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Collocated Shape Regulation for Soft Robots
Pietro Pustina, Ebrahim Shahabi, Daniel Feliu-Talegon, Alessandro De Luca, Cosimo Della Santina
TL;DR
Soft-robot shape control faces the challenge of underactuation and complex dynamics, while prior approaches can be limited to particular robot subclasses or lack performance guarantees. This paper develops a collocated-control framework with stability analyses for regulator families. Experiments report a transient-performance and stability advantage for PD regulators, while PID regulators achieve tracking errors averaging an order of magnitude smaller.
Problem
Shape regulation in soft robots is complicated by underactuation, and prior stable approaches have focused mainly on specific robot subclasses with differing assumptions.
Source: §1 · pp. 1–2Method
Results
PD regulators show superior transient performance and stability to PID regulators, while PID regulators achieve tracking errors averaging an order of magnitude smaller.
Source: §6 · p. 17Takeaways & Limitations
The framework and experiments compare regulator performance across proportional gains, payloads, and reduced-order models on a newly designed soft-robot platform.
Source: §6 · p. 17The paper excludes sophisticated controllers needed in some underactuated scenarios, focusing instead on general, relatively simple control methods.
Source: §3 · p. 7
Abstract
from arXiv · showhide
Controlling the shape of a continuum soft robot typically requires an accurate dynamic model and actuation of all degrees of freedom. We show that regulating only the actuated coordinates, through collocated shape control, achieves provably stable convergence of those coordinates and, under an explicit compatibility condition, of the entire robot shape. While collocated control is a cornerstone of high-performance motion control in rigid robotics, extending this formulation to continuum soft robots has remained challenging due to the complexity of their dynamics. We present the first general framework for collocated control of continuum soft robots and derive a unified family of controllers, including PD, PID, PsatID, and their counterparts with compensation and cancellation components. The framework unifies existing approaches while introducing new controller designs. In particular, we develop three classes of PD and PID like regulators with local, semi-global, and global stability guarantees, and provide rigorous convergence analyses for each. Extensive experimental validation demonstrates the effectiveness of the proposed methods across different model discretizations and controller parameters. The resulting framework provides practical design guidelines for selecting and implementing controllers with known stability guarantees, without requiring a complete dynamic model of the robot
1 Introduction
The introduction frames soft-robot shape regulation as a challenge of controlling underactuated systems while retaining stability guarantees. It proposes a unified collocated-control framework and evaluates regulators across models and discretizations.
1 Introduction
- Learning-based shape control avoids deriving complex dynamics but lacks performance guarantees, while underactuation and non-collocated outputs complicate model-based stability and control.Prior stable regulators largely target specific robot subclasses, leaving assumptions and applicability fragmented.Source: §1 · pp. 1–2
- The paper investigates shape-regulation strategies theoretically and experimentally, comparing their performance and stability to address the fragmented, assumption-dependent prior work.It also aims to demonstrate applicability independent of kinematic discretization and its order.Source: §1 · p. 2
- A unified collocated-control framework formalizes shape regulation and introduces PD- and PID-like regulators with local, semi-global, and global stability guarantees.The framework is presented as a way to develop relatively simple, robust solutions less reliant on model-dependent terms.Source: §1 · p. 2
- The majority of proposed regulators are experimentally validated on a novel platform, with performance evaluated across seven dynamic models and compared for shape regulation.The introduction also notes that controller implementation relies on an inverse-dynamics algorithm introduced in Pustina et al. (2025).Source: §1 · p. 2
2 Preliminaries
The preliminaries set out the reduced-order underactuated robot model and assumptions, then transform it into collocated coordinates. They define robot subclasses for global guarantees and specify when actuation-coordinate convergence should imply convergence of the full shape.
