Human-robot coupling dynamic modeling and analysis for upper limb rehabilitation robots

Abstract

BACKGROUND:

Upper limb rehabilitation robots have become an important piece of equipment in stroke rehabilitation. Human-robot coupling (HRC) dynamics play a key role in the control of rehabilitation robots to improve human-robot interaction.

OBJECTIVE:

This study aims to study the methods of modeling and analysis of HRC dynamics to realize more accurate dynamic control of upper limb rehabilitation robots.

METHODS:

By the analysis of force interaction between the human arm and the upper limb rehabilitation robot, the HRC torque is achieved by summing up the robot torque and the human arm torque. The HRC torque and robot torque of a 2-DOF upper limb rehabilitation robot (FLEXO-Arm) are solved by Lagrangian equation and step-by-step dynamic parameters identification method.

RESULTS:

The root mean square (RMS) is used to evaluate the accuracy of the HRC torque and the robot torque calculated by the parameter identification, and the error of both is about 10%. Moreover, the HRC torque and the robot torque are compared with the actual torque measured by torque sensors. The error of the robot torque is more than twice the HRC. Therefore, the HRC torque is more accurate than the actual torque.

CONCLUSIONS:

The proposed HRC dynamics effectively achieves more accurate dynamic control of upper limb rehabilitation robots.

Keywords

Human-robot coupling (HRC) dynamics dynamic model parameter identification rehabilitation robotics

1. Introduction

As many countries enter an aging society, the global incidence of stroke will increase, and the number of patients with limb dysfunction caused by hemiplegia will increase as well. Research shows that scientific and effective rehabilitation training plays a crucial role in the recovery of limb motor function of stroke patients. Compared with the one-on-one training method of therapists, the rehabilitation effect of the upper limb rehabilitation robot is more significant, because the upper limb rehabilitation robot not only provides adaptive training and highly repetitive movement but also enhances the self-care ability of patients with dysfunction [1, 2, 3]. So far, there have been many assistive or rehabilitation robotics, including ARMin III [4], LIMPACT [5], Alex [6], AAU exoskeleton [7], Harmony [8], CAREX-7 [9], and BONES [10]. The upper limb rehabilitation robot directly acts on the human body, and the human body and the rehabilitation robot are a closed loop. For different patients and patients in different periods, the human-robot interaction is different. Therefore, it is particularly important to model and analyze the dynamics of the human arm and upper limb rehabilitation robots, which can improve the accuracy of the rehabilitation robot and the effectiveness of rehabilitation training and assistance.

Some researchers only considered the mechanical dynamic model of the upper limb rehabilitation robot for convenience [11, 12, 13], and ignored the dynamic model of the human arm, which may lead to inaccurate control of rehabilitation training and assistance. Some researchers take the human as the research object, establish the human upper limb equivalent mechanism and derive its dynamic equations [14, 15, 16, 17]. It is of great significance to study the modeling and analysis of human dynamics, which can guide the design and control strategy of rehabilitation robots. There are also some researchers who proposed the impedance model of the human arm. By adjusting the control parameters of the impedance model, the sensitivity of the model to human-robot interaction can be adjusted and the compliance control of the system can be realized. It is suitable for rehabilitation training of different patients in different periods. The dynamics of post-stroke patients’ upper limbs as an impedance model, and an adaptive control scheme were proposed to control the rehabilitation robot to move stably and smoothly [18, 19]. Some previous studies divided the human-robot interaction control strategy into multiple modes, and realized the active assisting strategy. Zhang et al. proposed the human-robot interaction controller that has three modes, including human-dominant mode, robot-dominant mode and safety-stop mode [20]. Wang et al. evaluated the patient’s motor ability and divides the motor ability into initialization ability, moving ability and stop ability. The magnitude of the assistance in these three stages is different [21].

In the previous studies, the dynamic models of the upper limb rehabilitation robots are mostly modeled only for robots, the human arm, or the human-robot interaction. However, human-robot as a closed-loop system, it is very important to study the human-robot coupling (HRC) dynamics and explore the relationship between the HRC system and the human arm and the rehabilitation robot. In this paper, we model and analyze the HRC dynamics of a 2-DOF upper limb rehabilitation robot (FLEXO-Arm). The HRC torque and the robot torque are solved by Lagrangian equation and step-by-step parameter identification method. The effectiveness of the HRC dynamics is verified by experiments, which provides more reference for the human-robot control strategy.

