Gradient flow approximation for stabilization and motion planning in dynamic environments
Final Report Abstract
The project was aimed at developing novel effective solutions for a series of important control problems related to the stabilization and motion planning of nonholonomic systems in dynamic environments. The overall goal of the considered problems is to control a system in such a way that, starting from any initial position, it approaches an arbitrary small neighborhood of target point. A static (time-invariant) target position means the stabilizability property, while moving (changing with time) target gives rise to target tracking problems. If, additionally, it is necessary that the system avoids collisions with static or moving objects, then obstacle avoidance problems take place. The case of unknown target or obstacle positions is closely related to extremum seeking problems. The project was focusing mainly on the nonholonomic systems governed by driftless control-affine systems with the number of controls essentially less than the number of state variables. One of the key achievements of this project is the unifying framework for stabilization and motion planning of nonholonomic systems, both in static and dynamic environments. The main idea behind our control design approach is to guarantee that the trajectories of the resulting controlled system approximate a gradient flow of a certain time-varying potential function. With this purpose, we have defined the control functions as trigonometrical polynomials with state-dependent coefficients. These control laws are gradient-based, that is explicitly depend on the derivatives of a potential function. It is important to note that the control functions are constructed for general classes of first- and second degree nonholonomic systems, and our novel explicit and relatively simple formulas for the control coefficients allow a straightforward extension of this result to higher degree nonholonomic systems. Moreover, a variety of admissible potential functions ensures a broad range of applications of the developed control approach. In particular, we have used it to handle stabilization, trajectory tracking, obstacle avoidance, and extremum seeking problems. Up to our knowledge, these problems have not been treated in such generality for nonohlonomic systems so far. It is worth mentioning that we have carried out a thorough analysis of the asymptotic properties of the obtained closed-loop systems, which resulted in novel stability conditions for non-autonomous dynamical systems. Additionally to the intended results of the project, we have derived conditions for the partial asymptotic stability of nonholonomic systems. This result is of great importance in many applied scenarios, where the requirement of the asymptotic stability with respect to all system variables is redundant or cannot be satisfied at all. In many situations, complete information on the target, obstacles, and even system model may be unknown for control design. In these cases, gradient-free controls, which use only the values of the potential function, are of great importance. One of the major results of this project is an innovative approach for generating stabilizing gradient-free control laws for first degree nonholonomic systems. The invented control algorithm combines model-based stabilizing strategies for nonholonomic systems and gradient-free model-free extremum seeking controllers. This result provides a novel solution to the extremum seeking problem and yet can be treated as a novel approach for the dynamic stabilization of nonholonomic systems. Besides that, we have developed a powerful extremum seeking approach for a rather general class of nonlinear dynamic systems. Although not foreseen in the original research plan, this result has become one of the most striking achievements of the project. Thus, the goals of the project have been achieved and partially exceeded. We have developed a number of novel control design approaches, stability concepts, and techniques for the analysis of the asymptotic behavior of non-autonomous nonlinear systems. These results contribute to mathematical control and stability theory and can be also useful in related research areas and industrial applications. For example, the obtained control algorithms for nonholonomic systems can be potentially implemented in various industrial objects such as mobile robots, manipulators, wheeled, underwater and air vehicles, etc.
Publications
- Obstacle avoidance problem for second degree nonholonomic systems. In Proc. of the 2018 IEEE Conference on Decision and Control, pages 1500–1505, 2018
V. Grushkovskaya and A. Zuyev
(See online at https://doi.org/10.1109/CDC.2018.8619058) - On exponential stabilization of nonholonomic systems with time-varying drift. IFAC-PapersOnLine, 52(16):156–161, 2019
V. Grushkovskaya and A. Zuyev
(See online at https://doi.org/10.1016/j.ifacol.2019.11.771) - On stabilization of nonlinear systems with drift by time-varying feedback laws. In Proc. of the 12th International Workshop on Robot Motion and Control, pages 9–14, 2019
A. Zuyev and V. Grushkovskaya
(See online at https://doi.org/10.1109/RoMoCo.2019.8787353) - Partial stability concept in extremum seeking problems. IFAC-PapersOnLine, 52(16):682–687, 2019
V. Grushkovskaya and A. Zuyev
(See online at https://doi.org/10.1016/j.ifacol.2019.12.041) - Stabilization of non-admissible curves for a class of nonholonomic systems. In Proc. of the 18th European Control Conference, pages 656–661, 2019
V. Grushkovskaya and A. Zuyev
(See online at https://doi.org/10.23919/ECC.2019.8795948) - Stabilization of underactuated nonlinear systems to nonfeasible curves. PAMM, 19(1):e201900160, 2019
V. Grushkovskaya and A. Zuyev
(See online at https://doi.org/10.1002/pamm.201900160) - Extremum seeking approach for nonholonomic systems with multiple time scale dynamics. IFAC-PapersOnLine, 53(2):5392–5398, 2020
V. Grushkovskaya and A. Zuyev
(See online at https://doi.org/10.1016/j.ifacol.2020.12.1529) - Partial stabilization of nonholonomic systems with application to multi-agent coordination. In Proc. of the 2020 European Control Conference (ECC), pages 1665–1670, 2020
V. Grushkovskaya and A. Zuyev
(See online at https://doi.org/10.23919/ECC51009.2020.9143613) - Extremum seeking control of nonlinear dynamic systems using Lie bracket approximations. International Journal of Adaptive Control and Signal Processing, 35, pages 1233–1255, 2021
V. Grushkovskaya and C. Ebenbauer
(See online at https://doi.org/10.1002/acs.3152) - Stability analysis of nonholonomic extremum seeking systems via Lie brackets approximations. PAMM, 20(1):e202000024, 2021
V. Grushkovskaya
(See online at https://doi.org/10.1002/pamm.202000024)
