Prosthesis with adaptive speed and incline impedance control

Abstract

A powered knee-ankle prosthesis includes a controller configured to vary torque at the prosthetic joints as a function of impedance parameters that vary as a continuous function of stance phase progression and walking task condition. This is an improvement over conventional impedance controllers which are tuned to discrete sets of task conditions, such as multiple combinations of walking speeds and ground inclines. The disclosed impedance controller relies on models of each impedance parameter derived from able-bodied data and can self-adapt to changes in walking task conditions based on real-time estimates of gait phase and task conditions.

Description

[0001]PROSTHESIS WITH ADAPTIVE SPEED AND INCLINE IMPEDANCE CONTROL STATEMENT OF FEDERALLY SPONSORED RESEARCH This invention was made with government support under HD094772 awarded by the National Institutes of Health. The government has certain rights in the invention. TECHNICAL FIELD This disclosure is related to powered prostheses and control strategies intended to make artificial joint movement more natural. BACKGROUND The development of intuitive, natural, and safe control strategies for robotic prostheses is critical to their success, but challenges in tuning these control strategies for each user and activity continue to hinder their implementation. Impedance control is a promising control strategy because of its ability to replicate the natural dynamics of the healthy human body, provide compliant ground interaction, and maintain locally passive closed-loop behavior. Previous impedance-based controllers for powered knee-ankle prostheses have used finite state machines with dozens of user-specific parameters that require manual tuning by technical experts. Additionally, these parameters are only appropriate near the walking conditions at which they were tuned, necessitating many different parameter sets for variable-condition walking. Conventional methods of impedance control for lower-limb prostheses involve segmenting the gait cycle into discrete subphases, where each subphase has its own constant values of stiffness, damping, and equilibrium angle, which are used to calculate joint torque for a controller to provide at the desired prosthetic joint. Researchers manually tune the impedance parameters in each gait subphase until the observed gait is satisfactory for the particular user and incline for which the parameter is being tuned. Switching between subphases during walking is controlled by a finite state machine (FSM) with transition criteria based on sensor readings (e.g., elapsed time, leg loading, joint angles, etc.). Like the impedance parameters, these transition criteria are often experimentally tuned for an individual’s gait by a technical expert. Joint kinematics and kinetics vary based on the ground incline and walking speed—which are together termed the “task conditions.” Therefore, the necessary impedance parameters and state machine transition criteria also vary. For a standard FSM impedance controller to operate over a wide array of task conditions—i.e., different combinations of walking speed and ground incline—many tunable parameters are required. Such multi-modal impedance controllers have been known to require up to 140 tunable parameters for five ambulation modes. SUMMARY Embodiments of a powered prosthesis include a joint and a controller configured to vary torque at the joint as a function of one or more impedance parameters. At least one impedance parameter varies during stance phase of walking as a function of stance phase progression and at least one walking task condition. In various embodiments, the impedance parameters include joint stiffness, damping coefficient, and equilibrium angle. In various embodiments, joint stiffness, damping coefficient, and equilibrium angle vary during the stance phase as a function of stance phase progression and at least one walking task condition. In various embodiments, each impedance parameter function is a continuous function. In various embodiments, each impedance parameter function is a continuous function over the entire stance phase. In various embodiments, each impedance parameter function is non-linear. In various embodiments, the at least one walking task condition includes walking speed or incline. In various embodiments, the at least one walking task condition includes walking speed and incline. In various embodiments, the controller automatically adapts to changes in the at least one walking task condition based on real-time estimates of each walking task condition. In various embodiments, the controller automatically adapts to changes in walking speed and / or incline based on real-time estimates of each walking task condition. In various embodiments, the controller estimates stance phase progression in real-time based on a walking task condition-invariant phase variable. In various embodiments, the controller estimates a user gait phase based on the at least one walking task condition. In various embodiments, the gait phase estimates adapt to changes in the at least one walking task condition. In various embodiments, each impedance parameter function is based on able-bodied data for each impedance parameter at a plurality of combinations of walking task conditions. In various embodiments, each impedance parameter function is based on able-bodied data that includes non-steady-state data. The non-steady-state data may include perturbation responses or biological impedance estimates. In various embodiments, the combinations of walking task conditions include combinations of discrete walking speeds and inclines selected from ranges of walking speeds and inclines. In various embodiments, the controller is a hybrid kinematic and impedance controller additionally configured to vary joint angle as a continuous function of swing phase progression. In various embodiments, the joint is a knee joint and the prosthesis includes an ankle joint, whereby the prosthesis is a knee-ankle prosthesis. The controller is configured to vary torque at the ankle joint as a function of each impedance parameter. In various embodiments, the knee-ankle prosthesis include an upper leg member adapted for attachment to a user, a lower leg member coupled with the upper leg member at the knee joint, a foot member coupled with the lower leg member at the ankle joint, and first and second actuators for providing variable torque at the respective knee and ankle joints according to the impedance parameter functions. It is contemplated that any number of the individual features of the above-described embodiments and of any other embodiments depicted in the drawings or description below can be combined in any technically feasible combination to define an invention. BRIEF DESCRIPTION OF DRAWINGS FIG.1 is a schematic side view of a powered knee-ankle prosthesis depicted at mid-stance (solid), at the end of stance (dotted), and at the beginning of stance (phantom); FIG.2 is a block diagram of a hybrid kinematic-impedance controller (HKIC) for use with a powered and jointed prosthesis; FIGS.3A-3C are projections of knee joint impedance parameter functions derived from able- bodied data; FIGS.3D-3F are projections of ankle joint impedance parameter functions derived from able- bodied data; FIG. 4A is a plot of able-bodied thigh angle as a function of gait phase at various incline angles; FIG.4B is a plot of gait phase estimates relative to actual gait phase based on one estimating method; FIG. 4C is a plot of gait phase estimates relative to actual gait phase based on another estimating method; FIG.5A-5D are photographic images of powered prosthesis users during experiments; FIGS.6A and 6B illustrate various task condition sequences used during experiments with the powered prosthesis; FIG.7A illustrates kinematic and kinetic trajectories produced by the HKIC during constant- speed walking at various inclines; FIG.7B illustrates kinematic and kinetic trajectories produced by the HKIC during constant- incline walking at various walking speeds; FIG.8A illustrates error in observed kinematics relative to able-bodied data for the HKIC and a finite state machine controller (FSMC) during steady-state walking; FIG.8B illustrates error in observed kinetics relative to able-bodied data for the HKIC and the FSMC during steady-state walking; FIGS.9A-9C illustrate average cadence during steady-state walking as functions of speed at various incline angles; FIGS.10A-10C illustrate average work per stride at various incline angles for the knee joint, the ankle joint, and for both joints during steady-state walking; FIG.11A-11C illustrate average work per stride at various walking speeds for the knee joint, the ankle joint, and for both joints during steady-state walking; FIG.12 includes plots of average kinematic and kinetic error trajectories during continuously varying incline walking relative to able-bodied data; FIG.13 illustrates average phase estimate progression calculated in real-time by the HKIC during continuously varying task conditions; FIG.14 illustrates average able-bodied thigh angle as a function of gait phase on level ground; FIG.15 is a block diagram illustrating the overall structure of an FSM for controlling phase estimates; FIG.16 illustrates phase variable trajectories during various non-steady-state walking periods; FIG.17A and 17B are block diagrams depicting a comparative FSM impedance controller; and FIG.18 includes a table containing experimental FSM impedance parameter tuning data and a table containing experimental RMS error data for FSM and HKI controllers. DESCRIPTION OF EMBODIMENTS Described below is a powered knee-ankle prosthesis with a variable impedance controller employing impedance parameters that are functions of stance phase progression, speed, and incline and that automatically adapts to changes in those parameters. The impedance controller may be part of a hybrid kinetic-impedance controller (HKIC) by which continuously variable impedance control is employed during stance phase, and kinematic control is employed during swing phase. The controller sends torque commands to actuators of the prosthesis based on a data-driven model of variable joint impedance generated via convex optimization. A task condition-invariant phase variable has also been developed. The controller can use real-time estimates of walking speed and incline to enable autonomous adaptation to the current task conditions. The controller can also perform highly linear phase estimates and accurate task condition estimates, can produce biomimetic kinematic and kinetic trends as the task conditions vary, and can produce biomimetic joint work and cadence trends as the task conditions vary. The controller has demonstrated the ability to meet and / or exceed the performance of a benchmark finite state machine controller without the need for manual impedance tuning at discrete sets of task conditions. As used herein, a “variable impedance controller” is a controller that varies the output mechanical impedance at a prosthetic joint during user locomotion. Mechanical impedance is generally a measure of resistance to movement from an equilibrium position, which in this context is resistance to rotation about a joint axis. Resistance to rotation about a joint axis is a function of joint