Recognizing and identifying human locomotion is a critical step to ensuring fluent control of wearable robots, such as transtibial prostheses. In particular, classifying the locomotion mode and estimating the gait phase are key. In this work, a novel, interpretable, and computationally efficient algorithm is presented for simultaneously predicting locomotion mode and gait phase. Using able-bodied (AB) data and transtibial prosthesis (PR) data collected via a bypass adapter, seven locomotion modes are tested including slow, medium, and fast level walking (0.6, 0.8, and 1.0 m/s), ramp ascent/descent (5 degrees), and stair ascent/descent (20 cm height). Overall classification accuracy was 99.1% and 99.3% for the AB and PR conditions, respectively. The average gait phase error across all data was less than 4%. Exploiting the structure of the data, computational efficiency reached 2.91 µs per time step. The time complexity of this algorithm scales as O(N•M) with the number of locomotion modes M and samples per•gait cycle N. This efficiency and high accuracy could accommodate a much larger set of locomotion modes (~ 700 on the Open-Source Leg Prosthesis) to handle the wide range of activities pursued by individuals during daily living.

We revisit the concept of constraint embedding as a means for dealing with kinematic loop constraints during dynamics computations for rigid-body systems. Specifically, we consider the local loop constraints emerging from common actuation sub-mechanisms in modern robotics systems (e.g., geared motors, differential drives, and four-bar mechanisms). As a complementary perspective to prior work on constraint embedding, we present an analysis that generalizes the traditional concepts of joint models and motion/force subspaces between individual rigid bodies to generalized joint models and motion/force subspaces between groups of rigid bodies subject to loop constraints. We then use these generalized concepts to derive the constraint-embedded recursive forward dynamics algorithm using multi-handle articulated bodies. We demonstrate the broad applicability of the generalized joint concepts by showing how they also lead to the constraint-embedding-based recursive algorithm for inverse dynamics. Lastly, we benchmark our open-source implementation in C++ for the forward dynamics algorithm against state-of-the-art, sparsity-exploiting algorithms. Our alternative derivation is intended to make the constraint embedding methodology more accessible to the broader robotics community, while the benchmarking study clarifies the relative strengths and limitations of constraint embedding versus sparsity-exploiting methods. Indeed, our benchmarking validates that constraint embedding outperforms the non-recursive alternative in cases involving local kinematic loops.

This work introduces an optimization-based planning and control framework for real-time synthesis of whole-body motions for legged robots. At the core of the proposed framework is a cascaded-fidelity model predictive controller (Cafe-Mpc).Cafe-Mpcstrategically relaxes the planning problem along the prediction horizon (i.e., with descending model fidelity, increasingly coarse time steps, and relaxed constraints) for computational and performance gains. This problem is numerically solved with an efficient customized multiple-shooting iLQR solver that is tailored for hybrid systems. The action-value function fromCafe-Mpcis then used as the basis for a new value-function-based whole-body control (VWBC) technique that avoids additional tuning. In this respect, the proposed framework unifies whole-body MPC and more conventional whole-body quadratic programming, which have been treated as separate components in previous works. We study the effects of the cascaded relaxations inCafe-Mpcon the tracking performance and required computation time. We also show thatCafe-Mpc, if configured appropriately, advances the performance of whole-body MPC without necessarily increasing computational cost. Furthermore, we show the superior performance of VWBC over a conventional Riccati feedback controller in terms of constraint handling. The proposed framework enables accomplishing a gymnastic-style running barrel roll for the first time on quadruped hardware, whereCafe-Mpcruns at 50 Hz, and the solver spends on average 5.3 ms per iteration. Results are demonstrated in the accompanying video.

Powered lower-limb prostheses have shown promise in helping individuals with amputation regain functionality that passive prostheses cannot provide. However, the best method for controlling these devices in coordination with their users is still an open research topic. While powered devices can replicate normative joint kinematics and kinetics, active control also holds the potential to shape system-level characteristics such as the center of mass (CoM) that play an important role in balance. Controlling the prosthesis based on these system-level, or task-space, variables would further represent a new way of coordinating the user and their device.This paper explores the initial implementation of task-space control for a powered ankle prosthesis, characterizing the emergent outcomes of this new coordination strategy. One able-bodied subject walked using a bypass adapter while prosthesis torques were commanded based on reference ground reaction force (GRF) and CoM trajectories. The subject could walk comfortably and continuously at their preferred walking speed, achieving normative ankle torques and joint trajectories despite not tracking explicit joint-level references in stance.

