# This code block installs ffmpeg to your Deepnote Machine
# Allows Deepnote to write MP4
!apt update -y
!apt install ffmpeg -y
# Import Libraries:
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FFMpegWriter
from matplotlib.patches import Circle
from matplotlib.patches import Rectangle
from matplotlib.pyplot import *
from numpy import *
# Part b: ------------------------------------------------------------
# Simulate this system using Euler’s method (time step = 0.001
# seconds) in Python with initial conditions, x = -1m, dx/dt = 0m/s,
# over the course of 10 seconds
# Parameters
g = 9.81 # m/s^2
m = 1 # kg
# Time array
dt = 0.001
t_vec = np.arange(0,10,dt)
# Initialize a vector of zeros
vec_size = len(t_vec)
x_vec = np.zeros(vec_size)
dx_vec = np.zeros(vec_size)
y_vec = np.zeros(vec_size)
lam_vec = np.zeros(vec_size)
# Pole End effector Location for Animation:
x_pole = np.zeros(vec_size)
y_pole = np.zeros(vec_size)
# Initial Conditions
x_vec[0] = -1
y_vec[0] = 1
lam_vec[0] = 0
# Initial Pole End effector Location:
y_offset = 0 # The Y Location of where the Cart and Pole are connected
x_pole[0] = x_vec[0] #+ L * np.sin(theta_vec[0])
y_pole[0] = y_vec[0] #y_offset #- L * np.cos(theta_vec[0])
# Euler Simulation: Using Matrix Form (A * x = B)
# Initialize A and B:
A = np.array([[m, 0, 0], [0, m, 1], [0, 1, 0]])
b = np.array([[0], [-m*g], [0]])
for i in range (1, vec_size):
# Replace non constant values
A [0, 2] = -2 * x_vec[i-1]
A [2, 0] = -2 * x_vec[i-1]
b [2] = 2 * dx_vec [i-1]**2
# Solve for ddx, ddy, and lambda
[ddx, ddy, lam] = np.linalg.solve(A, b)
# Euler integration
x_vec [i] = x_vec[i-1] + dx_vec[i-1] * dt
dx_vec[i] = dx_vec[i-1] + ddx * dt
y_vec[i] = x_vec[i]**2
lam_vec[i] = lam
# x position vs time plotting
plt.plot(t_vec, x_vec)
plt.title("x position vs time")
plt.show()
# y position vs time plotting
plt.plot(t_vec, y_vec)
plt.title("y position vs time")
plt.show()
# constraint forces vs time plotting
plt.plot(t_vec, lam_vec)
plt.title("constraint forces vs time")
plt.show()
# Part c: ------------------------------------------------------------
# Animate the motion from Problem 1b, showing a bead sliding on a
# wire (at 20 frames per second).
# Setup Figure:
fig, ax = plt.subplots()
p, = ax.plot([], [], color='royalblue')
min_lim = -5
max_lim = 5
ax.axis('equal')
ax.set_xlim([min_lim, max_lim])
ax.set_ylim([min_lim, max_lim])
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_title('Bead on a string Simulation:')
title = "simulation"
# Setup Animation Writer:
FPS = 20
sample_rate = int(1 / (dt * FPS)) # Real Time Playback
dpi = 300
writerObj = FFMpegWriter(fps=FPS)
# Create Parabola
x = linspace(-1.5,1.5,5000)
y = x**2
plot(x,y)
xlabel("x axis")
ylabel("y axis")
# Initialize Patch: (Cart)
width = 0.4 # Width of Cart
height = width / 2 # Height of Cart
xy_cart = (x_vec[0], y_vec[0]) # Bottom Left Corner of Cart
r = Rectangle(xy_cart, width, height, color='cornflowerblue') # Rectangle Patch
ax.add_patch(r) # Add Patch to Plot
# Animate:
with writerObj.saving(fig, title + ".mp4", dpi):
for i in range(0, vec_size, sample_rate):
# Update Cart Patch:
r.set(xy =(x_vec[i]-0.2, y_vec[i]-0.1))
#c.set(xy =(x_vec[i], y_vec[i]), radius = 0.1)
# Update Drawing:
fig.canvas.draw()
# Save Frame:
writerObj.grab_frame()
from IPython.display import Video
Video("/work/simulation.mp4", embed=True, width=640, height=480)
Get:1 http://security.debian.org/debian-security buster/updates InRelease [65.4 kB]
Get:2 http://deb.debian.org/debian buster InRelease [122 kB]
Get:3 http://deb.debian.org/debian buster-updates InRelease [51.9 kB]
Get:4 http://security.debian.org/debian-security buster/updates/main amd64 Packages [314 kB]
Get:5 http://deb.debian.org/debian buster/main amd64 Packages [7,906 kB]
Get:6 http://deb.debian.org/debian buster-updates/main amd64 Packages [8,792 B]
Fetched 8,468 kB in 5s (1,803 kB/s)
5 packages can be upgraded. Run 'apt list --upgradable' to see them.
