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Path Planning

  • Reza N. Jazar

Abstract

A cubic path in joint space for the joint variable qi(t) between two points q i (t 0) and q i (t f ) is
$$ q_i (t) = a_0 + a_1 t + a_2 t^2 + a_3 t^3 $$
(13.1)
where
$$ \begin{gathered} a_0 = - \frac{{q_1 \left( {t_0^3 - 3t_0^2 t_f } \right) + q_0 \left( {3t_0 t_f^2 - t_f^3 } \right)}} {{\left( {t_f - t_0 } \right)^3 }} \hfill \\ - \frac{{q'_0 \left( {t_0 t_f^3 - t_0^2 t_f^2 } \right) + q'_1 \left( {t_0^2 t_f^2 - t_0^3 t_f } \right)}} {{\left( {t_f - t_0 } \right)^3 }} \hfill \\ \end{gathered} $$
(13.2)
and
$$ \begin{gathered} a_1 = \frac{{6q_0 t_0 t_f - 6q_1 t_0 t_f }} {{\left( {t_f - t_0 } \right)^3 }} \hfill \\ + \frac{{q'_0 \left( {t_f^3 + t_0 t_f^2 - 2t_0^2 t_f } \right) + q'_1 \left( {2t_0 t_f^2 - t_0^3 - t_0^2 t_f } \right)}} {{\left( {t_f - t_0 } \right)}} \hfill \\ \end{gathered} $$
(13.3)
$$ \begin{gathered} a_2 = - \frac{{q_0 \left( {3t_0 + 3t_f } \right) + q_1 \left( { - 3t_0 - 3t_f } \right)}} {{\left( {t_f - t_0 } \right)^3 }} \hfill \\ - \frac{{q'_1 \left( {t_0 t_f - 2t_0^2 + t_f^2 } \right) + q'_0 \left( {2t_f^2 - t_0^2 - t_0 t_f } \right)}} {{\left( {t_f - t_0 } \right)^3 }} \hfill \\ \end{gathered} $$
(13.4)
$$ a_3 = \frac{{2q_0 - 2q_1 + q'_0 \left( {t_f - t_0 } \right) + q'_1 \left( {t_f - t_0 } \right)}} {{\left( {t_f - t_0 } \right)^3 }} $$
(13.5)
and
$$ \begin{gathered} q_i \left( {t_0 } \right) = q_0 \hfill \\ \dot q_i \left( {t_0 } \right) = q'_0 \hfill \\ q_i \left( {t_f } \right) = q_f \hfill \\ \dot q_i \left( {t_f } \right) = q'_f . \hfill \\ \end{gathered} $$
(13.6)

Keywords

Path Planning Inverse Kinematic Transition Curve Transition Path Cartesian Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Asada, H., and Slotine, J. J. E., 1986, Robot Analysis and Control, John Wiley & Sons, New York.Google Scholar
  2. Murray, R. M., Li, Z., and Sastry, S. S. S., 1994, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, Florida.MATHGoogle Scholar
  3. Niku, S. B., 2001, Introduction to Robotics: Analysis, Systems, Applications, Prentice Hall, New Jersey.Google Scholar
  4. Spong, M. W., Hutchinson, S., and Vidyasagar, M., 2006, Robot Modeling and Control, John Wiley & Sons, New York.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Reza N. Jazar
    • 1
  1. 1.Department of Mechanical EngineeringManhattan CollegeRiverdale

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