A dynamic model of a device with a parallel-serial structure to support the human lower limb

Abstract

BACKGROUND:

Support systems designed for human lower limbs are usually characterized by a serial kinematic structure taking into account only one lower limb. To overcome the mobility range limitations, a new structure of the exoskeleton is proposed in this paper.

OBJECTIVE:

The design process of the dynamic model for the support structure characterized by a parallel-serial mechanism is presented in the paper. The structure works as an exoskeleton and is designed to assist motion of the human lower limb in the process of rehabilitation.

METHODS:

The structure of the support model was divided into linear (executive system) and nonlinear (the mechanical skeleton of the system) parts. The model of the executive system was designed and its parameters were estimated in the course of tests on a laboratory stand, as well as identification procedures. The nonlinear model was expressed by mathematical equations. The characteristic coefficients in the equation were determined based on a 3d CAD model.

RESULTS:

To analyze the behavior of the mechanism, a simulation of dynamic responses was compared with experimental results for a real system consisting of a mechatronic device, actuator drivers, a controller, and programmed software.

CONCLUSIONS:

The proposed new structure enables an increase of the range of rotation angles and can be fitted to an individual person. The derived model is in the analytical form and can also be easily adopted to the different versions of the exoskeleton and used in the design of control systems.

Keywords

Parallel-serial mechanism dynamic model electric linear drive exoskeleton

1. Introduction

The fundamental motoric activities performed by human lower limbs are running, walking, sitting down, and getting up. The walking process does not require the generation of large angular motions in the joints of the lower limbs. However, the process of sitting down and getting up imposes additional requirements that increase the range of motion in particular joints.

Support systems designed for human lower limbs are usually characterized by a serial kinematic structure taking into account only one lower limb. Different drives, i.e. electric motors with gearbox rotating [1], pneumatic cylinders, hydraulic [2, 3] and electric linear actuators, are used as actuated systems.

The most common solutions are structures with a serial mechanism actuated by electrical motors with a gearbox mounted in the joints [4]. The possibility of implementing large angular displacements (no limits on the range of drive motion) is an advantage of the solution. A disadvantage is the significant power limitation due to the limited weight and size of the reduction gears.

A serial structure with linear motion actuators is another solution [5]. The application of devices that generate more power is an advantage as far as the aforementioned solution. On the other hand, kinematic limitations in the form of a significantly narrower range of angular motion performed by the mechanism are an important disadvantage of the solution.

To overcome the limitations mentioned above, a new structure of the exoskeleton is proposed in this paper. The ability to automatically adapt the exoskeleton to a particular person was the main goal in the design of the support system for human lower limbs. Another goal was to ensure that the system has a large enough range of angular motion and sufficient power required to exercise vital human functions, performed by the support structure with a parallel-serial mechanism. To realize the aforementioned goals and develop the most optimal system, it was necessary to build a mathematical model of behavior of the kinematic and dynamic structure. The purpose of the model is to test the system through simulation of the mechanism for different loads, different people, and different motions.

The system was designed to support the physiotherapist during exercise required after mechanical injuries or strokes. The support system for human lower limbs was developed to force displacement in three joints (ankle, hip and knee). The input signal for the system are voltage signals with current (forces) limitations. Feedback signals are signals from the encoders mounted in the electrical actuators. The support system forces the lower limb of the patient to imitate movements generated by the rehabilitation instructor using the system for angular displacement measurements in the lower limb. The structure of the complex system is presented in Fig. 1. The two systems are integrated by the controller. The input signals for the controller are angular displacement signals generated by the instructor ( $\varphi_{1}$ , $\varphi_{2}$ , $\varphi_{3}$ ) and displacement signals ( $w_{1}$ , $w_{2}$ , $w_{3}$ ) of the actuator’s piston rods. Implemented in the controller were forward kinematic and invert kinematic algorithms of the system, and control algorithms with a MFC (Multifunction Controllers) structure, including a dynamic model of the support system, to generate control signals for the actuators. To calculate kinematic parameters and to generate a smooth trajectory within the declared range of angular displacement, velocity and acceleration, the system requests starting parameter, which also enable to automatically adjust the mechanical construction to the patient. The starting parameters in the system are: the characteristic dimensions of the patient’s lower limb, angular ranges in joints, and maximum velocity and acceleration.

The current version of the system does not yet collect feedback from the force exerted by the patient’s limb. Parallel studies carried out by the authors, presented in paper [14], intend to account for the force parameter in the evolution of the control system.

Figure 1.

The structure of a system to support movements in the human lower limb.

2. Structure design

To design and activate the exoskeleton, a measurement system was also implemented. Its application was twofold, i.e. it was used as a system for the estimation of the angular motion ranges in human lower limbs and it can be applied in the future for rehabilitation tasks. Fixed to the instructor’s body, it can generate command signals for the exoskeleton fixed to the patient’s body. Finally, the system can be considered as an instructor-patient or a leader-follower system.

2.1 Measurement system

As a result of human motion analysis, literature research, and a preliminary investigation on a laboratory stand [6, 7, 8], the mechanical structure to measure the angular displacements of human lover limbs was designed and applied in the stand. The structure of the measurement system consists of one element placed on the person’s back and two elements placed directly on their lower limbs. It can be fixed on the human body in five characteristic points such as: feet, back, and the segments between the hip joint and the knee joint (Fig. 2a). The structure of the measurement system has 20 degrees of freedom (Fig. 2b). In Fig. 1, variables $\varphi_{1},\varphi_{2},\varphi_{3}$ describe the angular displacements in the hip, knee and ankle joints, respectively. Dimensions $d_{1}$ , $d_{2}$ , $d_{3}$ , $d_{4}$ are the characteristic parameters describing the length of the human lower limb such as the length between the corresponding pairs of joints ( $d_{1}$ , $d_{2}$ ) and dimensions of the feet ( $d_{3}$ , $d_{4}$ ).

Figure 2.

