The above control law can be written in the following form:

where H z − 1 = h 1 − z − 1 and G z − 1 = g 0 + g 1 z − 1 . Note that the proportional and differential gains in Equation (8) are chosen by the following relation Equation (10).

An effective technique to design the polynomials H z − 1 and G z − 1 is based on the pole assignment concept (Goodwin and Sin, 1984).

Applying the controller Equation (8) and combining Equation (3) with Equation (9) yield the closed-loop Equation (11).

Let the closed-loop characteristic polynomial be defined as T z − 1 = t 0 + t 1 z − 1 + t 2 z − 2 , which has stable poles.

When A z − 1 and B z − 1 are completely known, to ensure the closed-loop stability, the polynomials H z − 1 and G z − 1 should be designed based on:

From Equations (2), (9), and (12), the coefficients are:

Further based on Equation (10), the proportional and differential gains k_p and k_I are designed as follows:

The above analysis is carried out based on the deterministic model. However, such an assumption is unrealistic for the SMA actuator. Actually, the parameters a₁ and b₁ of the gray-box model are uncertain, and it is difficult to offline choose fixed and appropriate k_p and k_I to ensure the closed-loop stability during the whole operating range. A more reasonable treatment seems to estimate A z − 1 and B z − 1 recursively, to update H z − 1 and G z − 1 online, and then to calculate k_p and k_I.

This subsection presents an online adaptation algorithm for uncertain parameters. Recursive least squares (RLS) algorithm has a fast convergence rate. However, it has high computational complexities, especially when it is applied to the Beckhoff IPC programming. On the other hand, recursive stochastic gradient (RSG) algorithm is more favorable to model adaptations, but it leads to a much slower convergence rate. To this end, a novel recursive estimator will be introduced, which has a similar form as the RSG algorithm, but possesses a similar convergence rate as the RLS algorithm.

The parameter identification will be carried out based on Equation (3). We first impose an assumption on this system.

Assumption 1: The unmodeled dynamics ζ(t) satisfies

where the bound Δ is user-designed.

Then the uncertain parameter estimation vector θ ^ t can be updated by the following modified recursive multi-innovation stochastic gradient identification algorithm (Zhang et al., 2008) with a novel dead-zone weighted factor:

where p is the dimension of the extended signals, which is designed by the user; r(0)=1; e(t) is the model error; E(t) is the extended model error; Φ(t) is the extended regressor vector; λ(t) is a nonnegative weighted factor; ε is a user-designed adaptation gain and satisfies 0 < ε ≤ 2 (Lemma 1 will explain the reason).

Based on Equation (7), the estimated polynomials at instant t can be defined as:

In order to adaptively design the proportional and differential gains k_p and k_I for the PI control Equation (8), a modified relationship is as follows:

where H ^ t z − 1 and G ^ t z − 1 are used in place of H z − 1 and G z − 1 . These two polynomials are defined as:

It is desired that the polynomials H ^ t z − 1 and G ^ t z − 1 satisfy the following relation:

where T z − 1 = t 0 + t 1 z − 1 + t 2 z − 2 is a pre-specified characteristic polynomial with stable poles.

Now that the estimates a ^ 1 t and b ^ 1 t are obtained, then the coefficients h ^ t , g ^ 0 t , and g ^ 1 t can be updated based on the relation Equation (28):

Similar to Equation (10), now the proportional and differential gains in Equation (8) are chosen by the following relation:

which means that k_p and k_I should be designed as:

From Equation (30), it is found that b ^ 1 t appears in the denominator. In order to ensure the smoothness of the control law, we impose a constrain on b ^ 1 t :

where b ⌢ is a pre-specified upper bound. It is noted that such treatment has no negative effect on the convergence or stability properties (Chen et al., 2001).

The proposed PI controller can be implemented as follows:

Step 1: Update a ^ 1 t and b ^ 1 t by Equations (16)–(21);

Step 2: Calculate h ^ t , g ^ 0 t and g ^ 1 t by Equation (28);

Step 3: Calculate k_p and k_I by Equation (30);

Step 4: Calculate u(t) by Equation (8);

Step 5: Let t =t + 1 and apply u(t) to the plant.

