trajectory analysis

[fabianromero]

Hello everyone,

I have the x, y coordinates (in meters) of an object. The frequency of capture is 30 frames per second. I need to calculate the speed and acceleration at each frame. any idea?

I'll appreaciate any help

  x                 y

|1|47.7241|26.2659|
|2|47.6629|26.0827|
|3|47.6016|25.8996|
|4|47.1948|25.6156|
|5|46.7879|25.3316|
|6|46.3811|25.0476|
|7|45.9742|24.7636|
|8|45.5674|24.4796|
|9|45.1605|24.1957|
|10|44.7537|23.9117|
|11|44.3468|23.6277|
|12|43.9399|23.3437|
|13|43.5331|23.0597|
|14|43.1262|22.7758|
|15|42.7194|22.4918|
|16|42.3125|22.2078|
|17|41.9057|21.9238|
|18|41.4988|21.6398|
|19|41.0920|21.3558|
|20|40.6851|21.0719|
|21|40.2783|20.7879|
|22|39.8714|20.5039|
|23|39.4646|20.2199|
|24|39.0577|19.9359|
|25|38.6509|19.6520|
|26|38.2440|19.3680|
|27|37.8372|19.0840|
|28|37.4303|18.8000|
|29|37.0234|18.5160|
|30|36.6166|18.2320|
|31|36.2097|17.9481|
|32|35.8029|17.6641|
|33|35.3960|17.3801|
|34|34.9892|17.0961|
|35|34.5823|16.8121|
|36|34.3687|16.6736|
|37|34.1550|16.5350|
|38|33.9414|16.3965|
|39|33.7277|16.2579|
|40|33.5141|16.1193|
41|33.3004|15.9808|
|42|33.0868|15.8422|
|43|32.8731|15.7037|
|44|32.6595|15.5651|
|45|32.4458|15.4265|
|46|32.2322|15.2880|
|47|32.0185|15.1494|
|48|31.8049|15.0109|
|49|31.5912|14.8723|
|50|31.3776|14.7338|
|51|31.1639|14.5952|
|52|30.9503|14.4566|
|53|30.7366|14.3181|
|54|30.5230|14.1795|
|55|30.3093|14.0410|
|56|30.0957|13.9024|
|57|29.8820|13.7638|
|58|29.6684|13.6253|
|59|29.4547|13.4867|
|60|29.2411|13.3482|
|61|29.0274|13.2096|
|62|28.8137|13.0710|
|63|28.6001|12.9325|
|64|28.3864|12.7939|
|65|28.1728|12.6554|
|66|27.9591|12.5168|
|67|27.7455|12.3783|
|68|27.5318|12.2397|
|69|27.3182|12.1011|
|70|27.1045|11.9626|
|71|26.8909|11.8240|
|72|26.6772|11.6855|
|73|26.4636|11.5469|
|74|26.2499|11.4083|
|75|26.0363|11.2698|
