FUNCTIONAL ELECTRICAL STIMULATION CYCLING DEVICE FOR PEOPLE WITH IMPAIRED MOBILITY

[0001] The present application claims priority under 35 U.S.C. §119(e) to U.S. Provisional Application Ser. No. 62/008,402, filed Jun. 5, 2014, which is hereby incorporated by reference in its entirety for all purposes.

[0002] Research leading to various aspects of the present invention was sponsored, at least in part, by the National Science Foundation under Grants No. CMMI-1161260 and CMMI-0547448, and by the Department of Defense, Air Force Office of Scientific Research, under Contract No. FA9550-11-C-0028. The U.S. Government has certain rights in the invention.

[0003] The present invention generally relates to functional electrical stimulation devices and associated methods, and, in particular, to functional electrical stimulation cycling devices.

[0004] Upper motor neuron lesions (UMNL), which may be caused by neural disorders such as stroke, cerebrovascular accident, spinal cord injury, cerebral palsy, and/or traumatic brain injury, cause disability and paralysis in millions of people. Since the lower motor neuron system and muscles are generally intact in those with UMNL, muscle contractions may be evoked by directly applying electrical stimulus to the muscles via one or more electrodes. This technique, which may be used for rehabilitation and restoration of motor function, may be referred to as neuromuscular electrical stimulation (NMES). It may also be referred to as functional electrical stimulation (FES) when it is applied to produce a functional outcome, such as standing, cycling, and/or walking.

[0005] A number of challenges have been associated with development of NMES devices and methods, including the nonlinear response of muscle to electrical stimulation, load changes during functional movement, unexpected muscle spasticity, time lag between electrical stimulation and muscle force output, uncertainties in muscle physiology (e.g., temperature, pH, and/or architecture), and muscle fatigue.

[0006] One particular application of FES is FES-induced cycling. Since the 1980s, FES-induced cycling has been investigated as a safe means of exercise and rehabilitation for people with lower-limb paresis or paralysis, and numerous physiological benefits (e.g., improved cardiovascular health, increased muscle mass and bone mineral density, decreased spasticity, lower limb function) and psychological benefits (e.g., improved self-image, independence, socialization) have been reported. However, despite the reported benefits, FES-induced cycling generally has been metabolically inefficient and has resulted in low power output compared to volitional cycling by able-bodied individuals.

[0007] Improved devices and methods for application of electrical stimulation to a human body to produce functional outcomes, such as cycling, would be desirable.

[0008] Embodiments described herein generally relate to providing functional electrical stimulation. In some embodiments, a method comprises delivering a first amount of electrical stimulation to at least a portion of a first set of quadriceps femoris muscles on a first leg of the person, wherein the first amount of electrical stimulation is delivered when a first torque transfer ratio between a knee of the first leg and a crank of a cycling device is negative. In certain cases, the method comprises delivering a second amount of electrical stimulation to at least a portion of a second set of quadriceps femoris muscles on a second leg of a person, wherein the second amount of electrical stimulation is delivered when a second torque transfer ratio between a knee of the second leg and the crank is negative. In some cases, the electrical stimulation causes the person to pedal the cycling device.

[0009] In some embodiments, an exercise device comprises: a crank, configured for rotation by at least one limb of a human; at least one sensor adapted to measure position and velocity of the crank; and a controller, coupled to the at least one sensor. In certain cases, the controller is programmed to generate functional electrical stimulation to a person using the exercise device, by: for each of at least one human limb, determining a torque transfer ratio between a portion of the human limb and the crank based on position and velocity of the crank measured by the at least one sensor; and, for each of the at least one human limbs, dynamically generating a control signal to control delivery of electrical stimulation to the at least one human limb based on the determined torque transfer ratio.

[0010] In some embodiments, a method of providing functional electrical stimulation to a person comprises delivering a first amount of electrical stimulation to at least a portion of a first set of gluteal muscles on a first leg of the person, wherein the first amount of electrical stimulation is delivered when a first torque transfer ratio between the hip of the person and a crank of a cycling device is positive. In certain cases, the method further comprises delivering a second amount of electrical stimulation to at least a portion of a second set of gluteal muscles on a second leg of the person, wherein the second amount of electrical stimulation is delivered when a second torque transfer ratio between the hip of the person and the crank is positive. In some cases, the electrical stimulation causes the person to pedal the cycling device.

[0011] In certain embodiments, a method of providing functional electrical stimulation to a person comprises delivering a first amount of electrical stimulation to at least a portion of a first set of hamstring muscles on a first leg of the person, wherein the first amount of electrical stimulation is delivered when a first torque transfer ratio between a knee of the first leg and a crank of a cycling device is positive. In some cases, the method further comprises delivering a second amount of electrical stimulation to at least a portion of a second set of hamstring muscles on a second leg of the person, wherein the second amount of electrical stimulation is delivered when a second torque transfer ratio between a knee of the second leg and the crank is positive. In some cases, the electrical stimulation causes the person to pedal the cycling device.

[0012] According to some embodiments, a method of providing functional electrical stimulation to a person comprises delivering a first amount of electrical stimulation to at least a portion of a first set of muscles on a first leg of the person for a first amount of time. In certain cases, the method further comprises delivering a second amount of electrical stimulation to at least a portion of a second set of muscles on a second leg of the person for a second amount of time. In certain embodiments, the method further comprises providing an amount of power to a crank of a cycling device for a third amount of time. In some cases, the first amount of time is during a first part of a trajectory of the crank, the second amount of time is during a second part of the trajectory of the crank, and the third amount of time is during a third part of the trajectory of the crank. In certain cases, at least a portion of the first, second, and third parts of the trajectory of the crank do not overlap. In some embodiments, the electrical stimulation causes the person to pedal the cycling device.

[0013] The subject matter of the present invention involves, in some cases, interrelated products, alternative solutions to a particular problem, and/or a plurality of different uses of one or more systems and/or articles.

[0014] Other advantages and novel features of the present invention will become apparent from the following detailed description of various non-limiting embodiments of the invention when considered in conjunction with the accompanying figures. In cases where the present specification and a document incorporated by reference include conflicting and/or inconsistent disclosure, the present specification shall control. If two or more documents incorporated by reference include conflicting and/or inconsistent disclosure with respect to each other, then the document having the later effective date shall control.

[0040] Hybrid orthotic devices configured to apply electrical stimulation to one or more body parts of a person are generally described. Control of hybrid orthotic devices to provide exercise and rehabilitation of injured limbs is also described, as are control techniques suitable for controlling an orthotic device to aid a wearer perform an activity such as cycling.

[0041] Some aspects relate to functional electrical stimulation (FES) cycling devices and associated methods. An FES cycling device may comprise a crank, two or more pedals connected to the crank, one or more sensors adapted to measure position and/or velocity of the crank, and one or more electrodes configured to deliver electrical stimulation to a person associated with the cycling device. In some cases, the FES cycling device further comprises a controller configured to receive input signals from the one or more sensors and deliver output signals to the one or more electrodes. In certain cases, the controller may dynamically generate a control signal to deliver an amount of electrical stimulation to a muscle group (e.g., quadriceps femoris, gluteal muscles, hamstring muscles) based on the value of a determined torque transfer ratio between a joint of the person and the crank of the cycling device. The electrical stimulation may, in some cases, cause the person to pedal the cycling device.

[0042] In some embodiments, a hybrid orthotic device comprises one or more sensors, one or more electrodes, and a controller configured to receive input signals from the one or more sensors and deliver output signals to the one or more electrodes. In some embodiments, the controller may be nonlinear in that stimulus signals generated by the controller may be different in different contexts, even for the same inputs to the controller. In certain cases, the nonlinear controller may be configured to store one or more programs, which may, when executed, implement algorithms as described herein. The one or more programs may be associated, in some embodiments, with functional activities (e.g., standing, sitting, and/or walking), and the programs may be configured to enable one or more body parts of a wearer to track a desired trajectory. In some embodiments, a program may implement a Robust Integral of the Sign of the Error (RISE) control method, which will be discussed in more detail below. It may be desirable, in some cases, to implement a RISE-based control method, at least in part because such methods may achieve asymptotic stability, even in the presence of unknown, additive disturbances and parametric uncertainties in a nonlinear system. A RISE-based control method does not require knowledge of a muscle model. It may also be desirable to implement a RISE-based control method because such methods may be continuous. In some cases, a program may further implement control methods to compensate for electromechanical delay and/or muscle fatigue. In certain embodiments, a program may further implement adaptive learning methods, such as neural network (NN)-based methods.

[0043] In some embodiments, the hybrid orthotic device comprises one or more sensors. In certain cases, at least one of the sensors may be an electrogoniometer. Generally, an electrogoniometer refers to an instrument for measuring angles. In some cases, the electrogoniometer may be attached to the hybrid orthotic device so as to measure the angle between a first body part of a person and a second body part of a person, coupled at a joint, and to which the hybrid orthotic device may be attached. In certain embodiments, at least one of the sensors may be a pressure sensor. In some such embodiments, the pressure sensor may, for example, measure pressure applied by a body part (e.g., a foot) on a surface (e.g., the ground). In some embodiments, at least one of the sensors may be configured to measure velocity of a joint.

[0044] The hybrid orthotic device may comprise one or more electrodes. In some embodiments, at least one of the electrodes may be configured to apply electrical stimulation to one or more body parts of a person. For example, at least one of the electrodes may be configured to apply electrical stimulation to the skin of a person and to one or more underlying skeletal muscles. The stimulation may cause the muscles to contract, when above a level, or allow the muscle to relax when below a level. Non-limiting examples of skeletal muscles and muscle groups that may be stimulated, based on configuration of the hybrid orthotic device, include the quadriceps femoris, hamstrings, gluteal muscles, triceps surae, tibialis anterior, and biceps femoris. For example, it may be desirable to apply electrical stimulation to portions of the quadriceps femoris muscle group to induce extension of a knee as part of a functional activity, such as standing, sitting, and/or walking. In some embodiments, the electrodes may be bipolar electrodes. In certain cases, the electrodes may be self-adhesive electrodes, and may be rigidly coupled to structural members of the hybrid orthotic device and/or separately attached to a wearer of the device. In some cases, the orthotic device may comprise at least two electrodes.

[0045] The hybrid orthotic device may further comprise a controller. In some embodiments, the controller may be a nonlinear controller configured to receive input signals from one or more sensors and deliver output signals to one or more electrodes. The controller may, in certain cases, generate output signals as a nonlinear function of the input signals based at least in part on a determined parameter indicative of muscle fatigue. In some embodiments, the controller may be connected to one or more sensors via electrical leads. In some cases, the electrical leads may comprise power and/or data lines. For example, in certain embodiments, power may be transferred from the controller to the one or more sensors, and data may be transferred from the one or more sensors to the controller. In some embodiments, the controller may be connected to one or more electrodes via electrical leads. In certain cases, the electrical leads may be one-way leads. For example, in some cases, signals may be transferred from the controller to the one or more electrodes, but no data may be transferred from the electrodes to the controller. In some cases, the controller may be connected to at least two electrodes. The controller may be configured to deliver interleaved output signals to the at least two electrodes. In some embodiments, the controller may comprise inputs and outputs for connecting with a computer, for example for programming and/or data logging. The controller may, in some cases, be configured to record data from previous activities.

[0046] In some cases, the nonlinear controller may be configured to store one or more programs. The one or more programs may be associated, in some embodiments, with one or more functional activities, and the programs may be configured to enable one or more body parts of the person to track a desired trajectory. In some embodiments, a program may implement a closed-loop control method. A program may, in certain cases, implement a continuous feedback control method. In some cases, a program may implement a control method that yields asymptotic stability for a nonlinear system (e.g., muscle model) in the presence of bounded nonlinear disturbances (e.g., muscle spasticity, electromechanical delays, muscle fatigue). In some embodiments, the program may implement a control method that tracks the time-varying trajectory of at least one body part of the person. The program may, in certain cases, implement a RISE-based control method. In some cases, a program may further implement control methods to compensate for electromechanical delay and/or muscle fatigue. For example, the control method may comprise a predictor-type method to compensate for electromechanical delay. In certain embodiments, a program may further implement adaptive learning methods. For example, a program may implement a control method comprising an adaptive feedforward method. In some embodiments, the adaptive learning methods may comprise neural network (NN)-based methods.

[0047] The programs, when executed, may cause the controller to receive and process outputs of one or more sensors and, based on those outputs and/or other data, derive one or more stimulus signals to apply to one or more electrodes. The stimulus signals may have one or more characteristics that change over time, such that the stimulus applied to muscles of a wearer is modulated. In some cases, the program may implement a voltage modulation scheme (e.g., with a fixed frequency and a fixed pulse width) and/or a pulse width modulation scheme (e.g., with a fixed voltage and a fixed frequency). In some cases, the program may implement a frequency modulation scheme (e.g., with a fixed voltage and a fixed pulse width).

