Analysis and trajectory planning for anti-swing on a one-stage cart pendulum

. In order to achieve smooth and anti-swing positioning process, a smooth trajectory planning scheme, which is mainly the linear combination of trigonometric functions, is proposed for a one-stage cart pendulum. Based on the internal dynamics of the cart pendulum system, the planning trajectory of the cart is derived from the anticipated swing angle of the pendulum bar, which satisfies the initial and terminal state constraints. Experiments are conducted to demonstrate the performance of anti-swing with the introduced planned trajectory.


Introduction
Pendulum on cart is a typical under-actuated system in physical applications. Within this system, the cart translation causes the pendulum to swing back and forth, which not only reduces the efficiency of the transferring but also leads to potential risks of hitting. The main control objectives of positioning in such system can be summarized in moving the cart to arrive at the desired position within a short time and suppressing the pendulum swing within a given domain for safety reasons.
Numerous researchers have studied the anti-swing control problem and proposed different control strategies for the under-actuated cranes described in [1]. The transition task is treated as a two-point boundary value problem in [2] to deal with the anti-swing and position control. In [3], a kinematic coupling-based off-line trajectory planning method is proposed to achieve smooth trolley transportation and small payload swing. Optimal sliding mode control method is applied in [4], while inverse transfer function method is taken into account in [5]. A control approach based on a model predictive control and a particle swarm optimizer is proposed for limiting the transient and residual swing in [6]. Closedloop signal shaping is utilized in [7] by placing the Smith predictor inside the feedback loop to improve manual control of a 3D crane. Trapezoidal acceleration spline curve programming and backstepping control strategy are applied for anti-swing and positioning in [8] and [9], respectively. In [10], a controller is designed for the under-actuated system with the knowledge of differential flatness. Meanwhile, optimal preview control is applied to complete the rapid positioning and swing angle suppression when utilizing the future information in [11].
Generally, trajectory planning and advanced control algorithm design are two available techniques to achieve the smooth and anti-swing transferring for the under-actuated systems. However, trajectories in the previous works were obtained only from the system kinematics, dynamics of the overall system was out of attendance when generating the trajectories. In this paper, focusing on the aspect of trajectory planning, a smooth trajectory planning scheme is proposed for a one-stage cart pendulum based on the internal dynamics of the physical system. In terms of the anticipated swing angle of the pendulum bar, the planning trajectory of the cart is derived from mostly the linear combination of sine and cosine functions, with specified initial and terminal state constraints. Simulation and experiments have been conducted to demonstrate the anti-swing performance of the introduced planned trajectory on a physical platform.

Model of the one-stage cart pendulum
As depicted in figure 1, the one-stage cart pendulum consists of a straight-line rail, a cart, a pendulum bar, and a driving unit, which drives the cart via a toothed belt by a synchronous motor. A pre-tuned velocity-loop controller of the synchronous motor is configured, such that an ideal control of the cart's speed can be presumed. In figure 1, XOY is the reference frame of the system, while the rest of notions are described in table 1. It is reasonable to assume that the mass of the pendulum bar is uniformly distributed. According to the Lagrange's equations, the equations of motion for the one-stage cart pendulum could be derived as Equation (1) represents the actuated part of the system, while equation (2) denotes the underactuated part, which describes the coupling dynamics of the cart displacement and pendulum swing angle. As the dynamics of the cart is fully controlled by the synchronous motor, the acceleration of the cart could then be chosen as a substitute for the new control x u =  3 Trajectory planning

Initial and terminal states in positioning
Taking the system of figure 1 into considered, a positioning movement represents the cart, taking with the pendulum bar, moves from one position to another setpoint position in a finite translation time. Once the desired position of the cart was achieved, the swing amplitude of the pendulum bar must be suppressed within an acceptable domain. That is, the pendulum bar is required to be anti-swing especially at the end of the cart motion. Table 2 gives the initial states and terminal states for the one-stage cart pendulum when performing point-to-point cart translation in a finite moving time from 0

Trajectory generation
Suppose x * and θ * were the solution for the internal dynamics, i.e., equation (4) Obviously, equation (7) and equation (8)    On the other hand, once p was determined, the maximal magnitude of the desired swing angle of the pendulum in equation (6) figure 2, the initial and terminal states meet the requirements in table 2. Figure 3 shows the corresponding discrete samples of the cart pendulum during the setpoint transition, where the pendulum angle changes smoothly.

Experiment
Experiments were performed on the one-stage cart pendulum, named GLIP2001 series, manufactured by Googol Paradox Technology to verify the anti-swing performance of the proposed trajectory planning approach. The experiment platform has a cart stroke of 0.6 meter, which is driven by a servo motor via a toothed belt. The servo motor, is operated in velocity-loop mode with an encoder adopting the resolution of 10000 pulses per revolution. Cart placement is measured through the encoder of the servomotor indirectly. That is, one pulse represents 0.46/37500.0 meter. Swing angle of the pendulum bar is obtained by an independent encoder with the resolution of 10000 pulses per revolution. Commands and ITM Web of Conferences 47, 02012 (2022) CCCAR2022 https://doi.org/10.1051/itmconf/20224702012 feedback signals are manipulated by the PC-based motion controller, GT-400-SV, provided by Googol Technology. Real-time controls, feedforward and feedback control schemes, are handled by means of the real-time toolbox in SIMULINK environment.   In the second set of experiments, terminal planning time was fixed as 2 seconds, while the target cart position was set from 0.2 meter to 0.5 meter. Experiment and numerical results are given in figure 5 and table 4, respectively. It could be found that, for a specific terminal planning time, the longer moving target position generates a larger pendulum swing angle.

Conclusion
In this paper, a smooth trajectory planning approach, which is mainly the linear combination of sine and cosine functions, is proposed for the anti-swing setpoint transferring process of a one-stage cart pendulum. Based on the internal dynamics of the cart pendulum system, the planning trajectory of the cart is derived from the anticipated swing angle of the pendulum bar, which satisfies the initial and terminal state constraints.
To illustrate the anti-swing performance of the introduced trajectory planning scheme, simulation and experiments are conducted and tested on a physical platform.