Abstract

The aim of the paper is to study the kinematics of the manipulator. The articulated robot with a spherical wrist has been used for this purpose. The Comau NM45 Manipulator has been chosen for the kinematic model study. The manipulator contains six revolution joints. Pieper’s approach has been employed to study the kinematics (inverse) of the robot manipulator. Using this approach, the inverse kinematic problem is divided into two small less complex problems. This reduces the time of analysing the manipulator kinematically. The forward and inverse kinematics has been performed, and mathematical solutions are detailed based on D-H (Denavit–Hartenberg) parameters. The kinematics solution has been verified by solving the manipulator’s motion. It has been observed that the model is accurate as the motion trajectory was smoothly followed by the manipulator.

1. Introduction

Locomotion is the process of causing a rigid body to move. The body needs force to move. Dynamics is the study of the motion of the body in which forces are modelled which helps the body to move, whereas kinematics is the geometrical study of the motion of the body without considering the forces that can affect the motion of the body.

Kinematics is the motion description of the rigid body. [1] Links are the connectivity body/member between joints. The kinematic chain is a grouping of links connected by joints, as illustrated in Figure 1. In the kinematic chain, the number of DoF (degree of freedom) is equal to the number of joints.

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(b)

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Figure 1 

Kinematics of the manipulator: (a) open chain; (b) closed chain [2].

Maintaining a strong connection between the two joints is called the kinematics function of a link. This connection can be described with the following factors:(i)a: link length(ii)α: link twist

Link length is measured along the line which is mutually perpendicular to both joints/axes. The perpendicularity in joints always exists except when both joints are parallel. Link twist is the angle of projection from the previous joint (i−1) to the next joint (i) onto the axis i−1 (previous joint); the projection line is parallel to the next joint (axis i). The relationship between link length and twist is described in Figure 2.(i)A joint axis is formed at the connection of two links. This joint will have two parameters (one for each link) connected to it. These parameters are as follows:(ii)d: distance between links(iii)ϴ: angle between links

Figure 2 

Kinematic function of link.

The relative position or distance between the links is called link offset. Figure 3 describes these parameters, in which the joint angle is the angle between the links.

Figure 3 

Link and joint parameters. Joint parameters are highlighted in red.

The four parameters demonstrated above are associated with each link. Axes can be aligned using these parameters. The parameters are also known as Denavit–Hartenberg link parameters. These are illustrated in Table 1 below:

The link numbering convention follows from the base of the arm till the last moving link. As mentioned in Figure 4, the first link is the connection between the base and first joint.

Figure 4 

A 7-DoF manipulator arm [3].

The parameters mentioned above in Table 1 are used for kinematic modelling of the robot. In kinematics, modelling the geometry of the robot is represented. Homogenous transformation (of the matrix) is commonly used as the definition of the kinematics model (particularly for chains mechanism). As described below,where n is the number of links, is the link transformation from the i^th joint, and is the final pose for end-effector relative to the base.

There are two main types of kinematic models: forward kinematics and inverse kinematics. In forward kinematics, the length of each link and angle of each joint is given, and through that, position of any point (x, y, z) can be found. In inverse kinematics, the length of each link and position of some points (x, y, z) is given, and the angle of each joint is needed to find to obtain that position.

Several models are developed for kinematic modelling, but the D-H (Denavit–Hartenberg) model [4] is the most popular model. Limitations of the D-H model are discussed, and CPC (completeness and parametric continuity) model and its mapping with the D-H model were proposed [5]. The parametric continuity of the CPC model was achieved by using singularity free line representation.

2. Forward Kinematics of Comau NM45

The Comau NM45 [6] is a medium-scale robot. It has 6 degree of freedom joints. It is an articulated arm with a spherical wrist. The wrist joint intersects at one point. Figure 5 shows the manipulator with its link length and working envelope.

Figure 5 

Comau NM45 (inline) operative area and link lengths.

Forward kinematics is the study of the manipulator to find out its tip or end-effector position and orientation by using joint values of the manipulator. The first step of performing the forward kinematics is to label link lengths. This step has been performed in Figure 5. The second step to find the forward kinematics of the manipulator is to assign the frames. The frame assignment is done in Figure 6.

Figure 6 

NM45 model with coordinates mapping.

The D-H parameters can be found based on the frame assignment. The modified DH convention has been used for the frame assignment and DH parameters [7, 8]. These parameters are illustrated in Table 2.

