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Functional electrical stimulation (FES) is used to externally excite muscles that would otherwise be uncontrollable to paraplegic patients.  A project has been proposed to develop a system that would restore gait to paraplegic patients by using a closed-loop control scheme to deliver FES to these muscles.  The idea is quite simple: deliver the proper electrical impulse at the correct time to the correct muscles that will render biped locomotion to a human subject.  But the theory and practical considerations are extremely complex; many of which have no known solution.

The purpose of this document is to address a few of the topics that need to be considered before such a system can be realized.  Background will be presented to familiarize the reader with what has been done and can be expected in the near future.  The control system as a whole will be dissected for possible problems.  A discussion on the feedback instrumentation will be presented, and practical limitations and foreseeable pitfalls will be explored.  The mechanical dynamics of such a complex system are so involved and complicated that this document can only hope to briefly cover some of the most crucial topics.

Since the time when the first pre-historic person began to walk on two legs, mankind has been given the gift of bipedal locomotion.  This very gift is what allowed humans to evolve into intelligent creatures by freeing up their hands to work and create with.  Tragically, many people cannot voluntarily move their own legs even when they have functional muscles.  But, even if their own nervous system cannot activate those muscles, external stimulators have been know to activate those very muscles.  In theory if the proper sequence of stimulations is given to those muscles, a person who had no voluntary control of their muscles would be able to walk from the external stimulations.  While the basic concept of this may be easy to understand, many of the details needed to develop such a system are truly complex.
 

The use of FES in the legs
A human spinal cord injury (SCI) can cut off the transmission of neural information between the central nervous system (CNS) above the lesion and the CNS and peripheral nerves below the lesion.  Such an injury results in muscular paralysis and lack of proproceptive and exeroceptive sensations.  Functional electrical stimulation (FES) can help in regaining limited locomotor activities in numerous humans with paralysis.  However, there are many unsolved questions relating to how and when to use FES for locomotion restoration, and what the real benefits of this technique are.

Neuromusculoskeletal questions arise from muscle fatigue, reduced joint torques generated through FES in comparison to CNS-activated torques in healthy subjects, modified reflex activities, spasticity, functional and joint contractures, and osteoporosis and stress fractures.  Recent FES systems stimulate motor units (synchronously) with stimulation frequencies above usual physiological values in normal subjects, affecting the duration of FES use because of muscle fatigue (Bigland-Ritchie el al., 1979).  From animal and human experiments it is evident that low-frequency chronic electrical stimulation is effective in increasing fatigue resistance.  This proves that muscle fatigue associated with low-frequency electrical stimulation is of peripheral origin, and that the loss of force is probably due to fatigue of fast-contracting glycolytic fatiguable type-II motor fibers, and is not caused by failure of neuromuscular transmission or conductance of the peripheral nerve (Kralj and Bajd, 1989).
 

Control methods
Many control FES algorithms for lower extremities have been developed in the last two decades.  The first control algorithms applied to FES systems for gait restoration were of the open-loop type.  Open-loop control assumes a complete knowledge of the system and its behavior in different environmental conditions.  Available data on gait performance of paralyzed individuals are often inadequate, which makes the application of the open-loop controllers in rehabilitation very difficult.  In addition to the difficulties mentioned above, individual properties such as muscle fatigue, spasticity, joint contractures, and muscle denervation must be modeled quantitatively for the synthesis of an open-loop controller.  Most of control systems adopted the principle of memorized and triggered FES sequences.  These sequences are based on the recorded average EMG patters in normal individuals (Marsolais and Kobetic, 1983; Thoma et al., 1983).  Direct, computer-controlled electrical stimulation was proposed by the Cleveland group (Chizeck et al., 1988), emphasizing muscular properties and a discrete event model with control goals for restoring locomotion functions.

Some investigators have introduced formal modeling to solve global issues such as standing stability, control strategies, feedback design, or synthesis of rhythmic joint trajectories for gait.  Kralj et al. (1990) proposed the synthesis of FES sequences on an execution level after the required joint torques were determined.  It is important to mention that the open-loop control relates to activation of the FES system, but the upper part of the body, including the hands and arms over the parallel bars, walker or crutches, actually works as a correction mechanism and provides feedback control.  The preserved voluntary motor control above the lesion allows the use of open-loop control, because the compensatory forces and movements can be accomplished.  A certain programming (feed-forward) control is required for fast movements and it is very convenient for cyclic motions.  However, the environmental changes in addition to muscle fatigue decrease the efficacy of the feed-forward control.

The second approach, which has been suggested and is used for some present FES systems, involves closed-loop control, which relies on the use of feedback.  When introducing feedback, questions arise about the nature and quality of sensors and their applicability in real time.  Two different feedback approaches exist.  One is based on natural sensors (e.g. myoelectric activity) as a source of control signals (Graupe and Kohn, 1988).  The second analytic closed-loop control method uses artificial sensory feedback (Crago el al., 1986).  What actually matters is not just the output of sensors, but their overall properties.  Man-machine systems are in great need of distributed matrix-type sensory systems with high resolution.  Several strategies were tested for FES systems.  At Wright State University, computer-controlled walking incorporating feedback principles was proposed (Petrofsky et al., 1984).
 

