This paper presents novel fourth- and sixth-order polynomials to solve the problem of joint-space trajectory generation with a via point. These new polynomials use a single-polynomial function rather than two-polynomial functions matched at the via point as in previous methods. The problem of infinite spikes in jerk is also addressed.

Joint-space trajectory generation is in common usage in robotics to provide smooth, continuous motion from one set of

There is an entire body of literature devoted to trajectory generation (aka motion planning and path planning) at the joint level. Paul and Zhong [

Apparently alone amongst all of the major robotics textbook authors, Angeles [

An early version of the current paper was presented by the author at a conference [

Standard joint-space trajectory generation assumes that two sets of

Polynomials are natural choices for providing smooth, continuous motion, with some level of continuous derivatives. In many robotics motion planning problems, the robot must pass through intermediate point(s) between the start and finish poses, such as for obstacle avoidance. The joint rates and accelerations need not go to zero at these so-called via points, but they must be matched between functions meeting at the via point(s).

Craig [

These two polynomials require eight constraints. Four constraints come from the initial and final time points:

We need four more constraints. We force the first polynomial to end at the via angle

The constraints yield eight linear equations in the eight unknowns (two third-order polynomials, four unknown coefficients each). Three of the unknown polynomial coefficients are found immediately, from the initial and via time constraints:

Use two third-order polynomials for smooth joint-space trajectory generation, plus motion through a via point, for one joint. Given

Two third-order polynomials with via point.

Note that the joint angle passes through the via point

The two third-order polynomials presented above, matched at a via point, appear in many robotics textbooks (e.g., [

The first original contribution of this paper is presented in this subsection. We can achieve the same goals as the two third-order polynomials meeting at a via point much more simply; let us use only one polynomial, forced to go through the via point. Here are the constraints for meeting the required angles with smooth motion (since we use a single continuous polynomial, the velocity and accelerations are guaranteed to match and be continuous at the via point):

Five linear equations in the five unknown polynomial coefficients

For the special case of

Use a single fourth-order polynomial for smooth joint-space trajectory generation, plus motion through a via point, for one joint. Given

Single fourth-order polynomial with via point.

The

The jerk still has an infinite spike at the start and end as we saw with the two third-order polynomials example, which is unacceptable. We now improve upon this with a single sixth-order polynomial in the next subsection; again, this next subsection is original work.

This section presents another original contribution. We can achieve the same goals as the single fourth-order polynomial with a via point and eliminate the infinite spikes in jerk at the start and end as follows. To the previous five constraints, add two more, for zero acceleration at the start and end of the single motion range. Here are the seven constraints:

Seven linear equations in the seven unknown polynomial coefficients

Use a single sixth-order polynomial for smooth joint-space trajectory generation, plus motion through a via point, for one joint. Given

Single sixth-order polynomial with via point.

Again, the

All joint-space trajectory generation methods presented by previous authors and the new ones proposed in this paper can easily be applied to multiple joints, simultaneously but independently.

This main original contribution of this paper is to present a new joint-space trajectory generation approach using a single polynomial for motion through a given via point. The robot does not need to stop at the via point, but the motion must be smooth and continuous through the via point. A single fourth-order polynomial was developed to achieve this smooth motion through a via point and to replace the two three-order polynomials matched at the via point in common usage today. With the new approach, this via point matching comes automatically and there is no need for two-polynomial functions.

An important secondary contribution of this paper is to expose a bad practice in common usage in joint-space trajectory generation in robotics today. We are not the first to notice this [

The novel single fourth-order polynomial introduced in this paper to go through a via point also suffers from this discontinuous acceleration. Therefore, to remedy this we introduced a novel sixth-order polynomial to move smoothly and continuously through a via point, with finite jerk throughout the entire motion, which is acceptable

For joint-space trajectory generation in robotics, the recommendations of this paper are simple; for all joint motions without a via point, use the standard fifth-order polynomial from Craig [