## “Introduction of b-splines to trajectory planning for robot manipulators”Authors: Per E. Koch and Kesheng Wang,
Affiliation: NTNU and SINTEF
Reference: 1988, Vol 9, No 2, pp. 69-80. |

**Keywords:**Robotics, trajectory planning, B-splines

**Abstract:**This paper describes how B-splines can be used to construct joint trajectories for robot manipulators. The motion is specified by a sequence of Cartesian knots, i.e., positions and orientations of the end effector of a robot manipulator. For a six joint robot manipulator, these Cartesian knots are transformed into six sets of joint variables, with each set corresponding to a joint. Splines, represented as linear combinations of B-splines, are used to fit the sequence of joint variables for each of the six joints. A computationally very simple, recurrence formula is used to generate the 8-splines. This approach is used for the first time to establish the mathematical model of trajectory generation for robot manipulators, and offers flexibility, computational efficiency, and a compact representation.

PDF (1059 Kb) DOI: 10.4173/mic.1988.2.2

**DOI forward links to this article:**

[1] Per Erik Koch (1992), doi:10.1016/B978-0-12-460510-7.50030-6 | |

[2] Ammar Alzaydi (2018), doi:10.1117/1.OE.57.12.120901 |

**References:**

[1] BOLLINGER, J. DIFFIE, N. (1979). Computer algorithms for high speed continuous path robot manipulator, Ann. CIRP 28, 391-395.

[2] COOK, C.C., HO, C.Y. (1982). The application of spline functions to trajectory generation for computer-controlled manipulators, Digital Systems for Industrial Automation, 1, 325-333.

[3] COX, M.G. (1972). The numerical evaluation of B-splines, J. Inst. Math. Applic., 10, 134-149 doi:10.1093/imamat/10.2.134

[4] CRAIG, J.J. (1986). Introduction to robotics mechanics and control, Addison-Wesley Publishing Company pp. 191-219.

[5] CURRY, H.B., SCHOENBERG, I.J. (1966). On polya frequency functions IV: The fundamental spline functions and their limits, J. dŽAnalyse Math. 17, 71-107 doi:10.1007/BF02788653

[6] DE BOOR, C. (1972). On calculating with B-splines, J. Approximation Theory, 6, 50-62 doi:10.1016/0021-9045(72)90080-9

[7] DE BOOR, C. (1977). Package for calculating with B-splines, SIAM Journal of Numerical Analysis 14, 441-472 doi:10.1137/0714026

[8] DE BOOR, C. (1978). A practical guide to splines, New York : Springer Verlag pp. 219 and 138.

[9] GORDON, W.J., RIESENFELD, R.F. (1974). B-spline curves and surfaces in Computer aided geometric design, edited by Barnhill, R. E. and Riesenfeld, R. F.,.Academic Press, 95-126.

[10] LUH, J.Y.S., LIN, C.S., CHANG, P.R. (1983). Formulation and optimization of cubic polynomial joint trajectory for industrial robots, IEEE Trans. Automatic Control, 28, 1066-1074 doi:10.1109/TAC.1983.1103181

[11] NEWMAN, W.M., SPROULL, R.F. (1983). Principles of interactive computer graphics, McGraw-Hill Inc, pp. 320-325.

[12] PAUL, R.C. (1981). Robot manipulators: mathematics, programming and control, Cambridge: MIT Press, 119-155.

[13] POWELL, M.J.D. (1981). Approximation theory and methods, Cambridge University Press, pp. 227-236.

[14] RAO, S.S. (1978). Optimization Theory and Applications, Wiley Eastern Limited pp. 307.

[15] SCHOENBERG, I.J. (1946). Contribution to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math., 4, 45-49, 112-141.

[16] SCHOENBERG, I.J., WHITNEY, A. (1953). On polya frequency functions III: The positivity of translation determinant with applications to the interpolation problem by spline curves, Trans. Amer. Math. Soc., 74, 246-259 doi:10.2307/1990881

[17] WANG, K., LIEN, T.K. (1987). The planning of straight line trajectory in robotics using interactive computer graphics, Modeling, Identification and Control, 8, 125-135 doi:10.4173/mic.1987.3.1

[18] WANG, K., LIEN, T.K. (1987). The solution with closed form for the inverse kinematics of PUMA robot manipulator, Proceedings of the International Conference on The Robotic, Paper No. 10, Yugoslavia.

**BibTeX:**

@article{MIC-1988-2-2,

title={{Introduction of b-splines to trajectory planning for robot manipulators}},

author={Koch, Per E. and Wang, Kesheng},

journal={Modeling, Identification and Control},

volume={9},

number={2},

pages={69--80},

year={1988},

doi={10.4173/mic.1988.2.2},

publisher={Norwegian Society of Automatic Control}

};