# Position Tracking of a Pneumatic-Muscle-Driven Rehabilitation Robot by a Single Neuron Tuned PID Controller

Authors: Jun Zhong; Yue Zhu; Chun Zhao; Zhenfeng Han; Xin Zhang
Journal: Complexity
 (2020)

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## Abstract

Pneumatic muscle actuators (PMAs) own compliant characteristics and are suitable for use in rehabilitation equipment. This paper introduces a rehabilitation robot driven by PMAs devised in the Rehabilitation and Medical Robot Laboratory. Considering high nonlinearities inside PMAs, a single neuron tuned PID controller is carefully designed. Experimental setup is built up and trials are performed. Results demonstrate the proposed advanced PID algorithm can achieve better capacity in position tracking than the conventional PID controller.


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## Body

## 1. Introduction

Features of physiological function recession in aging process include decreased limb flexibility, osteoporosis, muscle atrophy, and significant decline in loading capacity of bone tissue. The incidence of acute cardiovascular and cerebrovascular diseases and that of neurological diseases in the aged keep at a high level, and most of these patients have symptoms of hemiplegia [1, 2]. Quantity of patients with limb dyskinesia caused by other diseases, sports injury, and traffic accidents is increasing rapidly. For these patients, besides early surgical treatments and necessary medications, correct and scientific rehabilitation training plays an important role in restoration and improvement of limb motion functions [3]. Many patients suffer from muscle atrophy and lose limb mobility because of improper training methods and failure to get effective rehabilitation training. This unfortunately makes patients suffer tremendously and causes great burden to family and society [4, 5]. A lot of clinical practices have proved that, without scientific and sufficient rehabilitation training, many patients with impairment of limb motion functions cannot restore the walking capacity to normality and have to move in a typical asymmetric attitude. Thus, improvements in patients’ walking capacity and the ability to take care of themselves in lives make a lot of sense to both patients and society. However, dependence only on physiotherapists cannot meet requirements of patients’ rehabilitation because the amount of physiotherapists is small and one physiotherapist lacks the capacity of training several patients simultaneously. Another barrier is that different physiotherapists have different understandings about rehabilitation exercises and different evaluating indicators [6–10]. The above obstacles are harmful in achieving ideal training strength and effects. Besides, loss of necessary means of detection and control makes obstacles in meeting the needs of a large number of patients in modern society. Rehabilitation robots can help patients complete kinds of motion function rehabilitation training, such as arm restoring therapy and ankle rehabilitation [11–13]. Wu et al. designed a three-degrees-of-freedom lower rehabilitation robot involving hip, knee, and ankle joints and proposed an adaptive robust subcontroller for the robot to handle system uncertainties and disturbances from patients [14] Banala et al. developed a robot-assisted gait training algorithm and used a force-field controller to achieve more effective training [15]. Besides, other rehabilitation robots have been developed [16–29]. Fateh and Khoshdel presented a new voltage-based adaptive impedance force control for a lower limb rehabilitation robot, and gradient descent algorithm was adopted to tune impedance parameters to guarantee force controlling effect [30].Most of the presented rehabilitation robots adopted motors as actuators, which leads the lack of compliance during recovery. In this research, a compliant ankle-rehabilitation robot driven by PMAs is presented and an advanced PID algorithm is devised to handle nonlinearities and disturbances inside the robot. This paper is arranged as follows. Section2 introduces characteristics of the devised pneumatic muscle actuator-driven ankle rehabilitation robot. Section 3 proposes a single neuron adaptive PID controller for the robot; Section 4 establishes the experimental setup of the robot and performs the rehabilitation tests to validate the designed controller.


