# Dinámica y control de un doble péndulo

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• Publicado : 8 de julio de 2010

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1 Applied Robotics

J O I N T L E V EL C O N T R O L .

Applied Robotics.

SUMMARY: This paper presents the third midterm evaluation of Applied Robotics , the goal is to, find the dynamics of a 2DOF manipulator ( Double Pendulum ) , simulate the behavior of the unforced system, design 4-3-4 trajectories for both links, design torques to lead the robot using PID control, feedbacklinearization, and computed torque method, and also simulate the behavior of the controlled system.

2 Applied Robotics ABSTRACT: Analysis and simulation of the dynamics, and control for a two degree of freedom robot (double pendulum). The control of the 2DOF robot is using PID control, non linear feedback, and computed torque methods. . KEYWORDS: 2DOF dynamics, PID control, non-linear feedback, computedtorque.

1. INTRODUCTION. During the third partial of Applied Robotics, one of the topics was Manipulator Dynamics, that essentially are the mathematical equations describing the dynamic behavior of the robot (manipulator). And another key topic of the entire course was Manipulator Control, the theory for the behavior of dynamic systems, like a manipulator. Control is a very important part toany robot, because affects the desired behavior through the obtained references of the system. In the next paper we will explain the methodology followed to obtain both, the dynamics of the manipulator and the control using three different methods. 2. MANIPULATOR DYNAMICS. Manipulator dynamics relates joint torques with robot motion, producing a change in the acceleration, velocity and the position.According to the dynamics model of a n-link arm the equation that best describes the dynamics is as follows:

So, we can establish the dynamics for the 2DOF as:

3 Applied Robotics

block diagram of the system without forcing and initial conditions θ1(0) = θ2(0) = π. And the simulation of the θ1 and θ2:

3. SIMULATION OF THE BEHAVIOR. According to the specifications to simulate thebehavior of the unforced system for initial conditions θ1(0) = θ2(0) = π and null velocities, and using the parameters of the table:

Theta 1.

The equations for the dynamics needs to be expressed into simulink blocks, leading to the next block diagram with a lot of feedbacks:

Theta 2. This behavior is comprehensible since there is no friction considered in the equations, and the angles willcontinuing spinning until something stops them.

4 Applied Robotics 4. Path Planner. Trajectory planning is to interpolate, or approximate the desired path by a class of polynomial functions and generate a sequence of time-based control set points, for the control of the manipulator from an initial configuration to a destination. For generate this path planner the 4-3-4 polynomials and set ofrules were followed:

All this development is in annexes in a maple file, including the next graphics:

And finally:

5 Applied Robotics The path planner in simulink is shown below: 5. PID Control. A Proportional–Integral–Derivative controller (PID controller) is a generic control loop feedback mechanism widely used in robotic systems. A PID controller calculates an "error" value as thedifference between a measured process variable and a desired set point. The controller attempts to minimize the error by adjusting the process control inputs. For best performance, the PID parameters used in the calculation must be tuned according to the nature of the system – while the design is generic, the parameters depend on the specific system. The mathematical expression for a PID controlleris:

And the position, velocity and accelaration:

And the way it operates its describe in the next image:

So, by having the dynamic model, now is time to add the PID controller:

6 Applied Robotics The PID controller needs to to lead the robot from position (0,0) to position (π,π). 6. FEEDBACK LINEARIZATION. Feedback linearization is a control method in wich non linear systems, like a...