stiffness, joint damping coefficient, and joint angle relative to an equilibrium angle. Mechanical impedance at a prosthetic joint can be provided via an applied torque in a rotational direction about the joint axis. This torque can be applied by an actuator (e.g., an electric motor) coupled with a transmission or other suitable means. Impedance control is different from kinematic control in that impedance is a measure of resistance to movement relative to an equilibrium position in a freely rotatable joint, while kinematic control positively controls joint movement by rotating an object about the joint axis from one angular position to another with a particular angular speed function. FIG. 1 schematically illustrates an example of a powered knee-ankle prosthesis 10 that includes an upper leg member 12, a lower leg member 14, and a foot member 16. One end of the upper leg member 12 is adapted for attachment to the end of the residual leg of an above-knee amputee (AKA) user, and another end of the upper leg member is coupled with the lower leg member 14 at a knee joint 18. The knee joint 18 is a rotational joint that provides rotational movement of the lower leg member 14 relative to the upper leg member 12 about a knee axis 20. The lower leg member 14 extends from the knee joint 18 to an ankle joint 22 at which the foot member 16 is coupled with the lower member 14. The ankle joint 22 is a rotational joint that provides rotational movement of the foot member 16 relative to the lower leg member 14 about an ankle knee axis 24. The prosthesis 10 includes a first actuator 26 (e.g., a motor) configured to provide a torque ^^^at the knee joint 18 in a rotational direction and a second actuator 28 configured to provide a torque ^^^at the ankle joint 22 in a rotational direction. For example, the first actuator 26 may be rigidly mounted along the upper leg member 12, and its rotational output may be converted to the torque ^^^applied to the knee joint 18 via a transmission component rigidly attached to the lower leg member 14. Similarly, the second actuator 28 may be rigidly mounted along the lower leg member 14, and its rotational output may be converted to the torque ^^^applied to the ankle joint 22 via a transmission component rigidly attached to the foot member 16. Other arrangements are possible to provide torque at one or more of the prosthetic joints. The prosthesis 10 includes at least one controller 30 configured to store and employ a control scheme 32 according to which the controller operates each motor 26, 28 to provide the desired torque ^^^, ^^^at each joint 18, 22. The controller 30 is programable and is in communication with each actuator 26, 28 to control its output torque. The controller 30 may be in two-way communication with each actuator 26, 28 to receive one or more inputs from the actuators, such as a real-time encoder position which can be used to determine real-time angular velocities at each joint 18, 22, among other values. The controller 30 may receive additional information from one or more sensors 34 (e.g., an accelerometer or thigh angle sensor) to implement the control scheme 32. FIG.1 schematically illustrates the prosthesis 10 at different phases of a walking gait cycle. The gait cycle may be divided into two distinct phases, including a stance phase and a swing phase. The prosthesis 10 is illustrated in the stance phase in solid lines at approximately mid-stance (MS). The stance phase is defined as the phase of the gait cycle during which the foot member 16 is in contact with the ground. The prosthesis 10 is illustrated in the swing phase in broken lines, including the start of the swing phase (SS) in dotted lines, as the foot member 16 first leaves the ground at toe- off (TO), and the end of the swing phase (ES) in phantom lines, as the foot member 16 is about to contact the ground at heel-strike (HS) after swinging forward. The term “phase” may be used herein in two senses. In one sense, a complete gait cycle is divided into a stance phase and a swing phase as described above. These distinct phases may be referred to as the stance and swing portions of the gait cycle, or simply as “stance” and “swing” in some contexts. In another sense, the term “phase” indicates “phase progression,” which is indicative of how far into the stance phase or swing phase the user is at any moment during ambulation. Specifically, “stance phase progression” refers to the value of a phase variable along a continuous function between 0 and 1 during the stance portion of the gait cycle, where 0 is the beginning of stance phase (i.e., at heel strike) and 1 is the end of stance phase (i.e., at toe-off). Similarly, “swing phase progression” refers to the value of a phase variable along a continuous function between 0 and 1 during the swing portion of the gait cycle, where 0 is the beginning of swing phase (i.e., at toe-off) and 1 is the end of swing phase (i.e., at heel strike). The same can be applied to “gait phase progression” in reference to the value of a phase variable along a continuous function between 0 and 1 during an entire gait cycle, where 0 marks the beginning of the gait cycle from a reference position (e.g., at heel strike) and 1 marks the end of the gait cycle (e.g., at a subsequent heel strike). By way of example, when a phase variable representing stance phase progression is at 0.5, the user and prosthesis are halfway through the stance portion of a gait cycle. When a phase variable representing gait phase progression is at 0.8, the user and prosthesis are 80% through an entire gait cycle. The word “progression” may be omitted at times but can implied from context—e.g., when the value of a phase variable rather than a period of the gait cycle is being referenced. The control scheme 32 may be a hybrid control scheme using impedance control during one part of the gait cycle and kinematic control during another part of the gait cycle. For instance, impedance control may be used during the stance phase while kinematic control is used during the swing phase. While FIG.1 illustrates only one controller 30 and associated control scheme 32, some implementations include multiple controllers. The prosthesis 10 may for example include an impedance controller with an impedance control scheme and a separate kinematic controller with a kinematic control scheme. The prosthesis may include dedicated controllers for each joint as well. FIG. 2 is a block diagram of a hybrid kinematic-impedance controller (HKIC). Real-time estimates of gait phase ^^̂ and task conditions ^^̂ define desired joint impedance parameters ^^, ^^, ^^^^and joint angles ^^ௗusing data-driven models. Depending on whether the user and prosthesis are in stance or swing phase, the torque commands ^^ are generated using either an impedance controller or a position (kinematic) controller, respectively. In the HKIC, real-time phase and task condition estimates provide inputs to an impedance model during stance, as described below, and a conventional kinematic model during swing to provide reference joint behavior. Impedance and position controllers enforce the respective model outputs as described below. Once configured with a user’s mass and leg segment lengths, the controller can operate autonomously, requiring neither manual impedance tuning nor external knowledge of the terrain. The following sections discuss each component of the HKIC in turn. Variable Impedance Model for Stance A. Model Framework Discussed below is the development of a variable impedance model for use by the controller during the stance phase of the gait cycle. In an impedance control scheme, the controller 30 calculates joint torque ^^ ( ^^^or ^^^, above and in FIG.1) based on a joint angle θ and joint velocity ^^^according to: ^^ ൌ െ ^^൫ ^^ െ ^^^^൯ െ ^^ ^^^, (1)where K, B, and ^^ ^^are impedance equilibrium angle, respectively. To use impedance control in a continuous, phase-based control framework, the impedance parameter model is continuously parameterized by both gait phase ^^ and task conditions ^^ ൌ ^ ^^, ^^^. Here, the “task” is walking, and the “task conditions” are variable characteristics of walking. The task conditions used in this case are the current walking speed ^^ and ground incline angle ^^ over the ranges of interest, but the model could be parameterized using other or additional walking task conditions. A walking speed range of 0.8 ^ ^^ ^ 1.2 m / s and a ground incline range of െ10 ^ ^^ ^ 10 degrees is used in the illustrative model development described here. A model meeting these criteria can be constructed from a linear combination of phase-varying polynomials, where the linear combination weights vary with the task conditions. Polynomial functions of phase are useful to model parameter progression during stance because they are simply parameterized and can represent arbitrary aperiodic signals. Here, fourth order polynomials ( ^^ ൌ 4) were used, as they allow sufficient flexibility to model parameter behavior without overfitting. Once the appropriate polynomial functions were identified for individual task conditions in a dataset, bilinear interpolation was used to create a unified, continuous model with task condition and phase inputs. First, task condition-specific polynomial functions were defined representing how the parameters vary during stance for a set of fixed task conditions. For convenience, let ^^^௧be the stance phase (i.e., ^^^௧ൌ^^ൗ ^^்ை, where ^^்ைis the phase at toe-off). Then, the impedance parameters for the ^^௧^fixed task conditions ^^^are: ^^ఞ^ ൌ∑ௗ^ୀ^ ^^^^ ^^^^௧, ^^ఞ^ ൌ∑ௗ^ୀ^ ^^^^ ^^^^௧, ^^^^,ఞ^ ൌ∑ௗ^ୀ^ ^^^^ ^^^^௧, (2)where ^^ ఞ^ൌ … , defining the trajectories for arbitrary task conditions ( ^^, ^^) are calculated through bilinear interpolation of its four nearest neighboring task conditions ^^ఔ^,ఊ^, where ^^^∈ ^ ^^^, ^^ଶ^, ^^^∈ ^ ^^^, ^^ଶ^. For all j elements in ^^ఔఊ, this interpolation is: ^^^^^^ ^ఔ ିఔ ఔି௩ ^ఔ ఊ^ఔ ఊ^^ଶ െ ^^^^. (3)^ Finally, using ^^ఔఊan d equation (2) evaluated at the current stance phase ^^^௧, the impedance parameters are calculated. As a result, the model is fully defined once each task condition-specific set of coefficients ^^ఞ^is calculated. B. Model Fitting An optimization-based approach was used to fit the model to an able-bodied walking dataset. The dataset contains kinematic and kinetic joint trajectories recorded from 10 participants walking at steady-state at 15 distinct points in the task condition space, including combinations of five inclines (-10, -5, 0, 5, and 10 degrees) with three walking speeds (0.8, 1.0, and 1.2 m / s). While not included in the example presented here, the able-bodied walking dataset may include non-steady state gait data. This may include perturbation response data in addition to steady-state walking data. For example, walking data during sudden treadmill belt accelerations or decelerations could provide a more representative model of slip or trip responses to make the control model more robust to disturbances and more biomimetic. In another manner of collecting perturbation response data, able-bodied joint responses to torque pulses applied by a wearable exoskeleton may allow the model to converge to impedance parameters that are more representative of biology. It is further possible to directly include biological impedance estimates (i.e., estimates of the joint impedance rendered by the