The capstone project of the Modern Robotics: Mechanics, Planning, and Control Specialization involves several concepts from robotics put together to create a kinematic simulation of a mobile manipulator.
The robot to be kinematically simulated is the Kuka YouBot.
Fig: Kuka YouBot
This Robot has to perform a pick place operation where in the object to be moved is a cube of side 50mm. The entire operation has to be kinematically simulated by the program and a final CSV file of the timestamped robot states has to be generated for visualization in CoppeliaSim. The scene for visualization can be found here.
Fig: Kuka YouBot Visualised in coppeliaSim
- MATLAB
- coppeliaSim
The Project is divided into 3 milestones for simplicity.
The objective of this milestone is to write a program NextState.m. This program calculates the resultant robot configuration if a given configuration is subjected to a given velocity vector. It does so by integrating the velocity vector over the given time interval.
The robot configuration at any point of time is represented by the 13-vector:
chassis phi, chassis x, chassis y, J1, J2, J3, J4, J5, W1, W2, W3, W4, gripper state
The velocity Vector is likewise represented as:
js1, js2, js3, js4, js5, ws1, ws2 ws3 ws4
where: js1 is joint 1 velocity, ws1 is wheel 1 velocity.
The end-effector trajectory is a series of transforms in the world/space frame that represent the desired position for the end effector at each timestep. This is generated by the function TrajectoryGenerator.m.
There are two methods of producing the required trajectory.
Fig: Screw and cartesian trajectories
And two methods of time scaling the trajectory.
The motion of the end-effector through space is a continous screw motion from start to finish.
The motion of end-effector is a separate into a simultaneous rotation and straight line translation.
The trajectory's progression through time is represented by a cubic polynomial.
s = 0 at start. s = 1 at end. T = total time of motion
The trajectory's progression through time is represented by a quintic polynomial.
s = 0 at start. s = 1 at end. T = total time of motion
The trajectory is outputed in a matrix where each row represents a transformation matrix along with the gripper state (whether closed or open):
r11, r12, r13, r21, r22, r23, r31, r32, r33, px, py, pz, gripper_state
The purpose of this milestone is to write code to calculate the velocity vector which when applied to the robot configuration over the given time interval will minimize the error between the desired end-effector position and the actual end-effector position at the current timestamp.
This controller will use the control law:
Fig: feedforward + PI control
This equation will output the required end-effector twist that will take the end-effector to the desired position in the given time interval. It is implemented in the program FeedbackControl.m.
The twist has to be converted to the velocity vector by using a Jacobian pseudo inverse. This calculation is given by the equation:
Fig: Calculate wheel and joint speeds.
This is implemented in the program end_eff_twist_to_joint_wheel_velocities.m.
The various described components will be combined to form the final program. The overall flow of the program is described in the chart given below.
Fig: Program flow
