Design and Application of Hydraulic Inverted Pendulum

Abstract

This paper briefly describes the designing process of a hydraulic inverted pendulum including hardware and software design. First, the mechanical structure, including the components of the platform will be introduced. Second, the electrical system including controllers for receiving signals from sensors which measure the variables important for controlling inverted pendulum is about to be shown. Afterwards, the paper will present a mathematical model of the whole platform, then shows up an open loop simulation established by AMESim and Simulink in order to analysis its dynamic characteristic. By comparing the simulation result and reality, the rationality of mathematical model is finally verified.

1 Introduction

Closed-circuit hydraulic system has already been implemented not only in application of steering system for wheel loader but also in other construction vehicles [1, 2]. It shows greater efficiency and higher power density compared to opened-loop hydraulic system. In particular, Direct-Drive Volume Control (DDVC) system has gained sufficient attentions in studying its dynamic characteristics and control strategies [3, 4]. Compared with the traditional valve control system, the DDVC system has the technical advantages of a high power–gain ratio, high integration, environmental friendliness and high efficiency and energy saving [5, 6]. Since pump-controlled motor servo system is similar to conventional DDVC system since its actuator is hydraulic motor instead of hydraulic cylinder, its study of dynamic characteristic is still about to be explored. Meanwhile, the inverted pendulum has become a hot topic as a typical nonlinear and unstable system. It is a simple pendulum whose mass is located in the air. The system presents an unstable equilibrium in a vertical position. This position is maintained by the control of a movable cart [7]. Its control strategy varies from reinforcement learning to fuzzy control [8, 9], which mainly used to test the effectiveness of various control methods, such as their ability to handle nonlinear and unstable issues effectively etc.

In addition, the inverted pendulum has been applied in many different research fields, particularly in robotics. Since the balance control of robots has the similar objective compared to the inverted pendulum, the mathematical model of certain robots can be considered as a variation of inverted pendulum. For instance, the inverted pendulum robot [10] uses wheels in order to maintain the balance. By limiting the robot’s posture, the wheeled inverted pendulum can replace the tradition model which requires more complicated control methods to maintain the posture’s stability [11, 12]. This mathematical model has been applied not only in electric-driven robot [13] but also in hydraulic powered robot [14].

However, research combining hydraulic system and inverted pendulum is not as extensive as expected. Many researchers focus mainly on either the control strategies for the former one or the improvement of the control method towards the latter one powered by electric motor, even the control strategy for triple inverted pendulum has already used neural networks to control the complete system within 3.0 s [15]. Let alone building the actual platform in order to verify the control method. In this paper, hardware design for hydraulic inverted pendulum, the method of establishing mathematical model of the system will be provided afterwards. Then, the paper presents an open-loop control using co-simulation between AMESim and Simulink in order to study its dynamic characteristic under step disturbance. Finally, by comparing the result between simulation and field test of step response of the system the paper verifies the rationality of mathematical model.

2 Hardware Composition of Hydraulic Inverted Pendulum Platform

2.1 Mechanical Structure of Hydraulic Inverted Pendulum Platform

The mechanical structure of the proposed hydraulic inverted pendulum platform using pump-controlled motor servo system includes two parts: the pump station and the motor-inverted pendulum platform. The 3D models for both of the parts are showed in Figs. 1 and 2. The length of the whole platform is about 1000 mm at maximum, while its full width is about 215 mm and the height is approximately 128 mm.

Fig. 1

Mechanical structure of the pump station

Fig. 2

Mechanical structure of the motor-inverted pendulum platform

The pump station includes servo motor, oil tank, and valve blocks. The motor-inverted pendulum platform contains not only inverted pendulum itself, but also hydraulic motor and the valve blocks which are used to connect between hydraulic motor. Apart from that, there are connectors which are applied for installing encoders for measuring angle displacement and linear displacement for pendulum rod and slider.

One of the advantages of applying hydraulic closed circuit in driving inverted pendulum is that closed circuit provides higher power and efficiency compared to open hydraulic circuit. Furthermore, since the shaft speed of hydraulic motor is controlled eventually by the servo motor, transitions of the speed and torque produced by hydraulic motor can be accomplished directly through reducer, and pulleys in the guide rail. So, it’s obvious that the volume of the whole hydraulic system hardware is much smaller compared to the way of using double piston rod hydraulic cylinder and servo valve.

2.2 Hydraulic Circuit of the Hydraulic Inverted Pendulum Platform

The hydraulic circuit is the most critical part of the hydraulic inverted pendulum platform, which schematic diagram is shown in Fig. 3.