2. Methods

2.1 The prototype of an upper limb rehabilitation robot (FLEXO-Arm)

In this paper, the HRC model is established for a power assistive robot (FLEXO-Arm) for upper limb rehabilitation. The robot can not only provide continuous rehabilitation training with multiple degrees of freedom (DOF) for stroke patients in different rehabilitation stages but also has multiple movement modes, such as passive movement, active movement, and teaching movement, to meet the rehabilitation needs of different patients. The upper limb rehabilitation robot has two active DOF and eight passive DOF, as shown in Fig. 1. The green arrows denote active DOF, and the yellow arrows denote passive DOF. One active DOF (R1) is the flexion/extension of the shoulder joint. The other (R2) is the flexion/extension of the elbow joint. The flexion/extension of the shoulder and elbow joints are the most important movements in upper extremity training. The motors of the shoulder and elbow joints are MAXON flat motor EC-60 and EC-45, respectively. In order to meet the needs of patients with different height, sitting height and different sides of the affected limb, an adjustable length and left and right robotic arm exchange are designed. P1 is used to adjust the height of the robotic arm. P2 is used to adjust the horizontal position of the robotic arm and to exchange left and right hands. P3 and P4 are used to adjust the lengths of robotic arm. The robot has four passive rotating DOF to enlarge the upper limb motion space, protect the joints of patients, and do more active training. R1 and R2 are internal and external rotations for the robotic arm. R3 is the passive DOF of elbow flexion and extension in the horizontal plane. R4 is the flexion and extension of the wrist joint. The encoders are installed on six rotating joints (R1, R2, R1, R2, R3 and R4) to record the joint angles for position feedback and control.

Figure 1.

Mechanical structure and DOF distribution of FLEXO-Arm. The green arrows denote active DOF, and the yellow arrows denote passive DOF.

Figure 2.

The HRC system of FLEXO-Arm.

2.2 Upper limb HRC dynamics modeling

The upper limb rehabilitation robot and the human arm are regarded as one system when the HRC dynamics model was established. The joint angles, angular velocities and angular accelerations of the human are consistent with the robot while the human arm moves with the robotic arm, as shown in Fig. 2. The power of the HRC system is the muscle force of the human arm and the force contributed by the robot’s motors. So, the HRC torque is equal to the motor torque of the robot and the muscle torque of the human arm. The HRC torque is obtained as follows:

$\displaystyle\tau_{\textit{HRC}}=\tau_{\textit{motor}}+\tau_{\textit{muscle}}$ (1)

Here, $\tau_{\textit{HRC}}$ is the torque of HRC. $\tau_{\textit{motor}}$ is the torque of the robot motor. $\tau_{\textit{muscle}}$ is the torque of the human muscle.

For the robot system, the external force of the system is the force of the motor and the interaction force between the human arm and the robot. So, the robot torque is equal to the motor torque and the interaction torque of the robot is as follows:

$\displaystyle\tau_{\textit{robot}}=\tau_{\textit{motor}}+\tau_{\textit{% interaction\_robot}}$ (2)

Here, $\tau_{\textit{robot}}$ is the torque of the robot. $\tau_{\textit{interaction\_robot}}$ is the interaction torque that human exert on the robot.

For the human system, the external force of the system is the force of the muscle and the interaction force between the robot and the human arm. The torque of human arm is equal to the muscle torque and the interaction torque of the human is as follows:

$\displaystyle\tau_{\textit{human}}=\tau_{\textit{muscle}}+\tau_{\textit{% interaction\_human}}$ (3)

Here, $\tau_{\textit{human}}$ is the torque of human arm joint. $\tau_{\textit{interaction\_human}}$ is the interaction torque that the robot exerts on the human arm. We know that $\tau_{\textit{interaction\_robot}}$ and $\tau{}_{\textit{interaction\_human}}$ are action and reaction. $\tau_{\textit{interaction\_robot}}$ is the interaction torque that the human exerts on the robot. $\tau_{\textit{interaction\_human}}$ is the interaction torque that the robot exerts on the human arm. When the robot and human are regarded as a whole, $\tau_{\textit{interaction\_robot}}$ and $\tau_{\textit{interaction\_human}}$ are the internal forces. Therefore, we can get:

$\displaystyle\tau_{\textit{interaction\_robot}}=-\tau_{\textit{interaction\_% human}}$ (4)

With Eqs (1)–(4), we can find the relationship between the HRC torque, robot torque and human torque as follows:

$\displaystyle\tau_{\textit{HRC}}=\tau_{\textit{robot}}+\tau_{\textit{human}}$ (5)

It is known that the upper limb HRC torque is achieved by summing up the robot torque and the human torque through the force analysis of the interaction between the human arm and the robot. Obviously, it is not complete and accurate to only analyze the robot dynamics for the HRC dynamics, and ignoring the human dynamics will lead to inaccurate control of the upper limb rehabilitation robot. This way, the personalized training of patients developed by the therapists cannot be implemented, and the training effect of the patients is reduced. Next, we will solve and compare the robot torque and the HRC torque.

