Dynamic Locomotion (Running, Jumping, Parkour) — Robotics Reference

Scope. “Dynamic” locomotion is everything that violates quasi-static balance: every motion in which the centre of mass leaves the support polygon, or in which kinetic and elastic energy are first-class state variables rather than negligible. Running, hopping, bounding, galloping, leaping, somersaults, parkour. Where [[Robotics/humanoid-balance]] keeps a robot stably upright on its feet and [[Robotics/legged-robotics]] covers the static and quasi-static gait stack, this note covers the regime where flight phases, ballistic trajectories, spring–mass orbits, and torque saturation dominate the design. Cross-link: most of the analytical machinery comes from one model — SLIP (Spring-Loaded Inverted Pendulum) — and one principle: store elastic energy on touchdown, release it on take-off.

1. At a glance

A robot is in the dynamic locomotion regime when one or more of:

• A flight phase exists (all feet off the ground for some interval; Δt_flight > 0).
• The Froude number Fr = v² / (g L) ≳ 0.5 (where L is leg length); biology’s walk→run transition.
• Mechanical specific energy turns over rapidly: ½ v² ≳ g L per stride.
• Vertical CoM motion is comparable to leg length: Δh_CoM / L ≳ 0.05.

The reigning analytical model is the SLIP (Spring-Loaded Inverted Pendulum, Blickhan 1989) — point mass on a massless linear spring. Despite its crudeness it captures the essential dynamics of human running, horse galloping, and Cassie sprinting. Stance is a half-cycle of a spring–mass oscillator; flight is ballistic. The orbit closes if and only if leg-spring tuning matches the touchdown geometry and apex velocity — a Poincaré map condition that becomes the design target for hardware (spring stiffness, leg length, leg inertia) and control (touchdown angle, leg force during stance).

2026 highlights (named demonstrations, not vapour):

• Boston Dynamics Atlas (hydraulic, 2018–2024): parkour videos showing vaults, jumps, somersaults; retired April 2024 in favour of the electric variant which inherited the parkour skills.
• Boston Dynamics Spot 1.6 m/s gallop: the production commercial quadruped reaches a sustained 1.6 m/s (5.76 km/h) trot/gallop with payload (BD spec sheet, 2024).
• Unitree H1 sprint: 3.3 m/s biped run, March 2024 — fastest production humanoid as of late 2024.
• Cassie 5 km outdoor run (Agility Robotics, 2022): Siekmann/Hurst RSS 2021 controller — RL policy, blind, outdoor, 53 minutes.
• MIT Mini Cheetah backflip (Katz/Di Carlo 2019): 9 kg quadruped executing a 720° aerial rotation; the demonstration that opened the modern parkour-research wave.
• ANYmal Parkour (Hoeller 2024): ANYbotics quadruped jumping gaps, climbing 1 m boxes, scaling parallel bars.

First ask before applying: Is the motion periodic or aerial? Periodic → SLIP / centroidal MPC + WBC suffice; aerial → trajectory optimisation in flight + landing-impact handling. Energy-recoverable or not? If yes, design springs (parallel elastic, series elastic, or just the rotor inertia of QDD); if no, design for thermal limits and battery sag. Open or closed loop in flight? Once airborne, only angular momentum + body inertia matter; plan the body rotation in flight, foot placement at touchdown.

2. First principles

2.1 SLIP — Blickhan 1989

A point mass m is attached to a massless linear spring of rest length L₀ and stiffness k. The mass alternates between stance (spring in contact with ground at touchdown point r_TD) and flight (ballistic free-fall under gravity):

$\text{stance:}m\ddot{\mathbf{r}}=k(L-L_0)\hat{\mathbf{u}}-mg\hat{\mathbf{z}},L=\Vert \mathbf{r}-{\mathbf{r}}_{TD}\Vert$ $\text{flight:}m\ddot{\mathbf{r}}=-mg\hat{\mathbf{z}}$

The spring frequency ω = √(k/m) sets the stance dwell time t_st ≈ π/ω (half a spring period). The touchdown angle α (leg angle from vertical) parameterises the orbit. Periodic running orbits are fixed points of the apex-to-apex Poincaré map P(v, α) ↦ (v', α'). The remarkable empirical result (Geyer-Seyfarth-Blickhan 2006): the SLIP supports passively stable running gaits over a wide parameter range with no active control, simply by choosing the right α(v) law on each cycle (“Raibert hopping principle”, 1986).

For human-scale parameters (m = 80 kg, L₀ = 1.0 m, k = 20 kN/m):

• ω = √(20 000/80) = 15.8 rad/s
• t_st = π/ω ≈ 0.20 s
• Resonant gait frequency 2.5 Hz, period 0.40 s — matches human running.

