This project uses a Gazebo simulation platform to program a Kuka KR210 6 degree of freedom serial manipulator to pick up a cylinder from a shelf and drop it into a bin next to the manipulator.
The cylinder is located radnomly on the shlef, Thus the robot needs to figure out his motor and joint angles and speeds to reach to the can from Global reference frames to his local frames. This could be acheived by calculaing the inverse Kinematics of the Robot.
Kinematics is the a branch of mechanics that describes the motion of a system without considering the forces involved. Lots if the kinematics
After evaluating the the kr210.urdf.xacro, and comparing it with our DH parameter reference frames, we converted the Robot dimension information to the DH table as the following.
Img:1 DH Parameter Reference Frames
Img:2 URDF file reference frames
| Links | alpha(i-1)(rad) | a(i-1)(m) | d(i)(m) | theta(i)(rad) |
|---|---|---|---|---|
| 0->1 | 0 | 0 | 0.33 + 0.42= 0.75 | q1 |
| 1->2 | - pi/2 | 0.35 | 0 | -pi/2 + q2 |
| 2->3 | 0 | 1.25 | 0 | q3 |
| 3->4 | - pi/2 | 0 | 0.96 + 0.54=1.5 | q4 |
| 4->5 | pi/2 | 0 | 0 | q5 |
| 5->6 | - pi/2 | 0 | 0 | q6 |
| 6->EE | 0 | 0 | 0.193 + 0.11 = 0.303 | 0 |
2. Using the DH parameter table you derived earlier, individual transformation matrices were created about each joint. In addition, also a generalized homogeneous transform between base_link and gripper_link was generated using only end-effector(gripper) pose
Python was used to create the individual transformation matrices about each joint, a snippet of the code is shown:
### Create symbols for joint variables to use in DH table
q1, q2, q3, q4, q5, q6, q7 = symbols('q1:8')
d1, d2, d3, d4, d5, d6, d7 = symbols('d1:8')
a0, a1, a2, a3, a4, a5, a6 = symbols('a0:7')
alpha0, alpha1, alpha2, alpha3, alpha4, alpha5, alpha6 = symbols('alpha0:7')
# Dictionary (associative array) DH Parameters table, q is theta
s = {alpha0: 0, a0: 0, d1: 0.75,
alpha1: -pi/2, a1: 0.35, d2: 0, q2: -pi/2+q2,
alpha2: 0, a2: 1.25, d3: 0,
alpha3: -pi/2, a3: -0.054, d4: 1.5,
alpha4: pi/2, a4: 0, d5: 0,
alpha5: -pi/2, a5: 0, d6: 0,
alpha6: 0, a6: 0, d7: 0.303, q7: 0,}
#### Homogeneous Transforms of neighbouring links
T0_1 = Matrix([[ cos(q1), -sin(q1), 0, a0],
[ sin(q1)*cos(alpha0), cos(q1)*cos(alpha0), -sin(alpha0), -sin(alpha0)*d1],
[ sin(q1)*sin(alpha0), cos(q1)*sin(alpha0), cos(alpha0), cos(alpha0)*d1],
[ 0, 0, 0, 1]])
T0_1 = T0_1.subs(s)this gives the output:
T0_1=
⎡cos(q₁) -sin(q₁) 0 0 ⎤
⎢sin(q₁) cos(q₁) 0 0 ⎥
⎢ 0 0 1 0.75⎥
⎣ 0 0 0 1 ⎦
T1_2 =
⎡sin(q₂) cos(q₂) 0 0.35⎤
⎢ ⎥
⎢ 0 0 1 0 ⎥
⎢ ⎥
⎢cos(q₂) -sin(q₂) 0 0 ⎥
⎢ ⎥
⎣ 0 0 0 1 ⎦
T2_3 =
⎡cos(q₃) -sin(q₃) 0 1.25⎤
⎢ ⎥
⎢sin(q₃) cos(q₃) 0 0 ⎥
⎢ ⎥
⎢ 0 0 1 0 ⎥
⎢ ⎥
⎣ 0 0 0 1 ⎦
T3_4 =
⎡cos(q₄) -sin(q₄) 0 -0.054⎤
⎢ ⎥
⎢ 0 0 1 1.5 ⎥
⎢ ⎥
⎢-sin(q₄) -cos(q₄) 0 0 ⎥
⎢ ⎥
⎣ 0 0 0 1 ⎦
T4_5 =
⎡cos(q₅) -sin(q₅) 0 0⎤
⎢ ⎥
⎢ 0 0 -1 0⎥
⎢ ⎥
⎢sin(q₅) cos(q₅) 0 0⎥
⎢ ⎥
⎣ 0 0 0 1⎦
T5_6 =
⎡cos(q₆) -sin(q₆) 0 0⎤
⎢ ⎥
⎢ 0 0 1 0⎥
⎢ ⎥
⎢-sin(q₆) -cos(q₆) 0 0⎥
⎢ ⎥
⎣ 0 0 0 1⎦
###Notice becuse of the Gripper's reference frame has the same orientation as joint6
(the one before it) the transformation matrix in the rotation part is the identity matrix.
