Partial feedback linearization and sliding mode techniques for 2D crane control

Abstract

The overhead crane is an under-actuated system because its degree of freedom is larger than that of actuators. The three state variables of trolley motion, cargo lifting motion and cargo swing are controlled by two input signals composed of trolley driving and cargo lifting forces. In the present study, a novel non-linear control scheme for an overhead crane is proposed based on the combination of two control design techniques. The cargo swing vanishing mechanism is constructed using partial feedback linearization. Control of trolley and cargo tracking is designed based on the sliding mode technique. An anti-swing structure is then merged with the tracking scheme of the trolley and cargo hoisting motions to enable indirect control of the cargo swing angle. Both simulation and experimental results show that the combined controller not only stabilizes all trajectories of system states but also guarantees the robustness in which the shapes of system responses are consistently retained despite the wide variation in crane parameters.

Keywords

Feedback linearization Lyapunov function overhead cranes sliding mode control

Introduction

Overhead cranes are widely used in many industrial factories, such as shipyards and automotive factories. Cranes are required to have fast-paced operation to increase productivity. Speedy operation without control easily leads to cargo swing on the wire rope; cargo vibration increases with the speed of cargo transport. Cargo swings pose danger during operation, bringing about possible damage to the factory, crane and other equipment, and can even lead to accidents if the cargo swing angle is too large.

Many crane control techniques have been applied, from classical methods to modern control approaches. Many researchers have focused on traditional controls consisting of linear techniques (Kim, 2004; Kim et al., 2004; Sakawa and Sano, 1997; Sawodny et al., 2002), non-linear controls (Chwa, 2009; Fang et al., 2003; Hong et al., 2000; Kim and Hong, 2009; Lee, 2003, 2004, 2005; Le et al., 2013c) and optimal controls (Algarni et al., 1995; Wang, 2006). Furthermore, adaptive control approaches, such as self-turning control (Hua and Shine, 2007; Yang and Shen, 2010), gain-scheduling control (Corriga and Giua, 1998; Giua et al., 2001), multi-rate adaptive control (Mizumoto et al., 2007), iterative learning strategy (Sun et al., 2012), and model-reference adaptive control (Le et al., 2013b), are also popularly used. Intelligent control approaches for cranes have been proposed, such as fuzzy logic (Chang, 2006; Chang and Chiang, 2008; Chen et al., 2009), neural network control (Mahfouf et al., 2000; Suh et al., 2005; Yu et al., 2001) and input sharpening (Hong et al., 2003; Ngo and Hong, 2009; Singhose et al., 2000, 2008; Sorensen et al., 2007).

The following section reviews previous papers in relation to two symbolic types of non-linear control techniques composed of feedback linearization control (FLC) and sliding mode control (SMC). The general theory of partial feedback linearization (PFL) for under-actuated systems, such as overhead crane and under-actuated manipulator (Hong, 2002), was first developed by Spong (1994) and Fang and Kelkar (2001). Since then, papers on crane control using PFL have been published. Cheng and Chen (1996) proposed a robust controller that combined a feedback linearization approach and a time delay control scheme. Park et al. (2007) considered a controller for container cranes using PFL for tracking trolley and lifting payloads. Cho et al. (2008) presented a controller with components of feedback linearization and corrective control. Cho and Lee (2008) extended the PFL-based controller in the paper of Cho et al. (2008) to integrate an adaptive component. Chen et al. (2005) designed a non-linear control scheme for a gantry crane based on PFL. However, only local stability of the crane was proven in this study. Moustafa (2001) suggested using FLC for an overhead crane and used Lyapunov functions to prove the stability of a crane’s equilibrium point. Considering the crane as a linearized parameter-varying model, Giua et al. (1999) designed a controller and observer via the state-feedback stabilization technique for time-varying systems.

SMC is a robust control technique effectively used for non-linear systems. The general theory of SMC for under-actuated systems was first introduced by Lee et al. (1996), developed by Ashrafiuon and Erwin (2008) and completed by Sankaranarayanan and Mahindakar (2009). The application of SMC for overhead crane control can be found in many previous papers. Karkoub and Zribi (2001) introduced a variable structure controller in conjunction with a state feedback control scheme and a µ-synthesis control scheme. Bartolini et al. (2002) proposed a simple control scheme based on second-order sliding modes for payload–cart system with constant cable length. In another paper by Bartolini et al. (2003), they constructed two SMC schemes, namely, a PI-based controller and a linear observer-based time-varying feedback scheme. A combination of the fuzzy logic technique and SMC applied to an overhead crane was designed by Liu et al. (2004) based on the linearized mathematical model. To control cargo swing and trolley motion concurrently, Lee et al. (2006) formed an SMC controller derived from the analysis of sliding surface stability. They defined the sliding surface by linearly combining all state errors. Park et al. (2008) proposed an adaptive fuzzy SMC control for cargo anti-swing and trolley tracking in a simple crane model with constant cable length. Almutrairi and Zribi (2009) developed an SMC scheme from the study of Lee et al. (2006) for a 3D crane system. Furthermore, an observer was designed to eliminate the velocities of system outputs. These velocity components were used in an SMC scheme as a feedback states. Ngo and Hong (2012b) discussed an SMC controller for payload anti-sway of container cranes. By enhancing the study of Lee et al. (2006), Ngo and Hong (2012a) designed an adaptive SMC for container cranes that considered the switching gain of control law as a time-varying parameter.

The PFL technique mentioned in the papers of Cheng and Chen (1996), Park et al. (2007), Cho et al. (2008), Cho and Lee (2008), Chen et al. (2005), Moustafa (2001) and Giua et al. (1999) is rather simple and easy to design and implement. However, it is not useful for systems with uncertainties. PFL controllers do not guarantee the consistency of system responses in cases of varying system parameters. Previous papers on crane control (Almutairi and Zribi, 2009; Bartolini et al., 2002; Karkoub and Zribi, 2001; Lee et al., 2006; Liu et al., 2004; Ngo and Hong, 2012a,b; Park et al., 2008) applied the SMC technique that assures the robustness of crane systems. However, anti-swing control was designed incompletely. The anti-swing problem was addressed by simply adding the swing angle variable (or velocity of swing) into the sliding surface in which swing dynamics was not fully considered.

