Learning-based methods excel at robot motion generation but remain limited in contact-rich physical interaction. Impedance control provides stable and safe contact behavior but requires task-specific tuning of stiffness and damping parameters. We present Diffusion-Based Impedance Learning, a framework that bridges these paradigms by combining generative modeling with energy-consistent impedance control. A Transformer-based Diffusion Model, conditioned via cross-attention on measured external wrenches, reconstructs simulated Zero-Force Trajectories (sZFTs) that represent contact-consistent equilibrium behavior. A SLERP-based quaternion noise scheduler preserves geometric consistency for rotations on the unit sphere. The reconstructed sZFT is used by an energy-based estimator to adapt impedance online through directional stiffness and damping modulation. Trained on parkour and robot-assisted therapy demonstrations collected via Apple Vision Pro teleoperation, the model achieves sub-millimeter positional and sub-degree rotational accuracy using only tens of thousands of samples. Deployed in real-time torque control on a KUKA LBR iiwa, the approach enables smooth obstacle traversal and generalizes to unseen tasks, achieving 100% success in multi-geometry peg-in-hole insertion. The code for all experiments is publicly available on GitHub and videos of the experiments are available on the project website.

In this paper, a kinematically modular approach to robot control is presented. The method involves structures called Elementary Dynamic Actions and a network model combining these elements. With this control framework, a rich repertoire of movements can be generated by combination of basic modules. The problems of solving inverse kinematics, managing kinematic singularity and kinematic redundancy are avoided. The modular approach is robust against contact and physical interaction, which makes it particularly effective for contact-rich manipulation. Each kinematic module can be learned by Imitation Learning, thereby resulting in a modular learning strategy for robot control. The theoretical foundations and their implementation on a real robot are presented. Using a KUKA LBR iiwa robot, three tasks were considered: (1) generating a sequence of discrete movements, (2) generating a combination of discrete and rhythmic movements, and (3) a drawing and erasing task. The results obtained show that this modular approach has the potential to simplify the generation of a diverse range of robot actions.

Humans are skillful at manipulating objects that possess nonlinear underactuated dynamics, such as clothes or containers filled with liquids. Several studies suggested that humans implement a predictive model-based strategy to control such objects. However, these studies only considered unconstrained reaching without any object involved or, at most, linear mass-spring systems with relatively simple dynamics. It is not clear what internal model humans develop of more complex objects, and what level of granularity is represented. To answer these questions, this study examined a task where participants physically interacted with a nonlinear underactuated system mimicking a cup of sloshing coffee: a cup with a ball rolling inside. The cup and ball were simulated in a virtual environment and subjects interacted with the system via a haptic robotic interface. Participants were instructed to move the system and arrive at a target region with both cup and ball at rest, 'zeroing out' residual oscillations of the ball. This challenging task affords a solution known as 'input shaping', whereby a series of pulses moves the dynamic object to the target leaving no residual oscillations. Since the timing and amplitude of these pulses depend on the controller's internal model of the object, input shaping served as a tool to identify the subjects' internal representation of the cup-and-ball. Five simulations with different internal models were compared against the human data. Results showed that the features in the data were correctly predicted by a simple internal model that represented the cup-and-ball as a single rigid mass coupled to the hand impedance. These findings provide evidence that humans use simplified internal models along with mechanical impedance to manipulate complex objects.

Humans have an astonishing ability to extract hidden information from the movement of others. In previous work, subjects observed the motion of a simulated stick-figure, two-link planar arm and estimated its stiffness. Fundamentally, stiffness is the relation between force and displacement. Given that subjects were unable to physically interact with the simulated arm, they were forced to make their estimates solely based on observed kinematic information. Remarkably, subjects were able to correctly correlate their stiffness estimates with changes in the simulated stiffness, despite the lack of force information. We hypothesized that subjects were only able to do this because the controller used to produce the simulated arm's movement, composed of oscillatory motions driving mechanical impedances, resembled the controller humans use to produce their own movement. However, it is still unknown what motion features subjects used to estimate stiffness. Human motion exhibits systematic velocity-curvature patterns, and it has previously been shown that these patterns play an important role in perceiving and interpreting motion. Thus, we hypothesized that manipulating the velocity profile should affect subjects' ability to estimate stiffness. To test this, we changed the velocity profile of the simulated two-link planar arm while keeping the simulated joint paths the same. Even with manipulated velocity signals, subjects were still able to estimate changes in simulated joint stiffness. However, when subjects were shown the same simulated path with different velocity profiles, they perceived motions that followed a veridical velocity profile to be less stiff than that of a non-veridical profile. These results suggest that path information (displacement) predominates over temporal information (velocity) when humans use visual observation to estimate stiffness.

