On Robust Hybrid Force/Motion Control Strategies Based on Actuator Dynamics for Nonholonomic Mobile Manipulators

Abstract

Robust force/motion control strategies are presented for mobile manipulators under both holonomic and nonholonomic constraints in the presence of uncertainties and disturbances. The controls are based on structural knowledge of the dynamics of the robot, and the actuator dynamics is also taken into account. The proposed control is robust not only to structured uncertainty such as mass variation but also to unstructured one such as disturbances. The system stability and the boundness of tracking errors are proved using Lyapunov stability theory. The proposed control strategies guarantee that the system motion converges to the desired manifold with prescribed performance. Simulation results validate that not only the states of the system asymptotically converge to the desired trajectory, but also the constraint force asymptotically converges to the desired force.

1. Introduction

Mobile manipulators refer to robotic manipulators mounted on mobile platforms. Such systems combine the advantages of mobile platforms and robotic arms and reduce their drawbacks [1–4]. For instance, the mobile platform extends the arm workspace, whereas the arm offers much operational functionality. Applications for such systems could be found in mining, construction, forestry, planetary exploration, teleoperation, and military [5–11].

Mobile manipulators possess complex and strongly coupled dynamics of mobile platforms and manipulators [12–16]. A control approach by nonlinear feedback linearization was presented for the mobile platform so that the manipulator is always positioned at the preferred configurations measured by its manipulability [17]. In [14], the effect of the dynamic interaction on the tracking performance of a mobile manipulator was studied, and nonlinear feedback control for the mobile manipulator was developed to compensate the dynamic interaction. In [18], a basic framework for the coordination and control of vehicle-arm systems was presented, which consists of two basic task-oriented control: end-effector task control and platform self-posture control. The standard definition of manipulability was generalized to the case of mobile manipulators, and the optimization of criteria inherited from manipulability considerations were given to generate the controls of the system when its end-effector motion was imposed [19]. In [20], a unified model for mobile manipulator was derived, and nonlinear feedback was applied to linearize and decouple the model, and decoupled force/position control of the end-effector along the same direction for mobile manipulators was proposed and applied to nonholonomic cart pushing. The previously mentioned literature concerning with control of the mobile manipulator requires the precise information on the dynamics of the mobile manipulator; there may be some difficulty in implementing them on the real system in practical applications.

Different researchers have investigated adaptive controls to deal with dynamics uncertainty of mobile manipulators. Adaptive neural-network- (NN-) based controls for the arm and the base had been proposed for the motion control of a mobile manipulator [21, 22]; each NN control output comprises a linear control term and a compensation term for parameter uncertainty and disturbances. Adaptive control was proposed for trajectory/force control of mobile manipulators subjected to holonomic and nonholonomic constraints with unknown inertia parameters [23, 24], which ensures the state of the system to asymptotically converge to the desired trajectory and force. The principal limitation associated with these schemes is that controllers are designed at the velocity input level or torque input level, and the actuator dynamics are excluded.

As demonstrated in [25–27], actuator dynamics constitute an important component of the complete robot dynamics, especially in the case of high-velocity movement and highly varying loads. Many control methods have therefore been developed to take into account the effects of actuator dynamics (see, e.g., [28–30]). However, the literature is sparse on the control of the nonholonomic mobile manipulators including the actuator dynamics. In most of the research works for controlling mobile manipulators, joint torques are control inputs though in reality joints are driven by actuators (e.g., DC motors), and therefore using actuator input voltages as control inputs is more realistic. To this effect, actuator dynamics is combined with the mobile manipulator's dynamics in this paper.

This paper addresses the problem of stabilization of force/motion control for a class of mobile manipulator systems with both holonomic and nonholonomic constraints in the parameter uncertainties and external disturbances.

Unlike the force/motion control presented in [31–37], which is proposed for the mechanical systems subject to either holonomic or nonholonomic constraints, in our paper, the control is to deal with the system subject to both holonomic and nonholonomic constraints. After the dynamics based on decoupling force/motion is first presented, the robust motion/force control is proposed for the system under the consideration of the actuator dynamics uncertainty to complete the trajectory/force tracking. The paper has main contributions listed as follows.(i)Decoupling robust motion/force control strategies are presented for mobile manipulator with both holonomic and nonholonomic constraints in the parameter uncertainties and external disturbances, and nonregressor-based control design is developed in a unified manner without imposing any restriction on the system dynamics.(ii)The actuators (e.g., DC motor) dynamics of both the mobile platform and the arm are integrated with mobile manipulator dynamics and kinematics so that the actuator input voltages are the control inputs thus making the system more realistic.

Simulation results are described in detail that show the effectiveness of the proposed control law.

The rest of the paper is organized as follows. The system description of mobile manipulator subject to nonholonomic constraints and holonomic is briefly described in Section 2. Problem statement for the system control is given in Section 4. The main results of robust adaptive control design are presented in Section 5. Simulation studies are presented by comparison between the proposed robust control with nonrobust control in Section 6. Concluding remarks are given in Section 7.

2. System Description

Consider an 𝑛 DOF mobile manipulator with nonholonomic mobile base. The constrained mechanical system can be described as 𝑀(𝑞)𝑞̈+𝐶(𝑞,𝑞̇)𝑞̇+𝐺(𝑞)+𝑑(𝑡)=𝐵(𝑞)𝜏+𝑓,(2.1) where 𝑞=[𝑞₁,…,𝑞_𝑛]^𝑇∈𝑅^𝑛 denote the generalized coordinates; 𝑀(𝑞)∈𝑅^𝑛×𝑛 is the symmetric bounded positive definite inertia matrix; 𝐶(𝑞̇,𝑞)𝑞̇∈𝑅^𝑛 denotes the Centripetal and Coriolis torques; 𝐺(𝑞)∈𝑅^𝑛 is the gravitational torque vector; 𝑑(𝑡) denotes the external disturbances; 𝜏∈𝑅^𝑚 is the control inputs; 𝐵(𝑞)∈𝑅^𝑛×𝑚 is a full rank input transformation matrix and is assumed to be known because it is a function of fixed geometry of the system; 𝑓∈𝑅^𝑚 denotes the vector of constraint forces; 𝐽∈𝑅^𝑛×𝑚 is Jacobian matrix; 𝜆=[𝜆_𝑛,𝜆_ℎ]∈𝑅^𝑚 is Lagrange multipliers corresponding to the nonholonomic and holonomic constraints.

