MIT Cheetah Research Articles

Smooth Parameterization of Rigid-Body Inertia

In this letter, we propose a parameterization of the rigid-body inertia tensor that is singularity free, guarantees full physical consistency, and has a straightforward physical interpretation. Based on a version of log-Cholesky decomposition of the pseudo-inertia matrix, we construct a smooth isomorphic mapping from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbb {R}^{10}$</tex-math></inline-formula> to the set of fully physically consistent inertia tensors. This facilitates inertial estimation via unconstrained optimization on a vector space and avoids the non-uniqueness and singularities which we show are inherent to parameterizations of the inertia tensor based on eigenvalue decomposition and principal moments. The elements of our parameterization have straightforward physical meanings in terms of geometric transformations applied to a reference body. While adopting this new parameterization breaks the linear least squares structure of the system identification, theoretical results guarantee that all local optima of the resulting problems remain global optima. We compare the performance of three different parameterizations of inertia by performing inertial estimation on test data from a set of 1000 simulations of randomly sampled rigid bodies subject to an external time-varying wrench and measurement noise. We also investigate the performance of the log-Cholesky parameterization on a dataset from MIT Cheetah 3 to empirically demonstrate the theoretical results. Overall, the results indicate that the log-Cholesky parameterization achieves fast convergence while providing a simple and intuitive parametric description of inertia.

Jumping over obstacles with MIT Cheetah 2

This paper presents a planning framework for jumping over obstacles with quadruped robots. The framework accomplishes planning via a structured predictive control strategy that combines the use of heterogeneous simplified models over different prediction time scales. A receding multi-horizon predictive controller coordinates the approach before the jump using a kinematic point-mass model. Consideration of the optimal value function over different planning horizons enables the system to select an appropriate number of steps to take before jumping. The jumping motion is then tailored to the sensed obstacle by solving a nonlinear trajectory optimization problem. The solution of this problem online is enabled by exploiting the analyticity of the flow map for a planar bounding template model under polynomial inputs. By planning with this combination of models, MIT Cheetah 2 is shown to autonomously jump over obstacles up to 40 cm in height during high-speed bounding. Untethered results showcase the ability of the method to automatically adapt to obstacles of different heights and placements in a single trial.

Geometric Robot Dynamic Identification: A Convex Programming Approach

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-a-vis existing vector space methods while ensuring fast, guaranteed convergence to the global solution.

Online Gait Transitions and Disturbance Recovery for Legged Robots via the Feasible Impulse Set

Gaits in legged robots are often hand tuned and time based, either explicitly or through an internal clock, for instance, in the form of central pattern generators. This strategy requires trial and error to identify leg timings, which may not be suitable in challenging terrains. In this letter, we introduce new concepts to quantify leg capabilities for online gait emergence and adaptation, without fixed timings or predefined foothold sequences. Specifically, we introduce the Feasible Impulse Set, a notion that extends aspects of the classical wrench cone to include a prediction horizon into the future. By considering the impulses that can be delivered by the legs, quantified notions of leg utility are proposed for coordinating adaptive lift-off and touch-down of stance legs. The proposed methods provide push recovery and emergent gait transitions with speed. These advances are validated in experiments with the MIT Cheetah 3 robot, where the framework is shown to automatically coordinate aperiodic behaviors on a partially moving walkway.

Linear Matrix Inequalities for Physically Consistent Inertial Parameter Identification: A Statistical Perspective on the Mass Distribution

With the increased application of model-based whole-body control in legged robots, there has been a resurgence of research interest into methods for accurate system identification. An important class of methods focuses on the inertial parameters of rigid-body systems. These parameters consist of the mass, first mass moment (related to center of mass location), and rotational inertia matrix of each link. The main contribution of this paper is to formulate physical-consistency constraints on these parameters as Linear Matrix Inequalities (LMIs). The use of these constraints in identification can accelerate convergence and increase robustness to noisy data. It is critically observed that the proposed LMIs are expressed in terms of the covariance of the mass distribution, rather than its rotational moments of inertia. With this perspective, connections to the classical problem of moments in mathematics are shown to yield new bounding-volume constraints on the mass distribution of each link. While previous work ensured physical plausibility or used convex optimization in identification, the LMIs here uniquely enable both advantages. Constraints are applied to identification of a leg for the MIT Cheetah 3 robot. Detailed properties of transmission components are identified alongside link inertias, with parameter optimization carried out to global optimality through semidefinite programming.

