Online Relative Footstep Optimization for Legged Robots Dynamic Walking Using Discrete-Time Model Predictive Control

This paper investigates dynamic gait planning optimization and balance control of quadruped robots under external disturbance forces. First, a platform of quadruped walking robot with fourteen active degrees of freedom is designed. Then, a forward kinematic model of joints is built for quadruped robots based on Denavit-Hartenberg(D-H) method. The inverse kinematic equations are solved to result in joint values when the desired position and orientation are specified. A dynamic gait planning algorithm is proposed and tested on the quadruped robot. The planning function is established to create some point-to-point trajectories. The angle values of the joints can be calculated by using the inverse kinematics equations for every moment. Considering the external distribution a balance control approach is proposed to stabilize the robot based on the information from the attitude sensors. The walking is stabilized by a feedback control that uses a three-axis acceleration sensor. Experiments have been performed on the quadruped robot. The results showed that the proposed methods work well in dynamic gait planning and external disturbances of a quadruped bionic robot.

The ability to maintain balance and stability is a major challenge for quadruped robots in locomotion. For example, when walking in the unstructured environment, the robot not only needs to adjust its gait according to the terrain information, but also needs to tune the body posture based on the real-time body posture acquired from inertial measurement unit (IMU) to maintain stability. In this paper, we present a method that focuses on the design of the posture correction controller based on the IMU feedback for adaptive slope walking for a quadruped robot. To this end, the posture correction controller is integrated into the foot trajectory generator to maintain balance. We experimentally validated our approach both on simulation and real quadruped robot. Compared with the one only with the foot trajectory generator, the quadruped robot can walk more stable with the posture correction controller integrated into the foot trajectory generator and has a better locomotion ability. Our method increases locomotion stability when walking on the slopes.

To improve the adaptability and robustness of the quadruped robot to the environment, we design a novel multi-configuration quadruped robot with redundant degree-of-freedoms (DOFs). Compared with traditional quadruped robots, the presented robot can adapt to rugged terrain with optimal leg configurations. The quadruped robot can use insect-like or reptile-like configurations to pass through rough terrain. The transformation between different configurations is realized by redundant DOFs. The quadruped robot uses dynamic gait and optimal configuration to pass through complex terrain, instead of traditional crawling gait. To solve the redundant inverse kinematics, a simplified jacobian pseudoinverse (JP) algorithm based on quadratic integral programming is used. The algorithm based on spring loaded inverted pendulum (SLIP) is used to control the motion of the quadruped robot in various configurations. The multi-configuration motion of the robot is verified by dynamic simulation software Webots. The optimal configuration of the quadruped robot is explored by comparing the performance of each configuration in multiple scenarios.

Ground reaction forces (GRFs) are the primary interaction forces that enable a legged robot to maintain balance and perform locomotion. Most quadruped robot controllers estimate GRFs indirectly using joint torques and a kinematic model, which depend on assumptions and are highly sensitive to modeling errors. In contrast, direct sensing of contact forces at the feet provides more accurate and immediate feedback. Beyond force magnitude, tactile sensing also enables richer contact interpretation, such as detecting force direction and surface properties. In this work, we show how tactile sensor information can be used inside the feedback of the control loop to achieve compliance of legged robots during ground contact. The three main contributions are (i) a fast and computationally efficient 3D force reconstruction method tailored for spherical tactile sensors, (ii) a tactile admittance controller that adjusts leg motions to achieve the desired GRFs and compliance, and (iii) experimental validation on a quadruped robot, demonstrating enhanced load distribution and balance during external perturbations and locomotion. The results show that the peak ground reaction forces were reduced by 55% while balancing on a beam. During a locomotion scenario involving sudden touchdown after a fall, the tactile admittance controller reduced oscillations and regained stability compared to proportional–derivative (PD) control.

