An Adaptive Variable Structure Control Scheme for Underactuated Mechanical Manipulators

Abstract

Mechanical arms have been widely used in the industry for many decades. They have played a dominant role in factory automation. However the control performance, or even system stability, would be deteriorated if some of the actuators fail during the operations. Hence, in this study, an adaptive variable structure scheme is presented to solve this problem. It is shown that, by applying the control mechanism proposed in this paper, the motion of robot systems can maintain asymptotical stability in case of actuators failure. The control algorithms as well as the convergence analysis are theoretically proved based on Lyapunov theory. In addition, to demonstrate the validity of the controller, a number of simulations as well as real-time experiments are also performed for Pendubot robot and Furuta robot systems. The results confirm the applicability of the proposed controller.

1. Introduction

Stabilization and tracking control problems of the nonlinear uncertain underactuated systems are difficult to be dealt with because of their fewer independent control actuators than the degrees of freedom to be controlled. In addition to the lack of enough control inputs for system’s degrees of freedom, the dynamics of underactuated mechanical systems is generally characterized by uncertain nonholonomic constraints. And hence, the underactuated system is always a challenging problem in the field of control applications. Robot manipulators have been ubiquitously equipped in many industrial plants for manufacturing automation and most of them were fully actuated. In general, the controller is designed for fully actuated systems based on the properties such as linear controllability, feedback linearizability, and passivity. However, actuators failure might occur due to infrequent maintenance or limited life cycle, which could cause severe damages to the operators and products. In general, the parts with actuator failures would be unactuated. The underactuated system presents challenging control program since the control design must typically exploit some coupling between the unactuated and the actuated states to achieve the control objective. In this paper, an adaptive variable structure control is presented to obtain globally asymptotical stabilization for a class of underactuated mechanical systems.

Underactuated systems are known as the systems with fewer independent control actuators than the degree of freedom to be controlled. The mathematical representation can be as below: 𝑞̈=𝑓𝑞,𝑞̇+𝑏𝑞𝑢. (1.1) The system is said to be underactuated if the system satisfies the condition rank(𝑏)<dim(𝑞), where 𝑞 is the state vector of generalized coordinates on the configuration manifold, 𝑞̇ is the generalized velocity vector, and 𝑏(𝑞) is the input state vector.

The underactuated system generally exhibits nonminimum phase [1] and is subject to nonholonomic constraints [2]. In contrast to fully actuated systems controller design, the underactuated system is more difficult with less independent control actuators than the degrees of freedom to be controlled. Tracking control of nonminimum phase systems is a highly sophisticated problem encountered in many practical engineering applications such as rocket control [3], aircraft control [4], flexible link manipulator control [5], and inverted pendulum systems [6]. The nonminimum phase property has long been recognized to be a major obstacle in many control problems. It is well known that unstable zeros cannot be moved with state feedback while the poles can be arbitrarily placed (if completely controllable). For this reason, a growing interest is arising for the design of automatic control systems for the underactuated systems. A lot of literature discussing the control of underactuated systems models has been published in the past years. Wang and other researchers [7, 8] presented a new sliding mode controller for second order underactuated systems. This paper has shown that all sliding surfaces are asymptotically stable. In [9, 10] the authors introduce an IDA-PBC approach to the underactuated mechanical systems. In [11, 12] some nonlinear control schemes, such as feedback linearization, inverse dynamics, and adaptive controller design, have been proposed for the control of underactuated systems. In order to achieve better performance, possibly unknown dynamics must also be considered in control design. In [13], the sliding mode control is proposed to achieve the robustness of many systems such as power systems, electronic motors, and robot manipulators. In [14], the paper provides the state of the art of recent developments in sliding mode control systems with soft computing, examining key technical research issues and future perspectives. In [15], a nonlinear control design is presented, but it requires the solution of a Hamilton-Jacobi type differential equation. It renders the solutions to achieve stability of the system robustly asymptotically stable. The robust backstepping method is utilized to find a stabilizing controller in [16]. The adaptive backstepping controller evolves from [17] with unknown parameters appearing linearly in the state equation, and adaptation mechanism is included to cope with unknown parameters. In [18], the controller is designed for a general class of input-output linearizable systems without zero dynamics, which can be extendable to minimum phase systems. The control scheme uses the backstepping design technique and guarantees semiglobal stability. In paper [19], an adaptive fuzzy output feedback control approach is proposed for single-input–single-output nonlinear systems without the measurements of the states. It is shown that by applying the proposed adaptive fuzzy control approach, the closed-loop systems are semiglobally uniformly ultimately bounded. In [20], the backstepping technique and adaptive fuzzy backstepping-control approaches for a class of strict-feedback large-scale nonlinear systems with unmodeled dynamics, dynamic disturbances, unknown high-frequency gain signs, and unmeasured states are designed. It has been proved that the proposed adaptive fuzzy control approach can guarantee the uniformly ultimate boundedness of all the signals in the closed-loop system. In [21] a robust adaptive multiestimation-based scheme has been designed for robotic manipulators with force reflection and uncertainties. The closed-loop stability is guaranteed if a minimum residence time, which might be updated online when unknown, between different controller parameterizations is respected. In paper [22], the stable adaptive fuzzy sliding mode controller is developed for nonlinear multivariable systems with unavailable states. It is proved that uniformly asymptotic output feedback stabilization can be achieved. In [23] the authors utilize the coupled sliding mode control concept to achieve the stabilization of inverted pendulum systems. By properly designing the control parameters, the resulting zero dynamics is proved to be semiglobally stable in the presence of bound disturbance.