The experimental platform diagram of the SMA actuator is depicted in Figure 5, and the experimental set-up is presented in Figure 6. The structure of this SMA actuated system is similar to the one in Romano and Tannuri (2009) but without a cooling device. The SMA wire is the Flexinol actuator wire which is produced by Dynalloy, Inc. For this type of wire, the diameter is 0.25 mm, the length is 340 mm, the deformations are up to about 4%, and the Austenite start temperature is 90°C. In this experiment, the system output is the displacement (unit: m) and the input signal is the current (unit: A), which is constrained to the range 0 ~ 0.4. The control current applied to the SMA actuator is obtained from a V/I converter. The SMA wire then generates significant strains in response to the temperature changes caused by the current heating effect. The displacement of the SMA wire is measured by a high precision encoder. The Beckhoff EtherCAT terminals are used for the transformation and conversion of data, and the sample frequency is 200 Hz. The load is fixed as 500 g for the set-point tracking experiment, but varies for the other experiments.

To describe this nonlinear plant, two groups of models have been considered in previous studies (Nikdel et al., 2014; Pai et al., 2017; Pan et al., 2017), namely, mechanism models or neural networks models. However, there exist some inevitable drawbacks in each group. The objective of this work is to find an alternative way to simultaneously address the computational burden and the unmodeled dynamics issues.

The proposed adaptive PI controller is applied to this plant. Before the control implementation, some offline identifications have been carried out in the Matlab software. The purpose of the offline procedure is to probe the main dynamic properties of this nonlinear plant. Based on some groups of input–output data around different operating points, an RLS algorithm is used to identify the parameters of the model Equation (6). Then some groups of convergent parameter estimates are obtained. Based on these estimates and other input–output data, we have also conducted the model test experiment. Finally, the best prediction model is selected as y(t + 1) = 0.9923 y(t) + 0.001 u(t). Meanwhile, the obtained results are used as initial conditions for the controller design. For the proposed PI control method, the initialization is θ ^ 0 = − 0.9923 , 0.001 T , the multi-innovation length is p = 3, the gain is ε = 1, the bound is Δ = 0.00012, the characteristic polynomial is pre-specified as T z − 1 = 1 − 1.44 z − 1 + 0.445 z − 2 , and the constrain is b ⌢ = 0.001 .

As a comparison, the conventional fixed-gains PI controller is applied to this plant as well. The proportional and differential gains k_p and k_I are pre-specified as k_p = 500 and k_p = 5.

The load is fixed as 500 g in this test. Sinusoidal trajectory and square-wave trajectory are both considered. The set-point tracking results of these methods are shown in Figures 7 8.

The performance of the adaptive PI controller is better than the conventional PI one, especially for the milder control input. It is obvious that the adaptive PI controller can accurately track the reference trajectory with a slowly changing reference trajectory. In addition, the overshoot and oscillation of the adaptive PI controller are more satisfactory. Interestingly, the unmodeled dynamics has been gradually compensated by the adaptive PI controller, which can verify Theorem 1 in Appendix.

An additional load with 200 g is imposed on this actuator at 90th second, and removed at 105th second. Another heavier load with 300 g is added at 120th second, and removed at 135th second. The regulation results are shown in Figures 9 10.

The conventional PI controller leads to unattractive results under uncertainties induced by load variations. Worse still, the system becomes unstable after 135th second. Obviously, thanks to the online adaptation, the adaptive PI controller can ensure satisfactory robust stability despite of severe uncertainties.

We further test the disturbance rejection ability. An unknown instantaneous vertical force is suddenly imposed on the load at 160th second, and then an unknown instantaneous lateral force is suddenly added at 175th second. The disturbance rejection result of the proposed adaptive PI controller is shown in Figure 11.

The result shows that the proposed adaptive PI controller is reliable under non-Gaussian stochastic noise, which is ensured by the dead-zone weighted factor Equation (20). Though the control input varies a lot, the system output stays within a small region.

For robotic applications, plenty of issues (i.e., modeling error, load variations and stochastic noise) may cause uncertainties. The proposed adaptive PI controller can address these issues in a computationally efficient manner. During the whole operation, the proportional and differential gains k_p and k_I are updated according to the current working conditions, as shown in Figure 12. Most interestingly, it is seen that when the system suffers from severe uncertainties, especially around 135th and 160th seconds, the updated gains can address the negative effects timely.

