Kinematics Analysis of 6-DoF Articulated Robot with Spherical Wrist

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.


Introduction
Locomotion is the process of causing a rigid body to move. e 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. e 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.
Maintaining a strong connection between the two joints is called the kinematics function of a link. is 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. e 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). e relationship between link length and twist is described in Figure 2.
(i) A joint axis is formed at the connection of two links. is joint will have two parameters (one for each link) connected to it. ese parameters are as follows: (ii) d: distance between links (iii) W: angle between links e 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. e four parameters demonstrated above are associated with each link. Axes can be aligned using these parameters. e parameters are also known as Denavit-Hartenberg link parameters. ese are illustrated in Table 1 below: e 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. e 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, T i is the link transformation from the i th joint, and 0 n T is the final pose for end-effector relative to the base.
ere 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]. e parametric continuity of the CPC model was achieved by using singularity free line representation.

Forward Kinematics of Comau NM45
e Comau NM45 [6] is a medium-scale robot. It has 6 degree of freedom joints. It is an articulated arm with a spherical wrist. e wrist joint intersects at one point. Figure 5 shows the manipulator with its link length and working envelope.
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. e first step of performing the forward kinematics is to label link lengths.   is step has been performed in Figure 5. e second step to find the forward kinematics of the manipulator is to assign the frames. e frame assignment is done in Figure 6. e D-H parameters can be found based on the frame assignment. e modified DH convention has been used for the frame assignment and DH parameters [7,8]. ese parameters are illustrated in Table 2.
e transformation matrix for a link i is described as follows: A 1 is the transformation matrix T 0 1 : and α � 90, so T 0 1 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 A2 � T 1 2 , sin θ2 cos θ2 0 α2 sin θ2 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: sin θ2 cos θ2 0 0.75 sin θ2 For A3 � T 2 3 , 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: cos θ3 0 sin θ3 0.25 cos θ3 sin θ3 0 − cos θ3 0.25 sin θ3 For A4 � T 3 4 and α � 90, 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 A4 � T 4 5 , sin θ5 0 − cos θ5 α5 sin θ5 Mathematical Problems in Engineering 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 A5 � T 5 6 and α � 0, 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 A12 � T 0 2 � T 0 1 × T 1 2 , substitute values from equations (5) and (7): e resultant transformation between the base and joint 2 is illustrated as follows: Mathematical Problems in Engineering e resultant transformation between the base and joint 3 is illustrated as follows:

cos(u)cos(v + w) sin(u) cos(u)sin(v + w) m + 0.25 cos(u)cos(v + w)
e resultant transformation between joint 3 and joint 6 is illustrated as follows: Spherical wrist position can be extracted using the last column of T 0 6 transformation matrix, where T 0 6 � r 11 r 12 r 13 P x r 21 r 22 r 23 P y r 31 r 32 r 33 P z Which means the T 0 6 the transformation matrix can be represented in terms of R 0 6 and P 0 6 illustrated as follows: and spherical wrist position is illustrated as Transformation matrix from base to end-effector is 6 Mathematical Problems in Engineering

Inverse Kinematics of Comau NM45
Inverse kinematics is finding the joint values θ1, θ2, θ3, θ4, θ5, θ6 of the robot arm for the given position (p) and orientation (o). For inverse kinematics, the inverse orientation R and inverse position P are needed.
e Comau NM45 is an articulated arm with a spherical wrist. For finding anthropomorphic/articulated arm position and joint values θ1, θ2, θ3, the inverse position is needed.
Let O c be intersecting the last 3 joints. e motion of joints 4, 5, and 6 will not change the position of O c , as stated in Figure 7 [19] is is according to Pieper's approach [7], in which the manipulator is divided to analyse the inverse kinematics [10].
According to T 3 6 matrix from equation (22), the position P 3 6 is always as mentioned in the following equation: So, the position O will be as follows: where O is the position P 0 6 and R is the orientation which is R. e above equation can be written in terms of O 0 c as follows: e first three joints can be found in the following steps. ey will determine the position of the manipulator: For the orientation, the last three joints orientation is needed. e following equation shows the overall rotation of the manipulator in terms of R 0 3 and R 3 6 : Rearrangement of equation (31) will yield R 3 6 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.
Projection of wrist onto x 0 , y 0 plane has been shown in Figure 9.
is projection yields the triangle through which the angle value for θ1 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 (θ 2 , 2) and joint angle 3 (θ 3 , 3).
Law of cosines can be applied to obtain the joint angle 3 (θ 3 , 3), as follows: By substituting the values of r and s, these can be extracted by using Figure 8: As NM45 2.0 is inline, no shoulder is offset and, hence, d � 0. ∅ � a tan 2(r13, r23).
For the trajectory planning, the program was written in Matlab by using forward and inverse kinematics equations mentioned in the above sections. e planned trajectory required the smooth motion of the end-effector. e joint angles were calculated using the inverse kinematics equations. Figure 12 shows the trajectory planning and robot motion along with the trajectory for the Comau NM45. e robot was able to follow the trajectory smoothly. Figure 13 shows the mapping of the values of joints to move on 100 points to follow the trajectory. It shows that the achievement was smooth as there are no sudden spikes in the joint values.

Conclusion
e modified DH convention has been used to perform the forward kinematics for the manipulator. e kinematics decoupling has been used to perform the inverse kinematics. e manipulator was divided into two parts to make the inverse kinematics problem simpler. e first 3 joints were resolved by using a geometrical approach, whereas the last three joints were resolved using the algebraic approach. e resultant kinematics solution was applied on the manipulator, and it was able to follow a test trajectory successfully. A similar approach can be used while solving the articulated robot with a spherical wrist. e techniques are applied for the 6-DoF robot.

Data Availability
All the data used are presented as part of paper.

Conflicts of Interest
e authors declare that they have no conflicts of interest.