Adaptive Sliding Mode Control of Mobile Manipulators with Markovian Switching Joints

Abstract

The hybrid joints of manipulators can be switched to either active (actuated) or passive (underactuated) mode as needed. Consider the property of hybrid joints, the system switches stochastically between active and passive systems, and the dynamics of the jump system cannot stay on each trajectory errors region of subsystems forever; therefore, it is difficult to determine whether the closed-loop system is stochastically stable. In this paper, we consider stochastic stability and sliding mode control for mobile manipulators using stochastic jumps switching joints. Adaptive parameter techniques are adopted to cope with the effect of Markovian switching and nonlinear dynamics uncertainty and follow the desired trajectory for wheeled mobile manipulators. The resulting closed-loop system is bounded in probability and the effect due to the external disturbance on the tracking errors can be attenuated to any preassigned level. It has been shown that the adaptive control problem for the Markovian jump nonlinear systems is solvable if a set of coupled linear matrix inequalities (LMIs) have solutions. Finally, a numerical example is given to show the potential of the proposed techniques.

1. Introduction

The hybrid joint shown in Figure 1 was first proposed in [1–5], which is with one clutch and one brake. When the clutch is released, the link is free, and the passive link is directly controlled by the dynamic coupling of mobile manipulators; when it is on, the joint is actuated by the motor. Moreover, the passive link can be locked by the brake embedded in the joint as needed. The robot with hybrid joints is called the hybrid actuated robot.

Figure 1 

The hybrid joint.

One of the advantages of using hybrid actuated robots is that they may consume less energy than the fully-actuated ones. For example, hyperredundant robots, such as snake-like robots or multilegged mobile robots [6], need large redundancy for dexterity and specific task completion while underactuation structure allows a more compact design and much simpler control and communication schemes. The hybrid actuated robot concept is also useful for the reliability or fault-tolerant design of fully-actuated robots working in hazardous areas or with dangerous materials. If any of the joint actuators of such a device fails, one degree of freedom of the system would be lost. It is important, in these cases, that the passive (failed) joint can still be controlled via the dynamic coupling with the active ones, so the system can still make use of all of its degrees of freedom originally planned.

Hybrid actuated mobile manipulator is the robot manipulator consisting of hybrid joints mounting on a wheeled mobile robot, which first appeared in [1–5]. Hybrid actuated mobile manipulators are different from full-actuated mobile manipulators in [7–26], due to simultaneously integrating both kinematic constraints and dynamic constraints. For these reasons, increasing effort needs to be made towards control design that guarantee stability and robustness for hybrid actuated mobile manipulators with the consideration of joint switching. The hybrid joint is also with the characteristic of underactuated the joints [27–34], for example, the hybrid joints in the free mode, which can rotate freely, can be indirectly driven by the effect of the dynamic coupling between the active and passive joints. The zero torque at the hybrid joints results in a second-order nonholonomic constraint [35, 36].

The mobile manipulator using Markovian switching hybrid joint can be loosely defined as a system that involves the interaction of both discrete events (represented by finite automata) and continuous-time dynamics (represented by differential equations). The joint switching seems to be stochastic and the switching may appear in any joints of the robot which need to develop Markovian jump linear system (MJLS) [37] to incorporate abrupt changes in the joints of mobile manipulators and use the Markovian jumping systems to guarantee the stochastic stability. Therefore, the discrete part (switching part) can be regarded as a continuous-time Markov process representing the modes of the system and the continuous part represents the dynamics state of the system, which evolves according to the differential dynamic equation when the mode is fixed. The hybrid formulation provides a powerful framework for modeling and analyzing the systems subject to abrupt joint switching variations, which are partly due to the inherently vulnerability to abrupt changes caused by component failures, sudden environmental disturbances, abrupt variation of the operation point of mobile manipulator, and so on.

The joint switching seems to be stochastic and the switching may appear in any joints of the robot, while simple switching approach cannot handle all the possibility. In this paper, to avoid the necessity of stopping the robot as the joint switches, MJLS method used to model and analyze switching robotic systems is an effective but challenging work.

To our best knowledge, there are few works considering MJLS method used to model and analyze switching robotic systems. In this paper, we consider the problem of adaptive control for stochastic jump systems with matched uncertainties and disturbances. The jumping parameters are treated as continuous-time discrete-state Markov process. Note that adaptive control method is one of the most popular techniques of nonlinear control design [8]. However, adaptive control for stochastic nonlinear mechanical dynamics systems with Markovian switching has received relatively little attention. Therefore, this paper will be concerned with the design of adaptive control for mobile manipulators using Markovian switching joints. There exist parameter uncertainties, nonlinearities, and external disturbance in the systems and environments under consideration. First, we design a reduced model for the wheeled mobile manipulator with switching joints. After introducing continuous-time Markov chain, adaptive control is adopted to cope with the effect of Markovian switching and nonlinear dynamics uncertainty and drive wheeled mobile manipulators following the desired trajectory. The resulting closed-loop system is bounded in probability and the effect due to the external disturbance on the tracking error can be attenuated to any pre-assigned level. Moreover, unknown upper bounds of dynamics uncertainties and disturbances can be estimated by adaptive updated law. The mechanical system with matched disturbances and Markov jumping is solved in terms of a finite set of coupled LMIs. It has been shown that the adaptive control problem for the Markovian jump nonlinear systems is solvable if a set of coupled LMIs have solutions. Finally, a numerical example is given to show the potential of the proposed techniques.

The main contributions of this paper lie in: (i)developing a reduced model for mobile manipulators such that it could be transformed into the framework of MJLS with modeling system dynamics uncertainties; (ii)designing an adaptive sliding mode control (SMC) for wheeled mobile manipulators with hybrid joints with Markovian switching; (iii)the system with matched disturbances and Markov jumping is solved in terms of a finite set of coupled LMIs.

2. Preliminary

Lemma 2.1 (see [38]). Let 𝑒=𝐻(𝑠)𝑟 with 𝐻(𝑠) representing an (𝑛×𝑚)-dimensional strictly proper exponentially stable transfer function, 𝑟 and 𝑒 denoting its input and output, respectively. Then 𝑟∈𝐿^𝑚2⋂𝐿^𝑚∞ implies that 𝑒,𝑒̇∈𝐿^𝑛2⋂𝐿^𝑛∞, 𝑒 is continuous, and 𝑒→0 as 𝑡→∞. If, in addition, 𝑟→0 as 𝑡→∞, then 𝑒̇→0.

Lemma 2.2 (see [39]). For the matrix 𝐴 and 𝐵 with appropriate dimensions, if (𝐼+𝐴𝐵) is nonsingular, then (𝐼+𝐴𝐵)⁻¹=𝐼−𝐴(𝐼+𝐵𝐴)⁻¹𝐵.

Theorem 2.3. Given a Markov jump linear system with the system parameter matrices 𝐴_𝑖, 𝐵_𝑖, 𝐶_𝑖, 𝐷_𝑖, and 𝐼>𝜂²𝐷^𝑇𝑖𝐷_𝑖, for 𝜂≥0, Φ(𝑡) is unknown but satisfying ‖Φ(𝑡)‖≤𝜂, 𝑥̇=𝐴_𝑖+𝐵_𝑖𝐼−Φ(𝑡)𝐷_𝑖⁻¹Φ(𝑡)𝐶_𝑖𝑥,(2.1) if there exits 𝑃_𝑖>0 satisfies the following inequality for each 𝑖∈𝑆=1,2,…,𝑁, ⎡⎢⎢⎢⎢⎢⎢⎣𝑃_𝑖𝐴_𝑖+𝐴^𝑇𝑖𝑃_𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝜂𝑃_𝑖𝐵_𝑖𝐶^𝑇𝑖𝜂𝐵^𝑇𝑖𝑃^𝑇𝑖−𝐼𝜂𝐷^𝑇𝑖𝐶_𝑖𝜂𝐷_𝑖−𝐼⎤⎥⎥⎥⎥⎥⎥⎦<0,(2.2) then the system (2.1) is stable in the mean square sense.

