Key Takeaways & Executive Findings
- •• Introduces a unified dynamic modeling method for generalized space-deployable mechanisms using the local frame of the SE(3) Lie group, effectively handling both rigid and flexible links. • Derives equations of motion via Hamilton's principle, yielding a constant mass matrix in the local frame for compact and unified dynamic equations. • Employs a Lie group generalized-α time integration method to ensure numerical stability and efficiency for multibody systems with large rotations and deformations. • Demonstrates the formulation through two numerical examples, showing accurate motion responses under varying configurations and loading conditions, broadening the application of Lie group methods in space mechanisms.
Abstract
As space equipment become larger in size and more flexible, generalized mechanisms are being widely used in space-deployable structures. Dynamic modeling of large-scale generalized space-deployable mechanisms is challenging owing to the coupling between the deformation of flexible links and rigid body motion. This study develops a dynamic modeling method for generalized mechanisms using the local frame of the SE(3) Lie group. The model represents both rigid and flexible links within a unified Lie group setting. The expressions for the velocities of rigid links and deformation of flexible links are derived using the Lie algebra framework. The nonuniqueness of the degrees of freedom of generalized kinematic pairs is considered, and the velocity fields of kinematic pairs in different situations are expressed. The equations of motion are derived using Hamilton’s principle. Because the velocities are expressed in the local frame, the mass matrix in the equation is constant, which yields a compact and unified expression for the dynamic equation. A Lie group generalized-α time integration method is adopted to ensure numerical stability and efficiency in simulating multibody systems with large rotations and deformations. Two numerical examples are studied to demonstrate a formulation that reflects the motion responses under varying configurations and loading conditions. This study broadens the application of the local frame of the Lie group formulation in space mechanisms and provides a new concept for dynamic modeling of generalized mechanisms.
1. Introduction
With the continuous development of aerospace technology, space-deployable mechanisms, which are composed of the skeleton and actuators of space equipment, are increasing in size. Dynamic characteristics and responses of such mechanisms are complex because their motion process is affected by the deformation of links [1–3]. Large-size space-deployable mechanisms cannot be equivalent to rigid mechanisms or compliant mechanisms with flexibility because links such as springs, ropes, and flexible rods included in the mechanisms change the transmission forms of motions and forces. These mechanisms are generalized, and all their objects that can transmit motions and forces by their deformation can be employed as generalized links [4, 5].
Dynamic modeling of generalized space-deployable mechanisms inevitably considers the coupling between the motion of rigid links and deformation of flexible links in their motion process analysis [6–8]. Existing dynamic modeling methods can be classified into two categories according to the reference coordinate systems as follows: absolute and relative coordinate methods. A typical absolute coordinate method is the absolute nodal coordinate formulation (ANCF), which was proposed by Shabana (1996) [9]. Thus far, various ANCF solvers have been developed and widely applied to dynamic modeling of space-deployable structures [6, 10, 11]. A typical relative coordinate method is the geometrically exact beam formulation. It is implemented by fixing the local coordinate system on a flexible beam and describing the motion and deformation of the beam using inertial and floating coordinate systems [12]. The system dynamics equations of the ANCF have high dimensions and numerous constraints; hence, the computational efficiency is low. The traditional floating coordinate method describes the configuration of the local frame relative to that of the inertial frame using quaternions [13, 14] and Euler angles [15, 16], which may lead to singularities. Brüls et al. [17–20] described the motion of objects in a special Euclidean group, based on the Lie group and Lie algebra theory. Because this method defines an appropriate local frame when establishing the equations of motion, it is called the local frame of the Lie group (LFLG) formulation.
The Lie matrix group is a continuous matrix group for which the composition operation and inverse are smooth. Geometrically, this group is a differentiable manifold; therefore, the differential geometry can be used to operate on the group, which makes it well suited for dynamic modeling of complex mechanisms.
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Zijie Zeng, Tuanjie Li, Hangjia Dong (2025). Dynamics of Generalized Space-Deployable Mechanisms Based on the Local Frame of the SE(3) Group. Chinese Journal of Mechanical Engineering. https://doi.org/10.1186/s10033-025-01314-7
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Frequently Asked Questions
What is the main contribution of this paper?
The paper develops a dynamic modeling method for generalized space-deployable mechanisms using the local frame of the SE(3) Lie group, providing a unified framework for rigid and flexible links and yielding a constant mass matrix for compact equations of motion.
How does the proposed method handle the coupling between rigid and flexible links?
The method represents both rigid and flexible links within a unified Lie group setting, deriving velocities and deformations using Lie algebra, and uses Hamilton's principle to derive equations of motion that inherently account for the coupling.
What are the advantages of using the local frame of the SE(3) group?
Using the local frame results in a constant mass matrix, leading to a compact and unified dynamic equation. It also avoids singularities associated with traditional floating coordinate methods that use quaternions or Euler angles.
How is numerical stability ensured in the simulations?
A Lie group generalized-α time integration method is adopted, which ensures numerical stability and efficiency for multibody systems with large rotations and deformations.
What are the potential applications of this research?
The research broadens the application of Lie group formulations in space mechanisms, providing a new concept for dynamic modeling of generalized mechanisms, which can be used in the design and analysis of large-scale space-deployable structures.
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