Robot locomotion
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Robot locomotion refers to the methods by which robots move through their environment, with a particular focus on legged robots that walk, run, climb, and balance using discrete footholds rather than continuous contact like wheels or tracks. Legged locomotion is one of the oldest and most active areas of robotics research, spanning control theory, mechanical engineering, biomechanics, and machine learning. The field draws heavily from the study of animal and human movement, and its ultimate goal is to produce robots that can traverse the same unstructured, complex terrain that biological creatures navigate with ease.
While wheeled and tracked robots remain dominant in structured environments like factory floors and paved roads, legged locomotion offers clear advantages on rough, discontinuous, or cluttered terrain where wheels cannot operate. Stairs, rubble, rocky hillsides, forests, and human-built environments with narrow doorways and ladders all favor legs over wheels. This practical motivation has driven decades of research into bipedal (two-legged), quadrupedal (four-legged), and hexapod (six-legged) robot designs.
The history of legged robot locomotion stretches back to the early days of robotics, with steady progress from slow, statically stable walkers to today's agile, dynamically stable machines that run, jump, and perform acrobatics.
The theoretical foundations of legged locomotion were established in the late 1960s. In 1968, Miomir Vukobratovic and Davor Juricic introduced the concept that would later be called the zero moment point (ZMP), providing a mathematical criterion for the dynamic stability of bipedal walkers. This concept became central to bipedal robot control for the next several decades.
In 1973, Waseda University in Tokyo built WABOT-1, the first full-scale anthropomorphic robot capable of bipedal walking. Its movement was slow and limited to flat surfaces, but it demonstrated that a machine could coordinate legs, vision, and basic decision-making for locomotion.
The 1980s saw two major developments. Shigeo Hirose at the Tokyo Institute of Technology developed the TITAN series of quadruped robots, starting with TITAN III, which could climb stairs and obstacles using simple sensors. At MIT, Marc Raibert built a series of dynamically balancing hopping machines beginning in 1983. Raibert's one-legged hopper at the MIT Leg Laboratory demonstrated that a robot could balance and travel using active dynamic control, bouncing on a single leg like a pogo stick. His subsequent two-legged and four-legged hoppers established the principles of dynamic legged locomotion that would influence the entire field. Raibert later founded Boston Dynamics in 1992.
Honda began its humanoid robotics program in 1986 and spent over a decade developing a series of prototype bipedal walkers (the E-series and P-series) before unveiling ASIMO in 2000. ASIMO stood 130 cm tall and could walk, run (at up to 6 km/h in later versions), climb stairs, recognize voices and faces, and carry objects. ASIMO relied on ZMP-based control, carefully planning its footsteps so that the zero moment point always remained within its support polygon. This approach produced reliable but conservative walking, the kind of stiff, flat-footed gait that became synonymous with humanoid robots of that era.
Several other ZMP-based humanoids followed, including the HRP series developed by AIST in Japan and the HUBO robot at KAIST in South Korea.
Boston Dynamics shifted the field toward highly dynamic locomotion. In 2005, the company (in collaboration with NASA JPL and Harvard) unveiled BigDog, a quadruped robot funded by DARPA for military logistics. BigDog used hydraulic actuation and could walk over rough terrain, climb muddy hillsides, and recover from being kicked. It was the first robot to demonstrate truly animal-like locomotion over unstructured terrain.
Boston Dynamics followed BigDog with a series of increasingly capable platforms:
The MIT Biomimetic Robotics Lab, led by Sangbae Kim, developed a series of quadruped robots focused on speed and energy efficiency. The MIT Cheetah could run at approximately 22 km/h (14 mph) on a treadmill and was designed to rival the locomotion efficiency of animals at the same scale. In 2015, the MIT Cheetah 2 demonstrated autonomous jumping over obstacles up to 46 cm (18 inches) tall while maintaining a running speed of 8 km/h.
The MIT Mini Cheetah, a smaller and more affordable platform, became widely used in research. In 2022, researchers used reinforcement learning to train the Mini Cheetah to run at 3.9 m/s (roughly 14 km/h), discovering that the learning algorithm accumulated the equivalent of 100 days of real-world experience in just three hours of simulation time.
