Open-loop stable limit cycles are foundational to legged robotics, providing inherent self-stabilization that minimizes the need for computationally intensive feedback-based gait correction. While previous methods have primarily targeted specific robotic models, this paper introduces a general framework for rapidly generating limit cycles across various dynamical systems, with the flexibility to impose arbitrarily tight stability bounds. We formulate the problem as a single-stage constrained optimization problem and use Direct Collocation to transcribe it into a nonlinear program with closed-form expressions for constraints, objectives, and their gradients. Our method supports multiple stability formulations. In particular, we tested two popular formulations for limit cycle stability in robotics: (1) based on the spectral radius of a discrete return map, and (2) based on the spectral radius of the monodromy matrix, and tested five different constraintsatisfaction formulations of the eigenvalue problem to bound the spectral radius. We compare the performance and solution quality of the various formulations on a robotic swing-leg model, highlighting the Schur decomposition of the monodromy matrix as a method with broader applicability due to weaker assumptions and stronger numerical convergence properties. As a case study, we apply our method on a hopping robot model, generating open-loop stable gaits in under 2 seconds on an Intel®Core i7-6700K, while simultaneously minimizing energy consumption even under tight stability constraints.

For effective operation in challenging outdoor environments, mobile unmanned robots face stiff and competing demands including payload capacity, driving speed, range, as well as the ability to traverse rough terrain. To address these issues we introduce the hybrid wheel-leg quadrupedal robot WaLTER. WaLTER utilizes a unique combination of continuously rotating distal leg joints, actuated wheels, and a roll body DOF to efficiently drive on flat ground and effectively tumble over stairs and difficult, broken terrain. We developed an intuitive teleoperation scheme and employed deep reinforcement learning as proof of concept control techniques for the novel morphology. To test its capabilities, we constructed a multi-body simulation in MuJoCo and a 2.1 kg physical prototype for experimentation on traversability and energy economy. Our testing demonstrated the ability to traverse rougher terrain relative to larger-wheeled counterparts and reliable stair-climbing while maintaining a 4 km range on a 24.4 Wh battery (COT: 1.21).

Collision-free planning is essential for bipedal robots operating within unstructured environments. This paper presents a real-time Model Predictive Control (MPC) frame-work that addresses both body and foot avoidance for dynamic bipedal robots. Our contribution is two-fold: we introduce (1) a novel formulation for adjusting step timing to facilitate faster body avoidance and (2) a novel 3D foot-avoidance formulation that implicitly selects swing trajectories and footholds that either steps over or navigate around obstacles with awareness of Center of Mass (COM) dynamics. We achieve body avoidance by applying a half-space relaxation of the safe region but introduce a switching heuristic based on tracking error to detect a need to change foot-timing schedules. To enable foot avoidance and viable landing footholds on all sides of foot-level obstacles, we decompose the non-convex safe region on the ground into several convex polygons and use Mixed-Integer Quadratic Programming to determine the optimal candidate. We found that introducing a soft minimum-travel-distance constraint is effective in preventing the MPC from being trapped in local minima that can stall half-space relaxation methods behind obstacles. We demonstrated the proposed algorithms on multibody simulations on the bipedal robot platforms, Cassie and Digit, as well as hardware experiments on Digit.

Robots routinely encounter obstacles and rough terrain, but terrain curvature is seldom included in models for real-time algorithms. We present a closed-form dynamic model for rolling with two planar smooth curves, and apply it to sagittal-plane locomotion problems. We assumed that the body rolls without slip and maintains a single point of contact. Using an auxiliary coordinate system to define the rolling body and terrain as parametric curves, we derived rolling constraints and dynamic equations of motion for model-based control algorithms – specifically Operational Space Control. The formulation was used to simulate an arbitrarily curved rock rolling on undulating terrain and to generate control signals to stabilize it on parabolic terrain. The stabilization problem was solved as a quadratic program in < 3 ms which shows that our formulation is suitable for real-time control algorithms. We also applied this framework to dynamically balance an underactuated 2 degree-of-freedom leg on parabolic terrain and achieve prescribed locomotion tasks for a wheel-leg vehicle on sinusoidal terrain in simulation. A supplementary video is available at https://youtu.be/EtPQEzkqsK8.

