Efficient Search and Learning for Agile Locomotion on Stepping Stones

Legged robots have become capable of performing highly dynamic maneuvers in the past few years. However, agile locomotion in highly constrained environments such as stepping stones is still a challenge. In this paper, we propose a combination of model-based control, search, and learning to design efficient control policies for agile locomotion on stepping stones. In our framework, we use nonlinear model predictive control (NMPC) to generate whole-body motions for a given contact plan. To efficiently search for an optimal contact plan, we propose to use Monte Carlo tree search (MCTS). While the combination of MCTS and NMPC can quickly find a feasible plan for a given environment (a few seconds), it is not yet suitable to be used as a reactive policy. Hence, we generate a dataset for optimal goal-conditioned policy for a given scene and learn it through supervised learning. In particular, we leverage the power of diffusion models in handling multi-modality in the dataset. We test our proposed framework on a scenario where our quadruped robot Solo12 successfully jumps to different goals in a highly constrained environment.

Multi-contact Stochastic Predictive Control for Legged Robots with Contact Locations Uncertainty

Trajectory optimization under uncertainties is a challenging problem for robots in contact with the environment. Such uncertainties are inevitable due to estimation errors, control imperfections, and model mismatches between planning models used for control and the real robot dynamics. This induces control policies that could violate the contact location constraints by making contact at unintended locations, and as a consequence leading to unsafe motion plans. This work addresses the problem of robust kino-dynamic whole-body trajectory optimization using stochastic nonlinear model predictive control (SNMPC) by considering additive uncertainties on the model dynamics subject to contact location chance-constraints as a function of robot's full kinematics. We demonstrate the benefit of using SNMPC over classic nonlinear MPC (NMPC) for whole-body trajectory optimization in terms of contact location constraint satisfaction (safety). We run extensive Monte-Carlo simulations for a quadruped robot performing agile trotting and bounding motions over small stepping stones, where contact location satisfaction becomes critical. Our results show that SNMPC is able to perform all motions safely with 100% success rate, while NMPC failed 48.3% of all motions.

Visual-Inertial and Leg Odometry Fusion for Dynamic Locomotion

Implementing dynamic locomotion behaviors on legged robots requires a high-quality state estimation module. Especially when the motion includes flight phases, state-of-the-art approaches fail to produce reliable estimation of the robot posture, in particular base height. In this paper, we propose a novel approach for combining visual-inertial odometry (VIO) with leg odometry in an extended Kalman filter (EKF) based state estimator. The VIO module uses a stereo camera and IMU to yield low-drift 3D position and yaw orientation and drift-free pitch and roll orientation of the robot base link in the inertial frame. However, these values have a considerable amount of latency due to image processing and optimization, while the rate of update is quite low which is not suitable for low-level control. To reduce the latency, we predict the VIO state estimate at the rate of the IMU measurements of the VIO sensor. The EKF module uses the base pose and linear velocity predicted by VIO, fuses them further with a second high-rate IMU and leg odometry measurements, and produces robot state estimates with a high frequency and small latency suitable for control. We integrate this lightweight estimation framework with a nonlinear model predictive controller and show successful implementation of a set of agile locomotion behaviors, including trotting and jumping at varying horizontal speeds, on a torque-controlled quadruped robot.

Nonlinear Stochastic Trajectory Optimization for Centroidal Momentum Motion Generation of Legged Robots

Generation of robust trajectories for legged robots remains a challenging task due to the underlying nonlinear, hybrid and intrinsically unstable dynamics which needs to be stabilized through limited contact forces. Furthermore, disturbances arising from unmodelled contact interactions with the environment and model mismatches can hinder the quality of the planned trajectories leading to unsafe motions. In this work, we propose to use stochastic trajectory optimization for generating robust centroidal momentum trajectories to account for additive uncertainties on the model dynamics and parametric uncertainties on contact locations. Through an alternation between the robust centroidal and whole-body trajectory optimizations, we generate robust momentum trajectories while being consistent with the whole-body dynamics. We perform an extensive set of simulations subject to different uncertainties on a quadruped robot showing that our stochastic trajectory optimization problem reduces the amount of foot slippage for different gaits while achieving better performance over deterministic planning.

