Bipedal locomotion
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Bipedal locomotion is movement powered by two legs. In robotics it refers to the engineering and control problem of building machines that walk, run, balance, climb stairs, and recover from disturbances on two legs. The topic sits at the intersection of classical mechanics, control theory, biomechanics, and machine learning. It is the dominant gait of humanoid robots and is also studied in prosthetics and biomechanics, where the goal is to understand and reproduce the way humans and other bipeds move.
The field has been worked on continuously since the early 1970s. Waseda built the first full-scale walking humanoid in 1973, and Honda spent more than three decades developing ASIMO. Boston Dynamics, Agility Robotics, Tesla, Figure AI, Apptronik, Unitree, and several Chinese laboratories now ship or demonstrate humanoids whose walking quality has improved sharply since around 2018, driven largely by simulation-based reinforcement learning and faster electric actuators.
The practical argument for bipedal robots is that the world is built for human bodies. Doors, stairs, ladders, vehicle cabs, kitchens, warehouses, and factory floors are all sized for people. A wheeled robot can solve many tasks more cheaply, but it cannot climb the stairs in a typical house or step over a cable on a factory floor. A bipedal humanoid does not need the environment to change.
From a research point of view, bipedal locomotion is a long-standing motor-control benchmark. A walking biped is underactuated: the foot can push or rock, but it cannot pull on the ground, so the contact wrench is constrained. The system is hybrid (continuous dynamics interrupted by discrete impacts), high-dimensional, and sensitive to small modeling errors. For decades it was the canonical hard problem in legged robotics, and it has become the testbed for sim-to-real transfer of learned policies in the 2020s. Until roughly 2018 the field was dominated by model-based control built around the zero-moment point and reduced-order pendulum models. After learned controllers succeeded on quadrupeds (ANYmal, Spot, Unitree's quadruped line), the same recipe began to work on bipeds. Cassie, Digit, Atlas, and Unitree's H1 and G1 now run policies trained almost entirely in simulation, then deployed on the real hardware with little or no fine tuning.
A few quantities and reduced-order models recur in almost every bipedal control paper. The same vocabulary is used in biomechanics, so the field shares a common language with the prosthetics and gait-analysis communities.
The center of mass (CoM) is the mass-weighted average position of all the body's links. For a humanoid, the CoM lives roughly behind the navel and slightly above the hips. Almost every gait planner reasons about CoM trajectories rather than the full 30-plus-dimensional joint state, because the CoM captures the dominant dynamics of falling and catching.
The center of pressure (CoP) is the point on the support surface where the resultant ground reaction force can be considered to act. The zero-moment point (ZMP), introduced by Miomir Vukobratović in the late 1960s, is the point on the ground at which the net moment of inertia and gravity forces has no component along the horizontal axes. While the foot is fully in contact and not slipping, the CoP and the ZMP coincide. As long as the ZMP stays strictly inside the foot's support polygon, the foot does not rotate and the robot cannot fall by tipping. Most classical humanoid walking controllers, including the Honda ASIMO line, are organized around keeping the ZMP inside the support polygon.
The capture point, introduced by Pratt, Carff, Drakunov, and Goswami in 2006, is the point on the ground where a humanoid would have to step to come exactly to rest, given its current CoM position and velocity. The capture region generalizes this to all states from which a single step can stop the robot. Capture-point control is widely used for push recovery.
The linear inverted pendulum model (LIPM), formalized in three dimensions by Kajita and colleagues in 2001, treats the humanoid as a point mass on a massless leg, constrained so that the CoM moves on a horizontal plane. Under that constraint the dynamics become linear and decoupled in the sagittal and lateral directions, and the CoM trajectory has a closed-form solution given the ZMP trajectory. The LIPM is the workhorse model for ZMP-based gait generation.
The spring-loaded inverted pendulum (SLIP) is a point mass on a massless springy leg. SLIP captures the hopping and running dynamics of humans, kangaroos, and many birds with a single elastic element. It is the standard reduced-order model for running gaits and is the conceptual ancestor of Marc Raibert's hopping machines and of compliant humanoid legs like Cassie's.
A bipedal step is usually decomposed into three phases. In single support, one foot is on the ground and the other is swinging forward. In double support, both feet are on the ground, and weight transfers from the trailing leg to the leading leg. The swing phase is the portion of single support during which the swing leg is moving. Walking has a long double-support phase; running has no double support and instead has a flight phase in which neither foot touches the ground.
