This paper presents a Model Predictive Control (MPC) formulation for bipedal footstep planning based on the Linear Inverted Pendulum (LIP) model, ensuring recursive feasibility when navigating restricted regions. The proposed approach incorporates capturability and introduces a new constraint that forces the Divergent Component of Motion (DCM) into a finite-step capture region, adjusted between consecutive MPC calls. This constraint enables the MPC to anticipate beyond its prediction horizon, preventing collisions with the walking surface boundaries. We validate the approach through high-fidelity simulations with the bipedal robot Digit, demonstrating recursively feasible MPC footstep planning in restricted regions. Future efforts will extend the approach to general polytopic constraints, thereby facilitating footstep planning in cluttered environments while preserving the MPC's recursive feasibility.

Humanoid robots are expected to navigate in changing environments and perform a variety of tasks. Frequently, these tasks require the robot to make decisions online regarding the speed and precision of following a reference path. For example, a robot may want to decide to temporarily deviate from its path to overtake a slowly moving obstacle that shares the same path and is ahead. In this case, path following performance is compromised in favor of fast path traversal. Available global trajectory tracking approaches typically assume a given-specified in advance-time parametrization of the path and seek to minimize the norm of the Cartesian error. As a result, when the robot should be where on the path is fixed and temporary deviations from the path are strongly discouraged. Given a global path, this paper presents a Model Predictive Contouring Control (MPCC) approach to selecting footsteps that maximize path traversal while simultaneously allowing the robot to decide between faithful versus fast path following. The method is evaluated in high-fidelity simulations of the bipedal robot Digit in terms of tracking performance of curved paths under disturbances and is also applied to the case where Digit overtakes a moving obstacle.

This paper presents an adaptive control scheme for robotic systems that operate in the face of-potentially large-structured uncertainty. The proposed adaptive controller employs an on-line supervisor that utilizes logic-based switching among a finite set of controllers to identify uncertain parameters, and adapt the behavior of the system based on a current estimate of their value. To achieve this, the adaptive control approach in this paper combines on-line parameter estimation and feedback control while avoiding some of the inherent difficulties of classical adaptive control strategies. Furthermore, the proposed supervisory control architecture is modular as it relies on established "off-the-shelf" feedback control law and estimator design approaches, instead of cus-tomizing the overall design to the specific requirements of an adaptive control algorithm. We demonstrate the efficacy of the method on the problem of a dynamically-walking bipedal robot delivering a payload of unknown mass, and show that, by switching to the controller that is the "best" according to a current estimate of the uncertainty, the system maintains a low energy cost during its operation.

Complex robot motions are frequently generated by composing simpler primitive movements. We use this approach to formulate robot motion plans as sequences of primitives to be executed one after the other. When dealing with dynamical movement primitives, besides accomplishing the high-level objective, planners must also reason about the effect of the plan's execution on the safety of the platform. This task is exacerbated by the presence of disturbances, such as non-vanishing external forces. To address this issue, we present a framework that builds on rigorous control-theoretic tools to generate safely executable motion plans for externally excited robotic systems. We illustrate the proposed framework on adapting the motion of a 3D bipedal robot model to persistent external forcing by switching among dynamic movement primitives, each corresponding to a limit-cycle walking gait.

This paper presents a control method for steering three dimensional (3D) dynamically walking bipeds that are engaged in cooperative tasks such as object transportation. Towards achieving safe interaction with a leading human (or robot) collaborator, the walking biped is required to exhibit compliance at the port of interaction, while simultaneously adapting its walking pattern in response to the perceived interaction forces. To address these issues, we propose a method that fuses impedance control of the biped's arm with position control of its legs in a way that the biped adaptively modifies its stepping pattern according to the collaborator's intentions. The method is applied on a 3D bipedal robot that is driven in the workspace by a collaborator, whose intention is communicated to the biped through the interaction force.

