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#set page(width: 16cm, margin: 0.5em, height: auto)
#let definition(content) = box(fill: luma(92%), width: 100%, inset: 0.5em, stroke: black)[#content]
#let comment(body) = [#text(size: 0.8em)[(#body)]]
#let pr = $nu$
#let time = $t$
== Stroke end
The position modeling algorithm will lag behind the raw input by some distance. This algorithm iterates the previous
dynamical system a few additional time using the raw input position as the anchor to allow a `catch up` of the stroke
(though this prediction is only given by `predict`, so is not part of the `results` and becomes obsolete on the next
input).
*Algorithm* :Stroke end\
#[*Input*:
- Final anchor position $p["end"] = (x["end"],y["end"])$ #comment(text(black)[From the original input stream])
- final tip state $q_f ["end"] = (p_f ["end"]= (x_f ["end"],y_f ["end"]),v_f ["end"], a_f ["end"])$ #comment[#text(
fill: black,
)[returned from the physical modeling from the last section, $dot_f$ signifies that we are looking at the filtered output]]
- $K_"max"$ max number of iterations #comment[#text(fill: black)[`sampling_end_of_stroke_max_iterations`]]
- $Delta_"target"$ the target time delay between stroke #comment[#text(fill: black)[1/`sampling_min_output_rate`]]
- $d_"stop"$ stopping distance #comment[#text(fill: black)[`sampling_end_of_stroke_stopping_distance`]]
- $k_"spring"$ and $k_"drag"$ the modeling coefficients
]
#[*initialize* the vector $q_o$ with $q_0 [0] = lr((underbrace(p_f ["end"], p_o [0]),
underbrace(v_f ["end"], v_o [0]), underbrace(a_f ["end"], a_0 [0])), size: #1em)$]\
#[*initialize* $Delta t = Delta_"target"$]\
#[*for* $1<= k <= K_"max"$]\
- #[*calculate* the next candidate
$
a_c &= (p ["end"] - p_o ["end"])/(k_"spring") - k_"drag" v_0 ["end"]\
v_c &= v_o [0] + Delta t a_c\
p_c &= p_o [0] + Delta v_c
$
]\
- #[*if* $norm(p_c - p["end"])< d_"stop"$ #comment[#text(black)[further iterations won't be able to catch up and won't move closer to the anchor, we stop here]]],
- #[*return* $q_0$]\
- #[*endif*]\
- #[*if*
$angle.l p_c - p_o ["end"], p["end"] - p_o ["end"] angle.r < norm(p_c - p_o ["end"])$
#comment[#text(black)[we've overshot the anchor, we retry with a smaller step]]
]\
- #[$Delta t <- (Delta t)/2$],
- #[*continue* #comment[this candidate will be discarded, try again with a smaller time step instead]],
- #[*else*]\
- #[$q_0["end +1"] = (p_c, v_c, q_c)$ #comment[#text(black)[We append the result to the end of the $q_0$ vector]]]\
- #[*endif*]\
- #[*if* $norm(p_c - p["end"]) < d_"stop"$],
- #[*return*
#comment[We are within tolerance of the anchor, we stop iterating]],
- #[*endif*],
#[*Output* : ${q_o [k] = (s_o [k], v_o [k], a_o [k]), 0 <= k <= n (<= K_"max" - 1)}$]
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