In recent years, it has become well known that rational Bézier and B-spline curves in the space o... more In recent years, it has become well known that rational Bézier and B-spline curves in the space of dual quaternions correspond to rational Bézier and B-spline motions. However, the influence of weights of these dual quaternion curves on the resulting rational motions has been largely unexplored. In this paper, we present a thorough mathematical exposition on the influence of dual-number weights associated with dual quaternions for rational motion design. By deriving the explicit equations for the point trajectories of the resulting motion, we show that the effect of real weights on the resulting motion is similar to that of a rational Bézier curve and how the change in dual part of a dual-number weight affects the translational component of the motion. We also show that a rational Bézier motion can be reparameterized in a manner similar to a rational Bézier curve. Several examples are presented to illustrate the effects of the weights on rational motions.
Intelligent Robots and Computer Vision XXI: Algorithms, Techniques, and Active Vision
ABSTRACT This paper presents new class sensors for onboard direct measurement of the angular orie... more ABSTRACT This paper presents new class sensors for onboard direct measurement of the angular orientation of robotic mobile platforms relative to a fixed or moving coordinate system. The currently available sensors are either based on inertia, vision or optical means to measure the angular orientation of an object. The inertial based devices, however, generally suffer from drift and noise. The vision systems and optical sensors generally have relatively short range and require line-of-sight access. The novel class of sensors presented in this paper are wireless, are in the form of waveguides that are illuminated by polarized Radio Frequency sources. A mobile robotic platform equipped with three or more of such waveguide sensors can determine its 3D orientation relative to the ground or other mobile robotic platforms. The 3D orientation sensors require very low power for operation, may be located at relatively far distances from the ground source or the illuminating mobile platform, and can operate while out of line-of-sight of the illuminating source. In this paper, the design, operation, algorithms for calculating 3D angular orientation from the sensor output, and a number of experimental results of sensor performance are presented. In addition, a discussion of the methods to increase the performance of the sensor system and other related issues are provided.
Journal of Mechanisms, Transmissions, and Automation in Design
This paper uses the Euler parameters of the motion of the planet of a spherical epicyclic gear tr... more This paper uses the Euler parameters of the motion of the planet of a spherical epicyclic gear train to obtain a curve on the surface of a hypersphere in four dimensions. This curve, called the image curve, represents the rotational motion of the planet as it rolls without slipping on the fixed gear. Two image curves are obtained for two different choices of moving and fixed reference frames and it is shown that they are related by an orthogonal transformation in four dimensional space. The differential properties of the image curve are computed and it is found that the curvature and torsion are constant. A reference position is chosen and the canonical frame and instantaneous invariants of the motion are determined in terms of the dimensions of the gear train.
Based on the general degree-of-freedom (DOF) formula for spatial mechanisms proposed by the autho... more Based on the general degree-of-freedom (DOF) formula for spatial mechanisms proposed by the author in 2012, the early single open chain (SOC)-based composition principle for planar mechanisms is extended to general spatial mechanisms in this paper. First, three types of existing mechanism composition principle and their characteristics are briefly discussed. Then, the SOC-based composition principle for general spatial mechanisms is introduced. According to this composition principle, a spatial mechanism is first decomposed into Assur kinematic chains (AKCs) and an AKC is then further decomposed into a group of ordered SOCs. Kinematic (dynamic) analysis of a spatial mechanism can then be reduced to kinematic (dynamic) analysis of AKCs and finally to kinematic (dynamic) analysis of ordered SOCs. The general procedure for decomposing the mechanism into ordered SOCs and the general method for determining AKC(s) contained in the mechanism are also given. Mechanism's kinematic (dynam...
This paper examines the problem of geometric constraints acquisition of planar motion through a l... more This paper examines the problem of geometric constraints acquisition of planar motion through a line-geometric approach. In previous work, we have investigated the problem of identifying point-geometric constraints associated with a motion task which is given in a parametric or discrete form. In this paper, we seek to extend the point-centric approach to the line-centric approach. The extracted geometric constraints can be used directly for determining the type and dimensions of a physical device such as mechanical linkage that generates this constrained motion task.
