STEP5

Given a specific $x$ value, the Step5 function evaluates the $y$ value of a function that smoothly transitions from $(x_{0}, y_{0})$ to $(x_{1}, y_{1})$ This function has continuous first and second derivatives, but discontinuous third derivatives at the end points.

Format

$Step5 (x, x_{0}, y_{0}, x_{1}, y_{1})$

Arguments

$x$: The independent variable. For example, to specify time as the independent variable, specify $x$ as TIME.
$x_{0}$: The $x$ value at which the Step5 function begins.
$y_{0}$: The value of the Step5 function desired at $x_{0}$ .
$x_{1}$: The $x$ value at which the Step5 function ends.
$y_{1}$: The value of the Step5 function desired at $x_{1}$

Definition

(1)

\begin{array}{l} step5 = {\begin{cases} y_{0}, if x \leq x_{0} \\ y_{0} + Δ y {\tilde{x}}^{3} (10 - 15 \tilde{x} + 6 {\tilde{x}}^{2}), if x_{0} < x < x_{1} \\ y_{1}, if x \geq x_{1} \end{cases} \\ where, \\ Δ y = (y_{1} - y_{0}) \\ \tilde{x} = \frac{(x - x_{0})}{(x_{1} - x_{0})} \end{array}

Example

<Motion_Joint
     id                  = "104003"
     type                = "EXPRESSION"
     val_type            = "D"
     expr                = "STEP5(TIME,0,0,0.01,-50)"
     joint_id            = "104008"
     joint_type          = "T"
/>