Robot trajectory tracking control based on differential-algebraic equation (DAE) models is still a thorny issue, because the DAEs of such systems are inherently complex and unstable, such as the high index problem. In this paper, based on controlled DAEs, a symplectic instantaneous optimal control (IOC) method for robot trajectory tracking with input saturation is proposed. Based on the discrete variational principle and the canonical transformation, a symplectic discretization form for the controlled DAEs is first constructed. Then, the continuous trajectory tracking problem is approximated for a series of IOC problems at every time step, and the linear complementarity problem (LCP) can be derived for solving the IOC problems. Finally, the control inputs can be obtained by solving the corresponding standard LCP. The proposed method provides a unified framework for solving the trajectory tracking control problems of robot multibody dynamic systems. Numerical simulations and virtual experiments are conducted to verify the robustness and the efficiency of the proposed method, i.e., the input saturation constraints are satisfied at the discrete time points, and high accuracy tracking control results can be obtained at low computational cost.

Cable-driven continuum robots with hyper-redundant deformable backbones show great promise in applications, such as inspection in unstructured environments, where traditional rigid robots with discrete links and joints fail to operate. However, the motion of existing continuum robots is still constrained by their homogeneous backbones, and limited to environments with modest geometrical complexity. In this study, inspired by highly deformable elephant trunks, we presented a modular tensegrity structure with preprogrammable stiffness for continuum robots. Then we derived a mechanical model based on a positional formulation finite element method for predicting the configuration of the structure in different deformation scenarios. Theoretical predictions revealed that the curvature of each segment could be regulated by preprogramming their spring stiffness. Hence, our customizable design could offer an effective route for efficient robotic interactions. We further fabricated a continuum robot consisting of 12 modules, and showcased its deformation patterns under multiple scenarios. By regulating the distribution of spring stiffness, our robot could move through channels with varying curvatures, exhibiting its potential for applications where varying curvature, and conformal and efficient interactions are needed. Leveraging the inherent intelligence, this robotic system could simplify the complexity of the required actuation and control systems.

Purpose of review: In this review, we briefly summarize the numerical methods commonly used for the nonlinear dynamic analysis of soft robotic systems. The underlying mechanical principles as well as the geometrical treatment tailored for soft robots are introduced with particular emphasis on one-dimensional models. Additionally, the review encompasses three-dimensional frameworks, available simulation packages, and various types of interaction models, shedding light on the design, actuation, motion control, and internal and external forces of soft robots. Recent findings: Reduced-order models can offer high efficiency in characterizing nonlinear deformations, allowing convenient tailoring based on specific structural and material configurations. For pursuing high simulation accuracy and detailed mechanics, the finite element method proves to be a valuable tool through numerous off-the-shelf platforms. Furthermore, machine learning has emerged as a promising tool to effectively address the challenges within the mechanics community. Summary: A wide range of kinematic and dynamic numerical models is available for simulating the behaviors of soft robots, offering exceptional adaptability to different geometries and structures based on existing modeling theories and numerical solution algorithms. However, the trade-off between computational complexity and simulation accuracy remains a challenge in achieving fast, accurate, and robust control of soft robots in complex environments.