This is to certify that the thesis entitled "Modelling and robust PD compensation of two-link flexible manipulator" submitted by Anuraag Parida and Subhakanta Ranasingh in partial fulfillment of the requirements for the award of a bachelor of technology degree in Electrical Engineering at the National Institute of Technology, Rourkela (Respected University) is an authentic work carried out by them under my supervision and guidance. We are extremely privileged to be involved in such an exciting and challenging research project as, “Modeling and robust PD compensation of two-link flexible manipulator”. The Two Link Flexible manipulator (TLFM) is a two-input-two-output, highly nonlinear and unstable system.
For this purpose, first, to establish a nonlinear mathematical model of the TLFM, its kinematic and dynamic movements are analyzed using assumed state method. This presents a challenge to design control techniques that provide precise control of desired parameters in the system at the desired time, ability to cope with sudden changes in the bounded system parameters, and robust performance. In the last few years, modeling of two-joint flexible manipulators has been done through different modeling approaches such as assumed mode method (AMM) approach and finite element (FE) approach.
One of the advantages is that the AMM method describes the flexibility and vibration modes more descriptively than the FE method. The nonlinear model is linearized around operating points to obtain a linear model of the system. The ultimate goal of such robot designs is to accurately control the tip position, despite the flexibility, in a reasonable amount of time.
Many controller algorithms such as adaptive control, Neural Network (NN), fuzzy logic have been used for tip position control of two-joint flexible manipulators.
CHAPTER - 2
MODEL DESCRIPTION
- System Description
- Mathematical Modeling 2.3 State-space Modeling
- Mathematical Modeling
- Kinematic modeling
- Dynamic modeling
- State-space Modeling
The primary link is firmly clamped to the first drive (aka elbow) and carries at its end the second harmonic drive (also known as shoulder) to which another flexible link is attached. Each flexible link is equipped with one strain gauge sensor located at the clamped end of the link. The rigid joint rotation matrix Ai, and the rotation matrix Ei of the flexible link at the end point are given by [1],.
The dynamic equation of motion for the two-link flexible manipulator is derived through the Lagrangian approach following reference [1]. By calculating the kinetic energy "T" and the potential energy "U" of the system, the Lagrangian equation is formed which is given by, L = T-U. The kinetic energy associated with the charge of mass mp and moment of inertia Jp placed at the end of the second link is.
AiT Ai = S𝛳 i, EiT Ei = (I𝑦 i.e. + S)𝑦 i.e. (9) The potential energy (without taking gravity into account, i.e. horizontal plane motion) is given by The dynamic model of the two-link flexible manipulator is obtained using the Lagrange-Euler equations. As a result of this procedure, the equations of motion for a planar 2-joint flexible arm can be written in the familiar closed form.
M is the positive-definite symmetric inertia matrix, h is the vector of coriolis and centrifugal forces, K is the stiffness matrix, and Q is the input weighting matrix (i.e., a state-space model is constructed using a set of system variables which define the state of a process at any time In general, system behavior changes with time, and the information about this evolution of system state usually resides in the rate-of-change variables of a system, or in combinations of these variables and their derivatives.
These state variables are known as the state variables of the system and the set of state variables that describe the behavior of a system is called the system state. The state space modeling is done to make the system analysis simpler and to find the measure of the state w.r.t. Dynamic model of 2 link flexible manipulator is of 2nd order so for designing state space model we have adopted state variables as,.
CHAPTER - 3
LINEARIZATION
- Linearization of Dynamic Model 3.2 Linear State-space Model
- Transfer Function Model
- Linearization of Dynamic Model
- Linear State-space Model
The model is linearized around the operating point which is found by solving the following equations. So, for any small changes in the model, they should behave linearly with respect to these operating points. Using the formulas described in the previous section, the state space linear matrices A B C D are found to be, .
0 0 (24) The tip positions of both links of the two-link flexible manipulator are taken as output. A transfer function model is fitted using the matrices A, B, C, D obtained earlier and considering the two torques required to move the link as the two inputs and the measured tip positions of the two links as two exits.
