Zero Moment Point (ZMP)
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The Zero Moment Point (ZMP) is the point on the ground where the horizontal component of the moment generated by the contact reaction force vanishes, leaving only a vertical reaction. Equivalently, it is the point on the supporting surface at which the net moment of the inertial and gravity forces acting on a legged robot has no horizontal component [1][3]. In legged robotics, especially bipedal locomotion, the ZMP is the standard tool for reasoning about whether a humanoid robot will tip over while walking. If the ZMP stays inside the convex hull of the foot contact, called the support polygon, the robot is dynamically balanced. If the ZMP would have to leave that polygon to satisfy the equations of motion, the foot rolls onto its edge and the robot falls.
The concept was introduced in 1968 by Miomir Vukobratović and Davor Juričić at the Mihailo Pupin Institute in Belgrade, originally to design powered exoskeletons for paraplegic patients. It was first presented to the legged-locomotion community in January 1968 at the Third All-Union Congress of Theoretical and Applied Mechanics in Moscow, and the first published statement appeared in 1969 in the IEEE Transactions on Bio-Medical Engineering paper "Contribution to the synthesis of biped gait" [1]. The term "zero-moment point" itself was coined slightly later, in works from roughly 1970 to 1972 [3]. Six decades later, ZMP is still the textbook entry point for legged-robot control and was the principle behind every walking demo that made the public take humanoids seriously, including Honda ASIMO, HRP-2, HUBO, and the early NAO and Pepper platforms. The strongest case for ZMP is also its biggest limitation: it gives a stability test that is provably correct for a flat-foot, rigid-contact, point-mass model, and it gets you a stable robot quickly. Once you ask a robot to run, jump, climb stairs faster than a slow shuffle, or recover from a shove, ZMP-only methods start to creak. The modern frontier has shifted toward whole-body control, model predictive control, capture-point methods, and end-to-end reinforcement learning policies, with ZMP relegated to a sanity check or a baseline.
| Concept introduced | January 1968 (Moscow), first journal publication 1969 [1][3] |
| Originators | Miomir Vukobratović, Davor Juričić |
| Home institution | Mihailo Pupin Institute, Belgrade |
| Defining paper | Vukobratović and Juričić, "Contribution to the synthesis of biped gait", IEEE Trans. Bio-Med. Eng. 16(1), 1969 [1] |
| First dynamically balanced ZMP biped | WL-10RD, Waseda University (Ichiro Kato lab), 1984, 16 years after the concept [3] |
| Key follow-up | Vukobratović and Borovac (2004), "Zero-moment point: thirty five years of its life", Int. J. Humanoid Robotics 1(1) [3] |
| Standard textbook | Kajita, Hirukawa, Harada, Yokoi (2014), Introduction to Humanoid Robotics [8] |
| Primary use | Dynamic stability criterion for bipedal and legged robots |
| Equivalent (rigid foot) | Center of pressure on the contact surface [11] |
| Most used control method | Preview control of ZMP (Kajita et al. 2003 ICRA) [7] |
| Replaced or complemented by | Capture point, divergent component of motion, MPC, RL policies |
The technical definition is short, but it has subtleties that have caused decades of arguments. Vukobratović and Borovac, in their 2004 retrospective, are blunt that the idea was never axiomatized at the start: "the notion of ZMP has never been introduced in the form of a formal definition" [3]. What everyone agrees on is the physical picture. The foot of a walking robot cannot be controlled directly, only indirectly through the dynamics of the body above it, so the useful quantity is the single point on the ground where the influence of all forces acting on the mechanism can be replaced by one resultant force with no horizontal tipping moment. That point is the ZMP [3].
Vukobratović and Borovac state the role of the concept plainly: "Recognition of the significance and role of ZMP in the biped artificial walk was a turning point in gait planning and control" [3]. They are equally clear that it is not only a planning device but a control signal: "The ZMP is also indispensable in biped control, for establishing the practically unavoidable feedback with respect to dynamic ground reaction forces" [3].
The ZMP belongs to a small family of so-called ground-reference points (the related center of pressure, foot rotation indicator, and capture point are compared later in this article). The defining condition assumes the contact area is planar and has high enough friction to keep the foot from sliding [3]. Two situations recur in a walking cycle: a statically stable double-support phase, when both feet are on the ground, and a statically unstable single-support phase, when only one foot is in contact while the other swings forward [3]. The ZMP is what lets a controller reason about balance across both.
