Legged Locomotion — Bipedal, Quadrupedal, ZMP, MPC, RL Policies

Scope. This note covers legged robot locomotion as a specialty: hopping, bipedal, and quadrupedal platforms; the classical control stack (ZMP, capture point, LIP, MPC, whole-body QP); the modern learned stack (PPO/SAC sim-to-real, RMA, parkour, diffusion policies); the underlying actuator choices (tendon-driven, SEA, direct-drive); and the production fleets — Boston Dynamics, ANYbotics, Agility, Unitree, Ghost Robotics, plus the 2023–2025 humanoid wave (Figure, 1X, Apptronik, Tesla Optimus, Unitree H1/G1, Fourier GR-1). See [[Robotics/legged-robotics]] for an overview, [[Robotics/dynamic-locomotion]] for high-speed gait dynamics, and [[Robotics/humanoid-balance]] for the balance subset.

1. Why legged

Wheeled and tracked robots dominate flat industrial environments because rolling contact is energetically cheap and kinematically simple. Legged systems are worth their large added control + mechanical complexity in three regimes:

1. Rough terrain. Rocks, rubble, soft soil, vegetation — wheels stall or sink. Legs select discrete footholds and step over obstacles.
2. Vertical features. Stairs, ladders (Atlas, Cassie, Digit), construction debris, oil-platform gratings, nuclear-decommissioning floors with hose runs and trip hazards.
3. Anthropomorphic environments. Spaces built for humans — narrow doors, light switches at hip height, ladders, tool handles — favor a body geometry roughly human or dog-shaped. A wheeled robot in a warehouse stair cell is stuck; a quadruped or biped is not.

The biological precedent is overwhelming: every terrestrial vertebrate with body mass above ~10 g uses legs. Insects do too (RHex, OctoRoACH are the canonical hexapods that exploit this).

2. Hopping — one-leg simplification

Before bipedal, the field cracked the dynamics of one leg. Marc Raibert at MIT Leg Lab (1980s) built planar and 3D hoppers and decomposed control into three quasi-independent loops:

1. Hop height — controlled by the spring-loaded leg force at stance push-off.
2. Forward velocity — controlled by foot placement at touchdown (foot lands at the neutral point — the projection of the hip at midstance — to maintain current velocity; ahead to decelerate, behind to accelerate).
3. Body attitude — controlled by hip torque during stance (with the foot pinned by friction, hip torque rolls the body).

Raibert’s 1986 book Legged Robots That Balance (MIT Press) codified this. The principle — that locomotion control can be decomposed across orthogonal axes — survives in modern bipedal control. See [[Robotics/raibert-hopper]] for the planar hopper math.

3. Bipedal platforms — deployed and developmental

3.1 The Honda lineage → ASIMO (1986–2022)

Honda’s secret Wako-Honda Lab began the P-series in 1986 (P1 1993, P2 1996, P3 1997). ASIMO (Advanced Step in Innovative MObility) launched 2000; final v3 (2011) measured 130 cm, 54 kg, 9 km/h running, with a 51-DOF body. Control was offline-planned ZMP gait — every footstep + body trajectory computed in advance from CAD-known kinematics. ASIMO was retired in 2022. The ZMP-offline-plan paradigm dominated bipedal robotics from 1997 to roughly 2015.

3.2 Boston Dynamics PETMAN, Atlas, electric Atlas

• PETMAN (2009–2013): hydraulic biped for chemical suit testing; tethered.
• Atlas (hydraulic) (2013, DARPA Robotics Challenge): 1.5 m, 89 kg, ~3.5 mph; ~28 DOF; iconic parkour + backflip demos 2017–2023. Hydraulics gave high torque density at the cost of leaks, weight, and noise.
• Atlas (electric) (April 2024): same envelope, all-electric joints; rotational ranges exceed human and reverse-kneed at hip; designed for industrial deployment. Hyundai Motor Group (owner of BD since 2021) plans factory deployment 2025–2026.

