New time delay estimation-based virtual decomposition control for n-DoF robot manipulator

One of the most efficient approaches to control a multiple degree-of-freedom robot manipulator is the virtual decomposition control (VDC). However, the use of the regressor technique in the conventionnal VDC to estimate the unknown and uncertaities parameters present some limitations. In this paper, a new control strategy of n-DoF robot manipulator, refering to reorganizing the equation of the VDC using the time delay estimation (TDE) have been investigated. In the proposed controller, the VDC equations are rearranged using the TDE for unknown dynamic estimations. Hence, the decoupling dynamic model for the manipulator is established. The stability of the overall system is proved based on Lyapunov theory. The effectiveness of the proposed controller is proved via case study performed on 7-DoF robot manipulator and compared to the conventionnal Regressor-based VDC according to some evalution criteria. The results carry out the validity and efficiency of the proposed time delay estimationbased virtual decomposition controller (TD-VDC) approach.

INTRODUCTION In many robotic applications, the principal technical challenges arise in control implementations, specialy when its subject to hight number of degree-of-freedom (DoF). Indded, the robotic systems with high DoF, can be modeled by a set of coupled highly nonlinear differential equations, with various uncertainties and disturbances, which increases the complexity of their control.

SYSTEM MODELING 2.1. Desription of system
The equation of motion of an n-DOF robot manipulators are described according to the Euler-Lagrange theory [3], as (1): τ = M (q)q + C(q,q) + G(q) where M (q), C(q,q), G(q) and G(q,q) are respectively the manipulator's mass matrix, the Coriolis and centrifugal terms vector, the gravity terms vector, and the torque friction vector. In the VDC controller, the dynamic equations of the system can be expressed as link subsystems and joint subsystems, where Virtual Power Flow (VPF) [21] is used to characterize the coupling dynamic interactions among subsystems. The Figure 1 represents the virtual decomposition of serial robot manipulator to i links , where i =

Link dynamics
The dynamic equation of the i th rigid-link subsystem following its fixed frame can be expressed as (2) [21]: where Bi F * denote the net force/moment vectors applied from the lower (i − 1) th link to the i th link expressed in frame B i . Bi V is the generalized linear/angular velocity, and M Bi , C Bi , G Bi represent respectively the inertial, Centrifugal/Coriolis, and gravitational terms, respectively. Using an iterative process computation, the vector of the total generalized force (forces/moments) acting on the i th rigid body can be computed as (3): where Bi F is the generalized force exerted by body i + 1 on body i, Bi+1 F is the generalized force exerted by bodyi + 1 on body i, Bi V r is the velocity of body i. Bi U T Bi+1 is the transformation matrix, defined as (4): where Bi R Bi+1 represents the rotation matrix from frame B i to the frame B i+1 , 0 3x3 is the null matrix, S is the skew-symmetric matrix operator performing the cross product between two vectors, and Bi r Bi+1 denotes a vector from the origin of frame B i to the frame B i+1 , expressed in frame B i .

Joint dynamics
The dynamic equation of the i th joint subsystem expressed in its fixed frame is given by the following (5) [21]: where I mi and k ci denote the moment of inertia, and the Coulomb friction coefficient of the i th joint respectively.
Finally the control torque of the global system can be expressed by (6): τ ij and τ i denote the net torque and the control torque applied to the i th joint respectively. τ il denotes the output torque of the i th joint toward the links, which can be computed in magnitude by the torque projected from the links, expressed in (7).

CONTROLLER DESIGN
The main control objective in VDC approach is to track a required trajectory such that the joint tracking error between the actual and required velocity converges asymptotically to zero in finite-time, with high accuracy even in presence of uncertainties and external disturbances. In order to design the VDC controller some vectors must be defined related to this approach.