The formal idea of the Zero Moment Point first appeared in print in 1969 in two related papers from Belgrade. The most cited is Vukobratović and Juričić, "Contribution to the synthesis of biped gait", published in the IEEE Transactions on Bio-Medical Engineering [1]. Both came out of the Mihailo Pupin Institute, where Miomir Vukobratović led a group working on active exoskeletons for people with lower-limb paralysis. Vukobratović himself dates the underlying idea to 1968, when he and Juričić presented it at the Third All-Union Congress of Theoretical and Applied Mechanics in Moscow; the journal version appeared the following year [1][3].
The motivation was practical, not academic. Belgrade had a serious clinical interest in rehabilitation engineering at the time, and the team was trying to design a wearable exoskeleton that could carry a paraplegic patient through a normal walking cycle. They needed a way to specify reference trajectories for the joints that would not topple the patient over. The ZMP gave them a sufficient condition: pick joint trajectories so that the resulting ZMP, computed from the dynamic model, stays under the foot. Vukobratović and Borovac later named this the seminal method for gait synthesis, a "semi-inverse method" that "remained for a long time the only procedure for biped gait synthesis" [3].
A second paper with Stepanenko in 1972 sharpened the stability principle and showed that the ZMP could be used as a synthesis tool, not only as a post-hoc check [2]. The same year, Vukobratović published the first book-length treatment of biped locomotion under the ZMP framework. The Belgrade group built and demonstrated several active exoskeletons through the 1970s, sometimes with the patient strapped in, that walked using ZMP-based reference trajectories. These were not the cinematic humanoids that came later, but they worked, and they established the idea that bipedal walking could be approached as a constrained dynamics problem.
The first practical realization of a fully dynamically balanced ZMP-based biped gait came not in Belgrade but in Japan. Vukobratović and Borovac credit the WL-10RD robot, built in Ichiro Kato's laboratory at Waseda University, with the first dynamically balanced biped gait in 1984, which they note was reported 16 years after the appearance of the ZMP concept [3]. Kato's group had been building bipedal mechanisms since 1968 and had pushed cycle time down from roughly 45 seconds per step on the early WL-5 to about 1.3 seconds per step on WL-10RD [3].
Vukobratović kept publishing on the topic for the rest of his career. His 2004 retrospective with Branislav Borovac, "Zero-moment point: thirty five years of its life" in the International Journal of Humanoid Robotics, is the standard reference for the early history [3]. Vukobratović died in 2012, by which point the field had absorbed the ZMP into its standard toolkit and was already starting to move past it.
ZMP entered the broader Japanese humanoid program through several routes after Kato's Waseda work. Honda's secret E-series project, started in 1986 with the E0 chassis and continuing through E1 to E6, used ZMP-based control as the basis for its walking gait. The program was made public in 1996 with the P2 reveal [9]. The published descriptions of the controller (Hirai, Hirose, Haikawa, Takenaka 1998 ICRA [9], and the ASIMO papers by Sakagami and colleagues at IROS 2002 [10]) describe a layered scheme: a desired ZMP trajectory is planned in advance, the body is commanded to follow a center-of-mass trajectory consistent with that ZMP, and a foot landing position is adjusted online to keep the actual ZMP inside the support polygon when disturbances push the body off the planned trajectory. The third item, called "foot landing point modification" in Honda's papers, is what makes ASIMO recover from small shoves. It is also the part that does not generalize easily once the ground is uneven or the disturbance is large.
The 2000 ASIMO unveiling was the first time many people saw a humanoid take a step that was not a shuffle. By the 2005 model, ASIMO could run at 6 km/h and climb stairs, with walking speed raised to 2.7 km/h, still inside the ZMP framework with extensions for flight phases [10].
The other big Japanese influence was the work led by Shuuji Kajita, first at the Mechanical Engineering Laboratory and later at AIST. Kajita and Tanie's 1991 paper introduced the linear inverted pendulum model, which became the standard simplified dynamics for ZMP-based gait generation [5]. Kajita and colleagues then published the 2003 ICRA paper "Biped walking pattern generation by using preview control of zero-moment point", the most-cited ZMP paper of the 2000s [7]. The textbook Introduction to Humanoid Robotics (Kajita, Hirukawa, Harada, Yokoi 2014) is the cleanest single source for the math [8]. AIST's HRP-2P prototype, demonstrated in 2002 with Kawada Industries, put Kajita's controller into a sellable platform, and it spread to laboratories around the world.