The 3D extension adds an azimuth angle and a lateral foot placement. Stability properties survive but require active steering — Raibert’s three-part decomposition (forward velocity, body attitude, hopping height) generalises naturally.

2.2 Spring-mass running and elastic energy

Real running stores and releases elastic energy in tendons (humans: Achilles + plantar fascia), spring-loaded prosthetic blades, parallel elastic elements (springy linkage feet on MIT Cheetah, Cassie), or series-elastic actuators (Digit, Apollo). The fraction of mechanical energy that round-trips through elastic storage is the elastic efficiency η_el:

${\eta }_{el}=\frac{E_{\text{stored}}-E_{\text{dissipated}}}{E_{\text{stored}}}$

Human Achilles: η_el ≈ 0.93. Carbon-fibre prosthetic blade: η_el ≈ 0.95. Cassie’s leaf spring: η_el ≈ 0.85. MIT Cheetah QDD (no separate spring; uses rotor inertia + low gearing): η_el ≈ 0.6 effective. The metabolic / electrical cost of transport scales inversely with η_el.

2.3 Stance + flight as hybrid dynamics

Dynamic locomotion is the canonical hybrid system:

flight → touchdown event (foot z = 0, vertical velocity < 0)
       → stance dynamics
       → lift-off event (spring force = 0)
       → flight again

The hybrid Poincaré map composes the flight half (ballistic, closed-form) with the stance half (nonlinear ODE, integrated). Local stability is the spectral radius of the Jacobian of this composed map at a fixed point. See [[Robotics/legged-robotics]] §2.1 for the manipulator-equation form and contact constraint handling.

2.4 Froude number

The dimensionless speed:

$Fr=\frac{v^2}{gL}$

with L taken as standing leg length (hip to ground). Biology observed transitions:

• Walk → run at Fr ≈ 0.5 (Alexander 1989).
• Trot → gallop in quadrupeds at Fr ≈ 2.
• Top sustained running speed in mammals: Fr ≈ 2–3 for most, up to 17 for cheetahs (under brief sprint).

For a 1.0 m-leg humanoid: walk/run transition at v = √(0.5 × 9.81 × 1.0) ≈ 2.2 m/s. For Unitree H1 (L ≈ 0.85 m): transition at v ≈ 2.0 m/s; observed top sprint 3.3 m/s corresponds to Fr ≈ 1.3.

2.5 Cost of Transport

$\mathrm{CoT}=\frac{P_{\text{mech}}}{mgv}\text{(or use electrical input }{P}_{\text{el}})$

Dimensionless energy per unit weight per unit distance. Benchmarks:

• Walking humans: CoT ≈ 0.20 (mechanical), 0.40 (metabolic).
• Boston Dynamics BigDog: 1.0 (gas-engine).
• Honda ASIMO: 1.6 (electrical).
• ANYmal: ≈ 0.40.
• Spot: 0.30 at 1.0 m/s.
• MIT Cheetah 3: 0.45.
• Cassie: 0.30.

The CoT vs. speed curve is U-shaped: high at low speeds (overhead dominates), minimum near design speed, rising again at sprint where dissipation in motors and air-drag dominate.

2.6 Centroidal angular momentum in steady running

Orin-Goswami 2008 showed that the centroidal angular momentum L_G is approximately zero during straight-line steady running — leg swings cancel arm swings, torso rotations are small. This near-conservation is the analytical basis for biped running gait design: the angular-momentum task in WBC is the zero vector during straight running.

For aerial somersaults, by contrast, L_G is set at lift-off and conserved through the flight (no external torque). The somersault rate ω_body is determined by L_G / I_body. Backflip design: pick a desired flight rotation Δθ (e.g. 360°, 720°, 1080°); compute required L_G; solve for take-off torque profile.

2.7 Capture-point + step planning extended to aerial phase

The capture-point logic ([[Robotics/humanoid-balance]] §2.4) extends naturally to gaits with flight: compute the flight-foot-touchdown point r_TD as the projection of the CoM along the diverging eigenvector of the next stance’s LIP/SLIP, accounting for flight-phase ballistic offset Δr = v_xy · t_flight. This is what every dynamic biped controller does implicitly. For 3D running, r_TD is a 2-vector with both forward and lateral components, the lateral one tuned to reject roll perturbations.

3. Worked examples

Example A — SLIP apex Poincaré

A 30 kg quadruped trunk (one leg pair as effective SLIP). Parameters:

• m = 30 kg
• k_leg = 5000 N/m
• L₀ = 0.30 m
• Touchdown velocity v_TD = 2.0 m/s (horizontal), vertical down −1.0 m/s.
• Touchdown angle α = 60° from vertical.