3.3 Agility Robotics — Cassie + Digit

Spin-out from Jonathan Hurst’s lab at Oregon State (2017). Cassie is the legs-only research platform: 1 m tall ostrich-style reverse-kneed legs, point feet, no upper body. Digit (v1 2019, v4 2024) adds a torso, arms, and a perception head; ~1.75 m, 65 kg, payload 16 kg. Customers: Amazon warehouse trial 2023 at BFI1 in Sumner WA (tote moving), Ford partnership (last-mile delivery proof), GXO Logistics deployments 2024–2025. Agility raised **$150MSeriesB\ast \ast 2024valuingat$1 B. Manufacturing facility “RoboFab” in Salem OR opened 2023 for 10k units/yr.

3.4 The 2023–2025 humanoid wave

Company	Robot	Date	Notes
Figure AI	Figure 01 / 02	2023 / Aug 2024	$675 M Series B Feb 2024 (Microsoft, OpenAI, NVIDIA, Bezos); BMW Spartanburg pilot starting 2024 Q2
1X Technologies	NEO (humanoid, 2024); EVE (wheeled, 2023)	2024	NEO uses soft drivetrain (cable-driven) for human-safe forces
Apptronik	Apollo	Aug 2023	Mercedes-Benz pilot at Berlin-Marienfelde 2024; ~1.7 m, 73 kg, 25 kg payload
Sanctuary AI	Phoenix	2023	Magna automotive pilot Canada; emphasizes “carbon-based AI” Carbon system
Unitree	H1 (Oct 2023), G1 (May 2024)	2023–2024	H1: ~$90k,3.3m/s;G1:$16k consumer, 1.32 m, 35 kg, 23 DOF
Tesla	Optimus Gen 2 / Gen 3	Dec 2023 / 2024–2025	In-house Tesla actuators; demo’d folding shirts, walking outdoor
Fourier Intelligence	GR-1	2023	China; ~1.65 m, 55 kg, 40 DOF; medical rehab adjacency
XPeng	Iron	2024	Chinese EV-maker pivot; rotor-driven hand
Boston Dynamics	Atlas (electric)	Apr 2024	All-electric; Hyundai factory target

Investment in humanoids 2023–2024 exceeded $1.5 B (CB Insights). The bet underwriting it: that end-to-end learned policies (see §6.5) eliminate the per-task engineering cost that historically made humanoids uneconomic.

4. Quadrupedal platforms

4.1 Boston Dynamics quadruped lineage

• BigDog (2005, DARPA funded): 109 kg, gasoline-engine + hydraulics, ~30 kW peak; carried 154 kg payload over rubble.
• LS3 (2012, Legged Squad Support System): scaled BigDog, retired due to engine noise unacceptable for dismounted infantry.
• AlphaDog / WildCat / Cheetah-style prototypes.
• Spot (commercial Sep 2019): 32.5 kg, 1.5 hr endurance, electric, ~$74.5k base; modular payload (Spot Arm, Spot CAM+, gas sensors, LiDAR). Used by BP Mad Dog Gulf platform, Shell Nyhamna LNG, Statkraft, plus police/fire test programs. As of 2024, ~1,500 Spots deployed globally.

4.2 MIT Cheetah lineage (Sangbae Kim group)

• Cheetah 1 (2009): trot, ~13 mph.
• Cheetah 2 (2014): bound, jumped over obstacles.
• Cheetah 3 (2018): blind locomotion, stairs, backflip — proved control without vision through proprioception + MPC.
• Mini-Cheetah (2019): 9 kg, $10k BOM, the first open-source design with proprioceptive direct-drive actuators (low-gear, low-inductance BLDC motors with current-loop torque control); spawned Unitree A1 and most subsequent quadrupeds. Top speed ~2.5 m/s.

