Dynamic control simulation of a new joint model with energy recovery during walking, using magnetorheological fluids, for lower limb prosthesis

Abstract

Researchers can now utilize new materials to create innovative models for lower limb prostheses and explore novel ways to use them for efficient dynamic control. To achieve user-friendliness, one area of research focuses on recovering and reusing kinetic walking energy for dynamic control. This paper proposes a new design for a magnetorheological (MR) valve, along with a rotary actuator which offers a dynamic control for a lower limb prosthesis. The design will allow the storage of the energy during heel and mid-foot contact phases and to utilize it during toe support to lift the foot off the ground and establish a balance for the lower limb prosthesis. The energy is transferred through a magnetorheological hydraulic circuit and stored using a pneumatic system. The speed of energy transfer is regulated by magnetorheological valves. A series of MR valve designs were proposed and evaluated experimentally, which allowed the identification of the most suitable variant in the targeted application context. The design of the lower limb prosthesis was simulated using SolidWorks, and its dynamic behavior was analyzed in ANSYS.

1 Introduction

One of the most essential factors in human evolution is bipedal rectilinear walking, which distinguishes humans from other mammals. Although the positioning of the foot gives the direction of movement, the joint ankle provides this positioning of the foot. The complete physiological function of an amputated ankle can be substituted with a prosthetic device. In recent years, researchers have explored improving the functionality of lower limb prostheses [1,2,3]. There is a wide range of leg prostheses from which lower limb amputees can choose the optimal one to regain mobility and carry out their daily activities. Light plastics, carbon fiber, composite elements and metal alloys have successfully replaced wood and steel materials used initially in the construction of shields [4]. The design of the elements of the current prostheses is conducted so it allows the storage of the energy generated by the deformation of elastic components, and, when the ankle is pushed for forward movement, the use of the stored energy to actuate the motion.

In the case of existing mechanical prostheses on the market, where the ankle never participates in energy recovery, they cannot faithfully replicate the physiological movements of the ankle.

Most ankle prostheses currently available on the market are based on passive leaf springs that absorb and release energy during walking, reducing the impact of ground reaction forces that occur during walking and converting some of the absorbed energy into usable energy to propel the body forward [5,6,7]. However, the stored energy is much less needed to move the body forward during push. As a result, people who use these protectors with only passive elements tend to walk more slowly and use more energy than usual. Since the musculoskeletal complex of the ankle not only absorbs energy but also generates more energy than it absorbs, for its complete replacement, it becomes imperative to use active prosthetic components that allow the variation of the ankle step, thus reproducing the function performed by the muscles and extract more energy from the ankle than was supplied in its deflection [8, 9].

Current research addresses the need for more energy during push-offs to propel the body forward and is directed toward ankle prostheses, in which active components achieve this [10,11,12]. However, the downside to these models is the degree of increase in ankle size and weight is generally determined by the amount of power the actuator provides to propel the body forward. To minimize this problem, most of the behavior of the prosthesis should be achieved using passive elements and a limited contribution of active components (actuators).

The objective of this study is to present a novel design of dynamically controllable prosthesis that allows energy storage during the foot in balance—heel phase contact and heel contact plus mid-foot phase—and uses the energy stored on the support only on the front of the sole and resting on the toes (toes), lifting the foot off the ground and entering the balance phase.

2 Materials and methods

2.1 Stepping phases

Human walking represents a cyclic locomotor movement, which is realized by successive positioning of a pelvic limb in front of the other, each of the two lower limbs having propellant and support functions. The simple step is the distance between the heel of the foot in contact with the ground and the tip of the propulsion leg, and the length of the simple step is greater in men than in women; in men, it is about 60 cm, and in women, it is approximately 50 cm. The basic unit of walking is given by a cycle of steps. This cycle consists of a double step represented by the interval of time that elapses between two touches of the ground by the heel of the same lower member. The duration of each phase in which the lower limb is the support pillar or pendulous element depends on the displacement speed.

The two types of movement define stepping phases (Fig. 1).

• A. Stance Phase

• 1. Initial contact The initial contact of the foot with the ground.

• 2. Response to loading From initial contact to lifting the opposite limb from the ground.

• 3. Middle phase From the end of the load response until both ankles are aligned in the frontal plane.

• 4. Terminal position The period from the intermediate position to just before the initial contact of the opposite limb.

• B. Swing phase

• 1. Pre-balance Interval from the initial contact of the opposite limb to just before lifting the stance leg.

• 2. Initial swing Raise the extremity to the period of maximum knee wear.

• 3. Mid-balance The period from maximum knee flexion until the tibia is vertical.

• 4. Final balance The end of the intermediate period to the period immediately before the first contact.

Fig. 1

Stepping phases

The position or contact phase is decisive because the foot is in contact with the ground and actively participates in moving, supporting and balancing the body. The swing phase repositions the leg for the next stance phase and balances the body during walking. If we refer only to the foot, the swing phase is only used to position the foot for the next step.

Joint control is essential in both phases of stepping: stance and swing. Because the foot anatomically has a complex structure that cannot yet be replicated in prostheses, the ankle and foot are considered to include three orthogonally superimposed rotation joints in the ankle, plus a rotation joint at the base of the toes.

2.2 Natural movements of the human ankle

The ankle joint has 3 degrees of freedom in movement, as illustrated in Fig. 2. Table 1 also illustrates the moving ranges of these moves.

Fig. 2

Degrees of freedom in terms of movement for the ankle joint

Table 1 Ankle range of motion

According to Fig. 3, we have the following joints:

• 1. Joint 1 in the ankle provides plantar/dorsiflexion movement (raising/lowering the foot). In the stance phase, a resistive moment is induced in the joint for step control. This resistive moment is due to the weight of the body and the moments of inertia due to a fast walk or a sudden stop. It has significant values. In the swing phase, a motor moment is induced in the joint to position the foot for the next step. This motor moment acts only on foot, so due to its small weight (relative to the entire body mass), it has small values. Control of both resistive torque and motor torque is required.

• 2. Joint 2 in the ankle provides the movement of inversion/eversion (lateral foot rotation). This movement is necessary to keep the foot parallel to the ground during the step. It also improves posture when walking on inclined or uneven ground, especially with a swing (specifically for people with prosthetic limbs). In joint 2, a resistive moment occurs (in the stance phase) and a motor moment in the swing phase. Only resistive moment control is required. After a rotational movement of the foot around joint 2, it is necessary to return the foot to the initial position. Following the study of walking [15], it was found that there is no need to orientate the foot in a position other than that of relaxation. In very few cases (sloping terrain with a steep slope, very rough terrain), controlling the foot relative to the ground is necessary. Control of this type is too complicated (it includes equipment for detecting the terrain topography) and does not justify the complexity/quality ratio. A system based on elastic elements returns the foot to its initial position/relaxation (motor moment is small).

• 3. Joint 3 in the ankle provides adduction/abduction movement (foot rotation around the tibia). This rotational movement is necessary to change the direction of walking. Since a predictive control of the change of direction is challenging to achieve, this joint (if it exists in prostheses) is of the elastic type.