T6_G =
⎡1 0 0 0 ⎤
⎢ ⎥
⎢0 1 0 0 ⎥
⎢ ⎥
⎢0 0 1 0.303⎥
⎢ ⎥
⎣0 0 0 1 ⎦
##From-Base Link to End-Effector Transformation Matrix To Determine the complete homogeneous transform between the base link and the gripper link (end-effector) using just the end-effector pose (position ==(Px,Py,Pz)==+rotation ==(Roll-Pitch-Yaw== angles of the end-effector)).
T BL_EE =
⎡ Px ⎤
⎢ ⎥
⎢ RT Py ⎥
⎢ ⎥
⎢ Pz ⎥
⎢ ⎥
⎣0 0 0 1 ⎦
Where RT (Rotation transform) can be the result of the intrinsic rotations of the end gripper around the given roll, pitch, yaw angles=Rx(rol)*Ry(yaw)*Rz(pitch).
Note: this gives the same result of T6_G after substituting the Theta angles the Kuka210
T BL_EE =
⎡ Px ⎤
⎢ ⎥
⎢ RT Py ⎥
⎢ ⎥
⎢ Pz ⎥
⎢ ⎥
⎣0 0 0 1 ⎦
Substituting the Thetas with Zeros gives the cumulative distance of the robot from Base Link to End Effector:
T_EE=
[ 1.0, 0, 0, 2.153],
[ 0, 1.0, 0, 0],
[ 0, 0, 1.0, 1.946],
[ 0, 0, 0, 1]]
##Inverse Kinametics
Inverse Kinematics is computing the joint angles when knowing the end effector's position and orientation. Therefore, when knowing the target location, IK computes the amount of movement and direction that needs to be executes to reach that location, making it an important chapter in Robotics.
Because the Kuka KR210 joints (4 5 and 6) satisfies that its:
Three neighboring joint axes intersect at a single point
The inverse Kinematic problem it is solvable in closed-form with wrist center at joint 5.
which is able to rotate the end effector to the exact orientation required for the task at hand. The first three joints move the wrist center to a position where it has enough mobility to orient itself to the angle required.
I hope you got something from all the graphics above: 😏
The inverse kinematic code was tested with the IK_debug.py file. it tests the code against known solutions.
Finally, the IK_server.py file contains the code that links into the ROS/Gazebo/Rviz simulator.
To run the IK_server.py code, install this Udacity Kinematics Project code. Then change the inverse kinematics flag to false.
To launch the simulator, run: first you need to run
cd ~/catkin_ws/
catkin_make
./src/RoboND-Kinematics-Project/kuka_arm/scripts/safe_spawner.shYou should see both Gazebo and Rviz launching. Gazebo should have a living room environment with the arm, bookshelf, bin and blue can all visible. Rviz should have the same, along with text above the scene indicating its status.
To run the inverse kinematic code, run the following code in a new terminal:
rosrun kuka_arm IK_server.pyYou must now press Next in the Rviz window to have each step proceed. If everything worked, the arm should eventually move to pick up the can, grab it, then move so the arm will drop the can into the bin on the ground.