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Model Based Control of Soft Robots: A Survey of the State of the Art and Open Challenges

Cosimo Della Santina, Christian Duriez, Daniela Rus

arXiv:2110.01358v2eess.SYcs.RO

TL;DR

The paper addresses how to control continuum soft robots despite continuum dynamics, elastic effects, and diverse sensing and actuation strategies. It surveys model-based control through a unified robotics framework, showing how finite-dimensional models support analysis and how existing controllers can be adapted while identifying unresolved underactuation and actuator-dynamics challenges.

  • Problem

    Continuum soft-robot control is difficult because the systems combine continuum or multibody dynamics, elastic potential fields, and widely varying sensing and actuation strategies.

  • Method

    The paper surveys model-based soft-robot control using common terminology and finite-dimensional approximations that connect continuum models with standard robotics formulations.

  • Results

    The survey shows that model-based controllers can address posture regulation, trajectory tracking, task-space control, and iterative learning across soft-robot models, while several stability problems remain open.

  • Takeaways & Limitations

    A unified model-based framework makes similarities with rigid and flexible robots apparent and supports transferring some established control results to soft robots.

  • Takeaways & Limitations

    Provably stable task-space dynamic control with strong underactuation remains an open problem, and stabilization through dynamic couplings is largely unexplored.

Abstract

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Continuum soft robots are mechanical systems entirely made of continuously deformable elements. This design solution aims to bring robots closer to invertebrate animals and soft appendices of vertebrate animals (e.g., an elephant's trunk, a monkey's tail). This work aims to introduce the control theorist perspective to this novel development in robotics. We aim to remove the barriers to entry into this field by presenting existing results and future challenges using a unified language and within a coherent framework. Indeed, the main difficulty in entering this field is the wide variability of terminology and scientific backgrounds, making it quite hard to acquire a comprehensive view on the topic. Another limiting factor is that it is not obvious where to draw a clear line between the limitations imposed by the technology not being mature yet and the challenges intrinsic to this class of robots. In this work, we argue that the intrinsic effects are the continuum or multi-body dynamics, the presence of a non-negligible elastic potential field, and the variability in sensing and actuation strategies.

Finite Dimensional Models for Control Purposes

Soft-robot models begin with continuum descriptions but are commonly reduced to finite-dimensional coordinates for control. Rod-based models describe backbone strains, while standard Lagrangian mechanics yields ODEs containing multibody, elastic, dissipative, and gravitational effects.

  • Finite-dimensional modeling: Finite-dimensional ODE approximations can describe continuum soft-robot behavior with tractable and sufficiently precise models for control.Exact continuum formulations use PDEs, whereas recent approximations use standard ODEs.
  • Modeling for control: Simulation and control models use different compromises: simulation favors accuracy, whereas control models must be lower-dimensional, interpretable, and structurally analyzable.Soft-robot simulation models are typically more sophisticated than models used for control.
  • Rod models: Rod models represent posture through a spatial backbone curve and local strains including curvature, twist, elongation, and shear.The backbone posture is recovered by integrating the strain field, analogous to forward kinematics.
  • Finite-dimensional modeling: Finite-dimensional strain models approximate ξ as a function of configuration q, enabling concepts from discrete robotics to be recast for continuum robots.Piecewise-constant and functional parameterizations provide the main reduction strategies.
  • Dynamic formulation: The resulting dynamics combine inertia, Coriolis and centrifugal effects, gravity, elasticity, and damping through a standard ODE structure.The robot state is formed by (q, ˙q), with actuation mapped into configuration space by A(q).

Piecewise Constant Strain Approximations

Piecewise constant strain models reduce continuum robots by assuming strains are constant between fixed nodes. PCC models provide a particularly simple case, while extensions incorporate three-dimensional bending, additional strains, and rigid-link approximations.

  • Piecewise Constant Strain Approximations: Piecewise constant strain models assume ξ is constant between fixed points along the rod called nodes.The resulting finite-dimensional coordinates collect the strain values associated with the segments.
  • Piecewise Constant Strain Approximations: Planar PCC models retain one piecewise-constant curvature per segment, producing connected circular arcs and n degrees of freedom for nS segments.They extend serial revolute manipulators by distributing angle changes along segments.
  • Three-dimensional PCC models: PCC kinematics remain a sequence of arcs in three dimensions, where changing bending planes introduces two degrees of freedom per segment.The usual orientation-based representation can introduce singularities and discontinuities.
  • Alternative approximations: Rigid-link approximations model rods as links with independent joints and parallel lumped springs representing impedance.This corresponds to strains concentrated at a finite set of points rather than distributed along the rod.
  • Generalized strain models: Piecewise constant strain models extend PCC by including shear and twist, with q ∈ R6ns and full inertial coupling among the six strain components.The elastic potential is usually diagonal or block diagonal.

