Passivity-based adaptive trajectory control of an underactuated 3-DOF overhead crane

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Abstract

This paper presents a robust passivity-based adaptive control method for payload trajectory tracking of a three degree-of-freedom overhead crane with a varying-length rigid hoist cable. A reformulation of the overhead crane’s equations of motion is used and an adaptive control law is developed to ensure the system has an output strictly passive (OSP) input–output mapping. The Passivity Theorem is used to prove that the closed-loop system is input–output stable when an OSP negative feedback controller is implemented. Convergence and boundedness of the payload trajectory tracking error is proven with two candidate OSP feedback controllers, including a constant gain and a strictly positive real (SPR) controller. Numerical simulations and experiments are performed with the proposed control method and two state-of-the-art control laws from the literature. The results demonstrate the practicality of the proposed control method and the improved performance of the proposed SPR controller in the experiments compared to the controllers from the literature.

Introduction

Overhead cranes are ubiquitous with the construction of modern infrastructure owing to their high payload capacity and operational flexibility. Accurately controlling the motion of an overhead crane’s payload when performing quick trajectories is a challenging task (Abdel-Rahman et al., 2003, Ramli et al., 2017). This is predominantly due to the fact that overhead cranes are underactuated, meaning that they have less control inputs than generalized coordinates. Traditionally, manipulation of an overhead crane relies heavily on the experience and skill of a human operator. Relying on human operators is limiting, as overhead cranes are sometimes required to perform repetitive tasks over long duration, which potentially leads to operator fatigue, increased risk, and a degradation in crane performance. Furthermore, the use of overhead cranes is sometimes required in environments that are unsafe for a human operator. Autonomous control systems are proposed as a replacement for human operators in these scenarios. A number of control strategies are found in the literature that can be used to augment the performance of a human operator or an autonomously operated crane (Abdel-Rahman et al., 2003, Ramli et al., 2017). Open-loop feedforward control methods that rely on input-shaping of the reference commands sent to the crane are well-studied (Blackburn et al., 2010, Kim and Singhose, 2010, Singhose et al., 2000, Sorensen et al., 2007, Sung and Singhose, 2009, Tho et al., 2020, Vaughan et al., 2010), and are particularly useful in preventing sway of the crane’s hoist cable caused by a human operator. An overview of the concept behind input shaping and a comparison of different input-shaping methods is presented in Vaughan, Yano, and Singhose (2008). Offline motion planning methods that involve preliminary trolley/payload trajectory design to smooth out the motion of the payload sway (Blajer and Kołodziejczyk, 2007, Chen et al., 2016, Sun et al., 2014, Sun et al., 2012, Wu and Xia, 2014) are also useful in autonomous and periodic crane operation. Although input-shaping techniques and offline motion planning algorithms have shown great promise, this paper focuses on closed-loop control overhead crane control, where it is assumed that a desired payload trajectory is provided to the lower-level closed-loop controller.

Advanced nonlinear closed-loop control methods, including gain scheduling (Takagi & Nishimura, 1999), predictive control (Ileš, Matuško, & Kolonić, 2018), sliding-mode control (Almutairi and Zribi, 2009, Bartolini et al., 2002, Ngo and Hong, 2012, Ouyang et al., 2019, Tuan et al., 2014, Tuan et al., 2013, Xi and Hesketh, 2010, Zhang et al., 2018, Zhang, Zhang, Chen et al., 2019, Zhang, Zhang and Cheng, 2019, Zhang, Zhang, Ouyang et al., 2020), and Lyapunov-based methods (Sun et al., 2015, Sun et al., 2016, Yang et al., 2021) have been proposed for the trajectory tracking control of crane payloads over a range of operating conditions. Only a small subset of these methods account for a hoist cable with varying length (Almutairi and Zribi, 2009, Bartolini et al., 2002, Sun et al., 2015, Takagi and Nishimura, 1999, Tuan et al., 2014, Tuan et al., 2013). A significant challenge in designing closed-loop controllers for cranes is ensuring that the controller is robust to model uncertainty and unknown external disturbances. Disturbances can be rejected and unmodeled dynamics can be accounted through the use of an observer (Yang et al., 2021, Zhang et al., 2018). A common solution to account for unmodeled and uncertain crane dynamics is through the use of adaptive control techniques (Ngo and Hong, 2012, Ouyang et al., 2019, Sun et al., 2015, Sun et al., 2016, Tuan et al., 2014, Tuan et al., 2013, Zhang, Zhang, Ouyang et al., 2020), where parameters of the crane dynamics are adaptively estimated. Only three of these adaptive control techniques apply to overhead cranes with varying length hoist cables (Sun et al., 2015, Tuan et al., 2014, Tuan et al., 2013), two of which use sliding-mode control (Tuan et al., 2014, Tuan et al., 2013), while the other uses a Lyapunov-based controller (Sun et al., 2015). The limited number of closed-loop control methods for overhead cranes that explicitly account for model uncertainty and a varying length hoist cable highlights the challenge of developing robust controllers for this class of cranes. It is also worth noting that the controller in Sun et al. (2015) does not actively damp out payload sway, which severely limits its ability to perform high-acceleration payload tracking maneuvers.