• 4. Joint 4 at the base of the toes allow bending the toes in the terminal stance phase. Generally, during normal walking, the toes perform plantarflexion and dorsiflexion (toe up/down) movements. These movements help maintain balance and transfer body weight while walking. Also, sometimes abduction or adduction of the toes can be performed while walking. These movements can help adjust balance and prevent slipping or falling. From the point of view of prostheses, it is only possible to control dorsiflexion during the step phase. An elastic element is used in most cases to bring the toes to the relaxation/initial position (during the swing phase). In joint 4, a resistive moment appears in the stance phase and a motor moment in the swing phase. The resistive moment has significant values and controls the position of the toes during stepping. The motor torque has the role of returning the toes to the resting/relaxed position. With enough energy, the toes can contribute to the propulsive force during walking as they push into the ground and help propel the body forward during movement. It is an essential component for fast walking. It also helps to quickly detach the foot from the ground to avoid hooking the foot/toes to the ground at the beginning of the swing phase (which leads to imbalance).

• 5. Although there is no rotation joint at the base of the heel in human anatomy, some prostheses include this joint. Let us call it joint 5. During the step, when the foot contacts the ground and when the weight is transferred to this foot, there is a damping of the vertical movement of the body. During walking, the body also has a vertical direction (up/down), with an amplitude dependent on the type of movement and the characteristics of the person being studied (male/female, age, weight/height, level of prosthesis). In humans, this cushioning is ensured mainly by the knee joint. However, some of this cushioning is provided by the elasticity/cushioning of the heel and the normal curvature of the medial longitudinal arch of the foot, which causes the sole to flatten or bend inwards. Joint 5 can replace these elements not included in some prostheses. In addition, this joint has a controlled resistive moment and a motor moment. An elastic element can provide motor torque during the swing phase. However, it is of little value, acting on parts of the foot, so lightweight. The resistive moment appears in the stance phase and has significant value, especially when standing on one leg (Heel off).

Fig. 3

The five joints are used in movement

2.3 Rheological controlled prosthesis with energy recovery

2.3.1 General description

The prosthesis proposed in the paper is a dynamically controllable prosthesis with energy recovery for the ankle and includes the five joints listed previously. Joints 1, 2, 4 and 5 are rotary joints formed by controllable double-stroke hydraulic cylinders. The hydraulic working fluid is magnetorheological (MR). The rotation/moment of the joints is controlled using MR valves with the possibility of controlling the dynamics of the movement. Joint 3 is an elastic type (rubber drum) rotary joint. As shown in Fig. 3, joints 1, 2 and 3 replace the human ankle joint, which is orthogonal.

2.3.1.1 Magnetorheological fluid (MR)

MR fluids are fluids that change their viscosity under the action of an excitation field. They are composed of solid (micrometric) suspended particles distributed in a liquid medium. They also contain surfactants that prevent agglomeration or sedimentation of the solid particles. When immersed in a magnetic field, solid particles form chains of particles oriented longitudinally between the magnetic poles. If an MR fluid flows through a magnetic field, which has the field lines transverse to the direction of flow, the chains of particles formed impede the flow, resulting in its slowing down (until the flow stalls) (Fig. 4). This behavior is like the increase in fluid viscosity.

Fig. 4

MR behaviors in the exciting electromagnetic field: a unexcited, b mild excited c medium excited d highly excited e maximally excited, when large clusters develops from discrete chains, resulting in stopping the flow entirely. We thank the authors of the image for accepting its publication in our work [13]

2.3.1.2 MR valve

One of the main applications of MR fluids is the stop valve (Fig. 5). It consists of a flow channel through which rheological fluid flows perpendicular to the lines of the excitation field. The fluid's viscosity is directly proportional to the intensity of the excitation field [13]. Many factors, including fluid composition, geometric characteristics of the flow channel, viscous flow parameters, rheological flow parameters, etc., influence the dependence. By controlling the excitation field's intensity, we control the fluid's viscosity, thus controlling dynamics through the flow channel. When exposed to a significant pressure difference between the ends of the flow channel (the portion exposed to the excitation field), some particle chains break (while others reform). By increasing the pressure, the number of particle chains decreases, so the flow rate increases.

Fig. 5

MR valve principle. The fluid flows from the left to the right. The elements generating the magnetic field are represented as blue plates on the image. On the left side of the flow channel, the solid particles are homogenous distributed in the MR liquid. By approaching the magnetic field generating elements (the flow channel), the solid particles start to agglomerate. By reaching the magnetic field generating elements, the solid particles start to form transversal chains/large clusters, depending on the magnetic field intensity

Similarly, increasing the excitation field's intensity increases the number of particle chains. Hence, the flow velocity decreases (until the flow is blocked for a good design of the stop valve). This type of valve is energy efficient and has low power consumption, especially if the frequency of the excitation field is matched with the hysteresis frequency of the MR fluid.

2.3.2 MR control of joints

This prosthetic model consists of five joints. Joint 3 is elastic type, uncontrollable and designed according to body weight. It is a passive joint that allows abduction/adduction movement when a moment occurs in the joint and returns to the initial position/relaxation when this moment disappears.

Joints 1, 2, 4 and 5 are similar in construction and function. However, they differ dimensionally (diameter/length), depending on the position and location within the prosthesis. Each is designed to fit into the prosthesis and take the moments corresponding to the joint it replaces.

2.3.2.1 MR rotary joint

The joint (Fig. 6) comprises a rotating, double-stroke hydraulic cylinder. The cylinder liner and its piston are connected to the two moving parts of the joint, thus allowing a controlled rotational movement. The working fluid is magnetorheological. The two chambers of the cylinder are interconnected using an MR valve. When changing the angular increment, the MR fluid passes through the valve from one chamber of the cylinder to the other.

Fig. 6

MR rotary joint. (1) and (2)—cylinder liners; (3)—piston; (4) and (5)—internal chambers of the main cylinder volume; (6) and (7)—internal chambers of the secondary cylinder volume; (8)—fluid connector for chamber (4); (9)—Fluid connector for chamber (5); (10)—fluid connector for chamber (6); (11)—fluid connector for chamber (7)

In general, hydraulic cylinders perform mechanical work due to the pressure difference between the chambers. So, the motion is induced by the hydraulic fluid. In the case of cylinders in the prosthesis structure, the rotation movement of the cylinder piston is caused mechanically, not hydraulically, by the rotation of the prosthesis elements. Hydraulic fluid control leads to control of the angular movement of the cylinder. By controlling the intensity of the magnetic field in the valve structure, we control the flow rate of the fluid through it. Thus, we control the fluid flow rate between the chambers of the hydraulic cylinder. So, we control the angular dynamics of the cylinder.

During the stance phase of walking (foot on the ground), angular moments are induced in the joints of the prosthesis, under the action of body weight and due to the kinematics of the prosthesis/foot. Depending on the step phase, the angular dynamics of the joints, respectively, of the hydraulic cylinders must be controlled by resistive moments. This control is achieved using magnetorheological valves.

After completing the position phase and moving to the swing phase (so leg in the air), the articulations, respectively, the rotating hydraulic cylinders, return to their initial position. The return to the original relaxation position of the leg and, therefore, of the joints, respectively, of the hydraulic cylinders, is carried out under the action of elastic elements connected in parallel with the hydraulic cylinders. The moment these elastic elements develop is smaller than the resistive moments in the hydraulic cylinders. While the hydraulic cylinders control the total body movement and the moments induced by the body weight, the elastic elements control only elements of the prosthesis with minimal weight.

