# Development of Task-Space Nonholonomic Motion Planning Algorithm Based on Lie-Algebraic Method
## 3 Simulations
Length and energy of resulting paths strongly depend on initial values of \(\mathbf{p}_{c}\), especially when the basic Newton algorithm is used. The Newton algorithm, being local and iterative, has to generate trajectory-dependent results (the next iteration output depends on the output of the current iteration) and it tends to get into a desired location as fast as possible. It may happen that the basic algorithm provides better results than the optimized one. However, a smaller variance of results was observed in the energy-optimized version. In this case, the search space is searched more thoroughly. 2. Trajectories generated with the optimized version of the Newton algorithm usually form a few clusters corresponding to local minima. This feature was not observed in the basic algorithm, as trajectories are not constrained by the energy minimization. 3. Although the null-space optimization Newton algorithm does not necessarily guarantee the best accuracy, still much more often it promptly reaches the target. 4. The failure ratio, cf. Equation (26), is much bigger for non-redundant parameterizations. What is interesting, in some tasks, parameterizations with the same number of parameters, cf. (23) and (24), generate significantly different results. Sometimes a task is solvable in one parameterization, while it is not in the other. 5. The optimization within the null space significantly increases computational costs. In order to decrease the costs, this version of the Newton algorithm might be run after the target point was reached using the basic algorithm. 6. For the unicycle robot and the identity output function, the quality of the solutions was generally better than that for tasks with smaller dimensional task-spaces, likely due to simplicity of the model and a small number of local minima. However, the length of a path and an energy of motion is smaller for tasks with low-dimensional output functions as it is less restricted. 7. Obviously, the higher dimensional output space is, the more difficult task is generated. Additionally, a high redundancy in a parameter space is more computationally costly, but gives more flexibility as well. Thus, a compromise between flexibility and costs has to be found. 8. Computational costs are smaller for the identity output function than for other output functions, as there is no need to transfer data between configuration and task spaces. 9. Short length steps, characterized by small values of \(\delta\), are easier to control. 10. There are a few sources of inaccuracies in motion planning using Lie-algebraic methods: the truncation of the Lie series (5), evaluation of vector fields at \(q_{c}\) although a trajectory surrounds the point (quite well visible for long-range motions) numeric properties of the Newton algorithm. All of them can be controlled by the right choice of parameters. 11. A planning accuracy varies substantially with the current configuration \(q_{c}\), as vector fields in the Lie-algebraic methods are evaluated at the point, cf. Equation (5), and vector fields exhibit different variability around different \(\mathbf{q}_{c}\). 12. In almost all \(xy\)-projection plots one can notice characteristic cusps, which are typical in Lie-algebraic motion planning. At a close vicinity of a cusp a tangent line to the trajectory remains almost the same, while direction of motion is changed. It means that due to particular values of controls and generators, some components of velocity \(\vartheta\) in Equation (1) change their signs while passing through zero. An illustrative example of such behavior can be constructed by setting: \(\mathbf{X}\) long, \(\mathbf{Y}\) short, \(u_{2}\) small, and \(u_{1}\) control passing though zero. Therefore, one can expect that the number of cusps strongly and positively correlates with frequencies of controls. For mobile robots tested the explanation is even simpler, as the cusps result from linear velocity sign changes. 13.