Spring natural frequency:

$\omega =\sqrt{k/m}=\sqrt{5000/30}=12.91\text{rad/s}$

Stance half-period ≈ π/ω = 0.243 s. Apex height above stance foot (energy conservation, ignoring losses):

$h_{\text{apex}}=L_0\cos \alpha +\frac{v_z^2}{2g}+\frac{v_{TD}^2-v_{LO}^2}{2g}$

For a symmetric orbit v_LO = v_TD and the third term is zero. With α = 60°:

• L₀ cos α = 0.30 × 0.5 = 0.15 m
• v_z²/(2g) = 1/(19.62) ≈ 0.051 m
• Apex ≈ 0.20 m above the floor.

The Poincaré map iterates (v_apex, α_TD) → (v'_apex, α'_TD). For a fixed α_TD = 60° and the given parameters, a periodic orbit exists at v_apex* ≈ 2.0 m/s with stride length 2 L₀ sin α × t_flight/t_st = 2 × 0.30 × sin 60° × (0.20 s / 0.24 s) ≈ 0.43 m. Step frequency 1 / (t_st + t_flight) ≈ 2.3 Hz.

The Jacobian of the apex-to-apex map at this fixed point has spectral radius ≈ 0.6: the orbit is passively stable — small perturbations decay over a few strides without any active control beyond holding α_TD constant. This is Raibert’s foundational observation.

Example B — MIT Mini Cheetah backflip (Katz/Di Carlo 2019)

The Mini Cheetah is a 9 kg quadruped. Backflip target: 720° aerial rotation (two flips) with the body landing on its feet.

• Mass m = 9 kg
• Body inertia about pitch axis I_yy ≈ 0.13 kg·m² (longitudinal length 0.4 m, lumped)
• Peak motor torque (knee): τ_max = 17 N·m (T-motor U8 KV85 + 6:1 planetary)
• Peak motor speed: ω_max = 27 rad/s

Backflip plan:

1. Crouch. Lower CoM to h_0 = 0.10 m to load springs.
2. Take-off. Extend legs vertically for Δt ≈ 0.1 s; impart vertical impulse J_z = m v_TO and angular impulse J_θ = I_yy ω_body.
3. Flight. Ballistic free-fall + pure rotation. Required peak height for Δt_flight = 0.5 s:

$h_{\text{peak}}=\frac{1}{2}g(\Delta t_{flight}/2{)}^2=\frac{1}{2}\times 9.81\times 0.0625=0.31\text{m}$

Take-off vertical velocity v_TO = g · Δt_flight/2 = 9.81 × 0.25 = 2.45 m/s. Required vertical impulse J_z = 9 × 2.45 = 22 N·s delivered in ≈ 0.1 s → mean vertical force ≈ 220 N (about 2.5× body weight).

Angular momentum for 720° in 0.5 s:

${\omega }_{\text{body}}=2\times 2\pi /0.5=25.1\text{rad/s}$ $L_G=I_{yy}{\omega }_{\text{body}}=0.13\times 25.1=3.3{\text{kg m}}^2/\text{s}$

Imparted by an asymmetric leg push (front and rear legs differ in radial-velocity profile) over the 0.1 s take-off → required mean torque ≈ 33 N·m about the body pitch axis, sourced as a couple between front and rear leg foot-forces (lever arm ≈ 0.4 m). Foot-force differential of about 80 N — comfortably within the actuator envelope.

Reality: Katz-Di Carlo precomputed the trajectory offline with TROC (a custom DDP solver), and tracked it at runtime with a whole-body controller. Landing impact (~ 700 N at the foot for 5 ms) absorbed in the leg compliance + WBC torque limit.

Example C — Spot galloping, energy budget

Boston Dynamics Spot at sustained 1.6 m/s gallop. Mass m = 32.5 kg.

Mechanical kinetic energy per stride (treating the body as a point mass for the first-pass estimate):

$E_k=\frac{1}{2}m{v}^2=\frac{1}{2}\times 32.5\times 1.6^2=41.6\text{J}$

At a stride frequency of f_stride ≈ 2.5 Hz, the energy delivered per stride is repeatedly accelerated and decelerated; per the SLIP picture, ideal elastic storage means zero net power. Real losses come from:

• Rotor heat (i² R). Per leg ≈ 30 W average → 120 W for four legs.
• Friction in gearboxes (planetary η ≈ 0.92) ≈ 20 W per leg, 80 W total.
• Damping in feet + minor air drag: ≈ 40 W.
• Computer + electronics + sensors: 80 W (NVIDIA Jetson + IMU + cameras).

