Active Orthosis for Ankle Articulation Pathologies

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1 Active Orthosis for Ankle Articulation Pathologies Carlos André Freitas Vasconcelos IST, Universidade Técnica de Lisboa Av. Rovisco Pais, Lisboa, Portugal Abstract This work addresses the analysis, simulation, and control of the ankle joint during gait, with the goal of designing an active ankle-foot orthosis (AAFO) to assist individuals without motor control at the ankle joint. An elastic foot contact model was developed in SimMechanics, allowing that with knee and leg kinematics prescribed, the correct movement of ankle joint during gait cycle could be achieved by the controller. Three control strategies were implemented: proportional-derivative (P-D) control and linear quadratic regulator (LQR) for reference following, and impedance control (IC) for mimic the human control by considering an ankle stiffness and damping during gait cycle. An AAFO was designed for assisting the ankle movement during gait with the characteristics on biomechanical requirements for an individual with a body mass of 70kg. A series elastic actuator (SEA) provides the active control of the AAFO. The SEA was designed to be lightweight, compact, and with enough power to provide the ankle movement during gait. The presented AAFO is expected to be autonomous and assist the ankle movement in all phases of the gait cycle. Keywords: Active Ankle-Foot Orthosis (AAFO), Series Elastic Actuator (SEA), Ankle-Foot Complex Simulation and Control, Impedance Control, Human Gait. 1. INTRODUCTION Standing or walking is something people take it for granted, however, every year thousands of people are prevented of doing it. Facing various gait disabilities, several approaches of active systems have been made on trying to improve the patient s life, such as assist people actively with robotic solutions. It is in this context that the concept of wearable robots has emerged, where the robotic counterparts of orthoses are robotic exoskeletons (1). The function of the exoskeleton have been widely developed, since it can be applied not only on restoring handicapped functions or rehabilitation, but also on the improvement of human body performance. As a part of the locomotor unit, the ankle and foot are the final segments that provide support to the body by distributing gravitational and inertial loads. Thus, this dissertation investigates in particular the disabilities at the ankle joint, considering the worst scenario where the individual has very low or any motor control of the ankle joint. Assist this pathology, which can occur due to weakness in dorsiflexor and plantar flexor muscles, allows the full functioning of the ankle joint and faces the challenge of interrelating biomechanical principles with engineering concepts. Several techniques on assisting ankle pathologies have been developed for the different pathologies (2), however, ankle-foot orthoses (AFOs) are the most used systems. AFO is usually an orthosis that covers the foot, spans the ankle joint and covers the lower leg. This passive lower limb orthotic device can be divided into categories of metal, plastic, and more recently carbon (3), where its recent ankle joints allow much motion as possible while blocks unwanted movement (4). With the necessity of assisting ankle pathologies more actively, the AFOs have been endowed of active systems, generally denominated of active ankle-foot orthoses (AAFOs). Several active ankle-foot orthosis have been developed in the last decade for rehabilitation and gait assist purposes. The objective of this work is to develop a lightweight and autonomous system capable of providing ankle movement during gait cycle. A passive ankle-foot orthosis will be the basis of the system, which should be endowed of sensing devices that identify the different phases of the gait, and an actuator with enough power to provide the ankle movement during the gait cycle. Several control strategies will be developed to the system in order to control the angular position of the ankle joint during the gait cycle. 1