Measurement system: a) laboratory stand for angular displacement measurement in lower limbs during human motion, b) kinematic scheme of the mechanical structure, c) dimensions of the system in the sagittal plane [6].

2.2 Support system

The scheme of the designed structure for the support of the lower limb is shown in Fig. 3. The mechanical mechanism has five elements with a fixed length and three elements with variable lengths (linear electric actuators I, II, II). The structure allows to move the human lower limb in the sagittal plane perpendicular to the knee joint rotation axis. Hence, the mechanism consists of eight moveable elements ( $n=$ 8), which create eleven fifth class kinematic pairs. Therefore, the movability of the mechanism equals 3. This means that three independent drives in the mechanism (electrical cylinders with linear movement) are needed.

The device is attached to the human body in two places: to the foot and in the hip section (Fig. 3). There are no connections in the knee section, which allows to adapt the structure to the person, based on characteristic dimensions $d_{1}$ , $d_{2}$ , $d_{3}$ , $d_{4}$ of the lower limb. The structure is adapted to the person through the introduction and fixing of the neutral position of pistons in the motors. Piston displacements are calculated with the use of the relationship between the kinematics of the support system and the kinematics of the human lower limb.

In order to increase the usable range of the support system, we have proposed the parallel-serial mechanism. Such a solution allows to realize angular displacements in particular joints of the lower limb in a wider range, in comparison with serial mechanisms with linear drives. The kinematics of such a structure allows for the normal human gait, sitting down, standing up, and squat. Table 1 shows exemplary capabilities of the structure in terms of maximum displacements. The term ‘exemplary’ is related to the person to which structure was attached, whereas different maximum ranges of displacements depend on the person’s height. The displacement values were compared for similar structures (parallel-serial, serial) with the same mechanical parameters, and for the same person.

As is evident from the data presented in Table 1, the angular ranges are wider in the case of the parallel-serial structure. Therefore, such a structure should be used in the exoskeletons.

Table 1
Ranges of angular movements in joints allowed by structures

The displacement of the hip joint $\varphi_{1}$ [deg]			The displacement of the knee joint $\varphi_{2}$ [deg]			The displacement of the ankle joint $\varphi_{3}$ [deg]
w*	I*	II*	w*	I*	II*	w*	I*	II*
min	65	65	min	0	0	min	$-$ 23	$-$ 23
max	114
min	46,8			38,4
max	102,7
min	31			65,5
max	84
min	65			87,6
max	15,8
min	$-$ 2,6
max	57,7	114	max	110	110	max	28	28

where: I* – the structure with a parallel-serial structure, II* – the structure with a serial structure, w* – minimum/maximum displacement value.

Figure 3.

Scheme of the support system, where: I, II, III – in-line electric actuators.

3. Dynamic model

The input signals in the proposed exoskeleton are as follows: voltages in the electrical cylinders $U_{1}$ , $U_{2}$ , $U_{3}$ and pressure forces generated by the human foot $F_{z1}$ , $F_{z2}$ , $F_{z3}$ . The model outputs are: displacements, velocities, and accelerations of pistons in actuators $w_{1}$ , $w_{2}$ , $w_{3}$ , $\dot{w}_{1}$ , $\dot{w}_{2}$ , $\dot{w}_{3}$ , $\ddot{w}_{1}$ , $\ddot{w}_{2}$ , $\ddot{w}_{3}$ . After the analysis, the structure of the model was divided into linear and nonlinear parts (Fig. 4). The linear model shows dependences between the movement parameters of drive cylinders (their displacements, velocities, and accelerations) as functions of control voltages and the forces generated by piston rods. The nonlinear model relates pressure forces from the foot acting on the actuator with variables of actuator motion.

Figure 4.

Scheme of the support system of the dynamic model.

3.1 Linear drive model

This model of the executive system was designed and its parameters were estimated in the course of tests on a laboratory stand and identification procedures described in [10, 11]. The structure of the linear electric drive is shown in Fig. 5.

Figure 5.

Cross-section of the linear drive.

The motion equations of a single electrical drive [6] are shown below:

$\left\{{{\begin{array}[]{l}{U=R\cdot i+L\frac{di}{dt}+k_{e}\omega}\\ {k_{m}i=J\frac{d\omega}{dt}+B\omega+W\cdot F}\\ {\dot{{w}}=K_{1}K_{2}\omega}\\ \end{array}}}\right.$ (1)

where: $U$ – control input in the form of a signal with constant (24 [V]) voltage and different PWM (Pulse Whit Modulation) duty cycle, $F$ – force acting on piston, $w$ – output of the object in the form of piston displacement, $R$ – rotor winding resistance, $L$ – rotor winding inductance, $K_{1}$ – planetary gear constant, $K_{2}$ – nut constant – guide screw pair reduction, $W$ – ratio allowing to determinate the braking torque of the motor shaft by the force exerted on the piston rod, $k_{e}$ – electrical constant, $B$ – reduced coefficient of friction, $J$ – reduced moment of inertia, $k_{m}$ – mechanical constant, $\omega$ – velocity of the motor shaft, $i$ – current.

The motion equations of piston (1) after differentiation lead to a single matrix state equation formulated for the electrical actuator:

$\displaystyle\dot{w}_{n}(t)=A\left[\begin{array}[]{c}w_{n}\left(t\right)\\ \dot{{w}}_{n}\left(t\right)\\ \ddot{{w}}_{n}\left(t\right)\\ \end{array}\right]+B\left[\begin{array}[]{c}\dot{{F}}_{n}\left(t\right)\\ U_{n}\left(t\right)\\ F_{n}\left(t\right)\\ \end{array}\right],y_{n}\left(t\right)=C\left[\begin{array}[]{c}w_{n}\left(t% \right)\\ \dot{{w}}_{n}\left(t\right)\\ \ddot{{w}}_{n}\left(t\right)\\ \end{array}\right]+D\left[\begin{array}[]{c}\dot{{F}}_{n}\left(t\right)\\ U_{n}\left(t\right)\\ F_{n}\left(t\right)\\ \end{array}\right]$ (2)

where:

