Inverse Kinematics Analysis for Manipulator Robot with Wrist Offset Based On the Closed-Form Algorithm

Mohammed Z. Al-Faiz & Mohammed S. Saleh
International Journal of Robotics and Automation (IJRA), Volume (2) : Issue (4) : 2011 256
Inverse Kinematics Analysis for Manipulator Robot With Wrist
Offset Based On the Closed-Form Algorithm
Mohammed Z. Al-Faiz ,MIEEE mzalfaiz@ieee.org
Computer Engineering Dept.
Nahrain University
Baghdad, Iraq
Mohammed S.Saleh mohselman@gmail.com
Electronics Engineering Dept
Diyala University
Diyala, Iraq
Abstract
This paper presents an algorithm to solve the inverse kinematics for a six degree of freedom (6
DOF) manipulator robot with wrist offset. This type of robot has a complex inverse kinematics,
which needs a long time for such calculation. The proposed algorithm starts from find the wrist
point by vectors computation then compute the first three joint angles and after that compute the
wrist angles by analytic solution. This algorithm is tested for the TQ MA2000 manipulator robot as
case study. The obtained results was compared with results of rotational vector algorithm where
both algorithms have the same accuracy but the proposed algorithm saving round about 99.6% of
the computation time required by the rotational vector algorithm, which leads to used this
algorithm in real time robot control.
Keyword: Manipulator Robot, Inverse Kinematics, DOF.
1. INTRODUCTION
A robotic manipulator (arm) consists of a chain of links interconnected by joints. There are
typically two types of joints; revolute joint (rotation joint) and prismatic joint (sliding). It would be
desirable to control both the position and orientation of a end-effecter or work piece, located at
the tip of the manipulator, in its three-dimensional workspace. The end-effecter can be
programmed to follow a planned trajectory, provided relationships between joint variables and
position and the orientation of the end-effecter are formulated. This task is called the direct
kinematics problem [1].
There are some difficulties to solve the inverse kinematics (IK) problem when the kinematics
equations are coupled, multiple solutions and singularities exist. There are mainly two types of IK
solution: analytical solution and numerical solution. In the first type, the joint variables are solved
analytically according to given configuration data. In the second type of solution, the joint
variables are obtained based on the numerical techniques. However, they are slow in practical
applications [2]. An Artificial Neural Network (ANN) using backpropagation algorithm is applied to
solve inverse kinematics problems of industrial robot manipulator. 6R robot manipulator with
offset wrist was chosen as industrial robot manipulator because geometric feature of this robot
does not allow solving inverse kinematics problems analytically [3].
2. DENAVIT-HARTENBERG (DH) PARAMETERS
A general arm equation that represents the kinematic motion of the manipulator can be obtained
by systematically assigning coordinate frames for each link. The parameters associated with joint
k are defined with respect to zk-1, which is aligned with the axis of joint k. The first joint parameter,
θk is called the joint angle. It is the rotation about zk-1 needed to make xk-1 parallel to xk. The
second joint parameter, dk is called the joint distance. It is the translation along zk-1 needed to
make xk-1 intersect with xk see Fig.1. Note that for each joint, it will always be the case that one of

these parameters is fixed and the other is variable. The variable joint parameter depends on the
type of joint.
FIGURE 1: Joint angle θ and joint distance d
For instance, for a revolute joint, the joint angle θk is variable while the joint distance dk is fixed
while for a prismatic joint, the joint distance dk is variable and the joint angle θk is fixed [1]. See
Fig.2.
FIGURE 2: DH parameters defined for two links of a serial manipulator
3. FORWARD KINEMATIC ANALYSIS
These industrial robots are basically composed by rigid links, connected in series by joints
(normally six joints), having one end fixed (base) and another free to move and perform useful
work when properly end-effecter. As with the human arm, robot manipulators use the first three
joints (arm) to position the structure and the remaining joints (wrist, composed of three joints in
the case of the industrial manipulators) are used to orient the end-effecter. There are five types of
arms commonly used by actual industrial robot manipulators: Cartesian, cylindrical, polar, SCARA
and revolution [4]. The TQ MA2000 is considered in this work, it has 6 degree of freedom (DOF),
one per each joint which means that each joint can be represented by a independent variable θn
given that each joint is actuated, i.e. in this case each link has its own servo motor acting on it.
Robot manipulator is an open kinematic chain whose joint positions can be defined by a vector of
six single variables θn and the number of joints equals the number of degrees of freedom. Recall
that the number of DOF is the number of independent position variables that should be specified