4.3 ANYbotics — ANYmal

ETH Zurich Robotic Systems Lab spin-out (2016). ANYmal uses torque-controlled series-elastic actuators (SEA) — direct elasticity (spring) in series with the geared motor gives shock tolerance + accurate force control. Customers: Helsinki metro tunnel inspection, BASF Ludwigshafen, Yokohama Pier (Japan harbor), Petronas Malaysia LNG, offshore wind ops. Quietly the most “industrialized” quadruped: ATEX Zone 1 certified (explosive atmospheres), -20 to 40 °C ops, IP67. ~$140k unit.

4.4 Unitree (Hangzhou, founded 2016)

The price disruptor. Founder Wang Xingxing’s PhD work on Mini-Cheetah-style direct-drive actuators productized aggressively:

Model	Year	Mass	Price	Notes
Laikago	2017	22 kg	$30k	First commercial product
A1	2019	12 kg	~$10k	Education + research
Go1	2021	12 kg	~$2.7k	Consumer; popularized quadrupeds
Aliengo / B1	2020 / 2022	21 kg / 50 kg	$40k/$100k	Industrial payload
Go2	2024	15 kg	$1.6k–$3.5k	LiDAR option, LLM API (OpenAI integration), 4 m/s
B2	2024	60 kg	$100k+	120 kg payload, climbing

Estimated ~25k units sold 2024. Unitree’s pricing pressure is the dominant commercial dynamic in quadruped robotics 2023–2025.

4.5 Other players

• Ghost Robotics (Philadelphia, 2015): Vision 60 + Vision 70; US Air Force base patrol, Tyndall AFB 2020 deployments; Israeli + US tactical fielding. Direct-drive, water-immersible.
• Xiaomi CyberDog (2021) / CyberDog 2 (2024): hobbyist-grade.
• Sony Aibo (1999 — first consumer; relaunched 2018 ERS-1000): toy/companion.
• Deep Robotics (Hangzhou): X20, Lite3 — competitor to Unitree.

4.6 Bio-inspired hexapods

• RHex (Uluc Saranli, Howie Choset CMU + UPenn, 2001): six C-shaped legs; runs on rubble with no proprioceptive state estimation beyond motor encoders.
• OctoRoACH (UC Berkeley Biomimetic Millisystems): 35 g, 8 legs, electrostatic-clutch payload steering.

These exploit passive dynamic stability: the leg shape and compliance handle the disturbances without explicit feedback.

5. Kinematics + actuation

5.1 Leg topology

• Serial 3-DOF (hip flexion/abduction + knee): the dominant pattern (Spot, ANYmal, Unitree, Cheetah). Three motors per leg, 12 total for a quadruped.
• Parallel (e.g., MIT Cheetah variants — coaxial hip motors driving knee through a four-bar): higher effective bandwidth, lower leg inertia at the cost of design complexity.
• Avian / reverse-kneed (Cassie, Digit): the “knee” is actually the ankle; the structural advantage is that the bird-leg’s spring-loaded shank stores + releases energy.

5.2 Actuator choices

Approach	Examples	Trade-off
Hydraulic	BigDog, Atlas (hydraulic)	Extreme torque density; leaks, weight, noise, inefficiency
Tendon-driven	Boston Dynamics electric Spot Arm, 1X NEO	Lightweight distal mass (motor at base, cable to joint); compliance + backlash
Series-elastic (SEA)	ANYmal, Cassie/Digit ankles	Accurate torque control via spring deflection; bandwidth limit ~50 Hz
Direct-drive / quasi-direct	MIT Cheetah, Unitree, most 2020+ quadrupeds	Low gear ratio (~6:1 vs traditional 100:1); current ≈ torque, ~1 kHz bandwidth, backdrivable
Cable-driven	Boston Dynamics (some prototypes), 1X NEO	Compliance + distal mass reduction, complex routing

Series-elastic actuators (SEA) were formalized by Gill Pratt + Matt Williamson at MIT (1995). The deliberate spring decouples motor inertia from output torque, making force control trivial: measure spring deflection → multiply by stiffness → that is the output torque. SEA dominated 1995–2015. Direct-drive returned with low-inductance BLDC motors (T-Motor, Unitree, the Mini-Cheetah open design) — high-bandwidth proprioceptive torque from current control alone.