2.3 Analysis of 2-DOF dynamic model

The upper limb rehabilitation robot (FLEXO-Arm) has two active DOF, including shoulder flexion/extension and elbow flexion/extension. The two movements are in the same plane. So the robot can be reduced to a 2-DOF link to analyze its dynamics. The second kind of Lagrange equation is suitable for complete systems, and it can get the dynamic equation of complex systems in the simplest form. The dynamics of a 2-DOF rehabilitation robot can be calculated by the second kind of Lagrange equation, and the following equation is obtained as follows:

$\displaystyle\tau_{1}=\frac{d}{dt}\left({\frac{\partial L}{\partial\dot{\theta% }_{1}}}\right)-\frac{\partial L}{\partial\theta_{1}}=[{m_{1}d_{1}^{2}+m_{2}(l_% {1}^{2}+d_{2}^{2})+2m_{2}l_{1}d_{2}\cos\theta_{2}+I_{1}+I_{2}}]\ddot{\theta}_{% 1}{}+(m_{1}d_{1}+m_{2}l_{1})g\sin\theta_{1}+m_{2}gd_{2}\sin(\theta_{1}+\theta_% {2})+(m_{2}d_{2}^{2}+m_{2}l_{1}d_{2}\cos\theta_{2}+I_{2})\ddot{\theta}_{2}{}-2% m_{2}l_{1}d_{2}\sin\theta_{2}\cdot\dot{\theta}_{1}\dot{\theta}_{2}-m_{2}l_{1}d% _{2}\sin\theta_{2}\cdot\dot{\theta}_{2}^{2}$ (6) $\displaystyle\tau_{2}=\frac{d}{dt}\left({\frac{\partial L}{\partial\dot{\theta% }_{2}}}\right)-\frac{\partial L}{\partial\theta_{2}}=({m_{2}d_{2}^{2}+m_{2}l_{% 1}d_{2}\cos\theta_{2}+I_{2}})\ddot{\theta}_{1}+({m_{2}d_{2}^{2}+I_{2}})\ddot{% \theta}_{2}{}+m_{2}l_{1}d_{2}\sin\theta_{2}\cdot\dot{\theta}_{1}^{2}+m_{2}gd_{% 2}\sin({\theta_{1}+\theta_{2}})$ (7)

Here, $\tau_{1}$ is the torque of the shoulder joint, $\tau_{2}$ is the torque of the elbow joint, $\theta_{1}$ and $\theta_{2}$ are the angles of the shoulder and elbow joints. $\dot{\theta}_{1}$ and $\dot{\theta}_{2}$ are the angular velocities of the shoulder and elbow joints. $\ddot{\theta}_{1}$ and $\ddot{\theta}_{2}$ are the angular accelerations of the shoulder and elbow joints. $m_{1}$ and $m_{2}$ are the masses of the upper arm and forearm. $l_{1}$ is the length of the upper arm. $d_{1}$ and $d_{2}$ are the distances from the mass center to the rotation center of the upper arm and forearm. $I_{1}$ and $I_{2}$ are the moment of inertia of the shoulder and elbow joints.

Therefore, the 2-DOF robot dynamic model is proposed as follows:

$\displaystyle M(q)\ddot{q}+C({q,\dot{q}})\dot{q}+G(q)=\tau$ (8)

Here, $\tau=[\tau_{1},\tau_{2}]$ is the shoulder and elbow joints torque. $\ddot{q},\dot{q},q$ are the angular acceleration, angular velocity and angle of the joint, respectively. $M(q)$ is the mass matrix, $C({q,\dot{q}})$ is the centrifugal and Coriolis forces, and $G(q)$ is the gravity compensation matrix. The motor of the robot is affected by friction. The joint friction model is a complex nonlinear function, which is difficult to model accurately. Synthetically considering the effects of cost, efficiency, precision and other factors, only the viscous friction and Coulomb friction are considered in the parameter identification [22]. So, the 2-DOF upper limb rehabilitation robot dynamic model is proposed as follows:

$\displaystyle M(q)\ddot{q}+C({q,\dot{q}})\dot{q}+B\dot{q}+D\text{sgn}(\dot{q})% +G(q)=\tau$ (9)

Here, $B$ is the viscous friction term of the motor and $D(\dot{q})$ is the Coulomb friction term of the motor. According to the friction terms ( $B, D$ ) introduced in Eq. (9), combined with Eqs (6) and (7), we simplify the torque of the shoulder and elbow joints to the following equations:

$\displaystyle\left[\begin{array}[]{l}\tau_{1}\\ \tau_{2}\\ \end{array}\right]=\left[\begin{array}[]{ll}M_{1}+2R\cos(\theta_{2})&M_{2}+R% \cos(\theta_{2})\\ M_{2}+R\cos(\theta_{2})&M_{2}\\ \end{array}\right]\left[\begin{array}[]{l}\ddot{\theta}_{1}\\ \ddot{\theta}_{2}\\ \end{array}\right]+\left[\begin{array}[]{l}-R\sin(\theta_{2})\dot{\theta}_{2}^% {2}-2R\sin(\theta_{2})\dot{\theta}_{1}\dot{\theta}_{2}\\ R\sin(\theta_{2})\dot{\theta}_{1}^{2}\\ \end{array}\right]$ (10) $\displaystyle\quad+\left[\begin{array}[]{ll}B_{1}&0\\ 0&B_{2}\\ \end{array}\right]\left[\begin{array}[]{l}\dot{\theta}_{1}\\ \dot{\theta}_{2}\\ \end{array}\right]+\left[\begin{array}[]{l}D_{1}\text{sgn}(\dot{\theta}_{1})\\ D_{2}\text{sgn}(\dot{\theta}_{2})\\ \end{array}\right]+\left[\begin{array}[]{l}G_{1}\text{sgn}(\theta_{1})+G_{2}% \sin(\theta_{1}+\theta_{2})\\ G_{2}\sin(\theta_{1}+\theta_{2})\\ \end{array}\right]$ $\displaystyle M_{1}=2m_{1}d_{1}^{2}+2m_{2}d_{2}^{2}+m_{2}l_{2}^{2}$ $\displaystyle M_{2}=2m_{2}d_{2}^{2}$ $\displaystyle R=m_{2}d_{2}l_{1}$ (11) $\displaystyle G_{1}=(m_{1}d_{1}+m_{2}l_{1})g$ $\displaystyle G_{2}=m_{2}gd_{2}$