2 2. ANKLE FOOT COMPLEX Biomechanics of the human movement can be defined as the interdiscipline which describes, analyzes and assesses human movement (5). Understanding human movement is essential when developing systems capable of assisting human body, requiring the study of anatomy and physiology of the human body. In particular for the ankle movement, two movements are possible in the sagittal plane: dorsiflexion and plantar flexion. Figure 2.1: Movements of the foot in the sagittal plane(6). The kinematic variables evaluated in gait analysis are usually linear and angular displacements, velocities, and accelerations (5). Limb angles in the spatial reference system are defined using counterclockwise from the horizontal as positive. Thus angular velocities and accelerations are also positive in a counterclockwise direction in the plane of movement, which is essential for consistent use in subsequent kinetic analyses. Kinetics in human gait represents the forces and torques that cause the motion of the body (7), where both internal and external forces are included. Internal forces come from muscle activity, ligaments or friction in the muscles and forces, while external forces come from the ground or from external loads (5 p. 10). Figure 2.2: Schematic of the lower leg during gait - free body diagram of the foot showing the ankle moment, weight of the foot ( ), and ground reaction force (2 p. 59), The moment of force acting through ankle joint,, becomes (2.1) where is the ankle rotational inertia due to the mass of the foot, the ankle angular acceleration, and the position vectors of the center of mass (COM) and center of pressure (COP) relative to ankle joint center, respectively. 3. SIMULATION AND CONTROL When a passive system is endowed of active system, it is necessary to provide some kind of control strategy to have the desired behavior. When endowing an ankle-foot orthosis (AFO) with an actuator to assist ankle pathologies, several control strategies can be implemented. Three control strategies were proposed in this work. With rehabilitation purposes, a variation of a Proportional-Integral-Derivative (PID) controller and a Linear Quadratic Regulator (LQR),where the system is controlled by reference following. With a control behavior similar to the human control, Impedance Control (IC) is the control strategy for assisting the AAFO during free walking where the controller tries to mimic the human control Control System A control system consists of subsystems and processes (or plants) assembled for the purpose of controlling the output of the processes (8). These processes are characterized by its inputs and outputs. The system input is the manipulated variable, which is the condition that is varied by the controller to affect the controlled variable. The controlled variable is the system output, which is the condition that is measured and controlled (9). Manipulated and controlled variables are the key concept for the entire designing, since a bad choice of variables can affect the possibility of controlling the system. The control scheme presented in Figure 3.1 is a closed-loop control system, where the difference in the desired and actual condition creates a correction control command to remove the error (10). The controlled variable involved in this control scheme is the ankle angle sensor output, while the manipulated variable is the ankle torque that causes the rotation of the ankle joint. In fact, the ankle torque results from the output force of a series elastic actuator (SEA), this is a linear actuator in series with a spring that causes a torque through the ankle joint due to the offset of the application point with respect to the joint center. The existence of a spring in the actuator provides some mechanical compliance, acting 2

3 also as an indirect force sensor by measuring the deflection of the spring. It also makes the system more robust to the application of sudden external forces and more close to its biological counterpart, improving the overall response of the system in the correction of pathological gait. Figure 3.2: Scheme for the multibody ankle-foot model with the representation of kinematics prescribed at the knee joint. Figure 3.1: Diagram of a simplified control scheme for controlling the ankle movement with the AAFO. The control action of the AAFO is executed by an actuator (series elastic actuator), which causes a torque about the ankle joint with the goal of assisting the ankle movement during gait Mathematical Ankle-Foot Model A basic approximation for the mathematical ankle-foot model can passes by the application of Newton s second law for rotational motion (11), where the ankle torque,, is calculated. (3.1) where is the ankle rotational inertia due to foot mass, and the ankle angular acceleration. This equation can be written as a transfer function using the Laplace variables, assuming that ankle torque is the input, and the angular displacement of the ankle is the output. (3.2) The resulting transfer function represents a second order system, unstable, characterized by the two poles at the origin Multibody Ankle-Foot Model Instead of deriving and programming equations, biomechanical models can be developed in multibody simulation tools. Multibody systems are used to model the dynamic behavior of interconnected rigid or flexible bodies that have their relative motion