The current study analyses overhead crane control in accordance with another direction. A novel non-linear controller is constructed by combining PFL and SMC. Specifically, an anti-swing mechanism for cargo angle is designed by fully considering swing dynamics (un-actuated dynamics) using PFL. The control scheme of the trolley and cargo lifting motions is then considered using the SMC technique. A combined control scheme is then proposed by integrating the anti-swing structure into an SMC-based algorithm. Given the kinematic and geometric constraints between trolley travelling/cargo lifting and cargo swing, the actuators not only drive trolley moving and cargo hoisting directly but also control the cargo swing angle indirectly. Therefore, control from the sliding mode tracking structure and the effect of the anti-swing mechanism can stabilize the cargo swing and simultaneously track the trolley and cargo to the desired positions accurately.

Compared with previous controllers in PFL-based papers (Chen et al., 2005; Cheng and Chen, 1996; Cho and Lee, 2008; Cho et al., 2008; Giua et al., 1999; Moustafa, 2001; Park et al., 2007) and SMC based papers (Almutairi and Zribi, 2009; Bartolini et al., 2002; Karkoub and Zribi, 2001; Lee et al., 2006; Liu et al., 2004; Ngo and Hong, 2012a,b; Park et al., 2008), the proposed controller has the following strengths. 1) It retains the robustness of tracking control component because the tracking component is designed based on the SMC technique. 2) The anti-swing component of the combined controller is designed by fully considering un-actuated dynamics (in which non-linearities of un-actuated dynamics are taken into full account). As a result, the structure of the combined controller is perfect than that of the SMC controllers. 3) The control structure has more gains compared with others, leading to more opportunities in fine-tuning.

The organization of the paper is as follows. Next, a non-linear mathematical model of an overhead crane is constructed; then the design process of the combined controller is introduced, along with the anti-swing mechanism design, the proposal for the tracking control structure and the analysis of system stability. To present a comparison with the proposed controller, two conventional non-linear controllers are briefly re-designed. The simulation of system responses, experimental study and results of the analysis and comparison are given and concluding remarks are discussed.

System dynamics

Crane dynamics are constructed in the case of a simultaneous combination of trolley and cargo hoisting motions. The crane system presented in Figure 1 has three masses: m_t, m_c and m_l. Cargo mass m_c and trolley mass m_t are considered to have point masses concentrated at their centres. m_l denotes the equivalent mass of all rotating components of the cargo lifting mechanism. The generalized coordinates of the system include x(t), l(t) and θ(t), which are trolley displacement, cable length and cargo swing angle, respectively. Furthermore, frictions of trolley moving and cargo hoisting are characterized by b_t and b_r, respectively. Forces of the driving motors of trolley travelling and cargo lifting u_t and u_l are designed to move the trolley and hoist the cargo from the starting points to their destinations as fast as possible under minimized cargo swing.

For convenience, the following assumptions are given. 1) Mass and elastics of the wire rope are neglected. 2) No disturbance is caused by the wind outside the factory floor because the overhead crane is usually operated indoors. 3) Motions of all system components are considered in a plane. 4) The cargo swing angle is bounded, $| θ | \leq π / 2,$ and cable length is $l > 0 \forall t > 0 .$

Using the virtual work principle and Lagrange’s equation, the motion equations describing the system dynamics (Le et al., 2012) can be derived as follows:

\begin{matrix} (m_{t} + m_{c}) \ddot{x} - m_{c} \sin θ \ddot{l} - m_{c} l \cos θ \ddot{θ} \\ + b_{t} \overset{\cdot}{x} - 2 m_{c} \cos θ \overset{\cdot}{l} \overset{\cdot}{θ} + m_{c} l \sin θ {\overset{\cdot}{θ}}^{2} = u_{t} (t) \end{matrix}

(1)

- m_{c} \sin θ \ddot{x} + (m_{c} + m_{l}) \ddot{l} + b_{r} \overset{\cdot}{l} - m_{c} l {\overset{\cdot}{θ}}^{2} - m_{c} g \cos θ = u_{l} (t)

(2)

- \cos θ \ddot{x} + l \ddot{θ} + 2 \overset{\cdot}{l} \overset{\cdot}{θ} + g \sin θ = 0

(3)

which can be rewritten in matrix form as

M (q) \ddot{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) = F

(4)

where

\begin{matrix} M (q) = [\begin{matrix} m_{t} + m_{c} & - m_{c} \sin θ & - m_{c} l \cos θ \\ - m_{c} \sin θ & m_{c} + m_{l} & 0 \\ - m_{c} l \cos θ & 0 & m_{c} l^{2} \end{matrix}] \\ denotes the mass matrix, \end{matrix}

\begin{matrix} C (q, \overset{\cdot}{q}) = [\begin{matrix} b_{t} & - m_{c} \cos θ \overset{\cdot}{θ} & m_{c} l \sin θ \overset{\cdot}{θ} - m_{c} \cos θ \overset{\cdot}{l} \\ 0 & b_{r} & - m_{c} l \overset{\cdot}{θ} \\ 0 & m_{c} l \overset{\cdot}{θ} & m_{c} l \overset{\cdot}{l} \end{matrix}] \\ denotes the damping and centrifugal matrices, \end{matrix}

G (q) = {[\begin{matrix} 0 & - m_{c} g \cos θ & m_{c} \lg \sin θ \end{matrix}]}^{T} is the gravity vector,

\begin{matrix} F = {[\begin{matrix} u_{t} (t) & u_{l} (t) & 0 \end{matrix}]}^{T} \end{matrix}

denotes a vector of the control forces of driving motors, and $\begin{matrix} q = {[\begin{matrix} x (t) & l (t) & θ (t) \end{matrix}]}^{T} \end{matrix}$ indicates the generalized coordinate vector.

The overhead crane is an under-actuated system the dynamics of which is composed of two parts. Actuated dynamics is calculated by Equations (1) and (2) corresponding to the actuated state $q_{a} = {[\begin{matrix} x (t) & l (t) \end{matrix}]}^{T}$ . The un-actuated one described by Equation (3) corresponds to the un-actuated state $q_{u} = θ (t) .$ Swing dynamics in Equation (3) clearly displays the close kinematic relationship between the actuated and un-actuated states. This property is useful in creating a non-linear combined controller, as presented in the next section.

