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代写MCEN90028-Assignment 4代做留学生R语言、R语言程序帮写

MCEN90028 Robotics Systems 
Assignment 4 (34 marks) 
Due date: Friday 29 May 2020 5pm (Week 11) 
5 June 2020 5pm (Week 12) 
1 General Instructions 
1. This is an individual assignment. 
2. You are expected to construct you own Matlab codes in this assignment and 
not call Peter Corke’s toolbox anywhere in your codes. 
3. Please read the submission instructions carefully before submitting your as- 
signment. 
2 Assignment Description and Tasks 
Task 1. Construction of a Robot Dynamic Simulator (8 
marks) 
In the first task, construct a Matlab code that represents a 2 link RR (revolute- 
revolute) robot as shown in Figure 1. 
Kinematically, 
• q1 = 0 when link 1 is aligned with the inertial X axis. 
• q2 = 0 when the robot is straight. 
• the axes of rotation for both joints (i.e. zˆ1 and zˆ2 are defined to point out of 
the page. 
• q1 and q2 are not bounded by any joint limits in Tasks 1 and 2. Joint limits 
are introduced in Task 3 onwards. 
The robot has link lengths of L1, L2 as indicated, with masses m1,m2 as shown and 
angular moments of inertia: 
I1 = 
 Ixx1 0 00 Iyy1 0 
0 0 Izz1 
 
I2 = 
 Ixx2 0 00 Iyy2 0 
0 0 Izz2 
 
about their centres of mass. 
The values for the dynamics parameters are: 
• m1 = 2kg, m2 = 1kg. 
• Izz1 = 0.5kgm2, Izz2 = 0.3kgm2. Ixx1 = Ixx2 = Iyy1 = Iyy2 = 0. 
• L1 = 1m, L2 = 0.6m, rC1 = 0.5m, rC2 = 0.3m 
• gravity acts in the negative Y direction. 
Figure 1: Planar Robot to be simulated. Note that gravity is acting in the negative 
Y direction. 
Download the Matlab script files provided for the Assignment: 
• Example RP robot.m 
• runrobot.m 
Run the main script file Example RP robot.m. It will call runrobot.m. The runrobot.m 
file contains the robot dynamics (the equation of motion). It takes the input of the 
current robot state y = [q, q˙]T and outputs y˙ = [q˙, q¨]T . This gradient is used by the 
numerical integrator ode45 in the main file, that takes the joint torque and the robot 
state, performs the numerical integration to produce the motion of the robot [q, q˙]T 
at the next time instance. 
The Example RP robot.m will produce a plot of the movement of the robot vs time. 
Note that the duration of runtime tf has been set to 5s in the example and the 
sampling period dt has been set to 0.1s. 
For Task 1, perform the necessary calculations to obtain the equation of motion for 
the robot described in Figure 1, using the provided sample code Example RP robot.m 
and runrobot.m as a starting point. 
For the report, for Task 1: 
1. produce the derivation of the equation of motion of the robot in this assignment 
(from DH frame assignments, DH parameter, Jacobian matrices to the centres of 
mass, and finally the matrices A, b, G). (This essentially reviews the kinematics 
content for the entire subject to date.) 
2. produce the plots q1 vs t and q2 vs t for tf = 5s, from the initial displacement 
of q1 = q2 = 0 at rest, with sampling period dt = 0.01s and joint torque 
τ1 = τ2 = 0 throughout. Set friction coefficient to friction coeff = 0.1. Please 
take note that the friction coefficient in the example has been set to zero. (Go 
ahead and play around with different settings to familiarise yourself with the 
code and the robotic system, but please submit only the plot requested to 
assist with marking). 
3. produce the plots q1 vs t and q2 vs t for tf = 5s, exactly the same as the 
previous point, but instead of setting τ1 = τ2 = 0, get the robot to compensate 
for the gravity forces acting on the robot. Explain how this is achieved in 
your report. Your result should show that the robot is balanced perfectly 
against gravity and therefore is not moving. Go ahead and play with the 
resulting code, for example, intentionally mis-estimate the mass, for example, 
underestimate the masses and watch as the robot floats like a balloon. Discuss 
the performance of the proposed method based on your observation. 
For your own enjoyment, produce an XY robot animation plot, i.e. the plot of link 
1 and link 2 (and the end-effector) of the robot movement in space (like in Figure 
1) using Matlab plot function, such that way you can watch the movement of the 
robot that you produce visually. This animation can be produced by combining the 
plot function and the pause function. Dictate the values of the axes so Matlab 
does not adaptively re-adjust the axes. 
For example: 
1 for t=0:dt:tf 
2 plot(.......) 