$\displaystyle A=\left[\begin{array}[]{ccc}0&1&0\\ 0&0&1\\ 0&-\frac{k_{e}k_{m}+RB}{LJ}&-\frac{RJ+LB}{LJ}、、\end{array}\right],B=\left[% \begin{array}[]{ccc}0&1&0\\ 0&0&1\\ \frac{WK_{1}K_{2}}{J}&{\frac{k_{m}K{}_{1}K_{2}}{LJ}}\hfill&{-\frac{K{}_{1}K_{2% }RW}{LJ}}\\ \end{array}\right],$ $\displaystyle C=\left[\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ 0&0&1\\ \end{array}\right]D=\left[\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ 0&0&1\\ \end{array}\right],$

$n$ -n-th actuator,

3.2 Nonlinear model of the mechanical device

The nonlinear dynamics of the device (Fig. 3) are derived below. In order to take into consideration the motion variables of the piston, the structure parameters, and loads, a mathematical description is provided in three steps (Fig. 6) [12].

Figure 6.

Scheme of signal flow in the mathematical model.

Step 1

In the first step, geometrical dependencies of angular displacements, velocities, and accelerations as functions of linear displacements, velocities, and accelerations of piston rods are obtained (Fig. 7).

Figure 7.

Kinematic scheme of the device.

The kinematic parameters are expressed as follows.

Angular displacements:

$\displaystyle\varphi_{1{\rm w}}\left(t\right)=f\left({\varphi_{2w}\left(t% \right),w_{2}\left(t\right)}\right)=\pi-{\rm acos}\left({\frac{D_{1}^{2}+D_{2}% ^{2}+D_{6}^{2}+\left({D_{3}-D_{6}}\right)^{2}-\left({s_{2}+w_{2}(t)}\right)^{2% }+\sigma_{2}}{2{\mskip 1.0mu }\sqrt{D_{1}^{2}+D_{2}^{2}}{\mskip 1.0mu }\sigma_% {1}}}\right)-{\rm acos}\left({\frac{2{\mskip 1.0mu }\left({D_{3}-D_{6}}\right)% ^{2}+\sigma_{2}}{2{\mskip 1.0mu }\left({D_{3}-D_{6}}\right){\mskip 1.0mu }% \sigma_{1}}}\right)-{\rm atan}\left({\frac{D_{2}}{D_{1}}}\right),$ (3)

$\displaystyle\varphi_{2w}\left({{\rm t}}\right)=f\left({w_{1}\left({{\rm t}}% \right)}\right)=\text{atan}\left(\frac{D_{5}}{D_{4}}\right)+\text{acos}\left(% \frac{D_{4}^{2}+D_{5}^{2}+D_{13}^{2}-(s_{2}+w_{1}(t))^{2}}{2D_{13}\sqrt{D_{4}^% {2}+D_{5}^{2}}}\right),$ (4)

$\displaystyle\varphi_{3w}\left({\rm t}\right)=f\left(w_{3}\left({\rm t}\right)% \right)=\text{atan}\left(\frac{D_{8}}{D_{7}}\right)-\frac{\pi}{2}+\text{acos}% \left(\frac{D_{7}^{2}+D_{8}^{2}+D_{10}^{2}-\left(s_{3}+w_{3}(t)\right)^{2}}{2D% _{10}\sqrt{D_{7}^{2}+D_{8}^{2}}}\right),$ (5)

$\displaystyle\varphi_{5{\rm w}}\left(t\right)=f\left({w_{1}\left(t\right),w_{2% }\left(t\right),\phi_{2{\rm w}}\left(t\right)}\right)=\pi-{\rm atan}\left({% \frac{D_{1}}{D_{2}}}\right)-\text{acos}\left(\right),$ (6)

$\displaystyle\varphi_{57{\rm w}}\left(t\right)=f\left({w_{1}\left(t\right),% \varphi_{1w}\left({{\rm t}}\right)}\right)=\pi-\text{acos}\left(-\right)$ (7)

$\displaystyle\varphi_{\rm 42w}\left(t\right)=f\left({w_{1}\left(t\right)}% \right)=\pi-\text{acos}\left(-\frac{(D_{4}^{2}+D_{5}^{2}-D_{13}^{2}-(s_{1}+w_{% 1}(t))^{2})}{2D_{13}(s_{1}+w_{1}(t))}\right),$ (8)

$\displaystyle\varphi_{\rm 63w}\left(t\right)=f(w_{3}(t))=\pi-\text{acos}\left(% -\frac{(D_{7}^{2}+D_{8}^{2}-D_{10}^{2}-(s_{3}+w_{3}(t))^{2})}{2D_{10}(s_{3}+w_% {3}(t))}\right),$ (9)

where:

$\displaystyle\sigma_{1}=\sqrt{D_{6}^{2}+\left({D_{3}-D_{6}}\right)^{2}+\sigma_% {2}},$ $\displaystyle\sigma_{2}=2D_{6}{\mskip 1.0mu }\cos\left({\varphi_{2w}(t)}\right% )\left({D_{3}-D_{6}}\right),$

Angular velocities:

$\displaystyle\dot{\varphi}_{\rm 1w}(t)=\frac{\partial\varphi_{1}(t)}{\partial t% }=f(w_{1}(t),\dot{w}_{1}(t),w_{2}(t),\dot{w}_{2}(t)),$ (10) $\displaystyle\dot{\varphi}_{\rm 2w}(t)=\frac{\partial\varphi_{2}(t)}{\partial t% }=f(w_{1}(t),\dot{w}_{1}(t)),$ (11) $\displaystyle\dot{\varphi}_{\rm 3w}(t)=\frac{\partial\varphi_{3}(t)}{\partial t% }=f(w_{3}(t),\dot{w}_{3}(t)),$ (12) $\displaystyle\dot{\varphi}_{\rm 5w}(t)=\frac{\partial\varphi_{5}(t)}{\partial t% }=f(w_{1}(t),\dot{w}_{1}(t),w_{2}(t),\dot{w}(t)),$ (13)

