Developing feasible body trajectories for legged systems on arbitrary terrains is a challenging task. In this paper, we present a paradigm that allows to design feasible Center of Mass (CoM) and body trajectories in an efficient manner. In our previous work [1], we introduced the notion of the 2D feasible region, where static balance and the satisfaction of joint torque limits were guaranteed, whenever the projection of the CoM lied inside the proposed admissible region. In this work we propose a general formulation of the improved feasible region that guarantees dynamic balance alongside the satisfaction of both joint-torque and kinematic limits in an efficient manner. To incorporate the feasibility of the kinematic limits, we introduce an algorithm that computes the reachable region of the CoM. Furthermore, we propose an efficient planning strategy that utilizes the improved feasible region to design feasible CoM and body orientation trajectories. Finally, we validate the capabilities of the improved feasible region and the effectiveness of the proposed planning strategy, using simulations and experiments on the 90 kg Hydraulically actuated Quadruped (HyQ) and the 21 kg Aliengo robots.

We present a unified model-based and data-driven approach for quadrupedal planning and control to achieve dynamic locomotion over uneven terrain. We utilize on-board proprioceptive and exteroceptive feedback to map sensory information and desired base velocity commands into footstep plans using a reinforcement learning (RL) policy. This RL policy is trained in simulation over a wide range of procedurally generated terrains. When run online, the system tracks the generated footstep plans using a model-based motion controller. We evaluate the robustness of our method over a wide variety of complex terrains. It exhibits behaviors that prioritize stability over aggressive locomotion. Additionally, we introduce two ancillary RL policies for corrective whole-body motion tracking and recovery control. These policies account for changes in physical parameters and external perturbations. We train and evaluate our framework on a complex quadrupedal system, ANYmal version B, and demonstrate transferability to a larger and heavier robot, ANYmal C, without requiring retraining.

We present a control architecture for real-time adaptation and tracking of trajectories generated using a terrain-aware trajectory optimization solver. This approach enables us to circumvent the computationally exhaustive task of online trajectory optimization, and further introduces a control solution robust to systems modeled with approximated dynamics. We train a policy using deep reinforcement learning (RL) to introduce additive deviations to a reference trajectory in order to generate a feedback-based trajectory tracking system for a quadrupedal robot. We train this policy across a multitude of simulated terrains and ensure its generality by introducing training methods that avoid overfitting and convergence towards local optima. Additionally, in order to capture terrain information, we include a latent representation of the height maps in the observation space of the RL environment as a form of exteroceptive feedback. We test the performance of our trained policy by tracking the corrected set points using a model-based whole-body controller and compare it with the tracking behavior obtained without the corrective feedback in several simulation environments, and show that introducing the corrective feedback results in increase of the success rate from 72.7% to 92.4% for tracking precomputed dynamic long horizon trajectories on flat terrain and from 47.5% to 80.3% on a complex modular uneven terrain. We also show successful transfer of our training approach to the real physical system and further present cogent arguments in support of our framework.

To dynamically traverse challenging terrain, legged robots need to continually perceive and reason about upcoming features, adjust the locations and timings of future footfalls and leverage momentum strategically. We present a pipeline that enables flexibly-parametrized trajectories for perceptive and dynamic quadruped locomotion to be optimized in an online, receding-horizon manner. The initial guess passed to the optimizer affects the computation needed to achieve convergence and the quality of the solution. We consider two methods for generating good guesses. The first is a heuristic initializer which provides a simple guess and requires significant optimization but is nonetheless suitable for adaptation to upcoming terrain. We demonstrate experiments using the ANYmal C quadruped, with fully onboard sensing and computation, to cross obstacles at moderate speeds using this technique. Our second approach uses latent-mode trajectory regression (LMTR) to imitate expert data—while avoiding invalid interpolations between distinct behaviors—such that minimal optimization is needed. This enables high-speed motions that make more expansive use of the robot’s capabilities. We demonstrate it on flat ground with the real robot and provide numerical trials that progress toward deployment on terrain. These results illustrate a paradigm for advancing beyond short-horizon dynamic reactions, toward the type of intuitive and adaptive locomotion planning exhibited by animals and humans.