Performance of FES systems
Over the past three decades, many research groups have shown that partial restoration of function to paralyzed limbs can be achieved through neural prostheses systems that use functional electrical stimulation (FES) to activate the dormant muscles.  FES systems use controlled electrical currents to activate muscles that have lost their connection to the spinal cord and higher-level control centers, but still have their peripheral nerve supply intact.  In many cases, this form of stimulation can provide a more natural movement than can be achieved through traditional mechanical orthosis.  The added advantage of stimulation is that it provides dynamic assistance, rather than the fixed stability provided by orthosis.  However, despite a great deal of research, the methods by which FES systems can be used to meet practical needs of paralyzed subjects has seen limited success outside the laboratory.  Much work remains before FES motor function restoration can aid the large number of disabled individuals outside the research laboratory.  Baker et al. (Baker et al., 1993) have presented the state of the art in the clinical use of FES and have identified future research needs that must be explored before FES can be used for gait assistance or other functional tasks.

The major factors limiting the use of FES are:  1) the current delivery system, electrodes; 2) lack of availability of the control systems driving the multiple channel stimulation systems: the inconsistencies associated with the desired outcome and the stimulated response in the open loop systems and the difficulties in sensing feedback signals to be used by the closed-loop controller; 3) the proper interface system to inform (feedback) the subject of the state of the system; 4) the inadequacy of present system performance, i.e., speed of stimulated gait and the required energy expenditure during FES gait; 5) lack of understanding of how to deal with kinematic and kinetic redundancy of the system and how to generate the optimal excitation patterns to drive the skeletal system.  Although the challenges are many, the present study considers only some of the control issues that will contribute to the realization of a more efficient FES walking system once the other technical difficulties are adequately addressed.  By developing models and strategies to control the system, the future performance of FES systems can be studied through simulation, without the constraints imposed by our current technology.

One of the constraints for wider usage of FES systems is lack of efficient control.  In FES systems, in home or clinical applications, a set of switches must be controlled volitionally for up to six channels of stimulation, or preprogrammed sequence of stimulation patterns is applied to as many as 48 muscles (Kobetic et al., 1994).  Tuning the stimulation patterns is “hand-crafted” for each user.  Several more sophisticated control strategies are presented in the literature, involving open-loop, closed-loop, or nonanalytical control, but none is yet sufficiently practical to be widely used.

The lower extremities of a walking human can be represented as a complex, multiactuator, redundant, mechanical system.  Many methods have been employed to simulate the movement of human limbs, but the biomechanical models are generally very difficult to customize for a given individual, due to their complexity and poor user interface.  Hatze (Hatze, 1980) used the traditional Lagrangian approach to derive a mathematical model of the total human musculoskeletal system.  The model contained a linked set of ordinary first-order differential equations that describe the dynamics of the segments and muscles respectively.  With this model Hatze (Hatze, 1980) simulated the model with 17 segments and 46 muscles.

Zajac et al. (Zajac, 1989; Yamaguchi et al., 1990) developed a planar computer model to investigate paraplegic standing induced by FES.  Yamaguchi and Zajac (Yamaguchi et al., 1990) tried to determine the minimal set of muscle that could approximate able-bodied gait trajectories without requiring either higher levels of force or precise control of muscle activation.  They suggested that gait was more sensitive to changes in the on/off timing of the muscle stimulus than to its amplitude.  The process of adjusting the muscle activation levels was critically dependent upon accurately understanding the effect of each muscle on the dynamic response of the system.
 

Prospective
Because of primitive control solutions the obtained quality of FES enabled movements, functions, endurance, and practicality is very limited.  Therefore, in spite of the present state of technology and constant patient demands, practical and daily utilization of complex FES system is unlikely to happen until the control dilemmas and problems are solved adequately.

Advances in control theory and computer sciences give new hope that important breakthroughs into the understanding of life phenomena may be accomplished.  However in spite of many successful multidisciplinary efforts, the basic issues of man-machine relation have remained unsolved.  No evidence has been produced that the control theory relying on analytical and computer tools is capable to explain the motor control or the performance of the nervous system.  Complexity of biological entities is beyond the read of the mathematical control theory in its present form.  Dynamic systems whose variables are linked by a fixed functional relation have the great advantage that their past and future behavior may be determined analytically or by computer procedures once the mapping operator is known.  However, living systems do not belong to this class.  In this case, the mathematical modeling produces best results only when applied to limited time periods and structures.  The advent of the computer was instrumental in establishing new directions in the study of man-machine relation.  The enormously increased computational potential of the mankind was, directly and indirectly, the main factor leading to new approaches in many fields.  In this context, however, two basic contributions must be mentioned.  Great progress in the understanding of similarities and differences between the machine and the biological systems has been made possible when ways to transfer knowledge from the man to the machine were developed.  Symbol processing as used in the artificial intelligence to extend computer power in the cognitive direction and neural networks that simulate the functions of the nervous system by connectivism both contribute to the advancement of the use of FES for locomotion.  Opinions about the computer capabilities to reproduce activities of the nervous system are currently quite divided.  Both optimistic and pessimistic forecasts are found in the scientific community.  Such discussions are to a large extent arbitrary unless the full understanding of the nervous system exists and adequate execution organs can be applied.

The control of an assistive FES system is a problem in man-machine interaction.  The synthesis of a decomposed hierarchical controller for this multi-task and multivariable system is essential.  At the highest level, the subject directly interacts with the control system through a command interface comprising of manual switches, EMG control, or ideally through neural recording above the lesion.  Each locomotion mode is controlled with reference to a finite state model of the process.  This model serves to change the control strategy, as required, at different stages or phases of the locomotion cycle.  At the lowest level an associated predetermined control strategy comprising of a number of independent open or closed loop actuator controllers are required.  The actuator control level directly interfaces with the multichannel FES stimulator and external brace actuators if required.  Some of the actuator control loops can be based on artificial reflexes in addition to the described more traditional control system techniques.