After verifying the driving principle of SMA and the proposed adaptive PI control algorithm, we designed an actuator mechanism based on SMA and integrated it into a hand rehabilitation robot system to form an SMA-based hand rehabilitation robot system platform, which is suitable for hand rehabilitation training of hemiplegic patients. In this hand rehabilitation robot, each finger is controlled by an individual SMA actuator, and the entire robot comprises five identical SMA actuators. The SMA actuator is primarily comprised of six components, as shown in the left part of Figure 13A. This includes the installation of a pulley device on the main plate of the actuator, winding a shape memory alloy wire around the pulley, connecting the shape memory alloy wire to the output wire and the preloaded pulley through connecting members, and incorporating a wiring mechanism on the actuator’s main plate for ease of wiring. Additionally, a displacement feedback mechanism is established to enhance control over the shape memory alloy wire. A prototype SMA actuator was fabricated and assembled using 3D printing technology, as shown in the right part of Figure 13A. To prevent short-circuiting of the wiring mechanism with the shape memory alloy filament, a layer of Teflon tape with insulating and high-temperature-resistant properties was applied to the copper sheet of the wiring mechanism.

After the hand rehabilitation robot system based on SMA actuator is built, the movement of the hand rehabilitation robot is controlled in the form of sending commands from the upper computer to the lower computer, so as to assist the patient in rehabilitation training. The framework of the hardware system is shown in Figure 13B.

In this subsection, the control core utilizes the Raspberry Pi, and the SMA is subjected to heating signals dispatched to the controller, causing it to contract and deform, thereby propelling the movement of the rehabilitation hand. In this experiment, the SMA actuator of the index finger part of the hand rehabilitation robot is selected as the control object, and based on the adaptive PI control algorithm proposed in this paper to track the position response curve of the SMA actuator under the step signal as well as the sinusoidal signal. For comparative analysis, the PID control law (Khalil, 1996) is utilized as a reference algorithm. Meanwhile, in order to be able to visually compare and analyze the control effects of the two control algorithms, the errors of the SMA actuator-based hand rehabilitation robotic system will be compared when it reaches the steady state under the two control algorithms, respectively. The actual results of the robot system tracking the step and sinusoidal signals and the steady state error results are shown in Figure 14.

From the experimental results in Figure 14A, it can be seen that under the adaptive PI control algorithm, the desired value of the hand rehabilitation robot system is set to 4 mm at 2 s, and the system responds at 2.4 s, reaches the desired position at about 3 s, and maintains stability thereafter, with almost no deviation. Meanwhile, the response times of the two control algorithms are basically the same, but the hand rehabilitation robotic system does not produce overshooting and has a smaller steady state error when the step signal is tracked under the adaptive PI control algorithm. Consequently, for reference trajectories represented by step signals, the hand rehabilitation robot system demonstrates superior control performance under the adaptive PI control algorithm proposed in this paper. Examining the experimental outcomes in Figure 14B, it is observed that the hand rehabilitation robotic system adeptly tracks sinusoidal signals. While the response times of the SMA actuator system remain consistent under both algorithms, the adaptive PI algorithm proposed in this paper achieves more accurate position tracking with less error when tracking sinusoidal signals. Thus, for various signal amplitudes, the methodology presented in this paper enables the SMA actuator-based hand rehabilitation robotic system to approach the target position with reduced overshooting and a smaller steady-state error. These experiments substantiate the reliability and accuracy of the proposed methodology, affirming the safety of the SMA actuator-based hand rehabilitation robot in assisting subjects during the rehabilitation training process.

We oriented the SMA actuator-based hand rehabilitation robotic system platform to conduct the hand passive rehabilitation training experiments on subjects with different gestures, and the training process is shown in Figure 15. The hand rehabilitation exercises are divided into five movements, which are thumb extension/flexion, index extension/flexion, index and middle finger extension/flexion, three fingers extension/flexion and hand open/close. During the hand gesture rehabilitation training experiment with the SMA actuator-based hand rehabilitation robot system, a complete single flexion-extension training cycle takes a total of 12 s. Throughout this process, spanning from 0 to 4 s, the SMA contracts upon heating and powering, propelling the fingers to their maximum extended position. Subsequently, from 4 to 12 s, the SMA undergoes cooling facilitated by a fan on the outer shell of the hand rehabilitation robot, causing the hand to return to its initial state. Importantly, this mechanism satisfies the requirements of passive rehabilitation training for multiple gestures in patients with hand hemiplegia, demonstrating an optimal control effect. This experiment effectively establishes the reliability and precision of the SMA actuator-based hand rehabilitation robotic system for subject-specific rehabilitation training under the adaptive PI control strategy. It is worth noting that, due to space limitations, our experiments only focused on the low-level robust adaptive control of hand rehabilitation robots based on SMA actuators. We did not conduct experiments related to neural rehabilitation control involving mid-level and high-level controllers. This aspect will be addressed in our future research.