Proof. If there exists a positive definite matrix 𝑃_𝑖 satisfying Lyapunov inequality (2.3), then the indefinite system (2.1) is asymptotically stable: 𝑃_𝑖𝐴_𝑖+𝐴^𝑇𝑖𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝐼−Φ(𝑡)𝐷_𝑖⁻¹Φ(𝑡)𝐶_𝑖+𝐴_𝑖+𝐵_𝑖𝐼−Φ(𝑡)𝐷_𝑖⁻¹Φ(𝑡)𝐶_𝑖^𝑇𝑃^𝑇𝑖+^𝑁_𝑗=0𝜋_𝑖𝑗𝑃_𝑗<0.(2.3) Let 𝑝_𝑖=[𝐼−Φ(𝑡)𝐷_𝑖]⁻¹Φ(𝑡)𝐶_𝑖𝑥, then 𝑝_𝑖 can be represented as 𝑝_𝑖=Φ(𝑡)[𝐶_𝑖𝑥+𝐷_𝑖𝑝_𝑖]. Then, from the inequality ‖Φ(𝑡)‖≤𝜂, we can achieve 𝑝^𝑇𝑖𝑝_𝑖≤𝜂²[𝐶_𝑖𝑥+𝐷_𝑖𝑝_𝑖]^𝑇[𝐶_𝑖𝑥+𝐷_𝑖𝑝_𝑖]. Since 2𝑥^𝑇𝑃_𝑖𝐵_𝑖𝐼−Φ(𝑡)𝐷_𝑖⁻¹Φ(𝑡)𝐶_𝑖𝑥=2𝑥^𝑇𝑃_𝑖𝐵_𝑖𝑝_𝑖≤2𝑥^𝑇𝑃_𝑖𝐵_𝑖𝑝_𝑖+𝐶_𝑖𝑥+𝐷_𝑖𝑝_𝑖^𝑇𝐶_𝑖𝑥+𝐷_𝑖𝑝_𝑖−𝜂⁻²𝑝^𝑇𝑖𝑝_𝑖=𝑥^𝑇𝐶^𝑇𝑖𝐶_𝑖𝑥+2𝑥^𝑇𝑃_𝑖𝐵_𝑖+𝐶^𝑇𝑖𝐷_𝑖𝑝_𝑖−𝜂⁻²𝑝^𝑇𝑖𝐼−𝜂²𝐷^𝑇𝑖𝐷_𝑖𝑝_𝑖.(2.4) Assume that 𝑎^𝑇𝑖=𝑥^𝑇(𝑃_𝑖𝐵_𝑖+𝐶^𝑇𝑖𝐷_𝑖),𝑏_𝑖=𝑝_𝑖,𝑊_𝑖=𝜂²(𝐼−𝜂²𝐷^𝑇𝑖𝐷_𝑖)⁻¹, using inequality (2.4) and Lemma 2.1, there is 2𝑥^𝑇𝑃≤_𝑖𝐵𝑥_𝑖𝑇𝐼−Φ(𝑡)𝐷_𝑖⁻¹Φ(𝑡)𝐶_𝑖𝑥𝐶^𝑇𝑖𝐶_𝑖𝑥+𝜂²𝑥^𝑇𝑃_𝑖𝐵_𝑖+𝐶^𝑇𝑖𝐷_𝑖𝐼−𝜂²𝐷^𝑇𝑖𝐷_𝑖⁻¹𝑃_𝑖𝐵_𝑖+𝐶^𝑇𝑖𝐷_𝑖^𝑇𝑥.(2.5) If the following inequality stands, then inequality (2.3) holds: 𝑃_𝑖𝐴_𝑖+𝐴^𝑇𝑖𝑃_𝑖+^𝑁_𝑗=0𝜋_𝑖𝑗𝑃_𝑗+𝐶^𝑇𝑖𝐶_𝑖+𝜂²𝑃_𝑖𝐵_𝑖+𝐶^𝑇𝑖𝐷_𝑖𝐼−𝜂²𝐷^𝑇𝑖𝐷_𝑖⁻¹𝑃_𝑖𝐵_𝑖+𝐶^𝑇𝑖𝐷_𝑖^𝑇<0.(2.6) With Schur Complement, it is easy to transfer (2.6) into (2.2), namely, the system (2.1) is stable. The proof is completed.

Definition 2.4. A stochastic process 𝜈(𝑡) is said to be bounded in probability if the random variables |𝜈(𝑡)| are bounded in probability uniformly in 𝑡, that is, lim_𝑟→∞sup_𝑡>0𝑃||𝜈(𝑡)||>𝑟=0.(2.7)

3. System Description

3.1. Dynamics

Consider an 𝑛_𝑎 DOF robotic manipulator mounted on a two-wheeled driven mobile platform, the dynamics can be described as: 𝑀(𝑞)𝑞̈+𝑉(𝑞,𝑞̇)𝑞̇+𝐺(𝑞)+𝑑(𝑡)=𝐵(𝑞)𝜏+𝑓,(3.1) where 𝑞=[𝑞^𝑇𝑣,𝑞^𝑇𝑎]^𝑇∈ℝ^𝑛 with 𝑞_𝑣=[𝑥,𝑦,𝜗]^𝑇∈ℝ^𝑛𝑣 denoting the generalized coordinates for the mobile platform and 𝑞_𝑎∈ℝ^𝑛𝑎 denoting the coordinates of the robotic manipulator joints, and 𝑛=𝑛_𝑣+𝑛_𝑎. The symmetric positive definite inertia matrix 𝑀(𝑞)∈ℝ^𝑛×𝑛, the Centripetal and Coriolis torques 𝑉(𝑞,𝑞̇)∈ℝ^𝑛×𝑛, the gravitational torque vector 𝐺(𝑞)∈ℝ^𝑛, the known input transformation matrix 𝐵(𝑞)∈ℝ^𝑛×𝑚, the control inputs 𝜏∈ℝ^𝑚, and the generalized constraint forces 𝑓∈ℝ^𝑛 could be represented as, respectively, 𝑀(𝑞)=⎡⎢⎢⎣𝑀^𝑣𝑀_𝑣𝑎𝑀_𝑎𝑣𝑀_𝑎⎤⎥⎥⎦,𝑉(𝑞,𝑞̇)=⎡⎢⎢⎣𝑉_𝑣𝑉_𝑣𝑎𝑉_𝑎𝑣𝑉_𝑎⎤⎥⎥⎦,𝑓=⎡⎢⎢⎣𝐽^𝑇𝑣𝜆_𝑛𝐺(𝑞)=⎡⎢⎢⎣𝐺_𝑣0⎤⎥⎥⎦,𝐺_𝑎⎤⎥⎥⎦,𝐵(𝑞)𝜏=⎡⎢⎢⎣𝜏_𝑣𝜏_𝑎⎤⎥⎥⎦,𝑑(𝑡)=⎡⎢⎢⎣𝑑_𝑣𝑑_𝑎⎤⎥⎥⎦,(3.2) where 𝑀_𝑣 and 𝑀_𝑎 describe the inertia matrices for the mobile platform and the links, respectively, 𝑀_𝑣𝑎 and 𝑀_𝑎𝑣 are the coupling inertia matrices of the mobile platform and the links; 𝑉_𝑣, 𝑉_𝑎 denote the Centripetal and Coriolis torques for the mobile platform, the links, respectively; 𝑉_𝑣𝑎, 𝑉_𝑎𝑣 are the coupling Centripetal and Coriolis torques of the mobile platform, the links. 𝐺_𝑣 and 𝐺_𝑎 are the gravitational torque vectors for the mobile platform, the links, respectively; 𝜏_𝑣 is the input vector associated with the left driven wheel and the right driven wheel, respectively; and 𝜏_𝑎 are the control input vectors for the joints of the manipulator; 𝑑_𝑣, 𝑑_𝑎 denote the external disturbances on the mobile platform, the links, respectively; 𝐽_𝑣∈ℝ^{𝑙×𝑛𝑣} is the kinematic constraint matrix related to nonholonomic constraints; 𝜆_𝑛∈ℝ^𝑙 is the associated Lagrangian multipliers with the generalized nonholonomic constraints. We assume that the mobile manipulator is subject to known nonholonomic constraints. A method of modeling the dynamics of wheeled robots considering wheel-soil interaction mechanics is presented in [40, 41]. For the reason of simplification, we can adopt the methods of producing enough friction between the wheels of the mobile platform and the ground.

3.2. Reduced System

When the system is subjected to nonholonomic constraints, the (𝑛−𝑚) nonintegrable and independent velocity constraints can be expressed as 𝐽_𝑣(𝑞)𝑞_𝑣̇=0.(3.3) The constraint (3.3) is referred to as the classical nonholonomic constraint when it is not integrable. In the paper, constraint (3.3) is assumed to be completely nonholonomic and exactly known.

