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Introduction of
Robot Kinematics
The robot
kinematics is the mixture of the robotics technology as well as the
trigonometric functions and the mathematical geometry. The robot kinematics is
dealing with the movements of the robot and telling that how the trigonometric
function will be included as well as are involved in the robotic because every
robot in the world has the common functionality and feature of moving. For the
movement, it definitely needs the technology but it also needs the mathematical
function such as geometric and trigonometric functions. The study is providing
the brief information related to the robot kinematics as well as it is also
dealing and providing the information related to the related ways which are:
forward kinematics and the inverse kinematics. The kinematic equation part is
providing the information related to the important equation and some information
on the kinematics, end effector location and many more. There are some
kinematic equations for robotics are also provided in this study with the
explanation. The brief information of the Jacobean matrix is also provided in
this study. Some related past studies and the related works in the form of
literature review also discussed.
Robot Kinematics
The Robot
Kinematics applies the geometry as well as trigonometric functions due to the
multi degree movements of freedom kinematic chains to develop a robot. As the
rigid bodies, frames as well as structures of the robots, it models the links
of the robots which is the mean of the geometry and the joints of the robotic
body are supposed to give the proper movement and rotation to the robotic arms,
legs or any other parts if developed. Many studies on the robotic kinematics provide
the information and shows the relations among the connectivity and dimensions of
chains of kinematic, as well as acceleration, position and the velocity of
every link in the system of robotics to measure and calculate the torque, and
the actuator forces, as well as to plan and control the movement. Furthermore, the
association among motion, inertia properties, torques, associated forces as
well as the mass of the robot are studied in robot dynamics.
Use of Kinematic Equation
of Robot Kinematics
The
kinematic equations are the basic conceptual technique and significant tool into
the robotic kinematics that refers to or showing the link with the kinematic
chains. To map the joint parameters, such kind of nonlinear equations are used as
well as they are also used to configure the system of robot. In the compute
animation of articulated character as well as the biomechanics of the skeleton,
the equations of kinematics are used. In this section, a brief information related
to the kinematic equation as well as its types which can be used in the robots,
will be comprehensively discussed. The kinematic equation has further two forms
which are: forward kinematics and the inverse kinematics, are also discussed
briefly in this section. The kinematics equations are used to compute and
calculate the position of end effector by the forward kinematics form assumed
or stated values for common parameters. On the other hand, the reverse process
of computation of joint parameters that able to obtain the defined end effector
position is known as the inverse kinematics. Moreover, the kinematic equations
which are used in the robotic kinematics, and the dimensions of the robot are
used to specify the space volume of the robot.
Furthermore,
the derivation of kinematic equations are involved by the robot kinematics for
the description of the effective analytical association among the tip or end
effector as well as the joint parameters of the robot. So, there are two ways
of deriving and generating the kinematic equations for the movement of the
robot. The two ways to derive the kinematics equation are: forward kinematics
and inverse kinematics given and explained below.
Froward Kinematics of Robot Kinematics
In the
forward kinematics section, it is described the comprehensive information related
to the robotic kinematics and the kinematic equations. The forward kinematics are
very important the effectively postulates the joint parameters of the robots as
well as it also effectively computes the chain configuration. The direct
substitution of the joint parameters in the equations of the forward kinematics
achieve the serial manipulators for the seral chain. The solution of the set of
the polynomial constraints are required for the parallel manipulators of joint
the parameters within the kinematics equations for the determination of the
possible locations the end effector set. Furthermore, the joint angle values which
given into the diagram showing the robotic arm movement. The end effector
location of the robot is calculated and computed by the forwards kinematics
equation into the coordinate space.
The above equation is the link coordinate transform from
link i to link i-1 of the 2-d planner robot as well as it also rotates about
the Zi-1 axis by 
, and then about the Xi
axis by
αi.
The equation which is given above is forward kinematics and
refers to the translational component to axes i-1.
And thus, the above equation is showing the Homogenous transform
Ti from frame i to the frame i-1.
Inverse kinematics of Robot Kinematics
The end
effector location is specified by the inverse kinematics as well as the inverse
kinematics has also the capability to compute the significant linked joint
angles. Furthermore, the solution is of the set of polynomials is also required
for the serial manipulators that will be obtained from the kinematics equations
as well as it also produces multiple configuration for the chain. Moreover, the
kinematics equations are simplified by the specification of the end effector location
for the parallel manipulator that generates the formulas for the joint
parameters.
