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1. A Distributed Layered Planning and Control Algorithm for Teams of Quadrupedal Robots: An Obstacle-Aware Nonlinear Model Predictive Control Approach

   https://doi.org/10.1115/1.4066632  

   Imran, Basit Muhammad; Fawcett, Randall T; Kim, Jeeseop; Leonessa, Alexander; Hamed, Kaveh Akbari (May 2025, Journal of Dynamic Systems, Measurement, and Control)

   Abstract This paper aims to develop a distributed layered control framework for the navigation, planning, and control of multi-agent quadrupedal robots subject to environments with uncertain obstacles and various disturbances. At the highest layer of the proposed layered control, a reference path for all agents is calculated, considering artificial potential fields (APF) under a priori known obstacles. Second, in the middle layer, we employ a distributed nonlinear model predictive control (NMPC) scheme with a one-step delay communication protocol (OSDCP) subject to reduced-order and linear inverted pendulum (LIP) models of agents to ensure the feasibility of the gaits and collision avoidance, addressing the degree of uncertainty in real-time. Finally, low-level nonlinear whole-body controllers (WBCs) impose the full-order locomotion models of agents to track the optimal and reduced-order trajectories. The proposed controller is validated for effectiveness and robustness on up to four A1 quadrupedal robots in simulations and two robots in the experiments.1 Simulations and experimental validations demonstrate that the proposed approach can effectively address the real-time planning and control problem. In particular, multiple A1 robots are shown to navigate various environments, maintaining collision-free distances while being subject to unknown external disturbances such as pushes and rough terrain. 

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2. Distributed Feedback Controllers for Stable Cooperative Locomotion of Quadrupedal Robots: A Virtual Constraint Approach*

   https://doi.org/10.23919/ACC45564.2020.9147673  

   Hamed, Kaveh Akbari; Kamidi, Vinay R.; Pandala, Abhishek; Ma, Wen-Loong; Ames, Aaron D. (July 2020, American Control Conference (ACC))

   This paper aims to develop distributed feedback control algorithms that allow cooperative locomotion of quadrupedal robots which are coupled to each other by holonomic constraints. These constraints can arise from collaborative manipulation of objects during locomotion. In addressing this problem, the complex hybrid dynamical models that describe collaborative legged locomotion are studied. The complex periodic orbits (i.e., gaits) of these sophisticated and high-dimensional hybrid systems are investigated. We consider a set of virtual constraints that stabilizes locomotion of a single agent. The paper then generates modified and local virtual constraints for each agent that allow stable collaborative locomotion. Optimal distributed feedback controllers, based on nonlinear control and quadratic programming, are developed to impose the local virtual constraints. To demonstrate the power of the analytical foundation, an extensive numerical simulation for cooperative locomotion of two quadrupedal robots with robotic manipulators is presented. The numerical complex hybrid model has 64 continuous-time domains, 192 discrete-time transitions, 96 state variables, and 36 control inputs. 

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3. Analytical solution to a time-varying LIP model for quadrupedal walking on a vertically oscillating surface

   https://doi.org/10.1016/j.mechatronics.2023.103073  

   Iqbal, Amir; Veer, Sushant; Gu, Yan (December 2023, Mechatronics)

   Sun, Weichao; Yao, Bin (Ed.)

   This paper introduces an analytically tractable and computationally efficient model for legged robot dynamics during locomotion on a dynamic rigid surface (DRS), along with an approximate analytical solution and a real-time walking pattern generator synthesized based on the model and solution. By relaxing the static-surface assumption, we extend the classical, time-invariant linear inverted pendulum (LIP) model for legged locomotion on a static surface to dynamic-surface locomotion, resulting in a time-varying LIP model termed as “DRS-LIP”. Sufficient and necessary stability conditions of the time-varying DRS-LIP model are obtained based on the Floquet theory. This model is also transformed into Mathieu’s equation to derive an approximate analytical solution that provides reasonable accuracy with a relatively low computational cost. Using the extended model and its solution, a walking pattern generator is developed to efficiently plan physically feasible trajectories for quadrupedal walking on a vertically oscillating surface. Finally, simulations and hardware experiments from a Laikago quadrupedal robot walking on a pitching treadmill (with a maximum vertical acceleration of 1 m/s ) confirm the accuracy and efficiency of the proposed analytical solution, as well as the efficiency, feasibility, and robustness of the pattern generator, under various surface motions and gait parameters. 

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4. Position Control and Variable-Height Trajectory Tracking of a Soft Pneumatic Legged Robot

   https://doi.org/10.1109/IROS51168.2021.9635966  

   Liu, Zhichao; Karydis, Konstantinos (September 2021, 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS))

   Soft pneumatic legged robots show promise in their ability to traverse a range of different types of terrain, including natural unstructured terrain met in applications like precision agriculture. They can adapt their body morphology to the intricacies of the terrain at hand, thus enabling robust and resilient locomotion. In this paper we capitalize upon recent developments on soft pneumatic legged robots to introduce a closed-loop trajectory tracking control scheme for operation over flat ground. Closed-loop pneumatic actuation feedback is achieved via a compact and portable pneumatic regulation board. Experimental results reveal that our soft legged robot can precisely control its body height and orientation while in quasi-static operation based on a geometric model. The robot can track both straight line and curved trajectories as well as variable-height trajectories. This work lays the basis to enable autonomous navigation for soft legged robots. 

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5. A Simple Controller for Omnidirectional Trotting of Quadrupedal Robots: Command Following and Waypoint Tracking

   https://doi.org/10.3390/robotics12020035  

   Bhounsule, Pranav A.; Yang, Chun-Ming (April 2023, Robotics)

   For autonomous legged robots to be deployed in practical scenarios, they need to perform perception, motion planning, and locomotion control. Since robots have limited computing capabilities, it is important to realize locomotion control with simple controllers that have modest calculations. The goal of this paper is to create computational simple controllers for locomotion control that can free up computational resources for more demanding computational tasks, such as perception and motion planning. The controller consists of a leg scheduler for sequencing a trot gait with a fixed step time; a reference trajectory generator for the feet in the Cartesian space, which is then mapped to the joint space using an analytical inverse; and a joint controller using a combination of feedforward torques based on static equilibrium and feedback torque. The resulting controller enables velocity command following in the forward, sideways, and turning directions. With these three velocity command following-modes, a waypoint tracking controller is developed that can track a curve in global coordinates using feedback linearization. The command following and waypoint tracking controllers are demonstrated in simulation and on hardware. 

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