Required vectors
The required velocity, is one of the concept related to VDC approach [21], wich can be expressed as q d and q d denote respectively the desired joint velocity and the desired joint angle, λ > 0 is a constant. The dynamic equations for the VDC controller design are based on the required joint velocity and the required linear/angular velocity vectors. The required linear/angular velocity of the link can be computed as (9): Adaptive control law of rigid-link subsystem

Adaptive control law of link subsystem
Referring to the dynamic link subsystem (2), and the required linear/angular velocity vector and its time-derivative, the required force/moment vectors are expressed as (10) [21]: Where Bi F r * the required net force/moment vectors of the subsystem links. Bi V the vector of the generalized velocities (i.e., linear and angular components), wich can be expressed as (11): Consider the linear parameterization form, the link subsystem (2) can be expressed as (12): where the Y li denotes the i th regressor matrix formed by the joint velocity, the linear/angular velocity and its time-derivative; and θ li denotes the i th parameters vector formed by the uncertainties parameter vector. Therefore, the control law of link subsystem is designed as (13): ❒ ISSN: 2089-4856 where K li is a diagonal matrix representing the gain of the feedback controller, and Bi e V is a measure of the tracking accuracy defined by (14): θ li is the estimate of the uncertainties parameter vector θ li Finally, the control law of link subsystem, can be computed by an iterative process, as (15): .

Adaptive control law of joint subsystem
For the control law of joint subsystem defined as, the required net torque τ ijr applied to the i th joint, is based on the required joint velocity vectors, as (16) [21]: According to the linear parameterization property, the required net torque τ * ir can be written in linear form as (17): where Y ji denotes the regressor matrix formed by the joint velocity and acceleration, and θ ji denotes the parameters vector formed by the physical dynamic parameters. Due to the difficulty in knowing the exact value of the physical parameters of the i th joint, they should be estimated. Then the estimation vector denoted byθ ji is used, and the equation of control becomes (18): where K ji is a diagonal matrix representing the gain of the feedback controller, and e q is a tracking joint error defined by (19): Finally, the total control torque is computed using the required output torque of the i th joint toward the links, and the required control torque of the i th joint as (20): where τ ijr denotes the control torque of the ith joint, and τ ilr the required output torque of the ith joint toward the links expressed with the required force/moment vectors as (21): The control based VDC approach is to resolve equation (20), where the vectors of parameters estimationθ li andθ ji are used. The parameter adaptation function should be chosen to ensure system stability.

Regressor-based VDC controller
In the conventionnal Regressor-based VDC controller, the uncertainties parameter vectorsθ ji andθ li for joint and link subsystems, are updated using the projection function P defined as a differentiable scalar function [21]. According to the link subsystem, the uncertainties parameter vector is estimated as (22): whereθ jiγ denotes the γth element ofθ ji , s iγ (t) denotes the γth element of s i (t) defined as (23) [21]: ρ iγ > 0 is a parameter update gain, and a iγ (t), b iγ (t) denote the lower and upper bounds of θ jiγ . The projection function P is a differentiable scalar function defined by its time derivative which is governed by (24) and (25): It is the same forθ li of the required net torque. The use of the projection function for the uncertainties parameter vectors estimation, requires to compute the derivative of the regressor matrix in every sampling cycle. However, the regressor matrix derivation is not unique, though the process is standardized. Furtheremore, it presents high complexity, then an additionnal computational burden