2.1 Dynamic model
- The preliminaries assume reduced-order, underactuated robot dynamics, with fewer inputs than configuration variables and boundedness and regularity conditions for controller design.The stated assumptions cover robot kinematics, elastic forces and energy, damping, and existence of the collocated form.Source: §2.1 · pp. 3–4
2.2 Collocated Form
- A coordinate transformation yields collocated dynamics in which inputs directly affect actuation coordinates, but the desired generalized actuation force must satisfy a structural constraint.The transformation addresses the difficulty of mapping desired configuration-space forces through the generally tall actuation matrix.Source: §2.2 · pp. 4–5
- The framework focuses on systems admitting a collocated form; without it, controllers must explicitly account for the actuation matrix, a challenge left open here.The condition for the form can be relaxed using an input transformation, as noted in Pustina (2025, Lemma 4.3.2).Source: §2.2 · pp. 4–5
2.3 Two interesting sub-classes of soft robots
- Elastic decoupling separates elastic forces on actuated and unactuated degrees of freedom, while elastic dominance requires unactuated stiffness to dominate gravity.For elastically dominated robots, the potential is convex in unactuated coordinates and actuation coordinates uniquely determine equilibrium.Source: §2.3 · p. 5
2.4 Control Objective
- The control objective regulates the desired actuation coordinates and characterizes conditions for unactuated coordinates to converge to a compatible equilibrium, yielding convergence of the full shape.Experiments also evaluate performance under different discretizations and gain tuning on a real system.Source: §2.4 · p. 6
3 Collocated shape regulation: definition and control goal
Collocated shape regulation targets configuration variables power-conjugate to the inputs, rather than directly regulating every degree of freedom. It can regulate the actuated coordinates, but convergence of those coordinates alone does not guarantee a desired full-robot shape.
3 Collocated shape regulation: definition and control goal
- Collocated variables are paired with actuation inputs in a power-conjugate sense, and the framework is independent of the chosen kinematic discretization.Non-collocated variables depend on discretization order and are stabilized indirectly through system potential forces.Source: §3 · pp. 6–7
- Collocation concerns where generalized forces are applied, not whether unactuated coordinates move, since inertial and dynamic coupling can make their motion depend on the input.The input is absent from non-collocated generalized-force equations, but may still affect their motion.Source: §3 · p. 6
- A collocated controller defines its tracking error using only collocated coordinates, their derivatives, and associated controller states, while non-collocated coordinates remain governed by closed-loop internal dynamics.Auxiliary states may include integrators, observers, or filters.Source: §3 · p. 6
- The control goal is asymptotic convergence of the collocated coordinates to an arbitrary desired reference, with constant references treated as regulation.Time-varying references are treated as trajectory tracking.Source: §3 · pp. 6–7
- Because full robot shape depends on all configuration variables, regulating actuated coordinates alone may not align passive parts; the framework derives conditions for steering the entire shape.Some cases, such as an unstable passive segment, may require more sophisticated strategies outside this work's scope.Source: §3 · p. 7
4 Shape Regulators
Section 4 develops PD/PID-like shape regulators incrementally, from local to global convergence, and summarizes their guarantees and implementation costs in Fig. 3.
