Guaranteed Control of Sampled Switched Systems using Semi-Lagrangian Schemes and One-Sided Lipschitz Constants

In this paper, we propose a new method for ensuring formally that a controlled trajectory stay inside a given safety set S for a given duration T. Using a finite gridding X of S, we first synthesize, for a subset of initial nodes x of X , an admissible control for which the Euler-based approximate trajectories lie in S at t $\in$ [0,T]. We then give sufficient conditions which ensure that the exact trajectories, under the same control, also lie in S for t $\in$ [0,T], when starting at initial points 'close' to nodes x. The statement of such conditions relies on results giving estimates of the deviation of Euler-based approximate trajectories, using one-sided Lipschitz constants. We illustrate the interest of the method on several examples, including a stochastic one

Optimality Condition-Based Sensitivity Analysis of Optimal Control for Hybrid Systems and Its Application

Gradient-based algorithms are efficient to compute numerical solutions of optimal control problems for hybrid systems (OCPHS), and the key point is how to get the sensitivity analysis of the optimal control problems. In this paper, optimality condition-based sensitivity analysis of optimal control for hybrid systems with mode invariants and control constraints is addressed under a priori fixed mode transition order. The decision variables are the mode transition instant sequence and admissible continuous control functions. After equivalent transformation of the original problem, the derivatives of the objective functional with respect to control variables are established based on optimal necessary conditions. By using the obtained derivatives, a control vector parametrization method is implemented to obtain the numerical solution to the OCPHS. Examples are given to illustrate the results

Notions, Stability, Existence, and Robustness of Limit Cycles in Hybrid Systems

This paper deals with existence and robust stability of hybrid limit cycles for a class of hybrid systems given by the combination of continuous dynamics on a flow set and discrete dynamics on a jump set. For this purpose, the notion of Zhukovskii stability, typically stated for continuous-time systems, is extended to the hybrid systems. Necessary conditions, particularly, a condition using a forward invariance notion, for existence of hybrid limit cycles are first presented. In addition, a sufficient condition, related to Zhukovskii stability, for the existence of (or lack of) hybrid limit cycles is established. Furthermore, under mild assumptions, we show that asymptotic stability of such hybrid limit cycles is not only equivalent to asymptotic stability of a fixed point of the associated Poincar\'{e} map but also robust to perturbations. Specifically, robustness to generic perturbations, which capture state noise and unmodeled dynamics, and to inflations of the flow and jump sets are established in terms of $\mathcal{KL}$ bounds. Furthermore, results establishing relationships between the properties of a computed Poincar\'{e} map, which is necessarily affected by computational error, and the actual asymptotic stability properties of a hybrid limit cycle are proposed. In particular, it is shown that asymptotic stability of the exact Poincar\'{e} map is preserved when computed with enough precision. Several examples, including a congestion control system and spiking neurons, are presented to illustrate the notions and results throughout the paper.Comment: 26 pages. Version submitted for revie

Actuation-Aware Simplified Dynamic Models for Robotic Legged Locomotion

In recent years, we witnessed an ever increasing number of successful hardware implementations of motion planners for legged robots. If one common property is to be identified among these real-world applications, that is the ability of online planning. Online planning is forgiving, in the sense that it allows to relentlessly compensate for external disturbances of whatever form they might be, ranging from unmodeled dynamics to external pushes or unexpected obstacles and, at the same time, follow user commands. Initially replanning was restricted only to heuristic-based planners that exploit the low computational effort of simplified dynamic models. Such models deliberately only capture the main dynamics of the system, thus leaving to the controllers the issue of anchoring the desired trajectory to the whole body model of the robot. In recent years, however, we have seen a number of new approaches attempting to increase the accuracy of the dynamic formulation without trading-off the computational efficiency of simplified models. In this dissertation, as an example of successful hardware implementation of heuristics and simplified model-based locomotion, I describe the framework that I developed for the generation of an omni-directional bounding gait for the HyQ quadruped robot. By analyzing the stable limit cycles for the sagittal dynamics and the Center of Pressure (CoP) for the lateral stabilization, the described locomotion framework is able to achieve a stable bounding while adapting to terrains of mild roughness and to sudden changes of the user desired linear and angular velocities. The next topic reported and second contribution of this dissertation is my effort to formulate more descriptive simplified dynamic models, without trading off their computational efficiency, in order to extend the navigation capabilities of legged robots to complex geometry environments. With this in mind, I investigated the possibility of incorporating feasibility constraints in these template models and, in particular, I focused on the joint torques limits which are usually neglected at the planning stage. In this direction, the third contribution discussed in this thesis is the formulation of the so called actuation wrench polytope (AWP), defined as the set of feasible wrenches that an articulated robot can perform given its actuation limits. Interesected with the contact wrench cone (CWC), this yields a new 6D polytope that we name feasible wrench polytope (FWP), defined as the set of all wrenches that a legged robot can realize given its actuation capabilities and the friction constraints. Results are reported where, thanks to efficient computational geometry algorithms and to appropriate approximations, the FWP is employed for a one-step receding horizon optimization of center of mass trajectory and phase durations given a predefined step sequence on rough terrains. For the sake of reachable workspace augmentation, I then decided to trade off the generality of the FWP formulation for a suboptimal scenario in which a quasi-static motion is assumed. This led to the definition of the, so called, local/instantaneous actuation region and of the global actuation/feasible region. They both can be seen as different variants of 2D linear subspaces orthogonal to gravity where the robot is guaranteed to place its own center of mass while being able to carry its own body weight given its actuation capabilities. These areas can be intersected with the well known frictional support region, resulting in a 2D linear feasible region, thus providing an intuitive tool that enables the concurrent online optimization of actuation consistent CoM trajectories and target foothold locations on rough terrains