Controller design

The crane dynamics expressed by three non-linear differential equations (Equations 1–3) shows the high kinematic constraints among trolley motion, cargo hoisting and cargo swing angle. The three system states that need to be controlled consist of x, l and θ. However, the crane system only has two control inputs, u_t and u_l, that directly drive trolley displacement and cargo hoisting motion, respectively. The main goals of the proposed controller are to track the trolley from its initial position to the desired position, to hoist the cargo with reference to the cable, and to suppress cargo swing as fast as possible. Two control techniques are used in designing the control scheme. First, a cargo anti-swing mechanism is calculated based on the PFL method. Second, the tracking control problems of trolley moving and cargo lifting are analysed using the SMC approach. The proposed control scheme not only forces actuated states x and l to follow the desired trajectories but also drives the un-actuated state θ asymptotically until it reaches zero. Therefore, the overall controller is determined by merging the anti-swing mechanism with an SMC-based structure.

Decoupling

In designing the anti-swing control and tracking control schemes, crane dynamics should be separated into actuated and un-actuated dynamics. As l>0 for any t>0, Equation (3) can be rewritten as

\ddot{θ} = \frac{\cos θ}{l} \ddot{x} - \frac{2 \overset{\cdot}{θ} \overset{\cdot}{l}}{l} - \frac{g \sin θ}{l}

(5)

This equation shows that cargo swing angle θ is directly affected by trolley motion x and wire rope length l. Substituting Equation (5) into Equation (1) and combining this with Equation (2) yields the actuated dynamics.

\begin{matrix} ((m_{t} + m_{c} - m_{c} \cos^{2} θ) \ddot{x} - m_{c} \sin θ \ddot{l} \\ + b_{t} \overset{\cdot}{x} + m_{c} l \sin θ \overset{\cdot}{θ} \overset{\cdot}{θ} + m_{c} g \sin θ \cos θ \end{matrix}) = u_{t} (t)

(6)

(- m_{c} \sin θ \ddot{x} + (m_{c} + m_{l}) \ddot{l} + b_{r} \overset{\cdot}{l} - m_{c} l \overset{\cdot}{θ} \overset{\cdot}{θ} - m_{c} g \cos θ) = u_{l} (t)

(7)

which can be rewritten in matrix equation form as

\bar{M} (q) {\ddot{q}}_{a} + B {\overset{\cdot}{q}}_{a} + \bar{C} (q, \overset{\cdot}{q}) {\overset{\cdot}{q}}_{u} + \bar{G} (q) = U

(8)

where

\bar{M} (q) = [\begin{matrix} (m_{t} + m_{c} \sin^{2} θ) & - m_{c} \sin θ \\ - m_{c} \sin θ & (m_{c} + m_{l}) \end{matrix}], B = [\begin{matrix} b_{t} & 0 \\ 0 & b_{r} \end{matrix}],

\bar{C} (q, \overset{\cdot}{q}) = [\begin{matrix} m_{c} l \sin θ \overset{\cdot}{θ} \\ - m_{c} l \overset{\cdot}{θ} \end{matrix}], \bar{G} (q) = [\begin{matrix} m_{c} g \sin θ \cos θ \\ - m_{c} g \cos θ \end{matrix}], U = [\begin{matrix} u_{t} (t) \\ u_{l} (t) \end{matrix}]

$\bar{M} (q)$ is a positive definite matrix. Actuated dynamics (8) can be represented in another form

{\ddot{q}}_{a} = A_{a} {\overset{\cdot}{q}}_{a} + B_{a} {\overset{\cdot}{q}}_{u} + C_{a} + D_{a} U

(9)

where

\begin{matrix} A_{a} = - {\bar{M}}^{- 1} (q) B, B_{a} = - {\bar{M}}^{- 1} (q) \bar{C} (q, \overset{\cdot}{q}), \\ C_{a} = - {\bar{M}}^{- 1} (q) \bar{G} (q), D_{a} = {\bar{M}}^{- 1} (q) \end{matrix}

Similarly, swing dynamics (5) can be rewritten as

{\ddot{q}}_{u} = A_{u} {\ddot{q}}_{a} + B_{u} {\overset{\cdot}{q}}_{a} + C_{u}

(10)

where

A_{u} = [\begin{matrix} \frac{\cos θ}{l} & 0 \end{matrix}], B_{u} = [\begin{matrix} 0 & - \frac{2 \overset{\cdot}{θ}}{l} \end{matrix}], C_{u} = - \frac{g \sin θ}{l}

Substituting Equation (9) into Equation (10) yields

{\ddot{q}}_{u} = (A_{u} A_{a} + B_{u}) \overset{\cdot}{q_{a}} + A_{u} B_{a} {\overset{\cdot}{q}}_{u} + A_{u} C_{a} + C_{u} + A_{u} D_{a} U

(11)

The dynamics of a closed-loop system is described in another form, including actuated dynamics (9) and un-actuated dynamics (11). This form shows that both the un-actuated and actuated states are directly related to controlling input U.

Cargo anti-swing control

Based on PFL, a cargo anti-swing scheme is proposed as follows:

U^{*} = M_{A} (\begin{matrix} - K_{pu} q_{u} - (A_{u} A_{a} + B_{u}) {\overset{\cdot}{q}}_{a} - (A_{u} B_{a} + K_{du}) {\overset{\cdot}{q}}_{u} \\ - A_{u} C_{a} - C_{u} + {\ddot{q}}_{ud} + K_{du} {\overset{\cdot}{q}}_{ud} + K_{pu} q_{ud} \end{matrix})

(12)

where K_du and K_pu are positive control gains, and the 2×1 matrix $M_{A} = {[\begin{matrix} m_{t} l / \cos θ & m_{l} l \tan θ \end{matrix}]}^{T}$ is derived from expression $A_{u} D_{a} M_{A} = 1$ .

