2.1 State equation of robot manipulators

The motion equations of an n degree-of-freedom manipulator can be derived using the Lagrange-Euler formulation as

where J(q(t),Ø)∈R^n×n is a symmetric positive definite inertia matrix, D(q(t). $\dot{q}$(t).Ø)∈Rⁿ is called a smooth generalized disturbance vector as follows:

including the centrifugal and Coriolis terms H(q(t), $\dot{q}$(t),Ø), Coulomb and viscous or any other frictions F(q(t), $\dot{q}$(t),Ø), gravity terms G(q(t),Ø), unknown pay loadand etc. where τ is an input vector, and q(t), $\dot{q}$(t), and $\ddot{q}$(t)∈Rⁿ are the generalized position, velocity, and acceleration vectors, respectively. The Ø is the vector composed of the parameters of robot manipulators (i.e. the masses, lengths, offset angles, and inertia of links). An exact modeling of physical robot dynamics is difficult because of the existence of parameter uncertainties, unknown frictions, and payload variations. In this study for the regulation problem, a desired reference q_d∈Rⁿ is given from a current state and $\dot{q}$(t)= $\ddot{q}$Z(t)=0 is satisfied. Let us define the state vector X(t)∈Rⁿ in the error coordinate system for the improved integral variable structure regulation controller as

where X₁(t) and X₂(t) are the trajectory errors and its derivative as

Then the state equation of robot systems for the regulation control becomes

where $X\left(0\right)=[q_d^T-q^T(0)-{\dot{q}}^T(0){]}^T$ is a given initial condition. For (5), a new improved integral variable structure regulation controller will be designed through the two steps, design of the integral sliding surface and choice of continuous control input. And some analysis about the relationship between the error to the sliding trajectory and the non-zero value of the sliding surface and the closed loop stability will be given in each step.

2.2 An integral sliding surface, its sliding trajectory, and error analysis

First of all, let's define an integral-augmented sliding surface vector s(t) be

where

where K_v and K_ρ are diagonal coefficient matrices and X₀(t) is the integral of the error with the special initial condition for removing the reaching phase by means of making the integral sliding surface be zero at t=0, i.e., s(0)=0. Thus this integral augmented sliding surface determines the ideal sliding mode dynamics to have an ideal second order dynamics exactly from a given initial condition to the origin in the error coordinate system without any reaching phase, not a straight line of the conventional sliding surface through the origin. If X₀(0)=0 in (7) such as previous works⁽²³⁾ on the integral variable structure systems, there exist still the reaching phase problems because s(t)≠0 at t=0 and an inevitable over shoot problems as the side effect because the integral state accumulated form the zero must re-converge to the zero. The sliding dynamics from a given initial condition to the origin defined by equation (6) is obtained from $\dot{s}$(t)=0 as follows:

Then rewrite equation (8) into the state equation form

where

The solution of the state equation of the sliding dynamics (9)$q_s^{*}$ and ${\dot{q_s}}^{*}$∈Rⁿ theoretically predetermines the ideal sliding trajectory from a given initial state q(0) to the desired reference q_d defined by (6). Since det[λI-Λ]=[λ²I+λK_v+K_ρ], K_v and K_ρ∈R^n×n can be chosen so that all the eigenvalues of Λ have the negative real parts, which guarantees the exponential stability of the system (9), then there exists the positive scalar constants K and κ such that

where ||·|| is the induced Euclidean norm.

Now, define $\overline{X_1}\left(t\right)$ and $\overline{X_2}\left(t\right)$ are the error from the ideal sliding trajectory and its derivative, respectively as

If the sliding surface is zero for all time, naturally this defined error and its derivative are also zeros. The sliding surface may be not exactly zero if the input of the improved integral variable structure regulation controller is continuous. Hence the effect of the non-zero value of the sliding surface to the error to the sliding trajectory is analyzed in the following Theorem 1 as a prerequisite to the main theorem.

