Abstract
The main motive for carrying out this research is to validate the proposed mathematical model while determining the optimal parameters for smooth and precise control of a 5-DOF (Degree of Freedom) upper limb rehabilitation robot. The proposed controlling parameters based on a validated mathematical model will help patients hit by a stroke perform passive type rehabilitation. The key parameter in passive-type rehabilitation is position control. To track the exercise trajectory, precise position control is required. Due to human coupling, considered external disturbances, creates complexities in controlling the position of the exercise. This may harm the patient’s muscles instead of recovering their range of motion. A mathematical model of a 5-DOF upper limb rehabilitation robot is developed using a Lagrangian approach to achieve the composite transfer function of the system. An extension to a widely used PID controller known as 2-DOF PID is used as an independent position controller for all five joints of the Robot. The critical issue is the selection of control parameters, as they play a vital role in controlling the Robot’s motion. A mathematical analysis technique is proposed to effectively control the rehabilitation robot to perform passive type rehabilitation. The suggested optimal control technique was applied on a 5-DOF RAX hardware. It was revealed that proposed system rejected unwanted external noise and disturbances and keeps the system stable under harsh conditions. This research paper has validated mathematical models to determine the PID parameters for optimization. Moreover, an approach named Ziegler Nichols was applied to validate the proposed mathematical modeling in terms of error. Ziegler Nichols is used to determine the optimal internal parameter for PID control.
Key Terms: Manipulator, DOF, upper extremity, Denavit-Hartenberg Convention, Forward Kinematics
Introduction
Rehabilitation Robotics has been defined as a branch of robotics which aims to eliminate most of the disadvantages of classical rehabilitation and focuses on technology which may assist people to recover from severe physical trauma or assist them in activities of daily living1. Utilizing robotics to design machines for rehabilitation increases the number of the training session with consistent repetitions which enables the opportunity to assign one therapist to train more patients at a time2. It also permits to assess the objective and quantitative evaluation of the patients during and after the therapy sessions which is not possible with conventional therapy. In addition, augmenting gaming and virtual reality with therapy creates an exaggerated environment which can intensify motivation during sessions3. The clinical studies involving rehabilitation robotics have supported the idea of implementation of such devices in the treatment of stroke and spinal cord injury patients. The efficacy of rehabilitation robotics has gained significant traction in recent years as they have been clinically used in restoring locomotor skills and upper extremity movements of stroke patients. Misfortune events such as traffic or disaster accidents or an unforeseen haemorrhage may lead to brain or musculoskeletal injuries which has influence over motor and cognitive functionalities. Globally, stroke is a third leading cause of mortality which results in 5 million deaths annually4. It has been predicted by the World Health Organization that by 2030, around 23.3 million individuals would suffer from chronic diseases. It is expected that by 2030, 25% of the people will be over the age of 65. The increase in over ageing population will cause a rise in the cost of health care and burden the rehab therapy centres and hospitals5. Elders are usually inclined to suffer muscle deterioration and porous bones which create difficulties in performing daily activities. Classical rehabilitation is labour-intensive and expensive therefore training sessions are shorter than required which is the factor that hinders achieving the ideal therapeutic outcomes. This is due to the consistency of the exercise and its benefits depend on the performance of the therapist. In the due course, it would be a good advancement if robots could assist in performing the treatment6. The rehabilitation robotics has then risen as an appealing alternative to fit the demand. Rehabilitation robotics can be classified into two groups which are end effector-based robots and exoskeleton-based robots as represented in Figure 1 shows a 2-DOF planar robot MIT-MANUS which has a planar workspace and is an end-effector based robot7’8’9’10. Examples of exoskeletons for upper limb extremity include 5-DOF RUPER11, 5-DOF MAHI Exo-skeleton12 6-DOF ARMin and 7 DOF CADEN-713 are shown in Figure 1. In recent years, rehabilitation engineering has gradually focused on the evaluation of motor abilities to obtain an objective evaluation of rehabilitation14.

