where Δl is the maximal tension length of the piston rod. c1 is a hydraulic damper coefficient, and it denotes damper structure. c2 is a spring resistance coefficient. c1 and c2 are constants. Furthermore, from Equation (5), we have,
Equation (5) shows how the torque at knee relates to needle valve opening and knee speed when the structure parameters of knee prosthesis are given.
DYNAMICS MODEL OF PROSTHETIC LEG
Generally, the inverse problem for the dynamics of the IPL is to solve generalized torques of active joints by given motion trajectory.13–15 Figure 4 shows the two rigid bodies model of the IPL. Suppose the mass of rigid body is focused on each center. m1 and m2 are the mass of legs, l1 and l2 are the length of legs, lG1 and lG2 are the mass center position of legs, θ1 and θ2 are the angle of legs, I1 and I2 are the moment of inertia for leg's mass center, τ1 and τ2 are the angular torques.
Assuming that there is no friction force, the dynamics model of IPL can be expressed as below by using Lagrange formula.
where D(θ)εR2×2 is the inertial matrix, C(θ,θ̇)εR2×2 contains Coriolis and centrifugal terms, Gs(qs)εR2×1 is the gravity vector, and ΓεR2×1 is the general torques matrix.
In fact, there is joint resistance in knee prosthesis. Because Lagrange formula is a typical function to dynamics of a holonomic constraint system,16 the dynamics of IPL considered nonlinear resistance can be established by Lagrange formula.
Selected un(k) and u(k) as generalized coordinates, the displacement of rigid body can be written as
The kinetic energy of IPL can be given by the following equation.
and m1 and m2 represent the masses of two rigid bodies, respectively, as shown in Figure 4, and I1 and I2 represent the central axial inertial moments of the two rigid bodies for the axes perpendicular to the plane of motion.
So, Equation (10) can be expressed in the following form.
The potential energy of IPL is
Considering the resistance at hip and knee joints, the nonpotential force can be written as follows:
where T2 is an active torque. In this article, passive knee prosthesis is studied. So, T2=0. f(θ̇2)=M2 is the resistance function of angular speed between hip and knee.
The dynamical model can be determined from Lagrange formula as follows:
where Lagrangian function is defined as L = T − V, including the total kinetic energy T and the total potential energy V.
Using Equations 8 and 11 to 14, Lagrange formula of IPL becomes
SIMULATIONS OF DYNAMICS
To verify the feasibility of nonlinear damper parameters and dynamics model, simulations were done by using Matlab.
SETTING OF SIMULATION PARAMETERS
In simulation experiments, the parameters and corresponding data of the dynamics model are listed in Table 1. The data were all just set as examples for simulation depending on the related researches performed by other authors and the design parameters used in this article.
The basic data in Table 1 were set by referring to the research done by Popović and Kalanović,7 which are explained as follows: m1 and l1, and m2 and l2 are the masses and lengths of the segments of the rigid bodies, lG1 and lG2 are the distances from the proximal joint to the center of the mass of thigh and shank, respectively. I1 and I2 are the central axial inertial moments for the axes perpendicular to the plane of motion. G is the total mass of the amputee. From the aforementioned data, I1 and I2 in the equations, which are the central axial inertial moments of the two rigid bodies, can also be calculated easily.
The other parameters were set according to the data used for hydraulic IPL design here, which can be explained as follows: M1 is the hip torque generated by the amputee while walking, l is the distance between the rotating center of knee joint and the piston rod of damping cylinder, Δl is the maximal tension length of piston rod, k is an elasticity coefficient of tension-assisting spring, and ρ is the density of hydraulic oil.
The taper section area of needle valve can be compared as
, and the parameters related to damp torque of knee joint Ci(i=1,2) can be calculated with Equations (6) and (7) as follows:
With the aforementioned simulation data, a simulation of dynamics was made with Matlab. A relationship between the needle valve opening and the rotating angle curve of knee joint was revealed. Figure 5 shows the simulation results of knee joint θ2 under different needle valve opening, including X = 1.20 mm, X = 0.80 mm, and X = 0.60 mm.
Coupling with the subsystem of the damping cylinder, a new dynamics model of IPL system simplified in two rigid bodies model was developed. The novel point of the dynamics model is that the direct relationship between the control parameters and the swing speed of knee joint was established for study of the control system and analysis of dynamics. Considering the lack of driving torque at IPL knee joint, the control of swing speed for IPL knee can only be realized by adjusting the damper instead of applying external power as robot. Therefore, it is necessary to set up the aforementioned dynamics model so as to identify the dynamic interaction between the swing speed and the opening of needle valve at the damper. The simulation results for the dynamics model proved the following facts: 1) swing angle of knee joint is affected by the needle valve opening of IPL to some degree; that is, the larger opening of the needle valve is adjusted, the smaller damping force is achieved, and the swing angle of knee joint becomes bigger. This is in accordance with the actual situation of a traditional prosthetic leg while walking; 2) the swing speed becomes faster along with the reduction of the needle valve opening, also in accord with the normal motion rule of traditional prosthetic; 3) although the average swing speed becomes slower and the swing amplitude become smaller when the needle valve opening diminishes, the actual time of period per swing cycle becomes shorter and swing frequency becomes higher. This shows a negative correlation existing between the swing frequency and the needle valve opening. It also proves that the swing speed of IPL can be increased by increasing damping.
Future researches in this area will concentrate on experimental verification of simulation results and application of dynamics model in IPL control studies.
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KEY INDEXING TERMS: intelligent prosthetic leg; hydraulic damper; knee joint; dynamics model© 2010 American Academy of Orthotists & Prosthetists