Total: ≈ 320 W electrical → with ≈ 30 % overhead at the battery → 420 W. Spot’s 605 Wh battery → ≈ 1.4 h runtime at this gait. Spec sheet: 90 min — consistent.

Cost of transport at this speed:

${\mathrm{CoT}}_{\text{el}}=\frac{P_{el}}{mgv}=\frac{420}{32.5\times 9.81\times 1.6}=0.82$

(Higher than the SLIP idealisation because Spot is geared, not springy.)

4. Gait categories

Class	Gait	Phase pattern	Flight?	Fr range	Use case
Quadruped	Walk	4-beat, 3 feet down	No	0 – 0.3	Quasi-static
Quadruped	Trot	2-beat, diag pairs	Brief	0.3 – 1	Default dynamic
Quadruped	Pace	2-beat, ipsilateral	Brief	0.4 – 0.8	Camels, MIT Cheetah
Quadruped	Bound	2-beat, in-phase pairs	Yes	0.8 – 2	Rabbits, Mini Cheetah
Quadruped	Gallop	4-beat asymmetric	Yes	1 – 3+	Horses, cheetahs, Spot top
Biped	Walk	Always 1 foot down	No	0 – 0.5	Humans below 2 m/s
Biped	Run	Flight phase every step	Yes	0.5 – 3	Humans, Cassie, H1
Biped	Sprint	Long flight, high stride	Yes	2 – 5	Top athletes; not yet robots
Biped	Hop	1-leg, 1-foot landing	Yes	Variable	Raibert 1986
Biped	Jump	Discrete ballistic event	Yes	Per-jump	Vault, leap
Hopper	1-leg planar	Spring–mass	Yes	—	Raibert canonical
Aerial	Backflip	360° rotation	Yes	—	Mini Cheetah, Atlas
Aerial	Twist	Yaw-rotation in flight	Yes	—	Atlas parkour
Aerial	Parkour	Compose vault/leap	Yes	—	Atlas, ANYmal

Gait transition logic. Hoyt-Taylor 1981 (Nature) measured horse metabolic CoT vs. speed: each gait has a U-shaped CoT(v), and the horse transitions exactly at the speed where the next gait becomes cheaper. Modern controllers can either:

• Discretely switch based on speed thresholds (Spot, ANYmal).
• Continuously blend via a learned periodic-reward policy (Siekmann 2021): a single recurrent policy spans walk through run by phase-shifting reward components.

5. Control architectures

5.1 Model-based: convex SRBD MPC

Di Carlo 2018 IROS, the foundational convex MPC for legged robots. State: centroidal pose + twist x ∈ ℝ¹². Input: foot-force vector u ∈ ℝ^{3·n_c} (n_c contacts). Horizon N = 10–20 nodes at 30–50 ms each (total 0.3–1.0 s). Once the gait scheduler fixes the contact schedule, the problem is a convex QP solved at 50–500 Hz with OSQP or qpSWIFT. Friction cones approximated as inscribed pyramids for linearity. Works for trot, pace, bound. Limitation: convexity is lost the moment the contact schedule itself is a decision variable (hybrid MIQP).

5.2 Nonlinear MPC + DDP for jumping

For aerial manoeuvres, the SRBD is not enough — vertical impulse and angular momentum must be planned through the take-off. Crocoddyl (Mastalli 2020) and OCS2 (ETH) use Differential Dynamic Programming with explicit handling of the impact + lift-off events. Box-FDDP variant adds joint-limit + torque-limit inequality constraints. Solves in < 10 ms per iteration on a single CPU core for a 0.5–1 s horizon.

5.3 Hybrid SRBD MPC + WBC

The MIT Cheetah 3 / Spot / Atlas Electric recipe. Outer convex MPC at 50–200 Hz commands contact wrenches. Inner whole-body QP at 1 kHz maps wrenches → joint torques while honouring secondary tasks (swing foot, body posture). The two layers communicate through a wrench reference. The split is the dominant 2024–2026 stack.

5.4 Reinforcement Learning (PPO, recurrent)

Hwangbo 2019 Science Robotics opened the modern pipeline: train a small (3-layer MLP or GRU, ~ 10⁵ parameters) policy in massively-parallel sim (initially RaiSim; now Isaac Lab, MuJoCo MJX, Genesis); randomise terrain, mass, friction, motor gain; observation = proprioception only (joint pos/vel, IMU). Lee 2020 Science Robotics added perception (depth + history). Margolis 2024 IJRR (Cassie) showed it works for high-speed biped running.