## 2. Structure of the Ankle-Rehabilitation Robot

### 2.1. Structure Description

The detailed structure of the ankle rehabilitation robot is addressed prudently and shown in Figure1. Decomposition diagram of the ankle joint mechanism is displayed in Figure 2. One of the crucial issues to be handled is that placing angle and height of the training device must own the capacity of adjustment according to different patients and circumstances. Thus, the robot framework is made of aluminum profiles whose installing grooves are in unique T-shape (shown in Figure 3). By this way, arrangement angle and height of the robot can be adjusted arbitrarily in the framework. Besides, robot mounting panel has slots and holes. Slots allow the robot to move along and be fixed by bolts. Holes help the robot rotate around to modulate the arrangement angles and guarantee comfort during rehabilitation exercises. The panel framework is made up of aluminum profiles, whilst the robot is fabricated by light and high tensile aluminum. Another important issue is to guarantee safety during training exercises. This is realized by the limitation of the allowable angle range of the executive mechanism. Actually, angle limiting device (shown in Figure 4) is devised in the robot to avoid hurt on the patients’ ankles. The range of the executive mechanism in the robot is (−30°, 30°).Figure 1
CAD model of the ankle rehabilitation robot.Figure 2
Decomposition diagram of the ankle joint mechanism.Figure 3
Regulating device for installing angle and height of the robot.Figure 4
Angle limited part.In traditional ankle rehabilitation robots, driving components are usually installed on one side of the ankle joint. This type of unilateral asymmetric structure (shown in Figure5(a)) may exert uncertain force to joint, produces harmful torque on the robot joint in extra directions, and has detrimental influence on mechanical structure stability. Bilateral structure is adopted in the ankle joint design in order to preventing from producing harmful torques in extra directions. This bilateral structure keeps balance in different components of ankle rehabilitation robot and eliminates harmful torques created by drive power and forces from the feet of patients. Power supplying mechanism and transmission mechanism are both installed bilaterally about the ankle joint, which makes identical forces on the two sides of the structure. The symmetrical configuration can further guarantee the stability and reliability of mechanical transmission, make good use of space, and improve compact layout of the robot (shown in Figure 5(b)).Figure 5
Comparison of two installing types of driving mechanism: (a) unilateral asymmetric structure; (b) symmetrical configuration.
(a)(b)


### 2.2. Prototype

The prototype of the robot (shown in Figure6) adopts aluminum alloy to make sure of light weight and high strength. Aluminum alloy profile frame is used to regulate posture of the robot for various patients. A torque sensor is installed in one side of the output shaft of the transmission mechanism to measure driving torque or the human ankle, and an incremental encoder is fixed with the other side of the identical output shaft to record the real time rotation angle.Figure 6
Prototype of the rehabilitation ankle robot driven by PMAs.


## 2.1. Structure Description

The detailed structure of the ankle rehabilitation robot is addressed prudently and shown in Figure1. Decomposition diagram of the ankle joint mechanism is displayed in Figure 2. One of the crucial issues to be handled is that placing angle and height of the training device must own the capacity of adjustment according to different patients and circumstances. Thus, the robot framework is made of aluminum profiles whose installing grooves are in unique T-shape (shown in Figure 3). By this way, arrangement angle and height of the robot can be adjusted arbitrarily in the framework. Besides, robot mounting panel has slots and holes. Slots allow the robot to move along and be fixed by bolts. Holes help the robot rotate around to modulate the arrangement angles and guarantee comfort during rehabilitation exercises. The panel framework is made up of aluminum profiles, whilst the robot is fabricated by light and high tensile aluminum. Another important issue is to guarantee safety during training exercises. This is realized by the limitation of the allowable angle range of the executive mechanism. Actually, angle limiting device (shown in Figure 4) is devised in the robot to avoid hurt on the patients’ ankles. The range of the executive mechanism in the robot is (−30°, 30°).Figure 1
CAD model of the ankle rehabilitation robot.Figure 2
Decomposition diagram of the ankle joint mechanism.Figure 3
Regulating device for installing angle and height of the robot.Figure 4
Angle limited part.In traditional ankle rehabilitation robots, driving components are usually installed on one side of the ankle joint. This type of unilateral asymmetric structure (shown in Figure5(a)) may exert uncertain force to joint, produces harmful torque on the robot joint in extra directions, and has detrimental influence on mechanical structure stability. Bilateral structure is adopted in the ankle joint design in order to preventing from producing harmful torques in extra directions. This bilateral structure keeps balance in different components of ankle rehabilitation robot and eliminates harmful torques created by drive power and forces from the feet of patients. Power supplying mechanism and transmission mechanism are both installed bilaterally about the ankle joint, which makes identical forces on the two sides of the structure. The symmetrical configuration can further guarantee the stability and reliability of mechanical transmission, make good use of space, and improve compact layout of the robot (shown in Figure 5(b)).Figure 5
Comparison of two installing types of driving mechanism: (a) unilateral asymmetric structure; (b) symmetrical configuration.
(a)(b)


## 2.2. Prototype

The prototype of the robot (shown in Figure6) adopts aluminum alloy to make sure of light weight and high strength. Aluminum alloy profile frame is used to regulate posture of the robot for various patients. A torque sensor is installed in one side of the output shaft of the transmission mechanism to measure driving torque or the human ankle, and an incremental encoder is fixed with the other side of the identical output shaft to record the real time rotation angle.Figure 6
Prototype of the rehabilitation ankle robot driven by PMAs.