The paper is organized as follows. In Section 2, an adaptive variable structure control scheme for a class of underactuated mechanical systems is developed. It is verified by computer simulations as well as experiments for the Pendubot system and the Furuta pendulum system, as shown in Sections 3 and 4, respectively. It is observed from both the simulations and the experiments that the effectiveness of the designed controller can be confirmed. Finally, some conclusions are given in Section 5.

2. Adaptive Variable Structure Controller Design

2.1. Problem Formulation

For convenience, the dynamic equations of a general underactuated mechanical manipulator system can be expressed as 𝐻𝑞𝑞̈+𝐷𝑞,𝑞̇𝑞̇+𝐺𝑞=𝑢, (2.1) where 𝑞=𝜃₁𝜃₂^𝑇,𝜏0𝑢=, (2.2) where 𝑞 is a vector of 𝑛 joint displacement, 𝑢 is the applied torque vector, 𝐻(𝑞)∈𝑅^𝑛×𝑛 is the effective moment of inertia matrix which is symmetric and positive definite, 𝜃₁∈𝑅^𝑚 is defined as the actuated joint, 𝜃₂∈𝑅^𝑛−𝑚 denotes the unactuated joint, and m is the number of actuated joint. The centripetal and Coriolis terms collected in the vector 𝐷(𝑞,𝑞)̇𝑞̇ and 𝐺(𝑞) represent the gravitational forces.

The dynamic system given in (2.1) exhibits the following properties that are utilized in the subsequent control development and stability analysis.(P1) The property of 𝐻(𝑞) matrices is positive definite and satisfies the following inequalities: 𝑄₁‖Ψ‖²≤𝜉^𝑇𝐻𝑞𝜉≤𝑄₂‖Ψ‖²,∀Ψ∈𝑅^𝑛, (2.3) where  𝑄₁,𝑄₂∈𝑅 are positive constants.(P2) The property of skew-symmetric matrices satisfies the following relationship: Ψ^𝑇12𝐻̇𝑞−𝐷𝑞,𝑞̇Ψ=0,∀Ψ∈𝑅^𝑛. (2.4)Equivalently, (2.1) can be rewritten in following partitioned form: ℎ₁₁ℎ₁₂ℎ₂₁ℎ₂₂𝜃̈₁𝜃̈₂+𝑑₁₁𝑑₁₂𝑑₂₁𝑑₂₂𝜃̇₁𝜃̇₂+𝑔₁𝑔₂=𝜏0. (2.5) The control objective in this section is to, for all arbitrary initial conditions, ensure the asymptotical stability of error signal, that is, lim_𝑡→∞‖𝑒‖=0. To achieve the tracking control objective, the controller is based on the assumption that 𝑞(𝑡), 𝑞(̇𝑡) are measurable.