Theorem 1. The anti-swing controller (12) guarantees the asymptotical stability of the cargo swing angle. Specifically, $lim_{t \to \infty} [\begin{matrix} θ (t) & \overset{\cdot}{θ} (t) & \ddot{θ} (t) \end{matrix}] = 0 .$ Speed of the damped motion of the cargo swing depends on the proper selection of control gains K_du and K_pu.

Proof. The tracking error of the un-actuated state is defined,

e_{u} = q_{u} - q_{ud} = q_{u} = θ

(12a)

To stabilize the cargo swing $q_{u},$ the un-actuated dynamics can be ‘linearized’ (11) as

{\ddot{q}}_{u} = V

(13)

where

V = (A_{u} A_{a} + B_{u}) {\overset{\cdot}{q}}_{a} + A_{u} B_{a} {\overset{\cdot}{q}}_{u} + A_{u} C_{a} + C_{u} + A_{u} D_{a} U

(14)

is the equivalent input. Selection of equivalent input V is based on the stability of the un-actuated state. Thus,

V = {\ddot{q}}_{ud} - K_{du} ({\overset{\cdot}{q}}_{u} - {\overset{\cdot}{q}}_{ud}) - K_{pu} (q_{u} - q_{ud})

(15)

where $K_{du}$ and $K_{pu}$ are positive constants. Equivalent input V leads to the stability of the cargo swing error as

{\ddot{e}}_{u} + K_{du} {\overset{\cdot}{e}}_{u} + K_{pu} e_{u} = 0

(16)

Substituting Equation (15) into Equation (14) yields the anti-swing control, as shown in (12). The anti-swing mechanism (12) stabilizes the cargo swing for any positive constants $K_{du}$ and $K_{pu}$ . If the control gains $K_{du}$ and $K_{pu}$ are selected so that damping ratio $ζ = K_{du} / 2 \sqrt{K_{pu}} \geq 1,$ the cargo angle and its derivatives exponentially approach to zero as t tends to infinity. In summary, trolley and cargo driving forces $U^{*} = {[\begin{matrix} u_{t} (t) & u_{l} (l) \end{matrix}]}^{T}$ described by Equation (12) can suppress the cargo swing angle $q_{u}$ indirectly. This result holds because of the kinematic relationship among all system state variables $q = {[\begin{matrix} x (t) & l (t) & θ (t) \end{matrix}]}^{T}$ shown in the equations on system dynamics (Equations 1–3).

Trolley and cargo hoisting motion control

In this section, the control problems of trolley tracking and cargo hoisting are considered. The main objective is to design the control scheme to drive the actuated states $q_{a} = {[\begin{matrix} x (t) & l (t) \end{matrix}]}^{T}$ to approach the desired trajectories $q_{ad} = {[\begin{matrix} x_{d} (t) & l_{d} (t) \end{matrix}]}^{T}$ based on the SMC technique. Defining the tracking error vector as $e_{a} = {[\begin{matrix} (x - x_{d}) & (l - l_{d}) \end{matrix}]}^{T}$ yields

{\overset{\cdot}{e}}_{a} = {\overset{\cdot}{q}}_{a} - {\overset{\cdot}{q}}_{ad} and {\ddot{e}}_{a} = {\ddot{q}}_{a} - {\ddot{q}}_{ad}

(17)

Next, a sliding surface is described as a linear combination of position and velocity errors of actuated states

\begin{matrix} s = [\begin{matrix} s_{1} \\ s_{2} \end{matrix}] = (\frac{d}{dt} + λ) e_{a} = {\overset{\cdot}{e}}_{a} \\ + λ e_{a} = ({\overset{\cdot}{q}}_{a} - {\overset{\cdot}{q}}_{ad}) + λ (q_{a} - q_{ad}) \end{matrix}

(18)

where $λ = diag (λ_{1}, λ_{2})$ is the design parameter.

Deriving the sliding surface s with respect to time leads to

\overset{\cdot}{s} = {\ddot{e}}_{a} + λ {\overset{\cdot}{e}}_{a} = ({\ddot{q}}_{a} - {\ddot{q}}_{ad}) + λ ({\overset{\cdot}{q}}_{a} - {\overset{\cdot}{q}}_{ad})

(19)

Substituting Equation (9) into Equation (19) and setting $\overset{\cdot}{s} = 0$ yield the equivalent input yields

\hat{U} = \bar{M} (q) (- (λ + A_{a}) {\overset{\cdot}{q}}_{a} - B_{a} {\overset{\cdot}{q}}_{u} - C_{a} + λ {\overset{\cdot}{q}}_{ad} + {\ddot{q}}_{ad})

(20)

To maintain the state trajectory of the system on the sliding surface, the switching action must be introduced into Equation (20). The sliding mode structure composed of equivalent control and switching action becomes

U^{* *} = \bar{M} (- (λ + A_{a}) {\overset{\cdot}{q}}_{a} - B_{a} {\overset{\cdot}{q}}_{u} - C_{a} + λ {\overset{\cdot}{q}}_{ad} + {\ddot{q}}_{ad}) - Ksgn (s)

(21)

where $K = diag (K_{1}, K_{2})$ denotes the positive switched gains. Design parameters $λ$ and $K$ are selected so that $s$ approaches zero as fast as possible.

A switching control usually causes chattering in the state trajectory on the switching surface. To reduce chattering, the sgn(s) function is replaced by a saturation function whose components have the following form

{[sat (s)]}_{i} = {\begin{matrix} 1 if s_{i} / ε > 1 \\ s_{i} / ε if - 1 < s_{i} / ε < 1 \\ 1 if s_{i} / ε < - 1 \end{matrix}

(22)

where ε is the constant indicating thickness of the boundary layer.

Stability of the sliding surface

Two control mechanisms have been designed based on the feedback linearization approach and the SMC technique. The following theorem shows the asymptotical stability of sliding surface.

Theorem 2. Under the action of SMC input (21), actuated states $q_{a}$ are attracted to the sliding surface s. In other words, the sliding surface (18) is asymptotically stable.

