Selected Publications
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Year |
Title |
Authors |
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1999 |
Ciblak, N. and Lipkin, H. |
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Citation:
1999 IEEE International Conference on Robotics & Automation, |
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Abstract: A new, systematic approach to the synthesis of Cartesian stiffness by springs is presented using screw (spatial vector) algebra. The space of solutions is fully characterized for all stiffnesses realizable by springs. The main result shows that a rank r stiffness can always be synthesized by r springs. Further, it can also be synthesized by an arbitrarily large number of springs greater than r. Synthesis algorithms are presented and numerical results support the theory. |
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1999 |
A Dynamic Quasi-Newton Method for
Uncalibrated Visual Servoing |
Piepmeier, J. A., McMurray, G. V., Lipkin, H |
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Citation:
1999 IEEE International Conference on Robotics & Automation, |
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Abstract:
Tracking of a moving target by uncalibrated model independent visual servo
control is achieved by developing a new ''dynamic'' quasi-Newton approach.
Model independent visual servo control is defined as using visual feedback to
control a robot without precisely calibrated kinematic and camera models. The
control problem is formulated as a nonlinear least squares optimization. For
the moving target case, this results in a time-varying objective function
which is minimized using a new dynamic |
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1999 |
Alexiou, J. |
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Citation: M.S. Thesis, Mechanical Engineering, Georgia Institute of Technology, March 1999, i-xiv, pp. 1-233. |
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Abstract:
In this thesis, it is proposed to solve for the accelerations and internal
forces of multiple connected rigid bodies using recursive formulations and
screw theory. Screw theory allows for compact notation of the equations of
motion and kinematics. The concept of a single articulated inertia modeling
the behavior of an entire subchain is used to simplify the equations such
that a recursive solution is possible for systems with no kinematic loops. In
planar cases, the graphical interpretation of the equations introduces the
connection between projective geometry and dynamics with screw theory.
Projective geometry uses subspace decompositions and projections to extract
useful information from problems. Similarly in dynamics, to find the reaction
forces on all the joints, the splitting of all the internal forces into
active and reactive subspaces is required. The splitting of body
accelerations into active and reactive parts is also needed in order to
understand how each solution affects the next recursion. Together the two
decompositions form a symmetric and dualistic set of projections that is used
for both single rigid bodies and multiple articulated rigid body chains.
These projections are applied to the equations of motion yielding a new set
of recursive equations with fewer steps. Each projected articulated equation
maintains the two separate parts of active and reactive components. Examining
the special cases presented when either of the two parts is zero for either
the forces or accelerations indicates the physical interpretation of the
parts. A similar non-recursive formulation is introduced to solve systems
with kinematic loops. Several planar and spatial examples are included to
illustrate solutions to projections, open and closed loop accelerations, and
articulated inertias. A\ MATLAB toolbox is developed, described, and used in
some of the examples implementing screw theory and projective articulated
dynamics. Suggestions on further
development are made that enhance the usability and understanding of
multibody dynamics with screw theory. |
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Year |
Title |
Authors |
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1998 |
Ciblak, N. and Lipkin, H. |
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Citation:
Paper no. DETC98/MECH-5878, CD-ROM Proceedings ASME 1998 Design Engineering
Technical Conferences, September 13-16, |
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Abstract: Orthonormal bases of isotropic vectors for indefinite square matrices are proposed and solved. A necessary and sufficient condition is that the matrix must have zero trace. A recursive algorithm is presented for computer applications. The isotropic vectors of 3 x 3 matrices are solved explicitly. Deviatoric stresses in continuum mechanics, the existence of isotropic vectors (particularly in screw space), and stiffness synthesis by springs are shown to be related to the isotropic vector problem. |
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1998 |
