Modelling of extended de-weight fuzzy control for an upper-limb exoskeleton

2.1. Kinematics and dynamics of the human arm

The development of the upper-limb exoskeleton is based on the kinematics and dynamics of the human arm during daily activities. Hence, it is important to know how movement occurs in the human body, specifically the human upper limbs. It is known that the musculoskeletal system comprising the connective tissues, muscles, and joints supports, stabilises, protects, and produces precise movements for the human body.

The human upper limb is designed as segments of linked bones that are linked by joints that have multiple degrees of freedom. The movements of the human upper limb are due to the existence of muscles. Muscles provide power or moment across the joints to enable movement. Some examples of human movements are shoulder flexion/extension, shoulder abduction/adduction, elbow flexion/extension, and wrist flexion/extension.

Fig. 1Shoulder movements: Initial position (Extreme left), (1a) Shoulder abduction, (1b) Shoulder adduction, (2a) Shoulder extension (2b), Shoulder flexion, (3a) Shoulder internal rotation, and (3b) Shoulder external rotation

In this study, the development of the exoskeleton is solely based on the daily upper-extremity activities, in which the activities usually comprise combinations of basic movements. The movement combinations are presented in Table 1 and shown in Figs. 1-3. In this study, the shoulder is attached to the body by a ball and socket joint, and this joint is responsible for shoulder flexion/extension, shoulder adduction/abduction, and shoulder internal/external movements (Fig. 1). The upper and lower arm are connected by a single rotating joint known as a revolute joint at the elbow (Fig. 2). The lower arm and the palm are also connected by a revolute joint at the wrist (Fig. 3). The region of rotation for each joint is shown in Table 2.

Fig. 2Initial position (Extreme left) (a) Elbow flexion (b) Elbow extension (c) Forearm pronation (d) Forearm supination

Fig. 3Initial position (Extreme left) (a) Wrist extension (b) Wrist flexion (c) Ulnar deviation (d) Radial deviation

2.2. Mechanical design of the upper-limb exoskeleton

The design of the exoskeleton used in this study was inspired by the TitanArm due to the simplicity of its design. Moreover, it is capable of powered use and data transmission on a mobile platform [9]. The material used to design the exoskeleton is aluminium as it is a low-density material with reasonable strength characteristics. As mentioned in Section 2.1, the shoulder joint in this study has three degrees of freedom, while the elbow and wrist joints have one degree of freedom, respectively (Fig. 4).

2.2.1. Kinematics of the exoskeleton

Fig. 5 shows the schematic diagram of the exoskeleton and the Denavit-Hartenberg Table for shoulder adduction/abduction movements. The $0_1$, $0_2$, $0_3$, and $0_4$ values represent the base for shoulder internal/external, shoulder extension/flexion, elbow extension/flexion, and the endpoint of the exoskeleton. The homogenous transformation matrix Eq. (1) is used to obtain the position and orientation of the end-effector ($0_4$), with respect to the fixed reference frame ($0_0$):

1

${Ti}_i^{i-1}=Rot\left(Z,{\theta }_i\right)Trains\left(Z,d_i\right)Trains\left(X,a_i\right)Rot\left(X,{\alpha }_i\right)$
$=\left[\begin{array}{c}\begin{array}{c}\cos {\theta }_i\sin {\theta }_i\cos {\alpha }_i\mathit{}\sin {\theta }_i\sin {\alpha }_i{a}_i\cos {\theta }_i\\ \sin {\theta }_i\cos {\theta }_i\cos {\alpha }_i-\cos {\theta }_i\cos {\alpha }_i{a}_i\sin {\theta }_i\end{array}\\ 0\sin {\alpha }_i\cos {\theta }_i{d}_i\\ 0001\end{array}\right].$

Fig. 4View of the exoskeleton from three perspectives

a) Front view

b) Right view

c) Perspective view

Fig. 5a) Schematic diagram of exoskeleton, b) Denavit-Hartenberg table

a)

b)

2.2.2. Dynamics of the exoskeleton

The dynamics of the upper-limb exoskeleton was developed using the Euler-Lagrange approach as it is frequently used for the modelling of rigid robotic systems. Euler Lagrange is derived from Lagrangian ($\mathcal{L}$). Lagrangian is the difference between the kinetic ($\mathcal{T}$) and potential ($\mathcal{V}$) energy:

2

$\mathcal{L}\left(q_i,{\dot{q}}_i\right)=\mathcal{T}\left(q_i,{\dot{q}}_i\right)-\mathcal{V}\left(q_i\right).$

The total kinetic and potential energy for the whole system is presented as follows:

3

$\mathcal{T}\left(q_i,{\dot{q}}_i\right)=\frac{1}{2}\left(m_1\left({\dot{x}}_1^2+{\dot{y}}_1^2\right)+m_2\left({\dot{x}}_2^2+{\dot{y}}_2^2\right)+m_3\left({\dot{x}}_3^2+{\dot{y}}_3^2\right)+m_4\left({\dot{x}}_4^2+{\dot{y}}_4^2\right)\right)$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+\frac{1}{2}(I_1{\dot{\theta }}_1^2+I_2({{\dot{\theta }}_1+{\dot{\theta }}_2)}^2+I_3({{\dot{\theta }}_1+{\dot{\theta }}_2+{\dot{\theta }}_3)}^2+I_4({{\dot{\theta }}_1+{\dot{\theta }}_2+{\dot{\theta }}_3+{\dot{\theta }}_4)}^2,$

4

$\mathcal{V}\left(q_i\right)=m_{1}g{y}_1+m_{2}g{y}_2\mathrm{}{+\mathrm{}m}_{3}g{y}_3{+\mathrm{}m}_{4}g{y}_4.$

By applying the partial derivative to Eqs. (3, 4), the dynamic of the whole system is obtained and presented as follows:

5

$M\left(q\right)\ddot{q}+C\left(q,\dot{q}\right)\dot{q}+G\left(q\right)=\tau .$

The $M\left(q\right)$ is the exoskeleton inertia matrix, $C\left(q,\dot{q}\right)$ is a matrix containing Coriolis and Centrifugal terms, $G\left(q\right)$ is a vector containing the gravity term, and $\tau$ is the vector of external torques acting on the actuated degree-of-freedom (DOF). The $q$, $\dot{q},$ and $\ddot{q}$ are positions, angular velocities, and angular accelerations of the revolute joints.