From a constructive point of view, each joint consists of two hydraulic cylinders with MR fluid (connected in series, with the same piston (rod)) that act as a unit. As can be seen in the figure, the cylinder is tripartite (divided into three sections). This constructive type allows the tripling of the piston section at the expense of reducing its angular stroke. Since the angular displacements in the leg joints are reduced (see the Table 1), we can use this constructive type to take over more significant moments. Consequently, we can reduce the dimensions of the joint. According to the figure, the hydraulic cylinder is formed by the cylinder liner (elements 1 and 2) and the piston 3. The cylinder liner is constructively formed by elements 1 and 2 to allow a simple assembly, respecting the conditions of use of the MR fluid (for example, the elimination of air bubbles). Each section of the cylinder has two chambers, 4 and 5 for one cylinder, respectively, 6 and 7 for the other. When the piston rotates in a counterclockwise direction, the MR fluid is admitted into chamber 4 through connector 8, and it is discharged from chamber 5 through connector 9, respectively.

Similarly, for the second cylinder, the fluid is admitted into chamber 6 and is removed from chamber 7, using connectors 10 and 11. For each cylinder in the position phase, the MR fluid is circulated between the chambers. The fluid circulation between the chambers uses an MR valve, which controls the flow rate. Since the MR fluid is incompressible, we thus control the dynamics of the joint.

The two cylinders have different roles depending on the phases of the passage. In the position phase (foot in contact with the ground), cylinder 1 (length/large section) controls the moments in the joints, developing resistive moments. The fluid circulates between the two chambers through the MR valve, thus controlling the dynamics of the joint rotation. The control block prescribes the resistive torque (respectively, the intensity of the exciting magnetic field of the MR valve) from a tabular correspondence for each joint. In the swing phase, this cylinder has no active role, circulating only the fluid between the two chambers (when repositioning the foot for the next step). On the other hand, cylinder 2 (length/small section) has the role of the actuator in the swing phase, repositioning the leg for the next step. This cylinder has no active role in the position phase, circulating only the fluid between the two chambers.

2.3.2.2 MR valve design

Fluid circulation between the chambers of the magnetorheological (MR) hydraulic cylinders in the leg joints is done through a magnetorheological valve. This controls the joint dynamics by regulating the flow of MR fluid through the valve. To choose the optimal valve, the design requirements for this type of valve must be met, and constructive parameters must be optimized: the flow channel section, the shape of the flow channel, the flow channel direction in relation to the direction of the excitation magnetic field lines and the type of material used (from a diamagnetic perspective). To achieve this goal, magnetorheological valves have been designed and constructed to allow the study and optimization of these parameters. Figure 7 presents some of the representative valves (designed and tested) for studying the influence of each parameter [14].

Fig. 7

MR valve design. (1)—Labyrinth flow channel, square section, 1.2 mm width, length in magnetic field 50 mm, magnetic field perpendicular to the flow direction; (2)—curved flow channel, square section, 1.2 mm width, length in magnetic field 15 mm, magnetic field perpendicular to the flow direction; (3)—linear flow channel, circular section: 1.5 mm diameter (lower valve), 2.5 mm diameter (middle valve), 3.3 mm diameter (upper valve), length in magnetic field 15 mm, magnetic field perpendicular to the flow direction; (4)—cylindrical core ferrite valve, linear flow channel, circular section: 1 mm diameter (center valve), 2.4 mm diameter (left valve), 4 mm diameter (right valve), length in magnetic field 20 mm (center) to 50 mm (left), magnetic field along the flow direction; (5)—linear flow channel, circular section, 13 mm diameter, length in magnetic field 15 mm, magnetic field along the flow direction

In their construction process, the following considerations have been taken into account:

• 1. The valve material should block/disturb the magnetic field as little as possible;

• 2. As much as possible, the magnetic field generator's magnet should be in direct contact with the magnetorheological fluid;

• 3. The chosen construction method should allow for easy change/ disassembly;

• 4. The cross-sectional area of the electrovalve should be identical to the cross-sectional area of the magnetic field generator.

A specially designed test platform was used to test the MR valves. The physical platform is shown in Fig. 8, and a selection of results is presented in Sect. 3.1.

Fig. 8

Physical platform used for test the MR valves: 1—acquisition board extension; 2, 3—Quanser power module; 4, 5—DC power sources; 6—oscilloscope; 7—signal generator; 8—connector board; 9—manual control board; 10—multimeter; 11—rheological valve + magnetic field generator; 12—magnetic field intensity measurement board; 13—magnetic field probe; 14—pressure sensor; 15—displacement sensor; 16—PWM LORD signal generator; 17—PC + acquisition system; 18—hydraulic piston; 19—pneumatic piston; 20—pneumatic electrovalve

Conceptually, the platform (Fig. 9) comprises the drive block of the MR fluid, the generation block of the excitation field and the solenoid valve itself.

Fig. 9

Platform from a conceptual point of view. 11—rheological valve + magnetic field generator; 14—pressure sensor; 15—displacement sensor; 17—PC + acquisition system; 18—hydraulic piston; 19—pneumatic piston; 20—pneumatic electrovalve

Fluid circulation between the chambers of the magnetorheological (MR) hydraulic cylinders in the leg joints is done through a magnetorheological valve. Thus, the dynamics of the joint are controlled, by controlling the dynamics of the MR fluid flow through the valve. For the optimal choice of the valve, the design requirements of this type of valve must be respected, and the constructive parameters must be optimized: the section of the flow channel, the shape of the flow channel, the direction of the flow channel relative to the direction of the excitation magnetic field lines, the type of material used (from a diamagnetic point of view). In order to achieve this goal, magnetorheological valves were designed and made that allow the study and optimization of these parameters. Figure 7 shows some of the representative valves (designed and tested) for studying the influence of each parameter.

The following considerations were taken into account during their construction:

• 1. The material of the valve should block/disrupt the magnetic field as little as possible;

• 2. As far as possible the core of the magnetic field generating magnetic field generator is in direct contact with the magnetorheological fluid;

• 3. The construction method chosen allows for easy change/disassembly;

• 4. The section of the solenoid valve in the magnetic field must be identical to the section of the magnetic field generator.

The platform ensures the circulation of the MR fluid through the test valve. Thus, for a complete displacement of the hydraulic cylinder pistons, the fluid is transferred from one cylinder to the other through the rheological valve. The displacement of the pistons of the hydraulic cylinders is induced by the pneumatic cylinders. The pistons of the pneumatic cylinders move in unison with those of the hydraulic cylinders (constructive). The working pressure (required by the pneumatic pistons) is provided by an external compressor, with a gas cylinder capacity of 10 l. The maximum working pressure allowed by the platform is 7 bar.

From several variants tested, we chose a magnetic field generator consisting of a closed-loop ferrite core magnetic field generator with two coils, based on the following considerations:

• 1. An iron core magnetic field generator saturates at frequencies higher than 200 Hz;

• 2. A closed-loop magnetic field path is necessary to avoid power losses in the field. The valve essentially represents the only interruption in the magnetic path;

• 3. We chose two coils to study the application of a magnetic field with two-step power levels: a lower power for studying the rheological fluid flow through the valve and a higher power for studying the rapid blocking of flow as well as intermittent actuation at high frequency, requiring a fast response time.