Angular accelerations:

$\displaystyle{{\ddot{\varphi}}_{1{\rm{w}}}}\left(t\right)=\frac{{{\partial^{2}% }{{\rm{\varphi}}_{1}}\left(t\right)}}{{\partial{t^{2}}}}f\left({{w_{1}}\left({% \rm{t}}\right),{{\dot{w}}_{1}}\left({\rm{t}}\right),{{\ddot{w}}_{1}}\left({\rm% {t}}\right),{w_{2}}\left({\rm{t}}\right),{{\dot{w}}_{2}}\left({\rm{t}}\right),% {{\ddot{w}}_{2}}\left({\rm{t}}\right)}\right),$ (14) $\displaystyle{\ddot{\varphi}_{2{\rm{w}}}}\left(t\right)=\frac{{{\partial^{2}}{% {\rm{\varphi}}_{2}}\left(t\right)}}{{\partial{t^{2}}}}=f\left({{w_{1}}\left({% \rm{t}}\right),{{\dot{w}}_{1}}\left({\rm{t}}\right),{{\ddot{w}}_{1}}\left({\rm% {t}}\right)}\right)$ (15) $\displaystyle{\ddot{\varphi}_{{\rm{3w}}}}\left(t\right)=\frac{{{\partial^{2}}{% {\rm{\varphi}}_{3}}\left(t\right)}}{{\partial{t^{2}}}}=f\left({{w_{3}}\left({% \rm{t}}\right),{{\dot{w}}_{3}}\left({\rm{t}}\right),{{\ddot{w}}_{3}}\left({\rm% {t}}\right)}\right)$ (16) $\displaystyle{{\ddot{\varphi}}_{5{\rm{w}}}}\left(t\right)=\frac{{{\partial^{2}% }{{\rm{\varphi}}_{5}}\left(t\right)}}{{\partial{t^{2}}}}=\left({{w_{1}}\left({% \rm{t}}\right),{{\dot{w}}_{1}}\left({\rm{t}}\right),{{\ddot{w}}_{1}}\left({\rm% {t}}\right),{w_{2}}\left({\rm{t}}\right),{{\dot{w}}_{2}}\left({\rm{t}}\right),% {{\ddot{w}}_{2}}\left({\rm{t}}\right)}\right).$ (17)

The full analytical form of the angular velocities and accelerations are given in the thesis [13].

Step 2

In the second step, the system is divided into four distinctive groups of elements (Fig. 8). Each group (except the third group) has one degree of freedom, controlled by cylinder motion. For a particular group, the formulae describing reduced local moments of inertia and coordinates of gravity centres as a displacement function of the cylinders are calculated. The designed functions for each group are presented below:

$\displaystyle I_{n}=f_{n}\left({w_{n}}\right)=A_{n}\cdot w_{n}+B_{n},$ $\displaystyle x_{n}=g_{n}\left({w_{n}}\right)=C_{n}\cdot w_{n}+D_{n},$ (18) $\displaystyle y_{n}=z_{n}\left({w_{n}}\right)=E_{n}\cdot w_{n}+F_{n}+H_{n},$

where: $w_{n}$ – displacement of n-th actuator, $I_{n}$ – reduced moment of inertia, $x_{n},y_{n}$ – coordinates of reduced center of mass, $A_{n},B_{n},C_{n},D_{n},E_{n}$ , $F_{n}$ , $H_{n}$ – coefficients estimated during identification procedures on the laboratory stand.

In the next stage, the mechanism is substituted by a simplified system (Fig. 9). In this system, the fourth group is replaced by forces generated by a given element. Reduced elements are treated as rod elements parameterized with mass, centre of mass, and moment of inertia. Three forces, generated by the human foot, are applied to the third group.

Figure 8.

Scheme of mechanism divided into groups.

Figure 9.

Scheme of simplified system.

Moments of forces for each group ( $M_{1}$ , $M_{2}$ , $M_{3}$ , $M_{5}$ ) were calculated as functions of the angular variable in joints as well as of external forces. It was performed with the assumption that forces acting on the third group equal zero. To find the aforementioned relationships, Lagrange’s equation, force equilibrium equations, and static moments were used.

Finally, the equations were derived in the following way:

$\displaystyle\left[{\begin{array}[]{c}{{M_{1}}}\\ {{M_{2}}}\\ {{M_{3}}}\\ {{M_{5}}}\end{array}}\right]=\left[{Z\left(t\right)}\right]\cdot\left[{\begin{% array}[]{l}{\begin{array}[]{l}{\begin{array}[]{l}{{{\ddot{\varphi}}_{1{\rm{w}}% }}\left(t\right)}\\ {\dot{\varphi}_{1{\rm{w}}}^{2}\left(t\right)}\\ {{{\dot{\varphi}}_{1{\rm{w}}}}\left(t\right)\cdot{{\dot{\varphi}}_{2{\rm{w}}}}% \left(t\right)}\end{array}}\\ {{{\dot{\varphi}}_{1{\rm{w}}}}\left(t\right)\cdot{{\dot{\varphi}}_{3{\rm{w}}}}% \left(t\right)}\\ {{{\ddot{\varphi}}_{2{\rm{w}}}}\left(t\right)}\\ {\dot{\varphi}_{2{\rm{w}}}^{{(t)}^{2}}}\end{array}}\\ {{{\dot{\varphi}}_{2{\rm{w}}}}\left(t\right)\cdot{{\dot{\varphi}}_{3{\rm{w}}}}% \left(t\right)}\\ {{{\ddot{\varphi}}_{3{\rm{w}}}}\left(t\right)}\\ {\begin{array}[]{l}{\dot{\varphi}_{3{\rm{w}}}^{{(t)}^{2}}}\\ {{F_{z1}}\left(t\right)}\\ {{F_{z2}}\left(t\right)}\\ {\begin{array}[]{l}{{F_{z3}}\left(t\right)}\\ {{{\ddot{\varphi}}_{5{\rm{w}}}}\left(t\right)}\\ 1\end{array}}\end{array}}\end{array}}\right],$ (19)

where:

