Trajectory Generation Cherevatsky Boris

Mathematical Fact Given n+1 values of a n-degree polynomial : i.e. if we have the values: we can compute the coefficients. This is true also if we are given derivatives till n-th order on some of the points (the same points) we can eliminate the need of another points. For example given:

Mathematical Fact If we have the values in 4 points of the polynomial we can find all the coefficients or if we have values in 2 points and the derivatives in these 2 points. This fact helps us to define a polynomial trajectory in a joint space from some starting point to an end point. Also we can pass through via points.

Trajectory for single joint Suppose we are given a simple robot We want to move the joint from to in 4 seconds and the trajectory should be a cubic polynomial.

Trajectory for single joint Let’s denote the initial velocity and the final velocity. We also know that, so if then we get that : By solving this 4 equations we can extract the coefficients of the polynomial:

When we have a via points Suppose we are given a trajectory as a series of time points and the joint parameter value on these points, i.e. and we want to move from the first to the last point and don’t have any speed constraint on the via points ! Solution: heuristic ! If we change the slope sign the speed will be 0, else it will be choose the average of the 2 slopes.

When we have a via points For example given the graph look like So