Sparse Control for Dynamic Movement Primitives

This paper describes the use of spatially-sparse inputs to influence global changes in the behavior of Dynamic Movement Primitives (DMPs). The dynamics of DMPs are analyzed through the framework of contraction theory as networked hierarchies of contracting or transversely contracting systems. Within this framework, sparsely-inhibited rhythmic DMPs (SI-RDMPs) are introduced to both inhibit or enable rhythmic primitives through spatially-sparse modification of the DMP dynamics. SI-RDMPs are demonstrated in experiments to manage start-stop transitions for walking experiments with the MIT Cheetah. New analytical results on the coupling of oscillators with diverse natural frequencies are also discussed.

Proprioceptive Actuator Design in the MIT Cheetah: Impact Mitigation and High-Bandwidth Physical Interaction for Dynamic Legged Robots

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.

High-speed bounding with the MIT Cheetah 2: Control design and experiments

This paper presents the design and implementation of a bounding controller for the MIT Cheetah 2 and its experimental results. The paper introduces the architecture of the controller along with the functional roles of its subcomponents. The application of impulse scaling provides feedforward force profiles that automatically adapt across a wide range of speeds. A discrete gait pattern stabilizer maintains the footfall sequence and timing. Continuous feedback is layered to manage balance during the stance phase. Stable hybrid limit cycles are exhibited in simulation using simplified models, and are further validated in untethered three-dimensional bounding experiments. Experiments are conducted both indoors and outdoors on various man-made and natural terrains. The control framework is shown to provide stable bounding in the hardware, at speeds of up to 6.4 m/s and with a minimum total cost of transport of 0.47. These results are unprecedented accomplishments in terms of efficiency and speed in untethered experimental quadruped machines.

Implementation of trot-to-gallop transition and subsequent gallop on the MIT Cheetah I

This paper presents a demonstration of the trot-to-gallop transition and subsequent stable gallop in a robotic quadruped. The MIT Cheetah I, a planar quadruped platform for high-speed running, achieves these tasks with a speed of 3.2 m/s (Froude number of 2.1) on a treadmill. The controller benefits from clues from biological findings and it incorporates (1) a gait pattern modulation that imposes predefined gait patterns with a proprioceptive touchdown feedback, (2) tunable equilibrium-point foot-end trajectories for four limbs that intentionally modulate ground reaction forces, and (3) programmable leg compliance that provides instantaneous reflexes to leg–ground interaction. An inertial measurement unit sensor is integrated with the controller in order to regulate leg angles of attack at touchdown. We reduce the dimension of the control parameters which describe temporal/spatial characteristics of quadruped locomotion, and the values are tuned via dynamic simulation and then experiment. Given a pre-defined virtual leg compliance and a desired angle of attack of legs, the equilibrium-point foot-end trajectories and phase relationships between four legs for stable trot and gallop gaits are found independently. We propose a simple throw-and-catch gait transition strategy which connects two stable limit cycles, the trot and the gallop, by linearly varying control parameters during the transition period. Successful gait transition is achieved in both simulation and experiment. Comprehensive analysis on the characteristics of the MIT Cheetah I experimental trot-to-gallop transition is provided. The phase portraits imply that stable limit cycles are achieved with the proposed controller in both trot and gallop, which enables the trot-to-gallop gait transition at high speed.