The following additional packages will be installed:
i965-va-driver intel-media-va-driver libaacs0 libaom0 libasound2
libasound2-data libass9 libasyncns0 libavc1394-0 libavcodec58 libavdevice58
libavfilter7 libavformat58 libavresample4 libavutil56 libbdplus0 libbluray2
libbs2b0 libcaca0 libcap2 libcdio-cdda2 libcdio-paranoia2 libcdio18
libchromaprint1 libcodec2-0.8.1 libcrystalhd3 libdc1394-22 libdrm-amdgpu1
libdrm-common libdrm-intel1 libdrm-nouveau2 libdrm-radeon1 libdrm2
libfftw3-double3 libflac8 libflite1 libgl1 libgl1-mesa-dri libglapi-mesa
libglvnd0 libglx-mesa0 libglx0 libgme0 libgsm1 libiec61883-0 libigdgmm5
libjack-jackd2-0 liblilv-0-0 libllvm7 libmp3lame0 libmpg123-0 libmysofa0
libnorm1 libnuma1 libogg0 libopenal-data libopenal1 libopenjp2-7 libopenmpt0
libopus0 libpciaccess0 libpgm-5.2-0 libpostproc55 libpulse0 libraw1394-11
librubberband2 libsamplerate0 libsdl2-2.0-0 libsensors-config libsensors5
libserd-0-0 libshine3 libslang2 libsnappy1v5 libsndfile1 libsndio7.0
libsodium23 libsord-0-0 libsoxr0 libspeex1 libsratom-0-0 libssh-gcrypt-4
libswresample3 libswscale5 libtheora0 libtwolame0 libusb-1.0-0 libva-drm2
libva-x11-2 libva2 libvdpau-va-gl1 libvdpau1 libvidstab1.1 libvorbis0a
libvorbisenc2 libvorbisfile3 libvpx5 libwavpack1 libwayland-client0
libwayland-cursor0 libwayland-egl1 libwebpmux3 libx11-xcb1 libx264-155
libx265-165 libxcb-dri2-0 libxcb-dri3-0 libxcb-glx0 libxcb-present0
libxcb-shape0 libxcb-sync1 libxcb-xfixes0 libxkbcommon0 libxshmfence1
libxss1 libxtst6 libxv1 libxvidcore4 libxxf86vm1 libzmq5 libzvbi-common
libzvbi0 mesa-va-drivers mesa-vdpau-drivers va-driver-all vdpau-driver-all
xkb-data
Suggested packages:
ffmpeg-doc i965-va-driver-shaders libasound2-plugins alsa-utils
libbluray-bdj firmware-crystalhd libfftw3-bin libfftw3-dev jackd2
libportaudio2 opus-tools pciutils pulseaudio libraw1394-doc lm-sensors serdi
sndiod sordi speex nvidia-vdpau-driver nvidia-legacy-340xx-vdpau-driver
nvidia-legacy-304xx-vdpau-driver
The following NEW packages will be installed:
ffmpeg i965-va-driver intel-media-va-driver libaacs0 libaom0 libasound2
libasound2-data libass9 libasyncns0 libavc1394-0 libavcodec58 libavdevice58
libavfilter7 libavformat58 libavresample4 libavutil56 libbdplus0 libbluray2
libbs2b0 libcaca0 libcap2 libcdio-cdda2 libcdio-paranoia2 libcdio18
libchromaprint1 libcodec2-0.8.1 libcrystalhd3 libdc1394-22 libdrm-amdgpu1
libdrm-common libdrm-intel1 libdrm-nouveau2 libdrm-radeon1 libdrm2
libfftw3-double3 libflac8 libflite1 libgl1 libgl1-mesa-dri libglapi-mesa
libglvnd0 libglx-mesa0 libglx0 libgme0 libgsm1 libiec61883-0 libigdgmm5
libjack-jackd2-0 liblilv-0-0 libllvm7 libmp3lame0 libmpg123-0 libmysofa0
libnorm1 libnuma1 libogg0 libopenal-data libopenal1 libopenjp2-7 libopenmpt0
libopus0 libpciaccess0 libpgm-5.2-0 libpostproc55 libpulse0 libraw1394-11
