Hayder F N Al-Shuka
email: hayder.al-shuka@rwth-aachen.de

R Song

Decentralized adaptive partitioned approximation control of high degreesof-freedom robotic manipulators considering three actuator control modes *

Keywords: Adaptive Approximation Control, Orthogonal Basis Functions, Actuator Dynamics, Electrically Driven Robots

Partitioned approximation control is avoided in most decentralized control algorithms; however, it is essential to design a feedforward control term for improving the tracking accuracy of the desired references. In addition, consideration of actuator dynamics is important for a robot with high-velocity movement and highly varying loads. As a result, this work is focused on decentralized adaptive partitioned approximation control for complex robotic systems using the orthogonal basis functions as strong approximators. In essence, the partitioned approximation technique is intrinsically decentralized with some modifications. Three actuator control modes are considered in this study: (i) a torque control mode in which the armature current is well controlled by a current servo amplifier and the motor torque/current constant is known, (ii) a current control mode in which the torque/current constant is unknown, and (iii) a voltage control mode with no current servo control being available. The proposed decentralized control law consists of three terms: the partitioned approximation-based feedforward term that is necessary for precise tracking, the high gain-based feedback term, and the adaptive sliding gain-based term for compensation of modeling error. The passivity property is essential to prove the stability of local stability of the individual subsystem with guaranteed global stability. Two case studies are used to prove the validity of the proposed controller: a two-link manipulator and a six-link biped robot.

Introduction

For complex robotic systems, such as humanoid robots or any robot having a number of degrees of freedom (DOFs) larger than 6-DOFs, difficulties are encountered in the implementation of the control algorithms. Therefore, over 30 years, the robotics researchers have focused on the problem of computational efficiency. Many efficient O (n) algorithms have been developed for inverse [START_REF] Luh | On-line computational scheme for mechanical manipulator[END_REF][START_REF] Hollerbach | A recursive Lagrangian formulation of manipulator dynamics and comparative study of dynamics formulation complexity[END_REF][START_REF] Walker | Efficient dynamic computer simulation of robotic mechanisms[END_REF][START_REF] Balafoutis | Efficient computation of manipulator, inertia matrices and the direct dynamics problem[END_REF][START_REF] Ma | An algorithm for the inverse dynamics of n-axis general manipulator using Kane's formulation of dynamical equations[END_REF][START_REF] Saha | Dynamics of serial multibody systems using the decoupled natural orthogonal complement matrices[END_REF] and forward dynamics [START_REF] Featherstone | The calculation of robot dynamics using articulated-body inertias[END_REF][START_REF] Brandle | A very efficient algorithm for the simulation of robots and similar multibody systems without inversion of the mass matrix[END_REF][START_REF] Mohan | A recursive numerically stable and efficient simulation algorithm for serial robots[END_REF][START_REF] Lee | A new perspective on O (n) mass-matrix inversion for serial revolute manipulators[END_REF] of robotic systems. For more details on the efficient dynamic algorithms, we refer to [START_REF] Featherstone | Robot dynamics: Equations and algorithms[END_REF][START_REF] Saha | Recursive dynamics algorithms for serial, parallel and closed-chain multibody systems[END_REF].