How Fine Should the Discretization Be?

Discretization resolution is a design choice balancing fidelity, computational cost, actuation structure, and control properties. Rod, functional, FEM, and reduced-order approaches offer different ways to represent the soft robot at finite dimension.

  • How Fine Should the Discretization Be?: Increasing the number of piecewise-constant segments approaches the continuum model, but simulation cost grows at best as O(ns^2).The designer selects segment number based on mechanics, application, and control requirements.
  • How Fine Should the Discretization Be?: Segment placement can alter structural properties such as actuation rank, and planar PCC models with tip torques may yield A(q) = I.Models are sometimes designed to be fully actuated by choosing the number and locations of segments.
  • Functional Parametrizations: Functional parametrizations reduce dimensionality by projecting strain or shape onto a finite basis whose weights serve as robot coordinates.Including constant basis functions makes the model extend constant-curvature or constant-strain descriptions.
  • Finite Element Models: FEM represents volumetric deformation with a mesh of nodes and is preferred when three-dimensional changes are not negligible relative to the backbone.The configuration collects node positions, and accuracy generally improves as node count increases.
  • Model order reduction: FEM models can use thousands of nodes for high accuracy, while reduced-order models project configurations onto nr << n directions, often reducing dimensions by roughly n/nr ≳ 10^2.Modal or data-derived bases also regularize high-frequency numerical vibrations.

Existence of Equilibria

The equilibrium equation characterizes configurations balancing actuation, stiffness, gravity, and other forces. Under suitable monotonicity or radial-unboundedness conditions, at least one equilibrium exists for each constant actuation, though multiple equilibria may occur.

  • Existence of Equilibria: Equilibrium configurations are states q̄ satisfying the static balance equation associated with a constant control input τ̄.The equilibrium set is defined from the robot’s dynamic model after setting the relevant steady-state terms.
  • Existence of Equilibria: If K + G is monotone and A is configuration independent, a single equilibrium exists for every actuation choice.The closed-form solution is q̄ = (K + G)^−1(Aτ̄) under these assumptions.
  • Existence of Equilibria: If stiffness K(q) is radially unbounded while G(q) is bounded, at least one equilibrium exists even when the actuation field is configuration dependent.Continuity and radial unboundedness provide the existence argument without requiring monotonicity.
  • Existence of Equilibria: Several equilibrium configurations may exist when the combined stiffness and gravity field is not monotone.The scalar proof uses radial unboundedness of (K + G)/A and continuity.
  • Existence of Equilibria: The existence result contrasts with classic rigid robots, where constant actuation cannot produce equilibrium unless gravity is involved.This difference follows from the soft robot’s elastic potential field.

Shape Control in the Fully Actuated Approximation

In the fully actuated approximation, soft-robot shape control uses feedforward and PD-like feedback strategies to regulate posture and track trajectories. Elastic and gravitational fields contribute to stability, while feedback gains provide additional local stabilization and tracking guarantees.

  • Posture regulation: Shape control regulates the full configuration q, with curvature, strain, or volume control representing model-dependent versions of the same task.
  • Posture regulation: A constant feedforward action creates a closed-loop structure equivalent to a nonlinear PD regulator through the robot’s elastic and gravitational dynamics.The physical system itself supplies the proportional-like action associated with the potential fields.
  • Posture regulation: Any constant control input is associated with an equilibrium, and Theorem 1 establishes asymptotic stability when a suitable neighborhood condition holds.The equilibrium is the desired state (q̄, 0), and the proof uses an energy-based Lyapunov argument.
  • Posture regulation: Adding proportional and derivative gains yields local asymptotic stability when the damping and neighborhood conditions in Corollary 1 hold.The neighborhood N(q̄) is then contained in the region of asymptotic stability.
  • Posture regulation: Large proportional gains can stiffen the soft robot, amplify noise, or excite neglected dynamics, motivating alternatives such as nonlinear integral actions or state-evaluated compensation.
  • Trajectory tracking: For trajectory tracking, feedforward compensation plus PD feedback provides local exponential stabilization when gains exceed bounds that grow with gravity gradients and reference velocity and acceleration.For slowly varying references or sufficiently high natural impedance, the feedback gains may be zero; generic trajectories generally require extra feedback.