Simulation and Control of Running Models

This work focuses on the locomotion of one-legged robots, with focus on approaches that stabilize passive limit cycles. Locomotion based on the socalled passive gaits promises to greatly reduce the actuation effort required for legged robots to move. In this work, the passive gaits of robots of varying complexity are characterized and stabilizing controllers are reviewed from the literature and newly formulated. The robots are modelled as hybrid dynamical systems and numerically simulated, thereby allowing to validate the proposed control strategies. Firstly, the vertical control through energy regulation of a one-dimensional hopper is considered. Secondly, the SLIP model is reviewed and then extended to the “pitchingSLIP”, with the aim of characterizing its passive gaits with somersaults. Two controllers based on energy and angular momentum regulation are then formulated to stabilize passive gaits with somersaults, making the control effort converge to zero. A further extension of the SLIP template, denominated “bodySLIP”, is then used to test the control approach on a more realistic model. The controllers shall be later extended to more complex cases, in which the somersaults are not necessarily present in the passive gaits. Thirdly, the locomotion of a one-legged robot with a body link is studied. Raibert’s control approach based on the foot placement algorithm is reviewed and compared to the non-dissipative touchdown controller of Hyon and Emura. The latter is then extended to be used with continuous torque profiles and to perform velocity tracking. Moreover, damping is added to the joints in order to study its effect on the controller, which was then modified to achieve stable running even in such conditions. The results obtained shall lay the foundations for a later test on hardware on DLR’s quadruped Bert

Design and Optimization of a Compass Robot with Subject to Stability Constraint

In the first part of this thesis, the design of a compass robot is explored by considering its components and their interaction with each other. Three components including robot's structure, gear and motor are interacting during design process to achieve better performance, higher stability and lower cost. In addition, the modeling of the system is upgraded by considering the torque-velocity constraint in the motor. Adding this constraint of DC motor make the interaction of different components more complicated since it affects the gear and walking dynamics. After achieving the design method, different actuators (motor+ gear+ batteries) are selected for a given structure and the their performance is compared in the terms of cost, efficiency and their effect on the walking stability. In the second part of the thesis, structural optimization of the compass robot with stability constraint is investigated. The stability of a compass robot as a hybrid system is analyzed by Poincare map. Including stability analysis in the optimization process, makes it very complicated. In addition, the objective function of the system has to be evaluated in the convergent limit cycle. Different methods are examined to solve this problem. Limit cycle convergence is the best solution among the existing methods. By adding convergence constraint to the optimization, in addition of making the stability analysis valid, it helps the optimization estimates the correct objective function in each iteration. Finally, the optimization process is improved in two steps. The first step is using a predictive model in the optimization which covers the stable domain so that one does not need to check the stability of walking in each iteration. The Support Vector Domain Description (SVDD) approach which is applied to establish the stable domain, improve the decreases the optimization time. Another important step to upgrade the optimization is developing a computational algorithm which obtains the convergent limit cycle and its fixed-point in a short time. This algorithm speeds up the optimization time tremendously and allows the optimization search in a broader area. Combining SVDD approach in combination with Fixed-Point Finder Algorithm improve the optimization in the terms of time and broader area for search