Proposed TDVDC controller
As demonstrated, regressor-based VDC method presents inherent limits regarding the use of the projection function to estimate the unknown parameter vectorsθ li andθ ji for link and joint subsystems respectively. To over come the issues with regressor-based VDC technique, a new control strategy combining the TDE and VDC approachs is proposed. The idea is refered to estimate the dynamic uncertainties and parameter vectors by the use of TDE. Refering to the dynamic link subsystem given in the (2), and the dynamique joint subsystem given in the (5), the dynamic uncertainties and unknown parameter vectors can be regrouped as (26) and (27): For the Joint subsystems τ ijr = I miqir + H ji (27) H li and H ji represents the dynamic uncertainties and unknown parameter vectors of the link and joint subsystems respectively, where (28) and (29): Therefore, the control law subsystems, are given by (30): m i and i i are a constant coefficients associated to M Bi , and I mi respectively. The determination of both constant coefficients m i and i i is discussed in [37], [38].Ĥ li andĤ ji represents respectively the estimate of H li and H ji .
According to the use of TDE [37], and if the assumption A2 is verified, we can estimate H li and H ji . Indeed the value of the function H li and H ji are considered at the present time t, very close to that at time (t − T ) in the past for a small time delay T in (31). For the link subsystemĤ therefore, using an iterative process, the estimate of the uncertainties parameter vector of the link substem H li (t) can be computed as: For the joint substem, the estimates of the uncertainties parameter vectorĤ ji (t) is given by (32) and (33): where T is the estimation time delay. The accuracy estimation ofĤ li (t) andĤ ji (t) improves for a small T . In practice, the smallest estimation time delay T is chosen to be the sampling period which means that the perfect parameters vector are identified every sampling period. Finally, the proposed control is obtained as (34)- (36): where and The closed-loop control system based on the proposed TDVDC technique is presented in Figure 2.

VIRTUAL STABILITY ANALYSIS
According to the virtual work approach, the global stability of the system's VDC is proven through the virtual stability of each subsystem. Indeed, using the definition 2.17 and theorem 2.1 in [21], the global system is stable in the sense of Lyapunov, if each subsystem is proved to be virtually stable. It will be proven that all the decomposed subsystems of the studied system with their respective control equations are virtually stable, leading to the stability of the entire system. Generally, the stability analysis, in the sense of the Lyapunov approach, refers to define a positive candidate function and then to show that its variation is a decreasing function. Considering the Lyapunov candidate function for the entire robot as summation of two functions for the link (V li ) and joint V ji )subsystems as (37)

Virtual stability of the ith link
Let consider the Lyapunov candidate function for the i th link as (38): Then from [21] and the dynamic equation of the i th link given in (13), the first derivative along time of V li can be given by (39):V where Bi e V T C Bi Bi e V = 0, since C Bi defined as skew-symmetric. According to the TDE use theV li becomes (40): Therefore from [21], (40)-(42), we obtain: where p B li and p T li represent the virtual power flows at the two cutting points of each link. As defined in [21], according to an open chaine structure, for p B l1 = 0 and p T ln = 0 the total virtual power flows is given by (44): Therefore the (43) becomes (45):

Virtual stability of the ith joint
The positive Lyapunov candidate function related to the joint dynamics can be chosen according to the joint dynamic and its control law, as (46): with the TDE use, and the dynamic equation of the i th joint given in (18), theV ji becomes (48): According to [21], (41), (48), and VPF definition, we obtain (49): As described in the above section, using VFP the (49) becomes (50)

Stability of the global system
The derivative of the global Lyapunov candidate function (37), is given as (51): TheV function is proved to be always decreasing based on the virtual power as the inner product of the linear angular velocity vector error and the force moment vector error presented in [21], and the choice of the parameter function adaptation, where (52): where δ j > 0 and δ l > 0 are the Lipschitz constants. SinceV < 0 where all gains are positive, the system is asymptotically stable in the sense of Lyapunov [21].

