S-curve-motion-planning.

The purpose of this material is to impart an understanding of polynomial transitions for a trajectory generator. The ideal constant jerk S-curve (jerk is the derivate of acceleration and a measure of impact) can be represented by a second-order polynomial in velocity (3rd in position). Its shape is governed by the motion conditions at the start and end of the transition.

An S-curve with an intermediate constant acceleration (lineair portion) is often used to reduce the time to make large speed changes. The jerk can be used to determine how much of the rise or fall period can be made under constant acceleration.

Implementation:

• Velocity up s-curve [acc period].
• Velocity down s-curve [dcc period].
• Acceleration begin value.
• Acceleration end value.
• Velocity begin.
• Velocity end.
• Max acceleration.
• Lineair stage.
• Scientific papers included.

Functions:

    //! Inputs:
    //! at_time=request displacment at a certain time stamp.
    //! vs=velocity max.
    //! cs=curve displacement
    RESULT scurve_lineair(double at_time, double vs, double cs);

    //! Inputs:
    //! sct=scurve type [0=acc, 1=dcc]
    //! vo=velocity start
    //! vs=velocity max
    //! ve=velocity end
    //! am=acceleration max
    //! acs=acceleration start
    //! ace=acceleration end
    //! Results:
    //! at_time=request curve at [t]
    RESULT scurve_acc_dcc(int sct, double vo, double ve, double am, double acs, double ace, double at_time);

    //! Inputs:
    //! vs=velocity max.
    //! am=acceleration max.
    //! vo=velocity begin.
    //! acs=acceleration begin.
    //! ltot=pathlenght.
    //! ve=velocity end.
    //! ace=acceleration end.
    //! at_time=curve request at time [t]
    RESULT motion(double vs, double am, double vo, double acs, double ltot, double ve, double ace, double at_time);

Result:

    struct RESULT{
        //! Displacement result.
        double sr=0;
        //! Velocity result.
        double vr=0;
        //! Acceleration result.
        double ar=0;
        //! Curve time.
        double ct=0;
        //! Curve displacment.
        double cs=0;
        //! Error.
        bool error=0;
    };    

Build in Logic:

• If maximum velocity can not be reached for a motion, curves are sampled to fit.
• Debug information when input values are out of scope.
• In the /gui_project is a simple path planner example.

Performance:

Time taken by function nanoseconds: ~1955 milliseconds: ~0.002

Graphics

~/gui_project/motion/

To use the opencascade graphics along with the gui project, follow these instructions : https://github.com/grotius-cnc/oce/releases/tag/1.0.1

Summary

Constant jerk S-curves are used to transition robot moves. The S-curve is used to eliminate discontinuities in acceleration that are detrimental to robot performance.

License

Released under MIT by @grotius-cnc.