4 Shape Regulators
- The section presents PD and PID-like regulators with stability guarantees that progress from local to global convergence.Fig. 3 summarizes their performance guarantees and implementation costs.Source: §4 · p. 7, Figure 3 · p. 7
4.1 PD Regulator
- For the basic PD regulator, trajectories converge to an equilibrium set, but the actuated equilibrium can differ from the target and approaches it as K_P increases.For elastically dominated robots, sufficiently large K_P also ensures a unique equilibrium, and convergence of actuated coordinates then determines a unique unactuated equilibrium.Source: §4.1 · pp. 7–8
4.2 PD with feedforward
- PD with feedforward makes the desired configuration an equilibrium and guarantees local asymptotic stability; stability is global when the robot is elastically dominated.The controller requires only measurements of the actuated coordinates.Source: §4.2 · p. 8
4.3 Regulators with local asymptotic stability properties
- A model-based term can make the desired actuated coordinates an equilibrium and locally asymptotically stable for sufficiently large K_P, subject to a sufficient local condition.The condition is less restrictive than the elastically dominated requirement, but the theorem does not specify a convergence region; elastically dominated robots also tolerate unmodeled DoF under sufficiently large K_P.Source: §4.3 · p. 9
4.4 Regulators with semi-global stability properties
- With suitable structural constraints, PD/model-based regulators can achieve semi-global stability, and increasing K_P and β can enlarge the convergence region arbitrarily.A saturated integral term improves robustness to constant disturbances but can be removed without affecting theoretical convergence; including it allows smaller proportional-gain requirements in the stated cases.Source: §4.4 · pp. 9–10
4.5 Regulators with global stability properties
- Under an additional condition on the model-based term, sufficiently large K_P yields bounded trajectories converging globally to the desired actuated coordinates.The section also treats global stability for elastically decoupled and elastically dominated robots.Source: §4.5 · p. 10
Elastically decoupled soft robots
For elastically decoupled robots, a model-based controller can regulate the actuated coordinates to the desired value with sufficiently large proportional gain. Multiple unactuated equilibria may still be compatible with that value.
Elastically decoupled soft robots
- For any desired actuated coordinate and sufficiently large K_P, the controller guarantees bounded trajectories converging to an equilibrium with θ_a = θ_ad.The guarantee applies to an elastically decoupled soft robot (Class 1).Source: Elastically decoupled soft robots · p. 10
- The model-based term in Corollary 1 is bounded because elastic compensation is constant and g remains bounded.This boundedness satisfies the conditions used to establish the convergence guarantee.Source: Elastically decoupled soft robots · p. 10
- The model-based term in (23) uses unactuated-coordinate feedback, needed because a fixed θ_ad can correspond to multiple unactuated equilibria.Figure 4 illustrates two such possible equilibria.Source: Elastically decoupled soft robots · pp. 10–11, Figure 4 · p. 11
Elastically dominated soft robots
For elastically dominated robots, simpler actuated-coordinate-only controllers offer global regulation, with convergence guarantees under a sufficiently large proportional gain.
Elastically dominated soft robots
- Two simpler controllers solve the regulation problem globally while using feedback from θ_a only.The paper presents them specifically for elastically dominated soft robots.Source: Elastically dominated soft robots · p. 11
- For an elastically dominated robot (Class 2), sufficiently large K_P makes θ_d globally asymptotically stable.The result is stated for the closed-loop system in (3)–(13).Source: Elastically dominated soft robots · p. 11
- The P-satI-D controller can regulate actuation coordinates from arbitrary initial conditions without system knowledge, and requires measuring only θ_a.The stated convergence result is for Class 2 robots with sufficiently large K_P; trajectories converge to (θ_d, 0_n).Source: Elastically dominated soft robots · p. 11
- Table 3 lets users select a controller by available measurements and model knowledge, then check its structure and performance guarantee.The guarantee may depend on the stated robot-class assumption.Source: Elastically dominated soft robots · p. 11
5 Experimental Results
The experiments validate regulators on one tendon-actuated soft arm and compare their performance across robot-model discretizations and controller conditions. The study also establishes a collocated form for the PCC and PC models used in the comparison.
5 Experimental Results
- The experiments provide a systematic comparison across discretizations of the same soft robot, testing most regulators discussed and three additional laws without formal stability proofs.The additional laws operate without feedback from unactuated degrees of freedom.Source: §5 · p. 11
5.1 Setup
- The planar Eco-Flex 20 arm uses two tendons; eight cameras track three marker rings, while control runs at 80 Hz, limited by actuation and measurement frequencies.The setup includes a slender continuum arm and a real-time Simulink controller.Source: Figure 5 · p. 12, §5.1 · p. 12
5.2 Kinematic model and state estimation
- Seven CC/PC curvature models are considered, while state estimation limits models to four DoF to avoid an under-constrained inverse-kinematics problem and support faster real-time convergence.CC models range from one to four bodies; PC models use one body with polynomial basis functions.Source: §5.2 · p. 12
5.3 Dynamic model
- Hooke-law elasticity is linear in q, and the PCC and PC actuation matrices are q-independent, yielding collocated coordinates equal to end-effector orientation.The input is modeled as distributed torque acting on the robot cross-sections.Source: §5.3 · p. 13
5.4 Controllers comparison
- Controllers are compared using tracking-error and time metrics while varying proportional gain, payload presence, and discretization.Tests include six step references from the straight configuration followed by six consecutive steps.Source: §5.4 · p. 14
PD regulators
Across the tested PD regulators, feedforward performs best on error metrics, while gain and discretization affect error and timing differently.
