Efficient rigid-body dynamics algorithms are instrumental in enabling high-frequency dynamics evaluation for resource-intensive applications (e.g., model predictive control, large-scale simulation, reinforcement learning), potentially on resource-constrained hardware. Existing recursive algorithms with low computational complexity are mostly restricted to kinematic trees with external contact constraints or are sensitive to singular cases (e.g., linearly dependent constraints and kinematic singularities), severely impacting their practical usage in existing simulators. This article introduces two original, low-complexity recursive algorithms: the loop-constrained articulated body algorithm (LCABA) and proxBBO, both based on a proximal dynamics formulation for forward simulation of closed-loop mechanisms. These algorithms are derived from first principles using non-serial dynamic programming, exhibit linear complexity in practical scenarios, and are numerically robust in the face of singular cases. They extend the existing constrained articulated body algorithm (constrainedABA) to handle internal loops and the pioneering BBO algorithm from the 1980 s to singular cases. Both algorithms have been implemented by leveraging the open-source Pinocchio library, benchmarked in detail, and demonstrate state-of-the-art performance for various robot topologies, including over$6\times$speed-ups compared to existing non-recursive algorithms for high degree-of-freedom systems with internal loops, such as recent humanoid robots.

Legged robots with closed-loop kinematic chains are increasingly prevalent due to their increased mobility and efficiency. Yet, most motion generation methods rely on serial-chain approximations, sidestepping their specific constraints and dynamics. This leads to suboptimal motions and limits the adaptability of these methods to diverse kinematic structures. We propose a comprehensive motion generation method that explicitly incorporates closed-loop kinematics and their associated constraints in an optimal control problem (OCP), integrating kinematic closure conditions and their analytical derivatives. This allows the solver to leverage the non-linear transmission effects inherent to closed-chain mechanisms, reducing peak actuator efforts and expanding their effective operating range. Unlike previous methods, our framework does not require serial approximations, enabling more accurate and efficient motion strategies. We also are able to generate the motion of more complex robots for which an approximate serial chain does not exist. We validate our approach through simulations and experiments, demonstrating superior performance in complex tasks such as rapid locomotion and stair negotiation. This method enhances the capabilities of current closed-loop robots and broadens the design space for future kinematic architectures.

Trajectory optimization has been a popular choice for motion generation and control in robotics for at least a decade. Several numerical approaches have exhibited the required speed to enable online computation of trajectories for real-time of various systems, including complex robots. Many of these said are based on the differential dynamic programming (DDP) algorithm—initially designed for unconstrained trajectory optimization problems—and its variants, which are relatively easy to implement and provide good runtime performance. However, several problems in robot control call for using constrained formulations (e.g., torque limits, obstacle avoidance), from which several difficulties arise when trying to adapt DDP-type methods: numerical stability, computational efficiency, and constraint satisfaction. In this article, we leverage proximal methods for constrained optimization and introduce a DDP-type method for fast, constrained trajectory optimization suited for model-predictive control (MPC) applications with easy warm-starting. Compared to earlier solvers, our approach effectively manages hard constraints without warm-start limitations and exhibits good convergence behavior. We provide a complete implementation as part of an open-source and flexible C++ trajectory optimization library calledaligator. These algorithmic contributions are validated through several trajectory planning scenarios from the robotics literature and the real-time whole-body MPC of a quadruped robot.

The promise of model-predictive control in robotics has led to extensive development of efficient numerical optimal control solvers in line with differential dynamic programming because it exploits the sparsity induced by time. In this work, we argue that this effervescence has hidden the fact that sparsity can be equally exploited by standard nonlinear optimization. In particular, we show how a tailored implementation of sequential quadratic programming achieves state-of-the-art model-predictive control. Then, we clarify the connections between popular algorithms from the robotics community and well-established optimization techniques. Further, the sequential quadratic program formulation naturally encompasses the constrained case, a notoriously difficult problem in the robotics community. Specifically, we show that it only requires a sparsity-exploiting implementation of a state-of-the-art quadratic programming solver. We illustrate the validity of this approach in a comparative study and experiments on a torque-controlled manipulator. To the best of our knowledge, this is the first demonstration of closed loop nonlinear model-predictive control with constraints on a real robot.