Since 𝐽_𝑣(𝑞)∈ℝ^{(𝑛𝑣−𝑚)×𝑛} introduce 𝐽_𝑎∈ℝ^{𝑛𝛼×𝑛}, and 𝐽=[𝐽_𝑣,𝐽_𝑎]^𝑇∈ℝ^{(𝑛−𝑚)×𝑛}, such that it is possible to find a 𝑚+𝑛_𝑎 rank matrix 𝑅(𝑞)∈ℝ^{𝑛×(𝑚+𝑛𝑎)} formed by a set of smooth and linearly independent vector fields spanning the null space of 𝐽(𝑞), that is, 𝑅^𝑇(𝑞)𝐽^𝑇(𝑞)=0,(3.4) where 𝑅(𝑞)=[𝑟₁(𝑞),…,𝑟_𝑚(𝑞),𝑟_𝑚+1(𝑞),…,𝑟_{𝑚+𝑛𝑎}(𝑞)]. Define an auxiliary time function 𝑧(̇𝑡)∈ℝ^{𝑚+𝑛𝑎}, and 𝑧(̇𝑡)=[𝑧̇₁(𝑡),…,𝑧̇_𝑚(𝑡),𝑧̇_𝑚+1(𝑡),…,𝑧̇_{𝑚+𝑛𝑎}(𝑡)]^𝑇 such that 𝑞̇=𝑅(𝑞)𝑧̇(𝑡)=𝑟₁(𝑞)𝑧̇₁(𝑡)+⋯+𝑟_𝑚(𝑞)𝑧̇_𝑚(𝑡)+𝑟_𝑚+1(𝑞)𝑧̇_𝑚+1(𝑡)+⋯+𝑟_{𝑚+𝑛𝑎}(𝑞)𝑧̇_{𝑚+𝑛𝑎}(𝑡).(3.5) Equation (3.5) is the kinematic model for the wheeled inverted pendulums. Usually, 𝑧(̇𝑡) has physical meaning, consisting of the angular velocity 𝜔, the linear velocity 𝑣, and the joint angle vector 𝜃_𝑎, that is, 𝑧(̇𝑡)=[𝑣𝜔𝜃̇^𝑇𝑎]^𝑇. Equation (3.5) describes the kinematic relationship between the motion vector 𝑞 and the velocity vector 𝑧(̇𝑡).

Differentiating (3.5) yields 𝑞̈=𝑅̇(𝑞)𝑧̇+𝑅(𝑞)𝑧.̈(3.6) From (3.5), ż can be obtained from 𝑞 and 𝑞̇ as 𝑧̇=𝑅^𝑇(𝑞)𝑅(𝑞)⁻¹𝑅^𝑇(𝑞)𝑞̇.(3.7) The dynamic equation (3.1), which satisfies the nonholonomic constraint (3.3), can be rewritten in terms of the internal state variable ż as 𝑀(𝑞)𝑅(𝑞)𝑧̈+𝑉^∗𝑧̇+𝐺(𝑞)+𝑑(𝑡)=𝐵(𝑞)𝜏+𝐽^𝑇(𝑞)𝜆,(3.8) with 𝑉^∗=[𝑀(𝑞)𝑅̇(𝑞)+𝑉(𝑞,𝑞̇)𝑅(𝑞)], 𝜆=[𝜆_𝑛,0]^𝑇.

Substituting (3.5) and (3.6) into (3.1), and then premultiplying (3.1) by 𝑅^𝑇(𝑞), the constraint matrix 𝐽^𝑇(𝑞)𝜆 can be eliminated by virtue of (3.4). As a consequence, we have the transformed nonholonomic system ℳ(𝑞)𝑧̈+𝒱(𝑞,𝑞̇)𝑧̇+𝒢(𝑞)+𝒟=𝒰,(3.9) where ℳ(𝑞)=𝑅^𝑇𝑀(𝑞)𝑅, 𝒱(𝑞,𝑞̇)=𝑅^𝑇[𝑀(𝑞)𝑅̇+𝑉(𝑞,𝑞̇)𝑅], 𝒢(𝑞)=𝑅^𝑇𝐺(𝑞), 𝒟=𝑅^𝑇𝑑(𝑡), 𝒰=𝑅^𝑇𝐵(𝑞)𝜏, which is more appropriate for the controller design as the constraint 𝜆 has been eliminated from the dynamics.

Remark 3.1. In this paper, we choose 𝑧=[𝜃_𝑟𝜃_𝑙𝜃₁𝜃₂,…,𝜃_𝑛𝑎]^𝑇, where 𝜃_𝑟, 𝜃_𝑙 denote the rotation angle of the left wheel and the right wheel of the mobile platform, respectively, and 𝜃₁,…,𝜃_𝑛𝑎 denote the joint angles of the link 1,2,…,𝑛_𝑎, respectively, and 𝜏=[𝜏_𝑟,𝜏_𝑙,𝜏₁,…,𝜏_𝑛𝑎].

Remark 3.2. The total degree of freedom for a two-wheeled driven mobile manipulator is 𝑛_𝑞=𝑛_𝑎+2.

3.3. Switching Dynamics

The hybrid joint is within each actuator of the wheels and links of the mobile manipulator, such that switching may appear in every joint independently. Since the left wheel and right wheel are symmetric, for simplification, we assume that the switching appears in the left wheel and each joint of the manipulator independently. Therefore, there are 2^𝑛𝑎+1 modes of operation, which are listed in Table 1 depending on which hybrid joint is in the active (actuated) or passive (underactuated) mode.

Table 1 

The modes of operation.

Let ℎ_𝑝 be the number of passive hybrid joints that have not already reached their set point in a given instant. If ℎ_𝑝>ℎ_𝑎, ℎ_𝑎 passive joints are controlled and grouped in the vector 𝑧_𝑝∈ℝ^ℎ𝑎, the remaining passive hybrid joints, if any, are kept locked by the brakes, and the active joints are grouped in the vector 𝑧_𝑎∈ℝ^ℎ𝑎. If ℎ_𝑝<ℎ_𝑎, the ℎ_𝑝 passive hybrid joints are controlled applying torques in ℎ_𝑎 active hybrid joints. In this case, 𝑧_𝑝∈ℝ^ℎ𝑝 and 𝑧_𝑎∈ℝ^ℎ𝑎. The strategy is to control all passive hybrid joints until they reach the desired position, considering the conditions exposed above, and then turn on the clutch. After that, all the active hybrid joints are controlled by themselves as a fully-actuated robot.

The dynamics (3.9) can be partitioned into two parts, the actuated part and the passive part, represented by “𝑎” and “𝑝,” respectively. Then we can rewrite the dynamics (3.9) as ⎡⎢⎢⎣ℳℳ_𝑎(𝜁)ℳ_𝑎𝑝(𝜁)_𝑝𝑎(𝜁)ℳ_𝑝(𝜁)⎤⎥⎥⎦⎡⎢⎢⎣𝑧𝑧̈̈^𝑎_𝑝⎤⎥⎥⎦+⎡⎢⎢⎣𝒱_𝑎𝒱_𝑎𝑝𝒱_𝑝𝑎𝒱_𝑝⎤⎥⎥⎦⎡⎢⎢⎣𝑧̇_𝑎𝑧̇_𝑝⎤⎥⎥⎦+⎡⎢⎢⎣𝒢_𝑎𝒢_𝑝⎤⎥⎥⎦+⎡⎢⎢⎣𝒟𝒟_𝑎(𝑡)_𝑝(𝑡)⎤⎥⎥⎦=⎡⎢⎢⎣𝒰𝒰^𝑎_𝑝⎤⎥⎥⎦,(3.10) where (i)ℳ_𝑎∈ℝ^{ℎ𝑎×ℎ𝑎}, ℳ_𝑝∈ℝ^{ℎ𝑝×ℎ𝑝}: the inertia matrices of the actuated parts and the passive parts, respectively; (ii)ℳ_𝑎𝑝∈ℝ^{ℎ𝑎×ℎ𝑝}, ℳ_𝑝𝑎∈ℝ^{ℎ𝑝×ℎ𝑎}: the coupling inertia matrices of the actuated parts and the passive parts, respectively; (iii)𝒱_𝑎∈ℝ^{ℎ𝑎×ℎ𝑎}, 𝑉_𝑝∈ℝ^{ℎ𝑝×ℎ𝑝}: the Centripetal and Coriolis torque matrices of the actuated parts and the passive parts, respectively; (iv)𝒱_𝑎𝑝∈ℝ^{ℎ𝑎×ℎ𝑝}, 𝒱_𝑝𝑎∈ℝ^{ℎ𝑝×ℎ𝑎}: the coupling Centripetal and Coriolis torques of the actuated parts and the passive parts, respectively; (v)𝒢_𝑎∈ℝ^ℎ𝑎, 𝒢_𝑝∈ℝ^ℎ𝑝: the gravitational torque vector for the actuated parts and the passive parts, respectively; (vi)𝒟_𝑎(𝑡)∈ℝ^ℎ𝑎, 𝒟_𝑝(𝑡)∈ℝ^ℎ𝑝: the bounded external disturbance from the environments on the actuated parts and the passive parts, respectively; (vii)𝒰_𝑎∈ℝ^ℎ𝑎: the control input torque vector for the actuated parts of the joints; (viii)𝒰_𝑝∈ℝ^ℎ𝑝: the control input torque vector for the passive parts of the joints satisfying 𝒰_𝑝=0.