However, the adaptive control algorithm that deals with controlling the robotic systems despite their uncertain parameters may decrease the computational efficiency of the dynamics O (n) algorithms. K. S. Fu et al. [13] have shown that the combined identification and control algorithms can be computed in O (n 3 ) time despite using recursive Newton-Euler (NE) formulation. Therefore, most researchers have used a local controller (decentralized controller) such as PID family for controlling complex dynamic systems [START_REF] Sandell | Survey of decentralized control methods for large scale systems[END_REF][START_REF] Spong | Robot dynamics and control[END_REF][START_REF] Boha | Decentralized control of redundant parallel robot construction[END_REF][START_REF] Ohri | Tracking control of robots using decentralized robust PID control for friction and uncertainty compensation[END_REF][START_REF] Xu | Integrating centralized and decentralized approaches for multi-robot coordination[END_REF][START_REF] Leena | A set of decentralized PID controller for an nlink robot manipulator[END_REF]. However, Liu [START_REF] Liu | Decentralized control of robot manipulators: nonlinear and adaptive approaches[END_REF] proved that decentralized PD control for robotic manipulator cannot ensure global stability. Therefore, the author proposed decentralized PD + nonlinear cubic term for ensuring global stability. A modified version of Liu's work has been proposed by Hsu and Fu. Yang et al. [START_REF] Yang | Decentralized adaptive robust control of robot manipulators using disturbance observers[END_REF] have used a disturbance observer and an adaptive sliding mode control strategy for compensation of low-pass and high-pass coupled uncertainties respectively. In effect, most available Apart from conventional robust control, the adaptive approximation control is a powerful tool to control robotic manipulator with unknown dynamics. There are two possible categories of adaptive approximation techniques: lumped approximation-based adaptive control [START_REF] Lewis | Neural net robot controller: structure and stability proofs[END_REF][START_REF] Lewis | Neural net robot controller with guaranteed tracking performance[END_REF][START_REF] Lewis | Multilayer neural net robot controller with guaranteed tracking performance[END_REF][START_REF] Liu | Radial basis function (RBF) neural network control for mechanical systems: design, analysis and Matlab simulation[END_REF][START_REF] Hfn | Adaptive hybrid regressor and approximation control of robotic manipulators in constrained space[END_REF]61,[START_REF] Hfn | Hybrid regressor and approximationbased adaptive control of piezoelectric flexible beams[END_REF] and partitioned approximation-based adaptive control [START_REF] Huang | Adaptive control of robot manipulators: a unified regressor-free approach[END_REF][START_REF] Ge | Adaptive neural network control of robotic manipulators[END_REF][START_REF] Hfn | On local approximation-based adaptive control with applications to robotic manipulators and biped robots[END_REF][START_REF] Hfn | Hybrid regressor and approximationbased adaptive control of robotic manipulators with contact-free motion[END_REF]. The former deals with collecting the uncertainty in one term and using the linear-in-the-parameters property to represent the uncertainty in terms of weighting and basis-function matrices. Then designing control law based on updating the weighting-coefficient matrix using Lyapunov stability. This strategy requires nominal (ideal) estimation for the unknown parameters. On the other hand, partitioned approximationbased adaptive control approximates each dynamic matrix of the equation of motion of the target robot separately using the linear-in-the-parameters property. The control law is then designed based on updating the weighting-coefficient matrices of the corresponding basis-function matrices using Lyapunov stability. To our knowledge, the partitioned approximation does not require nominal estimation for the dynamic coefficients that eases the control task. However, most decentralized approximation control [START_REF] Liu | Decentralized control of robot manipulators: nonlinear and adaptive approaches[END_REF][START_REF] Yang | Decentralized adaptive robust control of robot manipulators using disturbance observers[END_REF][START_REF] Jasim | Enhanced decentralized robust adaptive control of robots with arbitrarily-switched unknown constraints[END_REF][START_REF] Panagi | Decentralized adaptive approximation based control of a class of large-scale systems[END_REF][START_REF] Tan | Decentralized adaptive controller design of large-scale uncertain robotic systems[END_REF][START_REF] Fateh | Decentralized direct adaptive fuzzy control of robots using voltage control strategy[END_REF][START_REF] Li | Decentralized adaptive neural control of nonlinear interconnected large-scale systems with unknown time delays and input saturation[END_REF][START_REF] Yt | Decentralized adaptive fuzzy backstepping control of rigid-link electrically driven robots[END_REF][START_REF] Zhu | Decentralized adaptive neural network control for reconfigurable manipulators[END_REF][START_REF] Zhou | Decentralized adaptive fuzzy tracking control for robot finger dynamics[END_REF] have used lumped approximation term combined with either PD term or nominal values. Banagi and Polycarpou [START_REF] Panagi | Decentralized adaptive approximation based control of a class of large-scale systems[END_REF] used the decentralized neural network (NN) control for adaptive control of the large-scale system. The unknown dynamics and the unknown interconnections are approximated by using linear parameterized NN assuming that the interconnection term is bounded by summation of analytic functions in terms of error. The dynamics are formulated in state space model. Tan et al. [START_REF] Tan | Decentralized adaptive controller design of large-scale uncertain robotic systems[END_REF] used decentralized NN adaptive control for robotic manipulators. It is a model-free control strategy that is based on NN control, but the authors did not consider the actuator dynamics in their proposed strategy.

Consideration of actuator dynamics is of importance for a robot with high-velocity movement and highly varying loads [START_REF] Huang | Adaptive control of robot manipulators: a unified regressor-free approach[END_REF]. Fateh and Fateh [START_REF] Fateh | Decentralized direct adaptive fuzzy control of robots using voltage control strategy[END_REF] used decentralized fuzzy voltage control for robotic manipulators. The lumped uncertainty of interconnections at voltage level is approximated using fuzzy compensator. The authors considered the actuator dynamics using the voltage control mode. Therefore, Huang and Chen [START_REF] Huang | Adaptive control of robot manipulators: a unified regressor-free approach[END_REF] investigated the effect of actuator dynamics on the performance of the actuator dynamics assuming that the motor torque/current constant is known; however, their control strategy was fully centralized. Zhu [START_REF] Zhu | Virtual decomposition control, toward hyper degrees of freedom robots[END_REF] proposed three motor control modes using the distributed virtual decomposition control. According to his strategy, an electric motor can be in the motor torque control mode when the armature current is well controlled by a current servo amplifier and the motor torque/current constant is known. Otherwise, an electric motor should be in the motor current control mode when only the armature current is well controlled but the torque/current constant is unknown. Finally, an electric motor must be in the motor voltage control mode when no current servo control is available. For more details on actuators dynamics, the reader is referred to [START_REF] Kelly | Control of robot manipulators in joint space[END_REF]. In addition, for more work on decentralized approximation control techniques, see [START_REF] Li | Decentralized adaptive neural control of nonlinear interconnected large-scale systems with unknown time delays and input saturation[END_REF][START_REF] Yt | Decentralized adaptive fuzzy backstepping control of rigid-link electrically driven robots[END_REF][START_REF] Zhu | Decentralized adaptive neural network control for reconfigurable manipulators[END_REF][START_REF] Zhou | Decentralized adaptive fuzzy tracking control for robot finger dynamics[END_REF][START_REF] Vá Zquez | Decentralized identification and control in real-time of a robot manipulator via recurrent wavelet first-order neural network[END_REF].