Rigid-body dynamics algorithms have played an essential role in robotics development. By finely exploiting the underlying robot structure, they allow the computation of the robot kinematics, dynamics, and related physical quantities with low complexity, enabling their integration into chipsets with limited resources or their evaluation at very high frequency for demanding applications (e.g., model predictive control, large-scale simulation, reinforcement learning, etc.). While most of these algorithms operate on constraint-free settings, only a few have been proposed so far to adequately account for constrained dynamical systems while depicting low algorithmic complexity. In this article, we introduce a series of new algorithms with reduced (and lowest) complexity for the forward simulation of constrained dynamical systems. Notably, we revisit the so-called articulated body algorithm (ABA) and the Popov–Vereshchagin algorithm (PV) in the light of proximal-point optimization and introduce two new algorithms, called constrained ABA and proxPV. These two new algorithms depict linear complexities while being robust to singular cases (e.g., redundant constraints, singular constraints, etc.). We establish the connection with existing literature formulations, especially the relaxed formulation at the heart of the MuJoCo and Drake simulators. We also propose an efficient and new algorithm to compute the damped Delassus inverse matrix with the lowest known computational complexity. All these algorithms have been implemented inside the open-source framework Pinocchio and depict, on a wide range of robotic systems ranging from robot manipulators to complex humanoid robots, state-of-the-art performances compared to alternative solutions of the literature.

Optimization layers within neural network architectures have become increasingly popular for their ability to solve a wide range of machine learning tasks and to model domain-specific knowledge. However, designing optimization layers requires careful consideration as the underlying optimization problems might be infeasible during training. Motivated by applications in learning, control and robotics, this work focuses on convex quadratic programming (QP) layers. The specific structure of this type of optimization layer can be efficiently exploited for faster computations while still allowing rich modeling capabilities. We leverage primal-dual augmented Lagrangian techniques for computing derivatives of both feasible and infeasible QP solutions. More precisely, we propose a unified approach which tackles the differentiability of the closest feasible QP solutions in a classical $\ell_2$ sense. We then harness this approach to enrich the expressive capabilities of existing QP layers. More precisely, we show how differentiating through infeasible QPs during training enables to drive towards feasibility at test time a new range of QP layers. These layers notably demonstrate superior predictive performance in some conventional learning tasks. Additionally, we present alternative formulations that enhance numerical robustness, speed, and accuracy for training such layers. Along with these contributions, we provide an open-source C++ software package called QPLayer for differentiating feasible and infeasible convex QPs and which can be interfaced with modern learning frameworks.

Nonlinear model-predictive control has recently shown its practicability in robotics. However it remains limited in contact interaction tasks due to its inability to leverage sensed efforts. In this work, we propose a novel model-predictive control approach that incorporates direct feedback from force sensors while circumventing explicit modeling of the contact force evolution. Our approach is based on the online estimation of the discrepancy between the force predicted by the dynamics model and force measurements, combined with high-frequency nonlinear model-predictive control. We report an experimental validation on a torque-controlled manipulator in challenging tasks for which accurate force tracking is necessary. We show that a simple reformulation of the optimal control problem combined with standard estimation tools enables to achieve state-of-the-art performance in force control while preserving the benefits of model-predictive control, thereby outperforming traditional force control techniques. This work paves the way toward a more systematic integration of force sensors in model predictive control.

Designing robust algorithms in the face of estimation uncertainty is a challenging task. Indeed, controllers seldom consider estimation uncertainty and only rely on the most likely estimated state. Consequently, sudden changes in the environment or the robot’s dynamics can lead to catastrophic behaviors. Leveraging recent results in risk-sensitive optimal control, this paper presents a risk-sensitive Extended Kalman Filter that can adapt its estimation to the control objective, hence allowing safe output-feedback Model Predictive Control (MPC). By taking a pessimistic estimate of the value function resulting from the MPC controller, the filter provides increased robustness to the controller in phases of uncertainty as compared to a standard Extended Kalman Filter (EKF). The filter has the same computational complexity as an EKF and can be used for real-time control. The paper evaluates the risk-sensitive behavior of the proposed filter when used in a nonlinear MPC loop on a planar drone and industrial manipulator in simulation, as well as on an external force estimation task on a real quadruped robot. These experiments demonstrate the ability of the approach to significantly improve performance in face of uncertainties.