librubberband2 libsamplerate0 libsdl2-2.0-0 libsensors-config libsensors5
libserd-0-0 libshine3 libslang2 libsnappy1v5 libsndfile1 libsndio7.0
libsodium23 libsord-0-0 libsoxr0 libspeex1 libsratom-0-0 libssh-gcrypt-4
libswresample3 libswscale5 libtheora0 libtwolame0 libusb-1.0-0 libva-drm2
libva-x11-2 libva2 libvdpau-va-gl1 libvdpau1 libvidstab1.1 libvorbis0a
libvorbisenc2 libvorbisfile3 libvpx5 libwavpack1 libwayland-client0
libwayland-cursor0 libwayland-egl1 libwebpmux3 libx11-xcb1 libx264-155
libx265-165 libxcb-dri2-0 libxcb-dri3-0 libxcb-glx0 libxcb-present0
libxcb-shape0 libxcb-sync1 libxcb-xfixes0 libxkbcommon0 libxshmfence1
libxss1 libxtst6 libxv1 libxvidcore4 libxxf86vm1 libzmq5 libzvbi-common
libzvbi0 mesa-va-drivers mesa-vdpau-drivers va-driver-all vdpau-driver-all
xkb-data
0 upgraded, 128 newly installed, 0 to remove and 5 not upgraded.
Need to get 64.3 MB of archives.
After this operation, 380 MB of additional disk space will be used.
Get:1 http://deb.debian.org/debian buster/main amd64 libaom0 amd64 1.0.0-3 [1,165 kB]
Get:2 http://security.debian.org/debian-security buster/updates/main amd64 libavutil56 amd64 7:4.1.8-0+deb10u1 [243 kB]
Get:3 http://security.debian.org/debian-security buster/updates/main amd64 libswresample3 amd64 7:4.1.8-0+deb10u1 [111 kB]
Get:4 http://security.debian.org/debian-security buster/updates/main amd64 libavcodec58 amd64 7:4.1.8-0+deb10u1 [4,809 kB]
Get:5 http://deb.debian.org/debian buster/main amd64 libdrm-common all 2.4.97-1 [13.8 kB]
Get:6 http://deb.debian.org/debian buster/main amd64 libdrm2 amd64 2.4.97-1 [39.7 kB]
Get:7 http://deb.debian.org/debian buster/main amd64 libva2 amd64 2.4.0-1 [57.2 kB]
Get:8 http://deb.debian.org/debian buster/main amd64 libva-drm2 amd64 2.4.0-1 [16.6 kB]
Get:9 http://deb.debian.org/debian buster/main amd64 libva-x11-2 amd64 2.4.0-1 [21.3 kB]
Get:10 http://deb.debian.org/debian buster/main amd64 libvdpau1 amd64 1.1.1-10 [42.0 kB]
Get:11 http://deb.debian.org/debian buster/main amd64 libcodec2-0.8.1 amd64 0.8.1-2 [233 kB]
Get:12 http://deb.debian.org/debian buster/main amd64 libcrystalhd3 amd64 1:0.0~git20110715.fdd2f19-13 [45.9 kB]
Get:13 http://deb.debian.org/debian buster/main amd64 libgsm1 amd64 1.0.18-2 [27.7 kB]
Get:14 http://deb.debian.org/debian buster/main amd64 libmp3lame0 amd64 3.100-2+b1 [367 kB]
Get:15 http://deb.debian.org/debian buster/main amd64 libopenjp2-7 amd64 2.3.0-2+deb10u2 [158 kB]
Get:16 http://deb.debian.org/debian buster/main amd64 libopus0 amd64 1.3-1 [191 kB]
Get:17 http://deb.debian.org/debian buster/main amd64 libshine3 amd64 3.1.1-2 [23.6 kB]
Get:18 http://deb.debian.org/debian buster/main amd64 libsnappy1v5 amd64 1.1.7-1 [17.0 kB]
Get:19 http://deb.debian.org/debian buster/main amd64 libspeex1 amd64 1.2~rc1.2-1+b2 [54.8 kB]
Get:20 http://deb.debian.org/debian buster/main amd64 libsoxr0 amd64 0.1.2-3 [65.9 kB]
