1. Minor Project - Human Interaction Robot Arm
Master in Control Systems Engineering
Arnhem, 1 July 2015
Presenters: Okubanjo A. Ayodeji [ ]
Gumbie Tawanda [ ]
Prakash Parth [ ]

2. Outlines Control Goal Mathematical models The Robot Simulations
Conclusions

3. Control Goals
To determine a robot model joints angles given a specific position
By deriving joints angles from both kinematics and dynamics equation
Starting from 1 DOF to 2 DOF and extend the theory to 3DOF
Stabilizing the joints angle by designing a PID Controller.
Simulating the model and compare with mathematical result for validation.
Observing the model in the presence of obstacles.

4. Introduction
Robot arm dynamics deals with the mathematical formulation of the robot arm motion.
They are useful as :
An insight into the structure of the robot system.
A background for model based control systems
A background for computer simulations.
They also relate the motion of manipulator arises from torque applied to the actuators or from external forces applied to the manipulator.

5. Mathematical models
We developed mathematical equations to determine the forward and inverse kinematics for 1DOF and 2DOF.
Forward kinematics: mapping from joint variables to position and orientation of the end effector.
Inverse Kinematics: finding joints variables that satisfy a given position and orientation of the end effector

6. Mathematical models Forward Kinematic equation 2 DOF
Jacobain equation 2 DOF
Inverse Jacobain equation 2 DOF
Inverse Kinematic equation 2 DOF

7. Mathematical models-Dynamic Equation
This deals with time rate of change of robot configuration in relation to the joint torques exerted by the actuator.
Taking into account the:
effect of joints linkage
Inertial forces
Coriolis and Centripetal forces
Gravitational forces and Frictional forces
Moment of Inertial

8. Standard Dynamics Equation Form
Dynamics Equation 2 DOF

9. Robot arm 1 DOF and Simulations
1 DOF response with Load of 1Kg and Torque of 1Nm after 5 secs
1 DOF response with zero degree initial condition and load of 1 Kg without Torque
1 DOF response for desired angles pi/6 with Load [1Kg]
1 DOF response for desired angles pi/6 without Load

10. Robot Arm 2 DOF in Joint Space and Simulations

11. Angular Representation of Joint SpaceY
Th3’ = 180 Y
Th3’ = 180
Th3 = 90
Th2 = 180
Th2’ = 90
Th2’ = 60 -X X -X
Th2 = 30 X
Th3 = 90 -Y -Y
Case 1: Initial Condition : Th2 and Th3
Desired Condition : Th2’ and Th3’
Case 2:
Initial Condition : Th2 and Th3 Desired Condition : Th2’ and Th3’

12. Joint Space Coordinates responses
2DOF response of joint angle Theta2 & Theta3.
Desired angle - Theta2 (Pink) : 90 degree
Initial condition Theta 2(Pink) : 180 degree
Desired angle_Theta3 (Blue) : 180 degree
Initial condition Theta 3(Blue) : 90 degree
2DOF response of joint angle Theta2 & Theta3.
Desired angle – Theta 2 (Pink) : 60 degree
Initial condition Theta : 30 degree
Desired angle_ Theta3 (Blue) :180 degree
Initial condition Theta : 90 degree

13. Robot arm 2 DOF in Cartesian Coordinate and Simulations

14. Coordinate Representation of End Effector
( 0, 0.68 ) Y Y
Th3’ = 0
Th3 = 45
Th3 = 90
Th2’ = 90
Th2 = 45
Th2 = 0
(0.68,0)
Th3’ = 0 -X
Th2’ = 0 X -X X -Y -Y

15. Cartesian Space Coordinates Angle Responses
2DOF response of joint angle Theta2 & Theta3.
Desired position of End Effector : Px = Py = 0
Initial Arm2 Angle (Th2) (Blue) : 0 degree
Initial Arm3 Angle ( Th3) (red) : 90 degree
2DOF response of joint angle Theta2 & Theta3.
Desired position of End Effector :
Px = 0 Py = 0.68
Initial Arm 2 Angle(Th2) (Blue) : 45 degree
Initial Arm 3 Angle (Th3) (red) : 45 degree

16. Cartesian Space Coordinates Position Responses
2DOF response of joint angle Theta2 & Theta3.
Desired position of End Effector : Px (Blue) = Py (red) = 0
Initial Coordiates of End Effector :
Px (blue) = Py (red) = 0.34
2DOF response of joint angle Theta2 & Theta3.
Desired position of End Effector : Px (blue) = 0 Py (red) = 0.68
Initial Coordiates of End Effector :
Px (blue) = Py (red) = 0.57

17. Effect of External Force on the Joint Angles
TG start:

18. External Force and effect on Desired Position and Joint Angles
Theta 3 more affected(R) and theta 2 (B)
They are coupled
2DOF response of joint angle Theta2 & Theta3.
With External Force as input Disturbance
2DOF response of positions
With External Force as input Disturbance

19. Dynamic Decoupling Block
Concept for 3 - DOF Model
INV Kinematics
ROBOT 3 DOF Px
Theta 1 C1
Dynamic Decoupling Block
Theta 2 E C2
INVERSE Kinematics Py
The rotary part is neglected. Pz C3 E
Theta 3
C1 C2 and C3 = Controllers
E = Error on Angles

20. CONCLUSION and DEDUCTION
The 1 DOF and the 2 DOF in Joint Space works as expected.
Saturation needs to be considered on the angles the Arms can flex.
The Cartesian Coordinate implementation on 2-DOF gives a better control of a Robot Arm as one can just provide the position of the End - Effector in XY Coordinate values.
The angles or the Positions needs to be reasonably provided as the desired values.
Although PID Controller gives a satisfactory results, other controllers like a fuzzy based controller can be tested in future with 3 – DOF.

21. THANK YOU

22. REFERENCES :