5.5 Hierarchical RL + skill chaining (parkour)

Atlas’s parkour pipeline (Boston Dynamics 2018–2024 public talks; ANYmal Parkour, Hoeller 2024 Science Robotics) chains low-level “skill” policies (walk, jump, climb, vault) under a high-level planner that selects which skill to activate based on perception. Each skill is RL-trained in its specialised regime; the planner can be classical (behaviour tree) or learned (model-based RL).

5.6 Online trajectory optimisation for jumping

For one-shot manoeuvres (backflip, vault), full trajectory optimisation runs once offline (TROC, Crocoddyl, IPOPT) producing reference joint trajectories. At runtime a WBC tracks them with PD + feedforward. The harder problem is online re-optimisation of the take-off given perception of the obstacle — emerging area circa 2024–2026.

Architecture	Compute	Update	Strengths	Weaknesses	Used by
Convex SRBD MPC	x86 single core	50–500 Hz	Robust, interpretable	Fixed contact schedule	Cheetah 3, Spot, ANYmal
NL-MPC / DDP	x86 multi-core	30–100 Hz	Handles flight + impacts	Slower, harder to tune	Atlas Electric, Crocoddyl users
Hybrid MPC + WBC	x86 + ARM	50–200 Hz + 1 kHz	Modular, certifiable	Wrench/torque interface	All modern platforms
RL policy	NN inference on edge	50–500 Hz	Robust to terrain noise	No certificate, sim-to-real gap	H1, G1, Cassie blind
RL + perception	NN + depth on edge	50 Hz	Handles unstructured terrain	Latency, perception failure	ANYmal Parkour, Atlas Electric
Skill chaining (HRL)	NN + planner	Variable	Long-horizon parkour	Each skill needs training	Atlas parkour, ANYmal Parkour
Offline traj-opt + WBC	Offline CPU + edge tracking	Per-manoeuvre	Reaches the kinematic edge	Brittle to perturbation	Mini Cheetah backflip, Atlas demos

6. Real platforms

6.1 Production / commercial

• Boston Dynamics Atlas Electric (2024–). 89 kg, 1.5 m, electric custom actuators. Inherited the parkour skills from the hydraulic predecessor. Stack: SRBD MPC + WBC + learned residual policy.
• Boston Dynamics Atlas hydraulic (2013–2024). Iconic parkour demos (2018: parkour course; 2021: vault sequence; 2023: backflip with object pick-up). Hydraulic actuation: up to 5 kW peak per joint enabled aggressive aerial manoeuvres impractical for then-current electric actuators. Retired April 2024.
• Boston Dynamics Spot (2020–). 32.5 kg, 0.84 m at shoulder. Commercial since 2020. Sustained 1.6 m/s top speed. Stack: convex MPC (Di Carlo style) at 50 Hz + WBC at 1 kHz. Over 1500 units sold by end-2024.
• Unitree H1 (2024–). 1.8 m, 47 kg humanoid. QDD actuators, parallel mechanism. 3.3 m/s sprint demonstration March 2024 — fastest production humanoid.
• Unitree G1 (2024–). 1.32 m, 35 kg smaller humanoid. Same QDD family.
• Unitree H1 + Z1 arm. H1 base with a Z1 manipulator on the chest; same locomotion stack.
• Cassie (Agility Robotics, 2017–). Now superseded for production by Digit, but the research platform for Hurst / Margolis / Siekmann RL papers. 5 km outdoor run, 2022.
• Digit (Agility Robotics, V4, 2023+). 1.75 m, 45 kg humanoid. SEA + 4-bar tendon-spring legs. HZD + capture-point. Deployed in Amazon and GXO warehouses 2024. Walking gait, not aggressive parkour (yet).
• ANYbotics ANYmal X (2024). Industrial inspection quadruped, IP67 + ATEX explosion-protected. Hutter-lineage controller; SRBD MPC + WBC + learned residual.
• ANYbotics ANYmal C, D (2018–2023). Earlier industrial variants; D is the current shipping model.
• Unitree Go2 (Edu, Pro), 2023+. $1.5–3 k research quadruped. 12 DoF, ROS 2 SDK. Default RL gait. Widely used in academic labs.
• MIT Cheetah 3 (research, 2018). 32 kg, the SRBD MPC reference platform. Bledt 2018 IROS. Now retired but the architecture lives on in Spot and Cheetah-Humanoid.
• MIT Mini Cheetah (2019). 9 kg, the backflip platform. T-motor U8 KV85 + 6:1 planetary.
• MIT Humanoid (Wensing, 2023+). 24 kg, research humanoid; centroidal MPC + WBC.
• Stanford Doggo (2019). Open-design 5 kg quadruped; jumping demonstration via simple kinematic control. STL files publicly available.
• Apptronik Apollo (2024). 1.73 m, 73 kg. SEA + harmonic. Walking → running on the roadmap.
• Figure 02 (2024). 1.7 m, 70 kg. Pilot at BMW Spartanburg. Walking, not yet running.
• Tesla Optimus Gen 3 (2025). 1.73 m, 57 kg. WBC + RL.
• 1X NEO Gamma (2024). 1.65 m, 30 kg. Tendon-and-gear soft drive. Lighter mass enables jumping in roadmap.