constrained by kinematic joints that are acted by forces. Several foot models have been widely proposed as an attempt to calculate the ground reaction forces on the foot or the torque through the ankle joint. In the modeling of the ankle-foot complex, a currently accepted approach to quantify foot and ankle kinematics during gait is to represent the entire foot as a single rigid body with a revolute ankle joint, generally for sagittal plane studies (7). Thus, a simple foot contact model was considered for the modeling, adding to the foot (rigid body) elastic contacts. The developed foot model, presented in Figure 3.3, has two elastic contact points, considering for each the vertical and horizontal components of force. The contact forces,, are defined by the Hertz contact law (11) where (3.3) represents the foot penetration on the ground, and the stiffness of the contact point. The foot penetration is the displacement of the contact point below the ground (Y=0), i.e., when contact point is at negative values for the vertical coordinate (Y) the foot is actuated by the ground reaction forces. Figure 3.3: Elastic foot contact model for the multibody ankle-foot model. The software used in the development of multibody biomechanical model was SimMechanics. SimMechanics is a module from Simulink, which is a simulation tool from MATLAB (12) software. 3

4 For the configuration of the ankle-foot model, the development of the multibody system was realized in two phases. First phase was characterized by all joints having their kinematics prescribed, where the ankle torque was obtain through inverse dynamics. In the second phase, the ankle joint ceased to have prescribed kinematics, setting the ankle joint as a forward dynamics model. Figure 3.4: Multibody biomechanical model used in the simulation (SimMechanics ). With all joints kinematics prescribed, it was necessary to set the parameters of the ankle-foot model to use. The main goal was to assign a position and stiffness to each contact point on the foot, in order to replicate the ankle torque during gait cycle on the ankle joint due to reaction forces on the contact points. In an iterative procedure, the positions of the contact points on the foot were first chosen in an attempt to match some particular events during the stance phase with the intersection of the ground with the contact points. Figure 3.5: Dynamics of gait data and model during gait cycle: (a) ankle torque, (b) horizontal shear force, (c) vertical force. Achieving an acceptable ankle-foot model for the gait cycle, the ankle joint ceased to have prescribed kinematics. In this situation, the anklefoot model relies on the functional error for which this work has considered to assist, excessive dorsiflexion and plantar flexion. Thus, this model can be connected to a controller in order to assist the movement of the ankle joint during gait cycle System States Depending on the type of controllers, it may be required the definition of the state of the system, i.e., characterize the situation in which the system is. During the gait cycle, the ankle joint is subjected to a large movement and torque, which characterizes the system as nonlinear. Therefore, it is preferable to linearize or define different states for the system, allowing the controllers to have specific parameters for the different states. The linearization of the system provides a more efficient control of the system, demanding a lower control effort. A total of four states were defined for the gait cycle: state 1 as loading response, state 2 as stance, state 3 as pre swing, and state 4 as swing. Figure 3.6: Events that cause the transition of states. Sensors enabled in red and disabled in black P-D Control and LQR control The PID controller is the most commonly used controller (13 p. 216). In the basic PID control system, when the reference input is a step function, the presence of the derivative term in the control action, the manipulated variable,, involves an impulse function known as derivative kick (9). Derivative kick is a phenomenon that generally leads to the instability of the system, as also to the damage of physical components. To avoid the derivative kick phenomenon, it is necessary to operate the derivative action only in the feedback path so that differentiation occurs only on the feedback signal and not on the reference signal. The control 4

