Auke Jan Ijspeert

Guillaume Bellegarda, Auke Ijspeert

In this letter, we present a method for integrating central pattern generators (CPGs), i.e. systems of coupled oscillators, into the deep reinforcement learning (DRL) framework to produce robust and omnidirectional quadruped locomotion. The agent learns to directly modulate the intrinsic oscillator setpoints (amplitude and frequency) and coordinate rhythmic behavior among different oscillators. This approach also allows the use of DRL to explore questions related to neuroscience, namely the role of descending pathways, interoscillator couplings, and sensory feedback in gait generation. We train our policies in simulation and perform a sim-to-real transfer to the Unitree A1 quadruped, where we observe robust behavior to disturbances unseen during training, most notably to a dynamically added 13.75 kg load representing 115% of the nominal quadruped mass. We test several different observation spaces based on proprioceptive sensing and show that our framework is deployable with no domain randomization and very little feedback, where along with the oscillator states, it is possible to provide only contact booleans in the observation space. Video results can be found at https://youtu.be/xqXHLzLsEV4.

Milad Shafiee, Guillaume Bellegarda, Auke Ijspeert

Learning a locomotion policy for quadruped robots has traditionally been constrained to a specific robot morphology, mass, and size. The learning process must usually be repeated for every new robot, where hyperparameters and reward function weights must be re-tuned to maximize performance for each new system. Alternatively, attempting to train a single policy to accommodate different robot sizes, while maintaining the same degrees of freedom (DoF) and morphology, requires either complex learning frameworks, or mass, inertia, and dimension randomization, which leads to prolonged training periods. In our study, we show that drawing inspiration from animal motor control allows us to effectively train a single locomotion policy capable of controlling a diverse range of quadruped robots. The robot differences encompass: a variable number of DoFs, (i.e. 12 or 16 joints), three distinct morphologies, a broad mass range spanning from 2 kg to 200 kg, and nominal standing heights ranging from 18 cm to 100 cm. Our policy modulates a representation of the Central Pattern Generator (CPG) in the spinal cord, effectively coordinating both frequencies and amplitudes of the CPG to produce rhythmic output (Rhythm Generation), which is then mapped to a Pattern Formation (PF) layer. Across different robots, the only varying component is the PF layer, which adjusts the scaling parameters for the stride height and length. Subsequently, we evaluate the sim-to-real transfer by testing the single policy on both the Unitree Go1 and A1 robots. Remarkably, we observe robust performance, even when adding a 15 kg load, equivalent to 125% of the A1 robot's nominal mass.

Wanming Yu, Chuanyu Yang, Christopher McGreavy et al.

Robot motor skills can be learned through deep reinforcement learning (DRL) by neural networks as state-action mappings. While the selection of state observations is crucial, there has been a lack of quantitative analysis to date. Here, we present a systematic saliency analysis that quantitatively evaluates the relative importance of different feedback states for motor skills learned through DRL. Our approach can identify the most essential feedback states for locomotion skills, including balance recovery, trotting, bounding, pacing and galloping. By using only key states including joint positions, gravity vector, base linear and angular velocities, we demonstrate that a simulated quadruped robot can achieve robust performance in various test scenarios across these distinct skills. The benchmarks using task performance metrics show that locomotion skills learned with key states can achieve comparable performance to those with all states, and the task performance or learning success rate will drop significantly if key states are missing. This work provides quantitative insights into the relationship between state observations and specific types of motor skills, serving as a guideline for robot motor learning. The proposed method is applicable to differentiable state-action mapping, such as neural network based control policies, enabling the learning of a wide range of motor skills with minimal sensing dependencies.