Agility Robotics, a spin-off from Oregon State University founded in 2015, developed Cassie, a bipedal robot with an ostrich-like leg design featuring 3-degree-of-freedom hip joints. Cassie was funded by a $1 million DARPA grant and became a testbed for learning-based locomotion control.
In 2021, Cassie became the first bipedal robot to use machine learning to control a running gait on outdoor terrain, completing a 5K run on the Oregon State campus in 53 minutes on a single battery charge. In 2022, Cassie set a Guinness World Record for the fastest 100 meters by a bipedal robot, completing the distance in 24.73 seconds.
Agility Robotics subsequently developed Digit, a full humanoid robot (1.7 m tall, 42 kg) designed for logistics and warehouse work. Digit features bipedal locomotion with arms and grasping capabilities. By early 2026, Digit robots were working 8-hour shifts at a Schaeffler Group factory in South Carolina, hauling components between stamping presses and conveyor belts.
Unitree entered the humanoid robot market with two platforms. The H1, standing 1.8 m tall and weighing 47 kg, achieved bipedal running at 3.3 m/s (about 12 km/h), which was a world record for full-size humanoids at the time of its announcement, with the company claiming potential speeds exceeding 5 m/s. The smaller G1 (1.32 m, 35 kg) offers 23 to 43 degrees of freedom and supports walking speeds up to 2 m/s, stair climbing, and dynamic balance recovery including the ability to get up from a supine position. As of 2025, the G1 had shipped over 1,000 units, making it the best-selling humanoid robot on the market.
Bipedal (two-legged) locomotion is the most challenging form of legged robot movement because the robot has only two points of ground contact and must continuously maintain balance. A standing biped is inherently unstable, similar to an inverted pendulum, and must actively control its posture at all times.
Bipedal robots can be broadly divided into two categories based on their approach to stability:
| Category | Description | Examples |
|---|---|---|
| Static walkers | Keep the center of mass projected within the support polygon at all times. Produce slow, flat-footed gaits. | ASIMO, WABOT-1, NAO |
| Dynamic walkers | Allow the center of mass to move outside the support polygon temporarily, relying on momentum and active control to maintain balance. Produce faster, more natural gaits. | Atlas, Cassie, Digit, Unitree H1 |
Dynamic walking is more energy-efficient and faster than static walking, but it requires more sophisticated control systems.
Quadruped (four-legged) robots are inherently more stable than bipeds because they can maintain a wider support polygon. A quadruped walking with a static gait always keeps at least three feet on the ground, forming a stable tripod. At higher speeds, quadrupeds use dynamic gaits that include aerial phases (moments when no feet touch the ground).
Common quadrupedal gaits include:
| Gait | Description | Typical speed |
|---|---|---|
| Walk | Legs move one at a time in sequence (e.g., right hind, right front, left hind, left front). At least two or three feet remain on the ground at all times. | Slow |
| Trot | Diagonal leg pairs move together in antiphase. Two feet are on the ground at a time. | Medium |
| Pace | Legs on the same side move together in phase. | Medium |
| Bound | Front legs and hind legs move together, each pair in antiphase with the other. | Fast |
| Gallop | An asymmetric gait where legs contact the ground in sequence with brief aerial phases. | Fast |
| Pronk | All four legs move in phase, launching the robot into the air simultaneously (similar to a springbok). | Variable |
Quadruped animals transition smoothly between gaits depending on speed. Reproducing these transitions in robots has been an active research topic, with recent work using reinforcement learning to discover automatic gait transitions.
Six-legged (hexapod) robots draw inspiration from insects. They offer even greater static stability than quadrupeds; a hexapod can always keep three legs on the ground in an alternating tripod gait while moving the other three. This makes hexapods naturally suited for rough terrain where stability is more important than speed. Examples include the RHex robot from the University of Michigan and Boston Dynamics' early RiSE climbing robot.
Before the rise of learning-based methods, legged locomotion relied on analytical control techniques derived from dynamics and control theory. Several of these approaches remain in active use today, often combined with modern methods.
The zero moment point is the point on the ground where the net moment of all active forces (gravity, inertia, ground reaction) produces zero horizontal torque. For a legged robot to remain dynamically stable, the ZMP must stay within the convex hull of the foot contact points (the support polygon).