The technical definition is short, but it has subtleties that have caused decades of arguments.
When a foot is in contact with rigid, flat ground, the ground exerts a reaction wrench on the foot: a force vector $f$ and a moment vector $\tau$. Pick any point $P$ on the ground and resolve the moment about $P$. The ZMP is the unique point $P^* = (x_{zmp}, y_{zmp}, 0)$ at which the horizontal components of $\tau$ vanish:
$$\tau_x(P^) = 0, \quad \tau_y(P^) = 0.$$
The ground reaction at the ZMP looks like a single normal force plus a vertical (yaw) torque. There is no tipping moment at that point, hence "zero moment point". Vukobratović and Borovac describe exactly this construction: as long as the reaction force stays within the area covered by the foot, the horizontal moment components $M_x$ and $M_y$ are compensated by shifting the acting point of the vertical reaction, so at the ZMP only the vertical component $M_z$ survives [3].
For a rigid foot in flat contact, the ZMP is identical to the center of pressure (CoP), the point at which the integrated normal pressure has its centroid. This equivalence holds whenever the foot is not about to tip. If the dynamics demand a ZMP that lies outside the foot, the actual CoP saturates at the edge of the foot, the foot starts to rotate, and the equivalence breaks. Sardain and Bessonnet's 2004 IEEE SMC paper untangles the differences carefully [11].
The ZMP can be computed directly from the inertial state of the robot. Treat the robot as a tree of rigid bodies with total mass $M$, center of mass at $\mathbf{c} = (c_x, c_y, c_z)$, and angular momentum about the center of mass $\mathbf{L}$. Gravity is $g$ in the $-z$ direction. Then the ZMP coordinates on flat ground are
$$x_{zmp} = c_x - \frac{c_z , \ddot{c}_x + \dot{L}_y / M}{\ddot{c}_z + g},$$
$$y_{zmp} = c_y - \frac{c_z , \ddot{c}_y - \dot{L}_x / M}{\ddot{c}_z + g}.$$
The denominator is the total vertical force per unit mass. The numerator combines the horizontal acceleration of the center of mass with the rate of change of angular momentum about the center of mass. For a robot whose arms and torso are not flailing, the angular-momentum term is small and is often dropped. On a real robot the formula is rarely used for online computation. Instead the actual ZMP is read out at runtime from four-corner load cells in each foot, and the planned ZMP from a model is compared to the measured value as the basic feedback signal [3].
The full Newton-Euler form is awkward to plan with, because $c_z$ and $\dot{L}$ depend on the joint trajectory you have not chosen yet. The trick that made ZMP planning practical is to reduce the dynamics to a single point mass on a flat plane.
In the Linear Inverted Pendulum Model (LIPM) (Kajita and Tanie 1991, Kajita et al. 2001), the robot is modeled as a point mass at fixed height $z_c$ that swings about a moving pivot on the ground [5][6]. The angular momentum term is set to zero. The ZMP equations collapse to
$$x_{zmp} = c_x - \frac{z_c}{g} , \ddot{c}_x.$$
This is a linear second-order relationship between the center-of-mass trajectory and the ZMP trajectory, with constant coefficients. Pulling the height to a constant is what makes the model linear and time-invariant; that is also why a robot constrained to LIPM dynamics tends to walk with stiff, almost-straight legs.
Kajita's 2003 paper renamed the same equation the cart-table model [7]. Picture a cart of mass $M$ rolling on top of a table whose footprint is the support polygon. If the cart accelerates too far past the edge, the table tips. The ZMP is the projection of the cart's reaction force on the ground; the support polygon is the table top. The cart-table model is the LIPM in disguise, but it makes the planning structure clear: pick a desired ZMP trajectory inside the polygon, then solve for the center-of-mass trajectory that produces it.
The ZMP-based stability criterion, sometimes called the Vukobratović criterion, is short:
A bipedal gait is dynamically balanced if and only if the computed ZMP lies strictly inside the convex hull of the foot contact at every instant. [3]
This is a sufficient condition for not tipping. It is not a necessary condition for staying upright in the colloquial sense, since a robot can recover from a tip by stepping. It is also not the same as static balance, which only requires the projected center of mass to lie in the support polygon. ZMP collapses to the static condition only when the robot is not accelerating.