The position tracking error is defined as follows: ̇𝑒̇𝑆=𝑒+𝐾𝑒=¹+𝑘₁𝑒₁𝑒₂̇+𝑘₂𝑒₂=𝑠₁𝑠₂, (2.6) where 𝐾=^{𝑘1𝐼𝑚×𝑚0}_{0𝑘2𝐼(𝑛−𝑚)×(𝑛−𝑚)} with 𝑘₁, 𝑘₂ are some designated positive constants. 𝐼_𝑖×𝑖 is 𝑖×𝑖 identity matrix, 𝑒=^𝑒1_𝑒2=^𝜃1_𝜃2⁻₋^𝜃𝑑1_𝜃𝑑2∈𝑅^𝑛, and 𝜃^𝑑₁ and 𝜃^𝑑₂ are the desired trajectories with ‖𝜃^𝑑₁‖_∞≤𝜀₁ and ‖𝜃^𝑑₂‖_∞≤𝜀₂, where 𝜀₁,𝜀₂ are some bound positive values and the design of 𝜃^𝑑₁ and 𝜃^𝑑₂ has to satisfy the zero dynamics ℎ₂₁𝜃̈_1𝑑+ℎ₂₂𝜃̈_2𝑑+𝑑₂₁𝜃̇_1𝑑+𝑑₂₂𝜃̇_2𝑑+𝑔₂=0. In general, the desired trajectories are often chosen to be constant. Then, in view of (2.5), it is readily obtained that ℎ₁₁ℎ₁₂ℎ₂₁ℎ₂₂𝑠₁̇𝑠₂̇+𝑑₁₁𝑑₁₂𝑑₂₁𝑑₂₂𝑠₁𝑠₂+𝑘₃00𝑘₄𝑠₁𝑠₂=𝑊₁𝜙₁+𝑘₃𝑠₁𝑊+𝜏₂𝜙₂+𝑘₄𝑠₂ (2.7) or more compactly 𝐻𝑞̇𝑞̇W𝑆+𝐷𝑞,𝑆+𝐾𝑆=¹𝜙₁+𝑘₃𝑠₁W+𝜏₂𝜙₂+𝑘₄𝑠₂, (2.8) where 𝑊₁𝜙₁=ℎ₁₁−𝜃̈^𝑑₁+𝑥₁𝑒₁̇+ℎ₁₂−𝜃̈^𝑑₂+𝑥₂𝑒₂̇+𝑑₁₁−𝜃̇^𝑑₁+𝑥₁𝑒₁+𝑑₁₂−𝜃̇^𝑑₂+𝑥₂𝑒₂−𝑔₁,𝑊₂𝜙₂=ℎ₂₁−𝜃̈^𝑑₁+𝑥₁𝑒₁̇+ℎ₂₂−𝜃̈^𝑑₂+𝑥₂𝑒₂̇+𝑑₂₁−𝜃̇^𝑑₁+𝑥₁𝑒₁+𝑑₂₂−𝜃̇^𝑑₂+𝑥₂𝑒₂−𝑔₂, (2.9)𝐾=^{𝑘3𝐼𝑚×𝑚0}_{0𝑘4𝐼(𝑛−𝑚)×(𝑛−𝑚)} and 𝑘₃∈𝑅, 𝑘₄∈𝑅 are some positive constants.

2.2. Controller Design

In this subsection, an adaptive variable structure control scheme is presented to account for parameter uncertainty existing in𝑊₁𝜙₁ and𝑊₂𝜙₂. From the system dynamics of a robot manipulator with revolute joints, it is well known and can be easily verified that 𝑊₁𝜙₁≤^𝑙_𝑖=1𝛿_𝑖𝑄_𝑖𝑞̇,𝑊𝑞,₂𝜙₂≤^𝑟_𝑗=1𝜎_𝑗𝑄_𝑗𝑞,𝑞̇, (2.10) where 𝛿_𝑖, 𝜎_𝑗, 𝑖=1,2,…,𝑙, 𝑗=1,2,…,𝑟 are some unknown positive constants, and 𝑄_𝑖,𝑄_𝑗 are known positive functions of 𝑞 and 𝑞̇.

Based on the development of the aforementioned error dynamics, the control law is presented as follows: 𝜏=−𝑘₃𝑠₁−sgn𝑠₁^𝑙_𝑖=1𝛿_𝑖𝑄_𝑖𝑞,𝑞̇−sgn𝑠₁𝑠₂𝜉−𝑘₅sgn𝑠₁𝑠₂,𝜉̇(𝑡)=𝜉^1/(2𝑛+1)−𝑘₄𝑠₂²−𝑠₂^𝑚_𝑗=1𝜎_𝑗𝑄_𝑗𝑞,𝑞̇, (2.11)(2.12) where 𝑛 is some positive integer chosen by designer, 𝑘_𝑙,𝑙=3,4,5 are some positive constants, and sgn(⋅) is the sign function, which is defined as sgn(𝑥)=1𝑥>00𝑥=0−1𝑥<0. (2.13) In (2.11) and (2.12) 𝛿_𝑖,𝜎_𝑗 denote the parameter estimates, and the estimation errors are defined as follow: 𝛿_𝑖=𝛿_𝑖−𝛿_𝑖𝜎,∀𝑖=1,…,𝑙,_𝑗=𝜎_𝑗−𝜎_𝑗,∀𝑗=1,…,𝑟. (2.14)

Let the parameter estimation laws be designed as ̇𝛿_𝑖=Γ_𝑎𝑖𝑠₁𝑄_𝑖𝑞̇,𝜎̇𝑞,_𝑗=Γ_𝑏𝑗𝑠₁𝑄_𝑗𝑞,𝑞̇, (2.15)(2.16) where Γ_𝑎𝑖∈𝑅, Γ_𝑏𝑗∈𝑅, (forall𝑖=1,2,…,𝑙,𝑗=1,2,…,𝑟) are positive definite gain matrices.

Remark 2.1. From the robustness point of view, it would be better if additional feedback term −𝑘₅sgn(𝑠₁)‖𝑠₂‖ is included in the control law (2.11). With such an inclusion, the stabilization of 𝜃₂ subject to external disturbance can also be maintained as the 𝜃₁ arrived at zero. This can be easily checked from the stability proof given in the theorem.

