Modelling of Bound Estimation Laws and Robust Controllers for Robot Manipulators Using Functions and Integration Techniques

Some robust control methods have been developed in the past in order to increase tracking performance in the presence of parametric uncertainties. In the presence of parametric uncertainty, unmodelled dynamics and other sources of uncertainties, robust control laws are used. Corless-Leitmann [1] approach is a popular approach used for designing robust controllers for robot manipulators. In early application of Corless-Leitmann [1] approach to robot manipulators [2, 3], it is difficult to compute uncertainty bound precisely. Because, uncertainty bound on parameters depends on the inertia parameters, the reference trajectory and manipulator state vector. Spong [4] proposed a new robust controller for robot manipulators using the Lyapunov theory that guaranties stability of uncertain systems. In this approach, Leithmann [5] or Corless-Leithman [1] approach is used for designing the robust controller. One of the advantage of Spong’s approach [4] is that uncertainty on parameter is needed to derive robust controller and uncertainty bound parameters depends only on the inertia parameters of the robots. Yaz [6] proposed a robust control law based on Spong’s study [4] and global exponential stability of uncertain system is guaranteed. However, disturbance and unmodelled dynamics are not considered in algorithm of [4, 6]. Danesh at al [7] develop Spong’s approach [4] in such a manner that control scheme is made robust not only to uncertain inertia parameters but also to robust unmodelled dynamics and disturbances. Koo and Kim [8] introduce adaptive scheme of uncertainty bound on parameters for robust control of robot manipulators. In [8], upper uncertainty bound is not known as would be in robust controller [4] and uncertainty bound is estimated with estimation law in order to control the uncertain system. A new robust control approach is proposed by Liu and Goldenerg [9] for robot manipulators based on a decomposition of model uncertainty. Parameterized uncertainty is distinguished from unparameterized uncertainty and a compensator is designed for parameterized and unparameterized uncertainty. A decomposition-based control design framework for mechanical systems with model uncertainties is proposed by Liu [10]. In order to increases tracking performance of uncertain systems, design of uncertainty bound estimation functions are considered. For this purpose, some uncertainty bound estimation functions are developed [11-15] based on a Lyapunov function, thus, stability of


Introduction
Some robust control methods have been developed in the past in order to increase tracking performance in the presence of parametric uncertainties.In the presence of parametric uncertainty, unmodelled dynamics and other sources of uncertainties, robust control laws are used.Corless-Leitmann [1] approach is a popular approach used for designing robust controllers for robot manipulators.In early application of Corless-Leitmann [1] approach to robot manipulators [2,3], it is difficult to compute uncertainty bound precisely.Because, uncertainty bound on parameters depends on the inertia parameters, the reference trajectory and manipulator state vector.Spong [4] proposed a new robust controller for robot manipulators using the Lyapunov theory that guaranties stability of uncertain systems.In this approach, Leithmann [5] or Corless-Leithman [1] approach is used for designing the robust controller.One of the advantage of Spong's approach [4] is that uncertainty on parameter is needed to derive robust controller and uncertainty bound parameters depends only on the inertia parameters of the robots.Yaz [6] proposed a robust control law based on Spong's study [4] and global exponential stability of uncertain system is guaranteed.However, disturbance and unmodelled dynamics are not considered in algorithm of [4,6].Danesh at al [7] develop Spong's approach [4] in such a manner that control scheme is made robust not only to uncertain inertia parameters but also to robust unmodelled dynamics and disturbances.Koo and Kim [8] introduce adaptive scheme of uncertainty bound on parameters for robust control of robot manipulators.In [8], upper uncertainty bound is not known as would be in robust controller [4] and uncertainty bound is estimated with estimation law in order to control the uncertain system.A new robust control approach is proposed by Liu and Goldenerg [9] for robot manipulators based on a decomposition of model uncertainty.Parameterized uncertainty is distinguished from unparameterized uncertainty and a compensator is designed for parameterized and unparameterized uncertainty.A decomposition-based control design framework for mechanical systems with model uncertainties is proposed by Liu [10].In order to increases tracking performance of uncertain systems, design of uncertainty bound estimation functions are considered.For this purpose, some uncertainty bound estimation functions are developed [11][12][13][14][15] based on a Lyapunov function, thus, stability of