One consequence is that the support polygon during double support (both feet on the ground) is much larger than during single support, where it is just one footprint. ZMP-based walkers therefore tend to spend a noticeable fraction of the cycle in double support. Watching ASIMO walk, you can see this directly: the gait looks deliberate and slightly crouched, with both feet on the floor more often than a human's would be.
ZMP belongs to a family of ground-reference points used to reason about balance. They are not interchangeable; each is exact under different assumptions, and confusing them is a standard source of bugs in legged controllers [11][12].
| Point | Definition | Lives where | Coincides with ZMP when |
|---|---|---|---|
| Center of mass (CoM) / center of gravity (CoG) | Mass-weighted average of body positions | Inside the body | Never (different object) |
| Ground projection of CoM | Vertical projection of the CoM onto the ground | On the ground | Robot has zero acceleration (static balance) |
| Center of pressure (CoP) | Centroid of the normal-force distribution under the foot | On the foot, always inside the contact | Foot is in flat rigid contact and not tipping |
| Zero moment point (ZMP) | Point where horizontal moment of contact reaction vanishes | On the contact surface (in principle anywhere on the supporting plane) | (Defined as such) |
| Foot rotation indicator (FRI) | Point on the floor where the net moment from inertia and gravity would have to act to keep the foot stationary | Anywhere on the floor; can lie outside the foot | Foot is not tipping (FRI inside the foot) |
| Capture point / DCM | Position where stepping would bring the LIPM to rest | On the ground | Coincides with the ground projection of the CoM at rest |
The foot rotation indicator (FRI), introduced by Ambarish Goswami in 1999, is the most useful of these for understanding what happens when the ZMP demands of a planned trajectory exceed what the foot can provide [12]. The FRI is allowed to lie outside the support polygon. When it does, its distance from the foot edge tells you how badly the foot is tipping and which way. ZMP cannot say this, because by construction the ZMP saturates at the foot edge during tipping. Goswami's 1999 International Journal of Robotics Research paper is the cleanest treatment, and it makes the distinction explicit: the FRI point incorporates the robot's full dynamics and, in contrast with the ZMP, may leave the support area [12].
In practice, robotics engineers tend to use the terms "ZMP" and "CoP" interchangeably, because they are working in the regime where the foot is in good contact and the two coincide. The distinction matters when you analyze edge cases: tipping, multi-contact (a hand on a railing), or compliant ground.
In an online control loop, ZMP appears in two roles.
The planned ZMP is computed forward from a planned center-of-mass trajectory using the cart-table or full Newton-Euler equations. This is the reference signal for gait generation.
The measured ZMP is computed from foot-mounted force-torque sensors. The standard configuration is four load cells per foot, mounted at the corners of a rectangular sole. From the four normal forces and the moments they produce, the controller reconstructs the centroid of the pressure distribution under each foot, and combines the two feet into a global CoP that, on flat rigid ground, equals the ZMP. ASIMO, HRP-2, HUBO and TORO all use variants of this scheme [10].
The gap between the planned ZMP and the measured ZMP, integrated through a stabilizer (typically a PID controller on the foot or hip joints), is what keeps the robot upright when the model is wrong. Vukobratović and Borovac call this the "practically unavoidable feedback with respect to dynamic ground reaction forces" [3].
Kajita, Kanehiro, Kaneko, Fujiwara, Harada, Yokoi and Hirukawa published "Biped walking pattern generation by using preview control of zero-moment point" at ICRA 2003, and that paper changed how the field thought about ZMP planning [7]. The earlier approach was to design closed-form joint trajectories and check the ZMP afterward. Kajita's group reframed the problem: treat the LIPM as a linear time-invariant system whose output is the ZMP, treat the desired ZMP trajectory as a known reference, and solve for the center-of-mass trajectory using Katayama's preview controller, a finite-horizon optimal regulator that uses future reference values [7].
The practical recipe is
The horizon matters. If you only know one footstep into the future, the ZMP reference looks like a square wave and the optimal center-of-mass trajectory chatters at the support transitions. With about 1.5 seconds of preview, the cost function smooths out the transitions and you get a clean, almost sinusoidal sway. Most ZMP-based walkers built in the 2000s and 2010s used some variant of this scheme [7][8].