## 3. Single Neuron Tuned Adaptive PID Controller

### 3.1. Single Neuron Adaptive PID Algorithm

Proportional Integral Differential (i.e., PID) algorithm is widely used in various rehabilitation and medical robots because of concision, efficiency, and reliability. However, high nonlinearity and strong hysteresis of PMAs bring a tricky job in keeping high tracking capacity because of which conventional PID algorithm owns constant values of parametersP, I, and D. Considering single neuron strategy has the excellent capacity in regulating structural parameters, a single neuron tuned PID controller is devised, as shown in Figure 7. ω1, ω2, and ω3 are structural parameters of single neuron algorithm and represent P, I, and D of the PID controller, respectively. Values of ω1, ω2, and ω3 are online regulated according to some criterions, and this paper adopts a hybrid of supervisory delta learning rule and nonsupervisory Hebbian rule. A cost function is defined as the evaluation indicator of the tuning algorithm in the following equation:(1)Θk=12θrefk−θk2.Figure 7
Tracking controller of the robot in passive rehabilitation therapy.The principle is to reduceΘk by online tuning parameters ω1, ω2, and ω3. Gradient descent algorithm is used as follows:(2)ωik+1=ωik−ηi∂Θk∂ωik,i=1,2,3.According to chain rule of differential equations,∂Θ/∂ωik is calculated as follows:(3)∂Θ∂ωi=∂Θ∂θ∂θ∂upid∂upid∂ωi=−ek⋅∂Θ∂upid⋅xi.Usually,∂Θ/∂upid is difficult to calculate because of complexity of the actual system. To simplify the calculation, it is replaced by sgn∂θ/∂upid. xi (i = 1, 2, 3) are inputs of neuron units ω1, ω2, and ω3, respectively. In the single neuron PID controller, xi are assigned according to tracking errors of the system, i.e.,(4)x1=ek,x2=ek−ek−1,x3=ek−2ek−1+ek−2.The outputupid is as follows:(5)upid=K∑i=13xiωi∑i=13ωi.


### 3.2. Controller Design of the Rehabilitation Robot

The controller is devised using single neuron strategy tuned PID algorithm in Figure7. Two electrical proportional valves with type of ITV1050-212N made by SMC Corporation are used in the setup to control the pressure inside the pair of PMAs. This type valve has an output range of pressure in (0.005, 0.9) MPa and linear correspondent controlling voltage in (0, 5) V. u01 in Figure 8 is the initial input voltage of the first valve, i.e., the initial pressure inside the first PMA. Similarly, u02 is the initial input voltage of the second valve, i.e., the initial pressure inside the second PMA. Values of u01 and u02 are tuned by the trial and error. Pvalveu is the transfer function of the proportional valve and has the following form:(6)Pvalveu=0.179u+0.005.Figure 8
Tracking errors of passive rehabilitation therapy experiment under a half sinusoidal wave excitation of 0.002 Hz with amplitude 50°.Torque switch in the controlling block is used to guarantee the safety of patients by setting a range of torque. If the sampled torque value is larger than the upper limit Tmax, or less than lower limit Tmin, proportional valves will output zero to PMAs and the robot stops working. Tmax and Tmin are acquired by trial and error.


### 3.3. Stability Analysis of the Proposed Controller

Controlling parameters of single neuron PID algorithm vary along with the direction of negative gradient descent ofω1, ω2, and ω3. Stability of the single neuron PID controller is analyzed by Lyapunov principle. Firstly, a Lyapunov function is defined as follows:(7)Ek=Θk=12e2k.Variation ofEk in the self-learning process of the single neuron model is expressed:(8)ΔEk=12e2k+1−12e2k.Similarly, variation ofe can be acquired as follows:(9)ek+1=ek+∑i=13∂ek∂ωikΔωik.Considering(10)Δωik=−ηi∂Θk∂ek∂ek∂ωik=−ηiek∂ek∂ωik,then(11)Δek=e∑i=13∂ek∂ωikηiΔωik=−eΟ⌢TηΟ⌢,where η=diagη1η2η3, Ο⌢=∂ek∂ω1k∂ek∂ω2k∂ek∂ω3kT.Thus,(12)ΔEk=−12ekO⌢T2η−ηO⌢O⌢TηekO⌢.Several conclusions can be acquired by Lyapunov principle:(a)
Ek is positive definite(b)
If and only if2η−ηO⌢O⌢Tη>0, ΔEk is negative definite, which means stability of the system depends on the learning step(c)
Whenk approaches infinity, Ek approaches zeroObviously, proper values ofη make ΔEk < 0, which means the controlling system keeps stable.


