Intelligent bionics leg is a static instable, strong coupling, nonlinear, and flexible trajectory tracking control system.^{1–3} The tracking object is the angle at knee joint, and the control object is the damper at knee joint. So, the dynamics model of intelligent prosthetic leg (IPL) is very important for motion control. Some studies on the control of IPL were reported in past years.^{4–7} Although some of them focus on control arithmetic, the dynamics model is often ignored, especially the dynamics model considering the damping characters of IPL. A control model and dynamics model of prosthetic legs have been derived using Lagrange equation based on two or three rigid bodies.^{8,9} With the similar method, a detailed dynamic model of the residual limb-prosthesis system for a transfemoral amputee was developed for examining the influence of controls and design parameters on the limb system performance during the swing phase of gait.^{10} For the knee trajectory tracking control, the torque at hip, knee, and ankle was computed based on the inverse dynamics of prosthetic legs.^{10} However, the angular torques that affect leg gaits are controlled by nonlinear dampers, and the control torques cannot be tracked by computed results of microprocessor outputs. So, computed angular torques cannot be used to control the actual prosthetic leg. A man-machine dynamics model is needed based on nonlinear damper parameters and human hip torque. Considering the relationship between energy consumption of key performances and swing speed, this article proposes a dynamics model of swing phase, which describes the direct relationship between the knee damper control parameters and the dynamics parameters and can be used for study on adaptive control of the IPL.

In the following sections, this article first briefly introduces a structure of hydraulic damper in knee prosthesis, and then the development of a nonlinear dynamics model of IPL with regard to the knee damper is discussed. Simulation was also done to evaluate the proposed dynamics model. Finally, some conclusions are drawn.

#### HYDRAULIC DAMPER OF KNEE PROSTHESIS

A passive knee prosthesis controlled by microcomputer has been applied in clinical experiments all over the world. It has a wider step speed than a traditional prosthesis, which only has a limited resistance scope.^{1,11,12} The cylinder spring of a passive knee prosthesis controlled by microcomputer is usually used to simulate the damper stiffness of the human muscle and tendon. Figure 1 shows a traditional prosthetic knee joint controlled by a cylinder. Figure 2 shows a prosthetic knee with a double-piston damper designed by the author.

Figure 1 Image Tools |
Figure 2 Image Tools |

To establish the IPL dynamic model, it is assumed that IPL model is a double-pendulum mechanism with a hinge joint, and the hip joint is a floating pending point of the double-pendulum. To simplify calculation, the damper is invariable within work scope, and the hip moves only in horizontal direction. In this article, a knee prosthesis controlled by a hydraulic damper is proposed based on two rigid bodies. Figure 3 shows the hydraulic damper structure of the knee prosthesis and its enlarged needle valve.

Hydraulic knee joint has an outstanding stability during stance phase. Consequently, to simplify knee joint equipment, a knee prosthesis consists of a single-axis structure. The single axis knee prosthesis is a rotary joint, and the upper and lower body are linked by a rotary axis and thrust ball bearing. During stance phase, the stability of prosthesis is obtained by a postpositional bearing line. During swing phase, the motion of knee joint is controlled by an adjustable friction resistance and exchangeable tension spring.

From Figure 3, torque at knee prosthesis of hydraulic damper can be obtained as

where Δ*S* is the displacement of the piston rod, *k* is an elasticity coefficient of tension spring, and *l* is the distance between the rotating center of knee joint and the piston rod of the damping cylinder.

The resistance of the cylinder can be expressed as

where *A*_{2} is an effective section area of the cylinder and *p* is the pressure of the cylinder.

The liquid flow of the taper throttle can be expressed as

where *Cd* is the flow coefficient of the valve, *A* is the section area of needle valve passage, and ρ is the density of hydraulic oil.

The section area of the needle valve circuit can be calculated by the following equation

where *X* is a needle valve opening and φ is a taper angle of needle valve. *x*_{1} and *x*_{2} are the maximal radius and the minimal radius of taper hole, respectively. In this article, *x1=1.5* mm, *x2=2.5* mm, and φ = 60°.

From Equations (1) and (4), the torque at knee can be written as

Equation 1 Image Tools |
Equation 4 Image Tools |

where Δ*l* is the maximal tension length of the piston rod. *c*_{1} is a hydraulic damper coefficient, and it denotes damper structure. *c*_{2} is a spring resistance coefficient. *c*_{1} and *c*_{2} 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. *m*_{1} and *m*_{2} are the mass of legs, *l*_{1} and *l*_{2} are the length of legs, *l*_{G1} and *l*_{G2} are the mass center position of legs, θ_{1} and θ_{2} are the angle of legs, *I*_{1} and *I*_{2} are the moment of inertia for leg's mass center, τ_{1} and τ_{2} 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)εR*^{2×2} is the inertial matrix, *C(θ,θ̇)εR*^{2×2} contains Coriolis and centrifugal terms, *Gs(qs)εR*^{2×1} is the gravity vector, and *ΓεR*^{2×1} 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.

where

and *m*_{1} and *m*_{2} represent the masses of two rigid bodies, respectively, as shown in Figure 4, and *I*_{1} and *I*_{2} 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 *T*_{2} is an active torque. In this article, passive knee prosthesis is studied. So, *T*_{2}=0. f(θ̇_{2})=M_{2} 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

Equation 8 Image Tools |
Equation 11 Image Tools |
Equation 12 Image Tools |

Equation 13 Image Tools |
Equation 14 Image Tools |

where

and,

Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |

Equation (Uncited) Image Tools |

#### 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: *m*_{1} and *l*_{1}, and *m*_{2} and *l*_{2} are the masses and lengths of the segments of the rigid bodies, *l*_{G1} and *l*_{G2} are the distances from the proximal joint to the center of the mass of thigh and shank, respectively. *I*_{1} and *I*_{2} 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, *I*_{1} and *I*_{2} 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: *M*_{1} 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:

Equation 6 Image Tools |
Equation 7 Image Tools |
Equation (Uncited) Image Tools |

and

##### SIMULATION RESULTS

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 θ_{2} under different needle valve opening, including *X* = 1.20 mm, *X* = 0.80 mm, and *X* = 0.60 mm.

#### CONCLUSION

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.