The limitations are visible in the recipe. Step 1 assumes you have already chosen footprints, which is the hard part of recovering from a shove. Step 4 assumes inverse-kinematics with no contact constraints beyond the planned footfalls. The controller has nothing to say about whole-body redundancy, contact transitions, or compliant ground.
Later work replaced the preview controller with explicit model predictive control. Wieber's 2006 Humanoids paper formulated ZMP gait generation as a quadratic program with hard constraints that the ZMP lie inside the support polygon [14]. The QP version is what most modern ZMP-style controllers actually run, because it handles the polygon constraint cleanly and extends to step-timing adjustment. The basic dynamics are the same.
The LIPM dynamics decouple into a stable and an unstable mode. The unstable mode is called the capture point or divergent component of motion (DCM): the position where, if you placed your next foot, the LIPM would coast to a stop without further input.
For the LIPM at constant height, the capture point in 1D is
$$\xi = c + \sqrt{z_c / g} , \dot{c}.$$
This is just the center of mass plus a velocity-scaled lookahead. Pratt, Carff, Drakunov and Goswami formalized the capture-point concept in a 2006 Humanoids paper titled "Capture point: a step toward humanoid push recovery" [13]. Englsberger and colleagues at the German Aerospace Center (DLR) generalized it to the divergent component of motion in a series of papers from 2011 to 2015 [15].
DCM control is a strict generalization of ZMP preview control. You still plan in terms of footstep locations and an underlying ZMP/CoP, but the control law works directly on the unstable mode, which is one-step recoverable. Push recovery becomes cleaner: when shoved, compute where the capture point is now, and step there. ZMP, in this view, is one piece of the dynamics, not the whole controller.
The following table is a partial census of bipedal robots whose published controllers were ZMP-based at some point in their lives. Several have since moved to other paradigms. The earliest fully dynamically balanced entry, WL-10RD, achieved roughly 1.3 seconds per step in 1984 [3].
| Robot | Maker | First demo | Walking control |
|---|---|---|---|
| WL-10RD | Waseda (Kato lab) | 1984 | First dynamically balanced ZMP gait [3] |
| WL-12 | Waseda (Kato lab) | 1985 | Static and quasi-dynamic ZMP |
| WABIAN-2 | Waseda | 2005 | ZMP preview control |
| P2 / P3 / ASIMO | Honda | 1996 / 1997 / 2000 | ZMP with foot-landing modification [9] |
| H6, H7 | University of Tokyo (Inaba lab) | 2000 | Online ZMP planning |
| QRIO | Sony | 2003 | ZMP preview |
| HRP-2 Promet | Kawada and AIST | 2002 | Kajita preview control [7] |
| HRP-3 | Kawada and AIST | 2007 | Preview control with hand contact |
| HRP-4 | Kawada and AIST | 2010 | Preview control |
| HRP-4C Miim | AIST | 2009 | Preview control |
| HRP-5P | AIST | 2018 | Preview control plus multi-contact |
| KHR-1, KHR-2, HUBO | KAIST | 2002-2005 | ZMP feedback |
| HUBO 2, DRC-HUBO | KAIST | 2008 / 2015 | ZMP plus task-priority IK |
| LOLA | TU Munich | 2009 | Cart-table preview |
| iCub | IIT | 2008 | ZMP-based balancer |
| NAO | Aldebaran / SoftBank | 2008 | Simplified ZMP |
| Pepper | Aldebaran / SoftBank | 2014 | Wheeled, ZMP-style sway control |
| TORO | DLR | 2013 | DCM control over ZMP [15] |
| T-HR3 | Toyota | 2017 | ZMP teleoperation |
| Schaft S-One | Schaft Inc. (DRC) | 2013 | Preview control plus QP |
ASIMO is the public face of ZMP. The 2000 unveiling was the first time many people saw a humanoid take a step that was not a shuffle, and the engineering behind it was a layered ZMP controller of the kind described above [10]. By the 2005 model, ASIMO could run at 6 km/h and climb stairs, still inside the ZMP framework with extensions for flight phases. Honda retired ASIMO in 2018.
HRP-2 Promet is, in academic terms, the most influential ZMP humanoid. Built by Kawada Industries and AIST in 2002, it ran Kajita's preview controller in production and was sold to laboratories worldwide [7]. Many open-source whole-body controllers in use today, including early TSID and SOT codebases, were first developed on an HRP-2.