## 3.1. Single Neuron Adaptive PID Algorithm

Proportional Integral Differential (i.e., PID) algorithm is widely used in various rehabilitation and medical robots because of concision, efficiency, and reliability. However, high nonlinearity and strong hysteresis of PMAs bring a tricky job in keeping high tracking capacity because of which conventional PID algorithm owns constant values of parametersP, I, and D. Considering single neuron strategy has the excellent capacity in regulating structural parameters, a single neuron tuned PID controller is devised, as shown in Figure 7. ω1, ω2, and ω3 are structural parameters of single neuron algorithm and represent P, I, and D of the PID controller, respectively. Values of ω1, ω2, and ω3 are online regulated according to some criterions, and this paper adopts a hybrid of supervisory delta learning rule and nonsupervisory Hebbian rule. A cost function is defined as the evaluation indicator of the tuning algorithm in the following equation:(1)Θk=12θrefk−θk2.Figure 7
Tracking controller of the robot in passive rehabilitation therapy.The principle is to reduceΘk by online tuning parameters ω1, ω2, and ω3. Gradient descent algorithm is used as follows:(2)ωik+1=ωik−ηi∂Θk∂ωik,i=1,2,3.According to chain rule of differential equations,∂Θ/∂ωik is calculated as follows:(3)∂Θ∂ωi=∂Θ∂θ∂θ∂upid∂upid∂ωi=−ek⋅∂Θ∂upid⋅xi.Usually,∂Θ/∂upid is difficult to calculate because of complexity of the actual system. To simplify the calculation, it is replaced by sgn∂θ/∂upid. xi (i = 1, 2, 3) are inputs of neuron units ω1, ω2, and ω3, respectively. In the single neuron PID controller, xi are assigned according to tracking errors of the system, i.e.,(4)x1=ek,x2=ek−ek−1,x3=ek−2ek−1+ek−2.The outputupid is as follows:(5)upid=K∑i=13xiωi∑i=13ωi.


## 3.2. Controller Design of the Rehabilitation Robot

The controller is devised using single neuron strategy tuned PID algorithm in Figure7. Two electrical proportional valves with type of ITV1050-212N made by SMC Corporation are used in the setup to control the pressure inside the pair of PMAs. This type valve has an output range of pressure in (0.005, 0.9) MPa and linear correspondent controlling voltage in (0, 5) V. u01 in Figure 8 is the initial input voltage of the first valve, i.e., the initial pressure inside the first PMA. Similarly, u02 is the initial input voltage of the second valve, i.e., the initial pressure inside the second PMA. Values of u01 and u02 are tuned by the trial and error. Pvalveu is the transfer function of the proportional valve and has the following form:(6)Pvalveu=0.179u+0.005.Figure 8
Tracking errors of passive rehabilitation therapy experiment under a half sinusoidal wave excitation of 0.002 Hz with amplitude 50°.Torque switch in the controlling block is used to guarantee the safety of patients by setting a range of torque. If the sampled torque value is larger than the upper limit Tmax, or less than lower limit Tmin, proportional valves will output zero to PMAs and the robot stops working. Tmax and Tmin are acquired by trial and error.


## 3.3. Stability Analysis of the Proposed Controller

Controlling parameters of single neuron PID algorithm vary along with the direction of negative gradient descent ofω1, ω2, and ω3. Stability of the single neuron PID controller is analyzed by Lyapunov principle. Firstly, a Lyapunov function is defined as follows:(7)Ek=Θk=12e2k.Variation ofEk in the self-learning process of the single neuron model is expressed:(8)ΔEk=12e2k+1−12e2k.Similarly, variation ofe can be acquired as follows:(9)ek+1=ek+∑i=13∂ek∂ωikΔωik.Considering(10)Δωik=−ηi∂Θk∂ek∂ek∂ωik=−ηiek∂ek∂ωik,then(11)Δek=e∑i=13∂ek∂ωikηiΔωik=−eΟ⌢TηΟ⌢,where η=diagη1η2η3, Ο⌢=∂ek∂ω1k∂ek∂ω2k∂ek∂ω3kT.Thus,(12)ΔEk=−12ekO⌢T2η−ηO⌢O⌢TηekO⌢.Several conclusions can be acquired by Lyapunov principle:(a)
Ek is positive definite(b)
If and only if2η−ηO⌢O⌢Tη>0, ΔEk is negative definite, which means stability of the system depends on the learning step(c)
Whenk approaches infinity, Ek approaches zeroObviously, proper values ofη make ΔEk < 0, which means the controlling system keeps stable.