$Z(t)=\left[{{\begin{array}[]{*{20}c}{{\begin{array}[]{*{20}c}{{\begin{array}[]% {*{20}c}{{\begin{array}[]{*{20}c}{Z_{11}\left(t\right)}\hfill\\ 0\hfill\\ {Z_{12}\left(t\right)}\hfill\\ \end{array}}}\hfill\\ {Z_{13}\left(t\right)}\hfill\\ {Z_{14}\left(t\right)}\hfill\\ {Z_{15}\left(t\right)}\hfill\\ \end{array}}}\hfill\\ {Z_{16}\left(t\right)}\hfill\\ {Z_{17}\left(t\right)}\hfill\\ {{\begin{array}[]{*{20}c}{Z_{18}\left(t\right)}\hfill\\ {Z_{19}\left(t\right)}\hfill\\ {Z_{110}\left(t\right)}\hfill\\ {{\begin{array}[]{*{20}c}{Z_{111}\left(t\right)}\hfill\\ 0\hfill\\ {Z_{112}\left(t\right)}\hfill\\ \end{array}}}\hfill\\ \end{array}}}\hfill\\ \end{array}}}\hfill&{{\begin{array}[]{*{20}c}{{\begin{array}[]{*{20}c}{{\begin% {array}[]{*{20}c}{Z_{21}\left(t\right)}\hfill\\ {Z_{22}\left(t\right)}\hfill\\ 0\hfill\\ \end{array}}}\hfill\\ {Z_{23}\left(t\right)}\hfill\\ {Z_{24}\left(t\right)}\hfill\\ {Z_{25}\left(t\right)}\hfill\\ \end{array}}}\hfill\\ {Z_{26}\left(t\right)}\hfill\\ {Z_{27}\left(t\right)}\hfill\\ {{\begin{array}[]{*{20}c}{Z_{28}\left(t\right)}\hfill\\ {Z_{29}\left(t\right)}\hfill\\ {Z_{210}\left(t\right)}\hfill\\ {{\begin{array}[]{*{20}c}{Z_{211}\left(t\right)}\hfill\\ 0\hfill\\ {Z_{212}\left(t\right)}\hfill\\ \end{array}}}\hfill\\ \end{array}}}\hfill\\ \end{array}}}\hfill&{{\begin{array}[]{*{20}c}{{\begin{array}[]{*{20}c}{{\begin% {array}[]{*{20}c}{Z_{31}\left(t\right)}\hfill\\ {Z_{32}\left(t\right)}\hfill\\ {Z_{33}\left(t\right)}\hfill\\ \end{array}}}\hfill\\ 0\hfill\\ {Z_{34}\left(t\right)}\hfill\\ {Z_{35}\left(t\right)}\hfill\\ \end{array}}}\hfill\\ {Z_{36}\left(t\right)}\hfill\\ {Z_{37}\left(t\right)}\hfill\\ {{\begin{array}[]{*{20}c}{Z_{38}\left(t\right)}\hfill\\ {Z_{39}\left(t\right)}\hfill\\ {Z_{310}\left(t\right)}\hfill\\ {{\begin{array}[]{*{20}c}{Z_{311}\left(t\right)}\hfill\\ 0\hfill\\ {Z_{312}\left(t\right)}\hfill\\ \end{array}}}\hfill\\ \end{array}}}\hfill\\ \end{array}}}\hfill&{{\begin{array}[]{*{20}c}{{\begin{array}[]{*{20}c}{{\begin% {array}[]{*{20}c}0\hfill\\ 0\hfill\\ 0\hfill\\ \end{array}}}\hfill\\ 0\hfill\\ 0\hfill\\ 0\hfill\\ \end{array}}}\hfill\\ 0\hfill\\ 0\hfill\\ {{\begin{array}[]{*{20}c}0\hfill\\ 0\hfill\\ 0\hfill\\ {{\begin{array}[]{*{20}c}0\hfill\\ {Z_{51}\left(t\right)}\hfill\\ {Z_{52}\left(t\right)}\hfill\\ \end{array}}}\hfill\\ \end{array}}}\hfill\\ \end{array}}}\hfill\\ \end{array}}}\right]^{T},$

