Figure 2(a) shows the CAD model of the artificial knee mechanism for different knee angles. The knee mechanism consists generally of three linear hydraulic actuators. Two of the actuators are arranged symmetrically in the front of the prosthesis, while the third is arranged at the back of the prosthesis. Figure 2(b) illustrates the novel knee mechanism and a leg within the gait cycle (the kinematics of a gait cycle is shown in Figure 2). The left-top part of Figure 2 depicts how different angles are realized with the novel knee mechanism. To realize the knee of zero degrees, the front cylinders are at zero stroke and the back cylinder is at maximum stroke. To realize an angle, the forward cylinders are extended, the rearward cylinder is retracted. In this way, angles up to 120˚ can be realized. If the front cylinders and the rear cylinder are retracted, the length of the prosthesis is shortened, if all actuators are extended, the prosthesis is extended accordingly.

Figure 2(c) shows the hydraulic circuit diagram. The linear actuators are realized as hydraulic cylinders in a closed system. The volume of two front cylinders (indices 2 and 3) is equal to the volume of the rear cylinder (index 1). In this way, only the desired defined knee-angles can be realized. To implement a damping functionality, two electronically controllable throttle valves are integrated. To further implement the functionality of controlled leg length change, additional components have to be included. These components are described in Section 4. The change of the knee angle is driven by the external influences: hip movement, the patient weight force, and the ground reaction force. In the swing phase, the hip is moved forward and, thus, because of the inertia of the prosthesis, the prosthesis leg. In the stance phase the foot is fixed at the ground and the body is moved forward mainly by the hip movement. The ground reaction force and the movement of the hip generate a torque at the fork. So, in both phases of the gait, a counter direction movement of the rearward cylinder (cylinder 1) and the forward cylinders (cylinders 2 and 3) is generated. The fluid flows through the proportional valves 1 and 2 and the knee angle can be damped by adjusting the cross section of the valves to generate a sufficient torque in the stance phase and to control the correct knee angle in the swing phase. The damping value is set by proportional valves, depending on the current gait phase.

The invented knee mechanism is designed to sustain the mechanical stress, required by standard ISO10328 [10] . Most parts have been optimized using the Finite Element Method. If possible, lightweight materials have been used. The parts of the prosthesis, their materials and the corresponding mass are shown in Table 1. The knee mechanism has a weight of 1.115 kg while the whole prosthesis weighs 2.5 kg and is designed for patients with a maximum weight of 125 kg. A maximum knee angle of 136˚ is attainable.

The required damping value depends on the gait phase. During the stance phase, the knee has to provide enough resistance to generate a sufficient torque to accommodate the weight force of the patient. While in the swing phase, the damping has to be low but adaptive to generate a natural looking swinging movement. Thus, controlling a gait cycle requires the knowledge of the ground reaction forces in the stance phase, the current angle of the knee in the swing phase, and the desired movement of the knee. Any patient has his own pattern of movement, depending on the age, gender, body dimensions and, body proportions. Nevertheless, the general course of movement is nearly equal for any patient [11] [12] and called gait cycle. The kinematics of the knee and the hip joint within the gait cycle are depicted in Figure 3.

The shown cycle begins with the stance phase. In this phase, the hip is at around thirty degrees, the knee is strait, and the foot touches the ground. While the foot is touching the ground, the hip is rotated back to approx. zero degrees and the knee is bended to approx. eighteen degrees. In this phase, the body is moved forward. In the swing phase of the illustrated leg (the other leg would conduct the stance phase), the hip is rotated to the starting position, while the knee has to be rotated to around sixty degrees to avoid collision of the toes with the ground.

The gait cycle begins with the stance phase while the foot is in contact with the ground. Within the swing phase, however, there is no contact between the foot and the ground. The lower part of Figure 4 shows the horizontal and vertical ground reaction force, based on the average measuring results, presented in [12] -[18] . The upper part (right) shows a model of the location of the ground reaction force on the foot depending on the gait cycle which was used for the simulations in this contribution. The point of application of the force vector depends on the gait phase. To be able to implement the ground reaction force in the simulation model, it is modeled as a superposition of the three vectors at the points of application (, Figure 4(b). The according vectors are weighted by the time dependent variables, , and. The sum of these variables is 1 at any time:

At the beginning of the stance phase the heel has the first contact with the ground: is 1, and are equal to zero. Within the stance phase, the point of application moves to the toes. In this way, at the end of the stance phase, is 1 and and are equal to zero (Figure 4(a)).