6. Control approaches

Five families, deployed in roughly chronological order. Most fielded systems combine them.

6.1 Zero Moment Point (ZMP) — Vukobratović 1972

The ZMP is the point on the ground where the net moment from ground reaction forces is zero in the horizontal plane. If ZMP lies strictly inside the support polygon (convex hull of contact points), the robot does not topple. Control reduces to: plan a footstep sequence + body CoM trajectory that keeps ZMP within the support polygon. Used by Honda P-series + ASIMO + HRP-2/3/4 (AIST Japan, Kawada Industries).

Strengths: provable static-style stability. Weaknesses: assumes flat ground + known kinematics + low-bandwidth motion; produces the “robotic walk” with bent knees + small steps. Practical for slow industrial humanoids; useless for running.

6.2 Linear Inverted Pendulum + Capture Point

LIP (Kajita 2001): approximate the CoM as a point mass at constant height on a massless leg pivoting at the foot. The dynamics are linear, decouple per axis, and admit closed-form solutions.

Capture point (Pratt et al. 2006): the point on the ground where a single step would bring the LIP CoM to rest. Walking control reduces to placing each foot near the instantaneous capture point. Used at IHMC Pensacola for ATLAS DRC, and as a building block in Cassie/Digit’s planners.

6.3 Model Predictive Control (MPC)

For each step (or each $\sim$30 ms cycle), solve a finite-horizon optimal-control problem over the simplified dynamics. The convex MPC formulation for quadrupeds (Di Carlo, Wensing, Kim 2018 IROS) uses the single-rigid-body model (lumps the legs into massless springs of varying force), linearizes around current orientation, and solves a QP in tens of microseconds. Used in MIT Cheetah 3, Spot, ANYmal, most modern quadrupeds.

Whole-body QP (Kuindersma et al. 2014; Wensing-Orin 2013): on top of MPC, the lower stack solves another QP at 1 kHz mapping desired CoM wrench → joint torques subject to friction-cone + torque limits. The two-level structure (MPC trajectory + whole-body QP tracking) is now canonical.

6.4 Hybrid Zero Dynamics — Westervelt-Grizzle 2003

For bipedal walking under impact, the dynamics are hybrid: continuous swing-leg dynamics interrupted by instantaneous impact at foot-strike. HZD designs the controller so that the swing trajectory + impact map close to a limit cycle. Used in MABEL, ATRIAS, Cassie. Provides provable periodic gaits with limited use of online optimization.

6.5 Reinforcement Learning policies — sim-to-real

The dominant paradigm 2019–present. Train a neural-network policy in simulation (Isaac Gym, MuJoCo, Brax) on randomized terrain + dynamics; deploy zero-shot on hardware. Algorithm: PPO most common, SAC for actor-critic-with-replay.

Key milestones:

• Hwangbo et al. Science Robotics 2019 — first ANYmal RL policy walking outdoors; trained in 4 hours on a desktop GPU; no F/T sensor.
• Lee et al. 2020 Science Robotics — ANYmal teacher-student with proprioception-only deployment; “Learning quadrupedal locomotion over challenging terrain.”
• RMA (Rapid Motor Adaptation) — Kumar, Fu, Pathak, Malik 2021. Online estimator infers terrain parameters from recent history; policy conditions on the estimate.
• legged-gym — Margolis, Yang, Paigwar, Agrawal 2022 (ETH+NVIDIA); GPU-parallel training framework, 20+ minutes wall-clock from scratch to walking policy on RTX 4090.
• Walk These Ways — Margolis-Agrawal 2022: multi-skill (gait, cadence) policy conditioned on commands.
• Parkour for Legged Robots — Cheng, Shi, Agarwal, Pathak 2023; ANYmal/Unitree leaps over gaps + climbs boxes.
• Extreme Parkour — Zhuang et al. 2023, Caltech-CMU.
• Humanoid Locomotion as Next Token Prediction — Radosavovic et al. UC Berkeley 2024; transformer policy trained on sensor-motor sequences.
• Diffusion Policy — Chi, Florence, Song et al. RSS 2023; visuomotor policies as denoising; influential for manipulation, beginning to filter into locomotion.