The human world is full of risks that threaten failure of robotic tasks. Dynamic robots, such as agile drones and walking bipeds, are particularly susceptible to failure because their time to make critical decisions is short. This work seeks a control algorithm which adapts to failures and reprioritizes robot behavior automatically, all at real-time speeds. Our failure-adaptive control framework learns failure probabilities from in situ experience and minimizes the risk of future failures using fast online planners (i.e. model predictive control). By reasoning about probabilities of failure, more imminent risks are automatically prioritized by the framework without manually tuning weighting factors. Further, our low-order probability model is learned using fast convex optimizations, allowing for immediate learning from triggered failures during operation. We demonstrate the framework's capability to learn and plan in real time (< 20 ms) in highly dynamic scenarios with micro-aerial vehicles (i.e. drones). We conduct two experiments: a chase-avoid task, and a chase-avoid - track task. In both scenarios, a single failure causes a categorical shift in robot behavior and the drone will adapt, plan, and execute a non-failing strategy within one second post-failure.

In real-world settings, bipedal robots must avoid collisions with people and their environment. Further, a biped can choose between modes of avoidance: (1) adjust its pose while standing or (2) step to gain maneuverability. We present a real-time motion planner and multibody control framework for dynamic bipedal robots that avoids multiple moving obstacles and automatically switches between standing and stepping modes as necessary. By leveraging a reduced-order model (i.e. Linear Inverted Pendulum Model) and a half-space relaxation of the safe region, the planner is formulated as a convex optimization problem (i.e. Quadratic Programming) that can be used for real-time application with Model-Predictive-Control (MPC). To facilitate mode switching, we introduce center-of-pressure related slack-variables to the convex planning optimization that both shapes the planning cost function and provides a mode switching criterion for dynamic locomotion. Finally, we implement the proposed algorithm on a 3D Cassie bipedal robot and present hardware experiments showing real-time bipedal standing avoidance, stepping avoidance, and automatic switching of avoidance modes.

We want robots to complete assigned tasks even when unexpected task pressures arise, either from the robot or the environment. This paper presents a method of both learning sources of task failure in situ and rapidly planning new motions on-the-fly to accommodate them. This “risk-adaptive” approach to robot control uses a few encounters with a novel failure mode to generate a probabilistic failure model which we use to optimize a risk-mitigating motion plan. We demonstrate two toy problems, where risk-adaptive double-integrator agents are introduced to separate environments, each with their own tasks and modes of failure. The agents are not aware a priori of any risks the environments might present, but after one failure, the agents quickly adapt their motion plans and ensure task completion. We further conduct numerical experiments to characterize the algorithm's speed of adaptation with respect to environmental uncertainty. We see this framework as a natural extension for the myriad of robotic applications using model-based motion planners.

Tracks, wheels, and legs are all useful locomotion modes for Unmanned Ground Vehicles (UGVs), and ground robots that combine these mechanisms have the potential to climb over large obstacles. As robot morphologies include more degrees of freedom and obstacles become increasingly large and complex, UGVs will need to rely on automatic motion planning to compute the joint trajectories for traversal. This article presents a trajectory optimization formulation for multibody UGVs with combined wheel-leg and track-leg designs. We derive the dynamics and constraints for rolling wheels and circulating elliptical tracks. Using direct collocation, we formulate a model-based trajectory optimization where all constraints and objectives are written in closed-form with smooth and exact derivatives for tractable computation times with existing large-scale nonlinear optimization solvers (<1 minute). We demonstrate the trajectory optimization on numerous simulated planar wheel-leg and track-leg morphologies completing locomotion tasks, demonstrating full body dynamic coupling for the multibody system. Future work will extend this formulation to 3D and include contact planning.

Heterogeneous robotic systems in the field often encounter bodies of water with unknown traversability properties. One approach to measuring depth, current, soil composition, etc. is via an in situ underwater sensor being dragged by cable attached to a maneuvering airborne multicopter - which entails a novel motion planning and control problem with mixed resistive media. In this work we propose a framework to plan trajectories for future characterization sensors and payloads moving through mixed (air-water) media while considering uncertainty in the depth of the underwater ground surface. The methodology is applied to example underactuated systems with suspended payloads of increasing levels of complexity, including a cable robot and 4- and 8-DOF multicopter systems. Simulation studies employing trajectory optimization indicate that under certain payload configurations and task constraints, there are maneuvers in which it is more efficient to drag the payloads through water than through air. The paper also includes preliminary experiments with a testbed cable robot platform.