The generalized coordinates may be separated into two sets 𝑞=[𝑞_𝑣,𝑞_𝑎]^𝑇, where 𝑞_𝑣∈𝑅^𝑣 describes the generalized coordinates for the mobile platform, 𝑞_𝑎∈𝑅^𝑟 is the coordinates of the manipulator, and 𝑛=𝑣+𝑟.

Assumption 2.1 (see [38–40]). The mobile manipulator is subject to known nonholonomic constraints.

Assumption 2.2. The system (2.8) is subjected to 𝑘 independent holonomic constraints, which can be written as ℎ(𝑞)=0,ℎ(𝑞)∈𝑅^𝑘,(2.2) where ℎ(𝑞) is full rank, then 𝐽(𝑞)=𝜕ℎ/𝜕𝑞.

Remark 2.3. In actual implementation, we can adopt the methods of producing enough friction between the wheels of the mobile platform and the ground such that this assumption holds [41–43].

The vehicle is subjected to nonholonomic constraints, the 𝑙 nonintegrable and independent velocity constraints can be expressed as 𝐴𝑞_𝑣𝑞_𝑣̇=0,(2.3) where 𝐴(𝑞_𝑣)=[𝐴^𝑇1(𝑞_𝑣),…,𝐴^𝑇𝑙(𝑞_𝑣)]^𝑇∶𝑅^𝑣→𝑅^𝑙×𝑣 is the kinematic constraint matrix which is assumed to have full rank 𝑙. In the paper, the vehicle is assumed to be completely nonholonomic. The effect of the constraints can be viewed as a restriction of the dynamics on the manifold Ω_𝑛 as Ω_𝑛=𝑞_𝑣,𝑞_𝑣̇∣𝐴𝑞_𝑣𝑞_𝑣̇=0.(2.4)

The generalized constraint forces for the nonholonomic constraints can be given by 𝑓_𝑛=𝐴^𝑇𝑞_𝑣𝜆_𝑛.(2.5)

Assume that the annihilator of the codistribution spanned by the covector fields 𝐴₁(𝑞_𝑣),…,𝐴_𝑙(𝑞_𝑣) is a (𝑣−𝑙)-dimensional smooth nonsingular distribution Δ on 𝑅^𝑣. This distribution Δ is spanned by a set of (𝑣−𝑙) smooth and linearly independent vector fields 𝐻₁(𝑞_𝑣),…,𝐻_𝑣−𝑙(𝑞_𝑣); that is, Δ=span{𝐻₁(𝑞_𝑣),…,𝐻_𝑣−𝑙(𝑞)}, which satisfy, in local coordinates, the following relation: 𝐻^𝑇𝑞_𝑣𝐴^𝑇𝑞_𝑣=0,(2.6) where 𝐻(𝑞_𝑣)=[𝐻₁(𝑞_𝑣),…,𝐻_{𝑛𝑣−𝑙}(𝑞_𝑣)]∈𝑅^{𝑣×(𝑣−𝑙)}. Note that 𝐻^𝑇𝐻 is of full rank. Constraints (2.3) imply the existence of vector 𝜂̇∈𝑅^𝑣−𝑙 [44], such that 𝑞_𝑣̇=𝐻𝑞_𝑣𝜂̇.(2.7)

Considering the nonholonomic constraints (2.3) and its derivative, the dynamics of mobile manipulator can be expressed as ⎡⎢⎢⎣𝐻^𝑇𝑀_𝑣𝐻𝐻^𝑇𝑀_𝑣𝑎𝑀_𝑎𝑣𝐻𝑀_𝑎⎤⎥⎥⎦⎡⎢⎢⎣𝑞𝜂_𝑎̈̈⎤⎥⎥⎦+⎡⎢⎢⎣𝐻^𝑇𝑀_𝑣𝐻̇+𝐻^𝑇𝐶_𝑣𝐻𝐻^𝑇𝐶_𝑣𝑎𝑀_𝑎𝑣𝐻̇+𝐶_𝑎𝑣𝐻𝐶_𝑎⎤⎥⎥⎦⎡⎢⎢⎣𝑞𝜂_𝑎̇̇⎤⎥⎥⎦+⎡⎢⎢⎣𝐻^𝑇𝐺_𝑣𝐺_𝑎⎤⎥⎥⎦+⎡⎢⎢⎣𝐻^𝑇𝑑_𝑣𝑑_𝑎=⎡⎢⎢⎣𝐻_𝑇⎤⎥⎥⎦𝐵_𝑣𝜏_𝑣𝐵_a𝜏_𝑎⎤⎥⎥⎦+⎡⎢⎢⎣𝐽00_𝑣𝐽_𝑎⎤⎥⎥⎦^𝑇⎡⎢⎢⎣𝜆0_ℎ⎤⎥⎥⎦.(2.8)

From Assumption 2.2, the holonomic constraint force 𝑓_ℎ can be converted to the joint space as 𝑓_ℎ=𝐽^𝑇𝜆_ℎ. Hence, the holonomic constraint on the robot’s end effector can be viewed as restricting only the dynamics on the constraint manifold Ω_ℎ defined by Ω_ℎ={(𝑞,𝑞̇)∣ℎ(𝑞)=0,𝐽(𝑞)𝑞̇=0}. The vector 𝑞_𝑎 can be further rearranged and partitioned into 𝑞_𝑎=[𝑞^1𝑎,𝑞^2𝑎]^𝑇;  𝑞^1𝑎∈𝑅^𝑟−𝑘 describes the constrained motion of the manipulator, and 𝑞^2𝑎∈𝑅^𝑘 denotes the remaining joint variable. Then, 𝐽(𝑞)=𝜕ℎ𝜕𝜂,𝜕𝜕𝑞ℎ^1𝑎,𝜕ℎ𝜕𝑞^2𝑎.(2.9) From [45], it could be concluded 𝑞 is the function of 𝜁=[𝜂,𝑞^1𝑎]^𝑇, that is, 𝑞=𝑞(𝜁), and we have 𝑞̇=𝐿(𝜁)𝜁̇, where 𝐿(𝜁)=𝜕𝑞/𝜕𝜁, 𝑞̈=𝐿(𝜁)𝜁̈+𝐿̇(𝜁)𝜁̇, and 𝐿(𝜁), 𝐽¹(𝜁)=𝐽(𝑞(𝜁)) satisfy the relationship 𝐿^𝑇(𝜁)𝐽^1𝑇(𝜁)=0.(2.10) The dynamic model (2.8), when it restricted to the constraint surface, can be transformed into the reduced model: 𝑀¹𝐿(𝜁)𝜁̈+𝐶¹𝜁̇+𝐺¹+𝑑¹(𝑡)=𝑢+𝐽^1𝑇𝜆_ℎ,(2.11) where 𝑀¹=⎡⎢⎢⎣𝐻^𝑇𝑀_𝑣𝐻𝐻^𝑇𝑀_𝑣𝑎𝑀_𝑎𝑣𝐻𝑀_𝑎𝐶₁⎤⎥⎥⎦,=⎡⎢⎢⎣𝐻^𝑇𝑀_𝑣𝐻̇𝐻^𝑇𝑀_𝑣𝑎𝑀_𝑎𝑣𝐻𝑀_𝑎⎤⎥⎥⎦𝐿̇(𝜁)+⎡⎢⎢⎣𝐻^𝑇𝑀_𝑣𝐻̇+𝐻^𝑇𝐶_𝑣𝐻𝐻^𝑇𝐶_𝑣𝑎𝑀_𝑎𝑣𝐻̇+𝐶_𝑎𝑣𝐶_𝑎𝐺⎤⎥⎥⎦𝐿(𝜁),₁=⎡⎢⎢⎣𝐻^𝑇𝐺_𝑣𝐺_𝑎⎤⎥⎥⎦,𝑑₁(𝑡)=⎡⎢⎢⎣𝐻^𝑇𝑑_𝑣𝑑_𝑎𝑢=𝐵₁⎤⎥⎥⎦,𝜏,𝐵¹=⎡⎢⎢⎣𝐻^𝑇0𝐵_𝑣𝐵0_𝑎⎤⎥⎥⎦,𝜁=⎡⎢⎢⎣𝑞𝜂_1𝑎⎤⎥⎥⎦.(2.12)