With the knowledge of the external influences within a gait cycle a control scheme is developed (Figure 5). The knee mechanism, presented in this paper, damps the leg movement hydraulically using proportional valves. As a control algorithm a fuzzy controller with 25 fuzzy rules is used to calculate the optimal damping value depending on the gait phase and optimized by the optimization algorithm “active set” [19] [20] . The input values of the fuzzy controller are the pressures inside the knee mechanism (and), as well as the knee angle. It is necessary to convert the necessary damping value from the controller to the resulting damping values of the pistons. For this purpose, the direct kinematic is defined analytically according to the underlying geometric relations. The damping of the pistons is realized by adjusting the values of the hydraulic resistances (and) of the valves 1 and 2.

The calculation of the knee angle from the measured piston positions is based on the inverse kinematic, which is, however, very complex to deduce analytically. Instead, discrete values of the inverse kinematic are identified using the CAD Software ProEngineer^© and approximated by a fourth order Fourier transformation

[21] . The calculation of the cross section of the proportional valves from the damping values of the pistons is derived in [22] .

is the critical Reynolds number. is the fluid kinematic viscosity, is the fluid density and is the flow coefficient [23] . To investigate the presented control scheme the knee mechanism is built up in a test rig (Figure 6).

Two additional pneumatic actuators are implemented into the rig. In this way, it is possible to apply forces to simulate the ground reaction force and so, to change the knee angle while simultaneously control the desired movement of the gait cycle. Displacement sensors inside of the cylinders of the knee mechanism measure the piston positions that allow the calculation of the knee angle. The test rig is controlled with a dSpace rapid prototyping system. Sample results of a kinematic gait cycle simulation are shown in Figure 7. The figure shows the piston movement, as well as the desired and the measured knee angle in a gait cycle. The results generated by

the test rig show that a gait cycle can be performed with the developed control scheme. The piston movement occurs in counter direction as expected.

The patient’s center of gravity is not directly above the knee joint. Thus, the displacement between the patient’s center of gravity and the knee causes torque acting on the suspension fork of the knee mechanism (see Figure 8). This torque is amplified by long lever arms, e.g. if the patient stumbles.

The difference of the lateral forces, and, resulting from the torque leads to different movements of the pistons in the cylinders 2 and 3 and might jam those. Thus, the lateral forces must be absorbed, either mechanically or hydraulically by generating hydraulic forces which counteract the lateral forces and that way eliminate the torque. The presented knee mechanism is designed to counteract the lateral forces using a linear bearing inside the hydraulic cylinders, as shown in Figure 8. The linear bearing has a height of 10 mm, an inner diameter of 13 mm and the outer diameter is 17 mm. This bearing is very light and space-saving, but, in the case of very high lateral forces, it might be too small to counteract all forces. A taller and more robust bearing would remedy this problem, but would also increase the necessary space and weight. In particular additional components outside the cylinders would have a significant stake of the prosthesis weight. Further weight reduction is attainable by combining the mechanical support structure with a new concept to reduce lateral forces hydraulically. In that case, additionally to the mechanical bearings, the hydraulic valves are controlled to generate hydraulic forces to compensate the lateral forces. The lateral forces to be compensated, are calculated as shown in (3), and (4):

However, since valves are passive elements, only damping forces (forces against the moving direction of the pistons) can be generated. Generating a force in moving direction, however, would require additional actuators. The necessary direction of the compensation force depends on the current load case, as shown in (5), and (6):

If the distance between the line of action of the weight force is less than the distance between the pivot of the folk and the hydraulic cylinder, then both compensation forces (, and) would act against the moving direction of the pistons and the whole torque can be eliminated by controlling the valves. If, however, the compensation force would have to act in the same direction as the piston movement. Since a force in that direction cannot be generated in a passive system, is set to 0 and a combination of mechanical and hydraulic absorption of lateral forces is required (7).

is the torque, generated by the linear bearings. To estimate the actual lever arm, the standard ISO 10328 [10] is used. The different test cases expect a lever arm between 35 mm and 57 mm. The distance between the pivot of the fork and the cylinder is 34 mm. Thus, in the worst test case (d = 57 mm), only 60% of the torque can be absorbed hydraulically and the linear bearing has to be designed to withstand 40% of the torque.