The control of kinematically redundant robots is often approached using nullspace projection, which requires precise models and can be computationally challenging. Humans have many more degrees of freedom than are required to accomplish their tasks, but given neuromechanical limitations, it seems unlikely that biology relies on precise models or complex computation. An alternative biologically inspired approach leverages the compositionality of mechanical impedance. In theory, nullspace projection eliminates any conflict between two tasks. In contrast, superposition of task-space impedance and a full-rank joint-space impedance may impose a task conflict. This work compared nullspace projection with impedance superposition during unconstrained motion and forceful physical interaction. In practice, despite their theoretical differences, we did not observe a substantial influence of the nullspace projector weighting matrix. We found that nullspace projection and impedance superposition both resulted in measurable task conflict. Remarkably, when the dimensionality of the nullspace was increased, impedance superposition was comparable to nullspace projection.

Humans are capable of managing multiple tasks simultaneously. It is widely assumed that human motor control can be emulated by impedance control. To achieve human-like behavior, however, the impedance parameters of multiple tasks may vary during task execution. We propose an algorithm that shapes task impedance as a function of the robot’s time-varying inertial properties. These properties involve virtually constrained masses and virtually constrained inertias that counteract a task in order to comply with a given constraint. In this work, we not only detect task conflicts, but also show how to handle them. Our method is able to control kinematically redundant robots. We developed a damping-design method that does not interfere with our desired Cartesian task-space behavior. The control approach was verified in experiments on a real robot. We compared our impedance shaping method with two alternative control approaches: simple impedance superposition and nullspace projection. Our method preserved the passivity while improving the Cartesian task performance of an impedance controller. The method has computational advantages, beneficial to control robots with many degrees of freedom.

Prior experiments have shown that human gait synchronizes to periodic torque pulses applied about the hip and ankle joints by robotic exoskeletons. Importantly, entrainment occurred even when the pulse period differed slightly from the user’s preferred stride period, making it a viable approach to increase gait speed. As gait speed is an important outcome of gait therapy, gait entrainment to mechanical perturbations may serve as a promising new method of robot-aided therapy. Still, an understanding of the underlying neuromechanical processes that give rise to gait entrainment is needed to fully evaluate its therapeutic potential. To gain such insight, the goal of this paper was to evaluate whether an existing neuromechanical model of human locomotion exhibited entrainment behavior similar to that observed in the prior human experiments. Simulation results showed that the model entrained to pulses applied at both the ankle and hip joints. The convergence of relative phase between model gait and hip perturbations was similar to that observed with the human gait, but differed slightly for ankle perturbations. Thus, models that can more accurately describe neuromechanical interactions between human gait and robotic exoskeletons are still needed. Nevertheless, the simulation results support the notion that the limit-cycle behavior observed during locomotion does not require supra-spinal control or a self-sustaining oscillatory neural network, which has important implications for improving gait therapy.

A prominent challenge in the field of robotics is manipulation of flexible objects. One major factor that makes this task difficult is the complex dynamics emerging from its high-dimensional structure. This argues against the use of popular optimization-based approaches, which scale poorly with system dimension (the "curse of dimensionality"). Nevertheless, almost indifferent to this complexity, humans handle it on a daily basis, without any apparent difficulty.Inspired by human motor control, we propose that encoding movements based on dynamic primitives can simplify the task of manipulating flexible objects and provides a way around the curse of dimensionality. Using an extreme example — manipulating a whip — we tested in simulation whether targets at various locations could be reached with a whip by using a controller based on dynamic primitives. Regardless of the target location, this approach successfully managed the complexity of a 54 degree-of-freedom system (yielding a 108-dimensional state-space representation) and identified an upper-limb movement that achieved the task. This approach did not require a detailed model of the whip, which thereby significantly simplified the computational complexity of the control task. We believe that this approach may facilitate robotic manipulation of flexible materials, and in general afford a simplified way to control dynamically complex objects.