Therefore, the parameters to be solved in the simplified torque Eqs (2.3) and (2.3) include the following nine parameters ( $M_{1},M_{2},R,B_{1},B_{2},D_{1},D_{2},G_{1},G_{2}$ ). We can determine that the dynamic parameter set $p$ : $p=\{{M_{1},M_{2},R,B_{1},B_{2},D_{1},D_{2},G_{1},G_{2}}\}\in R^{9}$ .

For a 2-DOF link, it corresponds to a parameter set $p$ with nine parameters, and its joint torque can be calculated if it is required to the parameter set $p$ . When the human arm is placed on FLEXO-Arm and follows the robot, the human arm is consistent with the motion of the robot. The human arm is also limited to shoulder flexion/extension and elbow flexion/extension, while the movement of the human arm in other directions is constrained. In that case, the human arm can be simplified as a 2-DOF link. We noted that both FLEXO-Arm and the human arm can be simplified into a 2-DOF link, so the whole body that couples the human arm and the robot can also be a 2-DOF link. The HRC torque can be obtained if its parameters are required. Therefore, the HRC torque can be solved by Lagrangian equation.

2.3.1 Dynamics analysis of the upper limb rehabilitation robot

If we can find the dynamic parameters of FLEXO – Arm, the torque of the robot can be solved by Eqs (2.3) and (2.3). We can assume the robot dynamic parameters set $p_{r}$ :

$\displaystyle p_{r}=\{{M_{r1},M_{r2},R_{r1},B_{r},B_{r2},D_{r1},D_{r2},G_{r1},% G_{r2}}\}\in R^{9}$

The purpose of dynamic parameter identification is to accurately calculate the torque required for a robot to carry out a movement according to the identification value of dynamic parameters. So the next step is the identification of $p_{r}$ . In the experiments of the robot parameter identification, an optimized trajectory is given for each joint of the robot. In the process of the robot movement, the torque and angle of each joint are measured. The measured data are substituted into the identification model, and the value of dynamic parameters can be calculated through the identification algorithm. We use a step – by – step identification method to design experiments to identify the parameters of the shoulder and elbow joints. The first step is identifying the dynamic parameters of the elbow joint of the robot, and the second step is identifying the dynamic parameters of the shoulder joint of the robot.

We design three experiments (experiments 1–3) to identify the dynamic parameters of the robot. We can identify the elbow dynamic parameters through experiment 1 and the shoulder dynamic parameters through experiments 2 and 3.

Experiment 1: The shoulder joint of the robot is fixed at $q_{1}$ , and the elbow joint moves freely. The purpose of experiment 1 is to identify the parameters $\{{M_{r2},B_{r2},D_{r2},G_{r2}}\}$ . The corresponding dynamic equation of experiment 1 is shown in Eq. (12):

$\displaystyle M_{2}\ddot{q}_{2}(t)+B_{2}\dot{q}_{2}(t)+D_{2}\text{sgn}(\dot{q}% _{2}(t))+G_{2}\sin(q_{1}+q_{2})=[\begin{array}[]{*{20}c}{\ddot{q}_{2}(t)}&{% \dot{q}_{2}(t)}&{\text{sgn}(\dot{q}_{2}(t))}&{\sin(q_{1}(t)+q_{2}(t))}\\ \end{array}]\left[{\begin{array}[]{l}M_{r2}\\ B_{r2}\\ D_{r2}\\ G_{r2}\\ \end{array}}\right]=\tau_{2}(t)$ (12)

After FLEXO-Arm moved according to experiment 1, we can obtain the angle, angular velocity, angular acceleration and elbow joint torque $\{{q_{1}(t),q_{2}(t),\dot{q}_{2}(t),\ddot{q}_{2}(t),\tau_{2}(t)}\}$ of the robot and substitute them into Eq. (12). The parameters $\{{M_{r2},B_{r2},D_{r2},G_{r2}}\}$ can be obtained by solving the equation by least square method. The least square method is to find the optimal function matching of data by minimizing the squared sum of errors, and is also a common method to solve overdetermined equations.

