Nonlinear Model Predictive Control of a Robotic Soft Esophagus

AI summary

Overview

• Research area: Soft robotics control, data-driven nonlinear dynamics identification, and biomedical device testing (esophageal stent migration).
• Technical level: Advanced. The paper assumes familiarity with Model Predictive Control, discrete-time state-space models, sparse regression, and pneumatic actuation hardware.
• Scope (one sentence): The paper develops an embedded-sensing version of a soft robotic esophagus (RoSEv2.0) and applies a SINDYC-based nonlinear MPC framework to autonomously generate prescribed peristaltic waves and to run stent migration tests with different food bolus consistencies.

Note on completeness: the version of the paper supplied here is truncated partway through Section VII (Results). Quantitative results for the stent migration experiments are therefore not reported in the content available, and this summary flags that explicitly rather than supplying figures.

What This Paper Is About

The original Robotic Soft Esophagus (RoSE) is an in vitro platform built to test esophageal stents and study food viscosity, but it was driven in open loop because its conduit had no visibility and no embedded sensing. Without sensing, peristalsis could not be regulated against a target profile and could not be adapted to different wave shapes.

The goal of this work is to build RoSE version 2.0 (RoSEv2.0) with embedded Time of Flight (TOF) distance sensors and pressure sensors, then use data from those sensors to learn discrete-time dynamic models of the robot and close the loop with Model Predictive Control. Doing so lets the platform autonomously generate peristaltic waves of specified speed, wavelength, and amplitude, and then use those waves to study stent migration under different bolus consistencies.

Key Contributions

1. A sensing-enabled soft esophagus platform. RoSEv2.0 replaces the original outer casing with a transparent PDMS (SYLGARD 182, Dow, USA) casing and mounts an array of TOF sensors (VL6180X, STMicroelectronics, Switzerland) on top of it to measure conduit deformation laterally from the outside, solving the previously reported lack of visibility and embedded sensing.
2. First application of SINDYC-based MPC to a soft robot. The authors state that, to the best of their knowledge, the SINDYC-based MPC framework presented here has been applied for the first time in a soft robot, and that closed-loop control in the form of MPC has been implemented for the first time in RoSEv2.0.
3. A discrete-time SINDYC (DTSINDYC) modeling pipeline for soft robot control. Discrete-time differential equation (DTDE) models are discovered from TOF and pressure/valve data using a library of constant and polynomial candidate terms, avoiding derivative computation from noisy data and producing a model form directly usable inside MPC.
4. Demonstrated generality of the modeling and control methodology. The authors note the approach is not robot- or sensor-specific and can extend to any soft robotic system given input-output datasets, and they verify this by implementing MPC of RoSEv2.0 controlling chamber air pressure with the valve pressure sensor (VPS) using the same methodology as for TOF.