The efficient evaluation the dynamic stability of legged robots on non-coplanar terrains is important when developing motion planning and control policies. The inference time of this measure has a strong influence on how fast a robot can react to unexpected events, plan its future footsteps or its body trajectory. Existing approaches suitable for real-time decision making are either limited to flat ground or to quasi-static locomotion. Furthermore, joint-space feasibility constraints are usually not considered in receding-horizon planning as their high dimensionality prohibits this. In this paper we propose the usage of a stability criterion for dynamic locomotion on rough terrain based on the Feasible Region (FR) and the Instantaneous Capture Point (ICP) and we leverage a Neural Network (NN) to quickly estimate it. We show that our network achieves satisfactory accuracy with respect to its analytical counterpart with a speed up of three orders-of-magnitude. It also enables the evaluation of the stability margin's gradient. We demonstrate this learned stability margin in two diverse applications - Reinforcement Learning (RL) and nonlinear Trajectory Optimization (TO) for legged robots. We demonstrate on a full-sized quadruped robot that the network enables the computation of physically-realizable Center of Mass (CoM) trajectories and foothold locations satisfying friction constraints and joint-torque limits in a receding-horizon fashion and on non-coplanar terrains.

Simplified models are useful to increase the computational efficiency of a motion planning algorithm, but their lack of accuracy have to be managed. We propose two feasibility constraints to be included in a Single Rigid Body Dynamics-based trajectory optimizer in order to obtain robust motions in challenging terrain. The first one finds an approximate relationship between joint-torque limits and admissible contact forces, without requiring the joint positions. The second one proposes a leg model to prevent leg collision with the environment. Such constraints have been included in a simplified nonlinear non-convex trajectory optimization problem. We demonstrate the feasibility of the resulting motion plans both in simulation and on the Hydraulically actuated Quadruped (HyQ) robot, considering experiments on an irregular terrain.

In legged locomotion, the projection of the robot's Center of Mass (CoM) being inside the convex hull of the contact points is a commonly accepted sufficient condition to achieve static balancing. However, some of these configurations cannot be realized because the joint-torques required to sustain them would be above their limits (actuation limits). In this article, we rule out such configurations and define the feasible region, a revisited support region that guarantees both global static stability in the sense of tip-over and slippage avoidance and of existence of a set of joint-torques that are able to sustain the robot's body weight. We show that the feasible region can be employed for the online selection of feasible footholds and CoM trajectories to achieve statically stable locomotion on rough terrains, also in presence of load-intensive tasks. Key results of our approach include the efficiency in the computation of the feasible region using an Iterative Projection (IP) algorithm and the successful execution of hardware experiments on the HyQ robot, that was able to negotiate obstacles of moderate dimensions while carrying an extra 10-kg payload.

We present a unified control framework that generates dynamic walking motions for biped and quadruped robots with online relative footstep optimization. The footstep optimization is formulated as a discrete-time Model Predictive Control problem which determines future footstep locations. The framework has a hierarchical structure consisting of three layers: footstep planner, trajectory generator and whole-body controller. The footstep planner plans next footstep position based on Linear Inverted Pendulum (LIP) model. Relative footstep optimization is proposed to enable automatic footstep planning without the use of any predefined footstep sequences. The trajectory generator will generate CoM and feet trajectory given the next footstep placement. In order to generalize to quadruped robots, “virtual leg” concept has been used to coordinate leg pair movement. The whole-body inverse dynamic controller calculates joint torques to track given Cartesian reference trajectories. To include under-actuation into consideration, contact vertices formulation of ground reaction forces (GRFs) has been adopted. Generalized whole-body controller can handle biped robot with line feet as well as quadruped robots with point feet walking with dynamic gaits. Several simulations have been performed to demonstrate the robustness and generality of the proposed framework.

We propose a reactive locomotion strategy, called height reflex, that is useful to address big elevation changes in the terrain (e.g. when a quadruped robot has to step down from a high platform). In these cases the swing leg can lose mobility creating issues in the subsequent steps. The height reflex is a foot trajectory replanning strategy that redistributes the swing motion (in a smart way) to the stance legs to “lower” the whole trunk and to aid the foothold searching motion. To spread the motion we exploit a massless link model of the robot with virtual dampers at the joints, which is used to replan the feet trajectories. The proposed approach is able to incorporate kinematic limits, it is easy-to-tune, computationally efficient and suitable for real-time implementations. The reflex is implemented and experimentally evaluated on the 80 kg hydraulic quadruped HyQ. With our approach we were able to address high steps, up to 24 cm which is 30% of HyQ leg length and 53% of its retractable leg range.

The identification of inertial parameters is crucial to achieve high-performance model-based control of legged robots. The inertial parameters of the legs are typically not altered during expeditions and therefore are best identified offline. On the other hand, the trunk parameters depend on the modules mounted on the robot, like a motor to provide the hydraulic power, or different sets of cameras for perception. This motivates the use of recursive approaches to identify online mass and the position of the Center of Mass (CoM) of the robot trunk, when a payload change occurs. We propose two such approaches and analyze their robustness in simulation. Furthermore, experimental trials on our 80-kg quadruped robot HyQ show the applicability of our strategies during locomotion to cope with large payload changes that would otherwise severely compromise the balance of the robot.