In the following, two approaches for a control based compensation of the lateral forces are presented. The first approach is an open loop control scheme (feed forward), wherein the forces are detected by force sensors and the necessary cross sections of the valves are calculated by a model of the hydraulic mechanical couplings (Figure 9(a)). The second control scheme is a closed loop control (feed back), wherein the difference of the piston position is detected and corrected by the valves (Figure 9(b)).

To realize the feed forward approach, the mathematical derivation of the mechanical and the hydraulic system in Figure 2 is presented while the mechanical system is loaded with the three forces, , and.

The equations of motion of the cylinders are

where is the mass of the piston in cylinder 1 and is the mass of the pistons in cylinders 2 and 3., , , are the pressures in the cylinder chambers (Figure 2(c)). and are the piston surfaces. The forces, , and are the loads acting on the piston rods Figure 9(b)). Due to the incompressibility of the fluid, stiffness effects can be neglected. The volume flow can be calculated as follows

Therefore, the relation between position velocities is

The damping effect is provided by the valves 1 and 2. The mathematical description of the valves is analogous to a hydraulic resistor. Macia and Thaler describe a hydraulic resistor as the ratio of pressure difference and volume flow [24] .

Using Equations (8)-(13) the dynamic behavior of the undamped hydraulic mechanical system can be described in the following way

Based on that general system equation, the necessary relation between the resistors and of the valves to compensate the lateral forces is derived as follows.

The pressure balance is calculated by (9) and (10):

The pressure balance

As well as (13), and (15) lead to the dynamic behavior of the description of the dynamic coupling between the pistons.

The pistons are constrained to equal speed:

The requirement (18) as well as (12) and (17) gives the relation between and, which enables the compensation of lateral forces

As an alternative to the feed forward approach, a feed back approach is presented, which determines the necessary cross sections of the valves to compensate lateral forces from the difference of the piston positions of cylinders 2 and 3 by a PI control element. Therefore, the positions of the piston rods of the cylinders are measured by displacement sensors and the difference of the two positions is equal to the error to be compensated. Based on this error, the PI controller calculates the correcting variable to compensate the lateral forces. The demanded cross section of the valves is calculated from the correcting variable by the inverse model of the valves (IVM).

For the investigation of both approaches a Matlab/Simulink model of the knee mechanism has been developed. The mechanical part is implemented by the SimMechanics toolbox, the hydraulic part is realized by the SimHydraulics toolbox. In the simulation model the three pistons of the knee mechanism are loaded by different force steps. All force step occurs at the same time (after 0.07 s).

is loaded with 1000 N, is loaded 800 N, and is loaded with 200 N. These test signals are chosen, because they show a high difference between the lateral forces and. As a result, the three pistons move differently. The mechanical coupling of the pistons of cylinder 2 and 3 via the suspension fork as well as the mechanical support structure is not considered in these investigations.

To be able to compare and evaluate the approaches to compensate lateral forces, an assessment scheme is developed. The comparison values are the error surface and the maximum error (Figure 10). The goal is to minimize both values. Figure 11 shows the results. Analyzed are the feed forward approach (FF), the feedback approach (FB) with P and PI control element as well as combinations of both.

The column chart in Figure 11(a) a shows that the feed forward approach (FF) gives very accurate results.

The feed back approach (FB) based on a P-control element is less accurate. The closed loop approach based on a PI-controller starts as accurately as the P-control element which leads to almost the same maximum error (Figure 11(b)). The I-control element, however, eliminates the steady state error which leads to an error surface less than the P-control element. The best results are achieved by superposition of the open loop control and the closed loop control.

The open loop control approach shows better results for two reasons: On the one hand, the closed loop control approach because a model of the process is implemented. On the other hand, that approach includes additional process information given by the force sensors. Thus, the control is able to act before an error occurs. The closed loop control approach, however, reacts after an error has occurred which leads to higher error values. The presented investigations are based on ideal assumptions. To evaluate the actual system behavior, it is necessary to consider delays of the valves, model inaccuracies, measurement inaccuracies and inaccuracies in the valves adjustments. These evaluations are presented in Section 3.3.

The analyses in Section 3.2 assume ideal conditions. The actual valves, however, do not act immediately but delayed. This delay is approximated by a first order low pass characteristic. Figure 12 shows the maximum deviation in response to the time constant of the valves.