Fig. 3

Schematic diagram of the hydraulic system. 1: Oil tank, 2: hydraulic pump, 3: check valve, 4: relief valve, 5: hydraulic motor, 6: servo motor

The shaft speed of hydraulic motor is controlled directly by the pump, while the flow rate of the pump is dominated by the speed changing from the servo motor. The check valves are acquired to replenish oil from the oil tank since the pump and motor both have external leakage. The cracking pressure of the check valves is small enough for the hydraulic system in order to accomplish replenishing oil without using accumulator or slippage pump. Finally, the relief valves are applied to control the maximum pressure of the whole system.

2.3 Electrical System of Hydraulic Inverted Pendulum Platform

The electrical system includes sensors, motion controller, servo controller and PC, the components of which are shown in Fig. 4. The limit switches ensure the extreme displacement of the slider, giving protection to the guide rail. The motion controller is the key part of the electrical system, it receives signals not only from the encoders but also from the limit switches. The connection between PC and motion controller is accomplished by Ethernet, while the latter using Ether CAT to establish the communication to the servo controller. Last but not least, the power supply and the control signal towards the servo motor, were all provided by the servo controller.

Fig. 4

Components of electrical system used in hydraulic inverted pendulum

3 System Modelling and Dynamic Characteristics

3.1 Mathematical Model of Hydraulic Inverted Pendulum

Assuming that both the car and the pendulum are considered as rigid bodies, the conveyor belt is not retractable, and ignoring air resistance and friction resistance moment of driving components, the differential equations of the inverted pendulum are:

$$\left\{ {\begin{array}{*{20}l} {\ddot{x} = - \frac{{m^{2} l^{2} g}}{{(M + m)J + Mml^{2} }}\theta - \frac{{(J + ml^{2} )b}}{{(M + m)J + Mml^{2} }}\dot{x} + \frac{{J + ml^{2} }}{{(M + m)J + Mml^{2} }}F} \\ {\ddot{\theta } = \frac{mlb}{{(M + m)J + Mml^{2} }}\dot{x} - \frac{(m + M)mgl}{{(M + m)J + Mml^{2} }}\theta - \frac{ml}{{(M + m)J + Mml^{2} }}F} \\ \end{array} } \right.$$

(1)

where $x$ is the displacement of the slider, $\theta$ is the angle displacement of the rod, $M$ and $m$ are the masses of the rod and the slider, $F$ is the external force given to the slider, $J$ is the rotational inertia of the rod, and $l$ is the distance from the centre of mass of the rod to the axis of rotation.

As for the hydraulic system, ignoring the flow pulsation of hydraulic pump and hydraulic motor and pressure loss in pipeline and frictional resistance loss in the hydraulic system, The flow continuity of the hydraulic pump (high pressure side) can be described as:

$$\left\{ {\begin{array}{*{20}l} {q_{ph} = d_{p} n_{p} - C_{tp} p_{h} } \\ {q_{ph} - C_{tm} p_{h} = D_{m} \frac{{d\theta_{m} }}{dt} + \frac{{V_{h} }}{{\beta_{e} }}\frac{{dp_{h} }}{dt}} \\ \end{array} } \right.$$

(2)

Similarly, the flow continuity of the hydraulic pump (low pressure side) can be considered as follows:

$$\left\{ {\begin{array}{*{20}l} {q_{pl} = - d_{p} n_{p} - C_{tp} p_{l} } \\ {q_{pl} - C_{tm} p_{l} = - D_{m} \frac{{d\theta_{m} }}{{d_{t} }} + \frac{{V_{l} }}{{\beta_{e} }}\frac{{dp_{l} }}{dt}} \\ \end{array} } \right.$$

(3)

The balance equation of the shaft torque of hydraulic motor can be described as this:

$$D_{m} p_{h} = J_{t} \frac{{d^{2} \theta_{m} }}{dt} + B_{tm} \frac{{d\theta_{m} }}{dt} + T_{L}$$

(4)

Due to disregarding the frictional resistance moment, the load torque of the reducer is directly applied to the slider of the inverted pendulum through the external force converted by the pulley, so the equations from the load torque to the torque generated by hydraulic motor are:

$$\left\{ {\begin{array}{*{20}l} {T_{d} = F \cdot r} \\ {T_{m} - T_{d} = I\frac{{d\omega_{m} }}{dt}} \\ {T_{m} = n \cdot T_{L} } \\ {\dot{x} = \omega_{m} r} \\ \end{array} } \right.$$

(5)

Finally, the differential equation of the whole system can be described as follows:

$$\left\{ {\begin{array}{*{20}l} {\ddot{x} = Z\theta + V\dot{x} + O\dot{P}_{h} } \\ {\ddot{\theta } = N\theta + M\dot{x} + S\dot{P}_{h} } \\ \end{array} } \right.$$

