Cassie: Wetting State & Bipedal Robot

• Cassie is a composite wetting state where a liquid rests atop textured surfaces with trapped air, yielding high apparent contact angles and low adhesion.
• Researchers analyze the Cassie–Wenzel transition using free-energy landscapes, revealing gradual versus abrupt collapse mechanisms influenced by geometry and surface chemistry.
• In robotics, Cassie is a 20-DOF bipedal platform used for studying hybrid locomotion, trajectory optimization, reinforcement learning, and dynamic control across varied terrains.

“Cassie” denotes two distinct technical objects in contemporary research. In interfacial science, the term refers to the Cassie or Cassie–Baxter wetting state, a composite state in which a liquid sits on top of a textured surface while vapor or air remains trapped in the underlying corrugations; this state underlies much of the modern literature on superhydrophobicity, Cassie–Wenzel transitions, and transport over textured interfaces (Amabili et al., 2018). In robotics, Cassie is a 20-DOF, underactuated, human-scale biped developed by Agility Robotics, which has become a common experimental platform for research on hybrid locomotion, trajectory optimization, feedback control, reinforcement learning, wheeled riding, and expressive motion (Chen et al., 2019).

1. Cassie as a wetting state

In the wetting literature, the Cassie state is the composite-interface state in which the liquid rests on top of pillars, ridges, or other asperities, with vapor trapped below; the Wenzel state is the fully intruded state in which the liquid completely wets the corrugations (Amabili et al., 2018). On a rough composite surface with solid fraction $f_s$ and intrinsic Young angle $\theta_Y$, the classical Cassie–Baxter relation is

$\cos\theta^* = f_s(\cos\theta_Y + 1) - 1,$

whereas the Wenzel relation is

$\cos\theta_W = r\cos\theta_Y,$

with $r$ the roughness ratio (Omeje et al., 2021, Fang et al., 8 May 2026). In this formulation, the Cassie state is associated with large apparent contact angles, low adhesion, and enhanced slip, while the Wenzel state corresponds to the loss of the composite interface and the disappearance of most superhydrophobic properties (Amabili et al., 2018).

The classical Cassie–Baxter equation is explicitly static. It assumes equilibrium, a well-defined solid fraction, and no penetration into the asperities, and it does not describe spreading, recoil, infiltration, or contact-line dissipation (Omeje et al., 2021). This limitation becomes central once wetting is treated as a transition problem rather than a static geometry problem.

A rigorous two-scale homogenization treatment modifies the classical picture. For purely chemically inhomogeneous surfaces, the modified Cassie law becomes

$$\cos\theta_a = \int_0^1 \cos\bigl(\theta_s(Y)\bigr)\,dY,$$

which is a line-average of the local $\cos\theta_s$ along the contact line rather than an area-average over the substrate (Xu, 2016). In that framework, the modified equations correspond to local minimizers of the total interface energy and thereby account for hysteresis, whereas the classical area-averaged laws correspond to a global-minimizer idealization (Xu, 2016).

2. Free-energy landscapes and the Cassie–Wenzel transition

A central result of the modern Cassie literature is that the Cassie–Wenzel transition is not determined solely by a single Laplace-pressure threshold. For intrusion and extrusion in a single truncated or inverted truncated conical pore, the macroscopic capillary-thermodynamics description uses the grand potential

$$G = F - \Delta p\,V, \qquad F = \gamma_{lv}S_{lv} - \gamma_{lv}\cos\theta_Y\,S_{sl},$$

with $\Delta p = p_{\text{liquid}} - p_{\text{vapor}}$ and filling fraction $\zeta=z/H\in[0,1]$ (Iwamatsu, 2020). For the truncated-cone geometry, the excess grand potential relative to the dry Cassie state has the cubic form

$\theta_Y$0

so the transition is governed by the full one-dimensional free-energy landscape rather than by a local criterion alone (Iwamatsu, 2020).

The usual local-stability condition at $\theta_Y$1 yields a critical pressure

$\theta_Y$2

which reduces for a cylinder $\theta_Y$3 to the familiar $\theta_Y$4 (Iwamatsu, 2020). However, this criterion identifies only the loss of local stability of the dry state. It does not determine the location of a new interior minimum, the remaining barrier to full filling, or the reversibility of the pathway (Iwamatsu, 2020).