5 Ankle Angle (degrees) scheme arranged in this way is called the P-D control and was implemented in this work. The closed-loop transfer function of P-D controller can be compared with the typical transfer function of a second-order system. Hence, if and are given as design specifications, the following relations can be found (3.4) (3.5) LQR is an optimal control strategy. Given a system, optimal control objective s consists on finding a control law by recurring to a certain optimality criterion. The linear quadratic regulator (LQR) has the advantage of providing a systematic way of computing the state feedback control gain matrix. When designing an optimal control system, it is required the definition of a control decision, subjected to certain constraints, so as to minimize some measure of the deviation from ideal behavior (14 p. 566). In the LQR problem, given the system equation it is determined the gain matrix control vector so as to minimize the performance index (3.6) of the optimal (3.7). To find the values for the gain matrix in the control input, it is necessary to solve the algebraic Riccati equation. The P-D and LQR controls were implemented in the control of mathematical and multibody anklefoot model with the goal of performing reference following, where the reference consists of ankle angle during gait. Both models were considered with rehabilitation purposes, where the mathematical model would simulate the rehabilitation without ground contact, and the multibody model to simulate the rehabilitation with ground contact. The performance specifications in the control of both models, the desire was to have a following error inferior to three degrees Gait Cycle (%) Figure 3.7: Ankle angle of simulated and controlled multibody ankle-foot model during gait cycle. In general, the reference following was acceptable, not showing abrupt changes during the gait, even having parameters change during the states. Although the value of the angle error can have biomechanical acceptance, when analyzing the ankle torque of the controlled models, it is visible that the peak ankle torque was not achieved The ankle torque deficit is visible in the interval from 35% to 55% of GC, which includes the peak torque. In a realistic situation, this ankle torque deficit could cause some unrestrained tibial advancement and failure in propelling the body forward. Little discontinuities are also visible around 22% and 60% of GC in the ankle torque, which are caused by the changing in the parameters of the controllers. The use of same parameters for the entire gait cycle was tested, however, the control action at states 1 and 4 were very oscillatory Impedance Control Simulated Model P-D Controller LQR Controller Impedance control was implemented in the multibody ankle-foot model with the goal of assisting the ankle movement during gait by trying to mimic the human control of ankle joint. Impedance control uses a target reference in the control law in which the system stiffness and damping are related. Since there is no reference following, this controller requires a trigger in order to update the target reference, considering that these triggers cannot rely solely on the ankle angle. Therefore, the impedance control was only implemented with the multibody ankle-foot model, where the contact with the ground could provide the reference for the different states of the gait cycle (GC). 5

6 Amplitude Table 3.1: Parameters and characteristics of implemented impedance control of multibody anklefoot model during GC. States Loading Response ( 1 ) Stance ( 2 ) Pre Swing ( 3 ) Swing ( 4 ) Figure 3.8: The desired effect of impedance control represented by the use of a rotational mass-springdamper system. In the implementation of impedance control, it was necessary the tuning of four parameters: ankle angle target, ankle angular velocity target, ankle stiffness, and ankle damping. Considering the states triggers presented in section 3.4, the ankle angle target was set by the desired ankle angle in the end of the considered state, i.e., if in the end of state 2 (weight acceptance) the ankle is expected to be the neutral position, then zero degrees will be the ankle angle target. The same procedure was applied to the ankle angular velocity target. The other two parameters were set by trial and error, in order to achieve the following goals: State 1: Controlled plantar flexion to avoid foot slap; State 2: Avoid unrestrained tibial advancement and initiate the propulsion; State 3: Provide final propulsion, moving to peak angle plantar flexion; State 4: Provide toe clearance to avoid toe drag and finalize the swing period with the ankle smoothly dorsiflexed; Rotational Damping B (Nms/rad) Rotational Stiffness K (Nm/rad) Target ankle angle (deg) Target ankle angular velocity - (deg/s) Natural frequency (rad/s) Damping ratio In the overall, impedance control strategy presented good results in the control of multibody ankle-foot model. Disregarding some discontinuities in the states transitions, the controller proved to be capable of avoiding foot slap, provide a controlled tibial