Quadruped locomotion is challenging for many learning-based algorithms. This is because it requires tedious manual tuning to cope with different types of terrains and is difficult to deploy in reality due to the sim-to-real gap between the training and the testing scenarios. This paper proposes a quadruped robot learning system for agile locomotion which does not require any pre-training and works well in various terrains. We introduce a hierarchical framework that uses reinforcement learning as the high-level policy to adjust the low-level trajectory generator for a better adaptability to various terrains. We compact the observation and the action spaces of reinforcement learning to deploy the proposed framework on a host computer interfaced with the robot. Besides, we design an omnidirectional trajectory generator guided by robot posture, which generates omnidirectional foot trajectories to interact with the environment. Experimental results and the supplementary video demonstrate that our hierarchical framework only trained in simulation can be easily deployed in the real world, and also has the advantages of fast convergence and good terrain adaptability. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper presents a hierarchical framework for quadruped robots. It combines a high-level reinforcement learning controller with a posture-guided trajectory generator to adaptively generate omnidirectional motions. Our method is easy to train as it converges fast and does not need to adjust a dozen or so of rewards. The quadruped robot can be deployed in a real environment directly after being trained in simulation. With the trained hierarchical framework deployed on a remote host computer, the robot works well in a variety of real-world environments unseen in the simulation.

In this paper, we simplify the dynamics of a bounding quadrupedal robot by using a planar and conservative model which has a rigid body and two massless virtual spring legs to separately simulate the effects of the front and hind legs. The quantitative formulation of the model is derived by using the Lagrangian method. We proceed to search for stable operation points in state space, and the local system behaviors of which are then approximated by a trained neural network model. Next, a model predictive controller is utilized to stabilize the model. In simulation results, the cont vroller succeeded in balancing the model in a dynamic bounding gait without any form of energy input. This research may serve as a guideline for real quadrupedal robots with actuators to create energy conservative gaits.

Background Meniscal injury is a common knee disorder that may alter lower-limb loading patterns during walking and contribute to functional impairment and secondary joint degeneration. However, whether gait kinetic alterations differ across MRI-defined Stoller grades remains unclear. Objective To investigate the effects of isolated meniscal injuries of different Stoller grades on gait kinetics, focusing on ground reaction forces (GRF) and joint moments in both affected and contralateral limbs. Methods A total of 114 participants were categorized into three groups based on MRI findings: normal, Grade I–II, and Grade III. Three-dimensional gait analysis recorded GRF components and sagittal-plane joint moments. Multivariate analysis of covariance, adjusting for age and body mass index (BMI), followed by Bonferroni post hoc tests, was used to compare intergroup differences. Results There were significant differences in BMI and Lysholm scores between groups, and these factors were corrected in subsequent analyses. Compared with controls, patients with meniscal injury demonstrated altered GRF components (anterior, posterior, and lateral) and reduced ankle dorsiflexion, knee extension, and hip flexion/extension moments on the affected limb. Similar kinetic alterations were identified in the contralateral limb. Peak joint moments at the ankle, knee, and hip were significantly correlated with GRF parameters. No significant differences were detected between the Grade I–II and Grade III groups in any gait kinetic variables. Conclusion Gait kinetic alterations appear to be associated with the presence of meniscal injury rather than MRI-defined injury severity. These findings underscore the importance of functional gait assessment and support individualized rehabilitation strategies regardless of Stoller grade.

In this paper, in order to clarify the role of dynamics based control, we conducted research on dynamics of the mechanical system for a quadrupedal robot in jumping locomotion. A control strategy of the mechanical system for Patrush, our quadrupedal robot with compliant legs, was introduced and successfully applied in our research. In addition, we developed a tendon driven mechanism with a flexible pneumatic actuator, by means of which the quadrupedal robot is able to execute a dynamic running gait. MPEG footage of these experiments can be seen at: http://www.kimura.is.uec.ac.jp.

To realize the dynamic walking of quadruped robot in underwater environment, a stable controller based on the optimization of Ground Reaction Force (GRF) is proposed. First, we develop the integrated equations of motion for underwater quadruped robot. They describe the characteristics consisting of both hydrodynamic forces and multi-body model. The main highlight is that the GRF optimization is consistent with the particular underwater equations of motion, and it takes into account the force constraints employing Quadratic Program (QP) method. The inputs of the optimization model are the required acceleration commands, which are solved by state feedback controller according to the desired response performances. To achieve continuous walking, the gait planner is developed. It plans the motion instructions, and schedules the states between stance and swing for each leg. Simulation results show that the underwater quadruped robot can track the motion commands accurately with certain robustness to external disturbance force.

The foot and ground force directly affect the propulsion performance and stability of quadruped robot. At present, the typical robots’ foot, such as flat foot, cylindrical foot and spherical foot, are being widely used. However, the bionic foot almost haven’t got very good application. This paper presents a kind of bio-inspired equinus model and a mechanical single leg system. The three dimensional model of equinus had been conducted, and 3D printing technology is also been used to produce bio-inspired foot. To compare the ground reaction force on quadruped robot, this paper conducts tests of hemispherical foot and bio-inspired equinus. Experiments show that the propulsive force of bio-inspired equinus is more excellent than hemispherical foot; and the former makes the robot more stable during walking. The experimental result provides the experimental basis and references for the follow-up study on Bio-inspired foot of quadruped robot.

