scholarly journals A Tutorial for the Analysis of the Piecewise-Smooth Dynamics of a Constrained Multibody Model of Vertical Hopping

2018 ◽  
Vol 23 (4) ◽  
pp. 74 ◽  
Author(s):  
Roland Zana ◽  
Bálint Bodor ◽  
László Bencsik ◽  
Ambrus Zelei
Keyword(s):  
Degrees Of Freedom ◽  
Legged Locomotion ◽  
Rigid Bodies ◽  
Pattern Generation ◽  
Constrained Systems ◽  
Piecewise Smooth ◽  
Geometric Constraints ◽  
Mathematical Representation ◽  
Differential Algebraic Equation ◽  
Kinematic Loops

Contradictory demands are present in the dynamic modeling and analysis of legged locomotion: on the one hand, the high degrees-of-freedom (DoF) descriptive models are geometrically accurate, but the analysis of self-stability and motion pattern generation is extremely challenging; on the other hand, low DoF models of locomotion are thoroughly analyzed in the literature; however, these models do not describe the geometry accurately. We contribute by narrowing the gap between the two modeling approaches. Our goal is to develop a dynamic analysis methodology for the study of self-stable controlled multibody models of legged locomotion. An efficient way of modeling multibody systems is to use geometric constraints among the rigid bodies. It is especially effective when closed kinematic loops are present, such as in the case of walking models, when both legs are in contact with the ground. The mathematical representation of such constrained systems is the differential algebraic equation (DAE). We focus on the mathematical analysis methods of piecewise-smooth dynamic systems and we present their application for constrained multibody models of self-stable locomotion represented by DAE. Our numerical approach is demonstrated on a linear model of hopping and compared with analytically obtained reference results.

scholarly journals An Efficient Contact Model for the Simulation of Cargo Airdrop Extraction Phase

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Leiming Ning ◽  
Jichang Chen ◽  
Mingbo Tong
Keyword(s):  
Degrees Of Freedom ◽  
Friction Model ◽  
Rigid Bodies ◽  
Contact Dynamics ◽  
Moving Frame ◽  
Contact Friction ◽  
Practical Implementation ◽  
Contact Formulation ◽  
Efficient Code ◽  
Extraction Phase

A high-fidelity cargo airdrop simulation requires the accurate modeling of the contact dynamics between an aircraft and its cargo. This paper presents a general and efficient contact-friction model for the simulation of aircraft-cargo coupling dynamics during an airdrop extraction phase. The proposed approach has the same essence as the finite element node-to-segment contact formulation, which leads to a flexible, straightforward, and efficient code implementation. The formulation is developed under an arbitrary moving frame with both aircraft and cargo treated as general six degrees-of-freedom rigid bodies, thus eliminating the restrictions of lateral symmetric assumptions in most existing methods. Moreover, the aircraft-cargo coupling algorithm is discussed in detail, and some practical implementation details are presented. The accuracy and capability of the present method are demonstrated through four numerical examples with increasing complexity and fidelity.

Introducing a Projection-Based Method to Compare Three Approaches Computing the Accumulation of Geometric Variations

2018 ◽  
Author(s):  
Vincent Delos ◽  
Santiago Arroyave-Tobón ◽  
Denis Teissandier
Keyword(s):  
Degrees Of Freedom ◽  
Mechanical Design ◽  
Geometric Constraints ◽  
Computational Time ◽  
Manufacturing Defects ◽  
Design Tolerance ◽  
Polyhedral Objects ◽  
The Difference ◽  
Bounded Sets ◽  
Accumulation Of Defects