The transformation matrix for a link is described as follows:

A₁ is the transformation matrix :and , so will be as follows:whereas a1 = l1 = 0.4 and d1 = 0.75; by placing these values above, the following equation which is the resultant for the transformation between the base and joint 1 is obtained:

For ,whereas a2 = l2 = 0.75 and d2 = 0; by placing these values above, the following equation which is the resultant for the transformation between joint 1 and joint 2 is obtained:

For ,whereas a3 = l3 = 0.25 and d3 = 0; by placing these values above, the following equation which is the resultant for the transformation between joint 2 and joint 3 is obtained:

For and ,whereas a4 = 0; by placing the value of a4 in the above equation, the following equation which is the resultant for the transformation between joint 3 and joint 4 is obtained:

For ,whereas a5 = 0 and d5 = 0; by placing these values in the above equation, the following equation which is the resultant for the transformation between joint 4 and joint 5 is obtained:

For and ,whereas a6 = 0; by placing the value of a6 in the above equation, the following equation which is the resultant for the transformation between joint 5 and joint 6 is obtained:

2.1. Transformation

For simplification, the following has been substituted:

For , substitute values from equations (5) and (7):

The resultant transformation between the base and joint 2 is illustrated as follows:

For , for simplicity,

The resultant transformation between the base and joint 3 is illustrated as follows:

For ,

The resultant transformation between joint 3 and joint 6 is illustrated as follows:

Spherical wrist position can be extracted using the last column of transformation matrix, whereWhich means the the transformation matrix can be represented in terms of and illustrated as follows:and spherical wrist position is illustrated as

Transformation matrix from base to end-effector is

3. Inverse Kinematics of Comau NM45

Inverse kinematics is finding the joint values of the robot arm for the given position (p) and orientation (o). For inverse kinematics, the inverse orientation and inverse position are needed.

The Comau NM45 is an articulated arm with a spherical wrist. For finding anthropomorphic/articulated arm position and joint values , the inverse position is needed.

Let be intersecting the last 3 joints. The motion of joints 4, 5, and 6 will not change the position of , as stated in Figure 7 [19] This is according to Pieper’s approach [7], in which the manipulator is divided to analyse the inverse kinematics [10].

Figure 7 

Kinematics decoupling (Spong, Hutchinson, and Vidyasagar, 2006).

According to matrix from equation (22), the position is always as mentioned in the following equation:

So, the position will be as follows:where is the position and R is the orientation which is . The above equation can be written in terms of as follows:

The first three joints can be found in the following steps. They will determine the position of the manipulator:

For the orientation, the last three joints orientation is needed. The following equation shows the overall rotation of the manipulator in terms of and :

Rearrangement of equation (31) will yield below, through which the last three joint angles can be found:

For articulated manipulator, the first three joints tell the position, as illustrated in Figure 8.

Figure 8 

Comau NM45 (articulated arm) first three joints (Spong, Hutchinson, and Vidyasagar, 2006).

Projection of wrist onto plane has been shown in Figure 9.

Figure 9 

Projection of wrist centre onto x0, y0 plane (Spong, Hutchinson, and Vidyasagar, 2006).

This projection yields the triangle through which the angle value for can be found as follows:

If the wrist is rotated, then it will result in the following equation:

Another projection, as mentioned in Figure 10, on the plane formed with link 2 and link 3 can help to find the value of joint angle 2 and joint angle 3 .

Figure 10 

Projection on the plane (Link2 l2, Link3 l3) (Spong, Hutchinson, and Vidyasagar, 2006).

Law of cosines can be applied to obtain the joint angle 3 , as follows:

By substituting the values of and , these can be extracted by using Figure 8:

As NM45 2.0 is inline, no shoulder is offset and, hence, d = 0.

So,where according to DH labelled figure, and

Also,

The value of can be written in terms of and as follows:where is for elbow up and is for elbow down.

The projection in Figure 11 has been drawn onto the link 2 and link 3 plane to find :

Figure 11 

Projection onto l2 and l3 plane to find θ₂. can be written in terms of angles and .

So, the value of can be written as follows:

3.1. Finding Spherical Wrist Joint Values (Rotation) Using Euler Angles

The last three joint variables, , are the Euler angles. So,

These angles are concerning the coordinate frame . Now we need to calculate the transformation from to which is :where can be written in terms of and mentioned as follows:where is the position.