The project has two parts:
The control law was found using direct methods. Euler’s method was used for the propagation of dynamics in time. There are constraints on the states and the control action. The following section illustrates the energy-optimal solutions for various link length combinations.
The following videos show the path followed by the manipulator under their respective control law.
link-1: 0.35 m, link-2: 0.3 m
link-1: 0.55 m, link-2: 0.1 m
link-1: 0.1 m, link-2: 0.55 m
Since the controller is trying to minimise the control effort, we can see that the distal link is always pulled closer towards the proximal link to reduce the effective inertia against which the actuators work. Thus reducing the effort. Another interesting fact to note is that since actuating motor-2 (joining link-1 and link-2) to move link-2 would cause a reaction on link-1, which seems to be exploited by the controller. This can be observed in the initial phase of movement where the link-1 goes below the starting pose, where only the motor-2 was being actuated and once the link-2 is close enough motor-1 starts actuating. The same is observed at the end of the path where motor-1 need not work. Actuating only motor-2 (Not significant actuation of motor-1) would cause both the links to move towards the required goal.
The following videos show the path followed by the manipulator under their respective control law.
link-1: 1 kg, link-2: 8 kg
link-1: 4.5 kg, link-2: 4.5 kg
link-1: 8 kg, link-2: 1 kg
The difference is subtle in this case and can be seen by observing how much below the link-1 moves in cases 1 and 3. In the third case link-1 moves very little initially which is coherent to the case that it has the maximum inertia. In case 1, link-1 moves much more backwards before moving forward. It can also be observed that in case 1 since the distal link is heavier, it is maintained closer to the proximal link than it is in case 3. This makes sense if we need to minimise control energy required to complete the process.
The project was done in collaboration with my colleague R. Gautham for the course “Design, Analysis and Control of Serial Manipulators”, instructor, Prof. Sandipan Bandyopadhyay. For further information, please find the complete project report here.
Optimal Control of 2-R Manipulator