neuromuscular system) at specific points of phase progressions and specific task conditions based on reference data. While not included in this example, known biological impedance values could, for example, be incorporated as model bounds or as additional cost function terms with the aim of favoring biological impedance for known walking task conditions. For each set of task conditions ^^^, an optimization problem was constructed to identify the set of impedance parameter coefficients ^^ఞ∗^ that, when used in equations (1) and (2), best reproduced the mass-normalized joint torques ^^ in the dataset given the dataset kinematics ( ^^, ^^^) over all n data points at ^^^: ^^ఞ∗^ൌ arg min^^‖^^ െ ^^̂‖ଶଶ,(4)As written, equation the unknown parameters, and the overall objective function is non-convex. To avoid this issue, a similar, convex problem was solved and its solution was used to approximate a solution to equation (4). First, the product ^^ఞ^^^^^,ఞ^is combined into a new, higher order polynomial with independent coefficients ^^^^:^^ఞ^ ^^^^,ఞ^ ൌ∑ௗ^ୀ^ ^^^^ ^^^^௧∑ௗ^ୀ^ ^^^^ ^^^^௧ൌ∑ଶ^ୀௗ^^^^^ ^^^^௧ൌ ^^ఞ^ (5)By treating the ^^ linear in the unknown parameters ^^^^, ^^^^, and ^^^^. The modified optimization problem can then be written as a standard quadratic program (QP), defining a new argument vector ^^ ∈ ℝସௗାଷൈ^as: ^^ ൌ ^ ^^ , … , ^^ , ^^ , … , ^⊺^^ ௗ^ ^^^ௗ^, ^^^^, … , ^^ଶௗ^൧. (6) Let ^^ ∈ ℝସௗାଷൈ^be ^^ ൌ ^െ ^^ ^^^, … , െ ^^ ^^ௗ, െ^ ^ ^ ௗ ^ ଶௗ ⊺^^ ^ ^ ^ ^^^ ^^^ , … , െ ^^^ ^^^ , ^^^ , … , ^^^ ൧ (7)Then, the objective ^^ ^ ^^ఞ^^ ൌ^^‖^^ െ ^^̂‖ଶଶൌ^ ^∑ ^^ୀ^ ^^^ଶെ ^^⊺^^ ^^ ଶ ^^⊺^^ ^^, (8) where: ^^ ൌଶ^ ∑^^ୀ^^^^^^^⊺, ^^ ൌ ଶ ^ ∑^^ୀ^^^^^^^(9) To prevent added to ^^ to penalize the L2 norm of x. The ^^௧^diagonal entries in ^^ corresponding to the regularization weights on ki and bi were λn = 1e-5while λn = 1e-2for the δi terms. These hyperparameters were chosen prior to the experiments in order to produce a smooth model that captured general behavior instead of overfitting to the training dataset. Next, a constraint matrix A was added to ensure that ^^ఞ^^ ^^^ and ^^ఞ^^ ^^^ remained within ranges that were both physiologically realistic and feasible for the prosthesis to render in a stable manner. Namely, ^^ఞ^^ ^^^ was constrained above 1.5 Nm / rad / kg and ^^ఞ^^ ^^^ was constrained between 0.01 and 1.0 In addition, AKA participants in preliminary experiments noted that a low stiffness at heelstrike was unsettling, as they were accustomed to a locked knee during early stance with their take-home prostheses. Therefore, a minimum heelstrike stiffness constraint of 3.0 Nm / rad / kg was added to increase participants’ confidence that the prosthesis was ready for weight acceptance at heelstrike. To enforce these constraints, stance phase was discretized into njpoints in the range [0, 1]. A constraint matrix ^^ ∈ ℝଷ^ೕൈସௗାଷwas constructed from submatrices ^^^∈ ℝ^ೕൈௗା^as:^^^ ⋯ ^ ௗ^ ^^ െ ^^^ 0 0^^^ൌ ^⋮ ⋱ ⋮^ , ^^ ൌ ^0 െ ^^ 0൩. (10) ^^^^^ೕ⋯ ^^^ௗೕ 0 ^^^0 A column vector ^^ ∈ ℝଷ^ೕൈ^contained njcopies of the minimum stiffness and damping and maximum damping values, with the first term modified for the heelstrike constraint: ^^ ൌ െ^3.0, 1.5, … , 1.5, 0.01, … , 0.01, െ1.0, … , െ1.0^⊺. (11) While not included in this example, these constraints could include upper and / or lower bounds on stiffness and / or damping based on known ranges of biological impedance values at specific phases and task conditions. Finally, the full QP was arrived at, with the positive offset torque (sum-of-squares) in equation (8) neglected without loss of generality: mini ^⊺௫mizeଶ ^^^^^ ^ ^^^^^ െ ^^⊺^^ (12) This QP was solved for each subject and set of task conditions ^^^in the dataset (N = 150) using the MATLAB Optimization Toolbox (R2021b, MathWorks, MA, USA). Then, the solution to the original equation (4) was approximated by projecting the rational function ^^ఞ^^ ^^^௧^ൗ ^^ఞ^^ ^^^௧^ ൌ ^^^^,ఞ^^ ^^^௧^ onto a ^^௧^order polynomial. It was assumed the polynomial order the rational function ^^ఞ^^ ^^^௧^ൗ ^^ఞ^^^^^௧^without significant information loss. This assumption was validated by the error, detailed in the next section. Then, for each set of task conditions ^^^, the intersubject mean set of coefficients ^̅^ఞ^was calculated for use as the final model. Trials that did not well-represent the data, measured by a Variance Accounted For (VAF) below 75%, were discarded as outliers prior to averaging. C. Modeling Results FIGS. 3A-3F show the calculated impedance parameter model projected onto a speed of 1 m / s, which was produced by evaluating equations (2) and (3) with ^̅^ఞ^. FIGS.3A-3F includes plots of the resulting impedance parameter functions for stiffness ^^^ ^^^௧, ^^, ^^^, damping ^^^ ^^^௧, ^^, ^^^, and equilibrium angle ^^^^^ ^^^௧, ^^, ^^^ for the knee joint and the ankle joint. These surfaces show the approximated solution to the original optimization equation (4). For purposes of visual demonstration, FIGS.3A and 3F include bold traces along the multi-variable stiffness functions at endpoints of the incline range to illustrate continuity of the functions with respect to incline and with respect to stance phase progression. Analogous curves can be drawn with respect to the walking speed task condition. To quantify the impedance parameter model’s reconstruction error, ^^̂ was calculated for the knee joint and ankle joint over each trial in the dataset using the model: ^^̂ ൌ ^^^^^^௧, ^^, ^^^൫ ^^^^^^^^௧, ^^, ^^^െ ^^൯ െ ^^^^^^௧, ^^, ^^^^^^. (13)Then, the root mean squared error (RMSE) in joint torque was calculated over all subjects for each set of task conditions ^^^in the dataset and normalized by the dataset torque’s standard deviation for ^^^. This dimensionless metric, referred to herein as normalized reconstruction error ^^, describes how many standard deviations ^^̂ is from the mean dataset torque trajectories, on average. Normalized reconstruction errors below 1.0 indicate that the model is able to predict joint torque to accuracy levels similar to able-bodied inter-subject variation. Averaged over all task conditions, the knee and ankle normalized reconstruction errors were ^^k = 0.78 ± 0.11 and ^^a = 0.58 ± 0.09, respectively. Hybrid Kinematic Impedance Controller A. Task Condition-Invariant Phase Estimation An estimate of the user’s progression through the gait cycle is used to synchronize the control outputs with the user’s gait. One version of this estimate, referred to herein as a “phase variable,” increases from 0 to 1 at a constant rate between sequential heel strikes of the same foot. The HKIC’s phase variable ^^̂ is calculated using a piecewise-linear mapping of the user’s global thigh angle θth, which has a roughly sinusoidal trajectory, as illustrated in FIG.4A, which shows mean able-bodied thigh trajectories as a continuous function of gait phase at different inclines ranging from -10 to 10 degrees. This angle can be measured directly using an Inertial Measurement Unit (IMU, 3DM-CX5- 25, LORD Microstrain, Williston, VT) mounted to the proximal end of the prosthesis knee joint. Mounting the IMU to the prosthesis instead of the person ensures a rigid connection to prevent slipping and vibration, which are commonly associated with soft tissue connections. Proper alignment of the prosthesis by a prosthetist ensures correct alignment of the IMU. The thigh angle-based method of phase estimation may be preferable because it allows the user to start and stop the gait cycle at will and enables non-rhythmic behavior. However, previous iterations of thigh angle-based phase variables have not worked very well for variable-task condition locomotion because of assumptions made about the shape of the thigh angle trajectory. In previous attempts to use a thigh angle-based phase variable to determine gait phase, it was assumed that thigh angle trajectory could be divided into two monotonic sections. While this assumption holds fairly well for level ground and incline walking, it is invalid for steep declines, as illustrated in FIG.4A. Previous methods produced inaccurate, saturated phase estimates for such cases (FIG.4B). Further, previous methods did not account for periods of low thigh angular velocity (i.e., when the hip joint is most extended or most flexed), leading to pauses in the phase estimate and subsequent problems in controller behavior. Better phase estimates are obtained based on thigh angle by relaxing previous assumptions and adding flexibility to the phase variable to better parameterize the gait cycle based on the diverse thigh angle trajectories observed in locomotion with variable task conditions. First, short periods of feed-forward phase progression are introduced that allow ^^̂ to maintain a constant positive rate even when thigh angular velocity is low, which enables a powerful and biomimetic push-off. Second, states are added to account for thigh trajectories that have more than two monotonic sections (especially common during ramp descent) to prevent excessive phase saturation and gait desynchronization. Third, a technique is introduced to improve the linearity of ^^̂, correcting for previous steady-state nonlinearities and thus making it closer to an ideal phase estimate. Mathematical details for these improvements are provided below in Appendix A. To illustrate the benefits of the new phase variable over its predecessors, a simulation was conducted using the able-bodied thigh kinematic data used for FIG.4A. For each trial of treadmill walking in the dataset, the phase variable was calculated using both the new method (Appendix A) and a prior art method. For each incline, the phase trajectories were averaged over all strides, participants, and walking speeds. The results of the prior art method are shown in FIG.4B, and the results of the presently disclosed method are shown in FIG. 4C. Notably, the presently disclosed method eliminates the phase estimate pause associated with maximum hip extension that was observed with the prior art phase variable. The new method also reduced the early saturation seen in the prior art phase variable, which was particularly prominent at steep ramp declines. Finally, the new method demonstrated improved linearity, particularly during midstance. Compared to an ideal linear phase trajectory, the new method showed 6.25% RMSE with R2= 0.990 while the prior art method showed 7.48% RMSE and R2= 0.976 over all task conditions. B. Task Condition Estimation In addition to the gait phase estimate, the controller may use an estimate of the user’s current task conditions ^^̂, which is calculated at each toe-off (TO) during steady walking. The estimates here include a walking speed estimate and an incline estimate. The controller may update each estimate once per stride and filter with a moving average over a plurality of consecutive strides (e.g., three strides) to account for stride-to-stride variation. Although filtering introduces a time delay in the task condition estimates, experiments have demonstrated that this limitation does not prevent the user from continuing to walk while the estimates converge. 