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Industrial Automation, Communication and Data Management Prof. Cesana, Rocco, Tanca January xx, 2020 SOLUTION Industrial Automation, Communication and Data Management Prof. Matteo Cesana, Paolo Rocco and Letizia Tanca Consider the robot carrying a glass with liquid sketched in the picture: The robot should be moved in such a way to avoid spilling of the liquid from the container. In a first approximation, the sloshing dynamics of the liquid can be modelled with a pendulum, see the following picture: For simplicity, assume that the pendulum moves on a vertical plane. 1. Explain what are the direct and the inverse kinematics problems for a robotic manipulator. How many solutions does the inverse kinematics for an anthropomorphic manipulator have? The direct kinematics is the computation of the position and orientation of the end effector of the robot given the positions of its joints. Conversely, the inverse kinematics means to compute the joint positions given position and orientation of the end effector. For an anthropomorphic manipulator there are in general eight solutions to the inverse kinematics problem. 2. Write the equations of a dynamic system that describes the motion of the pendulum (when the robot is still). Since no torque is applied to the pendulum, the model of the system can be written as: ML 2 ¨α +MgL sinα = 0 Defining as x1 (first state variable) the angle α, x2 (second state variable) the angular velocity ˙α, the state equations can be written as: ˙x1 = x2 ˙x2 = −g L sin (x1) 3. Write the general equation that allows to find the equilibrium states in a dynamic system. Apply such formula to find the equilibrium states for the pendulum at hand. Given a dynamic system described by the state equations: ˙ x= f(x, u) assuming a constant input u = ¯ u, vectors ¯…