Geometrical Significance Of Trajectory Planning Through Polynomial Equation – Customized RPPPRR Model Robot

1. DEPARTMENT OF MECHANICAL
ENGINEERING
GIET UNIVERSITY
GUNUPUR
PRESENTATION MADE BY:-
G.AVINASH SHARMA
TEAM MEMBERS:-
 Kali Charan Rath
 Supriya Sahu
 Anshuman Nayak
 Suvikram Pradhan
 G.Avinash Sharma
 Patro Pankajkumar Munibara
INTERNATIONAL CONFERENCE ON ADVANCES IN
SIGNAL PROCESSING COMMUNICATIONS AND
COMPUTATIONAL INTELIGENCE(ASCCI 2K21)
CMR TECHNICAL CAMPUS

2. Geometrical
Significance Of
Trajectory
Planning
Through
Polynomial
Equation –
Customized
RPPPRR Model
Robot

3. Robotics
Robotics, design, construction, and use of machines (robots) to perform tasks done traditionally
by human beings. Robots are widely used in such industries as automobile manufacture to
perform simple repetitive tasks, and in industries where work must be performed
in environments hazardous to humans. Many aspects of robotics involve artificial intelligence;
robots may be equipped with the equivalent of human senses such as vision, touch, and the
ability to sense temperature. Some are even capable of simple decision making, and current
robotics research is geared toward devising robots with a degree of self-sufficiency that will
permit mobility and decision-making in an unstructured environment. Today’s industrial robots
do not resemble human beings; a robot in human form is called an android.
Application
 Collaborative Robots
 Robotic Painting
 Robotic Welding
Robotic Assembly
Part Transfer and Machine Tending
Material Removal

4. INTRODUCTION
 ISO 8373:2012 defines an business robot as follows: for use in commercial automation applications,
reprogrammable, repeatedly controlled, flexible manipulator with three or more axes that may be fixed in place
or cell.
 The phrases used inside the definition above are explained in more element under: Reprogrammable: intended
to allow the planned motions or secondary functions to be personalized without the need for any physical
modifications; Multipurpose: with physical modification, it can be converted to a different application purpose
Physical alteration: The physical system is modified, i.e. the mechanical device does not encompass storage
media, ROMs, etc.. Axis: An axis is a course that specify how a robotic moves in a linear or rotational mode
 Several approaches for generating trajectory have been investigated during the previous few decades. Many
authors [1–4] propose the same strategy to generate joint-space line or trajectory. Some papers [5–7] described
a different polynomial joint-space trajectory generation strategy . Researchers [8-10 ] developed advanced cubic
method to interrupt joint positions with related velocities The topic of optimizing a trajectory going through
given waypoints while adhering to kinematic restrictions was investigated [11-13]. Robotics has been employed
within a few enterprises for decades, but the blessings of robot automation have finally unfolded to different
industries. The "three D's" rule applies to figuring out which commercial jobs are maximum desirable for robots:
any tasks that is grimy (Dirty) , Dull, or risky ( Dangerous / Dicey) . Straightforward and cyclic tasks that insist
constant resources are typical robotic applications. The most common industrial robots are articulated robots.
Because they resemble a human arm, they are sometimes known as robotic arms or manipulator arms. The
articulated arms may additionally flow in an extensive variety of guidelines thanks to their more than one stage
of freedom articulations. Cartesian, SCARA, Cylindrical, Delta, Polar, and Vertically articulated are the six main
types of industrial robots. There are, but, numerous different styles of robot configurations. Every of these
varieties have a completely unique joint association. Axes are the joints that make up the arm.

5. Course development algorithms construct geometric path from starting point to the
target point, passing through intermediate points, either in the joint region or within the
robotics’ operational space, unlike curve planning, that uses an existing geometric path
and add temporal statistics to it, trajectory planning methods employ an existing
geometric path.
Inside the discipline of robotics and, more widely, within the field of automation, path
planning and trajectory planning are critical worries.
Indeed, in order to attain shorter production periods, robots and autonomous equipment
are increasingly operating at rapid speeds. Due to the fact, first-rate performances are
demanded from the actuators and manipulate the device; the high operating speed may
additionally hinder the precision and repeatability of the robotic movements. As a result,
creating a trajectory should be approached with caution.
Methodology for path planning assemble a numerical direction from an preliminary to
last target, passing via pre-described through-factors, both inside the joint space or within
the robot's operational area, while developing plans for a trajectory, algorithms consider
the existing geometric direction. and upload temporal statistics to it. In Robotics,
trajectory planning algorithms are vital due to the fact placing the instances of passing at
the via-points affects each the kinematic and dynamic components of the motion.
The approaches used to construct the geometric direction are often classified into three
categories: roadmap techniques, cell breakdown methods, and synthetic capability
techniques. The characteristics that are maximized are frequently termed after the
algorithms for trajectory planning, such as minimal time, minimal energy, and minimal
jerk. Polynomials are the choice for smooth, continuous motion with continuous
derivatives. Through intermediate point(s), the robot must have to travel among the start
and ending point in diverse robotics movement making plans challenges. A trajectory is
ROBOT TRAJECTORY