6.2 Aerial / parkour-specific

• Salto-1P (UC Berkeley, 2017). Single-leg jumping monoped, 0.1 kg. 1 m vertical jump from a standing start. Demonstrates spring-energy storage in a tiny system. Cited as the practical lower bound of jumping robots.
• MIT Mini Cheetah (Katz/Di Carlo 2019). 720° backflip.
• ANYmal Parkour (Hoeller 2024 Science Robotics). Gap jumping, box climbing, parallel-bar traversal.

7. Tools and software

• Isaac Lab (NVIDIA, 2024+). GPU-parallel RL training environment, the successor to Isaac Gym. Default for Unitree H1, Figure 02, Optimus development.
• Genesis (CMU / Sea AI Lab, 2024+). Differentiable, GPU-parallel sim claiming 100× MuJoCo throughput. Open-source; rapid uptake in academic labs.
• MuJoCo (Google DeepMind). Reference contact simulator; MuJoCo MPC for online optimal control; MJX (JAX/XLA) for GPU parallelism.
• Crocoddyl (LAAS-CNRS). DDP-based optimal control for legged robots; FDDP and Box-FDDP variants.
• OCS2 (ETH). Whole-body MPC framework; used on ANYmal, MIT Mini-Cheetah variants.
• Pinocchio (LAAS-CNRS). State-of-the-art floating-base rigid-body library, used as back-end for Crocoddyl, TSID, Aligator.
• legged_gym + rsl_rl (ETH / NVIDIA). Open-source RL training recipe; the de facto starting point for ANYmal-class platforms.
• Boston Dynamics SDK. Spot, Atlas API; gait + behaviour layer; no controller access below the WBC interface.
• Unitree SDK. H1/G1/Go2 low-level API: joint torque, joint position, IMU. Full controller access.
• Drake (TRI / MIT). Trajectory optimisation, contact solving, formal verification; used in TRI Punyo and various academic platforms.

8. Edge cases and failure modes

• Slip on icy / wet surfaces. Friction coefficient μ drops from 0.8 (dry rubber) to 0.1 (wet ice). The friction cone in the MPC becomes nearly degenerate; required tangential force is unachievable. Mitigation: estimate μ online (foot tangential velocity vs. force ratio at each contact) and re-tune the MPC margin; shorten step length; widen stance.
• Touchdown failure modes. Premature touchdown (foot hits ground earlier than expected): impact velocity higher than planned, rotor torque spike, possible encoder skip. Late touchdown (foot misses ground): CoM continues past stable region; one extra flight-phase before recovery. Off-target touchdown (foot lands in wrong location): may still recover via capture-point logic if within N-step capture region.
• Mid-air collision. Body or limb contacts an obstacle during flight phase. Catastrophic for control: collision wrench enters at an unscheduled contact; SRBD MPC has no model for it. Mitigation: perception-aware flight planner that maintains clearance margin.
• Torque saturation in jumping. Take-off impulse requires peak torque; if motor + driver cannot deliver it, lift-off velocity is insufficient and the manoeuvre fails. Mitigation: design margin (~ 2× peak vs. RMS), explicit torque-limit constraint in NL-MPC.
• Battery sag at sprint. Bus voltage droops under peak current; available torque drops. Mitigation: voltage feed-forward in motor controller, conservative torque limit at low SoC, supercapacitor in parallel for peak current bursts.
• Encoder slip on hard impact. Magnetic absolute encoders can momentarily lose synch under high acceleration; relative encoders accumulate error. Mitigation: redundant absolute + incremental encoders; immediate post-impact homing.
• Foot slip during turn. Lateral force during yawing exceeds friction limit; foot slides. Detected by tangential foot velocity. Mitigation: replan lateral foot placement to widen lateral support.
• Aerial flight-time miscalculation. Touch-down planned too early → robot is still ascending; too late → robot has already impacted with high vertical velocity. Mitigation: closed-loop on IMU vertical velocity during flight; trigger touch-down on v_z < 0 and z < threshold.
• Hard landing rotor heat. Peak landing torque can exceed continuous thermal rating for tens of milliseconds. Acceptable if duty cycle is low; failure mode is demagnetisation if winding temperature exceeds Curie point of magnets. Mitigation: thermal model in firmware; thermal cut-back; design for peak / continuous ratio ≥ 4.
• Wind disturbance. Outdoor humanoids face lateral wind loads > 50 N at 10 m/s wind, 0.5 m² frontal area. Significant relative to body weight at the lateral CoP margin. Mitigation: wind estimator + lateral CoP bias.
• Sim-to-real gap (high-energy). RL policies trained in sim with idealised actuators fail on hardware when impact dynamics are mis-modelled. Specific to dynamic regimes because of the high impulses. Mitigation: thorough actuator characterisation, domain randomisation on motor gain, latency, and gear backlash.
• Unmodelled actuator dynamics. Real motors have torque ripple, cogging, friction, back-EMF, bandwidth limits. Idealised model in sim → policy commands torques the motor cannot deliver. Mitigation: identify motor dynamics; include them in sim.
• Imperfect terrain perception. Depth camera mis-estimates step height by 1–2 cm; foot scuffs or over-steps. Mitigation: terrain-aware policy with depth history; fallback blind policy on perception loss.