Underactuation, Actuators Dynamics, and Task Execution

Underactuation, actuator dynamics, and environmental interaction make soft-robot control more complex, requiring assumptions, specialized models, and careful treatment of stability and compliance. Existing approaches address parts of these challenges, but provably stable task-space control and broader dynamic solutions remain open.

  • Underactuation: Underactuated equilibrium stabilization requires conditions linking actuation directions to the effective stiffness, with collocation determining whether local methods apply.The equilibrium can be stabilized only when actuation is collocated with directions where effective stiffness loses rank.
  • Underactuation: With constant actuation, underactuated impedance control can inherit fully actuated stability results when the actuation and gain conditions satisfy the stated hypotheses.The closed-loop structure replaces α and β with AαA⊤ and AβA⊤ under the constant-actuation assumptions.
  • Underactuation: For unstable equilibria or repulsive unactuated directions, stabilization must rely on feedback, less-local strategies, or dynamic couplings.The paper identifies stabilization through non-collocated feedback as an early direction, particularly for low-stiffness soft inverted pendulums.
  • Actuator dynamics and constraints: Actuator dynamics substantially shape soft-robot behavior, motivating singular perturbation or backstepping when actuator and robot time scales must be modeled explicitly.Backstepping avoids assuming separated time scales but produces a more complex control architecture.
  • Task-space regulation and tracking: Underactuated task-space control can generate fully actuated task-space dynamics under a full-rank condition, but full-state convergence remains unguaranteed.The required condition is that J(q)M^-1(q)A(q) be full rank; designing a provably stable task-space dynamic controller remains open.
  • Interaction with the environment: Environmental interaction can compensate for underactuation and input saturation, while compliant behavior creates a trade-off between low stiffness and open-loop stability.External wrenches can support the robot or enlarge its accessible space, but interaction control must balance stability against compliance.

leveraging data and machine learning in model-based control

The survey examines how data and machine learning can complement model-based control for soft robots, addressing incomplete models and application uncertainties. It organizes integration through adaptive feedback, learned models, iterative feedforward learning, and model-based simulation.

  • Incomplete models and changing sensors, actuators, and physical parameters motivate integrating data into model-based soft-robot control.The stated difficulties include first-principles actuator modeling, computationally expensive discretization, unknown environments, sensor and actuator unreliability, and parameter variation.
  • Adaptive control uses an online learning loop to update model parameters while a parameterized model-based controller generates the control action.The uncertainty is represented by unknown parameters p, whose updates are driven by a learning rule informed by the model structure.
  • Iterative learning control updates feedforward actions across repeated trials using task-execution errors and nominal-model knowledge.The survey reports applications to optimal-trajectory tracking, soft worms, spherical joints, soft fingers, and soft bending actuators.
  • Models can guide learning indirectly through virtual environments, including differentiable simulators that permit gradients to be backpropagated for machine learning.
  • Learning a model before using it in a model-based controller can improve explainability and closed-loop performance or stability relative to directly learning an end-to-end controller.The survey describes learned forward kinematics, inverse-problem solution through Jacobians, open-loop actions, and learned models used with MPC.

Conclusions

The survey unifies model-based control results for soft robots while highlighting both their intrinsic simplifications and unresolved challenges. It closes by connecting theoretical progress to the experimental goal of improving real soft robots’ motor capabilities.

  • A common terminology and discretization framework reveals similarities among rigid, flexible, and soft robots while exposing soft robots’ intrinsic underactuation.The survey identifies positive-definite elastic potential and strictly dissipative forces as structural features that simplify control despite many degrees of freedom.
  • Open questions include handling underactuation, deciding when non-actuated dynamics can be neglected, stabilizing generic unstable equilibria, and achieving compliant, fast, precise motion.
  • Theoretical advances are framed as contributing to the experimental ambition of giving real soft robots improved motor capabilities.

Sidebar: Dynamics of a Constant Curvature Segment

The constant-curvature sidebar derives a compact dynamic model for a planar soft segment and compares it with lumped rigid-link approximations. The models share structural properties, but their geometric and dynamic expressions diverge as curvature increases.