(6)

The value of Z, V, O, W, N, M, S, T are shown as follows:

$$\left\{ \begin{gathered} \begin{array}{*{20}l} {Z = \frac{{r^{2} m^{2} gl^{2} }}{\Theta }} \\ {O = \frac{{nrD_{m} (J + ml^{2} )}}{\Theta }} \\ {V = \frac{{(J + ml^{2} )(r^{2} b + n^{2} B_{tm} )}}{\Theta }} \\ \end{array} \hfill \\ S = \frac{1}{r\Omega }\left[ {\frac{{(n^{2} J_{t} + I)(J + ml^{2} )}}{\Theta } - 1} \right] \hfill \\ N = \frac{1}{\Omega }\left[ {(m + M)mgl - \frac{{(n^{2} J_{t} + I)m^{3} gl^{3} }}{\Theta }} \right] \hfill \\ M = \frac{1}{\Omega }\left[ {mlb - \frac{{ml(n^{2} J_{t} + I)(J + ml^{2} )(r^{2} b + n^{2} B_{tm} )}}{{r^{2} \Theta }} + \frac{{n^{2} mlB_{tm} }}{{r^{2} }}} \right] \hfill \\ \end{gathered} \right.$$

(7)

While:

$$\left\{ {\begin{array}{*{20}l} {\Omega = (M + m)J + Mml^{2} } \\ {\Theta = r^{2} \Omega + (n^{2} J_{t} + I)(J + ml^{2} )} \\ \end{array} } \right.$$

(8)

The meanings of the variables of the equations above are shown in Table 1.

Table 1 Meanings of variables from Eqs. (2) to (7)

3.2 Dynamic Characteristics of Hydraulic Inverted Pendulum

The form of co-simulation using AMESim and Simulink is shown in Fig. 5.

Fig. 5

Scheme of co-simulation for hydraulic inverted pendulum

The parameters of the system are showed in Table 2. Given a step input to the system (500 rpm), the corresponding relationship between the angle of the inverted pendulum rod and the displacement of the slider can be obtained in Figs. 6 and 7. And the simulation result of rotary velocity of servo motor and hydraulic motor are shown in Fig. 8.

Table 2 Parameters used for co-simulation

Fig. 6

Slider displacement under step input

Fig. 7

Angle displacement under step input

Fig. 8

Simulation result of rotary velocity of servo motor and hydraulic motor

From the simulation results, by giving a step input, it can be concluded that the inverted pendulum itself is an unstable system, and the hydraulic system has a certain degree of hysteresis compared to the reference step input. By combining the characteristic of inverted pendulum and hydraulic system, the traditional control strategy such as PID or LQR are no longer valid for hold the rod to maintain balance.

4 Field Test on Hydraulic Inverted Pendulum Platform

4.1 Platform Establishment

The actual platform in kind as mentioned before is displaced in Fig. 9, including pump station, inverted pendulum, and electrical system including motion controller and servo controller.

Fig. 9

Hydraulic inverted pendulum platform in kind

4.2 Program Design

In order to achieve hardware-in-loop control (HLP) of hydraulic inverted pendulum, the article implements level 2 MATLAB s-function to achieve programming for motion controller. The flow chart of the codes for controlling motion controller is shown in Fig. 10.

Fig. 10

Flow chart of the codes for controlling motion controller

4.3 Results and Discussions

By giving a step input (500 rpm), the comparison between simulation result, referenced input and actual rotary velocity of the hydraulic motor is shown in Fig. 11. According to the result, t because of the errors generated by the sensor, the actual value of the rotary velocity of hydraulic motor can’t remain constant. Also, some of the variables are actually ignored such as the moment of inertia of the joints connecting between hydraulic pump and servo motor or the joints connecting between the reducer and pulley etc. The rotary of hydraulic motor is able to stabilize at approximately 500 rpm, which can approximately fit into the simulation result presented in Fig. 11. It can be seen from this that the established mathematical model can basically conform to the actual system.

Fig. 11

Comparison between actual value and simulation result

5 Conclusion

In this paper, an introduction of the hardware design of hydraulic inverted pendulum is presented, including mechanical structure, hydraulic system and electrical system. The mathematical model of the hydraulic inverted pendulum is showed afterwards, from inverted pendulum to hydraulic system. The simulation result given by step input shows that the hydraulic system contains hysteresis when driving the inverted pendulum. Finally, the paper presents the field test in order to verify the characteristic of the model, from the result, it can be concluded that the mathematical model can generally fit into the actual system to show its characteristic. In the future, close loop control strategy will be established in order to achieve the swing up and balance control.