04. Principles of robotics. Programming a da Vinci surgical robot in simulated environment.

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Lecture

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Warning

Test 1 (ROS principles, publisher, subscriber. Python principles. Principles of robotics.) October 27.

Rigid body motion

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Def. Rigid body

A rigid body is defined as a body on which the distance between two points remains constant in time regardless of the force applied on it.

• Shape and the volume of the rigid bodies are also constant.
• The pose of a rigid body can be given by the three coordinates of three of its points that do not lie on the same straight line.

• The pose of a rigid body can be described in a more expressive way by the three coordinates of one of its points chosen arbitrarily position and the body's orientation.

• The motion of rigid bodies is composed by two elemental motions: translation and rotation.

• During translation, all points of the body move along straight, parallel lines.

• During rotation, the position of the points of the rotational axis are constant, and the other points of the body move along circles in planes perpendicular to the axis of rotation.

• The free motion of rigid bodies can always be expressed as the superposition of a translational motion and a rotation around a single axis.

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3D transformations

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• Position: 3D offset vector

• Orientation: 3 x 3 rotation matrix

  • further orientation representations: Euler-angles, RPY, angle axis, quaternion

• Pose: 4 × 4 (homogenous) transformation matrix

• Frame: origin, 3 axes, 3 base vectors, right hand rule
• Homogenous transformation: rotation and translation in one transfromation
  • e.g., for the rotation \(\mathbf{R}\) and translation \(\mathbf{v}\):

\[ \mathbf{T} = \left[\matrix{\mathbf{R} & \mathbf{v}\\\mathbf{0} & 1 }\right] = \left[\matrix{r_{1,1} & r_{1,2} & r_{1,3} & v_x\\r_{2,1} & r_{2,2} & r_{2,3} & v_y\\r_{3,1} & r_{3,2} & r_{3,3} & v_z\\\ 0 & 0 & 0 & 1 }\right] \]

• Homogenous coordinates:
  • Vector: extended with 0, \(\mathbf{a_H}=\left[\matrix{\mathbf{a} \\ 0}\right]=\left[\matrix{a_x \\ a_y \\ a_z \\ 0}\right]\)
  • Point: extended by 1, \(\mathbf{p_H}=\left[\matrix{\mathbf{p} \\ 1}\right]=\left[\matrix{p_x \\ p_y \\ p_z \\ 1}\right]\)
  • Applying transfomrations is much easier:

\[ \mathbf{q} = \mathbf{R}\mathbf{p} + \mathbf{v} \to \left[\matrix{\mathbf{q} \\ 1}\right] = \left[\matrix{\mathbf{R} & \mathbf{v}\\\mathbf{0} & 1 }\right]\left[\matrix{\mathbf{p} \\ 1}\right] \]

• Degrees of Freedom (DoF): the number of independent parameters.

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Principles of robotics

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• Robots are built of: segments (or links) és joints
• Task space (or cartesian space):
  • 3D space around us, where the task, endpoint trajectories, obstacles are defined.
  • TCP (Tool Center Point): Frame fixed to the end effector of the robot.
  • Base frame, world frame
• Joint space:
  • Properties or values regarding the joints.
  • Low-level controller.
  • Joint angles, joint velocities, accelerations, torques....

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Python libraries

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Numpy

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• Python library
• High dimension arrays and matrices
• Mathematical functions

import numpy as np

# Creating ndarrays
a = np.zeros(3)
a.shape
a.shape=(3,1)
a = np.ones(5)
a = np.empty(10)
l = np.linspace(5, 10, 6)
r = np.array([1,2])    # ndarray from python list
r = np.array([[1,2],[3,4]])
type(r)

# Indexing
l[0]
l[0:2]
l[-1]
r[:,0]

# Operations on ndarrays
r_sin = np.sin(r)
np.max(r)
np.min(r)
np.sum(r)
np.mean(r)
np.std(r)

l < 7
l[l < 7]
np.where(l < 7)

p = np.linspace(1, 5, 6)
q = np.linspace(10, 14, 6)

s = p + q
s = p * q
s = p * 10
s = p + 10
s = p @ q    # dot product
s = r.T

If not installed:

pip3 install numpy

Matplotlib

• Visualization in python
• Syntax similar to Matlab

import numpy as np
from matplotlib import pyplot as plt

X = np.linspace(-np.pi, np.pi, 256)
C, S = np.cos(X), np.sin(X)

plt.plot(X, C)
plt.plot(X, S)

plt.show()

If not installed:

pip3 install matplotlib

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Practice

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1. dVRK ROS 2 install

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The da Vinci Surgical System is used to perform minimally invasive surgeries by teleoperation. The da Vinci Research Kit (DVRK) is an open-source hardware and software platform, offers, amongst others, reading and writing all the joints of the da Vinci, and also simulators for each arm. The DVRK software can be built as follows.

1. Clone the dVRK (da Vinci Reserach Kit) using vcs into a new workspace, then build it:

   mkdir -p ~/dvrk2_ws/src
   cd ~/dvrk2_ws/src                
   vcs import --input https://raw.githubusercontent.com/jhu-dvrk/dvrk_robot_ros2/devel/dvrk-2.2.vcs --recursive    cd ~/dvrk2_ws
   cd ~/dvrk2_ws
   colcon build --symlink-install --cmake-args -DCMAKE_BUILD_TYPE=Release   
   source ~/dvrk2_ws/install/setup.bash
   

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2. Add the following line to the end of the .bashrc file:

   source ~/dvrk2_ws/install/setup.bash
   

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3. Run these commands in separate terminals to launch the simulation. Do not forget to push the Home button in the DVRK console.