On the Use of Torque Measurement in Centroidal State Estimation

State of the art legged robots are either capable of measuring torque at the output of their drive systems, or have transparent drive systems which enable the computation of joint torques from motor currents. In either case, this sensor modality is seldom used in state estimation. In this paper, we propose to use joint torque measurements to estimate the centroidal states of legged robots. To do so, we project the whole-body dynamics of a legged robot into the nullspace of the contact constraints, allowing expression of the dynamics independent of the contact forces. Using the constrained dynamics and the centroidal momentum matrix, we are able to directly relate joint torques and centroidal states dynamics. Using the resulting model as the process model of an Extended Kalman Filter (EKF), we fuse the torque measurement in the centroidal state estimation problem. Through real-world experiments on a quadruped robot with different gaits, we demonstrate that the estimated centroidal states from our torque-based EKF drastically improve the estimation of these quantities compared to direct computation.

BiConMP: A Nonlinear Model Predictive Control Framework for Whole Body Motion Planning

Online planning of whole-body motions for legged robots is challenging due to the inherent nonlinearity in the robot dynamics. In this work, we propose a nonlinear MPC framework, the BiConMP which can generate whole body trajectories online by efficiently exploiting the structure of the robot dynamics. BiConMP is used to generate various cyclic gaits on a real quadruped robot and its performance is evaluated on different terrain, countering unforeseen pushes and transitioning online between different gaits. Further, the ability of BiConMP to generate non-trivial acyclic whole-body dynamic motions on the robot is presented. Finally, an extensive empirical analysis on the effects of planning horizon and frequency on the nonlinear MPC framework is reported and discussed.

A unified framework for walking and running of bipedal robots

In this paper, we propose a novel framework capable of generating various walking and running gaits for bipedal robots. The main goal is to relax the fixed center of mass (CoM) height assumption of the linear inverted pendulum model (LIPM) and generate a wider range of walking and running motions, without a considerable increase in complexity. To do so, we use the concept of virtual constraints in the centroidal space which enables generating motions beyond walking while keeping the complexity at a minimum. By a proper choice of these virtual constraints, we show that we can generate different types of walking and running motions. More importantly, enforcing the virtual constraints through feedback renders the dynamics linear and enables us to design a feedback control mechanism which adapts the next step location and timing in face of disturbances, through a simple quadratic program (QP). To show the effectiveness of this framework, we showcase different walking and running simulations of the biped robot Bolt in the presence of both environmental uncertainties and external disturbances.

Rapid Convex Optimization of Centroidal Dynamics using Block Coordinate Descent

In this paper we explore the use of block coordinate descent (BCD) to optimize the centroidal momentum dynamics for dynamically consistent multi-contact behaviors. The centroidal dynamics have recently received a large amount of attention in order to create physically realizable motions for robots with hands and feet while being computationally more tractable than full rigid body dynamics models. Our contribution lies in exploiting the structure of the dynamics in order to simplify the original non-convex problem into two convex subproblems. We iterate between these two subproblems for a set number of iterations or until a consensus is reached. We explore the properties of the proposed optimization method for the centroidal dynamics and verify in simulation that motions generated by our approach can be tracked by the quadruped Solo12. In addition, we compare our method to a recently proposed convexification using a sequence of convex relaxations as well as a more standard interior point method used in the off- the-shelf solver IPOPT to show that our approach finds similar, if not better, trajectories (in terms of cost), and is more than four times faster than both approaches. Finally, compared to previous approaches, we note its practicality due to the convex nature of each subproblem which allows our method to be used with any off-the-shelf quadratic programming solver.

Model-free Reinforcement Learning for Robust Locomotion Using Trajectory Optimization for Exploration

In this work we present a general, two-stage reinforcement learning approach for going from a single demonstration trajectory to a robust policy that can be deployed on hardware without any additional training. The demonstration is used in the first stage as a starting point to facilitate initial exploration. In the second stage, the relevant task reward is optimized directly and a policy robust to environment uncertainties is computed. We demonstrate and examine in detail performance and robustness of our approach on highly dynamic hopping and bounding tasks on a real quadruped robot.