The supply voltage is 0–10Vdc with a current range of 3–5A for the case of DC supply.

For the supply voltage, the platform can provide an adjustable PWM voltage, manually or from software, both in frequency and pulse width.

In our search for the power supply, we have chosen for the excitation magnetic field generator the LORD Wonder Box^® generator, which is a controller accompanying all products of Lord's category of devices with magnetorheological fluids. The kit consists of the controller, a 12 V power supply and two connectors. The controller provides closed-loop control to compensate for load changes up to the upper limit of the power supply. It can operate as an interfacing device with PLCs or computers for controlling MR fluid devices, accepting an analogue input in the form of a continuous voltage ranging from 0 to 5 V, denoted as D. For manual control, the device is also equipped with a rotary potentiometer denoted as C. In the platform, it is used to generate the magnetic field either through manual control of the pulse width or through automatic control by applying a 0–5 V voltage to the D input of the device. The output voltage of the device is directly applied to the magnetic field generator of the rheological valve. The potentiometer adjusts the duty cycle of the PWM signal generated with constant frequency and an amplitude of 12 V.

The current flowing through the coil varies proportionally with the value of the prescribed voltage signal or the position of the potentiometer. The output current can be 0.0 A when the input signal is 0.4–0.6 V.

Whether it is manual control or through an external reference, the current can be canceled by using a switch.

Technical data:

• Pulse width modulated (PWM) frequency: 30 kHz;

• Output current value: max. 2A.

The sensory system consists of:

• 1. Pressure sensor;

• 2. Displacement sensor;

• 3. Magnetic field intensity sensor.

The GS4003 pressure sensor is manufactured by GENSPEC and is an absolute pressure sensor. It can measure pressures between 0 and 10 bar having as output a current of 4–20 mA for a supply voltage of 24 V. We have calculated the measuring resistance so that we get a voltage drop of 0–10 V for the measuring range of the sensor. The value obtained was 487 W. The sensor is connected to the system via a straight connection.

An absolute resistive sensor with a resistance value of 2 kW is used to measure the displacement. The voltage applied to its resistance is 10 V. The useful stroke of the potentiometer is 100 mm. It is rigidly mounted on the body of the pneumatic cylinder 2, determining its position, the mobile element of the sensor moving in unison with the piston of the cylinder.

To determine the intensity of the magnetic field acting on the magnetorheological valve, we preferred the experimental measurement of the field intensity instead of a calculation of ferrite-based magnetic field generators, a laborious and less accurate calculation in the case of our particular system. For this determination, I used a PICmicro MCU development board HP488-00-3 development system with the PIC microcontroller PIC16F87/88 as the computing core (Fig. 10).

Fig. 10

Acquisition system for determination of magnetic field strength with PIC microcontroller

As a transducer of magnetic field intensity, we used the Vernier magnetic probe, which has a measurement accuracy of 0.3 mT. The program created for this development system allows the reading of magnetic field intensity for two measurement ranges: low-intensity fields and high-intensity fields up to 5600 G.

The presented development system is based on the PIC16F87/88 microcontroller.

The main steps taken to measure the magnetic field strength are:

• 1. The system was calibrated using a permanent magnet with known parameters;

• 2. Several determinations of the magnetic field intensity generated by the magnetorheological valve's magnetic field generator were performed, without including it in the system, and the values were read on the development system's display;

• 3. The magnetic field generator was then included in the system, the probe was positioned in a convenient position, and the voltage provided by the development system (voltage resulting from the measurement of the magnetic field intensity) was introduced to one of the inputs of the Quanser acquisition system;

• 4. Thus, a continuous acquisition of the value of the magnetic field intensity was achieved throughout the system's evolution.

2.3.2.3 The acquisition and control model made in the Simulink development and simulation environment

The Quanser acquisition system allows the development of control programs in the Simulink development environment. To interact with the acquisition system, a PC is used, on which several specific software programs have been previously installed, to create the compilation of the models that will be run on this system.

We proposed a Simulink model that controls the system and the data acquisition process.

The created Simulink model (Fig. 11) ensures:

• 1. The acquisition of system parameter variation

• 2. The control for the movement of the pistons;

• 3. The adjustment of the intensity for the magnetic field.

Fig. 11

Block diagram of the Simulink model

Acquisition of system parameters—acquisition of information provided by the displacement sensor, pressure sensor and magnetic field strength sensor.

The information provided by the displacement sensor has the form of a voltage variation in the 0–10 Vdc interval. This voltage variation is directly picked up by the Quanser acquisition board.

The information provided by the pressure sensor is a current variation, in the form of a pressure value / current intensity relation. The current intensity is read through a resistor by picking up/reading the voltage drop around it. This voltage, which is within the measuring range of the Quanser board, is acquired directly on the board inputs.

The magnetic field strength sensor is part of the PICmicro MCU development board HP488-00-3 system. The development system assures the reading of the sensor data and converts it in the form of a voltage variation, compatible with the Quanser acquisition board inputs.

The control for the movement of the pistons is carried out by controlling the pneumatic solenoid valve. An output of the Quanser system is used to control a low-power relay, having the role of powering the pneumatic solenoid valve. This has the role of galvanic separation and power adaptation between the Quanser board and the pneumatic solenoid valve. It is obvious that when pressure is applied to one or another of the pneumatic cylinders, by means of the pneumatic solenoid valve, movement is generated. Since pneumatic cylinders implicitly use compressible gases, the movement will be complete (between the minimum and maximum positions), without the possibility of movement control. By coupling the pistons of the pneumatic cylinders with the pistons of the hydraulic cylinders, and implicitly with the rheological valve, the movement of these pistons will be controlled by means of the rheological valve. So, by supplying or not supplying the pneumatic solenoid valve, movement is generated in one direction or another by means of the pneumatic pistons. The moments of time in which the solenoid valve is commanded (and implicitly the direction of movement changes) correspond to the minimum and maximum positions of the pneumatic pistons. These positions are determined by position variation processing.

A complete acquisition of the parameters can be done during a single stroke of the pistons. Since this is a stand for experimental determinations, which must redo the determinations hundreds of times, the possibility of programming an appropriate number of runs automatically has been provided. For this, the movement between the extreme points of the movement is done repeatedly.

The signals that can be acquired in this form of the Simulink model are (Fig. 11):

• 1. Control voltage of the pneumatic solenoid valve.

• 2. System pressure.

• 3. Position.

• 4. Speed.

• 5. Acceleration.

• 6. The shock.

• 7. The triangular signal from the composition of the PWM signal.

• 8. The DC voltage signal from the composition of the PWM signal.

• 9. Supply voltage of the magnetic field generator of the rheological valve.

• 10. The four signals are generated when the pistons pass through the extreme positional points.

• 11. The intensity of the magnetic field in the area of the rheological valve.