It is important that the patient has complete intact control of his body above the lesion; the machine should only affect the part below the lesion.  At this point, an SCI patient can directly influence balance and posture by means of his preserved voluntary and reflex responses through his upper limbs and trunk musculature.  In the case of incomplete lesions, the influence is extended through preserved lower limb motor control and sensory pathways to a degree dependent on the particular pathways preserved.  The latter can be highly variable, almost individual, and enables the patient to learn to adapt, by means of compensatory movements.  This requires a certain degree of intelligence in the control.

The definition of the system to be studied includes the interactions between patient physiology, mechanical dynamics, controller, and the feedback loop.  Before a control system could be properly designed, a few things needed to be taken into consideration.
 

Biological considerations
Since contraction of skeletal muscle is a chemical process, contraction itself does not develop instantaneously but develops over tens of milliseconds until contractile power peaks.  After a muscle begins contracting, it takes a finite time before the muscle stops contracting due to the removal (by metabolically coupled pumps) of calcium from the inside of the sarcomeres.  When electrodes are applied to a motor nerve near the spinal cord, there is a considerably long latency period before the muscle actually contracts followed by a long period of tension development.  This simple model, however, vastly oversimplifies the complexities of the neuromuscular system in terms of actuation.

Sensors are required for the closed-loop control system, and the body has its own natural sensors.  These sensors are typified by the pacinian corpuscle, muscle spindle, and Golgi tendon organ.  Golgi tendon organs transmit information about force and the pacinian corpuscles transmit information about pressure to the central nervous system.  Using the pacinian corpuscle as an example, this is an onion-shaped device with a free nerve ending in the center.  Pressure on this device causes ionic leakage in the free never endings.  This results in trains of nerve impulses being developed.

However, most biological sensors are not ideal sensors.  In fact, biological sensors suffer from a phenomenon call accommodation.  If pressure is applied to the pacinian corpuscle over a long periods of time, the receptor will eventually change shape and adapt to the pressure so that no output will appear on the nerve (although the pressure is still being exerted).  Different sensors in the body adapt with different time constants. When the pressure is finally released, nerves will generally fire again until they accommodate to the new base line pressure. For this reason, accommodation is an important variable and certainly an important part of normal closed-loop control of the human body.  For example, pressure receptors from the skin show accommodation.  When we put our clothes on in the morning, we can feel their weight on our skin.  Over a period of time, the pressure receptors accommodate to the weight of the clothes and we no longer feel them.
 

Considerations for control
Accuracy of the control system increase with system gain.  However, if adequate measures are not taken to assure stable operation, the advantages of closed-loop control are useless.  Stability alone is not sufficient; one must have a system that demonstrates an adequate margin of stability.  Especially in neuromuscular control, the system must recover rapidly and smoothly from the stress of irregular inputs and severe disturbances.

The requirements of stability and accuracy are mutually incompatible.  To provide greater accuracy, the tolerable error for corrective action must be smaller and full corrective action must be initiated sooner.  This higher accuracy is acquired through higher system gain.  Time delays such as those occurring in neuromuscular excitation-contraction coupling which were not significant at low gain may become significant in a system having high levels of gain. Since there are significant time delays in the neuromuscular control problem, current closed-loop muscle control systems exhibit low gains (Crago, 1980).

Two categories of feedback control systems need to be defined when surveying this broad field as applied to muscle control.  The nature of the variable controlled seems to determine the choice of systems.

A regulator or regulating system is a feedback control system in which the reference input or command is constant, often for long periods of time.  The system is common when the controlled variable is isometric muscle force (i.e. under conditions of a constant muscle length).  Crago el al. (Crago, 1980) have demonstrated a family of isometric force plateaus using linear analog feedback control.  Petrofsky (Petrofsky, 1979) had earlier demonstrated a family of isometric force steps using nonlinear digital feedback control.  It is generally recognized that the muscle control problem is nonlinear.  Except for an occasionally homogeneous muscle, most muscles significant in gait are of mixed fast-twitch and slow-twitch populations (Ariano, 1973) and therefore nonlinear regarding their strength-recruitment profile.

Nonlinearities introduce significant complications to the automatic control problem.  First, the effective range over which the variable can be controlled is significantly reduced.  Second, and more important, standard stability criteria cannot be applied to nonlinear systems (Cosgriff, 1958).  For these reasons, researchers have often utilized piece-wise linear analysis to mathematically describe system variables.  This is rather straightforward when employing digital control, as it requires only the addition of appropriate software commands.

A servomechanism or servomotor system is a power amplifying feedback control system in which the controlled variable is mechanical position or a time derivative of position such as velocity or acceleration.  This system is common when the controlled variable is isotonic muscle length (i.e., under conditions of constant muscle force).  Servomechanisms are divided into type 0, 1, and 2.  The type number depends only on the number of poles at the origin of the open-loop transfer function, and not on the time constants of the function.  When the controlled variable is isotonic muscle length, then there is a type 0 servomechanism.  Partridge (Partridge, 1972) has developed and analyzed such a system using nonlinear analog feedback control.  When the controlled variable is isotonic muscle velocity, there is a type 1 servomechanism.

The preceding classification system is important to our understanding of the closed-loop muscle control problem.  By categorizing such feedback control systems into their appropriate category, we can then interpret the physiological experimental results in terms of control system science.  It is now accepted that control systems engineering spans not only the entire breadth of all engineering science, but the biological and social sciences as well.

Crago et al. (Crago, 1996) have recently proposed several new control strategies for neuroprosthesis systems.  Issues related to feedback for the patient have remained largely unanswered.  The major difficulty is that stimulation systems are incapable of effectively sensing the patient’s response such that closed-loop control of gait can be achieved.  Due to problems with feedback, clinical applications of electrical stimulation involving reciprocal gait have been restricted to open-loop with the aid of walker or crutches. With the orthosis providing stability a crude open-loop control is sufficient to propel the legs.
 