After some simple manipulation, we can further obtain 𝒰_𝑎=𝑀(𝑧)𝑧̈_𝑝+𝐻(𝑧,𝑧)̇+𝐷(𝑡),(3.11) where 𝑀=ℳ_𝑎−ℳ_𝑎ℳ^−1𝑝𝑎ℳ_𝑝,𝐻=𝑉₁𝑧̇_𝑎+𝑉₂𝑧̇_𝑝+𝒢_𝑎−ℳ_𝑎ℳ^−1𝑝𝑎𝒢_𝑝,𝐷(𝑡)=𝒟_𝑎−ℳ_𝑎ℳ^−1𝑝𝑎𝒟_𝑝,𝑉₁=𝑉_𝑎−ℳ_𝑎ℳ^−1𝑝𝑎𝑉_𝑝𝑎,𝑉₂=𝑉_𝑎𝑝−ℳ_𝑎ℳ^−1𝑝𝑎𝑉_𝑝.(3.12)

4. Control Design

4.1. Model-Based Control with Unmodeled Dynamics

Define the tracking errors as 𝑒𝑒̇==𝑧𝑧̇_𝑝𝑝−𝑧_𝑝𝑑,−𝑧̇_𝑝𝑑,(4.1) where 𝑧̈_𝑝𝑑, 𝑧̇_𝑝𝑑 and 𝑧_𝑝𝑑 denote the desired trajectories vectors of passive joint accelerations, velocities, and positions, respectively.

The parameters 𝑀, 𝐻, and 𝐷(𝑡) in dynamical model (3.11) are functions of physical parameters of mobile manipulators like links masses, links lengths, moments of inertial, and so on. The precise values of these parameters are difficult to acquire due to measuring errors and environment and payloads variations. Therefore, it is assumed that actual value 𝑀, 𝐻, and 𝐷(𝑡) can be separated as nominal parts denoted by 𝑀₀, 𝐻₀, and 𝐷₀(𝑡) and uncertain parts denoted by Δ𝑀, Δ𝐻, and Δ𝐷(𝑡), respectively. These variables satisfy the following relationships: 𝑀=𝑀₀+Δ𝑀,𝐻=𝐻₀+Δ𝐻,𝐷=𝐷₀+Δ𝐷.(4.2) Suppose that the dynamical models of robot manipulators are known precisely and unmodeled dynamics are excluded, that is, Δ𝑀, Δ𝐻, and Δ𝐷 in (3.11) are all zeros. At this time, dynamical models (3.11) can be converted into the following nominal models: 𝑀₀(𝑧)𝑧̈_𝑝+𝐻₀(𝑧,𝑧)̇+𝐷₀=𝒰₀.(4.3)

Consider the control law as 𝒰₀=𝑀₀(𝑧)𝑧̈_𝑝𝑑−𝐾_𝑣𝑒̇−𝐾_𝑝𝑒+𝐻₀(𝑧,𝑧)̇+𝐷₀,(4.4) where 𝐾_𝑣 and 𝐾_𝑝 are positive definite matrices. Substituting (4.4) into (4.3) yields 𝑒̈+𝐾_𝑣𝑒̇+𝐾_𝑝𝑒=0.(4.5) From Lemma 2.1, it is obvious that errors 𝑒̇ and 𝑒̈ will asymptotically if proportional gain 𝐾_𝑝 and derivative gain 𝐾_𝑣 are chosen in the favorable situation.

According to (4.4), the proposed control is effective based on the strong assumptions that exact knowledge of robot dynamics is precisely known and unmodeled dynamics has to be ignored, which is difficult to obtain in practices. Therefore, we need to approximate dynamics nonlinear functions. One can imagine that model-based control is used to control nominal system and another adaptive based control attaching to model-based control for uncertain system can be designed. In this way, applying (4.4) to original systems (3.11) yields Ξ=−𝑒̈+𝐾_𝑣𝑒̇+𝐾_𝑝𝑒=Ξ,(4.6)𝑀⁰⁻¹Δ𝑀𝑧̈_𝑝+Δ𝐻(𝑧,𝑧)̇+Δ𝐷(𝑡),(4.7) which Ξ is a function of joint variables, physical parameters, parameters variations, unmodeled dynamics, and so on and denotes the structured uncertainty and unstructured uncertainty.

Up to now, the control objective can be restated as: seek a control law based on nominal parameters and adaptive-based compensator such that joint motions of robotic systems (3.11) can follow desired trajectories. The overall control law can be written as 𝒰_𝑎=𝒰₀+𝒰_𝑐,(4.8) where 𝒰_𝑐 is an adaptive-based controller serving as a compensator for model-based control and designed later. Using control law (4.8), the closed-loop system becomes: 𝑒̈+𝐾_𝑣𝑒̇+𝐾_𝑝𝑒=𝑀⁰⁻¹𝒰_𝑐+Ξ.(4.9)

Supposed that the state vector is defined as 𝑥=[𝑒^𝑇,𝑒^𝑇̇]^𝑇, the state space equation has form as 𝐴=⎡⎢⎢⎣−0𝐾𝑥_𝑝̇=𝐼𝐴𝑥+𝐵𝑈,(4.10)−𝐾_𝑣𝑈=⎤⎥⎥⎦,𝐵=⎡⎢⎢⎣𝐼0⎤⎥⎥⎦,𝑀⁰⁻¹𝒰_𝑐+Ξ.(4.11)

4.2. Stochastic Control Design

Since the hybrid joints can be switched among different modes, considering the Markovian jumping, we can rewrite (4.10) by integrating Markovian jumping parameters as 𝑥̇(𝑡)=𝐴𝑟_𝑡𝑥(𝑡)+𝐵𝑟_𝑡𝑈,(4.12) where 𝑟_𝑡=𝑗, and 𝑗 is one of the Markovian jumping parameters in the limited set 𝑆={1,2,…,𝑁} with the mode transition rate matrix ∏=(𝜋_𝑗𝜄),(𝑘,𝜄∈𝑁). The jump transition probability can be defined as 𝑃𝑟_𝑡+Δ𝑡=𝜄∣𝑟_𝑡=𝑘=𝜋^𝑘𝜄1+Δ𝜋𝑡+_𝑘𝑘𝑜(Δ𝑡),𝑘≠𝜄,Δ𝑡+𝑜(Δ𝑡),𝜄=𝑘,(4.13) where ∑^{𝑁𝜄=1,𝜄≠𝑘}𝜋_𝑘𝜄=−𝜋_𝑘𝜄,𝜋_𝑘𝜄≥0,∀𝜄,𝑘∈Ω,𝜄≠𝑘. Here, Δ𝑡>0 and lim_Δ𝑡→0𝑜(Δ𝑡)/Δ𝑡=0. The model of the form (4.12) is a hybrid system in which one state 𝑥(𝑡) takes values continuously and another state 𝑟_𝑡, referred to as the mode or operating form, takes values discretely in 𝑆.

For 𝑉(𝑡,𝑥)∈𝐶¹, let us introduce the weak infinitesimal operator ℒ𝑉 of the process {𝑥(𝑡),𝜂_𝑡,𝑡≥0} at the point {𝑡,𝑥,𝑗},ℒ𝑉=𝜕𝑉𝜕𝑡+𝜕𝜕𝑉𝑥𝑥̇(𝑡)+^𝑁_𝑘=1𝜋_𝑘𝑗𝑉(𝑥,𝑗).(4.14)

For each possible value 𝑟_𝑡=𝑗, 𝑗∈𝑆, we will denote the system matrices associated with mode 𝑗 by 𝐴𝑟_𝑡=𝐴(𝑗)=𝐴_𝑗,𝐵𝑟_𝑡=𝐵(𝑗)=𝐵_𝑗,(4.15) where 𝐴_𝑗, 𝐵_𝑗 are known real constant matrices of appropriate dimensions which describe the nominal system.