Multiplying 𝐿^𝑇 by both sides of (2.11), we can obtain 𝑀_𝐿(𝜁)𝜁̈+𝐶_𝐿𝜁,𝜁̇𝜁̇+𝐺_𝐿+𝑑_𝐿(𝑡)=𝐿^𝑇𝐵¹𝜏.(2.13) The force multipliers 𝜆_ℎ can be obtained by (2.11): 𝜆_ℎ=𝑍(𝜁)𝐶¹𝜁,𝜁̇𝜁̇+𝐺¹+𝑑¹(𝑡)−𝐵¹𝜏,(2.14) where 𝑀_𝐿=𝐿^𝑇𝑀¹𝐿, 𝐶_𝐿=𝐿^𝑇𝐶¹, 𝐺_𝐿=𝐿^𝑇𝐺¹, 𝑍=(𝐽¹(𝑀¹)⁻¹𝐽^1𝑇)⁻¹𝐽¹(𝑀¹)⁻¹.

Property 1. The matrix 𝑀_𝐿 is symmetric and positive definite.

Property 2. The matrix 𝑀̇_𝐿−2𝐶_𝐿 is skew symmetric.

Property 3 (see [46]). For holonomic systems, matrices 𝐽¹(𝜁), 𝐿(𝜁) are uniformly bounded and uniformly continuous if 𝜁 is uniformly bounded and continuous, respectively.

Property 4. There exist some finite positive constants 𝑐_𝑖>0(1≤𝑖≤4) and finite nonnegative constant 𝑐_𝑖≥0(𝑖=5) such that for all 𝜁∈𝑅^𝑛, for all 𝜁̇∈𝑅^𝑛, ‖𝑀_𝐿(𝜁)‖≤𝑐₁, ‖𝐶_𝐿(𝜁,𝜁̇)‖≤𝑐₂+𝑐₃‖𝜁̇‖, ‖𝐺_𝐿(𝜁)‖≤𝑐₄, and sup_𝑡≥0‖𝑑_𝐿(𝑡)‖≤𝑐₅.

3. Actuator Dynamics

The joints of the mobile manipulators are assumed to be driven by DC motors. Consider the following notations used to model a DC motor: 𝜈∈𝑅^𝑚 represents the control input voltage vector; 𝐼 denotes an 𝑚-element vector of motor armature current; 𝐾_𝑁∈𝑅^𝑚×𝑚 is a positive definite diagonal matrix which characterizes the electromechanical conversion between current and torque; 𝐿_𝑎=diag[𝐿_𝑎1,𝐿_𝑎2,𝐿_𝑎3,…,𝐿_𝑎𝑚], 𝑅_𝑎=diag[𝑅_𝑎1,𝑅_𝑎2,𝑅_𝑎3,…,𝑅_𝑎𝑚], 𝐾_𝑒=diag[𝐾_𝑒1,𝐾_𝑒2,𝐾_𝑒3,…,𝐾_𝑒𝑚], 𝜔=[𝜔₁,𝜔₂,…,𝜔_𝑚]^𝑇 represent the equivalent armature inductances, resistances, back EMF constants, angular velocities of the driving motors, respectively; 𝐺_𝑟=diag(𝑔_𝑟𝑖)∈𝑅^𝑚×𝑚 denotes the gear ratio for 𝑚 joints; 𝜏_𝑚 are the torque exerted by the motor. In order to apply the DC servomotors for actuating an 𝑛-DOF mobile manipulator, assuming no energy losses, a relationship between the 𝑖th joint velocity 𝑞_𝑖̇ and the motor shaft velocity 𝜔_𝑖 can be presented as 𝑔_𝑟𝑖=𝜔_𝑖/𝑞_𝑖̇=𝜏_𝑖/𝜏_𝑚𝑖 with the gear ratio of the 𝑖th joint 𝑔_𝑟𝑖, the 𝑖th motor shaft torque 𝜏_𝑚𝑖, and the 𝑖th joint torque 𝜏_𝑖. The motor shaft torque is proportional to the motor current 𝜏_𝑚=𝐾_𝑁𝐼. The back EMF is proportional to the angular velocity of the motor shaft; then we can obtain 𝐿_𝑎𝑑𝐼𝑑𝑡+𝑅^𝑎𝐼+𝐾_𝑒𝜔=𝑣.(3.1) In the actuator dynamics (3.1), the relationship between 𝜔 and 𝜁̇ is dependent on the type of mechanical system and can be generally expressed as 𝜔=𝐺_𝑟𝑇𝜁̇.(3.2) The structure of 𝑇 depends on the mechanical systems to be controlled. For instance, in the simulation example, a two-wheel differential drive 2-DOF mobile manipulator is used to illustrate the control design. From [47], we have 𝑣=𝑟𝜃̇^𝑙+𝑟𝜃̇_𝑟𝜃̇=𝑟𝜃̇_𝑟2,−𝑟𝜃̇_𝑙𝜃̇2𝑙₁,𝜃̇₂=𝜃̇₁,=𝜃̇₂,(3.3) where 𝜃̇_𝑙 and 𝜃̇_𝑟 are the angular velocities of the two wheels, respectively, and 𝑣 is the linear velocity of the mobile platform, as shown in Figure 1. Since 𝑦̇=𝑣cos𝜃, we have 𝜃̇_𝑙𝜃̇_𝑟𝑇𝜃=̇₁⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜃̇₂1^𝑇=𝑇𝑦̇𝜃̇𝜃̇₁𝜃̇₂^𝑇,𝑟co1s𝜃𝑟𝑙00𝑟co00s𝜃−00𝑟𝑙100010⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(3.4) where 𝑟 and 𝑙 are shown in Figure 1.