3 axis([-5 5 -5 5]) %this sets the axes to avoid re-sizing of ... 
the view of the plot 
4 pause(.1) %this sets the display speed to 0.1 to correspond ... 
to real time dt 
5 end 
Listing 1: Example of avoiding re-sizing in Matlab plots 
For the Matlab files to be submitted for Task 1: 
• Name your main file as Task1 main.m. 
• Name your robot dynamics file as Task1 dynamics.m. 
• Label the figures produced in the code according to the subtasks. 
• Use Code Sections in Matlab to structure the subtasks. 
Task 2. PID Controller on robot in joint space (6 marks) 
After Task 1, the robot, represented by the ode45 function, takes the input of joint 
torque [τ1, τ2] 
T . The robot is therefore in “torque-controlled mode”. In Task 2, we 
wish to send joint velocity commands to the robot. This means, in the main file, we 
would send a joint velocity command at each sampling instance to a PID loop, which 
implements a closed loop control at sampling period dtPIDloop = 0.001s to regulate 
the torque sent to the robot joints so that the robot joint velocity would track the 
joint velocity command given. Note that dtPIDloop is different from dt, which is the 
main robot control loop. Tune your PID controller to your best efforts – feel free to 
go back to earlier subjects (such as ELEN90055 Control Systems) if you forgot how 
to do so. A keyword reminder of Zielger-Nichols may help job your memory, though 
there are many ways. Your tuning does not need to be perfect or optimised, as long 
as it performs reasonably well. A rough guide would be a settling time for 5%− 10% 
tolerance band (aim for 5%) of ≤ 0.01s for a δq˙ = 1°/s commanded step velocity for 
individual joints. 
The outcome of Task 2 is equivalent in practice to having servo motors for your 
robot, which accept velocity command. 
For the report, for Task 2: 
1. produce the plots q1 vs t, q2 vs t, q˙1 vs t, and q˙2 vs t; for t = 0 to tf = 5s, from 
the initial displacement of q1 = 30° and q2 = 75° at rest at t = 0, to arrive at 
the final joint displacement of q1 = −15° and q2 = 45°, with q˙1(tf ) = q˙2(tf ) = 0 
at tf = 5s. Reference trajectory is to be cubic polynomial of time in joint space. 
Sampling period dt = 0.01s (note again, dt is the sampling period of the main 
control loop, not the PID loop). Set friction coefficient to friction coeff = 0.1. 
2. produce the plots of the corresponding joint torque τ1 vs t and τ2 vs t (corre- 
sponding to the resulting robot motion in the first point of Task 2). 
3. produce the XY plot of the robot (like in Figure 1) which consist of link 1 and 
link 2 (and the end-effector, e.g. marked with a circle of triangle), every 0.5s. 
(meaning at t = [0, 0.5, 1, ..., 5]s) for the resulting robot motion in Task 2. 
For the Matlab files to be submitted for Task 2: 
• Name your main file as Task2 main.m. 
• Name your robot dynamics file as Task2 dynamics.m. 
• Label the figures produced in the code according to the subtasks. 
• Use Code Sections in Matlab to structure the subtasks. 
Hint 
Added on 20 May 2020. 
During the workshop today, it appeared that a lot of students were not able 
to get past Task 2, which was to construct a PID closed loop controller to 
allow the robot to receive a joint velocity command q˙C instead of the torque 
command. This is equivalent (in real life) to having a velocity servo drive for 
your motors, which accepts velocity command from your main robot loop. 
This velocity control loop is to run 10 times faster than the main robot 
control loop. This essentially allows the joint velocity controller 10 sampling 
instances to produce the velocity step response on the robot joints for every 
joint velocity command issued by the main loop. 
How is it done? 
This is a closed-loop system with q˙C as the reference input, a PID as the 
controller (that you are meant to design and tune), the robot as the plant (the 
robot as a plant takes in joint torque τ as the input and robot joint motion 
q and q˙C as outputs) and the output q˙ is used as a feedback to this velocity 
control loop. Therefore: 
evc = q˙C − q˙; 
τ = Kpe+Kde˙+KI 
∫ t 
edτ ; 
(1) 
where vc denotes “velocity control”, and Kp, Kd, KI are diagonal matrices 
containing the gains assigned to each joint, of size n × n, where n is the 
number of joints in the robot. 
Hope that helps. Have fun! 
Task 3. Joint space control of the velocity commanded robot 
(8 marks) 
Continuing from where we left off in Task 2, this time, we are going to get the 
end-effector of the robot to move in “task space”, while still utilising joint space 
control. That is, you are to generate a reference trajectory (using cubic polynomial) 
to describe the reference trajectory in (xref , yref ) for the end-effector, from an initial 
coordinate (xi, yi) to the final coordinate (xf , yf ) in a straight line. The reference tra- 
jectory is converted (using inverse kinematics) within at each robot sampling instance 
dt into a reference joint space trajectory, that is then used to command the veloc- 
ity commanded robot (i.e. as the input to the velocity PID control of the robot joints). 