For robotic transtibial prosthesis control, the global tibia kinematics can be used to monitor gait cycle progression and command smooth and continuous actuation. In this work, these global tibia kinematics define a phase variable impedance controller (PVIC), which is implemented as the nonvolitional base controller within a hybrid volitional control framework (PVI-HVC). The gait progression estimation and biomechanic performance of one able-bodied individual walking on a robotic ankle prosthesis via a bypass adapter are compared for three control schemes: benchmark passive controller, PVIC, and PVI-HVC. The different actuation of each had a direct effect on the global tibia kinematics, but the average deviation between the estimated and ground truth gait percentages were 1.6%, 1.8%, and 2.1%, respectively, for each controller. Both PVIC and PVI-HVC produced good agreement with able-bodied kinematic and kinetic references. As designed, PVI-HVC results were similar to those of PVIC when the user used low volitional intent, but yielded higher peak plantarflexion, peak torque, and peak power when the user commanded high volitional input in late stance. This additional torque and power also allowed the user to volitionally and continuously achieve activities beyond level walking, such as ascending ramps, avoiding obstacles, standing on tip-toes, and tapping the foot. In this way, PVI-HVC offers the kinetic and kinematic performance of the PVIC during level ground walking, along with the freedom to volitionally pursue alternative activities.

This manuscript primarily aims to enhance the performance of whole-body controllers(WBC) for underactuated legged locomotion. We introduce a systematic parameter design mechanism for the floating-base feedback control within the WBC. The proposed approach involves utilizing the linearized model of unactuated dynamics to formulate a Linear Quadratic Regulator(LQR) and solving a Riccati gain while accounting for potential physical constraints through a second-order approximation of the log-barrier function. And then the user-tuned feedback gain for the floating base task is replaced by a new one constructed from the solved Riccati gain. Extensive simulations conducted in MuJoCo with a point bipedal robot, as well as real-world experiments performed on a quadruped robot, demonstrate the effectiveness of the proposed method. In the different bipedal locomotion tasks, compared with the user-tuned method, the proposed approach is at least 12% better and up to 50% better at linear velocity tracking, and at least 7% better and up to 47% better at angular velocity tracking. In the quadruped experiment, linear velocity tracking is improved by at least 3% and angular velocity tracking is improved by at least 23% using the proposed method.

Model-based control for robots has increasingly depended on optimization-based methods like Differential Dynamic Programming (DDP) and iterative LQR (iLQR). These methods can form the basis of Model-Predictive Control (MPC), which is commonly used for controlling legged robots. Computing the partial derivatives of the robot dynamics is often the most expensive part of these algorithms, regardless of whether analytical methods, Finite Difference, Automatic Differentiation (AD), or Chain-Rule accumulation is used. Since the second-order derivatives of the robot dynamics result in tensor computations, they are often ignored, leading to the use of iLQR, instead of the full second-order DDP method. In this paper, we present analytical methods to compute the second-order derivatives of Inverse and Forward Dynamics for open-chain rigid-body systems with multi-DoF joints and fixed/floating bases. An extensive comparison of accuracy and run-time performance with AD and other methods is provided, including the consideration of code-generation techniques in C/C++ to speed up the computations. For the 36 DoF ATLAS humanoid, the second-order Inverse and Forward Dynamics derivatives take$\approx 200 \mu s$, and$\approx 2.1 ms$respectively, on a 12th Gen Intel i5-12400 processor with 2.5 GHz clock-speed, resulting in a$\approx 3.2 \times$and$\approx 3.8 \times$speedup respectively over the AD approach.

In a world designed for legs, quadrupeds, bipeds, and humanoids have the opportunity to impact emerging robotics applications from logistics, to agriculture, to home assistance. The goal of this survey is to cover the recent progress toward these applications that have been driven by model-based optimization for the real-time generation and control of movement. The majority of the research community has converged on the idea of generating locomotion control laws by solving an optimal control problem (OCP) in either a model-based or data-driven manner. However, solving the most general of these problems online remains intractable due to complexities from intermittent unidirectional contacts with the environment, and from the many degrees of freedom of legged robots. This survey covers methods that have been pursued to make these OCPs computationally tractable, with a specific focus on how environmental contacts are treated, how the model can be simplified, and how these choices affect the numerical solution methods employed. The survey focuses on model-based optimization while paving its way for broader combination with learning-based formulations to accelerate progress in this growing field.