In view of above, the partitioned approximation control is avoided in most decentralized control algorithms. However, it can be a strong solution to control high DOF robots since it is decentralized in nature but with some modifications. In this study, two important issues are considered: i) design of decentralized control law exploiting the feature of partitioned approximation technique, and ii) consideration of three actuator control modes: motor torque mode, motor current mode, and motor voltage mode. The proposed control law consists of three terms: partitioned approximation-based feedforward term, high-gain-based feedback term, and adaptive sliding mode term for compensating for modeling errors. The second issue coincides with virtual decomposition control proposed by Zhu [START_REF] Zhu | Virtual decomposition control, toward hyper degrees of freedom robots[END_REF]. These control modes are applied in sequence for adaptive control purposes without considering prior information of robot dynamics parameters, friction, actuator dynamics parameters, etc. In effect, this work is an extended version of the conference paper [START_REF] Hfn | Hybrid regressor and approximationbased adaptive control of robotic manipulators with contact-free motion[END_REF] in which the actuator dynamics are not considered and the focus was on torque control mode only.

The remainder of the paper is organized as follows. Section 2 introduces preliminaries and motivation for the problem. The methodology is presented in Section 3. Section 4 described simulation results and discussions. Section 5 concludes.

Preliminaries and Motivation

Preliminaries

As stated previously, the function approximation technique (FAT) is an essential tool for approximating the unknown parameters of a dynamic system for adaptive control purposes. In general, there are two essential adaptive approximation techniques: lumped approximation technique (global approximation) [START_REF] Lewis | Neural net robot controller: structure and stability proofs[END_REF][START_REF] Lewis | Neural net robot controller with guaranteed tracking performance[END_REF][START_REF] Lewis | Multilayer neural net robot controller with guaranteed tracking performance[END_REF][START_REF] Liu | Radial basis function (RBF) neural network control for mechanical systems: design, analysis and Matlab simulation[END_REF][START_REF] Hfn | Adaptive hybrid regressor and approximation control of robotic manipulators in constrained space[END_REF]61,[START_REF] Hfn | Hybrid regressor and approximationbased adaptive control of piezoelectric flexible beams[END_REF], and partitioned approximation (local approximation) techniques [START_REF] Huang | Adaptive control of robot manipulators: a unified regressor-free approach[END_REF][START_REF] Ge | Adaptive neural network control of robotic manipulators[END_REF][START_REF] Hfn | On local approximation-based adaptive control with applications to robotic manipulators and biped robots[END_REF][START_REF] Hfn | Hybrid regressor and approximationbased adaptive control of robotic manipulators with contact-free motion[END_REF]. The former approximation technique collects all uncertain parameters in one lumped vector term. The lumped vector can then be represented as linear combinations of basis functions using miscellaneous approximators (splines, orthogonal functions, intelligent techniques etc.). Its main drawback is the need for nominal (simplified) model. On the other hand, the partitioned approximation technique attempts to approximate (linearly parameterize) each dynamic matrix and vector (mass and Coriolis matrices, gravity vector, friction vector etc.) separately. In general, the procedure employed in constructing the FAT-based adaptive control includes a selection of a suitable approximator for the uncertain parameters, choosing an appropriate adaptation law for the weighting coefficient and designing the controller structure [START_REF] Hfn | On local approximation-based adaptive control with applications to robotic manipulators and biped robots[END_REF][START_REF] Hfn | Adaptive hybrid regressor and approximation control of robotic manipulators in constrained space[END_REF].

There are different classes of approximators such as polynomials, splines, radial basis functions, Cerebellar Model Articulation Controller (CMAC), multilayer perceptron, fuzzy approximation and wavelets. In this paper, we will focus on orthogonal functions due to their simplicity and capability of achieving minimum approximation errors [START_REF] Huang | Adaptive control of robot manipulators: a unified regressor-free approach[END_REF][START_REF] Ge | Adaptive neural network control of robotic manipulators[END_REF][START_REF] Hfn | On local approximation-based adaptive control with applications to robotic manipulators and biped robots[END_REF][START_REF] Hfn | Adaptive hybrid regressor and approximation control of robotic manipulators in constrained space[END_REF].