In this paper, we have presented a unified framework for generating planar four-bar motions for a... more In this paper, we have presented a unified framework for generating planar four-bar motions for a combination of poses and practical geometric constraints and its implementation in MotionGen app for…
Volume 7: 33rd Mechanisms and Robotics Conference, Parts A and B, 2009
ABSTRACT This paper explores the concept of kinematic convexity of planar displacements as an ext... more ABSTRACT This paper explores the concept of kinematic convexity of planar displacements as an extension of the projective convexity in computational geometry to planar kinematics. This is achieved with the help of planar quaternions which converts planar displacements into points in the space of planar quaternions called the image space. In this way, projective convexity of points in the image space is developed and used as a representation of kinematic convexity of planar displacements. To address the issue of distance metric for planar displacements, we explored the connection between planar quaternions and quaternions and formulated the concept of kinematic convexity in the space of quaternions where a bi-invariant metric exists. An example is provided in the end to illustrate the use of kinematic convexity for estimating the “closest distance” from a fixed body to a moving body undergoing a rational Bézier motion.
In this paper, we present an interactive, visual design approach for the dimensional synthesis of... more In this paper, we present an interactive, visual design approach for the dimensional synthesis of planar 6R single-loop closed chains for a given rational motion using constraint manifold modification. This approach is implemented in an interactive software tool that provides mechanism designers with an intuitive way to determine the parameters of planar mechanisms, and in the process equips them with an understanding of the design process. The theoretical foundation of this work is based on representing planar displacements with planar quaternions, which can be seen as points in a special higher dimensional projective space (called the image space), and on formulating the kinematic constraints of closed chains as algebraic surfaces in the image space. Kinematic constraints under consideration limit the positions and orientation of the coupler in its workspace. In this way, a given motion of a mechanism in the Cartesian space maps to a curve in the image space that is limited to sta...
ABSTRACT In this paper, we revisit the classical Burmester problem of the exact synthesis of a pl... more ABSTRACT In this paper, we revisit the classical Burmester problem of the exact synthesis of a planar four-bar mechanism with up to five task positions. Instead of assuming the joint type (revolute or prismatic) a priori, we seek to extract both the dimensions and joint types of a four-bar linkage from the given tasks. Kinematic mapping of plane kinematics has been used to formulate the Burmester problem as a manifold fitting problem in the image space. Instead of finding the design parameters of planar dyads directly, this paper seeks to determine a set of eight homogeneous coefficients for the constraint manifold in the null space associated with the five given tasks. Two additional constraints on these coefficients are then applied to finalize the synthesis process. The result is a novel algorithm that is simple and efficient and allows for task driven design of four-bar linkages with revolute and prismatic joints.
Volume 2: 32nd Mechanisms and Robotics Conference, Parts A and B, 2008
ABSTRACT In this paper, we study the problem of rational motion interpolation under kinematic con... more ABSTRACT In this paper, we study the problem of rational motion interpolation under kinematic constraints of spatial SS open chains. The objective is to synthesize a smooth rational motion that interpolates a given set of end effector positions and satisfies the kinematic constraints imposed by spatial SS open chains. The kinematic constraints under consideration define a constraint manifold representing all the positions available to the end effector. By choosing dual quaternion representation for the displacement of the end effector, the problem is reduced to designing a smooth curve in the space of dual quaternions that is constrained to lie inside the constraint manifold of the spatial SS open chain. An iterative numerical algorithm is presented that solves this problem effectively. The results presented in this paper are extension of our previous work on the synthesis of piecewise rational planar and spherical motions for open and closed chains under kinematic constraints.
This paper deals with smooth motion interpolation. It presents two geometric algorithms for synth... more This paper deals with smooth motion interpolation. It presents two geometric algorithms for synthesizing composite Be´zier motions with second-order geometric continuity (G2). The first one is a direct algorithm for constructing a G2 spline motion that approximates a set of displacements; the second one is an inverse design algorithm for a G2 spline motion that interpolates through a set of displacements. The results are useful for computer aided motion animation, and Cartesian trajectory generation for CNC machines and robot manipulators.
