In general, please report ALL the passages and not only the nal results.
1. [30 marks] The 2R planar manipulator’s initial and nal positions are shown in Figure 1. Two segments of the manipulator are identical aluminum hollow rods, l = 1 [m], m = 50 [kg]. Motors at both joints have the mass mM = 5 [kg] and the internal moment of inertia IM = 0.01 [kgm2].
Perform trajectory planning for both segments using the following methods
(a) linear segments with parabolic blends (nd angles θ1 and θ2 rst for the initial and nal position, and then plan the trajectory for both segments independently in the joint space) for two dierent movement strategies (calculate position, velocity and acceleration laws in joint space and plot them):
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i. Slow: The blending time is 0.6 [s], the maximum velocity has magnitude of 0.3 [rad/s]. ii. Fast: The blending time is 0.6 [s], the maximum velocity has magnitude of 1 [rad/s].
(b) 4-3-4 trajectories with starting/nal zero velocities and accelerations, positions at the waypoints equal to the initial and nal linear part of the parabolic blending (second trajectory of the parabolic blending). Moreover, t1 = 0.6 [s]. Plot the position, velocity, acceleration in the two cases
i. Slow: Final time equal to the nal time of the slow parabolic blending trajectory and t2 equal to the nal time of the linear portion of the parabolic trajectory.
ii. Fast: Final time equal to the nal time of the fast parabolic blending trajectory and t2 equal to the nal time of the linear portion of the parabolic trajectory.
Plot the position, velocity and acceleration for the two trajectories and perform a qualitatively comparison (on a representative case): what are the dierences among the two types of trajectories? What can be additionally impose with one trajectory that cannot be done with the other?
Bonus Does the robot encounter in singularities during the motion? Which are the singularities for this robot? Assume that the robot encounters in a singularity, is it possible to avoid them through the planning techniques mentioned above? How?
2. [12.5 marks] For the two cases above (of the parabolic blending), discuss the choice of motors for the two joints. What torques do they need to have (what inertia load each has to work with, what acceleration?) to perform these actions? Discuss the transmission (reduction) choices as well, assuming that you are using electric drives.
3. [12.5 marks] For the two cases above and the choice of motors, discuss the choice of sensors for the two joints. What resolution do you need for having a precision of at least 0.0015 rad, and where would you position them (motor/load side and why)?
4. [15 marks] The Gray-code optical encoder in Table 1 has 5 bits. Find the interconversion logic circuit which yields a binary-coded output word. Which is the precision (i.e., the maximum angular error) of this encoder? [Hint: try to nd relationships starting from y1 in function of x1. Then, y2 in function of y1,x1,x2, then y3 based on y2, y1, x1, x2, x3 and so on.]
Figure 1: Trajectory planning for two-segment manipulator: (a) initial and (b) nal conguration
Gray Binary
x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 00000000000 10000100001 20001000011 30001100010 40010000111 50010100110 60011000100 70011100101 80100001111 90100101110
10 0 1 0 1 0 0 1 1 0 0 11 0 1 0 1 1 0 1 1 0 1 12 0 1 1 0 0 0 1 0 0 0 13 0 1 1 0 1 0 1 0 0 1 14 0 1 1 1 0 0 1 0 1 1 15 0 1 1 1 1 0 1 0 1 0 16 1 0 0 0 0 1 1 1 1 1 17 1 0 0 0 1 1 1 1 1 0 18 1 0 0 1 0 1 1 1 0 0 19 1 0 0 1 1 1 1 1 0 1 20 1 0 1 0 0 1 1 0 0 0 21 1 0 1 0 1 1 1 0 0 1 22 1 0 1 1 0 1 1 0 1 1 23 1 0 1 1 1 1 1 0 1 0 24 1 1 0 0 0 1 0 0 0 0 25 1 1 0 0 1 1 0 0 0 1 26 1 1 0 1 0 1 0 0 1 1 27 1 1 0 1 1 1 0 0 1 0 28 1 1 1 0 0 1 0 1 1 1 29 1 1 1 0 1 1 0 1 1 0 30 1 1 1 1 0 1 0 1 0 0 31 1 1 1 1 1 1 0 1 0 1
Table 1: The conversion table between the Gray code (the inputs xi, i = 1, . . . , 5) and the Binary output code (the outputs yi,i = 1,…,5)
5. [30 marks] The 2R robot in Figure 1 is composed, as the name suggests, from two revolute joints. The former is actuated by a DC electric motor that is controlled voltage Va in the armature circuit. The motor has a friction coecient of Fm = 5 10−2 [mNm/(rad/s)], a resistance of the armature of Ra = 0.309 [Ohm] and kt = 7.88[mNm/A] = 7.88 10−3[Nm/A],kv = 7.88 10−3 [V/(rad/s)]. An harmonic drive is used as transmission. It has a reduction ratio n = −128. At steady state, the output shaft rotates with an angular velocity equal to ω = π [rad/s] and it rotates in counterclockwise direction.
What is the value of Va for the giving data?
Which is the rotation speed ωm of the motor side? [Hint] Take into account the sign of rotation
The harmonic drive has an input shaft and an output shaft. Typically, the input is mounted on the motor
and the output carries the load. In this case, the the reduction ratio is n = nflex , where nflex nflex −ncirc
are the number of teeth on the exspline and ncirc are the number of teeth on the circular spline. Find reasonable values for nflex,ncirc for obtaining n = −128. What represents the minus? What happens if the input shaft is connected to the load and the output to the motor side (with the same choice of the exspline and circular spline teeth)? What is the expression of the reduction ratio in this case and what changes w.r.t. the rst conguration? Why the rst conguration is typically used in robotics?
Bonus If an incremental encoder with quadrature detection is used on the motor side, how many bits should its internal counter use to obtain at least a resolution of ε = 10−3 degrees on the link side? How many pulses/turn should the optical disk have? How many pulses would be counted in total by the counter in one second when the motor is rotating at the steady-state speed ωm? Would the counter go up or down?
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