Theorem 4.1. If the linear matrix inequalities (4.16) have the solution 𝑋_𝑗 for given 𝐴_𝑗, 𝐵_𝑗, 𝑋_𝑗>0, and ⎡⎢⎢⎢⎢⎣𝐵00_𝑗𝐼00𝐼00⎤⎥⎥⎥⎥⎦^𝑇⎡⎢⎢⎢⎢⎢⎢⎣𝐴_𝑗𝑋_𝑗+𝑋_𝑗𝐴^𝑇𝑗+^𝑁_𝑘=1𝜋_𝑘𝑗𝑋_𝑗𝐴𝜂−∗𝐼𝜂∗𝐼_𝑗𝑋_𝑗𝜂𝐼−𝐼⎤⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝐵^𝑗00𝐼00𝐼00⎤⎥⎥⎥⎥⎦<0,(4.16) and define the sliding surface as 𝑆𝜎_𝑗=𝑆_𝑗𝑥_𝑗+𝛾^𝑡0𝑆_𝑗𝑥_𝑗𝑑𝑡,(4.17)_𝑗=𝐵^𝑇𝑗𝑋^𝑗−1𝐵_𝑗⁻¹𝐵^𝑇𝑗𝑋^𝑗−1.(4.18) Consider the adaptive control as 𝒰_𝑐=−𝑀₀𝐾𝜎_𝑗−𝑀₀𝑆_𝑗𝐴_𝑗𝑥_𝑗+𝛾𝑆_𝑗𝑥_𝑗−𝑀₀1𝑏⁵_𝑖=1𝜎_𝑗𝑐̂_𝑖Φ^2𝑖‖‖𝜎_𝑗‖‖Φ_𝑖+𝛿,(4.19) with the adaptive law 𝑐̇̂_𝑖=−𝜛_𝑖𝑐̂_𝑖+𝜔^𝑖Φ^2𝑖‖‖𝜎_𝑗‖‖²‖‖𝜎_𝑗‖‖Φ_𝑖+𝛿_𝑖,𝑖=1,…,5,(4.20) where 𝐶=[𝑐̂₁,…,𝑐̂₅]^𝑇, and Φ=[‖𝑧̈_𝑝𝑟‖,1,‖𝑧‖̇,1,‖𝑧‖̇]^𝑇, 𝜔_𝑖>0, 𝐾 is positive definite, 𝛿_𝑖>0 and 𝜛_𝑖>0 ( 1≤𝑖≤5) satisfying: lim_𝑡→∞𝛿_𝑖(𝑡)=0, ∫^∞0𝛿_𝑖(𝑠)𝑑𝑠=𝜌_𝑖𝛿<∞, lim_𝑡→∞𝜛_𝑖(𝑡)=0, ∫^∞0𝜛_𝑖(𝑠)𝑑𝑠=𝜌_𝑖𝜛<∞ with the constants 𝜌_𝑖𝛿 and 𝜌_𝑖𝜛, and 𝑏 will be defined later. Then, a stable sliding mode exists from the initial time, and the sliding dynamics is stable.

