We implemented our manipulation planner in C++ and ran knot-tying simulation experiments on a 1.5GHz Intel Xeon PC with 1GB RAM. The physical model of the DLO that we use takes into account the essential mechanical properties of a typical DLO such as stretching, compressing, bending and twisting, as well as the effect of gravity. It manages self-collisions efficiently and also accounts for the interaction of the DLO with other static and rigid objects in the environment. The planner took 10-15 minutes of CPU time to generate motions for the dual 6-DOF robots that tie common knots in simulation, such as bowline, stunsail, shoe-lace, and neck-tie.
Our real-life experimental set-up consists of two PUMA-560 robots at the Stanford Artificial Intelligence Lab. We tuned the parameters of the rope model such that the simulated rope ``visually'' behaved like a typical common-life rope, as one of its ends was being manipulated and the other end was kept fixed. Then we let our planner generate a manipulation plan for tying a bowline knot. Thin aluminium rods were used to simulate the needles and hand-placed, prior to the execution of the manipulation plan, at the positions determined by the planner. Using the generated plan, the two PUMA robots were able to co-operatively tie a bowline knot with a real rope (see figure below), successfully, even though the tuned rope model did not exactly represent the physics of the real rope.

The robustness of our planner to inaccuracies in the physical model of the rope was further ascertained when the robots were able to tie bowline knots with the same plan, but using four other ropes of different nature, i.e. ropes with different materials and thicknesses. However, the same plan failed to tie the knot for a plastic rope, because it was too stiff and the plan was originally generated for significantly less stiff ropes. The table in the figure below (left) lists the materials and thicknesses of the ropes used. The figure, in the right, below shows the final bowline shape attained by the ropes.

We have also been interested in quantifying the robustness of generated plans to inaccuracies in the rope model. But it is not feasible to do so from real experiments since rope manufacturers do not provide numerical values for the mechanical properties of ropes. So, we quantified the robustness of generated plans from computer simulations in the following manner. After generating a manipulation plan for tying a bowline using a particular rope model, we tested in simulation if the same manipulation plan still achieves a bowline after corrupting the rope model parameters with Gaussian noises. The table in the figure below lists the means and standard deviations (estimated numerically) for the Gaussian distributions from which 3 main parameters of the model were independently chosen, such that a bowline was achieved more than 90% of the time. The mean values correspond to the original model parameter values, used to generate the manipulation plan. The standard deviations and the corresponding mean values have comparable values, indicating a high degree of robustness.

Figure: Means and standard deviations of Gaussian distributions from which
the three main rope model parameters were chosen for the robustness analysis.