Lecture 12 : Trajectory planning I ( point to point and continuous trajectories)
Contd...
In this manner, we can fit a cubic polynomial between any two positions with prescribed velocities. Thus in general, if we have to move from an initial position to a final position through n via points at specified velocities, we can find (n+1) cubic segments. The via points and the corresponding velocities at these via points, may be more readily specified by the user in terms of the end effector motion. Corresponding joint coordinate positions and velocities need to be determined using inverse kinematics. In certain cases, inverse kinematics may not permit a solution. Thus it is the operator's responsibility to provide kinematically consistent data at all the via points. Instead, the operator may prefer to simply specify the via point positions but allow the system to choose appropriate velocities. Let us consider motion through three points i, j, k. Let the position and velocity be prescribed at i and k and only position is prescribed at j. Let the duration of motion between the points be prescribed as and . We can then fit two cubic segments as follows:
At j, we can require that the velocity and acceleration be continuous. i.e.
(12.2.11)
(12.2.12)
These eight equations can be solved for the eight unknowns (a0 - a3 ) and (b0– b3). Thus, in general, in moving from an initial position to a final position (with prescribed velocities at both positions), through n via points, we can fit (n+1) cubic segments with coefficients. The (4n+4) equations required to find these coefficients are written down as follows
Constraint
No. of Equations
Position and velocity constraints at initial position
2
Position and velocity constraints at final position
2
Position constraint at each via point (one for each Segment on either side of via-points)
2n
Velocity and acceleration continuity constraints at
2n
Total
4n+4
For example if we have motion specified from an initial position to a final position using 3 via-points, we can fit 4 cubic segments. Hence we need to determine 16 coefficients. The 16 constraints that can be used to generate the 16 equations necessary to determine all the coefficients are indicated in Fig.12.2.3. Thus the entire motion from the initial position to the final position through the via points can be fit using these 4 cubic polynomials.
and 
. We can then fit two cubic segments as follows: 
coefficients. The (4n+4) equations required to find these coefficients are written down as follows
