Real-Time Trajectory Generation for Soft Robot Manipulators Using Differential Flatness

TL;DR

This work tackles real-time trajectory generation for soft robot manipulators by exploiting differential flatness of piecewise constant curvature (PCC) dynamics. By proving that PCC soft-robot dynamics are differentially flat with flat outputs $y=q$, the authors enable algebraic computation of control inputs from end-effector trajectories, bypassing expensive nonlinear dynamical solutions. The method combines inverse kinematics to map task-space goals to PCC segment angles and a flatness-based input computation, using simple finite differences to obtain velocities and accelerations, producing dynamically-feasible trajectories with substantial speed advantages. Simulations on a planar two-segment PCC manipulator demonstrate ~23x real-time speedups at 100 Hz and micrometer-level tracking accuracy, suggesting potential for fast verifiable replanning in safety-critical environments and informing future hardware implementations. This approach offers a principled, first-principles alternative to data-driven or reduced-order methods for real-time soft-robot control, with clear paths toward 3D extension and obstacle-aware planning.

Abstract

Soft robots have the potential to interact with sensitive environments and perform complex tasks effectively. However, motion plans and trajectories for soft manipulators are challenging to calculate due to their deformable nature and nonlinear dynamics. This article introduces a fast real-time trajectory generation approach for soft robot manipulators, which creates dynamically-feasible motions for arbitrary kinematically-feasible paths of the robot's end effector. Our insight is that piecewise constant curvature (PCC) dynamics models of soft robots can be differentially flat, therefore control inputs can be calculated algebraically rather than through a nonlinear differential equation. We prove this flatness under certain conditions, with the curvatures of the robot as the flat outputs. Our two-step trajectory generation approach uses an inverse kinematics procedure to calculate a motion plan of robot curvatures per end-effector position, then, our flatness diffeomorphism generates corresponding control inputs that respect velocity. We validate our approach through simulations of our representative soft robot manipulator along three different trajectories, demonstrating a margin of 23x faster than real-time at a frequency of 100 Hz. This approach could allow fast verifiable replanning of soft robots' motions in safety-critical physical environments, crucial for deployment in the real world.

Figures (6)

• Figure 1: The trajectory planning method in this manuscript generates dynamically feasible trajectories of a piecewise constant curvature soft robot manipulator (purple). A series of tip positions (black dotted lines) are tracked using a combination of fast inverse kinematics and differential flatness.
• Figure 2: (a) A piecewise constant curvature soft robot manipulator has kinematics defined by its subtended angles $q_i$ and lengths $L_i$. (b) The center of mass dynamics of a PCC robot can be matched to a constrained C.o.M., $\mu$, of rigid manipulator per della2020imporved. Here, we use the RPPR representation.
• Figure 3: With a sufficiently-small $\Delta t$, our method calculates open-loop trajectories that track the three reference trajectories A, B and C with low error.
• Figure 4: Our procedure plans faster than real time whenever the average iteration time for Alg. 1 is less than the planning/control timestep. Our fastest real-time $\Delta t$ is 360 microseconds (green line), which to the authors' knowledge is significantly faster than any other soft robot trajectory generator with dynamic feasibility guarantees.
• Figure 5: We confirm that our results are not an artifact of our test setup by relying on the asymptotic stability of the PCC manipulator dynamics, and initializing the robot at a different set of joint angles, before executing the open-loop $u(0),\hdots,u(T)$ of Trajectory A. The simulation converges to the reference trajectory quickly despite no loop closure, validating that Alg. \ref{['alg:one']}'s calculated $\{x(t),u(t)\}_{0\hdots T}$ is feasible.

Theorems & Definitions (4)

• Theorem 1
• proof
• Remark 1
• Remark 2