In mechanical design, tolerance zones and contact gaps can be represented by sets of geometric constraints. For computing the accumulation of possible manufacturing defects, these sets have to be summed and/or intersected according to the assembly architecture. The advantage of this approach is its robustness for treating even over-constrained mechanisms i.e. mechanisms in which some degrees of freedom are suppressed in a redundant way. However, the sum of constraints, which must be computed when simulating the accumulation of defects in serial joints, is a very time-consuming operation. In this work, we compare three methods for summing sets of constraints using polyhedral objects. The difference between them lie in the way the degrees of freedom (DOFs) (or invariance) of joints and features are treated. The first method proposes to virtually limit the DOFs of the toleranced features and joints to turn the polyhedra into polytopes and avoid manipulating unbounded objects. Even though this approach enables to sum, it also introduces bounding or cap facets which increase the complexity of the operand sets. This complexity increases after each operation until becoming far too significant. The second method aims to face this problem by cleaning, after each sum, the calculated polytope to keep under control the effects of the propagation of the DOFs. The third method is new and based on the identification of the sub-space in which the projection of the operands are bounded sets. Calculating the sum in this sub-space allows reducing significantly the operands complexity and consequently the computational time. After presenting the geometric properties on which the approaches rely, we demonstrate them on an industrial case. Then we compare the computation times and deduce the equality of the results of all the methods.

scholarly journals An Asymptotically Optimal NMPC Core for Multi Degrees of Freedom Rigid Bodies

2012 ◽  
Vol 45 (17) ◽  
pp. 207-212
Author(s):  
Chyon Hae Kim ◽  
Shimon Sugawara ◽  
Shigeki Sugano
Keyword(s):  
Degrees Of Freedom ◽  
Rigid Bodies ◽  
Asymptotically Optimal

Sliding Mode Control of a Three Degrees of Freedom Anthropoid Robot by Driving the Controller Parameters to an Equivalent Regime

2000 ◽  
Vol 122 (4) ◽  
pp. 632-640 ◽  
Author(s):  
M. Onder Efe ◽  
Okyay Kaynak ◽  
Xinghuo Yu
Keyword(s):  
Degrees Of Freedom ◽  
Sliding Mode ◽  
Initial Conditions ◽  
Tracking Error ◽  
Variable Structure ◽  
Mathematical Representation ◽  
Dynamic Complexity ◽  
Variable Structure Systems ◽  
Sliding Motion ◽  
Three Degrees Of Freedom

Noise rejection, handling the difficulties coming from the mathematical representation of the system under investigation and alleviation of structural or unstructural uncertainties constitute prime challenges that are frequently encountered in the practice of systems and control engineering. Designing a controller has primarily the aim of achieving the tracking precision as well as a degree of robustness against the difficulties stated. From this point of view, variable structure systems theory offer well formulated solutions to such ill-posed problems containing uncertainty and imprecision. In this paper, a simple controller structure is discussed. The architecture is known as Adaptive Linear Element (ADALINE) in the framework of neural computing. The parameters of the controller evolve dynamically in time such that a sliding motion is obtained. The inner sliding motion concerns the establishment of a sliding mode in controller parameters, which aims to minimize the error on the controller outputs. The outer sliding motion is designed for the plant. The algorithm discussed drives the error on the output of the controller toward zero learning error level, and the state tracking error vector of the plant is driven toward the origin of the phase space simultaneously. The paper gives the analysis of the equivalence between the two sliding motions and demonstrates the performance of the algorithm on a three degrees of freedom, anthropoid robotic manipulator. In order to clarify the performance of the scheme, together with the dynamic complexity of the plant, the adverse effects of observation noise and nonzero initial conditions are studied. [S0022-0434(00)01704-4]

Control of Locomotion in Hexapods

2017 ◽  
pp. 422-438 ◽  
Author(s):  
Roy E. Ritzmann ◽  
Sasha N. Zill
Keyword(s):  
Control Systems ◽  
Local Control ◽  
Sensory Information ◽  
Legged Locomotion ◽  
Pattern Generation ◽  
Central Complex ◽  
Joint Movement ◽  
Control Circuits ◽  
Pattern Generators ◽  
The Brain

This article discusses legged locomotion in insects. It describes the basic patterns of coordinated movement both within each leg and among the various legs. The nervous system controls these actions through groups of joint pattern generators coupled through interneurons and interjoint reflexes in a range of insect species. These local control systems within the thoracic ganglia rely on leg proprioceptors that monitor joint movement and cuticular strain interacting with central pattern generation interneurons. The local control systems can change quantitatively and qualitatively as needed to generate turns or more forceful movements. In dealing with substantial obstacles or changes in navigational movements, more profound changes are required. These rely on sensory information processed in the brain that projects to the multimodal sensorimotor neuropils collectively referred to as the central complex. The central complex affects descending commands that alter local control circuits to accomplish appropriate redirected movements.