Recent progress in reinforcement learning (RL) in robotics has been obtained by training control policy directly in simulation. Particularly in the context of quadrupedal locomotion, astonishing locomotion policies depicting high robustness against environmental perturbations have been trained by leveraging RaiSim simulator. While it avoids introducing forces at distance, it has been shown recently that RaiSim does not obey the maximum dissipation principle, a fundamental principle when simulating rigid contact interactions. In this note, we detail these relaxations and propose an algorithmic correction of the RaiSim contact algorithm to handle the maximum dissipation principle adequately. Our experiments empirically demonstrate our approach leads to simulation following this fundamental principle.

Physics simulation is ubiquitous in robotics. Whether in model-based approaches (e.g., trajectory optimization), or model-free algorithms (e.g., reinforcement learning), physics simulators are a central component of modern control pipelines in robotics. Over the past decades, several robotic simulators have been developed, each with dedicated contact modeling assumptions and algorithmic solutions. In this article, we survey the main contact models and the associated numerical methods commonly used in robotics for simulating advanced robot motions involving contact interactions. In particular, we recall the physical laws underlying contacts and friction (i.e., Signorini condition, Coulomb's law, and the maximum dissipation principle), and how they are transcribed in current simulators. For each physics engine, we expose their inherent physical relaxations along with their limitations due to the numerical techniques employed. Based on our study, we propose theoretically grounded quantitative criteria on which we build benchmarks assessing both the physical and computational aspects of simulation. We support our work with an open-source and efficient C++ implementation of the existing algorithmic variations. Our results demonstrate that some approximations or algorithms commonly used in robotics can severely widen the reality gap and impact target applications. We hope this work will help motivate the development of new contact models, contact solvers, and robotic simulators in general, at the root of recent progress in motion generation in robotics.

Collision detection is a fundamental problem in various domains, such as robotics, computational physics, and computer graphics. In general, collision detection is tackled as a computational geometry problem, with the so-called Gilbert, Johnson, and Keerthi (GJK) algorithm being the most adopted solution nowadays. While introduced in 1988, GJK remains the most effective solution to compute the distance or the collision between two 3D convex geometries. Over the years, it was shown to be efficient, scalable, and generic, operating on a broad class of convex shapes, ranging from simple primitives (sphere, ellipsoid, box, cone, capsule, etc.) to complex meshes involving thousands of vertices. In this article, we introduce several contributions to accelerate collision detection and distance computation between convex geometries by leveraging the fact that these two problems are fundamentally optimization problems. Notably, we establish that the GJK algorithm is a specific sub-case of the well-established Frank-Wolfe (FW) algorithm in convex optimization. By adapting recent works linking Polyak and Nesterov accelerations to Frank-Wolfe methods, we also propose two accelerated extensions of the classic GJK algorithm. Through an extensive benchmark over millions of collision pairs involving objects of daily life, we show that these two accelerated GJK extensions significantly reduce the overall computational burden of collision detection, leading to computation times that are up to two times faster. Finally, we hope this work will significantly reduce the computational cost of modern robotic simulators, allowing the speed-up of modern robotic applications that heavily rely on simulation, such as reinforcement learning or trajectory optimization.

Reinforcement learning (RL) and trajectory opti-mization (TO) present strong complementary advantages. On one hand, RL approaches are able to learn global control policies directly from data, but generally require large sample sizes to properly converge towards feasible policies. On the other hand, TO methods are able to exploit gradient-based information extracted from simulators to quickly converge towards a locally optimal control trajectory which is only valid within the vicinity of the solution. Over the past decade, several approaches have aimed to adequately combine the two classes of methods in order to obtain the best of both worlds. Following on from this line of research, we propose several improvements on top of these approaches to learn global control policies quicker, notably by leveraging sensitivity information stemming from TO methods via Sobolev learning, and Augmented Lagrangian (AL) techniques to enforce the consensus between TO and policy learning. We evaluate the benefits of these improvements on various classical tasks in robotics through comparison with existing approaches in the literature.

