controlling a movement of robotic arm … · robotic arm with full dynamics is used to validate the...

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International Journal of Mechanical Engineering and Technology (IJMET) Volume 8, Issue 10, October 2017, pp. 597–607, Article ID: IJMET_08_10_065 Available online at http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=8&IType=10 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 © IAEME Publication Scopus Indexed

CONTROLLING A MOVEMENT OF ROBOTIC

ARM USING LINEAR FLAT CONTROL

METHOD

R.Hushein, M.Shri Harish

Assistant Professor, Department of Mechanical Engineering, Veltech Dr.RR & Dr.SR University, Chennai, Tamil Nadu, India

R.Hashein

Assistant Professor, Department of Mechanical Engineering, Loyola Institute of Technology, Chennai, Tamil Nadu, India

Syed Riyaz Ahammed

Assistant Professor, Department of Aeronautical Engineering, Veltech Dr.RR & Dr.SR University, Chennai, Tamil Nadu, India

ABSTRACT:

The highly coupled nonlinear dynamics of robotic arm (6-DOF) is decoupled by

with its variables as a function of a flat output and its derivative values. Hence the

detailed expression of the flat output for the robotic arm is presented. With the flat

output, each and every trajectories of the generalised coordinates are easily designed

with the derivative values and open loop control is made possible. Using

MATLAB/Simulink S-functions combined with the differential flatness property of the

robot, trajectory tracking is carried out in closed loop by using a linear flat controller.

The merit of this approach reduces the computational complexity of the robot

dynamics by allowing online computation of a high order system at a lower

computational cost. Using the same processor, the run time for tracking arbitrary

trajectories is reduced significantly to about 10 seconds as compared to 30 minutes in

the original study(1). The design is taken further by including a Jacobian

transformation for tracking of trajectories in Cartesian space. Simulations’ using the

robotic arm with full dynamics is used to validate the study.

Keywords: Robotic Arm, Matlab, Direct dynamics, Simulation platform, Matlab, SimMechanics

Cite this Article: R.Hushein, M.Shri Harish, R.Hashein and Syed Riyaz Ahammed, Controlling a movement of robotic arm using Linear Flat control Method, International Journal of Mechanical Engineering and Technology 8(10), 2017, pp. 597–607. http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=8&IType=10

R.Hushein, M.Shri Harish, R.Hashein and Syed Riyaz Ahammed


1. INTRODUCTION

The development of industrial robots has been advanced in the last decade. This is mainly due to increase in the complexity of tasks that they execute. The controller which is a major component of robots has received a lot of attention from robotic researchers. This is because it has a direct impact on their performance and could inhibit their deployability and applicability in certain areas (2; 3; 4). Many control techniques have been proposed for modern industrial robot manipulators including the classical PID, Computed torque, feedback linearization, inverse dynamics, neuro fuzzy,model predictive control etc. More recently, robot control methods have been model based which simply relies on the mathematical model of the robot. Model based control has been applied in trajectory tracking tasks(5; 6; 7). However, the computational requirements of these model based systems are quite high as it is required to solve very large equations in their controls. Despite fast computer processor speeds, effective control algorithms become very computationally expensive with high run times and in some cases impossible to achieve. Most researchers in dealing with trajectory robot tracking applications have had to work with lower order robotic dynamics such as in wheeled robots(8; 9; 10), parallel robots(11) and under actuated robots(12; 2; 9) in proposing their control algorithms. Under actuating robotic manipulators reduces the order from a high degree of freedom (DOF) to a lower one. As the number of degree of freedom increases, the computational complexity of computing the control also increases. In this study, a 6-dof robot is modeled using the Newton- Euler approach. The computations done with the MAPLE software resulted in very long dynamic equations, literally running into several pages. The mere size of the equations can be a challenge to the control engineer in terms of the computing time. The equations are also nonlinear which adds to the burden. In these scenarios, classical control techniques fail due to the fact that they require long times for solutions to be obtained. This study presents a technique of tracking a 6 DOF without the use of under actuation or such other assumptions that would have first reduced the complexity of the problem. One nonlinear control strategy that is gaining popularity among robotic researchers is the differential flatness based control(14; 15; 16; 17; 18; 19). It has been applied to control mobile robots(20; 10; 8), UAVs, UGVs(21), flexible robots(22), under actuated planar robot(23; 24; 12; 25) and so on. Differential flatness is known to be well suited for the problem of trajectory generation and tracking(20; 19; 26). With differential flatness, the trajectories (position, velocity, acceleration and jerk) of a nonlinear system can be easily interpolated by defining a smooth curve with initial and final conditions. The state and control variables can then be reconstructed without having to integrate the system equations (19).