1) Walking Speed: The user’s current walking speed may be estimated using a three-link leg model, comprising thigh, shank, and foot links equivalent to the upper leg member 12, lower leg member 14, and foot member 16 of FIG.1. Using forward kinematics and inputs from joint encoders and the thigh IMU, Cartesian locations of the heel and toe relative to the hip can be calculated, respectively given by ^^^^^^and ^^௧^^. At each TO event, the forward progression of the hip relative to the foot’s point of contact with the ground during the previous stance phase is calculated as: ^^^௧ൌ ฮ ^^௧^^െ ^^ுௌష^^^^ฮଶ, (14) where ^^ுௌష^^^^is the value of ^^^^^^from the previous heelstrike (HS). Similarly, by assuming a symmetric gait, the forward progression of the hip relative to the contralateral foot’s ground contact point over a swing phase is approximated at each HS as: ^^^௪ൌ ฮ ^^^^^^െ ^^்ைష௧^^ฮଶ, (15) where ^^்ைష^ likew ௧^ise is ^^௧^^from cycle can be calculated as ^^^௧^ ^^^௪^ ^^^^^௧, where ^^^^^௧is a constant accounting for the length of the prosthetic foot. Finally, walking speed is estimated by dividing forward progression by stride time. 2) Incline: The ground inclination can be estimated by the global angle of the foot ^^^when the foot is flat on the ground. To avoid adding an extra inertial sensor to the foot, this angle can be calculated from the thigh IMU using forward kinematics, along with a correction for foot bending. Prosthetic feet are often designed to deflect for energy storage, so foot deflection significantly impacts the incline estimate. Experiments with a prosthetic foot member (Lo Rider, 1E57, Ottobock, Duderstadt, Germany) showed that deflection correlated with the bending moment in the sagittal plane my. An on-board 6-axis load cell (M3564F, Sunrise Instruments, Nanning, China), located at the distal end of the ankle joint can measure this moment directly. Then, ^^^can be calculated as: ^^^ൌ ^^௧^െ ^^^^ ^^^^ ^^^^^ ^^^^^௬, (16) where ^^^is the linear bending coefficient, ^^^is the relative knee angle, and ^^^is the relative ankle angle, with all joint angles being measured positive in flexion and at zero when the user stands upright. The constant offset term ^^^^accounts for the angular difference between the prosthetic foot, the cosmesis, and the sole of the shoe. To determine when the foot member is flat on the ground, the center of pressure in the foot reference frame ^^ୡ୭୮is calculated using the load cell. The foot member may be considered flat when 7.5 ^ ^^ୡ୭୮^ 12 cm from the ankle joint, which corresponds to the ground reaction force acting between the middle and the ball of the foot. During this period, ^^^may be averaged to produce the incline estimate for the stride. C. Impedance and Kinematic Controllers 1) Stance Impedance Controller: During stance, a variable impedance controller is used to calculate joint torques. First, the stance phase estimate ^^̂^௧is calculated by: ^^̂^௧ ൌ ^^̂⁄^̅^்̂ை , (17)where ^̅^்̂ைis the expected value of the ^^̂^௧and ^^̂, joint stiffness ^^, damping coefficient ^^, and equilibrium angle ^^^^can be calculated from equations (2) and (3) and the above-developed model. Then, each joint torque during stance can calculated with the following impedance control law, scaled by user mass m: ^^^௧ ൌ ^^^ ^^^ ^^̂^௧, ^^̂^൫ ^^^^^ ^^̂^௧, ^^̂^ െ ^^൯ െ ^^^ ^^̂^௧, ^^̂^ ^^^^. (18)2) Swing ^^^and ^^ௗto directly track desired joint angle trajectories. This is in contrast to the equilibrium angles of the impedance controller, which do not necessarily align with the normative joint angles. A continuous model of able-bodied joint kinematics can provide the desired trajectories defined as: ^^ௗ^ ^^, ^^^ ൌ ∑ே^ୀ^^^^^ ^^^ ^^^^ ^^^ , (19) where ^^ ^^ ^^^ are Fourier series model, equation (19) is evaluated in real-time using ^^̂ and ^^̂. Then, the PD torque command during swing, ^^^௪, is given by ^^^௪ ൌ ^^^^ ^^ௗ െ ^^^ ^ ^^ௗ൫ ^^^ௗ െ ^^^൯. (20)3) Stance to Swing transition from impedance control to position control. Because impedance control may allow the joint angles to vary from their nominal trajectories depending on how the user loads the prosthesis, this smoothing is useful to avoid discrete changes in joint torque. At TO, ^^^௪increases from 0 to 1 over 0.25 s for the knee and 0.05 s for the ankle. The ankle smoothing is faster because close tracking of the ankle kinematics during early swing is important to avoid toe-stubbing. The actual output to the joint actuators is given by ^^ ൌ ^^^^௧ during stance^^ ^^ during s (21)^௪ ^௧ wing.Because the equilibrium angles at heelstrike are close to the kinematic references at the end of the gait cycle, smoothing may not be necessary for the swing-to-stance transition. Close examination of equation (21) shows that, for a brief period just following TO, minimal control action is applied to the joints. This is acceptable when the prosthesis uses low-impedance actuators, which allow the joints to continue moving along their current trajectories according to their passive dynamics without control input. Passive early swing knee and ankle dynamics have been shown to produce human-like gait, and these passive dynamics may contribute to the biomimetic behavior of the controller. Amputee Participant Experiments Experiments with two AKA participants were performed to investigate the ability of the hybrid controller to produce biomimetic gaits over variable task conditions. To benchmark the hybrid controller’s performance against another well-known controller, a standard, piecewise-constant FSM impedance controller was also implemented and tuned for each participant. The participants completed the experimental protocol once with each controller. FIGS.5A-5D are photographic images of AKA participants P1 and P2 walking with various task conditions with the hybrid controller during the experiments. FIGS.5A and 5B respectively show participant P1 during overground acclimation and while walking along a treadmill at a 7° incline and at 1 m / s. FIGS.5C and 5D respectively show participant P2 while walking along a treadmill at a 0° incline at 1 m / s and while walking along a treadmill at a 7° incline at 1 m / s. A. Comparative Example – FSM Impedance Controller As a comparative example, a Finite State Machine controller (FSMC) was designed based on a prior art variable-incline FSM impedance controller, with an additional stance state and modified transition criteria to improve performance (see Appendix B for details). This controller was chosen as a benchmark because of its simple construction, widespread use, and ability to create biomimetic walking gaits when appropriately tuned. While more sophisticated variants of the FSM impedance control paradigm have shown stronger results, such as those that modulate the impedance parameters based on joint angles or prosthesis axial force, the original version described by F. Sup et al. (“Upslope walking with a powered knee and ankle prosthesis: Initial results with an amputee subject,” IEEE Trans. Neural Syst. Rehabil. Eng., vol.19, no.1, pp.71–78, 2011) provides a valuable benchmark for comparing novel controllers because its performance and limitations are widely understood. Further, many modern controllers still use FSM impedance control in some if not all sections of the gait cycle, so understanding the HKIC’s performance relative to the FSMC is scientifically relevant. The FSMC had five discrete states throughout the gait cycle, each with its own set of constant impedance parameters and transition criteria. Similar to the methods discussed in the Background section, these parameters required hand-tuning by an expert researcher in order to produce the desired gait. To enable walking at various inclines, three sets of tunable impedance parameters and transition criteria were instantiated for each joint—i.e., one set for level ground, one set for declines, and one set for inclines. The controller selected between impedance parameter sets based on the estimated incline ^^^ (see FIG.17B). In total, the FSMC required 96 tunable parameters, including 45 impedance parameters per joint and 6 FSM transition criteria. B. Experimental Methods Two AKA individuals participated in the experiment, with attributes shown in TABLE I, below. TABLE I ID Sex Age Mass Height Years since Etiology al P2 Male 40 84 kg 1.8 m 23 Cancer sity o c gan ( ), an e par cpans wore a ce ng-moune sa ey arness while walking on the treadmill. For the experiments, the presented HKIC and the comparative FSMC were implemented on a backdrivable, powered knee-ankle prosthesis, shown in FIGS 5A-5D and consistent with the schematic view of FIG.1. The prosthesis was equipped with quasi-direct drive actuators that enable open-loop joint impedance control. A licensed prosthetist fit the prosthesis to the participants and ensured proper alignment. The participants were instructed on the expected high-level behavior of both controllers and given time to acclimate to each controller while walking overground within parallel bars. Importantly, the participants were not told which controller was expected to perform better during the experiment. Following this overground acclimation, five trials with each controller were conducted on an in- ground treadmill (Bertec, Columbus, Ohio, USA). For safety, instrumented handrails were provided on either side of the treadmill. The participants were encouraged to limit body weight support on the handrails to maximize the realism of the experiment, which was verified by handrail force data. Participant P1’s mean handrail usage was under 12% bodyweight and participant P2 frequently chose to use only one handrail (FIGS.5C and 5D). The first three trials investigated the performance of the HKIC and the FSMC during steady walking at different task condition (speed and incline) combinations. Each trial focused on a range of small task condition deviations (±2 degrees, ±0.2 m / s) around one of three baseline sets of task conditions: χ = (0 degrees, 1 m / s), χ = (5 degrees, 1 m / s), and χ = (-5 degrees, 1 m / s). These steady- state task condition trials are referred to as SS Level, SS Incline, and SS Decline, respectively. For the SS-Incline trial, speed was limited to 1.1 m / s to ensure that the participants could safely perform the trial. The steady-state task condition trials began with an acclimation period, where the participants walked at the baseline task conditions until feeling comfortable. During this time, the FSMC was tuned to produce a natural gait, incorporating feedback from the participants and the prosthetist. Tuning continued until the researchers, prosthetist, and participant were satisfied with the resulting gait. The time required to tune the FSMC was recorded. Notably, no tuning was performed with the HKIC. After tuning and acclimation, the participants walked on the treadmill as it cycled through each of the 5 task conditions near the baseline task condition, each commanded for 45 seconds. In these trials, true task condition feedback was provided to the controllers so that any errors in the task condition estimates did not affect the results. The tuning, acclimation and testing procedure above was repeated for each baseline task condition. These baseline task conditions were chosen to be far apart in the task condition space in order to sample a wide range of task conditions without deviating too far from any one of the FSMC’s tuning points. FIG.6A shows the recorded task condition space profiles from the treadmill for each trial, where the black dots indicate each commanded task condition. The SS Level trials proceeded through the following sequence of task condition combinations: 0° at 1 m / s, 2° at 1 m / s, 0° at 0.8 m / s, -2° at 1 m / s, and 0° at 1.2 m / s. The SS Incline trials proceeded through the following sequence of task condition combinations: 5° at 1 m / s, 7° at 1 m / s, 5° at 0.8 m / s, 3° at 1 m / s, and 5° at 1.1 m / s. The SS Decline trials proceeded through the following sequence of task condition combinations: -5° at 1 m / s, -7° at 1 m / s, -5° at 0.8 m / s, -3° at 1 m / s, and -5° at 1.2 m / s. The latter two trials consisted of more rapid task condition changes to investigate each controller’s behavior during continuous task condition variations rather than at steady-state and over a wider range of task conditions. Also, during these trials, the controllers received no real-time knowledge of the task conditions from the treadmill, investigating the autonomous capability of each controller to operate over variable task conditions. Both controllers utilized the same task condition estimation methods described above. The FSMC transitioned between the tuned impedance parameters sets based on the estimated incline (see FIG.17B). In these two trials, one with inclines (CV Incline) and the other with declines (CV Decline), the treadmill started at χ = (0 degrees, 1 m / s) and explored eight other points within the task condition space in the range of [0, 8] degrees and [0.6, 1.2] m / s. Each task condition point was commanded to the treadmill for 20 seconds. Because the treadmill required time to change task conditions, smooth task condition trajectories with continuous variations were generated, as shown in FIG.6B. B. Experimental Results 1) FSMC Tuning Time: For the two participants, the FSMC required on average 30 minutes of tuning to produce normative gaits for the three baseline task conditions. On average, the level ground task condition required 11 minutes of tuning, the incline task condition required 15 minutes of tuning, and the decline task condition required 5 minutes of tuning. Participant-specific tuning times and tuned FSM parameters are listed in Table IV of FIG.18. Trends in the tuned parameters included higher stiffness values during stance than in swing and highly varying knee equilibrium angles across task conditions. The observed gait was also noted to be quite sensitive to the tunable FSM transition criteria. A significant variance in the required tuning time for the different task conditions was also observed. 2) Steady-State Trials: The kinematic and kinetic trajectories produced by the HKIC during the steady-state trials highlight its ability to reproduce normative biomechanics over variable task conditions, as shown in FIGS.7A and 7B. FIG.7A illustrates results with variable inclines (-7 to 7 degrees) at 1 m / s, and FIG.7B illustrates results with variable speeds (0.8 to 1.2 m / s) on level ground. Able-bodied trajectories from the literature are also shown for reference. The HKIC produced smooth kinematic variations with incline changes as well as increasing knee flexion and ankle push-off torque with increased speed, resembling the able-bodied trajectories. Bilinear interpolation was used to generate the able-bodied reference trajectories for task conditions between those reported in the dataset. The observed HKIC trajectories show strong similarity to the able-bodied references, particularly at the ankle joint. Knee moments are the most different relative to able-bodied for both the HKIC and the FSMC. The separation and trends seen in the HKIC closely resemble those observed in the able-bodied data, suggesting appropriate adaptation in response to variable-task condition walking. The similarity between the observed and able-bodied trajectories during stance and swing is quantified in FIGS. 8A and 8B, indicating that the HKIC produced a low RMSE in most metrics. FIGS. 8A and 8B illustrate inter-participant RMSE in the observed kinematics and kinetics, respectively, relative to able-bodied walking data for both the HKIC and FSMC during the steady- state task condition trials. The error bars represent ±1 standard deviation over lumped participant strides. The HKIC demonstrated lower mean error than the FSMC in 7 of 8 metrics, with particular improvements at the ankle joint. Stance and swing were treated separately to isolate the performance of the novel impedance parameter model, as it was only used during stance phase. The first 15 seconds at each set of task conditions were neglected to allow time for the treadmill to reach steady-state. Unless otherwise specified, inter-participant averages are presented, and standard deviations are calculated using lumped participant strides. Individual RMSE values for each participant were similar to the inter- participant averages and are available in Table V of FIG.18. The low RMSE values suggest that, in addition to replicating normative trends as task conditions varied, the HKIC produced kinematics and kinetics that were close to the reference values. Further, the HKIC’s performance was as good as or better than the hand-tuned FSMC’s performance in seven of the eight metrics. The high knee kinematic error during swing can be attributed to the intentional early knee extension meant to improve user confidence and did not result in adverse gait effects. Inter-participant spatiotemporal gait metrics also showed similarity to able-bodied data. Both controllers elicited lower cadence gaits (equivalently longer stride length gaits) compared to able- bodied but show generally similar trends of increasing cadence with walking speed. FIGS.9A-9C illustrate inter-participant average cadence for the steady-state task condition trials as functions of speed for different ramp inclinations, including ramp descent (-5 deg.), level ground (0 deg.), and ramp ascent (5 deg.). Error bars represent ±1 standard deviation over lumped participant strides. Both controllers show similar cadence trends as the able-bodied (AB) reference calculated from the literature, with increasing step frequency with increasing speed. Overall, the participants preferred longer strides relative to able-bodied, which may be due to the larger mass of the powered prosthesis. Additionally, the stance time symmetry ratio ^^ௌ்ௌwas calculated using the ground reaction force data and is defined as the ratio between the average prosthetic stance time and the contralateral limb stance time. The mean and standard deviation over the steady-state trials of both participants were ^^ௌ்ௌ= 0.902 ± 0.017 for the HKIC and ^^ௌ்ௌ= 0.892 ± 0.016 for the FSMC. Both controllers produced a slightly more symmetric gait than average AKA participants with passive prostheses ( ^^ௌ்ௌ= 0.784, reported in the literature), but less symmetric gaits than able-bodied people ( ^^ௌ்ௌ= 1.02, reported in the literature). The HKIC also produced trends in joint work across variable task conditions that were consistent with able-bodied data. FIGS.10A-10C illustrate inter-participant average prosthesis work per stride over variable inclines, and FIGS. 10A-10C illustrate inter-participant average prosthesis work per stride over variable speeds during the steady-state task condition trials. Error bars represent ±1 standard deviation over lumped participant strides. An able-bodied reference (AB) calculated from the literature shows that the HKIC demonstrated biomimetic energy injection, particularly through a linear increase in ankle work as incline increased, corresponding to 100.6% of the able-bodied rate. Both controllers showed less energy absorption at the knee during steep declines, suggesting that the participants may have had habitual aversions to early stance knee flexion. One of the benefits of impedance control is the ability to control energy exchange with the environment, the HKIC should be able to replicate this biological behavior. The HKIC showed similar trends as the able-bodied data, with a linear increase in net work performed with increasing incline, particularly at the ankle (increase of 0.0337 J / kg / deg, R2= 0.982). For comparison, able-bodied ankle work increases linearly at 0.0335 J / kg / deg with R2= 0.987. The HKIC also increased total work with increasing speed in a manner consistent with able-bodied data, though the work differences between the slow and fast speeds are minor for the able-bodied reference. In contrast, the net work performed by the FSMC decreased with speed and appeared discretized to three levels with respect to inclines, corresponding to its tuned task conditions. Interestingly, the HKIC and FSMC showed less energy absorption at the knee during declines, which may reflect the habitual aversion to early stance knee flexion commonly observed in AKA populations. 3) Continuously Varying Trials: The continuously varying trials demonstrated the HKIC’s ability to autonomously adapt behavior to the sensed walking speed and ground incline. The kinematic and kinetic errors were calculated in a similar manner for the continuously varying task condition trials, though this time including strides that occurred during task condition transients. FIG. 12 includes plots of the inter-participant average kinematic and kinetic error trajectories in the continuously varying incline trial relative to able-bodied data. The knee data is shown on the left of FIG.12, and the ankle data is shown on the right of FIG.12. Shaded regions represent ± 1 standard deviation over lumped participant strides. Aside from intentional discrepancies in the late-swing knee kinematics, the HKIC showed low RMSE across the gait cycle throughout varying task conditions, suggesting appropriately adapting biomechanics. Table V of FIG.18 details the participant-specific stance and swing kinematic and kinetic RMSE for both the CV-Incline and CV-Decline trials. The magnitude of the FSMC error is larger than the HKIC error for most of the gait cycle, highlighting the improvement provided by the HKIC’s continuously adaptive nature. As both controllers received no external task condition input during these continuously varying trials, the task condition estimates (and the phase estimate for the HKIC) contributed to the kinematic and kinetic errors. The task condition estimate RMSE, averaged over each stride and participant, is shown in TABLE II for each trial. TABLE II Trial Controller Incline (deg) Speed (m / s) ers, FSMC showed higher incline estimate error, suggesting that differences in controller behavior may have impacted the incline estimate’s efficacy. Additionally, the average phase estimate trajectories produced by HKIC during the CV-Incline and CV-Decline trials were highly linear (mean R2= 0.989) and accurate (mean RMSE of 6.157%), even as speed and incline varied. FIG.13 illustrates average phase estimate progression calculated in real-time by the HKIC during the continuously varying task condition trials for participants P1 and P2. Shaded regions represent ± 1 standard deviation. The linearity and consistency of the trajectories illustrate the phase variable’s ability to adapt to continuous task condition variations and appropriately parameterize the gait cycle. However, the phase estimate saturated more often for participant P2 than participant P1, suggesting that participant P2’s thigh trajectory was less similar to able-bodied trajectories than participant P1’s. Discussion A. HKIC Performance This disclosure presents a data-driven, phase-based walking controller for a powered knee- ankle prosthesis that autonomously adapts its behavior across a continuous range of walking speeds