   # dVRK main console
   ros2 run dvrk_robot dvrk_console_json -j ~/dvrk2_ws/install/sawIntuitiveResearchKitAll/share/sawIntuitiveResearchKit/share/console/console-PSM1_KIN_SIMULATED.json
   

   # ROS 2 joint and robot state publishers
   ros2 launch dvrk_model dvrk_state_publisher.launch.py arm:=PSM1
   

   # RViz
   ros2 run rviz2 rviz2 -d ~/dvrk2_ws/install/dvrk_model/share/dvrk_model/rviz/PSM1.rviz
   

   # rqt_gui
   ros2 run rqt_gui rqt_gui
   

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2. PSM subscriber

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1. Create a new file named psm_grasp.py in the ~/ros2_ws/src/ros2_course/ros2_course folder. Add the new entry point to the setup.py, as usually.

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2. Check the topics and nodes of the simulator using the commands learned earlier (rostopic list, rosrun rqt_graph rqt_graph, etc.). PSM1 publishes the pose of the TCP and the angle of the jaws into the topics below. Subscribe to these topic in psm_grasp.py and store the current values into variables.

   /PSM1/measured_cp
   /PSM1/jaw/measured_js
   

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3. Build and run the node:

   cd ~/ros2_ws
   colcon build --symlink-install
   ros2 run ros2_course psm_grasp 
   

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3. Move the TCP along a linear trajectory

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1. PSM1 expects commands regarding the pose of the TCP and the angle of the jaws from the topics below. Create publishers to these topic in psm_grasp.py.

   /PSM1/servo_cp
   /PSM1/jaw/servo_jp
   

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2. Implement a method that moves the TCP to the desired position along a linear trajectory. Send the gripper to the position (0.0, 0.05, -0.12), leave the orientation as it is. Let the sampling time dt be 0.01s.

   def move_tcp_to(self, target, v, dt):
   

   Tip

   Use the function np.linspace(start, stop, num) to create the array of t values (T). This function can also be used to create the linear trajectory along the axes x, y, z in separate arrays X, Y and Z.

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3. Write a method that can open and close the gripper jaws, also along a linear trajectory.

   def move_jaw_to(self, target, omega, dt):
   

   

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4. Dummy marker

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1. Write a node that creates a virtual marker that can be grasped publishingvisualization_msgs/Marker messages. Create a new file named dummy_marker.py in the ~/ros2_ws/src/ros2_course/ros2_course folder. Add it to the setup.py, as usually. Copy the following code into the file dummy_marker.py:

   import rclpy
   from rclpy.node import Node
   from visualization_msgs.msg import Marker
   
   class DummyMarker(Node):
       def __init__(self, position):
           super().__init__('minimal_publisher')
           self.position = position
           self.publisher_ = self.create_publisher(Marker, 'dummy_target_marker', 10)
           timer_period = 0.1  # seconds
           self.timer = self.create_timer(timer_period, self.timer_callback)
           self.i = 0
           i = 0
   
       def timer_callback(self):
           marker = Marker()
           marker.header.frame_id = 'PSM1_psm_base_link'
           marker.header.stamp = self.get_clock().now().to_msg()
           marker.ns = "dvrk_viz"
           marker.id = self.i
           marker.type = Marker.SPHERE
           marker.action = Marker.MODIFY
           marker.pose.position.x = self.position[0]
           marker.pose.position.y = self.position[1]
           marker.pose.position.z = self.position[2]
           marker.pose.orientation.x = 0.0
           marker.pose.orientation.y = 0.0
           marker.pose.orientation.z = 0.0
           marker.pose.orientation.w = 1.0
           marker.scale.x = 0.008
           marker.scale.y = 0.008
           marker.scale.z = 0.008
           marker.color.a = 1.0 # Don't forget to set the alpha!
           marker.color.r = 0.0
           marker.color.g = 1.0
           marker.color.b = 0.0;
   
           self.publisher_.publish(marker)
           self.i += 1
   
   
   def main(args=None):
       rclpy.init(args=args)
       marker_publisher = DummyMarker([-0.05, 0.08, -0.12])
       rclpy.spin(marker_publisher)
   
       # Destroy the node explicitly
       # (optional - otherwise it will be done automatically
       # when the garbage collector destroys the node object)
       marker_publisher.destroy_node()
       rclpy.shutdown()
   
   if __name__ == '__main__':
       main()
   

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2. Build and run the node. Visualize the marker in RViz.

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5. Grasp the marker

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1. Subscribe to the topic with the marker position dummy_target_publisher the file psm_grasp.py.

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2. Implement a method in psm_grasp.py to grasp the generated marker with PSM1.

   Note

   Some values tends to stuck in the simulator. Thus, at the beginning of the program, it is a good idea to reset the arm:

   #Reset the arm
   psm.move_tcp_to([0.0, 0.0, -0.12], 0.01, 0.01)
   psm.move_jaw_to(0.0, 0.1, 0.01)
   

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Links

• Download and compile dVRK 2
• Marker examples
• Numpy vector magnitude
• Numpy linspace