The ZMP concept, introduced by Vukobratovic in 1968 to 1972, became the dominant framework for bipedal walking control through the 1990s and 2000s. The standard approach involves planning a desired ZMP trajectory, then computing the corresponding center-of-mass trajectory using a simplified model (typically the linear inverted pendulum model), and finally tracking these trajectories with joint-level controllers. Kajita et al. developed a preview control method in 2003 that used future ZMP reference values to generate smooth walking patterns, and this approach was widely adopted in humanoid robots including ASIMO and the HRP series.
The main limitation of ZMP-based control is that it requires the robot to keep its feet flat on the ground and maintain a conservative gait, which limits speed and agility.
The linear inverted pendulum model approximates a walking biped as a point mass atop a massless leg, constrained to move at a constant height. This simplification reduces the complex multi-body dynamics of a humanoid to a single linear differential equation, making real-time trajectory planning tractable.
The LIPM is used in conjunction with ZMP-based control: given a desired ZMP trajectory, the LIPM equations determine how the center of mass must move. Extensions of this model include the capture point (also called the divergent component of motion), which represents the point on the ground where the robot must step to avoid falling. Capture point-based controllers enable more reactive walking that can respond to pushes and disturbances in real time.
The spring-loaded inverted pendulum (SLIP) model extends the inverted pendulum concept by adding a compliant leg. When the foot contacts the ground, the leg compresses like a spring, storing and releasing energy. The SLIP model captures the fundamental dynamics of running gaits in both animals and robots, and it has been used to study bipedal, quadrupedal, and multi-legged locomotion. The MIT Cheetah robots used SLIP-inspired control to achieve efficient running.
Central pattern generators are neural circuits in animals that produce rhythmic motor patterns (such as walking or swimming) without requiring rhythmic sensory input. Researchers have built artificial CPGs using mathematical oscillators (such as Hopf oscillators, Matsuoka oscillators, or SO(2) oscillators) to generate coordinated leg movements for robots.
CPG-based controllers generate stable rhythmic patterns with low computational cost and can be modulated by simple high-level commands (speed, direction, gait type). They also recover quickly from perturbations due to their limit-cycle behavior. Notable examples include Ijspeert et al.'s salamander robot, which used a spinal cord CPG model to transition between swimming and walking gaits, and Kimura et al.'s quadruped that walked stably on irregular outdoor terrain using a CPG neural system with sensory feedback.
CPG controllers are typically integrated with sensory feedback to adapt to external disturbances. The feedback can be simple reflex rules inspired by biology or trained neural networks for more complex behaviors.
Hybrid zero dynamics, developed primarily by Jessy Grizzle and collaborators at the University of Michigan, provides a rigorous mathematical framework for designing stable walking controllers for underactuated bipedal robots. The approach uses virtual constraints (relationships between joint angles imposed by feedback control) to synchronize the robot's joints to an internal gait phasing variable. This reduces the full hybrid dynamics of walking (with its continuous motion phases and discrete impact events) to a low-dimensional zero dynamics manifold that captures the essential underactuated behavior.
HZD-based controllers have been implemented on several physical robots, including the MABEL robot, which demonstrated stable, fast running using this framework. The approach provides provable stability guarantees, which is an advantage over purely empirical methods.
Passive dynamic walkers exploit gravity and the natural pendulum-like swinging of legs to walk down slopes with zero energy input. Tad McGeer demonstrated in 1990 that a simple mechanical device with no motors or controllers could walk stably down a gentle slope, and this insight has influenced energy-efficient robot design ever since.
The Cornell Ranger, developed at Cornell University, demonstrated the practical potential of passive-dynamics-inspired design. In 2011, Cornell Ranger walked 65.17 km (40.5 miles) on a single battery charge without human intervention, setting a distance record for walking robots. Its total cost of transport (TCOT) was 0.19, far lower than any other legged robot at the time and comparable to the efficiency of biological locomotion.
Modern classical control for legged robots often combines model predictive control (MPC) with whole-body control (WBC) in a hierarchical architecture.