In the following theorem, it is shown that, by applying the adaptive variable structure controller designed above, the asymptotical stabilization of overall closed-loop system can be achieved.

Theorem 2.2. Consider the robot system (2.5) with imprecise system parameters. By applying the control laws (2.11)-(2.12) and estimate laws (2.15)-(2.16), the objective of global asymptotical stabilization can be achieved, that is, all signals inside the close-loop system are bounded and 𝑒₁,𝑒₂→0 asymptotically.

Proof. To prove the theorem, let the Lyapunov function candidate be defined as 1𝑉=2𝑆^𝑇1𝐻𝑆+2^𝑙_𝑖=1𝛿_𝑖^𝑇Γ⁻¹_𝑎𝑖𝛿_𝑖+12^𝑟_𝑗=1𝜎_𝑗^𝑇Γ⁻¹_𝑏𝑗𝜎_𝑗+2𝑛+1𝜉2𝑛^{2𝑛/(2𝑛+1)} (2.17) and take the time derivative of 𝑉 to get 𝑉̇=𝑆^𝑇𝐻𝑆̇+12𝑆^𝑇𝐻̇𝑆+^𝑙_𝑖=1𝛿^𝑇_𝑖Γ_𝑎𝑖𝛿_𝑖̇+^𝑟_𝑖=1𝜎^𝑇_𝑗Γ_𝑏𝑗𝜎_𝑖̇𝜉^{−1/(2𝑛+1)}𝜉̇=𝑆^𝑇−𝐷𝑆+𝜏+𝑊₁𝜙₁𝑊₂𝜙₂+12𝑆^𝑇𝐻̇𝑆+^𝑙_𝑖=1𝛿^𝑇_𝑖Γ_𝑎𝑖𝛿_𝑖̇+^𝑟_𝑗=1𝜎^𝑇_𝑗Γ_𝑏𝑗𝜎_𝑖̇+𝜉^{−1/(2𝑛+1)}𝜉̇=12𝑆^𝑇𝐻̇−2𝐷𝑆+𝑠^𝑇₁𝜏+𝑠^𝑇₁𝑊₁𝜙₁+𝑠^𝑇₂𝑊₂𝜙₂+^𝑙_𝑖=1𝛿^𝑇_𝑖Γ_𝑎𝑖𝛿_𝑖̇+^𝑟_𝑗=1𝜎^𝑇_𝑗Γ_𝑏𝑗𝜎_𝑖̇+𝜉^{−1/(2𝑛+1)}𝜉̇=𝑠^𝑇₁𝜏+𝑠^𝑇₁𝑊₁𝜙₁+𝑠^𝑇₂𝑊₂𝜙₂+^𝑙_𝑖=1𝛿^𝑇_𝑖Γ_𝑎𝑖𝛿_𝑖̇+^𝑟_𝑗=1𝜎_𝑗^𝑇Γ_𝑏𝑗𝜎_𝑖̇+𝜉^{−1/(2𝑛+1)}𝜉̇. (2.18) In the above derivations, (2.4) has been applied. Then, by using the boundedness of parameter uncertainties (2.10), the time derivative of 𝑉 can be further expressed as 𝑉̇≤𝑠^𝑇₁𝜏+𝑠₁𝑊₁𝜙₁+𝑠^𝑇₂𝑊₂𝜙₂+^𝑙_𝑖=1𝛿^𝑇_𝑖Γ_𝑎𝑖𝛿_𝑖̇+^𝑟_𝑗=1𝜎^𝑇_𝑗Γ_𝑏𝑗𝜎_𝑖̇+𝜉^{−1/(2𝑛+1)}𝜉̇≤𝑠^𝑇₁𝜏+𝑠₁^𝑙_𝑖=1𝛿_𝑖𝑄_𝑖𝑞,𝑞̇+𝑠^𝑇₂𝑊₂𝜙₂+^𝑙_𝑖=1𝛿^𝑇_𝑖Γ_𝑎𝑖𝛿_𝑖̇+^𝑟_𝑗=1𝜎^𝑇_𝑗Γ_𝑏𝑗𝜎_𝑖̇+𝜉^{−1/(2𝑛+1)}𝜉̇. (2.19) Thus, by applying the designed controller (2.11)-(2.12) and the adaptive laws (2.15)-(2.16), the following inequality holds: 𝑉̇≤−𝑘₃𝑠₁²+𝑠₁𝑠₂𝜉+𝑠₂𝑊₂𝜙₂+^𝑟_𝑗=1𝜎^𝑇_𝑗Γ_𝑏𝑗𝜎_𝑖̇+𝜉^{−1/(2𝑛+1)}𝜉̇−𝑘₅𝑠₁𝑠₂≤−𝑘₃𝑠₁²−𝑠₁𝑠₂𝜉+𝑠₂^𝑚_𝑗=1𝛿_𝑗𝑄_𝑗𝑞,𝑞̇+^𝑟_𝑗=1𝜎^𝑇_𝑗Γ_𝑏𝑗−Γ_𝑏𝑗𝑠₂𝑄_𝑗𝑞,𝑞̇+𝜉^{−1/(2𝑛+1)}𝜉̇−𝑘₅𝑠₁𝑠₂≤−𝑘₃𝑆₁²−𝑆₁𝑆₂𝜉+𝑆₂^𝑚_𝑗=1𝛿_𝑗𝑄_𝑗𝑞,𝑞̇+^𝑟_𝑗=1𝜎^𝑇_𝑗Γ_𝑏𝑗−Γ_𝑏𝑗𝑆₂𝑄_𝑗𝑞̇+𝑞,−𝑘₄𝑆₂²−𝑆₂^𝑚_𝑗=1𝜎𝑄_𝑗𝑞,𝑞̇−𝑘₅𝑠₁𝑠₂≤−𝑘₃𝑆₁²−𝑘₄𝑆₂²−𝑆₁𝑆₂𝜉−𝑘₅𝑠₁𝑠₂≤−‖𝑆‖^𝑇𝐾‖𝑆‖, (2.20) where 𝐾=^𝑘30_0𝑘4. Thus, it is clear that 𝑉̇(𝑡)≤0 if 𝑘₃,𝑘₄,𝑘₅>0. Then, according to Barbalat’s lemma [22], it is readily obtained that 𝑆→0 as 𝑡→∞ asymptotically and, hence, 𝑒,𝑒̇→0 as 𝑡→∞.

