Elastic Stability of Cosserat Rods and Parallel Continuum Robots
Abstract:
Classic theories in nonlinear elasticity have increasingly been used to obtain accurate and efficient models for continuum robots and other elastic structures. Numerically computed solutions of these models typically satisfy the first-order conditions necessary for equilibrium, but do not provide any information about the elastic stability of the solution. The inability to detect or avoid physically unstable model solutions poses a major hindrance to reliable model-based simulation, planning, design, and control. In this paper, we adapt results from optimal control to determine the stability of Kirchhoff rods and Cosserat rods subject to general end constraints, including coupled multirod models which describe parallel continuum robots.We formulate a sufficient condition for the stability of a solution, a numerical test for evaluating this condition, and a heuristic stability metric.We verify that our numerical stability test agrees with the classical results for the buckling of single columns with various end constraints and for multicolumn frames. We then validate our approach experimentally on a six degree-of-freedom parallel continuum robot
EXISTING SYSTEM:
Maintaining elastic stability has been recognized as a concern for many continuum robots, and prior research has investigated stability questions related to design and control. For cable-driven continuum robots, Li and Rahn [8] demonstrated buckling of the central backbone between two cable supports. The elastic stability of concentric-tube robots has also been studied extensively with analyses based on energy [7], [9], monotonicity, and slope of an input–output “S-curve” mapping [10]–[13], variational calculus [14], [15], and optimal control [16]. The recent concentric-tube work of Ha et al. [16] is similar to our efforts in that it applies established results in optimal control to formulate a numerical test for the stability of a concentrictube robot model solution. Bretl et al. have also recently studied stability for robotic manipulation of a single elastic Kirchhoff rod (a special case of Cosserat rods with no transverse shear or axial strain) [17]–[20]. Their approach rigorously uses geometric optimal control theory for problems defined on manifolds (since the state variable is a member of the group SE(3)) and considers the case of a fully constrained terminal state (pose) with no external loading. They have shown all static equilibrium configurations form a path-connected smooth manifold with a global chart. Aside from robotics applications, the stability of elastic rods is important in fields such as DNA modeling [21], [22] and computer graphics simulation [23]. Many rod stability problems have been studied in the continuum mechanics literature. Approaches include the use of variational calculus [24], [25], dynamic lumped parameter models [26], and finite element methods [27]. This field has also included studies of special cases such as branched rods [28] and rods with intrinsic curvature [29].
PROPOSED SYSTEM:
The optimal control approach is elegant, rigorous, and is minimally affected by discretization issues, in contrast to the lumped-parameter and finite element approaches. In this paper, we construct an approach based on optimal control which builds on the work above and provides some distinct contributions. First, in contrast to prior work, we consider the general problem of one or more elastic rods under loading and subject to arbitrary terminal constraints. Whereas Ha et al. [16] considers no terminal constraints (a free end), and Bretl and McCarthy [19] considers a fully constrained terminal pose (a fixed end), our approach can be used to assess the stability of elastic rods with partial constraints (e.g., constraints made by pinned joints, sliding joints, etc.), or a geometric coupling to another elastic system (as in the case of parallel continuum robots). The stability of planar tree-like rod structures studied in [28] is a related problem, but the connectivity graph of a parallel continuum robot can contain a closed cycle, which requires general terminal conditions not considered in [28]. Second, our approach examines the full Cosserat rod model in addition to the more restricted Kirchhoff model (no shear or axial strains) studied in [16] and [19]. While the differences are largely negligible for the slender rods used in our experiments, removing the Kirchhoff restrictions expands the generality of our approach, making it suitable for soft parallel robots with nonnegligible shear and extension strains, such as [30], [31], and in general for rods with a low slenderness ratio. Third, we provide a unique way of dealing with the spatial case (where the rod state variable is an element of the Euclidean group SE(3)) by using Euler–Poincar´ reduction following Holm’s treatment in [32], resulting in a minimal set of Lagrange multipliers, which simplifies the conditions for equilibrium and stability. This approach is perhaps more accessible than the geometric optimal control formalism in [19] while still obtaining a minimal model representation that takes advantage of group symmetry. Finally, we demonstrate that our stability test can be implemented at interactive rates. Experimental results validate the efficacy of instability prediction and highlight potential pitfalls. Contrasted against achievements of the prior literature, the main contribution of this paper is the consideration of general boundary conditions rather than a fixed end or a free end with an applied force. Although used here to couple rods, such a derivation would be useful in various scenarios such as touching a surface that applies a normal reaction force.
CONCLUSION:
Starting from the rich literature on optimal control and stability assessment, we have derived a sufficient numerical test for the stability of Cosserat rods with arbitrary terminal constraints, including the multirod structures of parallel continuum robots. We validated the approach in simulation by comparing our results to classical results in the special cases of straight column buckling and sway frame buckling.We have further implemented the test to assess stability of a six DOF prototype parallel continuum robot, and the experimental data supports the effectiveness of the test. Parallel continuum robot research and applications were previously hindered by the inability to recognize unstable model solutions. Our test provides this capability, which will enable robust model-based design, motion planning, and control in future work toward applications in robotic surgery and human–robot interaction.We also hope that other applications in robotics and elsewhere will benefit from our simple and general derivation of the first- and second-order conditions for spatial elastic rods with arbitrary terminal constraints. For example, our approach could be adapted to assess the stability of other continuum robots and long elastic objects in cases whenever partial end-pose constraints or coupling occur, such as contact with rigid fixtures or two robots manipulating the same object.
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