Starting from the two-dimensional crease pattern of the conventional Miura unit (Fig. a), we rearrange the peripheral creases so that the four corners become right angles, thereby transforming the Miura cell into a rectangular unit. As illustrated in Fig. b, the new crease pattern is tessellated from these rectangular units in an n×m array; joining the left and right edges of the pattern produces an origami tube. Each new unit is composed of two distinct right-angled trapezoids that together form a rectangle. The rectangle has an overall length of (a0+b0) and a center-line length of (a1+b1), with a width of 2h; within each unit, an inclined crease subtends an angle α with the horizontal axis.

Rolling the Miura-derivative rectangular array illustrated in Fig. b yields an origami tube whose cross-section is depicted in Fig. c. The tube forms a hollow n-sided frustum; the outer and inner polygons have edge lengths of (a0-b0) and (a1-b1), respectively, subtending a central angle θ. The fully folded tube is characterized by an inner radius r1 and an outer radius rd.

To ensure the rigor of geometric derivation, the following assumptions are adopted for the Miura-derived origami unit. (1) All creases are ideal hinges with no thickness or deformation. (2) All facet panels are rigid and do not undergo in-plane deformation during folding. (3) The origami tube is formed by symmetric tessellation of uniform rectangular units. (4) Folding motion follows a continuous and single-valued geometric mapping without self-intersection. The following geometric relationships are derived: h=a1-b12tan⁡θ2-a0-b02tan⁡θ2. In the equations, ai and bi denote the horizontal length parameters of the creases, and θ is the central angle of the regular polygon; the same notation is used hereafter. The inclined crease angle is defined as α=π-θ2, while the inner radius r1=a1-b12sin⁡θ/2 and outer radius rd=a0-b02sin⁡θ/2 characterize the geometric dimensions of the origami tube in terms of crease structural parameters.

By establishing the geometric parameters of the folding pattern, we derived closed-form expressions that relate the origami tube's inner and outer diameters to these parameters. This relationship quantifies how tube dimensions vary with design variables, providing a theoretical basis for the design of origami-based tubular space structures.

The three-dimensional configuration of the Miura-derived origami continuum satisfies the following. (1) The dihedral folding angle β is uniformly distributed along the axial direction. (2) All units deform synchronously without relative sliding or distortion. (3) The tubular structure maintains axisymmetric deformation during deployment and bending.

To investigate the spatial configuration of the Miura-origami-inspired continuum, we define the dihedral folding angle of a rectangular unit as β and its width as Δz during deployment. These two variables govern the degree of folding of the rectangular unit, as illustrated in Fig. a. After n rectangular units are tessellated into a tubular configuration, a direct relationship exists between the geometric parameters of the two-dimensional crease pattern and those of the three-dimensional folded state. As depicted in Fig. b, the inner radius r1 increases monotonically as the origami tube is deployed. Specifically, the rotation angle θ of each trapezoidal facet illustrated in Fig. c is θ=2πcos⁡β2/n. In the equation, β denotes the dihedral angle formed along the creases after folding into a spatial tubular structure.

As illustrated in Fig. c, the overall length H of the Miura-origami continuum can be expressed as 1H=mΔz=2mhsin⁡β2. In the equation, m denotes the number of unit rows in the crease pattern, and h corresponds to the height of the trapezoidal facet that constitutes the polygonal origami tube.

By tailoring the geometric parameters of the crease pattern, the structural deformation required for bending and compression can be precisely engineered and modulated . Moreover, the structure cannot achieve a fully folded state without simultaneous deformation of the origami facets; consequently, the strain energy under compression is stored not only in the creases but also within the compliant facets, thereby imparting enhanced elasticity that facilitates elastic recovery of the continuum . Therefore, by judiciously designing the crease parameters, continuum structures with tailored lengths and stiffnesses can be systematically realized.

To establish a rigorous mapping from crease geometry to continuum kinematics, the following assumptions are introduced. (1) The Miura-derived origami continuum deforms with constant curvature along its axis. (2) Bending motion is constrained within a single plane for capture tasks (ϕ=0). (3) The relationship between folding angle βi and cable actuation is monotonic and continuous. (4) The modified D–H method is valid under small-deformation and rigid-panel assumptions.