[0048] One problem that may arise during NMES is rapid onset of muscle fatigue during repeated contractions. The onset of muscle fatigue may be correlated with stimulation parameters such as intensity, frequency, and pattern of stimulation, with higher stimulation frequencies causing muscle to fatigue faster. The controller may be configured to generate stimulus signals based on passage of time or indicators of fatigue. In some embodiments, the program may implement a scheme comprising feedback-based frequency modulation to reduce fatigue. In certain cases, the program may implement a modulation scheme that adjusts frequency to maintain body part position tracking error within a specified range. For example, in some cases, the stimulation frequency may be progressively increased or decreased. Starting with a low frequency and changing to a high frequency when muscle output dictates higher demands may more closely mimic biological approaches than constant frequency schemes. In some cases, the program may implement a control method that increases and/or decreases the amplitude and/or frequency of the output signals according to a predefined rule. In certain embodiments, the program may implement a control method that modulates the amplitude and/or frequency of the output signals based at least in part on a value of a parameters indicative of muscle fatigue.

[0049] The hybrid orthotic device may comprise a bracing element configured to attach to a limb of a person. In some embodiments, the bracing element comprises one or more cuffs. In some embodiments, the bracing element may comprise a first cuff configured to attach to a first body part of a person and a second cuff configured to attach to a second body part of a person, where the first body part and second body part are coupled at a joint (e.g., knee, ankle, elbow, shoulder, wrist, hip). The cuffs may be adjustable and may be tightened or loosened (e.g., such that the cuff fits around a limb, such as a leg). In some cases, the cuffs may be substantially flexible (e.g., resilient), semi-rigid, or substantially rigid (e.g., non-resilient). The cuffs may comprise materials including, but not limited to, fabric (e.g., polyester, nylon, neoprene), plastic, and/or metal (e.g., aluminum). The cuffs may be fastened by any suitable fastener, including those now known in the art. For example, the cuffs may be fastened by one or more Velcro fasteners, buckles, and/or hook fasteners. In some embodiments, the bracing element may further comprise a sleeve. The sleeve may be substantially flexible (e.g., such that it can conform to the limb and/or joint). In certain embodiments, the sleeve may comprise a fabric. The bracing element may, in some cases, allow hinged movement of the joint (e.g., may not inhibit flexion and/or extension of the joint). In some embodiments, the bracing element may advantageously provide lateral and/or rotational stability to the joint. For example, the bracing element may prevent hyperextension and/or excessive rotation of the joint.

[0050] In some embodiments, the hybrid orthotic device may further comprise one or more elastic resistance elements. The elastic resistance element may comprise a first end secured to a first location on the bracing element and a second end secured to a second location on the bracing element. In certain cases, the elastic resistance element may comprise a spring and/or an elastic band. It may be desirable, in some cases, for the orthotic device to comprise an elastic resistance element for use in exercising one or more parts of the body.

[0051] FIG. 1 shows an exemplary schematic illustration of a hybrid orthotic device, according to some embodiments of the invention. In FIG. 1, hybrid orthotic device 100 comprises bracing element 102, which is configured to be worn on a leg. In some embodiments, the bracing element may comprise one or more cuffs. For example, in FIG. 1, bracing element 102 comprises two cuffs: one that can be positioned around the lower leg, and one that can be positioned around the thigh. In some embodiments, one or more sensors and their associated electrical leads may be embedded in the bracing element. For example, in FIG. 1, sensor 104 and electrical leads 106 are embedded in bracing element 102, with sensor 104 located at a hinge of bracing element 102 located between the two cuffs. Sensor 104 may, in some cases, be an electrogoniometer configured to measure joint angle. Electrical leads 106 may connect sensor 104 to a controller 108, which may also be attached to bracing element 102. The controller may comprise a portable power supply (e.g., batteries) and inputs and outputs for interfacing with sensors, electrodes, and/or external computers. For example, controller 108 may be configured to send power to and receive data (e.g., data about the joint angle) from sensor 104 via electrical leads 106. Controller 108 may also be configured to receive and store programs from and send data to an external computer. Additionally, controller 108 may also be configured to send output stimulation signals to electrodes 110 via electrical leads 112 to cause contractions of the muscles of the wearer to induce a desired joint motion. In some embodiments, leads 112 may be one-way leads (e.g., data may not be transmitted from electrodes 110 to controller 108). In FIG. 1, electrodes 110 and leads 112 are embedded in a sleeve 114. Sleeve 114, which may comprise a flexible material and conform to the joint and leg, may be positioned under bracing element 102. In some embodiments, electrodes 110 may be configured to be in contact with the skin of a person and to transmit electrical stimulation to underlying muscle. In some embodiments, electrodes 110 may be used to activate different muscle groups at the same time.

[0052] An exemplary schematic illustration of a hybrid orthotic device comprising elastic resistance elements is shown in FIG. 2. Like orthotic device 100 in FIG. 1, orthotic device 200 in FIG. 2 comprises a bracing element 202 and a sensor 204, which is connected to controller 208 via electrical leads 206. Sensor 204, electrical leads 206, and controller 208 are embedded in bracing element 202. In FIG. 2, controller 208 is connected to a cable 216. Controller 208 may be configured to receive power from an external power source via cable 216. Controller 208 may also be configured to communicate with an external computer in real time (e.g., for data logging and/or visualization) via cable 216. Additionally, controller 208 is connected to electrodes 210 via electrical leads 212. Electrodes 210 and electrical leads 212 are embedded in a sleeve 214. Orthotic device 200 also comprises elastic resistance element 218, which is attached to bracing element 202 via anchors 220. Elastic resistance element 218 may be a tension or compression element. In some cases, it may be desirable for an orthotic device to comprise one or more elastic resistance elements for exercising a joint. Anchors 220 may allow elastic resistance element 218 to be easily removed and replaced, facilitating exercise that calls for progressive increase and/or decrease of resistance.

[0053] FIG. 3 shows an exemplary schematic illustration of a hybrid orthotic device configured to induce movement about the knee and/or ankle joints, according to some embodiments. In FIG. 3, orthotic device 300, which is a knee-ankle-foot orthosis, comprises bracing element 302. Bracing element 302 may comprise one or more hinges and may be substantially flexible, semi-rigid, or rigid. In FIG. 3, bracing element 302 comprises a first adjustable cuff configured to attach to a lower leg and a second adjustable cuff configured to attach to a thigh. Bracing element 302 further comprises a foot piece designed to fit inside a shoe for a foot. A pressure sensor 308 may be in direct contact with the foot piece. Pressure sensor 308 may be configured to collect data relating to contact between the foot and an external surface (e.g., the ground) and to transmit the data to a controller 312 via electrical lead 310. In FIG. 3, additional sensors 304 are embedded in bracing element 302. In some embodiments, at least one of sensors 304 is an electrogoniometer. The electrogoniometer may measure, for example, knee and/or ankle angle. Sensors 304 may be configured to transmit data to controller 312 via electrical leads 306. In some cases, electrical leads 306 and 310 may comprise power and data lines. In FIG. 3, controller 312 is also connected to electrodes 314 via electrical leads 316. Electrodes 314 and electrical leads 316 may be embedded in sleeve 318. In some embodiments, electrodes 314 may be configured to apply stimulation to induce flexion, extension, or rotation of the knee and/or ankle.

[0054] FIG. 4 displays an exemplary schematic block diagram illustrating the interactions between components of a system comprising a hybrid orthotic device, according to some embodiments. It should be appreciated that FIG. 4 is a functional block diagram and the components represented in that block diagram may be configured and positioned in any suitable way, including as illustrated in any of FIGS. 1-3. In FIG. 4, system 400 comprises a body 402, sensors 404, electrodes 406, orthosis 408, and a closed-loop functional electrical stimulation (CL-FES) unit 410. Sensors 404 are in direct contact with body 402, and they transmit body state information to CL-FES unit 410. CL-FES unit 410 comprises a microcontroller, a stimulator, and a power supply. The power supply may supply power to the microcontroller and the stimulator, and may additionally supply power to sensors 404. The microcontroller receives body state information from sensors 404. Additionally, the microcontroller is configured to receive programming input from an external computer and to transmit data to the external computer. Though, in some embodiments, a connection to an external computer may be active while the device is in operation to enable the external computer to perform some or all of the control functions. The microcontroller can also send information to the stimulator, which transmits a stimulation signal to electrodes 406. Electrodes 406 are in direct contact with body 402, and the electrical stimulation may induce movement of body 402 such that a functional activity may be performed. CL-FES unit 410 is also in direct contact with an orthosis 408, which may provide stabilizing forces to body 402.

[0055] An exemplary schematic block diagram illustrating the interactions between components of a system comprising an externally-powered hybrid orthotic device, according to some embodiments, is shown in FIG. 5. In FIG. 5, CL-FES unit 510 comprises a stimulator and microcontroller, like CL-FES unit 410. However, unlike CL-FES unit 410, CL-FES unit 510 does not comprise a power source. Rather, power is supplied from external power supply 512. Additionally, in FIG. 5, CL-FES unit 510 is in direct contact with orthosis 508, which comprises one or more elastic resistance units. Orthosis 508 is therefore configured to provide not only stabilizing forces but also resistance forces to body 502.

[0056] FIG. 6 displays an exemplary schematic block diagram illustrating the interactions between components of a controller, according to some embodiments. In FIG. 6, a state of user 602 is measured. The state may be represented in whole or in part by sensor outputs and may include, for example, joint angle. The measured state may then be compared to the state estimated by state estimator 604. The difference between the measured state and the estimated state may be sent to compensator 606, which utilizes a compensating method to compensate for differences between the measured and estimated states. The measured state may also be compared to the desired trajectory, and the tracking error may be sent to controller 608. Controller 608 utilizes a control method to minimize the tracking error and calculate the amplitude, frequency, and pulse width of output signals (which may serve as stimulus signals input to electrodes) to enable user 602 to most closely follow the desired trajectory. Output from compensator 606 and controller 608 may be sent to fatigue compensator 610, which compensates for muscle fatigue induced by electrical stimulation. Compensation, for example, may increase stimulation as increased fatigue is measured. The output signals may then be sent to electrodes to apply electrical stimulation to user 602.

[0057] Some embodiments relate to methods of providing functional electrical stimulation to a person. For example, a method may comprise the step of receiving signals from at least one sensor. Additionally, the method may comprise the steps of calculating an amount of electrical stimulation to apply to at least one body part of the person and delivering electrical stimulation to the at least one body part of the person.

[0058] In some embodiments, delivering electrical stimulation to at least one body part of a person assists the person in interacting with an external device. For example, functional electrical stimulation may be used to assist a person in pedaling a cycling device (e.g., a stationary bicycle, a mobile bicycle, a mobile tricycle). A cycling device may, in some cases, comprise a crank and two pedals, where each pedal is connected to an end of the crank. In some embodiments, the cycling device further comprises at least one wheel (e.g., two wheels, three wheels, four wheels). The cycling device may, in some cases, further comprise one or more sensors. In some cases, at least one sensor is configured to measure position and/or velocity of the crank of the cycling device. In some cases, at least one sensor is an electrogoniometer. According to certain embodiments, the cycling device optionally comprises a motor (e.g., an electric motor) connected to the crank (e.g., such that the motor may provide power to the crank). A person may be associated with the cycling device such that a first lower limb (e.g., a foot) of a person is in contact with a first pedal of the cycling device. In some cases, a second lower limb (e.g., a foot) of the person is in contact with a second pedal of the cycling device.

[0059] In certain embodiments, electrical stimulation is delivered to a first leg of a person via at least one electrode (e.g., two electrodes, three electrodes, four electrodes, or more), inducing pedaling of a cycling device. In some cases, the pedaling is forward pedaling. In some cases, the pedaling is backward pedaling. In certain embodiments, electrical stimulation is delivered to a second, different leg of the person via at least one electrode (e.g., two electrodes, three electrodes, four electrodes, or more), inducing pedaling (e.g., forward pedaling and/or backward pedaling) of the cycling device. In some cases, the pedaling may be substantially continuous for an amount of time. In some embodiments, substantially continuous pedaling occurs for a period of at least about 1 minute, at least about 5 minutes, at least about 10 minutes, at least about 15 minutes, at least about 20 minutes, at least about 30 minutes, at least about 60 minutes, or at least about 90 minutes. Substantially continuous pedaling may occur for a period ranging from about 1 minute to about 5 minutes, about 5 minutes to about 15 minutes, about 5 minutes to about 20 minutes, about 5 minutes to about 30 minutes, about 5 minutes to about 60 minutes, about 5 minutes to about 90 minutes, about 10 minutes to about 30 minutes, about 10 minutes to about 60 minutes, about 10 minutes to about 90 minutes, about 30 minutes to about 60 minutes, about 30 minutes to about 90 minutes, or about 60 minutes to about 90 minutes.

[0060] In some embodiments, a method comprises delivering a first amount of electrical stimulation to a first leg of a person for a first amount of time. In some embodiments, the first amount of electrical stimulation is applied during a first portion of a crank trajectory cycle. In certain cases, the electrical stimulation activates at least a portion of the quadriceps femoris muscles of the first leg. In some embodiments, the electrical stimulation activates at least a portion of the gluteal muscles, hamstring muscles, and/or calf muscles of the first leg. In certain embodiments, electrical stimulation is applied to two or more muscle groups of the first leg during the first amount of time. The same or different amounts of electrical stimulation may be applied to the two or more muscle groups of the first leg during the first amount of time.