in order to completely define a pose for the manipulator. It defines in how many ways the robot is
able to move. In practice, to cover the entire 3D workspace with a 3-component orientation
vector, six degrees of freedom are sufficient. Table I shows the DH parameters whole set of
Kinematic parameters for the TQ MA2000 Robot. See Fig.2 The graphic representation of the TQ
MA2000 robot is shown in Fig.3
Link i Length ai(cm) Twist Angle αi Offset di (cm) Joint angles θi
1 0 π/2 26 θ 1
2 23 0 0 θ2
3 24 0 0 θ3
4 0 π/2 5 θ4
5 0 π/2 4.4 θ5
6 0 0 8 θ6
TABLE 1: Complete DH parameter table for the TQ MA2000 6-DOF manipulator shown in Fig. 2
FIGURE 3: Graphic Represent of TQ MA2000 robot
The arm motion can be represented by transformation matrix as follow
Arm motion = (1)
Where, represent the transformation matrix from i
th
location to j
th
location. And ( ) is
the orientation and position of the arm with respect to the base coordinate frame, depend on the
first three joint angles . And the wrist motion can be represented by the transformation
matrix from wrist to end-effecter as follow.
Wrist motion = (2)
And ( ) is the orientation and position of the grip of the wrist with respect to the
third link coordinate frame, depend on the last three joint angles . Note that
maps tool-tip coordinates into roll coordinates, maps roll coordinates into yaw coordinates,
and maps yaw coordinates into wrist pitch coordinates. Thus the composite transformation
maps end-effecter coordinates into wrist coordinates. Similarly maps wrist
coordinates into elbow coordinates, maps elbow coordinates into shoulder coordinates,
and maps shoulder coordinates into base coordinates. Thus the composite transformation
maps wrist coordinates into base coordinates. The general solution can be
expressed as:

