Background
Principle of two-wheeled IP with an extended rod
System description
Advantages of using the proposed design with standard wheels over omnidirectional wheels
Description of the system DOFs
Ser. | Subtask | DOFs associated | Right wheel motor
\( \tau_{R} \)
| Left wheel motor
\( \tau_{L} \)
| Linear actuator I
\( F_{1} \)
| Linear actuator II
\( F_{2} \)
|
---|---|---|---|---|---|---|
a | Moving till the picking place |
\( _{{\delta_{R} }} \), \( _{{\delta_{L} }} \), \( _{\theta } \)
| ✓ | ✓ | × | × |
b | Extension of the IB |
\( _{{\delta_{R} }} \), \( _{{\delta_{L} }} \), \( _{\theta } \), \( _{{h_{1} }} \)
| ✓ | ✓ | ✓ | × |
c | Extension of the end-effector |
\( _{{\delta_{R} }} \), \( _{{\delta_{L} }} \), \( _{\theta } \), \( _{{h_{2} }} \)
| ✓ | ✓ | × | ✓ |
d | Reverse motion of the end-effector |
\( _{{\delta_{R} }} \), \( _{{\delta_{L} }} \), \( _{\theta } \), \( _{{h_{2} }} \)
| ✓ | ✓ | × | ✓ |
e | Contraction of the IB |
\( _{{\delta_{R} }} \), \( _{{\delta_{L} }} \), \( _{\theta } \), \( _{{h_{1} }} \)
| ✓ | ✓ | ✓ | × |
f | Placing of the object |
\( _{{\delta_{R} }} \), \( _{{\delta_{L} }} \), \( _{\theta } \), \( _{{h_{2} }} \)
| ✓ | ✓ | × | × |
Mathematical modelling
Parameter | Description | Value | Unit |
---|---|---|---|
\( m_{1} \)
| Mass of the chassis | 3.1 | kg |
\( m_{2} \)
| Mass of the linear actuators | 0.6 | kg |
\( m_{\rm w} \)
| Mass of wheel | 0.14 | kg |
\( g \)
| Gravitational acceleration | 9.81 | m/s2
|
\( l \)
| Distance of chassis’s centre of mass for wheel axle | 0.14 | m |
\( R \)
| Wheel radius | 0.05 | m |
\( J_{1} \)
| Rotation inertia of | 0.068 | kg m2
|
\( J_{2} \)
| Rotation inertia of | 0.0093 | kg m2
|
\( J_{\rm w} \)
| Rotation inertia of a wheel | 0.000175 | kg m2
|
\( \mu_{c} \)
| Coefficient of friction between chassis and wheel | 0.1 | Ns/m |
\( \mu_{\rm w} \)
| Coefficient of friction between wheel and ground | 0 | Ns/m |
\( \mu_{1} \)
| Coefficient of friction of vertical linear actuator | 0.3 | Ns/m |
\( \mu_{2} \)
| Coefficient of friction of horizontal linear actuator | 0.3 | Ns/m |
Deriving equations of motion
Modelling using Lagrange formulation
-
\( q_{i} \quad (i = 1,2, \ldots ,n) \) are generalized coordinates such as: \( q_{i} = \left[ {\begin{array}{*{20}c} {h_{1} } & {h_{2} } & \theta & {\delta_{\text{L}} } & {\delta_{\text{R}} } \\ \end{array} } \right] \)
-
\( f_{i} \) is generalized forces that contain all the given forces in the system acting along the coordinates such as: \( f_{i} = \left[ {\begin{array}{*{20}c} {F_{1} } & {F_{2} } & 0 & {\tau_{\text{L}} } & {\tau_{\text{R}} } \\ \end{array} } \right] \)
-
\( D \) is the dissipation function and illustrated as \( D = \tfrac{1}{2}bq_{i}^{2} \)
State space modelling
State space modelling
-
Right wheel displacement, \( \delta_{\text{R}} \)
-
Left wheel displacement, \( \delta_{\text{L}} \)
-
Chassis pitch angle, \( \theta \)
-
Vertical linear link displacement, \( h_{1} \)
-
Horizontal linear link displacement, \( h_{2} \)
-
Right wheel velocity, \( \dot{\delta }_{\text{R}} \)
-
Left wheel velocity, \( \dot{\delta }_{\text{L}} \)
-
Chassis angular velocity, \( \dot{\theta } \)
-
Vertical linear link velocity, \( \dot{h}_{1} \)
-
Horizontal linear link velocity, \( \dot{h}_{2} \)
-
\( \tau_{\text{R}} \) and \( \tau_{\text{R}} \) are the required torques for the right and left wheels,
-
\( F_{1} \) and \( F_{1} \) are the generated linear force by the linear actuator for moving the payload in a vertical and horizontal direction, respectively.
Numerical simulation
Open-loop system response
Control scheme design
PID control without switching mechanisms
Payload free movement (h1 = h2 = 0)
Simultaneous horizontal and vertical motion (h1 and h2 ≠ 0)
Design of switching mechanisms
Payload vertical movement only
Payload horizontal movement only
Payload simultaneous horizontal and vertical movements
-
Checking the robustness of the developed control approach. In Fig. 19, the IB leans in the opposite direction to compensate for the change in the position of the COM due to extension of \( h_{2} \). Activating each individual actuator at a certain time tends to act as a sudden disturbance, in particular changing \( h_{2} \) to the system which already achieved a stability.
-
Adding switching mechanisms mimics real scenarios in practical applications where not all actuators work at the same time.
Conclusions
-
Testing the vehicle in confined space for path tracking and picking and placing an object, this will include consideration of additional weight of an object, tracking a pre-specified trajectory, picking and placing the object from a certain location, carrying it and placing in desired location.
-
Further investigation will include also workspace and kinematics analysis of the vehicle.
-
Implementation of various optimization tools including bacterial forging (BF), spiral dynamics (SD) and hybrid spiral dynamics bacterial chemotaxis (HSDBC) for better performance of the system and improved energy consumption.
-
Further investigation of the linear model of the system will be carried out while implementing various control approaches including fuzzy logic control (FLC).