Theorem 1 (Weierstrass Approximation Theorem). [START_REF] Panagi | Decentralized adaptive approximation based control of a class of large-scale systems[END_REF] Let 𝑓(𝑥) be a continuous function on [a, b], then for all 𝜖 > 0, there exists a sequence of polynomials 𝑝 𝛽 (𝑥) of degree < 𝛽 that converges uniformly to 𝑓(𝑥) on [a, b], i.e. ‖𝑓(𝑥) -𝑝 𝛽 (𝑥)‖ < 𝜖 where

𝑝 𝛽 (𝑥) = ∑ 𝑤 𝑗 𝛽 𝑗=1 𝜙 𝑗 (𝑥) (1)
where 𝑤 𝑗 and 𝜙 𝑗 are the weighting coefficients and basis functions respectively.

This theorem indicates that each continuous function on a compact interval can be approximated by polynomials with any degree. The scope of this paper focuses on orthogonal basis functions.

Definition 1. [START_REF] Huang | Adaptive control of robot manipulators: a unified regressor-free approach[END_REF] For any set of orthonormal functions {𝜙 𝑗 (𝑥)}, (∀ 𝑗 = 1, 2, … , 𝛽) on [a, b], an arbitrary function 𝑓(𝑥) can be approximated using Eq. (1) under conditions of Theorem 1 with the following weighting coefficients being calculated as

𝑤 𝑗 = ∫ 𝑓(𝑥) 𝑏 𝑎 𝜙 𝑗 (𝑥)𝑑𝑥 ∫ 𝜙 𝑗 2 (𝑥) 𝑏 𝑎 (2)
The following important points should be noted [START_REF] Huang | Adaptive control of robot manipulators: a unified regressor-free approach[END_REF][START_REF] Burden | Numerical analysis[END_REF]: 2) is computed using the orthogonal property

 Equation (
∫ 𝑓(𝑥) 𝑏 𝑎 𝜙 𝑖 𝜙 𝑗 𝑑𝑥 { = 0 𝑖 ≠ 𝑗 ≠ 0 𝑖 = 𝑗 
To guarantee convergence of the approximating series, the orthogonal set should be complete.



There are many orthogonal basis functions: Chebyshev polynomials, Legendre polynomials, Laguerre polynomials, etc.



The Chebyshev approximation formula is very close to the minimax polynomial since the approximation error is spread smoothly over the period [-1, 1]. Accordingly, this orthogonal approximation polynomials will be used as strong approximators in the current study.

Motivation

To motivate the core of this paper, let us consider the following equation of motion of 𝑛-joint robotic manipulators in free space

𝑴(𝒒)𝒒̈+ 𝑪(𝒒, 𝒒̇)𝒒̇+ 𝒈(𝒒) = 𝝉 𝑙 (3)
where 𝑴 ∈ ℝ 𝑛×𝑛 is the inertia matrix of the links, 𝑪 ∈ ℝ 𝑛×𝑛 is the Coriolis and centripetal matrix of the links, 𝒈 ∈ ℝ 𝑛×1 is the gravity vector, 𝒒 ∈ ℝ 𝑛×1 represents the displacement output joint variables after gear transmission, and 𝝉 𝑙 ∈ ℝ 𝑛×1 denotes to the output joint torques.

However, the detailed analysis of robotic manipulators dynamics considering actuator dynamics will be introduced in the next section.

The following properties assumptions are necessary for adaptive approximation control (centralized and decentralized version) [START_REF] Ge | Adaptive neural network control of robotic manipulators[END_REF][START_REF] Mulero-Martí Nez | Uniform bounds of the Coriolis/centripetal matrix of serial robot manipulators[END_REF].

Property 1. The inertia matrix, Coriolis and centrifugal matrix, and the gravity vectors are uniformly bounded.

The last property can be extended to include that the dynamic coefficients of the elements of each matrix/vector are uniformly bounded and satisfying theorem 1.

Property 2 (Passivity property).

The matrix 𝑵 = 𝑴 ̇-2𝑪 is a skew-symmetric matrix, i.e., 𝒔 𝑇 𝑵𝒔 = 0 , if 𝑪(𝒒, 𝒒̇) is defined using the Christoffel symbols.

According to the property 2, each diagonal element of 𝑵, with 𝑘, 𝑗 ∈ 𝑛, is equal to zero. This property is essential to prove the stability of the proposed decentralized control law.

Assumption 1. The state variables of the target robots are measurable and bounded.

Assumption 2. Each entry of dynamic matrices of 𝑴(𝒒), 𝑪(𝒒, 𝒒̇), and 𝒈(𝒒) satisfies conditions of Theorem 1, i.e., they can be represented as linear combinations of orthogonal basis functions.

The last assumption includes that each entry of the dynamic matrices can be bounded by analytic and continuous functions. Please for more details on properties of the dynamic matrices, the gravity vector and the residual dynamics, the reader is referred to [START_REF] Ge | Adaptive neural network control of robotic manipulators[END_REF][START_REF] Mulero-Martí Nez | Uniform bounds of the Coriolis/centripetal matrix of serial robot manipulators[END_REF][START_REF] Kelly | Control of robot manipulators in joint space[END_REF].