HUBO at KAIST under Jun-Ho Oh used an orthodox ZMP feedback controller. DRC-HUBO, the version that won the 2015 DARPA Robotics Challenge, added a task-priority inverse-kinematics layer for manipulation but still used ZMP for walking. HRP-5P, AIST's 2018 construction-task humanoid, is one of the most recent high-end ZMP-based walkers still in active research use. It is heavy, slow, conservative, and reliably does not fall over while installing plasterboard.
Early work on biped walking at Boston Dynamics with PETMAN and the original hydraulic Atlas used ZMP-style stability checks within a trajectory-optimization framework, although the company was always cagey about implementation details. The all-electric Atlas unveiled in April 2024 has been described as running model predictive control with whole-body dynamics and learned components on top, and is no longer a ZMP-first system.
Cassie and Digit from Agility Robotics started in this lineage but moved away. Cassie's early controllers were trajectory-optimization-based with ZMP-style stability checks; current Digit deployments use learned policies trained in simulation.
ZMP is exact for the model on which it is built, and the model is the catch. Four limits matter.
First, the model assumes a flat-foot rigid contact. Once the foot rolls onto its edge, or steps on something compliant, or makes contact with something other than the foot (a hand on a railing, a knee on the floor), the simple ZMP test breaks. "Multi-contact ZMP" generalizations exist (Caron, Pham, Nakamura 2017), but they require evaluating the contact wrench cone, which is most of the work [17]. Goswami's foot rotation indicator is the cleanest way to reason about the regime where the basic ZMP test has already failed [12].
Second, the criterion is only sufficient. The strict reading of the Vukobratović criterion is conservative: it forbids any motion that would briefly pull the ZMP onto the foot edge. In practice humans walk with toe-off and heel-strike phases that violate this. ZMP-only walkers look stiff because they have to stay in single-support with a centered ZMP, which forces a slow gait.
Third, angular momentum is treated as a small term. The Newton-Euler formula for ZMP includes the rate of change of angular momentum, but the LIPM and cart-table simplifications drop it. For agile motions (running, jumping, recovering from a hard shove) the angular-momentum term is not small, and ignoring it produces a controller that is right on average and wrong at the moments that matter. Centroidal-momentum control (Orin, Goswami, Lee 2013) keeps the angular-momentum term and generalizes cleanly to multi-contact [16].
Fourth, ZMP is a planning tool, not a contingency plan. It tells you whether a planned trajectory is balanced. It does not tell you what to do when the trajectory is no longer feasible, beyond "step somewhere else". DCM does that [15], model predictive control does it more flexibly [14], and reinforcement learning policies do it without thinking about ZMP at all.
The rough chronology of bipedal control over the last twenty years: ZMP preview was the standard from roughly 2003 to 2012 [7]. Capture point and DCM dominated about 2008 to 2018 [13][15]. From 2014 onward, whole-body torque control, with quadratic-program-based inverse dynamics solvers like SOT, TSID and OCS2, began to replace inverse kinematics on every research humanoid that mattered. In parallel, model-predictive trajectory optimization with full or centroidal dynamics started to take over the planning layer [16]. By 2020, a different lineage emerged: train a neural-network policy in simulation with reinforcement learning, domain-randomize the simulation, and run the policy on hardware with no analytic stability guarantees at all.
The RL approach took off because of three things at once. Simulators (MuJoCo, Isaac Sim, NVIDIA Isaac Lab) became fast enough to run thousands of walking-time-equivalent rollouts in parallel. Sim-to-real techniques, especially domain randomization and observation noise, became reliable enough that policies trained entirely in simulation worked on real hardware. And the hardware (Unitree H1 and G1, Atlas, Tesla Optimus, Figure 02, Apptronik Apollo) became proprioceptively rich and torque-controllable enough to run a learned policy.