The dynamic model of the whole system was validated by comparing the generated torques between Eq. (5) with the SimMechanics model (Fig. 6). The input to the system is the angular positions of the shoulder and elbow joints for both the flexion and extension movements. The results for the validation process are presented in Section 4.

Fig. 6Validation approach

3.1. Implementation of PID control

The PID control was selected as a reference point for comparison as it was simple to implement and reliable. The input to the PID controller is the error of the trajectory. The error is obtained by comparing the actual trajectory from the desired trajectory. The error is then fed into the controller and the necessary torque is generated by the controller. The generated torque is then sent to the motor joint to move or to achieve the desired trajectory (Fig. 8).

Fig. 8Implementation of the exoskeleton with PID control

3.2. Implementation of fuzzy-based PD control

The implementation of the fuzzy-based PD control for the exoskeleton is presented in Fig. 9. The combination of fuzzy-based PD was selected due to the ability of PD to minimise the steady-state error and the rise time, thus requiring less power consumption.

The inputs to the fuzzy-based PD control are the trajectory error ($e$) and the rate of change of error ($\dot{e}$). The error is obtained by measuring the difference between the actual trajectory and the desired trajectory, while the rate of change of error is calculated by the derivative of the error. The controller will generate the torque according to the inputs to ensure that the motor joint moves to the desired trajectory.

Fig. 9Implementation of the exoskeleton with fuzzy-based PD control

3.3. Implementation of an extended de-weight fuzzy control

The extended de-weight fuzzy controller consists of fuzzy-based PD and fuzzy-based de-weighting control (Fig. 10). The fuzzy-based PD assists the motor joint to move to the desired trajectory, whilst the fuzzy-based de-weighting is used to compensate the gravity torque to increase the smoothness of the movement. The inputs for the fuzzy-based PD is similar to those described in Section 3.2. The inputs for the fuzzy-based de-weight are position error ($e$) and the current position of the motor joint ($a$). The torques generated from the fuzzy-based PD and fuzzy-based de-weight are summed up and sent to the motor joint. For both fuzzy-based PD and extended de-weight fuzzy, a saturation block is included to ensure that the inserted torque is within the range of the human torque [22, 23].

Fig. 10Implementation of the exoskeleton with extended de-weight fuzzy control

4.1. Validation of the dynamic model

As previously described in Section 2.2.2, the inputs to the system were the angular positions of the shoulder and elbow joint. Flexion and extension movements were applied to both joints (Fig. 11).

Fig. 11Angular positions of flexion and extension for shoulder and elbow joints

Fig. 12(a-b) shows that although the movement patterns of the velocity and torque required for each joint were similar, they had different values. The torque generated from the mathematical representation (Simulink) was slightly higher than the value obtained from SimMechanics. This could be due to minor differences in the geometrical model built into SimMechanics and parameters described in the mathematical equation [10].

4.2. Observation of control strategies

As previously mentioned in Section 3, three types of controllers were used, and their performances were compared. During the experiments, a disturbance was included to validate the stability and robustness of the system. The external forces or disturbance at 1000 Nm were applied at the position shown in Fig. 13 from 2.5 s until 10 s. The external force applied at the forearm was expected to affect the movement of the elbow joint.

Fig. 12Flexion and extension movements: a) velocity and b) torque required

a)

b)

The control parameters used for PID, fuzzy-based PD, and extended de-weight fuzzy are shown in Table 3. These parameters were obtained by heuristic tuning. In general, all the controllers were able to track the desired movements even if a high disturbance was applied to the forearm of the exoskeleton (Figs. 14-16), thus indicating that all the controllers were stable and robust. As mentioned previously, joint 4 was the most affected as the disturbance was applied at the forearm. The observations were performed in two stages. Firstly, the PID and fuzzy-PD controllers in terms of root mean square (RMS) and maximum absolute error (MAE) (Table 4) were observed. As shown in Table 4, the RMS of Joint 4 for PID was slightly higher as compared to fuzzy-PD. In addition, the MAE of PID was almost 50 % higher than the fuzzy-PD. This indicates that the maximum difference between the desired and actual trajectory for fuzzy-PD is better than PID, despite the high disturbance applied to the system.

Secondly, the fuzzy-PD was compared with the control proposed in this study, namely the extended de-weight fuzzy.

Fig. 13Exoskeleton: a) exoskeleton movement and b) external disturbance

Fig. 14Performance of PID controller with external force (1000 Nm)

Based on Table 4, the RMS and MAE values for the extended de-weight fuzzy were lesser as compared to fuzzy-PD. This indicates that the combination of fuzzy-based PD and de-weight fuzzy provides a more stable and robust system as compared to the previous study [1, 2]. This observation could be due to the algorithm of the extended de-weight fuzzy itself, in which the value of the actual movements and the error is considered and subsequently added as inputs to the fuzzy. Therefore, when the disturbance was applied, the system could identify that there was a slight uncommon movement from the actual trajectory and hence, corrected the uncommon movement.

Fig. 15Performance of fuzzy-PD controller with external force (1000 Nm)

Fig. 16Performance of extended de-weight fuzzy controller with external force (1000 Nm)