# Objective

To study the working of an mechanical articulator via remote experimentation.

# Theory

Research towards an understanding of motor control needs to be studied different levels of abstraction, for instance, by examining the biochemical mechanisms of neuronal firing, the representational power of single and populations of neurons, neuroanatomical pathways, the biomechanics of the musculoskeletal system, the computational principles of biological feedback control and learning, or the interaction of action and perception.

For students laboratory studies, it is important to understand the robotic representation of movement information. This is one such experiment that allows students to understand motion in terms of extractable coordinates.

Robotic arm are the most useful mechanical devices which are used in the where accuracy and precision are needed. These things work similar to human arm and can be programmed as well. Traditional robotic arm will have n number of links, which are connected by n-1 joints with which links are rotate or translate. This links and joints form a Kinetic chain which has an End Effector at the last link.

Terms frequently used while constructing a robotic arm

1) Degree of freedom (DOF)

2) Forward Kinematics

3) Denavit-Hartenberg (DH) Parameters

## Degree Of Freedom (DOF)

In mechanics, the degrees of freedom of a body are the number of independent parameters that deﬁne the displacement and deformation of the body. That is the number of actuators in the Robotic arm is the number DOF of the robot, means each DOF is a joint where robot will rotate or translate. The figure2 is a robotic arm which has 6 DOF,which means there are 6 motors. As the degree of freedom increases in robot it will be difficult to control.

## Determining the Cartesian space co-ordinates of the end-eﬀector using forward kinematics

There are two scenarios that emerge in robotics. One is where the angles ofjoints are known and the other one is where the end eﬀector co-ordinates are known. In the former scenario, the requirement is to determine the end-eﬀector co-ordinates whereas in the latter scenario, the joint angles are determined.

The forward kinematics problem is to determine the position and orientation of the end-eﬀector, given the values for the joint variables of the robot. The joint variables are the angles between the links in the case of revolute or rotational joints.

As described earlier, a robot manipulator is composed of a set of links connected together by various joints. The objective of forward kinematic analysis is todetermine the cumulative eﬀect of the entire set of joint variables. Here, we will illustrate a set of conventions that provide a systematic procedure for performing this analysis. A robot manipulator with n joints will have n + 1 links, since each joint connects two links. We number the joints from 1 to n, and we number the links from 0 to n, starting from the base. By this convention, joint i connects link i − 1 to link i. We will consider the location of joint i to be ﬁxed with respect to link i − 1. When joint i is actuated, link i moves.

To perform the kinematic analysis, we rigidly attach a coordinate frame to each link. In particular, we attach oxyz(i) to link i. This means that, whatever motion the robot executes, the coordinates of each point on link are constant when expressed in the ith coordinate frame. Furthermore, when joint i is actuated, link I and its attached frame, oxyz(i), experience a resulting motion. The frame oxyz(0), which is attached to the robot base, is referred to as the inertial frame.

The next step is to determine homogenous transformation matrices for eachof the joints with respect to the previous joint. Suppose A(i) is the homogeneous transformation matrix that expresses the position and orientation of oxyz(i) with respect to oxyz(i − 1). The matrix A(i) is not constant, but varies as the conﬁguration of the robot is changed. However, the assumption that all joints are either revolute means that A(i) is a function of only a single joint variable, namely q(i).

In other words,

A(i) = A(i)(q(i))

**A diagrammatic representation of co-ordinate frames attached to each frame **

Now the homogeneous transformation matrix that expresses the position and orientation of oxyz(i) with respect to oxyz(j) is called, by convention, a transformation matrix, and is denoted by T(i).

T(i) w.r.t j= A(i+1)*A(i+2).A(j) if i < j

T(i) w.r.t j = I if i = j

T(i) w.r.t j = [T(i) w.r.t j ]T if i > j

By the manner in which we have rigidly attached the various frames to the corresponding links, it follows that the position of any point on the end-eﬀector, when expressed in frame n, is a constant independent of the conﬁguration of therobot.

Then the position and orientation of the end-eﬀector in the inertial frame are given by:

H = [T(0) w.r.t n] = A(q(1))....A(q(n)).

In principle, that is all there is to forward kinematics. Determining the functions A(q(i)), and multiplying them together as needed. However, it is possible to achieve a considerable amount of streamlining and simpliﬁcation by introducing further conventions, such as the Denavit-Hartenberg representation of a manipulator.

## Denavit Hartenberg parameters

It is possible to carry out all of the analysis for forward kinematics using an arbitrary frame attached to each link; it is helpful to be systematic in the choice of these frames. A commonly used convention for selecting frames of reference in robotic applications is the Denavit-Hartenberg, or D-H convention. In this convention, each homogeneous transformation A(i) is represented as a product of four basic transformations

### Transformation matrix

Where the four quantities o(i), a(i), d(i),α(i) are parameters associated with link i and joint i. The four parameters o(i), a(i), d(i),α(i) are generally given thenames link length, link twist, link oﬀset, and joint angle, respectively. A complete transformation matrix has been given below.[10]

Where R is the 3 _ 3 sub-matrix describing rotation and T is the 3 _ 1 sub-matrix describing translation

### Calculating DH parameters for a given kinematic chain

The calculation of D-H parameters for a robotic manipulator involves the selection of proper axes. The following are the thumb rules for selecting the properaxes:

1. The Z-axis is in the direction of the joint axis

2. The X-axis is parallel to the common normal between the chosen Z axis. If there is no unique common normal (parallel

Z-axes), then d is a free parameter.This occurs in the case of joints with parallel axes of motion.

3. The Y-axis follows from the X and Z-axis by choosing it to be a right-handed coordinate system.

The transformation is then described by the following four parameters are:

- d: oﬀset along previous to the common normal.
- o: angle about previous , from old to new.
- a: length of the common normal. Assuming a revolute joint, this is the radius about previous Z axis.
- α: angle about common normal, from old Z-axis to new Z-axis.

### An example to illustrate

An example representing the axes chosen for each joint.

The D-H parameters for ﬁgure 3.9 are represented by:

The D-H parameters for above example

O(1) and O(2) represent the joint angles and a1 and a2 represent the link length.