Get:21 http://deb.debian.org/debian buster/main amd64 libogg0 amd64 1.3.2-1+b1 [21.3 kB]
Get:22 http://deb.debian.org/debian buster/main amd64 libtheora0 amd64 1.1.1+dfsg.1-15 [169 kB]
Get:23 http://deb.debian.org/debian buster/main amd64 libtwolame0 amd64 0.3.13-4 [49.3 kB]
Get:24 http://deb.debian.org/debian buster/main amd64 libvorbis0a amd64 1.3.6-2 [93.5 kB]
Get:25 http://deb.debian.org/debian buster/main amd64 libvorbisenc2 amd64 1.3.6-2 [80.2 kB]
Get:26 http://deb.debian.org/debian buster/main amd64 libvpx5 amd64 1.7.0-3+deb10u1 [800 kB]
Get:27 http://deb.debian.org/debian buster/main amd64 libwavpack1 amd64 5.1.0-6 [86.8 kB]
Get:28 http://deb.debian.org/debian buster/main amd64 libwebpmux3 amd64 0.6.1-2+deb10u1 [97.8 kB]
Get:29 http://deb.debian.org/debian buster/main amd64 libx264-155 amd64 2:0.155.2917+git0a84d98-2 [528 kB]
Get:30 http://deb.debian.org/debian buster/main amd64 libnuma1 amd64 2.0.12-1 [26.2 kB]
Get:31 http://deb.debian.org/debian buster/main amd64 libx265-165 amd64 2.9-4 [1,041 kB]
Get:32 http://security.debian.org/debian-security buster/updates/main amd64 libavformat58 amd64 7:4.1.8-0+deb10u1 [1,018 kB]
Get:33 http://deb.debian.org/debian buster/main amd64 libxvidcore4 amd64 2:1.3.5-1 [241 kB]
Get:34 http://security.debian.org/debian-security buster/updates/main amd64 libpostproc55 amd64 7:4.1.8-0+deb10u1 [107 kB]
Get:35 http://security.debian.org/debian-security buster/updates/main amd64 libswscale5 amd64 7:4.1.8-0+deb10u1 [208 kB]
Get:36 http://deb.debian.org/debian buster/main amd64 libzvbi-common all 0.2.35-16 [64.3 kB]
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Get:38 http://security.debian.org/debian-security buster/updates/main amd64 libavfilter7 amd64 7:4.1.8-0+deb10u1 [1,053 kB]
Get:39 http://deb.debian.org/debian buster/main amd64 libasound2-data all 1.1.8-1 [59.6 kB]
Get:40 http://deb.debian.org/debian buster/main amd64 libasound2 amd64 1.1.8-1 [361 kB]
Get:41 http://security.debian.org/debian-security buster/updates/main amd64 libavdevice58 amd64 7:4.1.8-0+deb10u1 [129 kB]
Get:42 http://security.debian.org/debian-security buster/updates/main amd64 libavresample4 amd64 7:4.1.8-0+deb10u1 [108 kB]
Get:43 http://deb.debian.org/debian buster/main amd64 libraw1394-11 amd64 2.1.2-1+b1 [47.2 kB]
Get:44 http://deb.debian.org/debian buster/main amd64 libavc1394-0 amd64 0.5.4-5 [19.9 kB]
Get:45 http://deb.debian.org/debian buster/main amd64 libass9 amd64 1:0.14.0-2 [96.7 kB]
Get:46 http://deb.debian.org/debian buster/main amd64 libbluray2 amd64 1:1.1.0-1 [145 kB]
Get:47 http://deb.debian.org/debian buster/main amd64 libchromaprint1 amd64 1.4.3-3 [41.4 kB]
Get:48 http://security.debian.org/debian-security buster/updates/main amd64 ffmpeg amd64 7:4.1.8-0+deb10u1 [1,440 kB]