After conducting the tests and analyzing the results, we determined the optimal configuration which can be successfully used for the lower limb prosthesis with energy recovery during walking. Thus, we designed and built a valve which features a closed magnetic circuit in the form of a ring, ensuring a minimal volume for maximum fluid surface/pressure within the magnetic field. Additionally, the flow channel has a rectangular cross section and a circular shape, which represents an optimal solution for the flow of a viscous, rheological fluid. The resulting model of the MR valve is described in Sect. 3.1 Nevertheless, the descriptive mathematical model is complex and often indeterminate. For this reason, in such applications, a portion of the terms are approximated to cover constants whose values are experimentally determined specifically for the utilized design model. To achieve this goal, a second experimental test bench was designed and implemented for dimensional optimization of the valves and determination of the flow constants. (Fig. 12).

Fig. 12

Test platform designed to determine the flow constants of the proposed MR valve. (1)—double-stroke pneumatic piston; (2)—double-stroke hydraulic cylinder actuating the joint; (3)—pneumatic reversing solenoid valve; (4)—resistive linear sensor; (5)—pressure sensor; (6)—magnetorheological valve assembly—excitation magnetic field generator; (7)—command-and-control module; (8)—signal conditioning module; (9)—power stage module

The double-stroke pneumatic piston (element 1) generates the mechanical work required to circulate the magnetorheological fluid through the flow channel. It carries out the piston’s back-and-forth strokes (left/right). The piston of this pneumatic cylinder is firmly connected to the corresponding hydraulic piston of the joint and induces movements in the joint. Element 2 represents the hydraulic cylinder corresponding to the joint, having the magnetorheological fluid as the working fluid. It is driven by pneumatic cylinder 1. It is a double-stroke hydraulic cylinder. Unlike hydraulic pistons used in our proposed lower limb prosthesis joints, which are rotary/angular/circular hydraulic pistons (rotary stroke) with double stroke (as displayed in Fig. 6), this piston is linear (linear stroke). From the point of view of the section and the circulation of the magnetorheological fluid through the valve, and having in mind the intended goals for the tests, this aspect does not constitute an impediment. Element 3 is a pneumatic reversing solenoid valve required to change the direction of movement of the pneumatic cylinder piston. The resistive linear sensor (element 4) measures the hydraulic cylinder piston’s linear displacement. The pneumatic pressure measurement is performed using the pressure sensor 5. Correlating the sections of the two cylinders (pneumatic and hydraulic), the pressure in the hydraulic cylinder chambers is determined. Element 6 represents the magnetorheological valve assembly—excitation magnetic field generator. Element 7 is the command-and-control module formed around a microcontroller. Element 8 is the signal conditioning module—the electronic interface circuit (adaptation in voltage and power) between the platform elements and the control circuit. Element 9 constitutes the power stage module—the electronic interface to energize the magnetic field generator.

2.3.3 The prosthesis management algorithm

The command-and-control module prescribes the intensity of the magnetic field for each MR valve in the system, thus controlling the dynamics of each joint.

Through the implementation of a hydro-pneumatic system for energy recovery, storage and reuse, it becomes feasible to reposition the components of the prosthesis (i.e., joints actuating) and propel the foot forward during the swing phase. Consequently, the type of motion can be varied solely through command inputs (e.g., walking, brisk walking, running), without necessitating physical alterations to the prosthesis itself.

Figure 13 presents the diagram for one of the joints of the prosthesis.

Fig. 13

Diagram for one of the joints of the prosthesis. PA1 and PA2—piston accumulators; CV1 ÷ CV6—adjustable flow control MR valve; RA1 and RA2—double-acting rotary actuator (limited angle)

The elements in Fig. 13 are the PA—piston accumulator, RA—double-acting rotary actuator (limited angle) and CV—adjustable flow control MR valve as previously described.

PA1 is a piston accumulator with a working pressure of 5 bar. One of the chambers communicates with cylinders RA1 and RA2 through MR valves CV2 and CV5. The other chamber of this cylinder contains gas at a pressure of 5 bar. The second piston accumulator works similarly using MR valves CV3 and CV6. Its working pressure is 0.7 bar.

The rotary cylinders RA1 and RA2 have a standard rod (according to the previous description) and represent the two inserted hydraulic cylinders of the joint. The pressure at each cylinder inlet/outlet connector has a pressure sensor installed.

In the stance phase, the joint rotates trigonometrically under the action of the leg/body according to the kinematics/dynamics of the step. During this process, mechanical work is generated. The steps taken in this stage are:

• 1. The joint begins to increase its angular increment. Due to this, the fluid is circulated between the RA1 cylinder chambers through the CV1 valve. This valve is controlled by the command-and-control block, blocking the fluid flow and increasing the pressure difference between the chambers. So, the CV1 valve stops the flow creating a resistive moment in the RA1 cylinder. Similarly, the RA2 cylinder circulates fluid between the chambers through the CV4 valve. The CV4 valve is not commanded in this phase, being passive. So RA2 cylinder has no active role in the position phase. Valves CV5 and CV6 are closed.

• 2. For pressures lower than the working pressure of accumulator PA1, the fluid is circulated between chambers through CV1.

• 3. When the pressure is higher than the working pressure of the PA1 accumulator, the CV2 valve opens, discharging the MR fluid into the PA1 accumulator. Similarly, in the second chamber of the cylinder, fluid is introduced from accumulator PA2 through valve CV3. CV1 valve closes. Due to the gas chamber of the PA1, the fluid stored in the accumulator always has the working pressure.

In the swing phase, the joint rotates in the opposite trigonometric direction, under the control of the command-and-control block, according to the kinematics/dynamics of the step. During this process, the foot is repositioned to perform the next step. The steps taken in this stage are:

• 1. The joint begins to decrease its angular increment. This fact is due to the injection of the MR fluid from the PA1 reservoir into RA2 in chamber 2 through the CV5 valve.

• 2. The fluid from chamber 1 is ejected into accumulator PA2 through valve CV6, which is open.

• 3. The control of the dynamics of the movement is realized by controlling the dynamics of the CV5 valve.

• 4. The CV4 valve is closed.

The steps are repeated for each step.

The resistive moments in the joints are prescribed according to the step phase. The prosthetic person is equipped with a sensory system that provides information on the step phase from the point of view of the prosthesis. This information refers to the distribution of body weight on the sole during stepping, depending on the phase of the step. Also included are the angular displacements and their dynamics of the hip, knee, ankle and toe joints. The information acquired in real time, tabulated with the averaged information previously obtained for a significant batch of people, specifies the next phase of the step and its dynamics [15].

Using artificial intelligence techniques, the control program learns and adapts itself to the person wearing the prosthesis. Using voice recognition techniques, primary operating modes can be prescribed (slow walk, regular walk, fast walk, stop, etc.).

For a reasonable use of the prosthesis, the movement will always be initiated with the normal leg, whether starting, stopping or changing the walking regime. For cases where actual conditions do not allow this (e.g., space to take one more step), voice commands will be used.

3 Results

3.1 The resulting model for the MR valve

Next, the results for some types of valves are presented, as examples, using the platform for determinations. These results led to the constructive variant of the final variant of the MR valve.

Valve with the linear cylindrical flow channel, plastic. This type of valve is constructed from a plastic parallelepiped, with a circular passage channel, provided with a connection at the ends through which the MR fluid circulates (Fig. 7(3)).

Construction parameters:

• The thickness of the parallelepiped—8 mm.

