Collocated Shape Regulation for Soft Robots

Pietro Pustina; Ebrahim Shahabi; Daniel Feliu-Talegon; Alessandro De Luca; Cosimo Della Santina

cs.ROSep 23, 2026

Collocated Shape Regulation for Soft Robots

Authors: Pietro Pustina, Ebrahim Shahabi, Daniel Feliu-Talegon, Alessandro De Luca, Cosimo Della Santina

Organizations: Department of Computer, Control and Management Engineering, Sapienza University of Rome, Rome, Italy · Department of Cognitive Robotics, Delft University of Technology, 2628 CN Delft, The Netherlands · Institute of Robotics and Mechatronics, German Aerospace Center (DLR), 82234 Oberpfaffenhofen, Germany

Abstract

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

Figures & tables

Figure 1: Drive a soft robot’s shape by actuating only where you can - $collocated\ control$ - with guarantees that scale from model-free practical regulation to model-based global stability guarantees. This approach paves the way for shape regulation in soft robots with stability guarantees, without requiring a full dynamic model.

Symbol	Description
$\mathbb{R}^{n}$	Euclidean space of dimension $n$
$\mathbb{R}^{n\times m}$	Space of $n\times m$ matrices over $\mathbb{R}$
$\mathbb{R}^{+}$	Positive real numbers $n$
$\bm{0}_{n\times m}$	Zero matrix of dimension $n\times m$
$\bm{P}>0$	Symmetric positive definite matrix
${\left\\|{\bm{v}}\right\\|}$	Euclidean norm of $\bm{v}\in\mathbb{R}^{n}$

Table 1: Nomenclature

\small\left(\begin{array}[]{cc}\bm{M}_{{aa}}&\bm{M}_{{au}}\\ \bm{M}_{{au}}^{T}&\bm{M}_{{uu}}\end{array}\right)\left(\begin{array}[]{c}\ddot{\bm{\theta}}_{a}\\ \ddot{\bm{\theta}}_{u}\end{array}\right)+\left(\begin{array}[]{cc}\bm{C}_{{aa}}&\bm{C}_{{au}}\\ \bm{C}_{{au}}&\bm{C}_{{uu}}\end{array}\right)\left(\begin{array}[]{c}\dot{\bm{\theta}}_{a}\\ \dot{\bm{\theta}}_{u}\end{array}\right)+\left(\begin{array}[]{c}\bm{g}_{{a}}\\ \bm{g}_{{u}}\end{array}\right)+\left(\begin{array}[]{c}\bm{k}_{{a}}\\ \bm{k}_{{u}}\end{array}\right)+\left(\begin{array}[]{cc}\bm{D}_{{aa}}&\bm{D}_{{au}}\\ \bm{D}_{{au}}&\bm{D}_{{uu}}\end{array}\right)\left(\begin{array}[]{c}\dot{\bm{\theta}}_{a}\\ \dot{\bm{\theta}}_{u}\end{array}\right)=\left(\begin{array}[]{c}\bm{u}\\ \bm{0}_{n-m}\end{array}\right)

(3)

Table 3

Figure 2: Block diagram of the collocated shape-regulation framework, incorporating the coordinate transformation, $\bm{h}(\bm{q})$ , a collocated shape regulator, and a decomposition of the system dynamics into actuated and unactuated components.

Figure 3: Summary of the proposed controller families, their performance guarantees, and implementation costs.

Name	Expression of $\bm{u}_{m}$
Cancellation	$\bm{g}_{a}(\bm{\theta}_{a},\bm{\theta}_{u})+\bm{k}_{a}(\bm{\theta}_{a},\bm{\theta}_{u})$
Gravity cancellation and collocated elasticity compensation	$\bm{g}_{a}(\bm{\theta}_{a},\bm{\theta}_{u})+\bm{k}_{a}(\bm{\theta}_{a_{d}},\bm{\theta}_{u})$
Collocated gravity compensation and elasticity cancellation	$\bm{g}_{a}(\bm{\theta}_{a_{d}},\bm{\theta}_{u})+\bm{k}_{a}(\bm{\theta}_{a},\bm{\theta}_{u})$
Collocated gravity and elasticity compensation	$\bm{g}_{a}(\bm{\theta}_{a_{d}},\bm{\theta}_{u})+\bm{k}_{a}(\bm{\theta}_{a_{d}},\bm{\theta}_{u})$

Table 2: Possible model-based terms

\bm{u}_{m}

for ( 19 ) guaranteeing stability according to Theorem 2 .

Figure 4: Possible unactuated equilibria $\bm{\theta}_{u_{d_{1}}}$ and $\bm{\theta}_{u_{d_{2}}}$ compatible with the desired $\bm{\theta}_{a_{d}}$ value of the actuation coordinates.