Humans dexterously interact with a variety of objects, including those with complex internal dynamics. Even in the simple action of carrying a cup of coffee, the hand not only applies a force to the cup, but also indirectly to the liquid, which elicits complex reaction forces back on the hand. Due to underactuation and nonlinearity, the object's dynamic response to an action sensitively depends on its initial state and can display unpredictable, even chaotic behavior. With the overarching hypothesis that subjects strive for predictable object-hand interactions, this study examined how subjects explored and prepared the dynamics of an object for subsequent execution of the target task. We specifically hypothesized that subjects find initial conditions that shorten the transients prior to reaching a stable and predictable steady state. Reaching a predictable steady state is desirable as it may reduce the need for online error corrections and facilitate feed forward control. Alternative hypotheses were that subjects seek to reduce effort, increase smoothness, and reduce risk of failure. Motivated by the task of 'carrying a cup of coffee', a simplified cup-and-ball model was implemented in a virtual environment. Human subjects interacted with this virtual object via a robotic manipulandum that provided force feedback. Subjects were encouraged to first explore and prepare the cup-and-ball before initiating a rhythmic movement at a specified frequency between two targets without losing the ball. Consistent with the hypotheses, subjects increased the predictability of interaction forces between hand and object and converged to a set of initial conditions followed by significantly decreased transients. The three alternative hypotheses were not supported. Surprisingly, the subjects' strategy was more effortful and less smooth, unlike the observed behavior in simple reaching movements. Inverse dynamics of the cup-and-ball system and forward simulations with an impedance controller successfully described subjects' behavior. The initial conditions chosen by the subjects in the experiment matched those that produced the most predictable interactions in simulation. These results present first support for the hypothesis that humans prepare the object to minimize transients and increase stability and, overall, the predictability of hand-object interactions.

Restoring healthy kinematics is a critical component of assisting and rehabilitating impaired locomotion. Here we tested whether spatiotemporal gait patterns can be modulated by applying mechanical impedance to hip joints. Using the Samsung GEMS-H exoskeleton, we emulated a virtual spring (positive and negative) between the user's legs. We found that applying positive stiffness with the exoskeleton decreased stride time and hip range of motion for healthy subjects during treadmill walking. Conversely, the application of negative stiffness increased stride time and hip range of motion. These effects did not vary over long nor short repeated exposures to applied stiffness. In addition, minimal transient behavior was observed in spatiotemporal measures of gait when the stiffness controller transitioned between on and off states. These results suggest that changes in gait behavior induced by applying hip stiffness were purely a mechanical effect. Together, our findings indicate that applying mechanical impedance using lower-limb assistive devices may be an effective, minimally-encumbering intervention to restore healthy gait patterns.

Relatively little work in human and robot control has examined the control of underactuated objects with internal dynamics, such as transporting a cup of coffee, a task that presents little problems for humans. This study examined how humans move a `cup of coffee' with a view to identify principles that may be useful for robot control. The specific focus was on how humans choose initial conditions to safely reach a steady state. We hypothesized that subjects choose initial conditions that minimized the transient duration to reach the steady state faster, as it presented more predictable dynamics. In the experiment, the cup of coffee was reduced to a 2-D cup with a sliding ball inside which was simulated in a virtual environment. Human subjects interacted with this virtual object via a robotic manipulandum that provided haptic feedback. Participants moved the cup between two targets without losing the ball; they were instructed to explore different initial conditions before initiating the continuous interaction. Results showed that subjects converged to a small set of initial conditions that decreased their transient durations and achieved a predictable steady state faster. Simulations with a simple feedforward controller and inverse dynamics calculations confirmed that these initial conditions indeed led to shorter transients and less complex interaction forces. These results may inform robot control of objects with internal dynamics where the effects of initial conditions need further investigation.

This study examined strategies humans chose to manipulate an object with complex (nonlinear, underactuated) dynamics, such as liquid sloshing in a cup of coffee. The problem was simplified to the well-known cart-and-pendulum system moving on a horizontal line. This model was implemented in a virtual environment and human subjects manipulated the object via a robotic manipulandum. The task was to maneuver the system from rest to arrive at a target position such that no residual oscillations of the pendulum bob remained. Our goal was to test whether humans simplified control by employing dynamic primitives, specifically submovements. Experimental velocity profiles of the human movements were compared to those predicted by three different control models. Two models used continuous optimization-based control, the third control model was based on Input Shaping. Input Shaping is a method for controlling flexible objects by convolving a motion profile with impulses of appropriate amplitude and timing. To evaluate whether humans used Input Shaping, we decomposed the velocity profiles recorded from humans into submovements, as proxies for the convolved impulses. Comparing the motion profiles from the 3 models with the experimentally measured human profiles showed superior performance of the Input Shaping model. These initial results are consistent with our hypothesis that combining dynamic primitives, submovements, is a competent description of human performance and may provide a simpler alternative to computationally complex optimization-based methods of robot control.