and inclines. To achieve this without manual impedance tuning, an able-bodied dataset was used to optimize for continuous stiffness, damping, and equilibrium angle functions that reproduced biological stance joint torques, given biological kinematics. In an initial offline analysis, it was demonstrated that the optimized impedance parameter model produced joint torques with across-task condition average normalized RMSE values of 0.78 and 0.58 for the knee and ankle, respectively. The low normalized RMSE suggests that the model captures the essential joint dynamics of able- bodied walking. The subsequent experiments with two AKA participants demonstrated that the identified impedance parameter functions also rendered appropriate stance phase joint mechanics when used for real-time control in the HKIC. Other normative walking features were observed, such as increasing ankle work with increasing incline (FIGS. 10A-11C) and increasing cadence with walking speed (FIGS.9A-9C). Although the kinematic and kinetic profiles produced by the HKIC had small differences relative to able-bodied data (FIGS. 7A-8B), the participants exhibited qualitatively normal gait patterns over a wide array of task conditions. Kinematic and kinetic trends emerged with variable speeds and inclines that were consistent with able-bodied data (FIGS.7A-7B), including appropriately varying peak ankle moments, stance ankle kinematics, and knee stance kinematics. Knee swing kinematics showed the highest error for the HKIC, which was expected because the phase variable was intentionally allowed to saturate early to ensure full knee extension prior to heelstrike. Pilot testing showed that consistent full knee extension helped eliminate participants’ problematic instinctive compensations and promoted confidence that the prosthesis was ready to accept weight. Small phase shifts result in large swing kinematic errors due to the large knee range of motion, and although the error values appear large, they did not interfere with the participants’ gait or cause toe-stubbing. Appropriate kinematic and kinetic adaptation are both practically and clinically important for the user. For example, knee swing kinematic adaptations enable the prosthesis to have the proper configuration at heelstrike as incline varies. Without such adaptations, the user may, for example, toe- stub during swing when walking uphill with level ground kinematics, or vice-versa, experience too much flexion to enable heelstrike at the desired time. Further, kinetic adaptation during stance enables increasing peak ankle moments for propulsion as incline and speed increase (FIGS.7A-7B). Improper joint kinetics can cause improper ground reaction forces, which can affect user balance. Finally, appropriate kinematic and kinetic co-adaptation enables joint work adaptation, even in cases where both kinematics and kinetics deviate from able-bodied normative trajectories. For example, the HKIC’s peak ankle moment at a 7 deg incline is slightly smaller than able-bodied. However, a corresponding increase in peak plantarflexion angle allows the HKIC to maintain appropriate ankle work. Biomimetic energy injection is important to prevent compensations from other joints and additional health problems. Qualitative remarks by the participants also testified to the biomimetic task condition adaptation of the HKIC. Participant P1 remarked while walking at the seven-degree incline that he did not feel like he was walking uphill, suggesting appropriate joint dynamics and energy exchange. Participant P2 remarked that he did not even notice that the treadmill had transitioned to the two- degree decline and that he could “climb up much easier” while ascending steep inclines. These anecdotal remarks further support the claim that the HKIC adapts to changing task conditions to produce normative able-bodied biomechanics, which could result in many practical benefits for the user. For example, the biomimetic energy injection at steep inclines may allow users to walk uphill for longer before fatiguing. While the FSMC’s performance was not drastically worse than the HKIC’s in the tested metrics, it required on average 10 minutes of tuning per tuned task condition combination. Although only 3 task conditions were tuned for this study, practical deployment of the FSMC would likely require many more sets of task conditions to be tuned. For example, participant P2 noted that the FSMC was “kicking off way too hard” when going uphill at slow speeds, but was happy with its behavior at normal speeds, suggesting that more speed-specific impedance parameter sets could be beneficial. However, adding more tuning points is likely impractical in a clinical setting, especially without specialized equipment such as a variable-incline treadmill. Therefore, the HKIC’s potential to produce biomimetic behavior over varying task conditions without manual impedance tuning is a significant benefit. For online implementation of the continuous impedance parameter model, gait phase needed to be estimated in real time. The improved phase variable behavior observed in simulation was confirmed in the participant experiments. FIG. 13 shows how the phase variable eliminated the previously observed phase pause near push-off. The result of this monotonicity is visible in the kinematics of FIGS.7A-7B, as there is not a pause in the kinematic trajectories near push-off as has been reported previously in the literature. Further, the general linearity of the average phase trajectories in FIG. 13 (mean R2= 0.989) is improved compared to those reported in the literature. Because both the impedance and kinematic models in HKIC assume a perfectly linear phase estimate, the observed linearity keeps the model outputs of the controller synchronized with the user’s gait. Additionally, FIG.13 shows that participant P2’s phase variable saturates earlier in the gait cycle than participant P1. This occurs because the methods used to estimate the thigh trajectory features prioritize phase saturation (and subsequently full knee extension) to promote participant confidence. This early phase saturation suggests that P2 preferred for the knee to be fully extended earlier in swing, whereas P1 was satisfied with full knee extension occurring right before heelstrike, as it does in able-bodied data. The task condition estimates are important for walking over continuously-varying task conditions. As shown in Table II, the error in the speed estimate was fairly consistent over the trials, with RMSE between 0.10 and 0.12 m / s for both controllers. This error is likely due to a slightly asymmetric gait, which violates the assumptions made in the speed estimator’s formulation. Gait asymmetries may be the result of our participants’ habitual compensations, socket comfort, or the significant mass difference between the robotic prosthesis and participants’ passive prostheses. Interestingly, the incline estimate produced lower error with the HKIC (0.61 to 0.66 degrees) than the FSMC (1.03 to 1.81 deg). The higher error in the FSMC may be due to a feedback interaction between incline estimate errors and the impedance parameters. Due to the discrete switching behavior of the impedance parameters (FIG. 17B), a small incline estimate error can result in large changes in prosthesis behavior and may affect the θfand lcopprogressions. Therefore, the continuous nature of the HKIC may be preferable, as it does not display discrete changes in behavior with small changes in task condition inputs. Appendix A. Task Condition-Invariant Phase Variable Algorithm The new phase variable ^^̂ is calculated through a series of linear equations with ^^௧^as an input. An FSM controls when each equation is used. Although the FSM contains discrete states, the structure of the linear equations ensures that ^^̂ is continuous. Each equation is defined by quantitative features of the ^^௧^trajectory, which are measured in real-time. Table III lists the features’ definitions and notations. TABLE III Symbol Definitions 2 4 Δ ^^ Time since state transition ^^ Ti i hi i First, the rationale for ea ble equation is given. Then, methods to estimate the thigh trajectory features in real-time, as well as the steps taken to promote closed-loop stability of the phase estimate, are presented. 1) Phase Variable FSM: Consider the average ^^௧^trajectory for an able-bodied individual walking at 1 m / s on level ground, shown in FIG.14. The pertinent ^^௧^trajectory features used in the phase estimate are labeled, as well as the standard timing of the FSM states. The overall structure of the FSM used to control the phase estimate is shown in FIG.15. The FSM begins in S1, occurring just after a heelstrike (HS) event. During S1, ^^௧^is linearly scaled as the hip joint extends from ^^௧ு^ௌto ^^௧ெ^ுாsuch that ^^̂ increases and ^^̂ = ^^ெுாwhen ^^௧^= ^^௧ெ^ுா. Mathematically, this is given by ఏಹೄ^^̂ ൌ^^ ିఏ^^in S1, S2. (22) The FSM transitions to S2 at a ൌ to the point in the gait cycle where the θth trajectory In S2, ^^̂ is calculated using the same linear relationship as in S1 but is denoted as a distinct state because it represents a portion of the gait cycle where ^^௧^(and therefore ^^̂) has constant velocity. The average rate of change of ^^̂ for use in S3. The FSM transitions to S3 once ^^̂ଶ→ଷൌ 0.9 ^^ெுா, which typically corresponds to the end of the linear portion of the thigh trajectory, or if ^^^௧^^ 0. This second case rarely occurs during steady walking, but is an important path to S3 in the event of an unusually short stride. S3 occurs during the section of the gait cycle where ^^௧^reaches its minimum, and thus has aperiod of low angular velocity ^^^௧^. Previous work has shown that sections of low ^^^௧^ are problematicbecause they cause a pause in the phase variable trajectory. This pause violates the assumption that ^^̂ increases monotonically and at a constant rate, resulting in incorrect kinematic and impedance model outputs. Therefore, during S3, ^^̂ is decoupled from ^^௧^and it is instead assumed that phase continuesprogressing at ^^̂^ௌଶ:^^̂ ൌ ^^̂^௧ଶଷ ^ ^^ ^^̂^ௌଶ ^^ ^^ in S3. (23)This feedforward phase limits the user’s ability to stop phase progression during S3, such cases are unlikely because stopping would inhibit power delivery from the ankle during push-off. Moreover, the under-actuated dynamics of bipedal walking dictate that once the user’s gravity vector passes anterior of the stance foot, the user must continue the gait cycle until the contralateral foot lands. Therefore, the sacrifice in direct control of phase progression during this section of the gait cycle is expected to be negligible. After TO, the FSM transitions to S4, where phase is again estimated via a linear scaling of ^^௧^. This mapping is defined such that ^^̂ increases from ^^̂TO towards ^^ெுிas ^^௧^increases: ^^̂ ൌఏ ^ೀ^^ିఏ^^ఏಾಹಷିఏ^ೀ^^^ெுி െ ^^்̂ை^^ ^^்̂ை in S4. (24)The FSM transitions to corresponds to the end of this linear section of the thigh trajectory. Two problems typically occurred with previous phase variable methods when ^^௧^^ ^^௧ெ^ுி,which occurs during S5 in the new FSM. First, a pause in ^^̂ would occur as ^^^௧^ slowed and ^^௧^approached ^^௧ெ^ுி, similar to the effect seen in S3. Second, the previous methods assumed that ^^௧ெ^ுி= ^^௧ு^ௌ. In cases where ^^௧ெ^ுி> ^^௧ு^ௌ, such as the trajectory shown in FIG.14, the resulting ^^̂ would prematurely. Excessive saturation in the phase variable can cause desynchronization between the prosthesis and the user, leading to problems such as toe-stubbing. This effect was most exaggerated during declined walking, as the difference between ^^௧ெ^ுிand ^^௧ு^ௌwas most pronounced. To avoid both excessive saturation and a phase variable pause, a feedforward phase progression isagain enforced based on the average phase rate in S4, ^^̂^ௌସ:^^̂ ൌ ^^̂ସହ ^ ^^௧^^^̂ௌ^ସ ^^ ^^ in S5. (25)This feedforward phase rate continues until either a heelstrike occurs or ^^̂ = 1. If the user is walking consistently and the ^^௧^trajectory feature estimates are correct, ^^̂ = 1 should occur simultaneously with heelstrike, returning the FSM to S1. If ^^̂ = 1 prior to HS, the FSM transitions to S6. S6 is primarily encountered if the user pauses at the end of the gait cycle, so it does not appear in FIG.14. During S6, ^^̂ is again calculated using a linear scaling of ^^௧^, giving the user volitional control of ^^̂ through ^^௧^: ^^̂ ൌఏ ^ିఏಾಹಶ^ ^^ఏಹೄ ಾಹಶ^^ ିఏ^^^1 െ ^^ெுா^ ^ ^^ெுாin S6. (26) This volitional control during S6 is important because it allows movements such as kicking and non- steady leg swinging. As in S5, a HS event returns the FSM to S1. 