I need a servomotor to perform a movement that is not abrupt (organic movement). This type of movement is smooth just like in the attached video.
What's the question? The video appears to give you what you need.
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The ServoEasing library may be of interest.
All servos can move synchronized.Includes the following easing functions:Linear, Quadratic, Cubic and QuarticSine, Circular, Back, Elastic and BounceUser definedEach function supports the easing types In, Out, InOut and BouncingOutInTrim and reverse operations are supported
If you just want to control the servo speed there is the VarServoSpeed library or the MobaTools servo class will allow speed control.
If you want example code, you can look here: GitHub - montagdude/PingBot_firmware: Arduino code for my ping pong training robot
See the "setAngle" method in the MyServo class, which gives the option to specify a rotation speed. Or feel free to use it directly.
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