Imitation from animal motion capture — Peng, Ma, Tan, Mordatch et al. SIGGRAPH 2020, “Learning Agile Robotic Locomotion Skills by Imitating Animals.” Took mocap of a dog, retargeted to Laikago; reproduced bound, gallop, sidewalk.

6.6 Contact-implicit trajectory optimization

For motion through contact transitions (e.g., a quadruped climbing onto a ledge), the mode sequence (which feet are in contact when) is itself a decision variable. Posa et al. 2014 formulates this as an optimization with complementarity constraints (force × gap = 0). Slow to solve but produces motions classical MPC cannot (e.g., a biped climbing a stool with one hand).

7. State estimation + sensing

Proprioception:

• IMU (3-axis accel + gyro, often + magnetometer) — 200–1000 Hz; MEMS units like Microstrain 3DM-GX5 or Xsens MTi.
• Joint encoders — absolute (magnetic, MagPie, AksIM) + incremental (optical or magnetic); resolution to 20+ bits at the joint.
• Foot force sensing — increasingly proprioceptive only (estimate ground reaction from joint torques + dynamics). Hwangbo 2019 showed F/T sensors unnecessary for sim-to-real RL.
• Series-elastic spring deflection → joint torque.

Exteroception:

• 2D / 3D LiDAR — Velodyne VLP-16 / Puck, Ouster OS-0/OS-1, Livox Mid-360 (Spot, ANYmal Mission Computer, Unitree LiDAR variants).
• Stereo depth — Intel RealSense D435/D455, ZED 2i.
• ToF — Pmd, ams-OSRAM.
• RGB cameras for VIO + semantic perception.

State estimator: tightly-coupled factor-graph SLAM (GTSAM, Cassie’s SLAM stack), or invariant EKF (Hartley et al. 2018) for the proprioceptive base estimator. Perception runs on a separate compute (often Jetson AGX Orin) and feeds elevation maps to the locomotion stack at 5–20 Hz.

8. Power + endurance

Platform	Battery	Endurance
Spot	605 Wh Li-ion, 45 V	90 min
ANYmal D	900 Wh, 50 V	60–120 min
Unitree Go2	8000 mAh, 5 V (15.4 V pack)	2–4 hr standby, 30 min active
Digit	1.5 kWh	~4 hr
Atlas (electric)	undisclosed	undisclosed; estimated 60–90 min
Cassie	2 kg pack	4 hr standing, ~1 hr walking

Hot-swap battery packs are standard on Spot, ANYmal, Digit. Charging-dock + autonomous-mission patterns (Percepto-style) emerging for Spot.

9. Safety + certification

• ISO 13482:2014 — Robots and robotic devices — Safety requirements for personal care robots. Covers mobile servant robots (Spot, Digit operate in this regime).
• ISO 12100 — General machinery risk assessment.
• ANSI/RIA TR R15.806 — Mobile robot safety (more general).
• IEC 61508 / 62061 — Functional safety, SIL ratings.
• ATEX (EU) / IECEx — Explosive atmospheres (ANYmal X is ATEX Zone 1).

Practical deployment in warehouse / industrial settings: safety scanners (SICK MicroScan3, Hokuyo UST), e-stop chains, zoned operation, and often physical separation from humans during high-energy motion (e.g., Digit operates in fenced lanes at Amazon trials). Direct human collaboration remains rare and high-effort.

