Double Pendulum Trajectories

Model and units

double_pendulum_walk() models two point masses joined by rigid, massless rods in a vertical plane. There is no friction or external forcing. Independent normal perturbations of the initial angles create an ensemble of trajectories; after initialization, motion is deterministic. This is not a random-waiting-time walk. Large-angle motion can be chaotic, so small starting differences can grow.

Angles are in radians from downward vertical, angular velocities in radians per second, lengths in meters, masses in kilograms, and time in seconds. Both angles are absolute. The pivot is (0, 0) and positive y points upward.

The optional deSolve package is required for simulation. Install it yourself before running these examples if it is unavailable. Plotting uses ggplot2; animation additionally uses gganimate, and GIF rendering uses gifski.

Repeatable trajectories

set.seed(287)
walks <- double_pendulum_walk(.num_walks = 2, .n = 101)
head(walks)
#> # A tibble: 6 × 11
#>   walk_number step_number  time theta1 theta2 omega1     omega2    x1       y1
#>   <fct>             <int> <dbl>  <dbl>  <dbl>  <dbl>      <dbl> <dbl>    <dbl>
#> 1 1                     1  0      1.59   1.56  0      0         1.000  0.0145 
#> 2 1                     2  0.05   1.57   1.56 -0.490 -0.0000959 1.000  0.00223
#> 3 1                     3  0.1    1.54   1.56 -0.981 -0.000333  0.999 -0.0345 
#> 4 1                     4  0.15   1.48   1.56 -1.46  -0.0114    0.995 -0.0956 
#> 5 1                     5  0.2    1.39   1.55 -1.92  -0.0628    0.984 -0.180  
#> 6 1                     6  0.25   1.28   1.55 -2.33  -0.203     0.959 -0.283  
#> # ℹ 2 more variables: x <dbl>, y <dbl>
attr(walks, "initial_states")
#>        theta1   theta2 omega1 omega2
#> [1,] 1.585287 1.555304      0      0
#> [2,] 1.564327 1.566868      0      0

.n counts observations, including time zero. With .n = 101 and the default .delta_time = 0.05, the final observation is at five seconds. The default 401 observations cover 20 seconds. Sampling intervals do not constrain the adaptive solver’s internal integration steps.

x1, y1 locate the first bob; x, y locate the second. These are positions, not increments, so the generator does not add cumulative statistics.

fixed <- double_pendulum_walk(.num_walks = 2, .n = 21, .angle_sd = 0)

Zero angle noise produces identical trajectories for identical initial states and does not consume random numbers. Set .theta1, .theta2, .omega1, and .omega2 to choose other starts; masses, lengths, and gravity are configurable.

Plot and summarize

plot_double_pendulum(walks, .walk = 1)

The spatial plot shows the second bob’s path, with equal axis scaling and color for elapsed time. .walk selects a walk label, not its position in the data.

summarize_walks(walks, .value = x)
#> # A tibble: 1 × 17
#>   fns    fns_name dimensions   obs mean_val median range quantile_lo quantile_hi
#>   <chr>  <chr>         <int> <dbl>    <dbl>  <dbl> <dbl>       <dbl>       <dbl>
#> 1 doubl… Double …          2   101   -0.176 -0.648  3.99       -1.98        2.00
#> # ℹ 8 more variables: variance <dbl>, sd <dbl>, min_val <dbl>, max_val <dbl>,
#> #   harmonic_mean <dbl>, geometric_mean <dbl>, skewness <dbl>, kurtosis <dbl>
visualize_walks(walks, .pluck = c("x", "y"))

visualize_walks() shows coordinate traces against step number. Signed coordinates can produce undefined geometric means in the existing summary function; those statistics are not changed by this generator.

Animate and save a GIF

Construction returns a customizable gganim object and does not render or save files. Each sampled observation becomes a frame with both rods, bobs, a pivot, elapsed time, and the second bob’s recent trail. Set .trail_length = 0 to hide the trail; otherwise it counts positions including the current observation.

The following rendering example is intentionally not run during vignette builds:

animation <- animate_double_pendulum(walks, .walk = 1, .trail_length = 30)
gif <- gganimate::animate(
  animation,
  nframes = attr(walks, "n"),
  fps = 1 / attr(walks, "delta_time"),
  width = 500, height = 500,
  renderer = gganimate::gifski_renderer()
)
gganimate::anim_save("double-pendulum.gif", animation = gif)

Use every sampled observation and the reciprocal sampling interval as the frame rate to preserve simulated timing. Changing the frame rate changes playback speed. The display uses discrete states with fixed spatial limits and no tweening.

Double Pendulum Trajectories

Model and units

Repeatable trajectories

Plot and summarize

Animate and save a GIF

Numerical model