Design specifics
The objective of this specific project is the design and implementation of a control system using functional electrical stimulation that will allow paraplegic patients to regain gait.  The controller will be closed-loop, meaning feedback information will be incorporated in the control scheme to compensate for external disturbances.  The project is in its early stages of development and many of the details have not yet been worked out.

Feedback will be provided via sensor 5 sensor packs.  These sensor packs will be placed on the upper legs, lower legs, and one on the torso.  Each sensor pack will contain three accelerometers and three gyroscopes (piezoelectric vibratory gyroscopes; They will be discussed in section 4).  Six sensors in orthogonal parameter space should provide enough information (in theory) to uniquely determine the position and orientation of a solid object (e.g. linkage of leg or torso).  So, the five sensor packs with six sensors each should deliver raw enough information to the controller (microprocessor) to determine the kinematics of five-link model of patient (lower legs, upper legs, and torso).

Accelerometers will be responsible for predicting three degrees of freedom.  Acceleration data is numerically integrated twice to obtain position.  Initial position and velocity must be an input into the algorithm.  Integration of transducer data as it relates to accuracy is a double-edged sword.  On one hand the integration process tends to smooth out erratic signals (e.g. noise), and over a long time and if the noise is assumed Gaussian, then the noise is essentially eliminated.  On the other hand inaccuracies in the signal that do not have a mean at the actual value will carry throughout the integration process.  Integration is said to have memory because past values are incorporated into the current value.  Problems such as zero-drift and non-linearities would cause an integrated signal to become worse over time.  Once the initial values are set, there is no check to see if the integrated approximation is anywhere near where it should be.

Three orthogonal gyroscopes will provide the remaining three degrees of freedom.  They will be responsible for predicting angular position about the primary axes.  Gyroscopes output a voltage proportional to its angular velocity, so this signal will have to be integrated once to obtain an estimate of angular position.  The initial orientation of the three-dimensional object (e.g. shank) will be determined from the three accelerometers (assuming they are static).  Gyroscopes will also be responsible for keeping track of the direction of gravity.  The gravity vector must be known at all times so that the gravity vector can be subtracted from the acceleration vector that was deduced from the accelerometers.

The controller itself will be some sort of microprocessor / DSP unit capable of multi-input / multi-output processing.  The computation demand of the control system will call for a powerful processor.

Details of the control algorithm have not yet been worked out.  Designs range from bottom-up approaches such as using first principles to directly compute kinematics, to more top-down approaches that use routines such as neural networks to learn the pattern of inputs that will correspond to desired output.  It is not clear which method will be most suitable for this system, therefore, many alternatives will be explored.

Some of the more traditional controllers used in this field have been presented in section 1.3.  Other methods such as fuzzy logic and neural networks have found their way into some control designs.  Many of these techniques have just recently become practical because they often pay a hefty toll on computation.

Davoodi and Andrews (Davoodi, 1998) were able to simulate a closed-loop self-adaptive fuzzy logic controller based on reinforcement machine learning (FLC-RL).  The FLC-RL was able to recover from simulated disturbances approximating those encountered in FES assisted standing up in paraplegia.  Although the method appears to be promising, only the theoretical feasibility has been demonstrated, further work is required to demonstrate clinical feasibility.

Neural networks are another technique that has recently been very popular in non-linear control.  Despite its complex theory, neural networks are very simple to actually implement.  This is because the control algorithms don’t need to be derived from first principles.  Rather, the network learns the input/output relation.  Basically the network is given a certain operating domain of inputs and their corresponding outputs.  Software is then used that allows the network to develop patterns (similar to how the human brain learns).  These patterns from the point of view of the network represent the input /output relation (the transfer function).  Neural networks work very well for non-linear systems.  The actual input/output relation is in general extremely non-linear and often very difficult to write an analytic expression for.  Another problem with neural networks is that they begin to perform very badly if the input falls outside the domain that the network was trained on.  So, unforeseeable situations often give the network problems.

The motion of living organisms by means of legs, especially the locomotion of bipeds, has always been a challenging problem to scientists of different vocations: biologists, physiologists, medicine specialists, mathematicians, and engineers.  In spite of their efforts, however, this problem has not been solved yet in a satisfactory way.

Introduction
From the view point of mechanics the motion of living organisms can be interpreted as a result of changes in equilibrium conditions within the fields of forces in which the system finds itself.  The spontaneous motion due to the redistribution of tension in muscle groups modifies the relations between forces, bringing theses relations to equilibrium or taking them away from the equilibrium position.  Study of these systems and their motion requires certain simplification because the legged locomotion systems, and particularly the anthropomorphic mechanisms, represent extremely complex dynamic systems both from the aspect of mechanical-structural and control system complexity (Vukobratovic, 1975).  Nearly 350 muscle pairs are available to man for his complete skeletal activity.  Such a system involves great dynamical complexity, even if it is idealized to a system of rigid levers with simple torque generators acting at each joint.  In fact, mathematical analysis of the relationship between force and movement indicates that this form of interaction does not have a unique dependence.  This absence of uniqueness stem from the fact that the relationship between force and movement is generated in a biomechanical sense based on a second-order differential equation whose solution requires two initial values.  These constants of integration (initial position, initial velocity,…) can lead to quite different effects during the same initial innervation.  This complexity (mechanical and any other) is the reason why any attempt of practical realization of artificial control of biped locomotion is so difficult.