Specialized motions such as jumping are often achieved on quadruped robots by solving a trajectory optimization problem once and executing the trajectory using a tracking controller. This approach is in parallel with Model Predictive Control (MPC) strategies that commonly control regular gaits via online re-planning. In this work, we present a nonlinear MPC (NMPC) technique that unlocks on-the-fly replanning of specialized motion skills and regular locomotion within a unified framework. The NMPC reasons about a hybrid kinodynamic model, and is solved using a variant of a constrained Differential Dynamic Programming (DDP) solver. The proposed NMPC enables the robot to perform a variety of agile skills like jumping, bounding, and trotting, and the rapid transition between them. We evaluated the proposed algorithm with three challenging motion sequences that combine multiple agile skills, on two quadruped platforms, Unitree A1, and MIT Mini Cheetah, showing its effectiveness and generality.

Differential Dynamic Programming (DDP) is an efficient computational tool for solving nonlinear optimal control problems. It was originally designed as a single shooting method and thus is sensitive to the initial guess supplied. This work considers the extension of DDP to multiple shooting (MS), improving its robustness to initial guesses. A novel derivation is proposed that accounts for the defect between shooting segments during the DDP backward pass, while still maintaining quadratic convergence locally. The derivation enables unifying multiple previous MS algorithms, and opens the door to many smaller algorithmic improvements. A penalty method is introduced to strategically control the step size, further improving the convergence performance. An adaptive merit function and a more reliable acceptance condition are employed for globalization. The effects of these improvements are benchmarked for trajectory optimization with a quadrotor, an acrobot, and a manipulator. MS-DDP is also demonstrated for use in Model Predictive Control (MPC) for dynamic jumping with a quadruped robot, showing its benefits over a single shooting approach.

Phase variable control based on global tibia kinematics holds promise for predicting gait cycle progression to continuously control robotic transtibial prostheses. Calibration of the phase variable is critical to ensure its monotonic behavior, to approach a linear relationship with gait percentage, and to accurately predict the percentage of gait. This paper compares four calibration approaches using data from 22 able-bodied subjects walking at 14 speeds [1]. The typical pure centering (PC) approach employed for thigh-based phase variables is not viable, yielding monotonic phase progression in fewer than half of the cases. An optimization (OPT) approach found monotonic calibrations in 305/308 cases with high linearity (average R2 of 0.91). Critical point centering (CPC) approximates the OPT performance, with 274/308 monotonic calibrations and an average R2 of 0.85, whereas the related vertical weighted average (VWA) approach was only slightly better than PC. All four approaches are similarly accurate in predicting gait percentage, staying within 5% at least 92.7% of the time.

This article studies capturability and push recovery for quadruped locomotion. Despite the rich literature on capturability analysis and push recovery for legged robots, existing tools have been developed mainly with the requirement of reaching static or quasi-static balance following a push. In practice, this requirement commonly restricts capturability analysis to cases with simple dynamics and fails to encode the time dependence of capturable states for legged locomotion with time-based gaits. To address these issues, we apply switched systems to model quadruped locomotion and extend capturability notions through a novel specification ofdynamic balance. We also provide an explicit model predictive control (EMPC) scheme to compute the dynamic balance and capturable tubes and offer a way of using the capturable tube to synthesize push recovery controllers. Such a generalization allows for a rigorous characterization of disturbance timing on the capturability of quadrupedal locomotion and opens the door of disturbance-timing-aware push recovery control strategies. Extensive simulation and hardware experiments illustrate the necessity of considering dynamic balance for quadrupedal push recovery, reveal how disturbance timing affects capturability, and demonstrate the significant improvement in disturbance rejection with the proposed strategy. Hardware experimental validations on a replica of the Mini Cheetah quadruped further verify that the proposed approach performs statistically better than the state-of-the-art baseline considered.

Seemingly in defiance of basic physics, cats consistently land on their feet after falling. In this paper, we design a controller that lands the Mini Cheetah quadruped robot on its feet as well. Specifically, we explore how trajectory optimization and machine learning can work together to enable highly dynamic bioinspired behaviors. We find that a reflex approach, in which a neural network learns entire state trajectories, outperforms a policy approach, in which a neural network learns a mapping from states to control inputs. We validate our proposed controller in both simulation and hardware experiments, and are able to land the robot on its feet from falls with initial pitch angles between −90 and 90 degrees.

