A generalized exponential formula for forward and differential kinematics of open-chain multi-body systems

Chhabra, Robin; Emami, M. Reza

doi:10.1016/j.mechmachtheory.2013.09.013

Cited by 26 publications

(18 citation statements)

References 22 publications

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“…Combining equations (7) and (8), the homogeneous transformation matrix of the tool coordinate system relative to the workpiece coordinate system is Step 1…”

Section: (7)mentioning

confidence: 99%

“…Geometric error modelling is a key part of error compensation technology and also for the basis for decoupling identification [5,6]. Many beneficial explorations and studies on high-quality modelling have been conducted by researchers, and some effective methods have been proposed and applied to modelling, such as the product of exponentials [7,8], homogeneous transformation matrices (HTMs) method based on multibody system theory [9], differential transformation method [10,11], screw theory method [6,12,13], and parametric polynomial method [14]. Fu et al [15] established the rotation twist and rotation product of exponential formulas of rotary axes with a clear geometric meaning of twists to describe their positions and motions, and the corresponding POE formulas of squareness errors were established by analysing their geometric definition.…”

Section: Introductionmentioning

confidence: 99%

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A Recognition Methodology for the Key Geometric Errors of a Multi-Axis Machine Tool Based on Accuracy Retentivity Analysis

2019

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This paper proposes a recognition methodology for key geometric errors using the feature extraction method and accuracy retentivity analysis and presents the approach of optimization compensation of the geometric error of a multiaxis machine tool. The universal kinematics relations of the multiaxis machine tool are first modelled mathematically based on screw theory. Then, the retentivity of geometric accuracy with respect to the geometric error is defined based on the mapping between the constitutive geometric errors and the time domain. The results show that the variation in the spatial error vector is nonlinear while considering the operation time of the machine tool and the position of the motion axes. Based on this aspect, key factors are extracted that simultaneously consider the correlation, similarity, and sensitivity of the geometric error terms, and the results reveal that the effect of the position-independent geometric errors (PIGEs) on the error vectors of the position and orientation is greater than that of the position-dependent geometric errors (PDGEs) of the linear and rotary axes. Then, the fruit fly optimization algorithm (FOA) is adopted to determine the compensation values through multiobjective tradeoffs between accuracy retentivity and fluctuation in the geometric errors. Finally, an experiment on a four-axis horizontal boring machine tool is used to validate the effectiveness of the proposed approach. The experimental results show that the variations in the precision of each test piece are lower than 25.0%, and the maximum variance in the detection indexes between the finished test pieces is 0.002 mm when the optimized parameters are used for error compensation. This method not only recognizes the key geometric errors but also compensates for the geometric error of the machine tool based on the accuracy retentivity analysis results. The results show that the proposed methodology can effectively enhance the machining accuracy.

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“…Combining equations (7) and (8), the homogeneous transformation matrix of the tool coordinate system relative to the workpiece coordinate system is Step 1…”

Section: (7)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Recognition Methodology for the Key Geometric Errors of a Multi-Axis Machine Tool Based on Accuracy Retentivity Analysis

2019

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“…Therefore, if D i is left-invariant, i.e., D i (r i ) = T e i L r i (D i (e i )), ∀r i ∈ P i , involutivity of D i coincides with the closedness of the Lie bracket on D i (e i ) as a linear subspace of Lie(P i ), and T e i Q i = D i (e i ) becomes a Lie sub-algebra of Lie(P i ). As the result, the integral manifold of D i , denoted by Q i , is a unique d i -dimensional connected Lie subgroup of P i with the Lie algebra Lie(Q i ) = D i (e i ) [36].…”

Section: Holonomic Displacement Subgroupsmentioning

confidence: 99%

“…We identify different types of displacement subgroups by the connected Lie subgroups of SE(3), up to conjugation, which are tabulated in Table 1 [36]. From this table, we can observe that the displacement subgroups consist of the six lower kinematic pairs, i.e., revolute, prismatic, helical, cylindrical, planar and spherical joints, and combinations of them.…”

Section: Holonomic Displacement Subgroupsmentioning

confidence: 99%

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Symplectic reduction of holonomic open-chain multi-body systems with constant momentum

Chhabra¹,

Emami

2015

Journal of Geometry and Physics

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a b s t r a c tThis paper presents a two-step symplectic geometric approach to the reduction of Hamilton's equation for open-chain, multi-body systems with multi-degree-of-freedom holonomic joints and constant momentum. First, symplectic reduction theorem is revisited for Hamiltonian systems on cotangent bundles. Then, we recall the notion of displacement subgroups, which is the class of multi-degree-of-freedom joints considered in this paper. We briefly study the kinematics of open-chain multi-body systems consisting of such joints. And, we show that the relative configuration manifold corresponding to the first joint is indeed a symmetry group for an open-chain multi-body system with multidegree-of-freedom holonomic joints. Subsequently using symplectic reduction theorem at a non-zero momentum, we express Hamilton's equation of such a system in the symplectic reduced manifold, which is identified by the cotangent bundle of a quotient manifold. The kinetic energy metric of multi-body systems is further studied, and some sufficient conditions are introduced, under which the kinetic energy metric is invariant under the action of a subgroup of the configuration manifold. As a result, the symplectic reduction procedure for open-chain, multi-body systems is extended to a two-step reduction process for the dynamical equations of such systems. Finally, we explicitly derive the reduced dynamical equations in the local coordinates for an example of a six-degree-of-freedom manipulator mounted on a spacecraft, to demonstrate the results of this paper.

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A Unified Approach to Input-output Linearization and Concurrent Control of Underactuated Open-chain Multi-body Systems with Holonomic and Nonholonomic Constraints

Chhabra¹,

Emami

2015

J Dyn Control Syst

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A generalized exponential formula for forward and differential kinematics of open-chain multi-body systems

Cited by 26 publications

References 22 publications

A Recognition Methodology for the Key Geometric Errors of a Multi-Axis Machine Tool Based on Accuracy Retentivity Analysis

A Recognition Methodology for the Key Geometric Errors of a Multi-Axis Machine Tool Based on Accuracy Retentivity Analysis

Symplectic reduction of holonomic open-chain multi-body systems with constant momentum

A Unified Approach to Input-output Linearization and Concurrent Control of Underactuated Open-chain Multi-body Systems with Holonomic and Nonholonomic Constraints

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