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--- robotic_manipulators [2016/07/20 17:02] – joaomatos
+++ robotic_manipulators [2016/08/06 23:28] (current) – joaomatos
@@ Line 1: / Line 1: @@
 ===== Robotic Manipulators Basics =====
+**Author:** Joao Matos Email: <jcunha@id.uff.br> <!-- replace with your email address -->
+\\
+**Date:** Last modified on 6/8/2016
+\\
+**Keywords:** Robotic Manipulator , Serial Arm, Denavit Hartenberg, Inverse kinematics
+\\
  This page is to introduce the theory behind the robotic manipulators. It will be used the Denavit-Hartenberg parameters notation to describe the geometry of a serial chain of links and joints (Serial-Link).Using the robotic toolbox developed by Peter Corke (RVCtoolbox) we can visualize and understand more about the Denavit-Hartenberg parameters and how the process of the inverse kinematics works.
@@ Line 45: / Line 52: @@
 **Four steps and rules to define the frame on each joint **
-  - You will define the z axes as the joints rotation axes. Start drawing only the z axis on all your joints, starting from z(0) at the base and going to z(....) at the end effector . DON'T FORGET that the z axis of the end effector must be in the same direction of the z axis of the last joint.
+  - You will define the z axes as the joints rotation axes (CCW). Start drawing only the z axis on all your joints, starting from z(0) at the base and going to z(....) at the end effector . DON'T FORGET that the z axis of the end effector must be in the same direction of the z axis of the last joint.
-  - Now you will draw the x axes , the x axis must be perpendicular to both z(j-1) and z(j). Do it for every joint. If you have more than one direction that satisfy this condition , you can choose the direction that goes from z(j-1) to z(j) axes. DON'T FORGET that the x axis of the end effector must be in the same direction of the x axis from the last joint
+  - Now you will draw the x axes , the x axis must be perpendicular to both z(j-1) and z(j). Do it for every joint. If you have more than one direction that satisfy this condition , choose the direction that goes from z(j-1) to z(j) axes. DON'T FORGET that the x axis of the end effector must be in the same direction of the x axis from the last joint
   - Now you will draw the y axes.The y axis must follow the right hand rule , from z axis to x axis. And don't forget that the y axis of the end effector must be in the same direction of the y axis of the last joint.
   - Now that you drawn the three axis on every joint , you must check if the x(j) axis intersect the z(j-1) axis. To your DH notation be right , every x(j) axis must intersect the z(j-1) axis. If you don't have this , you must translate your (j) frame , in order to guarantee this intersection.
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-  * **(aj)**: Is the displacement along the xj axis between the center of the frame (j-1) and frame (j).
+  * **(aj)**: Is the displacement along the xj axis DIRECTION between the center of the frame (j-1) and frame (j).
   * **(αj)**: Is the Rotation around the xj axis that make the axis z(j-1) and z(j) to match each other.
-  * **(dj)**: Is the displacement along the z(j-1) between the center of the frame (j-1) and frame (j).
+  * **(dj)**: Is the displacement along the z(j-1) DIRECTION between the center of the frame (j-1) and frame (j).
   * **(θj)**: If you have only revolute joints , you can call them q1,q2,q3,q4.......
@@ Line 74: / Line 81: @@
 {{youtube>SZP1KQ2qSEA?medium}}
+----
+===== Denavit - Hartenberg notation for DASL Serial Arm =====
+ Following the step by step and rules presented previously, lets calculate the DH parameter for the serial arm available on the DASL at UNLV.
+{{ ::novos_frames.jpg?direct |}}
+ ** Defining the Frames **
+-> drawing the z axes.
+{{ ::ded1.jpg?direct |}}
+-> drawing the x axes.
+{{ ::ded2.jpg?direct |}}
+-> drawing the y axes.
+{{ ::ded3.jpg?direct |}}
+-> checking if every x(n) axis intersect the z(n-1) axis.
+{{ ::ded4.jpg?direct |}}
+-> Change the axis that don't follow rule 4.
+{{ ::ded5.jpg?direct |}}
+-> Create the table and get the DH parameters.
+{{ ::ded6.jpg?direct |}}
 ----
@@ Line 248: / Line 280: @@
 ----
-=====Visualizing the Denavit-Hartenberg Parameters =====
+=====Creating the DASL Serial Arm and solving IK on the RVC toolbox =====
- There is an interesting tool on the RVC toolbox that is the 'teach'  plot. There we can adjust manually the joints angle values and see the robot arm moving - this is useful to test if you created your robot arm right ( if you used the right DH parameters). For example , if you used the wrong alpha parameter your link will rotate on the wrong direction , and this will interfere on the foward/inverse kinematics calculations.