Figure 1 demonstrated grounded and ungrounded exoskeleton for upper limb rehabilitation. Moreover, exoskeleton equipment can be further segmented into two more classes which are grounded and ungrounded robots. Usually Ungrounded robots are acknowledged as portable in nature and can be worn by a subject and also called wearable robots. Such kind of wearable robots have a big workstations and space also made the person capable to move more natural movements. It is an electrically actuated system with 14 DOF’s and provides close precise behaviour to the human movements functions. Despite their exceeded movement capacity, ungrounded devices offer limited torque capabilities. Devices like this are normally fabricated keeping in the mind the light weight of the device as subject needs to wear that device all the time during the sessions of rehabilitation process. Due to this light weight constraint while designing this device the actuator size is relatively small and the resulted torque is limited. On the other hand the grounded rehabilitation robots are considered as the more flexible in terms of the appropriate choice of actuators because of their structure. Moreover, they provide simple mechanism and functionality framework in comparison with wearable robots. It has been observed that the actuator types has a great direct significant impact on the torque, force, size and weight as these are control parameters are needed for robotic systems. Electrical actuation, hydraulic actuation and pneumatic actuation systems are the most frequently types are available to trigger or activate the robotics movements and other functionalities. Each type has its own merits and de-merits depending on the technology16. Whereas, pneumatic actuators offer high power to weight ratio that allows them to be used in lightweight application such as ungrounded robots. Usually, in rehabilitation devices, electric actuators are used as they allow the implementation of control easier17. A transmission system is also required in such systems so that the motion or movement can be transmitted from the actuation system to the specific part of the system where the movement needs to be performed. By introducing such a system, generated torque can be enhanced and the motion velocity is reduced. Consider the device that is being utilized with the impairments of neurological devices. The statement would be acknowledged very feasible that the low speed precise control is needed in controlling the movements of upper limb rehabilitation robots. Therefore in these type of devices there is no need of high speed design specifications based robotic application18’19. Gaing introduced a new approach to fine-tune PID controller settings for an Automatic Voltage Regulator (AVR) system using Particle Swarm Optimization (PSO). His research showed that this method significantly improved system performance and efficiency compared to the Genetic Algorithm (GA)20. Later, researchers built on this work by optimizing a similar system with a PID controller using the Artificial Bee Colony (ABC) algorithm. Their findings suggested that ABC outperformed PSO for AVR-type systems21. Similarly, Ayas applied Particle Swarm Optimization (PSO) to optimize PID controller parameters for a two-degree-of-freedom (2-DOF) rehabilitation robot22. In his study, he evaluated performance using the Integral Squared Error metric, helping to refine control strategies for robotic rehabilitation. Many AI based approaches have been applied for the optimal values of PID tuning and precise movement can control robotics. The existing approaches are not upto the mark due to some inadequate model and inadequate algorithms. In this research model has been proposed.
The research objective of this study is to propose an accurate, robust and stable control for upper limb rehabilitation robot. Mathematical model formulation and controller design are the criteria to achieve desired control objectives. This section describes the mathematical modelling and the controller design of the rehabilitation robot.
Joint | Therapeutic Exercise | Workspace |
Shoulder | Flexibility/ extendibility | 0 o /91 o |
Rotation (Internal/External) | -40 o /90 o | |
Abduction-adduction | 0 o /91 o | |
Elbow | Flexibility/ extendibility | 0o/150o |
Internal / External rotation | -20 o / 90 o | |
Wrist | Flexibility/ extendibility | -70 o /80 o |
Research Gap
Rehabilitation robots have very direct and immediate interaction with humans due to which they are also known as human-coupled exoskeletons. The best optimal parameters are required to cancel out the noise and unwanted signals or disturbances for the smooth and precise control of robotic arm. Existing approaches lack the precision and accuracy due to inaccurate algorithms and mathematical validation model. Mathematical expressions were developed to validate the mathematical model for the smooth and precise controlling parameters of controller.
Methodology
Proposed Mathematical Model for System
Commonly mathematical modelling is used to do the motion analysis of the Robot. Mathematical modelling of the robots consists of two parts, kinematics and dynamics. Kinematics modelling is the study of motion whereas kinetics (DYNAMICS) has to do with the forces that cause the motion. Forward kinematics and dynamics have been discussed in the sections below. The kinematics of a robotic end effector and manipulator are referred to the mathematical equations which explain the reverse and forward relationships among the variables of joints i.e. the angular position of the revolute joint and the Cartesian coordinates of position for the manipulator end-effector position. The kinematic equations are considered as very reliable tools in the mapping procedure of the robots required trajectory from joint space to Cartesian space and vice versa. The proposed model used the 5 DOF as a prototype research then further research will be extended to 6-DOF and 7-DOF in future work. The proposed model can be acknowledged as novel as it comprises of customized 5 DOF optimized design, integration for forward kinematics with dynamic modelling, torque and force estimation using langrangian mathematical mechanics and real world implementation testing on CAD and MATLAB.
Forward Kinematics
In this research, a 5-DOF open-chain robot is used. Where the degree of freedom (DOF) is calculated using Gruebler’s Formula:
(1)
Where = number of DOF in space; 3 for planner; 6 for 3-dimensional space, N= Number of links, j = Number of Joints,
of a permitted to each joint.
