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

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.

Benchmark description — Planners Benchmarking documentation (planners-benchmarking.readthedocs.io)

• 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.

Another suggestion was offered in this forum post: How to measure smoothness of a time series in R? - Cross Validated (stackexchange.com)

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?

Answer 1

[Amrtanshu Raj]

Hi Timling,

Great to see your intererst in calculating smoothness for 3 dimensional path. We are actively working on this.

Thanks

Amrtanshu