Nicolas Mansard

Acting in cluttered environments requires predicting and avoiding collisions while still achieving precise control. Conventional optimization-based controllers can enforce physical constraints, but they struggle to produce feasible solutions quickly when many obstacles are present. Diffusion models can generate diverse trajectories around obstacles, yet prior approaches lacked a general and efficient way to condition them on scene structure. In this paper, we show that combining diffusion-based warm-starting conditioned with a latent object-centric representation of the scene and with a collision-aware model predictive controller (MPC) yields reliable and efficient motion generation under strict time limits. Our approach conditions a diffusion transformer on the system state, task, and surroundings, using an object-centric slot attention mechanism to provide a compact obstacle representation suitable for control. The sampled trajectories are refined by an optimal control problem that enforces rigid-body dynamics and signed-distance collision constraints, producing feasible motions in real time. On benchmark tasks, this hybrid method achieved markedly higher success rates and lower latency than sampling-based planners or either component alone. Real-robot experiments with a torque-controlled Panda confirm reliable and safe execution with MPC. An open-source implementation is provided here.

Path Integral methods have demonstrated remarkable capabilities for solving non-linear stochastic optimal control problems through sampling-based optimization. However, their computational complexity grows linearly with the prediction horizon, limiting long-term reasoning, while constraints are merely enforced through handcrafted penalties. In this work, we propose a unified and efficient framework for enabling long-horizon reasoning and constraint enforcement within Model Predictive Path Integral (MPPI) control. First, we introduce a practical method to incorporate a terminal value function, learned offline via temporal-difference learning, to approximate the long-term cost-to-go. This allows for significantly shorter roll-outs while enabling infinite-horizon reasoning, thereby improving computational efficiency and motion performance. Second, we propose a discount modulation strategy that adjusts the return of sampled trajectories based on constraint violations. This provides a more interpretable and effective mechanism for enforcing constraints compared to traditional cost shaping. Our formulation retains the flexibility and sampling efficiency of MPPI while supporting structured integration of long-term objectives and constraint handling. We validate our approach on both simulated and real-world robotic locomotion tasks, demonstrating improved performance, constraint-awareness, and generalization under reduced computational budgets.

The promise of model-predictive control (MPC) 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 (QP) achieves state-of-the-art MPC. 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 QP 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 MPC with constraints on a real 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.

Model Predictive Control has emerged as a popular tool for robots to generate complex motions. However, the real-time requirement has limited the use of hard constraints and large preview horizons, which are necessary to ensure safety and stability. In practice, practitioners have to carefully design cost functions that can imitate an infinite horizon formulation, which is tedious and often results in local minima. In this work, we study how to approximate the infinite horizon value function of constrained optimal control problems with neural networks using value iteration and trajectory optimization. Furthermore, we experimentally demonstrate how using this value function approximation as a terminal cost provides global stability to the model predictive controller. The approach is validated on two toy problems and a real-world scenario with online obstacle avoidance on an industrial manipulator where the value function is conditioned to the goal and obstacle.

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 called ALIGATOR. 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.

Full-dynamics model predictive control (MPC) has recently been applied to quadrupedal locomotion in semi-unstructured environments. These advances have been fueled by the availability of efficient trajectory optimization (TO) algorithms and inexpensive computational power. The main advantages of full-dynamics MPC are (i) enabling complex locomotion manoeuvres, (ii) considering actuation limits, and (iii) improving robot stability. However, to make the TO problem sufficiently simple to be solved at run time, reference swing foot trajectories are usually tracked in the MPC formulation. These trajectories are often computed independently of the motion of the joints, limiting the approach generality and capability. To address this limitation, we present a full-dynamics MPC formulation that does not require reference swing-foot trajectories, featuring a novel cost function targeting swing foot motion and considering environmental information. Removing the need for reference swing foot trajectories, our approach can also automatically adjust footstep locations, as long as the contact surfaces are predefined. We have validated our MPC formulation through simulations and experiments on the ANYmal B robot. Our approach has similar computational efficiency to state-of-the-art formulations, while displaying superior push-recovery capabilities on various terrains.

In a world designed for legs, quadrupeds, bipeds, and humanoids have the opportunity to impact emerging robotics applications from logistics, to agriculture, to home assistance. The goal of this survey is to cover the recent progress toward these applications that have been driven by model-based optimization for the real-time generation and control of movement. The majority of the research community has converged on the idea of generating locomotion control laws by solving an optimal control problem (OCP) in either a model-based or data-driven manner. However, solving the most general of these problems online remains intractable due to complexities from intermittent unidirectional contacts with the environment, and from the many degrees of freedom of legged robots. This survey covers methods that have been pursued to make these OCPs computationally tractable, with a specific focus on how environmental contacts are treated, how the model can be simplified, and how these choices affect the numerical solution methods employed. The survey focuses on model-based optimization while paving its way for broader combination with learning-based formulations to accelerate progress in this growing field.

In a world designed for legs, quadrupeds, bipeds, and humanoids have the opportunity to impact emerging robotics applications from logistics, to agriculture, to home assistance. The goal of this survey is to cover the recent progress toward these applications that has been driven by model-based optimization for the real-time generation and control of movement. The majority of the research community has converged on the idea of generating locomotion control laws by solving an optimal control problem (OCP) in either a model-based or data-driven manner. However, solving the most general of these problems online remains intractable due to complexities from intermittent unidirectional contacts with the environment, and from the many degrees of freedom of legged robots. This survey covers methods that have been pursued to make these OCPs computationally tractable, with specific focus on how environmental contacts are treated, how the model can be simplified, and how these choices affect the numerical solution methods employed. The survey focuses on model-based optimization, covering its recent use in a stand alone fashion, and suggesting avenues for combination with learning-based formulations to further accelerate progress in this growing field.

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.