Proof. Define the transfer matrix 𝑇_𝑗 and the related vector 𝜈, we have 𝑇_𝑗=⎡⎢⎢⎣𝐵^𝑗𝑋_𝑗𝐵_𝑗𝐵^𝑇𝑗𝐵^𝑇𝑗𝑋^𝑗−1𝐵_𝑗⁻¹𝐵^𝑇𝑗𝑋^𝑗−1𝜈=𝜈₁⎤⎥⎥⎦,(4.21)𝜈₂^𝑇=𝑇_𝑗𝑥_𝑗,(4.22) where 𝜈₁∈𝑅^𝑛−𝑚, and 𝜈₂∈𝑅^𝑚, 𝐵_𝑗 is any basis of the null space of 𝐵^𝑇𝑗, that is, 𝐵_𝑗 is an orthogonal complement of 𝐵_𝑗, Note that given any 𝐵_𝑗, 𝐵_𝑗 is not unique. Moreover, 𝑇^𝑗−1=_𝑋𝑗𝐵_𝑗𝐵𝑗.
Consider (4.22), it is easy to have 𝜎̇_𝑗=𝜈̇₂+𝛾𝜈₂.(4.23)
From the definition of 𝜎_𝑗, we have 𝜎̇_𝑗=𝑆_𝑗𝐴_𝑗𝑥_𝑗+𝐵_𝑗𝑈+𝛾𝑆_𝑗𝑥_𝑗.(4.24) Consider (4.11) and (4.18), we can rewrite (4.24) as 𝜎̇_𝑗=𝑆_𝑗𝐴_𝑗𝑥_𝑗+𝑆_𝑗𝐵_𝑗𝑀⁰⁻¹𝒰_𝑐+𝑆_𝑗𝐵_𝑗Ξ+𝛾𝑆_𝑗𝑥_𝑗=𝑆_𝑗𝐴_𝑗𝑥_𝑗+𝑀⁰⁻¹𝒰_𝑐+Ξ+𝛾𝑆_𝑗𝑥_𝑗(4.25) with 𝑆_𝑗𝐵_𝑗=𝐼.
Consider (4.7), and 𝑧̈_𝑝=𝑧̈_𝑟+𝜎̇_𝑗 with 𝑧̈_𝑟=𝑧̈_𝑝𝑑−𝛾𝜈₂ and 𝜎̇_𝑗=𝜈̇₂+𝛾𝜈₂, we can rewrite it as Ξ==−−𝑀⁰⁻¹Δ𝑀𝑧̈_𝑝+Δ𝐻(𝑧,𝑧)̇+Δ𝐷(𝑡)𝑀⁰⁻¹Δ𝑀𝑧̈_𝑟−𝑀⁰⁻¹Δ𝑀𝜎̇_𝑗−𝑀⁰⁻¹Δ𝐻(𝑧,𝑧)̇−𝑀⁰⁻¹Δ𝐷.(4.26)
Let Γ=(𝐼+𝑀⁰⁻¹Δ𝑀)⁻¹, then we have 𝜎̇_𝑗=Γ𝑆_𝑗𝐴_𝑗𝑥_𝑗+𝛾𝑆_𝑗𝑥_𝑗+𝑀⁰⁻¹𝒰_𝑐−𝑀⁰⁻¹Δ𝑀𝑧̈_𝑟−𝑀⁰⁻¹Δ𝐻(𝑧,𝑧)̇−𝑀⁰⁻¹Δ𝐷.(4.27)
Let us consider the Lyapunov function as 𝑉₁=𝜎^𝑇𝑗𝜎_𝑗.(4.28) Taking the derivative (4.28) and integrating (4.27), we have 𝑉₁̇=𝜎^𝑇𝑗𝜎̇_𝑗+𝜎̇^𝑇𝑗𝜎_𝑗=2𝜎^𝑇𝑗Γ𝑆_𝑗𝐴_𝑗𝑥_𝑗+𝛾𝑆_𝑗𝑥_𝑗+𝑀⁰⁻¹𝒰_𝑐−𝑀⁰⁻¹Δ𝑀𝑧̈_𝑟−𝑀⁰⁻¹Δ𝐻(𝑧,𝑧)̇−𝑀⁰⁻¹Δ𝐷.(4.29)
Substituting (4.19) into (4.29), we have 𝑉₁̇=×−2𝜎−^𝑇𝑗Γ1𝐾𝜎_𝑗+2𝜎^𝑇𝑗Γ𝑏⁵_𝑖=1𝜎_𝑗𝑐̂_𝑖Φ^2𝑖‖‖𝜎_𝑗‖‖Φ_𝑖+𝛿−=−2𝜎_𝑇𝑗𝑀⁰⁻¹Δ𝑀𝑧̈_𝑟−𝑀⁰⁻¹Δ𝐻(𝑧,𝑧)̇−𝑀⁰⁻¹Δ𝐷Γ𝐾𝜎_𝑗−2𝜎^𝑇𝑗Γ1𝑏⁵_𝑖=1𝜎𝑐̂_𝑖Φ^2𝑖‖‖𝜎_𝑗‖‖Φ_𝑖+𝛿−2𝜎^𝑇𝑗Γ𝑀⁰⁻¹Δ𝑀𝑧̈_𝑟≤−−22𝜎𝜎^𝑇𝑗_𝑇𝑗Γ𝑀⁰⁻¹Δ𝐻(𝑧,𝑧)̇−2𝜎^𝑇𝑗Γ𝑀⁰⁻¹Δ𝐷Γ𝐾𝜎_𝑗−2Γ1𝑏⁵_𝑖=1‖‖𝜎_𝑗‖‖²𝑐̂_𝑖Φ^2𝑖‖‖𝜎_𝑗‖‖Φ_𝑖+𝛿+2‖‖𝜎^𝑗+2‖‖𝜎_𝑗‖‖‖Γ‖‖‖𝑀⁰⁻¹Δ𝑀‖‖‖‖𝑧̈_𝑟‖‖‖‖‖Γ‖‖‖𝑀⁰⁻¹‖‖‖Δ𝐻(𝑧,𝑧)̇‖+2‖‖𝜎_𝑗‖‖‖Γ‖‖‖𝑀⁰⁻¹‖‖‖Δ𝐷‖.(4.30)
Assumption 4.2. There exist some finite positive constants 𝑐_𝑖>0(1≤𝑖≤5) such that ∀𝑧∈𝑅^𝑛−𝑙, ∀𝑧̇∈𝑅^𝑛−𝑙, ‖Γ‖‖𝑀⁰⁻¹Δ𝑀‖≤𝑐₁, ‖Γ‖‖𝑀⁰⁻¹‖‖Δ𝐻(𝑧,𝑧)̇‖≤𝑐₂+𝑐₃‖𝑧‖̇, ‖Γ‖‖𝑀⁰⁻¹‖‖Δ𝐷‖≤𝑐₄+𝑐₅‖𝑧‖̇.Remark 4.3. For simplification, we assume that Δ𝑀>0. There exist the minimum and maximum eigenvalues 𝜆_min(Γ) and 𝜆_max(Γ), such that forall𝑥∈𝑅^{(𝑛−𝑙−𝑛𝑝)}, there exists the known positive parameter 𝑏 satisfying 0<𝑏≤𝜆_min(Γ), that is, 𝑥^𝑇𝑏𝐼𝑥≤𝑥^𝑇𝜆_min(Γ)𝐼𝑥. Remark 4.4. In reality, these constants 𝑐_𝑖,1≤𝑖≤5 cannot be obtained beforehand. Although any fixed large 𝑐_𝑖 can guarantee good performance, it is not practical as large 𝑐_𝑖 imply, in general, high noise amplification and high cost of control. Therefore, it is necessary to develop an adaptive law which can approximate the knowledge of 𝑐_𝑖,1≤𝑖≤5.
Choose the Lyapunov function candidate 𝑉₃=𝑉₁+𝑉₂ with 𝑉₂=𝐶^𝑇Ω⁻¹𝐶,(4.31) where 𝐶=𝐶−𝐶, and Ω=diag[𝜔_𝑖], 𝑖=1,…,5, therefore, we have 𝑉₃≤−2𝜎^𝑇𝑗Γ𝐾𝜎_𝑗−2Γ1𝑏⁵_𝑖=1‖‖𝜎_𝑗‖‖²𝑐̂_𝑖Φ^2𝑖‖‖𝜎_𝑗‖‖Φ_𝑖+𝛿+2‖‖𝜎^𝑗+2‖‖𝜎_𝑗‖‖‖Γ‖‖‖𝑀⁰⁻¹Δ𝑀‖‖‖‖𝑧̈_𝑟‖‖‖‖‖Γ‖‖‖𝑀⁰⁻¹‖‖‖Δ𝐻(𝑧,𝑧)̇‖+2‖‖𝜎_𝑗‖‖‖Γ‖‖‖𝑀⁰⁻¹‖‖‖Δ𝐷‖+2𝐶̇^𝑇Ω⁻¹𝐶.(4.32) Integrating (4.20) into (4.32), we have 𝑉₃≤−2𝜎^𝑇𝑗Γ𝐾𝜎_𝑗−2Γ1𝑏⁵_𝑖=1‖‖𝜎_𝑗‖‖²𝑐̂_𝑖Φ^2𝑖‖𝜎_𝑗‖Φ_𝑖+𝛿+2‖‖𝜎^𝑗+2‖‖𝜎_𝑗‖‖‖Γ‖‖‖𝑀⁰⁻¹Δ𝑀‖‖‖‖𝑧̈_𝑟‖‖+2₅‖‖‖Γ‖‖‖𝑀⁰⁻¹‖‖‖Δ𝐻(𝑧,𝑧)̇‖+2‖‖𝜎_𝑗‖‖‖Γ‖‖‖𝑀⁰⁻¹‖‖‖Δ𝐷‖_𝑖=1𝑐̂^𝑇𝑖𝜛_𝑖𝜔^𝑖−1𝑐̃_𝑖−2⁵_𝑖=1𝑐̃_𝑖Φ^2𝑖‖‖𝜎_𝑗‖‖²‖‖𝜎_𝑗‖‖‖‖Φ_𝑖‖‖+𝛿_𝑖≤−2𝜎^𝑇𝑗Γ𝐾𝜎_𝑗+2⁵_𝑖=1𝑐̂^𝑇𝑖𝜛_𝑖𝜔^𝑖−1𝑐̃_𝑖+⁵_𝑖=12𝛿_𝑖≤−2𝜆_min(Γ𝐾)‖‖𝜎_𝑗‖‖²+⁵_𝑖=1𝜛_𝑖2𝜔_𝑖𝑐^2𝑖+⁵_𝑖=12𝛿_𝑖,(4.33) with 𝑐̃_𝑖𝑐̂_𝑖=−(𝑐̂_𝑖−(1/2)𝑐_𝑖)²+(1/4)𝑐^2𝑖. Therefore, 𝑉₃̇≤−𝜆_min(Γ𝐾)‖𝜎_𝑗‖²+∑^5𝑖=1(𝜛_𝑖/2𝜔_𝑖)𝑐^2𝑖+∑^5𝑖=12𝛿_𝑖. Since ∑^5𝑖=1(𝜛_𝑖/2𝜔_𝑖)𝑐^2𝑖+∑^5𝑖=12𝛿_𝑖 is bounded, there exists 𝑡>𝑡₁, ∑^5𝑖=1(𝜛_𝑖/2𝜔_𝑖)𝑐^2𝑖+∑^5𝑖=12𝛿_𝑖≤𝜌₁ with the finite constant 𝜌₁, when ‖𝜎_𝑗‖≥√𝜌₁/𝜆_min(Γ𝐾), then 𝑉₃̇≤0. For ‖𝜎_𝑗‖≥√𝜌₁/𝜆_min(Γ𝐾), and 𝜎_𝑗 will converge to a compact set denoted by Υ_𝑗∶=𝜎_𝑗∶||𝜎_𝑗||≤𝜌₁𝜆_min(Γ𝐾).(4.34) From all the above, 𝜎_𝑗 converges to a small set containing the origin as 𝑡→∞. Moreover, 𝜎_𝑗→0 as 𝑡→∞ because of lim_𝑡→∞𝛿_𝑖(𝑡)=0, lim_𝑡→∞𝜛_𝑖(𝑡)=0, therefore, Υ_𝑗 converges to the origin, there 𝜎_𝑗→0, therefore, 𝜈̇₂→0 and 𝜈₂→0.
Consider (4.21) and (4.22), we have 𝜈̇=𝑇_𝑗𝑥̇_𝑗.(4.35) Consider Theorem 2.3 and (4.23), and let 𝜎̇=0, it is easy to have 𝜈̇=⎡⎢⎢⎣𝜈̇¹𝜈̇₂⎤⎥⎥⎦=𝐴_𝑗𝜈,(4.36) where 𝐴_𝑗=⎡⎢⎢⎣Λ^𝑗1Λ_𝑗20−𝛾𝐼⎤⎥⎥⎦=⎡⎢⎢⎣𝐵^𝑇𝑗𝑋𝐵_𝑗⁻¹𝐵^𝑇𝑗𝐴_𝑗𝑋_𝑗𝐵𝐵^𝑇𝑗𝑋𝐵_𝑗⁻¹𝐵^𝑇𝑗𝐴_𝑗𝐵_𝑗0−𝛾𝐼⎤⎥⎥⎦,(4.37) Therefore, we can partition the state equation as as 𝜈̇₁=𝜈Λ̇_𝑗12𝜈₁+Λ_𝑗2𝜈₂,=−𝛾𝜈₂,(4.38) Since 𝜈₂→0 and 𝜈̇₂→0, we only consider the stability of 𝜈̇₁=Λ_𝑗1𝜈₁. It is easy to obtain that if there exists a positive-define matrix 𝑃_𝑗=𝐵^𝑇𝑗𝑋_𝑗𝐵_𝑗 enabling the following inequality to hold: ⎡⎢⎢⎢⎢⎢⎢⎣𝑃_𝑗Λ_𝑗1+Λ^𝑇𝑗1𝑃_𝑗+^𝑁_𝑘=1𝜂𝐵𝜋_𝑘𝑗𝑃_𝑗∗∗_𝑗𝐶−𝐼∗_𝑗𝜂𝐼−𝐼⎤⎥⎥⎥⎥⎥⎥⎦<0,(4.39) where 𝐶_𝑗=𝐴_𝑗𝑋_𝑗𝐵_𝑗, then the system 𝜈̇₁=Λ_𝑗1𝜈₁(4.40) is asymptotically stable, where 𝑋 is a solution matrix to the LMIs (4.16), which implies that the sliding-mode dynamics (4.36) is asymptotically stable. This implies that (4.39) holds if the matrix inequality shown in (4.16) holds.