Model predictive control solves an optimization problem at each control step, computing the optimal sequence of actions over a finite time horizon given a simplified dynamic model of the robot. For legged robots, MPC typically uses a single rigid body model to compute optimal ground reaction forces that will move the robot's body along a desired trajectory while respecting friction cone constraints and contact schedules.
MPC runs at a relatively low frequency (often 20 to 50 Hz) because of the computational cost of solving the optimization problem, but it provides a principled way to plan dynamically feasible motions over a longer time horizon.
Whole-body control takes the desired body motion and ground reaction forces from the MPC layer and computes the joint torques, positions, and velocities needed to achieve them, accounting for the full rigid-body dynamics of the robot. WBC typically runs at a higher frequency (200 to 1000 Hz) and uses prioritized task-space optimization to handle conflicting objectives (e.g., tracking a desired body trajectory while maintaining contact constraints).
This MPC + WBC architecture has been used successfully on platforms including the MIT Mini Cheetah, ANYmal, and various humanoid robots. It enables dynamic maneuvers including running with aerial phases, jumping, and recovering from pushes.
More recent work uses nonlinear MPC, which retains the full nonlinear dynamics of the robot rather than relying on a linearized model. NMPC can optimize more complex gaits and highly nonlinear trajectories in real time, though at greater computational cost. Hybrid WBC frameworks that combine task prioritization with weight-based coordination have been developed to work with NMPC for agile quadrupedal locomotion.
Since roughly 2019, reinforcement learning (RL) has transformed legged robot locomotion. RL-based controllers now match or exceed the performance of hand-tuned classical controllers on many tasks, and they can discover locomotion behaviors that human engineers would not have designed.
RL locomotion policies are almost always trained in physics simulation rather than on real hardware, because training requires millions or billions of steps of trial-and-error experience. Modern GPU-accelerated simulators like NVIDIA Isaac Sim, MuJoCo, and PyBullet can simulate thousands of robot instances in parallel, accumulating years of equivalent experience in hours of wall-clock time.
The dominant training algorithm for locomotion is Proximal Policy Optimization (PPO), a policy gradient method published by OpenAI in 2017. PPO is sample-efficient, stable during training, and scales well to the high-dimensional observation and action spaces of legged robots.
A typical RL locomotion policy takes as input the robot's joint positions, joint velocities, body orientation (from an IMU), and a velocity command, and outputs desired joint positions or torques at each control step.
Sim-to-real transfer is the process of deploying a policy trained in simulation on a physical robot. The main challenge is the "reality gap": differences between the simulated and real physics (friction, actuator dynamics, sensor noise, contact modeling) that can cause a policy that works perfectly in simulation to fail on hardware.
Two primary techniques address this gap:
| Technique | Description |
|---|---|
| Domain randomization | During training, the simulation randomly varies physical parameters (mass, friction, joint damping, motor strength, sensor noise, terrain properties) across episodes. The policy learns behaviors that are robust to a wide range of dynamics, increasing the likelihood that they transfer to the real robot. |
| System identification | The simulation is carefully calibrated to match the real robot's dynamics as closely as possible, reducing the gap that the policy must bridge. This can include identifying motor transfer functions, contact parameters, and sensor characteristics. |
In practice, most successful sim-to-real locomotion systems use both techniques. Domain randomization provides robustness, while system identification reduces the range of variation the policy must handle.
A 2025 framework for systematic sim-to-real transfer integrated physics-grounded energy models for actuators with reinforcement learning, achieving a 32% reduction in the cost of transport on ANYmal robots and successful deployment across thirteen different legged platforms.
A common training paradigm, pioneered by ETH Zurich's Robotic Systems Lab for the ANYmal robot, uses a two-stage teacher-student approach:
The student policy is then deployed on real hardware with zero-shot transfer (no further fine-tuning on the real robot). This approach has enabled ANYmal to traverse muddy trails, snowy slopes, dense vegetation, and streams without manual tuning or pre-mapping of the environment.
RL-trained policies can learn to adapt their gait to terrain without explicit terrain classification. By training on randomized terrain in simulation (stairs, slopes, rough ground, gaps, stepping stones), the policy implicitly learns to estimate terrain properties from proprioceptive signals and adjust its foot placement and body posture accordingly.