We present a real time algorithm for humanoid 3D walking and/or running based on a Model Predictive Control (MPC) approach. The objective is to generate a stable gait that replicates a footstep plan as closely as possible, that is, a sequence of candidate footstep positions and orientations with associated timings. For each footstep, the plan also specifies an associated reference height for the Center of Mass (CoM) and whether the robot should reach the footstep by walking or running. The scheme makes use of the Variable-Height Inverted Pendulum (VH-IP) as a prediction model, generating in real time both a CoM trajectory and adapted footsteps. The VH-IP model relates the position of the CoM to that of the Zero Moment Point (ZMP); to avoid falling, the ZMP must be inside a properly defined support region (a 3D extension of the 2D support polygon) whenever the robot is in contact with the ground. The nonlinearity of the VH-IP is handled by splitting the gait generation into two consecutive stages, both requiring to solve a quadratic program. Thanks to a particular triangular structure of the VH-IP dynamics, the first stage deals with the vertical dynamics using the Ground Reaction Force (GRF) as a decision variable. Using the prediction given by the first stage, the horizontal dynamics become linear time-varying. During the flight phases, the VH-IP collapses to a free-falling mass model. The proposed formulation incorporates constraints in order to maintain physically meaningful values of the GRF, keep the ZMP in the support region during contact phases, and ensure that the adapted footsteps are kinematically realizable. Most importantly, a stability constraint is enforced on the time-varying horizontal dynamics to guarantee a bounded evolution of the CoM with respect to the ZMP. Furthermore, we show how to extend the technique in order to perform running on tilted surfaces. We also describe a simple technique that receives input high-level velocity commands and generates a footstep plan in the form required by the proposed MPC scheme. The algorithm is validated via dynamic simulations on the full-scale humanoid robot HRP-4, as well as experiments on the small-sized robot OP3.

In this paper, optimal cyclic reference trajectories are designed for three walking gaits of a quadruped robot, the curvet, the amble, the trot, taking into account the actuators characteristics and for a given motion robot velocity. The objectives of this study are multiple: • Automatic design of admissible reference trajectories, • Definition of the maximal speed of the robot for given actuators, • Study of the evolution of the torque cost for various motion velocities, • Comparison of three different gaits with respect to a sthenic criterion. For the gaits studied, during some phases the system is underactuated. The design of the cyclic gaits takes this fact into account. The optimization problem is solved with an algebraic optimization technique. The actuated joint evolution is chosen as a polynomial function of time. The coefficients of the polynomial functions are optimization parameters. The evolution of the passive joint is defined using the dynamic model to have admissible trajectory. During the optimization process, the constraints on the ground reactions, the validity of impact, the torques, the joints velocities of the robot are taken into account. Simulation results are presented for the three gaits. All motions are realistics. Curvet is the less efficient gait with respect to the criterion studied. For slow motion, trot is the more efficient gait. But amble permits the fastest motion with the same actuators.

Gait kinetics impact shoe tread wear rate

In this paper, a compliance control strategy based on ZMP preview control for quadruped walking gait locomotion under unperceived disturbances is proposed. The preview control is designed to plan the CoM trajectory to realize the desired ZMP. When the robot suffers from interference of environmental disturbances, we introduce an adjusted expected ZMP to suppress the change of the ZMP. The force distribution module calculate desired contact force by eliminating interaction force between the stance legs and the force controller make the contact force track the desired force. Simulation results demonstrate the effectiveness of our compliance control strategy based on ZMP preview control as the quadruped robot successfully walking under unperceived disturbances.

Aiming at the control problem of quadruped robot in fast and stable running state, a backward control method of quadruped robot based on CPG was proposed in this paper. Rhythm signal produced by central pattern generator (CPG) to control the height of the body, pitching angle, and the periodic generation and switching of foot trajectory can be realized. At the same time, the impedance control was used to realize TD detection and feedback the parameter signal about the stability to the robot. The virtual force generated by PD control of trajectory deviation was fed back to the system to form a closed-loop backward control. Simulation results were given to show that this method can make the quadruped robot run at a stable and high speed.