Get:49 http://deb.debian.org/debian buster/main amd64 libgme0 amd64 0.6.2-1 [121 kB]
Get:50 http://deb.debian.org/debian buster/main amd64 libmpg123-0 amd64 1.25.10-2 [140 kB]
Get:51 http://deb.debian.org/debian buster/main amd64 libvorbisfile3 amd64 1.3.6-2 [25.5 kB]
Get:52 http://deb.debian.org/debian buster/main amd64 libopenmpt0 amd64 0.4.3-1+deb10u1 [627 kB]
Get:53 http://deb.debian.org/debian buster/main amd64 libssh-gcrypt-4 amd64 0.8.7-1+deb10u1 [200 kB]
Get:54 http://deb.debian.org/debian buster/main amd64 libbs2b0 amd64 3.1.0+dfsg-2.2 [12.2 kB]
Get:55 http://deb.debian.org/debian buster/main amd64 libflite1 amd64 2.1-release-3 [12.8 MB]
Get:56 http://deb.debian.org/debian buster/main amd64 libserd-0-0 amd64 0.28.0~dfsg0-1 [40.8 kB]
Get:57 http://deb.debian.org/debian buster/main amd64 libsord-0-0 amd64 0.16.0~dfsg0-1+b1 [23.0 kB]
Get:58 http://deb.debian.org/debian buster/main amd64 libsratom-0-0 amd64 0.6.0~dfsg0-1 [16.6 kB]
Get:59 http://deb.debian.org/debian buster/main amd64 liblilv-0-0 amd64 0.24.2~dfsg0-2 [44.2 kB]
Get:60 http://deb.debian.org/debian buster/main amd64 libmysofa0 amd64 0.6~dfsg0-3+deb10u1 [38.2 kB]
Get:61 http://deb.debian.org/debian buster/main amd64 libfftw3-double3 amd64 3.3.8-2 [733 kB]
Get:62 http://deb.debian.org/debian buster/main amd64 libsamplerate0 amd64 0.1.9-2 [949 kB]
Get:63 http://deb.debian.org/debian buster/main amd64 librubberband2 amd64 1.8.1-7 [86.0 kB]
Get:64 http://deb.debian.org/debian buster/main amd64 libvidstab1.1 amd64 1.1.0-2 [37.6 kB]
Get:65 http://deb.debian.org/debian buster/main amd64 libnorm1 amd64 1.5.8+dfsg2-1 [287 kB]
Get:66 http://deb.debian.org/debian buster/main amd64 libpgm-5.2-0 amd64 5.2.122~dfsg-3 [158 kB]
Get:67 http://deb.debian.org/debian buster/main amd64 libsodium23 amd64 1.0.17-1 [158 kB]
Get:68 http://deb.debian.org/debian buster/main amd64 libzmq5 amd64 4.3.1-4+deb10u2 [246 kB]
Get:69 http://deb.debian.org/debian buster/main amd64 libslang2 amd64 2.3.2-2 [507 kB]
Get:70 http://deb.debian.org/debian buster/main amd64 libcaca0 amd64 0.99.beta19-2.1 [346 kB]
Get:71 http://deb.debian.org/debian buster/main amd64 libcdio18 amd64 2.0.0-2 [199 kB]
Get:72 http://deb.debian.org/debian buster/main amd64 libcdio-cdda2 amd64 10.2+0.94+2-4 [26.4 kB]
Get:73 http://deb.debian.org/debian buster/main amd64 libcdio-paranoia2 amd64 10.2+0.94+2-4 [27.8 kB]
Get:74 http://deb.debian.org/debian buster/main amd64 libusb-1.0-0 amd64 2:1.0.22-2 [55.3 kB]
Get:75 http://deb.debian.org/debian buster/main amd64 libdc1394-22 amd64 2.2.5-1 [106 kB]
Get:76 http://deb.debian.org/debian buster/main amd64 libglvnd0 amd64 1.1.0-1 [48.6 kB]
Get:77 http://deb.debian.org/debian buster/main amd64 libglapi-mesa amd64 18.3.6-2+deb10u1 [66.3 kB]
Get:78 http://deb.debian.org/debian buster/main amd64 libx11-xcb1 amd64 2:1.6.7-1+deb10u2 [191 kB]