The locomotion and balance capabilities of bipedal robots have greatly improved in recent years. However, maintaining balance on difficult terrain still poses a significant challenge. In this paper, we examined how humans maintain mediolateral balance when standing on a narrow beam with bare feet and wearing rigid soles. Our results show that foot-beam interaction dynamics critically influence balancing behavior. Importantly, this suggests that differences in human balancing behavior across different support surfaces may not solely result from changes in their neural control strategy. They may also result from changes in foot-ground interaction. Thus, the altered foot-ground interaction dynamics must be considered to accurately capture changes in the human controller across different support surfaces. A simplified model of foot-beam interaction was added to a double inverted pendulum model for human balancing. This extended model could replicate the change in human behavior across different foot contact conditions (bare feet vs. rigid feet). A better understanding of how humans coordinate whole-body behavior across a range of conditions may inform the development of balance controllers for bipedal robots.

While rehabilitation of upper-limb motor function with human-interactive robots has been met with success, robot-aided locomotor rehabilitation has proven challenging. To inform more effective approaches to robotic gait therapy, it is important to understand neuro-mechanical dynamics and control of unimpaired locomotion. Our previous studies reported that human gait entrained to periodic mechanical perturbations at the ankle when the perturbation period was close to preferred walking cadence. Moreover, entrainment was accompanied by synchronizing the perturbations to a constant gait phase, the same for all subjects, where they provided mechanical assistance. To test the generality of entrainment-based assistance, the present study evaluated the behavior of live unimpaired subjects who walked overground while wearing a hip exoskeleton robot. Periodic torque pulses were applied to the subjects' hips, with a period different from, but close to, their preferred stride cadence. Results indicated that unimpaired subjects entrained their gait to periodic mechanical perturbations at the hip. Convergence of relative phase between gait and perturbations was observed, but clustered around two distinct gait phases, in contrast to the single converged phase observed in entrainment to periodic ankle torques. These entrainment studies quantify important aspects of the nonlinear neuro-mechanical dynamics underlying the control of walking, which will inform the development of effective approaches to robotic walking therapy.

Theoretical studies suggest and experimental evidence confirms that maintaining and changing human joint stiffness by coactivated antagonistic muscles are metabolically expensive, even if muscles do not perform net mechanical work. Based on this observation, we posit that effective human augmentation can be achieved by actuators operated in parallel to human joints, even if these actuators only supplement joint stiffness without doing net mechanical work. In this article, we present a prototype variable-length leaf-spring actuator capable of large-range stiffness modulation. The key feature of the actuator is that it provides intrinsically low-energy-cost stiffness modulation even for large output deflection, by keeping the force on the driving motor low. Variable stiffness actuators use two motors to provide both stiffness and equilibrium position modulation as they are designed to do net mechanical work. The proposed actuator conceptually differs from variable stiffness actuators because first, it uses a single motor to only provide stiffness modulation, second, it does not provide equilibrium position modulation, and third, unless externally loaded, it cannot do net mechanical work. Using this actuator, we demonstrate stiffness augmentation during human-machine collaboration in challenging postural stabilization and weight-bearing tasks. Our results indicate that the proposed actuator can be used to complement a biological system by restoring or extending its functionality with low energy cost, and that variable stiffness spring actuators could effectively augment humans by doing no or a limited amount of mechanical work.

This study examines human control of physical interaction with objects that exhibit complex (nonlinear, chaotic, underactuated) dynamics. We hypothesized that humans exploited stability properties of the human-object interaction. Using a simplified 2D model for carrying a "cup of coffee", we developed a virtual implementation to identify human control strategies. Transporting a cup of coffee was modeled as a cart with a suspended pendulum, where humans moved the cart on a horizontal line via a robotic manipulandum. The specific task was to transport the cart-pendulum system to a target, as fast as possible, while accommodating assistive and resistive perturbations. To assess trajectory stability, we applied contraction analysis. We showed that when the perturbation was assistive, humans absorbed the perturbation by controlling cart trajectories into a contraction region prior to the perturbation. When the perturbation was resistive, subjects passed through a contraction region following the perturbation. Entering a contraction region stabilizes performance and makes the dynamics more predictable. This human control strategy could inspire more robust control strategies for physical interaction in robots.