For short time constants the open loop control approach shows the best results. In this case, the process model is very accurate. For higher time constants the error increases rapidly, because the model of the process does not consider the valve behavior. However, for higher time constants, the closed loop approach shows better results due to the subsequent elimination of the additional error, caused by the dynamic behavior of the valves. The best results are achieved by superposition of both approaches. Current high response valves achieve a switching frequency of 125 Hz and a switching time of less than 3 ms, which is enough to generate good results as shows in Figure 12.

In summary it has been shown that the hydraulic concept is able to absorb lateral forces, if the pistons of cylinder 2, and cylinder 3 are loaded with forces with a positive sign (,). In combination with the mechanical bearings, all loads, based on [10] can be eliminated.

The active variation of the leg length offers advantages for the patient. After an amputation the amputee is not able to lift his foot anymore. With the active variation of the leg length the patient is able to obtain more space between the moving foot and the ground. Thus, risks of collision and stumble can be reduced. The necessary energy to change the leg length is generated by the gait cycle. Thus, no external source of energy is required.

As depicted in Section 2 the movement of the pistons inside of the forward and the rearward cylinders in op-

posite direction changes the knee angle. Uniform movement of all three pistons, however, leads to the variation of the leg length. To enable uniform movement, the hydraulic scheme, depicted in Figure 2(c) needs to be expanded by an additional volume chamber to compensate the volume of the piston rods. The expanded hydraulic scheme is shown in Figure 13. In order to simplify the following calculations the valves 1 and 2 are summarized to the valve b. In that case the pressures in the chambers as well as the piston rods in the forward cylinders are equal.

This hydraulic scheme enables piston movements in opposite direction to change the knee angle as well as uniform motion to change the leg length. Superimposed motion of the angle and leg length is also possible. The directions of the movement of the pistons are adjusted by the two valves and by adjusting the cross section of the valves.

The shortening of the leg length happens in the stance phase and is driven by the weight of the patient. All three pistons move in negative direction and the hydraulic fluid fills the chamber (compensation element). Additionally, the weight of the patient compresses a spring energy storage. The stored energy is used to extend the leg length in the end of the swing phase.

The valve b damps the opposite movement of the pistons of the backward and the forward cylinders and that way it damps the knee angle, while the valve a damps the uniform motion of the cylinders and controls the leg length variation. The mathematical description of the expanded hydraulic scheme is shown as follows:

where and are the hydraulic resistors of the valves and. The necessary value of depends on the demanded motion of and is calculated by the Equations (21)-(22)

The necessary value of is calculated by the Equation (20), and Equation (23) as follows:

The Equations (24)-(25) enable the calculation of the values of the hydraulic resistances, and. These values lead to the demanded damping of the knee angle and the demanded leg length variation at the same time. The relation between hydraulic resistor and the adjustable cross section of the valve as well as the relation between positions and the knee angle is depicted in Section 2.2.

The scheme, presented in Figure 13 is investigated with a simulation model. The control scheme as well as the

inverse and direct kinematics are realized by Matlab/Simulink. This model allows simulating a whole gait cycle as depicted in Figure 14.

The actual ground reaction force and the influence of the torque from the hip are considered in this simulation model. The simulation shows that a gait cycle with active leg length variation can be realized with the presented knee mechanism. Figure 14 shows a gait cycle with leg length variation. The leg is shortened within the stance phase using the patient’s weight and it is lengthened within the swing phase using the force of the spring.

In the presented case, the leg length is varied by 10 mm. However, the curve of the leg length is just an example to show, that the actual leg length follows the desired leg length very well. The necessary curve of leg length variation is an anatomic challenge which has not been investigated yet.

Alternative to the presented hydraulic scheme, it is possible to realize leg length variation without an additional energy store. In that case, within the stance phase, only the piston inside the backward cylinder is moving and that way the knee angle as well as the leg length is changing, while a volume chamber is filled. To extend the leg length within the swing phase, the kinetic energy of the leg is used.

The presented simulation assumes ideal conditions. Additionally, the influence of the low pass behavior of the valves is investigated using the assessment scheme, presented in Section 3.2. Here, the influence of the accuracy of the leg length variation of 10 mm is explored. The knee angle is not considered because the leg length variation is more sensitive than the knee angle damping.

Figure 15 shows the influence of the low pass behavior. The error of the leg length increases approximately linearly. A time constant of 0.02 seconds leads to an error of the leg length of 1 mm or 10% of the desired variation.

In general the influence of the valve behavior of the gait cycle as well as of the leg length variation is less than of the compensation of the lateral forces, because the lateral forces appear immediately.