MinJerk 2D-3Links

Planar movement of 3-segment arm moving smoothly from initial to final position.

• End-effector motion minimizes the sum of the squared jerk along its trajectory.
• Link3 has a fixed orientation (phi).
• Link1 and Link 2 orientation are computed so that the end effector reaches the requested 2D position.

Usage

• (Clone or) download the repository
• On your computer :
  • Open and run main.m with matlab
  • In the figure, clic (in the reachable zone) to move the hand to a novel position.
    

Code description

• main is the entry point
• UpdateMinJerk takes care of the animation of the arm following a clic

Minimum jerk movement

• MinJerkTrajectory simulates a linear movement that follows a min jerk in 2D

• MinJerkPositionTimeSeries produces a min jerk motion in 1D. The output timeseries is sampled at 100hz, and is generated using the following equation:
  
  Where :

  • Time t is in seconds
  • Position x depends on what is measured (meter, pixel, radian, etc).
  • Initial conditions are (t₀, x₀) and final conditions are (t_f, x_f).

• See also: Mika's coding blog and Shadmehr's paper and for more Philip Freeman's PhD

Basic geometry in 2D

The goal is to produce a minimum-jerk trajectory of the end effector of an N-Links arm, with a simple geometrical approach, which is easy for a non-redundant system.

For a 2-links arm

With 2 links in a 2D space, the system is not redundant: a unique solution exists.

• Cart2Ang computes the angles of a 2 links segment in 2D
• Ang2Cart computes the coordinates of a 2 links segment in 2D

For a 3-links arm

With one more link, the system is redundant: no unique solution exists... Yet, if the orientation of one link is invariant, the problem is back to a 2 degrees of freedom problem.

• Link3 orientation = constant => constant position shift from Link2 to Link3
• Link2 and Link1 are free => back to a 2-links arm

The graphical representation of the arm shows the definitions of angles and coordinates.