Milad Shafiee, Guillaume Bellegarda, Auke Ijspeert

Quadruped animal locomotion emerges from the interactions between the spinal central pattern generator (CPG), sensory feedback, and supraspinal drive signals from the brain. Computational models of CPGs have been widely used for investigating the spinal cord contribution to animal locomotion control in computational neuroscience and in bio-inspired robotics. However, the contribution of supraspinal drive to anticipatory behavior, i.e. motor behavior that involves planning ahead of time (e.g. of footstep placements), is not yet properly understood. In particular, it is not clear whether the brain modulates CPG activity and/or directly modulates muscle activity (hence bypassing the CPG) for accurate foot placements. In this paper, we investigate the interaction of supraspinal drive and a CPG in an anticipatory locomotion scenario that involves stepping over gaps. By employing deep reinforcement learning (DRL), we train a neural network policy that replicates the supraspinal drive behavior. This policy can either modulate the CPG dynamics, or directly change actuation signals to bypass the CPG dynamics. Our results indicate that the direct supraspinal contribution to the actuation signal is a key component for a high gap crossing success rate. However, the CPG dynamics in the spinal cord are beneficial for gait smoothness and energy efficiency. Moreover, our investigation shows that sensing the front feet distances to the gap is the most important and sufficient sensory information for learning gap crossing. Our results support the biological hypothesis that cats and horses mainly control the front legs for obstacle avoidance, and that hind limbs follow an internal memory based on the front limbs' information. Our method enables the quadruped robot to cross gaps of up to 20 cm (50% of body-length) without any explicit dynamics modeling or Model Predictive Control (MPC).

Milad Shafiee, Guillaume Bellegarda, Auke Ijspeert

Quadruped animals seamlessly transition between gaits as they change locomotion speeds. While the most widely accepted explanation for gait transitions is energy efficiency, there is no clear consensus on the determining factor, nor on the potential effects from terrain properties. In this article, we propose that viability, i.e. the avoidance of falls, represents an important criterion for gait transitions. We investigate the emergence of gait transitions through the interaction between supraspinal drive (brain), the central pattern generator in the spinal cord, the body, and exteroceptive sensing by leveraging deep reinforcement learning and robotics tools. Consistent with quadruped animal data, we show that the walk-trot gait transition for quadruped robots on flat terrain improves both viability and energy efficiency. Furthermore, we investigate the effects of discrete terrain (i.e. crossing successive gaps) on imposing gait transitions, and find the emergence of trot-pronk transitions to avoid non-viable states. Compared with other potential criteria such as peak forces and energy efficiency, viability is the only improved factor after gait transitions on both flat and discrete gap terrains, suggesting that viability could be a primary and universal objective of gait transitions, while other criteria are secondary objectives and/or a consequence of viability. Moreover, we deploy our learned controller in sim-to-real hardware experiments and demonstrate state-of-the-art quadruped agility in challenging scenarios, where the Unitree A1 quadruped autonomously transitions gaits between trot and pronk to cross consecutive gaps of up to 30 cm (83.3 % of the body-length) at over 1.3 m/s.

Mengze Tian, Qiyuan Fu, Chuanfang Ning et al.

Amphibious legged robots inspired by salamanders are promising in applications in complex amphibious environments. However, despite the significant success of training controllers that achieve diverse locomotion behaviors in conventional quadrupedal robots, most salamander robots relied on central-pattern-generator (CPG)-based and model-based coordination strategies for locomotion control. Learning unified joint-level whole-body control that reliably transfers from simulation to highly articulated physical salamander robots remains relatively underexplored. In addition, few legged robots have tried learning-based controllers in amphibious environments. In this work, we employ Reinforcement Learning to map proprioceptive observations and commanded velocities to joint-level actions, allowing coordinated locomotor behaviors to emerge. To deploy these policies on hardware, we adopt a system-level real-to-sim matching and sim-to-real transfer strategy. The learned controller achieves stable and coordinated walking on both flat and uneven terrains in the real world. Beyond terrestrial locomotion, the framework enables transitions between walking and swimming in simulation, highlighting a phenomenon of interest for understanding locomotion across distinct physical modes.

Peizhuo Li, Hongyi Li, Ge Sun et al.