Standing on a beam is a challenging motor skill that requires the regulation of upright balance and stability. In this paper, we analyzed the behavior of humans balancing on a narrow beam without footwear. The results revealed high anti-correlation between lumped upper- and lower-body angular momentum. Despite differences in gross measures of balance, interlimb coordination was consistent between the novice and expert subjects, suggesting that both performances could be described with the same balance controller. By simulating a double inverted pendulum model utilizing different balancing controllers described in the robotics literature, we identified that the whole behavior observed from humans standing on a beam was best replicated with controllers that predominantly utilized hip actuation.

For robotic systems to interact with or learn from the actions of surrounding humans, it is important that they can accurately interpret the intention driving human motor actions. Making such interpretations, however, requires the ability to perceive the relevant feature(s) from the observed human behavior. With visual sensing alone, robots are typically limited to perceiving only the human's overt motion in the form of joint angles and positions. Ideally, robots designed to interface with humans would also be able to infer information as to how the human is controlling itself from that overt motion. In this study, we investigated if and how humans might be able to visually sense changes in limb mechanical impedance of others. Results indicated that humans can visually perceive changes in joint stiffness from the motion of a two-link planar arm, suggesting that humans can extract information regarding how humans control limb impedance from kinematic information. These findings have important implications for applications where robots must interpret the motor actions of humans, such as during robot imitation learning and human-robot physical interaction.

Dynamic behavior of legged robots is strongly affected by ground impedance. Empirical observation of robot hardware is needed because ground impedance and foot-ground interaction is challenging to predict in simulation. This paper presents experimental data of the MIT Super Mini Cheetah robot hopping on hard and soft ground. We show that controllers tuned for each surface perform better for each specific surface type, assessing performance using measurements of 1.) stability of the robot in response to self-disturbances applied by the robot onto itself and 2.) the peak accelerations of the robot that occur during ground impact, which should be minimized to reduce mechanical stress. To aid in controller selection on different ground types, we show that the robot can measure ground stiffness and friction in-situ by measuring its own interaction with the ground. To motivate future work in variable-terrain control and in-situ ground measurement, we show preliminary results of running gaits that transition between hard and soft ground.

Energy efficient compliant actuation is the missing ingredient and key enabler of next-generation autonomous systems, domestic robots, prosthetic devices, orthotic devices, and wearable exoskeletons, to name a few. For all these devices, one would wish to develop actuators enabling wide range impedance modulation with low energy cost. Using conventional and biologically-inspired compliant actuation, previous research led to functional devices but with high energy cost. Here we introduce a minimalistic compliant actuator to realize impedance modulation with low energy cost. Using this actuator we demonstrate stiffness augmentation in human-machine collaboration. We argue that the non-biologically-inspired actuation concept presented here may effectively complement a biological system, by restoring or extending its functionality, with negligible energy cost.

Managing the trade-off between performance and stability is a crucial issue in physical human-robot interaction, and this has become more important than ever apace with growing needs for physically human-interactive robots in many fields, such as assistive robotics and rehabilitation robotics. In this paper, we present two essential considerations for design and control of robots physically interacting with humans: energetic passivity and mechanical impedance. Characterization of passive, dissipative, and active behavior of the human neuromuscular system is essential to ensure and control coupled stability in physical human-robot interaction. In addition, characterization of human mechanical impedance at the interaction port provides detailed quantitative information to describe interactive dynamics. The importance of these characterizations is demonstrated by simple examples and the authors' previous research on the human ankle. Implications for quantitative guidelines for robot design and control are discussed.

Numerous legged robots have demonstrated the effectiveness of tuned leg impedance to achieve dynamically stable running and hopping. However, selecting appropriate impedance values for new machines remains challenging. This paper investigates the effect of joint impedance selection on locomotion stability and efficiency by analyzing a simulation model of the MIT Cheetah quadruped robot performing a trot gait. An exhaustive search of impedance parameters of the knee and hip shows that locomotion stability is highly sensitive to knee impedance and insensitive to hip impedance. Inspection of simulations operating during a ground-height disturbance reveals why: During a disturbance response most of the variation in work performed in the legs occurs in the knee joints. Mechanical work data from the MIT Cheetah exhibits close experimental agreement with the simulation predictions. The exhaustive search also reveals that, within the range of impedance parameters that can achieve stable locomotion, joint impedance values do not have a significant effect on the mechanical cost of transport. These results indicate that the dynamic response of the leg-extension degree of freedom is of primary importance to achieving dynamically stable running, and that robust stability may be achieved with minimal compromise of locomotion efficiency.

