Knotted trajectory

Do you see why this knot corresponds to p = 3, <7 = 27 Follow the knotted trajectory in Figure 8.6.5, and count the number of revolutions made by during the time that 0, makes one revolution, where 0, is latitude and 0 is longitude. Starting on the outer equator, the trajectory moves onto the top surface, dives into the hole, travels along the bottom surface, and then reappears on the outer equator, two-thirds of the way around the torus. Thus 62 makes two-thirds of a revolution while 0 makes one revolution hence p = 3, q = 2. 

When plotted on the torus, the same trajectory gives. .. a trefoil knot Figure 8.6.5 shows a trefoil, alongside a top view of a torus with a trefoil wound around it. 

In fact the trajectories are always knotted if p, q>2 have no common factors. The resulting curves are called p. q torus knots. 

To find a joint trajectory that approximates the desired path closely, the Cartesian path points are transformed into N sets of joint displacements, with one set for each joint. Application of Bezier polynomial will provide trajectories that are smooth and have small overshoot of angular displacement between two adjacent knot points. The continuity conditions for joint displacement, velocity, and acceleration must be satisfied on the entire trajectory for the Cartesian robot path. 

A simple and comprehensive menu-driven computer-based method for trajectory planning and force analysis in a planar robot is developed in the first paper. The robot designer is able to vary parameters and study their effect on the robot performance. In the second paper, a simple method to analyze the effect of torque and force on the first three links of a PUMA robot has been determined. Minimum time trajectory and bang-bang control with discontinuity points and knot points smoothed by parabolic blend are used. The workspace of a robotic arm using the Articulated Total Body model is calculated in the third paper. Computation of the workspace of the end effector is important in determining the effectiveness of a robot. 

---