⚡ electrical engineering

Rigid-Flexible Coupling Dynamic Modeling of Truss Deployable Antenna Based on a Non-Orthogonal Joint Coordinate System

This paper proposes a non-orthogonal joint coordinate system for fully parameterized ANCF beams to eliminate joint axis direction errors caused by cross-sectional deformation in truss deployable antennas, thereby significantly improving the accuracy of dynamic modeling and stress prediction compared to traditional orthogonal methods and commercial software.

Original authors: Xiaoyu Hu, Yongsheng Zhao, Zongke Liu, Chencheng Zhang, Fei Hu, Bo Han, Jiantao Yao

Published 2026-10-15

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CC BY 4.0

Original authors: Xiaoyu Hu, Yongsheng Zhao, Zongke Liu, Chencheng Zhang, Fei Hu, Bo Han, Jiantao Yao

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Space is a place of extreme precision. Satellites orbiting Earth rely on massive antennas to send and receive signals, acting as the ears and eyes of our global communication network. To fit inside a rocket, these antennas must be folded into a compact package and then unfolded once they reach orbit. This unfolding process, known as deployment, is a delicate mechanical dance where rigid nodes and flexible rods must move in perfect unison. If the structure is too stiff, it might break; if it is too flexible, it might wobble or fail to lock into the correct shape. Engineers have long used computer simulations to predict how these structures will behave, but a specific problem has persisted: when the flexible parts of the antenna bend and twist, the mathematical models used to describe their joints often lose their way, leading to inaccurate predictions of stress and motion.

Researchers at Yanshan University and the China Academy of Space Technology have addressed this issue by developing a new way to model the joints in these deployable structures. Their work focuses on a specific type of mathematical description called the absolute nodal coordinate formulation, which is excellent at tracking large movements and deformations. However, when a beam bends, its cross-section—the shape you would see if you cut the beam in half—can distort. In traditional models, the computer forces the coordinate system describing this cross-section to remain perfectly square, or orthogonal, even when the material itself has warped. This forced straightening creates a subtle but significant error in how the model defines the axis of the hinge connecting the parts. The researchers realized that instead of forcing the coordinate system to stay square, they should let it follow the natural, warped shape of the material. By introducing a non-orthogonal frame that moves with the deformation, they created a model that respects the actual geometry of the bending beam.

To test this idea, the team built a digital simulation of a basic module from a truss deployable antenna, a structure made of interconnected rods and nodes. They compared their new method against a standard approach that forces the coordinate system to stay orthogonal, and also against a high-fidelity reference simulation created in commercial software. The results were striking. When the antenna module deployed and folded, the traditional model showed a growing error in the direction of the hinge axis, which was directly linked to a specific type of shearing deformation in the beam. In contrast, the new model, which allowed the coordinate system to follow the material's natural distortion, kept the hinge axis aligned with the physical reality of the structure. This difference had a profound impact on the predicted stress within the material. The standard model produced a variation in stress predictions with a standard deviation of 26.3 megapascals compared to the reference, whereas the new model reduced this error to just 2.9 megapascals. This represents an approximately 89.0 percent reduction in error, meaning the new method predicts the internal forces on the antenna with far greater accuracy.

The researchers also examined how the structure moved, tracking its position, speed, and acceleration during a full cycle of deployment and folding. They tested three different configurations of torsion springs, which provide the energy to unfold the antenna. In all cases, the new model successfully reproduced the main movements and the sequence of events, including the initial deployment, the shock of reaching the fully open position, and the subsequent folding back. While small differences remained in the high-frequency vibrations during the most chaotic parts of the motion, the overall behavior was captured with high fidelity. The study confirms that by allowing the mathematical description of a joint to adapt to the actual shape of the deforming material, engineers can create much more reliable simulations. This advancement offers a powerful tool for designing the next generation of space antennas, ensuring they will deploy correctly and withstand the harsh environment of orbit without the need for expensive and difficult ground-based testing.

Technical Summary: Rigid-Flexible Coupling Dynamic Modeling of Truss Deployable Antenna Based on a Non-Orthogonal Joint Coordinate System

Problem Statement
The accurate dynamic modeling of spaceborne truss deployable antennas requires a rigorous rigid-flexible coupling approach. While the Absolute Nodal Coordinate Formulation (ANCF) effectively describes large displacements and deformations of flexible beams by retaining transverse gradient vectors, it allows for cross-sectional deformation, including shear. A critical issue arises when modeling revolute joints in such fully parameterized ANCF beams: the standard approach often employs an orthogonal tangent frame (constructed via cross-products of gradient vectors) to define joint axes. However, under $xy$-shear deformation, the material directions of the beam cross-section become non-orthogonal. Reconstructing an orthogonal frame in this state introduces a directional error in the representation of the revolute joint axis, as the orthogonalized frame deviates from the actual material-defined axis embedded in the deforming cross-section. This discrepancy can lead to inaccuracies in kinematic constraints and stress predictions, particularly in high-precision deployment scenarios where ground testing is limited by gravity compensation challenges.