We present a real-time motion planner that avoids multiple moving obstacles without knowing their dynamics or intentions. This method uses convex optimization to generate trajectories for linear plant models over a planning horizon (i.e. model-predictive control). While convex optimizations allow for fast planning, obstacle avoidance can be challenging to incorporate because Euclidean distance calculations tend to break convexity. By using a half-space convex relaxation, our planner reasons about an approximated distance-to-obstacle measure that is linear in its decision variables and preserves convexity. Further, by iteratively updating the relaxation over the planning horizon, the half-space approximation is improved, enabling nimble avoidance maneuvers. We further augment avoidance performance with a soft penalty slack-variable for-mulation that introduces a piecewise quadratic cost. As a proof of concept, we demonstrate the planner on double-integrator models in both single-agent and multi-agent tasks-avoiding multiple obstacles and other agents in 2D and 3D environments. We show extensions to legged locomotion by bipedally walking around obstacles in simulation using the Linear Inverted Pendulum Model (LIPM). We then present two sets of hardware experiments showing real-time obstacle avoid-ance with quadcopter drones: (1) avoiding a 10m/s swinging pendulum and (2) dodging a chasing drone.

As we build our legged robots smaller and cheaper, stable and agile control without expensive inertial sensors becomes increasingly important. We seek to enable versatile dynamic behaviors on robots with limited modes of state feedback, specifically proprioceptive-only sensing. This work uses model-based trajectory optimization methods to design open-loop stable motion primitives. We specifically design running gaits for a single-legged planar robot, and can generate motion primitives in under 3 seconds, approaching online-capable speeds. A direct-collocation-formulated optimization generated axial force profiles for the direct-drive robot to achieve desired running speed and apex height. When implemented in hardware, these trajectories produced open-loop stable running. Further, the measured running achieved the desired speed within 10% of the speed specified for the optimization in spite of having no control loop actively measuring or controlling running speed. Additionally, we examine the shape of the optimized force profile and observe features that may be applicable to open-loop stable running in general.

Out in the field, bipedal robots need to travel on terrain that is uneven, non-rigid, and sometimes moving beneath their feet. We present a force-based double support balancing controller for such dynamic terrain scenarios for bipedal robots, and test it on the robotic bipedal platform "Tallahassee Cassie." The presented controller relies on minimal information about the robot model, requiring its kinematics and overall weight, but not inertias of individual links or components. The controller is pelvis-centric, commanding pelvis positions in Cartesian space, which a model-free PD controller converts to motor torques in joint space. By commanding forces, torques, and a frontal center of pressure in this fashion, Tallahassee Cassie is capable of balancing on a variety of scenarios, from a lifting/sliding platform, to soft foam, to a sudden drop. These results show the potential for bipedal control to balance successfully despite minimal model information, the presence of large dynamic impacts-e.g., falling through trap door, and soft series-spring deflections. These results motivate future work for walking and running controllers on dynamic terrain with relatively low reliance on modeling information.

Robots encounter many risks that threaten the success of practical locomotion tasks. Legs break, electrical components overheat, and feet can unexpectedly slip. When all risks cannot be completely avoided, how does a robot decide its best action? We present a method for planning robot motions by reasoning about risk-of-failure probabilities instead of applying cost-penalty functions or inflexible path constraints. This work develops a risk-constrained formulation that can be straightforwardly included in existing motion planning optimizations. The risk constraints scale tractably with many risk sources, and in some cases, only add linear constraints to the optimization problem and are therefore compatible with model-predictive control techniques. We present a toy "Puck World" proof-of-concept example and a practical implementation on a planar monopod robot that runs at 3.2 m/s when permitted to take high-risk maneuvers. We believe this risk approach can be used to optimize robot behaviors under numerous conflicting task pressures and model risk-conscious behaviors in animals.

This paper describes the design, control and initial experimental results of the quadruped robot LLAMA. Designed to operate in a human-scale world, this 67kg-class, all-electric robot is capable of rapid motion over a variety of terrains. Thanks to a unique leg configuration and custom high-torque, low gear-ratio motors, it can move omnidirectionally at speeds over 1 m/s. A hierarchical reactive control scheme allows for robust and efficient motion even under variable payloads. This paper describes the structure of the controller and outlines simulation results that probe the performance envelope of the robot suggesting payload capacities up to one third of its body weight. Initial testing shows robust motion over loose debris and a variety of ground slopes. Videos of the robot may be seen at https://tinyurl.com/llama-robot.