As illustrated in Fig. , actuation cables are routed along its lateral edges and anchored to each segment, and the magnitude of bending is characterized by the spatial folding angles βi (i=1,2). The pose of the continuum axis can be fully described by three independent parameters: the arc length s along the continuum, its curvature κ, and the curvature angle ϕ. A mapping is subsequently derived that relates the spatial folding angle β of the origami tube to the continuum's pose variables.

Subsequently, by mapping the continuum variables s, κ, and ϕ to the Denavit–Hartenberg (D–H) parameters, the end-effector coordinates (x, y) of the Miura-origami-derived continuum can be derived. The derivation of the forward kinematics is formulated as follows: x,y⟵D-Hθ,d⟵f1s,κ,ϕ⟵f2βi, where x and y denote the Cartesian coordinates of the end-effector; θ and d are the parameters obtained via the D–H convention; and the functions f1 and f2 are defined in Eqs. (12) and (8), respectively.

The equivalent model of the Miura-origami tube continuum aims to establish a mapping between the actuating cable lengths and the pose of the continuum robot. Based on the established relationship between the spatial folding angles βi of the Miura-origami tube and the shape variables s, κ, and ϕ of the continuum robot joints, and in accordance with the operational requirements of the capture robot – wherein the mechanism bends within a single plane – the curvature angle is set to zero (ϕ=0). Consequently, the corresponding functional relationship is obtained as follows: 2κ,s,ϕT=f2β1,β2=sin⁡β22-sin⁡β12rdsin⁡β12+sin⁡β22,mhsin⁡β12+sin⁡β220T. In the equation, m denotes the number of unit rows in the crease pattern, h is the height of the trapezoidal facet forming the origami-tube polygon, and rd represents the circumscribed radius of the origami-tube polygon.

The Miura-origami-derived continuum comprises one revolute joint and one prismatic joint, which collectively enable its bending degree of freedom. Its kinematic model can therefore be reduced to a virtual rigid-link robot consisting of a revolute joint at the base, a prismatic joint at the mid-span and a second revolute joint at the distal end, as illustrated in Fig. .

The coordinates of the continuum end point are first determined by the base revolute joint and the mid-span prismatic joint. A subsequent rotation of the distal revolute joint aligns the local coordinate frame with the tangent vector at the continuum tip, thereby defining the full pose of the Miura-origami continuum robot. The robot kinematics are formulated using the modified D–H convention; the corresponding parameters are listed in Table . Owing to the kinematic symmetry, the joint variables of the base and top revolute joints are related by θ1=θ3, enabling the derivation of the homogeneous transformation matrix i0T.

The kinematic equivalence between the origami crease geometry and the virtual rigid-link robot is established under the condition that the bending curvature generated by folding angles β1,β2 is uniquely mapped to the D–H joint variables θ1,d2,θ3. Due to kinematic symmetry, θ1=θ3 holds throughout the motion. The prismatic variable d2 represents the effective chord length corresponding to the origami continuum bending.

Applying the serial-chain kinematic transformation, the generalized form is obtained as follows: 3ii-1T=cos⁡θi-sin⁡θi0ai-1sin⁡θicos⁡αi-1cos⁡θicos⁡αi-1-sin⁡αi-1-disin⁡αi-1sin⁡θisin⁡αi-1cos⁡θisin⁡αi-1cos⁡αi-1dicos⁡αi-10001.

The end-effector pose of the Miura-origami-derived continuum with respect to the fixed frame is obtained by sequentially right multiplying the homogeneous transformation matrices of each joint 30T=10T21T32T. Substituting the D–H parameters yields the following explicit homogeneous transformation matrix: 430T=c1c3-s1s3-c1s3-c3s10s1d20010-c1s3-c3s1s1s3-s1c30c1d20001. In the expression, c denotes cos⁡, and s denotes sin⁡.

By exploiting the established relationships between the D–H joint variables θi and di of the Miura-origami-derived continuum robot and its shape variables s and κ, the corresponding functional relations are obtained as follows: 5θ1d2θ3T=f1s,κ=π-κs2,2κsin⁡κs2,π-κs2T.

