Scenario
For this analysis, we consider a simple arm manoeuvre, which is to pick up a sample from the ground.
The steps involved in the arm manoeuvre are as follows:
- The arm base should rotate right to reach the desired position.
- The hinge joint should be lowered to reach the sample.
- The gripper should close to pick up the sample.
As determined in rotational dynamics we calculate the rotational inertia of the arm which is used to then determine the torque required to move the arm.
Using the rotational kinematics we can determine the angular acceleration of the steps involved in the arm manoeuvre.
\[\alpha_{\text{required}} = \frac{2}{t^2} \cdot \Delta \theta\] \[\tau = I \alpha\]Work Done
The work done by the arm to perform the sample collection is given by:
\[W_{\text{arm}} = \tau \cdot \Delta \theta\]where:
- \(\tau\) is the torque required to move the arm.
- \(\Delta \theta\) is total the angular displacement.
Power Requirements
The power required to perform the arm manoeuvres is given by:
\[P_{\text{arm}} = \frac{W_{\text{arm}}}{t}\]where:
- \(W\_{\text{arm}}\) is the work done by the arm.
- \(t\) is the time taken to perform the task.
Energy Requirements
The energy required to perform the arm manoevers is given by:
\[E_{\text{arm}} = W_{\text{arm}}\]where:
- \(E\_{\text{arm}}\) is the total energy required.
- \(W\_{\text{arm}}\) is the work done by the arm.
Efficiency
In a real-world setting, the system’s components might show inefficiencies, which can be accounted for by introducing an efficiency factor. The efficiency factor is given by:
\[E_{\text{source\_arm}} = \frac{E_{\text{arm}}}{\eta}\]where:
- \(\eta\) is the efficiency factor of the system.