Control Lyapunov Functions for Underactuated Soft Robots

Huy Pham; Zach J. Patterson

cs.ROMar 5, 2026

Control Lyapunov Functions for Underactuated Soft Robots

Authors: Huy Pham, Zach J. Patterson

Organizations: Mechanical and Aerospace Engineering, Case Western Reserve University

Abstract

Soft and soft-rigid hybrid robots are inherently underactuated and operate under tight actuator limits, making task-space control with stability guarantees challenging. Common nonlinear strategies for soft robots (e.g., those based on PD control) often rely on the assumption of full actuation with no actuator limits. This paper presents a general control framework for task-space regulation and tracking of underactuated soft robots under bounded inputs. The method enforces a rapidly exponentially stabilizing control Lyapunov function as a convex inequality constraint while simultaneously satisfying underactuated full-body dynamics and actuator bounds. We validate the approach in simulation on several platforms spanning increasing underactuation: a simple two link tendon-driven "finger", a trimmed helicoid manipulator, and a highly underactuated spiral robot. We compare against a number of baseline methods from the literature. Results show improved task-space accuracy and consistent Lyapunov convergence under input limits, achieving superior set-point and trajectory-tracking performance.

Figures & tables

Figure 1: Results of our proposed Soft ID-CLF-QP control approach operating on various soft robot architectures to solve the task-space control problem. The robot platforms are: (a) Finger (L = 0.24 m), (b) Helix (L = 0.45 m), and (c) SpiRob (L = 0.50 m).

Robot	Controller	$K_{p}$	$\epsilon$	$w_{1}$	$w_{2}$	$w_{3}$	$w_{4}$	$\rho$
Finger	CLF-QP	500.0	0.05	1	–	–	–	1000
	ID-CLF-QP	500.0	0.03	1	0.2	0.2	0.1	1000
	IC	500.0	0.03	–	–	–	–	–
	UIC	500.0	0.03	–	–	–	–	–
	IC-QP	1500.0	0.03	1	0.02	0.2	–	–
Helix	CLF-QP	500.0	0.05	1	–	–	–	1000

Table I : Controller parameter settings for each robot platform.

Figure 2: Lyapunov functions over time for set points tracking experiment.

Figure 3: Lyapunov functions over time for trajectory tracking experiment.

Figure 4: Task-space convergence of the SpiRob during a set-point regulation experiment in the

x

–

z

plane. The variables

x_{d}

and

z_{d}

denote the target point coordinates.

Figure 5: Task-space trajectory tracking performance during the second cycle across controllers for Finger (a) and Helix (b) for

\omega=0.2\pi

rad/s.

Robot	Controller	Final Error (SP) (cm)	TT–MSE ( $\mathbf{cm^{2}}$ )
Finger	CLF-QP	0.75 ± 1.05	6.90 ± 0.80
	ID-CLF-QP (ours)	0.08 ± 0.09	0.53 ± 0.26
	IC	0.08 ± 0.14	1.26 ± 0.73
	UIC	0.22 ± 0.39	1.00 ± 0.51
	IC-QP	0.07 ± 0.12	1.26 ± 0.71
Helix	CLF-QP	Failed Convergence	Failed Convergence

Table II : Combined benchmark comparison for set point (SP) and trajectory tracking (TT) across three robot platforms.

Robotic Control

Robotic control studies how machines use sensing and computation to produce reliable movement and interaction in changing environments.

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Soft robots are well suited for contact-rich tasks due to their compliance, yet this property makes accurate and tractable modeling challenging. Planning motions with dynamically-feasible trajectories requires models that capture arbitrary deformations, remain computationally efficient, and are compatible with underactuation. However, existing approaches balance these properties unevenly: continuum rod models provide physical accuracy but are computationally demanding, while reduced-order approximations improve efficiency at the cost of modeling fidelity. To address this, our work introduces a control-oriented reformulation of Discrete Elastic Rod (DER) dynamics for soft robots, and a method to generate trajectories with these dynamics. The proposed formulation yields a control-affine representation while preserving certain first-principles force-deformation relationships. As a result, the generated trajectories are both dynamically feasible and consistent with the underlying actuation assumptions. We present our trajectory generation framework and validate it experimentally on a pneumatic soft robotic limb. Hardware results demonstrate consistently improved trajectory tracking performance over a constant-curvature-based baseline, particularly under complex actuation conditions.

Abstract

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