# Theory

## Robot manipulator theory.

In this section, some theoretical concepts about robot manipulators are presented. Some content has been developed in collaboration with Sebastián Tornil-Sin.

T1_MATH_Tranformations_V3.pdf

# 1_Mathematical Transformation

The robot manipulator can be modeled as a set of joints and links from the base frame to the end effector. In this section we define a frame for each joint and the relationship between close joints. The algebra lineal allows us define homogeneous transformations (Translation, Rotation) and define the pose of the robot manipulator in free space.

Robot Kinematics_V6.pdf

# 2_Kinematics

The coordinates of a point in terms of the base frame allow us the obtention of the kinematic chain. The movement possibilities of the robot manipulator are related to base frame while robot tasks are related to the movement possibilities of the end effector.

Inverse Kinematics_V2.pdf

# 3_Inverse Kinematics

In this section the inverse kinematics problem is presented. How arrive to an specific point in free space using diferent robot configurations? Computational methods are required to solve inverse kinematic (numerical or iterative methods).

Jacobian_V5.pdf

# 4_Jacobian

The Jacobian function is useful to stablish the relationship between different sets of coordinates. Give us information about robot singularities and is useful in robot control.

Trajectories_3.pdf

# 5_Trajectories

From a point to the target point a polynomial path interpolation is considered in this section with the aim to explain robot trajectories.

Calibration_3.pdf

# 6_Calibration

How define a home position for our robot manipulator? this is a basic step when the human operator learn how work with a new robot. Other associated problem is understand which is the level of accuracy of the robot manipulator when the end effector arrives to a target point.

Robot Dynamics_V4.pdf

# 7_Dynamics

Related to kinematic control (trajectories) in this case we aply the Lagrange-Euler formalism for the obtention of the dynamics of the robot manipulator (Inertia, Coriolis, Gravity). The dynamic's behaviour is useful when you have in mind the goal of study robot control.

Actuator Dynamics_V4.pdf

# 8_Actuator Dynamics

This section is focused in electric drive systems and how the relationship between engine, gearboxs and brake defines an effective subsystem.

Robot Control_V4.pdf

# 9_Robot Control

The application of control theory of a robot manipulator take into account how reduce the error tracking. The dynamic previously explained (Lagrange-Euler formalism, actuator) derives into the control of electric engines with pertorbations. The PID control law is the core of control strategies (Computed troque, for instance) for robotic systems.

Robot_Controller_V2.pdf

# 10_Robot Controller

Internal architecture of a robot cabinet is presented in this section. The knowledge of the connectivity of the robot with external sensors, external PLC controller and communications networks is trhe previous step for the integration of the robot in the factory.