advancement and propel the body forward. Although a large ankle angle error occurred in the initial swing phase, the toe drag was avoided and the swing period ended with the foot slightly dorsiflexed State 1 State 2 State 3 State Time (s) Figure 3.10: Responses to a unit step input of a closed-loop system with mathematical ankle-foot model and impedance control. Figure 3.9: Ankle angle during gait cycle from gait data and controlled multibody ankle-foot model with impedance control. In basic second-order systems, a damping ratio superior to the unity correspond to overdamped systems, verifying no overshoot on the step response. However, in this particular system, the controller adds a zero to the system which causes the overshoot in the step responses (Figure 3.10), even for damping ratios superior to the unity. This event, present when the system is in the second, third, and fourth states, allows a 6

7 decrease in either in the rise time as in the overshoot, increasing the control possibilities. In a physical implementation of this controller, the chosen parameters may not lead to the same type of responses, since it depends on the characteristics of the physical actuator in use. In a general comparison between the gains of the three control strategies, the impedance control strategy required less control effort. However, the ankle angle error in impedance control was substantially larger than in the other two control strategies. 4. ACTIVE ANKLE-FOOT ORTHOSIS Provide a full assistance during different gait phases cannot be provided by a common anklefoot orthosis (AFO). These common orthoses are passive systems, acting like energy storing systems, which generally take the foot back to its neutral position. Aiming for a system capable of assisting the ankle movement during gait led to the development of an active ankle-foot orthosis (AAFO) AAFO Design As design parameter, it was defined that the AFO would be actuated by system based on linear motion. In the calculation of the linear quantities, it was take in account the fact that the arm does not have a constant value, because the line of action of the actuator is practically vertical and point of force application rotates through the ankle axis. The corresponding linear quantities were calculated considering that the arm between the ankle axis and point of force application,, had 0.08m of length, with the results presented in Table 4.1. Table 4.1: Linear quantities required for an actuator to assist the ankle movement. Quantity Linear Displacement - (mm) Linear Velocity - (mm.s -1 ) Linear Acceleration - (mm.s -2 ) Maximum negative Maximum positive Linear Force - (N) Power - (W) Components of the AAFO Figure 4.1: Active ankle-foot orthosis 3D model: (a) left view, (b) right view. AFO The AFO chosen for the system was a standard polypropylene AFO, with approximately 5 millimeters of thickness, and articulated aluminum joints. This joints allow motion in the sagittal plane and restrict the motion in the others planes. The motion in the sagittal plane is limited to the ankle angle range relative to the leg during gait, acting as a safety device. Some modifications on the AFO are required for the adaptation of the ankle angle sensor and fixation of the SEA. Ankle Angle Sensor The selected rotary potentiometer was a Bourns 6639S kω. To ensure accurate measures of the ankle between the leg and the foot (ankle angle), the potentiometer rotation axis has must agree with the AFO joints rotation axis. Footswitches Footswitches, often called by event switches, are generally used for acquiring the timing of gait. The data from footswitches allows determining the time in stance period, as also the transition between phases. The placement of the foot sensors is crucial for detecting correctly the desired states transition. Series Elastic Actuator (SEA) The SEA consists of a brushless DC motor coupled laterally to a ball screw shaft by a gear drive. The nut of the ball screw shaft is connected to a set of springs placed in series with carbon bars that transmit the forces to the output. These springs are responsible for the low impedance of the actuator. Further details of the actuator are presented in next section. 7