|76|25.8226|11.1312|
|77|25.6090|10.9927|
|78|25.3953|10.8541|
|79|25.1817|10.7155|
|80|24.9680|10.5770|
|81|24.7544|10.4384|
|82|24.5407|10.2999|
|83|24.3271|10.1613|
|84|24.1134|10.0228|
|85|23.8998|9.8842|
|86|23.6861|9.7456|
|87|23.4725|9.6071|
|88|23.2588|9.4685|
|89|23.0368|9.5760|
|90|22.8147|9.6835|
|91|22.5926|9.7910|
|92|22.3706|9.8984|
|93|22.1485|10.0059|
|94|21.9264|10.1134|
|95|21.7044|10.2209|
|96|21.4823|10.3284|
|97|21.2602|10.4358|
|98|21.0382|10.5433|
|99|20.8161|10.6508|
|100|20.5940|10.7583|
|101|20.3720|10.8657|
|102|20.1499|10.9732|
|103|19.9278|11.0807|
|104|19.7058|11.1882|
|105|19.4837|11.2957|
|106|19.2617|11.4031|
|107|19.0396|11.5106|
|108|18.8175|11.6181|
|109|18.5955|11.7256|
|110|18.3734|11.8331|
|111|18.1513|11.9405|
|112|17.9293|12.0480|
|113|17.7072|12.1555|
|114|17.4851|12.2630|
|115|17.2631|12.3705|
|116|17.0410|12.4779|
|117|16.8189|12.5854|
|118|16.5969|12.6929|
|119|16.3748|12.8004|
|120|16.1527|12.9079|
|121|15.9307|13.0153|
|122|15.7086|13.1228|
|123|15.4865|13.2303|
|124|15.2645|13.3378|
|125|15.0424|13.4452|
|126|14.8203|13.5527|
|127|14.5983|13.6602|
|128|14.3762|13.7677|
|129|14.1541|13.8752|
|130|13.9321|13.9826|
|131|13.7100|14.0901|
|132|13.4880|14.1976|
|133|13.2659|14.3051|
|134|13.0438|14.4126|
|135|12.8218|14.5200|
|136|12.5997|14.6275|
|137|12.3776|14.7350|
|138|12.1556|14.8425|
|139|11.9335|14.9500|
|140|11.7114|15.0574|
|141|11.4894|15.1649|
|142|11.2673|15.2724|
|143|11.0452|15.3799|
|144|10.8232|15.4874|
|145|10.6011|15.5948|
|146|10.3790|15.7023|
|147|10.1570|15.8098|
|148|9.9349|15.9173|
|149|9.7128|16.0247|
|150|9.4908|16.1322|
|151|9.2687|16.2397|
|152|9.0466|16.3472|
|153|8.6554|17.1089|
|154|8.2642|17.8707|
|155|7.8730|18.6324|
|156|7.4818|19.3942|
|157|7.0906|20.1559|
|158|6.6994|20.9177|
|159|6.3082|21.6794|
|160|5.9170|22.4412|