[0061] According to some embodiments, the first amount of electrical stimulation is applied to a muscle group (e.g., quadriceps femoris, gluteal muscles, hamstring muscles, calf muscles) when a torque transfer ratio for the muscle group has a certain sign (e.g., positive, negative). For example, in some cases, forward pedaling about a crank of a cycling device (e.g., clockwise in FIGS. 7 and 13) may be desirable, and a muscle group may be activated when it yields a forward torque about the crank. A muscle group may yield a forward torque when the torque transfer ratio is positive or negative, depending on the muscle group. In certain embodiments, the first amount of electrical stimulation is applied when a torque transfer ratio between the hip or the knee of the first leg and the crank of the cycling device is negative. In some cases, the first amount of electrical stimulation is applied to at least a portion of the quadriceps femoris when the torque transfer ratio for the quadriceps femoris is negative. In certain embodiments, the first amount of electrical stimulation is applied when a torque transfer ratio between the hip or the knee of the first leg and the crank of the cycling device is positive. In some embodiments, the first amount of electrical stimulation is applied to at least a portion of the gluteal muscles when the torque transfer ratio for the gluteal muscles is positive. In some embodiments, the first amount of electrical stimulation is applied to at least a portion of the hamstring muscles when the torque transfer ratio for the hamstring muscles is positive.

[0062] In certain embodiments, the first amount of electrical stimulation is applied to a muscle group when the magnitude of the torque transfer ratio for the muscle group is above a certain threshold (e.g., at least about 0.1). In some cases, electrical stimulation of a muscle group only when its torque transfer ratio is above a certain threshold may enhance efficiency by avoiding stimulation that would result in relatively low power output at the crank of the cycling device.

[0063] In some embodiments, a method comprises delivering a second amount of electrical stimulation to a second, different leg for a second amount of time. In some embodiments, the second amount of electrical stimulation is applied during a second portion of a crank trajectory cycle. In certain cases, the electrical stimulation activates at least a portion of the quadriceps femoris muscles of the second leg. In some embodiments, the electrical stimulation activates at least a portion of the gluteal muscles, hamstring muscles, and/or calf muscles of the second leg. In certain embodiments, electrical stimulation is applied to two or more muscle groups of the second leg during the second amount of time. The same or different amounts of electrical stimulation may be applied to the two or more muscle groups of the second leg during the second amount of time.

[0064] In some embodiments, the second amount of electrical stimulation is applied to a muscle group (e.g., quadriceps femoris, gluteal muscles, hamstring muscles, calf muscles) when a torque transfer ratio for the muscle group has a certain sign (e.g., positive, negative). In certain embodiments, the second amount of electrical stimulation is applied when a torque transfer ratio between the knee of the second leg and the crank is negative. In some cases, the second amount of electrical stimulation is applied to at least a portion of the quadriceps femoris when the torque transfer ratio for the quadriceps femoris is negative. In certain embodiments, the second amount of electrical stimulation is applied when a torque transfer ratio between the hip or the knee of the first leg and the crank of the cycling device is positive. In some embodiments, the second amount of electrical stimulation is applied to at least a portion of the gluteal muscles when the torque transfer ratio for the gluteal muscles is positive. In some embodiments, the second amount of electrical stimulation is applied to at least a portion of the hamstring muscles when the torque transfer ratio for the hamstring muscles is positive.

[0065] In certain embodiments, the second amount of electrical stimulation is applied to a muscle group when the magnitude of the torque transfer ratio for the muscle group is above a certain threshold (e.g., at least about 0.1).

[0066] In some cases, the first amount of electrical stimulation and second amount of electrical stimulation are substantially the same. In some cases, the first amount of electrical stimulation and second amount of electrical stimulation are different. In certain embodiments, the first amount of time and second amount of time are substantially the same. In some embodiments, the first amount of time and second amount of time are different. In some embodiments, electrical stimulation is not applied to the first leg and the second leg at the same time (e.g., electrical stimulation is alternately applied to a first leg, then a second leg, then the first leg, etc.). In some embodiments, the first portion of the crank trajectory cycle and the second portion of the crank trajectory cycle do not overlap.

[0067] According to some embodiments, a motor (e.g., an electric motor) provides power to the crank of the cycling device during at least a portion of the crank trajectory cycle. In some cases, a motor may advantageously supplement the ability of a person's muscles to provide power to the crank. In certain embodiments, the motor may promote stability. However, in order to promote efficiency, it may be desirable for the motor to provide assistance only during certain portions of the crank trajectory cycle. For example, it may be advantageous, in some cases, for the motor to provide power to the crank when the kinematic effectiveness of one or more stimulated muscle groups is relatively low. In some cases, during a first portion of a crank trajectory cycle, only motor input may be received (e.g., power is generated by the motor and no electrical stimulation is applied to the person's muscles). In some cases, during a second portion of the crank trajectory cycle, only functional electrical stimulation (FES) input may be received (e.g., electrical stimulation is applied to one or more muscle groups of the person and no power is generated by the motor). In certain embodiments, during a third portion of the crank trajectory cycle, both motor and FES input may be received. A crank trajectory may comprise any combination of the first, second, and third portions described above. In some embodiments, at least one sensor and/or at least one electrode is connected (e.g., electrically connected) to a controller. In some cases, the controller is adapted to receive an input signal from at least one sensor and deliver an output signal to at least one electrode. In some cases, the first amount of electrical stimulation, second amount of electrical stimulation, first amount of time, second amount of time, first portion of the crank trajectory cycle, and/or second portion of the crank trajectory cycle are calculated using a control method. In certain cases, the controller may dynamically generate a signal to control delivery of electrical stimulation based on a torque transfer ratio between a joint of the person and the crank of the cycling device. The control method may, in some cases, be a nonlinear control method. The control method may be discontinuous or continuous. In some cases, the control method is exponentially stable. In certain embodiments, the control method is a switched sliding mode control method.

[0068] In some embodiments, electrical stimulation of one or more muscle groups of a person induces cycling at a relatively high cadence (e.g., pedal velocity). In some embodiments, the mean pedal cadence is at least about 5 revolutions per minute (rpm), at least about 10 rpm, at least about 15 rpm, at least about 20 rpm, at least about 30 rpm, at least about 50 rpm, at least about 70 rpm, or at least about 100 rpm. In some cases, the mean pedal cadence is in the range of about 5 rpm to about 10 rpm, about 5 rpm to about 20 rpm, about 5 rpm to about 50 rpm, about 5 rpm to about 100 rpm, about 10 rpm to about 20 rpm, about 10 rpm to about 50 rpm, about 10 rpm to about 100 rpm, about 20 rpm to about 50 rpm, about 20 rpm to about 100 rpm, or about 50 rpm to about 100 rpm.

[0069] In some embodiments, the mean cadence tracking error (e.g., the numerical average of the differences between desired cadence and actual cadence achieved) is relatively small. In some cases, the mean cadence tracking error is about 10 rpm or less, about 5 rpm or less, about 2 rpm or less, about 1 rpm or less, about 0.5 rpm or less, about 0.4 rpm or less, about 0.3 rpm or less, about 0.2 rpm or less, about 0.1 rpm or less, about 0.05 rpm or less, about 0.02 rpm or less, or about 0.01 rpm or less. In some embodiments, the mean cadence tracking error is in the range of about 0 rpm to about 0.1 rpm, about 0 rpm to about 0.2 rpm, about 0 rpm to about 0.3 rpm, about 0 rpm to about 0.4 rpm, about 0 rpm to about 0.5 rpm, about 0 rpm to about 0.6 rpm, about 0 rpm to about 0.7 rpm, about 0 rpm to about 0.8 rpm, about 0 rpm to about 0.9 rpm, about 0 rpm to about 1 rpm, about 0 rpm to about 2 rpm, about 0 rpm to about 5 rpm, or about 0 rpm to about 10 rpm.

[0070] In certain embodiments, the standard deviation of the cadence tracking error is relatively small. In some cases, the standard deviation of the cadence tracking error is about 10 rpm or less, about 5 rpm or less, about 2 rpm or less, about 1 rpm or less, about 0.5 rpm or less, about 0.4 rpm or less, about 0.3 rpm or less, about 0.2 rpm or less, about 0.1 rpm or less, about 0.05 rpm or less, about 0.02 rpm or less, or about 0.01 rpm or less. In some embodiments, the standard deviation of the cadence tracking error is in the range of about 0 rpm to about 0.1 rpm, about 0 rpm to about 0.2 rpm, about 0 rpm to about 0.3 rpm, about 0 rpm to about 0.4 rpm, about 0 rpm to about 0.5 rpm, about 0 rpm to about 0.6 rpm, about 0 rpm to about 0.7 rpm, about 0 rpm to about 0.8 rpm, about 0 rpm to about 0.9 rpm, about 0 rpm to about 1 rpm, about 0 rpm to about 2 rpm, about 0 rpm to about 5 rpm, or about 0 rpm to about 10 rpm.

[0071] In certain cases, the root mean square (RMS) of the cadence tracking error is relatively small. In some cases, the RMS of the cadence tracking error is about 10 rpm or less, about 5 rpm or less, about 2 rpm or less, about 1 rpm or less, about 0.5 rpm or less, about 0.4 rpm or less, about 0.3 rpm or less, about 0.2 rpm or less, about 0.1 rpm or less, about 0.05 rpm or less, about 0.02 rpm or less, or about 0.01 rpm or less. In some embodiments, the RMS of the cadence tracking error is in the range of about 0 rpm to about 0.1 rpm, about 0 rpm to about 0.2 rpm, about 0 rpm to about 0.3 rpm, about 0 rpm to about 0.4 rpm, about 0 rpm to about 0.5 rpm, about 0 rpm to about 0.6 rpm, about 0 rpm to about 0.7 rpm, about 0 rpm to about 0.8 rpm, about 0 rpm to about 0.9 rpm, about 0 rpm to about 1 rpm, about 0 rpm to about 2 rpm, about 0 rpm to about 5 rpm, or about 0 rpm to about 10 rpm.

[0072] In some embodiments, electrical stimulation is applied to a person with a disorder affecting movement of one or more limbs (e.g., legs). In certain cases, electrical stimulation is applied to a person completely lacking motor control of one or more limbs (e.g., a person having a complete spinal cord injury). In such cases, the person may be a completely passive rider on the cycling device. In certain cases, electrical stimulation is applied to a person having diminished motor control of one or more limbs. Non-limiting examples of disorders that may result in diminished motor control of one or more limbs (e.g., legs) include Parkinson's disease, incomplete spinal cord injury, cerebral palsy, multiple sclerosis, stroke, and/or traumatic brain injury.

[0073] It may be desirable, in some cases, to model the total dynamics of a joint to facilitate development of a controller that could enable movement about the joint to follow a desired trajectory. In some embodiments, the total dynamics of a joint can be modeled as:

M_r+M_e+M_g+M_v+τ_d=τ  (1)

where M_I represents inertial effects of one or more body parts about the joint, M_e represents elastic effects due to joint stiffness, M_g represents gravitational effects, M_v represents viscous effects due to damping in the musculotendon complex, τ_d represents one or more unknown bounded disturbances (e.g., muscle spasticity, dynamic fatigue, signal and response delays, changing muscle geometry, and/or load changes during functional tasks), and T represents the torque produced at the joint. The first and second time derivatives of unknown disturbance τ_d can be assumed to exist and to be bounded.

[0074] Inertial effects M_I can be modeled, in some cases, as:

M_I({umlaut over (q)}(t))=J{umlaut over (q)}(t)  (2)

where {umlaut over (q)}(t) denotes the angular acceleration of the one or more body parts about the joint, and/denotes the unknown inertia of the one or more body parts.

[0075] In certain cases, elastic effects M_e can be modeled as:

M_e(g(t))=−k₁e^−k2q(t)(q(t)−k₃)  (3)

where q (t) denotes the angular position of the one or more body parts about the joint, and k₁, k₂, and k₃ are unknown positive coefficients.

[0076] Gravitational effects M_g may be modeled as:

M_g(q(t))=mgl sin(g(t))  (4)

where m denotes the unknown combined mass of the body parts, l is the unknown distance between the joint and the center of mass of the body parts, and g denotes the gravitational acceleration. Gravitational acceleration generally refers to the value of about 9.8 meters per second squared (m/s²).

[0077] Viscous effects M_v may be modeled as:

M_v({dot over (q)}(t))=B₁ tan h(−B₂{dot over (q)}(t))−B₃{dot over (q)}(t)  (5)

where ė(t) denotes the angular velocity of the body parts about the joint and B₁, B₂, and B₃ are unknown positive constants.

[0078] Torque τ(t) about the joint may be produced through muscle tendon forces F_T (t), such as those generated through electrical stimulation of one or more muscles. Torque τ(t) may be related to musculotendon force F_T (t) through the relation:

τ(t)=ζ(q(t))F_T(t)  (6)

where ζ(q(t)) represents a positive moment arm that changes with extension and movement of the body parts. The moment arm may have unique values for a given range of motion. The unique values may be considered continuously differentiable, positive, monotonic, and bounded. The unique values may also be considered to have a bounded first time derivative. The total musculotendon force may be a net sum of active forces generated by contractile, elastic, and viscous elements. The forces may have dynamic characteristics. For example, in some cases, a passive elastic element may increase with increasing muscle length. In some cases, the musculotendon force generated at the tendon may be the projection of the net sum of the elements along the line parallel to the tendon.