Development of visual display with anti-vibration system for camera on MIT Cheetah robot

Development of visual display with anti-vibration system for camera on MIT Cheetah robot

Design Principles for Energy-Efficient Legged Locomotion and Implementation on the MIT Cheetah Robot

This paper presents the design principles for highly efficient legged robots, the implementation of the principles in the design of the MIT Cheetah, and the analysis of the high-speed trotting experimental results. The design principles were derived by analyzing three major energy-loss mechanisms in locomotion: heat losses from the actuators, friction losses in transmission, and the interaction losses caused by the interface between the system and the environment. Four design principles that minimize these losses are discussed: employment of high torque-density motors, energy regenerative electronic system, low loss transmission, and a low leg inertia. These principles were implemented in the design of the MIT Cheetah; the major design features are large gap diameter motors, regenerative electric motor drivers, single-stage low gear transmission, dual coaxial motors with composite legs, and the differential actuated spine. The experimental results of fast trotting are presented; the 33-kg robot runs at 22 km/h (6 m/s). The total power consumption from the battery pack was 973 W and resulted in a total cost of transport of 0.5, which rivals running animals' at the same scale. 76% of the total energy consumption is attributed to heat loss from the motor, and the remaining 24% is used in mechanical work, which is dissipated as interaction loss as well as friction losses at the joint and transmission.

Quadrupedal galloping control for a wide range of speed via vertical impulse scaling

This paper presents a bio-inspired quadruped controller that allows variable-speed galloping. The controller design is inspired by observations from biological runners. Quadrupedal animals increase the vertical impulse that is generated by ground reaction forces at each stride as running speed increases and the duration of each stance phase reduces, whereas the swing phase stays relatively constant. Inspired by this observation, the presented controller estimates the required vertical impulse at each stride by applying the linear momentum conservation principle in the vertical direction and prescribes the ground reaction forces at each stride. The design process begins with deriving a planar model from the MIT Cheetah 2 robot. A baseline periodic limit cycle is obtained by optimizing ground reaction force profiles and the temporal gait pattern (timing and duration of gait phases). To stabilize the optimized limit cycle, the obtained limit cycle is converted to a state feedback controller by representing the obtained ground reaction force profiles as functions of the state variable, which is monotonically increasing throughout the gait, adding impedance control around the height and pitch trajectories of the obtained limit cycle and introducing a finite state machine and a pattern stabilizer to enforce the optimized gait pattern. The controller that achieves a stable 3 m s−1 gallop successfully adapts the speed change by scaling the vertical ground reaction force to match the momentum lost by gravity and adding a simple speed controller that controls horizontal speed. Without requiring additional gait optimization processes, the controller achieves galloping at speeds ranging from 3 m s−1 to 14.9 m s−1 while respecting the torque limit of the motor used in the MIT Cheetah 2 robot. The robustness of the controller is verified by demonstrating stable running during various disturbances, including 1.49 m step down and 0.18 m step up, as well as random ground height and model parameter variations.

High speed trot-running: Implementation of a hierarchical controller using proprioceptive impedance control on the MIT Cheetah

This paper presents implementation of a highly dynamic running gait with a hierarchical controller on the MIT Cheetah. The developed controller enables high-speed running of up to 6 m/s (Froude number of Fr ≈ 7.34) incorporating proprioceptive feedback and programmable virtual leg compliance of the MIT Cheetah. To achieve a stable and fast trot gait, we applied three control strategies: (a) programmable virtual leg compliance that provides instantaneous reflexes to external disturbance and facilitates the self-stabilizing shown in the passive dynamics of locomotion; (b) tunable stance-trajectory design, intended to adjust impulse at each foot-end in the stance phase in a high speed trot-running according to the equilibrium-point hypothesis; and (c) a gait-pattern modulation that imposes a desired cyclic gait-pattern taking cues from proprioceptive TD feedback. Based on three strategies, the controller is hierarchically structured. The control parameters for forward speeds, a specific gait-pattern, and desired leg trajectories are managed by a high-level controller. It consists of both a gait-pattern modulator with proprioceptive leg TD detection and a leg-trajectory generator using a Bèzier curve and a tunable amplitude sinusoidal wave. Instead of employing physical spring/dampers in the robot’s leg, the programmable virtual leg compliance is realized using proprioceptive impedance control in individual low-level leg controllers. To verify the developed controller, a robot dynamic simulator is constructed based on the model parameters of the MIT Cheetah. The controller parameters are tuned with the simulator to achieve self-stability, and then applied to the MIT Cheetah in an experimental environment. Using leg kinematics and applied motor current feedbacks, the MIT Cheetah achieved a stable trot-running gait in the sagittal plane.