Ciblak, N. and Lipkin, H. |
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Citation:
Paper no. DETC98/MECH-5879, CD-ROM Proceedings ASME 1998 Design Engineering
Technical Conferences, September 13-16, |
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Abstract: A new, systematic approach to the synthesis of stiffness by springs is presented using screw (spatial vector) algebra. The space of solutions is fully characterized for all stiffnesses realizable by springs. The main result shows that a rank r stiffness can always be synthesized by r springs. Further, a stiffness can be synthesized by an arbitrarily large number of springs greater than r. Algorithms and numerical results support the theory. |
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1998 |
Application of Stiffness Decompositions to Synthesis by Springs |
Ciblak, N. and Lipkin, H. |
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Citation: Paper no. DETC98/MECH-5880, CD-ROM Proceedings ASME 1998 Design Engineering Technical Conferences, September 13-16, Atlanta. |
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Abstract: Stiffness decompositions originating from two distinct eigenvalue problems are applied to stiffness synthesis by springs. The resulting free-vector and line-vector syntheses provide a clear geometric explanation of the methods. Numerical examples and algorithms are also provided to verify the results. |
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1998 |
Analysis of Cartesian Stiffness and Compliance with Applications |
Ciblak, N. |
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Analysis of Cartesian Stiffness and Compliance with Applications - Presentation |
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Analysis of Cartesian Stiffness and Compliance with Applications - Presentation(html) |
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Citation: Ph.D. Thesis, Mechanical Engineering, Georgia Institute of Technology, May 1998, i-xiii, pp. 1-399. |
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Abstract:
Many elastic systems can be modelled by a 6x6 Cartesian stiffness or
compliance matrix. Using spatial vector (screw) algebra, spatial stiffness
and compliance are defined. Investigation of the elastic behavior is achieved
by analyzing the geometric and constitutive properties of the stiffness and
compliance matrices. The results are applicable in the analysis, design and
control of elastic systems such as serial and parallel robotic manipulators,
robotic grasp problems, assembly automation devices, spatial structures, and
so on. The geometric and constitutive properties of an elastic system can be
understood in terms of suitable eigenvalue problems. However, construction of
physically and geometrically intuitive eigenvalue problems for stiffness and
compliance in screw space is neither unique nor straightforward. First,
a set of singular eigenvalue problems from earlier studies is shown to be related
to free-vectors. Closed form solutions for the location of centers of
elasticity, stiffness and compliance are found in terms of quantities related
to free-vector eigenvalue problems. Then, the constitutive nature and other
properties of the centers of stiffness and compliance are presented, which
were previously unknown. The centers of elasticity, stiffness and compliance
are shown to be geometrically related. Considering line-vectors, instead of
free-vectors, a new set of singular eigenvalue problems is proposed and
solved. Every point in space generates a distinct set. Similar to the
free-vector case, line-vector decompositions of stiffness and compliance are
found and co-centers of elasticity are identified. The free-vector and
line-vector results lead to generalized definitions of compliant axes and a
refined compliance hierarchy. The
stiffness matrix of parallel spatial connections with line and torsional
springs is found in closed form. The skew-symmetric part of stiffness for
line springs is described completely, which explains previously observed
asymmetry. The observation in earlier studies that the stiffness of line
springs is symmetric in a special reference frame is explained. There exist
infinitely many such frames forming a 2-parameter family. In contrast, there
is no such frame for torsional springs. A
theory is developed to determine orthogonal sets of isotropic vectors of a
symmetric matrix, which, together with the stiffness equation for spring
systems, leads to the synthesis of stiffness by springs. A general synthesis
solution had not been found until now. The necessary and sufficient condition
is that the off-diagonals of stiffness matrix have a zero trace . Algorithms
and examples support the theory. The
free-vector and line-vector results are applied to rotational symmetry
devices such as the remote center of compliance (RCC) device used in
automated assembly operations. Previously unavailable and more accurate
design equations are determined. Optimum device configurations are demonstrated.