Remark 4.5. Note that Theorem 4.1 provides a solution to the problem of adaptive control for mechanical nonlinear systems with Markovian jump parameters. It is worth mentioning that the work conducted in this paper is the attempt to overcome the dynamics uncertainty arising in the sliding mode control for dynamics nonlinear systems with Markovian jump parameters and adopt adaptive control for dynamics nonlinear systems with Markovian jump parameters. The results obtained could be extended to general dynamics systems.

4.3. Switching Stability

For the system switching stability between the two different modes, we give the following theorems.

Theorem 4.6. Consider the switching system (4.13) if the system is both stable before and after the switching phase using the control law (4.19). Assume that there exists no external impacts during the switching, the system is also stable during the switching phase.

Proof. Since 𝕍₁ and 𝕍₂ are decreasing from Theorem 4.1, we know the system is stable no matter the hybrid joint is either actuated or underactuated. In the preceding, we have shown that the Lyapunov function is nonincreasing during the switching. Let 𝕍⁻¹²=(1/2)(𝜁̇⁻−𝜁̇)𝒟(𝜁̇⁻−𝜁̇) and 𝕍⁺¹²=(1/2)(𝜁̇⁺−𝜁̇)𝒟(𝜁̇⁺−𝜁̇) denote the Lyapunov function before and after the switching, and 𝜁̇⁺ and 𝜁̇⁻ represent the post- and preswitch velocities, respectively. The Lyapunov function change during the switching can be simplified as follows: Δ𝕍=𝕍⁺−𝕍⁻=12𝜁̇⁺−𝜁̇𝒟𝜁̇⁺−𝜁̇−12𝜁̇⁻−𝜁̇𝒟𝜁̇⁻−𝜁̇.(4.41) There is no external impact during the switching, which means that there are no extra energy injected into the system. Since the inertia properties of the switching joint and link exist, during the switching joint, if the switching joint is switched from the active mode to the passive mode without considering the friction, the motion of the link should be continuous, that is, 𝜁̇⁺=𝜁̇⁻=𝜁̇. Therefore, during the switching, the Lyapunov function is nonincreasing. If considering the friction, the Lyapunov function is decreasing, that is, Δ𝕍≤0, the motion is stable during the switching. Similarly, if the switching joint is switched from the passive mode to the active mode, although the joint torque is added, since the motion of the system is continuous because of the inertia, that is, Δ𝕍≤0, the motion of the system is also stable.

5. Simulation Studies

To verify the effectiveness of the proposed control algorithm, let us consider a wheeled mobile underactuated manipulator shown in Figure 2.

Figure 2 

The wheeled mobile manipulator in the simulation.

The following variables have been chosen to describe the vehicle (see also Figure 2), (i)𝜏_𝑙,𝜏_𝑟: the torques of two wheels; (ii)𝜏₁: the torques of joint 1; (iii)𝜃_𝑙, 𝜃_𝑟: the rotation angle of the left wheel and the right wheel of the mobile platform; (iv)𝑣: the forward velocity of the mobile platform; (v)𝜃: the direction angle of the mobile platform; (vi)𝜔: the rotation velocity of the mobile platform, and 𝜔=𝜃̇; (vii)𝜃₁: the joint angle of the underactuated link; (viii)𝑚₁,𝐼₁,𝑙₁: the mass, the inertia moment, and the length for the link; (ix)𝑟: the radius of the wheels; (x)𝑙: the distance of the wheels; (xi)𝑙_𝐺: the distance between the wheel and joint 1; (xii)𝑚: the mass of the mobile platform; (xiii)𝐼: the inertia moment of the mobile platform; (xiv)𝐼_𝑤: the inertia moment of each wheel; (xv)𝑔: gravity acceleration.

The mobile underactuated manipulator is subject to the following constraint: 𝑥̇cos𝜃−𝑦̇sin𝜃+𝜃̇𝑙_𝐺=0. Using the Lagrangian approach, we can obtain the dynamic model with 𝑞=[𝜃_𝑙,𝜃_𝑟,𝜃₁]^𝑇, then we could obtain 𝑀(𝑞)=⎡⎢⎢⎢⎢⎣𝑚₁₁𝑀(𝑞)𝑞̈+𝐶(𝑞,𝑞̇)𝑞̇+𝐺(𝑞)=𝐵𝜏,∗(𝑞)𝑚𝑚₁₂₂₂(𝑞)𝑚₁₃(𝑞)∗∗(𝑞)𝑚𝑚₂₃₃₃(𝑞)(𝑞)⎤⎥⎥⎥⎥⎦,𝐶(𝑞,𝑞̇)=⎡⎢⎢⎢⎢⎣𝑐¹𝑐(𝑞,𝑞̇)₂𝑐(𝑞,𝑞̇)₃𝐵=⎡⎢⎢⎢⎢⎣100010001⎤⎥⎥⎥⎥⎦,(𝑞,𝑞̇)⎤⎥⎥⎥⎥⎦,(5.1) where 𝑚₁₁(𝑞)=2𝑝₁+2𝑝³𝑙²𝑚₁₂tan²𝜃_𝑟,(𝑞)=𝑝⁴𝑙tan𝜃^𝑟𝑚₁₃−𝑝₈sin𝜃_𝑙,𝑚₂₃(𝑞)=0,𝑚₂₂(𝑞)=2𝑝₂,𝑐₁(𝑞)=𝑝₆,𝑚₃₃(𝑞)=2𝑝₅,(𝑞,𝑞̇)=4𝑝³𝑙²tan𝜃_𝑟sec²𝜃_𝑟𝜃̇^2𝑟𝜃̇_𝑙+𝑝⁴𝑙sec𝜃^𝑟𝜃̇^2𝑟−2𝑝₇sec𝜃_𝑟sin𝜃₁𝜃̇_𝑙𝜃̇^2𝑟−2𝑝𝑐₇₂sec²𝜃_𝑟cos𝜃₁𝜃̇₁𝜃̇_𝑙−𝑝₈cos𝜃₁𝜃̇₁𝜃̇_𝑟,(𝑞,𝑞̇)=𝑝⁴𝑙sec²𝑐₃𝜃_𝑟𝜃̇_𝑙𝜃_𝑟−𝑝₈cos𝜃₁𝜃_𝑙𝜃₁,𝑝₁(𝑞,𝑞̇)=𝑝₇tan𝜃_𝑟cos𝜃₁𝜃̇_𝑙𝜃̇₁+𝑝₈cos𝜃₁𝜃̇_𝑙𝜃̇_𝑟𝜃̇₁,=12𝑚+𝑚¹+𝐼^𝑤𝑟²,𝑝²=12𝐼+𝐼^𝑚+𝑚₁𝑙²¹+𝐼₁,𝑝₃=12𝐼^𝑚𝑝₄+𝐼_𝑤,=𝐼_𝑚,𝑝₅=12𝑚¹𝑙²¹+𝐼₁,𝑝₆=𝑚₁𝑙²¹+𝐼₁,𝑝₇=𝑚¹𝑙₁𝑙₁,𝑝₈=𝑚₁𝑙₁.(5.2)

As discussed in Section 2, we set the fully operational configuration represented by OOO while three possible configurations can occur: 𝐴𝐴𝑃, 𝐴𝑃𝐴, and 𝐴𝑃𝑃, where 𝐴 represents actuated joints and 𝑃 represents passive joints. For example, if we find that a switching occurs in 𝜏_𝜃1, then the switching configuration to validate the proposed methodology is the 𝐴𝐴𝑃 configuration. We consider a workspace with a positioning domain which range from −8° to 12^∘, with the velocities set to 1^∘/s, and use 2 sectors of position in each joint, denoted as I(−8^∘∶2^∘) and II(2^∘∶12^∘), to map the mobile manipulator workspace. The linearization points with respect to I and II are chosen as −3° and 7°, respectively. Then, according to Section 3.2, 8 linearization points with 32 modes are found. For simplification, we select the 8 modes in simulation, which are shown in Table 2. There exist 8 modes for the simulation example, which means an 8×8 dimension transition rate matrix Π is needed, so Π is defined asΠ=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−0000..0000.12.27622−000..0000012.3.527−0000.00002...22628−0000.0000..2.22812−00000....0002221.766648−00000...000.222.176668−00000...000222..176668−00000...000222..176668⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(5.3) The system parameters are chosen as 𝐺=0kg, 𝐵=𝐼, 𝑚=10.0kg, 𝑚₁=2.0kg, 𝐼=1.0kg⋅𝑚², 𝐼₁=1.0kg⋅𝑚², 𝐼_𝑚=2.0kg⋅𝑚², 𝐼_𝑤=2.0kg⋅𝑚², 𝑙=1.0𝑚, 𝑙₁=1.0𝑚, 𝑟=0.5𝑚.

