Spatial (2-dimensional) double inverted pendulum (DIP) system is a simplified model that can represent many real life scenarios, such as standing human or a launching rocket. It is an inherently unstable and nonlinear system, and thus is an ideal benchmark problem to demonstrate the validity of newly-developed control theory, including advanced nonlinear, adaptive and robust control. To develop a cartesian moving platform that physically balance a DIP, I worked closely with two master of engineer students and performed a wide variety of tasks. We modelled and analyzed both single and double inverted pendulum systems using MATLAB and Simulink. In the same software environment, we designed feedback controllers and confirmed the systems’ feasibility. Following the simulation process, we designed and fabricated physical platform and pendulum hardware and then performed identification process to obtain the plant’s actual physical parameters. Lastly, the feedback controller was implemented using the dSPACE microcontroller.
Spatial Double Inverted Pendulum
2 DOF planar inverted pendulum successfully balanced using X-Y table, where the control algorithms was embedded into a dSPACE microcontroller
2 DOF Single Inverted Pendulum Simulation
State-space dynamic model of 2 DOF (2 rotations at the base) spatial inverted pendulum was built in Simulink environment. We then designed an LQR (linear quadratic regulator) controller to balance it in simulation. The example animation shows the platform bringing the pendulum to vertical steady state from a slightly tilted state in the beginning.
4 DOF Double Inverted Pendulum Simulation
State-space dynamic model of 4 DOF (2 rotations at the base and 2 rotations between the two rods) spatial inverted pendulum was built by expanding the 2 DOF case. Similarly, an LQR controller was specifically designed to balance it in simulation. As the animation shows, the successful simulation proved the overall hardware feasibility. The simulation was also helpful in specifying dimension and mass properties of the pendulum and estimating the required actuator outputs.
Evolution of Balancing Platform Hardware
Our initial balancing platform was in CoreXY configuration, which used a timing belt and two motors to drive X- and Y-direction carriages using parallel kinematics. With CoreXY configuration, the two motors (the two largest sources of inertia) were fixated to the platform base and kept stationary, and thus rapid acceleration was possible. Indeed, it was successful in balancing spatial single inverted pendulum.
However, as we converted the CoreXY platform into the double inverted pendulum mode, several factors that were unaccounted for in simulation started to bite. Namely, the disturbance from the sensor cable, stretchiness of the timing belt, and added vibration mode from the pendulum protection cage.
Therefore, a new platform using regular X-Y table configuration was built. The new design used a stiffer alternative carriage driving mechanism and were given a larger range of movement in case future experimenters need it. The upper pendulum universal joint was also revised to eliminate the need of a protection cage.
Modular and Configurable Inverted Pendulum Joint
As a laboratory bench mark platform, the system needs to be re-configured to represent different system plant scenarios. Thus I designed a modular universal pendulum joint (with built-in angular sensors) that could be removed or installed at experimenters’ will, making the conversion between single and double pendulum states simple.
Furthermore, each universal joint module has one of its axes lockable. Thereby, experimenters could select from planar (1-dimensional) and spatial (2-dimensional) settings without replacing any parts from the pendulum sub-assembly.
Steel Cable Drives
Steel cable drive mechanism was chosen to replace the timing belt used in the former CoreXY table to reduce the deformation of the transmission durig operation. The steel cable drive is inherently backlash-less and is quite during operation.
For one of the axes, two sets of cable drive are arranged in parallel and are synchronized using a transmission rod. This is to ensure the wide carriage does not experience yaw moment due to unbalanced pulling action from the two parallel cable drives.
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Cable Carriers
When directly placed on the ground, sensor cables’ weights and reaction forces acting on the X-Y table carriages vary from one carriage location to another and thus are very difficult to model and identify. Inspired from my previous work experience at a flatbed inkjet printer company, where cable carriers are used to achieve smooth ink delivery from tanks to the moving printhead carriage, cable carriers are included in the X-Y table to mitigate the issue.
Moreover, twistable robot cables were also used to reduce the resistance during bending. These cables also guarantee longer life under action when compared to regular products.
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