$\displaystyle Z_{11}(t)=\left({x_{1}^{2}+y_{1}^{2}}\right)m_{1}+\left({\begin{% array}[]{l}D_{3}^{2}+2c_{2}D_{3}D_{9}\\ -2c_{2}D_{3}x_{2}+2s_{2}D_{3}y_{2}\\ +D_{9}^{2}-2D_{9}x_{2}+\\ +x_{2}^{2}+y_{2}^{2}\\ \end{array}}\right)m_{2}+\left({\begin{array}[]{l}D_{3}^{2}++2c_{2}D_{3}D_{9}% \\ +2s_{23}D_{3}x_{3}\\ +2c_{23}D_{3}y_{3}\\ +D_{9}^{2}+2s_{3}D_{9}x_{3}\\ +2c_{3}D_{9}y_{3}+x_{3}^{2}+y_{3}^{2}\\ \end{array}}\right)m_{3}+I_{1}+I_{2}+I_{3},$ $\displaystyle{\rm Z}_{12}(t)=\left({2D_{3}\left({\begin{array}[]{l}y_{2}c_{2}% \\ -D_{9}s_{2}\\ +x_{2}s_{2}\\ \end{array}}\right)}\right)m_{2}+\left({-2D_{3}\left({\begin{array}[]{l}y_{3}s% _{23}\\ -x_{3}c_{23}\\ +D_{9}s_{2}\\ \end{array}}\right)}\right)m_{3},$ $\displaystyle Z_{13}(t)=\left({2D_{3}x_{3}c_{23}-2D_{3}y_{3}s_{23}+2D_{9}x_{3}% c_{3}-2D_{9}y_{3}s_{3}}\right)m_{3},$ $\displaystyle Z_{14}(t)=\left({\begin{array}[]{l}D_{9}^{2}-2D_{9}x_{2}+\\ +D_{3}c_{2}D_{9}+\\ +x_{2}^{2}-D_{3}c_{2}x_{2}+y_{2}^{2}+\\ +D_{3}s_{2}y_{2}\\ \end{array}}\right)m_{2}+\left({\begin{array}[]{l}D_{9}^{2}+2s_{3}D_{9}x_{3}+% \\ +2c_{3}D_{9}y_{3}+\\ +D_{3}c_{2}D_{9}+\\ +x_{3}^{2}+D_{3}s_{23}x_{3}+\\ +y_{3}^{2}+D_{3}c_{23}y_{3}\\ \end{array}}\right)m_{3}+I_{2}+I_{3},$ $\displaystyle{\rm Z}_{15}(t)=\left({y_{2}c_{2}-D_{9}s_{2}+x_{2}s_{2}}\right)D_% {3}m_{2}-\left({y_{3}s_{23}-x_{3}c_{23}+D_{9}s_{2}}\right)D_{3}m_{3},$ $\displaystyle{\rm Z}_{16}(t)=\left({2D_{3}\left({x_{3}c_{23}-y_{3}s_{23}}% \right)+2D_{9}\left({x_{3}c_{3}-y_{3}s_{3}}\right)}\right)m_{3},$ $\displaystyle{\rm Z}_{17}(t)=\left({x_{3}^{2}+y_{3}^{2}+D_{3}\left({y_{3}c_{23% }+x_{3}s_{23}}\right)+D_{9}\left({y_{3}c_{3}+x_{3}s_{3}}\right)}\right)m_{3}+I% _{3},$ $\displaystyle{\rm Z}_{18}(t)=\left({D_{3}x_{3}c_{23}-D_{3}y_{3}s_{23}+D_{9}x_{% 3}c_{3}-D_{9}y_{3}s_{3}}\right)m_{3},{\rm Z}_{19}(t)=x_{31}-D_{3}s_{23}-D_{9}s% _{3},$ $\displaystyle{\rm Z}_{110}(t)=x_{32}-D_{3}s_{23}-D_{9}s_{3},{\rm Z}_{111}(t)=y% _{31}-D_{3}c_{23}-D_{9}c_{3},$ $\displaystyle{\rm Z}_{112}(t)=\left({G\left({\begin{array}[]{l}x_{1}c_{1}+\\ +y_{1}s_{1}\\ \end{array}}\right)}\right)m_{1}+\left({G\left({\begin{array}[]{l}D_{9}c_{12}-% \\ -x_{2}c_{12}+\\ +D_{3}c_{1}+\\ +y_{2}s_{12}\\ \end{array}}\right)}\right)m_{2}+\left({G\left({\begin{array}[]{l}D_{9}c_{12}+% \\ +D_{3}c_{1}+\\ +y_{3}c_{123}+\\ +x_{3}s_{123}\\ \end{array}}\right)}\right)m_{3},$ $\displaystyle Z_{21}(t)=\left({\begin{array}[]{l}D_{9}^{2}-2D_{9}x_{2}+\\ +D_{3}c_{2}D_{9}+\\ +x_{2}^{2}-D_{3}c_{2}x_{2}+y_{2}^{2}+\\ +D_{3}s_{2}y_{2}\\ \end{array}}\right)m_{2}+\left({\begin{array}[]{l}D_{9}^{2}+2s_{3}D_{9}x_{3}+% \\ +2c_{3}D_{9}y_{3}+\\ +D_{3}c_{2}D_{9}+x_{3}^{2}+\\ +D_{3}s_{23}x_{3}+y_{3}^{2}+D_{3}{\mskip 1.0mu }c_{23}y_{3}\\ \end{array}}\right)m_{3}+I_{2}+I_{3},$ $\displaystyle Z_{22}(t)=D_{3}\left({\left({y_{3}s_{23}-x_{3}c_{23}+D_{9}s_{2}}% \right)m_{3}-\left({y_{2}c_{2}-(D_{9}+x_{2})s_{2}}\right)m_{2}}\right),$ $\displaystyle Z_{23}(t)=\left({2D_{9}\left({x_{3}{\mskip 1.0mu }c_{3}-y_{3}s_{% 3}}\right)}\right)m_{3},$ $\displaystyle Z_{24}(t)=\left({\begin{array}[]{l}D_{9}^{2}-2D_{9}x_{2}+\\ +x_{2}^{2}+y_{2}^{2}\\ \end{array}}\right)m_{2}+\left({\begin{array}[]{l}D_{9}^{2}+2s_{3}D_{9}x_{3}+% \\ +2c_{3}D_{9}y_{3}+x_{3}^{2}+y_{3}^{2}\\ \end{array}}\right)m_{3}+I_{2}+I_{3},$ $\displaystyle Z_{25}(t)=\left({2D_{9}\left({x_{3}c_{3}-y_{3}s_{3}}\right)}% \right)m_{3},Z_{26}(t)=\left({x_{3}^{2}+y_{3}^{2}+D_{9}\left({c_{3}y_{3}+s_{3}% x_{3}}\right)}\right)m_{3}+I_{3},$ $\displaystyle Z_{27}(t)=\left({D_{9}\left({x_{3}c_{3}-y_{3}s_{3}}\right)}% \right)m_{3},{\rm Z}_{28}\left({{\rm t}}\right)={\rm x}_{31}-D_{9}\text{sin}% \left({{\varphi}_{3{\rm w}}}\right),$ $\displaystyle{\rm Z}_{29}\left({{\rm t}}\right)={\rm x}_{32}-D_{9}{\rm sin}% \left({{\rm\varphi}_{3{\rm w}}}\right),{\rm Z}_{210}\left({{\rm t}}\right)={% \rm y}_{31}-D_{9}{\rm cos}\left({{\rm\varphi}_{3{\rm w}}}\right),$ $\displaystyle{\rm Z}_{211}\left({{\rm t}}\right)=G\left({\left({D_{9}c_{12}-x_% {2}c_{12}+y_{2}s_{12}}\right)m_{2}+\left({D_{9}c_{23}+y_{3}c_{123}+x_{3}s_{123% }}\right)m_{3}}\right),$ $\displaystyle{\rm Z}_{31}\left({{\rm t}}\right)=\left({x_{3}^{2}+y_{3}^{2}+D_{% 3}y_{3}c_{23}+D_{3}x_{3}s_{23}+D_{9}y_{3}c_{3}+D_{9}x_{3}s_{3}}\right)m_{3}+I_% {3},$ $\displaystyle{\rm Z}_{32}\left({{\rm t}}\right)=\left(D_{3}y_{3}s_{23}-D_{3}x_% {3}c_{23}-D_{9}x_{3}c_{3}+D_{9}y_{3}\right)m_{3},$ $\displaystyle{\rm Z}_{33}\left({{\rm t}}\right)=\left({-2D_{9}\left({x_{3}c_{3% }-y_{3}s_{3}}\right)}\right)m_{3},{\rm Z}_{34}\left({{\rm t}}\right)=\left({x_% {3}^{2}+D_{9}s_{3}x_{3}+y_{3}^{2}+D_{9}c_{3}y_{3}}\right)m_{3}+I_{3},$ $\displaystyle{\rm Z}_{35}\left({{\rm t}}\right)=\left({-D_{9}\left({x_{3}c_{3}% -y_{3}s_{3}}\right)}\right)m_{3},{\rm Z}_{36}\left({{\rm t}}\right)=\left({x_{% 3}^{2}+y_{3}^{2}}\right)m_{3}+I_{3},\quad{Z_{37}}\left(t\right)={x_{31}},$ $\displaystyle{Z_{37}}\left(t\right)={x_{31}},{Z_{39}}\left(t\right)={y_{31}},{% Z_{310}}\left(t\right)=\left({G\left({{y_{3}}{c_{123}}+{x_{3}}{s_{123}}}\right% )}\right){m_{3}},$ $\displaystyle{Z_{51}}\left(t\right)={m_{5}}x_{5}^{2}+{I_{5}},{Z_{52}}\left(t% \right)={G}{m_{5}}{x_{5}}{s_{5}}$