Practical use of robots in diverse domains requires programming for, or adapting to, each domain and its unique characteristics. Failure to do so compromises the ability of the robot to achieve task-relevant objectives. Here we describe how the learned terrain reaction force profiles of a hopping robot serve the additional objectives of classifying terrain and quickly learning control strategies to accomplish a jumping task on novel terrain. We show that the reaction forces experienced during closed-loop jumping are sufficient to discriminate between three different terrain types (granular, trampoline, and rigid) when using the learned models as discriminators. Building on this, we show that applying the classification to unknown terrain types leads to faster task completion, where the task objective is to meet a specific jump height. The classification experiments, utilizing real-world jumping data, achieve 95% prediction accuracy. The online learning experiments leverage simulation as there is more control over the terrain properties. Terrain-informed learning achieves the target hop heights more than 2x faster than without terrain knowledge when the prediction is correct, and 1.5x faster when the prediction is incorrect. Thus, applying the closest approximately known terrain knowledge facilitates low shot learning when hopping on unknown terrain.

Mobile robots of all shapes and sizes move through the air, water, and over ground. However, few robots can move through the ground. Not only are the forces resisting movement much greater than in air or water, but the interaction forces are more complicated. Here we propose a soft robotic device that burrows through dry sand while requiring an order of magnitude less force than a similarly sized intruding body. The device leverages the principles of both tip-extension and granular fluidization. Like roots, the device extends from its tip; the principle of tip-extension eliminates skin drag on the sides of the body, because the body is stationary with respect to the medium. We implement this with an everting, pressure-driven thin film body. The second principle, granular fluidization, enables a granular medium to adopt a dynamic fluid-like state when pressurized fluid is passed through it, reducing the forces acting on an object moving through it. We realize granular fluidization with a flow of air through the core of the body that mixes with the medium at the tip. The proposed device could lead to applications such as search and rescue in mudslides or shallow subterranean exploration. Further, because it creates a physical conduit with its body, electrical lines, fluids, or even tools could be passed through this channel.

Hybrid zero dynamics (HZD) has emerged as a popular framework for dynamic walking but has significant implementation difficulties when applied to the high degrees of freedom humanoids. The primary impediment is the process of gait design-it is difficult for optimizers to converge on a viable set of virtual constraints defining a gait. This paper presents a methodology that allows for fast and reliable generation of dynamic robotic walking gaits through the HZD framework, even in the presence of underactuation. Specifically, we describe an optimization formulation that builds upon the novel combination of HZD and direct collocation methods. Furthermore, achieving a scalable implementation required developing a defect-variable substitution formulation to simplify expressions, which ultimately allows us to generate compact analytic Jacobians of the constraints. We experimentally validate our methodology on an underactuated humanoid, DURUS, a spring-legged machine designed to facilitate energy-economical walking. We show that the optimization approach, in concert with the HZD framework, yields dynamic and stable walking gaits in hardware with a total electrical cost of transport of 1.33.

Bipedal robotic running remains a challenging benchmark in the field of control and robotics because of its highly dynamic nature and necessarily underactuated hybrid dynamics. Previous results have achieved bipedal running experimentally with a combination of theoretical results and heuristic application thereof. In particular, formal analysis of the hybrid system stability is given based on a theoretical model, but due to the gap between theoretical concepts and experimental reality, extensive tuning is necessary to achieve experimental success. In this paper, we present a formal approach to bridge this gap, starting from theoretical gait generation to a provably stable control implementation, resulting in bipedal robotic running. We first use a large-scale optimization to generate an energy-efficient running gait, subject to hybrid zero dynamics conditions and feasibility constraints which incorporate practical limitations of the robot model based on physical conditions. The stability of the gait is formally guaranteed in the hybrid system model with an input to state stability (ISS) based control law. This implementation improves the stability under practical control limitations of the system. Finally, the methodology is experimentally realized on the planar spring-legged bipedal robot, DURUS-2D, resulting in sustainable running at 1.75m/s. The paper, therefore, presents a formal method that takes the first step toward bridging the gap between theory and experiment.