Proof. The Lyapunov lower bounded function, $V = \frac{1}{2} s^{T} s > 0,$ is considered. Its derivative is determined as

\overset{\cdot}{V} = s^{T} \overset{\cdot}{s} = s^{T} (({\ddot{q}}_{a} - {\ddot{q}}_{ad}) + λ ({\overset{\cdot}{q}}_{a} - {\overset{\cdot}{q}}_{ad}))

(23)

Substituting (9) and (21) into (23) and simplifying lead to

\overset{\cdot}{V} = s^{T} (- {\bar{M}}^{- 1} (q) Ksgn (s))

(24)

Functions $s_{i}$ and $sgn (s_{i})$ have the same sign, but their amplitudes are different. Therefore, the sign consideration of $\overset{\cdot}{V}$ is equivalent to that of the function

V_{1} = - s^{T} {\bar{M}}^{- 1} (q) Ks

(25)

The matrix

\begin{matrix} {\bar{M}}^{- 1} (q) K = \frac{1}{m_{t} (m_{c} + m_{l}) + m_{c} m_{l} \sin^{2} θ} \\ [\begin{matrix} K_{1} (m_{c} + m_{l}) & K_{2} m_{c} \sin θ \\ K_{1} m_{c} \sin θ & K_{2} (m_{t} + m_{c} \sin^{2} θ) \end{matrix}] \end{matrix}

is symmetric, where

a_{11} = \frac{K_{1} (m_{c} + m_{l})}{m_{t} (m_{c} + m_{l}) + m_{c} m_{l} \sin^{2} θ} > 0

and its determinant is

\begin{matrix} det ({\bar{M}}^{- 1} (q) K) = \frac{K_{1} K_{2} (m_{c} + m_{l}) m_{t} + K_{1} K_{2} m_{l} m_{c} \sin^{2} θ}{m_{t} (m_{c} + m_{l}) + m_{c} m_{l} \sin^{2} θ} \\ > 0 \forall K_{2}, K_{1} > 0 \end{matrix}

${\bar{M}}^{- 1} (q) K$ is positive definite for every control gains $K_{2}, K_{1} > 0$ . Therefore,

V_{1} = - s^{T} {\bar{M}}^{- 1} (q) Ks \leq 0 \forall s \in R^{2}

(26)

which leads to $\overset{\cdot}{V} \leq 0$ . Thus, $V (t) \leq V (0)$ . For this reason, $s$ must be bounded. Alternatively, the derivative of $\overset{\cdot}{V}$ is

\ddot{V} = ({\overset{\cdot}{s}}^{T} \overset{\cdot}{s} + s^{T} \ddot{s})

(27)

with

\overset{\cdot}{s} = - {\bar{M}}^{- 1} (q) Ksgn (s)

(28)

and

\begin{matrix} \ddot{s} = - ({\overset{\cdot}{\bar{M}}}^{- 1} (q) Ksgn (s) + {\bar{M}}^{- 1} (q) K [\frac{dsgn (s_{i})}{d s_{i}} {\overset{\cdot}{s}}_{i}]) \\ = - {\overset{\cdot}{\bar{M}}}^{- 1} (q) Ksgn (s) \forall s \neq 0 \end{matrix}

(29)

As the matrices ${\bar{M}}^{- 1} (q)$ and ${\overset{\cdot}{\bar{M}}}^{- 1} (q)$ are bounded, and $K$ is a finite positive diagonal matrix, $\overset{\cdot}{s}$ and $\ddot{s}$ are bounded vectors. Therefore, $\ddot{V}$ is bounded, leading to $\overset{\cdot}{V}$ , which is uniformly continuous in time. The application of Barbalat’s lemma (Slotine and Li, 1991) indicates that $lim_{t \to \infty} \overset{\cdot}{V} = 0$ , which yields $lim_{t \to \infty} s = 0 .$ Hence, the sliding surface s is asymptotically stable. Based on Slotine and Li (1991), the condition of sliding surface stability is expressed as follows:

\overset{\cdot}{V} \leq - s^{T} η sgn (s)

(30)

where $η = diag (η_{1}, η_{2})$ is a diagonal positive matrix. Comparing Equations (24) and (30), $K$ can be determined as

K = {η \bar{M} (q) |}_{θ \to 0} = {[\begin{matrix} η_{1} & 0 \\ 0 & η_{2} \end{matrix}] [\begin{matrix} (m_{t} + m_{c} \sin^{2} θ) & - m_{c} \sin θ \\ - m_{c} \sin θ & (m_{c} + m_{l}) \end{matrix}] |}_{θ \to 0}

which implies that

K_{1} = η_{1} m_{t} and K_{2} = η_{2} (m_{c} + m_{l})

(31)

Hence, control gain $K$ , selected according to Equation (31), guarantees the stability of the sliding surface.

Combined control scheme

To stabilize all state variables of the overhead crane system, both the anti-swing mechanism and control scheme of cargo lifting and trolley motions should be merged. Therefore, a combined control structure is proposed as follows:

U = Γ U^{*} + Λ U^{* *}

(32)

where $Γ = diag (Γ_{1}, Γ_{2})$ and $Λ = diag (Λ_{1}, Λ_{2})$ are weighting matrices. Substituting Equations (12) and (21) into Equation (32), the combined control scheme is determined by

\begin{matrix} U = \\ (\begin{matrix} - Γ M_{A} K_{pu} q_{u} - (Γ M_{A} (A_{u} A_{a} + B_{u}) + Λ \bar{M} (λ + A_{a})) {\overset{\cdot}{q}}_{a} \\ - (Γ M_{A} (A_{u} B_{a} + K_{du}) + Λ \bar{M} B_{a}) {\overset{\cdot}{q}}_{u} - Γ M_{A} A_{u} C_{a} - Λ \bar{M} C_{a} - Γ M_{A} C_{u} \\ + Γ M_{A} ({\ddot{q}}_{ud} + K_{du} {\overset{\cdot}{q}}_{ud} + K_{pu} q_{ud}) + Λ \bar{M} (λ {\overset{\cdot}{q}}_{ad} + {\ddot{q}}_{ad}) \end{matrix}) \\ - Λ Ksign (s) \end{matrix}

(33)

Theorem 3 shows that the combined control scheme (33) forces both actuated and un-actuated states to reach the desired positions asymptotically despite constraints on control gains.

