Problem 5:Consider the motion of a thin flat disk rolling on a plane shown in Figure: Problem 3. Theconfiguration space of the system is parameterized by the xy location of the contact point of the...

robotics


Problem 5: Consider the motion of a thin flat disk rolling on a plane shown in Figure: Problem 3. The configuration space of the system is parameterized by the xy location of the contact point of the disk with the plane, the angle 0 that the disk makes with the horizontal line, and the angle ¢ of a fixed line on the disk with respect to the vertical axis. Figure: Problem 3: Disc Rolling on a Plane ‘We assume that the disk rolls without slipping. Writing these equations with q = (6, x, y, ©) we have the corresponding kinematic equations of motion z— pcosf =0 §— psinfé =0, where, p > 0 is the radius of the disk. ---------Continued on next page Page 1 of 2 ‘Writing the above equations in the corresponding control system format as 0 ree ba]. ul] _ _ a0=|3|=lo1 921] = 6@u=g@u + gu: ¢ : Here, g1 and g; are 4x1 column vectors; control inputs u; = ¢ = the rate of rolling, and uz = 6 = rate of turning about the vertical axis. 1) Find the column vectors gi and ga. Using Lie algebra, 2) Evaluate g3 = [g1, g2] and ga = [g2, g3]. 3) Check if the complete span {g1, g2, g3, g4} is of full rank for the controllability of the system. Problem1: For trajectory generation of an industrial robot, evaluate the coefficients of the fifth-order polynomial s(t) = ast’ + ast* + ast’ + at? + ast + ao with time scaling that satisfies s(T) = 1 and s(0) = $(0) =5(0) = $(T) =8(T) = 0, where the initial time t = 0 and the final time T =1. Problem 2: For a one degree-of-freedom (DOF) robotic system of the form mi + bx + kx =f, where fis the control force and m = 4 kg, b=2 Ns/m, and £ = 0.1 N/m, obtain a proportional — derivative (PD) controller that yields critical damping and a 2% settling time of 0.01 sec.