Sensorimotor Systems

Bastien Berret, Frédéric Jean

TL;DR

This work introduces a framework that extends neighboring optimal control by enabling the feedforward plan to explicitly account for feedback uncertainties and latencies, and applies this framework to human neuromechanics, demonstrating that muscle co-contraction emerges as an optimal adaptation to task demands.

Abstract

Robust control of complex engineered and biological systems hinges on the integration of feedforward and feedback mechanisms. This is exemplified in neural motor control, where feedforward muscle co-contraction complements sensory-driven feedback corrections to ensure stable behaviors. However, deriving a general continuous-time framework to determine such optimal control policies for partially observable, stochastic, nonlinear, and high-dimensional systems remains a formidable computational challenge. Here, we introduce a framework that extends neighboring optimal control by enabling the feedforward plan to explicitly account for feedback uncertainties and latencies. Using statistical linearization, we transform the stochastic problem into an approximately equivalent deterministic optimization within a tractable, augmented state space that retains critical nonlinearities, offering both mechanistic interpretability and theoretical guarantees on approximation fidelity. We apply this framework to human neuromechanics, demonstrating that muscle co-contraction emerges as an optimal adaptation to task demands, given the characteristics of our sensorimotor system. Our results provide a computational foundation for neuromotor control and a generalizable tool for the control of nonlinear stochastic systems.

Stochastic Optimal Feedforward-Feedback Control for Partially Observable Sensorimotor Systems

TL;DR

This work introduces a framework that extends neighboring optimal control by enabling the feedforward plan to explicitly account for feedback uncertainties and latencies, and applies this framework to human neuromechanics, demonstrating that muscle co-contraction emerges as an optimal adaptation to task demands.

Abstract

Robust control of complex engineered and biological systems hinges on the integration of feedforward and feedback mechanisms. This is exemplified in neural motor control, where feedforward muscle co-contraction complements sensory-driven feedback corrections to ensure stable behaviors. However, deriving a general continuous-time framework to determine such optimal control policies for partially observable, stochastic, nonlinear, and high-dimensional systems remains a formidable computational challenge. Here, we introduce a framework that extends neighboring optimal control by enabling the feedforward plan to explicitly account for feedback uncertainties and latencies. Using statistical linearization, we transform the stochastic problem into an approximately equivalent deterministic optimization within a tractable, augmented state space that retains critical nonlinearities, offering both mechanistic interpretability and theoretical guarantees on approximation fidelity. We apply this framework to human neuromechanics, demonstrating that muscle co-contraction emerges as an optimal adaptation to task demands, given the characteristics of our sensorimotor system. Our results provide a computational foundation for neuromotor control and a generalizable tool for the control of nonlinear stochastic systems.
Paper Structure (16 sections, 1 theorem, 53 equations, 4 figures)

This paper contains 16 sections, 1 theorem, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Discussion
  4. Methods
  5. Acknowledgments

Key Result

Theorem 2

For $u^\mathrm{ol} \in \mathcal{L}^2$, set where $(m^\mathrm{ol},P,S)(\cdot)$ is the trajectory of the following control system associated with $u^\mathrm{ol}(\cdot)$: Then, through statistical linearization of the process $\bf{x} = (x,\hat{x})$, the stochastic optimal control problem $\min_{u \in \mathcal{U}} \mathbb{J}(u)$ is approximated by the deterministic optimal control problem $\min_{u^

Figures (4)

  • Figure 1: Optimal strategy depending on sensory noise and delays. A: Evolution of mean stiffness with respect to variations of sensory noise and delay (average value of $k_s(u_1+u_2)$ across 100 samples). B: Mean positional control feedback gain with respect to variations of sensory noise and delays ($L_{1,1}(t)$ is depicted). C: Mean absolute net torque computed (average value of $k_n(u_1-u_2)$ across 100 samples). D: Optimal expected cost with respect to variations of sensory noise and delay. Default values for parameters, corresponding to the 1x condition, were as follows: $\rho_p=0.05\,s$, $\rho_v=0.10\,s$, $\eta_p=5.0\,^\circ\sqrt{s}$ and $\eta_p=1.0\,^\circ\sqrt{s}$.
  • Figure 2: Optimal trajectories for different sensory noise magnitudes. A: Position and velocity traces for a sensory noise of 0.2x the reference level. Thick traces depict the mean behavior and thin traces depict a single trial. B,C: Same information for the different noise magnitudes, 1x and 5x. D: Net torque and muscle inputs for a sensory noise of 0.2x the reference level. Flexor torques are depicted with positive values and extensor torques are depicted with negative values for clarity. E,F: Same information for the different noise magnitudes, 1x and 5x. The mean co-contraction, defined as the mean of $u_{1}+u_{2}$ across the trial, is reported above the graphs.
  • Figure 3: Optimal strategy depending on sensory noise and force field magnitudes. A. Optimal muscle co-contraction. Co-contraction was calculated by averaging values across muscle groups, which were organized into three antagonist pairs (shoulder, elbow, and biarticular) and across time. B. Optimal lateral endpoint stiffness. This stiffness along the $\mathrm{x}$-axis was averaged along the trajectory, using the joint stiffness and arm's Jacobian matrix to compute it. C. Optimal feedback gain. Only the mean positional gain averaged across the 6 muscles and 2 joints and across time is depicted. For all variables, grand mean values for the NF and DF conditions are reported, both with and without vision.
  • Figure 4: Optimal trajectories for different sensory noise and force field magnitudes. A. Endpoint trajectories (mean and 5 samples depicted) in NF and DF conditions, both with and without vision. Green traces depict trajectories in DF but using the optimal control policy of the NF condition (i.e., a non-adapted scenario). B. Optimal muscle inputs (mean and same 5 samples depicted) for shoulder, elbow and biarticular muscles in the NF and DF conditions, with or without vision.

Theorems & Definitions (6)

  • Remark 1
  • Theorem 2
  • Remark 3
  • proof : Proof of Theorem \ref{['th:main']}
  • Remark 4
  • Remark 5