Mechanical complexity of locomotion systems is only one of the characteristics that make the study very complicated.  There are some other features to be mentioned, which also determine basic characteristics of the behavior of locomotion systems.

Presence of an unpowered degree of freedom (d.o.f.) is the most important characteristic of locomotion mechanisms, especially of bipeds, because of its crucial influence on the system stability.

Another fundamental characteristic of every legged motion is a certain repeatability of movements that generate it.  For a two-leg locomotion, the period on which movements are repeated over and over again is one step.  That means that the positions and velocities at the beginning and at the end of each step are the same, and only the motion satisfying theses conditions is acceptable for bipeds.  They are known as the repeatability conditions (Vukobratovic, 1975).  These conditions result from the nature of legged motion and impose additional constraints on the possible solution in the process of motion synthesis.  Also, it may happen that a particular gait type is required.

A third characteristic of two-leg locomotion is a permanent change of situations when the mechanism is supported on one foot and when the both feet are in contact with the ground.  When the kinematic chain playing the role of legs is in contact with the ground only at one end, while the other is in the swing phase, the situation corresponds to a single-support phase.  If, however, both ends are in contact with the supporting surface we speak of double-support phase.  Each of these two cases is characterized by quite different dynamic situations and is to be studied separately.

The presence of a closed kinematic chain lowers the system order, and the mathematical equations describing it are much more complicated than those for the open kinematic chain used to describe the single-support phase.  Additionally, when the system is in the double-support phase, there is not a unique set of driving torques and reaction forces that can be associated with the motion performed.  In fact, the closed chain can support the internal torques which act in opposition but do not contribute to the mechanism motion, and which are only useless load to joints.  In order to overcome this, it is necessary either to make some assumptions about the way in which the ground reaction forces are divided between the feet during a double-support phase, or to directly measure the forces acting on the feet.

There are two approaches to studying locomotion activity: experimental, by investigating the motion of living organisms, and theoretical, by mathematical modeling.

As already mentioned, the motion of a complex mechanism such as the human skeleton involves great dynamic complexity.  Consequently, its mathematical description results in a high order system of nonlinear differential equations.  However, the writing of such a large set of differential equations is always associated with the possibility of making mistakes.  Another, more serious, problem is the impossibility to solve them in analytic way.  This problem can be overcome in two ways.  One of them is the simplification of the mechanical model used to represent the locomotion system, up to the level when only the characteristics of interest are preserved.  For example, the human stance behavior is well approximated by a single inverted pendulum model, controlled by torques applied at the pendulum base.  Then, the whole body is approximated by one single massive link and the controlling torque corresponds to the ankle torque of the human body.  Of course, more complex motions require more complex models.  A further simplification is concerned with the system of differential equations.  This is usually achieved by linearization, though some other methods have been employed, too.  However, some very important features of locomotion systems may be lost by simplification, and for this reason it should be used with additional care.  Another approach to mathematical modeling (apart from simplification of either the mechanism, or the mathematical model) is transferring the task of forming and solving the model to a computer.  Then, the mathematical complexity is no more a limiting factor, and it can be chosen to match in the best way all requirements of the motion under investigation.  If an appropriate software package is available, all changes, including the structural ones, can be realized by changing simply the input data (Stepanenko, 1976).  Of course, the solution is obtained in numerical form, which is not convenient for further analysis.  However, some software packages have offered the solution in analytic form (Vukobratovic, 1985), evening the case of closed kinematic chains.  The analytic models are very convenient if real time computation of dynamics is required.
 

Artificial gait synthesis
Studies of human locomotion by measuring all its characteristics of interest can answer the question how man walks.  However, the question why he walks in such a way remains still unanswered.  As stated by Bernstein (Berstein, 1947), the human walk is a most automated motion, and no central nervous system is involved in its steady regime.  Thus, the locomotion process appears to be similar to a certain algorithm that is executed over and over again when no disturbances occur.

Some authors have tried to define the motion of an artificial biped mechanism, but in all cases they had to introduce certain simplifications.  Most often they assume massless legs.  As a consequence, the motion of the legs does not influence the body motion, which is of great importance in preserving a low order of model complexity.  Such a simplification, introduced by Beletskii (Beletskii, 1984), implies the reduction of the system of differential equations describing the body motion to one vector equation.  If the motion is restricted to one plane, this is a differential equation of second order, but if the torso moves in space then three such differential equations are necessary to define it. If the hip trajectory is prescribed, and the position of the supporting points of legs on the ground is known, then the legs’ positions are uniquely defined (Beletskii, 1984).  However, only the legs angles at the end of each half-step are thus defined, while the legs trajectories between these two terminal positions remain unknown.  Therefore, the assumption of massless legs can hardly be accepted as adequate for the actual system.

A quite different approach to the gait synthesis, base on optimal programming, has been proposed by Chow and Jacobson (Chow, 1970).  They have considered a seven-link planar mechanism consisting of two three-link legs (the shank and thigh are massive, whereas the foot is massless) and the torso modeled as a heavy inverted pendulum.  All geometrical and inertial data are supposed to be known.