This paper presents a method for navigating 3D dynamically walking bipedal robots amidst obstacles. Our framework relies on composing gait primitives corresponding to limit-cycle locomotion behaviors and it produces nominal motion plans that are compatible with the system's dynamics and can be tracked with high fidelity. The low-level controllers of the biped are designed within the Hybrid Zero Dynamics (HZD) framework. Exploiting the dimensional reduction afforded by HZD and properties of invariant sets of switching systems among multiple equilibria, we obtain polynomial approximations of a reduced order Poincaré map and of the net change of the center of mass location over a stride. These polynomials are then incorporated in a high-level Rapidly Exploring Random Tree (RRT) planner to generate nominal plans which are tracked by the biped with drastically low drifting errors, without adversely affecting the time for computation.

This paper presents a method to achieve online gait adaptation of a dynamically walking biped when collaborating with an external agent-either a human or a robot-acting as a leader. Adaptation occurs without any explicit information on the leader's intended motion; only implicit information is used through the interaction force developed between the leader and the biped. An adaptive supervisory control scheme is proposed and guarantees for boundedness of the state despite switching under external force are provided. The supervisory controller leverages the availability of a library of exponentially stable limit-cycle gaits, and orchestrates switching among them in an online fashion to achieve adaptation. As a result, the range of leader speeds that the biped can adapt to is drastically enlarged while the leader's effort is reduced.

This paper reports on a model-based control strategy for miniature legged robots tasked with navigation in cluttered environments. Our approach uses a new model for crawling locomotion to derive closed-form expressions of state propagation. The latter enable the development of a feedback control navigation strategy. The strategy consists of a waypoint tracking controller that steers the system along desired paths and an outer control loop that updates the reference path to account for uncertainty. This strategy allows noise-resilient navigation for miniature legged robots and is experimentally validated on an eight-legged robot that navigates in obstacle-cluttered environments.

This paper approaches from an optimal control perspective the problem of fixed-time detection of mobile radioactive sources in transit by means of a collection of mobile sensors. Under simplifying assumptions on the geometry of the source, sensors, and surrounding environment, and with knowledge of the source's trajectory, it is shown that the optimal control problem admits an intuitive, analytic closed-form solution. This is facilitated by the availability of analytic expressions for bounds on the probabilities of detection and false alarm during a Neyman-Pearson detection test. The intuition derived from this analytic solution leads to a motion control law that steers suboptimally the sensors to an arbitrarily small neighborhood of the suspected source, while navigating among stationary obstacles in their environment, making a step closer to a practical physical implementation.

This paper examines the problem of transitioning between gaits using a sagittal-plane reduced-order model of quadrupedal running, which has compliant elements in its torso and legs. First, periodic motions that correspond to pronking and two variations of bounding are generated. Analysis of the torso pitch motion in these gaits reveals basic differences that determine whether transitions between these gaits can be achieved. This observation is then incorporated in control design to enhance the stability of the generated motions. The domain of attraction of the closed-loop systems is estimated through simulation, showing that transition between pronking and bounding can be realized as a sequence of switchings between fixed points. The results in this work can be regarded as the first step towards synthesizing controllers for gait transitions within template models for quadrupedal running gaits.

Inspired by biological systems, compliant elements are introduced in the driving train of legged robots with the purpose of recycling energy. This paper presents the design and control concept of the monopedal robot SPEAR, driven by a novel implementation of a Switchable Parallel Elastic Actuator (S-PEA). At the stance phase, the parallel spring in S-PEA works with the actuator to support body weight and recover energy. The spring is removed from the system at the flight phase to gain precise joint control. The stiffness of the whole leg is also adjustable online either by active control of the motor or by changing the knee angle at touchdown. Experimental results show that the spring recycles part of energy during hopping, reducing the peak torque and peak power of the motor.

This paper presents a method for integrating a cooperative manipulation task in the design of dynamic walking motions for an underactuated bipedal robot. Applications that involve physical interaction between a walking biped and a leading human (or robot) collaborator, require that the biped exhibits compliance at the port of interaction with the collaborator, while at the same time be capable of adjusting its stepping pattern in response to the interaction forces developed. To achieve these objectives, the proposed method combines impedance control of the biped's arm with position control of its legs in a way that the closed-loop system adapts its stepping pattern in accordance with the collaborator's intentions. The method is applied in the case of a bipedal robot model walking over flat ground and up and down stairs of known geometry under the influence of a trajectory that is unknown to the biped and represents the intention of a collaborator.