[Total motion] = [arm motion] [wrist motion] (3)
Using the D-H method, the end-effector position can be written as: [1].
= (4)
The homogenous transformation matrix of the which include the rotation matrix (3x3) and
position vector (3x1) which given by equation (5) may be given as
(5)
The upper left 3x3 matrix, R, specifies the orientation of the end-effecter, while the 3 x 1 upper
right sub matrix P specifies the position of the end-effecter. Thus, in the direct kinematic solution,
for any given value of the joint angles θ, the arm matrix can be evaluated.
4. INVERSE KINEMATIC ANALYSIS
One of the most fundamental and ever present problems in robotics and computer animation is
the inverse kinematics (IK). This problem may be stated as follows: given a desired hand position
and orientation (posture), and the forward kinematics map, find the set of all configurations (joint
angle vectors) of the robot or animation character that satisfy the forward kinematics map. The IK
mapping is in general one to many involves complex inverse trigonometric functions, and for most
manipulators and animation figures has no closed form solutions. Extremely fast IK computation
is required in computer animation, for real-time applications in fast moving manipulator. Inverse
kinematics deals with the problem of finding the required joint angles to produce a certain desired
position and orientation of the end-effecter. Finding the inverse kinematics solution for a general
manipulator can be a very tricky task. Generally they are non-linear equations. Close-form
solutions may not be possible and multiple, infinity, or impossible solutions can arise.
Nevertheless, special cases have a closed-form solution and can be solved [5].
5. PROPOSED ALGORITHM FOR IK
The proposed algorithm can be applied directly to robots with shoulder and elbow offset.
However, wrist offset makes the hand position dependent on all joint angles rather than just the
first four. This requires dealing with position and orientation simultaneously and entails the
creation of 6-dimensional workspace which although possible, would substantially increase the
off-line processing and storage requirements. Manipulators can often be considered to be
composed of two parts. The first 3 joints form a regional structure whose primary purpose is to
position the wrist in space. The last 3 joints form the orienting structure whose purpose is to orient
the hand or grasped object.
5.1 Decompose the Manipulator at the Wrist
This is the single most important step of the procedure. The spherical wrist makes this possible.
Find the position of the wrist point, which subsequently is used to solve for the first 3 joint angles.
The wrist joint angles are then solved to give the correct orientation.
Fig.3 show the robot at reset position (all the joint angles equal zero), the proposed algorithm
start with specify the required position in (x, y, z) coordinate and orientation angles of pitch, yaw,
and roll (φp, φy, and φr) angles respectively. Then change the wrist joint angles
from the zero (reset state) to new values that equal the required orientation angles of pitch, yaw,
and roll (φp, φy, and φr) angles respectively, the arm joint angles remain unchanged (at reset
state, i.e. equal zero), then compute the foreword kinematics to compute the resulted position of
the arm end point (O4) and position of end-effecter (O6) for the robot, find the vector between
these two points as in equation (6)

(6)
To compute the suitable position of the arm end point to reach the required position use equation
(6) with that the vector from the origin point to the required position is equal to the vector
from origin base frame to the arm end point position sum with the vector V which is from
to the required position as in equation (7). as shown in Fig. 4
(7)
Where V is the vector compute by equation (6), and is the vector from the origin to the
required position. The position of the arm end point is the end of the vector V04. A planar two-
link manipulator often makes up the last two links of the regional structure.
FIGURE 4: Wrist with setting its angles
The link 4 parameters have been included in the regional structure because they locate the wrist
point , which will be known relative to . Equation (8) as shown in fig.(5).
(8)
FIGURE 5: Vector from
Project the position of the wrist point into the x0, y0 plane, i.e., just select the coordinates of
.

5.2 Solution for Θ1
The wrist offset makes the calculation of more complex, because the axis is no longer in the
vertical plane of the upper arm and forearm Figure. When and the wrist point are projected
onto the , plane, then the angle from to is no longer the same as the angle from
to the projected wrist point. The projection on the , plane is redrawn in Figure for clarity. The
position of the arm end point at ( , see Fig.5. Then compute as in equation (9)
and (10).
(9)
(10)
Where , and hence for right move then will be as follow
(11)
And for lefty move
(12)
5.3 Solution For Θ2 And Θ3
For each solution of find two set of solution for , elbow up and elbow down, therefore
will compute four set of solution of the first three angles that is (left, right, elbow up, elbow down):
(13)
Where , and
The cosine law for link 2 and link 3:
(14)
(15)
Where (16)
Then compute and
(17)
(18)
5.4 The Solution of the Last Three Joints
It is possible to obtain the following equations from the forward kinematics problem:
(19)
(20)