## 4. Experimental Validations and Discussion

The entire robot system, including controlling component and air compressor, is shown in Figure9. An AD/DA card of USB3102A type from Beijing Art Technology Development Co., Ltd., is adopted to control the robot and samples all status information from sensors. An air compressor is employed as the compressed air supplier. Main components in the robot are listed in Table 1.Figure 9
The PMAs driven rehabilitation ankle robot: (a) schematic drawing; (b) prototype of the experimental setup.
(a)(b)Table 1
Components of the PMA-driven rehabilitation ankle robot.
NameTypeManufacturerMain parametersPMADMSP-40-300N-RMCMFESTO corporationNominal inner diameter: 40 mmNominal length: 300 mmPressure range: 0–0.6 MPaMaximum force: 6000 NMaximum contractile ratio: 25%ValveITV1050-212NSMC corporationControlling voltage range: 0–5 VOutput pressure range: 0.005–0.9 MPaIncremental encoderZMK60Working voltage: DC 5 VMaximum frequency response: 300 kHzTorque sensorZNNT-FCHINO SENSORLoad range: 2–200 NmOutput signal: 0–5 V, 0–10 V, 4–20 mA, 0–10 mAComprehensive accuracy: 0.2%AD/DA cardUSB3102AART technology corporationAD module:Resolution: 16 bitMaximum sampling speed: 250 KspsRange: ±10 V, ±5 V, ±2 V, ±1 VDA module:Resolution: 16 bitMaximum sampling speed: 100 KspsComputerINS14-3476DELL corporationProcessor: Intel i5-7200UStorage disk: 500 GRAM: 4 GBAir compressorDET750-30LDAERTUO corporationWork power: 750 WExhaust air capacity: 120 L/minMaximum pressure: 0.8 MPaTo validate the effectiveness of the proposed single neuron tuned PID algorithm, several trials are conducted under different excitation. Experiments are performed on a male with weight of 77 kg and height of 176 mm in this study (shown in Figure10). It is generally known that passive rehabilitation therapy is used for serious dyskinesia of limbs and must operate at a very low speed. A half sinusoidal wave excitation of 0.004 Hz with amplitude 50° is applied to the robot, and tracking responses and errors are plotted in Figures 11 and 12, respectively. Relative large errors occur at the zero position due to assembly errors of the robot and creep and high nonlinearities inside PMA. Figure 11(a) shows that the maximum tracking error of the PID controller is 5.905° at the time of 548.3 s, whilst that of single neuron tuned PID is 4.541°. Figure 11(b) shows an enlarged part for the portion A in Figure 11(a), from where the conclusion that single neuron tuned PID controller achieves smaller tracking errors than conventional PID controller can be drawn. Figure 12 shows tracking errors of passive rehabilitation therapy in Figure 11, which further proves the better capacity of single neuron regulated PID than classic PID. Figure 13 displays a half sinusoidal wave excitation of 0.002 Hz with amplitude 50° and different responses of the adaptive PID controller and classic PID controller. Figure 13(a) shows a maximum tracking error of 3.364° from PID controller responses and a maximum tracking error of 2.603°. Figure 13(b) shows an enlarged part for the portion A in Figure 13(a), which further proves that the single neuron tuned PID controller achieves smaller tracking errors than the conventional PID controller. Figure 8 shows the corresponding tracking error performances in Figure 13, and comparison between response curves of the two controllers demonstrates better capacity of single neuron tuned PID algorithm in overcoming nonlinearities and disturbances of the robot.Figure 10
Experiment validations on a male with weight of 77 kg and height of 176 mm.Figure 11
Passive rehabilitation therapy experiment under a half sinusoidal wave excitation of 0.004 Hz with amplitude 50°: (a) tracking performance; (b) enlarged view for portion A in (a).
(a)(b)Figure 12
Tracking errors of passive rehabilitation therapy experiment under a half sinusoidal wave excitation of 0.004 Hz with amplitude 50°.Figure 13
Passive rehabilitation therapy experiment under a half sinusoidal wave excitation of 0.002 Hz with amplitude 50°: (a) tracking performance; (b) enlarged view for portion A in (a).
(a)(b)


## 5. Conclusions

Compliant ankle rehabilitation robot in this research is realized by employing pneumatic muscle actuators as the power source. A pair of PMAs is arranged in antagonistic form and provides driving torque through crank-slider mechanism. Prototype is manufactured and assembled. A single neuron tuned PID controller with torque safety switch is designed for the robot. Passive rehabilitation experiments are conducted and effectiveness of the adaptive controller is validated. Conclusion of experiments demonstrates that the devised controlling algorithm can improve accuracy of position tracking of the robot.



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