Five DOF robot consists of a 2-DOF shoulder joint and 2-DOF elbow joint and a joint of wrist is constrained to 1-DOF. The shoulder joint is capable of performing flexibility and extendibility, abduction/adduction and horizontal rotation for internal external whereas elbow joint is able to perform extension/ flexion and horizontal internal motion. The wrist joint is constrained for performing extension/flexion only. All the joints in this Robot are revolute joints. Robot in planner specification can be described in three parameters (x,y,). But in real-world application robots are three dimensional, so six parameters are needed (x,y,z, yaw, pitch, roll) to represent orientation and position of Robot in space.
The CAD model for the 5-DOF robot arm designed for upper limb rehabilitation can be seen in Figure 2.
Dynamic Model
The dynamics of the system plays a vital role in making a fundamental base for understanding of the robot motion control. For building a control mechanism for controlling the motion of the Robot, the relationship between force and torque that must be incorporated to the joints and the angular acceleration, velocities and position of the joints must be established. The motion control strategy for any robot is based on the generation of torque produced by actuator to control the movements of joints to the desired space configuration. The problem concerned with establishing the relationship can be solved using the manipulator’s dynamics equations.
There are two approaches to obtain the mathematical expression for forward dynamics of a robotic end effector and manipulator i.e. Newton Euler and Euler-Lagrange method. First, the Newton-Euler method and the equations of motion determined are known as Newton-Euler Equations of Motion. While on the other hand, an approach due to Lagrange, and the resulting equations are called Lagrange’s Equations of Motion. Newton-Euler approach is based on the fundamental idea of Newton’s second law of motion to determine the dynamics equation of any robot. While Euler-Lagrange mechanics, on the other hand, there is no requirement for explicit expressions for accelerations. It depends on the estimation of the total kinetic energy and potential energy for the Robot and then its utilization to determine Lagrangian parameter of the complete whole system which then can be utilized to estimate the torque for all 5-DOF’s. It is mandatory to determine what the forces are and torques. But in terms of kinematic quantities, only need to drive expressions for velocities. It is because kinetic energy essentially goes as half times mass time velocities squared while potential energy is a function of position. Assumptions are defined as the robot links are considered rigid bodies, meaning they do not deform under applied forces or torques. The actuators provide perfect torque/force output without any delays, saturation, or backlashes. Joint and link friction (static and dynamic) and viscous damping effects are considered negligible. The robot operates in an ideal environment without external forces, vibrations, or payload variations. The kinematic and dynamic equations perfectly describe the robot’s motion and do not have any modeling errors. These assumptions make it easier to analyze, design, and control the robotic system, but they also mean that real-world performance may differ from theoretical predictions. To bridge this gap, engineers often add friction compensation, adaptive control, and real-world testing to refine the model and improve accuracy.
A mathematical equation that is based on closed form can be used to determine and estimate the values for torque and force that were applied on the individual joints of a robotic manipulator is mentions as:
(2)
Where ‘L’ represents the Lagrangian of the manipulator of Robot
The kinetic and potential energies associated with one link are usually calculated at the center of mass of that particular link. The mathematical expressions to obtain the derivation for the total kinetic and potential energies of the 5-DOF robotic arm is mentioned as follows.
The following equation describes the kinetic energy computed at the center of mass of the ith link of the robotic arm.
(3)
Where, denotes the mass of ith link,
is the linear velocity vector at the centre of the link,
denotes angular velocity vector at the centre of link,
is the inertial matrix calculated at the centre of mass for link with respect to the inertial robot frame. The calculation for the inertial matrix of the link relies on the dimensions of geometry and the frame coordinate with respect to which the calculation was performed. Hence, to estimate initial tensor at the centre of the link, the geometric dimension of the link must be calculated, and frame coordinates should be linked at centre of the link as shown in Figure 4. Here, all the links are considered to have the cubic geometric shape. Other geometric shapes may be taken into account but the estimation for the inertial matrix vary accordingly.
Moment of inertia is calculated with respect to the centre of mass for each link
;
(4)
is the density of the material (how heavy it is per unit volume).
a, b, c are the dimensions of the link (length, width, and height).
Each term inside the matrix represents the moment of inertia about a specific axis (x, y, or z).
Substituting eq (5) and (11) in eq (12) yields the kinetic energy at the centre of the link. For the evaluation of the eq (12):
(5)
Finally, all the kinetic energies are summed up together to give total kinetic energy.
(6)
The following equations are used to determine the potential energy at the centre of mass of the link of the arm.
