Problem 1 (14 points). Figure 1 shows a robotic system in the plane (convoy system), comprising...

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Problem 1 (14 points). Figure 1 shows a robotic system in the plane (convoy system), comprising a mobile vehicle Bo pulling two carts, B and B2. Each body in the system is Y B Yo a 00 Y1 S Y2 100 x1 To B 100 80 X2 02 0 Figure 1: A convoy lead by a planar mobile robot equipped with a set of wheels, where the coordinates of the midpoint of the wheel's axis with respect to a fixed reference frame are denoted by (xi, yi) R, i = 0, 1, 2. The orientation of each body B is specified in terms of the angle 0; between the main axis of the body, and the x axis of the fixed fame, as shown in Figure 1. The convoy structure requires that Bi, i = 1,2, is rigidly attached to Bi-1 via a link of length 1 connected to a revolute joint in Bi-1. Ignoring the wheels (that is, assuming that each vehicle is represented by its chassis only): 1. Determine the number of degrees of freedom of the convoy system. 2. Determine the configuration space of the convoy system using an explicit representa- tion. 3. Determine the configuration space of the convoy system using the implicit represen- tation. Each pair of wheels introduces a constraint on the velocity, namely the component of the velocity of each body along the wheel's axis is zero (equivalently, the body's velocity is directed along the main axis of the vehicle). 4. Write the Pfaffian constraints that correspond to the constraint on the velocities imposed by the wheels. 5. Determine whether the Pfaffian constraints are holonomic or non-holonomic. Justify your answer analytically. Attach a body-fixed frame {i} = {i, i} to each vehicle Bi, i = 0, 1, 2, as seen in Figure 2. Y Yo Y S Y $ Bo To Xo 0 92 Figure 2: Body-fixed coordinate frames 0 X 6. Determine the rotation matrices R01 SO(2) and R02 SO(2) describing respec- tively the orientation of B and B with respect to Bo as a function of the configuration variables of the system. Problem 1 (14 points). Figure 1 shows a robotic system in the plane (convoy system), comprising a mobile vehicle Bo pulling two carts, B and B2. Each body in the system is Y B Yo a 00 Y1 S Y2 100 x1 To B 100 80 X2 02 0 Figure 1: A convoy lead by a planar mobile robot equipped with a set of wheels, where the coordinates of the midpoint of the wheel's axis with respect to a fixed reference frame are denoted by (xi, yi) R, i = 0, 1, 2. The orientation of each body B is specified in terms of the angle 0; between the main axis of the body, and the x axis of the fixed fame, as shown in Figure 1. The convoy structure requires that Bi, i = 1,2, is rigidly attached to Bi-1 via a link of length 1 connected to a revolute joint in Bi-1. Ignoring the wheels (that is, assuming that each vehicle is represented by its chassis only): 1. Determine the number of degrees of freedom of the convoy system. 2. Determine the configuration space of the convoy system using an explicit representa- tion. 3. Determine the configuration space of the convoy system using the implicit represen- tation. Each pair of wheels introduces a constraint on the velocity, namely the component of the velocity of each body along the wheel's axis is zero (equivalently, the body's velocity is directed along the main axis of the vehicle). 4. Write the Pfaffian constraints that correspond to the constraint on the velocities imposed by the wheels. 5. Determine whether the Pfaffian constraints are holonomic or non-holonomic. Justify your answer analytically. Attach a body-fixed frame {i} = {i, i} to each vehicle Bi, i = 0, 1, 2, as seen in Figure 2. Y Yo Y S Y $ Bo To Xo 0 92 Figure 2: Body-fixed coordinate frames 0 X 6. Determine the rotation matrices R01 SO(2) and R02 SO(2) describing respec- tively the orientation of B and B with respect to Bo as a function of the configuration variables of the system.