The table below sketches the controller stacks of the most-discussed humanoids as of 2025, as far as published work allows.
| Robot | Walking control |
|---|---|
| HRP-5P | ZMP preview |
| NAO | ZMP-based stabilizer |
| HUBO 2 | ZMP plus task-priority IK |
| TORO | DCM |
| Atlas (electric) | MPC with whole-body dynamics, RL components |
| Atlas (hydraulic, retired 2024) | Trajectory optimization plus QP whole-body controller |
| Digit | RL policy in production |
| Unitree H1, G1 | RL policy with reference motion |
| Tesla Optimus | RL policy, mocap-bootstrapped |
| Figure 02 | RL policy with neural visual policy |
| Apptronik Apollo | Hybrid: optimization-based locomotion, learned manipulation |
The split is not as clean as the table makes it look. Most modern systems blend several approaches. Atlas's locomotion stack has been described as MPC over a centroidal model with learned components on top, but Boston Dynamics has been deliberately quiet about the details. Optimus and Figure are more openly RL-driven, and the demos look it: the gait is fluid and doesn't have ASIMO's tell-tale double-support pause, but it is also harder to verify formally. Even the RL pipelines often initialize policies on ZMP-based reference trajectories or include ZMP-style penalty terms in the reward, so the old criterion lingers as a regularizer rather than a hard constraint.
What does ZMP still do for you in 2025? Three things.
It remains the easiest correct framework for teaching legged robotics. Kajita's textbook is still a sensible first read, and the cart-table model is the simplest non-trivial dynamic balance problem you can set up [8]. For an introductory course or a starter project, ZMP preview control is hard to beat for the ratio of insight to effort.
It is a sanity check. A learned policy that produces a foot trajectory you can compute the ZMP of is a policy you can audit against the same support-polygon criterion that ASIMO satisfied. Sometimes that is the only formal handle anyone has on what an RL controller is doing.
And in low-speed, conservative deployments (factory humanoids, exhibition robots, walking aids) ZMP-based controllers are still the safest engineering choice. They do not surprise their operators and they are easy to certify. HRP-5P is not slower than the electric Atlas by accident.
The direction of travel is clear. If your humanoid has to look natural, recover from real disturbances, or run, pure ZMP will not get you there. If your humanoid has to install plasterboard without injuring anybody, it might.
Imagine balancing a broom upright on your open palm. As long as you keep your hand under the broom, it stays up; the moment your hand has to slide past the broom's lean to keep it from falling, it tips. A walking robot's feet are like that palm, and the zero moment point is the single spot under the foot that the robot's whole body "pushes down on" at each instant. If that spot stays inside the area covered by the foot (the support polygon), the robot is balanced. If the math says the spot would have to be off the edge of the foot, the foot rolls over and the robot falls. ZMP control is just the rule "plan your steps so that spot always lands inside your foot."
| Year | Authors | Contribution |
|---|---|---|
| 1968 | Vukobratović and Juričić | First presentation of the ZMP idea, Moscow All-Union Congress [3] |
| 1969 | Vukobratović and Juričić | Introduces the ZMP concept (IEEE Trans. Bio-Medical Engineering) [1] |
| 1972 | Vukobratović and Stepanenko | Sharpens the stability principle; book-length treatment [2] |
| 1984 | Waseda (Kato lab) | WL-10RD, first dynamically balanced ZMP biped gait [3] |
| 1986 | Honda | Start of the secret E-series biped program |
| 1991 | Kajita and Tanie | Linear inverted pendulum model for biped gait [5] |
| 1996 | Hirai et al. (Honda) | P2 reveal, first public demo of a ZMP-controlled adult-size humanoid [9] |
| 1999 | Goswami | Foot rotation indicator (IJRR) [12] |
| 2000 | Honda | ASIMO public unveiling |
| 2002 | Sakagami et al. | The intelligent ASIMO, IROS; HRP-2P first demo [10] |
| 2003 | Kajita et al. | ZMP preview control, ICRA, the standard reference for the 2000s [7] |
| 2004 | Vukobratović and Borovac | "Zero-moment point: thirty five years of its life", retrospective [3] |
| 2004 | Sardain and Bessonnet | ZMP versus center of pressure, IEEE SMC [11] |
| 2006 | Pratt, Carff, Drakunov, Goswami | Capture point, Humanoids [13] |
| 2006 | Wieber | QP-based ZMP gait generation, Humanoids [14] |
| 2011-2015 | Englsberger et al. | Divergent component of motion, DLR [15] |
| 2013 | Orin, Goswami, Lee | Centroidal momentum control [16] |
| 2014 | Kajita, Hirukawa, Harada, Yokoi | Introduction to Humanoid Robotics textbook [8] |
| 2017 | Caron, Pham, Nakamura | Multi-contact ZMP and contact wrench cone [17] |
| 2018 | AIST | HRP-5P plasterboard demo; Honda retires ASIMO |
| 2024 | Boston Dynamics | All-electric Atlas, hydraulic Atlas retired |