• The diameter of the passage hole—1.5, 2.5 and 3.3 mm for the three solenoid valves.

• Proper length—20 mm.

• The length of the valve in the magnetic field—15 mm.

• The field lines cross the pass channel transversely.

Characteristics of the magnetic field:

• The magnetic field was created with a ferrite frame wound with Cu wire with a diameter of 0.6 mm, 500 turns X 2, powered at a maximum of 10 Vdc, 5A. The solenoid valve inserted into the magnetic circuit interrupts the square-shaped ferrite frame. The two coils are on two opposite sides of the frame.

• The magnetic field generator is powered by the Quanser UPM2405 power supply by direct control from the Quanser board. The actuation of an external switch allows the magnetic field to be generated or not. The voltage variation across the magnetic field generator coil dictates the strength of the magnetic field. The change in output voltage is directly proportional to an input voltage read on an external resistive potentiometer. By manipulating it, we can manually increase or decrease the voltage in the magnetic field generator coil, thus varying the intensity of the magnetic field.

The premises of the determinations:

• Data acquisition is done in 10 s. During this time, the pistons repeatedly move between the minimum and maximum positions.

• The generation of the magnetic field is done approximately after the 6th second.

• The position is calculated according to the prescribed values.

Results obtained:

To eliminate operational or reading errors, it is necessary for the MR fluid to be homogeneous (it settles after a period of time). For this reason, several full movements of the pistons are performed before each measurement. Also, improper operation of the fluid while filling the platform, or changing/inserting the valve, leads to the introduction of air into the system. Because air is compressible, displacement errors appear, as shown in Fig. 14.

Fig. 14

Highlighting the errors induced by the presence of air bubbles in the MR fluid

Air removal is achieved by circulating the fluid through air separator filters, making several tens of complete strokes of the cylinders.

Another factor that can influence the accuracy of measurements is the variation of pneumatic pressure (mechanical work generators). In our case, the ratio between the air volume of the compressor and the volumes of the pneumatic cylinders reduces this variation below 0.8%. If high accuracy is desired, the pressure variation during the process (measured quantity otherwise) can be sampled and entered into the calculation.

Figure 15 shows the pressure variation during the acquisition process (series 1). The figure also shows the variation of the control voltage of the pneumatic valve for changing the direction of movement of the pistons (series 2). The variation of the acquired position is shown in Fig. 16.

Fig. 15

Air pressure variation during an acquisition

Fig. 16

Variation of the position during an acquisition

The parameter acquisition for this round section and linear channel valve was made for various values of the voltage feeding the excitation magnetic field generator. So, it was ordered the magnetic field generator device with the values of 2 V, 4 V, 6 V, 8 V and 10 V (Fig. 17). As can be seen in the figure, two full strokes of the pistons (back and forth) were made, after which the magnetic field was applied, approximately after the 6th second. As can be seen, the displacement (thus flow) stopped completely (or slowed down a lot) for 10 V, 8 V, 6 V, and 4 V. In the case of supplying the magnetic field generator with 2 V, the displacement decreases its value. For each of the five cases, it is observed when the pneumatic solenoid valve that determines the direction of movement of the pistons is energized (active).

Fig. 17

Position evolution for different values of voltage magnetic field generator: 2 V, 4 V, 6 V, 8 V and 10 V

In Fig. 18, the evolution of the position when applying the magnetic field is detailed. It is also marked by a vertical line of time when the magnetic field generator is energized, the visibility and by the evolution of the position after the activation of the magnetic field generator. In order to be able to obtain a comparative representation of the five cases, the moments of time for powering the pneumatic solenoid valve were synchronized and started from the same moment of time relative to them.

Fig. 18

Position evolution for different values of magnetic field generator: 2 V, 4 V, 6 V, 8 V and 10 V. Detail the time of magnetic field generator actions

Figure 19 illustrates the variation in velocity upon magnetic field activation for the same acquisition.

Fig. 19

Velocity variation when the magnetic field generator is activated: 2 V, 4 V, 6 V, 8 V and 10 V. Detail the time of magnetic field generator actions

After analyzing the results, it was established:

• 1. This type of flow channel is most convenient from the point of view of flow and descriptive mathematical models (cylindrical channel). Bearing in mind that we have a viscous, rheological (so in layers) flow of a composite fluid (with solid particles immersed component), any irregularity of the flow channel induces flow vortices (and other viscous flow phenomena) that cannot be modeled mathematically. Although the flow channel of the valve is ideal, from this point of view, it is observed from the experimental data that the path of the fluid between the two hydraulic cylinders induces, in the flow, all the disturbances mentioned. Basically, we have a type of viscous, rheological flow, to which are added the effects of the excitation magnetic field in the rheological valve. Also, the rigidity of the platform elements influences the acquisition. For the observed velocities (in the case of the studied prosthesis or even other applications), these disturbances are far below the error threshold and can be removed by simple averaging. A real-time correction of the acquired data (e.g., inclusion of pneumatic pressure variation) can also be addressed.

• 2. As can be seen from the position and velocity evolution, the valve is very efficient, blocking the flow for relatively small values of the magnetic field strength. So, the desired flow blocking for reduced volume/weight of the excitation magnetic field generator is achieved. The blocking efficiency increases if the magnetic circuit is not interrupted (the magnetic field generator is in direct contact with the fluid MR, in the ideal case).

Figure 20 shows the evolution of the position for an exciting magnetic field generated by a PWM voltage with a frequency of 1 Hz. The applied PWM voltage is 2 V, 4 V, 6 V and 8 V.

Fig. 20

Position evolution for different values of PWM voltage magnetic field generator: 2 V, 4 V, 6 V and 8 V. Frequency 1HZ

As can be seen in the figure, the acquisition has no relevance from the point of view of the motion-blocking command. Instead, it is relevant for the behavior of the MR fluid in a magnetic field generated with a certain frequency. It is observed that for PWM voltage pulse width the motion is blocked/slowed. But for the remainder of the signal period, the fluid does not instantly return to its original flow rate. Because MR fluid hysteresis occurs at approximately 30 kHz (according to the fluid data sheet) and does not influence flow resumption. This behavior can be attributed more to the viscous locking moments induced by the hydraulic circuit when the flow is stopped/started.

The influence of the value of the frequency of the magnetic field is shown in Fig. 21.

Fig. 21

Position evolution for different values of PWM frequency magnetic field generator: 1 Hz, 5 Hz, 10 Hz 100 Hz and 700 Hz. Voltage 5 V

The previous determinations were made for the valve with a cylindrical channel, made of plastic, with the direction of flow perpendicular to the direction of the magnetic field lines (Fig. 7(3)).

The same type of valve made of steel was studied (Fig. 7(5)), with the diameter of the flow channel section of 1 mm, 2 mm and 3 mm. The position variation is shown in Fig. 22.

Fig. 22

Position evolution for steel valve. Voltage 40 V, current 2A, diameter 2 mm

As can be seen in the figure, the magnetic field closes through the metal walls of the valve, almost not influencing the MR fluid flow (even if the magnetic field values are 10 times higher than in the previous measurements). The influence of the diameter of the flow channel section is insignificant.