CASE STUDY 5.1. Simulation description
To illustrate the effectiveness of the proposed control strategy, in this section a case study is performed for tracking trajectory of 7-DoF robotic manipulator using the proposed TDVDC. The simulation routine is conducted following the control architecture given in Figure 2, wich includes the desired trajectory given in joint space. The desired joint velocity and the desired joint acceleration are obtained from the derivation of the desired joint position.The equation of motion for each link and joint (i) (i = 1, . . . , 7) subsystems is derived with respect to a local frame {B i } as shown in Figure 1. The mass, Coriolis, and the gravity termes of the link (i) can be described by (53)-(55): where the physical parameters of the using robot system are represented in Table 1. The numerical simulations are conducted for the proposed TDVDC controller and compared to the conventional regressor-based VDC in order to prove the effectiveness of the prposed approach. During the trajectory tracking, a disturbances was added to the torque input representing 5% of maximum value of the torque after t = 10s. In addition an uncertainty function U (t) was injected to the robot dynamic model to validate the effectiveness of the proposed control strategy in (56). U (q i , t) = q i * sin(t) + 0.5 * sin(500 * pi * t);  For the proposed TDVDC approach, the target robot is controlled following the closed-loop given in Figure 2. It concerns the use of TDE for the estimation terms defining the unknown and uncertainties parameter vectors of the robot. The required linear/angular velocity and its time derivative is computed using λ constant. The constant coefficients m i and i i are chosen according to the assumption A3. A suitable choose of these constants influence the stability and attenuation of measurement noise. These constants are conducted by the trial and error method. The time delay T is fixed as sampling time. The gain parameters of the feedback controller K j and K l for joint and link subsystems respectivelyare are fixed ensuring the stability condition. These parameters values must be adjusted in order to obtain the optimum performance. For the conventional regressor-based VDC approach, the parameters estimation is based on projection function presented in (22) which requires the derivation of the regressor matrix in every sampling time, as discussed previously. To accomplish the simulation routine, in addition to the gains feedback controller K j , K l and λ, the parameters ρ, a, b are used for the projection function.

Simulation results
The obtained simulation results of the tracking trajectory and the traking errors for the proposed TDVDC and the conventional Regressor-based VDC strategies are shown in Figure 3, Figure 4, and Figure  5 respectively It is shown that the system response converges for the conventional regressor-based VDC and the proposed TDVDC approaches, with insensitivity to uncertainties and disturbances. However, for the proposed TDVDC the signal response present some delay at the begining before t = 0.2s. The obtained joint torque for the TDVDC and Regressor-based VDC are given respectively in Figure 6 and Figure 7. It should be noted that the feedback gains should carefully be selected to get an accurate tracking.

Discussion
The goal of the proposed controller is to track the desired trajectory rather than convergence of physically uncertain parameters. For the two numerical simulations conducted using the proposed TDVDC and the conventionnal regressor-based VDC, the robot can track the desired references but with relatively a tracking error joint difference. In order to carry out an overall quantitative evaluation of the used controllers, the following evaluation criteria are adopted: Root mean square (RMS) of the tracking position error, defined by (57): where e i (t) = q ci − q di Computing time (CT) defined as a period from the start to the end moment of run simulation. Maximum torque (MT) defined as the largest torque input to the joint actuators. We can conclude that the proposed TDVDC presents the low tracking error compared to the conventionnal Regressor-based VDC. For the generated torque, the proposed controller present the low MT for the second joint q 2 , comparing to the regressor-based VDC for the same joint. In the otherhand, the CPU time with the proposed VDTDC approach is equals to 2.486009(s) and for the conventionnal VDC is equals to 4.721131(s), using MATLAB program. Therefore, in addition to the gain in terms of error the proposed VDTDC approach presents gain in terms of generated torque and also the computation time.

CONCLUSION
To sum up, in this paper, a new aproach has been developed of nonlinear control of n-DoF robot manipulator. The proposed controller is a modified VDC introducing the TDE technique to estimate unknown and uncertainties vector parameters.The system modeling, and the VDTDC controller design were presented. Hence, the stability of overall system was proved based on the Lyapunov theory. Simulation results performed with 7-DoF robot manipulator for the proposed TDVDC controller, shows the capability of the controller for ISSN: 2089-4856 ❒ 205 tracking the predefined trajectory. Comparatively to the conventionnel regressor-based VDC, the TDVDC presents the lower tracking error, with the lower value of maximum torque, and finally lower computation time, wich proves the effectiveness of the propsal. As futur work, the proposed controller were be tested experimentally to another kind of application.