Main Findings

• Peristaltic wave generation succeeded at specified parameters: Peristalsis waves of speed 20 mm.s⁻¹, wavelength 75 mm, and amplitudes 5, 7.5, and 10 mm were successfully generated by the MPC.
• Two distinct DTSINDYC models were built: Model M₁ was identified from TOF data with conduit displacement (mm) as the state representation, and model M₂ was identified from VPS data with pressure (kPa) as the state representation. Both used three state variables and peristaltic control actuation.
• Sensor placement was justified by biology: At least three RoSEv2.0 layers are required to match human esophagus peristalsis wavelengths, so layers L₅, L₆, and L₇ were chosen for sensing and control, with emphasis on TOF measurement accuracy.
• The model form is a sparse polynomial map: The learned function g(.) consists of first- and second-order polynomial terms plus constant terms, mapping from state and input to the next state, and provides the predictive step used by the MPC.
• Hardware operating envelope is bounded: Valves are rated for a set pressure range of 1–500 kPa and an input signal range of 0–10 V, giving a resolution of 50 kPa.V⁻¹. Because the 8-bit DACs have 0.02 V.step⁻¹ resolution, each digital value applied to the DAC-valve assembly generates 1 kPa.step⁻¹. Complete occlusion of the conduit requires a maximum pressure of 47 kPa, which is treated as the upper limit for MPC-generated control signals.
• End layers are deliberately unactuated: Layers L₁ and L₁₂ are not actuated because the bolus feeder pipes coincide with those conduit locations, and keeping the pipe orientation unchanged was required. Generated control values are stored in the L₂, L₃, and L₄ accumulator indices and right-shifted by three each iteration; it takes 3q+1 shifts for a command to reach L₅, L₆, and L₇, where q = ⌊t_d/T_s⌋.
• Timing example from the design: For c = 20 mm/s, x_d = 15 mm, w = 120 mm, and T_s = 0.1 s, the paper computes t_d = 0.75 s, f_d = 1/6 s⁻¹, and q = 7.
• A stability argument is provided: Theorem 1 states that if the plant model remains unchanged and stable throughout the control law computation, and settling to the reference occurs within the prediction horizon, then finite-horizon MPC is closed-loop stable. The proof links the infinite-horizon cost to a discrete-time Lyapunov equation with Q̃ = Σ (Aᵀ)ⁱ Q Aⁱ.
• Stent migration testing was conducted: RoSEv2.0 with the MPC was used to perform stent migration testing with various food bolus consistencies, and stent migration data for a candidate stent were recorded for various peristalsis speeds and bolus swallow conditions. Specific migration distances, forces, or statistical comparisons are not reported in the truncated content available.
• Reported limitation of the control scheme: The approach makes the MPC adaptable to various peristalsis waveform speeds and wavelengths, but the paper notes the downside is difficulty in achieving higher wave speeds if T_s increases. T_s is the time for one MPC loop cycle and can vary with reference signal type, model complexity, prediction horizon, and input bounds.

Methodology in Plain English

The researchers built a soft robotic esophagus in twelve identical layers (L₁ to L₁₂), each containing four air chambers arranged symmetrically around the conduit axis. Each layer's ring of four chambers is driven by one electro-pneumatic pressure valve (ITV-0030-3BS, SMC, USA), giving twelve independent inputs. The body is cast from Ecoflex 0030 RTV silicone rubber (Smooth-on, USA) using custom molds and housings, and a transparent PDMS outer layer is bonded on with Sil-Poxy adhesive (Smooth-On, USA). TOF sensors are mounted over that transparent layer so the conduit wall's motion can be measured from outside. The conduit has no rigid skeletal boundary.

The valves are driven from a Raspberry Pi 4 Model B through an ADC and DAC board, with custom firmware written in Python 3.7. Before use, the TOF sensors were calibrated for range offset and crosstalk through the PDMS, then further calibrated against a webcam (C922, Logitech, Switzerland) using the Python OpenCV CSRT tracker, with two 4 mm hemispherical retroreflective markers — one for tracking and one as a reference. TOF and webcam data were captured simultaneously during peristaltic inflation and compared to derive additional offsets and scaling factors.

The robot was actuated in open loop with time-shifted staircase waveforms of varying amplitude and timing at layers L₅, L₆, and L₇, and the resulting sensor data were collected. The authors define three discrete states — conduit displacement at the centers of L₅, L₆, and L₇ of side S₁ — and simplify the mechanics by assuming conduit thickness stays constant during deformation, adjacent chambers in a layer deform symmetrically, and displacement along the y and z axes is negligible.

Those data become the input-output dataset for DTSINDYC. The algorithm builds a library matrix of constant and polynomial candidate terms, including products of states with each other and products of states with inputs, then solves a sparse regression problem so that only a few terms survive. The result is a compact discrete-time equation predicting the next state from the current state and the pressure command — exactly the form MPC needs, and one that avoids differentiating noisy measurements.

For control, the MPC minimizes a cost function at each time step that penalizes the predicted trajectory's deviation from the reference wave (including a terminal cost) plus the magnitude of the control signal and the size of control moves, subject to bounds on the pressure commands. The reference itself comes from a sinusoid-based model of biological peristalsis, expressed in discrete time and instantiated as three time-shifted versions for L₅, L₆, and L₇ so that the wave travels down the conduit. Only the first control value in the optimized sequence is applied, then the whole computation repeats as the horizon moves forward one step.