The varied and complex dynamics of deformable terrain are significant impediments toward real-world viability of locomotive robotics, particularly for legged machines. We explore vertical jumping on granular media (GM) as a model task for legged locomotion on uncharacterized deformable terrain. By integrating (Gaussian process) GP-based regression and evaluation to estimate ground forcing as a function of state, a one-dimensional jumper acquires the ability to learn forcing profiles exerted by its environment in tandem to achieving its control objective. The GP-based dynamical model initially assumes a baseline rigid, non-compliant surface. As part of an iterative procedure, the optimizer employing this model generates an optimal control to achieve a target jump height while respecting known hardware limitations of the robot model. Trajectory and forcing data recovered from evaluation on the true GM surface model simulation is applied to train the GP, and in turn, provide the optimizer a more richly informed dynamical model of the environment. After three iterations, predicted optimal control trajectories coincide with execution results, within 1.2% jumping height error, as the GP-based approximation converges to the true GM model.

Hybrid zero dynamics (HZD) has emerged as a popular framework for dynamic and underactuated bipedal walking, but has significant implementation difficulties when applied to the high degrees of freedom present in humanoid robots. The primary impediment is the process of gait design-it is difficult for optimizers to converge on a viable set of virtual constraints defining a gait. This paper presents a methodology that allows for the fast and reliable generation of efficient multi-contact robotic walking gaits through the framework of HZD, even in the presence of underactuation. To achieve this goal, we unify methods from trajectory optimization with the control framework of multi-domain hybrid zero dynamics. By formulating a novel optimization problem in the context of direct collocation and generating analytic Jacobians for the constraints, solving the resulting nonlinear program becomes tractable for large-scale nonlinear programming solvers, even for systems as high-dimensional as humanoid robots. We experimentally validated our methodology on the spring-legged prototype humanoid, DURUS, showing that the optimization approach yields dynamic and stable 3D walking gaits.

Humanoid robots are designed with dozens of actuated joints to suit a variety of tasks, but walking controllers rarely make the best use of all of this freedom. We present a framework for maximizing the use of the full humanoid body for the purpose of stable dynamic locomotion, which requires no restriction to a planning template (e.g. LIPM). Using a hybrid zero dynamics (HZD) framework, this approach optimizes a set of outputs which provides requirements for the motion for all actuated links, including arms. These output equations are then rapidly solved by a whole-body inverse-kinematic (IK) solver, providing a set of joint trajectories to the robot. We apply this procedure to a simulation of the humanoid robot, DRC-HUBO, which has over 27 actuators. As a consequence, the resulting gaits swing their arms, not by a user defining swinging motions a priori or superimposing them on gaits post hoc, but as an emergent behavior from optimizing the dynamic gait. We also present preliminary dynamic walking experiments with DRC-HUBO in hardware, thereby building a case that hybrid zero dynamics as augmented by inverse kinematics (HZD+IK) is becoming a viable approach for controlling the full complexity of humanoid locomotion.

This paper presents the methodology used to achieve efficient and dynamic walking behaviors on the prototype humanoid robotics platform, DURUS. As a means of providing a hardware platform capable of these behaviors, the design of DURUS combines highly efficient electromechanical components with “control in the loop” design of the leg morphology. Utilizing the final design of DURUS, a formal framework for the generation of dynamic walking gaits which maximizes efficiency by exploiting the full body dynamics of the robot, including the interplay between the passive and active elements, is developed. The gaits generated through this methodology form the basis of the control implementation experimentally realized on DURUS; in particular, the trajectories generated through the formal framework yield a feedforward control input which is modulated by feedback in the form of regulators that compensate for discrepancies between the model and physical system. The end result of the unified approach to control-informed mechanical design, formal gait design and regulator-based feedback control implementation is efficient and dynamic locomotion on the humanoid robot DURUS. In particular, DURUS was able to demonstrate dynamic locomotion at the DRC Finals Endurance Test, walking for just under five hours in a single day, traveling 3.9 km with a mean cost of transport of 1.61-the lowest reported cost of transport achieved on a bipedal humanoid robot.

This work demonstrates fast motion planning for robot locomotion that is optimized for terrain with complex dynamics, specifically, rapid penetration of granular media. Gait planning is critical for many legged locomotion control approaches, but they typically assume rigid ground contact. We aim to extend these planning methods to include terrain dynamics we see in the natural world, like sand and dirt, which can both deform and fluidize. Using an added-mass description of collective grain motion, we formulated a model of hydrostatic and hydrodynamic terrain effects that is both principled and representable with closed-form dynamics. As a result, we present a model and fast optimization formulation which solves accurate motion plans on granular media with tractable solving times (6.4-3.8 seconds). For validation, we optimized open-loop motor trajectories for a testbed jumping robot to jump to a target apex height from a bed a loosely packed poppy seeds, a model granular medium. While jumps optimized for rigid ground were anemic on granular media, terrain-aware trajectories hit within 6% of their target. This demonstrates the potential for robot locomotion which meets practical task demands, all while being aware of the terrain beneath it.

