Dynamic Motion Control of Multi-Segment Soft Robots Using Piecewise Constant Curvature Matched with an Augmented Rigid Body Model

Compliant members in flexible link mechanisms undergo large deflections when subjected to external loads. Because of this fact, traditional methods of deflection analysis do not apply. Since the nonlinearities introduced by these large deflections make the system comprising such members difficult to solve, parametric deflection approximations are deemed helpful in the analysis and synthesis of compliant mechanisms. This is accomplished by representing the compliant mechanism as a pseudo-rigid-body model. A wealth of analysis and synthesis techniques available for rigid-body mechanisms thus become amenable to the design of compliant mechanisms. In this paper, a pseudo-rigid-body model is developed and solved for the tip deflection of flexible beams for combined end loads. A numerical integration technique using quadrature formulae has been employed to solve the large deflection Bernoulli-Euler beam equation for the tip deflection. Implementation of this scheme is simpler than the elliptic integral formulation and provides very accurate results. An example for the synthesis of a compliant mechanism using the proposed model is also presented.

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A Simple and Accurate Method for Determining Large Deflections in Complaint Mechanisms Subjected to End Forces and Moments

Volume 1B: 25th Biennial Mechanisms Conference ◽

10.1115/detc98/mech-5883 ◽

1998 ◽

Author(s):

A. Saxena ◽

Steven N. Kramer

Keyword(s):

Rigid Body ◽

Numerical Integration ◽

Compliant Mechanisms ◽

Compliant Mechanism ◽

Beam Equation ◽

Integration Scheme ◽

Large Deflections ◽

Body Model ◽

Rigid Body Model ◽

Analysis And Synthesis

Abstract Compliant members in flexible link mechanisms undergo large deflections when subjected to external loads for which, traditional methods of deflection analysis do not apply Nonlinearities introduced by these large deflections make the system comprising such members difficult to solve Parametric deflection approximations are then deemed helpful in the analysis and synthesis of compliant mechanisms This is accomplished by seeking the pseudo-rigid-body model representation of the compliant mechanism A wealth of analysis and synthesis techniques available for rigid-body mechanisms thus become amenable to the design of compliant mechanisms In this paper, a pseudo-rigid-body model is developed and solved for the tip deflection of flexible beams for combined end loads with positive end moments A numerical integration technique using quadrature formulae has been employed to solve the nonlinear Bernoulli-Euler beam equation for the tip deflection Implementation of this scheme is relatively simpler than the elliptic integral formulation and provides nearly accurate results Results of the numerical integration scheme are compared with the beam finite element analysis An example for the synthesis of a compliant mechanism using the proposed model is also presented.

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Evaluation of Equivalent Spring Stiffness for Use in a Pseudo-Rigid-Body Model of Large-Deflection Compliant Mechanisms

23rd Biennial Mechanisms Conference: Mechanism Synthesis and Analysis ◽

10.1115/detc1994-0219 ◽

1994 ◽

Author(s):

Larry L. Howell ◽

Ashok Midha

Keyword(s):

Rigid Body ◽

Large Deflection ◽

Compliant Mechanisms ◽

Compliant Mechanism ◽

Spring Stiffness ◽

Model Concept ◽

Analysis And Design ◽

Body Model ◽

Rigid Body Model ◽

Body Joints

Abstract Compliant mechanisms gain some or all of their mobility from the flexibility of their members rather than from rigid-body joints only. More efficient and usable analysis and design techniques are needed before the advantages of compliant mechanisms can be fully utilized. In an earlier work, a pseudo-rigid-body model concept, corresponding to an end-loaded geometrically nonlinear, large-deflection beam, was developed to help fulfill this need. In this paper, the pseudo-rigid-body equivalent spring stiffness is investigated and new modeling equations are proposed. The result is a simplified method of modeling the force/deflection relationships of large-deflection members in compliant mechanisms. Flexible segments which maintain a constant end angle are discussed, and an example mechanism is analyzed. The resulting models are valuable in the visualization of the motion of large-deflection systems, as well as the quick and efficient evaluation and optimization of compliant mechanism designs.