Collision detection is an important component of many robotics applications, from robot control to simulation, including motion planning and estimation. While the seminal works on the topic date back to the 80s, it is only recently that the question of properly differentiating collision detection has emerged as a central issue, thanks notably to the ongoing and various efforts made by the scientific community around the topic of differentiable physics. Yet, very few solutions have been suggested so far, and only with a strong assumption on the nature of the shapes involved. In this work, we introduce a generic and efficient approach to compute the derivatives of collision detection for any pair of convex shapes, by notably leveraging randomized smoothing techniques which have shown to be particularly adapted to capture the derivatives of non-smooth problems. This approach is implemented in the HPP-FCL and Pinocchio ecosystems, and evaluated on classic datasets and problems of the robotics literature, demonstrating few micro-second timings to compute informative derivatives directly exploitable by many real robotic applications, including differentiable simulation.

This work aims at leveraging instructional video to guide the solving of complex multi-contact task-and-motion planning tasks in robotics. Towards this goal, we propose an extension of the well-established Rapidly-Exploring Random Tree (RRT) planner, which simultaneously grows multiple trees around grasp and release states extracted from the guiding video. Our key novelty lies in combining contact states, and 3D object poses extracted from the guiding video with a traditional planning algorithm that allows us to solve tasks with sequential dependencies, for example, if an object needs to be placed at a specific location to be grasped later. To demonstrate the benefits of the proposed video-guided planning approach, we design a new benchmark with three challenging tasks: (i) 3D re-arrangement of multiple objects between a table and a shelf, (ii) multi-contact transfer of an object through a tunnel, and (iii) transferring objects using a tray in a similar way a waiter transfers dishes. We demonstrate the effectiveness of our planning algorithm on several robots, including the Franka Emika Panda and the KUKA KMR iiwa.

Trajectory optimization methods have become ubiquitous for the motion planning and control of underactuated robots for e.g., quadrupeds, humanoids etc. While they have been extensively used in the case of serial or tree type robots, they are seldomly used for planning and control of robots with closed loops. Series-parallel hybrid topology is quite commonly used in the design of humanoid robots, but they are often neglected during trajectory optimization and the movements are computed for a serial abstraction of the system and then the solution is mapped to the actuator coordinates. As a consequence, the full capability of the robot cannot be exploited. This paper presents a case study of trajectory optimization for series-parallel hybrid robot by taking into account all the holonomic constraints imposed by the closed kinematic loops present in the system. We demonstrate the advantages of this consideration with a weightlifting task on RH5 Manus humanoid in both simulation and experiments.

Over the past decade, the Differential Dynamic Programming (DDP) method has gained in maturity and popularity within the robotics community. Several recent contributions have led to the integration of constraints within the original DDP formulation, hence enlarging its domain of application while making it a strong and easy-to-implement competitor against alternative methods of the state of the art such as collocation or multiple-shooting approaches. Yet, and similarly to its competitors, DDP remains unable to cope with high-dimensional dynamics within a receding horizon fashion, such as in the case of online generation of athletic motion on humanoid robots. In this paper, we propose to make a step towards this objective by reformulating classical DDP as an implicit optimal control problem, allowing the use of more advanced integration schemes such as implicit or variational integrators. To that end, we introduce a primal-dual proximal Lagrangian approach capable of handling dynamical and path constraints in a unified manner, while taking advantage of the time sparsity inherent to optimal control problems. We show that this reformulation enables us to relax the dynamics along the optimization process by solving it inexactly: far from the optimality conditions, the dynamics are only partially fulfilled, but continuously enforced as the solver gets closer to the local optimal solution. This inexactness enables our approach to robustly handle large time steps (100 ms or more), unlike other DDP solvers of the state of the art, as experimentally validated through different robotic scenarii.

Trajectory optimization is an efficient approach for solving optimal control problems for complex robotic systems. It relies on two key components: first the transcription into a sparse nonlinear program, and second the corresponding solver to iteratively compute its solution. On one hand, differential dynamic programming (DDP) provides an efficient approach to transcribe the optimal control problem into a finite-dimensional problem while optimally exploiting the sparsity induced by time. On the other hand, augmented Lagrangian methods make it possible to formulate efficient algorithms with advanced constraint-satisfaction strategies. In this paper, we propose to combine these two approaches into an efficient optimal control algorithm accepting both equality and inequality constraints. Based on the augmented Lagrangian literature, we first derive a generic primal-dual augmented Lagrangian strategy for nonlinear problems with equality and inequality constraints. We then apply it to the dynamic programming principle to solve the value-greedy optimization problems inherent to the backward pass of DDP, which we combine with a dedicated globalization strategy, resulting in a Newton-like algorithm for solving constrained trajectory optimization problems. Contrary to previous attempts of formu-lating an augmented Lagrangian version of DDP, our approach exhibits adequate convergence properties without any switch in strategies. We empirically demonstrate its interest with several case-studies from the robotics literature.

