Upright position control

A number of researchers have studied the problem of stabilising inverted pendulums with passive joints at the upright position (Sahba 1983; Furut et al. 1984; Meier Farwig and Unbehauen 1990; Arai and Tachi 1991; Larcombe 1992; Medrano-Cerda et al. 1995; Fer and Enns 1996; Eltohamy and Kuo 1998; Lakshmi 2007; Zhai et al. 2007;

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Farmanbordar et al. 2011; Wu et al. 2011; Vinodh Kumar and Jerome 2013). The problem of stabilising and controlling the attitude of a triple inverted pendulum was also studied by Furut et al. (1984). In this system, the lower link was hinged to the ground and, to support the control of the whole system, horizontal bars were fixed to the links to increase their moments of inertia.

In the same vein, the problem of stabilising control of a double inverted pendulum mounted on a single parallel bar was presented by Hauser and Murray (1990). The researchers proposed a nonlinear control method of approximate linearisation to cause the system to move along the set of inverted equilibrium positions. In the suggested design, a braking mechanism was supplied to unactuated joints to reduce the coupling between the linkages and to simplify the control problem,. The simulation results show that this strategy functioned well but required slow motion.

Arai and Tachi (1991) published a paper in which they described the method of control of two degrees of freedom systems. The constructed system consisted of an active joint with an actuator and a passive joint with a holding brake instead of an actuator. The proposed controller was based on using the coupling physical characteristics of manipulator dynamics (Arai and Tachi 1991).

Medrano-Cerda et al. (1995) designed a robust computer control system for balance and position control of double and triple link inverted pendulums (TLIPs). The controllers were based on linearised models of the pendulums and included integral actions and optimal state feedback implemented via functional observers.

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In a similar study, Fer and Enns (1996) discussed the difficulty of utilising a single control input to stabilise a triple inverted pendulum on a cart which moves on a rail. The system has four degrees of freedom, so the stabilisation problem is more complicated than in some other studies (Furut et al. 1984; Meier Farwig and Unbehauen 1990). Fer and Enns (1996) used two types of controller to achieve a solution. The first was the LQR method (Anderson and Moore 1990) and the second was the multiple time-scale approximation (Bugajski and Enns 1992) with a nonlinear dynamic inversion approach.

Eltohamy and Kuo (1998) used a numerical optimisation algorithm for the controller design, which included a globally convergent numerical technique. Their designed controller was produced as an optimisation problem which accounted for the constraints of physical boundaries, system stability conditions, and the nonlinear infinite dimensional difference.

Bogdanov (2004) presented a comparison between various optimal controller algorithms for use in a Double Inverted Pendulum Controller (DIPC). In this research a feedback gain matrix was used to stabilise the system. The gain matrix manipulation was achieved via employing the LQR, State Dependent Riccati Equation (SDRE), Neural Network (NN) and NN+LQR/SDRE control schemes.

Delibasi (2007) described the stabilisation and disturbance rejection of an inverted double pendulum, which was achieved experimentally. In this research, the state feedback controller idea was dependent on the use of Proportional Integral Derivative (PID) controller technology.

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Wongsathan and Sirima (2009) described how to find the optimal values of the controller to stabilise an inverted pendulum system using a genetic algorithm. There were significant simulated results obtained from the preliminary analysis of the system response with the optimised controller.

Sehgal and Tiwari (2012) used a continuous LQR optimal control system to understand the mechanisms involved in balancing a triple inverted pendulum. The simulation results showed that the controller successfully stabilised the system with good performance. In the same manner Gupta et al. (2014) utilised the LQR system to stabilise a triple link inverted pendulum on a moving cart. The presented results show that the proposed controller could achieve the balancing of a system of upside down pendulums.

Yadav (2012) investigated the stablisation of a single input and multiple outputs Double Inverted Pendulum (DIP) on a cart. They presented the LQR to maintain stabilisation about the upright equilibrium position. For the same system structure, Singh and Yadav (2012) compared the optimal LQR controller with a PID controller based on a pole placement technique.

For trajectory tracking and balancing of a single inverted pendulum on a cart, Kumar et al. (2013) proposed two ways to stablise the system by utilising traditional and optimal control techniques. The controller was introduced using a proportional-integral- derivative and optimal state variable feedback using an optimal control LQR. The reported results show that the system was swung up and was then stabilised using the

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two suggested balancing controllers. The LQR controller was found to be a more robust controller than the PID controller.

Analysis of the stablisation and tracking control for a triple link structure Robogymnast (Eldukhri and Kamil 2013) was first carried out by Kamil et al. (2014). In this research the author combined a discrete-time linear quadratic regulator (DLQR) controller and an integral control action to satisfy the required performance of the control system. The simulation results showed that the overshoot of angular positions of the first, second and third links were satisfactory, and the Robogymnast could be settled in the upright position for an acceptable amount of time (Kamil et al. 2014).