The mechanism motion is described by five nonlinear differential second-order Lagrange’s equations.  Then, the problem of gait synthesis can be defined as: how to determine generalized forces such to perform a periodical walk.  Because of the great complexity of the problem, the authors introduced the following simplifications.  The first simplification is to prescribe the hip trajectory, which allows splitting the system into three independent parts: the torso and two legs.  The second assumption is that all partial derivatives of the hip trajectory are zero.  This enables the hip joint to be considered as the origin of a local coordinate frame that performs the desired motion.  In this way, the so-called “simplified dynamics”, is defined.  By additional studies of the foot trajectory in the support and deploy phase, and after some simplification related to the force which act on the foot, the model is reduced to two d.o.f. with kinematic constraints and the force and torque at the ankle joint.  In this way the model is prepared for the optimization procedure.  The authors prescribed the boundary conditions ensuring repeatability, so that the problem of gait synthesis can be restated as: how to define the torques at the hip and knee joints to perform the system’s periodical trajectories, and, at the same time, to minimize the desired criterion.  After numerous simplifications, the system of differential equations reduces to four equations of the first order.  To obtain the solution, the maximum principle was employed.  As a result, the authors arrived at a two-point boundary value problem that they solved numerically.

This work can serve as proof of how difficult the optimization task is by itself.  On the basis of the criterion of minimum energy consumption, the authors tried to obtain the driving torques at the hip and knee, and the appropriate “optimal” trajectories of the corresponding joints.  But, to make this optimization problem solvable, they made some assumptions that have considerably simplified the initial locomotion problem.  Thus, the motion problem was reduced to the sagittal plane.  Here, the problem of dynamic equilibrium was not solved, and the problem of the unknown dynamic reactions was avoided by using experimental data.  Given these simplifications, it would be rather difficult to expect that result of the optimization of such a model could present some realistic and truly optimal results.

Another possible approach is to obtain a direct solution of the model with some additional constraints imposed on the system.  A very good example of this is the work of Formal’skii (Formal’skii, 1982).  He considers a five-link planar model with all heavy links in the single-support phase only.  The problem is how to define driving torques at the mechanism’s joints such to ensure the mechanism performs a periodical walk.  In addition, Formal’skii assume the driving torques act for a very short time in the beginning of each half-step (the so-called impulse control).  The constraints of repeatability conditions are also imposed.  The solution is obtained in two ways.  The first approach is to linearize the model around the vertical position and then solve it.  In the second approach, a nonlinear model is solved by using an iterative procedure.  Two solutions are obtained: the symmetric – which is very close to the linearized case, and non-symmetric – resembling the way in which man walks.

In this way, by a direct solving of the whole model, only the repeatability conditions are satisfied.  However, the very important problem of the overall mechanism of stability is not taken into account, and neither is given the possibility to choose a desired gait type.  If these results are supposed to be applied in rehabilitation, this would be a serious drawback.

These problems can be overcome by using the synergy method.  Simply speaking, to the one part of the system dynamics is prescribed, whereas from the rest of the system such “compensating” dynamics is required to bring the system to equilibrium with respect to the task specifications and corresponding conditions of dynamics connections. It should be pointed out that in this way it comes to a specific procedure of dimensionality reduction which is neither a system’s simplification, nor its mathematical linearization.  By prescribing various synergies for the one part of the system an adequate set of compensating synergies can be obtained, and thus replace all dynamic possibilities of the system by those dynamic forms being of interest for the example considered.
 

Control and stability
Because of the presence of unpowered degrees of freedom (d.o.f.), the most serious problem that has to be solved is the overall system stability.  This is the reason why the control synthesis at two stages has been adopted.  At the first stage, the stage of nominal regimes, such control has to be synthesized to ensure the system’s motion in the absence of any disturbance along the exact nominal trajectories calculated in advance.  It should be derived in such a way to satisfy the conditions of both the desired gait type and overall system equilibrium.  At the second stage, the stage of perturbed regimes, only deviation of the actual state vector from its nominal value is considered, and additional control is applied to force the system state to its nominal.  However, the movement thus realized, can induce an additional inertial force that, on the other hand, can produce rotation of the whole system around the foot edge.  The movement should not make the situation worse, by producing some additional inertial forces.  As the nominal system motion is synthesized under the condition of the overall system equilibrium, the best was to realize the system’s return from a disturbed to it nominal regime is to prevent the excursion of the system state out of a certain finite region.  Deviation of the zero-moment point from its nominal position is adopted to be the measure of the system’s overall deviation from nominal trajectories.

The well-known aggregation-decomposition method of stability analysis via Lyapunov’s vector function in finite regions of state space can be used for stability analysis.  In this method, which is very convenient for computer use, it is assumed that each d.o.f. is powered by its “own” actuator.  However, in the case of the d.o.f. that are formed in the contact of the foot and ground surface, such actuator cannot be applied, and the method is not directly applicable.  This problem, however, can be solved by the appropriate modeling of the mechanism.  With manipulation robots each subsystem model usually contains one powered d.o.f.  In order to include unpowered d.o.f., the system should be modeled in such a way to unite the model of some powered and some unpowered d.o.f. into one subsystem model.  The subsystem thus modeled has its own input and the stability is investigated for the whole “composite subsystem”.

A system that will allow paraplegic patients to acquire biped locomotion through closed-loop control of functional electrical stimulation (FES) has been proposed.  The contents of this document present vital information that needs to be considered before such a system can be constructed.  While this document does not claim to cover everything that needs to be taken into consideration, a few specific topics are covered in detail.
 

Performance of current FES systems
Over the past three decades, many research groups have shown that partial restoration of function to paralyzed limbs can be achieved through neural prostheses systems that use functional electrical stimulation (FES) to activate the dormant muscles.  However, despite a great deal of research, the methods by which FES systems can be used to meet practical needs of paralyzed subjects has seen limited success outside the laboratory.  Much work remains before FES motor function restoration can aid the large number of disabled individuals outside the research laboratory.  The major factors limiting the use of FES are:  1) the current delivery system, electrodes; 2) lack of availability of the control systems driving the multiple channel stimulation systems: the inconsistencies associated with the desired outcome and the stimulated response in the open loop systems and the difficulties in sensing feedback signals to be used by the closed-loop controller; 3) the proper interface system to inform (feedback) the subject of the state of the system; 4) the inadequacy of present system performance, i.e., speed of stimulated gait and the required energy expenditure during FES gait; 5) lack of understanding of how to deal with kinematic and kinetic redundancy of the system and how to generate the optimal excitation patterns to drive the skeletal system.
 