Passivity-based control methods are widely used for the motion control of robotic manipulators (Ortega et al., 1998, Spong et al., 2020), where the control inputs (e.g., joint motor torques) are typically colocated with the measurement outputs (e.g., joint angular rates), resulting in a passive input–output mapping. In contrast to Lyapunov-based control methods that focus on asymptotic stability of the closed-loop system or sliding-mode control methods that often emphasize finite-time stability, passivity-based control is centered around the notion of closed-loop input–output stability using the Passivity Theorem (Brogliato, Lozano, Maschke, & Egeland, 2020). A particular type of input–output stability known as input–output L2 stability (referred to simply as input–output stability in this paper) is often considered, where the system outputs are guaranteed to be in the inner product space L2 provided the system inputs are in L2 (Brogliato et al., 2020). An advantage of passivity-based control is that passivity of a system’s input–output mapping, and as a result, closed-loop input–output stability, can often be proven without precise knowledge of the system’s parameters (e.g., mass, stiffness, damping properties) or explicitly defining the system’s internal states. This makes passivity-based controllers robustly stabilizing, which has been leveraged to develop robust control methods for other cable-driven systems (Caverly and Forbes, 2014, Caverly and Forbes, 2018, Caverly et al., 2015, Godbole et al., 2019) and gantry robots (Christoforou & Damaren, 2011) that share similarities with overhead cranes. The manner in which passivity-based controllers guarantee input–output stability through the Passivity Theorem enables a natural extension to passivity-based adaptive control (Damaren, 1996, Damaren, 1999, Godbole et al., 2019, Ortega and Spong, 1989), where uncertain model parameters can be estimated to improve closed-loop performance. A preliminary study on the use of passivity-based control for overhead cranes was performed in Shen and Caverly (2020), where both the unactuated payload sway and the axial vibrations of an elastic hoist cable are accounted for by the feedback control law. Although the method developed in Shen and Caverly (2020) is shown to perform well in certain situations, the control formulation relies on exact knowledge of the crane’s inertial properties and an assumption that the sway angle and derivative of the sway angle remain small. These restrictive assumptions severely limit the practicality of this passivity-based control method. Other work in the literature acknowledges that overhead cranes feature passive input–output mappings, but does not directly make use of this passivity property and the Passivity Theorem to develop a controller that guarantees closed-loop input–output stability (Chen et al., 2019, Fang et al., 2003, Zhang, Zhang, Ji et al., 2020).

The work presented in this paper uses passivity-based control techniques to develop a novel, practical, and robust three degree-of-freedom (DOF) overhead crane payload trajectory tracking controller that does not rely on exact knowledge of the system’s parameters. This is achieved by using a modified tracking error output inspired by the sliding-mode control method devised in Tuan et al. (2013) and an adaptive control law similar to that implemented in Damaren, 1996, Damaren, 1999 and Ortega and Spong (1989) for robotic manipulators to establish a passive input–output mapping from the system’s force and torque inputs to the modified tracking error output. The proposed adaptive control law is substantially different from the one used in Tuan et al. (2013), as it does not rely on a specific form of the crane dynamics and does not require any knowledge of the crane’s mass properties. The proposed passivity-based control approach is also significantly different from the preliminary passivity-based overhead crane control work in Shen and Caverly (2020), as no linearizing assumptions on the payload motion are required. The Passivity Theorem provides a guarantee of closed-loop input–output stability when an output strictly passive (OSP) negative feedback controller is implemented with the proposed passivity-based adaptive control method. Two OSP feedback controllers are proposed, including a constant gain controller and a strictly positive real (SPR) controller.