In the next section, the Pendubot system and the Furuta pendulum system are studied to verify the above theoretical results.

3. Applications to the Pendubot System

In this section, the adaptive sliding mode control scheme developed previously is applied to the Pendubot system. The Pendubot system is a two link planner manipulator in which the first link is driven by an actuator but the second link is unactuated, as shown in Figure 1. Link 1 and link 2 are connected by revolute joint, and link 1 can freely rotate about 360 degrees in the horizontal plane. In this section, the kinematic and dynamic models of a Pendubot system are presented.

(a)

(b)

Figure 1 

The Pendubot system.

The equation of motion is derived using Lagrange’s equation [24] as 𝑑𝑑𝑡𝜕𝐿𝜕𝑞̇𝑞,𝑞̇−𝜕𝐿𝜕𝑞̇𝑞,𝑞̇=𝜏. (3.1) The Lagrange’s function is given by 𝐿=𝐾−𝑃, (3.2) where 𝐾 is the kinetic energy of the system and 𝑃 is The potential energy of the system.

From the Lagrange’s equation, it is shown that the equation of motion for this system can be represented in the following form: 𝑀𝑞𝑞̈+𝐶𝑞,𝑞̇𝑞̇=𝜏, (3.3)𝑀(𝑞)∈𝑅^𝑛×𝑛 is the inertia matrix, 𝐶(𝑞,𝑞)̇∈𝑅^𝑛×𝑛 is the centripetal-Coriolis matrix.

Where 𝑞(𝑡)∈𝑅² is the joint position defined as follows: 𝑞=𝜃₁𝜃₂, (3.4) where 𝜃₁∈𝑅¹ and 𝜃₂∈𝑅¹ denote the angles of link 1 and link 2, respectively. 𝑀(𝑞)∈𝑅^2×2, 𝐶(𝑞,𝑞)̇∈𝑅^2×2, and control input vector 𝜏 are defined as follows: 𝜃𝑞=¹𝜃₂𝑞=𝑚,𝑀¹¹𝑚₁₂𝑚₂₁𝑚₂₂𝑞̇=𝑐,𝐶𝑞,¹¹𝑐₁₂𝑐₂₁𝑐₂₂𝑢0,𝜏=, (3.5) where 𝑚₁₁=𝑚₁𝑐₁²+𝑚₂𝑙₁²+𝑚₂𝑐₂²+2𝑚₂𝑙₁𝑐₂cos𝜃₂+𝐽₁+𝐽₂,𝑚₁₂=𝑚₂𝑐₂²+𝑚₂𝑙₁𝑐₂cos𝜃₂+𝐽₂,𝑚₂₁=𝑚₂𝑐₂²+𝑚₂𝑙₁𝑐₂cos𝜃₂+𝐽₂,𝑚₂₂=𝑚₂𝑐₂²+𝐽₂,𝑐₁₁=−𝑚₂𝑙₁𝑐₂𝜃̇₂sin𝜃₂,𝑐₁₂=−𝑚₂𝑙₁𝑐₂𝜃̇₁sin𝜃₂−𝑚₂𝑙₁𝑐₂𝜃̇₂sin𝜃₂,𝑐₂₁=𝑚₂𝑙₁𝑐₂𝜃̇₁sin𝜃₂,𝑐₂₂=0. (3.6) From (3.1), as mentioned above, the inertia and centripetal-Coriolis matrices satisfy the following condition: Ψ^𝑇12𝑀̇𝑞−𝐷𝑞,𝑞̇Ψ=0, (3.7) where 𝑀̇𝑞−2𝐷𝑞,𝑞̇=0𝑚₂𝑙₁𝑐₂sin𝑞₂2𝑞̇₁+𝑞̇₂−𝑚₂𝑙₁𝑐₂sin𝑞₂2𝑞̇₁+𝑞₂0 (3.8) which is a skew-symmetric matrix. The system variables and parameters are defined in Table 1.