$\displaystyle s_{1}=\sin\left({\varphi_{1w}}\right),c_{1}=\cos\left({\varphi_{% 1}}\right),s_{2}=\sin\left({\varphi_{2w}}\right),c_{2}=\cos\left({\varphi_{2}}% \right),s_{3}=\sin\left({\varphi_{3w}}\right),c_{3}=\cos\left({\varphi_{3w}}% \right),s_{5}=\sin\left({\varphi_{5w}}\right),s_{12}=\sin\left({\varphi_{1w}+% \varphi_{2w}}\right),c_{12}=\cos\left({\varphi_{1w}+\varphi_{2w}}\right),s_{23% }=\sin\left({\varphi_{2w}+\varphi_{3w}}\right),c_{23}=\cos\left({\varphi_{2w}+% \varphi_{3w}}\right),s_{123}=\sin\left({\varphi_{1w}+\varphi_{2w}+\varphi_{3w}% }\right),c_{123}=\cos\left({\varphi_{1w}+\varphi_{2w}+\varphi_{3w}}\right),$

In Eq. (19) the parameters such as:

$G$ variable – represent the gravitational acceleration,

$m_{1}$ , $m_{2}$ , $m_{3}$ , $m_{5}$ variables – represent weights of groups 1, 2, 3, 4 shown in Fig. 8,

$I_{1}$ , $I_{2}$ , $I_{3}$ , $I_{5}$ variables – represent the moment of inertia of groups 1, 2, 3, 4, shown in Fig. 8.

Step 3

In the last step, the equations describing the forces generated by the actuators are obtained from moments of forces derived for the substitute system. The forces and moments of forces that act on the mechanical structure are shown in Fig. 10.

Figure 10.

Scheme of the structure with forces acting on piston rods and moments of forces in the substitute system.

Figure 11.

Scheme of the dynamic model.

The forces generated by actuators I, II and III that counterbalance the moments derived for the substitute system are given by the equitation:

$\displaystyle\left[{{\begin{array}[]{*{20}c}{F_{1}\left(t\right)}\\ {F_{2}\left(t\right)}\\ {F_{3}\left(t\right)}\\ \end{array}}}\right]=\left[{{\begin{array}[]{cccc}{Z_{F1}\left(t\right)}&{Z_{F% 2}\left(t\right)}&0&{Z_{F3}\left(t\right)}\\ {Z_{F4}\left(t\right)}&{Z_{F5}\left(t\right)}&0&{Z_{F6}\left(t\right)}\\ 0&0&{Z_{F7}\left(t\right)}&0\\ \end{array}}}\right]\cdot\left[{{\begin{array}[]{*{20}c}{M_{1}\left(t\right)}% \\ {M_{2}\left(t\right)}\\ {M_{3}\left(t\right)}\\ {M_{5}\left(t\right)}\\ \end{array}}}\right],$ (20)

where:

