Boolean Algebras from Trace Automata

We consider trace automata. Their vertices are Mazurkiewicz traces and they accept finite words. Considering the length of a trace as the length of its Foata normal form, we define the operations of level-length synchronization and of superposition of trace automata. We show that if a family F of trace automata is closed under these operations, then for any deterministic automaton H in F, the word languages accepted by the deterministic automata of F that are length-reducible to H form a Boolean algebra. We show that the family of trace suffix automata with level-regular contexts and the subfamily of vector addition systems satisfy these closure properties. In particular, this yields various Boolean algebras of word languages accepted by deterministic vector addition systems

Unfolding of finite concurrent automata

We consider recognizable trace rewriting systems with level-regular contexts (RTL). A trace language is level-regular if the set of Foata normal forms of its elements is regular. We prove that the rewriting graph of a RTL is word-automatic. Thus its first-order theory is decidable. Then, we prove that the concurrent unfolding of a finite concurrent automaton with the reachability relation is a RTL graph. It follows that the first-order theory with the reachability predicate (FO[Reach] theory) of such an unfolding is decidable. It is known that this property holds also for the ground term rewriting graphs. We provide examples of finite concurrent automata of which the concurrent unfoldings fail to be ground term rewriting graphs. The infinite grid tree (for each vertex of an infinite grid, there is an edge from this vertex to the origin of a copy of the infinite grid) is such an unfolding. We prove that the infinite grid tree is not a ground term rewriting graph. We have thus obtained a new class of graphs for with a decidable FO[Reach] theory

Automates infinis et traces de Mazurkiewicz

We introduce the notion of level-regularity for Mazurkiewicz trace languages and we consider recognizable trace rewriting systems with level-regular contexts (RTLsystem). We prove that an automaton for which the underlying graph is the rewriting graph of a RTL system and for which the sets of initial vertices and final vertices are level-regular (RTL automaton), is word-automatic. In particular, the first-order theory of a RTL automaton is decidable. Then, we prove that, enriched with thereachability relation, an automaton for which the underlying graph is the concurrent unfolding of a finite concurrent graph, and for which the sets of initial vertices and final vertices are level-regular, is RTL. In particular, the first-order theory with the reachability predicate of such an automaton is decidable. Besides, it is known that this property also holds for ground term rewriting graphs (GTR graph). We highlight various concurrent unfoldings of finite concurrent graphs that are not GTRgraphs. The infinite quarter grid tree is such an unfolding. The class of concurrent unfoldings of finite concurrent graphs is therefore a class of word-automatic graphs for which the first-order theory with the reachability predicate is decidable and that contains some non GTR graphs. We define the operations of level-length synchronization and level-length superposition of trace automata (automata for which vertices are Mazurkiewicz traces) and we prove that if a family F of trace automata is closed under these operations, then for any deterministic trace automaton H 2 F, the languages accepted by the deterministic trace automata belonging to F and that are length-reducible to H, form a Boolean algebra; the length of a trace being the length of its Foata normal form, a trace automaton G is length-reducible to a trace automaton H if there exists a length-preserving morphism from G to H. Then, we show that the family of trace suffix automata with level regular contexts, the extension of word suffix automata to Mazurkiewicz traces, satisfies these closure properties. We define a generalized Petri net as a trace suffix automaton over a dependence alphabet for which the dependence is reduced to the equality and we show that the subfamily of generalized Petri nets also satisfies the closure properties above. In particular, this yields various Boolean algebras of word languages accepted by deterministic generalized Petri nets.Nous introduisons la notion de régularité par niveaux pour des langages de traces de Mazurkiewicz et nous considérons des systèmes reconnaissables de réécriture de traces, à contextes réguliers par niveaux (RTL). Nous prouvons qu’un automate dont le graphe sous-jacent est le graphe de réécriture d’un système RTL et dont les ensembles de sommets initiaux et finaux sont réguliers par niveaux (automate RTL), est mot-automatique. En particulier, la théorie du premier ordre d’un automate RTL est décidable. Ensuite, nous prouvons que, enrichi de la relation d’accessibilité, un automate dont le graphe sous-jacent est déplié concurrent d’un graphe fini concurrent et dont les ensembles de sommets initiaux et finaux sont réguliers par niveaux, est RTL. En particulier, la théorie du premier ordre avec accessibilité d’un tel automate est décidable. Par ailleurs, il est bien connu que la théorie du premier ordre avec accessibilité du graphe de réécriture suffixe d’un système de réécriture de termes clos (graphe GTR) est décidable. Nous mettons en évidence divers dépliés concurrents de graphes finis concurrents qui ne sont pas des graphes GTR. L’arbre du quart de la grille infinie est un exemple de tel déplié. La classe des dépliés concurrents des graphes finis concurrents constitue ainsi une classe de DAG mot-automatiques, dont la théorie du premier ordre avec accessibilité est décidable et qui contient des graphes non GTR. Nous définissons pour les automates de traces (automates dont les sommets sont des traces de Mazurkiewicz) deux opérations que sont la synchronisation par niveaux et la superposition par niveaux et nous montrons que si une famille F d’automates de traces est fermée pour ces opérations, alors pour tout automate déterministe H 2 F, les langages acceptés par les automates déterministes de F qui sont longueur-réductibles en H forment une algèbre de Boole ; la longueur d’une trace étant donnée par la longueur de sa forme normale de Foata, un automate de traces G est longueur-réductible dans un automate de traces H, s’il existe un morphisme de G dans H préservant la longueur. Ensuite, nous montrons que la classe des automates suffixes de traces à contextes réguliers par niveaux, qui n’est que l’extension aux traces de Mazurkiewicz des automates suffixes de mots, satisfait ces propriétés de fermeture. Nous appelons réseau de Petri généralisé un automate suffixe de traces sur un alphabet de dépendance pour lequel la dépendance est réduite à l’égalité. Nous montrons alors que la sous-famille des réseaux de Petri généralisés satisfait également les propriétés de fermeture ci-dessus. Cela conduit notamment à 5 diverses algèbres de Boole de langages acceptés par des réseaux de Petri généralisés déterministes