2) Thigh Trajectory Feature Prediction: The ^^௧^features used in equations (22)-(26) vary from stride-to-stride with changes in speed, incline, and natural gait variation. Some of these features are used in the phase estimate calculation before they occur in the gait cycle, specifically ^^௧ு^ௌ, ^^௧ெ^ுா, ^^௧ெ^ுி, ^^ெுா, and ^^ெுி. For example, ^^௧ெ^ுாis used to calculate ^^̂ during S1 and S2, but it does not typically occur until S3. Therefore, we predict these features in real-time based on observations from recent strides. At controller initialization, estimates of the thigh trajectory features are calculated using able-bodied data and updated as new strides became available. Bounds are enforced on all estimated feature values to reject atypical strides and avoid stride-to-stride oscillation in the estimates. Previous work showed that care must be exercised when predicting features of the thigh trajectory to prevent unwanted interaction between the prediction algorithms and the user’s gait progression. For example, it has been observed that if a simple moving average is used to calculate ^^௧ெ^ுா, a divergent behavior occurs that results in the user taking progressively larger strides. To avoid this behavior, the kinematic features ^^௧ு^ௌ, ^^௧ெ^ுா, and ^^௧ெ^ுிare estimated with moving average filters. These filters record the feature the previous five strides and average the median three for ^^௧ு^ௌand the minimum three for ^^௧ெ^ுாand ^^௧ெ^ுி. These filters were chosen to best reject non- representative strides, and the five-stride window balanced between filter response time and variance rejection. Another closed-loop interaction was observed during pilot studies regarding the predictions of ^^ெுாand ^^ெுி. In cases when the feature predictors were updating following a rapid change in task conditions, rare strides were observed where ^^̂ underestimated the true phase at the end of the gait cycle, causing the knee joint to not fully extend before heelstrike. Participants instinctively responded by asymmetrically extending the late swing portion of the gait cycle to try to force the knee to full extension. Moving average estimates of ^^ெுாand ^^ெுி, like those used for the kinematic features, caused of ^^ெுாand ^^ெுிto decrease, resulting in further underestimation of ^^̂ on the subsequent stride. It is suspected that participants behaved this way because they were accustomed to passive prostheses, which will collapse upon loading if the knee is not fully extended. Therefore, new prediction methods were developed for ^^ெுாand ^^ெுிthat favored ^^̂ saturation over underestimation to combat this instinctive behavior. Let ^^^̂ୀ^be the first time during the stride that ^^̂ = 1. Then, the ^^ெுாand ^^ெுிestimates were calculated as ௧ ି௧ ௧ ି௧ ^^ெுாൌ^ଶ ൬ಾಹಶ బ௧^ି௧బ^ಾಹಶ బ௧ೞ^సభି௧^, బ ) The first quot in each line second quotient is an upper bound on this true phase. We average the two so favors saturation and full knee extension in late swing, avoiding the potential unstable feedback loop with the user’s instinctive compensations. The results of equation (27) were likewise low-pass filtered with an infinite impulse response (IIR) filter to reject stride-to-stride variation and to prevent step estimate changes. Finally, to calculate the stance phase ^^̂^௧, the expected value of ^^̂ at TO, ^̅^்̂ை, must be estimated. This was calculated with a minimum moving average filter of ^^்̂ைobserved during previous strides, similar to the thigh trajectory features. The average window was nine strides long, as the toe-off phase exhibits slow changes with task conditions. Like the thigh trajectory features, ^^்̂ைwas initialized from able-bodied data. Note that some minor aspects of the thigh trajectory feature estimation algorithms were modified after P1’s experiment to better accommodate adaptation for users with thigh kinematics that differ significantly from able-bodied, such as P2. Namely, the feature estimate bounds were added, the ^^௧ெ^ுாand ^^௧ெ^ுிfilters were changed from moving average to moving minimum filters, and 5 the filter was changed from an IIR filter to a moving minimum filter. A post-hoc simulation of P1’s data before and after the minor adjustments showed only a 1.89% mean absolute difference in phase estimate between methods, suggesting that the changes would not have had an appreciable effect on his results. Further, no distinguishable effects were observed in the able-bodied simulation (FIG.4C). 3) Phase Variable Linearization: The phase variable described above produces a consistent phase estimate trajectory over each stride during steady walking. This consistency allows a linearization map to be formed in order to further improve the phase estimate. Once the ^^௧^feature predictions converged to steady values, the average progression of ^^̂ was recorded for each steady walking stride and low-pass filtered to produce an average trajectory, ^^̂. The time constant of the IIR low-pass filter was chosen to be sufficiently slow (19 strides) such that the transients of the ^^௧^feature predictors were rejected. As a further precaution, any saturated portions of ^^̂ were discarded prior to averaging, as they diminish as the ^^௧^trajectory feature predictions converge. The average phase was written as a function of true phase, given by ^^̂ ൌ ^^^ ^^^. Although the shape of the thigh trajectory may cause ^^^ ^^^ to be nonlinear, it is monotonic during normal walking. This implies that an inverse relationship ^^ ൌ ^^ି^^ ^̅^̂^ exists, which can be applied to correct for nonlinearities in ^^̂. First, ^^^ ^^^ was fit with a 6th order polynomial ^ത^^ ^^^ that was constrained with a minimum slope of 0.2. This minimum slope ensured strict monotonicity and numerical stability of the inverse. At each HS event, ^ത^^ ^^^ was recalculated to incorporate the previous stride’s effect on ^^̂. Then the final, linearized phase estimate was calculated by applying the inverse map ^ത^ି^to the results of equations (22)-(26). 4) Phase Variable Results: FIG.13 highlights the new phase variable algorithm’s ability to parameterize variable-incline walking, as consistent phase trajectories were produced for both participants during the continuously varying task condition trials. The feedforward states S3 and S5 allowed for a positive phase rate, even when thigh velocity was low. Additionally, the thigh trajectory features were appropriately estimated, allowing consistent phase estimates that were independent of the variable thigh trajectories seen with varying inclines. The phase linearization algorithm ensured highly linear estimates, with mean R2= 0.997 for participant P1 and mean R2= 0.982 for participant P2. Finally, the volitional start / stop behavior of the phase variable was preserved. FIG.16 shows the phase variable trajectory for four non-steady bouts during the overground acclimation for participant P1, confirming its ability to parameterize non-rhythmic motion. B. Comparative FSM Impedance Controller The Finite State Machine controller (FSMC) used in the above experiments was constructed to provide a benchmark with which to compare the HKIC. The flow of the FSMC’s state machine is depicted in FIG.17. A tunable center of pressure threshold, ^^^∗^^, controlled the transition from S1 to S2. Then, after a tunable duration, t2^3, the FSM transitioned to S3. Next, a TO event triggered the transition to S4. Finally, knee extension ( ^^^^< 0) caused a transition to S5, where the FSM remained until returning to S1 at HS. the impedance parameters were rate-limited to prevent step changes in torque. In the FSMC, the torque command was given by equation (1), where K, B, and θeq depended on the current FSM state (given in Table IV of FIG.18). Many methods have been proposed for deciding when to switch between sets of impedance parameters for different task conditions, including simple threshold methods and more complex machine learning methods. We employed a strategy where the prosthesis directly estimated the ground incline as described above. Then a secondary FSM was used to select between parameter sets based on the estimated incline ^^^. To prevent rapid switching between parameters at the boundaries, overlap was included in the switching thresholds (FIG.17B). C. Additional Detailed Results Table IV of FIG. 18 lists the results from the impedance parameter tuning for the FSMC, including the tuned impedance parameters, transition thresholds, and tuning times. The stiffness K, damping coefficient B, and equilibrium angle ^^^^were tuned by the research team for each of the five states (S1, S2, S3, S4, and S5) at three baseline task conditions for each participant. Table V of FIG. 18 shows the kinematic and kinetic RMSE values for each participant during both the steady-state and continuously varying trials, each separated by stance and swing phases. It is to be understood that the foregoing is a description of one or more embodiments of the invention. The invention is not limited to the particular embodiment(s) disclosed herein, but rather is defined solely by the claims below. Furthermore, the statements contained in the foregoing description relate to particular embodiments and are not to be construed as limitations on the scope of the invention or on the definition of terms used in the claims, except where a term or phrase is expressly defined above. Various other embodiments and various changes and modifications to the disclosed embodiment(s) will become apparent to those skilled in the art. All such other embodiments, changes, and modifications are intended to come within the scope of the appended claims. As used in this specification and claims, the terms "e.g.," “for example,” “for instance,” “such as,” and “like,” and the verbs “comprising,” “having,” “including,” and their other verb forms, when used in conjunction with a listing of one or more components or other items, are each to be construed as open-ended, meaning that the listing is not to be considered as excluding other, additional components or items. Other terms are to be construed using their broadest reasonable meaning unless they are used in a context that requires a different interpretation.