A method for derivation of bound estimation laws
In the absence of friction or other disturbances, the dynamic model of an n-link manipulator can be written as [16] M(q)q C(q,q)q G(q) τ      where q denotes generalised coordinates, is the n-dimensional vector of applied torques (or forces), M(q) is a positive definite mass matrix, C(q,q)q  is the n-dimensional vector of centripetal and Coriolis terms and G(q) is the n-dimensional vector of gravitational terms.Equation (1) can also be expressed in the following form.Y(q,q,q ) where π is a p-dimensional vector of robot inertia parameters and Y is an nxp matrix which is a function of joint position, velocity and acceleration.For any specific trajectory, the desired position, velocity and acceleration vectors are q d , d q  and d q  .The measured actual where  is a positive definite matrix.Then the following nominal control law is considered: 00 r0 r 0 rr 0 M (q)q C (q,q)q G (q) K Y(q,q,q ,q ) K where π 0 R p represents the fixed parameters in dynamic model and K is the vector of PD action.The corrected velocity error  is given as The control input is defined in terms of the nominal control vector 0 as 0r r r r 0 Y(q,q,q ,q )u(t) Y(q,q,q ,q )( u Where u(t) is the additional robust control input.It is assumed that there exists an unknown bound on parametric uncertainty such that 0 Since R +p is assumed to be unknown, should be estimated with the estimation law to control the system properly. (t)  ˆshows the estimate of and (t) Substituting ( 6) into (1) and after some algebra yields rr M(q) C(q,q) K Y(q,q,q ,q )( u(t)) By taking into account above parameters and control algorithm, the Lyapunov function candidate is defined as [15,16].
where BR nxn is a positive diagonal matrix, (t)  is chosen as a pxp dimensional diagonal matrix changes in time.The time derivative of V along the trajectories is Taking 2 BK  , using the property   [17,18], and taking time derivative of V of system ( 9) is Equation ( 12) is arranged as Equation ( 14) can be arranged as Consequently, a suitable expression for the time derivative of V is obtained.
where TT -q Kq-q ΛKΛq0      .If the rest of Equation ( 16) is zero, system will be stable.Remaining terms in Equation ( 16) are [( -(t)] ˆis considered as a common multiplier then Hence, we look for the conditions for which the equation 18) can be written as Equation ( 20) is arranged as Integration both side of Equation (21) yields Then, a general equation for derivation derivation of bound estimation law is developed as [14,15] -1 The Equation ( 23) is a general equation for derivation of the bound estimation law and it is derived from Lyapunov function.As a result, ˆ(t)  all derived from Equation (23) guarantess stability of uncertain system.However, (t) -1 and ˆ(t)  are unknown and ˆ(t)  is derived depending on the function (t) -1 .For derivation, selection of (t) -1 and integration techniques are very important.There is no certain rule for selection of (t) -1 and integration techniques for this systems.System state parameters and mathematical insight are used to search for appropriate function of (t) -1 as a solution of the Equation (23).

First choice of (t) -1
For the first derivation of ˆ(t) Substituting Equation ( 24) into (23) yields TT 11 11 After integration, the result is Then TT T 11 11 11   For the second derivation of Substituting Equation ( 29) into (23) yields After integration, the result is After multiplication by  is taken as initial condition, constant C is equivalent to -arctan (1).So, the estimation law for the uncertainty bound is derived as.

Third choice of (t) -1
For the third derivation of ˆ(t) After integration, the result is If we substitute ,   , and ˆ(t)   into Equation ( 16), the right terms of Equation ( 16) will be always zero and the derivation of the Lyapunov function will become a negative semidefinite function such that TT V-q K q -q ΛKΛq0        (38) So, the system is stable for all ˆ(t)  derived from Equation (23).

Design of robust contol laws
Based on the uncertainty bound estimation laws derived in section 2, and in [15], it is possible to develop robust control inputs.

Robust control law 1
In order to define first robust control input, the following theorem is proposed.

Theorem:
Additional control input in control law ( 6) is Where ε>0.If the control input (39) is substituted into the control law (6) for the control of the model manipulator, then, the control law ( 6) is continuous and the closed-loop system is uniformly ultimate bounded.

