Trajectory Generation for Underactuated Soft Robot Manipulators using Discrete Elastic Rod Dynamics

Beibei Liu; Akua K. Dickson; Ran Jing; Andrew P. Sabelhaus

cs.ROMar 23, 2026

Trajectory Generation for Underactuated Soft Robot Manipulators using Discrete Elastic Rod Dynamics

Authors: Beibei Liu, Akua K. Dickson, Ran Jing, Andrew P. Sabelhaus

Organizations: Division of Systems Engineering, Boston University, Boston MA, USA. · Department of Mechanical Engineering, Boston University, Boston MA, USA.

Abstract

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.

Figures & tables

Fig. 1: This work transforms a physics-based model of interconnected elastic rods into a control-affine representation for underactuated soft robots by introducing an explicit actuator-to-force mapping. The soft limb (a) executes a motion profile generated by our resulting model to track a desired trajectory. The Discrete Elastic Rod (DER) model uses a set of point masses to approximate the rod’s backbone, as illustrated in (b).

Fig. 2: DER uses a physically interpretable state representation based on point masses and edges, and therefore reduces the dynamics to the state

\{\mathbf{q},\bm{\phi}\}

, where

\mathbf{q}_{i}

denotes positions and

\phi_{i}

denotes twist angles. The bending angles

\theta

, material frames

\mathbf{m}

, and reference frames

\mathbf{a}

can be represented by

\{\mathbf{q},\bm{\phi}\}

Fig. 3: (a) PCC-like DER segment used to illustrate the internal force localization property in Thm. 1 , proving bending-induced forces concentrated at segment boundaries. (b) Illustration of the connecting nodes between adjacent segments in Cor. 2 , proving the colinearity induces the independence for segment-wise force decoupling.

Fig. 4: Trajectory generation using (a) the proposed DER formulation and (b) the PCC baseline. Both begin with inverse kinematics to obtain $\{\bar{\bm{q}}(t)\}_{t=0}^{T}$ from $\{\bar{\bm{r}}(t)\}_{t=0}^{T}$ [ 33 ] . The DER-based approach directly computes actuation inputs $u(t)$ through control-affine dynamics, producing dynamically feasible state–input trajectories $\{\bm{q}(t),\bm{u}(t)\}_{t=0}^{T}$ . In contrast, PCC approach computes virtual torques $\bm{\tau}(t)$ that are subsequently mapped to inputs $\bm{u}(t)$ via a calibrated mapping $\Lambda$ .

Fig. 5: Architecture and hardware overview. Our setup uses (a) computer vision for ground truth measurements of the robot’s pose and (b) a cascade control architecture where the high-level DER model generates desired pressure setpoints $\bar{u}$ , which are then tracked by a low-level feedback controller with proportional valves and sensors.

Fig. 6: Comparison of trajectories generated using two dynamics models, DER (red) and PCC (blue), against Case 1 (Asynchronous Bending)’s reference poses (yellow). Orange indicates poses where DER aligns with reference (overlap of red and yellow). The figure shows tracking performance over time, demonstrating that DER achieves closer agreement with the reference and overall better accuracy.

Fig. 7: Tracking error (

x,y

) and MSE for three distinct motion profiles (Cases 1–3, Fig. 6 ), with five trials per profile. Solid lines and shaded regions represent the mean error

\mu(e)

and its standard deviation, respectively. Our method consistently outperforms the PCC baseline across all setups.

Scenario	Model	Mean Error E (Total / x / y)	Std (Total / x / y)	Max Error (Total)
Case 1	DER	0.64 / 0.19 / 0.60	0.11 / 0.04 / 0.10	1.90
	PCC	1.59 / 0.54 / 1.44	0.26 / 0.08 / 0.25	4.81
Case 2	DER	0.29 / 0.13 / 0.23	0.05 / 0.01 / 0.05	0.98
	PCC	1.10 / 0.53 / 0.91	0.26 / 0.09 / 0.25	3.15
Case 3	DER	0.84 / 0.12 / 0.83	0.06 / 0.01 / 0.06	1.65
	PCC	1.03 / 0.12 / 1.01	0.04 / 0.01 / 0.04	2.83

TABLE I: Tracking performance comparison under different motion scenarios in cm for PCC model [ 33 ] and DER model (this paper)

Robot Motion Generation

Robot motion generation creates feasible, task-directed trajectories for robots to move, manipulate objects, and interact with people and environments.

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Control Lyapunov Functions for Underactuated Soft Robots

Huy Pham, Zach J. Patterson

Mechanical and Aerospace Engineering, Case Western Reserve University

Robotic Control Soft Robotics Lyapunov-Based Control Robot Trajectory Tracking

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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.

Abstract

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