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Parametric Deflection Approximations for Initially Curved, Large-Deflection Beams in Compliant Mechanisms

Volume 2A: 24th Biennial Mechanisms Conference ◽

10.1115/96-detc/mech-1215 ◽

1996 ◽

Cited By ~ 1

Author(s):

Larry L. Howell ◽

Ashok Midha

Keyword(s):

Rigid Body ◽

Large Deflection ◽

Compliant Mechanisms ◽

Curved Beam ◽

Analysis And Design ◽

Body Model ◽

Rigid Body Model ◽

Fundamental Type ◽

Applied Forces ◽

Flexible Member

Abstract The analysis of systems containing highly flexible members is made difficult by the nonlineararities caused by large deflections of the flexible members. The analysis and design of many such systems may be simplified by using pseudo-rigid-body approximations in modeling the flexible members. The pseudo-rigid-body model represents flexible members as rigid links, joined at pin joints with torsional springs. Appropriate values for link lengths and torsional spring stiffnesses are determined such that the deflection path and force-deflection relationships are modeled accurately. Pseudo-rigid-body approximations have been developed for initially straight beams with externally applied forces at the beam end. This work develops approximations for another fundamental type of flexible member, the initially curved beam with applied force at the beam end. This type of flexible member is commonly used in compliant mechanisms. An example of the use of the resulting pseudo-rigid-body approximations in compliant mechanisms is included.

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Compliant Pantographs via the Pseudo-Rigid-Body Model

Volume 1B: 25th Biennial Mechanisms Conference ◽

10.1115/detc98/mech-5930 ◽

1998 ◽

Author(s):

Andrew J. Nielson ◽

Larry L. Howell

Keyword(s):

Rigid Body ◽

Compliant Mechanisms ◽

Compliant Mechanism ◽

Test Results ◽

Rigid Link ◽

Body Model ◽

Design Flexibility ◽

Model Predictions ◽

Rigid Body Model ◽

Classical Mechanism

Abstract This paper uses a familiar classical mechanism, the pantograph, to demonstrate the utility of the pseudo-rigid-body model in the design of compliant mechanisms to replace rigid-link mechanisms, and to illustrate the advantages and limitations of the resulting compliant mechanisms. To demonstrate the increase in design flexibility, three different compliant mechanism configurations were developed for a single corresponding rigid-link mechanism. The rigid-link pantograph consisted of six links and seven joints, while the corresponding compliant mechanisms had no more than two links and three joints (a reduction of at least four links and four joints). A fourth compliant pantograph, corresponding to a rhomboid pantograph, was also designed and tested. The test results showed that the pseudo-rigid-body model predictions were accurate over a large range, and the mechanisms had displacement characteristics of rigid-link mechanisms in that range. The limitations of the compliant mechanisms included reduced range compared to their rigid-link counterparts. Also, the force-deflection characteristics were predicted by the pseudo-rigid-body model, but they did not resemble those for a rigid-link pantograph because of the energy storage in the flexible segments.

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The Rotating Rigid Body Model Based on a Non-twisting Frame

Journal of Nonlinear Science ◽

10.1007/s00332-020-09648-3 ◽

2020 ◽

Vol 30 (6) ◽

pp. 3199-3233 ◽

Cited By ~ 1

Author(s):

Cristian Guillermo Gebhardt ◽

Ignacio Romero

Keyword(s):

Kinetic Energy ◽

Rigid Body ◽

Conservation Laws ◽

Rotation Rate ◽

Governing Equations ◽

New Model ◽

Body Model ◽

Integration Schemes ◽

Rigid Body Model ◽

Unit Vectors

Abstract This work proposes and investigates a new model of the rotating rigid body based on the non-twisting frame. Such a frame consists of three mutually orthogonal unit vectors whose rotation rate around one of the three axis remains zero at all times and, thus, is represented by a nonholonomic restriction. Then, the corresponding Lagrange–D’Alembert equations are formulated by employing two descriptions, the first one relying on rotations and a splitting approach, and the second one relying on constrained directors. For vanishing external moments, we prove that the new model possesses conservation laws, i.e., the kinetic energy and two nonholonomic momenta that substantially differ from the holonomic momenta preserved by the standard rigid body model. Additionally, we propose a new specialization of a class of energy–momentum integration schemes that exactly preserves the kinetic energy and the nonholonomic momenta replicating the continuous counterpart. Finally, we present numerical results that show the excellent conservation properties as well as the accuracy for the time-discretized governing equations.

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