PD regulators
- The PD with feedforward (u2) performs best across error metrics; most model-based PD errors fall below 0.1 rad for kP ≥ 0.3 N m.The standard PD approaches the other regulators’ error performance when kP ≥ 0.5 N m; higher-order discretizations improve model-based regulators, with the PC model at four DoF performing best overall.Source: PD regulators · p. 14
- Overshoot is comparable, but PD with unactuated-variable compensation (u5) has the poorest timing; higher gains lengthen settling and transient times as control becomes more oscillatory.With PC models, time metrics increase as DoF increases and are smaller under a fully actuated approximation.Source: PD regulators · p. 14
- For the PC model with four DoF at kP = 0.5 N m and no tip mass, the most notable time-evolution differences are between standard PD (u1) and the other regulators.The cited comparison is illustrated in Fig. 7.Source: PD regulators · p. 14
PID regulators
Across discretizations, PID integral action improves tracking error over PD, while PD tends to offer better transient performance and stability. The results also show that controller behavior varies with model discretization, and compensating for unactuated degrees of freedom can worsen performance.
PID regulators
- PID (u7) and P-satI-D (u8) perform better on overshoot, while P-satI-D (u8) has better settling and transient times than the other PID regulators.Adding the model-based term um does not improve the time metrics; the integral term dominates the model-based components.Source: PID regulators · p. 15
- Compensating for unactuated degrees of freedom (u12) tends to worsen performance, especially for PC models and systems with more DoF.Controllers that exceed 5 s or are absent have not settled within the control interval and continue oscillating around the setpoint.Source: PID regulators · p. 15
5.5 PD vs PID
- PID tracking errors are approximately an order of magnitude smaller on average, while PD regulators show superior transient performance and stability.Across the tested settings, PD controllers have smaller overshoot and shorter settling and transient times; PID instability is more pronounced at larger proportional gains.Source: §5.5 · p. 15
- For two target shapes, combining integral action with model compensation reduces steady-state error and improves accuracy, but increases settling time from 0.4 s for PD+ to 0.85 s for P-satI-D+.The PD controller without model-based components does not provide sufficient accuracy in this example.Source: §5.5 · p. 16
6 Conclusions
- Increasing discretization order can adversely affect shape-related metrics, possibly because sensor locations influence configuration reconstruction accuracy.Three-DoF discretizations show reduced orientation performance relative to two-DoF models, but less variability in position metrics than models with fewer DoF.Source: §5.6 · p. 17
5.6 Discretizations comparison
- Among the tested configurations, the PC model with 4 DoF slightly outperforms other underactuated descriptions overall, while average task error remains consistent across discretizations.The fully actuated CC model with 1 DoF performs worst; 2 DoF improves configuration description, while 4 DoF models better capture shape, especially orientation.Source: §5.6 · p. 17
6 Conclusions
The paper presents a unified collocated-control framework for continuum soft-robot shape regulation, with PD- and PID-like regulator families offering distinct stability properties. Experiments compare controller performance and model discretizations, finding a trade-off between PD transient performance and PID tracking error.