In view of the partitioned approximation technique introduced in [27-29], Eq. ( 3) can be reformulated as

𝑾 𝑀 𝑇 ∅ 𝑀 𝒒̈+ 𝑾 𝐶 𝑇 ∅ 𝐶 𝒒̇+ 𝑾 𝑔 𝑇 𝝋 𝑔 + 𝝐 = 𝝉 𝑙 (4)
where 𝑾 𝑀 ∈ ℝ 𝑛𝛽×𝑛 , 𝑾 𝐶 ∈ ℝ 𝑛𝛽×𝑛 and 𝑾 𝑔 ∈ ℝ 𝑛𝛽×𝑛 are the weighting matrices, while ∅ 𝑀 ∈ ℝ 𝑛𝛽×𝑛 , ∅ 𝐶 ∈ ℝ 𝑛𝛽×𝑛 , and 𝝋 𝑔 ∈ ℝ 𝑛𝛽 are the basis-function matrices. 𝝐 ∈ ℝ 𝑛 is the accumulated modeling error vector. For a 2-DOF robot, Eq. ( 4) can be written as

[ 𝒘 𝑚11 𝑇 𝝋 𝑚11 𝒘 𝑚12 𝑇 𝝋 𝑚12 𝒘 𝑚21 𝑇 𝝋 𝑚21 𝒘 𝑚22 𝑇 𝝋 𝑚22 ] 𝒒̈+ [ 𝒘 𝑐11 𝑇 𝝋 𝑐11 𝒘 𝑐12 𝑇 𝝋 𝑐12 𝒘 𝑐21 𝑇 𝝋 𝑐21 𝒘 𝑐22 𝑇 𝝋 𝑐22 ] 𝒒̇+ [ 𝒘 𝑔1 𝑇 𝝋 𝑔1 𝒘 𝑔2 𝑇 𝝋 𝑔2 ] = 𝝉 𝑙 (5)
According to Eq. ( 5), the followings are noted:  The decoupled dynamics of each DOF can be represented as

𝒘 𝑚11 𝑇 𝝋 𝑚11 𝑞̈1 + 𝒘 𝑐11 𝑇 𝝋 𝑐11 𝑞̇1 + 𝒘 𝑚12 𝑇 𝝋 𝑚12 𝑞̈2 + 𝒘 𝑐12 𝑇 𝝋 𝑐12 𝑞̇2 + 𝒘 𝑔1 𝑇 𝝋 𝑔1 ⏟ 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 𝑡𝑜𝑟𝑞𝑢𝑒=𝜏 𝑐1 = 𝜏 𝑙1 𝒘 𝑚22 𝑇 𝝋 𝑚22 𝑞̈2 + 𝒘 𝑐22 𝑇 𝝋 𝑐22 𝑞̇2 + 𝒘 𝑚21 𝑇 𝝋 𝑚21 𝑞̈1 + 𝒘 𝑐21 𝑇 𝝋 𝑐21 𝑞̇1 + 𝒘 𝑔2 𝑇 𝝋 𝑔2 ⏟ 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 𝑡𝑜𝑟𝑞𝑢𝑒=𝜏 𝑐2 = 𝜏 𝑙2 (6) 
The whole dynamics of the target robot is intrinsically decoupled by using the partitioned approximation technique, i.e.

These equations can recursively be solved; however, this technique results in a number of equations that are equal to 𝑛 2 which can be computationally tedious.



The partitioned approximation technique has flexibility in decentralization, e.g., a 6-DOF robot can be decoupled into two 3-DOF subsystems and so on.

Methodology

Dynamics of contact-free motion robots

The dynamics of electrically driven 𝑛-joint robot consist of three subsystems: the link dynamics, the joint dynamics, and armature controlled DC motor dynamics (Fig. 1). The classical Euler-Lagrangian formulation of the whole system dynamics can be expressed as

𝑴𝒒̈+ 𝑪𝒒̇+ 𝒈 = 𝝉 𝑙 (8a) 𝑰 𝑚 𝒒̈𝑚 + 𝝉 ′ 𝑓 = 𝝉 𝑚 ′ -𝑮𝝉 𝑙 (8b) 𝑳𝜾̇+ 𝑹𝜾 + 𝑩 ′ 𝒒̇𝑚 = 𝒖 (8c)
where 𝒒 𝑚 ∈ ℝ 𝑛 is the angular motor displacement before the gear transmission and equal to 𝑮 -1 𝒒 with 𝑮 ∈ ℝ 𝑛×𝑛 referring to the diagonal gear-ratio matrix, 𝑰 𝑚 ∈ ℝ 𝑛×𝑛 is a diagonal effective inertia matrix of the actuator, 𝝉 ′ 𝑓 is the joint friction torques, and 𝝉 𝑚 ′ is the motor torque vector which is equal to 𝑲 𝜏 𝜾 , with 𝑲 𝜏 ∈ ℝ 𝑛×𝑛 denoting to the torque constant, and 𝜾 ∈ ℝ 𝑛 referring to the armature current, 𝑳 ∈ ℝ 𝑛×𝑛 is a diagonal inductance matrix, 𝑹 ∈ ℝ 𝑛×𝑛 is a diagonal resistance matrix, 𝑩 ′ ∈ ℝ 𝑛×𝑛 is a diagonal matrix representing the EMF constant, and 𝒖 ∈ ℝ 𝑛 is the input voltage control. Substituting Eqs. (8b) into Eq. (8a) leads to the simplified full dynamics of robotic manipulators