To summarize, the novel contributions of the control method proposed in this paper that make it distinct from existing results, including (Chen et al., 2019, Damaren, 1996, Damaren, 1999, Fang et al., 2003, Ortega and Spong, 1989, Shen and Caverly, 2020, Tuan et al., 2013, Zhang, Zhang, Ji et al., 2020) are (1) an overhead crane adaptive control law that does not rely on a specific structure of the system dynamics or exact knowledge of the crane’s mass properties, (2) explicit use of the Passivity Theorem to prove robust closed-loop input–output stability when any OSP negative feedback controller is implemented, and (3) an SPR feedback control law that is capable of filtering high-frequency measurement noise while maintaining robust closed-loop input–output stability, which is demonstrated through experimental results. To the best of the knowledge of the authors, this work is the first to experimentally validate a passivity-based overhead crane control method that explicitly makes use of the Passivity Theorem for closed-loop input–output stability.

The remainder of the paper proceeds as follows. Section 2 provides a summary of the notation and passivity theory used throughout the paper. The equations of motion of the overhead crane system are introduced in Section 3, including a reformulation of the equations of motion required in the control formulation developed in Section 4. Passivity and closed-loop stability analyses are also included in Section 4. Numerical simulation results are presented in Section 5, while experimental results using the Quanser 3 DOF crane (Quanser, 2013) are found in Section 6. Concluding remarks are given in Section 7.

Section snippets

Preliminaries

This section briefly summarizes important notation and passivity theory concepts that are key to the control formulations and proofs presented in this paper.

System dynamics

Consider the two-dimensional overhead crane with three DOFs, shown in Fig. 2. The crane has trolley mass m1, winch mass mw, payload mass m2, and a flexible hoist cable with cross-sectional area A, and density ρc. The cable is wrapped around the winch drum and attached to the payload. The winch’s center of mass is colocated with the trolley’s center of mass, neglecting any shift in center of mass due to the winding of the cable. An external horizontal force f in the a1 direction is applied to

Control formulation

A passivity-based adaptive control method inspired by Tuan et al. (2013) is derived and analyzed in this section. The key differences between the control approach presented in this section and the methods described in Tuan et al. (2013) include an adaptive control law that does not depend on the specific structure of the system dynamics or knowledge of system parameters and the use of passivity-based control, which allows for an entire class of feedback controllers to be used. Section 4.1

Numerical results

Numerical simulation are performed using the overhead crane equations of motion in (2) to demonstrate the control law formulated in Section 4. Numerical values of the system parameters used in the simulation are given in Table 1. The constants δθ,ρ=750 and δϕ,ρ=1.5×103 are chosen to satisfy the inequality in (17) based on upper bounds on the eigenvalues of M̄ρρθ and M̄ρρϕ, as shown in Lemma 1. Fig. 5 illustrates how these bounds are numerically determined in the range π2<θ<π2 and 0<ϕ<Lr.

Experimental results

The proposed control method presented in Section 4 is validated experimentally in this section using a small-scale tower crane built by Quanser, shown in Fig. 15.

Details of the experiment setup are provided in Section 6.1, while the experimental results are presented in Section 6.2.

Conclusion

The passivity-based adaptive overhead crane control method presented in this paper is shown to render the closed-loop system input–output stable and guarantee the payload tracking error converges to zero in the vertical direction and remains bounded in the horizontal direction. Numerical simulations and experiments are performed to demonstrate the proposed control law’s performance and novel features, which include an adaptive control law that does not require knowledge of the system’s mass

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported in part by a study grant from Chung-Cheng Institute of Technology, National Defense University, Taiwan (R.O.C.) .

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