Trajectory Generation

In trajectory planning, the two major portions we deal here are discussed. The first part is the trajectory generation where we generated the trajectory and optimised for the minimum jerk. To generate the trajectory for minimum jerk conditions, through the calculus of variational problem to minimize the integral and the solution obtained are given.

Suppose RALS needs to move about two-three waypoints int the workspace apart from its initial position. Then we are required to write two or more equations that include these points through the path provided. The below figure shows the cubic spline that generated for the given set of points.

So, for n-waypoints, a total of n+2 points, we have n+1 equations needed to be generated. for which we have 6(n+1) coefficient for which we require the boundary conditions to solve for the generation of trajectory. These shown below

For example, for two points i.e., a total of four points, we require three equations that mean 18*3 (x, y and z) coefficients thus generated a needed total of 18*3 equations to solve for these coefficients to solve for the trajectory. This is better solved using the above-mentioned boundary conditions and the basic knowledge of linear algebra. If we can put all the 18 equations into one algebraic equation, then we can solve it for these 18 coefficients. This is all put together into the simple equation given below.