𝑫𝒒̈+ 𝑪𝒒̇+ 𝒈 + 𝝉 𝑓 = 𝝉 𝑚 (9a)
𝝉 𝑚 = 𝑯𝜾 (9b)

𝑳𝜾̇+ 𝑹𝜾 + 𝑩𝒒̇= 𝒖 (9c) with 𝑫 = 𝑴 + 𝑮 -2 𝑰 𝑚 , 𝝉 𝑓 = 𝑮 -1 𝝉 ′ 𝑓 , 𝑯 = 𝑮 -1 𝑲 𝜏 , 𝑩 = 𝑩 ′ 𝑮 -1
Remark 1. Equation ( 9) neglects the coupling effects between rotor and link motion [START_REF] Sciavicco | Lagrange and Newton-Euler dynamic modeling of a gear-driven robot manipulator with inclusion of motor inertia effects[END_REF]. This assumption has been made for electrically driven robots provided with high gear ratio. Due to the high gear ration, the motor rotor moves faster along its axis than other directions and hence its motion is a pure rotation with respect to an inertial frame [START_REF] Zhu | Virtual decomposition control, toward hyper degrees of freedom robots[END_REF].

Remark 2. It is well-known that the electric time constant is smaller than the mechanical time constant leading to neglecting the inductance matrix. This assumption leads to simplified robot dynamics with reduced actuator dynamics. In other words, the full dynamics of robots can be simplified to one equation, please see chapter 7 of [START_REF] Spong | Robot dynamics and control[END_REF] for more details.

Decomposition approach

Based on Eq. ( 9), the 𝑘-th subsystem of the target robot, in which 𝑘 = 1, 2, … , 𝑛, can be expressed as

Equation ( 12) represents the standard dynamic representation that will be adopted to solve the problem of decoupled control.

Assumption 3. The interconnection/disturbance term ∆ 𝑘 (𝑞, 𝑞̇) is bounded with analytic and continuous functions satisfying theorem 1.

Remark 3. The interconnection/disturbance term is assumed bounded with smooth analytic functions, e.g., see [START_REF] Panagi | Decentralized adaptive approximation based control of a class of large-scale systems[END_REF][START_REF] Tan | Decentralized adaptive controller design of large-scale uncertain robotic systems[END_REF]. Accordingly, this assumption is enough to decompose this term as a linear combination of basis functions without using the inverse analysis proposed in most work.

Let us recall the partitioned approximation technique described in details in [START_REF] Hfn | On local approximation-based adaptive control with applications to robotic manipulators and biped robots[END_REF]. Accordingly, the approximationbased dynamics can be expressed as

[ 𝒘 𝐷11 𝑇 𝝋 𝐷11 ⋯ 𝒘 𝐷1𝑛 𝑇 𝝋 𝐷1𝑛 ⋮ ⋱ ⋮ 𝒘 𝐷𝑛1 𝑇 𝝋 𝐷𝑛1 ⋯ 𝒘 𝐷𝑛𝑛 𝑇 𝝋 𝐷𝑛𝑛 ] 𝒒̈+ [ 𝒘 𝐶11 𝑇 𝝋 𝐶11 ⋯ 𝒘 𝐶1𝑛 𝑇 𝝋 𝐶1𝑛 ⋮ ⋱ ⋮ 𝒘 𝐶𝑛1 𝑇 𝝋 𝐶𝑛1 ⋯ 𝒘 𝐶𝑛𝑛 𝑇 𝝋 𝐶𝑛𝑛 ] 𝒒̇+ [ 𝒘 ∆1 𝑇 𝝋 ∆1 ⋮ 𝒘 ∆𝑛 𝑇 𝝋 ∆𝑛 ] + 𝝐 = 𝝉 (13)
where 𝒘 (.) ∈ ℝ 𝛽 , 𝝋 (.) ∈ ℝ 𝛽 represent the weightingcoefficient and basis-function vectors respectively, and 𝝐 ∈ ℝ 𝑛 denotes to the accumulated modeling error vector.

According to Eq. ( 13), the partitioned approximation technique has inherently decentralization features that can be exploited for decentralized control. In view of properties 1,2, assumptions 1,2, and Eq. (13), Eq. (12a) can be reformulated as with 𝜖 (.) being denoted to the corresponding modeling error. Accordingly, the dynamics of the coupled robotic system is transformed to decoupled subsystems based on linear combinations of basis function with constant weighting coefficients that should be updated using the Lyapunov's theory; the details are next.

