Quang-Cuong Pham

A Contraction Theory Approach to Stochastic Incremental Stability

We investigate the incremental stability properties of Ito stochastic dynamical systems. Specifically, we derive a stochastic version of nonlinear contraction theory that provides a bound on the mean square distance between any two trajectories of a stochastically contracting system. This bound can be expressed as a function of the noise intensity and the contraction rate of the noise-free system. We illustrate these results in the contexts of nonlinear observers design and stochastic synchronization.

The formation of trajectories during goal-oriented locomotion in humans. I. A stereotyped behaviour

Human locomotion was investigated in a goal‐oriented task where subjects had to walk to and through a doorway starting from a fixed position and orientation in space. The door was located at different positions and orientations in space, resulting in a total of 40 targets. While no specific constraint was provided to subjects in terms of the path they were to follow or the expected walking speeds, all of them generated very similar trajectories in terms of both path geometry and velocity profiles. These results are reminiscent of the stereotyped properties of the hand trajectories observed in arm reaching movements in studies over the last 20 years. This observation supports the hypothesis that common constraining mechanisms govern the generation of segmental and whole‐body trajectories. In contrast, we observed that the subjects placed their feet at different spatial positions across repetitions, making unlikely the hypothesis that goal‐oriented locomotion is planned as a succession of steps. Rather, our results suggest that common planning and/or control strategies underlie the formation of the whole locomotor trajectory during a spatially oriented task.

The formation of trajectories during goal‐oriented locomotion in humans. II. A maximum smoothness model

Despite the theoretically infinite number of possible trajectories a human may take to reach a distant doorway, we observed that locomotor trajectories corresponding to this task were actually stereotyped, both at the geometric and the kinematic levels. In this paper, we propose a computational model for the formation of human locomotor trajectories. Our model is adapted from smoothness maximization models that have been studied in the context of hand trajectory generation. The trajectories predicted by our model are very similar to the experimentally recorded ones. We discuss the theoretical implications of this result in the context of movement planning and control in humans. In particular, this result supports the hypothesis that common principles, such as smoothness maximization, may govern the generation of very different types of movements (in this case, hand movements and whole‐body movements).

2008 Special Issue: Where neuroscience and dynamic system theory meet autonomous robotics: A contracting basal ganglia model for action selection

Action selection, the problem of choosing what to do next, is central to any autonomous agent architecture. We use here a multi-disciplinary approach at the convergence of neuroscience, dynamical system theory and autonomous robotics, in order to propose an efficient action selection mechanism based on a new model of the basal ganglia. We first describe new developments of contraction theory regarding locally projected dynamical systems. We exploit these results to design a stable computational model of the cortico-baso-thalamo-cortical loops. Based on recent anatomical data, we include usually neglected neural projections, which participate in performing accurate selection. Finally, the efficiency of this model as an autonomous robot action selection mechanism is assessed in a standard survival task. The model exhibits valuable dithering avoidance and energy-saving properties, when compared with a simple if-then-else decision rule.

A General, Fast, and Robust Implementation of the Time-Optimal Path Parameterization Algorithm

Finding the time-optimal parameterization of a given path subject to kinodynamic constraints is an essential component in many robotic theories and applications. The objective of this paper is to provide a general, fast, and robust implementation of this component. For this, we give a complete solution to the issue of dynamic singularities, which are the main cause of failure in existing implementations. We then present an open-source implementation of the algorithm in C++/Python and demonstrate its robustness and speed in various robotics settings.

Leveraging Cone Double Description for Multi-contact Stability of Humanoids with Applications to Statics and Dynamics

We build on previous works advocating the use of the Gravito-Inertial Wrench Cone (GIWC) as a general contact stability criterion (a “ZMP for non-coplanar contacts”). We show how to compute this wrench cone from the friction cones of contact forces by using an intermediate representation, the surface contact wrench cone, which is the minimal representation of contact stability for each surface contact. The observation that the GIWC needs to be computed only once per stance leads to particularly efficient algorithms, as we illustrate in two important problems for humanoids: “testing robust static equilibrium” and “time-optimal path parameterization”. We show, through theoretical analysis and in physics simulations, that our method is more general and/or outperforms existing ones.

Time-Optimal Path Parameterization for Redundantly Actuated Robots: A Numerical Integration Approach

Time-optimal path parameterization (TOPP) under actuation bounds plays a fundamental role in many robotic theories and applications. This algorithm was first developed and perfected for classical serial robotic manipulators whose actuation is nonredundant. Yet, redundantly actuated systems, such as parallel manipulators or humanoid robots in multicontact tasks, are increasingly common in all fields of robotics. Here, we extend the classical algorithm of TOPP (a.k.a. numerical integration approach) to the case of redundantly actuated systems. As illustration, we present an application to multicontact trajectory planning for a humanoid robot.

Stability of surface contacts for humanoid robots: Closed-form formulae of the Contact Wrench Cone for rectangular support areas

Humanoids locomote by making and breaking contacts with their environment. Thus, a crucial question for them is to anticipate whether a contact will hold or break under effort. For rigid surface contacts, existing methods usually consider several point-contact forces, which has some drawbacks due to the underlying redundancy. We derive a criterion, the Contact Wrench Cone (CWC), which is equivalent to any number of applied forces on the contact surface, and for which we provide a closed-form formula. It turns out that the CWC can be decomposed into three conditions: (i) Coulomb friction on the resultant force, (ii) CoP inside the support area, and (iii) upper and lower bounds on the yaw torque. While the first two are well-known, the third one is novel. It can, for instance, be used to prevent the undesired foot yaws observed in biped locomotion. We show that our formula yields simpler and faster computations than existing approaches for humanoid motions in single support, and assess its validity in the OpenHRP simulator.

Invariance of locomotor trajectories across visual and gait direction conditions

We studied the influence of vision (walking with or without vision) and of gait direction (walking forward or backward) on goal-oriented locomotion in humans. Subjects had to walk, in a free environment, from a given position and orientation towards a distant arrow which constrained their final position and orientation. We found that the average trajectories were mostly similar across the tested conditions, which suggests that locomotor trajectories are generated at a high cognitive level and, to some extent, independently of the detailed sensory and motor implementation levels. The variability profiles around the average trajectories were similar across the gait direction conditions but differed greatly across the visual conditions, indicating the existence of motor-independent and vision-dependent control mechanisms. Taken together, our observations argue further in favour of a top-down implementation of goal-oriented locomotion, where the control of locomotion is specified at the level of whole-body trajectories and then implemented through specific motor strategies.

Time-optimal Path Parameterization for critically dynamic motions of humanoid robots

Planning collision-free, dynamically-balanced movements for humanoid robots is a challenging problem. An effective approach consists of first planning a motion satisfying geometric and kinematic constraints (such as collision avoidance, joint angle limits, velocity limits, etc.) and, in a second stage, modifying this motion so that it respects dynamic balance criteria, such as those relative to the Zero Moment Point (ZMP). However, this approach currently suffers from the issue that the modified motion may give rise to new collisions with respect to the original motion, which can be very costly to deal with, especially for systems with many degrees of freedom and cluttered environments. Here we present an algorithm to modify the motions of humanoid robots under ZMP constraints without changing the original motion path, making thereby new collision checks unnecessary. We do so by adapting the minimum-time path parameterization under torque constraints algorithm of Bobrow et al. to the case of ZMP constraints. In contrast with a previous approach based on finite differences and iterative optimization to find the optimal path parameterization under ZMP constraints, our Bobrow-based algorithm finds this optimal parameterization in a single pass. We demonstrate the efficiency of this algorithm by simulations.