The conditions for the construction of RCC devices with beams and springs are
found. Definitions of RCC-like devices are generalized. The
theory for the elastic systems is shown to be applicable in the dynamics of
single rigid body. The mass matrix replaces the stiffness matrix. The
free-vector and line-vector eigenvalue problems are explicitly solved for the
mass matrix. Special axes resulting from the line-vector case explains the
center of percussion phenomenon. A practical optimum design of sport
equipment involving the center of percussion is presented. Combination of the
elastic and kinetic cases leads to the determination of the necessary and
sufficient conditions for the existence of special free vibration modes. Numerical examples are provided for each topic to verify the theoretical results. |
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1998 |
Blanchet, P. and Lipkin, H |
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Citation: Paper no. DETC98/MECH-5868, CD-ROM Proceedings ASME 1998 Design Engineering Technical Conferences, September 13-16, Atlanta. |
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Abstract: Recent discoveries about the structure of stiffness and inertia are combined to analyze free linear vibrations. It is shown that the dual to a pure translation mode is a pure couple mode. The analysis is performed for the general case and then specialized to more simple problems. The results also develop the conditions for the existence of pure translation and couple modes. Examples illustrate the results. |
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1998 |
Blanchet, P. |
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Citation: Ph.D. Thesis, Mechanical Engineering, Georgia Institute of Technology, May 1998, i-x, pp. 1-187. |
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Abstract:
A novel analysis is proposed for the vibration of an elastically suspended rigid
body. This new analysis is based on recent discoveries about the structure of
stiffness and inertia. For a single planar body, vibration centers are used
to describe the modes shapes and are shown to be constrained to regions
specified by the center of elasticity, center of mass, and stiffness
principal directions. Responses are classified by the number of pure
translation modes and conditions for existence are given. Necessary or
sufficient conditions for the existence of pure translation and pure couple
modes are also given for spatial articulated and rigid bodies. It is shown
that these two types of modes can be used to design a multi-degree of freedom
vibration absorber. For
non-proportionally damped planar vibrations of a single rigid body, the modes
shapes are shown to be rotations about a point which is either stationary or
traveling along a straight line depending on the type of damping (undamped,
underdamped, critically damped, overdamped). Similarly, the spatial mode
shapes are rotations and parallel translations about an axis which is either
stationary or traveling along a cylindroid. An explanation of the transition
between types of damping is also given. Analysis
of the forced (damped and undamped) vibrations for planar and spatial motion shows
how to avoid exciting a particular mode. Other
results include some properties associated with the stiffness matrix and a
decomposition of the damping matrix based on two singular eigenvalue
problems. The research makes significant contributions to understanding the relationship between constitutive properties (stiffness and inertia) and modal characteristics (natural frequencies and mode shapes). Finally, the application of the results to the design of mode shapes and to the inverse mode shape problem are outlined. Numerical examples illustrate the results. |
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1998 |
Tracking a Moving Target with Model Independent Visual Servoing: a Predictive Estimation Approach |
Piepmeier, J. A., McMurray, G. V., Lipkin, H. |
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Citation: pp. 2652-2657, 1998 IEEE International Conference on Robotics & Automation, Leuven, Belgium. May 1998 |
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Abstract: Target tracking by model independent visual servo control is achieved by augmenting quasi-Newton trust region control with target prediction. Model independent visual servo control is defined as using visual feedback to control the robot without precise kinematic and camera models. While a majority of the research assumes a known robot and camera model, there is a paucity of literature addressing model independent control. In addition, that research has focused primarily on static targets. The work presented here demonstrates the use of predictive filters to improve performance of the control algorithm for linear and circular target motions. The results show a performance of the same order of magnitude as compared to some model based visual servo control research. Certain limitations to the algorithm are also discussed. |
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Year |
Title |
Authors |
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1997 |
Blanchet, P. and Lipkin, H |
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New Geometric Properties For Modelled Planar Vibration - Presentation |
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Citation: Paper no. DETC97/VIB-4176, CD-ROM Proceedings ASME 1997 Design Engineering Technical Conferences, September 14-17, 1997, Sacramento, California. |
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Abstract: A novel analysis is used to identify new geometric proper-ties for a seemingly well-known model - the linear vibration of an elastically suspended rigid body in planar motion. Vibration centers describe the modes shapes and are shown to be constrained to regions specified by the center-of-elasticity, center-of-mass, and stiffness principal directions. Responses are classified by the number of pure translational modes and conditions for existence are given. Vibration centers are also applied to forced vibration problems where there is a simple relation between an applied force and the resulting mode amplitudes. Numerical examples illustrate the results. |
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1997 |
Cam Synthesis for Cable-Drive Steering on Articulated Wheeled Vehicles |
Frayard, C., Kordestani, A., and Lipkin, H. |
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Citation: 5th National Applied Mechanisms & Robotics Conference, vol. 1, pp. 012.1-7, Cincinnati, October 12-15, 1997. |
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Abstract: A new iterative algorithm is presented for the design synthesis of cam-cable systems. A cam is synthesized for a specified transmission ratio between the cam and a rotating body that is pinned to the end of the cable. It is used to design a steering system with a constant mechanical advantage for use in a three degrees-of-freedom, articulated, off-road vehicle. A numerical example illustrates the results. |
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1997 |
Biro, R., McMurray, G., and Lipkin |
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Citation: |
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Abstract: Visually servoing a robot to track a moving workpiece has been demonstrated in literature using specially customized equipment. In this work, a simple modular architecture is presented using off-the-shelf components with serial line interfaces. The method has widespread application for existing industrial robots whose capabilities can be upgraded without altering proprietary original equipment proven to be reliable. As proof of concept, an ADEPT robot is visually servoed to track workpieces on a conveyor belt moving up to 400 in/min. Simulation results are shown to compare reasonably well with experimental data. Advantages and limitations of the implementation are discussed including the crucial effect of delays. |
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1997 |
Alexiou,
J. and Lipkin, H. |
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Citation:
5th National Applied Mechanisms & Robotics Conference, vol. 1, pp.
