The inverse kinematic problem at the position level for a th

The inverse kinematic problem at the position level for a three-link robot is to find the joint angles theta_1, theta_2 and theta_3 that allow the end-effector (P) at the thud link to be at a specified point characterized by the coordinates(x,y,phi), where (x,y) is Cartesian position and phi the angle of the end-effector axis with respect to the x-axis In this example, we assume that the link lengths are a_1 = 6, a_2 = 5 and a_3 = 4 (cm) The first link makes an angle theta_2 with the horizontal axis, the second link makes an angle theta_2 with the direction defined by the first link, and the thud link makes an angle theta_2, with the direction defined by the second link as depicted in Fig. 1. By following the schematic representation illustrated in Fig. 1. give the expression of the coordinates(x,y,phi) as a function of the joint angles theta_1, theta_2 and theta_3. Determine the Jacobian of the system of three equations found in (1). Use the Newton-Raphson formula in vector form to find the values of the requested joint angles theta_1, theta_2 and theta_3, for the case of the arm completely extended, with initial joint angles of theta_1 = 0.5, theta_2 = 0.7 and theta_3 = 0.6 (rad) Perform up to three iterations. Under which the conditions the Jacobian of the robot is singular Give a physical interpretation of the singularity conditions of the robot. We wish to find the angles so that the arm will move to the position(x,y) = (14,4) (cm) with an orientation phi = 0.4 (rad). while starting with initial joint angles of theta_1 = 0.5, theta_2 = 0.7 and theta_3 = 0.6 (rad). Use the Newton-Raphson formula to find the values of the requested joint angles theta_1, theta_2 and theta_3. To proceed, write a subprogram implementing the Newton-Raphson algorithm. Perform up to six iterations or until obtaining an estimated error delta = squareroot |deta theta_1|^2 + |delta theta_2|^2 + |delta theta_3|^2 less than 0.001.

Solution

Our customer required a means of controlling and monitoring flow of sea water from a remote location. We had very specific details of how the valve should integrate with existing pipework and we provided advice with regards to a suitable actuated valve, flow meter and digital receiver.

All pipework was fabricated from stainless steel. The ball valve is operated by a Comparato Compact Pro electric actuator which features an IP67 enclosure, anti-condensation heater, manual override and position confirmation switches for feedback to the remote PLC. A Flow X3 paddlewheel flow meter / transmitter provides a 4-20mA signal to the Flow X3 digital receiver. Flow rate and batch flow can be read from the LCD display and fully integrated with the existing PLC.