ETH Zurich demonstrated this capability with ANYmal parkour, where the quadruped learned to perform agile navigation including jumping onto and off of obstacles, using reinforcement learning trained entirely in simulation. The team also taught ANYmal to climb ladders by developing custom hook-like paws and training with RL, achieving a 90% success rate in real-world tests.
Traversing rough or unknown terrain requires robots to perceive their surroundings and adapt their locomotion accordingly.
Some locomotion controllers operate without any exteroceptive sensors (cameras or lidar), relying solely on proprioception: joint positions, joint velocities, body orientation, and contact forces. These "blind" controllers use the history of proprioceptive observations to implicitly estimate terrain properties (e.g., whether the ground is soft, slippery, or sloped) and adapt in real time.
Blind locomotion controllers are robust because they do not depend on visual perception, which can fail in poor lighting, rain, fog, or dust. ETH Zurich's work on blind quadrupedal locomotion over challenging terrain, published in Science Robotics, showed that a proprioceptive RL policy could navigate rocky trails, forest undergrowth, and construction sites without any vision.
Perceptive locomotion uses depth cameras, lidar, or other exteroceptive sensors to build a local terrain map and plan footstep placements. This is necessary for tasks like stair climbing, gap crossing, and navigating discrete footholds where the robot cannot simply react to terrain after contact.
Recent approaches include:
Stair climbing is one of the most practically important terrain adaptation tasks, since stairs are ubiquitous in human environments. Both quadrupedal and bipedal robots have demonstrated stair climbing capabilities:
Modern humanoid and quadruped robots increasingly need to coordinate locomotion with other tasks such as manipulation, carrying objects, or recording video.
Loco-manipulation refers to performing manipulation tasks (grasping, carrying, pushing) while walking. This requires coordinating the legs for balance and locomotion while the arms perform the manipulation task. The challenge is that manipulation forces and carried loads shift the robot's center of mass and create disturbance torques that the locomotion controller must compensate for.
Figure 02, a humanoid robot developed by Figure AI, demonstrated loco-manipulation during 10-hour autonomous shifts at BMW facilities in 2025. The robot walked at 1.2 m/s while carrying its full 20 kg payload capacity, maintaining stability through IMU, gyroscope, and force sensor arrays.
In 2025, Carnegie Mellon University researchers introduced SoFTA, a two-agent control system for keeping humanoid robot hands stable while walking. A slow agent controls leg movement at 50 Hz, while a fast agent stabilizes the upper body at 100 Hz. Tested on the Unitree G1, this approach reduced hand acceleration by 50 to 80%, allowing the robot to carry water without spilling and record smooth video while walking.
The energy efficiency of locomotion is quantified using the cost of transport (COT), a dimensionless metric defined as:
COT = P / (m * g * v)
where P is power consumption, m is mass, g is gravitational acceleration, and v is forward velocity. Lower values indicate more efficient locomotion.
| System | COT | Notes |
|---|---|---|
| Human walking (1.0 m/s) | ~0.41 (total metabolic) | Mechanical COT ~0.05 |
| Cornell Ranger | ~0.19 (total electrical) | Record for legged robots (2011) |
| MIT Cheetah | ~0.5 | Comparable to animals of similar size |
| Typical humanoid robot | 2 to 5+ | Much less efficient than biological locomotion |
| Wheeled robot | ~0.01 to 0.1 | Wheels are far more efficient on flat ground |
The large gap between biological and robotic locomotion efficiency remains an open problem. Passive dynamic principles, compliant actuators, and energy-storing mechanisms (like springs in series-elastic actuators) are all approaches to closing this gap.
The following table summarizes notable legged robots that have demonstrated significant locomotion capabilities as of early 2026.