065.1-7, |
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Abstract: The space of wrenches and the space of twists are each decomposed into two meaningful subspaces based on the kinematic constraint of a joint. The decomposition is faciliated by the introduction of the absolute acceleration of a body about a joint. This is in distinction to employing the usual relative joint acceleration. Most importantly, this enables a symmetric and dualistic formulation of dynamic equations and quantities. |
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Year |
Title |
Authors |
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1996 |
Lipkin, H. |
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Citation:
Paper no. 96-DETC-MECH-1166, CD-ROM Proceedings ASME 1996 Design Engineering
Technical Conferences, August 18-22, |
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Abstract: An effective inertia of a serially jointed chain of bodies, such as a robotic arm, is often referred to as the articulated or operational space inertia. Six principal axes and six principal values of inertia are proposed to decouple articulated inertia into geometric and constitutive properties respectively. In the case of a single rigid body, these concepts naturally coincide with familiar descriptions of inertia. |
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1996 |
Ciblak, N. and Lipkin, H. |
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Citation:
Paper no. 96-DETC-MECH-1167, CD-ROM Proceedings ASME 1996 Design Engineering
Technical Conferences, August 18-22, |
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Abstract: Remote center-of-compliance devices have long been used to facilitate insertion for assembly operations. A new analysis with a more accurate model better characterizes the interesting properties of remote compliance. The center location and stiffnesses are presented as nondimensional ratios that are only functions of geometrical properties and Poisson's ratio. Important results show: i) the high sensitivity of the center location and ii) the inverse relationship between linear and angular lateral stiffnesses. |
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Year |
Title |
Authors |
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1994 |
Centers of Stiffness, Compliance, and Elasticity in the Modelling of Robotic Systems |
Ciblak, N. and Lipkin, H. |
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Citation:
ASME Design Technology Conferences, DE-Vol. 26, pp. 185-195, |
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Abstract: In the modelling of elastic suspensions between rigid bodies there are three identified points: the centers of elasticity, stiffness, and compliance. Previously, physical properties for the latter two have been unknown. Principal results are: 1) if a compliant axis exists, it must pass through all three centers, and 2) if two compliant axes exist, the three centers coalesce. Additional physical properties that characterize stiffness and the centers are presented. The theory is applied to an RCC device and a dexterous robotic hand. |
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1994 |
Asymmetric Cartesian Stiffness for the Modelling Compliant Robotic Systems |
Ciblak, N. and Lipkin, H. |
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Citation:
ASME Design Technical Conferences, DE-Vol. 72, pp. 197-204, |
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Abstract: Models for compliant robotic systems often use a symmetric 6 x 6 stiffness matrix. However, when subjected to external loads, the stiffness actually becomes asymmetric. For a compliant system modelled using line springs, a new and important theorem is presented that represents the skew-symmetric part in its simpliest form: the skew-symmetric part of the stiffness matrix is negative one-half the externally applied load expressed as a spatial cross product operator. Several corollaries follow including the obvious result that the stiffness matrix is symmetric if and only if it is at an unloaded equilibrium. |
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Year |
Title |
Authors |
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1992 |
Geometric Properties of Modelled Robot Elasticity: Part I, Decomposition |
Lipkin, H. and Patterson, T. |
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Citation:
ASME Design Technical Conferences, |
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Abstract: A new geometric decomposition is introduced that diagonalizes the 6 x 6 stiffness and compliance matrices which model robot elasticity. Using screw theory, a congruence transformation is developed from the three orthogonal wrench-compliant axes and the three orthogonal twist-compliant axes. The diagonal elements are the stationary values of linear and rotational compliance and stiffness. This generalizes and is analogous to principal axes and principal values for stress, strain, and rotational inertia. It is proved that the decomposition always exists for both the nonsingular and singular cases. |
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1992 |