The workspace is rigorously derived based on the geometric constraints of crease parameters and the allowable range of folding angles. All boundary conditions of the workspace are consistent with the physical folding limits of the Miura-derived origami structure, ensuring no invalid or non-physical configuration is included.

As illustrated in Fig. , the position of the Miura-origami-derived continuum end-effector center is fully determined by the arc length S and the curvature κ; consequently, only the joint variables of the first two joints are required to locate this point. Thus, the reachable positional workspace is obtained once the homogeneous transformation matrix (20T) of the end-effector center with respect to the fixed frame has been derived.

This matrix (20T) is obtained by sequentially right multiplying the homogeneous transformation matrices of the first three joints: 630T=10T21T=c10s1s1d20100-s10c1c1d20001. In the expression, c denotes cos⁡, and s denotes sin⁡.

Based on the analysis of a single Miura-origami-derived continuum, the admissible ranges of the Denavit–Hartenberg (D–H) joint variables are determined from the prescribed bounds of the shape variables s, κ, and ϕ via their established functional relationships.

Employing the Monte Carlo method to sample the continuum end-effector center yields the workspace of the Miura-origami-derived continuum, depicted in Fig. . A three-dimensional visualization of the origami-continuum workspace, generated via Monte Carlo sampling, further elucidates the reachable domain and deformation characteristics, as shown in Fig. .

To further enhance the depth and rigor of the workspace analysis, a deep analysis of grasping workspace is supplemented herein, focusing on the dependence of workspace characteristics on key design parameters and its relevance to the intended on-orbit capture task.

Based on the geometric and kinematic models established in the former section, a systematic sensitivity analysis is conducted to quantify the effects of core design parameters on the workspace of the Miura-derived origami continuum. The key design parameters investigated include the number of finger units “fn”, the height of the finger “fL”, the radius of the finger “fr”, and the outer circumscribed radius “bR”, which are critical to the structural deformation and workspace performance.

The quantitative relationships between each parameter and workspace characteristics are derived as follows. (1) The workspace volume V is positively correlated with “fn” and “fL”, following the trend V∝fn⋅fL within the allowable structural constraint, which is attributed to the increased deformable length and cross-sectional expansion of the continuum with larger “fn” and “fL”. (2) The maximum bending angle θmax⁡ is primarily determined by the radius of the finger “fr” and the geometric constraints of the modified Miura crease pattern. (3) The outer circumscribed radius “bR” affects the lateral reach of the workspace, with a larger “bR” leading to a wider lateral coverage but a slight reduction in the maximum bending angle due to increased structural rigidity. Comparative simulations are performed by varying each parameter while keeping others constant, and the results are visualized in the revised Fig.  to intuitively present the impact of each parameter on the workspace.

The workspace of the Miura-derived origami continuum is tailored to meet the practical demands of on-orbit non-cooperative target capture, and its relevance to the intended task is elaborated as follows. First, the workspace size is designed to match the typical size range of small- to medium-sized space debris, ensuring effective enveloping of targets with varying dimensions. Second, the flexible bending capability and continuous deformability of the workspace enable the gripper to adapt to irregular target geometries (e.g., spherical, cubic, irregular fragments) by passively conforming to the target surface, which is a key advantage for capturing non-cooperative targets without predefined grappling interfaces. Furthermore, the Monte Carlo sampling-based workspace analysis is supplemented with statistical evaluations of the reachable pose distribution. The adjustable reachable range of the workspace facilitates obstacle avoidance in complex orbital environments, where the gripper can adjust its posture within the workspace to avoid collisions with other orbital objects or spacecraft components.

Upon completion of the forward-kinematic analysis of the Miura-origami-derived continuum, the pose of the continuum robot and the position of its end-effector can be determined for any specified set of origami parameters.

To facilitate robot control, the target coordinates must serve as the primary input; hence, given the end-effector position of the continuum, the corresponding actuation magnitudes βi must be determined. Consequently, an inverse-kinematics analysis of the Miura-origami-derived continuum is conducted.