Infinite automata and Mazurkiewicz traces

Nous introduisons la notion de régularité par niveaux pour des langages de traces de Mazurkiewicz et nous considérons des systèmes reconnaissables de réécriture de traces, à contextes réguliers par niveaux (RTL). Nous prouvons qu’un automate dont le graphe sous-jacent est le graphe de réécriture d’un système RTL et dont les ensembles de sommets initiaux et finaux sont réguliers par niveaux (automate RTL), est mot-automatique. En particulier, la théorie du premier ordre d’un automate RTL est décidable. Ensuite, nous prouvons que, enrichi de la relation d’accessibilité, un automate dont le graphe sous-jacent est déplié concurrent d’un graphe fini concurrent et dont les ensembles de sommets initiaux et finaux sont réguliers par niveaux, est RTL. En particulier, la théorie du premier ordre avec accessibilité d’un tel automate est décidable. Par ailleurs, il est bien connu que la théorie du premier ordre avec accessibilité du graphe de réécriture suffixe d’un système de réécriture de termes clos (graphe GTR) est décidable. Nous mettons en évidence divers dépliés concurrents de graphes finis concurrents qui ne sont pas des graphes GTR. L’arbre du quart de la grille infinie est un exemple de tel déplié. La classe des dépliés concurrents des graphes finis concurrents constitue ainsi une classe de DAG mot-automatiques, dont la théorie du premier ordre avec accessibilité est décidable et qui contient des graphes non GTR. Nous définissons pour les automates de traces (automates dont les sommets sont des traces de Mazurkiewicz) deux opérations que sont la synchronisation par niveaux et la superposition par niveaux et nous montrons que si une famille F d’automates de traces est fermée pour ces opérations, alors pour tout automate déterministe H 2 F, les langages acceptés par les automates déterministes de F qui sont longueur-réductibles en H forment une algèbre de Boole ; la longueur d’une trace étant donnée par la longueur de sa forme normale de Foata, un automate de traces G est longueur-réductible dans un automate de traces H, s’il existe un morphisme de G dans H préservant la longueur. Ensuite, nous montrons que la classe des automates suffixes de traces à contextes réguliers par niveaux, qui n’est que l’extension aux traces de Mazurkiewicz des automates suffixes de mots, satisfait ces propriétés de fermeture. Nous appelons réseau de Petri généralisé un automate suffixe de traces sur un alphabet de dépendance pour lequel la dépendance est réduite à l’égalité. Nous montrons alors que la sous-famille des réseaux de Petri généralisés satisfait également les propriétés de fermeture ci-dessus. Cela conduit notamment à 5 diverses algèbres de Boole de langages acceptés par des réseaux de Petri généralisés déterministes.We introduce the notion of level-regularity for Mazurkiewicz trace languages and we consider recognizable trace rewriting systems with level-regular contexts (RTLsystem). We prove that an automaton for which the underlying graph is the rewriting graph of a RTL system and for which the sets of initial vertices and final vertices are level-regular (RTL automaton), is word-automatic. In particular, the first-order theory of a RTL automaton is decidable. Then, we prove that, enriched with thereachability relation, an automaton for which the underlying graph is the concurrent unfolding of a finite concurrent graph, and for which the sets of initial vertices and final vertices are level-regular, is RTL. In particular, the first-order theory with the reachability predicate of such an automaton is decidable. Besides, it is known that this property also holds for ground term rewriting graphs (GTR graph). We highlight various concurrent unfoldings of finite concurrent graphs that are not GTRgraphs. The infinite quarter grid tree is such an unfolding. The class of concurrent unfoldings of finite concurrent graphs is therefore a class of word-automatic graphs for which the first-order theory with the reachability predicate is decidable and that contains some non GTR graphs. We define the operations of level-length synchronization and level-length superposition of trace automata (automata for which vertices are Mazurkiewicz traces) and we prove that if a family F of trace automata is closed under these operations, then for any deterministic trace automaton H 2 F, the languages accepted by the deterministic trace automata belonging to F and that are length-reducible to H, form a Boolean algebra; the length of a trace being the length of its Foata normal form, a trace automaton G is length-reducible to a trace automaton H if there exists a length-preserving morphism from G to H. Then, we show that the family of trace suffix automata with level regular contexts, the extension of word suffix automata to Mazurkiewicz traces, satisfies these closure properties. We define a generalized Petri net as a trace suffix automaton over a dependence alphabet for which the dependence is reduced to the equality and we show that the subfamily of generalized Petri nets also satisfies the closure properties above. In particular, this yields various Boolean algebras of word languages accepted by deterministic generalized Petri nets