Energy for Biomimetic Robots: Challenges and Solutions

nimals and autonomous robots need to carry theirown fuel (unlike plants, they do not generate usableenergy from their surroundings). Animals typically exceedthe normal endurance and range of all our current untetheredrobots. As an obvious example, humans have tremendousburst speed (less than 10 seconds to run 100 meters) andendurance (running a 26-mile marathon), and they can con-tinue to do everyday activities without refueling (eating) forseveraldays.Thetypicalcostoftransportforhumansisabout0.2. In comparison, most robots operate for less than 1 houron their carried fuel; the cost of transport is 15 or 20 timesmore than that for animals. An intriguing insight is thatpassive dynamic walkers can approach the human cost oftransport (the Cornell Ranger can walk nonstop for 65km),but this is a single optimized task (walking) with none of theversatilityofananimalthatcanstepoverobjectsandoperateonvariedterrain.Whatitdoesillustrateisthatstructures(andby extension, material properties) can be exploited to ‘‘getthe most’’ out of a given fuel source. Surely, this is whatanimals do on a continuous basis. What do we need to do togive our robots similar capabilities? In particular, what arethe special demands, advantages, and limitations of fuelstorage and usage in soft robots? To begin exploring some ofthese issues and to also stimulate a larger dialog in the robotcommunity, the following discussion has been compiledfrom a series of questions posed to the participants.—Barry TrimmerSoft Robotics: What are the special problems in providingfuel (energy) for biomimetic and soft robots?William Messner: Biomimetic and soft robots in particularare limited because it is very difficult to convert efficientgenerators of high-speed rotary motion (e.g., turbines orelectric motors) to low-speed/high-force motion with softcomponents. Are there alternatives such as planetary orharmonic gear trains? While biological muscles are not par-ticularlyefficientfromathermodynamicperspective,theydonotsufferfromthehugelossesofminiaturizedgearreductionsystems, and they are soft!Inaddition,heatenginesscaleverypoorly—thermallossesand friction losses reduce their efficiency enormously atsmall scales. Thus, highly efficient turbines and even mod-estly efficient reciprocating heat engines do not deliver goodperformance at centimeter scales and below. Existing batte-ries on the other hand deliver poor energy density on a per-mass and per-volume basis.BarryTrimmer:Yes,Icaninserthereafewnumbersthatwekeep coming back to. The energy density of nuclear fuel suchas nuclear uranium is 80,620,000 MJ/kg; by comparison, hy-drogen (compressed at 70MPa) provides 123 MJ/kg and hy-drocarbons such as LPG, gas, diesel, and fat range from 35 to47 MJ/kg. Our best readily available batteries (Li and Li ion)are around 0.3–1.8 MJ/kg. Even if we could get around thesafety and security issues associated with radioactivity, cap-turing that energy requires a lot of physical plant and is un-suitablefor softmobilemachines.Clearly,hydrocarbonshavethe advantage.The obvious drawbackisthatour conventionaluse of hydrocarbons involves combustion and the productionofextremeheat,pressure,andpollutants.Thelow-temperatureoxidationpathwaysusedbylivingtissueswouldseemtomakemost sense for robots used around humans.Sangbae Kim: Most of human-made electric energy storagedevices are relatively rigid or semirigid (lithium-ion poly-mer). It will impede compliant behavior if electric energy isrequired to be part of the structure. Fuel in the form of liquidwould be ideal. A possible question to answer is how to dealwiththevolumechangeoftheliquid asthefuelisconsumed.In a rigid robot, like MIT Cheetah, we need better power-density fuel cells. The power density of the current fuel cellsis not high enough to use in a robot less than 50kg.Robert Shepherd: One of the major benefits that living or-ganismshaveoverrobotsistheirabilitytoharvestenergyfromtheenvironment:eating.Whilewecandeveloprobotsthatcango to charging or fuelstations autonomously, these waypointsmust be predetermined. Thus, even though the internal com-bustion engineismoreefficientthanmuscle, a praying mantiscan catch prey when it must refuel at many locations.William Messner: Making structural components out ofenergy storage devices would be a way to increase the range

The MIT Cheetah, an Electrically-Powered Quadrupedal Robot for High-speed Running

The MIT Cheetah, an Electrically-Powered Quadrupedal Robot for High-speed Running