The distinction between gradual and abrupt collapse follows directly from the geometry of the free-energy profile. For a narrowing pore $\theta_Y$5 with a hydrophobic wall, when $\theta_Y$6 rises above $\theta_Y$7 the meniscus slides gradually from $\theta_Y$8 to a finite-depth minimum and only at a larger pressure $\theta_Y$9 does the pore fill completely. For an inverted widening pore $\cos\theta^* = f_s(\cos\theta_Y + 1) - 1,$0, the Cassie state loses local stability abruptly at $\cos\theta^* = f_s(\cos\theta_Y + 1) - 1,$1 and the meniscus jumps to the bottom at a spinodal (Iwamatsu, 2020). The same model also predicts metastable minima under both compression and stretching, together with irreversibility driven by bottom-wall adsorption (Iwamatsu, 2020).

A related capillary result for spherical cavities shows that the free-energy barrier from complete drying to complete wetting is attained when the meniscus becomes flat; the free-energy maximum is given by an analytic formula, and the effect of line tension is largest at that maximum (Iwamatsu, 2016). Positive line tension raises the barrier and stabilizes the Cassie state, whereas negative line tension lowers the barrier and destabilizes it (Iwamatsu, 2016).

3. Microscopic, dynamic, and non-equilibrium extensions

At nanometric scales, atomistic and mesoscale calculations replace the single filling coordinate by field-based descriptions. In a submerged hydrophobic cavity, the atomistic string method in the coarse-grained density field finds a most probable wetting path in which a meniscus initially pinned at the top corners bends and slides downward, then breaks symmetry near the bottom by nucleating a liquid finger on one side, leaving a vapor bubble in the opposite corner before final collapse to Wenzel (Giacomello et al., 2014). In the Lennard-Jones cases reported there, the free-energy barrier is $\cos\theta^* = f_s(\cos\theta_Y + 1) - 1,$2 for a square groove and $\cos\theta^* = f_s(\cos\theta_Y + 1) - 1,$3 for a rectangular groove, with the larger barrier associated with the larger aspect ratio (Giacomello et al., 2014).

The choice of collective variables materially changes the computed mechanism. On pillared substrates, using only the overall liquid density in the interpillar volume produces discontinuous, non-physical meniscus evolution, spurious kinks in the free-energy profile, and substantial errors in both state stability and barrier height; in a $\cos\theta^* = f_s(\cos\theta_Y + 1) - 1,$4 pillared system at $\cos\theta^* = f_s(\cos\theta_Y + 1) - 1,$5, fine-grid CVs on an 864-cell grid give $\cos\theta^* = f_s(\cos\theta_Y + 1) - 1,$6 and $\cos\theta^* = f_s(\cos\theta_Y + 1) - 1,$7, whereas a single-CV description shifts these values to $\cos\theta^* = f_s(\cos\theta_Y + 1) - 1,$8 and $\cos\theta^* = f_s(\cos\theta_Y + 1) - 1,$9 (Amabili et al., 2018). Finite-size effects are similarly strong: a $\cos\theta_W = r\cos\theta_Y,$0 periodic-pillar system overestimates the barrier by $\cos\theta_W = r\cos\theta_Y,$1–$\cos\theta_W = r\cos\theta_Y,$2 and suppresses the translational-symmetry breaking of the meniscus (Amabili et al., 2018).

Microscopic DFT studies of nanoscale corrugation show that macroscopic force-balance remains qualitatively useful, but only if the Young angle is replaced by an effective wall angle $\cos\theta_W = r\cos\theta_Y,$3 that encodes finite interface thickness, finite-range solid–liquid forces, and line-tension corrections (Tretyakov et al., 2016). For nanostripes and square nanopillars, $\cos\theta_W = r\cos\theta_Y,$4 converges to a plateau about $\cos\theta_W = r\cos\theta_Y,$5–$\cos\theta_W = r\cos\theta_Y,$6 below the corresponding Young angle even for corrugation periods of order $\cos\theta_W = r\cos\theta_Y,$7, implying impalement pressures smaller than predicted by a purely macroscopic theory (Tretyakov et al., 2016). For binary liquid mixtures on nano-pitted surfaces, the transition cannot be predicted from a single contact angle parameter; composition, $\cos\theta_W = r\cos\theta_Y,$8–$\cos\theta_W = r\cos\theta_Y,$9 interaction strength, wall affinities, and pit dimensions all shift both the intrusion threshold and the Cassie/Wenzel free-energy difference (Singh et al., 2019).