CHAPTER - 4
CONTROLLER DESIGN
Controller Design 4.2 Robustness Analysis
Controller Design
Positive values of Kp and Kd are chosen to ensure the stability of the closed loop system. The two-link flexible manipulator system requires two controlled input torques to be applied to both links to control tip position. So two PD controllers have been designed for the system based on root locus loop shaping technique.
This technique involves analyzing the pole-zero diagram and placing the open-loop poles at appropriate locations in the s-plane so that the open-loop transfer function meets certain sensitivity and robustness criteria. The closed-loop control of the two-loop flexible manipulator consists of a negative feedback with loop 1 gain and a feed-forward PD controller together with a rotation transfer function. The reduced transfer function of the manipulator model is given by G(s), as seen in the previous section, and the transfer function of the controller is given by C(s), (27).
Since G(s) of the manipulator model is known and fixed, the transfer function C(s) of the controller should be chosen to form the loop function response L(s) in the frequency domain for better sensitivity and robustness, i.e.
Robustness Analysis
For robustness analysis, let us start with the definition of the so-called “infinite norm” of any stable, proper, rational transfer function T(s). The infinite norm of a rational function T(s), which is both analytic and bounded in the half-plane Re(s) > 0, is defined by the relation: [2]. Minimum limits for both the gain margin and the phase margin of a system characterized by.
22 functions of S in light of Figure 4.3, where Figure 4.2 has been decorated with an arc of radius 1, centered at the origin, to define the angle θ. Therefore, the distance l = 1− S -1 shown in Figure 4.3 represents the lower limit of the gain reserve of the system in the sense that. Furthermore, the angle θ shown in Figure 4.3 represents the lower limit of the phase edge of the system in the sense that.
𝑇 ∞ (38) Therefore, our requirement is to design a controller that gives || Δ ||∞ < 2 for robust stability and performance. By using "sigma[S:T]" command in MATLAB, the mixed sensitivity norm || Δ ||∞ for the TLFM system is plotted. From the above mixed sensitivity and root locus plots, it is observed that for PD gains [2.7, 2.7].
CHAPTER - 5
SIMULATION RESULTS
- Open Loop Responses of Nonlinear Model 5.2 Open Loop Responses of Linear Model 5.2 Open Loop Responses of Linear Model
- Closed Loop Responses of PD Compensated Model
- Open Loop Responses of Nonlinear Model
- Open Loop Responses of Linear Model
- Closed Loop Responses of PD Compensated Model
The nonlinear state space model of the dynamic equation of the two-connection flexible manipulator, described in Chapter 2, is simulated using the 'ODE45' command in MATLAB. By simulating the nonlinear state space model for 2 seconds using different initial conditions in MATLAB, the following results were obtained. Now the output responses of the linear state space model, obtained in section 3.2, are superimposed on those of the nonlinear state space model with initial conditions as δ21(0)= 0.1 and δ and a simulation time of 2 sec with an integration step of 10 msec;
A Simulink model is made using MATLAB with three different models; non-linear state-space model, transfer function model & linear state-space model and two loop PD controllers for each of the models. The point positions of both the links of the two-link flexible manipulator for PD controller gains obtained from robust analysis (Kp = 2.7 and Kd = 2.7) are plotted in the following figures. The PD controller was initially designed for transfer function model through sensitivity and robust stability study.
Furthermore, to study the robustness, the peak positions of both links are plotted for a different value of the peak mass. From fig. 5.6 it can be seen that the reactions almost correspond to the one in fig. 5.5, which ensures the robust stability of the system.
CHAPTER - 6
CONCLUSION
Modeling is the essence of interpreting a physical system in a mathematical form, which further facilitates system analysis and controller design. A physical model of a two-link flexible manipulator (TLFM) was analyzed and kinematic and Lagrangian modeling was performed for the dynamic model development system. From this model, an equivalent state space model is framed and then linearized with respect to the equilibrium points.
The resulting model is the linear model, which responded well to the operating points and its neighborhood. But it is observed that the plant already has two poles at the origin, so the goal becomes to design a PD controller rather than a PID controller. So to achieve the gains of the controller, root locus loop shaping technique has been adopted.
Robustness and stability criterion for the plant with PD controller is verified and positive inference is made. Finally, this controller is applied to the nonlinear model and tip responses and robustness are analyzed.