Generalized Abbe Principle: Position Error Propagation in Machine Elements

1996 ◽  
Author(s):  
Joseph Pegna
Keyword(s):  
Error Propagation ◽  
Degrees Of Freedom ◽  
Rigid Bodies ◽  
Statistical Characterization ◽  
Physical Limit ◽  
Potential Applications ◽  
Machine Elements ◽  
Tolerance Verification ◽  
Precision Machines

Abstract In the quest for ever finer levels of technology integration, mechanical linkages reach their precision limits at about 5micrometers per meter of workspace. Beyond this physical limit, all six dimensional degrees of freedom need to be precisely ascertained to account for mechanical imperfections. This paper substantiates Wu’s vision of “precision machines without precision machinery.” A formulation and statistical characterization of position and orientation error propagation in rigid bodies are presented for two extreme models of measurement. It is shown that error distribution is uniquely dependent upon the design of the measurement plan. The theoretical foundations presented were evolved in the course of designing precision machinery. Other potential applications include: fixture design, metrology, and geometric tolerance verification.

Spontaneous Symmetry-Breaking in a Network Model for Quadruped Locomotion

2017 ◽  
Vol 27 (14) ◽  
pp. 1730049 ◽  
Author(s):  
Ian Stewart
Keyword(s):  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
Legged Locomotion ◽  
Phase Shifts ◽  
Pattern Generation ◽  
Complex Conjugate ◽  
Connection Matrix ◽  
Specific Rate ◽  
Rate Model ◽  
Two Phase

Spontaneous symmetry-breaking proves a mechanism for pattern generation in legged locomotion of animals. The basic timing patterns of animal gaits are produced by a network of spinal neurons known as a Central Pattern Generator (CPG). Animal gaits are primarily characterized by phase differences between leg movements in a periodic gait cycle. Many different gaits occur, often having spatial or spatiotemporal symmetries. A natural way to explain gait patterns is to assume that the CPG is symmetric, and to classify the possible symmetry-breaking periodic motions. Pinto and Golubitsky have discussed a four-node model CPG network for biped gaits with [Formula: see text] symmetry, classifying the possible periodic states that can arise. A more specific rate model with this structure has been analyzed in detail by Stewart. Here we extend these methods to quadruped gaits, using an eight-node network with [Formula: see text] symmetry proposed by Golubitsky and coworkers. We formulate a rate model and calculate how the first steady or Hopf bifurcation depends on its parameters, which represent four connection strengths. The calculations involve a distinction between “real” gaits with one or two phase shifts (pronk, bound, pace, trot) and “complex” gaits with four phase shifts (forward and reverse walk, forward and reverse buck). The former correspond to real eigenvalues of the connection matrix, the latter to complex conjugate pairs. The partition of parameter space according to the first bifurcation, ignoring complex gaits, is described explicitly. The complex gaits introduce further complications, not yet fully understood. All eight gaits can occur as the first bifurcation from a fully synchronous equilibrium, for suitable parameters, and numerical simulations indicate that they can be asymptotically stable.

A Heuristic Scaling Strategy for Multi-Robot Cooperative Three-Dimensional Printing

2020 ◽  
Vol 20 (4) ◽  
Author(s):  
Laxmi Poudel ◽  
Chandler Blair ◽  
Jace McPherson ◽  
Zhenghui Sha ◽  
Wenchao Zhou
Keyword(s):  
3D Printing ◽  
Fused Deposition Modeling ◽  
Three Dimensional ◽  
Evaluation Framework ◽  
Geometric Constraints ◽  
Mathematical Representation ◽  
Printing Process ◽  
Fused Deposition ◽  
3D Printers ◽  
Printing Time