Table 1 

Definitions of system variables and parameters.

As described in the preceding section, the adaptive variable structure controller will be adopted to drive the first link to the desired position, and keep the second link to settle down at the original angle. From the dynamic equation shown above, the following bounding assumption can be made for 𝑊₁𝜙₁ and 𝑊₂𝜙₂: 𝑊₁𝜙₁≤𝛿₁𝜃̇²₁+𝜃̇²₂+𝛿₂𝑒₁²+𝑒₂²+𝛿₃,𝑊₂𝜙₂≤𝜎₁𝜃̇²₁+𝜃̇²₂+𝜎₂𝑒₁²+𝜎₃, (3.9) where 𝛿_𝑖, 𝑖=1,2,3 and 𝜎_𝑗, 𝑗=1,2,3 are unknown positive bounded constants. Before proceeding with the adaptive sliding mode control design, let the parameter estimation errors be defined as 𝛿_𝑖=𝛿_𝑖−𝛿_𝑖𝜎,𝑖=1,2,3,_𝑗=𝜎_𝑗−𝜎_𝑗,𝑗=1,2,3. (3.10) From the error dynamics (2.7), the adaptive controllers can be designed as 𝑢=−𝑘₃𝑠₁−sgn𝑠₁𝛿₁𝜃̇²₁+𝜃̇²₂+𝛿₂𝑒²₁+𝑒²₂+𝛿₃−sgn𝑠₁𝑠₂𝜉−𝑘₅sgn𝑠₁𝑠₂,𝜉̇(𝑡)=𝜉^1/(2𝑛+1)−𝑘₄𝑠²₂−𝑠₂𝜎₁𝜃̇²₁+𝜃̇²₂+𝜎₂𝑒²₁+𝜎₃, (3.11) with the adaptive laws ̇𝛿₁=Γ₁₁𝑠₁𝜃̇²₁+𝜃̇²₂,̇𝛿₂=Γ₁₂𝑠₁𝑒²₁+𝑒²₂,̇𝛿₃=Γ₁₃𝑠₁,𝜎̇₁=Γ₂₁𝑠₂𝜃̇²₁+𝜃̇²₂,𝜎̇₂=Γ₂₂𝑠₂𝑒²₁,𝜎̇₃=Γ₂₃𝑠₂, (3.12)(3.13)(3.14)(3.15)(3.16)(3.17) where Γ_𝑖𝑗, 𝑖=1,2, 𝑗=1,2,3 are positive constants.

3.1. Simulation Results

A number of simulations are performed for the adaptive sliding mode schemes proposed in this section. In this simulation, the initial positions of link 1 and link 2 are identically  0^∘. The desired final position for link 1 is 𝜃^𝑑₁=45^∘ and for link 2 is 𝜃^𝑑₂=0^∘, respectively.

The simulation results are shown in Figures 2–6. Figures 2 and 3 show the link 1 output angle and link 2 output angle responses, respectively. It is seen that link 1 and link 2 can converge to the target in 16 seconds. Figure 4 shows the convergence of the adaptive laws, while Figure 5 shows the control adaptation gain parameters 𝜉(𝑡). Finally, the control input is shown in Figures 6 and 12. The control gains used in the simulation are chosen to be 𝑘₁=0.3, 𝑘₂=20, 𝑘₃=1.3, and 𝑘₄=70, and the adaptive gains are chosen as Γ₁₁=0.1,Γ₁₂=1,Γ₁₃=0.01,Γ₂₁=1,Γ₂₂=1,andΓ₂₃=1.

Figure 2 

Link 1 angular displacement with adaptive algorithm.

Figure 3 

Link 2 angular displacement with adaptive algorithm.

Figure 4 

Adaptive control gains estimates for the controller.

Figure 5 

The control gain 𝜉(𝑡) estimate for the adaptive controller.

Figure 6 

Control input voltage for the adaptive controller.

3.2. Experimental Results

To validate the practical application of the proposed algorithm experiments on the underactuated mechanical system apparatus is also conducted, as shown in Figure 7. In the mechanical system the actuator is dc motor mounted on the arm (link 1) and coupled to links through the power MOSFET chopper amplifier. A 500 pulse/rev shaft encoder is used to sense the arm (link 1) position and pendulum position or link 2 position. A 12 bit A/D converter provides the required signal. The microcomputer used is a INTEL-P4-based system with 3 GHz clock.