Table 2 

Simulation modes.

Assume that the nominal models are obtained as: 𝐴𝐴₁=⎡⎢⎢⎢⎢⎢⎢⎣−00.0.000004407−00.0.000001120−100..00.00067051037−100..00.00067042078⎤⎥⎥⎥⎥⎥⎥⎦,𝐵₁=⎡⎢⎢⎢⎢⎢⎢⎣−00.0.000000033−00.3.000305240₂⎤⎥⎥⎥⎥⎥⎥⎦,𝐴=⎡⎢⎢⎢⎢⎢⎢⎣−00.0.000004407−00.0.000001120−100..00.00067051037−010..00.00607402708⎤⎥⎥⎥⎥⎥⎥⎦,𝐵₂=⎡⎢⎢⎢⎢⎢⎢⎣−00.0.00000003300..330033353₃4⎤⎥⎥⎥⎥⎥⎥⎦,𝐴=⎡⎢⎢⎢⎢⎢⎢⎣−00.0.000005674−00.0.000001141−100..00.00077029050−100..00.00067076064⎤⎥⎥⎥⎥⎥⎥⎦,𝐵₃=⎡⎢⎢⎢⎢⎢⎢⎣−00.0.000003355−00.3.000528429₄⎤⎥⎥⎥⎥⎥⎥⎦,𝐴=⎡⎢⎢⎢⎢⎢⎢⎣−00.0.000005674−00.0.000001141−100..00.00077029050−010..00.00607706604⎤⎥⎥⎥⎥⎥⎥⎦,𝐵₄=⎡⎢⎢⎢⎢⎢⎢⎣−00.0.00000335500..330035383₅2⎤⎥⎥⎥⎥⎥⎥⎦,𝐴=⎡⎢⎢⎢⎢⎢⎢⎣−00.0.000004428−00.0.000001163−100..00.00066029081−010..00.00606008606⎤⎥⎥⎥⎥⎥⎥⎦,𝐵₅=⎡⎢⎢⎢⎢⎢⎢⎣−00.0.00000222200..3000117558₆⎤⎥⎥⎥⎥⎥⎥⎦,𝐴=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−00.0.000004428−00.0.000001163−100..00.00066029081−010..00.00606008606⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐵₆=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−00.0.00000222200..3300137353₇⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐴=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−00.0.000005652−00.0.000001008−100..00.00078051039−100..00.00078010099⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐵₇=⎡⎢⎢⎢⎢⎢⎢⎣−00.0.000006688−00.3.000840785₈⎤⎥⎥⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎢⎢⎣−00.0.000005652−00.0.000001008−100..00.00078051039−010..00.00708100909⎤⎥⎥⎥⎥⎥⎥⎦,𝐵₈=⎡⎢⎢⎢⎢⎢⎢⎣−00.0.00000668800..3300383038⎤⎥⎥⎥⎥⎥⎥⎦.(5.4) The parameters in (4.19) are set as 𝐶(0)=[0.00002,…,0.00002]^𝑇, for 𝑖=1,2,…,8, 𝑀₀=𝐼, 𝜔_𝑖=0.5, 𝛼_𝑖=𝛿=1/(𝑡+1)², 𝐾=diag[1.0], 𝑏=1.0, 𝛾=1.0. The initial condition we used for simulation is 𝑥₀=[0.3,0.3,0.2,−0.1,0.1,−0.15]^𝑇. Via LMI optimization with the data 𝐴_𝑗,𝐵_𝑗, we can get the following solution to the LMIs (4.16) as: 𝑋₁=10⁴𝑋⎡⎢⎢⎢⎢⎢⎢⎣−1000...900.0600020115112−0100...090.0060002001023−−0100..00..000200061007230001....0000000201060127₂⎤⎥⎥⎥⎥⎥⎥⎦,=10³𝑋⎡⎢⎢⎢⎢⎢⎢⎣−−2000..70..00001091424582−−−2000.7...000610054004834−−−4000.8...000400090206420−−0400..08..000400190056₃30⎤⎥⎥⎥⎥⎥⎥⎦,=10³𝑋⎡⎢⎢⎢⎢⎢⎢⎣−−2000..40..00301012523038−−−2000.4...000910015008325−−−4000.8...000000070207580−−0400..08..000000270007₄20⎤⎥⎥⎥⎥⎥⎥⎦,=10³𝑋⎡⎢⎢⎢⎢⎢⎢⎣−−2000..20..00001071423856−−−2000.2...000610034009534−−−4000.7...000700040209460−−0400..07..000700140089₅30⎤⎥⎥⎥⎥⎥⎥⎦,=10³𝑋⎡⎢⎢⎢⎢⎢⎢⎣−−2000..80..00101011620603−−−2000.8...000610076009025−−−4000.8...000600050204530−−0400..08..000600150064₆20⎤⎥⎥⎥⎥⎥⎥⎦,=10³𝑋⎡⎢⎢⎢⎢⎢⎢⎣−−5000..20..00701033431097−−5000..30..00101050413390−−−5000.3...000200001302070−0050...003.000200300302₇0⎤⎥⎥⎥⎥⎥⎥⎦,=10³𝑋⎡⎢⎢⎢⎢⎢⎢⎣−−5000..20..00301064253534−−5000..30..00101030203037−−−5000.3...000100050503740−0050...003.000100450053₈0⎤⎥⎥⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎢⎢⎣−−−−0000....6000900020008000−−−−0000....0600090002000800−−−−0000....0060009000200080−−−−0000....0006000900020008⎤⎥⎥⎥⎥⎥⎥⎦.(5.5) So we can obtain the solution of 𝑆_𝑖, for 𝑖=1,2,…,8. Torque disturbances 𝐷(𝑡) are introduced to verify the robustness of the controllers ⎡⎢⎢⎢⎢⎣𝑑𝑑_𝑟(𝑡)_𝑙𝑑(𝑡)₁(𝑡)⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣0.007s0in.0(233𝑡)si+n0(.40𝑡0)9cos²0.015cos(5𝑡)𝑡⎤⎥⎥⎥⎥⎦.(5.6)The disturbance is turned off after the switching introduction in corresponding joint or wheel.

The system switches among the 8 modes randomly during operation. From Figure 3, we can see that firstly the system switches from mode 1 to mode 4, then from mode 4 to mode 1, finally, it switches from mode 1 to modes 4, 6, and 7. Figure 4 shows that the system is stabilized during operation. From Figures 5, 6, and 7, it can be noticed that the torque inputs are bounded. The simulation results demonstrate the tracking error decays to the equilibrium point under the designed mode-dependent controller.

Figure 3 

Mode jumping.

Figure 4 

States.

Figure 5 

The input control 𝑢1.

Figure 6 

The input control 𝑢2.

Figure 7 

The input control 𝑢3.

6. Conclusion

In this paper, we consider stochastic stability and sliding mode control for mobile manipulators using stochastic jumps switching joints. Adaptive parameter techniques are adopted to cope with the effect of the Markovian switching and nonlinear dynamics uncertainty and follow the desired trajectory for wheeled mobile manipulators. The resulting closed-loop system is bounded in probability and the effect due to the external disturbance on the tracking errors can be attenuated to any preassigned level. It has been shown that the adaptive control problem for the Markovian jump nonlinear systems is solvable if a set of coupled LMIs have solutions. Finally, a numerical example is given to show the potential of the proposed techniques.

7. Acknowledgment

This study was partially supported by the National Natural Science Foundation of China (Grant no. 61005080), Special Postdoctoral Foundation of China (Grant no. 201104405), Postdoctoral Foundation of China (20100480994), and the “111” Project (B07018). The authors would like to express their acknowledgment to Dr. Z. J. Li for his help in theory and literature.