+ If you followed the last topics on this page you already know how to create a serial Link based on its Denavit-Hartenberg parameters , so lets do this for the DASL serial arm. The code is:
-Lets create a mechanism with four links , and atribute random parameters at first , and then we will modify it.
 <Code>
-pi=3.14
+%Creating the DASL serial arm on the RVC toolbox.
+%L(n)=LINK([theta D A alpha Sigma(optional) Offset(optional)])
-%L(n)=LINK([theta D A alpha Sigma(optional) Offset(optional)])
+L1=11;
-%Denavit-Hartenberg parameters
+L2=15;
+L3=10;
+L4=21;
+L5=8;
+pi=3.1416;
+q0=[0 0 0 0 0];
-L(1)=Link([0 -1 0 pi/2]);
+L(1)=Link([0 -L1 0 -pi/2 0]);
-L(2)=Link([0 0  1 0]);
+L(2)=Link([0 0 L2 0 0]);
-L(3)=Link([0 0 1 pi/2]);
+L(3)=Link([0 0 L3 0 0]);
-L(4)=Link([0 0 0 0]);
+L(4)=Link([0 0 0 pi/2 0]);
+L(5)=Link([0 -(L4+L5) 0 0 0]);
-testing=SerialLink(L,'name','Testing');
+DaslArm=SerialLink(L,'name','DaslArm');
+DaslArm.teach(q0)
-%testing.teach()
-testing.plot([0 0 0 0],'nobase')
 </Code>
- You can change the '%' to the testing.plot or testing.teach. If you want to see only the robot arm you can hide the testing.teach inserting '%' before it.
+{{ ::novoteach.jpg?direct |}}
- This configuration will create the following robot arm:
+The teach command will create a window that allow you to change the q1,q2,q3,q4,q5 rotation and see how the arm moves , it's good to change this rotation by hand to check if your Denavit-Hartenberg parameters are OK.
-{{ ::8_manip.jpg?direct |}}
+ If everything is moving as the arm moves , we can use the IK solver available on the RVC toolbox to get the joints angles given a desired end effector pose , you just need to add this code after defining your serial link object :
- Remeber that by convention the link #0 is the base that is fixed, so we start to work with Link 1.
+<Code>
-Changing the offset value  **(d)** (second argument on the Link Function) lets see what happens.
- {{ ::9_manip.jpg?direct |}}
+%END EFFECTOR POSITION
+% trotx(angle) = rotate (angle) about the x axis
+% troty(angle) = rotate (angle) about the y axis
+% trotz(angle) = rotate (angle) about the z axis
+% transl(x,y,z) = pure translation of x , y , z
+% The end effector pose will be a combination of tranls and trot
- As can be seen , **CHANGING THE LINK 1 OFFSET (D)** value , we change the position on the z-axis of the joint 2 .
+%For example lets use a pure translation .
+T=transl(20,35,-30);
- Now changing the **link's length value (a):**
+%IK SOLVER
+%Initial pose (all rotations = 0)
+q0=[0 0 0 0 0];
+%Mask to be used on the IK solver (x y z rotx roty rotz)
+m=[1 1 1 0 0 0];
- {{ ::10_manip.jpg?direct |}}
+%Solving the IK and converting to degrees
+q_ikine=DaslArm.ikine(T,q0,m);
+q_degrees=q_ikine*(180/pi);
- As can be seen , the joint 2 got dislocated because the LINK 1 length changed.
+%Creating a trajectory from q0 to the IK solution
+%Time variable
+t=[0:0.05:4];
+%trajectory
+q_TRAJ=jtraj(q0,q_ikine,t);
- Now changing the **link's twist (alpha).**
+%Plot the result
+DaslArm.plot(q_TRAJ)
- {{ ::11_manip.jpg?direct |}}
+</Code>
- Note that the joint 2 got twisted, this will change the direction of rotation of the LINK 2. Use the teach command now (using link1 alpha=pi/2 and then alpha=0 ) and see how the direction will go.
+ This code will open a window to show the animation of the serial arm , going from an initial position q0 ( we defined as all the joint angles =0 ) to a final position - which is the IK solution. Is important to note that some end effector poses cannot be reached by the arm , so when you input these values , you will get some errors saying that the IK solver took more than 1000 iterations to find a solution and you will get an approximated solution. This IK solver available on the toolbox is not 100% accurate , the best choice is to program the IK solver by yourself , you can easily find a lot of algorithm to solve an IK problem on the internet.