The static Cassie–Baxter average also fails for dynamic imbibition. Starting from the Lucas–Washburn description with spatially varying wettability, the crossing time depends on $r$0 segment by segment, leading in the many-segment limit to the effective contact-angle relation

$r$1

rather than the arithmetic average of the classical Cassie–Baxter law (Fricke et al., 17 Jan 2025). Ordered arrangements are likewise nontrivial: the “more hydrophobic-first” configuration minimizes the crossing time (Fricke et al., 17 Jan 2025).

Non-equilibrium thermal effects provide an additional route to Cassie stabilization. On supercooled square-pillar silicon surfaces with $r$2, $r$3, and $r$4, localized freezing at the droplet bottom can arrest penetration and produce a final Wenzel-to-Cassie transition after impact (Fang et al., 8 May 2026). The regime map reported there places the Cassie regime at low $r$5 and high $r$6, specifically low $r$7 and high $r$8, with partial Wenzel and full Wenzel occupying the complementary regions (Fang et al., 8 May 2026). This suggests that freezing kinetics can act as a control parameter alongside geometry and wettability.

4. Hydrodynamics and heat transfer in the Cassie state

The Cassie state is not only a wetting morphology but also a transport regime. For shear flow over periodic rectangular grooves filled with an immiscible secondary fluid, eliminating the subphase dynamics yields a locally varying Navier slip condition on the meniscus,

$r$9

with $$\cos\theta_a = \int_0^1 \cos\bigl(\theta_s(Y)\bigr)\,dY,$$0 determined by groove geometry and the viscosity ratio $$\cos\theta_a = \int_0^1 \cos\bigl(\theta_s(Y)\bigr)\,dY,$$1 (Schönecker et al., 2013). Closed-form expressions for the effective slip length show that the surface response is anisotropic: longitudinal flow experiences larger local and effective slip than transverse flow because the trapped-fluid vortex structure differs between the two orientations (Schönecker et al., 2013). The analysis further shows that for water–air systems $$\cos\theta_a = \int_0^1 \cos\bigl(\theta_s(Y)\bigr)\,dY,$$2, groove aspect ratios $$\cos\theta_a = \int_0^1 \cos\bigl(\theta_s(Y)\bigr)\,dY,$$3–$$\cos\theta_a = \int_0^1 \cos\bigl(\theta_s(Y)\bigr)\,dY,$$4 already recover more than $$\cos\theta_a = \int_0^1 \cos\bigl(\theta_s(Y)\bigr)\,dY,$$5 of the maximal slip, so very deep grooves are unnecessary (Schönecker et al., 2013).

For laminar, fully developed internal flow through Cassie-state microchannels textured with isoflux ridges, matched asymptotics yields both hydrodynamic slip lengths and Nusselt numbers in terms of the small parameter $$\cos\theta_a = \int_0^1 \cos\bigl(\theta_s(Y)\bigr)\,dY,$$6 and solid fraction $$\cos\theta_a = \int_0^1 \cos\bigl(\theta_s(Y)\bigr)\,dY,$$7 (Kane et al., 2022). With ridges parallel to the flow, the dimensionless slip length satisfies

$$\cos\theta_a = \int_0^1 \cos\bigl(\theta_s(Y)\bigr)\,dY,$$8

and the new closed-form Nusselt result is accurate to $$\cos\theta_a = \int_0^1 \cos\bigl(\theta_s(Y)\bigr)\,dY,$$9 for any solid fraction, correcting earlier $\cos\theta_s$0 expressions that break down as $\cos\theta_s$1 (Kane et al., 2022). With ridges transverse to the flow, the leading inertial correction to the slip length is $\cos\theta_s$2, and the Nusselt-number accuracy depends additionally on $\cos\theta_s$3, $\cos\theta_s$4, and $\cos\theta_s$5 (Kane et al., 2022). The inner thermal problem can be summed in terms of polylogarithms, linking Cassie-state microchannel heat transfer to classical spreading-resistance theory (Kane et al., 2022).