This paper considers the problem of designing legged robots for traversing uneven terrain, wherein terrain characteristics represent uncertainty for the design process. When this process encompasses a wider variety of terrains, the likelihood of the designed robot falling in the real world should decrease. However, computational scalability limits the number of terrains that can be taken into account during design. The proposed framework uses the Alternating Direction Method of Multipliers (ADMM) to solve large-scale concurrent design (co-design) problems. The ADMM coordinates the solution of small-size sub-problems and enforces constraints to reach a consensus on the best design. The framework uses stochastic programming (SP) to account for terrain uncertainty and trajectory optimization (TO) to co-optimize a nominal trajectory alongside hardware parameters and a feedback controller. Case studies demonstrate application for a monopod and a quadruped. For the monopod, ADMM facilitated an increase in the number of terrains considered within co-design by 400% compared to SP alone, which contributed to robustifying the design and decreasing its failure probability to under 1% in an anticipated operating space. A multi-scenario co-design implementation for the quadruped had previously been intractable due to scalability limitations. The ADMM framework, by contrast, shows tractability running with 30 terrain types, opening the horizon for designing more complex systems.

Reinforcement Learning (RL) has witnessed great strides for quadruped locomotion, with continued progress in the reliable sim-to-real transfer of policies. However, it remains a challenge to reuse a policy on another robot, which could save time for retraining. In this work, we present a framework for zero-shot policy retargeting wherein diverse motor skills can be transferred between robots of different shapes and sizes. The new framework centers on a planning-and-control pipeline that systematically integrates RL and Model Predictive Control (MPC). The planning stage employs RL to generate a dynamically plausible trajectory as well as the contact schedule, avoiding the combinatorial complexity of contact sequence optimization. This information is then used to seed the MPC to stabilize and robustify the policy roll-out via a new Hybrid Kinodynamic (HKD) model that implicitly optimizes the foothold locations. Hardware results show an ability to transfer policies from both the A1 and Laikago robots to the MIT Mini Cheetah robot without requiring any policy re-tuning.

Optimization-based robot control strategies often rely on first-order dynamics approximation methods, as in iLQR. Using second-order approximations of the dynamics is expensive due to the costly second-order partial derivatives of the dynamics with respect to the state and control. Current approaches for calculating these derivatives typically use automatic differentiation (AD) and chain-rule accumulation or finite-difference. In this paper, for the first time, we present analytical expressions for the second-order partial derivatives of inverse dynamics for open-chain rigid-body systems with floating base and multi-DoF joints. A new extension of spatial vector algebra is proposed that enables the analysis. A recursive algorithm with complexity of$\mathcal{O}(Nd^{2})$is also provided where N is the number of bodies and d is the depth of the kinematic tree. A comparison with AD in CasADi shows speedups of 1.5-3 x for serial kinematic trees with N > 5, and a C++ implementation shows runtimes of$\approx \mathbf{5 1} \mu \mathrm{s}$for a quadruped.

This paper studies push recovery for humanoid robots based on a variable-height inverted pendulum (VHIP) model. We first develop an approach for treating zero-step capturability of the VHIP with a novel methodology based on Hamilton-Jacobi (HJ) reachability analysis. Such an approach uses the sub-zero level set of a value function to encode capturability of the VHIP, where the value function is obtained by numerically solving a HJ variational inequality offline. Based on this analysis, a simple and effective method for adjusting foothold locations is then devised for cases where the VHIP state is not zero-step capturable. In addition, the HJ reachability analysis naturally induces an optimal control law that allows for rapid planning with the VHIP during push recovery online. To enable use of the strategy with a position-controlled humanoid robot, an associated differential inverse kinematics based tracking controller is employed. The effectiveness of the overall framework is demonstrated with the UBTECH Walker robot in the MuJoCo simulator. Simulation validations show a significant improvement in push robustness as compared to the methods based on the classical linear inverted pendulum model.