6. Algorithm for robot trajectory in Cartesian space
● This research questions the widely held belief in robotics that
knowing a robot's kinematic information, such as the array of
links and joints, link size, and joint geometry, is required to
operate it.
● The ideas of kinematics and dynamics of expressed rigid
bodies form the underpinnings of modern robotics. Maximum
robotics textbook starts off evolved with an outline of robotic
design the use of joint angles, and then actions on to
kinematics, dynamics, and manage ( control ).
● The implicit assumption that knowing a robot's kinematic
information, such as the display of links and joints, link scope,
and joint geometry, are required to control it is a major
consequence of this. Through joint angles robot can be
controlled by keeping link dimensions as constant. A linkage is
a mechanical device made up of stiff structures known as links
and pin joints or sliding joints that connect them. DOF means
the number of independent variables that are required to fully
characterize the mechanical Robot configuration.

7. Step 1 : Path calculation from starting point to the target point
Step 2 : Assign a total time T(path) to pass through the path
Step 3 : Discretize the points in time and space
Step 4 : Between these points, combine a continuous time function
Step 5 : Solve the simulation model by inverse kinematics
ALGORITH FOR TRAJECTORY IN THE
OPERATIONAL SPACE
ALGORITHM FOR TRAJECTORY IN THE
JOINT SPACE
Step 1 : Calculate inverse kinematics answer from preliminary point to the very
last factor.
Step 2 : Allocate overall time T(path) using maximal joint velocities.
Step 3 : Discretize the individual joint trajectories in respect to time.
Step 4 : Merge a continuous function between these point.

8. Quinticpolynomialtrajectory
The Cubic Polynomial Trajectory method cannot specify accelerations at each point, hence acceleration will be discontinuous at each
point for a set of points. Because of the discontinuity in acceleration, the derivative of acceleration (jerk) at each through point is
infinite, as a result, the robot's movements takes an rapid jerk.
To prevent this, three restrictions must be given at each point: position, velocity, and acceleration. A fifth order polynomial fits the trajectory between two intermediate points .
p(𝑡0) = 𝑎0 + 𝑎1𝑡0 + 𝑎2𝑡0
2 + 𝑎3𝑡0
3 + 𝑎4𝑡0
4 + 𝑎5𝑡0
5 ------------------ (8)
𝑝 𝑡0 = 𝑣 𝑡0 = 𝑎1 + 2 𝑎2𝑡0 + 3𝑎3𝑡0
2 + 4𝑎4𝑡0
3 + 5𝑎5𝑡0
4 ------------------- (9)
𝑝 𝑡0 = 𝑎 𝑡0 = 2 𝑎2 + 6𝑎3𝑡0 + 12𝑎4𝑡0
2 + 20𝑡0
3 ------------------ (10)
p(𝑡𝑓) = 𝑎0 + 𝑎1𝑡𝑓 + 𝑎2𝑡𝑓
2 + 𝑎3𝑡𝑓
3 + 𝑎4𝑡𝑓
4 + 𝑎5𝑡𝑓
5 ------------------ (11)
𝑝 𝑡𝑓 = 𝑣 𝑡𝑓 = 𝑎1 + 2 𝑎2𝑡𝑓 + 3𝑎3𝑡𝑓
2 + 4𝑎4𝑡𝑓
3 + 5𝑎5𝑡𝑓
4 ------------------- (12)
𝑝 𝑡𝑓 = 𝑎 𝑡𝑓 = 2 𝑎2 + 6𝑎3𝑡𝑓 + 12𝑎4𝑡𝑓
2 + 20𝑡𝑓
3 ------------------ (13)
1 𝑡0 𝑡0
2 𝑡0
3
𝑡0
4 𝑡0
5
0 1 2𝑡0 3𝑡0
2
4𝑡0
3 5𝑡0
4
0 0 2 6𝑡0 12𝑡0
2
20 𝑡0
3
1 𝑡𝑓 𝑡𝑓
2 𝑡𝑓
3
𝑡𝑓
4 𝑡𝑓
5
0 1 2𝑡𝑓 3𝑡𝑓
2
4𝑡𝑓
3 5𝑡𝑓
4
0 0 2 6𝑡𝑓 12𝑡𝑓
2
20 𝑡𝑓
3
𝑎0
𝑎1
𝑎2
𝑎3
𝑎4
𝑎5
=
𝑝0
𝑝0
𝑝0
𝑝𝑓
𝑝𝑓
𝑝𝑓