From the forward-kinematic relationships given in former equations, we obtain 7xyT=f1(β1,β2)=rdsin⁡β12+sin⁡β22cos⁡mhrdsin⁡β22-sin⁡β12-1sin⁡β22-sin⁡β12rdsin⁡β12+sin⁡β22sin⁡mhrdsin⁡β22-sin⁡β12sin⁡β22-sin⁡β12.

In the equation, m denotes the number of unit rows in the crease pattern, h is the height of the trapezoidal facet forming the origami-tube polygon, and rd represents the circumscribed radius of the origami-tube polygon.

Consequently, the spatial folding angle βi can be expressed explicitly in terms of the Cartesian coordinates x and y as follows: βi=Finv(x,y).

To validate the proposed theory, a Miura-origami-inspired continuum robot prototype actuated by dual servomotors with antagonistic tendons routed along both lateral sides was fabricated. A straight-line trajectory was selected; the corresponding spatial folding angles βi were computed from the prescribed coordinates and subsequently validated on the prototype, as reported in Table .

The computed spatial folding angles βi were transmitted to the host computer, which actuated the tendons to regulate the configuration of the Miura-origami continuum robot, guiding its end-effector to the desired coordinates, as illustrated in Fig. . Grippers with varying bend angles maintain a constant height by adjusting β1 and β2; the MATLAB motion-simulation image in Fig. b confirms that, for the bending angles listed in Table , the end-effector center of the continuum robot remains at a uniform elevation. Figure a demonstrates that the Miura-origami-inspired continuum robot traces a rectilinear trajectory, thereby corroborating the accuracy of its inverse kinematics and validating the viability of the proposed path-planning strategy.

To further validate the reliability of the proposed kinematic model, quantitative evaluations of positioning accuracy and repeatability are conducted via repeated positioning tests. Figure  presents the experimental setup and measurement results of the repeated positioning test, where Fig. a shows the test device and Fig. b illustrates the comparison between measured and theoretical coordinates.

As shown in Fig. a, the end-effector position P(xp,yp) is measured under a fixed tendon actuation configuration, and 30 repeated measurements are performed to obtain the trajectory tracking data. Figure b presents the measured coordinate values (blue fitted curves) and the theoretical values predicted by the kinematic model (red dashed lines). The theoretical coordinates are x=51.7449 and y=85.8737 mm, while the measured results exhibit small fluctuations around these values.

The maximum absolute deviation in the x direction is less than 3 mm, and the maximum absolute deviation in the y direction is less than 2 mm, corresponding to a relative error of less than about 5.8 % and 2.3 %, respectively. These results demonstrate that the proposed kinematic model achieves high positioning accuracy and excellent repeatability under the same actuation input, confirming its reliability for trajectory prediction and control.

As depicted in Fig. , the prototype employs tendon-driven actuation: three origami continuum segments are individually actuated by a single servomotor that tensions or releases the tendons via a pulley mechanism, thereby controlling the motion of each segment. Low-stretch Kevlar tendons were selected for the drive lines, and bus-type ZP20D servomotors were used for actuation.

All prototype experiments and performance validations presented in this study are conducted under ground-based atmospheric and 1 g gravitational conditions, which serve as a proof-of-concept demonstration to verify the feasibility of the proposed Miura-derived origami continuum grasping mechanism. It is necessary to clearly distinguish between such ground-based verification and the actual requirements for real on-orbit space deployment.

In practical space applications, the gripper will face a series of critical challenges, including microgravity dynamics, extreme thermal vacuum environment, space radiation and long-term structural stability, as well as the capture of tumbling, non-cooperative and high-speed moving targets. In addition, the current prototype uses conventional engineering materials, which cannot meet the aerospace requirements of low outgassing, high-temperature resistance and radiation resistance. The actuation system, control strategy and reliability design also need to be further upgraded for on-orbit service.

This work focuses on the conceptual design, kinematic modeling and functional verification of the origami continuum space manipulator. Subsequent research will carry out space-adapted optimization, including aerospace-grade material selection, environmental adaptability testing and dynamic control for non-cooperative targets, so as to bridge the gap between the ground prototype and practical on-orbit applications.