PID Controller Design for Human Elbow Therapy

A controller design for mechatronic system which capable of doing passive therapeutic exercises of patients who have upper extremity limitation is presented in this paper. Expectation from controller is it should produce torque values can exactly repeat degree values depended on time which were taken from first therapy exercises of patients. The designed controller tested with real angle values which was taken from during elbow therapy. Simulation results showed that the proposed control system has good performance at tracking the therapy trajectory. Also that control system may be used for mechatronic upper limb therapy system which can be produced.


Introduction
Autonomous rehabilitation machines advantages realized by people day by day.Controllers are most important part of that machines so controller design is the most important topic.Several studies about human motion tracking, rehabilitation machines and controller design for that machines are outlined in this section.Zhou et al. compared many human motion tracking methods with each other.Their conclusion is inertial sensors is the best method for human motion tracking in terms of ease of use and data accuracy [1][2][3][4][5][6].Continuous passive motion device for shoulder is designed by Rasyid et al..The device can be used for joint motion limitation [7].Mihelj et al. used ARMin exoskeleton robot at their work and they proposed new patient-cooperative control strategy for upper limb rehabilitation device.They aimed provide support to patient with minimum intervention.In this way patient can use trajectory what he or she wants while reaching the destination point [8].Birch et al. designed rehabilitation device can be used either continuous passive motion (CPM) or continuous active motion (CAM) for human hand rehabilitation.Device actively resisting the movement at CAM mode and at CPM mode they used PD control [9].Saputra et al. used microcontroller to control automatically working CPM device.That device use for knee joint rehabilitation.When patients feel pain, DC motor overloaded so device stops movement [10].Dong et al. developed intelligent controller and prototype rehabilitation device for human joints [11].Hassani et al. developed device that perform passive and active motions.That devices main objective is helping health staff [12].Zhang et al. designed intelligent neural network controller for active rehabilitation device.They used BP neural network for estimation of human knee joint angle change [13].Prashant et al. designed parallel rehabilitation robot for human ankle movement.They used kinematic analysis and genetic algorithm for optimization [14].Rehabilitation robot for human ankle, knee and hip joints designed by Wang et al..They used swarm algorithm for optimization problem [15].Lee et al. placed artificial mechanism at human knee joint and they gave mathematical model for that artificial mechanism [16].Chua et al. measured angle change of human hip and knee joints and they designed rehabilitation robot [17].Yildirim and Eski designed neural network analyzer for human hip and knee joints and they used vibration data of human hip and knee joints [18].In this paper, we designed controller with data's which are taken by using inertial sensors.This controller is tracking exercise trajectories for elbow joint.

Dynamic of Human Upper Limbs
There were several different modeling's including human upper limbs in the literature.In this paper, the assumptions were made that human arm consist of three rigid limbs and have three-degree of freedom.Each of the joints was modeled as one-degree of freedom joint.Relationship between the external forces and displacements generated by external forces could be expressed by linear transfer function which generally called as mechanic impedance or admittance.Basic linear expression of one DOF (single joint) musculoskeletal system in Laplace domain: where Ɵ is the joint angle, I is the inertia moment, B is the joint viscosity, K is the joint stiffness, T m is the torque of muscle and T e is the external torque.Here, the visco-elastic joint features depending on joints itself, visco-elastic features of passive component of muscles and visco-elastic features of activated muscles.The muscles visco-elastic features can be divided into intrinsic system and reflexive system.B and K at Eq. 1 is include intrinsic system features but not include reflexive system.The muscles reflexive torque can be modeled as: where β 0 is the position feedback gain, β 1 is the velocity feedback gain, τ is the loop delay and α is the time constant.While using Eq. 1 for calculation of the human dynamics, if the reflexive torque is too small or muscle activation dynamics can be neglected, the joint dynamic equation will be second-order system [19].
Considering muscles are in fully relaxed condition and muscle activity will be nearly zero during the passive therapy exercises, the muscle reflexive torque T m can be neglected because it will be nearly zero.According to this in Eq. 1 angle only depends on external torque and intrinsic system as Milner et al. [20] and Morita et al. [21] used at their work.So dynamic equations for human joints separately modeled as second-order system is given by: ̈+  ̇+  =  (3)

System Modelling
For designing controller, firstly we need the dynamic model of shoulder, elbow and wrist joint.We use Eq. 3 and Fig. 1 for modelling human joints.The dynamics of shoulder, elbow and wrist joint is given below: ( 2 +  3 ) ̈2 +  2  ̇2 +  2  2 =  2 (5) As you seen in the Fig. 1, Eq. 4 expressed for shoulder joint, Eq. 5 for elbow joint and Eq. 6 for wrist joint.State-space model for Eq.4-6 is: [ For shoulder, elbow and wrist, dynamic parameters, which for fully relaxed condition and with minimum muscle activity of human extremities, given in Table 1 [19,20,[22][23][24].

Experimental and Simulation Results
At this section, data which are taken from patients has limitation on elbow joint and control techniques developed based on these data results are given.According to taken data from patients elbow joint, PID (Tune) and PID (ZN) control systems are designed.Matlab's PID tuning algorithm and Ziegler-Nichols algorithm are used to adjustment of PID's gain parameters and gain parameter given in Table 2 Transient state responses of control structures for step input are shown for the elbow joint at Fig. 2, As seen in Table 3, the PID (ZN) control structure has given best results on rise time and settling time but overshoot.As seen in figures, PID (ZN) control system has minimum steady-state error and giving better results on adapting comparing with the PID (Tune) control systems.

Conclusions
In this paper, control structure designed according to taken data from patients who has limitation elbow joint and different physical specifications.According to experimental and simulation results, the PID (ZN) control system is better than PID (Tune) on adapting and it has minimum steady -state error.Although, PID (ZN) gives better results than PID (Tune), for future work other intelligent control structures will be simulated and compared with PID (Tune) and PID (ZN) control systems.

Figure 1 .
Figure 1.Angle and torques descriptions for joints of the human upper limb

Figure 2 .Table 3 .
Figure 2. Control structures response for the elbow joint using unit step input signalTable 3. Transient response parameters of the control structures for the elbow joint Elbow joint Control Structures Rise Time Settling Time Overshoot SS Error PID (Tune) 0.102 0.513 %5.33 0 PID (ZN) 0.0467 0.421 %27.7 0

Table 1 .
Transient response parameters of the control structures for shoulder, elbow and wrist joints

Table 2 .
Control structures parameters for the elbow joint