2. KINEMATIC MODEL

The forward kinematics is used to find the position of the manipulator given the joint positions. This is easy once the DH parameter of the robot is known. For this study, we use the DH parameters as given in(28). Given the actual end effector pose of the robot, the inverse kinematics solves for the corresponding joint positions. The inverse kinematics problem has many solutions so it is not always unique: the same end effector pose can be reached in several configurations, corresponding to distinct joint position vectors. For robot manipulators with six DOF, with the last three joints intersecting at a wrist, such as the Irb140 robot, it is common practice to decouple the inverse kinematics problem into two simpler problems. The problem is defined in terms of the inverse position kinematics and inverse orientation kinematics.

Controlling a movement of robotic arm using Linear Flat control Method


The DH parameter of the robotic arm as described by (28) is shown in table 1:

Table 1 DH Parameters of Robotic Arm

A geometrical solution for the inverse kinematics as derived by (11) is described here: The Transformation matrix from frame 6 to frame 1 with respect to the base frame 0 defines the cartesian position and orientation of the end effector as:

Pxw, Pyw, Pzw describes the position of the wrist center. c1 = cos(q1), s12 = sin(q1 + q2), etc. The solution for the first angle q1 from figure 2 is: q1 = atan(Pyw, pxw) The expressions for the remaining joints q2 to q6 will be included in the appendix. With

the inverse kinematics, we can determine the joint positions required for joint controls given a reference Cartesian trajectory.



Figure 1 Robotic arm Locomotion control

3. HARDWARE AND SOFTWARE ROBOTIC ARM CONTROL

INTERFACE

In order to control the arm, both hardware and software, a simple Matlab interface was made using GUIDE toolbox, Brian et al (2001). The interface is presented in figure 4 and allows both hardware and software control using the blockdiagram presented in figure 5. The “Connect” button initiates communication between the real leg and Matlab. The connection status is displayed below. The “Off-Line” button allows only simulation of the leg. The controls in the left allow modification of the leg tip coordinates. The system proposed by the authors (Figure 4) is similar to the system xPCTarget component of Matlab. The software that make possible the communication between Arduino board and Matlab has been released with the last version of Matlab.

4. DYNAMIC MODEL

The purpose of robot dynamics is to determine the generalized forces required not only to overcome the self-inertial loads (due to the weight of the links of the robot) but also to produce the desired input motions. In order to obtain a more precise model we divided the mass of each link in two (Mi- servomotor mass, mi-link mass, Mi > mi) (Fig. 6). Lagrange equation is an analytical approach, in which an operator called the Euler – Lagrange operator is applied on the difference between the kinetic and potential energy. The first step of the Lagrange denoted by L algorithm is to calculate the kinetic and potential energies of the system EC respectively EP.



Figure 2 Kinematic Structure

The above mentioned figure developing dynamic model of the arm Mechanical systems of a robotic arms are having two important properties: The potential energy is a function of generalized coordinates only The kinetic energy is a quadratic function of the derivates of the generalized coordinates. The kinetic and potential energy for each component is composed from two parts:

Figure 3 Robotic arm dynamic model implementation

Translations

�� 2

Rotation:

Where: v – is the translation velocity of the body ω – is the angular velocity of the body I – inertia matrix of the body



In this case the potential energy is the gravitational potential energy and has the form.

� � ��

Where: h – height of the body. Summing the above we obtain:

Considering the generalized coordinates vector q = [q1, q2,q3]T the generalized vector

forces can be computed using the below equation:

Considering the above equation we can develop the generalized forces are: Y0 = h(x) = L0*f. h(x) Y1= L1f h(x) + Lgh(x)u = L1fh(x), Lgh(x) = 0 Y2 = L2fh(x) + LgL1f h(x)u = v The vector ‘u’can be expended for the individual joints as follows u1 = m1(x1) ¨x1 + C1(x1, ˙x1) + G1(x1) u2 = m2(x1) ¨x1 + C2(x1, ˙x1) + G2(x1) u3 = m3(x1) ¨x1 + C3(x1, ˙x1) + G3(x1) u4 = m4(x1) ¨x1 + C4(x1, ˙x1) + G4(x1) u5 = m5(x1) ¨x1 + C5(x1, ˙x1) + G5(x1) u6 = m6(x1) ¨x1 + C6(x1, ˙x1) + G6(x1) It can be seen that these control inputs are all expressed in terms of the flat output x1.

5. TRAJECTORY PLANNING

Trajectory planning for nonlinear systems requires an iterative solution by numerical methods to find control such that the initial and final time maneuvers for the system dynamics are satisfied. This process is often plagued by input saturation, nonlinearities and singularities.

