Cubic spline algorithms for orientation interpolation

Kang, Ingab; Park, F. C.

doi:10.1002/(sici)1097-0207(19990910)46:1<45::aid-nme662>3.0.co;2-k

Cited by 44 publications

(21 citation statements)

References 19 publications

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“…Piecewise interpolating functions with high continuity and/or geometrically continuous splines are adequate tools in generating smooth motion of the robotic manipulators when the manipulator kinematics (velocity, acceleration, and/or jerk) or dynamics (force and/or torque) is considered . Such approach should reduce resonant frequency excitation and generate smoother trajectory profile.…”

Section: Introductionmentioning

confidence: 99%

“…Such approach should reduce resonant frequency excitation and generate smoother trajectory profile. The interpolation of smooth curves (twice differentiable and cubic in the parametrized coordinates) invariant with respect to the fixed/moving frame represents an excellent approach to minimize angular acceleration . A new planning approach of a manipulator along a set of nodal points for a collection of established kinematical requirements is presented in the work of du Plessis and Snyman .…”

Section: Introductionmentioning

confidence: 99%

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A combined polar and Cartesian piecewise trajectory generation and analysis of a robotic arm

Dupac

2019

Comp and Math Methods

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In this paper, a combined polar‐Cartesian approach to generate a smooth trajectory of a robotic arm along priori defined via‐points is presented. Due to the characteristics/geometry of the robotic arm, cylindrical coordinates are associated with the trajectory of motion. Possible trajectories representing the system dynamics are generated by mix‐matching higher order polar piecewise polynomials used to devise the radial trajectory and Cartesian piecewise polynomials used to calculate the related height in a normal plane unfolded along the radial trajectory of the motion. To describe the kinematic properties of the end effector, a moving noninertial orthonormal Frenet frame is considered. Using the Frenet frame, the components of the velocity and acceleration along the frame unit vectors are calculated. Numerical simulations are performed for two different configurations in order to validate the approach.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A combined polar and Cartesian piecewise trajectory generation and analysis of a robotic arm

Dupac

2019

Comp and Math Methods

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“…The trajectories used in path planning can have different representations, among them, piecewise interpolating curves, parametric and/or geometric continuous splines,() or uniform cubic B‐spline with parametric and geometric continuity, have been usually considered. () The interpolation of smooth curves for generating orientation trajectories with minimal angular acceleration is presented in Courchamp et al A new interpolation methodology for the path planning of an industrial robot and a set of prescribed kinematical requirements is presented in Gregory and Courchamp . Algebraic‐trigonometric Hermite polynomial curves that are very practical in generating smooth and continuous manipulator motion are considered in Kramer and Drake .…”

Section: Introductionmentioning

confidence: 99%

A path planning approach of 3D manipulators using zenithal gnomic projection and polar piecewise interpolation

Dupac

2018

Math Methods in App Sciences

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In this paper, the mathematical modeling and trajectory planning of a 3D rotating manipulator composed of a rotating‐prismatic joint and multiple rigid links is considered. Possible trajectories of the end effector of the manipulator—following a sequence of 3D target points under the action of 2 external driving torques and an axial force—are modeled using zenithal gnomic projections and polar piecewise interpolants expressed as polynomial Hermite‐type functions. Because of the geometry of the manipulator, the time‐dependent generalized coordinates are associated with the spherical coordinates named the radial distance related to the manipulator length, and the polar and azimuthal angles describing the left and right and, respectively, up and down motion of the manipulator. The polar trajectories (left and right motion) of the end effector are generated using a inverse geometric transformation applied to the polar piecewise interpolants that approximate the gnomic projective trajectory of the 3D via‐points. The gnomic via‐points—located on a projective plane situated on the northern hemisphere—are seen from the manipulator base location, which represents the center of rotation of the extensible manipulator. The related azimuthal trajectory (up and down motion) is generated by polar piecewise interpolants on the azimuthal angles. Smoothness of the polygonal trajectory is obtained through the use of piecewise interpolants with continuous derivatives, while the kinematics and dynamics implementation of the model is well suited to computer implementation (easy calculation of kinematics variables) and simulation. To verify the approach and validate the model, a numerical example—implemented in Matlab—is presented, and the results are discussed.

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“…There have been many attempts to solve similar interpolation problems on Lie groups in terms of the coordinates of the embedding space [10], [13]. The idea is to find a suitable mapping in order to express the information in a linear space, solve the interpolation problem there and pull the trajectory back to a manifold.…”

Section: Introductionmentioning

confidence: 99%

“…Popular SO(3) interpolation algorithms adopt various re-parametrizations of the rotation matrices (e.g. rotation axes and angles, unit quaternions) and perform cubic spline interpolation based on such representations [10]. These algorithms, however, often do not generalize to higher-dimensional manifolds.…”

Section: Introductionmentioning

confidence: 99%

Interpolation in special orthogonal groups

Shingel¹

2008

IMA Journal of Numerical Analysis

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The problem of constructing smooth interpolating curves in non-Euclidean spaces finds applications in different areas of science. In this paper we propose a scheme to generate interpolating curves in Lie groups, focusing on a special orthogonal group SO(n). Our technique is based on the exponential representation of elements of the group, which allows to transfer the problem to the corresponding Lie algebra so (n) and benefit from the linearity of this space. Due to the exponential representation we can obtain a high degree of smoothness of an interpolating curve at relatively low costs. The underlying problem is challenging because the standard SO(n) −→ so(n) map is multi-valued.

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Cubic spline algorithms for orientation interpolation

Cited by 44 publications

References 19 publications

A combined polar and Cartesian piecewise trajectory generation and analysis of a robotic arm

A combined polar and Cartesian piecewise trajectory generation and analysis of a robotic arm

A path planning approach of 3D manipulators using zenithal gnomic projection and polar piecewise interpolation

Interpolation in special orthogonal groups

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