Figure 1 

The 2-DOF mobile manipulator.

Eliminating 𝜔 from the actuator dynamics (3.1) by substituting (3.2), one obtains 𝐿^𝑇𝐵𝜆¹𝐺_𝑟𝐾_𝑁𝐼=𝑀_𝐿(𝜁)𝜁̈+𝐶_𝐿𝜁,𝜁̇𝜁̇+𝐺_𝐿+𝑑_𝐿(𝑡),(3.5)_ℎ=𝑍(𝜈𝜁)=𝐶𝐿_2𝑎𝜁̇+𝐺₂+𝑑₂(𝑡)−𝐵¹𝐺_𝑟𝐾_𝑁𝐼,(3.6)𝑑𝐼𝑑𝑡+𝑅^𝑎𝐼+𝐾_𝑒𝐺_𝑟𝑇𝜁̇.(3.7)

Until now we have brought the kinematics (2.3), dynamics (3.5), (3.6) and actuator dynamics (3.7) of the considered nonholonomic system from the generalized coordinate system 𝑞∈𝑅^𝑛 to feasible independent generalized velocities 𝜁∈𝑅^{𝑛−𝑙−𝑘} without violating the nonholonomic constraint (2.3).

4. Problem Statement

Since the system is subjected to the nonholonomic constraint (2.3) and holonomic constraint (2.2), the states 𝑞_𝑣, 𝑞^1𝑎, 𝑞^2𝑎 are not independent. By a proper partition of 𝑞_𝑎, 𝑞^2𝑎 is uniquely determined by 𝜁=[𝜂,𝑞^1𝑎]^𝑇. Therefore, it is not necessary to consider the control of 𝑞^2𝑎.

Given a desired motion trajectory 𝜁^𝑑(𝑡)=[𝜂^𝑑𝑞^1𝑎𝑑]^𝑇 and a desired constraint force 𝑓_𝑑(𝑡), or, equivalently, a desired multiplier 𝜆_ℎ(𝑡), the trajectory and force tracking control is to determine a control law such that for any (𝜁(0),𝜁̇(0))∈Ω, 𝜁, 𝜁̇, 𝜆 asymptotically converge to a manifold Ω_𝑑 specified as Ω where Ω_𝑑=𝜁,𝜁̇,𝜆_ℎ∣𝜁=𝜁_𝑑,𝜁̇=𝜁̇_𝑑,𝜆=𝜆_𝑑.(4.1)

The controller design will consist of two stages: (i) a virtual adaptive control input 𝐼^𝑑 is designed so that the subsystems (3.5) and (3.6) converge to the desired values, and (ii) the actual control input 𝜈 is designed in such a way that 𝐼→𝐼^𝑑. In turn, this allows 𝜁−𝜁^𝑑 and 𝜆−𝜆^𝑑 to be stabilized to the origin.

Assumption 4.1. The desired reference trajectory 𝜁^𝑑(𝑡) is assumed to be bounded and uniformly continuous and has bounded and uniformly continuous derivatives up to the second order. The desired Lagrangian multiplier 𝜆_𝑑(𝑡) is also bounded and uniformly continuous.

5. Robust Control Design

5.1. Kinematic and Dynamic Subsystems

Let 𝑒_𝜁=𝜁−𝜁^𝑑,  𝜁̇_𝑟=𝜁̇^𝑑−𝑘_𝜁𝑒_𝜁,  𝑟=𝑒_𝜁̇+𝑘_𝜁𝑒_𝜁 with 𝑘_𝜁>0,  𝑒_𝛽=𝜆−𝜆^𝑑. A decoupled control scheme is introduced to control generalized position and constraint force separatively.

Consider the virtual control input 𝐼 is designed as 𝐼=𝐾^𝑁−1𝐺^𝑟−1𝐵¹⁻¹𝜏.(5.1) Let the control 𝑢 be as the form 𝑢𝑢=_𝑎𝐿^+𝑇𝑢_𝑎−𝐽^1𝑇𝑢_𝑏,𝑢_𝑏=𝐵¹𝐺_𝑟𝐾_𝑁𝑎𝐼_𝑎,=𝐵¹𝐺_𝑟𝐾_𝑁𝑏𝐼_𝑏,(5.2) where 𝑢_𝑎, 𝐼_𝑎∈𝑅^{𝑛−𝑙−𝑘} and 𝑢_𝑏, 𝐼_𝑏∈𝑅^𝑘 and 𝐿^+𝑇=(𝐿^𝑇𝐿)⁻¹𝐿^𝑇. Then, (2.13) and (2.14) can be changed to 𝑍(𝜁)𝐶₁𝑀_𝐿(𝜁)𝜁̈+𝐶_𝐿𝜁,𝜁̇𝜁̇+𝐺_𝐿+𝑑_𝐿(𝑡)=𝐵¹𝐺_𝑟𝐾_𝑁𝑎𝐼_𝑎,(5.3)𝜁,𝜁̇𝜁̇+𝐺¹+𝑑¹(𝑡)−𝐿^+𝑇B¹𝐺_𝑟𝐾_𝑁𝑎𝐼_𝑎+𝐵¹𝐺_𝑟𝐾_𝑁𝑏𝐼_𝑏=𝜆_ℎ.(5.4) Consider the following control laws: 𝐵¹𝐺_𝑟𝐾_𝑁𝑎𝐼^𝑑𝑎=−𝐾_𝑝𝑟−𝐾_𝑖Φ=𝐶_𝑇𝑟𝑑𝑡−Φ𝛾(‖𝑟𝑟Φ‖²)+𝛿,(5.5)𝐵Ψ,(5.6)₁𝐺_𝑟𝐾_𝑁𝑏𝐼^𝑑𝑏=𝜒²𝜒+𝛿+𝜆^𝑑ℎ𝜒=𝑐₁−𝐾_𝑓𝑒_𝜆,(5.7)‖𝑍(𝜁)‖‖‖𝐿^+𝑇‖‖‖‖‖𝑑𝑑𝑡𝜁̇^𝑑‖‖‖,(5.8) where 𝐶=𝑐₁𝑐₂𝑐₃𝑐₄𝑐₅; Ψ=‖(𝑑/𝑑𝑡)[𝜁̇_𝑟]‖‖𝜁̇_𝑟‖‖𝜁̇‖‖𝜁̇_𝑟‖11^𝑇; 𝐾_𝑝,𝐾_𝑖,𝐾_𝑓 are positive definite. 𝛾(‖𝑟‖) can be defined as follows: if ‖𝑟‖≤𝜌, 𝛾(‖𝑟‖)=𝜌, else 𝛾(‖𝑟‖)=‖𝑟‖, 𝜌 is a small value, 𝛿(𝑡) is a time-varying positive function converging to zero as 𝑡→∞, such that ∫^𝑡0𝛿(𝜔)𝑑𝜔=𝑎<∞. There are many choices for 𝛿(𝑡) that satisfies the condition.