Abstract While three-dimensional (3D) printing has been making significant strides over the past decades, it still trails behind mainstream manufacturing due to its lack of scalability in both print size and print speed. Cooperative 3D printing (C3DP) is an emerging technology that holds the promise to mitigate both of these issues by having a swarm of printhead-carrying mobile robots working together to finish a single print job cooperatively. In our previous work, we have developed a chunk-based printing strategy to enable the cooperative 3D printing with two fused deposition modeling (FDM) mobile 3D printers, which allows each of them to print one chunk at a time without interfering with the other and the printed part. In this paper, we present a novel method in discretizing the continuous 3D printing process, where the desired part is discretized into chunks, resulting in multi-stage 3D printing process. In addition, the key contribution of this study is the first working scaling strategy for cooperative 3D printing based on simple heuristics, called scalable parallel arrays of robots for 3DP (SPAR3), which enables many mobile 3D printers to work together to reduce the total printing time for large prints. In order to evaluate the performance of the printing strategy, a framework is developed based on directed dependency tree (DDT), which provides a mathematical and graphical description of dependency relationships and sequence of printing tasks. The graph-based framework can be used to estimate the total print time for a given print strategy. Along with the time evaluation metric, the developed framework provides us with a mathematical representation of geometric constraints that are temporospatially dynamic and need to be satisfied in order to achieve collision-free printing for any C3DP strategy. The DDT-based evaluation framework is then used to evaluate the proposed SPAR3 strategy. The results validate the SPAR3 as a collision-free strategy that can significantly shorten the printing time (about 11 times faster with 16 robots for the demonstrated examples) in comparison with the traditional 3D printing with single printhead.

Theory and Method of Dynamic Modeling for Multilayer Vibration Isolation System

2000 ◽  
Author(s):  
Liao Dao-Xun ◽  
Lu Yong-Zhong ◽  
Huang Xiao-Cheng
Keyword(s):  
High Speed ◽  
Degrees Of Freedom ◽  
Vibration Isolation ◽  
Rigid Bodies ◽  
Design Calculation ◽  
Isolation System ◽  
Dry Land ◽  
Vibration Isolation System ◽  
Six Degrees Of Freedom ◽  
Floating Raft

Abstract The multilayer vibration isolation system has been widely applied to isolate vibration in dynamic devices of ships, high-speed vehicles forging hammer and precise instruments. The paper is based on the coordinate transformation of space general motion for mass blocks (rigid bodies) and Lagrangian equation of multilayer vibration isolation system. It gives a strict mathematical derivation on the differential equation of the motion for the system with six degrees of freedom of relative motion between mass blocks (including base). The equations are different from the same kind of equations in the reference literatures. It can be used in the floating raft of ships in order to isolates vibration and decrease noise, also used in design calculation of the multilayer vibration isolation for dynamic machines and precise instruments on the dry land.

Contact-space resolution model for a physically consistent view of simultaneous collisions in articulated-body systems: theory and experimental results

2020 ◽  
Vol 39 (10-11) ◽  
pp. 1239-1258
Author(s):  
Shameek Ganguly ◽  
Oussama Khatib
Keyword(s):  
System Dynamics ◽  
Rigid Body ◽  
Degrees Of Freedom ◽  
Null Space ◽  
Rigid Bodies ◽  
Computationally Efficient ◽  
Space Resolution ◽  
Resolution Model ◽  
Space Dynamics ◽  
Contact Space

Multi-surface interactions occur frequently in articulated-rigid-body systems such as robotic manipulators. Real-time prediction of contact-interaction forces is challenging for systems with many degrees of freedom (DOFs) because joint and contact constraints must be enforced simultaneously. While several contact models exist for systems of free rigid bodies, fewer models are available for articulated-body systems. In this paper, we extend the method of Ruspini and Khatib and develop the contact-space resolution (CSR) model by applying the operational space theory of robot manipulation. Through a proper choice of contact-space coordinates, the projected dynamics of the system in the contact space is obtained. We show that the projection into the dynamically consistent null space preserves linear and angular momentum in a subspace of the system dynamics complementary to the joint and contact constraints. Furthermore, we illustrate that a simultaneous collision event between two articulated bodies can be resolved as an equivalent simultaneous collision between two non-articulated rigid bodies through the projected contact-space dynamics. Solving this reduced-dimensional problem is computationally efficient, but determining its accuracy requires physical experimentation. To gain further insights into the theoretical model predictions, we devised an apparatus consisting of colliding 1-, 2-, and 3-DOF articulated bodies where joint motion is recorded with high precision. Results validate that the CSR model accurately predicts the post-collision system state. Moreover, for the first time, we show that the projection of system dynamics into the mutually complementary contact space and null space is a physically verifiable phenomenon in articulated-rigid-body systems.

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