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

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.

This paper introduces a general Lie group framework for modeling continuum soft robots, employing Cosserat rod theory combined with cumulative parameterization on the Lie group SE(3). This novel approach addresses limitations present in current strain-based and configuration-based methods by providing geometric local control and eliminating unit quaternion constraints. The paper derives unified analytical expressions for kinematics, statics, and dynamics, including recursive Jacobian computations and an energy-conserving integrator suitable for real-time simulation and control. Additionally, the framework is extended to handle complex robotic structures, including segmented, branched, nested, and rigid-soft composite configurations, facilitating a modular and unified modeling strategy. The effectiveness, generality, and computational efficiency of the proposed methodology are demonstrated through various scenarios, including large-deformation rods, concentric tube robots, parallel robots, cable-driven robots, and articulated fingers. This work enhances modeling flexibility and numerical performance, providing an improved toolset for designing, simulating, and controlling soft robotic systems.

Continuum soft robots are inherently underactuated and subject to intrinsic input constraints, making dynamic control particularly challenging, especially in hybrid rigid-soft robots. While most existing methods focus on quasi-static behaviors, dynamic tasks such as swing-up require accurate exploitation of continuum dynamics. This has led to studies on simple low-order template systems that often fail to capture the complexity of real continuum deformations. Model-based optimal control offers a systematic solution; however, its application to rigid-soft robots is often limited by the computational cost and inaccuracy of numerical differentiation for high-dimensional models. Building on recent advances in the Geometric Variable Strain model that enable analytical derivatives, this work investigates three optimal control strategies for underactuated soft systems -- Direct Collocation, Differential Dynamic Programming, and Nonlinear Model Predictive Control -- to perform dynamic swing-up tasks. To address stiff continuum dynamics and constrained actuation, implicit integration schemes and warm-start strategies are employed to improve numerical robustness and computational efficiency. The methods are evaluated in simulation on three Rigid-Soft and high-order soft benchmark systems -- the Soft Cart-Pole, the Soft Pendubot, and the Soft Furuta Pendulum -- highlighting their performance and computational trade-offs.