5. Cassie as a bipedal robot platform

In robotics, Cassie is a human-scale biped developed by Agility Robotics. In the rigid-body model used for much of the locomotion literature, it is a 20-DOF, underactuated robot with a 6-DOF unactuated pelvis and two 7-DOF legs, each leg having five actuated joints and two passive leaf-spring joints (Gong et al., 2018). In the hovershoe study, the leg coordinates are identified as hip roll, hip yaw, hip pitch, knee pitch, shin pitch, tarsus pitch, and toe pitch, with the pelvis coordinates collected in the generalized vector

$\cos\theta_s$6

(Chen et al., 2019). A full-body compliant model retains spring dynamics and uses $\cos\theta_s$7 DOF, with constrained equations of motion of the form

$\cos\theta_s$8

within each contact domain (Reher et al., 2019).

A dominant model-based control line for Cassie combines virtual constraints, hybrid gait libraries, and direct-collocation optimization. One implementation regulates nine universal outputs

$\cos\theta_s$9

where $$G = F - \Delta p\,V, \qquad F = \gamma_{lv}S_{lv} - \gamma_{lv}\cos\theta_Y\,S_{sl},$$0 is parameterized by spline coefficients and $$G = F - \Delta p\,V, \qquad F = \gamma_{lv}S_{lv} - \gamma_{lv}\cos\theta_Y\,S_{sl},$$1 is a phase variable running from 0 to 1 over a nominal step time $$G = F - \Delta p\,V, \qquad F = \gamma_{lv}S_{lv} - \gamma_{lv}\cos\theta_Y\,S_{sl},$$2 (Gong et al., 2018). Seven periodic gaits were optimized offline for average sagittal speeds from $$G = F - \Delta p\,V, \qquad F = \gamma_{lv}S_{lv} - \gamma_{lv}\cos\theta_Y\,S_{sl},$$3 to $$G = F - \Delta p\,V, \qquad F = \gamma_{lv}S_{lv} - \gamma_{lv}\cos\theta_Y\,S_{sl},$$4, then interpolated online as a function of filtered forward velocity (Gong et al., 2018). On hardware, this controller enabled standing in place and walking over sidewalks, grass, snow, sand, and burning brush; the standing controller also enabled Segway riding at speeds up to $$G = F - \Delta p\,V, \qquad F = \gamma_{lv}S_{lv} - \gamma_{lv}\cos\theta_Y\,S_{sl},$$5 (Gong et al., 2018).

Trajectory generation at full order has been accelerated by FROST and C-FROST. For a 20-DOF floating-base Cassie-series model, the hybrid direct-collocation problem is transcribed over right- and left-stance phases with impact resets and periodicity constraints (Hereid et al., 2018). C-FROST’s C++ backend and multi-threaded sparse Jacobian evaluation reduce a single walk-in-place optimization from $$G = F - \Delta p\,V, \qquad F = \gamma_{lv}S_{lv} - \gamma_{lv}\cos\theta_Y\,S_{sl},$$6 in MATLAB/FROST to $$G = F - \Delta p\,V, \qquad F = \gamma_{lv}S_{lv} - \gamma_{lv}\cos\theta_Y\,S_{sl},$$7 in single-threaded C-FROST, and large batches of gaits can be generated at average times of $$G = F - \Delta p\,V, \qquad F = \gamma_{lv}S_{lv} - \gamma_{lv}\cos\theta_Y\,S_{sl},$$8 per gait on a 72-thread AWS EC2 configuration (Hereid et al., 2018). That framework was used to compute 1,331 gaits over a three-parameter grid of forward speed, lateral speed, and step height (Hereid et al., 2018).

A complementary line incorporates passive compliance directly into the dynamics and controller design. “Dynamic Walking with Compliance” uses full-body optimization together with task-space PD and approximate gravity compensation, and reports stable stepping in place over $$G = F - \Delta p\,V, \qquad F = \gamma_{lv}S_{lv} - \gamma_{lv}\cos\theta_Y\,S_{sl},$$9 with close agreement between optimized and experimental center-of-mass, joint, and spring-deflection trajectories (Reher et al., 2019). The same controller was reported to work over indoor tile floors, outdoor grass, gravel, and mild slopes (Reher et al., 2019).