8 Input and Output devices Communicating and controlling with the sensors and actuator in a physical prototype of the AAFO requires the use of data acquisition unit (DAQ) and controller for the rotary motor. The DAQ unit establishes the communication between the sensors and actuator, and the overall processing unit, the laptop. The motor controller, is a dedicate controller for the rotary DC motor and also communicates with the main controller by the DAQ unit. This is a basic DAQ unit with 8 analog inputs, 2 analog outputs, 12 digital I/O, and a 32-bit counter. Power Supply Aiming for an autonomous system, the energy to provide the system cannot be external. Two elements in the system require the majority of the energy: laptop and actuator. As laptop has its own battery, it was necessary to choose a battery to provide energy to the actuator. A LiPo battery was chosen. With the three pack configuration, the total energy capacity, kJ, has around three times the required energy for 12,000 steps in normal cadence. The main goal of this work is presented in Figure 4.2, an autonomous AAFO assisting an individual during gait. The rest of the unit includes wires, batteries, controller, DAQ, and processing unit. This unit is expected to have approximately 6kg, by considering a processing unit (laptop) with 2.5kg of mass, wires with a total mass of 1.5kg, batteries with a total mass of 1.6kg, controller with 0,25kg of mass, and the DAQ with 0,08kg of mass. 5. ACTUATING UNIT The demanding task of providing power to the ankle joint during gait requires an actuating unit with high power density. The fact of being coupled to an ankle-foot orthosis (AFO) and thus to human leg, implies the actuating unit to be smallest, and lightest as possible. The main idea is to have a system capable of supplying enough energy to provide the ankle movement during gait, without neglecting the size and weight Series Elastic Actuator Design A linear series elastic actuator (SEA) is an actuator that has an elastic element in series with the motor and the ball screw. A sensor measures the displacement of the elastic element and force is implied by Hooke s Law. By placing a spring in series with the output of an electric motor, the force control performance is improved. The motor is isolated from shock loads, and the effects of torque ripple, friction, and backlash are filtered by the elastic element (15 p. 77) Components Selection The design possibilities for a SEA are very large. Besides topology and geometry, there are six major components to take in account for the design: motor, amplifier, transmission, elasticity, sensor, and controller. Figure 4.2: Autonomous AAFO with the individual carrying the batteries and processing unit in a backpack. As a physical prototype, the proposed AAFO has the sensor-actuating unit in the leg-ankle-foot, and the controlling unit, with the respective power supply in a backpack. As a physical prototype, the proposed AAFO has the sensor-actuating unit in the leg-ankle-foot, and the controlling unit, with the respective power supply in a backpack. It is expected that the sensor-actuating unit of the AAFO will have approximately 1.3kg of mass. Figure 5.1: Series Elastic Actuator three dimensional model. DC Motor and Encoder The motor choice fell on the maxon EC-4pole W, due to its compact size and low weight. This rotary motor can develop a peak power of 8

9 Amplitude near 400W, satisfying the peak power occurred during gait cycle. Ball Screw When choosing the ball screw, the choice was made on the ball screw with the overall low peak angular velocity and low rotational inertia, leading to the ball screw with the nut BD 10x4 R Gears To have a compact actuator, the motor was coupled laterally to the screw, requiring the use of gears to transmit the power from the motor to the screw. This choice relies on the commitment between the actuator efficiency and its compactness. Motor Controller Amplifier To ensure compatibility between the motor and the controller, only controllers recommended by the motor manufacturer (16) were considered. With the interest of driving the system in current control, the controller choice fell on the EPOS2 50/5 Positioning Controller. Springs The spring stiffness is related with the mechanical impedance of the SEA, as also with the desired force output. Facing this, three different springs were chosen, differing on the stiffness. Linear Sensor For measuring the spring displacement, a linear potentiometer was chosen, having the sensor approximately 10g of mass SEA Control The close-loop transfer function for the control of the SEA is given by (5.1) where is the proportional gain, is the derivative gain, is the damping term, is the lumped mass, and is the total spring stiffness. The implementation of the SEA in the control of the ankle joint did not affect those systems. However, the selection of parameters for the SEA controller can be improved so the effect of the SEA in the human body control can be present better results SEA Characteristics With the parameters present in Table 5.1, the closed-loop transfer function of the SEA was subjected to a unitary step, with the response presented in Figure 5.2. As expected, the step response presents a considerable overshoot, representing