FAMILY	MEASURES	MODEL	INTEGRAL	CLASS	GUARANTEE	THEOREM
PD	$\bm{\theta}_{a}$	none	—	—	bounded / practical	—
PD + feedforward	$\bm{\theta}_{a}$	$\bm{F}_{c_{d}}$	—	under class	local / global ∗	T1
PD / PID + cancellation	full $\bm{\theta}$	full potential forces	optional	under class	semi-global ∗	T2–3
P-satI-D	$\bm{\theta}_{a}$	none	saturated integral	elastically dominated	global ∗	T4

Table 3: Summary of controller families and their guarantees. Select the row that matches the available measurements and model knowledge, then read across to identify the applicable controller structure and its corresponding performance guarantee. ∗ Guarantee holds under the stated class assumption; (Sec. 4 ).

Controller	Provably stable
$\bm{u}_{1}=\bm{K}_{P}\tilde{\bm{\theta}}_{a}-\bm{K}_{D}\dot{\bm{\theta}}_{a}$	yes
$\bm{u}_{2}=\bm{K}_{P}\tilde{\bm{\theta}}_{a}-\bm{K}_{D}\dot{\bm{\theta}}_{a}+\bm{g}_{a}(\bm{\theta}_{d})+\bm{k}_{a}(\bm{\theta}_{d})$	yes
$\bm{u}_{3}=\bm{K}_{P}\tilde{\bm{\theta}}_{a}-\bm{K}_{D}\dot{\bm{\theta}}_{a}+\bm{g}_{a}(\bm{\theta})+\bm{k}_{a}(\bm{\theta}_{d})$	yes
$\bm{u}_{4}=\bm{K}_{P}\tilde{\bm{\theta}}_{a}-\bm{K}_{D}\dot{\bm{\theta}}_{a}+\bm{g}_{a}(\bm{\theta})+\bm{k}_{a}(\bm{\theta})$	yes
$\bm{u}_{5}=\bm{K}_{P}\tilde{\bm{\theta}}_{a}-\bm{K}_{D}\dot{\bm{\theta}}_{a}+\bm{g}_{a}(\bm{\theta}_{a},\bm{\theta}_{u_{d}})+\bm{k}_{a}(\bm{\theta}_{d})$	no
$\bm{u}_{6}=\bm{K}_{P}\tilde{\bm{\theta}}_{a}-\bm{K}_{D}\dot{\bm{\theta}}_{a}+\bm{g}_{a}(\bm{\theta}_{a_{d}},\bm{\theta}_{u})+\bm{k}_{a}(\bm{\theta}_{d})$	yes

Table 4: Regulators considered in the experiments. For each controller, we report if some form of stability has been proven.

Figure 5: The experimental setup consists of a slender continuum soft arm of Eco-Flex 20 silicone actuated by a pair of tendons tensioned by Dynamixel XM430-W210-T motors. Eight Optitrack Prime x 13 cameras measure the position and orientation of three rings of markers placed along the robot body.

Figure 6: Robot kinematic models considered in the experiments.

Model	Curvature
CC 1 DoF	$\small\kappa(s,q)=\frac{1}{0.3}\,q_{1}$
PCC 2 DoF	$\small\kappa(s,{\bm{q}})=\frac{1}{0.15}\left\{\begin{array}[]{l}q_{1};\hskip 9.24994pts\in[0,0.15)\\ q_{2};\hskip 9.24994pts\in[0.15,0.3]\end{array}\right.$
PCC 3 DoF	$\small\kappa(s,{\bm{q}})=\frac{1}{0.1}\left\{\begin{array}[]{l}q_{1};\hskip 9.24994pts\in[0,0.1)\\ q_{2};\hskip 9.24994pts\in[0.1,0.2)\\ q_{3};\hskip 9.24994pts\in[0.2,0.3]\end{array}\right.$
PCC 4 DoF	$\small\kappa(s,{\bm{q}})=\frac{1}{0.075}\left\{\begin{array}[]{l}q_{1};\hskip 9.24994pts\in[0,0.075)\\ q_{2};\hskip 9.24994pts\in[0.075,0.15)\\ q_{3};\hskip 9.24994pts\in[0.15,0.225]\\ q_{4};\hskip 9.24994pts\in[0.225,0.3]\\ \end{array}\right.$
PC 2 DoF	$\small\kappa(s,{\bm{q}})=\frac{1}{0.3}\,(q_{1}+\frac{s}{0.3}q_{2})$
PC 3 DoF	$\small\kappa(s,{\bm{q}})=\frac{1}{0.3}\,(q_{1}+\frac{s}{0.3}q_{2}+\left(\frac{s}{0.3}\right)^{2}q_{3})$

Table 5: Curvature models used in the experiments. The variable

s

denotes the curvilinear abscissa.

Panel of Figure 7 — Figure 7: Time evolution of the actuation coordinate and the corresponding reference (black dashed lines) for the PD regulators under the PC discretization with $4$ DoF, a proportional gain $k_{P}=0.5\penalty\ [$ \mathrm{N}\text{,}\mathrm{m} $]$ and no mass at the tip.