10. Applications

10.1 Inspection (the dominant near-term ROI)

• Oil and gas: BP Mad Dog Gulf of Mexico (Spot), Shell Nyhamna LNG terminal Norway (Spot, ANYmal), Petronas Malaysia (ANYmal), Aker BP North Sea, Ineos.
• Nuclear decommissioning: Sellafield UK (Spot trials), Fukushima Daiichi (multiple quadrupeds in test).
• Power generation: National Grid UK (Spot), Statkraft hydropower Norway (ANYmal).
• Mining: Newmont, Anglo American underground inspection trials.

10.2 Construction monitoring

• HoloBuilder + DroneDeploy integrations on Spot — daily 360-photo capture, BIM comparison.
• Boston Dynamics + Trimble partnership (2022) for the Trimble Spot Site Scan.

10.3 Search + rescue, defense

• Ghost Robotics Vision 60 with US Air Force Tyndall AFB (2020) for base patrol.
• Israeli MoD and various US units evaluating armed-quadruped concepts (controversial — see SWORD Defense Systems’ SPUR rifle on Vision 60, 2021).
• Centibots project (SRI + Stanford 2003) and successor SAR work — multi-robot search in collapsed buildings; mostly research.

10.4 Logistics (the humanoid wedge)

• Amazon BFI1 Sumner WA — Digit tote-pickup pilot 2023; expanded to multiple FCs 2024.
• Mercedes-Benz Berlin-Marienfelde — Apollo (Apptronik) pilot 2024.
• BMW Spartanburg SC — Figure 02 pilot Q3 2024.
• GXO Logistics — Digit deployments across multiple sites 2024–2025.

10.5 Entertainment

• Boston Dynamics Spot dance videos (“Do You Love Me?” 2020, “Uptown Funk” 2018) — engineered choreography, became viral marketing → contributed to $1.1 B Hyundai acquisition valuation.
• CES + trade-show demos — Unitree, Xiaomi.

11. Open-source + simulation stacks

• MuJoCo — DeepMind open-sourced 2022; default for RL training of legged robots; MJX (GPU) variant 2023.
• Isaac Gym / Isaac Lab — NVIDIA; 4096+ parallel envs on a single GPU; legged-gym builds on this.
• PyBullet / Bullet — older but still widely used.
• Gazebo / Gazebo Sim (Ignition) — physics + sensor sim; better for system-level multi-robot than for high-fidelity contact.
• Brax — Google JAX-native physics, fully differentiable.
• legged_control (Open Robotics) — open MPC implementation for quadrupeds.
• Cheetah-Software — MIT Mini-Cheetah codebase (BSD).
• OCS2 (ETH RSL) — optimal-control library, used in ANYmal stack.
• Drake (TRI) — multibody + optimization; Atlas era.

12. Open problems

1. Long-horizon autonomy beyond inspection routes — open-ended manipulation + locomotion combined.
2. Dexterous + locomotive coupling — picking up objects without falling over (humanoid problem).
3. Energy efficiency — humans walk at ~0.2 W/kg specific cost of transport; current humanoids exceed 1 W/kg by 5–10×.
4. Safety certification for unfenced human-robot operation at full motion bandwidth.
5. Failure recovery — getting up from falls is hard for tall bipeds; risk of cascade damage.
6. Simulation gap — contact + soft-terrain physics remain imperfect; RL policies degrade on materials not represented in sim (grass, mud, ice).

Adjacent

• [[Robotics/legged-robotics]] — broader survey of legged robots (parent / Tier 1 ref).
• [[Robotics/dynamic-locomotion]] — high-speed gaits, ground-reaction modeling, energetics.
• [[Robotics/humanoid-balance]] — balance subproblem in detail (CoM, ZMP, capture point).
• [[Robotics/impedance-control]] — compliance + force control underlying SEA + admittance.
• [[Robotics/kinematics-dh]] — leg kinematics, Denavit-Hartenberg parameters.
• [[Compute/reinforcement-learning]] — PPO, SAC, sim-to-real (RL policy machinery).
• [[Math/optimal-control]] — MPC, QP, contact-implicit optimization.