This paper presents a control strategy for quadruped robots to hop on their rear legs in three-dimensional space. The proposed approach generates nominal center of mass (CoM) trajectories based on a template spring-loaded inverted pendulum (SLIP) model. Tracking this reference remains a challenge due to the underactauted nature of balance with point feet. To address this challenge, a control-Lyapunov function based quadratic programming (CLF-QP) controller is proposed, which modulates nominal ground reaction forces (GRFs) to balance the torso while considering friction limits. The CLF construction is guided by a variational-based linearization (VBL) applied to a reduced-order single-rigid-body (SRB) model, and treats underactuation via solving a Riccati equation to obtain the CLF. A new balance control approach is presented that effectively decouples sagittal plane control (via re-planning) with lateral and rotational control (via the CLF and VBL). The proposed approach shows more robust balancing performance than the conventional CLF-QP approach. Simulations of the Mini Cheetah demonstrate in-place hopping with up to a 0.71m apex height.

We present a novel control strategy for dynamic legged locomotion in complex scenarios that considers information about the morphology of the terrain in contexts when only on-board mapping and computation are available. The strategy is built on top of two main elements: first a contact sequence task that provides safe foothold locations based on a convolutional neural network to perform fast and continuous evaluation of the terrain in search of safe foothold locations; then a model predictive controller that considers the foothold locations given by the contact sequence task to optimize target ground reaction forces. We assess the performance of our strategy through simulations of the hydraulically actuated quadruped robot HyQReal traversing rough terrain under realistic on-board sensing and computing conditions.

This paper shows how CasADi’s state-of-the-art implementation of algorithmic differentiation can be leveraged to formulate and efficiently solve gait optimization problems, enabling rapid gait design for high-dimensional biped robots. Comparative studies on a 7-DOF planar biped show that CasADi generates optimal gaits 4 times faster than another existing advanced optimization package. The framework is also applied to simultaneously generate a gait and a feedback controller for 2 spatial bipeds: a 12-DOF model and a 20DOF model. Results suggest that CasADi’s unprecedented efficiency could provide a practical path toward real-time gait optimization for high-dimensional biped robots.

This paper outlines steps toward a framework for model-based user intent detection to enable fluent human-robot interaction in assistive exoskeletons. An interacting multi-model (IMM) estimation scheme is presented to address state estimation for lower-extremity exoskeletons and to handle their hybrid dynamics. The proposed IMM scheme includes new approaches that enable it to estimate states of hybrid systems with dynamics that are unique to each phase. Traditional IMMs only consider the probabilistic likelihood of being in each phase, while the implementation in this work has been modified to consider physical likelihood as well. The IMM compares exoskeleton sensor readings to multiple candidate gaits from a template model of walking. Candidate gaits are generated using a numerical optimization procedure applied to a Bipedal Spring-Loaded Inverted Pendulum (B-SLIP) model. The framework was tested with sensor data acquired from walking trials in an Ekso GT exoskeleton, and was used to estimate gait phase and center of mass velocity. It is shown that the standard IMM filtering approach results in incorrect estimates of gait phase, while the proposed addition to the IMM estimator using physical likelihood improves the estimates. Results with human subject data further show the ability to estimate gait phase and speed in experimental settings.

Recent work has shed light on the often unreliable performance of constrained least-squares estimation methods for robot mass-inertial parameter identification, particularly for high degree-of-freedom systems subject to noisy and incomplete measurements. Instead, differential geometric identification methods have proven to be significantly more accurate and robust. These methods account for the fact that the mass-inertial parameters reside in a curved Riemannian space, and allow perturbations in the mass-inertial properties to be measured in a coordinate-invariant manner. Yet, a continued drawback of existing geometric methods is that the corresponding optimization problems are inherently nonconvex, have numerous local minima, and are computationally highly intensive to solve. In this paper, we propose a convex formulation under the same coordinate-invariant Riemannian geometric framework that directly addresses these and other deficiencies of the geometric approach. Our convex formulation leads to a globally optimal solution, reduced computations, faster and more reliable convergence, and easy inclusion of additional convex constraints. The main idea behind our approach is an entropic divergence measure that allows for the convex regularization of the inertial parameter identification problem. Extensive experiments with the 3-DoF MIT Cheetah leg, the 7-DoF AMBIDEX tendon-driven arm, and a 16-link articulated human model show markedly improved robustness and generalizability vis-à-vis existing vector space methods while ensuring fast, guaranteed convergence to the global solution.