Furthermore, the end effector location of the robot which is
given in the diagram, the joint angles are calculated by the inverse kinematics
equations as well as required for the movement of end effector to the location.
The diagram is illustrating two links of robotic arm along with the anticipated
end effector location as well as the angels which are: 
.
As the
joint angles of the robot are computed as well as calculated by using the kinematic
equations, it can generate the new profile of motion by using the Jacobian
Matrix for effective movement of the end effector from the initial to the next
or final location. Furthermore, the Jacobian Matrix is very helpful to define the
relationship among the end effector velocities as well as the joint parameters of
the robots.
Figure 1: Bending movement of
robot arm
Jacobian Method of Robot Kinematics
In the Jacobean method of robot, the Jacobean of robot is
produced by the time derivation of the kinematics equation that associates with
the rates of the joint parameters of the robot to the angular velocity as well
as the linear velocity of the end effector tool location in the robot. Furthermore,
it is also shown by the virtual work principle that the relationship among the
joint torques and the resultant force are also provided by the Jacobian method
or Jacobian robotic matrix. Moreover, the torque applied by the end effector
location of the robot. The Jacobean matrix for robot is given below.
Velocity Kinematics
The results
of the Jacobean method of robot is the linear equation set that associated with
the rates of joint parameters of the robot to the six vectors. Furthermore, the
it forms the six vectors form the linear as well as angular velocity of the end
effector current location to the future location which is known as the twist. Postulating
the rates of joint parameters of the robot produces the twist in the end
effector location directly. Furthermore, the rates of joint parameter of the
robot are sought by the problem in the inverse velocity which provide the
specified twist in the end effector. The inversion of the Jacobean matrix
provides the effective solution for such kind of end effector twist and provides
the better solution for the inverse velocity problem. Furthermore, it may also
happen that the robot is in the configuration where an inverse is not had by
the Jacobean matrix. Thus, the singular configurations of the robot term such
kind of inverses.
Analysis of the
Static Force
The set of
linear equations are made by the principle of virtual work that associate and
link with the six-vector force torque and it is also known as wrench that acts
on the end effector to the joint robotic joint torques. Furthermore, the joint
torques are yielded by the direct calculation when the end effector wrench will
be known. Therefore, the wrench of end effector which is associated with the
given joint torques set, are sought by the inverse static problem, as well as the
inverse of the Jacobean matrix is also required. It cannot solve the problem at
the singular configuration in the inverse velocity analysis case. Although, it
is resulted in the large end effector wrench by small actuators torques near
similarities. Therefore, the configurations of the robot near singularity have
also the large mechanical advantage.
Literature Review
of Robot Kinematics
As
described by Liu & Brown (2006) that they did propose the fuzzy qualitative
trigonometric extensiont that takes the derivate of to FQT (fuzzy qualaitative
trignometry). First of all, the researchers of this research study visit and
observe the fuzzy qualitative trigonometry critically then they they were able
to derive the extension derivatives. Further on the derivation of the extension
derivative, the role of trigonometry was replaced in the robotic kinematic by
using the fuzzy qualitativ trigonometry as well as the proposed extension tha
is leading to the robotic kinematics of the general version. Furthermore, the
fuzzy qualitative transformation discusses as well as derives the velocity and
the position or location of the robot. Finally, the effect of the derived
methodology to the robotics especially the problems of minimizing the gaps
among the symbolic cognitive functions as well as the control tasks and the low
level sensing are also highlighted in this research which is the important open
issues for the artificially intelligent report. To support the theoretical
improvement, the simulation results are also provided in this research
As stated
by Liu, Brown, & Coghill, Fuzzy Qualitative Robot Kinematics (2008), the
fuzzy qualitative version of the robot kinematics is proposed with goal to make
bridge on the gap among the control tasks, numerical sensing as well as the
qualitative functions or the symbolic functions for the intelligent robotics. The
researchers of this study revisited the fuzzy qualitative trigonometry in the
very first phase and then they focus to derive the results, so the derivative extension
of the fuzzy qualitative trigonometry was derived. In the next step, the trigonometry
role is replaced by using the proposed derivation extension as well as the fuzzy
qualitative trigonometry in the kinematics that leads to the robotic
kinematic’s fuzzy qualitative version. They also discussed and derived the
velocity, location, as well as the transmission of FQ of the serial robot kinematics.