Despite recent advances in learning-based controllers for legged robots, deployments in human-centric environments remain limited by safety concerns. Most of these approaches use position-based control, where policies output target joint angles that must be processed by a low-level controller (e.g., PD or impedance controllers) to compute joint torques. Although impressive results have been achieved in controlled real-world scenarios, these methods often struggle with compliance and adaptability when encountering environments or disturbances unseen during training, potentially resulting in extreme or unsafe behaviors. Inspired by how animals achieve smooth and adaptive movements by controlling muscle extension and contraction, torque-based policies offer a promising alternative by enabling precise and direct control of the actuators in torque space. In principle, this approach facilitates more effective interactions with the environment, resulting in safer and more adaptable behaviors. However, challenges such as a highly nonlinear state space and inefficient exploration during training have hindered their broader adoption. To address these limitations, we propose SATA, a bio-inspired framework that mimics key biomechanical principles and adaptive learning mechanisms observed in animal locomotion. Our approach effectively addresses the inherent challenges of learning torque-based policies by significantly improving early-stage exploration, leading to high-performance final policies. Remarkably, our method achieves zero-shot sim-to-real transfer. Our experimental results indicate that SATA demonstrates remarkable compliance and safety, even in challenging environments such as soft/slippery terrain or narrow passages, and under significant external disturbances, highlighting its potential for practical deployments in human-centric and safety-critical scenarios.

Pembe Gizem Özdil, Chuanfang Ning, Jasper S. Phelps et al.

Computational models are critical to advance our understanding of how neural, biomechanical, and physical systems interact to orchestrate animal behaviors. Despite the availability of near-complete reconstructions of the Drosophila melanogaster central nervous system, musculature, and exoskeleton, anatomically and physically grounded models of fly leg muscles are still missing. These models provide an indispensable bridge between motor neuron activity and joint movements. Here, we introduce the first 3D, data-driven musculoskeletal model of Drosophila legs, implemented in both OpenSim and MuJoCo simulation environments. Our model incorporates a Hill-type muscle representation based on high-resolution X-ray scans from multiple fixed specimens. We present a pipeline for constructing muscle models using morphological imaging data and for optimizing unknown muscle parameters specific to the fly. We then combine our musculoskeletal models with detailed 3D pose estimation data from behaving flies to achieve muscle-actuated behavioral replay in OpenSim. Simulations of muscle activity across diverse walking and grooming behaviors predict coordinated muscle synergies that can be tested experimentally. Furthermore, by training imitation learning policies in MuJoCo, we test the effect of different passive joint properties on learning speed and find that damping and stiffness facilitate learning. Overall, our model enables the investigation of motor control in an experimentally tractable model organism, providing insights into how biomechanics contribute to generation of complex limb movements. Moreover, our model can be used to control embodied artificial agents to generate naturalistic and compliant locomotion in simulated environments.

Emmanouil Angelidis, Emanuel Buchholz, Jonathan Patrick Arreguit O'Neil et al.

Central Pattern Generators (CPGs) models have been long used to investigate both the neural mechanisms that underlie animal locomotion as well as a tool for robotic research. In this work we propose a spiking CPG neural network and its implementation on neuromorphic hardware as a means to control a simulated lamprey model. To construct our CPG model, we employ the naturally emerging dynamical systems that arise through the use of recurrent neural populations in the Neural Engineering Framework (NEF). We define the mathematical formulation behind our model, which consists of a system of coupled abstract oscillators modulated by high-level signals, capable of producing a variety of output gaits. We show that with this mathematical formulation of the Central Pattern Generator model, the model can be turned into a Spiking Neural Network (SNN) that can be easily simulated with Nengo, an SNN simulator. The spiking CPG model is then used to produce the swimming gaits of a simulated lamprey robot model in various scenarios. We show that by modifying the input to the network, which can be provided by sensory information, the robot can be controlled dynamically in direction and pace. The proposed methodology can be generalized to other types of CPGs suitable for both engineering applications and scientific research. We test our system on two neuromorphic platforms, SpiNNaker and Loihi. Finally, we show that this category of spiking algorithms shows a promising potential to exploit the theoretical advantages of neuromorphic hardware in terms of energy efficiency and computational speed.

