Theoretical Modeling and Bio-inspired Trajectory Optimization of A Multiple-locomotion Origami Robot

TL;DR

The paper tackles the lack of theoretical grounding for design and control of soft, bio-inspired multi-locomotion origami robots by developing a math-based framework for crawling and swimming. It couples a discrete dynamic crawling model with friction μ and a DH-based swimming kinematics model for a three-tower origami arm, together with a heuristic A*-based gait-planning approach. Key contributions include: (i) a friction-influenced crawling model and design guidance; (ii) a forward-kinematics model for 3-DoF origami towers enabling end-effector trajectory planning; and (iii) a graph-search gait optimization that yields human-like swimming gaits, with thrust and drag considerations expressed via $F_d=\tfrac{1}{2} \rho C_d A_p V^2$ and A* heuristics. Validation through simulations and experiments demonstrates improved locomotion efficiency and provides a transferable framework for other soft, multi-jointed devices.

Abstract

Recent research on mobile robots has focused on increasing their adaptability to unpredictable and unstructured environments using soft materials and structures. However, the determination of key design parameters and control over these compliant robots are predominantly iterated through experiments, lacking a solid theoretical foundation. To improve their efficiency, this paper aims to provide mathematics modeling over two locomotion, crawling and swimming. Specifically, a dynamic model is first devised to reveal the influence of the contact surfaces' frictional coefficients on displacements in different motion phases. Besides, a swimming kinematics model is provided using coordinate transformation, based on which, we further develop an algorithm that systematically plans human-like swimming gaits, with maximum thrust obtained. The proposed algorithm is highly generalizable and has the potential to be applied in other soft robots with multiple joints. Simulation experiments have been conducted to illustrate the effectiveness of the proposed modeling.

Figures (9)

• Figure 1: The diagram of the origami bio-inspired robot, retrieved from dong.
• Figure 2: The origami sheet for the octagon-origami structure(B). The black lines and the red dot lines represent mountain and valley folding patterns, respectively in (A). The force sketches of crawling captured at the time circle $t-1$ for the first half (B) and t for the second half(C). The sign ‘+’ represents the positive direction. $h$ denotes the height for the specified leg and $d$ is the horizontal distance between the reference point and the front foot for the given leg, respectively; $M_a$ is the equivalent actuation torque. The middle points (red points)of the ends of the octagon-origami structure are considered reference points.
• Figure 3: The geometrical and crawling sketches of the proposed robot. (A) The leg configuration with dimensions $h^i, H^i, b^i, r^i$, $i=l$ for the front leg and $i=r$ for the rear leg; (B), the projection lengths $m^i$ (along $x$-axis) and $n^i$ (along $y$-axis) directions; (C) The sketch of the proposed robot with geometrical parameters; (D-E) The status simplified sketches of the robot crawling in the first half and the second half. $\alpha_j^{{i}'}$ and $l_j^i$ symbolize the equivalent angle and the leg's length. $j$ being 0 or 1, denoting the first or second half. The subscripts 0 and 1 of the circumference angle $\theta$, the chord length $a$, and the chord inclination $\beta_k$ point to the sample time for a motion cycle. The robotic height can be mainly determined by the equivalent angle $\alpha_j^{{i}'}$ and leg length $l_j^f$ of the front leg.
• Figure 4: The Three Stages of Breaststroke locomotion(A), The sketch of the proposed robot(B); the front view(C-1), top-down view(C-2) and the corresponding crease pattern(C-3) of the origami twisted tower, where the solid black lines represent the mountain folds and the dashed black line denote the valley folds, respectively; the coordinate transformations for the first, second, third joint of the robotic arm(D); the top view of an origami joint(E). 1, 2, and 3 represent the lifting, outreaching, and rotating actions, and 4 indicates a flapping plate. $x_i, y_i$ and $z_i$ denote the coordinate reference frame of the $i$-th joint.
• Figure 5: The dimensions of the proposed robot(A) and the first joint of the proposed robot lifts arms in the water(B). The red dashed line indicates the dimension baseline(A). The blue dash lines represent the water(B). A blown-up view of the robotic foot is provided, which has two frictional surfaces.