Comparison of predictive controllers for locomotion and balance recovery of quadruped robots

International audienceAs locomotion decisions must be taken by considering the future, most existing quadruped controllers are based on a model predictive controller (MPC) with a reduced model of the dynamics to generate the motion, followed by a second whole-body controller to follow the movement. Yet the choice of the considered reduction in the MPC is often ad-hoc or decided by intuition. In this article, we focus on particular MPCs and analyze the effect of the reduced models on the robot behavior. Based on existing formulations, we offer additional controllers to better understand the influence of the reductions in the controller capabilities. Finally, we propose a robust predictive controller capable of optimizing the foot placements, gait period, center-of-mass trajectory and corresponding ground reaction forces. The behavior of these controllers is statistically evaluated in simulation. This empirical study is a basis for understanding the relative importance of the components of the optimal control problem (variables, costs, dynamics), that are sometimes arbitrarily emphasized or neglected. We also provide a qualitative study in simulation and on the real robot Solo

Implementation of a Reactive Walking Controller for the New Open-Hardware Quadruped Solo-12

International audienceThis paper aims at showing the dynamic performance and reliability of the low-cost, open-access quadruped robot Solo-12, which is developed within the framework of Open Dynamic Robot Initiative. It presents the implementation of a state-of-the-art control pipeline, close to the one that was previously implemented on Mini Cheetah, which implements a model predictive controller based on the centroidal dynamics to compute desired contact forces in order to track a reference velocity. Different contributions are proposed to speed up the computation process, notably at the level of the state estimation and the whole body controller. Experimental results demonstrate that the robot closely follow the reference velocity while being highly reactive and able to recover from perturbations

Real time footstep planning and control of the Solo quadruped robot in 3D environments

Quadruped robots have proved their robustness to cross complex terrain despite little environment knowledge. Yet advanced locomotion controllers are expected to take advantage of exteroceptive information. This paper presents a complete method to plan and control the locomotion of quadruped robots when 3D information about the surrounding obstacles is available, based on several stages of decision. We first propose a contact planner formulated as a mixed-integer program, optimized on-line at each new robot step. It selects a surface from a set of convex surfaces describing the environment for the next footsteps while ensuring kinematic constraints. We then propose to optimize the exact contact location and the feet trajectories at control frequency to avoid obstacles, thanks to an efficient formulation of quadratic programs optimizing Bezier curves. By relying on the locomotion controller of our quadruped robot Solo, we finally implement the complete method, provided as an open-source package. Its efficiency is asserted by statistical evaluation of the importance of each component in simulation, while the overall performances are demonstrated on various scenarios with the real robot

Improved Control Scheme for the Solo Quadruped and Experimental Comparison of Model Predictive Controllers

International audienceThis paper presents significant improvements to the nominal control scheme of the open-access Solo-12 quadruped and an experimental comparative study of different Model Predictive Controllers (MPC) that were implemented and tested on the robot. The modifications of the controller formulation that improved the nominal behavior and reduced tracking oscillations are first described. Thanks to them, the maximum reachable velocity of the robot was doubled and the foot placement was improved. On this basis three MPCs of increasing complexity were compared to evaluate the validity of different assumptions and heuristics. They range from a simplified linear centroidal model with contact points fixed by a heuristics, to a nonlinear one that also optimizes the contact points locations. Experimental results show that, thanks to the fast solver Crocoddyl and the proposed formulation, similar performances can be obtained with a MPC that optimizes both the center of mass trajectory and the foot placement, while taking into accounts the nonlinearity of the centroidal model, than with a modular scheme using a heuristic for foot placement and considering a linearized model. This paves the way for future work such as leveraging information about the environment to improve footsteps placement and timings

Real time footstep planning and control of the Solo quadruped robot in 3D environments

Quadruped robots have proved their robustness to cross complex terrain despite little environment knowledge. Yet advanced locomotion controllers are expected to take advantage of exteroceptive information. This paper presents a complete method to plan and control the locomotion of quadruped robots when 3D information about the surrounding obstacles is available, based on several stages of decision. We first propose a contact planner formulated as a mixed-integer program, optimized on-line at each new robot step. It selects a surface from a set of convex surfaces describing the environment for the next footsteps while ensuring kinematic constraints. We then propose to optimize the exact contact location and the feet trajectories at control frequency to avoid obstacles, thanks to an efficient formulation of quadratic programs optimizing Bezier curves. By relying on the locomotion controller of our quadruped robot Solo, we finally implement the complete method, provided as an open-source package. Its efficiency is asserted by statistical evaluation of the importance of each component in simulation, while the overall performances are demonstrated on various scenarios with the real robot