As legged robots are sent into unstructured environments, the ability to robustly manage contact transitions will be a critical skill. This paper introduces an approach to probabilistically fuse contact models, managing uncertainty in terrain geometry, dynamic modeling, and kinematics to improve the robustness of contact initiation at touchdown. A discrete-time extension of the generalized-momentum disturbance observer is presented to increase the accuracy of proprioceptive force control estimates. This information is fused with other contact priors under a framework of Kalman Filtering to increase robustness of the method. This approach results in accurate contact detection with 99.3 % accuracy and a small 4-5ms delay. Using this new detector, an Event-Based Finite State Machine is implemented to deal with unexpected early and late contacts. This allows the robot to traverse cluttered environments by modifying the control actions for each individual leg based on the estimated contact state rather than adhering to a rigid time schedule regardless of actual contact state. Experiments with the MIT Cheetah 3 robot show the success of both the detection algorithm, as well as the Event-Based FSM while making unexpected contacts during trotting.

This paper studies collaboration through the cloud in the context of cooperative adaptive control for robot manipulators. We first consider the case of multiple robots manipulating a common object through synchronous centralized update laws to identify unknown inertial parameters. Through this development, we introduce a notion of Collective Sufficient Richness, wherein parameter convergence can be enabled through teamwork in the group. The introduction of this property and the analysis of stable adaptive controllers that benefit from it constitute the main new contributions of this work. Building on this original example, we then consider decentralized update laws, time-varying network topologies, and the influence of communication delays on this process. Perhaps surprisingly, these nonidealized networked conditions inherit the same benefits of convergence being determined through collective effects for the group. Simple simulations of a planar manipulator identifying an unknown load are provided to illustrate the central idea and benefits of Collective Sufficient Richness.

This paper introduces a new robust, dynamic quadruped, the MIT Cheetah 3. Like its predecessor, the Cheetah 3 exploits tailored mechanical design to enable simple control strategies for dynamic locomotion and features high-bandwidth proprioceptive actuators to manage physical interaction with the environment. A new leg design is presented that includes proprioceptive actuation on the abduction/adduction degrees of freedom in addition to an expanded range of motion on the hips and knees. To make full use of these new capabilities, general balance and locomotion controllers for Cheetah 3 are presented. These controllers are embedded into a modular control architecture that allows the robot to handle unexpected terrain disturbances through reactive gait modification and without the need for external sensors or prior environment knowledge. The efficiency of the robot is demonstrated by a low Cost of Transport (CoT) over multiple gaits at moderate speeds, with the lowest CoT of 0.45 found during trotting. Experiments showcase the ability to blindly climb up stairs as a result of the full system integration. These results collectively represent a promising step toward a platform capable of generalized dynamic legged locomotion.

This paper presents an implementation of model predictive control (MPC) to determine ground reaction forces for a torque-controlled quadruped robot. The robot dynamics are simplified to formulate the problem as convex optimization while still capturing the full 3D nature of the system. With the simplified model, ground reaction force planning problems are formulated for prediction horizons of up to 0.5 seconds, and are solved to optimality in under 1 ms at a rate of 20-30 Hz. Despite using a simplified model, the robot is capable of robust locomotion at a variety of speeds. Experimental results demonstrate control of gaits including stand, trot, flying-trot, pronk, bound, pace, a 3-legged gait, and a full 3D gallop. The robot achieved forward speeds of up to 3 m/s, lateral speeds up to 1 m/s, and angular speeds up to 180 deg/sec. Our approach is general enough to perform all these behaviors with the same set of gains and weights.

This paper introduces a new policy-regularized model-predictive control (PR-MPC) approach to automatically generate and stabilize a diverse set of quadrupedal gaits. Model-predictive methods offer great promise to address balance in dynamic robots, yet require the solution of challenging nonlinear optimization problems when applied to legged systems. The new proposed PR-MPC approach aims to improve the conditioning of these problems by adding regularization based on heuristic reference policies. With this approach, a unified MPC formulation is shown to generate and stabilize trotting, bounding, and galloping without retuning any cost-function parameters. Intuitively, the added regularization biases the solution of the MPC towards common heuristics from the literature that are based on simple physics. Simulation results show that PR-MPC improves the computation time and closed-loop outcomes of applying MPC to stabilize quadrupedal gaits.