We introduce RoboPose, a method to estimate the joint angles and the 6D camera-to-robot pose of a known articulated robot from a single RGB image. This is an important problem to grant mobile and itinerant autonomous systems the ability to interact with other robots using only visual information in non-instrumented environments, especially in the context of collaborative robotics. It is also challenging because robots have many degrees of freedom and an infinite space of possible configurations that often result in self-occlusions and depth ambiguities when imaged by a single camera. The contributions of this work are three-fold. First, we introduce a new render & compare approach for estimating the 6D pose and joint angles of an articulated robot that can be trained from synthetic data, generalizes to new unseen robot configurations at test time, and can be applied to a variety of robots. Second, we experimentally demonstrate the importance of the robot parametrization for the iterative pose updates and design a parametrization strategy that is independent of the robot structure. Finally, we show experimental results on existing benchmark datasets for four different robots and demonstrate that our method significantly outperforms the state of the art. Code and pre-trained models are available on the project webpage [1].

Estimating the centroidal dynamics of legged robots is crucial in the context of multi-contact locomotion of legged robots. In this paper, we formulate the estimation of centroidal dynamics as a maximum a posteriori problem and we use a differential dynamic programming approach for solving it. The soundness of the proposed approach is first validated on a simulated humanoid robot, where ground truth data is available, enabling error analysis, and then compared to other alternatives of the state of the art, namely an extend Kalman filter and a recursive complementary filter. The results demonstrate that, compared to other approaches, the proposed method reduces the estimation error on the centroidal state in addition to ensuring the dynamics consistency of the state trajectory. Finally, the effectiveness of the proposed method is illustrated on real measurements, obtained from walking experiments with the HRP-2 humanoid robot.

Trajectory optimization is an important tool in task-based robot motion planning, due to its generality and convergence guarantees under some mild conditions. It is often used as a post-processing operation to smooth out trajectories that are generated by probabilistic methods or to directly control the robot motion. Unconstrained trajectory optimization problems have been well studied, and are commonly solved using Differential Dynamic Programming methods that allow for fast convergence at a relatively low computational cost. In this paper, we propose an augmented Lagrangian approach that extends these ideas to equality-constrained trajectory optimization problems, while maintaining a balance between convergence speed and numerical stability. We illustrate our contributions on various standard robotic problems and highlights their benefits compared to standard approaches.

Self-collision detection and avoidance are essential for reactive control, in particular for dynamics robots equipped with legs or arms. Yet, only few control methods are able to handle such constraints, and it is often necessary to rely on path planning to define a collision-free trajectory that the controller would then track. In this paper, we introduce a combination of two lightweight, conservative and smooth models to generically handle self-collisions in robot control. For pairs of bodies that are far from one another on average (e.g. segments of distinct legs), we rely on a standard forward kinematics approach, using simplified geometries for which we provide analytical derivatives. For bodies that are moving close to one another, we propose to use a data-driven approach, with datasets generated thanks to a standard collision library. We then build a simple torque-based controller that can be implemented on top of any control law to prevent unexpected self-collision. This controller is meant to be implemented as a low-level protection, directly on the robot hardware. We also provide an open-source library to generate ANSI-C code for any robot model, experimented on the real quadruped Solo.

Identifying an effective model of a dynamical system from sensory data and using it for future state prediction and control is challenging. Recent data-driven algorithms based on Koopman theory are a promising approach to this problem, but they typically never update the model once it has been identified from a relatively small set of observation, thus making long-term prediction and control difficult for realistic systems, in robotics or fluid mechanics for example. This paper introduces a novel method for learning an embedding of the state space with linear dynamics from sensory data. Unlike previous approaches, the dynamics model can be updated online and thus easily applied to systems with non-linear dynamics in the original configuration space. The proposed approach is evaluated empirically on several classical dynamical systems and sensory modalities, with good performance on long-term prediction and control.