The MIT Cheetah demonstrated a stable 6 m/s trot gait in the sagittal plane utilizing the self-stable characteristics of locomotion. This paper presents a numerical analysis of the behavior of a quadruped robot model with the proposed controller. We first demonstrate the existence of periodic trot gaits at various speeds and examine local orbital stability of each trajectory using Poincar`e map analysis. Beyond the local stability, we additionally demonstrate the stability of the model against large initial perturbations. Stability of trot gaits at a wide range of speed enables gradual acceleration demonstrated in this paper and a real machine. This simulation study also suggests the upper limit of the command speed that ensures stable steady-state running. As we increase the command speed, we observe series of period-doubling bifurcations, which suggests presence of chaotic dynamics beyond a certain level of command speed. Extension of this simulation analysis will provide useful guidelines for searching control parameters to further improve the system performance.

This paper presents a new method to characterize human ankle mechanical impedance during treadmill locomotion with a wearable ankle robot, Anklebot. An ensemble-based system identification method was used to investigate the time-varying behavior of ankle mechanical impedance in two degrees of freedom, both in the sagittal and frontal planes. We also provide solutions to overcome the limitations of original ensemble-based methods in practical applications. A pilot study of three human subjects demonstrated the efficacy of our approach. Analysis results showed clear time-varying behaviors of ankle impedance across the gait cycle except in the mid- and terminal-stance phases, and these behaviors were accurately approximated as a second-order model with stiffness, damping, and inertia components. Interestingly, all three subjects showed similar time-varying behaviors in both degrees of freedom: impedance increased around heel-strike and decreased significantly at the end of the stance phase.

Robotic studies have suggested a contribution of limit-cycle oscillation of the neuro-mechanical periphery to human walking by demonstrating stable bipedal robotic gaits with minimal actuation and control. As behavioral evidence of limit-cycle oscillation in human walking, we recently reported entrainment of human gaits to mechanical perturbations. We observed synchronization of human walking with mechanical perturbation only when the perturbation period was close to the original walking period. In addition, the entrainment was always accompanied by phase locking at the end of double-stance. A highly-simplified state-determined walker reproduced these salient features: 1) entrainment to periodic perturbations with a narrow basin of entrainment and 2) phase-locking at the end of double stance. Importantly, the model required neither supra-spinal control nor an intrinsic self-sustaining neural oscillator (like a rhythmic central pattern generator), which suggests that prominent features of human walking may stem from simple afferent feedback processes that produce limit-cycle oscillation of the neuro-mechanical periphery without significant involvement of the brain or rhythmic central pattern generators. One limitation of that model was that it entrained only to perturbations faster than the unperturbed walking period. In the study reported here, we modified the model to have two independent steps per stride. The revised model reproduced entrainment to perturbations both slower and faster than the unperturbed cadence, as observed experimentally in human walking.

Analysis of motor performance variability in tasks with redundancy affords insight about synergies underlying central nervous system (CNS) control. Preferential distribution of variability in ways that minimally affect task performance suggests sophisticated neural control. Unfortunately, in the analysis of variability the choice of coordinates used to represent multi-dimensional data may profoundly affect analysis, introducing an arbitrariness which compromises its conclusions. This paper assesses the influence of coordinates. Methods based on analyzing a covariance matrix are fundamentally dependent on an investigator's choices. Two reasons are identified: using anisotropy of a covariance matrix as evidence of preferential distribution of variability; and using orthogonality to quantify relevance of variability to task performance. Both are exquisitely sensitive to coordinates. Unless coordinates are known a priori, these methods do not support unambiguous inferences about CNS control. An alternative method uses a two-level approach where variability in task execution (expressed in one coordinate frame) is mapped by a function to its result (expressed in another coordinate frame). An analysis of variability in execution using this function to quantify performance at the level of results offers substantially less sensitivity to coordinates than analysis of a covariance matrix of execution variables. This is an initial step towards developing coordinate-invariant analysis methods for movement neuroscience.

In this paper, we present the design and characterization of a novel ankle robot developed at the Massachusetts institute of technology (MIT). This robotic module is being tested with stroke patients at Baltimore Veterans administration medical center. The purpose of the on-going study is to train stroke survivors to overcome common foot drop and balance problems in order to improve their ambulatory performance. Its design follows the same guidelines of our upper extremity designs, i.e., it is a low friction, backdriveable device with intrinsically low mechanical impedance. Here, we report on the design and mechanical characteristics of the robot. We also present data to demonstrate the potential of this device as an efficient clinical measurement tool to estimate intrinsic ankle properties. Given the importance of the ankle during locomotion, an accurate estimate of ankle stiffness would be a valuable asset for locomotor rehabilitation. Our initial ankle stiffness estimates compare favorably with previously published work, indicating that our method may serve as an accurate clinical measurement tool.