Theorem 3. The combined control scheme $U = {[\begin{matrix} u_{t} & u_{l} \end{matrix}]}^{T}$ described by (33) locally stabilizes all state trajectories $q = {[\begin{matrix} x & l & θ \end{matrix}]}^{T}$ determined from overhead crane dynamics (4) around the equilibrium points $q = q_{d}$ and $\overset{\cdot}{q} = 0$ if the following expressions are held:

(G + H + Γ_{1} K_{du}) > 0

(34a)

Γ_{1} (K_{pu} (G + H) + K_{du} GH) > Γ_{1} F (K_{du} H + K_{pu})

(34b)

Γ_{1} K_{pu} (G - F) H > 0

(34c)

where

F = (\frac{b_{t}}{m_{t}} (1 - Γ_{1}) + Λ_{1} (λ_{1} - \frac{b_{t}}{m_{t}}))

G = Λ_{1} λ_{1} - \frac{b_{t}}{m_{t}} (Λ_{1} + Γ_{1} - 1)

H = Λ_{2} λ_{2} - \frac{b_{r}}{m_{c} + m_{l}} (Λ_{2} + Γ_{2} - 1)

Proof. The overall control input (33) consists of the equivalent component and the switched action. The equivalent input takes charge of attracting all system states to the sliding surface. Meanwhile, the sign component keeps state trajectories on the sliding surface. The constraint conditions of control gains are found to stabilize all system outputs on the sliding surface. Substituting the ‘equivalent’ part of control law (33) into the mathematical model composed of Equations (9) and (11) leads to the following closed-loop dynamics:

{\ddot{q}}_{a} = (\begin{matrix} ((I - Γ - Λ) A_{a} - Λ Λ - Γ D_{a} M_{A} B_{u}) {\overset{\cdot}{q}}_{a} + ((I - Γ - Λ) B_{a} - Γ D_{a} M_{A} K_{du}) {\overset{\cdot}{q}}_{u} \\ - Γ D_{a} M_{A} K_{pu} q_{u} + C_{a} - (Λ + Γ) C_{a} - Γ D_{a} M_{A} C_{u} \\ + Γ D_{a} M_{A} ({\ddot{q}}_{ud} + K_{du} {\overset{\cdot}{q}}_{ud} + K_{pu} q_{ud}) + Λ (λ {\overset{\cdot}{q}}_{ad} + {\ddot{q}}_{ad}) \end{matrix}) = f (q)

(35)

{\ddot{q}}_{u} = (\begin{matrix} (A_{u} A_{a} + B_{u} - A_{u} Γ A_{a} - A_{u} Γ D_{a} M_{A} B_{u} - A_{u} Λ (λ + A_{a})) {\overset{\cdot}{q}}_{a} - A_{u} Γ D_{a} M_{A} K_{pu} q_{u} \\ + (A_{u} B_{a} - A_{u} Γ B_{a} - A_{u} Γ D_{a} M_{A} K_{du} - A_{u} Λ B_{a}) {\ddot{q}}_{u} + A_{u} C_{a} - A_{u} (Λ + Γ) C_{a} \\ + C_{u} - A_{u} Γ D_{a} M_{A} C_{u} + A_{u} Γ D_{a} M_{A} ({\ddot{q}}_{ud} + K_{du} {\overset{\cdot}{q}}_{ud} + K_{pu} q_{ud}) + A_{u} Λ (λ {\overset{\cdot}{q}}_{ad} + {\ddot{q}}_{ad}) \end{matrix}) = g (q)

(36)

The stability of a closed-loop dynamics comprising Equations (35) and (36) is analysed using Lyapunov’s linearization theorem (Slotine and Li, 1991). By setting four state variables

\begin{matrix} z_{1} = q_{a} - q_{ad}, z_{2} = {\overset{\cdot}{q}}_{a} - {\overset{\cdot}{q}}_{ad} \\ z_{3} = q_{u} - q_{ud}, z_{4} = {\overset{\cdot}{q}}_{u} - {\overset{\cdot}{q}}_{ud} \end{matrix}

the closed-loop dynamics can be rewritten in the first order as

{\overset{\cdot}{z}}_{1} = z_{2}

(37)

{\overset{\cdot}{z}}_{2} = (\begin{array}{l} ((I - Γ - Λ) A_{a} - Λ λ - Γ D_{a} M_{A} B_{u}) z_{2} - Γ D_{a} M_{A} K_{p u} z_{3} \\ + ((I - Γ - Λ) B_{a} - Γ D_{a} M_{A} K_{d u}) z_{4} + C_{a} - (Λ + Γ) C_{a} - Γ D_{a} M_{A} C_{u} \\ + Γ D_{a} M_{A} ({\ddot{q}}_{u d} + K_{d u} {\overset{\cdot}{q}}_{u d} + K_{p u} q_{u d}) + Λ (Λ {\overset{\cdot}{q}}_{a d} + {\ddot{q}}_{a d}) \end{array}) = f (z)

(38)

{\overset{\cdot}{z}}_{3} = z_{4}

(39)

{\overset{\cdot}{z}}_{4} = (\begin{matrix} (A_{u} A_{a} + B_{u} - A_{u} Γ A_{a} - A_{u} Γ D_{a} M_{A} B_{u} - A_{u} λ (λ + A_{a})) z_{2} - A_{u} Γ D_{a} M_{A} K_{pu} z_{3} \\ + (A_{u} B_{a} - A_{u} Γ B_{a} - A_{u} Γ D_{a} M_{A} K_{du} - A_{u} λ B_{a}) z_{4} + A_{u} C_{a} - A_{u} (λ + Γ) C_{a} \\ + C_{u} - A_{u} Γ D_{a} M_{A} C_{u} + A_{u} Γ D_{a} M_{A} ({\ddot{q}}_{ud} + K_{du} {\overset{\cdot}{q}}_{ud} + K_{pu} q_{ud}) + A_{u} λ (λ {\overset{\cdot}{q}}_{ad} + {\ddot{q}}_{ad}) \end{matrix}) = g (z)

(40)

where $z = {[\begin{matrix} z_{1} & z_{2} & z_{3} & z_{4} \end{matrix}]}^{T} \in R^{6}$ denotes a state vector. The non-linear dynamics (37)–(40) is asymptotically stable around the equilibrium point $z = 0$ ( $q = q_{d}$ and $\overset{\cdot}{q} = 0$ ) if the corresponding linearized system is strictly stable. Linearizing the closed-loop dynamics around $z = 0$ yields the linear system of state-space form