$\displaystyle Z_{F1}\left(t\right)=\frac{-D_{6}\text{sin}\left({{\rm\varphi}_{% 57{\rm w}}}\right)}{D_{13}\text{sin}\left({{\rm\varphi}_{42{\rm w}}}\right){% \rm sin}\left({{\rm\varphi}_{2{\rm w}}+{\rm\varphi}_{57{\rm w}}}\right)\left({% D_{3}-D_{6}}\right)},$ $\displaystyle Z_{F2}\left(t\right)=\frac{D_{6}{\rm sin}\left({{\rm\varphi}_{57% {\rm w}}}\right)+D_{3}{\rm sin}\left({{\rm\varphi}_{2{\rm w}}+{\rm\varphi}_{57% }}\right)-D_{6}{\rm sin}\left({{\rm\varphi}_{2{\rm w}}+{\rm\varphi}_{57{\rm w}% }}\right)}{D_{13}\text{ sin}\left({{\rm\varphi}_{42{\rm w}}}\right){\rm sin}\left({{\rm\varphi}_{2{\rm w% }}+{\rm\varphi}_{57{\rm w}}}\right)\left({D_{3}-D_{6}}\right)},$ $\displaystyle Z_{F3}\left(t\right)=\frac{D_{6}{\rm sin}\left({{\rm\varphi}_{2}% }\right)}{D_{13}{\rm sin}\left({{\rm\varphi}_{42{\rm w}}}\right){\rm sin}\left% ({{\rm\varphi}_{2{\rm w}}+{\rm\varphi}_{57{\rm w}}}\right)\left({s_{2}+{\rm w}% _{2}}\right)},$ $\displaystyle Z_{F4}\left(t\right)=\frac{-1}{{\rm sin}\left({{\rm\varphi}_{2{% \rm w}}+{\rm\varphi}_{57{\rm w}}}\right)\left({D_{3}-D_{6}}\right)},$ $\displaystyle Z_{F5}\left(t\right)=\frac{1}{{\rm sin}\left({{\rm\varphi}_{2{% \rm w}}+{\rm\varphi}_{57{\rm w}}}\right)\left({D_{3}-D_{6}}\right)},$ $\displaystyle Z_{F6}\left(t\right)=\frac{-{\rm cos}\left({{\rm\varphi}_{2{\rm w% }}+{\rm\varphi}_{57{\rm w}}}\right)}{{\rm sin}\left({{\rm\varphi}_{2{\rm w}}+{% \rm\varphi}_{57{\rm w}}}\right)\left({s_{2}+{\rm w}_{2}}\right)},$ $\displaystyle Z_{F7}\left(t\right)=\frac{-{\rm sin}\left({{\rm\varphi}_{63{\rm w% }}}\right)}{D_{10}}.$

4. Verification of the obtained model

Verification of the system model dynamics was performed through computer simulations and tests on a laboratory stand. A simulated answer to the excitation of the designed dynamic model of the support system of the lower limb was obtained in a Matlab/Simulink environment. Voltage signals with modulation (PWM) were used as excitation signals. The structure of the dynamic model includes the mechanical model, electric actuator models, and motors drivers (Fig. 11).

The obtained simulation model was verified on a laboratory stand prepared for the rehabilitation of the human lover limb in the sagittal plane. The laboratory stand (Fig. 12) consists of a mechanical structure with linear actuators, a real time controller, motors drivers, a and computer with data acquisition software.

In both methods of verification (simulation and laboratory), parameters such as $F_{z1}$ , $F_{z2}$ , $F_{z3}$ are not used. In the simulation, the forces equaled 0. In the future, the parameters in the system will be measured using a sensor for feet pressure force distribution. The initial studies that enable measurements of the forces generated by the human foot are presented in article [14].

The initial conditions of the experiments are presented in Table 2. The % fulfilment due to PWM duty of control signal in time. More information about the tests is given in [6].

Table 2
Initial conditions for the measurements

[dir=NW]Actuator number Number of test	Percent of the pulse-width modulation [%]			Initial displacement the cylinder [m]
	I	II	III	I	II	III
1	50	0	0	0	0.05	0
2	0	50	0	0	0.00	0
3	0	100	0	0	0.05	0

Figure 12.

Laboratory stand for the rehabilitation of the human lover limb in the sagittal plane.

A comparison of responses of the real model and the mathematical model is shown in Figs 13–15.

Figure 13.

Simulation results and measurement of actuator displacement during test 1.

Figure 14.

Simulation results and measurement of actuator displacement during test 2.

Figure 15.

Measurement and simulation results of actuator displacement during test 3.

5. Summary and conclusions

This paper presents investigations of a new structure of an exoskeleton for human lower limbs. The exoskeleton has a parallel-serial mechanism with electric linear drivers. It can be automatically adapted to a particular person. Moreover, the exoskeleton mechanism has a large range of angular motions and sufficient power required to exercise vital human functions. To develop a quasi-optimal system, it was necessary to build a mathematical model of the behaviour of the kinematic and dynamic structure. The purposed model allows to test the system through simulation of the mechanism for different loads, people, and motions. After analytical considerations, the measurement system and exoskeleton for human lower limbs were designed and tested on a laboratory stand. Such a system can be used for rehabilitation in the instructor-patient tandem. The system enables precise forcing of limb movements in selected joints with specific displacement and velocity ranges. Retrofitting the system with current measurements and proper calibration will enable moments of forces exerted on the joints. All of the above, in addition to a system of displacement settings that can be remembered and reconstructed, is an interesting product for rehabilitation centres.

A comparison between signals from the real mechanism and its mathematical model shows that the derived model with the applied simplifications does not significantly affect proper results of system behaviour. The obtained results confirm the correctness in the separation of the two groups of equations. Such grouping allows to reduce dynamic model and simplify its design. Furthermore, the model has a fully analytical form, thus it can be easily used for further modifications of the exoskeleton.

Footnotes

Acknowledgments

The work has been accomplished under research project No. S/WM/1/2016 financed by the Bialystok University of Technology.

Conflict of interest

None to report.

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A dynamic model of a device with a parallel-serial structure to support the human lower limb

Abstract

BACKGROUND:

OBJECTIVE:

METHODS:

RESULTS:

CONCLUSIONS:

Keywords

1. Introduction

2.1 Measurement system

Table 1 Ranges of angular movements in joints allowed by structures

Table 2 Initial conditions for the measurements

Footnotes

Acknowledgments

Conflict of interest

References

Table 1
Ranges of angular movements in joints allowed by structures

Table 2
Initial conditions for the measurements