[andresrcs]

If I understand you correctly, this would give you instant velocity on each "frame" (I'm not sure what you mean with this), following the same logic you can calculate the variation of speed over time (acceleration).

library(dplyr)

sample_df <- data.frame(
    x = c(47.7241,47.6629,47.6016,47.1948,46.7879,
          46.3811,45.9742,45.5674,45.1605,44.7537,44.3468,43.9399,
          43.5331,43.1262,42.7194,42.3125,41.9057,41.4988,41.092,
          40.6851,40.2783,39.8714,39.4646,39.0577,38.6509,38.244,
          37.8372,37.4303,37.0234,36.6166,36.2097,35.8029,35.396,
          34.9892,34.5823,34.3687,34.155,33.9414,33.7277,33.5141,
          33.3004,33.0868,32.8731,32.6595,32.4458,32.2322,32.0185,
          31.8049,31.5912,31.3776,31.1639,30.9503,30.7366,30.523,
          30.3093,30.0957,29.882,29.6684,29.4547,29.2411,29.0274,
          28.8137,28.6001,28.3864,28.1728,27.9591,27.7455,27.5318,
          27.3182,27.1045,26.8909,26.6772,26.4636,26.2499,26.0363,
          25.8226,25.609,25.3953,25.1817,24.968,24.7544,24.5407,
          24.3271,24.1134,23.8998,23.6861,23.4725,23.2588,23.0368,
          22.8147,22.5926,22.3706,22.1485,21.9264,21.7044,21.4823,
          21.2602,21.0382,20.8161,20.594,20.372,20.1499,19.9278,19.7058,
          19.4837,19.2617,19.0396,18.8175,18.5955,18.3734,18.1513,
          17.9293,17.7072,17.4851,17.2631,17.041,16.8189,16.5969,
          16.3748,16.1527,15.9307,15.7086,15.4865,15.2645,15.0424,
          14.8203,14.5983,14.3762,14.1541,13.9321,13.71,13.488,
          13.2659,13.0438,12.8218,12.5997,12.3776,12.1556,11.9335,
          11.7114,11.4894,11.2673,11.0452,10.8232,10.6011,10.379,
          10.157,9.9349,9.7128,9.4908,9.2687,9.0466,8.6554,8.2642,
          7.873,7.4818,7.0906,6.6994,6.3082,5.917),
    y = c(26.2659,26.0827,25.8996,25.6156,25.3316,
          25.0476,24.7636,24.4796,24.1957,23.9117,23.6277,23.3437,
          23.0597,22.7758,22.4918,22.2078,21.9238,21.6398,21.3558,
          21.0719,20.7879,20.5039,20.2199,19.9359,19.652,19.368,
          19.084,18.8,18.516,18.232,17.9481,17.6641,17.3801,
          17.0961,16.8121,16.6736,16.535,16.3965,16.2579,16.1193,15.9808,
          15.8422,15.7037,15.5651,15.4265,15.288,15.1494,15.0109,
          14.8723,14.7338,14.5952,14.4566,14.3181,14.1795,14.041,
          13.9024,13.7638,13.6253,13.4867,13.3482,13.2096,13.071,
          12.9325,12.7939,12.6554,12.5168,12.3783,12.2397,12.1011,
          11.9626,11.824,11.6855,11.5469,11.4083,11.2698,11.1312,
          10.9927,10.8541,10.7155,10.577,10.4384,10.2999,10.1613,
          10.0228,9.8842,9.7456,9.6071,9.4685,9.576,9.6835,9.791,
          9.8984,10.0059,10.1134,10.2209,10.3284,10.4358,10.5433,
          10.6508,10.7583,10.8657,10.9732,11.0807,11.1882,11.2957,
          11.4031,11.5106,11.6181,11.7256,11.8331,11.9405,12.048,
          12.1555,12.263,12.3705,12.4779,12.5854,12.6929,12.8004,
          12.9079,13.0153,13.1228,13.2303,13.3378,13.4452,13.5527,
          13.6602,13.7677,13.8752,13.9826,14.0901,14.1976,14.3051,
          14.4126,14.52,14.6275,14.735,14.8425,14.95,15.0574,15.1649,
          15.2724,15.3799,15.4874,15.5948,15.7023,15.8098,15.9173,
          16.0247,16.1322,16.2397,16.3472,17.1089,17.8707,18.6324,
          19.3942,20.1559,20.9177,21.6794,22.4412)
)

sample_df %>% 
    mutate(vel = sqrt((lag(x) - x) ^ 2 + (lag(y) - y) ^ 2) / (1/30)) %>% 
    as_tibble() # Just for friendly printing
#> # A tibble: 160 x 3
#>        x     y   vel
#>    <dbl> <dbl> <dbl>
#>  1  47.7  26.3 NA   
#>  2  47.7  26.1  5.79
#>  3  47.6  25.9  5.79
#>  4  47.2  25.6 14.9 
#>  5  46.8  25.3 14.9 
#>  6  46.4  25.0 14.9 
#>  7  46.0  24.8 14.9 
#>  8  45.6  24.5 14.9 
#>  9  45.2  24.2 14.9 
#> 10  44.8  23.9 14.9 
#> # … with 150 more rows

^{Created on 2020-03-26 by the reprex package (v0.3.0.9001)}

For future posts, please provide a proper REPRoducible EXample (reprex) illustrating your issue.

[fabianromero]

thank you, @andresrcs

that's what I needed. I didn't know the lag function.
I guess the acceleration formula is this, isn't it?

mutate(acc = sqrt((lag(vel) - vel) ^ 2) / (1/30))

[andresrcs]

I'm afraid not, this seems like a homework question so I'll let you figure it out, as a hint, that formula was for calculating distance between two points on a plane.

[fabianromero]

Thank you. I`ll use you information to do it.