This paper investigates the ability of dynamically walking bipeds to adapt their motion to persistent exogenous forcing. Applications that involve physical interactions between a bipedal robot and other robots (or humans), require that the robot adjust its stepping pattern in response to externally applied force signals. In our setting, an underactuated bipedal robot model walks under the influence of an external force. First, the hybrid zero dynamics method is used to design controllers that stabilize periodic walking motions in the absence of the external force. Then, conditions are derived analytically under which these (unforced) periodic gaits are modified to new (possibly aperiodic) stepping patterns that are consistent with the external force. It is deduced that underactuation holds the key to the ability of our model to adapt to external forces.

This paper examines the influence of torso compliance on the efficiency of quadrupedal running with a bounding gait. Two sagittal-plane models, one with and one without torso compliance, are developed in a template setting. The models feature non-trivial leg mass, and are coupled with a simple leg recirculation controller to generate bounding motions. Despite their simplicity, the proposed models are sufficiently expressive to capture the energetics of bounding motions and to assess the contribution of torso flexibility to gait efficiency measured by the cost of transport. Comparisons reveal that torso compliance promotes locomotion performance by reducing energy consumption only at speeds that are sufficiently high.

We present our findings on the first application of motion planning methodologies to the recently introduced Sprawl Tuned Autonomous Robot (STAR). The reported results provide a first glimpse on the capabilities of this novel, 3D-printed robot in performing autonomously non-trivial motion planning tasks in environments populated with obstacles. We employ methods from sampling-based motion planning under nonholonomic constraints, and implement in open loop the generated path on the physical robot for various environments of increasing complexity.

The paper suggests a new method for statistically validating, and selecting the parameters of a model for a miniature eight-legged robot. It is based on a novel adaptation of concepts and techniques originally developed in the context of robust control design using randomized algorithms. The proposed approach is data driven and offers probabilistic guarantees of model fidelity and descriptive capacity, checking models against experimental data. In principle, this method applies to a large class of physical processes, the available models of which may be in a variety of forms including sets of differential equations.

This paper studies the stability and control of passively generated bounding motions for a class of quadrupedal template models that incorporate compliant elements in their segmented torso and legs. First, the existence of self-stable cyclic bounding gaits in the presence of torso flexibility is examined. Based on numerical studies of the associated Poincaré return map in a dimensionless context, it is deduced that self-stable bounding gaits can indeed be generated through appropriate combinations of the stiffness of the torso and the stiffness of the legs. Next, the implications of this result to control design are discussed, and a hybrid controller is proposed that enhances the stability of the passively generated bounding gaits.

This paper examines the effect of torso flexibility on the dynamics of quadrupedal running with a bounding gait. A reduced-order passive and conservative model with a segmented flexible torso and compliant legs is introduced to study torso-leg coordination. Numerical return map studies reveal that a large variety of cyclic bounding motions can be realized passively, as a natural mode of the system. Despite the simplicity of the model, the resulting motions correspond to torso bending movements that resemble those in galloping mammals without explicit reliance on the fine structural and morphological details. This way, the proposed model offers a unifying description of the task-level locomotion behavior, and can be used to inform feedback control synthesis by serving as a behavioral target for the control system.

We propose a new model to describe the horizontal motion of an eight-legged bio-inspired miniature robot. The model does not include compliance, and can capture the kinematics of the observed locomotion behavior that corresponds to an alternating tetrapod gait. We exploit symmetries and synergies to reduce the eight-legged robot to a mechanism that is composed by two switching four-bar linkages, each representing the collective effect of a tetrapod in contact with the ground. Notwithstanding its apparent simplicity, the resulting model reproduces on average the motion of the robot. In addition, by properly tuning a family of physically-relevant parameters - including touchdown and sweep angles - different motion primitives corresponding to circular and forward motions can be realized. This model represents a first step toward developing reduced-order kinematic representations of legged robots that can be used for motion planning and feedback control purposes.