[0079] In some cases, the torque at the joint may be delayed. In some such cases, torque τ(t) may be represented by u(t−t_τ), where t_τ represents the electromechanical delay. The relation between the torque and musculotendon force may then be written as:

u(t−t_τ)=ζ(q(t))F_T(t−t_τ)  (7)

[0080] In certain cases, the musculotendon force F_T (t) may be modeled as:

F_T=F cos a(q(t))  (8)

where a(q(t)) is the pennation angle between the tendon and the muscle. For example, for the human quadriceps muscle, the pennation angle may change monotonically during contraction and may be a continuously differentiable, positive, monotonic, and bounded function with a bounded first time derivative.

[0081] In some embodiments, there may be asynchronous electrical stimulation of one or more muscles. In some such cases, there may be N stimulation channels in the system. Since each channel can activate different sets of muscle units, the corresponding dynamics may be different depending on the switching signal. For example, the dynamics may be modeled as:

M_I+M_e+M_g+M_vτ_d=τ_i+τ_j+ . . . +τ_N  (9)

where τ_i and τ_j represent the torque produced by stimulation of the i^th and j^th subsystems, respectively. The torque produced by stimulation of the i^th subsystem may be related to the musculotendon force as:

τ_i(t)=ζ_i(q(t))F_T,i(t)  (10)

where musculotendon force F_T,i=F_i cos a_i (q) and F_i is the force produced by the recruited muscle in the i^th subsystem.

[0082] In some embodiments, the relationship between musculotendon force F_T and the applied stimulation V(t) can be expressed as:

F_T(t)=η(t)V(t)  (11)

where η(t) is an unknown nonlinear function that is continuously differentiable, nonzero, positive, monotonic, and bounded, with a bounded first time derivative. In this example, V(t) may represent a voltage. However, the specific form in which the stimulus is applied is not critical to the invention, and the stimulus may be applied alternatively or additionally as a current or in any other suitable form. It may, in some cases, be desirable to introduce unknown nonlinear function η(t), since it may enable the muscle contraction to be considered under general dynamic conditions in the subsequent control development. The unknown, uncertain function η(t) can capture the dynamic characteristics of muscle recruitment, muscle force-length, and muscle force-velocity relationships, as well as active and passive characteristics. If electromechanical delay is taken into account, the relation may be written as:

F_T(t−t_τ)=η(q(t),{dot over (q)}(t))V(t−t_τ)  (12)

to capture the latency present between the application of stimulation and production of force.

[0083] In some embodiments, a controller may enable one or more body parts coupled at a joint to track a desired trajectory q_d (t), where q_d (t) and its i^th derivatives q_d (t) are bounded and within the range of motion of the joint for i=1, 2, 3, and 4. In some cases, desired trajectory q_d (t) may be based on joint kinetics and/or joint kinematics. In some embodiments, desired trajectory q_d (t) may be based on body part position. Alternatively, the desired trajectory may comprise continuous signals, periodic signals, step functions, and/or sinusoidal functions. In some cases, the periodic signals may have different frequencies and/or the step functions may have changes in the dynamic load.

[0084] In some embodiments, a position tracking error may be defined as:

e₁(t)=q_d(t)−q(t)  (13)

Two filtered tracking errors e₂ (t) and r (t) may be introduced to facilitate closed-loop error system development and stability analysis. Filtered tracking error e₂ (t) may be expressed as:

e₂(t)ė₁(t)+α₁e₁(t)  (14)

where α₁ is a positive constant control gain.

[0085] Filtered tracking error r (t) may be expressed as:

r(t)ė₂(t)+α₂e₂(t)  (15)

where a₂ is a positive constant control gain. In some cases, filtered tracking error r (t) may not be used in the controller due to a dependence on angular acceleration measurements. Multiplying Equation 16 by J and using the above expressions, the following expression can be obtained:

Jr=W−ΩW+τ_d  (16)

where auxiliary signal W is defined as:

W=j({umlaut over (q)}_d+α₁ė₁+α₂e₂)+M_e+M_g+M_v  (17)

and auxiliary function Ω is defined as:

Ω(q,t)ζ(q(t))η(t)cos a(q(t))  (18)

Multiplying Equation 16 by Ω⁻, the following expression can be obtained:

J_Ωr=W_Ω−V+τ_dΩ  (19)

where J_Ω=Ω^{31 1}J, W_Ω=Ω⁻¹W=J_Ω({umlaut over (q)}_d+α₁ė₁+α₂e₂)+Ω⁻¹(M_e+M_g+M_v), and τ_dΩ=Ω⁻¹τ_d.

[0086] In some cases, the following open-loop error system for Equation 19 may be obtained:

$\begin{array}{cc}{J}_{\Omega }\stackrel{.}{r}=-\frac{1}{2}{\stackrel{.}{J}}_{\Omega }r+N-\stackrel{.}{V}-e_2& \left(20\right)\end{array}$

where N denotes the unmeasureable auxiliary term:

$\begin{array}{cc}N={\stackrel{.}{W}}_{\Omega }+e_2-\frac{1}{2}{\stackrel{.}{J}}_{\Omega }r+{\stackrel{.}{\tau }}_{d\Omega }\left(q,t\right)& \left(21\right)\end{array}$

In some embodiments, it may be desirable to obtain the above expressions for Equations 20 and 21 to facilitate stability analysis. Another unmeasureable auxiliary term N_d may be defined as:

$\begin{array}{cc}{N}_d={\stackrel{.}{J}}_{\Omega }\left(q_d\right){\ddot{q}}_d+J_{\Omega }\left(q_d\right){\ddot{q}}_d+{\stackrel{.}{M}}_{e\Omega }\left(q_d\right)+{\stackrel{.}{M}}_{g\Omega }\left(q_d\right)+{\stackrel{.}{M}}_{v\Omega }\left(q_d\right)+{\stackrel{.}{\tau }}_{d\Omega }\left(q_d,t\right)& \left(22\right)\\ \mathrm{where}M_{e\Omega }=\frac{M_e}{\Omega },M_{g\Omega }=\frac{M_g}{\Omega },\mathrm{and}M_{v\Omega }=\frac{M_v}{\Omega }.& \end{array}$

The open-loop error system can then be expressed as:

$\begin{array}{cc}{J}_{\Omega }\stackrel{.}{r}=-\stackrel{.}{V}-e_2+\tilde{N}+N_d-\frac{1}{2}{\stackrel{.}{J}}_{\Omega }r& \left(23\right)\end{array}$

where unmeasureable auxiliary term Ñ(t) is defined as Ñ(t)=N−N_d. In some cases, it may be desirable to express the open-loop error system as in Equation 23 to segregate the uncertain nonlinearities and disturbances from the model into terms that are bounded by state-dependent bounds and terms that are upper bounded by constants. For example, in certain cases, the mean value theorem may be applied to upper bound unmeasureable auxiliary term Ñ(t) by state-dependent terms as:

∥{tilde over (N)}∥≦ρ(∥z∥)∥z∥  (24)

where z(t)[e₁^T e₂^T r^T]^T and ρ(∥z∥) is a positive, globally invertible, nondecreasing function. In some cases, the fact that q_d(t), g_dⁱ(t)εL_∞∀i=1, 2, 3, 4 can be used to upper bound N_d as:

∥N_d∥≦ζ_Nd∥{dot over (N)}_d∥≦ζ_{{dot over (N)}d}  (25)

where ζ_Nd and ζ_{{dot over (N)}d} are known positive constants.

[0087] Based on the joint dynamics, the control input V(t) may be designed as:

V(t)(k_s+1)e₂(t)−(k_s+1)e₂(0)+∫₀^t[(k_s+1)α₂e₂(τ)+βsgn(e₂(τ))]dτ  (26)

where k_s and β are positive constant adjustable control gains, and sgn denotes the signum function. In some cases, the time derivative of Equation 26 looks like a discontinuous sliding mode controller. Sliding mode control may be desirable because it is a method that can be used to reject the additive bounded disturbances present in the muscle dynamics (e.g., muscle spasticity, load changes, electromechanical delays) while still obtaining an asymptotic stability result. In some cases, the controller in Equation 26 can be implemented as a continuous controller (i.e., the unique integral of the sign of the error) while still yielding an asymptotic stability result. A controller using the input of Equation 26 may be referred to as a robust integral of the sign of the error (RISE) based controller. Equation 26 can ensure that all system signals are bounded under closed-loop operation. The position tracking error may be regulated in the sense that:

∥e₁(t)∥→0 as t→0  (27)

The controller may yield semi-global asymptotic tracking provided the control gain K_s, is selected sufficiently large and is selected according to the sufficient condition:

$\begin{array}{cc}\beta >\left({\zeta }_{N_d}+\frac{1}{{\alpha }_2}{\zeta }_{{\stackrel{.}{N}}_d}\right)& \left(28\right)\end{array}$

The asymptotic stability can be seen in the following Lyapunov-based proof. In some cases, D is a domain in R³⁺¹ containing y(t)=0, where y(t) is defined as:

y(t)[z^T√{square root over (P(t))}]^T  (29)

and auxiliary function P(t) is defined as:

P(t)β∥e₂(0)∥−e₂(0)^TN_d(0)−∫₀^tL(τ)dτ  (30)

The auxiliary function L(t) can be defined as:

L(t)r^T(N_d(t)−βsgn(e₂)).  (31)

The derivative of P(t) can be expressed as:

{dot over (P)}(t)=−L(t)r^T(N_d(t)−βsgn(e₂).  (32)

The following inequality can then be obtained:

∫₀^tL(τ)dτ≦β∥e₂(0)∥−e₂(0)^TN_d(0).  (33)

Equation 33 can be used to conclude that P(t)≧0.

[0088] In certain cases, V_L(y,t) can be a continuously differentiable, positive definite function defined as:

$\begin{array}{cc}{V}_L\left(y,t\right)\stackrel{\Delta }{=}e_1^Te_1+\frac{1}{2}e_2^Te_2+\frac{1}{2}r^TJ_{\Omega }r+P& \left(34\right)\end{array}$

that satisfies the inequalities:

U₁(y)≦V_L(y,t)≦U₂(y)  (35)

where U₁ and U₂ are continuous, positive definite functions. The derivative of V_L can be expressed as:

{dot over (V)}_L(y,t)=−(2α₁−1)e₁^Te₁=(α₂−1)e e₂^Te₂−r^Tr+r^TÑ−k_sr^Tr.   (36)

The unique integral signum term in the RISE controller can be used to compensate for the disturbance terms included in N_d, provided the control gain β is selected according to Equation 28. The term r^T(t)Ñ(e₁, e₂, r, t) can be upper bounded by:

∥r_N∥≦ρ(∥z∥)∥z∥∥r∥  (37)

to obtain:

{dot over (V)}_L(y,t)≦−min{2α₁−1,α₂−1,1}∥z∥²+[ρ(∥z∥)∥z∥∥r∥k_s∥r∥²].  (38)

Completing the squares for the bracketed terms yields:

$\begin{array}{cc}{\stackrel{.}{V}}_L\left(y,t\right)\le -\min \left\{2{\alpha }_1-1,{\alpha }_2-1,1\right\}{z}^2+\frac{{\rho }^2\left(z\right){z}^2}{4k_s}.& \left(39\right)\end{array}$

The following expression can be obtained from Equation 39:

{dot over (V)}_L(y,t)≦−U(y)  (40)

where U(y) is a continuous positive definite function, provided k_s, is selected sufficiently large based on the initial conditions of the system. The region of attraction can be made arbitrarily large to include any initial conditions by increasing the control gain k_s, (i.e., a semi-global type of stability result). Semi-global asymptotic tracking can therefore be achieved.

[0089] In some embodiments, a controller may be configured to store a plurality of programs. A program may be selected and executed based on a determined user activity. In some embodiments a program may be selected based on direct user input. In other embodiments, the program may be selected based on sensing user behavior or from other context information. Some or all of the programs may implement an algorithm compensating for electromechanical delay (EMD) during neuromuscular electrical stimulation. EMD generally refers to a biological artifact that arises due to a time lag between application of electrical stimulation and tension development in a muscle, and it may cause degraded performance and instability during neuromuscular electrical stimulation. Without wishing to be bound by any particular theory, EMD may be caused at least in part by the finite propagation time of chemical ions in muscle, cross-bridge formation between actin-myosin filaments, stretching of the series elastic components in response to the external electrical input, and/or synaptic transmission delays. In some cases, EMD may vary with the input signal frequency.

[0090] In some cases, the algorithm may comprise a predictive term that actively accounts for EMD. In certain cases, filtered error signal e₂ (t) may be expressed as:

e₂=ė₁+αe₁−B∫_t-tτ^t,V(θ)dθ  (41)

where α and B are known control gain constants. Auxiliary function fl may be defined as:

Ω(q,t)=ζ(q(t))η(t)  (42)

where ζ(q(t)) represents a positive moment arm and η(t) represents an unknown nonlinear function. The error ξ between B and J_Ω⁻¹ may be defined by:

$\begin{array}{cc}\xi =B-\frac{\Omega }{J}& \left(43\right)\end{array}$

where ξ satisfies the relation |ξ|≦ξ and ξ is a known constant.