9. Case studies

9.1 MIT Mini Cheetah backflip (Katz/Di Carlo, ICRA 2019)

The first robot to perform a full 720° aerial rotation under its own power. Platform: 9 kg, 12-DoF QDD quadruped (T-motor U8 KV85 + 6:1 planetary). Stack: TROC offline trajectory optimisation (custom DDP) producing reference joint trajectories for the 1.2 s manoeuvre; WBC at 1 kHz tracking the reference; PD + feedforward at the motor controller. Lift-off vertical velocity 2.5 m/s; flight time 0.5 s; angular velocity 25 rad/s. Robust to mis-tuning: succeeded 90% of attempts; failure mode was usually a touch-down with insufficient leg-folding for energy absorption. The result opened the modern parkour-research wave; downstream work includes the Cheetah 3 (Bledt 2018) and Cheetah Humanoid lines.

9.2 Cassie blind 5 km outdoor run (Siekmann 2021 RSS; Margolis 2024 IJRR)

Cassie ran a 5-km Berkeley outdoor course in 53 minutes (May 2022), with a recurrent neural policy trained entirely in simulation. Siekmann’s 2021 RSS paper introduced periodic reward composition: each gait component (single-support, double-support, flight) is rewarded in its own phase of the cycle; the policy learns to satisfy each in turn. The result: a single GRU policy spans walk through run; gait transitions are emergent. Margolis 2024 IJRR extended this to “rapid locomotion” — sub-second adaptation to terrain change, push recovery, gait change — using domain randomisation on motor dynamics and terrain friction. Together these works established the modern blueprint for RL-based biped running, now widely replicated.

9.3 Boston Dynamics Atlas parkour (2018–2024)

Public demonstrations across six years: 2018 parkour course (jumps + box ascents), 2021 vault sequence, 2023 backflip-with-object-pickup. Platform: hydraulic Atlas (89 kg, 1.5 m, peak joint power ~ 5 kW). Stack: SRBD MPC + WBC + offline-optimised reference trajectories + later RL residual policies. The hydraulic actuators allowed peak impulses inaccessible to contemporary electric drives; the 2024 transition to Atlas Electric kept the algorithms and rebuilt the actuation around custom high-bandwidth electric motors. The parkour-skill catalogue (walk, vault, jump, climb, somersault) is the prototype of the skill-library architecture now used at Figure, 1X, Apptronik, and Tesla.

10. Cross-references

• [[Robotics/legged-robotics]] — quadruped + biped quasi-static and dynamic gait stack
• [[Robotics/humanoid-balance]] — companion note: ZMP / capture-point / DCM for the upright regime
• [[Robotics/motors-electric]] — QDD high-bandwidth actuation that enables dynamic gaits
• [[Robotics/dynamics-rigid-body]] — floating-base manipulator equation, contact constraints
• [[Robotics/impedance-control]] — Hogan impedance for contact transitions and landing impact
• [[Robotics/rl-for-control]] — Isaac Lab, MJX, sim-to-real pipeline
• [[Robotics/trajectory-generation]] — DDP, NL-MPC, B-spline parameterisation
• [[Robotics/multirotor-design]] — parkour cross-link for aerial vs. legged trade-off; multirotor as flight-only contrast
• [[Robotics/state-space-lqr]] — LQR / MPC fundamentals
• [[Engineering/mpc-control]] — generic MPC theory
• [[Engineering/vibration-dynamics]] — structural modes, joint elasticity

