# Are there smoothness evaluation methods/algorithms of 3 dimensional cartesian trajectorys in the field of robotics

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timling on 11 Sep 2021
Answered: Amrtanshu Raj on 12 Apr 2022
I was wondering if there are any methods of determining 3 dimensional cartesian trajectory smoothness for the end effector of a robot for example.
I have heard that in the ROS MoveIt framework a smoothness metric is proposed which looks at the angle formed between three consecutive waypoints for 2 dimensional paths.
• Smoothness:Looks at three consecutive waypoints and the angle formed between them. Value is calculated as a square of sum of all the angles calculated that way. Formula:S = sumi=2n-1{(2 * (pi - arccos((di-2,i-12+di-1,i2-di-2,i2)/(2 * di-2,i-1 * di-1,i))))2}, where:n - number of waypoints on the path,dx,y - distance between waypoints with index x and y.Aligned points result in S = 0.The lower the value, the smoother plan.
It seems however that, this calculation method is limmited to 2 dimensional paths.
A proposed way of calculating smoothness is to look at the standard deviation of the time series data.
Could this suggestion be extended to the x, y, z, roll, pitch, yaw representation of trajectories by perhaps adding the standard deviations together?

Amrtanshu Raj on 12 Apr 2022
Hi Timling,
Great to see your intererst in calculating smoothness for 3 dimensional path. We are actively working on this.
Thanks
Amrtanshu

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