Methodology
To address the axis direction error induced by orthogonalization, the authors propose a modeling framework based on a non-orthogonal cross-section frame. The methodology proceeds as follows:

Component Modeling: Flexible rods are discretized using fully parameterized ANCF beam elements, which utilize absolute nodal positions and gradient vectors. Rigid nodes are modeled with standard six-degree-of-freedom coordinates.
Joint Coordinate System Formulation:
- Tangent Frame (Baseline): Defined by orthogonalizing the gradient vectors (e.g., $\mathbf{t} = \mathbf{r}_{,x}$ , $\mathbf{b} = \mathbf{r}_{,y} \times \mathbf{t}$ ). This frame enforces orthogonality but alters the joint axis direction under shear.
- Cross-Section Frame (Proposed): Defined directly by the material gradient vectors ( $\mathbf{t} = \mathbf{r}_{,y} \times \mathbf{r}_{,z}$ , $\mathbf{b} = \mathbf{r}_{,z}$ , $\mathbf{n} = \mathbf{r}_{,y}$ ). This frame is non-orthogonal but preserves the material-defined joint axis direction regardless of shear deformation.
Constraint and Force Formulation: The authors derive the geometric relationship between the two frames, proving that the tangent frame's axis deviation is directly proportional to the $xy$-shear angle. They then formulate the revolute joint constraints and the generalized forces (and their Jacobians) for driving torsion springs consistently within the non-orthogonal cross-section frame. This ensures that both kinematic constraints and torque transmission follow the local deformation of the cross-section.
System Assembly: Using graph theory and associative arrays, the component and joint formulations are assembled into system-level dynamic equations (Lagrange equations of the first kind) for a basic deployable module of a truss antenna.
Numerical Solution: The resulting index-3 differential-algebraic equations are solved using the Newmark- $\beta$ method combined with Newton's iterative scheme, including constraint equation rescaling to improve conditioning.

Key Results
The proposed model was validated against an Adams/View reference simulation (using finite element modal reduction) and compared against the standard tangent frame approach:

Joint Axis Accuracy: The cross-section frame maintained alignment with the material-defined joint axis, resulting in near-zero direction error. In contrast, the tangent frame exhibited an axis direction error that tracked the $xy$-shear deformation angle.
Stress Prediction: The cross-section frame significantly improved stress prediction accuracy. Compared to the reference simulation, the standard deviation of the von Mises stress residual was reduced from 26.3 MPa (tangent frame) to 2.9 MPa (cross-section frame), representing an 89.0% reduction. The tangent frame showed large deviations (up to 140 MPa) during periods of high shear.
Strain Energy: Both formulations yielded similar strain energy responses, indicating that the local axis discrepancy has a limited effect on global energy metrics but a significant effect on local stress and strain components (specifically the $E_{12}$ shear strain).
System Dynamics: Under three different torsion spring layouts, the proposed model successfully reproduced the principal position, velocity, and acceleration responses, as well as the deployment-shock-folding sequence. While the model showed high agreement during the deployment phase, larger discrepancies (measured by NRMSE) were observed during the shock and folding stages, attributed to the sensitivity of high-frequency elastic vibrations to modeling and numerical integration differences.

Significance and Claims
The paper claims that the primary contribution is a joint modeling formulation based on a non-orthogonal cross-section frame specifically designed for fully parameterized ANCF beams. The significance of this work lies in:

Eliminating Orthogonalization Artifacts: By retaining the material-defined joint axis direction, the proposed frame avoids the directional errors introduced by orthogonalizing the tangent frame under shear deformation.
Consistent Formulation: It enables the consistent formulation of both revolute joint constraints and torsion spring driving forces within the same deformation-following coordinate system.
Improved Local Accuracy: The framework provides a more effective approach for predicting local stress and strain responses in flexible deployable structures, which is critical for dynamic design evaluation where ground testing is insufficient.

The authors note that while the model improves local accuracy, discrepancies in high-frequency responses during transient stages suggest that future work should focus on numerical convergence and higher-fidelity transient analyses. Additionally, the current formulation assumes ideal revolute joints and specific axis orientations, leaving room for future extension to include joint friction, clearance, and arbitrary axis locations.

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