6 Conclusions
- The framework uses collocated form, which reformulates robot dynamics so that only some degrees of freedom are directly influenced by system inputs.It unifies three families of PD- and PID-like regulators with distinct stability properties; most were validated on a newly designed soft robotic platform.Source: §6 · p. 17
- Experiments indicate that PD regulators have superior transient performance and stability, while PID regulators achieve tracking errors approximately an order of magnitude smaller on average.Underactuated models generally provide a more accurate description of robot shape.Source: §6 · p. 17
- Increasing discretization order can adversely affect shape-related metrics, possibly because sensor locations influence configuration reconstruction accuracy.Source: §6 · p. 17
A Proof of theorem 1
- The local stability analysis is centered on the equilibrium θd, and its positive-definiteness condition can be enforced by choosing KP sufficiently large.Source: A Proof of theorem 1 · p. 17
B Proof of theorem 2
Theorem 2 uses a Lyapunov argument to establish convergence from a specified region, while Theorem 3 extends the analysis to a saturated integral action. In both proofs, sufficiently large proportional gain supports the required stability conditions.
B Proof of theorem 2
- Theorem 2's proof uses a Lyapunov-like function with z as the state associated with the saturated integral action.Source: B Proof of theorem 2 · pp. 17–18
- For Theorem 2, sufficiently large KP makes the Lyapunov function non-negative and radially unbounded, and trajectories starting in Ω0 converge to the largest invariant set where Vdot = 0.The proof invokes LaSalle's theorem after establishing the conditions for its application.Source: B Proof of theorem 2 · pp. 18–19
C Proof of theorem 3
- Theorem 3's proof establishes non-negativity of its Lyapunov function by showing that V3 is non-negative when KP is sufficiently large.Source: C Proof of theorem 3 · p. 19
- For Theorem 3, sufficiently large KP makes Vdot negative everywhere, and convergence follows from LaSalle's theorem.Source: C Proof of theorem 3 · p. 20
D Proof of Theorem 4
Theorem 4 analyzes a PID-like controller with model-based terms using a Lyapunov argument. Under gain conditions, the proof establishes negative definiteness of the Lyapunov derivative and concludes convergence by LaSalle's principle.
D Proof of Theorem 4
- Theorem 4's controller combines proportional and derivative feedback, integral action, and model-based terms evaluated at the desired configuration.Its stated control law is u = KP θ̃a − KD θ̇a + KI z + g(θd) + ka(θd).Source: D Proof of Theorem 4 · p. 20
- The proof makes the Lyapunov function positive definite under gain conditions including KI > 0 and KP > 2ν1KI, then uses sufficiently large ν1 to make Vdot negative definite and apply LaSalle's principle.Source: D Proof of Theorem 4 · pp. 20–21
E Block scheme of the closed-loop system
- The closed-loop system's block scheme is illustrated in Fig. 11.Source: E Block scheme of the closed-loop system · p. 21
F Identification Experiment for Gain Tuning of the controllers
The authors identify a second-order linear plant from a step response, then use it to tune PID gains and select experimental gain values. The appendix also provides detailed results across experimental settings.
F Identification Experiment for Gain Tuning of the controllers
- The identification experiment models robot dynamics as a second-order linear system using input u and directly measured actuated coordinates θa; other experiments estimate θ from curvature variables q.Figure 11 shows the closed-loop system block scheme.Source: F Identification Experiment for Gain… · pp. 21–22, Figure 11 · p. 22
- A step input of u = −0.3 [N m] yields identified parameters AP = 7.5432, Tw = 0.20737, and ζ = 0.88971.Figure 10(a) compares the measured and identified step responses for that input.Source: F Identification Experiment for Gain… · p. 22
- The PID tuning targets a crossover frequency of 6 [rad s−1] and phase margin π/2 [rad], with experimental kD = 0.039 [N m s], kI = 0.689 [N m s−1], and kP from 0 to 0.9 [N m].The initial estimated gains were kP = 0.294 [N m], kD = 0.0313 [N m s], and kI = 0.689 [N m s−1].Source: F Identification Experiment for Gain… · p. 22
G Comprehensive Results of the Experiments
- Figures 12–22 provide detailed experimental results across different settings for comparison and analysis.Source: G Comprehensive Results of the… · p. 22