The use of feedback control
Although feasibility has been demonstrated, current lower extremity FES technology is deficient in ways that have delayed and limited its clinical use.  In general, FES systems need to be easier to use, more reliable, and safer; they also must provide a greater level of function to the user.

Advances in control theory and computer sciences give new hope such that important breakthroughs into the understanding of life phenomena may be accomplished.  However in spite of many successful multidisciplinary efforts, the basic issues of man-machine relation have remained unsolved.  No evidence has been produced that the control theory relying on analytical and computer tools is capable to explain the motor control or the performance of the nervous system.

Although the implementation of open-loop control is less problematic than closed-loop control, it is difficult to achieve truly functional restoration without sensory feedback. The efforts often are tempered by the lack of models witch can adequately describe the input-output behavior of skeletal muscles.  Inconsistencies between the simulation input and the functional outcomes frequently are problems with open-loop systems.  The problem is further compounded by a lack of knowledge concerning normal muscle activation patterns, recruitment, fatigue, and its effects on muscle response (Giat, 1993), and appropriate adjustments required for the artificial excitation through FES.  The problems involving coordination, timing and interaction between different neural, muscular, and skeletal structures also must be addressed.  Another major problem with the design of an adequate controller is the extreme nonlinear nature of the plant (the system to be controlled).

Control theory predicts that a pure time delay in a closed-loop control system can be catastrophic.  Typical phase lags between controller and plant are inherent in any system.  But, a pure time is independent of frequency and magnitude of the control signal.  Unaccounted for time delays often cause the system to go unstable.

Stability alone is not sufficient; one must have a system that demonstrates an adequate margin of stability.  Especially in neuromuscular control, the system must recover rapidly and smoothly from the stress of irregular inputs and severe disturbances.

The requirements of stability and accuracy are mutually incompatible.  To provide greater accuracy, the tolerable error for corrective action must be smaller and full corrective action must be initiated sooner.  This higher accuracy is acquired through higher system gain.  Time delays such as those occurring in neuromuscular excitation-contraction coupling, which were not significant at low gain, may become significant in a system having high levels of gain.  Some time delays to watch for are:
· A delay occurs at the junction between the nerve and muscle (the neuromuscular junction).  The electrochemical nerve impulse is transduced into the release of a chemical substance called a neurotransmitter which then must diffuse across to the sarcolemma of skeletal muscle.
· Since contraction of skeletal muscle is a chemical process, contraction itself does not develop instantaneously but develops over tens of milliseconds until contractile power peaks.
· The finite time to compute the control signal from the feedback sensors could be significant.

In order to truly provide externally powered biped locomotion through FES, a feedback loop must be incorporated into the control scheme.  Only feedback compensation will be able to provide the vital information needed to sustain a stable plant in the presence of unaccounted disturbances.  But, many of the difficulties that are coupled with the use of closed-loop control may be impractical to overcome with present technology.
 

Man / machine interface
Complexity of biological entities is beyond the read of the mathematical control theory in its present form.  Dynamic systems whose variables are linked by a fixed functional relation have the great advantage that their past and future behavior may be determined analytically or by computer procedures once the mapping operator is known.  However, living systems do not belong to this class.  In this case, the mathematical modeling produces best results only when applied to limited time periods and structures.  The advent of the computer was instrumental in establishing new directions in the study of man-machine relation.  The enormously increased computational potential of the mankind was, directly and indirectly, the main factor leading to new approaches in many fields.  In this context, however, two basic contributions must be mentioned.  Great progress in the understanding of similarities and differences between the machine and the biological systems has been made possible when ways to transfer knowledge from the man to the machine were developed.  Symbol processing as used in the artificial intelligence to extend computer power in the cognitive direction and neural networks that simulate the functions of the nervous system by connectivism both contribute to the advancement of the use of FES for locomotion.  Opinions about the computer capabilities to reproduce activities of the nervous system are currently quite divided.  Both optimistic and pessimistic forecasts are found in the scientific community.  Such discussions are to a large extent arbitrary unless the full understanding of the nervous system exists and adequate execution organs can be applied.

The control of an assistive FES system is a problem in man-machine interaction.  The synthesis of a decomposed hierarchical controller for this multi-task and multivariable system is essential.  At the highest level, the subject directly interacts with the control system through a command interface comprising of manual switches, EMG control, or ideally through neural recording above the lesion.  Each locomotion mode is controlled with reference to a finite state model of the process.  This model serves to change the control strategy, as required, at different stages or phases of the locomotion cycle.  At the lowest level an associated predetermined control strategy comprising of a number of independent open or closed loop actuator controllers are required.  The actuator control level directly interfaces with the multichannel FES stimulator and external brace actuators if required.  Some of the actuator control loops can be based on artificial reflexes in addition to the described more traditional control system techniques.
 

Instrumentation
In experiments with human test subjects where accelerations of selected body segments ought to be measured, accelerometers may be mounted in various ways and at different locations to the segment of interest.  However, this segment consists of rigid and soft tissue.  For such measurements several questions are of interest such as:
· Which acceleration should be determined: the acceleration of a specific rigid part of the segment, the acceleration of a specific soft tissue part, or an average of rigid and soft tissue?
· How well does the measured acceleration correspond to the actual acceleration of interest?