(7)
(8)
(9)
(10)
Where
By adding potential energies of the links, total potential energy can be found,
(11)
After the estimation for kinetic energies and potential energies of the 5 DOF robot, the Lagrangian of the arm can be evaluated as follows:
(12)
By using Euler-Lagrange’s equation described in (11), torques applied to all the revolute joints of a 5-DOF robot arm for specified angular position , angular velocities
, and angular accelerations
can be computed.
Evaluation of the torque equation is computed in Appendix B. The model was validated with the CAD model developed on SolidWorks and imported to MATLAB.
In this research, Lagrange’s method helps model the robot’s motion in relation to external forces and internal system states (like joint velocities, accelerations, and positions). This makes the model accurate for simulating real-world conditions and optimizing control strategies.
Newton-Euler provided a more computationally efficient approach compared to Lagrange’s method, especially for robots with multiple rigid links. Zeigler-Nichols is used for optimizing the PID controller, ensuring quick and stable response in robot motion.
Euler-Lagrange’s equations presented a dynamic model of the robot, helping to derive the required torques and forces based on the robot’s energy state. Lagrange’s Equations of Motion was solved for precise motion analysis considering the robot’s kinetic and potential energy. Newton-Euler equations are adopted for a computationally efficient way to calculate the robot’s joint torques and forces in real-time.
Results and Discussion
Robot manipulators require actuators to produce the needed amount of torques at all the joints. Electrical energy is converted to rotational mechanical energy using actuators. In this research study, the DC motor is used as an actuator. DC motors are being widely used in robotics as an actuator because of high torque, speed controllability.
Figure 4 portrayed the DC motor model that comprised of mainly two parts. An electrical portion consisting of an armature circuit which includes armature inductance and armature resistance
. While
is the input voltage and
represents the armature current.
The mechanical part of the motor comprised of inertia motor shown by J and the motor damper . Further, the inertia load was connected to inertia motor by using a train gear with a reduction. The gear ratio and motor damper affect the torque amount required for the motor to operate.
A system in mathematical terms can be defined by its input/output relationship. The input and output relationship is usually defined by the term known as transfer function. The overall transfer function robot arm merged with the motor model can be drafted as:
(13)
Figure 5 showed the system model diagram. After deriving the motor model of the Maxon Motor EC-90. Robot manipulator transfer function combines with motor transfer function to give the complete composite transfer function of the system based on Figure 5.
Table 3 demonstrated the Zeigler-Nichols(ZN) technique for the optimization of control parameters of PID. Starting with Zeigler-Nichols (ZN) which is a classical criterion to find the gains for PID-ZN requires only two parameters to determine the gains of PID. The parameters of ZN are and
and are given in Table 2.
Controller | Parameters | Values |
Zeigler Nichols | Ku | 4.5 |
Zeigler Nichols | Tu | 2.1 |
Zeigler Nichols | Kp | 2.7 |
Zeigler Nichols | Ki | 2.5714 |
Zeigler Nichols | Kd | 0.70875 |
Zeigler Nichols | b | 1 |
Zeigler Nichols | c | 1 |
The robustness of the controller has been analyzed by evaluating its performance with different cost functions. The cost functions for the Zeigler Nichols are defined as follows:
(14)
(15)
Controller | |||||||
ISA | IAE | ||||||
O. S (%) | R. T (s) | S. T (s) | O. S (%) | R. T (s) | S. T (s) | ||
Zeigler Nichols | 55.3 | 0.252 | 5.87 | 55.3 | 0.252 | 5.87 |
Table 3 portrayed that Zeigler-Nichols tuning method is great for quick responses, but it sacrifices stability and precision due to its high overshoot and longer settling time. While it may work well in scenarios where speed is more important than smoothness, it’s not ideal for applications that require steady, controlled motion. Additional fine-tuning or alternative methods may be needed to improve performance in such cases. Other Methods will be performed in future work for the comparative analysis.
Conclusions
Easiness, simple mechanism, robustness, reliability and vast utilization makes the PID controller as the most preferred choice and can be considered as the most frequently used controllers. It included many other better characteristics for the controlling parameter compared to other controllers. The PID controller does the continuous evaluation to determine the difference between the required set point and estimated process variable and applies the correction as a self-healing process in terms of integral, proportional and derivative terms. In this research paper mathematical models have been validated to evaluate the parameters of PID for the optimization. Moreover an approach named as Ziegler Nichols was applied to validate the proposed mathematical modelling in terms of error. Ziegler Nichols is used to determine the optimal internal parameter for PID control. In the next phase the parameters would be optimized with the better optimization algorithms as a future work.
Data Availability
No dataset involved in this research.
Conflicts of Interest
There is no conflict of interest.
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