𝒘

Controller design

The objective of the proposed controller is to control the kth subsystem individually considering the possibly accompanied uncertainties. Accordingly, let us consider the case that 𝑑 𝑘𝑘 , 𝑐 𝑘𝑘 , ∆ 𝑘 , ℎ 𝑘𝑘 , 𝐿 𝑘𝑘 , 𝑅 𝑘𝑘 , 𝐵 𝑘𝑘 are unknowns. In addition, consideration of actuator dynamics is important for a robot with high-velocity movement and highly varying loads. Three possible actuator control modes are possible for modeling and control of the motor [START_REF] Zhu | Virtual decomposition control, toward hyper degrees of freedom robots[END_REF]: (i) the torque control mode in which the armature current is well controlled by a current servo amplifier and the motor torque/current constant is known, (ii) current control mode in which only the armature current is well controlled but the torque/current constant is unknown, and (iii) voltage control mode in which no current servo control is available. The solution steps can be summarized in Fig. 2. Assumption 2. For controller design, it is assumed that there is no servo current amplifier and the current-to-torque constant is unknown. Accordingly, the mentioned three control modes will be introduced in sequence.

Motor torque control mode

The aim of this mode is to generate the desired output torque with guaranteed precise tracking for the desired angular joint displacements. The control law for this mode (see Eq. (12a)) can be designed as where 𝐾 𝑘 and 𝛬 𝑘 are positive feedback gains, Γ ̂𝑘(𝑡) denotes to the adaptive sliding gain that should be updated in order to avoid the demand of modeling error bounds. Please for more details on adaptive sliding mode control see [START_REF] Utkin | Adaptive sliding mode control[END_REF], and

𝜏 𝑑𝑘 =
𝑠𝑔𝑛(𝑠) = |𝑠| 𝑠 (16)
The mathematical relationship of Eq. ( 16) is very useful in the stability proof discussed later. Substituting Eq. ( 15) into Eq. ( 14) leads to the output position closed loop dynamics

𝑑
From Eq. ( 17), the closed-loop control system is stable if 𝑤 ̃(.) → 0 , 𝜏 𝑘𝑑 -𝜏 𝑘 → 0 , 𝜖 𝑘 → 0 , and Γ ̂𝑘(𝑡) → 𝛿 𝑘 ≥ |𝜖 𝑘 | . This requires finding suitable update laws for the weighting coefficients, and the adaptive sliding gain. Let us select the following updating adaptive laws for the weighting vectors and adaptive sliding gain. where 𝑸 (.) ∈ ℝ 𝛽×𝛽 is a positive-definite adaptation matrix, and 𝜌 𝑘 is a positive adaptation gain.

𝒘 ̂̇𝐷𝑘 = -𝑸

Theorem 2. The k-th subsystem of the robotic manipulator (𝑘 = 1,2, … , 𝑛) described by Eq. (12a) combined with the desired applied torque control law of Eq. ( 15), and with the adaptation laws of the weighting vectors, and the adaptive sliding gain of Eq. ( 18), subject to

𝜏 𝑑𝑘 = 𝜏 𝑘 (19)
is stable in the sense of Lyapunov stability [START_REF] Spong | Robot dynamics and control[END_REF][START_REF] Utkin | Adaptive sliding mode control[END_REF][START_REF] Ioannou | Adaptive control tutorial[END_REF].

Proof.

Consider the following Lyapunov-like function candidate (𝑉 𝜏 ) along the trajectory of Eq. ( 17)

𝑉 𝜏𝑘 = 1 2 𝑑 𝑘𝑘 𝑠 𝑘 2 + 1 2 𝒘 ̃𝐷𝑘 𝑇 𝑸 𝐷𝑘 -1 𝒘 ̃𝐷𝑘 𝑇 + 1 2 𝒘 ̃𝐶𝑘 𝑇 𝑸 𝐶𝑘 -1 𝒘 ̃𝐶𝑘 𝑇 + 1 2 𝒘 ̃∆𝑘 𝑇 𝑸 ∆𝑘 -1 𝒘 ̃∆𝑘 𝑇 + 1 2 𝜌 𝑘 Γ ̃𝑘2 (20)
Taking the derivative of the last equation leads to

𝑉 ̇𝜏𝑘 = 𝑑 𝑘𝑘 𝑠 𝑘 𝑠̇𝑘 + 1 2 𝑑 ̇𝑘𝑘 𝑠 𝑘 2 -𝒘 ̃𝐷𝑘 𝑇 𝑸 𝐷𝑘 -1 𝒘 ̂̇𝐷𝑘 -𝒘 ̃𝐶𝑘 𝑇 𝑸 𝐶𝑘 -1 𝒘 ̂̇𝐶𝑘 - 𝒘 ̃Δ𝑘 𝑇 𝑸 Δ𝑘 -1 𝒘 ̂̇Δ𝑘 -𝜌 𝑘 Γ ̃𝑘(𝑡)Γ ̂k (𝑡) (21)
By substituting Eq. ( 17) into Eq. ( 21), using the adaptation laws of Eq. ( 18), and applying the passivity property 2 lead to