the low impedance of the system. Table 5.1: Properties of the modeled actuator used in the simulation control. Some of the values are calculated from motor, and screw literature. Parameter Value Units Maximum Force 1515 N Maximum Speed 0.31 m/s Intermittent Power 476 W Actuator Mass 0.75 kg Dynamic Mass kg Spring Constant kn / m Damping Ns / m Operational bandwidth Hz Natural frequency Hz Damping ratio No units Nominal Voltage U 48 V Maximum current (peak) 10 A Gear reduction screw-nut - Gear reduction motor-nut No units 5495 No units Time (s) Figure 5.2: Unit step response of the closed-loop transfer function of SEA. The implementation of the SEA in the control of the ankle joint did not affect those systems. However, the selection of parameters for the SEA controller can be improved so the effect of the SEA in the human body control can be present better results. The proposed design for the SEA is expected to have a total mass of 0.75kg, provide a maximum force of 1515 N, and a maximum velocity of 9

10 0.31m/s. Although the losses in gear reductions have not been taken in account, it is expected a great potential from the designed SEA. 6. CONCLUSIONS With the mathematical ankle-foot model it was not possible to extract the dynamics involved in the movement of the ankle. By contrast, the multibody ankle-foot model exhibits an acceptable behavior when comparing to the data. With the multibody model, to maintain the following error, the control effort was much higher. A general comparison between the gains of the three control strategies, the impedance control strategy required less control effort. However, the ankle angle error in impedance control was substantially larger than in the other two control strategies. All control strategies presented high parameters values, which in a physical implementation may not be possible to achieve. The designed AAFO and SEA are expected to provide a full assistance during gait in an autonomous way. With the selected parts for measuring the gait and provide energy, as the redesign of the SEA that made it more compact, light and with more power output, the construction of the prototype is the next step Future Work It would be good to use a more realistic foot contact model so that the dynamics through the ankle joint could be more close to the gait tables. Thus, with full modeling and control of the locomotor unit, it would be possible to study of functional errors of the ankle joint affect the gait. In the design of AAFO, a further study in systems capable of absorbing external forces to output later during the peak torque would allow having smaller motor and thus, smaller battery. For instance, coupling parallel rotational springs in the ankle joint could absorb the energy during loading response phase and output the energy during the mid stance phase. A solution for having a lighter and compact system for the AAFO could pass by connecting the ankle joint and the actuator application force point with a beam. By setting gages in the beam, the deflection of the beam could be obtained and thus the corresponding force applied by the actuator. 7. ACKNOWLEDGMENTS This work is inserted in the FCT DACHOR project Multibody Dynamics and Control of Hybrid Orthoses (MIT-Pt/BS-HHMS/0042/2008). 8. REFERENCES 1. Pons, J. Wearable Robots: Biomechatronic Exoskeletons. s.l. : John Wiley & Sons, Ltd, Rose, J. and Gamble, J. Human Walking, 3rd Edition. s.l. : Lippincott Williams & Wilkins, Cooper, G, [ed.]. Essencial Physical Medicine and Rehabilitation. New York : Humana Press, Össur. Össur. Össur UK. [Online] [Cited: August 16, 2010.] 5. Winter, D. Biomechanics and Motor Control of Human Movement. Hoboken, NJ, USA : John Wiley & Sons, Ltd, Whittle, M. Gait Analysis: An Introduction. Philadelphia : Elsevier, Harris, G., Smith, P. and Marks, R. Foot and Ankle Motion Analysis: Clinical Treatment and Technology. New York : Taylor & Francis Group, Nise, N. Control Systems Engineering. 4th Edition. Hoboken, NJ, USA : John Wiley & Sons, Inc, Ogata, K. Modern Control Engineering. New Jersey : Prentice Hall, Bishop, Robert H. Mechatronic Systems, Sensors, and Actuators: Fundamentals and Modeling. Texas, Austin : CRC Press, Tipler, P. and Mosca, G. Physics for Scientists and Engineers, 5th edition. New York : W. H. Freeman and Componany, MathWorks. MathWorks - MATLAB and Simulink for Technical Computing. [Online] [Citação: 18 de August de 2010.] Levine, W. The Control Handbook. New York : CRC Press, Inc, Vol. I. 14. Ogata, K. Discrete-time control systems, 2nd edition. New Jersey : Prentice Hall, Williamson, M. M. Series Elastic Actuators. Master's thesis. Cambridge : Massachusetts Institute of Technology, maxon motor ag. maxon global. [Online] [Cited: May 20, 2010.] 10