T. P. Jensen, S. Tata, A. J. Ijspeert et al.

Human beings and other vertebrates show remarkable performance and efficiency in locomotion, but the functioning of their biological control systems for locomotion is still only partially understood. The basic patterns and timing for locomotion are provided by a central pattern generator (CPG) in the spinal cord. The cerebellum is known to play an important role in adaptive locomotion. Recent studies have given insights into the error signals responsible for driving the cerebellar adaptation in locomotion. However, the question of how the cerebellar output influences the gait remains unanswered. We hypothesize that the cerebellar correction is applied to the pattern formation part of the CPG. Here, a bio-inspired control system for adaptive locomotion of the musculoskeletal system of the mouse is presented, where a cerebellar-like module adapts the step time by using the double support interlimb asymmetry as a temporal teaching signal. The control system is tested on a simulated mouse in a split-belt treadmill setup similar to those used in experiments with real mice. The results show adaptive locomotion behavior in the interlimb parameters similar to that seen in humans and mice. The control system adaptively decreases the double support asymmetry that occurs due to environmental perturbations in the split-belt protocol.

Salman Faraji, Philippe Muellhaupt, Auke J. Ijspeert

We present a new walking foot-placement controller based on 3LP, a 3D model of bipedal walking that is composed of three pendulums to simulate falling, swing and torso dynamics. Taking advantage of linear equations and closed-form solutions of the 3LP model, our proposed controller projects intermediate states of the biped back to the beginning of the phase for which a discrete LQR controller is designed. After the projection, a proper control policy is generated by this LQR controller and used at the intermediate time. This control paradigm reacts to disturbances immediately and includes rules to account for swing dynamics and leg-retraction. We apply it to a simulated Atlas robot in position-control, always commanded to perform in-place walking. The stance hip joint in our robot keeps the torso upright to let the robot naturally fall, and the swing hip joint tracks the desired footstep location. Combined with simple Center of Pressure (CoP) damping rules in the low-level controller, our foot-placement enables the robot to recover from strong pushes and produce periodic walking gaits when subject to persistent sources of disturbance, externally or internally. These gaits are imprecise, i.e., emergent from asymmetry sources rather than precisely imposing a desired velocity to the robot. Also in extreme conditions, restricting linearity assumptions of the 3LP model are often violated, but the system remains robust in our simulations. An extensive analysis of closed-loop eigenvalues, viable regions and sensitivity to push timings further demonstrate the strengths of our simple controller.

Salman Faraji, Auke Jan Ijspeert

We present a new framework to generate human-like lower-limb trajectories in periodic and non-periodic walking conditions. In our method, walking dynamics is encoded in 3LP, a linear simplified model composed of three pendulums to model falling, swing and torso balancing dynamics. To stabilize the motion, we use an optimal time-projecting controller which suggests new footstep locations. On top of gait generation and stabilization in the simplified space, we introduce a kinematic conversion method that synthesizes more human-like trajectories by combining geometric variables of the 3LP model adaptively. Without any tuning, numerical optimization or off-line data, our walking gaits are scalable with respect to body properties and gait parameters. We can change various parameters such as body mass and height, walking direction, speed, frequency, double support time, torso style, ground clearance and terrain inclination. We can also simulate the effect of constant external dragging forces or momentary perturbations. The proposed framework offers closed-form solutions in all the three stages which enable simulation speeds orders of magnitude faster than real time. This can be used for video games and animations on portable electronic devices with a limited power. It also gives insights for generation of more human-like walking gaits with humanoid robots.