NOTE: In Task 3, construct a joint limit of −60° ≤ q1 ≤ 60° and −90° ≤ q2 ≤ 90° 
in your code. 
Two reference trajectories for the end-effector are to be generated and tracked by 
the robot: 
• Trajectory A: initial end-effector position: (xi, yi) = (0.71, 1.08)m at (x˙i, y˙i) = 
(0, 0)m/s, final end-effector position: (xf , yf) = (1.485, 0.041)m and velocity 
at the final position (x˙f , y˙f ) = (0, 0)m/s. Trajectory duration: tf = 5s. 
• Trajectory B: initial end-effector position: (xi, yi) = (0.71, 1.08)m at rest 
and comes to a complete stop at final end-effector position: (xf , yf) = 
(1.36,−0.59)m. Trajectory duration: tf = 5s. 
For the report, for Task 3: 
1. produce the plots of the end-effector position x vs t and y vs t for Trajectory 
A. Superimpose the reference trajectory xref and yref on the plot. 
2. produce the XY plot of the robot (like in Figure 1) which consist of link 1 and 
link 2 (and the end-effector, e.g. marked with a circle or triangle), every 0.5s, 
(meaning at t = [0, 0.5, 1, ..., 5]s) for the resulting robot motion in Trajectory 
A. Comment on the difference between the resulting motion and that produced 
in Task 2. 
3. produce the XY plot of the robot (like in Figure 1) which consist of link 1 and 
link 2 (and the end-effector, e.g. marked with a circle or triangle), every 0.5s, 
(meaning at t = [0, 0.5, 1, ..., 5]s) for the resulting robot motion in Trajectory 
B. 
Note: 
• Trajectory B is designed so that it passes outside the reachable workspace 
dictated by the joint limits of the robot. 
• Superimpose the rough plot of the reachable workspace to aid your think- 
ing. 
• Discuss your results and observations. 
For the Matlab files to be submitted for Task 3: 
• Name your main file as Task3 main.m. 
• Name your robot dynamics file as Task3 dynamics.m. 
• Label the figures produced in the code according to the subtasks. 
• Use Code Sections in Matlab to structure the subtasks. 
Task 4. Task space control of the velocity commanded robot 
(6 marks) 
In Task 4, we would like to implement Task Space control, which means: our error 
term in the main loop ought to be in the task space. Using end-effector Trajectory 
A in Task 3, construct the error signal ex = xref − xr and ey = yref − yr at every 
sampling instace dt, where xref and yref are the reference position of the end-effector 
and xr and yr are the actual end-effector of the robot, obtained by performing 
forward kinematics of the output of the robot simulator (q, q˙). Generate task space 
velocity command (x˙C , y˙C) through a P-controller of the error (or PI controller – state 
what you chose to implement in your report). Use the inverse of the Jacobian ma- 
trix to obtain the joint velocity command to be sent to the velocity commanded robot. 
In Task 4, the joint limit of −60° ≤ q1 ≤ 60° and −90° ≤ q1 ≤ 90° introduced in 
Task 3 is retained. 
For the report, for Task 4: 
1. produce the plots of the end-effector position x vs t and y vs t for Trajectory 
A. Superimpose the reference trajectory xref and yref on the plot. 
2. produce the XY plot of the robot (like in Figure 1) which consist of link 1 and 
link 2 (and the end-effector, e.g. marked with a circle of triangle), every 0.5s, 
(meaning at t = [0, 0.5, 1, ..., 5]s) for the resulting robot motion in Trajectory 
A. Comment on any observation you had between the resulting motion and 
that produced in Task 3. 
For the Matlab files to be submitted for Task 4: 
• Name your main file as Task4 main.m. 
• Name your robot dynamics file as Task4 dynamics.m. 
• Label the figures produced in the code according to the subtasks. 
• Use Code Sections in Matlab to structure the subtasks. 
Task 5. Task space control of the velocity commanded robot 
with reactive obstacle avoidance (6 marks) 
Task 5 continues from Task 4, where the robot is to execute Trajectory A with Task 
Space control of the velocity commanded robot. However, a circular obstacle appears 
in the path. The robot does not know where it is beforehand (it is not a known 
obstacle from the start as the one you encountered in Assignment 3). However, 
you have been given a wonderful sensor, mounted at the end-effector. This sensor 
provides one set of measurement at each time instance. It can be called at each time 
instance by: 
1 [d, theta] = sensor t5(obs init, xr, yr) 
Inputs: 
obs init A variable to handle the generation of a new obstacle. Set it to 1 if a 
new obstacle needs to be generated, 0 if not initialising the obstacle. An 
example code is provided below shown in Listing 2. 
xr, yr The robots end-effector coordinate in the inertial frame at the time the 
function is called. 