11. Citations

• Raibert, M. (1986). Legged Robots That Balance. MIT Press. The canonical text on dynamic locomotion; introduces the three-part Raibert hopper decomposition.
• Blickhan, R. (1989). “The Spring-Mass Model for Running and Hopping.” Journal of Biomechanics 22(11–12): 1217–1227. The SLIP model.
• Hoyt, D., Taylor, C. (1981). “Gait and the Energetics of Locomotion in Horses.” Nature 292: 239–240. Gait transitions by metabolic CoT.
• Alexander, R. McN. (1989). “Optimization and Gaits in the Locomotion of Vertebrates.” Physiological Reviews 69(4): 1199–1227. Froude-number gait classification.
• Geyer, H., Seyfarth, A., Blickhan, R. (2006). “Compliant Leg Behaviour Explains Basic Dynamics of Walking and Running.” Proc. R. Soc. B 273(1603): 2861–2867. SLIP supports both walking and running gaits passively.
• Wensing, P., Wang, A., Seok, S., Otten, D., Lang, J., Kim, S. (2017). “Proprioceptive Actuator Design in the MIT Cheetah.” IEEE T-RO 33(3): 509–522. QDD design.
• Bledt, G., Powell, M., Katz, B., Di Carlo, J., Wensing, P., Kim, S. (2018). “MIT Cheetah 3: Design and Control of a Robust, Dynamic Quadruped Robot.” IROS 2018.
• Di Carlo, J., Wensing, P., Katz, B., Bledt, G., Kim, S. (2018). “Dynamic Locomotion in the MIT Cheetah 3 Through Convex Model-Predictive Control.” IROS 2018. Convex SRBD MPC reference paper.
• Katz, B., Di Carlo, J., Kim, S. (2019). “Mini Cheetah: A Platform for Pushing the Limits of Dynamic Quadruped Control.” ICRA 2019. Mini Cheetah backflip.
• Hwangbo, J., Lee, J., Dosovitskiy, A., Bellicoso, D., Tsounis, V., Koltun, V., Hutter, M. (2019). “Learning Agile and Dynamic Motor Skills for Legged Robots.” Science Robotics 4(26). ANYmal RL foundation.
• Lee, J., Hwangbo, J., Wellhausen, L., Koltun, V., Hutter, M. (2020). “Learning Quadrupedal Locomotion over Challenging Terrain.” Science Robotics 5(47). ANYmal perceptive locomotion.
• Siekmann, J., Godse, Y., Fern, A., Hurst, J. (2021). “Sim-to-Real Learning of All Common Bipedal Gaits via Periodic Reward Composition.” RSS 2021. Cassie blind RL running.
• Margolis, G., Kim, S., Pulkit, A. (2024). “Rapid Locomotion via Reinforcement Learning.” IJRR 43(4): 572–595. Cassie rapid locomotion.
• Hutter, M. et al (2016). “ANYmal — A Highly Mobile and Dynamic Quadrupedal Robot.” IROS 2016. ANYmal foundational paper.
• Mastalli, C., Budhiraja, R., Merkt, W., Saurel, G., Hammoud, B., Naveau, M., Carpentier, J., Righetti, L., Vijayakumar, S., Mansard, N. (2020). “Crocoddyl: An Efficient and Versatile Framework for Multi-Contact Optimal Control.” ICRA 2020.
• Orin, D., Goswami, A. (2008). “Centroidal Momentum Matrix of a Humanoid Robot.” ICRA 2008. CMM definition; near-zero angular momentum during steady running.
• Hoeller, D., Rudin, N., Sako, D., Hutter, M. (2024). “ANYmal Parkour: Learning Agile Navigation for Quadrupedal Robots.” Science Robotics.
• Westervelt, E., Grizzle, J., Chevallereau, C., Choi, J., Morris, B. (2007). Feedback Control of Dynamic Bipedal Robot Locomotion. CRC Press. HZD textbook.
• Carpentier, J. et al (2019). “The Pinocchio C++ Library.” SII 2019.
• Boston Dynamics (2018–2024). Atlas parkour public demonstrations (YouTube technical commentary 2021, 2023, 2024).
• Boston Dynamics (2024). “Atlas Electric — A New Era of Humanoid Robots.” Whitepaper, April 2024.
• Agility Robotics (2022). Cassie 5-km outdoor run (Berkeley, May 2022) — press release + technical brief.
• Yim, J., Singh, B., Wang, E., Featherstone, R., Fearing, R. (2018). “Precision Robotic Leaping and Landing using Stance-Phase Balance.” RA-L 3(2): 819–826. Salto-1P single-leg jumper.