9. Cubicpolynomialtrajectory
Cubic Polynomial Trajectory mathematical model for the path between two points p(𝑡0) and p(𝑡𝑓) expressed as :
p(t) = 𝑎0 + 𝑎1𝑡 + 𝑎2𝑡2 + 𝑎3𝑡3 ------------------ (1)
𝑝 𝑡 = 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝑣 𝑡 = 𝑎1 + 2 𝑎2𝑡 + 3𝑎3𝑡2 ------------------ (2)
𝑝 𝑡 = 𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑎 𝑡 = 2 𝑎2 + 6𝑎3𝑡 ------------------- (3)
The velocity and acceleration are determined by the first and second derivatives of equation-1 respectively.
Equation-3 indicates that acceleration is continuous and linearly varies with time; as a result, infinite accelerations are not required for the trajectory. To answer the unknowns, four
constraints must be specified: a0, a1, a2, and a3. The start and finish coordinates are obviously two limitations, and the final two velocities are the starting and end velocities.
p(𝑡0) = 𝑎0 + 𝑎1𝑡0 + 𝑎2𝑡0
2 + 𝑎3𝑡0
3 ------------------- (4)
𝑝 𝑡0 = 𝑣 𝑡0 = 𝑎1 + 2 𝑎2𝑡0 + 3𝑎3𝑡0
2 ------------------- (5)
p(𝑡𝑓) = 𝑎0 + 𝑎1𝑡𝑓 + 𝑎2𝑡𝑓
2 + 𝑎3𝑡𝑓
3 ------------------- (6)
𝑝 𝑡𝑓 = 𝑣 𝑡𝑓 = 𝑎1 + 2 𝑎2𝑡𝑓 + 3𝑎3𝑡𝑓
2 ------------------- (7)
Matrix representation of the above equation can be written as :
1 𝑡0 𝑡0
2 𝑡0
3
0 1 2𝑡0 3𝑡0
2
1 𝑡𝑓 𝑡𝑓
2 𝑡𝑓
3
0 1 2𝑡𝑓 3𝑡𝑓
2
𝑎0
𝑎1
𝑎2
𝑎3
=
𝑝0
𝑣0
𝑝𝑓
𝑣𝑓

10. Analysis has done by keeping in mind, base
frame to end effector.
Cubic trajectory execution by customized robot model
Isometric view Right side view
Front side view

11. EE configuration
matrix
FIG:-X- value for Link 1,2,3,4,5 & 6 FIG:-Y- value for Link 1,2,3,4,5 & 6
FIG:-Z- value for Link 1,2,3,4,5 & 6 FIG:-Joint-1: Joint value and joint velocity

12. Fig:-Joint-1: relationship between- Joint value, velocity, acceleration &
force / Torque

13. CONCLUSION
The cubic polynomial interpolation and quintic polynomial interpolation
methods are used to plan the motion of the manipulator without load from the
start point to the goal point, respectively. The angle, velocity and acceleration
curve is represented by the concerned graph. The start point and target point
of both approach are zero and the acceleration of the start point and target
points is zero too
Kinematics study of a 6-DOF customized robot model is examined in this
work. The use of cubic and quintic polynomials in the trajectory planning of 6-
DOF robot model is highlighted. The required matching curves were validated
by simulation of the robot model. The quintic polynomial method has a more
visible effect on the trajectory planning of the 6-DOF robotic arm, according to
the results.
When compared to the cubic polynomial, the quintic polynomial produces a
smooth and continuous trajectory, demonstrating that this method is feasible
for trajectory planning of similar 6-DOF robotic arms. When compared to the
cubic polynomial, the quintic polynomial produces a smooth and continuous
trajectory, demonstrating that this method is feasible for trajectory planning
of similar 6-DOF robotic arms.

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