We, the undersigned, having checked the above-mentioned thesis and the student's official record book(s), declare our approval of the thesis submitted in partial fulfillment of the requirements for the degree of Master of Technology in Industrial Designation National Institute of Technology Rourkela. This is to prove that the work presented in the thesis titled Off-Line Optimal Trajectory Planning of a 6 DOF Robotic Manipulator Using Multiple Objective Functions with Kinematic Constraints submitted by Amit Bhowmick, Roll Number 215ID1172, is a research record our original. supervision and guidance in partial fulfillment of the Master of Technology in Industrial Design degree requirements. Offline trajectory planning means planning the robot's motion before implementing it in the real world.
During operation, when a shock occurs, the robot deflects the end effector from its predetermined path. To reduce jerking, it is important to ensure continuity of speed and acceleration at each of the selected points.
Nomenclature
Introduction
- Background
- Objectives
- Motivation
- Outline of the thesis
Path planning refers to the generation of the geometric path based on Cartesian space or joint space coordinates without considering the kinematic or dynamic parameters. In other words, here the main objective is to optimize the total travel time of the robot. To obtain the feasible optimal trajectory, it is also important to consider the physical limitations of the robot.
Here are the specific objectives: (i) Formation of the optimal trajectory considering various parameters responsible for producing jerks; (ii) development of the optimal algorithm; and (iii) experimentally validating that the robot's performance can be improved using the development method. In short, it can be said that motion planning of the robot is important for the performance of the robot to increase production with high accuracy.
Literature Survey
Off-line trajectory planning
- Total operation time minimization problem
- Energy optimization problem
- Jerk optimization problem
Optimization problems based on time minimization are important for industrial robots because they increase productivity by increasing the speed of the robot. Here, a kinematic approach is used and a complete minimization problem is derived based on the physical constraints of the robot manipulator. It was found that although the time optimization technique produces a faster path, it reduces the smoothness of the path.
Actuator torque is assumed as objective function and it is assumed that energy during movement of the robot. As stated in the previous sections, offline trajectory planning involves planning the end-effector trajectory before deploying the robot to the real world.
On-line trajectory planning
Authors describe in [51] that jerk optimization trajectory planning is similar to the human arm movements. Here interval analysis based min-max optimization technique is used to solve the optimization problem. In another literature [52], authors described the correlation between jerk-minimized trajectory and human arm movement.
Another type of path planning is online path planning, where planning is done to track the path followed by the robot.
Problem Description
Methodology
Terminologies
- Robot manipulator
- Wrist of the robot
- End-effector
Like the human body, they are used as muscles to help the robot change its configuration.
Spline interpolation
Description of the Robot
- Robot kinematics
- Estimation of joint displacements
- ΞΈ ) (rad) 1
- ΞΈ ) (rad) 1
- ΞΈ ) (rad) 1
- ΞΈ ) (rad) 1
- ΞΈ ) (rad) 1
- ΞΈ ) (rad) 1
To optimize the entire problem, it is important to define the maximum values ββof the kinematic constraints. Using them, the origin oi of the frame Bi can be moved to the frame Biο«1. It is also the distance between the xiο1 and xi axes. e) ο±i is the joint angle and it is measured along the ziο1 axis.
The second approach is to first determine the Cartesian coordinates of the position of the end effector on the desired path. First the user must define the transformation matrix in terms of the position and orientation of the end effect. Before performing the inverse kinematics problem, the user must define the T60 vector in terms of orientation and end effector positions.
Since the closure of the path is required, it makes sense to define an additional point which is exactly equal to the starting point. These are the coordinates of the selected points on the predefined path as mentioned in Figure-4. Estimating the end effect positions, they are plotted in MATLAB 2015a as shown in Figure-5 to verify that the points are selected correctly.
The next task is to use inverse kinematics to obtain joint displacements at selected points. Before using inverse kinematics, it is important to define the correct transformation matrix containing the orientation. Joint movements for 64 points obtained after using inverse kinematics are given in the following table.
Time Intervals and Kinematic Constraints Calculation
Time intervals calculation
As the end effector moves from point 1 to point 2, each of the six joints will simultaneously change its initial position. If hi ο½ο¨tiο«1οtiο© represents the time interval for moving from point i to point iο1 on a predetermined route, there will be a total of ο¨nο1ο© time intervals. Since the robot has a total of six joints, the time taken by the end effector to reach from point 1 to point 2 is equal to the maximum time taken by each joint to move from the initial position to the next predetermined position as indicated in table-4.5.
Where H is the set of initial time intervals, this can also be called as an upper bound.
Kinematic constraints calculation
- Velocity constraints
Now to obtain the expression for the velocity, it is required to differentiate equation (5.12) with respect to time and the expression comes as. For example, if point 1, point 2 and point 3 are considered on the road, the time instances at these three points are t1,. To ensure continuity, point 2 of the first polynomial must coincide with point 2 of the second polynomial.
Where A is the matrix of coefficients of the matrix Qο’ο’ and the matrix B contains the right-hand side of the above equations. Array A contains only time intervals, while array B contains values ββof positions, time intervals, initial and final values ββof velocity and acceleration. Therefore, the derivative of a second-order polynomial function, or acceleration, is a linear function of time.
Optimization Process
Optimization problem definition
Optimization procedure
Here, Kt and Kj are the weights of the total time and total square hook functions in the objective functions. In this step, the Taylor series is applied to all KKT terms to determine the relationship between . Now the second-order partial derivative of the Lagrange function with respect to the time interval h, also known as the Hessian matrix, is given by .
Now the other four derivatives of Lagrange function with respect to hand ο‘ are given as below. Now these equations can be interpreted as the exact KKT terms of a quadratic programming problem. If an incremental or adverse change, the time interval meets a predefined tolerance value, the solution is terminated and an optimal solution is obtained.
Define KKT conditions and apply Taylor series to them, which is equivalent to a quadratic programming problem (QP).
Simulation and Experimental Results
Results before optimization process
The initial time responses at each of the 64 points can be calculated from the table above.
Joint 2 Joint 3 Joint 4 Joint 5 Joint 6
- Results after optimization
- Discussion
- Comparison of results
This table shows the time intervals and time responses can be calculated from this table. Following tables show the calculated values ββof the velocities, accelerations and jerk values ββof all joints and at all points selected on the predetermined path. In table 7.2 and table 7.6 it can be easily seen that velocities at each point for each joint are increased after optimization.
The diagrams in Figure 7.1 through Figure 7.6 show the velocity, acceleration and shock value pattern of all joints after optimization. Each of the six diagrams shows that the acceleration is a linear function of time and that the jerk remains constant for each segment of the spline curve. It was previously mentioned that the main task here is to optimize the total operating time and the total shock value.
From figure 7.7 to figure 7.12 a clear comparison can be made for jerk before optimization and after optimization.
COMPERASION OF JERK VALUES FOR ALL JOINTS
Experimental results
In the experiment, it is found that the experimentally obtained path differs from the theoretically obtained path. The total travel time was also around 18 seconds, which is more than the theoretically optimal time obtained.
Conclusion and Future Scope
Future scope
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