However, with the flat output: for the variables x1(t) given by the curve t 7→ x1(t), there exist a trajectory

� → �� , ��, �� … … , ��, ��, �� … … , ��

That satisfies the robot manipulator system equations as follows;

� → ��

��

These trajectories are obtained without integrating any system of differential equations. The robot states x(t) and inputs u(t) are parameterized by the flat output x1(t) using (31). Hence the motion planning problem can be defined as finding a trajectory.To track the reference trajectory with added disturbances or uncertainty, we define a function (19) fd ∈ Tx1 given a reference trajectory t 7→ x∗(t) of (x, f ). We find a feedback law x 7→ u(x) and the error x − x∗ denoted by e such that

e (t) = f*d(e(t)+ x1∗(t), u(e(t)+ x1∗(t)))− f (x1∗(t), u∗(t))



is asymptotically stable for all disturbances. There are many methods of generating these trajectories. They include, Bezier

interpolation, fourier series and polynomials. Newton interpolation is used in this study to generate the coefficients of the flat output polynomials.

Figure 4 Flatness based Controller Design in joint space

The robot control problem is seen as a determination of torques or forces generated by the actuators for the purpose of accomplishing a desired task. For the robot in study, we desire to accurately execute a planned trajectory using the end effector (e.g. in laser welding applications). These kinds of tasks require robust controllers to achieve minimum errors. The design of the controller is model based. Hence with the computed dynamics described in section 2 and the flat output of the robot, the controller parameters are determined. The inertias, gravity, coriolis and centrifugal forces are all considered in the design. The control is done in two stages: first is the open loop control which involves the generation of nominal flat feed forward trajectories. These explicit trajectories allows for a more precise tracking of the robot. In the second stage, we take advantage of flatness property of the robot in designing a feedback linearizing control that will stabilize the robot in the presence of disturbances. The approach to flatness based control is known as the two degrees of freedom control(16). One aspect deals with feedforward nominal control using trajectories and the second facet deals with feedback in the presence of disturbances. We consider both scenarios in this study for both joint space and task space control.

5.1 TRAJECTORY GENERATION IN JOINT SPACE

The problem of generating trajectories involves parametising the flat outputs and their derivatives over instants in time t1 to t2. The flatness property enables easy mapping between the trajectories in the nonlinear system and that of the flatness space.

Figure 5 PID control diagram in simulink



Terminal conditions for the flat outputs and their derivatives are made equivalent to the initial and final states of the complex manipulator dynamics over a specified period of time. Using the boundary conditions at t1 = 0 and t2 = 5s for the 6 flat outputs and their derivatives, the terminal conditions for the 12 states are also obtained. A 5th degree polynomial is used to determine the coefficients of the trajectories. The boundary conditions for these trajectories are set bearing in mind the joint limits of the robot model.

Steering the robot about the individual joints, a reference was defined for the joint coordinate and the following were obtained for reference trajectories: For a reference coordinate from origin (0, 0, 0, 0, 0, 0) to (−pi/2, 0, pi/2, −pi, pi/3, pi), we obtained

x11 = −0.1257Q3 + 0.0377Q4 − 0.0030Q5; x12 = 0; x13 = 0.1257Q3 − 0.0377Q4 + 0.0030Q5; x14 = −0.2513Q3 + 0.0754Q4 − 0.0060Q5; x15 = 0.0838Q3 − 0.0251Q4 + 0.0020Q5; x16 = 0.2513Q3 − 0.0754_4 + 0.0060Q5; To obtain derivative functions of x1, we differentiate above mentioned equation up to

twice.

5.2 TASK SPACE CONTROL

Using the inverse kinematics equations of section II and the robot jacobian, trajectories were generated for the end effector and converted to joint angles which was applied to the flat controller loop. The output from the control is then converted back to task space trajectories using the forward kinematics of the robot. The block diagram of the control in Simulink is shown in figure 6.

Figure 6 Task Space control in Cartesian space

6. SIMULATION, RESULTS AND DISCUSSION

Extensive computer simulations were carried out in Simulink/Matlab environment. Firstly, the open loop dynamics of the robotic arm was simulated under several scenarios. The robot was simulated under gravity and no gravity conditions. The open loop simulations were done using desired torque values that were computed based on the computed dynamics of the robot. Then a feedback flat controller was designed to simulate and stabilize the system in closed



loop. The reference trajectories and the complete dynamics of the robotic arm are generated using Matlab based S-functions. This enables online computations and updating of its functions with every time step in the simulations.

Figure 7 (a) Joint Arm control result in both system and model

Figure 7 (b) Joint Arm trajecories in both system and model



Figure 8 Joint trajectories using flat control (using various RMSE)

7. CONCLUSIONS

This article has described the flatness based tracking control of a 6-DOF industrial robot manipulator with full dynamics in joint and task space. The flatness based approach to trajectory control offers a fast alternative to PID control for such high dimensional robots. Compared to PID, the gains of the flat controller can be increased to reduce tracking errors without any fear of saturation. Having determined the flat output of the robot, trajectory control was achieved with reasonable accuracy within seconds as compared to when using the PID control. Therefore the proposed control is fast and gives accurate trajectory tracking. This makes it well suited for online practical implementation. In future work, more complex trajectories will be developed and tested with the flatness based controller.

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controlling a movement of robotic arm … · robotic arm with full dynamics is used to validate the...

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