# 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

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