  • A constant-curvature segment is represented by a single scalar q, the curvature or bending angle, and compared with a revolute-joint rigid-link approximation.
  • The continuum and lumped models can overlap near the straight configuration but progressively differ geometrically as |q| grows.The sidebar notes that the rigid approximation neglects motion of the lower half of the robot, producing a smaller straight-configuration inertia than the continuum model.
  • The constant-curvature model has configuration-dependent inertia, whose value decreases as |q| increases because curvature changes produce progressively smaller shape changes.
  • Combining the derived terms yields scalar second-order dynamics with the same structural properties as a lumped joint with parallel impedance but more complex expressions.
  • In the illustrated cases, the constant-curvature and rigid-with-parallel-springs evolutions are qualitatively similar, whereas the rigid model without springs differs.

Better than Rigid Robots: Exploiting Softness in Model-Based Control

Softness can act as a distributed low-level feedback mechanism and store or release energy during dynamic tasks. These properties can simplify control and support behaviors beyond equivalent rigid robots in some settings.

  • Physical impedance provides decentralized feedback throughout the soft body without requiring additional sensors or actuators.Elastic and dissipative fields act across the infinite-dimensional structure, while standard actuation reaches only m directions in an n-dimensional configuration space.
  • Elastic potential can store and release energy during dynamic tasks, enabling velocity or force objectives beyond those achievable by an equivalent-inertia rigid robot.The cited work associates these behaviors with lumped joint-spring-link systems and notes that continuum structures may exhibit more extreme effects through multistability and buckling.
  • The survey directs readers seeking a deeper treatment of intelligent behaviors generated by soft bodies to a separate paper on embodied intelligence.

Sidebar: Model-based Perception of Shape and Forces

Soft-robot models help infer shape and forces from limited sensing, but direct measurement of the full configuration is generally unavailable. Static force–configuration relationships support both posture extraction and force estimation.

  • Perception challenge: Most soft-robot models lack sensors capable of directly measuring the configuration q.Available measurements may instead be nonlinear combinations h(q) of state variables through forward kinematics.
  • Perception challenge: Models connect finite sensor measurements to the virtually infinite degrees of freedom of soft robots.
  • Static force–configuration relationships: The persistent potential field K(q) + G(q) links forces and configurations, particularly at steady state.
  • Static force–configuration relationships: Static inversion of a rigid-link approximation can extract posture information from a six-axis force/torque sensor at the robot base.

Sidebar: Robust control

Soft-robot models are uncertain because material and manufacturing variability combine with state discretization that omits part of the dynamics. Controllers therefore account for uncertainty through learning, intrinsic robustness, or explicit robust-control design.

  • Sources of uncertainty: Soft-robot models always contain uncertainty from material and manufacturing variability and from discretization that ignores part of the dynamics.
  • Sources of uncertainty: Model improvements may reduce uncertainty but are unlikely to eliminate it completely.
  • Robust-control strategies: Controllers address uncertainty through learning loops, intrinsically robust control loops, or explicit robust-control design.

Sidebar: Infinite Dimensional Control

Infinite-dimensional control avoids state-space discretization but requires more complex analysis. Finite-dimensional approximations simplify controller design, while PDE-based approaches can prevent spillover and remain an active research area for nonlinear continuum dynamics.

  • Sidebar: Infinite Dimensional Control: Finite-dimensional approximations simplify control design while capturing important dynamics to a desired level of precision.
  • Sidebar: Infinite Dimensional Control: Avoiding state-space discretization is the only way to exclude control spillover from high-order dynamics excited by finite-dimensional controllers.Spillover can degrade performance and eventually bring instability.
  • Sidebar: Infinite Dimensional Control: Infinite-dimensional analysis can yield more compact and interpretable solutions than high-dimensional ODE-based approaches.
  • Sidebar: Infinite Dimensional Control: Most PDE-control applications for continuum mechanics address linear beam models, while fully nonlinear extensions remain under active research.
  • Sidebar: Infinite Dimensional Control: Nonlinear beam-control studies include passivity-based stabilization, practical boundary regulation, and boundary feedback for large deformations.

Authors Biography

The authors’ biographies span robotics, deformable-object simulation, haptic rendering, artificial intelligence, data science, and research leadership across universities and research institutes.

  • Authors Biography: Cosimo Della Santina is an assistant professor at TU Delft and research scientist at DLR with a robotics Ph.D. from the University of Pisa.
  • Authors Biography: Christian Duriez is a robotics researcher whose work includes interactive simulation of deformable objects and haptic rendering at INRIA.
  • Authors Biography: Daniela Rus is an MIT professor, CSAIL director, and researcher in robotics, artificial intelligence, and data science.
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