Theorem 1: If the sliding surface defined by equation (6) satisfies ||s(t)|| $\le$γ for any t $\ge$0 and || $\hat{X}$(0)|| $\le$ γ/κ is satisfied at the initial time, then

is satisfied for all t $\ge$0 where ε₁ and ε₂ are the positive constants defined as follows:

Proof: Let us define a new error vector as

The sliding surface can be re-written as

and can be re-expressed in a differential matrix from as

In (17), the sliding surface may be considered as the bounded disturbance input, ||s(t)|| $\le$γ. The solution of (17) is expressed as

From the boundness of the sliding surface and (11), the Euclidean norm of the vector $\hat{X}$ becomes

for all time, t $\ge$0. Since $\left|\right|\widehat{X_1}\left|\right|\le \left|\right|\widehat{X\left|\right|}$, the following equation is obtained

From the sliding surface, one can be simply obtained as

If the norm operation is taken on both sides, (21) becomes

which completes the proof of Theorem 1.

The above Theorem 1 implies that the error from the ideal sliding trajectory and its derivative are uniformly bounded provided the sliding surface is bounded for all time t $\ge$0. Using this result of Theorem 1, we can give the specifications on the error from the ideal sliding trajectory being dependent upon the value of the integral sliding surface, (6). In the next section, we will design a variable structure regulation controller with the efficient compensation which can guarantee the boundedness of s(t), i.e., ||s(t)|| $\le$γ for a given γ, then the error to the ideal sliding trajectory is bounded by ε₁ in virtue of Theorem 1.

2.3 Continuous input and its stability analysis

Robot manipulators activated by several servo motors are subject to a variety of disturbances and uncertainties. The robust control of highly nonlinear robot manipulators is essential for developing robotics. It is often noted that the generalized nonlinear disturbances, D(q(t), $\dot{q}$(t), Ø), must be compensated for improving the control performance. As an ideal control input in the sliding mode control, the equivalent control of the augmented sliding surface (6) for the robot system (5) is obtained from equation (8)

The smooth generalized disturbance D(q(t), $\dot{q}$_q(t), ø) is included in an equivalent control, τ_eq(t). Since generally this smooth generalized disturbance is very complex, a direct calculation of the smooth generalized disturbance from the model of robot manipulators results in the long sampling time, limitations of the control performance, and difficulties of the controller design for highly nonlinear robot manipulators.

In this paper, using the efficient compensation method, so called disturbance observer⁽²²⁾, we consider the following continuous control input, τ(t)

where τ_c(t) is the compensation term for the smooth generalized disturbance as well as the error of the nominal inertia matrix, is not the direct calculation from $\hat{D}$(q(t), $\dot{q}$_q(t), ø) in the model but the efficient estimation of the generalized disturbance, D(q(t), $\dot{q}$_q(t), ø), only using the nominal inertia matrix, J_N of the model (1) and an available acceleration information which can be calculated from the speed information by means of the Euler method

where $\tilde{\ddot{q}}\left(t\right)$, ΔJ(q(t),Ø), Δ $\ddot{q}\left(t\right)$, Δτ(t) and are defined by

respectively, where ΔJ(q(t),Ø) is the deviation between the real inertia matrix and its nominal value, Δ $\ddot{q}\left(t\right)$ is the acceleration information error to the real acceleration value, Δτ(t) is the control input delay error resulted from the digital control, and h is the sampling time for digital implementation. If the sampling time is sufficiently small and control input is continuously implemented, then the acceleration information error Δ $\ddot{q}\left(t\right)$ and the control input delay error Δτ(t) can be small. This disturbance observer fails at the initial time because τ(t-h) is unknown, hence only τ_c(0) is once calculated by using the model of robots with off-line in advance. The detail features of disturbance observer is explained in the work of Komoda in [22]⁽²²⁾. The second term in the right hand side of the equation (24) is defined as

where $\widetilde{{\tau }_{eq}}\left(t\right)$ is the modified equivalent control for the compensated dynamics of equation (1), and is designed so that the error dynamics of the controlled system has the sliding surface dynamics defined by equation (9), which is defined as

As can be seen in (31), $\widetilde{{\tau }_{eq}}\left(t\right)$ is determined directly according to the design of the integral sliding surface. The τ_χ(t) is the continuous feedback term of the integral sliding surface for correcting the small compensation error as follows:

where k_χ1, k_χ2, and δ are the suitable positive constants as the design parameters for the continuous control input. After effectively compensating a almost part of nonlinear dynamics of robot manipulators based on the disturbance observer for avoiding a heavy computation burden, the sliding control input is totally continuously implemented. As the function of the disturbance observer, the effective compensation for highly nonlinear generalized disturbances and modeling errors of the inertia matrix will be studied. If we apply the continuous input control torque given by equation (24)-(32) to the robotic system (5), the following equation is obtained