Salman Faraji, Hamed Razavi, Auke J. Ijspeert

In this paper, we present a simple control framework for on-line push recovery with dynamic stepping properties. Due to relatively heavy legs in our robot, we need to take swing dynamics into account and thus use a linear model called 3LP which is composed of three pendulums to simulate swing and torso dynamics. Based on 3LP equations, we formulate discrete LQR controllers and use a particular time-projection method to adjust the next footstep location on-line during the motion continuously. This adjustment, which is found based on both pelvis and swing foot tracking errors, naturally takes the swing dynamics into account. Suggested adjustments are added to the Cartesian 3LP gaits and converted to joint-space trajectories through inverse kinematics. Fixed and adaptive foot lift strategies also ensure enough ground clearance in perturbed walking conditions. The proposed structure is robust, yet uses very simple state estimation and basic position tracking. We rely on the physical series elastic actuators to absorb impacts while introducing simple laws to compensate their tracking bias. Extensive experiments demonstrate the functionality of different control blocks and prove the effectiveness of time-projection in extreme push recovery scenarios. We also show self-produced and emergent walking gaits when the robot is subject to continuous dragging forces. These gaits feature dynamic walking robustness due to relatively soft springs in the ankles and avoiding any Zero Moment Point (ZMP) control in our proposed architecture.

Salman Faraji, Philippe Muellhaupt, Auke J. Ijspeert

We present a new walking controller based on 3LP, a 3D model of bipedal walking that is composed of three pendulums to simulate falling, swing and torso dynamics. Taking advantage of linear equations and closed-form solutions of 3LP, the proposed controller projects intermediate states of the biped back to the beginning of the phase for which a discrete LQR controller is designed. After the projection, a proper control policy is generated by this LQR controller and used at the intermediate time. The projection controller reacts to disturbances immediately and compared to the discrete LQR controller, it provides superior performance in recovering intermittent external pushes. Further analysis of closed-loop eigenvalues and disturbance rejection strength show strong stabilization properties for this architecture. An analysis of viable regions also show that the proposed controller covers most of the maximal viable set of states. It is computationally much faster than Model Predictive Controllers (MPC) and yet optimal over an infinite horizon.

Salman Faraji, Auke J. Ijspeert

In this paper, we present a new walking controller based on 3LP model. Taking advantage of linear equations and closed-form solutions of 3LP, the proposed controller can project the state of the robot at any time during the phase back to a certain event for which, a discrete LQR controller is designed. After the projection, a proper control policy is generated by the expert discrete controller and used online. This projecting architecture reacts to disturbances with minimal delay and compared to discrete controllers, it provides superior performance in recovering intermittent external pushes. Further analysis of closed-loop eigenvalues and disturbance rejection shows that the proposed time-projecting controller has strong stabilization properties. Controllable regions also show that the projecting architecture covers most of the maximal controllable set of states. It is computationally much faster than model predictive controllers, but still optimal.

Salman Faraji, Auke J. Ijspeert

In this paper, we present a new model of biped locomotion which is composed of three linear pendulums (one per leg and one for the whole upper body) to describe stance, swing and torso dynamics. In addition to double support, this model has different actuation possibilities in the swing hip and stance ankle which could be widely used to produce different walking gaits. Without the need for numerical time-integration, closed-form solutions help finding periodic gaits which could be simply scaled in certain dimensions to modulate the motion online. Thanks to linearity properties, the proposed model can provide a computationally fast platform for model predictive controllers to predict the future and consider meaningful inequality constraints to ensure feasibility of the motion. Such property is coming from describing dynamics with joint torques directly and therefore, reflecting hardware limitations more precisely, even in the very abstract high level template space. The proposed model produces human-like torque and ground reaction force profiles and thus, compared to point-mass models, it is more promising for precise control of humanoid robots. Despite being linear and lacking many other features of human walking like CoM excursion, knee flexion and ground clearance, we show that the proposed model can predict one of the main optimality trends in human walking, i.e. nonlinear speed-frequency relationship. In this paper, we mainly focus on describing the model and its capabilities, comparing it with human data and calculating optimal human gait variables. Setting up control problems and advanced biomechanical analysis still remain for future works.