5.2. Control Design at the Actuator Level

Till now, we have designed a virtual controller 𝐼 and 𝜁 for kinematic and dynamic subsystems. 𝜁 tending to 𝜁^𝑑 can be guaranteed, if the actual input control signal of the dynamic system 𝐼 be of the form 𝐼^𝑑 which can be realized from the actuator dynamics by the design of the actual control input 𝜈. On the basis of the above statements we can conclude that if 𝜈 is designed in such a way that 𝐼 tends to 𝐼^𝑑, then (𝜁−𝜁^𝑑)→0 and (𝜆−𝜆^𝑑)→0.

Defining 𝐼=𝑒_𝐼+𝐼^𝑑 and substituting 𝐼 and 𝜁̇ of (3.7) one gets 𝐿_𝑎𝑒_𝐼̇+𝑅_𝑎𝑒_𝐼+𝐾_𝑒𝐺_𝑟𝑇𝑒_𝜁̇=−𝐿_𝑎𝐼̇^𝑑−𝑅_𝑎𝐼^𝑑−𝐾_𝑒𝐺_𝑟𝑇𝜁̇^𝑑+𝜈.(5.9)

The actuator parameters 𝐾_𝑁, 𝐿_𝑎, 𝑅_𝑎, and 𝐾_𝑒 are considered unknown for control design; however, there exist 𝐿₀, 𝑅₀, and 𝐾_𝑒0, such that ‖‖𝐿_𝑎−𝐿₀‖‖≤𝛼₁,‖‖𝑅_𝑎−𝑅₀‖‖≤𝛼₂,‖‖𝐾_𝑒−𝐾_𝑒0‖‖≤𝛼₃.(5.10)

Consider the robust control law 𝜈=𝜈₀−³_𝑖=1𝑒_𝐼𝜇^2𝑖‖‖𝑒_𝐼‖‖𝜇_𝑖+𝛿−𝐾^𝑑𝑒_𝐼,(5.11) where 𝜈₀=𝐿𝜇₀₁𝐼̇^𝑑+𝑅₀𝐼^𝑑+𝐾_𝑒0𝐺_𝑟𝑇𝜁̇^𝑑,=𝛼₁‖‖‖𝑑𝑑𝑡𝐼^𝑑𝜇₂‖‖‖,𝜇₃=𝛼₂‖‖𝐼^𝑑‖‖,=𝛼₃‖‖‖𝑑𝑑𝑡𝜁^𝑑‖‖‖.(5.12)

5.3. Stability Analysis for the System

Theorem 5.1. Consider the mechanical system described by (2.1), (2.3), and (2.2); using the control law (5.5) and (5.7), the following hold for any (𝑞(0),𝑞̇(0))∈Ω_𝑛∩Ω_ℎ:(i)𝑟 and 𝑒_𝐼 converge to a set containing the origin with the convergence rate as 𝑡→∞; (ii)𝑒_𝑞 and 𝑒_𝑞̇ asymptotically converge to 0 as 𝑡→∞;(iii)𝑒_𝜆 and 𝜏 are bounded for all 𝑡≥0.