Claims

CLAIMS 1. A powered prosthesis comprising a joint and a controller configured to vary torque at the joint as a function of one or more impedance parameters, wherein at least one impedance parameter varies during a stance phase of walking as a function of stance phase progression and at least one walking task condition.

2. The powered prosthesis of claim 1, wherein the impedance parameters include joint stiffness, damping coefficient, and equilibrium angle.

3. The powered prosthesis of claim 2, wherein each impedance parameter varies during the stance phase of walking as a function of stance phase progression and the at least one walking task condition.

4. The powered prosthesis of claim 1, wherein each impedance parameter function is a continuous function.

5. The powered prosthesis of claim 1, wherein each impedance parameter function is a continuous function over the entire stance phase.

6. The powered prosthesis of claim 1, wherein each impedance parameter function is non-linear.

7. The powered prosthesis of claim 1, wherein the at least one walking task condition includes walking speed or incline.

8. The powered prosthesis of claim 1, wherein the at least one walking task condition includes walking speed and incline.

9. The powered prosthesis of claim 1, wherein the controller automatically adapts to changes in the at least one walking task condition based on real-time estimates of each walking task condition.

10. The powered prosthesis of claim 9, wherein the at least one walking task condition includeswalking speed and / or incline.

11. The powered prosthesis of claim 1, wherein the controller estimates stance phase progression in real-time based on a walking task condition-invariant phase variable.

12. The powered prosthesis of claim 1, wherein the controller estimates a user gait phase based on the at least one walking task condition.

13. The powered prosthesis of claim 12, wherein the gait phase estimates adapt to changes in the at least one walking task condition.

14. The powered prosthesis of claim 1, wherein each impedance parameter function is based on able-bodied data for each impedance parameter at a plurality of combinations of walking task conditions.

15. The powered prosthesis of claim 14, wherein the able-bodied data includes non-steady-state data.

16. The powered prosthesis of claim 15, wherein the non-steady-state data includes perturbation responses or biological impedance estimates.

17. The powered prosthesis of claim 14, wherein the combinations of walking task conditions include combinations of discrete walking speeds and inclines selected from ranges of walking speeds and inclines.

18. The powered prosthesis of claim 1, wherein the controller is a hybrid kinematic and impedance controller additionally configured to vary joint angle as a continuous function of swing phase progression.

19. The powered prosthesis of claim 1, wherein said joint is a knee joint and the prosthesis further comprises an ankle joint, whereby the prosthesis is a knee-ankle prosthesis, the controller being configured to vary torque at the ankle joint as a function of each impedance parameter.

20. A knee-ankle prosthesis according to claim 19, further comprising: an upper leg member adapted for attachment to a user; a lower leg member coupled with the upper leg member at the knee joint; a foot member coupled with the lower leg member at the ankle joint; and first and second actuators for providing variable torque at the respective knee and ankle joints according to the impedance parameter functions.

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