Get:79 http://deb.debian.org/debian buster/main amd64 libxcb-dri2-0 amd64 1.13.1-2 [101 kB]
Get:80 http://deb.debian.org/debian buster/main amd64 libxcb-dri3-0 amd64 1.13.1-2 [100 kB]
Get:81 http://deb.debian.org/debian buster/main amd64 libxcb-glx0 amd64 1.13.1-2 [116 kB]
Get:82 http://deb.debian.org/debian buster/main amd64 libxcb-present0 amd64 1.13.1-2 [99.1 kB]
Get:83 http://deb.debian.org/debian buster/main amd64 libxcb-sync1 amd64 1.13.1-2 [103 kB]
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Get:85 http://deb.debian.org/debian buster/main amd64 libxxf86vm1 amd64 1:1.1.4-1+b2 [20.8 kB]
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Get:87 http://deb.debian.org/debian buster/main amd64 libpciaccess0 amd64 0.14-1 [53.5 kB]
Get:88 http://deb.debian.org/debian buster/main amd64 libdrm-intel1 amd64 2.4.97-1 [69.8 kB]
Get:89 http://deb.debian.org/debian buster/main amd64 libdrm-nouveau2 amd64 2.4.97-1 [26.3 kB]
Get:90 http://deb.debian.org/debian buster/main amd64 libdrm-radeon1 amd64 2.4.97-1 [31.1 kB]
Get:91 http://deb.debian.org/debian buster/main amd64 libllvm7 amd64 1:7.0.1-8+deb10u2 [13.1 MB]
Get:92 http://deb.debian.org/debian buster/main amd64 libsensors-config all 1:3.5.0-3 [31.6 kB]
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Get:94 http://deb.debian.org/debian buster/main amd64 libgl1-mesa-dri amd64 18.3.6-2+deb10u1 [6,685 kB]
Get:95 http://deb.debian.org/debian buster/main amd64 libglx-mesa0 amd64 18.3.6-2+deb10u1 [180 kB]
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Get:98 http://deb.debian.org/debian buster/main amd64 libiec61883-0 amd64 1.2.0-3 [31.2 kB]
Get:99 http://deb.debian.org/debian buster/main amd64 libjack-jackd2-0 amd64 1.9.12~dfsg-2 [299 kB]
Get:100 http://deb.debian.org/debian buster/main amd64 libopenal-data all 1:1.19.1-1 [170 kB]
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Get:107 http://deb.debian.org/debian buster/main amd64 libxtst6 amd64 2:1.2.3-1 [27.8 kB]
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Get:112 http://deb.debian.org/debian buster/main amd64 xkb-data all 2.26-2 [681 kB]
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Get:119 http://deb.debian.org/debian buster/main amd64 i965-va-driver amd64 2.3.0+dfsg1-1 [311 kB]
Get:120 http://deb.debian.org/debian buster/main amd64 libigdgmm5 amd64 18.4.1+ds1-1 [78.7 kB]
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Get:125 http://deb.debian.org/debian buster/main amd64 mesa-va-drivers amd64 18.3.6-2+deb10u1 [1,972 kB]
Get:126 http://deb.debian.org/debian buster/main amd64 mesa-vdpau-drivers amd64 18.3.6-2+deb10u1 [2,101 kB]
Get:127 http://deb.debian.org/debian buster/main amd64 va-driver-all amd64 2.4.0-1 [13.0 kB]
Get:128 http://deb.debian.org/debian buster/main amd64 vdpau-driver-all amd64 1.1.1-10 [20.8 kB]
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