[0091] In some embodiments, the open-loop tracking error system may be developed by multiplying J_Ω by the time derivative of e₂ to obtain:

$\begin{array}{cc}{J}_{\Omega }{\stackrel{.}{e}}_2=J_{\Omega }{\ddot{q}}_d+M_{e\Omega }+M_{g\Omega }+M_{v\Omega }+{\tau }_{d\Omega }+\alpha J_{\Omega }{\stackrel{.}{e}}_1-V-J_{\Omega }\xi \left[V-V_{\tau }\right]& \left(44\right)\\ \mathrm{where}M_{e\Omega }=\frac{M_e}{\Omega },M_{g\Omega }=\frac{M_g}{\Omega },\mathrm{and}M_{v\Omega }=\frac{M_v}{\Omega }.& \end{array}$

In some such embodiments, auxiliary term N_d can be defined as:

N_d=J_Ωd{umlaut over (q)}_d/M_vΩd+M_gΩd+M_eΩd  (45)

where the notation J_Ωd, M_eΩd, M_gΩd, and M_vΩd represent J_Ω, M_eΩ, M_gΩ, and M_vΩ expressed in terms of desired limb position and velocity.

[0092] In some embodiments, control input V(t) may be expressed as:

V=k_be₂  (46)

where k_b is a known control gain that can be expanded as k_b=k_b1+k_b2+k_b3, and k_b1 k_b2, and k_b3 are known constants.

[0093] In certain cases, the closed-loop error system can be determined by adding and subtracting auxiliary term N_d to J_Ωė₂ and using other relations to arrive at:

$\begin{array}{cc}{J}_{\Omega }{\stackrel{.}{e}}_2=-\frac{1}{2}{\stackrel{.}{J}}_{\Omega }e_2+\tilde{N}+S-e_1-k_be_2-k_bJ_{\Omega }\xi \left[e_2-e_{2\tau }\right]& \left(47\right)\end{array}$

where the auxiliary terms Ñ, N, and S are defined as:

$\begin{array}{cc}N=\frac{1}{2}{\stackrel{.}{J}}_{\Omega }e_2+J_{\Omega }{\ddot{q}}_d+M_{e\Omega }+M_{g\Omega }+M_{v\Omega }+\alpha J_{\Omega }e_2-{\alpha }^2J_{\Omega }e_1+e_1+\alpha J_{\Omega }B{\int }_{t-t_{\tau }}^tV\left(\theta \right)d\theta & \left(48\right)\\ \tilde{N}=N-N_d& \left(49\right)\\ S=N_d+d_{\Omega }& \left(50\right)\end{array}$

where Ñ can be upper bounded as:

{tilde over (N)}≦ρ(∥z∥)∥z∥  (51)

where ρ(∥z∥) is a positive, globally invertible, nondecreasing function and z is defined as z=[e₁ e₂ e_z]^T, where e_z is defined as:

e_z=∫_t-τ^tV(θ)dθ  (52)

S can be upper bounded as:

∥S∥≦ε  (53)

where ε is a known constant.

[0094] Lyapunov-Karsovskii functionals P and Q can be defined as:

$\begin{array}{cc}P=\omega {\int }_{t-t_{\tau }}^t\left({\int }_s^t{V\left(\theta \right)}^2d\theta \right)\mathrm{ds}& \left(54\right)\\ Q=\frac{\stackrel{\_}{\xi }J_2k_b}{2}{\int }_{t-t_{\tau }}^t{e_2\left(\theta \right)}^2d\theta & \left(55\right)\end{array}$

where ω is a known constant.

[0095] The controller given in Equation 46 can ensure semi-global uniformly ultimately bounded tracking:

|e₁(t)|≦ε₀exp(−ε₁t)+ε₂  (56)

where ε₀, ε₁, and ε₂ are constants, provided control gains α and k_b are selected according to sufficient conditions:

$\begin{array}{cc}\alpha >\frac{B^2\tau }{2\omega },k_{b_3}>\frac{2\stackrel{\_}{\xi }J_2\left(k_{b_1}+k_{b_2}\right)+\omega k_b^2\tau }{1-2\stackrel{\_}{\xi }J_2}& \left(57\right)\end{array}$

[0096] The sufficient condition for k_b3 can be satisfied by selecting ω, k_b1, and k_b2 sufficiently small and k_b3 sufficiently large, provided 1−2ξJ₂>0.

[0097] In some embodiments, the controller comprises a neural network (NN) based feed forward controller that is augmented with a continuous robust feedback term to yield an asymptotic result despite an uncertain, nonlinear muscle response. Such a controller may be implemented as part of a program or in any other suitable way. In some cases, it may be desirable to use NN-based controllers with an adaptive element that can adjust to unstructured, nonlinear disturbances in the muscle model. Use of NN-based controllers may, in some cases, reduce the tracking error for a given stimulation input. The ability of neural networks to learn uncertain and unknown muscle dynamics may be complemented by the ability of RISE to compensate for additive system disturbances and NN approximation error.

[0098] In some cases, the open-loop tracking error system may be developed by multiplying the expression for filtered tracking error r (t) in Equation 15 by J/Ω to obtain:

J_Ωr=J_Ω(α₂e₂+α₁ė₁+{umlaut over (q)}_d)+M_eΩ+M_gΩ+M_vΩV+τ_dΩ  (58)

[0099] Equation 58 may be rewritten as:

J_Ωr=ƒ_d+S−V+τ_dΩ  (59)

where ƒ_d may be defined as:

ƒ_d=M_eΩd+M_gΩd+M_vΩd+J_Ωd{umlaut over (q)}_d  (60)

and S may be defined as:

S=J_Ω(α₂e₂+α₁ė₁)+J_Ω{umlaut over (q)}_d−J_Ωd{umlaut over (q)}_d+M_eΩ+M_gΩ+M_vΩ+(M_eΩd+M_gΩd+M_vΩd)  (61)

[0100] In some embodiments, S may be a compact, simply connected set of R⁴, and C(S) may be defined as the space where ƒ_d is continuous. In some such embodiments, there exist weights and thresholds such that the function ƒ_d may be represented by a three-layer neural network as:

ƒ_d=W^Tσ(U^Tx_d)+ε(x_d)  (62)

where x_d (t)=[1 q_d(t) {dot over (q)}_d(t) {umlaut over (q)}_d (t)]^T. UεR⁴¹ and WεR^N1+1 are bounded constant ideal weight matrices for the first-to-second and second-to-third layers, respectively, where N₁ is the number of neurons in the hidden layer. σ is the sigmoid activation function σ(•): R⁴→R^N1+1 and ε(x_d) is the functional reconstruction error. In some cases, the additional term “1” in the input vector x_d and activation term σ may allow for thresholds to be included as the first columns of the weight matrices, such that any estimation of W and U includes estimation of the thresholds. In some cases, a three-layer NN approximation for ƒ_d may be given as:

{circumflex over (ƒ)}_d=Ŵ^Tσ(Û^Tx_d)  (63)

where Ŵ(t) and Û(t) are estimates of the ideal weight matrices. The estimate mismatches may be defined as Ũ=U−Û and {tilde over (W)}=W−Ŵ. The mismatch for the hidden-layer output error may be defined as:

{tilde over (σ)}=σ−{circumflex over (σ)}=σ(U^Tx_d)−σ(U^Tx_d)  (64)

The ideal weights may be assumed to exist and to be bounded by positive values such that:

∥U∥_F²=tr(U^TU)=vec(U)^Tvec(U)≦Ū_B  (65)

∥W∥_F²=tr(W^TW)=vec(W)^Tvec(W)≦W_B  (66)

where ∥•∥_F is the Frobenius norm of a matrix and tr(•) is the trace of a matrix.

[0101] Based on the assumption that the desired trajectory is bounded, the following inequalities may hold:

|ε(x_d)≈ε_b1|ε({dot over (x)}_d)|≦ε_b²|ε({umlaut over (x)}_d)|≦ε_b³  (67)

where ε_b1, ε_b2, and ε_b3 are known positive constants.

[0102] In some embodiments, a closed-loop error system may be developed. In some cases, the NN structure may be developed in terms of the desired trajectories. This may be desirable, in certain cases, in order to avoid the use of acceleration measurements. In some cases, strategic separation and regrouping of terms may be needed to overcome the challenge that while the NN estimates may be upper bounded by constants, the time derivatives of the terms may be state-dependent.

[0103] In some cases, the control input may be:

V={circumflex over (ƒ)}_d+μ  (68)

where μ is the RISE feedback term defined as:

μ(k_s+1)e₂(t)−(k_s+1)e₂(0)+v  (69)

where k_s is a positive constant adjustable control gain and v(t) is the generalized solution to:

{dot over (v)}=(k_s+1)α₂e₂(t)+β₁sgn(e₂(t)),v(0)=0  (70)

where β₁ is a positive constant adjustable control gain. The estimates for the NN weights may be generated using a projection algorithm as:

{dot over ({circumflex over (W)})}=proj(Γ₁{circumflex over (σ)}′Û^T{umlaut over (x)}_de₂^T)  (71)

{dot over ({circumflex over (U)})}=proj(Γ₂{dot over (x)}_d({circumflex over (σ)}′^TŴe₂)^T)  (72)

where Γ₁ and Γ₂ are constant, positive definite, symmetric gain matrices. It should be noted that the expressions for estimated NN weights may vary depending on the desired tasks and other characteristics of the controller.

[0104] The NN-based feedforward component {circumflex over (ƒ)}_d may be used to approximate the desired musculoskeletal dynamics ƒ_d. The NN component may approximate the desired function through adaptive weight estimates that are adjusted online via the adaptive law given in Equation 72. In some cases, the RISE feedback controller μ(t) has implicit learning characteristics that maintain the robustness of the system in the presence of additive disturbances and residual function approximation error. One role of the RISE feedback controller may be to keep the system stable while the NN approximates the system dynamics.

[0105] The closed-loop tracking error system may be developed by substituting Equation 68 into Equation 59 as:

J_Ωr={tilde over (ƒ)}_d+S−μ+τ_dΩ  (73)

where {tilde over (ƒ)}_d=ƒ_d−{circumflex over (ƒ)}_d.

[0106] To facilitate subsequent closed-loop stability analysis, the time derivative of Equation 73 may be determined as:

J_Ω{dot over (r)}=−{dot over (J)}_Ωr+{dot over ({tilde over (ƒ)})}_d+{dot over (S)}−{dot over (μ)}+{dot over (τ)}_dΩ  (74)

Although the control input V(t) is present in the open-loop error system, an additional derivative may be taken to facilitate the design of the RISE-based feedback controller. The closed-loop system may be expressed as:

$\begin{array}{cc}{J}_{\Omega }\stackrel{.}{r}=-{\stackrel{.}{J}}_{\Omega }r+W^T{\sigma }^{\prime }\left(U^Tx_d\right)U^T{\stackrel{.}{x}}_d-{\stackrel{.}{\hat{W}}}^T\sigma \left({\hat{U}}^Tx_d\right)-{\hat{W}}^T{\sigma }^{\prime }\left({\hat{U}}^Tx_d\right){\hat{U}}^T{\stackrel{.}{x}}_d-{\hat{W}}^T{\sigma }^{\prime }\left({\hat{U}}^Tx_d\right){\stackrel{.}{\hat{U}}}^Tx_d+\stackrel{.}{\epsilon }\left(x_d\right)+\stackrel{.}{S}-\stackrel{.}{\mu }+{\stackrel{.}{\tau }}_{d\Omega }& \left(75\right)\\ \mathrm{where}{\sigma }^{\prime }\left({\hat{U}}^Tx_d\right)=\frac{d\sigma \left(U^Tx_d\right)}{d\left(U^Tx_d\right)}U^Tx_d={\hat{U}}^Tx_d.& \end{array}$

The following expression may then be obtained:

J_Ω{dot over (r)}=−{dot over (J)}_Ωr+Ŵ^T{circumflex over (σ)}′Ũ^T{dot over (x)}_d−{tilde over (W)}^T{circumflex over (σ)}′Û^T{dot over (x)}_d−Ŵ^T{circumflex over (σ)}′Ũ^T{dot over (x)}_d−W^T{circumflex over (σ)}′Û^T{dot over (x)}_d+W^Tσ′U^T{dot over (x)}_d+ε(x_d)−Ŵ^T{circumflex over (σ)}′{dot over ({circumflex over (U)})}^Tx_d−{dot over ({circumflex over (W)})}^Tσ+{dot over (S)}−{dot over (μ)}+{dot over (τ)}_dΩ  (76)

Using the NN weight tuning laws, the expression may be rewritten as:

$\begin{array}{cc}{J}_{\Omega }\stackrel{.}{r}=-\frac{1}{2}{\stackrel{.}{J}}_{\Omega }r+\tilde{N}+N-e_2-\left(k_s+1\right)r-\beta \mathrm{sgn}\left(e_2\right)& \left(77\right)\end{array}$

where unmeasureable auxiliary term Ñ may be defined as:

$\begin{array}{cc}\tilde{N}=-\frac{1}{2}{\stackrel{.}{J}}_{\Omega }r+\stackrel{.}{S}+e_2-{\mathrm{proj}\left({\Gamma }_1{\hat{\sigma }}^{\prime }{\hat{U}}^T{\stackrel{.}{x}}_de_2^T\right)}^T\hat{\sigma }-{\hat{W}}^T{\hat{\sigma }}^{\prime }{\mathrm{proj}\left({\Gamma }_2{{\stackrel{.}{x}}_d\left({\hat{\sigma }}^{\prime T}\hat{W}e_2\right)}^T\right)}^Tx_d& \left(78\right)\end{array}$

and unmeasureable auxiliary term N may be defined as:

N=N_B+N_d  (79)

N_d may be defined as:

N_d=W^Tσ′U^T{dot over (x)}_d+ε(x_d)+τ_dΩ  (80)

and N_B may be defined as:

N_B=N_B1+N_B2  (81)

where N_B1 and N_B2 may be given by:

N_B1=−Ŵ^Tσ′Ũ^T{dot over (x)}_d−W^T{circumflex over (σ)}′Û^T{dot over (x)}_d  (82)

N_B2=Ŵ^T{circumflex over (σ)}′Ũ^T{dot over (x)}_d+{tilde over (W)}^T{circumflex over (σ)}′Û^T{dot over (x)}_d  (82)

In some cases, it may be desirable to define the terms in such a way in order to segregate terms that are bounded by state-dependent bounds and terms that are upper bounded by constants for the development of the NN weight update laws and the subsequent stability analysis. The auxiliary term N_B may be further segregated to develop gain conditions in the stability analysis. The mean value theorem may be applied to upper bound Ñ as:

|{tilde over (N)}|≦ρ(∥z∥)∥z∥  (84)

where z[e₁ e₂ r]^T and the bounding function p(∥z∥) is a positive, globally invertible, nondecreasing function. The following inequalities may be developed:

|N_d|≦ζ₁,|N_B|≦ζ₂,|{dot over (N)}_d|ζ₃, and |{dot over (N)}_B|≦ζ₄+ζ₅|e₂|  (85)

where for ζ_i=1, 2, . . . 5 are positive known constants.

[0107] The composite NN and RISE controller can ensure that all system signals are bounded under closed-loop operation and that the position tracking error is regulated in the sense that:

|e₁(t)|→0 as t→∞  (86)

provided the control gains are selected according to the following sufficient conditions:

$\begin{array}{cc}\begin{array}{c}\begin{array}{cc}{\alpha }_1>\frac{1}{2},& {\alpha }_2>{\beta }_2+1\end{array}\\ \begin{array}{cc}{\beta }_1>{\zeta }_1+{\zeta }_2+\frac{1}{{\alpha }_2}{\zeta }_3+\frac{1}{{\alpha }_2}{\zeta }_4,& {\beta }_2>{\zeta }_5\end{array}\end{array}\hspace{1em}& \left(87\right)\end{array}$

and control gain k_s, is selected sufficiently large based on the initial conditions of the error system.

[0108] NMES can sometimes lead to early onset of muscle fatigue during stimulation, limiting the duration that functional tasks can be performed in assistive devices. In some embodiments, a controller may apply asynchronous stimulation, which may reduce NMES-induced fatigue. Asynchronous stimulation generally refers to use of multiple stimulation channels and electrodes to target different muscles or different groups of motor units within a given muscle. Without wishing to be bound by a particular theory, asynchronous stimulation may lead to lower rates of fatigue due to reduced duty cycle (i.e., a lower average stimulation frequency) for the recruited motor units compared to conventional stimulation.

[0109] In some cases, switching between different muscles may introduce discontinuities in the open-loop dynamics. These discontinuities may be a consequence of the fact that the response to a given stimulus will differ for each subsystem (i.e., each subsystem activates a different number and/or type of muscle fibers). In certain embodiments, pulses may be delivered in an interleaved fashion to each stimulation channel. Utilizing interleaved pulses rather than sequentially delivering pulse trains may be advantageous, in some cases, at least in part because interleaving the pulses across the stimulation channels may provide an averaging effect due to the temporal summation of the force. The extent of the averaging effect may depend upon the stimulation frequency; higher stimulation frequencies may smooth the force output, but a primary motivation for utilizing interleaved pulses is to achieve decreased rates of fatigue through the use of low frequency stimulation.

[0110] As suggested by Equation 9 above, the joint dynamics during stimulation of two subsystems may be modeled as:

M_I+M_e+M_g+M_v+τ_d=τ_i+τ_j  (88)

τ_i and τ_j represent the torque produced by stimulation of the i^th and j^th subsystems, and the inertial, gravitational, elastic, and viscous components are common to all subsystems since all subsystems act on the same joint. The unknown bounded disturbance torque τ_d may also be modeled as being common to all subsystems.

[0111] In some cases, the open-loop dynamics for asynchronous stimulation may be obtained as:

Jr=W−τ_i−τ_j+τ_d,i≠j  (89)

where J is the same inertia for each subsystem since each subsystem acts on the same joint complex. W(ė₁, e₂, t) denotes an auxiliary term defined as in Equation 17. The open-loop dynamics may be expressed as:

Jr=W−V_iΩ_i−V_jΩ_j+τ_d  (90)

where V_i denotes the stimulation applied to the i^th subsystem by electrical stimulation, and Ω_i is a positive auxiliary term that relates the stimulation applied to the i^th subsystem to the torque produced by the i^th subsystem, defined as:

Ω_iζ_iη_i cos a_i(q)  (91)

The first time derivative of Ω_i is continuous and bounded, and the second time derivative of Ω_i is bounded. In some cases, σ may denote a piecewise constant signal that selects the subsystem to be activated at time t. The instances when the value of σ changes are called the switching times, t_i. Immediately following each switching time, there may be a transition period Δt during which the input is transitioned from one subsystem to another. To facilitate the subsequent analysis, the closed-loop dynamics of the switched system and the Lyapunov candidate function may be made continuous through the signal X(t), where X is designed such that the transition period from one subsystem to another results in a continuous torque output due to stimulation. X denotes the percentage of the input delivered to one of the two selected subsystems, and the remaining percentage of the input (1−X), is delivered to the other subsystem. V(t) can denote the input to the system such that V=V_i+V_j, where

V_i=XV,V_j=(1−X)V.  (92)

The closed loop dynamics may be expressed as:

Jr=W−VΩ₁+τ_d,t₀≦t<t₁

Jr=W−XVΩ₁−(1|X)V−Ω₂+τ_d,t₁≦t<t₂

Jr=W−XVΩ₃(1−X)VΩ₂+τ_d,t₂≦t<t₃

Jr=W−XVΩ₃−(1−X)VΩ₄+τ_d,t₃<t<t₄  (93)

In some cases, an example definition of X may be given by:

$\begin{array}{cc}X\stackrel{\Delta }{=}\{\begin{array}{cc}\frac{1-\sin \left(\alpha \left(t-t_{2k-1}\right)-\frac{\pi }{2}\right)}{2},& t\in \left[t_{2k-1},t_{2k-1}+\Delta t\right)\\ 0,& t\in \left[t_{2k-1},+\Delta t,t_{2k}\right)\\ \frac{1+\sin \left(\alpha \left(t-t_{2k-1}\right)-\frac{\pi }{2}\right)}{2},& t\in \left[t_{2k},t_{2k}+\Delta t\right)\\ 1,& t\in \left[t_0,t_1\right)\bigcup \left[t_{2k}+\Delta t,t_{2k+1}\right)\end{array}.& \left(94\right)\end{array}$

where X and its first time derivative are bounded and continuous and the second time derivative is bounded. The transition period in Equation 94 is defined as

$\Delta t\stackrel{\Delta }{=}\frac{\pi }{a}$

arbitrarily short through the choice of a.

[0112] In some cases, it may be desirable to obtain the time derivative of Equation 93 to facilitate stability analysis. The time derivative may be expressed as:

$\begin{array}{cc}J\stackrel{.}{r}=-\frac{1}{2}\stackrel{.}{J}r+\tilde{N}+N_{d1}-e_2+\hspace{1em}\left[{\mathrm{XN}}_{\mathrm{d2},i}+\left(1-X\right)N_{\mathrm{d2},j}\right]-\left[X{\Omega }_i+\left(1-X\right){\Omega }_j\right]\stackrel{.}{V}+\hspace{1em}\left[\stackrel{.}{X}N_{\mathrm{d3},i}-\stackrel{.}{X}N_{\mathrm{d3},j}\right]V& \left(95\right)\end{array}$

where the following auxiliary terms may be defined as:

$\begin{array}{cc}N\stackrel{\Delta }{=}\stackrel{.}{W}+e_2-\frac{1}{2}{\stackrel{.}{J}}_r+{\stackrel{.}{\tau }}_d& \left(96\right)\\ N_{\mathrm{di}}\stackrel{\Delta }{=}J{\ddot{q}}_d+J{\stackrel{⃛}{q}}_d+{\stackrel{.}{M}}_e+{\stackrel{.}{M}}_g+{\stackrel{.}{M}}_v+{\stackrel{.}{\tau }}_d& \left(97\right)\\ N_{d2,i}\stackrel{\Delta }{=}-{\stackrel{.}{\Omega }}_iV& \left(98\right)\\ N_{d3,i}\stackrel{\Delta }{=}-{\Omega }_i& \left(99\right)\\ \tilde{N}\stackrel{\Delta }{=}N-N_{d1}.& \left(100\right)\end{array}$

In some cases, it may be desirable to express the open-loop error system as in Equation 95 to separate the model into groups that are bounded by state-dependent bounds or by constants. By applying the Mean Value Theorem, Kr may be upper bounded by state-dependent terms as:

|{tilde over (N)}|≦ρ(∥z∥)∥z∥  (101)

where z(t) [e₁^T e₂^T r^T]^T and ρ(∥z∥) is a positive, globally invertible, nondecreasing function. The desired trajectory may be used to prove the upper bounds:

|N_d1|≦ζ_Nd1  (102)

|{dot over (N)}_d1|≦ζ_{{dot over (N)}d1}  (103)

|N_d2,i|≦ρ_Nd2,i(∥z∥)|V(t)|(104)

|{dot over (N)}_d2,i|≦ρ_{1,{dot over (N)}d2,i}(∥z∥)|V(t)|+ρ_{2,{dot over (N)}d2,i}(∥z∥)  (105)

|N_d3,i|≦ρ_Nd3,i(∥z∥)  (106)

|{dot over (N)}_d3,i|≦ρ_{{dot over (N)}d3,i}(∥z∥)  (106)

where ζ_Nd1 and ζ_{{dot over (N)}d1} are known positive constants and ρ_Nd2,i (∥z∥), ρ_{1,{dot over (N)}d2,i} (∥z∥), ρ_{2,{dot over (N)}d2,i} (∥z∥), ρ_Nd3,i (∥z∥), and ρ_{{dot over (N)}d3,i} (∥z∥) are positive, globally invertible, nondecreasing functions.

[0113] To facilitate stability analysis, Equation 95 may be expressed as:

$\begin{array}{cc}J\stackrel{.}{r}=-\frac{1}{2}\stackrel{.}{J}r+\tilde{N}+N_{d1}-e_2+{\stackrel{\_}{N}}_{d2}-\stackrel{\_}{\Omega }\stackrel{.}{V}+\stackrel{.}{X}{\stackrel{\_}{N}}_{d3}V& \left(108\right)\end{array}$

where N_d2, Ω, and N_d3 are continuous functions defined as:

N_d2[XN_d2,i+(1−X)N_d2,j]  (109)

N_d3[N_d3,i−N_d3,j]  (110)

Ω[XΩ_i+(1−X)Ω_j]  (111)

The auxiliary terms may be bounded as:

|N_d2|≦ρ_Nd2(∥z∥)|V(t)|  (112)

|{dot over (N)}_d2|≦ρ_{1,{dot over (N)}d2}(∥z∥)|V(t)|+ρ_{2,{dot over (N)}d2}(∥z∥)  (112)

|N_d3|≦ρ_Nd3(∥z∥)|V(t)  (114)

|{dot over (N)}_d3|≦ρ_{{dot over (N)}d3}(∥z∥)  (115)

|Ω|≦ρ_Ω(∥z∥)  (116)

|{dot over (Ω)}|≦ρ_{{dot over (Ω)}}(∥z∥)  (117)

[0114] The RISE-based voltage controller may be designed as:

$\begin{array}{cc}V\left(t\right)\stackrel{\Delta }{=}\frac{1}{\epsilon }\left(k_s+1\right)\left(e_2\left(t\right)-e_2\left(0\right)\right)+v\left(t\right)& \left(118\right)\end{array}$

where v(t) is the generalized Fillipov solution to

$\begin{array}{cc}\stackrel{.}{v}\left(t\right)=\frac{1}{\epsilon }\left(k_s+1\right){\alpha }_2e_2\left(t\right)+(\beta +{\beta }_VV\mathrm{sgn}\left(e_2\left(t\right)\right)& \left(119\right)\end{array}$

where α₂, k_s, β, β_T are positive, constant control gains.