A biped is statically balanced if the projection of its CoM onto the ground stays inside the support polygon at all times. Static walking is slow and energy-hungry, but it is easy to plan and was the basis of WABOT-1's gait. Dynamic balance allows the CoM to leave the support polygon momentarily, as long as the next foot placement catches the fall. Almost all natural-looking humanoid walking is dynamically balanced.
Tad McGeer's 1990 paper Passive Dynamic Walking showed that a class of two-legged machines, given the right geometry and inertia, will walk down a shallow slope with a steady, human-like gait without any actuators or controllers. Passive walkers are powered only by gravity and tuned by mass distribution; they walk because they are essentially set up to fall forward, with the swing leg arriving in time to catch the fall. The result suggests that much of human walking is the natural mode of the mechanics rather than something the nervous system has to compute. Powered passive walkers from Cornell, MIT, and Delft add small amounts of actuation to keep a passive gait going on level ground at energy costs comparable to humans.
Bipedal walking robots have been built for more than half a century. The history below focuses on machines that were both real hardware and clearly influential.
| Year | Robot | Builder | Notes |
|---|---|---|---|
| 1973 | WABOT-1 | Waseda University (Ichiro Kato) | First full-scale anthropomorphic biped; quasi-dynamic walking, vision and speech, very slow |
| 1986 | E0 | Honda | First robot in Honda's E series, 1986 to 1993; experimental static walking |
| 1996 | P2 | Honda | First fully self-contained, untethered humanoid biped |
| 1997 | P3 | Honda | First completely autonomous bipedal humanoid; 160 cm, 130 kg |
| 2000 | ASIMO (original) | Honda | 120 cm, 52 kg; ZMP-based control; updated through 2011 |
| 2002 | HRP-2 | Kawada Industries and AIST | Designed by Yutaka Izubuchi; can lift heavy objects and stand back up |
| 2009 | PETMAN | Boston Dynamics | First Boston Dynamics humanoid; built to test chemical-protective suits |
| 2010 | HRP-4 | Kawada Industries and AIST | 150 cm, 39 kg; lightweight research platform |
| 2013 | Atlas (DRC) | Boston Dynamics for DARPA | 188 cm, 28 hydraulic DoF; built for the DARPA Robotics Challenge |
| 2016 | Atlas (next-gen) | Boston Dynamics | Smaller, electric-and-hydraulic redesign; first viral parkour videos |
| 2017 | Cassie | Agility Robotics (Oregon State spinout) | Bird-like legs, no upper body; first low-cost RL-friendly biped |
| 2018 | HRP-5P | AIST | 182 cm, 101 kg; designed for heavy labor and construction tasks |
| 2019 | Digit | Agility Robotics | Cassie legs plus a torso, arms, and a perception head |
| 2022 | ASIMO retired | Honda | Honda formally retires ASIMO in March 2022 |
| 2022 | Cassie 100m record | Oregon State | 24.73 seconds over 100 m, Guinness record (May 2022) |
| 2023 | Apollo | Apptronik | 173 cm, 73 kg; commercial humanoid built around custom linear actuators |
| 2023 | Phoenix | Sanctuary AI | 170 cm, 70 kg; 20-DoF hands; pilot-assisted operation |
| 2023 | Optimus Gen 2 | Tesla | 173 cm, 57 kg; unveiled in December 2023 |
| 2023 | H1 | Unitree | 178 cm, 70 kg; 27 DoF; 3.3 m/s running speed |
| 2023 | GR-1 | Fourier Intelligence | 165 cm, 55 kg; 40 DoF; 50 kg payload |
| 2024 | Electric Atlas | Boston Dynamics | All-electric replacement for hydraulic Atlas; announced one day after retirement of HD Atlas |
| 2024 | Figure 02 | Figure AI | 168 cm, 70 kg; deployed at BMW Spartanburg |
| 2024 | G1 | Unitree | About 132 cm, 35 kg; 23 DoF base, up to 43 DoF in EDU configuration; base price near $16,000 |
| 2024 | NEO Beta | 1X Technologies | 165 cm, 30 kg; tendon-driven, designed for the home |
| 2025 | Beijing humanoid half marathon | Multiple Chinese labs | Tien Kung Ultra finishes 21 km in about 2 h 40 min (April 2025) |
| 2026 | Beijing humanoid half marathon | Honor "Lightning" | Reported half-marathon time of 50 min 26 s; Honor sweeps the top six places (April 2026) |
Most of these machines used a model-based stack until around 2020. Cassie was the first widely deployed biped where simulation-trained RL policies became the default control approach, partly because its compliant legs and bird-like geometry simulate well.