Panel of Figure 8 — Figure 8: Time evolution of the actuation coordinate and the corresponding reference (black dashed lines) for the PID/P-satI-D regulators under the PC discretization with $4$ DoF, a proportional gain $k_{P}=0.5\penalty\ [$ \mathrm{N}\text{,}\mathrm{m} $]$ and no mass at the tip.

Figure 9: Time evolution of three controllers (PD ( $u_{1}$ ), PD+ ( $u_{2}$ ), and P-satI-D+ ( $u_{9}$ )) for two different target shapes using PC discretization with 4 DoF and a proportional gain $k_{P}=0.5\penalty\ [$\mathrm{N}\text{\,}\mathrm{m}$]$ . — Figure 9: Time evolution of three controllers (PD ( $u_{1}$ ), PD+ ( $u_{2}$ ), and P-satI-D+ ( $u_{9}$ )) for two different target shapes using PC discretization with 4 DoF and a proportional gain $k_{P}=0.5\penalty\ [$ \mathrm{N}\text{,}\mathrm{m} $]$ .

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 10 — Figure 10: (a) Measured and identified step response of the open-loop system, and (b) estimated closed-loop response of the linear plant $P(s)$ under a PID regulator with $k_{P}=0.294\penalty\ [$ \mathrm{N}\text{,}\mathrm{m} $]$ , $k_{D}=0.0313\penalty\ [$ \mathrm{N}\text{,}\mathrm{m}\text{,}\mathrm{s} $]$ , and $k_{I}=0.689\penalty\ [$ \mathrm{N}\text{,}\mathrm{m}\text{,}{\mathrm{s}}^{-1} $]$ .

Figure 11: Block scheme of the closed-loop system.

Figure 12: RMSE, absolute value of the steady state error, and its percentage for the PD regulators. The solid lines represent the mean value for each metric of the PD regulator. For each controller, gain, and metric, the plots show the median, lower and upper quartiles, as well as the minimum and maximum values, and with a colored dot the mean value.

Figure 13: Overshoot, settling time and transient time for the PD regulators. The solid lines represent the mean value for each metric of the PD regulator. For each controller, gain, and metric, the plots show the median, lower and upper quartiles, as well as the minimum and maximum values, and with a colored dot the mean value.

Figure 14: RMSE, absolute value of the steady state error and its percentage for the PID regulators. The solid lines represent the mean value for each metric of the PID regulator. For each controller, gain, and metric, the plots show the median, lower and upper quartiles, as well as the minimum and maximum values, and with a colored dot the mean value.

Figure 15: Overshoot, settling time and transient time for the PID regulators. The solid lines represent the mean value for each metric of the PID regulator. For each controller, gain, and metric, the plots show the median, lower and upper quartiles, as well as the minimum and maximum values, and with a colored dot the mean value.

Figure 16: Mean RMSE, absolute and percentage steady-state errors, overshoot, settling time, and transient time for the PD regulator (left side of the figure) and the PID regulator (right side of the figure). The lines represent the metric values averaged across all model discretizations.

Figure 17: Mean of the RMSE [left], absolute value of the steady state error [center] and its percentage [right] for the analyzed regulators with varying proportional gain (in [Nm]).

Figure 18: Mean value of the overshoot [left], settling time [center] and transient time [right] for the analyzed regulators with varying proportional gain (in [Nm]).

Figure 19: Mean of the RMSE [left], absolute value of the steady state error [center] and its percentage [right] respectively) [center] and its percentage [right] with varying proportional gain (in [Nm]) in the presence of a payload of $0.09\penalty\ [$\mathrm{kg}$]$ attached to the robot tip. — Figure 19: Mean of the RMSE [left], absolute value of the steady state error [center] and its percentage [right] respectively) [center] and its percentage [right] with varying proportional gain (in [Nm]) in the presence of a payload of $0.09\penalty\ [$ \mathrm{kg} $]$ attached to the robot tip.

Figure 20: Mean value of the overshoot [left], settling time [center] and transient time [right] for the analyzed regulators with varying proportional gain (in [Nm]) in the presence of a payload of $0.09\penalty\ [$\mathrm{kg}$]$ attached to the robot tip. — Figure 20: Mean value of the overshoot [left], settling time [center] and transient time [right] for the analyzed regulators with varying proportional gain (in [Nm]) in the presence of a payload of $0.09\penalty\ [$ \mathrm{kg} $]$ attached to the robot tip.

Panel of Figure 21 — Figure 21: Steady state and RMS values of the Cartesian, angular, and task errors for the considered discretizations.

Figure 22: Snapshots of the robot and corresponding dynamic model for the closed-loop system under the P-satI-D controller with $k_{P}=0.5\penalty\ [$\mathrm{N}\text{\,}\mathrm{m}$]$ . — Figure 22: Snapshots of the robot and corresponding dynamic model for the closed-loop system under the P-satI-D controller with $k_{P}=0.5\penalty\ [$ \mathrm{N}\text{,}\mathrm{m} $]$ .

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