Reasoning about 3D scenes from their 2D image projections is one of the core problems in computer vision. Solutions to this inverse and ill-posed problem typically involve a search for models that best explain observed image data. Notably, images depend both on the properties of observed scenes and on the process of image formation. Hence, if optimization techniques should be used to explain images, it is crucial to design differentable functions for the projection of 3D scenes into images, also known as differentiable rendering. Previous approaches to differentiable rendering typically replace non-differentiable operations by smooth approximations, impacting the subsequent 3D estimation. In this paper, we take a more general approach and study differentiable renderers through the prism of randomized optimization and the related notion of perturbed optimizers. In particular, our work highlights the link between some well-known differentiable renderer formulations and randomly smoothed optimizers, and introduces differentiable perturbed renderers. We also propose a variance reduction mechanism to alleviate the computational burden inherent to perturbed optimizers and introduce an adaptive scheme to automatically adjust the smoothing parameters of the rendering process. We apply our method to 3D scene reconstruction and demonstrate its advantages on the tasks of 6D pose estimation and 3D mesh reconstruction. By providing informative gradients that can be used as a strong supervisory signal, we demonstrate the benefits of perturbed renderers to obtain more accurate solutions when compared to the state-of-the-art alternatives using smooth gradient approximations.

We introduce Crocoddyl (Contact RObot COntrol by Differential DYnamic Library), an open-source framework tailored for efficient multi-contact optimal control. Crocoddyl efficiently computes the state trajectory and the control policy for a given predefined sequence of contacts. Its efficiency is due to the use of sparse analytical derivatives, exploitation of the problem structure, and data sharing. It employs differential geometry to properly describe the state of any geometrical system, e.g. floating-base systems. Additionally, we propose a novel optimal control algorithm called Feasibility-driven Differential Dynamic Programming (FDDP). Our method does not add extra decision variables which often increases the computation time per iteration due to factorization. FDDP shows a greater globalization strategy compared to classical Differential Dynamic Programming (DDP) algorithms. Concretely, we propose two modifications to the classical DDP algorithm. First, the backward pass accepts infeasible state-control trajectories. Second, the rollout keeps the gaps open during the early "exploratory" iterations (as expected in multipleshooting methods with only equality constraints). We showcase the performance of our framework using different tasks. With our method, we can compute highly-dynamic maneuvers (e.g. jumping, front-flip) within few milliseconds.

Synthesizing legged locomotion requires planning one or several steps ahead (literally): when and where, and with which effector should the next contact(s) be created between the robot and the environment? Validating a contact candidate implies a minima the resolution of a slow, nonlinear optimization problem, to demonstrate that a center of mass (CoM) trajectory, compatible with the contact transition constraints, exists. We propose a conservative reformulation of this trajectory generation problem as a convex 3-D linear program, named convex resolution of centroidal dynamic trajectories (CROC). It results from the observation that if the CoM trajectory is a polynomial with only one free variable coefficient, the nonlinearity of the problem disappears. This has two consequences. On the positive side, in terms of computation times, CROC outperforms the state of the art by at least one order of magnitude, and allows to consider interactive applications (with a planning time roughly equal to the motion time). On the negative side, in our experiments, our approach finds a majority of the feasible trajectories found by a nonlinear solver, but not all of them. Still, we demonstrate that the solution space covered by CROC is large enough to achieve the automated planning of a large variety of locomotion tasks for different robots, demonstrated in simulation and on the real HRP-2 robot, several of which were rarely seen before. Another significant contribution is the introduction of a Bezier curve representation of the problem, which guarantees that the constraints of the CoM trajectory are verified continuously, and not only at discrete points as traditionally done. This formulation is lossless, and results in more robust trajectories. It is not restricted to CROC, but could rather be integrated with any method from the state of the art.

In this paper, we introduce a method to automatically reconstruct the 3D motion of a person interacting with an object from a single RGB video. Our method estimates the 3D poses of the person and the object, contact positions, and forces and torques actuated by the human limbs. The main contributions of this work are three-fold. First, we introduce an approach to jointly estimate the motion and the actuation forces of the person on the manipulated object by modeling contacts and the dynamics of their interactions. This is cast as a large-scale trajectory optimization problem. Second, we develop a method to automatically recognize from the input video the position and timing of contacts between the person and the object or the ground, thereby significantly simplifying the complexity of the optimization. Third, we validate our approach on a recent MoCap dataset with ground truth contact forces and demonstrate its performance on a new dataset of Internet videos showing people manipulating a variety of tools in unconstrained environments.