There are several distinct control philosophies, and most modern humanoid stacks combine elements of all of them.
| Approach | Key idea | Representative work | Strengths | Weaknesses |
|---|---|---|---|---|
| ZMP-based gait generation | Plan a CoM trajectory that keeps the ZMP inside the support polygon | Honda ASIMO; Kajita et al. 2003 preview control | Stable, well-understood, predictable; deployable on real hardware since the 1990s | Conservative gaits, limited push recovery, looks robotic |
| Capture point and divergent component of motion | Use the capture point as a feedback variable for foot placement | Pratt et al. 2006; later DRC humanoids | Good push recovery, principled handling of disturbances | Still relies on simplified pendulum dynamics |
| Reduced-order models | Plan with LIPM or SLIP, then track the plan with a whole-body controller | Kajita et al. 2001 (3D LIPM); SLIP literature for running | Cheap to compute; closed-form solutions | Ignores arms, swing-leg dynamics, and contact richness |
| Whole-body control (WBC) | Solve a quadratic program at every control tick to produce torques that satisfy contact and task constraints | Used on HRP-series, Atlas DRC, many research humanoids | Handles many tasks at once, respects torque limits | Needs an accurate dynamics model and good state estimation |
| Trajectory optimization through contact | Optimize state, control, and contact forces jointly with complementarity constraints | Posa, Cantu, Tedrake 2014 (Drake-style direct collocation) | Discovers contact patterns automatically; powerful for offline planning | Solving the optimization is expensive and brittle |
| Model-predictive control (MPC) | Re-solve a short-horizon optimization at high frequency | Many recent humanoid stacks | Reactive, handles disturbances | Real-time solver design is hard |
| Reinforcement learning | Learn a policy in simulation by trial and error, then deploy it on hardware | Siekmann et al. 2021 (Cassie); Radosavovic et al. 2024 (Digit, H1, Atlas) | Robust, expressive, easy to combine with vision | Requires good simulators and domain randomization; safety guarantees are weaker |
| Hybrid model-based plus learned | Use a learned residual on top of an MPC plan, or a learned reward model with a planner | ETH RSL group, Boston Dynamics blog posts | Combines the structure of model-based control with the flexibility of RL | More moving parts, harder to debug |
The Kajita 2003 ICRA paper Biped Walking Pattern Generation by using Preview Control of Zero-Moment Point is the canonical reference for ZMP preview control. It models the humanoid as a cart on a table and designs a tracking servo for a desired ZMP, using a preview window of about 1.6 seconds of future reference. That paper underlies a generation of humanoid stacks at AIST and elsewhere.
The Posa, Cantu, and Tedrake 2014 paper, in the International Journal of Robotics Research, formulated trajectory optimization for rigid bodies as a mathematical program with complementarity constraints. The method does not require a pre-specified mode sequence: the optimizer figures out when each foot should make or break contact. This approach is implemented in MIT's Drake toolbox and influenced the planners used at the DARPA Robotics Challenge.
The pivot to learned controllers happened in the late 2010s. Tan et al. (2018) showed that simulation-trained policies could control real quadrupeds when training included randomized dynamics, latency, and motor models. The same recipe transferred to bipeds. Siekmann, Godse, Fern, and Hurst (ICRA 2021) showed that a single parametric reward, built from probabilistic periodic costs, lets a learned policy produce all the standard bipedal gaits (standing, walking, hopping, running, skipping) on Cassie. The 2022 Cassie 100 m record used the same lineage of policies and ran without external sensors.
Radosavovic, Xiao, Zhang, Darrell, Malik, and Sreenath published Real-World Humanoid Locomotion with Reinforcement Learning in Science Robotics in April 2024. They trained a causal transformer on a history of proprioceptive observations and actions in simulation, then deployed it zero-shot on a Digit humanoid for outdoor walking on grass, gravel, sidewalks, and packing materials. Follow-up work extended the approach to challenging terrain, and similar controllers now run on Unitree H1 and the electric Atlas. Most humanoid RL today runs on top of NVIDIA's Isaac Lab and ETH's open-source RSL-RL framework; training a humanoid walking policy on one workstation now takes hours rather than weeks because thousands of parallel environments run on a GPU.
The practical question for any learned controller is whether it survives the transfer from a simulator to a noisy, friction-varying physical robot. The standard answer is domain randomization: train across a distribution of simulator parameters (mass, inertia, latency, friction, motor backlash) so the policy learns to be robust to mismatch. Combined with fast parallel simulation, this is what made sim-to-real work for both quadrupeds and bipeds.