(21)
Where Pw is the wrist position and Pa position of arm end point then the rotation part of equation
(5).
(22)
where R is the required orientation of the robot. Translation part of equation (5)
(23)
Then (24)
Where , and (25)
(26)
Solving for orientation angles:
, and (27)
5.5 Case Study
In [6] they use the relative orientation representation by the rotation vector are based on the Euler
theorem which states that (a displacement of a rigid body with one fixed point can be described
as a rotation about some axis). And applied this algorithm to solve the inverse kinematics for the
TQ MA2000 manipulator robot to compute the sets of angles. Consider the required position is (-
25, -10, 50) cm as (x, y, z) with respect to the base coordinate frame. The orientation of the
gripper rotates (180o, 225o, 135o) degree as the rotating angle of the gripper about
pitch , yaw, and roll rotating axis respectively. There are four set of the joint variables (solution) of
the inverse kinematics of the robot.
Set = [ θ1 , θ2 , θ3 , θ4 , θ5 , θ6]
Set 1 = [-162.6436, -7.8977, 79.7788, 108.1189, 27.6436, 135]
Set 2 = [-162.6436, 75.8702, -79.7788, -176.0914, 27.6436, 135]
Set 3 = [ -1.2296,187.8977, -79.7788, 71.8811, -133.7704, 135]
Set 4 = [ -1.2296, 104.1298, 79.7788, 356.0914, -133.7704, 135]
The structure of the robot for all solution sets(1, 2, 3, 4) are shown in Fig.6 (a- right elbow down,
b- right elbow up, c- lefty elbow down, and d- lefty elbow up) respectively.
The proposed algorithm is solving the same case steady and finds the same results with the
same accuracy. By using DELL laptop computer with Central Processing Unit (CPU) Intel®
Core™ 2 Duo with 2.2GHz; the computation time for the proposed algorithm is equal 0.016
second and the computation time for rotation vectors is equal 3.984 second these less
computation time is suitable for used the proposed algorithm with real time manipulator robot
control system.

6. CONCLUSION
By camper this algorithm with the vector rotation algorithm that is used for solve the inverse
kinematics of the same manipulator robot [6]. They have same accuracy but the computation time
by proposed method is less than that for the first method to compute all the angles. The
computation time for the solving of inverse kinematics of the manipulator robot by the proposed
algorithm is about 0.4% of the computation time that required by use the rotational vector
algorithm, therefore this algorithm is best to used for real time manipulator robot control system.
a b
c d
FIGURE 6: Drawing of the robot according to set of solution a- set 1 b-set 2 c- set 3 d- set 4
7. REFERENCES
[1] A. N. Aljaw, A. S. Balamesh, T. D. Almatrafi and M. Akyurt ”Symbolic Modeling Of Robotic
Manipulators” the 6th Saudi engineering conference, KFUPM, vol. 4, December 2002
[2] S. Kucuk and Z. Bingul “Inverse Kinematics Solutions of Fundamental Robot Manipulators
with Offset Wrist” in Proc. International Conference on Mechatronics, 2005, pp 197-202.
[3] Z. Bingul, H. M. Ertunc and C. Oysu “Comparison of Inverse Kinematics Solutions Using
Neural Network for 6R Robot Maipulator with Offset” In ICSC Congress on Computational
intelligence, Turkey, 15-17 December 2005, pp. 5–10
[4] J.N. Pires. Industrial Robots Programming: Building Applications for The Factories of The
Future. Springer Science Business Media, 2007,pp 36

[5] C.F. Rose, P-P. J. Sloan and M.F. Cohen, “Artist-Directed Inverse Kinematics Using Radial
Basis Function Interpolation” EUROGRAPHICS, vol.20, no. 3, pp. 1-12, 2001
[6] M. Z. Al-Faiz, M. Z. Othman, and B. B. Al-Bahri ” An Algorithm to Solve the Inverse
Kinematics Problem of a Robotic Manipulator Based on Rotation Vectors” 3rd IEEE-FCC
2006, Bahrain

Inverse Kinematics Analysis for Manipulator Robot with Wrist Offset Based On the Closed-Form Algorithm

More Related Content

What's hot (20)

Viewers also liked (20)

Similar to Inverse Kinematics Analysis for Manipulator Robot with Wrist Offset Based On the Closed-Form Algorithm (20)

More from Waqas Tariq (16)

Recently uploaded (20)

Inverse Kinematics Analysis for Manipulator Robot with Wrist Offset Based On the Closed-Form Algorithm