It is nowadays well-established that locomotion can be written as a large and complex optimal control problem. Yet, current knowledge in numerical solver fails to directly solve it. A common approach is to cut the dimensionality by relying on reduced models (inverted pendulum, capture points, centroidal). However it is difficult both to account for whole-body constraints at the reduced level and also to define what is an acceptable trade-off at the whole-body level between tracking the reduced solution or searching for a new one. The main contribution of this paper is to introduce a rigorous mathematical framework based on the Alternating Direction Method of Multipliers, to enforce the consensus between the centroidal state dynamics at reduced and whole-body level. We propose an exact splitting of the whole-body optimal control problem between the centroidal dynamics (under-actuation) and the manipulator dynamics (full actuation), corresponding to a re-arrangement of the equations already stated in previous works. We then describe with details how alternating descent is a good solution to implement an effective locomotion solver. We validate this approach in simulation with walking experiments on the HRP-2 robot.

Locomotion of legged robots on arbitrary terrain using multiple contacts is yet an open problem. To tackle it, a common approach is to rely on reduced template models (e.g., the linear inverted pendulum). However, most of existing template models are based on some restrictive hypotheses that limit their range of applications. Moreover, reduced models are generally not able to cope with the constraints of the robot complete model, such as the kinematic limits. In this paper, we propose a complete solution relying on a generic template model, based on the centroidal dynamics, able to quickly compute multicontact locomotion trajectories for any legged robot on arbitrary terrains. The template model relies on exact dynamics and is thus not limited by arbitrary assumption. We also propose a generic procedure to handle feasibility constraints due to the robot's whole body as occupancy measures, and a systematic way to approximate them using offline learning in simulation. An efficient solver is finally obtained by introducing an original second-order approximation of the centroidal wrench cone. The effectiveness and the versatility of the approach are demonstrated in several multicontact scenarios with two humanoid robots both in reality and in simulation.

We present an optimization framework for the design and analysis of underactuated biped walkers, characterized by passive or actuated joints with rigid or non-negligible elastic actuation/transmission elements. The framework is based on optimal control, dealing with geometric constraints and various dynamic objective functions, as well as boundary conditions, which helps in selecting optimal values both for the actuation and the transmission parameters. Solutions of the formulated problems are shown for different kinds of bipedal architectures, and comparisons drawn between traditional rigid robots and compliant ones show the energy-efficiency of compliant actuators in the context of locomotion.

This paper presents a generic and efficient approach to generate dynamically consistent motions for under-actuated systems like humanoid or quadruped robots. The main contribution is a walking pattern generator, able to compute a stable trajectory of the center of mass of the robot along with the angular momentum, for any given configuration of contacts (e.g. on uneven, sloppy or slippery terrain, or with closed-gripper). Unlike existing methods, our solver is fast enough to be applied as a model-predictive controller. We then integrate this pattern generator in a complete framework: an acyclic contact planner is first used to automatically compute the contact sequence from a 3D model of the environment and a desired final posture; a stable walking pattern is then computed by the proposed solver; a dynamically-stable whole-body trajectory is finally obtained using a second-order hierarchical inverse kinematics. The implementation of the whole pipeline is fast enough to plan a step while the previous one is executed. The interest of the method is demonstrated by real experiments on the HRP-2 robot, by performing long-step walking and climbing a staircase with handrail support.

This paper discusses the problem of estimating the position of the center of mass for a polyarticulated system (e.g., a humanoid robot or a human body), which makes contact with its environment. The only sensors providing measurements on this point are either interaction force sensors or kinematic reconstruction applied to a dynamic model of the system. We first study the observability of the center-of-mass position using these sensors and we show that the accuracy domain of each measurement can be easily described through a spectral analysis. We finally introduce an original approach based on the theory of complementary filtering to efficiently merge these input measurements and obtain an estimation of the center-of-mass position. This approach is extensively validated in simulations by using a model of a humanoid robot through which we confirm the spectral analysis of the signal errors and show that the complementary filter offers a lower average reconstruction error than the classical Kalman filter. Some experimental applications of this filter on real signals are also presented.