Geometric Properties of Modelled Robot Elasticity: Part II, Center-of-Elasticity |
Lipkin, H. and Patterson, T. |
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Citation:
ASME Design Technical Conferences, |
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Abstract: The elastic characteristics of many robot systems can be modeled by a 6 x 6 stiffness or compliance matrix. Several new and important results are presented via screw theory: i) A generalized center-of-elasticity is proposed based on Ball's (1900) principal screws and its properties are investigated. ii) If a compliant axis exists, it is shown to pass through the center. iii) The perpendicular vectors from the center to the wrench-compliant axes are coplanar and sum to zero. A similar result holds for the twist-compliant axes. iv) Linear and rotational properties are characterized by dual ellipsoids in three-dimensional space. These elements simplify the understanding of complex elastic properties. |
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Year |
Title |
Authors |
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1991 |
Enumeration of Singular Configurations for Robotic Manipulators |
Lipkin, H., Pohl, E. |
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Citation: ASME Journal of Mechanical Design, vol. 113, pp. 272-279, September 1991. |
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Abstract:
Kinematic singularities are important considerations in the design and
control of robotic manipulators. For six degree-of-freedom manipulators, the
vanishing of the determinant of the Jacobian yields the conditions for the
primary singularities. Examining the vanishing of the minors of the Jacobian
yields further singularities which are special cases of the primary ones. A
systematic procedure is presented to efficiently enumerate all possible
singular configurations. Special geometries of representative manipulators
are exploited by expressing the Jacobian in terms of vector elements. In
contrast to using a joint-angle space approach, the resulting expressions
yield direct physical interpretations. |
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Year |
Title |
Authors |
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1990 |
Lipkin, H. |
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Citation:
Proceedings of the NSF Design and Manufacturing Conference, pp. 345-354, |
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Abstract:
The weighted pseudoinverse is examined for invariant properties under general
equivalence transformations. A key result is that the weighting matrices must
undergo an induced congruence transformation. It is demonstrated that, under
certain conditions, a noninvariant pseudoinverse can yield an invariant
solution. Two representative robotics applications are presented, inverse
velocity kinematics and load distribution for hands and multi-robot systems.
For these problems, it is shown that a constant positive definite weighting
matrix for the least squares problem does not exist. Two numerical examples
show the problem of noninvariance with respect to change of coordinates and
change of basis. |
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1990 |
Patterson, T. and Lipkin, H. |
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Citation: ASME Design
Technology Conferences - 21th Biennial Mechanisms Conference, DE-Vol. 26, pp.
315-322, Chicago, September 17-19, 1990. |
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Abstract:
An alternative treatment of robot compliance
is developed by applying screw theory to the compliance matrix eigenvalue
problem. The compliance and stiffness matrix eigenvalue problems are shown to
be equivalent. The eigenscrews are demonstrated to be Ball’s (1900) principal
screws of the potential. Several new propositions are presented
characterizing the compliance matrix eigenstructure. In a companion paper,
Patterson and Lipkin (1990) the results are used to classify general
compliance matrices. Using a novel formulation, a second eigenvalue problem
is developed. It is used to generalize the wrench-compliant axes of
Dimentberg (1965) to include twist-compliant axes. Together these two types
of compliant axes are shown to define conjugate screw systems of the
potential. |
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1990 |
Patterson, T. and Lipkin, H. |
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Citation: ASME Design
Technology Conferences - 21th Biennial Mechanisms Conference, DE‑Vol.
26, pp. 307-314, |
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Abstract:
The concept of compliant axes is
developed from the compliance matrix eigenvalue problem, Patterson and Lipkin
(1990). The resulting formulation leads to the necessary and sufficient
conditions for the existence of compliant axes, i.e. two collinear
eigenscrews with eigenvalues of equal magnitude and opposite sign. It is also
shown that if one compliant axis exists, the remaining four eigenscrews must
intersect that axis at right angles. These developments lead to a new classification
of compliance matrices based on the number of compliant axes produced.