This paper explores systematic control strategies for the stabilization of running on compliant robots with nontrivial torso pitch dynamics. The Spring Loaded Inverted Pendulum (SLIP) embedding controller is revisited and its pertinence to more general legged robot models is investigated. It is first deduced that - in the context of an asymmetric hopper - the existence of a SLIP embedding control law requires nominal running motions in which the torso is kept at a constant angle. To remove this overly restrictive condition, a new method is proposed here that retains the advantage of generating control inputs acting in concert with the compliant dynamics of the plant without explicit reliance on the SLIP. To illustrate the enhanced control authority afforded by the proposed method, a minimalist setting is considered, in which a three-degree-of-freedom asymmetric hopper is controlled by a single actuator located at the hip.

A hybrid controller that induces stable running gaits on a monopedal robot is developed. The robot features a rigid leg with a revolute knee and a heavy torso with center of mass located far from the hip. The torso houses a novel powertrain that provides series compliance in the compression direction of the leg. The proposed control law is developed within the hybrid zero dynamics framework and it acts on two levels. On the first level, continuous within-stride control asymptotically imposes (virtual) holonomic constraints reducing the dynamics of the robot to a lower-dimensional hybrid subsystem. On the second level, event-based control stabilizes the resulting hybrid subsystem. The controller achieves the dual objectives of working harmoniously with the system's natural dynamics and inducing provably exponentially stable running motions, while all relevant physical constraints are respected.

Two feedback controllers that induce stable running gaits on a three-degree-of-freedom asymmetric hopper, termed the asymmetric spring loaded inverted pendulum (ASLIP), see Fig. 1, are compared in terms of their steady-state and transient behaviors. In each case, feedback is used to create a lower-dimensional hybrid subsystem that determines the existence and stability properties of periodic motions of the full-dimensional closed-loop system. The first controller creates a one degree-of-freedom subsystem through imposing two suitably selected (virtual) holonomic constraints on the configuration variables of the ASLIP. The second controller asymptotically imposes a single (virtual) holonomic constraint to create a two-degree-of-freedom subsystem that is diffeomorphic to a standard spring loaded inverted pendulum (SLIP). The two controllers induce identical steady-state behaviors. Under transient conditions, however, the underlying compliant nature of the SLIP allows significantly larger disturbances to be accommodated, with less actuator effort, and without violation of the unilateral constraints between the leg end and the ground.

In this paper the authors discuss control and modeling issues significant for constructing simple running controllers and building experimentally validated simulation models for dynamically stable robots. A bounding controller resulting in dynamically stable running up to 1.3 m/s for the Scout II quadruped is presented. Simulation models are built and compared with experimental data to test the viability of various simplifying assumptions common in the literature for running robots. The authors demonstrate the need for including motor saturation and nor rigid torque transmission characteristics into simulation models. Similar issues are likely to be important in other legged robots as well. An extensive suite of experimental results documents the robot's performance and validates simulation models.

This paper discusses recent galloping results obtained using the Scout II quadrupedal robot, an underactuated robot with minimal sensing. The rotary variation of the gallop has been implemented, resulting in a motion which demonstrates both emergent stability and a circular trajectory, as predicted in earlier simulation studies by other researchers. It also demonstrates phase relationships in the leg touchdown pattern which resemble results from biology.

In this paper, we study the passive dynamics of quadrupedal bounding, based on a simplified model of our Scout II quadruped robot. Surprisingly, numerical return map studies reveal that passive generation of a large variety of cyclic bounding motion is possible. Most strikingly, local stability analysis shows that the dynamics of the open loop passive system alone can confer stability of the motion. Stability improves at higher speeds, which is in agreement with recent results from biomechanics. These results can be used in developing a general control methodology for legged robots, resulting from the synthesis of feed-forward and feedback models that take advantage of the mechanical system, and might explain the success of simple, open loop bounding controllers on our experimental robot.

A planning methodology for nonholonomic mobile manipulators that employs smooth and continuous functions such as polynomials is developed. The method decouples kinematically the manipulator from the platform by constructing admissible paths that drive it to a final configuration and is based on mapping the nonholonomic constraint to a space where it can be trivially satisfied. In addition, the method allows for direct control over the platform orientation. The developed transformation also maps Cartesian space obstacles to transformed ones and allows for obstacle avoidance by increasing the order of the polynomials that are used in planning trajectories. The additional parameters required are computed systematically. It is shown how the method can be extended for avoiding obstacles of any number.