Morris’ paper (Morris, 1973) claimed to have shown that accelerometers could be used to provide sufficient information to define the movement of a segment of the body.  The validity of this statement comes down to the definition of sufficient.  While results showed that he gathered acceleration data and then did a double integration to approximate position, the position data was never validated by an independent measure.  The study claimed to have used cinephotography to recognize unusual features in gait, but they never compared the two methods at all.

Smidt et al (Smidt, 1977) did show that they were able to estimate the movement of an accelerometer by validating the results with an independent measure for the test case.  When considering the results of this study keep in mind the nature of the movement.  It was a smooth linear movement with mild accelerations and decelerations.  While the results look promising for this idealized motion, who knows what types of results would be obtained by a more jerky motion (e.g., the motion of leg hitting the ground).

Each sensor pack was equipped with three orthogonal gyroscopes that would be used to approximate the three-dimensional angular position of its corresponding segment.  Experiments were conducted in attempt to calibrate them.  Due to unknown reasons, the results of the attempt were deemed unsatisfactory.  Some possible reasons were proposed, but none could be confirmed.  The magnitude of the inaccuracies in the calibration experiment was quite alarming.  Attempts of any calibration between the sensor data and the Kintrac data fail miserably.

Special care must be taken when designing the data acquisition system.  In order for this type of system to work properly, accurate measures of angular and spatial are necessary.  Small errors in measurement can yield catastrophic results for the control system.  Some errors to especially look out for include:
· Mechanical damping from soft tissue
· Accurate tracking of the gravity vector so that gravity components can be subtracted from their respective direction
· Signal artifacts due to phenomena such as vibrations induced from “heel-strike” or from external sources in accelerometers
· Manufacturing imperfections of gyroscopes that will cause departures from the ideal mass, stiffness and damping distributions and therefore effect the resonator dynamics.
· The changes that occur in the dynamics of the resonator of a gyroscope due to temperature changes and aging with time
· Nonlinearities including hysteresis, bias (zero offset), random drift (due to aged sensors), and dead band
· Errors due to accelerometers responding to rotation and gyroscopes responding to acceleration (g-sensitivities and anisoelasticity)

The resolution of state-of-the-art sensors does allow for enough accuracy in measurement to properly track the kinematics of human segment to within enough precision necessary for the controller.  But, the inaccuracies found in the feedback signals are usually due to other factors and not caused by the resolution of an ideal sensor.  These inaccuracies are due to factors discussed in sections 3, 4, and 5.
 

Complex dynamics
The motion of living organisms by means of legs, especially the locomotion of bipeds, has always been a challenging problem to scientists of different vocations: biologists, physiologists, medicine specialists, mathematicians, and engineers.  In spite of their efforts, however, this problem has not been solved yet in a satisfactory way.  The lower extremities of a walking human can be represented as a complex, multiactuator, redundant, mechanical system.  Many methods have been employed to simulate the movement of human limbs, but the biomechanical models are generally very difficult to customize for a given individual, due to their complexity and poor user interface.

Mechanical complexity of locomotion systems is only one of the characteristics that makes the study very complicated.  There are some other features to be mentioned, which also determine basic characteristics of the behavior of locomotion systems.  Presence of an unpowered degree of freedom (d.o.f.) is the most important characteristic of locomotion mechanisms, especially of bipeds, because of its crucial influence on the system stability.  Another fundamental characteristic of every legged motion is a certain repeatability of movements that generate it.  For a two-leg locomotion, the period on which movements are repeated over and over again is one step.  That means that the positions and velocities at the beginning and at the end of each step are the same, and only the motion satisfying theses conditions is acceptable for bipeds.  A third characteristic of two-leg locomotion is a permanent change of situations when the mechanism is supported on one foot and when the both feet are in contact with the ground.  When the kinematic chain playing the role of legs is in contact with the ground only at one end, while the other is in the swing phase, the situation corresponds to a single-support phase.  If, however, both ends are in contact with the supporting surface we speak of double-support phase.  Each of these two cases is characterized by quite different dynamic situations and is to be studied separately.

Because of the presence of unpowered degrees of freedom (d.o.f.), the most serious problem that has to be solved is the overall system stability.  This is the reason why the control synthesis at two stages has been adopted.  At the first stage, the stage of nominal regimes, such control has to be synthesized to ensure the system’s motion in the absence of any disturbance along the exact nominal trajectories calculated in advance.  It should be derived in such a way to satisfy the conditions of both the desired gait type and overall system equilibrium.  At the second stage, the stage of perturbed regimes, only deviation of the actual state vector from its nominal value is considered, and additional control is applied to force the system state to its nominal.  However, the movement thus realized, can induce an additional inertial force that, on the other hand, can produce rotation of the whole system around the foot edge.
 

Concluding remarks
A reoccurring theme of this entire document revolves around the notion that the proposed project is impossible to do.  While the validity of this statement cannot be dismissed, the reader of this document must keep two things in mind:
1. While the contents of this document were compiled in September, October, and November of 2000, many of the references used were a few years, and up to ten years old.
2. A fundamental doctrine of research suggests that every innovation has its niche in time.  Or, stated alternatively, technology begets technology.

Certain technologies need to be improved before the system in question can be properly constructed.  Low noise techniques need to be improved before an accurate enough signal (needed by the control algorithm) can be acquired.  The hardware required to perform such demanding processing must be powerful enough to satisfy the computational requirements, yet be portable enough to carry on your person.

The objectives of this project can only be met by applying cutting-edge technology to a brilliant system design.  Such a system is not impossible, but will require tremendous work effort to complete.
 

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