Figure 7 

Experimental setup for Pendubot system.

To demonstrate the effectiveness of our proposed controller, a comparison with the results in [23] is made. The controller gains are chosen to be 𝑘=10, 𝑐_𝑥=1, and 𝑐_𝜃=2 and the results are depicted in Figures 8 and 9. In the experiment, the desired final position for link 1 is 𝜃^𝑑₁=45^∘, and for link 2 is 𝜃^𝑑₂=0^∘. From Figure 10 it is seen that link 1 could converge to 23 degrees at nearly 5 seconds but link 2 couldn’t keep at 0 degree as indicated in Figure 11. From the experimental results, it is clearly observed that even though the exact dynamics of the nonlinear model is available but with the system parameter unknown, the control precision is still a problem needed to be solved.

Figure 8 

Link 1 angular displacement (compared sliding mode controller).

Figure 9 

Link 2 angular displacement (compared sliding mode controller).

Figure 10 

Link 1 angular displacement with adaptive algorithm.

Figure 11 

Link 2 angular displacement with adaptive algorithm.

Figure 12 

Control input voltage for the adaptive controller.

For the experiment by using the controller proposed in this paper, the same conditions as above are set, 𝜃^𝑑₁=45^∘ for link 1 and 𝜃^𝑑₂=0 for link 2. For the controller proposed in this paper, the following controlled gains are chosen for experiments: 𝑘₁=10, 𝑘₂=4, 𝑘₃=1, and 𝑘₄=1, and the adaptive gains are set to be Γ₁₁=Γ₁₂=Γ₁₃=Γ₂₁=Γ₂₂=Γ₂₃=0.1. Figures 10 and 11 depict the position response of link 1 and link 2, and from the figures it is observed that link 1 converges to the target nearly in 2 seconds, while the convergence of link 2 is completed in 6 seconds. The control input is described in Figure 14. It is seen that the adaptive controller can provide stable and better performance over a wide range of parameter variations in comparison with the conventional sliding mode controller.

4. Applications to the Furuta Pendulum System

In this section, the Furuta pendulum system [1] is considered for demonstration. Figure 13 illustrates the mechanical system.

(a)

(b)

Figure 13 

The Furuta pendulum system.

Figure 14 

Control Structure.

The equation of motion can be written in the following general form: 𝑀𝑞̈𝑞̇̇𝑞=𝜏𝑞+𝐷𝑞,𝑞+𝐺^𝑙0, (4.1) where 𝑀𝑞=𝐼₀+𝑚₁𝐿²₀+𝑙²₁sin²𝜃₂𝑚₁𝐿₀𝑙₁cos𝜃₂𝑚₁𝑙₁𝐿₀cos𝜃₂𝐽₁+𝑚₁𝑙²₁,𝐷𝑞̇=1𝑞,2𝑚₂𝑙²₁𝜃̇₂sin2𝜃₂−𝑚₁𝑙₁𝐿₀𝜃̇₂sin𝜃₂+12𝑚₁𝑙²₁𝜃̇₁sin2𝜃₂−12𝑚₁𝑙²₁𝜃̇₁sin2𝜃₂0,𝐺𝑞=0−𝑚₁𝑙₁𝑔sin𝜃₂. (4.2) Form (3.14)-(3.15), it is straightforward to show that the following relationship is satisfied: Ψ^𝑇12𝑀̇𝑞−𝐷𝑞,𝑞̇Ψ=0. (4.3) Clearly, 𝑀̇(𝑞)−2𝐷(𝑞,𝑞)̇=^{0−𝑚1𝑙1(𝑙1sin(2𝜃2)̇𝜃1)−𝐿0sin𝜃2̇𝜃2}_{𝑚1𝑙1(𝑙1sin(2𝜃2)̇𝜃1)−𝐿0sin𝜃2̇𝜃20} is a skew-symmetric matrix. The system variables and parameters are defined in Table 2.

Table 2 

Definitions of system variables and parameters for Furuta pendulum system.

Stage 1 (Swing up the Pendulum). The control objective here is to move the pendulum from the stable equilibrium position to the unstable equilibrium position. In order to balance the pendulum to the unstable equilibrium point, a swing-up controller should be first applied to make the pendulum inside some acceptable region around the vertical line. Then, the adaptive controller will be activated to keep the pendulum upright at the unstable equilibrium point.
The swing-up control law can be written as 𝑢_𝑠(𝑡)=𝑐0,𝑡=0,₁,𝑡_𝑖+1>𝑡_𝑖>0,−𝑐₂,𝑡_𝑖+2>𝑡_𝑖+1, (4.4) where 𝑐₁,𝑐₂ are positive constants.
As shown in Figure 14, when the pendulum is within ±0.8 rad of the vertical position, the balancing controller is activated. Otherwise, the system operates under swing-up control.