Another material studied for the same type of valve was ferrite (Fig. 7(5)). Constructively, this valve is different from the previously presented versions. It consists of a ferrite cylinder with an inner diameter (diameter of the flow channel) of 1 mm. The cylinder is coiled, and in this way, the valve incorporates the excitation magnetic field generation system (so the magnetic field generator shown previously is not used). An advantage of this valve is the material (ferrite) which becomes a magnetic field conductor for certain frequencies of the supply voltage (depending on the characteristics of the material and the geometrical characteristics). A major disadvantage is the fluid flow direction, the same as the magnetic field lines. The magnetic circuit is not closed, which is another disadvantage. The generated magnetic field to block the flow is half the field value for the plastic cylindrical channel valve. However, the flow is slowed down as shown in Fig. 23. Although the material (ferrite) presents clear advantages, the impossibility of creating these valves manually (not designed according to the existing elements in the profile catalogues) has led to the abandonment of this type of valve.

Fig. 23

Position evolution for ferrite valve

The last type of valve studied has a plastic body and channel with a circular section, but the flow channel is not linear but serpentine or labyrinthine (Fig. 7(1) (2)). The main purpose is to expose the fluid as much as possible to the excitation magnetic field. The pressure difference applied to the MR valve (between the inlet and outlet of the valve) depends proportionally on the length of the flow channel. The previously described magnetic field generator (also used in earlier plastic valves) was used to block the flow. It was supplied with a voltage with a frequency of 700 Hz, with a nominal voltage of 2 V, 4 V, 6 V and 8 V. The magnetic field was activated approximately after second 4. Figure 24 shows the evolution of the flow for these determinations.

Fig. 24

Position evolution for labyrinthic valve. Voltage 2 V, 4 V, 6 V and 8 V. Frequency 700 Hz

For the general verification of the behavior of the fluid exposed to the magnetic field (blocking/resumption of flow) a manual (random) variation of the supply voltage of the magnetic field generator was made, between 10 and 0 V. The result is shown in Fig. 25.

Fig. 25

Position evolution for labyrinthic valve. Voltage random 10 V-0 V

Using this type of valve, the LORD Wonder Box^® generator was used to provide the supply voltage to the magnetic field generator. It can generate a variable voltage with a frequency of about 30 kHz. At this frequency, the magnetic hysteresis of the fluid occurs. Basically, the process of creating/breaking chains of particles (which block the flow) when the magnetic field appears/disappears occurs under this frequency. This reduces the electrical power consumed because the magnetic field is not generated permanently (DC voltage). However, this advantage is countered by the difficulty of making ferrite valves (or similar materials).

For the general verification of the behavior of the fluid exposed to the magnetic field (blockage/resumption of flow) a manual (random) variation of the voltage generated by the LORD Wonder Box^® was made to power the magnetic field generator. The evolution of the fluid flow exposed to a field generated by a PWM voltage was followed. Contrary to the results obtained for low frequencies (Fig. 19), the action of the magnetic field generated with this frequency is similar to the action of a magnetic field generated by a direct voltage. The result is shown in Fig. 26.

Fig. 26

Position evolution for labyrinthic valve. Voltage generates with LORD Wonder Box^®

After analyzing the obtained results, the following requirements for the design, implementation and operating conditions of an MR valve can be formulated:

• Requirement having a high impact:

  • 1. The valve must be structurally integrated into the magnetic circuit. The coil core should be in contact with the MR fluid to maximize the magnetic field intensity through the fluid. It is known that the magnetic field intensity decreases with the cube of the distance, so not adhering to this requirement leads to unjustified increases in the volume and weight of the magnetic field generator;

  • 2. The magnetic field circuit must be a closed circuit, such as next forms of magnetic core: O, C + C, C + I, E + I, E + E, etc. Not following this requirement leads to unjustified increases in the volume and weight of the magnetic field generator;

  • 3. Only non-magnetically conductive materials should be used for constructing the valve. Not following this requirement leads to the closure of magnetic field lines through the valve body and only partially through the MR fluid, drastically reducing the magnetic field intensity in the fluid;

  • 4. If the material used allows for a magnetic field generated with a frequency in the tens of kHz range, then the magnetic field frequency must be equal to the hysteresis frequency of the fluid (as specified by the manufacturer), to reduce energy consumption;

  • 5. High-efficiency magnetic materials (e.g., neodymium) should be used for the magnetic field generator core;

  • 6. The flow must occur in a plane perpendicular to the magnetic field lines. Not adhering to this requirement leads to the formation of particle chains along the flow direction, resulting in minimal effect on flow blocking;

  • 7. The shape of the flow channel (through the valve) should minimize the occurrence of irregular fluid flow effects (vortices, etc.). Not adhering to this requirement necessitates an increase in mechanical work (increasing the pressure gradient of the valve) required for fluid manipulation/ movement in the absence of the magnetic field, resulting in additional energy losses;

  • 8. If the device, which includes the MR fluid, remains unused for a longer period (more than the sedimentation period), the fluid must be re-homogenized. Failure to comply with this condition can, in the best-case scenario, affect the initial operating cycles, but it may lead to complete blockage of the device (for significant sedimentation).

  • 9. Requirement having a medium impact:

  • 10. If flow dynamics (achieved through varying the magnetic field intensity) is done by supplying PWM voltage, its frequency must be higher than the mechanical hysteresis frequency of the device. Not adhering to this requirement may lead to jerky movement, inducing pronounced wear (occurrence of inertia moments due to the flow of viscous fluid during flow changes or stoppage);

  • 11. Periodic filtering of the fluid (removal of air bubbles), at least during the initial operation period;

  • 12. As much as possible, equalize the flow sections (or flow areas) both outside and inside the valve. Not following this requirement leads to power losses due to irregular flow induced by the behavior of viscous fluids when the flow section (area and/or shape) varies.

  • 13. Other requirements:

  • 14. Using a real-time, adaptive control algorithm, possibly based on AI techniques (from machine learning to reinforcement learning or expert systems) or tabular correspondence;

  • 15. Determining the functional parameters of the valve under conditions that closely resemble the actual (subsequent) operation.

As a result of the valve testing, the following valve model (Fig. 27) was chosen for the prosthesis.

Fig. 27

Resulting model for MR valve. (1) and (2)—inlet/outlet ports; (3)—ferromagnetic core location; (4)—supports for the magnetic field generators; (5)—MR fluid flow channel; (6)—signal conditioning module; (7) and (8)—supports for the magnetic field generators

As shown in Fig. 27, the valve consists of a circular flow channel with a rectangular section through which MR fluid flows. Two pot-type magnetic field generators provide an exciting magnetic field. Figure 27A and B (section) shows the valve with inlet/outlet 1, outlet/inlet 2 and area 3 for the ferromagnetic core required to close the magnetic circuit and flow channel 5. These figures show the valve in the SolidWorks design environment. Figure 27C, D and E shows the MR valve assembly with fluid inlets/outlets: 1 and 2—excitation magnetic field generator, consisting of two pot-type magnetic field generators, 4—supports for magnetic field generators, 7 and 8—electronic interface circuits of the magnetic field generator control 6. As shown in Fig. 27D, the support for magnetic field generator 8 is adjustable to allow valves to connect with different sections of the flow channel. In Fig. 27F, the type of magnetic circuit can be seen, formed by the two magnetic core type E (in section) of the pot-type magnetic field generators, with the closure of the magnetic circuit by the valve, respectively, by the magnetorheological fluid in the flow channel.