Experiment 2: The elbow joint of the robot is fixed at $q_{21}$ , and the shoulder joint moves freely. The purpose of experiment 2 is to identify the parameters $\left\{{M_{r1},B_{r1},D_{r1},G_{r1}}\right\}$ . The corresponding dynamic equation of experiment 2 is shown as follows:

$\displaystyle(M_{1}+2Rr\cos(q21))\ddot{q}_{1}(t)+B_{1}\dot{q}_{1}(t)+D_{1}% \text{sgn}(\dot{q}_{1}(t))+G_{1}\sin(q_{1})+G_{2}\sin(q_{1}+q_{21})=[{\begin{% array}[]{*{20}c}{\ddot{q}_{1}(t)}&{\dot{q}_{1}(t)}&{\text{sgn}(\dot{q}_{1}(t))% }&{\sin(q_{1}(t))}&{\sin(q_{1}(t)+q_{2}(t))}\\ \end{array}}]\left[{\begin{array}[]{l}M_{r11}\\ B_{r1}\\ D_{r1}\\ G_{r1}\\ G_{r2}\\ \end{array}}\right]=\tau_{1}(t)$ (13)

The data $\{{q_{1}(t),q_{2}(t),\dot{q}_{1}(t),\ddot{q}_{1}(t),\tau_{1}(t)}\}$ is obtained from experiment 2, and the parameters $\{M_{r11},\linebreak B_{r1},D_{r1},G_{r1}\}$ can be calculated by the least square method.

Experiment 3: The elbow joint of the robot is fixed at $q_{22}$ , and the shoulder joint moves freely. The purpose of experiment 3 is to identify the parameter $R_{r}$ .

The corresponding dynamic equation of experiment 3 is shown as follows:

$\displaystyle(M_{1}+2R_{r}\cos(q_{22}))\ddot{q}_{1}(t)+B_{1}\dot{q}_{1}(t)+D_{% 1}\text{sgn}(\dot{q}_{1}(t))+G_{1}\sin(q_{1})+G_{2}\sin(q_{1}+q_{22})=[{\begin% {array}[]{*{20}c}{\ddot{q}_{1}(t)}&{\dot{q}_{1}(t)}&{\text{sgn}(\dot{q}_{1}(t)% )}&{\sin(q_{1}(t))}&{\sin(q_{1}(t)+q_{2}(t))}\\ \end{array}}]\left[{\begin{array}[]{l}M_{r12}\\ B_{r1}\\ D_{r1}\\ G_{r1}\\ G_{r2}\\ \end{array}}\right]=\tau_{1}(t)$ (14)

The formula of $R_{r}$ can be found from Eqs (13) and (14) as follows:

$\displaystyle R_{r}=\frac{Mr_{21}-Mr_{11}}{2\cos(q_{22}-q_{21})}$ (15)

The data $\{{q_{1}(t),q_{2}(t),\dot{q}_{1}(t),\ddot{q}_{1}(t),\tau_{1}(t)}\}$ is obtained from experiment 3, and the parameter $R_{r}$ can be solved.

The nine dynamic parameters of the robot can be obtained through the three experiments (experiments 1–3). The torque of the robot joints can be calculated by substituting these parameters into Eq. (2.3).

2.3.2 Upper limb HRC dynamics analysis

We know that the whole body couples the human arm and the robot, which can also be simplified as a 2-DOF link, and the HRC torque can be solved by Eq. (2.3). We can assume the HRC dynamic parameters set $p_{c}$ :

$\displaystyle p_{c}=\{{M_{c1},M_{c2},R_{c},B_{c1},B_{c2},D_{c1},D_{c2},G_{c1},% G_{c2}}\}\in R^{9}$

The identification method of the HRC dynamic parameters is the same as that of the robot, except that in the experiment the human and robot move together.

Experiment 4: The shoulder joint of the robot is fixed at $q^{\prime}_{1}$ , and the elbow joint moves freely. During the experiment, the human arm and the robot arm move together. The purpose of experiment 4 is to identify the parameters $\{{M_{c2},B_{c2},D_{c2},G_{c2}}\}$ .

Experiment 5: The elbow joint of the robot is fixed at $q^{\prime}_{21}$ , and the shoulder joint moves freely. During the experiment, the human arm and the robot arm move together. The purpose of experiment 5 is to identify the parameters $\{{M_{c1},B_{c1},D_{c1},G_{c1}}\}$ .

Experiment 6: The elbow joint of the robot is fixed at $q^{\prime}_{22}$ , and the shoulder joint moves freely. During the experiment, the human arm and the robot arm move together. The purpose of experiment 6 is to identify the parameters $R_{c}$ .

Figure 3.

Experiment 1: The shoulder joint of the robot is fixed at 2 ${}^{\circ}$ , and the elbow joint moves alone. A. The data $\{q_{1}(t),\linebreak q_{2}(t),\dot{q}_{2}(t),\ddot{q}_{2}(t),\tau_{2}(t)\}$ of experiment 1. B. The comparison of experimental torque and actual torque of experiment 1.

Figure 4.