Outputs: 
d The distance (in metre) of the end-effector from the nearest detected point 
(of the obstacle) (therefore it is a scalar value, not a vector). 
theta The angle (in radians) to the same point from the end-effector, measure 
from the inertial positive X axis. 
1 newObs=1; % create a new obstacle in the environment. 
2 for ... 
3 ... 
4 [d, th]=sensor t5(newObs,x,y); 
5 newobs=0; % avoid changing the obstacle during the motion. 
6 ... 
7 end 
Listing 2: Example of initialising the obstacle 
The function sensor t5.p is provided for this assignment. The sensor output variables 
d and θ are shown in Figure 2. It is assumed that there is only ONE (circular) 
obstacle in the scenario. 
The sensor has a maximum range of 0.35m, beyond which it will not detect the 
obstacle and will output the value d = 1000 and theta = 1000. This means that you 
will not see the obstacle until it is within 0.35m of the end-effector. The strategy you 
Figure 2: The sensor outputs: d and θ shown in this diagram. The obstacle (Obs) 
is assumed to be always a circle. Its location is recorded inside the function of the 
sensor. 
need to come up with is therefore described as being “reactive”, or in the literature, 
it is often called “online obstacle avoidance”. 
You therefore initialise Task 5 driving the robot end-effector to track Trajectory A. 
In the effort to avoid the obstacle, you are to get your robot to track the desired 
trajectory as much as possible and only deviate from it where it is necessary to avoid 
the obstacle. In the main loop, you will call the sensor t5(obs init, xr, yr) function at 
each sampling instance and decide what to do. This has not been explicitly covered 
in our lectures, but it is part of the open-ended brain teaser to the class. There is 
plenty in the literature in the 80s - 90s on these type of problems if you wish to read 
up. Focus on the reactive strategies. 
In Task 5, the joint limit of −60° ≤ q1 ≤ 60° and −90° ≤ q1 ≤ 90° introduced in 
Task 3 is retained. 
For the report, for Task 5: 
• In less than 500 words, explain the outline of your strategy in guiding the 
robot end-effector through Trajectory A while avoiding the obstacle. Please do 
not use ready made functions, such as those in the Matlab Robotics Toolbox. 
Include references if you use material from the literature. 
IMPORTANT 
If you utilised any methods taken from the literature (from a paper, textbook, 
etc), provide a reference to the source. Add a section at the end of the re- 
port titled ”references”. Learn how to cite literature source appropriately in 
your own time at: https://students.unimelb.edu.au/academic-skills/explore- 
our-resources/referencing 
At this stage of your master degree, ignorance in how to appropriately ac- 
knowledge your source is not a valid excuse if you were found to plagiarise 
external material. If you have not already done so, please educate yourself as 
to the boundary of what constitute acceptable and non-acceptable practice. 
Those boundaries include the cases of how to appropriately report material 
that you may share with your peers in class. Know the difference between 
group discussion and helping each other to learn material and copying someone 
else’s work. 
• As part of your explanation above, produce the necessary plots to demonstrate 
your result implementing a Task Space control to the velocity commanded robot 
to traverse Trajectory A, with the adjustments made to avoid the obstacle. 
Include in the results relevant variables over t such as: the reference trajectory, 
the actual trajectory traversed by the robot to avoid the obstacle, sensor 
readings during the operation, and other variables relevant to demonstrating 
the effectiveness or in explaining the inner working of the strategy of your 
choice. Refer to these plot when explaining your answer and explain the plots. 
3 Submission 
In your report, detail the steps and understanding in producing your answers. Do not 
include the Matlab codes in the body of the report. Include only the pseudo-code 
if necessary and explain the logic. 
You need to submit one report per person plus the necessary Matlab codes. 
The report should be in an appropriate engineering report format and submitted as 
a PDF. The report should be no more than 30 pages (everything included) with 
12pt font size. 
Compress your report and all your relevant Matlab files as a .zip file with folder 
name “Assignment4 [#]”, where you replace [#] with your student ID (eg. As- 
signment4 123456) and submit to LMS. 
Note: only the last submission will be assessed. 
 
Submission checklist: 
Report is no more than 30 pages 
Report has 12pt font size 
Report saved as PDF 
Titles of files are in the right format 
Compressed all files as a single .zip file 
4 Academic Integrity 
We take academic integrity seriously. Please note that while students may discuss 
and help each other in the thinking process, you should work on your assignments 
separately. 
Details about academic integrity can be found on MCEN90028 Canvas page (under 
Subject Overview) or at http://academicintegrity.unimelb.edu.au/. Please 
check with the tutors or the lecturer if you are in doubt. Ignorance is not a valid 
reason for academic misconducts. 
 
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