Robots intended for high-force interaction with humans face particular challenges to achieve performance and stability. They require low and tunable endpoint impedance as well as high force capacity, and demand actuators with low intrinsic impedance, the ability to exhibit high impedance (relative to the human subject), and a high ratio of force to weight. Force-feedback control can be used to improve actuator performance, but causes well-known interaction stability problems. This paper presents a novel method to design actuator controllers for physically interactive machines. A loop-shaping design method is developed from a study of fundamental differences between interaction control and the more common servo problem. This approach addresses the interaction problem by redefining stability and performance, using a computational approach to search parameter spaces and displaying variations in performance as control parameters are adjusted. A measure of complementary stability is introduced, and the coupled stability problem is transformed to a robust stability problem using limited knowledge of the environment dynamics (in this case, the human). Design examples show that this new measure improves performance beyond the current best-practice stability constraint (passivity). The controller was implemented on an interactive robot, verifying stability and performance. Testing showed that the new controller out-performed a state-of-the-art controller on the same system

Robots for high-force interaction with humans face particular challenges to achieve performance and coupled stability. Because available actuators are unable to provide sufficiently high force density and low impedance, controllers for such machines often attempt to mask the robots physical dynamics, though this threatens stability. Controlling for passivity, the state-of-the-art means of ensuring coupled stability, inherently limits performance to levels that are often unacceptable. A controller that imposes passivity is compared to a controller designed by a new method that uses limited knowledge of human dynamics to improve performance. Both controllers were implemented on a testbed, and coupled stability and performance were tested. Results show that the new controller can improve both stability and performance. The different structures of the controllers yield key differences in physical behavior, and guidelines are provided to assist in choosing the appropriate approach for specific applications

In this paper, we present a retrospective and chronological review of our efforts to revolutionize the way physical medicine is practiced by developing and deploying therapeutic robots. We present a sample of our clinical results with well over 300 stroke patients, both inpatients and outpatients, proving that movement therapy has a measurable and significant impact on recovery following brain injury. Bolstered by this result, we embarked on a two-pronged approach: 1) to determine what constitutes best therapy practice and 2) to develop additional therapeutic robots. We review our robots developed over the past 15 years and their unique characteristics. All are configured both to deliver reproducible therapy but also to measure outcomes with minimal encumbrance, thus providing critical measurement tools to help unravel the key question posed under the first prong: what constitutes "best practice"? We believe that a "gym" of robots like these will become a central feature of physical medicine and the rehabilitation clinic within the next ten years.

In this paper, we present considerations for a novel robot to rehabilitate the ankle following stroke. This module has been developed at MIT and should commence pilot testing with stroke patients at the Baltimore Veterans Administration Medical Center by the end of Summer 2004. The purpose of the module is to train stroke survivors to overcome the common foot-drop problem. Its design follows the same guidelines as our upper-extremity designs, i.e., it is a low friction, backdriveable device with intrinsically low mechanical impedance.

Both in designing teleoperators or haptic interfaces and in fundamental biological motor control studies, it is important to characterize the motor commands and mechanical impedance responses of the operator (or subject). Although such a characterization is fundamentally impossible for isolated movements when these two aspects of motor behavior have similar time scales (as is the case with humans), it is nonetheless possible, if we are dealing with repeated movements, to measure a trajectory which is analogous to the current source in Norton-equivalent electrical circuits. We define the attractor trajectory to be this equivalent source and show that it rigorously embodies the notion of the attractor point of a time-evolving system. We demonstrate that most previous attempts to test a controversial motor control hypothesis known as the "equilibrium point" or "virtual trajectory" hypothesis are based on inadequate models of the neuromuscular system, and we propose a model-independent means of testing the hypothesis based on a comparison of measurable attractor trajectories at different levels of the motor system. We present and demonstrate means of making such measurements experimentally and of assigning error bounds to the estimated trajectories.