\overset{\cdot}{z} = Az

(41)

where

A = {[\begin{matrix} 0_{2 \times 2} & I_{2 \times 2} & 0_{2 \times 1} & 0_{2 \times 1} \\ \frac{\partial f (z)}{\partial z_{1}} & \frac{\partial f (z)}{\partial z_{2}} & \frac{\partial f (z)}{\partial z_{3}} & \frac{\partial f (z)}{\partial z_{4}} \\ 0_{1 \times 2} & 0_{1 \times 2} & 0 & 1 \\ \frac{\partial g (z)}{\partial z_{1}} & \frac{\partial g (z)}{\partial z_{2}} & \frac{\partial g (z)}{\partial z_{3}} & \frac{\partial g (z)}{\partial z_{4}} \end{matrix}]}_{z = 0}

(42)

is a Jacobian matrix. After modification, components of the state matrix are obtained as follows:

{\frac{\partial f (z)}{\partial z_{1}} |}_{z = 0} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]

\begin{matrix} {\frac{\partial f (z)}{\partial z_{2}} |}_{z = 0} = {((I - Γ - λ) A_{a} - Λ λ - Γ D_{a} M_{A} B_{u})}_{z = 0} \\ = [\begin{matrix} \frac{b_{t}}{m_{t}} (Λ_{1} + Γ_{1} - 1) - Λ_{1} λ_{1} & 0 \\ 0 & \frac{b_{r}}{m_{c} + m_{l}} (Λ_{2} + Γ_{2} - 1) - Λ_{2} λ_{2} \end{matrix}] \end{matrix}

{\frac{\partial f (z)}{\partial z_{3}} |}_{z = 0} = {(- Γ D_{a} M_{A} K_{pu})}_{z = 0} = [\begin{matrix} - Γ_{1} K_{pu} l_{d} \\ 0 \end{matrix}]

{\frac{\partial f (z)}{\partial z_{4}} |}_{z = 0} = {((I - Γ - λ) B_{a} - Γ D_{a} M_{A} K_{du})}_{z = 0} = [\begin{matrix} - Γ_{1} K_{du} l_{d} \\ 0 \end{matrix}]

{\frac{\partial g (z)}{\partial z_{1}} |}_{z = 0} = [\begin{matrix} 0 & 0 \end{matrix}]

\begin{matrix} {\frac{\partial g (z)}{\partial z_{2}} |}_{z = 0} = A_{u} {(A_{a} + B_{u} D_{a} M_{A} - Γ A_{a} - Γ D_{a} M_{A} B_{u} - λ (λ + A_{a}))}_{z = 0} \\ = [\begin{matrix} \frac{b_{t} (Λ_{1} + Γ_{1} - 1)}{m_{t} l_{d}} - \frac{Λ_{1} λ_{1}}{l_{d}} & 0 \end{matrix}] \end{matrix}

{\frac{\partial g (z)}{\partial z_{3}} |}_{z = 0} = {(- A_{u} Γ D_{a} M_{A} K_{pu})}_{z = 0} = - Γ_{1} K_{pu}

{\frac{\partial g (z)}{\partial z_{4}} |}_{z = 0} = A_{u} {(B_{a} - Γ B_{a} - Γ D_{a} M_{A} K_{du} - λ B_{a})}_{z = 0} = - Γ_{1} K_{du}

The linear system (41) is stable if A is a Hurwitz matrix. Based on Routh–Hurwitz’s criterion and after some calculation, the Hurwitz condition of matrix A leads to the relationship expressions (34a–34c). Therefore, the closed-loop dynamics of the crane system (35)–(36) is locally stable if the constraint conditions (34a–34c) are satisfied. The selection of parameters of the combined controller (33) must classify the given conditions (34a–34c).

In summary, the effect of this constraint and the action of the anti-swing mechanism lead to the successful design of the combined controller because the actuated dynamics (9) used for constructing the SMC scheme (21) contains the kinematic constraint received from un-actuated dynamics (10).

SMC and PFL

SMC and feedback linearization are two control techniques proposed to design the non-linear controller in previous papers (Almutairi and Zribi, 2009; Bartolini et al., 2002; Chen et al., 2005; Cheng and Chen, 1996; Cho and Lee, 2008; Cho et al., 2008; Giua et al., 1999; Karkoub and Zribi, 2001; Lee et al., 2006; Liu et al., 2004; Moustafa, 2001; Ngo and Hong, 2012a,b; Park et al., 2008; Park et al., 2007; Le et al., 2013a). To compare the results of the proposed novel combined controller with those of the previous controllers, two symbolic control schemes published in the literature are re-designed in the present paper.

Conventional SMC

This type of control scheme was first proposed by Lee et al. (2006) and was developed by Almutairi and Zribi (2009). In this scheme, both actuated and un-actuated states are integrated on the sliding surface, and the control law is designed based on the stability of the sliding surface. Based on the works by Lee et al. (2006) and Almutairi and Zribi (2009), the conventional SMC scheme is re-proposed as follows:

U = (B - \bar{M} (q) λ) {\overset{\cdot}{q}}_{a} - \bar{M} (q) {\overset{\cdot}{q}}_{u} + \bar{G} (q) - Ksign (s)

(43)

where $s = {\overset{\cdot}{q}}_{a} + λ (q_{a} - q_{ad}) + β q_{u}$ is the sliding surface, $λ = diag (λ_{1}, λ_{2}),$ $β = {[\begin{matrix} β_{1} & 0 \end{matrix}]}^{T},$ $K = diag (K_{1}, K_{2}) .$

PFL control

In study of Le et al. (2012), a non-linear control scheme is designed based on the PFL technique. This controller is re-described as follows:

\begin{matrix} U = - \bar{M} (q) ((K_{d 1} + A_{a}) {\overset{\cdot}{q}}_{a} + (α K_{d 2} + B_{a}) {\overset{\cdot}{q}}_{u} + K_{p 1} (q_{a} - q_{ad}) + α K_{p 2} q_{u} + C_{a}) \end{matrix}

(44)

where $K_{d 1} = diag (K_{d 11}, K_{d 12})$ and $K_{p 1} = diag (K_{p 11}, K_{p 12})$ are positive matrices, and $α = {[\begin{matrix} α_{1} & 0 \end{matrix}]}^{T},$ α₁, K_p₂ and K_d₂ are control parameters satisfying $α_{1} K_{d 2} > 0,$ $α_{1} K_{p 2} > 0 .$

Simulation and experiment

To investigate the quality of the designed controllers, both numerical simulation and experimental study are applied not only for the new combined controller (33) but also for the SMC and PFL schemes (43) and (44). These processes are implemented in the complicated operating condition in which trolley travelling and cargo lifting mechanisms are simultaneously operated. To show the robustness of a combined controller with uncertainties, its simulation is carried out in two cases.