Experiment 2: The elbow joint of the robot is fixed at 28 ${}^{\circ}$ , and the shoulder joint moves alone. A. The data $\{q_{1}(t),\linebreak q_{2}(t),\dot{q}_{1}(t),\ddot{q}_{1}(t),\tau_{1}(t)\}$ of experiment 2. B. The comparison of experimental torque and actual torque of experiment 2.

3. Experimental results

3.1 Identification of upper limb rehabilitation robot dynamic parameters

In order to solve the dynamic parameters of the robot, we designed experiments 1–3. The data of these experiments is collected by the robot in one minute, as shown in Figs 3A, 4A and 5A. The two pictures above are the motor torque (T) and joint angle (Angle) are collected by the torque sensor and encoder at the robot joints. The following two figures are the joint angular velocity and angular acceleration. The angular velocity is derived from the joint angle and time, and the angular acceleration is derived twice from the joint angle and time.

Table 1
The dynamic parameters of the robot

$M_{r1}$	$M_{r2}$	$R_{r}$	$B_{r1}$	$B_{r2}$	$D_{r1}$	$D_{r2}$	$G_{r1}$	$G_{r2}$
$-$ 3.4822	$-$ 2.2687	$-$ 3.1695	17.2255	2.9054	0.8552	0.7286	15.7884	4.1252

We can solve a set of dynamic parameters of the robot through experiments 1–3, of which the results are shown in Table 1. After obtaining the values of the parameters, we can solve the experimental torques of the shoulder joint and elbow joint of the robot in reverse as, shown in Eq. (3.1). The actual torque is the elbow joint torque of the robot measured by the torque sensor. The experimental torque is solved by the robot dynamic parameters. In Figs 3B, 4B and 5B, the pink dotted line is the actual robot joint torque without filtering, the red curve is the actual robot joint torque after filtering, and the blue curve represents the experimental torque. In order to quantify the effect of dynamic parameters identification, the root mean square (RMS) evaluation of the experimental and actual values of torque is used [22], as shown in Eq. (17). $\tau_{e}$ is the experimental torque, $\tau_{a}$ is the actual torque without filtering, and $K$ is the number of torques.

Table 2

Torque RMS of experiments 1–6

Experiment	Joint	RMS (N.m)
1	Elbow	0.7608
2	Shoulder	2.7999
3	Shoulder	2.6704
4	Elbow	0.9159
5	Shoulder	3.6425
6	Shoulder	2.7001

Figure 5.

Experiment 3: The elbow joint of the robot is fixed at 85 ${}^{\circ}$ , and the shoulder joint moves alone. A. The data $\{q_{1}(t),\linebreak q_{2}(t),\dot{q}_{1}(t),\ddot{q}_{1}(t),\tau_{1}(t)\}$ of experiment 3. B. The comparison of experimental torque and actual torque of experiment 3.

The torque RMS of experiments 1–3 is shown in Table 2. The torque RMS of elbow joint is 0.7608 N.m, the maximum of actual torque measured in experiment 1 is about 7.5 N.m, and the error is 10.1%. The torque RMS of experiment 2 is 2.7999 N.m, the maximum of actual torque measured in experiment 2 is about 35.0 N.m, and the error is 8.0%. The torque RMS of experiment 3 is 2.6704 N.m, the maximum of actual torque measured in experiment 3 is about 28.0 N.m, and the error is 9.5%. The error of shoulder joint is kept within 10%.

$\displaystyle\left[\begin{array}[]{*{20}c}\tau_{\textit{robot1}}\\ \tau_{\textit{robot2}}\\ \end{array}\right]=\left[\begin{array}[]{ll}-3.4822-{6.3390}\cos\theta_{2}&-2.% 2687-3.1695\cos\theta_{2}\\ -2.2687-3.1695\cos\theta_{2}&-2.2687\\ \end{array}\right]\left[\begin{array}[]{l}\ddot{\theta}_{1}\\ \ddot{\theta}_{2}\\ \end{array}\right]+\left[\begin{array}[]{l}0.8552\text{sgn}(\dot{\theta}_{1})% \\ 0.7286\text{sgn}(\dot{\theta}_{2})\\ \end{array}\right]$ $\displaystyle\quad{}+\left[\begin{array}[]{l}15.7884\sin(\theta_{1})+4.1252% \sin(\theta_{1}+\theta_{2})\\ 4.1252\sin(\theta_{1}+\theta_{2})\\ \end{array}\right]+\left[\begin{array}[]{l}3.1695\sin(\theta_{2})\dot{\theta}_% {2}^{2}-{6.3390}\sin(\theta_{2})\dot{\theta}_{1}\dot{\theta}_{2}\\ -3.1695\sin(\theta_{2})\dot{\theta}_{1}^{2}\\ \end{array}\right]$ (16) $\displaystyle\quad{}+\left[\begin{array}[]{ll}17.2255&0\\ 0&2.9054\\ \end{array}\right]\left[\begin{array}[]{l}\dot{\theta}_{1}\\ \dot{\theta}_{2}\\ \end{array}\right]$ $\displaystyle\Delta\tau_{\textit{RMS}}=\sqrt{\frac{1}{K}\sum_{k=1}^{K}% \varepsilon_{k}^{2}}=\sqrt{\frac{1}{K}\sum_{k=1}^{K}(\tau_{e}(k)-\tau_{a}(k))^% {2}}$ (17)

3.2 Identification of HRC dynamic parameters

In order to solve the HRC dynamic parameters, we designed experiments 4–6. The data of these experiments is collected by the robot in one minute, as shown in Figs 6A, 7A and 8A. The data acquisition and processing of HRC parameters identification are the same as those of the robot parameters identification. We can solve the HRC parameters through experiments 4–6, of which the results are shown in Table 3. After obtaining the values of the parameters, we can solve the experimental torques of the shoulder joint and elbow joint of HRC in reverse, as shown in Eq. (3.2).