Proof
It is assumed that there exists an unknown bound on parametric uncertainty such that 0   and 0 are substituded into (13), the time derivative of the Lyapunov function ( 13) is written as [14,15]. Where From the Cauchy-Schawartz inequality and our assumption on .If ||Y T ||<ε then  is bounded.The rest of the proof can be seen in [4, 8].

Robust control law 4
From Equations ( 43) and (44), it is ease to define the following control law.

Simulation results
For illustration, a two-link robot manipulator is given in Figure 1 [4].Parameterisation of this robot is given by Fig. 1.Two-link planar robot [4].
With this choice of nominal parameter vector  0 and uncertainty range given by (54), it is an easy matter to calculate the uncertainty bound  as follows: As a measure of parameter uncertainty on which the additional control input is based, can be defined as Having a single number to measure the parameter uncertainty may lead to overly conservative design, higher than necessary gains, etc.For this purpose, different "weights" or gains to the components of u may be assigned.This can be done as follows: Supposing that a measure of uncertainty for each parameter i  can be defined separately as:  denote the itch component of the vector T Y  , i  choose as the itch component of , and consequently the itch component of the control input u p is defined as [4].For computer simulation, a fifth order polynomial function is used as a reference trajectory for both joints.In order to analyse performance of the proposed controllers, each control law with the same control parameters K and Ʌ is applied to the same model system using same trajectory.The control matrices Ʌ and K are chosen to be identical as Ʌ=diag(10 10) and K=diag(30 30) for all controllers.The obtained results are plotted in Figures 2-4.As shown in Figures 2-4, tracking error is small and tracking performance changes according to uncertainty bound estimation laws.

Conclusion
In the past, some robust controllers are developed for robot manipulators.Corless-Leitmann [1] approach is a popular approach used for designing robust controllers for robot manipulators.Spong [4] proposed a new robust controller for robot manipulators and Leithmann [5] or Corless-Leithman [1] approach is used for designing the robust controller.In [4], uncertainty bound on parameter is needed to derive robust controller and uncertainty bound parameters depends only on the inertia parameters of the robots.However, constant uncertainty bound parameters cause pure tracking performances.In order to increase tracking performance of the uncetain system, uncertainty bound estimation laws are developed [11][12][13].Uncertainty bound estimation laws are updated as a function of exponential [11,12], logarithmic [13] and [14] functions depending robot kinematics parameters and tracking error.A first order differential equation function is developed for derivation of control parameters and only a single derivation of uncertainty bound estimation law is possible.A new method for derivation of a bound estimation law is not proposed in [11][12][13], because, definition of a new variable function for other derivation is diffucult.In the study [14], a general equation is developed from Lyapunov function and uncertainty bound estimation laws depending on trigonometric functions are developed.However, a general method for derivation of uncertainty bound estimation laws is not proposed.In a recent study [15], a general method for derivation of bound estimation laws based on the Lyapunov theory is proposed.In this method, functions and integration techniques are used for derivation of uncertainty bound estimation laws.Then, relations between the bound estimation laws and control inputs are established and four new robust control inputs are designed depending on each bound estimation law.It is possible to derive other different uncertainty bound estimation laws from general equation (23) if appropriate functions and integration techniques are defined.In this work, three different variable functions are defined and integration techniques are used in order to derive ˆ(t)  and relations between the uncertainty bouns and robust control laws are established.There is no distinct rule for definition of the (t) and integration techniques in order to derive ˆ(t)  .We use system state parameters and mathematical insight to search for appropriate function of (t) as a derivtion of ˆ(t)  .This study also shows that robust controllers are not limited with these derivations.It will be also possible to derive another bound estimation laws from Equation (23) if appropriate function (t) and integration techniques are chosen.

Table 1
[4]he ith component of π obtained by means of Equation (51) are given in Table2.It is assumed that the parameters m 2 , l c2 and I 2 are changed in the intervals Choosing the mean value for the range of possible  i in Equation (54) yields the nominal parameter vector and the computed values for ith component of π 0 is shown in Table3[4].

Table 4 .
[4]Since extended algorithm (56) is used, the uncertainty bounds for each parameter are shown separately in Table4.The uncertainty bounds  i in Table4are simply the difference between values given in Table3 and Table 2, and the value of is the Euclidean norm of the vector with components i[4].Uncertainty bound[4].