and the dynamics of s(t) is expressed in the following simple form

where n₁(t)∈Rⁿ is the resulting disturbance vector given by

From the equation (34), the 2n-th order original regulation control problem is converted to the 2n-th stabilization problems with three degree of freedoms k_χ1, k_χ2, and δ against the resultant disturbance n₁(t) by means of the proposed algorithm which implies the robustness problems in the design of controllers. For some positive constants ε₁ and ε₂ defined in (14), let the constant N be defined as follows:

where the matrix norm is defined as the induced Euclidean norm, and for a positive number ε $>$0 and a vector λ∈Rⁿ the boundary set defined by as

In equation (35), the resultant disturbances are mainly dependent on the acceleration information error and the control input computation delay error and not system uncertainties or modeling error of robot manipulators. The disturbance observer can compensate for modeling errors of the inertia matrix besides the smooth generalized disturbance (2). Thus the SMC design is independent of the maximum bound of modeling errors in the parameter space, but dependent on only the resultant disturbance composed of the acceleration information error and the control time delay due to the digital implementation.

The stability property of the system (5) with the control laws (24)-(32) will be stated in the next theorem:

Theorem 2: Consider the robot system with the control algorithm given by the equations (24)-(32). Assume that for some positive γ $>$0, ||s(0)|| $\le$γ and || $\hat{X}$(0)|| $\le$γ/κ are satisfied at the initial time t=0, and if the gains k_χ1 and k_χ2 satisfy

for a given δ $>$0 and N in (36), then the closed loop control system is uniformly bounded(i.e. the solution $\hat{X}$ is uniformly bounded at the origin in the error coordinate state space) for all time t $\ge$0 until ||s(t)|| $\le$η where η is defined by

Proof: The proof is straightforward, first take Lyapunov candidate function as

and differentiate with respect to time, it leads to

By the matrix inequality, (41) becomes

If the gains k_χ1 and k_χ2 satisfy the inequality (38)

at all t $\ge$0 as long as ||s(t)|| $\ge$η , which completes the proof of Theorem 2.

Theorem 2 guarantees the uniform bounded stability of the proposed continuous improved integral variable structure regulation controller for robot manipulators. The smaller δ in control algorithm (32), the lower value of η. The η can be decreased by an increase of k_χ1 for a given δ and N so that η is sufficiently smaller than γ the bound of the sliding surface in Theorem 1(η $<$γ). If the initial value of the sliding surface is small(||s(0)|| $\le$γ) which is reasonable in case of the known initial state of robot manipulators, the feedback control (24)-(32) designed by Theorem 1 and Theorem 2 maintains the bounded stability of the system with the prescribed performance:

which implies guaranteeing the prescribed tracking error ε₁ to the ideal sliding trajectory $q_s^{*}\left(t\right)$ predetermined by the integral sliding surface from a given initial condition q(0) to q_d, in other words, guaranteeing the predetermined output response with the prescribed accuracy. Fig. 1 shows the structure of the proposed algorithm composed of the compensation term, modified equivalent term, and continuous feedback term of the integral sliding surface which is relatively simple because of avoidance of large computation burden and nature of the VSS. Thus the sampling time can be as small as possible so that the acceleration information calculated by Euler method and the delayed control input are almostly exact to each real value, therefore, the maximum value N can be small. Therefore, a new SMC can be realized effectively. And robot manipulators are controlled to follow the predetermined sliding trajectory from q(0) to q_d. The sliding trajectory is obtained by the solution of the sliding dynamics of (9) in advance. Hence the output is predictable with ε₁ accuracy. The design procedure of the proposed sliding mode controller to guarantee the predetermined output with prescribed accuracy is as follows: First, choose the desired sliding surface defining the desired sliding dynamics (9) which means the determination of the coefficients, K_v and K_ρ and calculate the ideal sliding trajectory off-line (performance design phase). Second, find the constants and satisfying the equation (11). Third, determine the bound of the sliding surface, γ using (14) in Theorem 1 for a given the accuracy of the tracking error to the sliding trajectory, ε₁. And finally design the gains, k_χ1 and k_χ2, in equation (32) based on Theorem 2 so that the η is smaller than γ(robustness design phase). In the whole procedure, the design does not need the information of maximum bound of system parameter variations or uncertainties because of the efficient on-line compensation.