Simulation case 1

Parameters of the crane system and its controllers in this case are given in Table 1. At initial time, the cargo is handled on the cable with length l₀=0.5 m; the cable is perpendicular to the ground ( $θ_{0} = 0^{0}$ ). The combined control inputs (33), SMC inputs (43) and PFL controller (44) are created so that the cargo is lifted by 0.2 m (the desired cable length), and the trolley moves 0.3 m (the desired destination: $l_{d} = 0.2, x_{d} = 0.3$ ). Hoisting the cargo and moving the trolley must be commenced at the same initial time.

Table 1.

Control system parameters.

System dynamics	Combined controller	SMC controller	PFL controller
m_c=0.85 kg	λ₁= 1; λ₂=1.4; ε=0.05	λ₁=0.5	K_d1 = diag(10, 10)
m_t=5 kg	K_du=1.2; K_pu=0.5	λ₂=0.5	K_p1 = diag(1.3, 13)
m_l=2 kg	η=diag(0.8,1.4)	β₁=6	K_d₂ = 2
b_t=20 N.s/m	Γ=diag(−1.5,0)	ε=0.05	K_p₂ = 1.8
b_r=50 N.s/m	Λ=diag(1,1)	K = diag(1200, 400)	α = [1 0]^T
g=9.81 m/s²

SMC, sliding mode control; PFL, partial feedback linearization.

Simulation case 2: system with uncertainties

Many crane parameters are uncertainties that are changeable in connection with operating case and environment. For example, the crane transports the cargo m_c of various weights and volumes; the friction factors (characterized by b_t and b_r) vary in terms of temperature and environment condition. To investigate the robustness of the proposed controller, the crane system is simulated by increasing the cargo mass Δm_c=400% and decreasing the damped factors b_t=b_r=−10%. Only the system parameters are changed; the controller parameters are retained as in the simulation case 1.

Furthermore, to verify the quality of the simulating responses, an experimental study is carried out using a laboratory crane system (Figure 2) having the parameters of simulation case 1. The overhead crane consists of two DC motors that drive the trolley and hoist the cargo. Three incremental encoders measure trolley displacement, cargo hoisting and cargo swing. The real-time crane system is controlled by a host PC using MATLAB and SIMULINK environments with xPC Target solution. In this system, two interfacing cards are attached to the target PC. One is the NI PCI 6025E multifunction card, which sends direction control signals to the motor amplifiers, and the other is the NI PCI 6602 card, which acquires pulse signals from the encoders and sends PWM signals to the amplifiers. Both simulation and experimental results are described in Figures 3 –8.

The responses of the trolley motion are shown in Figures 3 and 4. Both responses do not have maximum overshoots. The simulation responses of the combined control are no smoother than those of SMC and PFL but achieve the steady state faster. Figures 5 and 6 present the paths of the cargo lowering motion. Both responses show ideal qualities, with the responses approaching the reference with smooth motion without maximum overshoot. The fastest motion is seen in the combined control responses. For the simulation responses of the combined controller in Figures 3 and 5, no large difference is found between the curves of simulation cases 1 and 2 despite the wide variation in system parameters. Thus, the combined controller achieves robust behaviour despite uncertainties.

Payload swings during the transport process are illustrated in Figures 7 and 8. The responses of both systems indicate that cargo swing is kept in a small boundary at a transient state. The smallest boundary can be observed at the PFL swing trajectories. PFL responses also show the best shape in this case. The simulation response of the combined control achieves the steady state after three oscillation periods. The simulation curves of FBL and SMC are completely suppressed after only one vibration period. All simulation swing responses are totally eliminated at steady state. However, the experiment swing responses show remaining small steady-state errors.

In summary, both controllers, from previous conventional schemes (43) and (44) to the proposed novel combining control law (33), stabilize all state variables of the crane system. Compared with conventional controllers, the combined controller is the best, with the smallest setting time. However, in terms of smooth motion, the control responses of the combined controller are no better than those of PFL and SMC. The combined controller shows robustness under varying system parameters.

Conclusion

The present study proposed a novel non-linear combined controller for an overhead crane based on the combination of PFL and SMC. The control scheme was successfully designed for the complicated operation of an overhead crane, simultaneously combining the control of cargo lifting, trolley moving and cargo swing elimination. To compare the results, simulation and real-time experiments were implemented not only for the proposed combined controller (33) but also for existing conventional control laws (43) and (44). Both the simulation and experimental results showed that the proposed controller stabilized all system responses asymptotically: cargo swing was kept small during the transfer process and completely disappeared at the payload destination; trolley motion and cargo lifting/lowering accurately reached the desired positions. The combined controller kept the shape of system responses consistent despite the wide variation in system parameters.

In future studies, the authors will enhance the control problem with an adaptive combined controller, in which the weighed matrices are considered time-varying parameters. Another future direction is to upgrade the combined control scheme for 3D overhead cranes.

Footnotes

Acknowledgements

This work was supported by the R&D program of MOTIE/KEIT [No. 10041109, Development of monitoring robot system at nuclear power plant] and the Technology Innovation Program of the Knowledge economy of MKE [No. 10041834, Technology development of service robot’s performance and standardization for movement/manipulation/HRI/networking]. Also, this research was supported by a grant of the SMART Highway Agency from Construction Technology Innovation Program (No. 10CCTI-A050948-04) funded by Ministry of Land, Transportation and Maritime Affairs(MLTM) of Korean government.

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