Dynamic humanoid locomotion is a challenging control problem, and running is especially difficult to achieve, given the underactuation inherent to aerial domains. Previous work developed a gait-generating optimization framework for dynamic locomotion in the context of hybrid zero dynamics, producing stable 3D walking on the humanoid hardware platform DURUS. Here, we demonstrate that this optimization method also extends to stable 3D running. Gaits generated from the optimization, which utilizes the dynamics of all 23 degrees of freedom to maximize energy economy, results in stable running in a DURUS simulation model. Notably, the presented running is underactuated in all domains, due to DURUS' spring-legged design. Further, we generate 25 different running gaits, over a range of speeds (1.5-3.0 m/s), to demonstrate the reliability of solving the large-scale nonlinear program. We report statistical performance of the optimization in successfully generating stable running (average computation time: 323 seconds) in an effort to establish a benchmark for large-scale gait generation. We inspected this array of gaits across speeds, noting recognizable trends in optimized strategies from prior studies on lower-order models-e.g., both increased step frequency and step length with speed-along with the first reported cost-of-transport curve for a 3D humanoid running model. We consider this result an important step toward humanoid running on the DURUS hardware platform.

Hybrid zero dynamics (HZD) has emerged as a popular framework for the stable control of bipedal robotic gaits, but typically designing a gait's virtual constraints is a slow and undependable optimization process. To expedite and boost the reliability of HZD gait generation, we borrow methods from trajectory optimization to formulate a smoother and more linear optimization problem. We present a multiple-shooting formulation for the optimization of virtual constraints, combining the stability-friendly properties of HZD with an optimization-conducive problem formulation. To showcase the implications of this recipe for improving gait generation, we use the same process to generate periodic planar walking gaits on two different robot models, and in one case, demonstrate stable walking on the hardware prototype, DURUS-R.

We investigate the task-optimality of legged limit cycles and present numerical evidence supporting a simple general locomotion-planning template. Limit cycles have been foundational to the control and analysis of legged systems, but as robots move toward completing real-world tasks, are limit cycles practical in the long run? We address this question both figuratively and literally by solving for optimal strategies for long-horizon tasks spanning as many as 20 running steps. These scenarios were designed to embody practical locomotion tasks, such as evading a pursuer, and were formulated with minimal constraints (complete the task, minimize energy cost, and don't fall). By leveraging large-scale constrained optimization techniques, we numerically solve the trajectory for a reduced-order running model to optimally complete each scenario. We find, in the tested scenarios in flat terrain, that near-limit-cycle behaviors emerge after a transient period of acceleration and deceleration, suggesting limit cycles may be a useful, near-optimal planning target. On rough terrain, enforcing a limit cycle on every step only degrades gait economy by 2-5% compared to optimal 20-step look-ahead planning. When perturbing the scenario with a single “bump” in the road, the model converged in a manner giving the appearance of an exponentially stable orbit, despite not explicitly enforcing exponential stability. Further, we show that the transient periods of acceleration and deceleration may be near-optimally approximated by planning with a simple “sliding mass” template. These results support the notion that limit cycles can be useful approximations of task-optimal behavior, and thus are useful near-term targets for long-term planning.

Legged robots enjoy kilohertz control rates but are still making incremental gains towards becoming as nimble as animals. In contrast, bipedal animals are amazingly robust runners despite lagged state feedback from protracted neuromechanical delays. Based on evidence from biological experiments, we posit that much of disturbance rejection can be offloaded from feedback control and encoded into feed-forward pre-reflexive behaviors called preflexes. We present a framework for the offline numerical generation of preflex behaviors to optimally stabilize legged locomotion tasks in the presence of response delays. By coupling directly collocated trajectory optimizations, we optimize the preflexive motion of a simple bipedal running model to recover from uncertain terrain geometry using minimal actuator work. In simulation, the optimized preflex maneuver showed 30-77% economy improvements over a level-ground strategy when responding to terrain deviating just 2-4cm from the nominal condition. We claim this “preflex-and-replan” framework for designing efficient and robust gaits is amenable to a variety of robots and extensible to arbitrary locomotion tasks.