Proof. (i) By combing (3.5) with (5.5), the closed-loop system dynamics can be rewritten as 𝑀_𝐿𝑟̇=𝐵¹𝐺_𝑟𝐾_𝑁𝑎𝐼^𝑑𝑎+𝐵¹𝐺_𝑟𝐾_𝑁𝑎𝑒_𝐼−𝑀_𝐿𝜁̈_𝑟+𝐶_𝐿𝜁̇_𝑟+𝐺_𝐿+𝑑_𝐿−𝐶_𝐿𝑟.(5.13)
Substituting (5.5) into (5.13), the closed-loop dynamic equation is obtained: 𝑀_𝐿𝑟̇=−𝐾_𝑝𝑟−𝐾_𝑖𝑟𝑑𝑡−Φ𝛾(‖𝑟𝑟Φ‖²)+𝛿−𝜇−𝐶^𝐿𝑟+𝐵¹𝐺_𝑟𝐾_𝑁𝑎𝑒_𝐼,(5.14) where 𝜇=𝑀_𝐿𝜁̈_𝑟+𝐶_𝐿𝜁̇_𝑟+𝐺_𝐿+𝑑_𝐿.
Consider the function 𝑉𝑉₁=𝑉₁+𝑉₂,=12𝑟^𝑇𝑀_𝐿𝑟+12𝑟𝑑𝑡^𝑇𝐾_𝑖𝑟𝑑𝑡+𝑒^𝑇𝜁𝑉₂𝑘_𝜁𝐾_𝑁𝑎𝐾_𝑝𝑒_𝜁,=12𝑒^𝑇𝐼𝐾_𝑁𝑎𝐿_𝑎𝑒_𝐼.(5.15) Then, differentiating 𝑉₁ with respect to time, we have 𝑉₁̇=𝑟^𝑇𝑀^𝐿𝑟̇+12𝑀̇^𝐿𝑟+𝐾_𝑖𝑟𝑑𝑡+2𝑒^𝑇𝜁𝑘_𝜁𝐾_𝑁𝑎𝐾_𝑝𝑒_𝜁̇.(5.16) From Property 1, we have (1/2)𝜆_min(𝑀_𝐿)𝑟^𝑇𝑟≤𝑉≤(1/2)𝜆_max(𝑀_𝐿)𝑟^𝑇𝑟. By using Property 2, the time derivative of 𝑉 along the trajectory of (5.14) is 𝑉₁̇=−𝑟^𝑇𝐾_𝑝𝑟−𝑟^𝑇𝜇−‖𝑟‖²Φ²Φ𝛾(‖𝑟‖)+𝛿+2𝑒^𝑇𝜁𝑘_𝜁𝐾_𝑁𝑎𝐾_𝑝𝑒_𝜁̇+𝑟^𝑇𝐵¹𝐺_𝑟𝐾_𝑁𝑎𝑒_𝐼≤−𝑟^𝑇𝐾_𝑝𝑟−‖𝑟‖²Φ²Φ𝛾(‖𝑟‖)+𝛿+‖𝑟‖Φ+2𝑒^𝑇𝜁𝑘_𝜁𝐾_𝑁𝑎𝐾_𝑝𝑒_𝜁̇+𝑟^𝑇𝐵¹𝐺_𝑟𝐾_𝑁𝑎𝑒_𝐼≤−𝑟^𝑇𝐾_𝑝𝑟−‖𝑟‖²Φ²−𝛾(‖𝑟‖)Φ²‖𝑟‖−‖𝑟‖Φ𝛿Φ𝛾(‖𝑟‖)+𝛿+2𝑒^𝑇𝜁𝑘_𝜁𝐾_𝑁𝑎𝐾_𝑝𝑒_𝜁̇+𝑟^𝑇𝐵¹𝐺_𝑟𝐾_𝑁𝑎𝑒_𝐼,(5.17) when ‖𝑟‖≥𝜌; therefore, 𝑉₁̇≤−𝑟^𝑇𝐾_𝑝𝑟+𝛿+2𝑒^𝑇𝜁𝑘_𝜁𝐾_𝑁𝑎𝐾_𝑑𝑟−2𝑒^𝑇𝜁𝑘_𝜁𝐾_𝑁𝑎𝐾_𝑝𝑘_𝜁𝑒_𝜁+𝑟^𝑇𝐵¹𝐺_𝑟𝐾_𝑁𝑎𝑒_𝐼.(5.18) Differentiating 𝑉₂(𝑡) with respect to time, using (3.7), one has 𝑉₂̇=−𝑒^𝑇𝐼𝐾_𝑁𝑎𝐿^𝑎𝐼^𝑑𝑎̇+𝑅_𝑎𝐼^𝑑𝑎+𝐾_𝑒𝐺_𝑟𝑇𝜁̇^𝑑+𝑅_𝑎𝑒_𝐼+𝐾_𝑒𝐺_𝑟𝑇𝑒_𝜁̇−𝜈.(5.19) Substituting 𝜈 in (5.19) by the control law (5.11), one has 𝑉₂̇=−𝑒^𝑇𝐼𝐾_𝑁𝑎𝐾_𝑑+𝑅_𝑎𝑒_𝐼−𝑒^𝑇𝐼𝐾_𝑁𝑎𝐾_𝑒𝐺_𝑟𝑇𝑒_𝜁̇−𝑒^𝑇𝐼𝐾_𝑁𝑎𝐿_𝑎−𝐿₀𝐼̇^𝑑−𝑒^𝑇𝐼𝐾_𝑁𝑎𝑅_𝑎−𝑅₀𝐼^𝑑−𝑒^𝑇𝐼𝐾_𝑁𝑎𝐾_𝑒−𝐾_𝑒0𝐺_𝑟𝑇𝜁̇^𝑑−𝑒^𝑇𝐼𝐾_3𝑁𝑎_𝑖=1𝜇^2𝑖𝑒_𝐼≤−𝑒_𝑇𝐼‖‖𝑒_𝐼‖‖𝜇_𝑖+𝛿+𝛼₂𝐾_𝑁𝑎𝐾_𝑑+𝑅_𝑎𝑒_𝐼−𝑒^𝑇𝐼𝐾_𝑁𝑎𝐾_𝑒𝐺_𝑟𝑇𝑒_𝜁̇+𝛼₁𝐾_𝑁𝑎‖‖𝑒_𝐼‖‖‖‖𝐼̇^𝑑‖‖𝐾_𝑁𝑎‖‖𝑒_𝐼‖‖‖‖𝐼^𝑑𝑎‖‖+𝛼₃𝐾_𝑁𝑎𝐺_𝑟𝑇‖‖𝑒_𝐼‖‖‖‖𝜁^𝑑‖‖−𝐾_3𝑁𝑎_𝑖=1‖‖𝑒_𝐼‖‖²𝜇^2𝑖≤−𝑒_𝑇𝐼‖‖𝑒_𝐼‖‖𝜇_𝑖+𝛿𝐾_𝑁𝑎𝐾_𝑑+𝑅_𝑎𝑒_𝐼−e^𝑇𝐼𝐾_𝑁𝑎𝐾_𝑒𝐺_𝑟𝑇𝑒_𝜁̇+𝐾_3𝑁𝑎_𝑖=1=−𝑒_𝑇𝐼𝛼_𝑖𝛿𝐾_𝑁𝑎𝐾_𝑑+𝑅_𝑎𝑒_𝐼−𝑒^𝑇𝐼𝐾_𝑁𝐾_𝑒𝐺_𝑟𝑇𝑟+𝑒^𝑇𝐼𝐾_𝑁𝑎𝐾_𝑒𝐺_𝑟𝑇𝑘_𝜁𝑒_𝜁+𝐾_𝑁𝑎𝛿³_𝑖=1𝛼_𝑖.(5.20)
Integrating (5.18) and (5.20), 𝑉̇ can be expressed as 𝑉̇≤−𝑟^𝑇𝐾_𝑝𝑟+𝛿+2𝑒^𝑇𝜁𝑘_𝜁𝐾_𝑁𝑎𝐾_𝑝𝑟−2𝑒^𝑇𝜁𝑘_𝜁𝐾_𝑁𝑎𝐾_𝑝𝑘_𝜁𝑒_𝜁+𝑟^𝑇𝐵¹𝐺_𝑟𝐾_𝑁𝑎𝑒_𝐼−𝑒^𝑇𝐼𝐾_𝑁𝑎𝐾_𝑑+𝑅_𝑎𝑒_𝐼−𝑒^𝑇𝐼𝐾_𝑁𝑎𝐾_𝑒𝐺_𝑟𝑇𝑟+𝑒^𝑇𝐼𝐾_𝑁𝑎𝐾_𝑒𝐺_𝑟𝑇𝑘_𝜁𝑒_𝜁+𝐾_𝑁𝑎𝛿³_𝑖=1𝛼_𝑖.(5.21)