[0115] In some cases, the designed switching signal σ may have a finite number of discontinuities on any bounded time interval. Any two consecutive switching times, t₁ and t_i+1 may satisfy t_i+Δt<t_i+1, and the switching signal may remain constant for tε[t₁,t_i+1).

[0116] The controller designed in Equation 108 may yield semi-global asymptotic tracking in the sense that |e₁ (t)|→0 as t>∞under any switching signal satisfying the condition above, provided that the control gains α₁, α₂, k_s, β, β_V are selected according to the sufficient conditions:

$\begin{array}{cc}\begin{array}{cc}{\alpha }_1>\frac{1}{2},& k_s>\frac{{\rho }^2y\left(0\right)}{4\lambda }\end{array}& \left(120\right)\\ {\beta }_v>{\max }_{i\in S}\left\{\frac{1}{\epsilon }\left[{\rho }_{N_{d2,i}}+\left(\frac{\alpha }{2}\right)\rho N_{d3,i}+1\right]{}_{z=y\left(0\right)}\right\}& \left(121\right)\\ \beta >\frac{1}{\epsilon }\left[{\zeta }_{N_{d1}}+1\right]& \left(122\right)\\ {\alpha }_2>\max \{\left[{\rho }_{1,{\stackrel{.}{\stackrel{\_}{N}}}_{d2}}+{\rho }_{2,{\stackrel{.}{\stackrel{\_}{N}}}_{d2}}{\beta }_V+{\rho }_{\stackrel{\_}{\Omega }}{\beta }_V^2+{\rho }_{\stackrel{.}{\stackrel{\_}{\Omega }}}{\beta }_V+\left(\frac{\alpha }{2}\right){\rho }_{{\stackrel{\_}{N}}_{d3}}\left[1+\alpha +{\beta }_V\right]+\left(\frac{\alpha }{2}\right){\rho }_{{\stackrel{.}{\stackrel{\_}{N}}}_{d3}}\right]{}_{z=y\left(0\right)},\hspace{1em}\left[{\zeta }_{{\stackrel{.}{N}}_{d1}}+\left({\rho }_{\stackrel{.}{\stackrel{\_}{\Omega }}}+{\rho }_{2,{\stackrel{.}{\stackrel{\_}{N}}}_{d2}}+{\rho }_{\stackrel{\_}{\Omega }}{\beta }_V+\left(\frac{\alpha }{2}\right){\rho }_{{\stackrel{\_}{N}}_{d3}}\right)\beta +\frac{1}{\epsilon }\left(k_s+1\right)y\left(0\right)\left({\rho }_{2,{\stackrel{.}{\stackrel{\_}{N}}}_{d2}}+{\rho }_{\stackrel{\_}{\Omega }}{\beta }_V+\left(\frac{\alpha }{2}\right){\rho }_{{\stackrel{\_}{N}}_{d3}}\right)\right]{}_{z=y\left(0\right)},2\}& \left(123\right)\\ \mathrm{where}\lambda \stackrel{\Delta }{=}\min \left\{2{\alpha }_1-1,{\alpha }_2-1,1\right\}.& \end{array}$

[0117] In some embodiments, the neural network may be used to estimate acceleration of a limb. A controller using such a neural network can achieve a controller using only position and velocity information.

[0118] This Example describes the use of functional electrical stimulation (FES) to activate lower limb muscles to pedal a cycling device (e.g., a stationary bicycle, a mobile bicycle, a mobile tricycle). FES-induced cycling could, for example, be used as a means of exercise and/or rehabilitation for individuals with dysfunctional lower limb muscles (e.g., individuals with neuromuscular disorders). In some embodiments, control as described in this example may be implemented in a hybrid orthotic device as described herein. However, it should be appreciated that the device providing functional electrical stimulation may be packaged in any suitable way, including using a controller in an exercise bicycle to compute functional electrical stimulation. Such a device may include sensors to provide measured values of control inputs to the controller. For example a sensor, or sensors, may provide measurements of position, velocity and/or other parameters of motion of a crank of an exercise device.

[0119] Previous FES-induced cycling efforts described in the literature found that FES-induced cycling was metabolically inefficient and yielded low power output at the cycle crank compared to able-bodied cycling. In this Example, to improve muscle control and thereby improve efficiency and increase power output, a stimulation pattern for quadriceps femoris-only FES-induced cycling was derived based on the kinematic effectiveness of knee joint torque in producing forward pedaling throughout the crank cycle. The stimulation pattern was designed such that stimulation was not applied in regions near the dead points of the cycle in order to bound the stimulation voltage, while everywhere else in the crank cycle either the right or left quadriceps were stimulated to produce forward pedaling. This led to partitioning of the crank cycle into controlled and uncontrolled regions. Stimulation of additional muscle groups (e.g., the gluteal muscles) could have eliminated uncontrolled regions, but also could have resulted in overlapping controlled regions and over-actuation. Therefore, the stimulation pattern only stimulated the quadriceps.

[0120] In some embodiments, control was switched such that only one leg was stimulated at one time. Accordingly, in this Example, in the controlled regions, a switched sliding mode controller was designed that guaranteed exponentially stable tracking of a desired crank trajectory, provided certain gains conditions were satisfied. In the uncontrolled regions, the tracking error proved to be ultimately bounded, provided a reverse dwell-time condition, which required that the system not dwell in the uncontrolled region for an overly long time interval, was satisfied. The reverse dwell-time condition was shown to be satisfied provided sufficient desired cadence conditions were satisfied. Stability was derived through Lyapunov methods for switched systems. Experimental results demonstrated the performance of the switched control system under typical cycling conditions.

[0121] A cycling device (e.g., comprising a seat, a crank, and two pedals) and a two-legged rider were modeled as a single degree-of-freedom system, which was expressed as:

M{umlaut over (q)}+V{dot over (q)}+G+τ_d=τ_crank  (124)

where qεQ⊂ denoted the crank angle (e.g., the clockwise angle between the ground and the right crank arm), Mε denoted inertial effects, Vε represented centripetal and Coriolis effects, Gε represented gravitational effects, τ_dε represented an unknown, time-varying, bounded disturbance (e.g., spasticity and/or changes in load), and τ_crankε represented the torque applied about the crank axis. The rider's legs were modeled as planar rigid-body segments with revolute hip and knee joints. More complex models of the knee joint had been found to have a negligible effect on the linkage kinematics. The ankle joint was assumed to be fixed in accordance with common clinical cycling practices for safety and stability. When the rider's feet were fixed to the pedals of the cycle, the resulting system could be completely described by the crank angle (or any other single joint angle measured with respect to ground) and the kinematic parameters of the limb segments. FIG. 7 depicts one side of the cycle-rider system described by Equation 124. As shown in FIG. 7, q_tε was the measurable constant trunk angle, q_hε was the hip angle, q_kε was the knee angle, l_xε_≧0 was the measureable constant horizontal distance between the hip and crank joint axes, l_yε_≧0 was the measureable constant vertical distance between the hip and crank joint axes, l_tε_>0 was the measureable constant thigh segment length, l_lε_>0 was the measureable constant shank segment length, and l_cε_>0 was the measureable constant crank segment length. The trunk, hip, and knee angles were geometric functions of l_x, l_y, l_t, and/or l_c.

[0122] The crank torque was composed of local torques (e.g., friction in the crank bearings) and torques from other joints that were translated to the crank axis (e.g., torques produced by the rider's muscles and/or ligaments). Taking into account, inter alia, viscous damping in the crank joint, passive hip and knee joint torques, and active torque produced at the knees by the quadriceps femoris muscle groups, the crank torque was expressed as:

$\begin{array}{cc}{\tau }_{\mathrm{crank}}={\tau }_b+\sum _{j\in }\sum _{s\in }B_j^s{\tau }_j^s& \left(125\right)\end{array}$

where τ−c{dot over (q)}, with cε_>0 as the unknown constant damping coefficient, B_j^sε was the Jacobian element relating torque from a given joint to the crank torque, and τ_j^sε was the net torque at a joint. The joint set was defined as {h,k}, where h and k indicated the hip and knee joints, respectively. The set {R,L} denoted the right leg (indicated by R) or left leg (indicated by L). For simplicity of notation, the superscript s indicating leg side was generally omitted unless it added clarity. The elements B_j, which were joint torque transfer ratios, were defined as:

$\begin{array}{cc}{B}_h\stackrel{\Delta }{=}-\frac{\partial q_h}{\partial q},B_k\stackrel{\Delta }{=}\frac{\partial q_h}{\partial q}-\frac{\partial q_k}{\partial q}& \left(126\right)\end{array}$

[0123] Each joint torque τ_j could be decomposed into active (i.e., from active muscle contractions) and passive (i.e., from viscoelastic tissue properties) torque as:

τ_j=τ_j,a+τ_j,p  (127)

where the subscript a indicated active torque and the subscript p indicated passive torque. Taking only active contractions of the quadriceps femoris muscle groups into account (e.g., neglecting the biarticular effects of the rectus femoris), the following expression for net active joint torque was obtained:

$\begin{array}{cc}\sum _{j\in }\sum _{s\in }{\tau }_{j,a}^s=\sum _{s\in }{\tau }_{\mathrm{quad}}^s& \left(128\right)\end{array}$

where τ_quadε was defined as the active torque produced by the quadriceps muscle group about the knee joint. τ_quad could be expressed as:

τ_quad=Ωu  (129)

where u ε was a control input (e.g., a voltage applied across the quadriceps muscle group) and Ωε was defined as:

Ωλη cos(a)  (130)

where λε_>0, was an uncertain moment arm of the quadriceps muscle force about the knee, ηε was an uncertain nonlinear function relating stimulation voltage to muscle fiber force, and a ε was an uncertain pennation angle of the quadriceps muscle fibers.

[0124] The passive viscoelastic effects in the hip and knee joints were modeled as:

τ_j,p=k_j,1e^−kj,2γj(γ_j−k_j,3)+b_j,1 tan h(−b_j,2{dot over (γ)}_j)−b_j,3{dot over (γ)}_j,jε  (131)

where k_j,i, b_j,iε_>0, iε{1, 2, 3} were unknown constant coefficients, and γ_iε were the relative hip and knee joint angles defined as:

γ_hq_h−q_t  (132)

γ_kq_k−q_h+π(133)

Substituting Equations 127-129 into Equation 125 yielded:

$\begin{array}{cc}{\tau }_{\mathrm{crank}}={\tau }_b+P+\sum _{s\in }B_k^s{\Omega }^su^s& \left(134\right)\end{array}$

where the auxiliary term Pε captured the passive joint torque effects on the crank and was defined as:

$\begin{array}{cc}P\stackrel{\Delta }{=}\sum _{j\in }\sum _{s\in }B_j^s{\tau }_{j,p}^s& \left(135\right)\end{array}$

The equation of motion for the total cycle-rider system with electrical stimulation of the quadriceps muscles was obtained by substituting Equation 134 into Equation 124 as:

$\begin{array}{cc}M\ddot{q}+V\stackrel{.}{q}+G+{\tau }_d-{\tau }_b-P=\sum _{s\in }B_k^s{\Omega }^su^s& \left(136\right)\end{array}$

The model in Equation 136 had the following properties:
Property 1. c_m≦M≦c_M, where c_m, c_Mε_>0, were known constants.
Property 2. |V|≦c_v|{dot over (q)}|, where c_vε_>0, was a known constant.
Property 3. |G|≦c_G, where c_Gε_>0, was a known constant.
Property 4. |τ_d|≦c_d, where c_dε_>0, was a known constant.
Property 5. |B_j^s|≦c_B∀sε, jε, where C_Bε_>0 was a known constant.
Property 6. |P|≦c_p1+c_p2|{dot over (q)}|, where c_p1, c_p2ε_>0 were known constants.
Property 7. c_Ωn≦Ω^s≦c_Ω2 •sε, where c_Ω1, c_Ω2ε_>0 were known constants.

[0125] $\frac{1}{2}\stackrel{.}{M}-V=0$

[0126] The torque transfer ratios B_k^s, provided insight into how the quadriceps muscles of each leg should be activated during the crank cycle. As shown in FIG. 7, the quadriceps produced a counter-clockwise torque about the knee joint that acted to extend the knee. Multiplication of the active knee torque by B_k transformed the counter-clockwise torque produced by the quadriceps to a resultant torque about the crank. Therefore, as forward pedaling required a clockwise torque about the crank, the quadriceps muscles should only be activated when they produced a clockwise torque about the crank (e.g., when B_k was negative). The torque transfer ratio B_k was negative definite for half of the crank cycle, and the sign of the torque transfer ratio of one leg was generally opposite from the other leg (e.g., B_k^R B_k^L≦0 ∀q), provided the crank arms were offset by π radians. To pedal using only the quadriceps muscles, the controller alternated between legs, using the right quadriceps when B_k^R<0 and the left quadriceps when B_k^L<0. The controller switched between legs when B_k=0, which occurred at the so-called “dead points” q*εQ*⊂Q, where

$Q^{*}\stackrel{\Delta }{=}\left\{q\in Qq=\arctan \left(\frac{l_y}{l_x}\right)+n\pi \right\},n\in \mathbb{Z}.$

[0127] Since the torque transfer ratios were minimal near the dead points, stimulation applied close to the dead points was inefficient in the sense that large knee torques generally yielded small crank torques. Therefore, the uncontrolled region, in which no stimulation was applied, was defined as Q_u {qεQ|−B_K^s(q)≦Σ}Q, where εε_>0 was a scalable constant that could be increased to reduce the portion of the cycle trajectory where stimulation was applied. Specifically, ε<max (−B_k^s). Also, the sets Q^s ⊂Q were defined as the regions where the right and left quadriceps muscle groups were stimulated, as

Q^S{qεQ|−B_k^s(q)>ε}  (137)

The set Q_c∪_sε Q^s was denoted as the controlled region, where Q_c ∪Q_u=Q and Q^R ∩Q^L=ø (e.g., stimulation voltage was generally not applied to both legs at the same time). The static stimulation pattern was then completely defined by the cycle-rider kinematics (e.g., B_k) and selection of ε. Smaller values of ε yielded larger stimulation regions, while larger values of ε yielded smaller stimulation regions. It had been suggested that the stimulation region should be made as small as possible while maximizing stimulation intensity to optimize metabolic efficiency, motivating the selection of large values of ε.