Selected matrices from the literature are used to illustrate both the
compliant axis concept and the new classification method. |
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1990 |
Smith, David R. |
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Citation:
Ph.D. Thesis, Mechanical Engineering, Georgia Institute of Technology, May
1990, i-x, pp. 1-174. |
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Although
much effort has been expended in finding closed form solutions for robots,
there has been less investigation of the structure of the solutions. Due to
this, lower order robots are frequently described as having special geometry
without explanation of what particular types of geometry qualify as
“special.” The notable exception to this is the well-known result that any 6R
robot with three consecutive joint axes intersecting or parallel has a fourth
order solution. However, even robots of these types are frequently made
kinematically simpler through the use of additional special geometries
without examination of the underlying structure. In this thesis, a new
approach to investigating the inverse kinematic solutions of fourth order
(i.e., solvable) robots is introduced. The technique maps the problem into a
one parameter family, or pencil, of conic sections where the four points
defining the pencil represent the solutions of a joint angle. This
representation allows properties of conics to be exploited in order to gain
new insights into the structure of the kinematics. There are many advantages
gained by this geometric representation. For example, it provides a method of
developing and analyzing lower order robots as well as their workspace region
boundaries. In
order to illustrate the technique, a detailed examination of all 6R robots
with wrists is conducted. The results include five new designs of degenerate
regional manipulators. Degenerate robots are particularly useful because they
have a second order inverse kinematic solution, which is advantageous to real
time control. Additionally, the workspace boundary features of regional
robots are examined and new interpretations are made in terms of the tonics.
Specifically, singular points of the region boundary curve are found to
correspond to certain types of contact Between the conics. These results
indicate that the method is a powerful new tool in kinematic analysis and
design of solvable manipulators. |
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Year |
Title |
Authors |
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1988 |
Lipkin, H. and Duffy, J. |
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Citation:
ASME Journal of Mechanisms, Transmissions, and Automation in Design, vol.
110, pp. 138-144, June 1988. |
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Abstract:
Three necessary conditions derived from classical geometry are proposed to
evaluate formulations for the simultaneous twist and wrench control of rigid
bodies, and for any theory to be meaningful it must be invariant with respect
to (1) Euclidean collineations, (2) change of (Euclidean) unit length, and
(3) change of basis. It is demonstrated in this paper that a previously
established theory of hybrid control for robot manipulators is in fact based
on the metric of elliptic geometry and is thus noninvariant with respect to
(I) and (2). A new alternative invariant formulation based on the metric of
Euclidean geometry and an induced metric of projective geometry is presented
in terms of screw theory. An example of insertion illustrates both the
invariant and noninvariant methods. |
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Year |
Title |
Authors |
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1985 |
Lipkin, H. and Duffy, J. |
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Citation: ASME Journal of Mechanisms, Transmissions, and Automation in Design, vol. 107, pp. 377-387, September 1985. |
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Abstract: The nature and invariant properties of the elliptic polarity of screws in Euclidean space is established using a novel series of mappings, central to which is a quaternion representation. The role of the elliptic polarity in modeling constrained motion is detailed and illustrated by way of an example which has application in the control of robot manipulators. Ball’s planar representation of the two system of screws is generalized and is shown to be a representation on a complex plane where the elliptic polarity is a conformal mapping. |
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1985 |
Geometry and Mappings of Screws with Applications to the Hybrid Control of Robotic Manipulators (7.5MB). |
Lipkin, H. |
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Citation:
Ph.D. Thesis, Mechanical Engineering, |
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Abstract:
The nature and invariant properties of the elliptic polarity of screws in
Euclidean space is established using a new series of mappings, central to
which is a quaternion representation. The mappings are used to generalize
Ball's planar representation of the two-system of screws which is shown to be
a mapping on a complex plane where the elliptic polarity induces a conformal
mapping. Screw theory is applied to the hybrid control of robotic
manipulators where it is demonstrated that a current theory based on "orthogonal"
projection yields noninvariant results. New invariant methods of hybrid
control are detailed by introducing invariant kinestatic filters. |
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