Stage 2 (Balance the Pendulum). Form the dynamic equation shown above, it is found that bounds of 𝑊₁𝜙₁ and 𝑊₂𝜙₂ can be written as 𝑊₁𝜙₁≤𝛿₁𝜃̇²₁+𝜃̇²₂+𝛿₂𝑒²₁+𝑒²₂+𝛿₃,𝑊₂𝜙₂≤𝜎₁𝜃̇²₁+𝜃̇²₂+𝜎₂𝑒₁²+𝜎₃. (4.5)

The controller and adaptive gain are designed as follows: 𝑢=−𝑘₃𝑠₁−sgn𝑠₁𝛿₁𝜃̇²₁+𝜃̇²₂+𝛿₂𝑒²₁+𝑒²₂+𝛿₃−sgn𝑠₁𝑠₂𝜉−𝑘₅𝑠₂,𝜉̇(𝑡)=𝜉^1/(2𝑛+1)−𝑘₄𝑠²₂−𝑠₂𝜎₁𝜃̇²₁+𝜃̇²₂+𝜎₂𝑒²₁+𝜎₃. (4.6)

The adaptive laws is as follows: ̇𝛿₁=Γ₁₁𝑠₁𝜃̇²₁+𝜃̇²₂,̇𝛿₂=Γ₁₂𝑠₁𝑒²₁+𝑒²₂,̇𝛿₃=Γ₁₃𝑠₁,𝜎̇₁=Γ₂₁𝑠₂𝜃̇²₁+𝜃̇²₂,𝜎̇₂=Γ₂₂𝑠₂𝑒₁^2,𝜎̇₃=Γ₂₃𝑠₂, (4.7) where Γ_𝑖𝑗,𝑖=1,2,𝑗=1,2,3 are positive constants.

4.1. Simulation Results

The simulations are performed using MATLAB and SIMULINK to show the validity of the controller. The results are indicated in Figures 15–19.

Figure 15 

Link 1 angular displacement with adaptive algorithm.

Figure 16 

Control input voltage for the adaptive controller.

Figure 17 

Link 2 angular displacement with adaptive algorithm.

Figure 18 

Adaptive control gains estimates for the controller.

Figure 19 

The control gain 𝜉(𝑡) estimate for the adaptive controller.

In the simulations, the initial positions of 𝜃₁ and 𝜃₂ are chosen identically as 0^∘. The desired arm position and pendulum position are chosen as 𝜃^𝑑₁=0.4 rad, as 𝜃^𝑑₂=0 rad, respectively. As illustrated in Figure 15 the steady state position tracking error can approach to zero in 6 seconds. As the results in Figure 17 show, the tracking error of unactuated link can converge stably in a fast manner, and it is observed that all the parameter estimates trajectories are convergent, as shown in Figure 18. The control input is shown in Figure 16. Figure 19 plots the trajectory of 𝜉(𝑡).

4.2. Experimental Results

Similarly, the results obtained from the sliding mode controller [23] and the adaptive controller proposed in this paper are also performed and compared in Figures 22–24. From Figures 20 and 21 it is seen that, although the pendulum converge to target, the arm could not precisely settle down to its desired position but oscillates within some bounded region by using the sliding mode controller. It means that the poor closed-loop system response might arise due to system parameters.

Figure 20 

Link 1 angular displacement (sliding mode controller).

Figure 21 

Link 2 angular displacement (sliding mode controller).

Figure 22 

Link 1 angular displacement with adaptive algorithm.

Figure 23 

Link 2 angular displacement with adaptive algorithm.

Figure 24 

Control input voltage for the adaptive controller.

As for the experiment of adaptive controller, the controlled gains are chosen as 𝑘₁=1, 𝑘₂=0.5, 𝑘₃=10, and 𝑘₄=1. The corresponding adaptive gains are set to be Γ₁₁=Γ₁₂=Γ₁₃=Γ₂₁=Γ₂₂=Γ₂₃=0.01. In this experiment, the control gains are difficult to determine due to the unknown uncertainties in practical applications and are generally chosen as a compromise between the stability and the control performance. Figures 22 and 23 depict the position response of arm and pendulum. It is observed from the figures that the arm could converge to the desired position at 6 sec with the pendulum swinging up to top position in 5 seconds. Finally the control input is described in the Figure 24.

5. Conclusion

In this paper, the adaptive control for underactuated mechanical systems with parameter uncertainty is discussed. By utilizing Lyapunov-based stability analysis asymptotical stabilization of such systems can be guaranteed. The control schemes are also implemented in the Pendubot system and Furuta pendulum system to verify the performance of the proposed adaptive sliding controller. It is shown that the tracking error can be made asymptotically stable undergoing such controller design.