[0128] Cycling was achieved by switching between the left and right limbs, where stimulation of the quadriceps occurred outside of Q_u to avoid the dead points and their neighborhoods, where pedaling was inefficient. Specifically, the switched control input u^s was designed as

$\begin{array}{cc}{u}^s\stackrel{\Delta }{=}\{\begin{array}{cc}{v}^s& \mathrm{if}q\in Q_c\\ 0& \mathrm{if}q\in Q_u\end{array}& \left(138\right)\end{array}$

where v^sε was the stimulation (e.g., voltage) applied across the quadriceps muscle group. Substitution of Equation 138 into Equation 136 yielded a switched system with autonomous state-dependent switching:

$\begin{array}{cc}M\ddot{q}+V\stackrel{.}{q}+G+{\tau }_d-{\tau }_b-P=\{\begin{array}{cc}{B}_k^s{\Omega }^sv^s& \mathrm{if}q\in Q_c\\ 0& \mathrm{if}q\in Q_u\end{array}& \left(139\right)\end{array}$

Assuming that q started within controlled region Q_c, the known sequence of switching states was defined as {q_n^on, q_n^off}, n ε{0, 1, 2, . . . }, where q₀^on was the initial crank angle and the switching states which followed were precisely the limit points of Q_u. The corresponding sequence of switching times {t_n^on,t_n^off}, which were unknown a priori, were defined such that each on-time t_n^on and off-time t_n^off occurred when q reached the corresponding on-angle q_n^on and off-angle q_n^off, respectively. An exemplary schematic illustration of controlled and uncontrolled regions throughout a crank cycle is shown in FIG. 8. The controlled regions are indicated as lightly shaded regions, and the uncontrolled regions are indicated as darkly shaded regions. The controlled region in which the right quadriceps were stimulated is labeled as Q^R, and the controlled region in which the left quadriceps were stimulated is labeled as Q^L. Additionally, the dead points are labeled as q_n* and q_n+1*, and the switching states are labeled as q_n^on, q_n+1^on, q_n^off, and q_n+1^off.

[0129] The control objective was to track a desired crank trajectory with performance quantified by the tracking error signals e₁, e₂ε, which were defined as:

e₁q_d−q  (140)

e₂ė₁+αe₁  (141)

where q_dε was the desired crank position, designed such that its derivatives existed and {dot over (q)}_d, {umlaut over (q)}_dε—_∞, and αε_>0, was a selectable constant. q_d was generally designed to monotonically increase (e.g., backpedaling was not desired). Taking the time derivative of Equation 141, multiplying by M, and using Equations 139-141 yielded the following open-loop error system:

$\begin{array}{cc}M{\stackrel{.}{e}}_2=\chi -V_{e_2}-\{\begin{array}{cc}{B}_k^s{\Omega }^sv^s& \mathrm{if}q\in Q_c\\ 0& \mathrm{if}q\in Q_u\end{array}& \left(142\right)\end{array}$

where the auxiliary term x ε was defined as:

χM({umlaut over (q)}_d+αė_i)+V({dot over (q)}_d+αe₁)+G+τ_d−τ_b−P  (143)

Based on Equation 143 and Properties 1-6, χ could be bounded as:

|χ|≦c₁+c₂∥z∥+c₃∥z∥²  (144)

where c₁, c₂, c₃ε_>0, were known constants. The error vector z ε² was defined as:

z[e₁e₂]^T  (145)

[0130] Based on Equation 142 and the subsequent stability analysis, the control input (e.g., voltage) was designed as:

v^s−k₁e₂−(k₂+k₃∥z∥+k₄∥z∥²)sgn(e₂)  (146)

where sgn(•) denoted the signum function and k₁, k₂, k₃, k₄ε_>0, were constant control gains. After substituting Equation 146 into the open-loop error system in Equation 142, the following switched closed-loop error system was obtained:

$\begin{array}{cc}M{\stackrel{.}{e}}_2=\chi -V_{e_2}-\{\begin{array}{cc}{B}_k^s{\Omega }^s\left[k_1e_2+\left(k_2+k_3z+k_4{z}^2\right)\mathrm{sgn}\left(e_2\right)\right]& \mathrm{if}q\in Q_c\\ 0& \mathrm{if}q\in Q_u\end{array}& \left(147\right)\end{array}$

The controller in Equation 146 could have been designed to include (B_k^s)⁻¹ to cancel the preceding B_k^s, since B_k^s, was known, and the result would have had less strict gain conditions. However, doing so would have caused sharp increases of the control input near the uncontrolled regions, depending on the choice of ε, resulting in large magnitude stimulation of the muscles in regions where their effectiveness was low.

[0131] V_L: ²→denoted a continuously differentiable, positive definite, radially unbounded, common Lyapunov-like function defined as:

$\begin{array}{cc}{V}_L\stackrel{\Delta }{=}\frac{1}{2}z^T\mathrm{Wz}& \left(148\right)\end{array}$

where the positive definite matrix Wε^2×2 was defined as:

$\begin{array}{cc}W\stackrel{\Delta }{=}\left[\begin{array}{cc}1& 0\\ 0& M\end{array}\right]& \left(149\right)\end{array}$

V_L satisfied the following inequalities:

λ₁∥z∥²≦V_L≦λ₂∥z∥²  (150)

where λ₁, Δ₂ε_>0, were known constants defined as:

$\begin{array}{cc}{\lambda }_1\stackrel{\Delta }{=}\min \left(\frac{1}{2},\frac{c_m}{2}\right),{\lambda }_2\stackrel{\Delta }{=}\max \left(\frac{1}{2},\frac{c_M}{2}\right)& \left(151\right)\end{array}$

Theorem 1. For qεQ_c, the closed-loop error system in Equation 147 was exponentially stable in the sense that:

$\begin{array}{cc}z\left(t\right)\le \sqrt{\frac{{\lambda }_2}{{\lambda }_1}}z\left(t_n^{\mathrm{on}}\right)e^{-\frac{{\gamma }_1}{2{\lambda }_2}\left(t-t_n^{\mathrm{on}}\right)}\forall t\in \left(t_n^{\mathrm{on}},t_n^{\mathrm{off}}\right),\forall n& \left(152\right)\end{array}$

where γ₁ε_>0 was defined as:

$\begin{array}{cc}{\gamma }_1\stackrel{\Delta }{=}\min \left(\alpha -\frac{1}{2},\epsilon c_{\Omega 1}k_1-\frac{1}{2}\right)& \left(153\right)\end{array}$

provided the following gain conditions were satisfied:

$\begin{array}{cc}\alpha >\frac{1}{2},k_1>\frac{1}{2\epsilon c_{\Omega 1}},k_2\ge \frac{c_1}{\epsilon c_{\Omega 1}},k_3\ge \frac{c_2}{\epsilon c_{\Omega 1}},k_4\ge \frac{c_3}{\epsilon c_{\Omega 1}}& \left(154\right)\end{array}$

Proof: Let z(t) for tε(t_n^on,t_n^off) be a Filippov solution to the differential inclusion żεK [h](z), where h: ×→² was defined as:

$\begin{array}{cc}h\stackrel{\Delta }{=}\left[\begin{array}{c}{\stackrel{.}{e}}_1\\ {\stackrel{.}{e}}_2\end{array}\right]=\hspace{1em}\left[\begin{array}{c}{e}_2-\alpha e_1\\ M^{-1}\left\{\chi -{\mathrm{Ve}}_2+B_k^s{\Omega }^s\left[k_1e_2+\left(k_2+k_3z+k_4{z}^2\right)\mathrm{sgn}\left(e_2\right)\right]\right\}\end{array}\right]& \left(155\right)\end{array}$

[0132] The time derivative of Equation 148 existed almost everywhere (a.e.) for almost all tε(t_n^on,t_n^off), and {dot over (V)}_L(z)ε{dot over (V)}_L, (z), where {dot over (V)}_L (z) was the generalized time derivative of Equation 148 along the Filippov trajectories of ż=h(z) and was defined as:

$\begin{array}{cc}{\stackrel{\stackrel{.}{~}}{V}}_L\stackrel{\Delta }{=}\bigcap _{\xi \in \delta V_L\left(z\right)}{\xi }^TK\left[\begin{array}{c}h\left(z\right)\\ 1\end{array}\right]& \left(156\right)\end{array}$

where δV_L was the generalized gradient of V_L. Since V_L was continuously differentiable in z, δV_L={∇V_L}, and:

$\begin{array}{cc}{\stackrel{\stackrel{.}{~}}{V}}_L⋐-\alpha e_1^2+e_1e_2+\chi e_2+\frac{1}{2}\left(\stackrel{.}{M}-V\right)e_2^2+B_k^s{\Omega }^s\left(k_1e_2^2\right)+B_k^s{\Omega }^s\left(k_2+k_3z+k_4{z}^2\right)K\left[\mathrm{sgn}\right]\left(e_2\right)e_2,& \left(157\right)\\ \mathrm{where}K\left[\mathrm{sgn}\right]\left(e_2\right)=\mathrm{SGN}\left(e_2\right)\mathrm{and}\text{:}& \\ \mathrm{SGN}\left(e_2\right)\stackrel{\Delta }{=}\{\begin{array}{cc}1& \mathrm{if}e_2>0\\ \left[1,1\right]& \mathrm{if}e_2=0\\ -1& \mathrm{if}e_2<0\end{array}& \left(158\right)\end{array}$

Using Property 8 and the fact that B_k^s(q)<0 ∀qεQ_c, Equation 157 was rewritten as:

{dot over ({tilde over (V)})}_L⊂−αe₁²+e₁e₂+χe₂−|B_k^s|Ω^s(k₁e₂²)−|B_k^s|Ω^s(k₂+k₃∥z∥+k₄∥z∥²)SGN(e₂)e₂  (159)

Since almost everywhere {dot over (V)}_L (z) ε{dot over ({tilde over (V)})}_L (z), Equation 159 demonstrated that, almost everywhere:

{dot over (V)}_L=αe₁²+e₁e₂+χe₂−|B_k^s|Ω^s(k₁e₂²)−|B_k²|Ω^s(k₂+k₃∥z∥+k₄∥z∥²)|e₂|  (160)

By using Young's inequality, equation 144, Property 7, and the fact that ε<max(−B_k^s), Equation 159 could be upper bounded, almost everywhere, as:

$\begin{array}{cc}{\stackrel{.}{V}}_L\le -\left(\alpha -\frac{1}{2}\right)e_1^2-\left(\epsilon c_{\Omega 1}k_1-\frac{1}{2}\right)e_2^2+\left(c_1-\epsilon c_{\Omega 1}k_2\right)e_2+\left(c_2-\epsilon c_{\Omega 1}k_3\right)ze_2+\left(c_3-\epsilon c_{\Omega 1}k_4\right){z}^2e_2& \left(161\right)\end{array}$

Provided the gain conditions in Equation 154 were satisfied, Equation 150 could be used to rewrite Equation 161 as:

$\begin{array}{cc}{\stackrel{.}{V}}_L\le -\frac{{\gamma }_1}{{\lambda }_2}V_L& \left(162\right)\end{array}$

where γ₁ was defined in Equation 153. The inequality in Equation 162 could be rewritten as:

$\begin{array}{cc}{e}^{\frac{{\gamma }_1}{{\lambda }_2}\left(t-t_n^{\mathrm{on}}\right)}\left({\stackrel{.}{V}}_L+\frac{{\gamma }_1}{{\lambda }_2}V_L\right)\le 0& \left(163\right)\end{array}$

which was equivalent to the following expression:

$\begin{array}{cc}\frac{}{t}\left(V_Le^{\frac{{\gamma }_1}{{\lambda }_2}\left(t-t_n^{\mathrm{on}}\right)}\right)\le 0.& \left(164\right)\end{array}$

[0133] Taking the Lebesgue integral of Equation 164 and recognizing that the integrand on the left-hand size was absolutely continuous allowed the Fundamental Theorem of Calculus to be used to yield:

$\begin{array}{cc}{V}_Le^{\frac{{\gamma }_1}{{\lambda }_2}\left(t-t_n^{\mathrm{on}}\right)}\le C& \left(165\right)\end{array}$