A rotational displacement of the coupler of a spherical four-bar linkage can be mapped to a point... more A rotational displacement of the coupler of a spherical four-bar linkage can be mapped to a point with coordinates given by the Euler parameters of the rotation. The set of rotational movements available to the coupler defines a curve in this three-dimensional projective space (four homogeneous coordinates). In this paper, we determine the generalized eigenvalues and eigenvectors of the pencil of quadrics that pass through this curve and examine their properties. The result is an algebraic classification of the image curves that parallels the well-known classification of spherical four-linkages. In addition, we find that the characteristic polynomial of the system yields Grashof’s criterion for the rotatability of cranks.
With the introduction of generalized function sets (GF set) to represent the characteristics of t... more With the introduction of generalized function sets (GF set) to represent the characteristics of the end-effectors of parallel mechanisms, two classes of GF sets are proposed. The type synthesis of parallel mechanisms having the second class GF sets and two dimensional rotations, including 2-, 3-, and 4DOF parallel mechanisms, is investigated. First, the intersection algorithms for the GF sets are established via the axiom of two dimensional rotations. Second, the kinematic limbs with specific characteristics are designed according to the axis movement theorem. Finally, several parallel mechanisms having the second class GF sets and two dimensional rotations have been illustrated to show the effectiveness of the proposed methodology.
This paper deals with the problem of designing dynamically compensated cam profiles to minimize r... more This paper deals with the problem of designing dynamically compensated cam profiles to minimize residual vibrations in high-speed cam-follower systems. The traditional Polydyne method is modified and extended to achieve significant improvement in residual vibration characteristics. First, cam displacement curves are represented by Bernstein-Be´zier harmonic curves as opposed to polynomial curves. These recently developed harmonic curves are low in harmonic content and therefore the resulting cam profiles are less prone to induce resonant vibrations in the follower system. Second, the design procedure is expanded such that the residual vibrations of the resulting cam-follower system is not only extinguished at the design speed but also made insensitive to speed variations. Numerical examples are given in the end.
In recent years, it has become well known that rational Bézier and B-spline curves in the space o... more In recent years, it has become well known that rational Bézier and B-spline curves in the space of dual quaternions correspond to rational Bézier and B-spline motions. However, the influence of weights of these dual quaternion curves on the resulting rational motions has been largely unexplored. In this paper, we present a thorough mathematical exposition on the influence of dual-number weights associated with dual quaternions for rational motion design. By deriving the explicit equations for the point trajectories of the resulting motion, we show that the effect of real weights on the resulting motion is similar to that of a rational Bézier curve and how the change in dual part of a dual-number weight affects the translational component of the motion. We also show that a rational Bézier motion can be reparameterized in a manner similar to a rational Bézier curve. Several examples are presented to illustrate the effects of the weights on rational motions.
Intelligent Robots and Computer Vision XXI: Algorithms, Techniques, and Active Vision
ABSTRACT This paper presents new class sensors for onboard direct measurement of the angular orie... more ABSTRACT This paper presents new class sensors for onboard direct measurement of the angular orientation of robotic mobile platforms relative to a fixed or moving coordinate system. The currently available sensors are either based on inertia, vision or optical means to measure the angular orientation of an object. The inertial based devices, however, generally suffer from drift and noise. The vision systems and optical sensors generally have relatively short range and require line-of-sight access. The novel class of sensors presented in this paper are wireless, are in the form of waveguides that are illuminated by polarized Radio Frequency sources. A mobile robotic platform equipped with three or more of such waveguide sensors can determine its 3D orientation relative to the ground or other mobile robotic platforms. The 3D orientation sensors require very low power for operation, may be located at relatively far distances from the ground source or the illuminating mobile platform, and can operate while out of line-of-sight of the illuminating source. In this paper, the design, operation, algorithms for calculating 3D angular orientation from the sensor output, and a number of experimental results of sensor performance are presented. In addition, a discussion of the methods to increase the performance of the sensor system and other related issues are provided.