3.2 Static finite element analysis of the joint

In the design stage of the joints, static finite element analysis of the joints was used for reasonable dimensioning. The ANSYS Workbench 19.2 software was used for this analysis.

The static finite element analysis with the loads shown in Fig. 28 is proposed for this model. These loads are represented by the forces applied to the rotor blades, which numerically have a value of Fn = 800N, with n = (1, 3) ̅, the equivalent of a human body weight of approximately 79.5 kg.

Fig. 28

The schematization of loads and establishing boundary conditions for the assembly will be analyzed

The equation underlying this algorithm is:

$$\sqrt {0,5\;\left[ {(S_{x} - S_{y} )^{2} + (S_{y} - S_{z} )^{2} + (S_{z} - S_{x} )^{2} + 3(S^{2}_{xy} + S^{2}_{yz} + S^{2}_{zx} )^{2} } \right]}$$

(1)

where S_x, S_y and S_z represent the axial stresses applied in the three directions of the global axis system, and S_xy, S_yz and S_xz, represent the shear stresses. Equation (1) is a general equation of this algorithm, and the 0.5 value represents a parametric form which depends on the cross section of the chosen finite element (Fig. 29).

Fig. 29

Establishing the contact between the landmarks in the structure of the actuator

Actuator components shall be considered to be made of 1C60 (OLC60) steel materials. The mechanical characteristics of the piston are:

• 1. For the benchmarks analyzed: 2C60

• 2. Yield strength: 400 [Mpa]

• 3. Breaking strength: 700 [Mpa]

• 4. Hardness: 255 [HB]

• 5. Longitudinal modulus of elasticity: 2.1 105 [Mpa]

• 6. Transverse contraction coefficient: υ = 0.28 [-].

• 7. Density: 7850 [kgm³].

To carry out the analysis with finite elements, geometrically constrained landmarks are found as a whole, being in contact with different surfaces. For this reason, it is necessary to identify the contacts between the landmarks and specify the type of contact, according to Fig. 34. Thus, the type of contact was chosen as "bounded" with the limits imposed and controlled by the program.

The finite elements that discretized the virtual model were of the SOLID type—tetrahedral and hexahedral.

For better accuracy, discretization was achieved by parameterizing the size of the finite element. As a result, the size of the finished element is 0.2 mm.

The forces specified schematically in Fig. 28 are applied in the nodes related to the surfaces of the rotor blades, according to Fig. 30. Thus, the value for the corresponding force applied on the surface of each rotor blade will have a value of F = 272.5N, these being 3 in number. Also, the direction of this force was adopted by changing the Cartesian coordinate system to the cylindrical system.

Fig. 30

Method of loading the actuator with force F in static mode in the cylindrical coordinate system

The degrees of freedom of the virtual model were canceled in the 3 directions concerning the global axis system, in the area of the outer cylinder, on the 6 related surfaces.

Based on the static finite element analysis, the following results were generated for the assembly in question according to figures from Figs. 31, 32, 33 and 34.

Fig. 31

von Mises maximum equivalent stresses at the joint level

Fig. 32

von Mises maximum equivalent deformations at the joint level

Fig. 33

Maximum total displacements at the joint level

Fig. 34

Maximum unidirectional displacements (x-axis) at the joint level

Thus, numerical results correspond to the stresses, displacements and deformations proposed by analyzing the assembly with finite elements in a static regime according to the von Mises method.

4 Discussion

The need to have a natural gait and minimize the metabolic cost led to the development of new prostheses, especially those with energy recovery.

Energy storage and release occur due to the elements' flexibility during stepping, resulting in improved thrust. As a result, energy recovery prostheses have many advantages over conventional ones; they allow people with lower limb amputations to cover long distances while walking without causing fatigue, change direction quickly and achieve a symmetrical gait.

Most studies on the design of ankle prostheses published in the specialized literature compare the effectiveness of energy-storing and return (ESAR) prostheses to Solid Ankle Cushioned Heel (SACH) prostheses. By comparison, Ossur's Flex-Foot and Seattle Light (ESAR prostheses) generate nearly 3 and 2 times more energy, respectively, compared to a SACH prosthesis [16]. Furthermore, in [17] and [18], it was shown that with a prototype using a controlled ESAR prosthesis, the increased pushing power contributed to a reduction in metabolic cost, and with the help of an advanced ESAR prosthesis (such as the Flex-Foot of Ossur), the increase in walking speeds can be auto-selected.

Recent technologies in these prostheses include hydraulic ankle–foot units [1, 19, 20], microprocessor-controlled ankle/feet [10, 21, 22] and continuous external propulsion ankle/feet [23, 24]. In terminology, these prostheses are powered (PWR) ankle prostheses.

In the push-off stage, the ESAR ankle prostheses offer less power than the PWR ones but more power than the SACH ones, which are composed of a solid core and elastic material and whose dynamics are based on the rebound of the flexible material both in heels as well as on top of them. In ESARs that contain a screw directly on the heel, energy storage is achieved by deforming the composite material from the tip to the middle of the sole. The energy required for propulsion is released by releasing the elastic structure, contributing to the push in the pre-swing phase. On the other hand, in PWRs adding external energy to the kinetic chain increases the potential energy available at pre-swing.

This paper presented a new idea of the ankle prosthesis with energy recovery, which differs from those previously introduced. This prosthesis is an active prosthesis with MR-actuated joints, without the need to submit other external power sources and does not contain any elastic elements that, through deformation, realize the pushing phase. This new design aims to accumulate energy during the foot in balance—contact with the heel phase and contact with the heel plus the middle part of the sole phase—and use the energy stored on the support only on the forefoot of the sole stage and supporting the toes, lifting the foot off the ground and entering the balance phase. The energy is transferred using a magnetorheological fluid hydraulic circuit and stored using a pneumatic system. The energy transfer rate (accumulation and consumption) is regulated using magnetorheological valves.

5 Conclusions

After analyzing the experimental results, the concept's validity was confirmed. This type of recuperative prosthesis has controllable movement dynamics, unlike recuperative prostheses based on elastic elements. By controlling the dynamics of each joint, the prosthesis can be used for distinct types of movement (slow walking, regular walking, brisk walking, walking on uneven ground, walking on slopes, etc.).

The constructive type of the prosthesis allows its concealment in an artificial leg, something impossible for current recuperative prostheses (with elastic elements). Furthermore, the prosthesis can be adapted for different types of shoes (varying height of the heel and different elasticity of the sole) by changing the command-and-control circuit parameters.

As a further development, the authors propose the control of the prosthesis through artificial intelligence techniques based on large volumes of pre-purchased data. Likewise, the data acquisition system (weight distribution on the sole and the dynamics of the movement of the elements (foot contract)) must be supplemented with a system for acquiring the ground pushing forces exerted during the stepping phases.

The proposed solution represents an advanced type of prosthesis, controllable without an external power source for actuators. This solution is a continuation of the research conducted within the framework of the Experimental fluid-based intelligent ankle prosthesis (ESAP-SMAM) project, in collaboration with orthopedic specialists.