Table 3
The dynamic parameters of HRC

$M_{c1}$	$M_{c2}$	$R_{c}$	$B_{c1}$	$B_{c2}$	$D_{c1}$	$D_{c2}$	$G_{c1}$	$G_{c2}$
$-$ 10.9229	$-$ 4.7493	$-$ 8.6157	19.8486	3.3915	1.9679	0.9241	21.4468	5.8234

Figure 6.

Experiment 4: The shoulder joint of the robot is fixed at 2 ${}^{\circ}$ , and the elbow joint moves alone. A. The data $\{q_{1}(t),\linebreak q_{2}(t),\dot{q}_{2}(t),\ddot{q}_{2}(t),\tau_{2}(t)\}$ of experiment 1. B. The comparison of experimental torque and actual torque of experiment 4.

Figure 7.

Experiment 5: The elbow joint of the robot is fixed at 28 ${}^{\circ}$ , and the shoulder joint moves alone. A. The data $\{q_{1}(t),\linebreak q_{2}(t),\dot{q}_{1}(t),\ddot{q}_{1}(t),\tau_{1}(t)\}$ of experiment 5. B. The comparison of experimental torque and actual torque of experiment 5.

In Figs 6B, 7B and 8B, the pink dotted line is the actual HRC torque without filtering, the red curve is the actual HRC torque after filtering, and the blue curve represents the experimental HRC torque. The torque RMS of experiments 4–6 is shown in Table 2. The torque RMS of elbow joint is 0.9159 N.m, the maximum of actual torque measured in experiment 4 is about 10.0 N.m, and the error is 9.0%. The torque RMS of experiment 5 is 3.6425 N.m, the maximum of actual torque measured in experiment 5 is about 44.0 N.m, and the error is 8.3%. The torque RMS of experiment 6 is 2.7001 N.m, the maximum of actual torque measured in experiment 6 is about 37.0 N.m, and the error is 7.3%. The error of shoulder joint is kept within 10%.

$\displaystyle\left[\begin{array}[]{l}\tau_{\textit{coupling1}}\\ \tau_{\textit{coupling2}}\\ \end{array}\right]=\left[\begin{array}[]{ll}-10.9229-{17.2314}\cos\theta_{2}&-% 4.7493-8.6157\cos\theta_{2}\\ -4.7493-8.6157\cos\theta_{2}&-4.7493\\ \end{array}\right]\left[\begin{array}[]{c}\ddot{\theta}_{1}\\ \ddot{\theta}_{2}\\ \end{array}\right]$ $\displaystyle\quad{}+\left[\begin{array}[]{ll}19.8486&0\\ 0&3.3915\\ \end{array}\right]\left[\begin{array}[]{l}\dot{\theta}_{1}\\ \dot{\theta}_{2}\\ \end{array}\right]+\left[\begin{array}[]{l}8.6157\sin(\theta_{2})\dot{\theta}_% {2}^{2}+17.2314\sin(\theta_{2})\dot{\theta}_{1}\dot{\theta}_{2}\\ -8.6157\sin(\theta_{2})\dot{\theta}_{1}^{2}\\ \end{array}\right]$ (18) $\displaystyle\quad{}+\left[\begin{array}[]{l}1.9679\text{sgn}(\dot{\theta}_{1}% )\\ 0.9241\text{sgn}(\dot{\theta}_{2})\\ \end{array}\right]+\left[\begin{array}[]{l}21.4468\sin(\theta_{1})+5.8234\sin(% \theta_{1}+\theta_{2})\\ 5.8234\sin(\theta_{1}+\theta_{2})\\ \end{array}\right]$