Journal of Mechanisms, Transmissions, and Automation in Design
This paper uses the Euler parameters of the motion of the planet of a spherical epicyclic gear tr... more This paper uses the Euler parameters of the motion of the planet of a spherical epicyclic gear train to obtain a curve on the surface of a hypersphere in four dimensions. This curve, called the image curve, represents the rotational motion of the planet as it rolls without slipping on the fixed gear. Two image curves are obtained for two different choices of moving and fixed reference frames and it is shown that they are related by an orthogonal transformation in four dimensional space. The differential properties of the image curve are computed and it is found that the curvature and torsion are constant. A reference position is chosen and the canonical frame and instantaneous invariants of the motion are determined in terms of the dimensions of the gear train.
Based on the general degree-of-freedom (DOF) formula for spatial mechanisms proposed by the autho... more Based on the general degree-of-freedom (DOF) formula for spatial mechanisms proposed by the author in 2012, the early single open chain (SOC)-based composition principle for planar mechanisms is extended to general spatial mechanisms in this paper. First, three types of existing mechanism composition principle and their characteristics are briefly discussed. Then, the SOC-based composition principle for general spatial mechanisms is introduced. According to this composition principle, a spatial mechanism is first decomposed into Assur kinematic chains (AKCs) and an AKC is then further decomposed into a group of ordered SOCs. Kinematic (dynamic) analysis of a spatial mechanism can then be reduced to kinematic (dynamic) analysis of AKCs and finally to kinematic (dynamic) analysis of ordered SOCs. The general procedure for decomposing the mechanism into ordered SOCs and the general method for determining AKC(s) contained in the mechanism are also given. Mechanism's kinematic (dynam...
This paper examines the problem of geometric constraints acquisition of planar motion through a l... more This paper examines the problem of geometric constraints acquisition of planar motion through a line-geometric approach. In previous work, we have investigated the problem of identifying point-geometric constraints associated with a motion task which is given in a parametric or discrete form. In this paper, we seek to extend the point-centric approach to the line-centric approach. The extracted geometric constraints can be used directly for determining the type and dimensions of a physical device such as mechanical linkage that generates this constrained motion task.
In this paper, we have presented a unified framework for generating planar four-bar motions for a... more In this paper, we have presented a unified framework for generating planar four-bar motions for a combination of poses and practical geometric constraints and its implementation in MotionGen app for…
Volume 7: 33rd Mechanisms and Robotics Conference, Parts A and B, 2009
ABSTRACT This paper explores the concept of kinematic convexity of planar displacements as an ext... more ABSTRACT This paper explores the concept of kinematic convexity of planar displacements as an extension of the projective convexity in computational geometry to planar kinematics. This is achieved with the help of planar quaternions which converts planar displacements into points in the space of planar quaternions called the image space. In this way, projective convexity of points in the image space is developed and used as a representation of kinematic convexity of planar displacements. To address the issue of distance metric for planar displacements, we explored the connection between planar quaternions and quaternions and formulated the concept of kinematic convexity in the space of quaternions where a bi-invariant metric exists. An example is provided in the end to illustrate the use of kinematic convexity for estimating the “closest distance” from a fixed body to a moving body undergoing a rational Bézier motion.
In this paper, we present an interactive, visual design approach for the dimensional synthesis of... more In this paper, we present an interactive, visual design approach for the dimensional synthesis of planar 6R single-loop closed chains for a given rational motion using constraint manifold modification. This approach is implemented in an interactive software tool that provides mechanism designers with an intuitive way to determine the parameters of planar mechanisms, and in the process equips them with an understanding of the design process. The theoretical foundation of this work is based on representing planar displacements with planar quaternions, which can be seen as points in a special higher dimensional projective space (called the image space), and on formulating the kinematic constraints of closed chains as algebraic surfaces in the image space. Kinematic constraints under consideration limit the positions and orientation of the coupler in its workspace. In this way, a given motion of a mechanism in the Cartesian space maps to a curve in the image space that is limited to sta...