Designing an actuator system for highly dynamic legged robots has been one of the grand challenges in robotics research. Conventional actuators for manufacturing applications have difficulty satisfying design requirements for high-speed locomotion, such as the need for high torque density and the ability to manage dynamic physical interactions. To address this challenge, this paper suggests a proprioceptive actuation paradigm that enables highly dynamic performance in legged machines. Proprioceptive actuation uses collocated force control at the joints to effectively control contact interactions at the feet under dynamic conditions. Modal analysis of a reduced leg model and dimensional analysis of DC motors address the main principles for implementation of this paradigm. In the realm of legged machines, this paradigm provides a unique combination of high torque density, high-bandwidth force control, and the ability to mitigate impacts through backdrivability. We introduce a new metric named the “impact mitigation factor” (IMF) to quantify backdrivability at impact, which enables design comparison across a wide class of robots. The MIT Cheetah leg is presented, and is shown to have an IMF that is comparable to other quadrupeds with series springs to handle impact. The design enables the Cheetah to control contact forces during dynamic bounding, with contact times down to 85 ms and peak forces over 450 N. The unique capabilities of the MIT Cheetah, achieving impact-robust force-controlled operation in high-speed three-dimensional running and jumping, suggest wider implementation of this holistic actuation approach.

This paper presents a method for the generation of dynamic walking gaits with a 3D Dual-SLIP model and its application to a simulated Hubo+ based humanoid. Previous approaches with the Dual-SLIP model have only focused on the planar case, wherein self-stable gaits can be found. When extended to 3D here, this model has not been found to exhibit self-stable gaits, requiring new methods for gait optimization and control. By taking advantage of a newly discovered symmetry condition for the Dual-SLIP model, this work proposes a quarter period (half step) optimization process to find periodic walking gaits in 3D. An LQR controller is developed to regulate the state of the model at leg midstance (MS) based on its return map dynamics. The Dual-SLIP model is extended by introducing a bio-inspired leg length actuation scheme in order to describe high-speed walking gaits (up to 2 m/s for human-compatible parameters). Finally, the CoM trajectory and footstep positions from the 3D Dual-SLIP are used as a reference in a task-space controller with a Hubo+ based humanoid model. By tracking these references, the methods successfully produce human-like dynamic walking gaits in simulation which are robust to disturbances. The whole-body control system for walking can handle uneven terrain with variation up to 10% of its leg length. This represents the first humanoid dynamic walking approach based on a 3D Dual-SLIP model.

The Dual-SLIP model has been proposed as a walking template that inherently encodes a rich set of human-like features. Previous work has used the 3D Dual-SLIP with bio-inspired leg actuation to generate a human-like dynamic walking gait over a wide range of speeds. The work presented in this paper extends the 3D Dual-SLIP walking strategy to uneven terrain. With nonlinear optimization based on a multiple-shooting formulation, actuated Dual-SLIP walking gaits over uneven terrain are identified that handle 1-step elevation changes up to ±10 cm. Moreover, this Dual-SLIP actuation strategy enables a constant center of mass (CoM) forward speed at leg midstance to be maintained. The resultant gaits have revealed a leg lengthening/shortening strategy that is similar to that adopted by a human when walking over prepared, uneven terrain. Results demonstrate that the CoM trajectories and ground reaction force patterns found with the approach are comparable to the human data found in the biomechanics literature. The trajectories generated by the Dual-SLIP model are also demonstrated to orchestrate a dynamic walking motion with an anthropomorphic humanoid model in simulation over uneven terrain.

This paper presents new methods to develop a running long jump for a simulated humanoid robot. Starting from a steady-state running motion, a new spring loaded inverted pendulum (SLIP) based 3D template model for a running jump is presented. The use of this model is motivated by a simpler model from biomechanics which describes the dynamics of human long jumpers in the sagittal plane. While previously only used to describe the thrust step of a long jump, this type of model is also shown to generate useful Center of Mass (CoM) trajectories to return to steady-state running upon landing. A principled optimization approach for this new template is described to generate reference CoM trajectories for maximum span which are able to be kinematically and dynamically retargeted to the humanoid. The key features of an optimal long jump are highlighted, and a task-space control approach to realize the motion on a humanoid is summarized. A video attachment to this paper shows an optimal long jump for a 6 m/s approach speed, where the humanoid is able to clear a large gap, and highlights the effects of non-optimal takeoff-velocity angles.

