Minor Project - Human Interaction Robot Arm Master in Control Systems Engineering Arnhem, 1 July 2015 Presenters: Okubanjo A. Ayodeji [ 544187 ] Gumbie Tawanda [ 542702 ] Prakash Parth [ 557179 ]

Outlines Control Goal Mathematical models The Robot Simulations Conclusions

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.

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.

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

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

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

Standard Dynamics Equation Form Dynamics Equation 2 DOF

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

Robot Arm 2 DOF in Joint Space and Simulations

Angular Representation of Joint Space Y 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’

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 2 : 30 degree Desired angle_ Theta3 (Blue) :180 degree Initial condition Theta 3 : 90 degree

Robot arm 2 DOF in Cartesian Coordinate and Simulations

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

Cartesian Space Coordinates Angle Responses 2DOF response of joint angle Theta2 & Theta3. Desired position of End Effector : Px = 0.68 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

Cartesian Space Coordinates Position Responses 2DOF response of joint angle Theta2 & Theta3. Desired position of End Effector : Px (Blue) = 0.68 Py (red) = 0 Initial Coordiates of End Effector : Px (blue) = 0.34 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) = 0.24 Py (red) = 0.57

Effect of External Force on the Joint Angles TG start:

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

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

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.

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