Why This Matters

Impact on research. The paper argues that evidence about which stent design performs better is currently limited to analytical and numerical modeling studies, because in vivo stent testing raises major ethical concerns. A sensing-enabled, closed-loop soft esophagus gives researchers a physical testbed that can reproduce controlled peristaltic conditions. The work also has a methodological claim: it shows a data-driven sparse dynamics model can serve as the predictive model inside MPC for a soft robot whose behavior resists first-principles modeling because of continuous, highly compliant deformation with effectively infinite degrees of freedom. The authors contrast SINDYC with black-box approaches such as artificial neural networks, noting it requires less training data, less execution time, and avoids overfitting.

Real-world applications.

• Pre-clinical evaluation of esophageal stent designs and migration behavior under realistic peristaltic motion.
• Food science and dysphagia research, studying how bolus viscosity interacts with peristalsis and implanted devices.
• Development and validation of medical devices for the gastrointestinal tract that must survive repeated peristaltic loading.
• General transfer of the DTSINDYC-MPC recipe to other pneumatically or fluidically driven soft robots where input-output data can be collected.

Industry relevance. The paper is published in IEEE Transactions on Industrial Electronics, and it situates MPC among the advanced control methods that have seen dramatically increasing industrial use since the 1980s, including in power systems and robotics. The contributions relevant to industry are the emphasis on respecting actuator constraints in real time, the modest data and compute requirements relative to neural-network surrogates, and the claim that the modeling pipeline is not robot- or sensor-specific.

Future Directions

• Increase achievable wave speed. The authors identify that if T_s (the time for one MPC loop) increases, achieving higher peristalsis speeds becomes difficult — a direct target for faster solvers or reduced model/loop complexity.
• Complete and report the stent migration results. The paper states that stent migration data were recorded for various peristalsis speeds and bolus swallow conditions, but the available content does not include those quantitative outcomes; reporting and analyzing them is the natural next step.
• Broaden the transfer demonstration. The paper verifies generality by re-running the methodology with the VPS for chamber air pressure instead of TOF for displacement. Extending this to other soft robotic systems and other sensor types is the stated generalization path, given that the methodology is not robot- or sensor-specific.
• Explore the limits of the learned model form. The DTSINDYC models use first- and second-order polynomial terms with a three-state reduction and simplifying assumptions about constant conduit thickness, symmetric adjacent chamber deformation, and negligible y- and z-axis displacement. Testing richer libraries or relaxed assumptions is an open question for fidelity at larger deformations.

Target Audience

This paper is most valuable to soft robotics researchers and control engineers working on model-based control of compliant, high-degree-of-freedom systems, particularly those facing the absence of governing equations and looking for data-driven alternatives to neural-network models. It also serves biomedical device engineers and gastroenterology researchers involved in esophageal stent design, stent migration, and dysphagia palliation, as well as food scientists studying bolus viscosity. The stability proof and hardware detail make it useful to graduate students and practitioners implementing MPC on pneumatic soft actuators.

Authors’ abstract

Strictures caused by esophageal cancer can narrow down the esophageal lumen, leading to dysphagia. Palliation of dysphagia has driven the development of a Robotic Soft Esophagus (RoSE), which provides a novel in vitro platform for esophageal stent testing and food viscosity studies. In RoSE, peristaltic wave generation and control were done in an open-loop manner since the conduit lacked visibility and embedded sensing capability. Hence, in this work, RoSE version 2.0 (RoSEv2.0) is designed with embedded Time Of Flight (TOF) and pressure sensors to measure conduit displacement and air pressure, respectively, for modeling and control. Model Predictive Control (MPC) of RoSEv2.0 is implemented to govern the peristalsis and air pressure profile autonomously. The implemented MPC used Sparse Identification Nonlinear Dynamics with Control (SINDYC) models to estimate the future states of ROSEv2.0. The dynamic models are discovered from the TOF and pressure sensor data. Peristalsis waves of speed 20 mm/s, wavelength 75 mm, and amplitudes 5, 7.5, and 10 mm were successfully generated by the MPC. Additionally, RoSEv2.0 with the MPC was employed to perform stent migration testing with various food bolus consistencies. The major contribution claimed in this paper is the application of SINDYC-based MPC to solve the closed-loop control problem of RoSE for achieving desired peristaltic waves.