In the last, an aggregation operator was also proposed to extract the behaviors
of the robot with the effect of proposed method highlight to the intelligent
robots. In the XTRIG MATLAB toolbox, it has integrated the proposed methods as
well as the case study on robots of PUMA has also been implemented to validate their
effectiveness
Dixon
(2007) described that the robot kinematics are the common assumptions in the
controller of previous robots as well as the Jacobean manipulator are also
perfectly known. Furthermore, the actuators of robot have also the capability to
generate the essential input torques level. With the uncertainty in the
kinematic as well as dynamic models, it develops the controller for the torque
input amplitude limited in this study for the robot manipulators revolution. Although,
the task space setpoint error’s semiglobal asymptotic regulation is yielded by
the adaptive controller. The ability to actively and effectively compensate for
unknow effects parametrically into the kinematic and the dynamic model included
by the advantages of proposed controller. Furthermore, the ability to making
sure the constraints of actuators are not ruptured through computing the needed
maximum torque
Kim & Kumar (1990) had also described that the robot kinematics
is the widely reached subject as well as several kinds of technique that are
well developed to analyze the spatial mechanism as well as the robotic
manipulator. Several and various technique as well as methodologies are
completely based on the transformations point. Moreover, far less research is
in proof for line changes. Lines and screws are progressively major to speed
examination and thus line changes are accepted to be more qualified for the
kinematic and static investigation of controllers. In this article, the
kinematics for sequential chain controllers utilizing line changes is
researched. Double number quaternions are utilized as spatial change
administrators since they permit minimal portrayal of the two revolutions and
interpretations. Although, they also described the inverse kinematics equations
as well as the computation of the forward kinematics equations by using the number
of quaternions dually
Conclusion of
Robot Kinematics
It is
concluded that the robot kinematics is dealing with the movements of the robot
and telling that how the trigonometric function will be included as well as are
involved in the robotic. The Robot Kinematics applies the geometry as well as trigonometric
functions due to the multi degree movements of freedom kinematic chains to
develop a robot. The kinematic equations are the basic conceptual technique and
significant tool into the robotic kinematics that refers to or showing the link
with the kinematic chains. The kinematic equation has further two forms which
are: forward kinematics and the inverse kinematics, are also discussed briefly in
the use of kinematics equation section. The kinematic equations, which are used
in the robotic kinematics, and the dimensions of the robot are used to specify
the space volume of the robot. The direct substitution of the joint parameters
in the equations of the forward kinematics achieve the serial manipulators for
the seral chain.
The joint angle values which given
into the diagram showing the robotic arm movement. The end effector location of
the robot is calculated and computed by the forwards kinematics equation into
the coordinate space. The end effector location is specified by the inverse
kinematics as well as the inverse kinematics has also the capability to compute
the significant linked joint angles. Two links of robotic arm along with the
anticipated end effector location as well as the angels which are: . The Jacobean of robot is
produced by the time derivation of the kinematics equation that associates with
the rates of the joint parameters of the robot to the angular velocity. The
results of the Jacobean method of robot is the linear equation set that
associated with the rates of joint parameters of the robot to the six vectors. Therefore,
the wrench of end effector which is associated with the given joint torques
set, are sought by the inverse static problem, as well as the inverse of the
Jacobean matrix is also required.
References of Robot Kinematics
Dixon, W. E. (2007). Adaptive Regulation of Amplitude
Limited Robot Manipulators With Uncertain Kinematics and Dynamics. IEEE Transactions on Automatic Control,
488-493.
Kim, J.‐H., & Kumar, V. R. (1990). Kinematics of
robot manipulators via line transformations. Journal of Robotic Systems.
doi:https://doi.org/10.1002/rob.4620070408
Liu, H., & Brown, D. J. (2006). An Extension to
Fuzzy Qualitative Trigonometry and Its Application to Robot Kinematics. In 2006 IEEE International Conference on
Fuzzy Systems, 1111-1118.
Liu, H., Brown, D. J., & Coghill, G. M. (2008).
Fuzzy Qualitative Robot Kinematics. IEEE
Transactions on Fuzzy Systems, 808-822.
Liu, H., Coghill, G. M., & Barnes, D. P. (2009). Fuzzy qualitative trigonometry. International Journal of Approximate Reasoning, 71-88.
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