We can obtain 𝑉̇≤−𝑟^𝑇𝑒_𝜁𝑒_𝐼𝑄⎡⎢⎢⎢⎢⎣𝐾^𝑁𝑎0𝐾0_𝑁𝑎000𝐾0_𝑁𝑎⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝑒𝑟_𝜁𝑒_𝐼⎤⎥⎥⎥⎥⎦,(5.22) where 𝑄=⎡⎢⎢⎢⎢⎢⎣𝐾^𝑝−𝐾_𝑝𝑘_𝜁12𝐺^𝑟𝐾_𝑒−𝑘_𝜁𝑇−𝐵¹𝐾_𝑝2𝑘_𝜁𝐾_𝑝𝑇𝑘_𝜁−12𝐾^𝑒𝐺_𝑟𝑇𝑘_𝜁12𝐺^𝑟𝐾_𝑒𝑇−𝐵¹−12𝐾^𝑒𝐺_𝑟𝑇𝑘_𝜁𝐾_𝑑+𝑅_𝑎⎤⎥⎥⎥⎥⎥⎦.(5.23) The term 𝑄 on the right-hand side (5.22) can always be negative definite by choosing suitable 𝐾_𝑝 and 𝐾_𝑑. Since [𝐾_𝑛𝑎] is positive definite, we only need to choose 𝐾_𝑝 and 𝐾_𝑑 such that 𝑄 is positive definite. Therefore, 𝐾_𝑑 and 𝐾_𝑝 can always be chosen to satisfy 𝐾_𝑑+𝑅>𝐾^𝑝−112𝐺^𝑟𝐾_𝑒𝑇−𝐵¹−12𝐾^𝑒𝐺_𝑟𝑇𝑘_𝜁⎡⎢⎢⎣2𝐼𝑘^𝜁−1𝑘^𝜁−1𝑘^𝜁−1𝑇⁻¹𝑘^𝜁−1⎤⎥⎥⎦⎡⎢⎢⎣12𝐺^𝑟𝐾_𝑒−1𝑇−𝐵¹2𝐾^𝑒𝐺_𝑟𝑇𝑘_𝜁⎤⎥⎥⎦.(5.24)
If ‖𝑟‖≤𝜌, it is easy to obtain 𝑉̇≤0. 𝑟, 𝑒_𝜁, and 𝑒_𝐼 converge to a set containing the origin with 𝑡→∞.
(ii) 𝑉 is bounded, which implies that 𝑟∈𝐿^{∞𝑛−𝑘}. From 𝑟=𝑒_𝜁̇+𝑘_𝜁𝑒_𝜁, it can be obtained that 𝑒_𝜁,𝑒_𝜁̇∈𝐿^{∞𝑛−𝑘}. As we have established 𝑒_𝜁,𝑒_𝜁̇∈𝐿_∞, from Assumption 4.1, we conclude that 𝜁(𝑡),𝜁̇(𝑡),𝜁̇_𝑟(𝑡),𝜁̈_𝑟(𝑡)∈𝐿^{∞𝑛−𝑘} and 𝑞̇∈𝐿^𝑛∞.
Therefore, all the signals on the right hand side of (5.14) are bounded, and we can conclude that 𝑟̇ and therefore 𝜁̈ are bounded. Thus, 𝑟→0 as 𝑡→∞ can be obtained. Consequently, we have 𝑒_𝜁→0, 𝑒_𝜁̇→0 as 𝑡→∞. It follows that 𝑒_𝑞,𝑒_𝑞̇→0 as 𝑡→∞.
(iii) Substituting the control (5.5) and (5.7) into the reduced order dynamic system model (5.4) yields 1+𝐾_𝑓𝑒_𝜆=𝑍(𝜁)𝐶¹𝜁,𝜁̇𝜁̇+𝐺¹+𝑑¹(𝑡)−𝐿^+𝑇𝐺_𝑟𝐾_𝑁𝑎𝐼_𝑎+𝐵¹𝐺_𝑟𝐾_𝑁𝑏𝐼^𝑑𝑏+𝐵¹𝐺_𝑟𝐾_𝑁𝑏𝑒_𝐼=−𝑍(𝜁)𝐿^+𝑇𝑀_𝐿(𝜁)𝜁̈+𝜒²𝜒+𝛿+𝐵¹𝐺_𝑟𝐾_𝑁𝑏𝑒_𝐼.(5.25) Since 𝜁̇=0 when 𝐼∈𝑅^𝑘, (3.7) could be changed as 𝐿_𝑎𝑑𝐼_𝑏𝑑𝑡+𝑅^𝑎𝐼_𝑏=𝜈_𝑏.(5.26) Therefore, 𝑟=0 and 𝑒_𝜁=0 in the force space; (5.20) could be changed as 𝑉₂̇=−𝑒^𝑇𝐼𝐾_𝑁𝑏𝐾_𝑑+𝑅𝑒_𝐼+𝐾_𝑁𝑏𝛿³_𝑖=1𝛼_𝑖.(5.27) Since 𝐾_𝑁𝑏 is bounded, 𝑉̇<0, we can obtain 𝑒_𝐼→0 as 𝑡→∞. The proof is completed by noticing that 𝜁̈, 𝑍(𝑞), 𝐾_𝑁𝑏 and 𝑒_𝐼 are bounded. Moreover, 𝜁→𝜁^𝑑, and −𝑍(𝜁)𝐿^+𝑇𝑀_𝐿(𝜁)(𝜁̈^𝑑)+𝜒²/(𝜒+𝛿)≤𝛿, 𝑒_𝐼→0, the right-hand side terms of (5.25), tend uniformly asymptotically to zero; then it follows that 𝑒_𝜆→0, then 𝑓(𝑡)→𝑓_𝑑(𝑡).
Since 𝑟, 𝜁, 𝜁̇, 𝜁_𝑟, 𝜁̇_𝑟, 𝜁̈_𝑟, 𝑒_𝜆 and 𝑒_𝐼 are all bounded, it is easy to conclude that 𝜏 is bounded from (5.2).