ABSTRACT In this paper, we revisit the classical Burmester problem of the exact synthesis of a pl... more ABSTRACT In this paper, we revisit the classical Burmester problem of the exact synthesis of a planar four-bar mechanism with up to five task positions. Instead of assuming the joint type (revolute or prismatic) a priori, we seek to extract both the dimensions and joint types of a four-bar linkage from the given tasks. Kinematic mapping of plane kinematics has been used to formulate the Burmester problem as a manifold fitting problem in the image space. Instead of finding the design parameters of planar dyads directly, this paper seeks to determine a set of eight homogeneous coefficients for the constraint manifold in the null space associated with the five given tasks. Two additional constraints on these coefficients are then applied to finalize the synthesis process. The result is a novel algorithm that is simple and efficient and allows for task driven design of four-bar linkages with revolute and prismatic joints.
Volume 2: 32nd Mechanisms and Robotics Conference, Parts A and B, 2008
ABSTRACT In this paper, we study the problem of rational motion interpolation under kinematic con... more ABSTRACT In this paper, we study the problem of rational motion interpolation under kinematic constraints of spatial SS open chains. The objective is to synthesize a smooth rational motion that interpolates a given set of end effector positions and satisfies the kinematic constraints imposed by spatial SS open chains. The kinematic constraints under consideration define a constraint manifold representing all the positions available to the end effector. By choosing dual quaternion representation for the displacement of the end effector, the problem is reduced to designing a smooth curve in the space of dual quaternions that is constrained to lie inside the constraint manifold of the spatial SS open chain. An iterative numerical algorithm is presented that solves this problem effectively. The results presented in this paper are extension of our previous work on the synthesis of piecewise rational planar and spherical motions for open and closed chains under kinematic constraints.
This paper deals with smooth motion interpolation. It presents two geometric algorithms for synth... more This paper deals with smooth motion interpolation. It presents two geometric algorithms for synthesizing composite Be´zier motions with second-order geometric continuity (G2). The first one is a direct algorithm for constructing a G2 spline motion that approximates a set of displacements; the second one is an inverse design algorithm for a G2 spline motion that interpolates through a set of displacements. The results are useful for computer aided motion animation, and Cartesian trajectory generation for CNC machines and robot manipulators.
A rotational displacement of the coupler of a spherical four-bar linkage can be mapped to a point... more A rotational displacement of the coupler of a spherical four-bar linkage can be mapped to a point with coordinates given by the Euler parameters of the rotation. The set of rotational movements available to the coupler defines a curve in this three-dimensional projective space (four homogeneous coordinates). In this paper, we determine the generalized eigenvalues and eigenvectors of the pencil of quadrics that pass through this curve and examine their properties. The result is an algebraic classification of the image curves that parallels the well-known classification of spherical four-linkages. In addition, we find that the characteristic polynomial of the system yields Grashof’s criterion for the rotatability of cranks.
With the introduction of generalized function sets (GF set) to represent the characteristics of t... more With the introduction of generalized function sets (GF set) to represent the characteristics of the end-effectors of parallel mechanisms, two classes of GF sets are proposed. The type synthesis of parallel mechanisms having the second class GF sets and two dimensional rotations, including 2-, 3-, and 4DOF parallel mechanisms, is investigated. First, the intersection algorithms for the GF sets are established via the axiom of two dimensional rotations. Second, the kinematic limbs with specific characteristics are designed according to the axis movement theorem. Finally, several parallel mechanisms having the second class GF sets and two dimensional rotations have been illustrated to show the effectiveness of the proposed methodology.
This paper deals with the problem of designing dynamically compensated cam profiles to minimize r... more This paper deals with the problem of designing dynamically compensated cam profiles to minimize residual vibrations in high-speed cam-follower systems. The traditional Polydyne method is modified and extended to achieve significant improvement in residual vibration characteristics. First, cam displacement curves are represented by Bernstein-Be´zier harmonic curves as opposed to polynomial curves. These recently developed harmonic curves are low in harmonic content and therefore the resulting cam profiles are less prone to induce resonant vibrations in the follower system. Second, the design procedure is expanded such that the residual vibrations of the resulting cam-follower system is not only extinguished at the design speed but also made insensitive to speed variations. Numerical examples are given in the end.
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