6. Learning, multimodal locomotion, and expressive behavior

Deep reinforcement learning has been used to synthesize Cassie feedback controllers directly from a realistic model. In one PPO-based formulation, the state has dimension 80, the action is a 10-dimensional offset to reference joint angles, and the reward is an exponential imitation objective combining joint tracking, pelvis position, pelvis orientation, and spring tracking (Xie et al., 2018). Relative to a hand-designed reference controller, the learned policy remains stable on sinusoidal terrain up to amplitude $\Delta p = p_{\text{liquid}} - p_{\text{vapor}}$0 rather than $\Delta p = p_{\text{liquid}} - p_{\text{vapor}}$1, and withstands pelvis pushes of $\Delta p = p_{\text{liquid}} - p_{\text{vapor}}$2 forward, $\Delta p = p_{\text{liquid}} - p_{\text{vapor}}$3 backward, and $\Delta p = p_{\text{liquid}} - p_{\text{vapor}}$4 lateral (Xie et al., 2018). Interpolating between speed-specific policies further improves terrain robustness, to $\Delta p = p_{\text{liquid}} - p_{\text{vapor}}$5 in the reported tests (Xie et al., 2018).

An iterative-design variant introduces Deterministic Action Stochastic State (DASS) tuples,

$\Delta p = p_{\text{liquid}} - p_{\text{vapor}}$6

as a mechanism for reward-function redefinition, policy regularization, and distillation across architectures (Xie et al., 2019). The mixed objective combines policy-gradient updates with supervised imitation on DASS data, allowing successive redesign of gait style while constraining deviation from a previously working controller (Xie et al., 2019). Hardware results include stable walking from $\Delta p = p_{\text{liquid}} - p_{\text{vapor}}$7 to $\Delta p = p_{\text{liquid}} - p_{\text{vapor}}$8, transfer from simulation without dynamics randomization, and compact variable-speed policies represented by 5–10k tuples, with a final small-network compression reported at 6k tuples (Xie et al., 2019).

Cassie has also served as a platform for multimodal locomotion. In autonomous riding of hovershoes, the robot is modeled jointly with two independent wheeled platforms, and the feedback architecture augments a nominal balancing controller with PD loops for foot alignment, forward velocity, and turning (Chen et al., 2019). The perception stack combines Intel RealSense T265 stereo VIO, D435i depth sensing, RTAB-Map occupancy mapping, Dijkstra global planning, and Timed Elastic Band local planning (Chen et al., 2019). Reported experimental capabilities include step velocity commands $\Delta p = p_{\text{liquid}} - p_{\text{vapor}}$9 with RMSE $\zeta=z/H\in[0,1]$0, yaw-rate steps $\zeta=z/H\in[0,1]$1 with RMSE $\zeta=z/H\in[0,1]$2, slopes up to $\zeta=z/H\in[0,1]$3 and down to $\zeta=z/H\in[0,1]$4, and stair descent over single steps at $\zeta=z/H\in[0,1]$5 and $\zeta=z/H\in[0,1]$6 of the hovershoe wheel radius (Chen et al., 2019).

The platform has also been used for affective robotics. “Animated Cassie” constructs an animation-to-hardware pipeline in which keyframed motions created in Blender are converted to dynamically feasible standing and walking motions by direct-collocation optimization and then executed through a finite-state machine and high-rate controllers (Li et al., 2020). The robot was used to perform “tired,” “happy,” and “curious” motions, as well as a stand-to-walk-to-stand “laser chase” story, with the optimized standing motion time-stretched by about $\zeta=z/H\in[0,1]$7 relative to the raw animation and with reported $\zeta=z/H\in[0,1]$8 success in mode transitions and less than $\zeta=z/H\in[0,1]$9 velocity error in walking tests (Li et al., 2020). In the robotics literature, Cassie therefore functions both as a benchmark locomotion platform and as an experimental substrate for dynamic character animation.