6. Simulations

To verify the effectiveness of the proposed control algorithm, let us consider a 2-DOF manipulator mounted on two-wheels-driven mobile base [23] shown in Figure 1. The mobile manipulator is subjected to the following constraints: 𝑥̇cos𝜃+𝑦̇sin𝜃=0. Using Lagrangian approach, we can obtain the standard form with 𝑞_𝑣=[𝑥,𝑦,𝜃]^𝑇, 𝑞_𝑎=[𝜃₁,𝜃₂]^𝑇, 𝑞=[𝑞_𝑣,𝑞_𝑎]^𝑇, and 𝐴_𝑣=[cos𝜃,sin𝜃,0]^𝑇: 𝑀_𝑣=⎡⎢⎢⎢⎢⎢⎣𝑚^𝑝12+2𝐼^𝑤sin²𝜃𝑟²−2𝐼^𝑤𝑟²−2𝐼_𝑤sin𝜃cos𝜃−𝑚₁₂𝑑sin𝜃𝑟²sin𝜃cos𝜃𝑚_𝑝12+2𝐼^𝑤cos²𝜃𝑟²−𝑚₁₂𝑚₁₂𝑑cos𝜃𝑑sin𝜃𝑚₁₂𝑑cos𝜃𝑀¹¹¹𝑀₁₁₁⎤⎥⎥⎥⎥⎥⎦,=𝐼_𝑝+𝐼₁₂+𝑚₁₂𝑑²+2𝐼^𝑤𝐿²𝑟²𝑀_𝑣𝑎,𝑀_𝑎=diag𝐼₁₂,𝐼₂,=⎡⎢⎢⎢⎢⎣00𝐼..0000..00₁₂𝐵=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣sin𝜃0.0⎤⎥⎥⎥⎥⎦,−co𝑟s𝜃−sin𝑟𝜃0.00.0−𝑟𝑙co𝑟s𝜃0.00.0𝐶00..00𝑟00𝑟𝑙..00100...000010...00_𝑣0⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎢⎣2𝐼^𝑤𝑟²𝜃̇sin𝜃cos𝜃2𝐼^𝑤𝑟²−2𝐼_𝑤𝜃̇sin²𝜃−𝑚₁₂𝑑𝜃̇cos𝜃0.0𝑟²𝜃̇cos²𝜃2𝐼^𝑤𝑟²𝐶0.0𝜃̇s0in.0𝜃cos𝜃𝑚₁₂𝑑0𝜃.0̇cos𝜃00..00_𝑣𝑎⎤⎥⎥⎥⎥⎥⎦,=0.0,𝐶_𝑎=0.0𝜏,𝐻=𝐺_𝑣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−=1000t[a....00000n.0𝜃,000100......000000,000010.0.....00000]^𝑇00001,.....00000_𝑣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐺_𝑎=0.0,𝑚₂𝑔𝑙₂sin𝜃₂^𝑇,𝑚_𝑝12=𝜏_𝑙,𝜏_𝑟^𝑇,𝜏_𝑎=𝜏₁,𝜏₂^𝑇,=𝑚_𝑝+𝑚₁₂,𝑚₁₂=𝑚₁+𝑚₂,𝐼₁₂=𝐼₁+𝐼₂.(6.1)

Let the desired trajectory 𝑞_𝑑=[𝑥_𝑑,𝑦_𝑑,𝜃_𝑑,𝜃_1𝑑,𝜃_2𝑑]^𝑇 and the end effector be subject to the geometric constraint Φ=𝑙₁+𝑙₂sin(𝜃₂)=0, and 𝑦_𝑑=1.5sin(𝑡), 𝜃_𝑑=1.0sin(𝑡), 𝜃_1𝑑=𝜋/4(1−cos(𝑡)), 𝜆_𝑑=10.0𝑁.

The trajectory and force tracking control problem is to design control law 𝜏 such that (4.1) holds and all internal signals are bounded.

In the simulation, we assume the parameter 𝑚_𝑝=𝑚₁=𝑚₂=1.0, 𝐼_𝑤=𝐼_𝑝=1.0, 2𝐼₁=𝐼₂=1.0, 𝐼=0.5, 𝑑=𝐿=𝑅=1.0, 2𝑙₁=1.0, 2𝑙₂=0.6, 𝑞(0)=[0,2.0,0.6,0.5]^𝑇, 𝑞̇(0)=[0.0,0.0,0.0,0.0]^𝑇, 𝐾_𝑁=diag[0.01], 𝐺_𝑟=diag[100], 𝐿_𝑎=[0.005,0.005,0.005,0.005]^𝑇, 𝑅_𝑎=[2.5,2.5,2.5,2.5]^𝑇, and 𝐾_𝑒=[0.02,0.02,0.02,0.02]^𝑇. The disturbance on the mobile base is set 0.1sin(𝑡) and 0.1cos(𝑡). By Theorem 5.1, the control gains are selected as 𝐾_𝑝=diag[1.0,1.0,1.0], 𝑘_𝜁=diag[1.0,1.0,1.0], 𝐾_𝑖=0.0 and 𝐾_𝑓=0.995, 𝐶=[8.0,8.0,8.0,8.0,8.0]^𝑇, 𝐾_𝑁=0.1, 𝐾_𝑑=diag[10,10,10,10], 𝛼₁=0.008, 𝛼₂=4.0, 𝛼₃=0.03. The disturbance on the mobile base is set 0.1sin(𝑡) and 0.1cos(𝑡). The simulation results for motion/force are shown in Figures 2, 3, 4, 5, 6, 7, 8, and 9. The desired currents tracking and input voltages on the motors are shown in Figures 5, 6, 8, and 9. The simulation results show that the trajectory and force tracking errors asymptotically tend to zero, which validate the effectiveness of the control law in Theorem 5.1.

Figure 2 

The positions of the joints.

Figure 3 

The velocities of the joints.

Figure 4 

The torques of the joints.

Figure 5 

Tracking the desired currents.

Figure 6 

The input voltages.

Figure 7 

The constraint force.

Figure 8 

Tracking the desired current of the joint 2.

Figure 9 

The input voltage of the joint 2.

7. Conclusion

In this paper, effective robust control strategies have been presented systematically to control the holonomic constrained nonholonomic mobile manipulator in the presence of uncertainties and disturbances, and actuator dynamics is considered in the robust control. All control strategies have been designed to drive the system motion converge to the desired manifold and at the same time guarantee the boundedness of the constrained force. The proposed controls are nonregressor based and require no information on the system dynamics. Simulation studies have verified the effectiveness of the proposed controller.

Acknowledgment

The authors are thankful to the Ministry of Science and Technology of China as the paper is partially sponsored by the National High-Technology Research and Development Program of China (863 Program) (no. 2009AA012201).