A few results from 2022 to 2025 stand out. Cassie's 24.73-second 100 m run at OSU's Whyte Track and Field Center on May 11, 2022 was the first Guinness-recognized bipedal-robot 100 m record. Cassie started from a standing position, returned to it without falls, and ran a learned policy trained for the equivalent of a year of simulation. Boston Dynamics' Atlas parkour videos, released between 2018 and 2023, used an offline trajectory-optimization stack on the hydraulic Atlas; the 2024 transition to electric Atlas reset the platform with new actuators, new joint ranges (including fully rotational joints), and an explicit shift toward learned and data-driven control. Radosavovic et al. (2024) deployed a single Transformer policy to walk a Digit humanoid through the Berkeley campus. Tesla has shown Optimus Gen 2 walking, navigating stairs, and performing factory tasks at its Fremont and Austin sites starting in mid-2024.
In April 2025, Beijing hosted what was described as the world's first humanoid-robot half marathon; Tien Kung Ultra completed the 21 km course in about two hours and forty minutes. A second running in April 2026 produced a much faster reported time of 50 min 26 s by Honor's "Lightning" robot. These events involve heavy human supervision and frequent battery swaps, so comparisons to human marathon times need caveats.
Bipedal locomotion looks easy in a finished demo, and very hard when you try to build one. The recurring problems are:
Specifications are taken from manufacturers' published material and reputable robotics outlets. Numbers can change between hardware revisions, so the values below should be read as approximate.
| Robot | Manufacturer | Year | Height | Weight | DoF | Control approach |
|---|---|---|---|---|---|---|
| Atlas (electric) | Boston Dynamics | 2024 | About 1.5 m | About 89 kg | About 28 | Hybrid model-based and learned, all-electric actuators |
| Tesla Optimus Gen 2 | Tesla | 2023 | 173 cm | 57 kg | 28 (11 per hand) | Vertically integrated; simulation-heavy training |
| Figure 02 | Figure AI | 2024 | 168 cm | 70 kg | 16 per hand; full-body high count | End-to-end learned policies; deployed at BMW Spartanburg |
| Unitree H1 | Unitree | 2023 | 178 cm | 70 kg | 27 | Sim-to-real RL on top of stiff electric actuators |
| Unitree G1 | Unitree | 2024 | About 132 cm | 35 kg | 23 (base) up to 43 (EDU Ultimate) | Low-cost research and consumer platform |
| Apollo | Apptronik | 2023 | 173 cm | 73 kg | High count | Custom linear electric actuators |
| Phoenix | Sanctuary AI | 2023 | 170 cm | 70 kg | 20 per hand | Carbon AI control; pilot-assisted operation |
| Digit | Agility Robotics | 2019 to present | 175 cm | 65 kg | 30 | Sim-to-real RL plus task-specific scripts |
| Fourier GR-1 | Fourier Intelligence | 2023 | 165 cm | 55 kg | 40 | Hybrid; aimed at care and labor tasks |
| NEO Beta | 1X Technologies | 2024 | 165 cm | 30 kg | 55 | Tendon-driven actuation; learned home policies |
The field is in an unusual state: walking on flat ground in a controlled lab is more or less solved, while almost everything else is still an active research problem.
Foundation models are the most visible new direction. NVIDIA's Project GR00T, announced in 2024 and continued as Isaac GR00T N1 in 2025, is a vision-language-action foundation model for humanoid robots, with a System 2 vision-language module producing high-level intent and a System 1 diffusion transformer producing motor actions. Whether such models will subsume locomotion-specific policies, or sit on top of them as a high-level planner, is still being worked out. Training pipelines that mix real teleoperation data with synthetic trajectories generated in Isaac Lab are now standard.
Generalization is another open frontier. Most published demos work on a fixed terrain class (flat warehouse floor, paved sidewalk, mowed grass). Walking reliably on stairs, ladders, ice, soft sand, or piles of cables is still hard. Loco-manipulation, the joint problem of moving and manipulating, gets renewed attention every time a humanoid is supposed to do warehouse work. Figure's BMW deployment, Apptronik's logistics pilots, and 1X's home pilots all run into the same wall: walking is easy, walking while doing useful work is hard.
Long-horizon autonomy and hardware cost still gate deployment. Public demos are usually under five minutes and run with a safety tether, while battery life of one to four hours dominates the operating model. Custom planetary-roller-screw actuators (Atlas), tendon drives (NEO), and high-density permanent magnets are all attempts to push power density up and cost down at the same time.