| Robot | Developer | Type | Legs | Weight | Top speed | Notable locomotion capabilities |
|---|---|---|---|---|---|---|
| Atlas (electric) | Boston Dynamics | Humanoid | 2 | ~85 kg | Not disclosed | Parkour, backflips, gymnastics, dynamic whole-body movements |
| Spot | Boston Dynamics | Quadruped | 4 | ~32 kg | ~1.6 m/s | Autonomous navigation, stair climbing, self-righting, obstacle avoidance |
| Digit | Agility Robotics | Humanoid | 2 | ~42 kg | ~1.5 m/s | Warehouse operations, carrying loads, 8-hour autonomous shifts |
| Unitree H1 | Unitree | Humanoid | 2 | ~47 kg | 3.3 m/s (record) | Fast bipedal running, AI-powered gait generation |
| Unitree G1 | Unitree | Humanoid | 2 | ~35 kg | ~2 m/s | Stair climbing, dynamic recovery, kip-up from supine |
| ANYmal | ANYbotics / ETH Zurich | Quadruped | 4 | ~50 kg | ~1.5 m/s | Parkour, ladder climbing, blind rough terrain, industrial inspection |
| MIT Mini Cheetah | MIT | Quadruped | 4 | ~9 kg | 3.9 m/s | RL-trained running, backflips, research platform |
| Tesla Optimus | Tesla | Humanoid | 2 | ~73 kg | ~1.3 m/s | Natural heel-to-toe stride, jogging (Gen 2/3, in development) |
| Figure 02 | Figure AI | Humanoid | 2 | ~60 kg | 1.2 m/s | 10-hour autonomous shifts, RL-trained walking, load carrying |
| Cassie | Agility Robotics | Biped (legless torso) | 2 | ~31 kg | ~4 m/s | 5K run, 100m in 24.73s (world record), outdoor RL locomotion |
Despite rapid progress, several major challenges remain in robot locomotion.
Current locomotion controllers, whether classical or learned, still fail in situations that animals handle easily. Unexpected terrain (ice, deep mud, loose gravel), extreme slopes, strong wind gusts, and novel obstacles can all cause falls. Real-world deployment demands hundreds or thousands of hours of continuous operation without failure, and no current system reliably achieves this outside of structured environments.
Legged robots remain far less energy efficient than their biological counterparts. A typical humanoid robot has a cost of transport several times higher than a human walking at the same speed. Improving efficiency requires advances in actuator design (more efficient motors, series-elastic actuators, variable-stiffness mechanisms), lighter materials, and control algorithms that exploit passive dynamics.
While robots like Atlas can perform impressive acrobatics in demonstration settings, sustained high-speed running over rough terrain remains elusive. The fastest bipedal robots reach roughly 3 to 4 m/s, well below the 10+ m/s that humans can sprint. Quadrupedal robots are similarly limited compared to their animal counterparts.
The locomotion research community lacks widely accepted benchmarks for comparing robot performance. Different labs test on different terrains, at different speeds, with different success criteria. Proposals for standardized benchmarking, such as the work on HRP-2 humanoid benchmarking and the MTBench suite (featuring 20 locomotion tasks), aim to address this gap, but no single standard has been adopted across the field.
As humanoid robots move from research labs into warehouses and factories, they need to walk while simultaneously performing useful tasks with their arms. Coordinating locomotion and manipulation in real time, especially when carrying heavy or awkward loads, remains an area of active research.
Current locomotion controllers are largely reactive, responding to terrain within a few steps ahead. Navigating complex environments (a cluttered construction site, a multi-story building, outdoor terrain with distant obstacles) requires integrating locomotion control with higher-level path planning and semantic scene understanding.
Several academic and industrial labs have made sustained contributions to robot locomotion research:
| Lab/Group | Institution | Focus areas |
|---|---|---|
| Robotic Systems Lab | ETH Zurich | ANYmal, RL-based locomotion, sim-to-real, rough terrain |
| Biomimetic Robotics Lab | MIT | MIT Cheetah series, energy-efficient running, actuator design |
| AMBER Lab | Caltech / Georgia Tech | Hybrid zero dynamics, formal methods for bipedal walking |
| Hybrid Robotics Lab | UC Berkeley | Safety-critical locomotion control, MPC with control barrier functions |
| Dynamic Robotics Lab | Oregon State University | Cassie, Digit, dynamic bipedal walking |
| Boston Dynamics | (Industry) | Atlas, Spot, commercial legged robots |
| Unitree Robotics | (Industry) | Affordable quadruped and humanoid platforms |
| Agility Robotics | (Industry) | Digit, commercial bipedal robots |
| LAAS-CNRS | Toulouse, France | HRP humanoids, multi-contact planning |