This paper presents new methods to control humanoid turns while running, through the use of a 3D-SLIP template model with steering control. The work builds on a previous controller for straight-ahead running and describes the new methods that enable online humanoid steering for different speeds and turn rates. As opposed to previous research which has studied 3D-SLIP steering with a monopod model, motion optimization for the SLIP here enforces leg separation. This leg separation gives rise to body sway in forward running and allows the template to capture the unique roles that the inside and outside legs each play during a high-speed turn. The trajectory optimization approach for this template is given, and the resultant CoM trajectories are characterized. Modifications to a previous controller for straight-ahead running are shown to enable running turns in a simulated humanoid model. The methods allow the humanoid to change its turn rate and direction from step to step and enable execution of a high-speed turn with a radius that is one fourth that of a standard 400m track. A video attachment to this paper shows the humanoid turning while running at up to 4.0 m/s, and highlights its ability to maintain balance in spite of push disturbances.

This paper presents a new formulation of prioritized task-space control for humanoids that is used to develop a dynamic kick and dynamic jump in a 26 degree of freedom simulated system. The demonstrated motions are controlled through a real-time conic optimization scheme that selects appropriate joint torques and contact forces. More specifically, motions are characterized in appropriate task spaces, and the real-time optimizer solves the task-space control problem while accounting for user-defined priorities between the tasks. In contrast to previous solutions of the Prioritized Task-Space Control (PTSC) problem for humanoids, the solution presented here satisfies the ZMP constraint and ground friction limitations at all levels of priority, and is general to periods of flight as well as support. All generated motions include control of the system's centroidal angular momentum, which leads to emergent whole-body behaviors, such as arm-swing, that are not specified by the designer. In addition, compared to a previous quadratic programming solution of the PTSC problem, our approach gains a factor of 2 speedup in its required computational time. This speedup allows the control approach to operate at real-time rates of approximately 200 Hz.

This paper presents new methods to control highspeed running in a simulated humanoid robot at speeds of up to 6.5 m/s. We present methods to generate compliant target CoM dynamics through the use of a 3D spring-loaded inverted pendulum (SLIP) template model. A nonlinear least-squares optimizer is used to find periodic trajectories of the 3D-SLIP offline, while a local deadbeat SLIP controller provides reference CoM dynamics online at real-time rates to correct for tracking errors and disturbances. The local deadbeat controller employs common foot placement strategies that are automatically generated by a local analysis of the 3D-SLIP apex return map. A task-space controller is then applied online to select whole-body joint torques which embed these target dynamics into the humanoid. Despite the body of work on the 2D and 3D-SLIP models, to the best of the authors' knowledge, this is the first time that a SLIP model has been embedded into a whole-body humanoid model. When running at 3.5 m/s, the controller is shown to reject lateral disturbances of 40 N·s applied at the waist. A final demonstration shows the capability of the controller to stabilize running at 6.5 m/s, which is comparable with the speed of an Olympian in the 5000 meter run.

This paper provides a reduced-order algorithm, the Extended-Force-Propagator Algorithm (EFPA), for the computation of operational-space inertia matrices in branched kinematic trees. The algorithm accommodates an operational space of multiple end-effectors, and is the lowest-order algorithm published to date for this computation. The key feature of this algorithm is the explicit calculation and use of matrices that propagate a force across a span of several links in a single operation. This approach allows the algorithm to achieve a computational complexity of O(N +md+m2) where N is the number of bodies, m is the number of end-effectors, and d is the depth of the system's connectivity tree. A detailed cost comparison is provided to the propagation algorithms of Rodriguez et al. (complexity O(N + dm2)) and to the sparse factorization methods of Featherstone (complexity O(nd2+ md2+ m2d)). For the majority of examples considered, our algorithm outperforms the previous best recursive algorithm, and demonstrates efficiency gains over sparse methods for some topologies.

Current biped robots with articulated legs, even the most impressive to date, still lack the ability to execute dynamic motions such as jumping and running with comparable performance to biological systems. This work explores dynamic jumping with the planar biped prototype KURMET, which employs unidirectional series-elastic actuation at each of its joints. While this actuation scheme enables the performance of high-power dynamic movements like the jump, its presence complicates the jumping control problem and has prevented previous researchers from obtaining precise jump control in systems of reasonable complexity. To manage this problem, this paper develops a layered fuzzy control system for KURMET that realizes repeated dynamic hopping and accurate control of both torso height and velocity at each top of flight. An effective two-stage training approach is used for the fuzzy controller to learn the required, yet highly nonlinear, relationships between its inputs and outputs. Finally, the state machine employed at the lowest-level of control is used to achieve a maximal normalized jump height that outperforms most humans and can be sequenced with the hopping movement.