3.3 Comparison of the robot torque and HRC torque

In the teaching mode of FLEXO-Arm, the robot drives the human arms to move together, and the shoulder and elbow joints move freely. The joint angles are recorded for one minute and the obtained angular velocities and accelerations of the joints are calculated by the differential method. The data $\{{q_{1}(t),q_{2}(t),\dot{q}_{1}(t),\dot{q}_{2}(t),\ddot{q}_{1}(t),\ddot{q}_{2% }(t),\tau_{1}(t),\tau_{2}(t)}\}$ are substituted into Eqs (3.1) and (3.2), and the robot torque and HRC torque are calculated separately. The experimental results are shown in Fig. 9. The pink line is the actual joint torque measured by torque sensors without filtering, and the red curve is the actual torque after filtering. The blue curve is the HRC torque calculated by Eq. (3.2), and the green curve represents the robot torque calculated by Eq. (3.1). The torque RMS of the robot and HRC is shown in Table 4. The torque RMS of the robot shoulder is 8.0712 N.m, the maximum of actual torque is about 45.0 N.m, and the error is 17.9%. The torque RMS of the robot elbow is 1.8873 N.m, the maximum of actual torque is about 10.0 N.m, and the error is 18.9%. The torque RMS of the HRC shoulder is 3.3401 N.m, and the error is 7.4%. The torque RMS of the HRC elbow is 0.9105 N.m, and the error is 9.1%. The results show that the torque RMSs of the shoulder and elbow joints of the robot is more than twice that of HRC.

Table 4
Torque RMS of the comparison of the robot and HRC

Comparison	Joint	RMS (N.m)
Robot	Shoulder	8.0712
	Elbow	1.8873
HRC	Shoulder	3.3401
	Elbow	0.9105

Figure 8.

Experiment 6: The elbow joint of the robot is fixed at 85 ${}^{\circ}$ , and the shoulder joint moves alone. A. The data $\{q_{1}(t),\linebreak q_{2}(t),\dot{q}_{1}(t),\ddot{q}_{1}(t),\tau_{1}(t)\}$ of experiment 6. B. The comparison of experimental torque and actual torque of experiment 6.

Figure 9.

Comparison of the robot torque and HRC torque. A. The elbow torque. B. The shoulder torque.

4. Discussion

This paper proposes a method for the modeling and analysis of HRC dynamics of upper limb rehabilitation robots, and solves and compares the HRC torque and the robot torque of a 2-DOF upper limb rehabilitation robot (FLEXO-Arm). The errors of HRC and robot dynamic parameters identification are both around 10%, and the robot’s torque RMSs are more than twice HRC. The results show the effectiveness of the HRC dynamics proposed in this paper and the HRC dynamics should be analyzed for the analysis of upper limb rehabilitation robot dynamics, not just the robot dynamics or human dynamics.

Upper limb rehabilitation robots can be classified into end-effector-base and exoskeleton robots. The end-effector-base robot traction by connecting the end-effector of the robot to the human hand. For the dynamic modeling of the end-effector-base robot, the interaction force between the human hand and the end-effector of the robot is collected, and the impedance model of the human arm is established. The torque of each joint can be solved by the robot’s Jacobi equation and the human-robot interaction force [23, 24]. The exoskeleton robot is designed by imitating the joints of human arm based on the bionic principle, so that it has a similar kinematic structure with human arm. The exoskeleton robot is attached to the human upper limb at multiple locations, which makes the force difficult to measure [1]. Hence, many researchers established the equivalent mechanism of the human arm and analyzed the kinematics and dynamics of the model, which provided a reference for the exoskeleton dynamics model [14, 15]. Due to the complexity of upper limb movement, many researchers ignored the influence of the human arm when analyzing the dynamics of exoskeleton robots. In this paper, the HRC dynamics is proposed, which takes the human arm and robot as a whole. The method is not only applicable to passive motion of the rehabilitation robot, but also to assistive motion. The assistive control strategy will be completed in the follow-up work.

5. Conclusions

This paper analyzes the human-robot interaction and proposes that the HRC torque is equal to the robot torque plus the human arm torque. The HRC torque and the robot torque of a 2-DOF upper limb rehabilitation robot (FLEXO-Arm) are solved by Lagrange equation and parameter identification. The root mean square is used to evaluate the accuracy of the HRC torque and the robot torque calculated by the parameter identification, and the error of both is about 10%. Moreover, the HRC torque and the robot torque are compared with the actual torque measured by torque sensors. The error of the robot torque is more than twice the HRC. Therefore, the HRC torque is more accurate than the actual torque. The results show the effectiveness of the HRC dynamics proposed in this paper and the HRC dynamics should be analyzed for upper limb rehabilitation robot dynamics, not just robot dynamics or human dynamics.

Footnotes

Acknowledgments

The authors gratefully acknowledge the financial support by th National Key R&D Program of China (2018YFB1307200), and the National Natural Science Foundation of China (61803265).

Conflict of interest

None to report.

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Human-robot coupling dynamic modeling and analysis for upper limb rehabilitation robots

Abstract

BACKGROUND:

OBJECTIVE:

METHODS:

RESULTS:

CONCLUSIONS:

Keywords

1. Introduction

2. Methods

2.1 The prototype of an upper limb rehabilitation robot (FLEXO-Arm)

3.1 Identification of upper limb rehabilitation robot dynamic parameters

Table 1 The dynamic parameters of the robot

Table 3 The dynamic parameters of HRC

Table 4 Torque RMS of the comparison of the robot and HRC

5. Conclusions

Footnotes

Acknowledgments

Conflict of interest

References

Table 1
The dynamic parameters of the robot

Table 3
The dynamic parameters of HRC

Table 4
Torque RMS of the comparison of the robot and HRC