Interpolating Solid Orientations with a C 2-Continuous B-Spline Quaternion Curve

Ge, Wenbing; Huang, Zhangjin; Wang, Guoping

doi:10.1007/978-3-540-73011-8_58

Cited by 12 publications

(7 citation statements)

References 10 publications

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“…By synthesizing formula (14) and (16), practical formula of the cubic uniform B-spline curve can be obtained as the following formula [21]. Unless the other is specified, the following B-spline curves all refer to cubic uniform B-spline curves:…”

Section: Preliminaries Of B-spline Curvementioning

confidence: 99%

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C²-Continuous Orientation Planning for Robot End-Effector with B-Spline Curve Based on Logarithmic Quaternion

Shi

Lin

et al. 2020

Mathematical Problems in Engineering

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Smooth orientation planning is beneficial for the working performance and service life of industrial robots, keeping robots from violent impacts and shocks caused by discontinuous orientation planning. Nevertheless, the popular used quaternion interpolations can hardly guarantee C2 continuity for multiorientation interpolation. Aiming at the problem, an efficient quaternion interpolation methodology based on logarithmic quaternion was proposed. Quaternions of more than two key orientations were expressed in the exponential forms of quaternion. These four-dimensional quaternions in space S3, when logarithms were taken for them, could be converted to three-dimensional points in space R3 so that B-spline interpolation could be applied freely to interpolate. The core formulas that B-spline interpolated points were mapped to quaternion were founded since B-spline interpolated point vectors were decomposed to the product of unitized forms and exponents were taken for them. The proposed methodology made B-spline curve applicable to quaternion interpolation through dimension reduction and the high-order continuity of the B-spline curve remained when B-spline interpolated points were mapped to quaternions. The function for reversely finding control points of B-spline curve with zero curvature at endpoints was derived, which helped interpolation curve become smoother and sleeker. The validity and rationality of the principle were verified by the study case. For comparison, the study case was also analyzed by the popular quaternion interpolations, Spherical Linear Interpolation (SLERP) and Spherical and Quadrangle (SQUAD). The comparison results demonstrated the proposed methodology had higher smoothness than SLERP and SQUAD and thus would provide better protection for robot end-effector from violent impacts led by unreasonable multiorientation interpolation. It should be noted that the proposed methodology can be extended to multiorientation quaternion interpolation with higher continuity than the second order.

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Section: Preliminaries Of B-spline Curvementioning

confidence: 99%

“…Although there are also other several ways to construct the high-order quaternion curve [13][14][15][16][17][18], it is still difficult to tell which one method is better than another, and there is still no perfect solution so far. Relevant researchers have been striving to study and explore in this direction.…”

Section: Introductionmentioning

confidence: 99%

C²-Continuous Orientation Planning for Robot End-Effector with B-Spline Curve Based on Logarithmic Quaternion

Shi

Lin

et al. 2020

Mathematical Problems in Engineering

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“…[10] proves that SLERP only has CO continuity, and SQUAD only has C1 continuity. Reference [15] proposed two auxiliary formulas in S 3 space to solve the control points accurately and obtained the C2 continuous attitude curve by interpolation. To avoid the singularity using Euler angles and the non-smooth movement using SLERP, Kong et al 16 present a novel interpolation method based on unit quaternion which constructs the orientation movement through interpolation the angle of rotation and results in smooth and controllable velocity and acceleration of orientation movement combined with sine-jerk trajectory planning which could generate a C3 continuous angle curve.…”

Section: Introductionmentioning

confidence: 99%

Research on Attitude Interpolation and Tracking Control Based on Improved Orientation Vector SLERP Method

Dong¹,

Yao²,

Li³

et al. 2019

Robotica

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SummaryIn order to make the end of the three-axis platform follow the control command and achieve stable control of the end attitude, an improved orientation vector spherical linear interpolation (SLERP) method is proposed for the requirements, which specifically handles the position of the gimbal lock, so that the platform can move smoothly around the gimbal lock position. A three-axis platform with a camera at the end is set up for the validity of the proposed algorithm. At first, an adaptive speed measurement method based on incremental encoder is introduced, which can automatically adapt to high and low speed, and estimate the ultra-low speed to realize the speed measurement of large dynamic range, and this is used for the motion control of the three-axis platform. Then, the SLERP method for the quaternion interpolation on the starting and ending attitudes represented in quaternion is introduced in detail, and it is continuously improved in response to its existing problems for the platform. Finally, an orientation vector SLERP method is proposed, which uses viscosity factor and rejection factor to adjust the algorithm near the platform’s gimbal lock position. A tracking experiment was designed using the red ball as the following target detected by the designed target tracking algorithm using the camera, which verified the effectiveness of the attitude tracking control based on the proposed improved orientation vector SLERP.

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“…For this reason, the quaternion curve obtained by this method cannot be applied to the robots' OI directly. Based on this, Ge et al (2007) proposed a method to solve the quaternion control points accurately according to two additional rules. Miura (2000) proposed a quaternion integral curve where the variation of curvature is given by rather simple expressions.…”

Section: Introductionmentioning

confidence: 99%

C2-continuous orientation trajectory planning for robot based on spline quaternion curve

Xuejuan

2018

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Purpose To realize the smooth interpolation of orientation on robot end-effector, this paper aims to propose a novel algorithm based on the unit quaternion spline curve. Design/methodology/approach This algorithm combines the spherical linear quaternion interpolation and the cubic B-spline quaternion curve. With this method, a C2-continuous smooth trajectory of multiple teaching orientations is obtained. To achieve the visualization of quaternion curves on a unit sphere, a mapping algorithm between a unit quaternion and a point on the spherical surface is given based on the physical meaning of the unit quaternion. Findings Finally, the curvature analysis of a practical case shows that the orientation trajectory (OT) constructed by this algorithm satisfied the C2-continuity. Originality/value This OT satisfies the requirement of smooth interpolation among multiple orientations on robots in industrial applications.

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Interpolating Solid Orientations with a C 2-Continuous B-Spline Quaternion Curve

Cited by 12 publications

References 10 publications

C²-Continuous Orientation Planning for Robot End-Effector with B-Spline Curve Based on Logarithmic Quaternion

C²-Continuous Orientation Planning for Robot End-Effector with B-Spline Curve Based on Logarithmic Quaternion

Research on Attitude Interpolation and Tracking Control Based on Improved Orientation Vector SLERP Method

C2-continuous orientation trajectory planning for robot based on spline quaternion curve

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Interpolating Solid Orientations with a C 2-Continuous B-Spline Quaternion Curve

Cited by 12 publications

References 10 publications

C2-Continuous Orientation Planning for Robot End-Effector with B-Spline Curve Based on Logarithmic Quaternion

C2-Continuous Orientation Planning for Robot End-Effector with B-Spline Curve Based on Logarithmic Quaternion

Research on Attitude Interpolation and Tracking Control Based on Improved Orientation Vector SLERP Method

C2-continuous orientation trajectory planning for robot based on spline quaternion curve

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Product

Resources

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C²-Continuous Orientation Planning for Robot End-Effector with B-Spline Curve Based on Logarithmic Quaternion

C²-Continuous Orientation Planning for Robot End-Effector with B-Spline Curve Based on Logarithmic Quaternion