Development and evaluation of teleoperators and haptic virtual environment technologies require objective quantitative measures of haptic quality. In this paper a theoretical framework and experimental procedures are introduced for quantitative assessment of haptic perception. Human haptic perception of objects is distorted and uncertain as well. Perceptual distortion is defined to be a systematic bias of perception with respect to an objective standard. Perceptual uncertainty is defined to be a measure of statistical distribution of percepts. A set of mathematical tools is presented to quantify perceptual distortion and uncertainty. Two experiments are presented which apply these tools to investigate the fundamental structure of human haptic perception.>

The authors explore the sources of and solutions to robot contact instability associated with force feedback. The behavior of several linear robot models is analyzed in terms of the properties of their admittances. Using the novel technique of passive physical equivalents, an explanation for the often-observed instability of a force-controlled robot contacting a stiff surface is offered, and it is shown that a fundamental limit exists to the efficacy of any force feedback controller implemented on a robot with noncolocated actuators and sensors. Suggestions for improved force control involving both mechanical design and compensator design are also presented.>

Recent work on the dynamics of interactions between humans and machined is reviewed. An experimental apparatus for simulating virtual environments and investigating the man/machine interface is described. Relevant experiments on the mechanical impedance of the human arm are reviewed. Despite active neuromuscular feedback control, the human arm exhibits the impedance of a passive object. Preliminary results of some experiments on the adaptability of human arm impedance are presented. These results indicate limitation in the rapidity of parameter adaptation in humans.>

Problems arising when kinematically redundant manipulators are controlled using the Jacobian pseudoinverse are related to the nonintegrability of the standard pseudoinverse. The authors present a class of pseudoinverse which have the property of being integrable within any simply connected nonsingular region of the workspace. Integrability is obtained by deriving the equations which describe an externally imposed motion with the hypothesis that a compliance function is associated with each degree of freedom of the manipulator. The result is a weighted pseudoinverse containing a term which accounts for the nonlinear features of the forward kinematics. The relation of this integrable pseudoinverse to the standard Moore-Penrose and weighted pseudoinverses are discussed.>

Design in the physical domain is proposed as a means of integrating control systems design with mechanical systems design. It is shown how this philosophy can lead to a robot architecture (macro/micromanipulator) that is inherently stable and well suited for high-bandwidth endpoint position and force control. Experimental verification is presented. A force-control bandwidth of 60 Hz, 32 times higher than the first structural mode of the robot, was achieved against an environment that is five times stiffer than the robot structure. An endpoint-position-control bandwidth of 28 Hz, 15 times higher than the first structural mode of the robot, was also achieved. This is beneficial in regulating interface forces and modulating endpoint impedance.>

A robot's ability to maintain desired interface forces during constrained motion is governed by its driving-point impedance and primarily by its inertia. Negative force feedback is a means of reducing this driving-point impedance, but the system becomes unstable at high bandwidths. A macro/micro manipulator system, consisting of a large (macro) robot carrying a small (micro) high-performance robot, alleviates this problem by physically reducing the endpoint inertia as well as providing an inherently stable physical configuration for high bandwidth force control. A robust controller design based on physical equivalence and impedance matching is proposed. It is shown that interface force regulation at bandwidths higher than the structural frequencies of the macromanipulator can be achieved with only minimal knowledge of the structure.>

Manipulation requires contact with the object being manipulated, and the full potential of robots can only be realized when they are applied to contact tasks. One of the difficulties engendered by contact tasks is that they require intimate dynamic interaction between the robot and its environment. That interaction changes the performance of the robot and can jeopardize the stability of its control system. A discussion is presented of the problem of preserving the stability of a manipulator's control system during contact tasks. It will be shown that contact stability may be guaranteed if the control system provides the manipulator with an appropriately structured dynamic response to environmental inputs. Two aspects of one implementation of such a controller will be considered. Robustness to large errors in the manipulator kinematic equations and to unmodeled interface dynamics is shown.>

This paper presents an experimental evaluation of the performance of a nonlinear robot control algorithm on a contact task involving free motion, constrained motion and transitions between the two. The algorithm is an implementation of impedance control which uses end-point force feedback. Stable control of the force exerted on a rigid surface is achieved without recourse to a soft sensor. Motion control is achieved without inverse kinematic computations. It is unnecessary to switch between different modes of control at the moment of contact as the impedance controller is competent in all phases of the task.

Time-optimal control of robot motion for dynamically decoupled manipulators is described in terms of potential functions. Avoidance of moving obstacles is incorporated via protective potential functions. An energy interpretation of the potential functions leads to rules for construction of avoidance functions and logical operations among them. Simple expressions for combining obstacle fields with an obstacle-free time-optimal solution result in the minimum safe influence of obstacles. Simulation results are given demonstrating high-speed target interception in the presence of obstacles.