𝑉 ̇𝜏𝑘 = -𝐾 𝑘 𝑠 𝑘 2 + 𝑠 𝑘 𝜖 𝑘 -𝑠 𝑘 Γ ̂𝑘(𝑡)𝑠𝑔𝑛(𝑠 𝑘 ) -(𝛿 𝑘 - Γ ̂𝑘(𝑡)) |𝑠 𝑘 | -𝑠 𝑘 (𝜏 𝑑𝑘 -𝜏 𝑘 ) (22)
Substituting Eq. ( 19) into Eq. ( 22) and using Eq. ( 16) yield

𝑉 ̇𝜏𝑘 = -𝐾 𝑘 𝑠 𝑘 2 + 𝑠 𝑘 𝜖 𝑘 -𝛿 𝑘 |𝑠 𝑘 | < 0 (23)
Both 𝑠 𝑘 and Γ ̃𝑘(𝑡) reach zero in finite time resulting in position error approaching to zero provided that 𝛿 𝑘 ≥ |𝜖 𝑘 |.

Motor current control mode

As aforementioned, the motor current control mode is necessary if the motor current can be controlled directly, and the current-to-torque constant (ℎ 𝑘𝑘 ) is not available. Accordingly, the desired torque associated with the unknown ℎ 𝑘𝑘 can be expressed as

𝜏 𝑑𝑘 = ℎ ̂𝑘𝑘 𝜄 𝑑𝑘 (24)
The updating adaptive law for unknown ℎ ̂𝑘𝑘 can be designed as

ℎ ̂̇𝑘𝑘 = 𝑄 ℎ𝑘 𝜄 𝑑𝑘 𝑠 𝑘 ( 25
)
Theorem 3. The k-th subsystem of the robotic manipulator (𝑘 = 1,2, … , 𝑛) described by Eqs. (12a) and (12b) combined with the desired applied torque control law of Eq. ( 15), the adaptation laws of the weighting vectors, the adaptive sliding gain of Eq. ( 18), and with updating law of current-to-torque constant of Eq. ( 25), subject to

𝜄 𝑑𝑘 = 𝜄 𝑘 ( 26
)
is stable in the sense of Lyapunov stability [START_REF] Spong | Robot dynamics and control[END_REF][START_REF] Utkin | Adaptive sliding mode control[END_REF][START_REF] Ioannou | Adaptive control tutorial[END_REF].

Proof.

The Lyapunov-like function candidate along the trajectory of Eq. ( 17) can be expressed as

𝑉 𝑐𝑘 = 𝑉 𝜏𝑘 + 1 2 𝑄 ℎ𝑘 -1 ℎ ̃𝑘𝑘 2 (27)
Taking the time derivative of Eq. ( 27) and considering Eq. ( 22) lead to

𝑉 ̇𝑐𝑘 = -𝐾 𝑘 𝑠 𝑘 2 + 𝑠 𝑘 𝜖 𝑘 -𝑠 𝑘 Γ ̂𝑘(𝑡)𝑠𝑔𝑛(𝑠 𝑘 ) -(𝛿 𝑘 -Γ ̂𝑘)|𝑠 𝑘 | + 𝑠(𝜏 𝑑𝑘 -𝜏 𝑘 ) -𝑄 ℎ𝑘 -1 ℎ ̃𝑘𝑘 ℎ ̂̇𝑘𝑘 (28)
However, the fifth term can be manipulated as

𝜏 𝑑𝑘 -𝜏 𝑘 = ℎ 𝑘𝑘 (𝜄 𝑑𝑘 -𝜄 𝑘 ) -ℎ ̃𝑘𝑘 𝜄 𝑑𝑘 (29)
Substituting Eq. ( 29) into Eq. ( 28) and considering the simplification of the third and the fourth terms as made in Eq. ( 23) yield Substituting Eq. ( 25) and Eq. ( 26) into Eq. ( 30) results in the same stability results of Eq. ( 23).

Motor voltage control mode

The motor voltage control mode is a complementary control mode necessary if a servo current amplifier is unavailable. The intuitive control law for the armature current dynamics described in Eq. (12c) can be selected as

𝑢 𝑑𝑘 = 𝒘 ̂𝑢𝑘 𝑇 𝝋 𝑢𝑘 -ℎ ̂𝑘𝑘 𝑠 𝑘 -𝐾 𝑣𝑘 (𝜄 𝑘 -𝜄 𝑑𝑘 ) (31)
where 𝐿 𝑘𝑘 𝜄̇𝑑 𝑘 + 𝑅 𝑘𝑘 𝜄 𝑘 + 𝐵 𝑘𝑘 𝑞̇𝑘 = 𝒘 𝑢𝑘 𝑇 𝝋 𝑢𝑘 + 𝜖 𝑢𝑘 , ℎ ̂𝑘𝑘 can be calculated from Eq. ( 25), and

𝒘 ̂̇𝑢𝑘 = -𝑸 𝑢𝑘 𝝋 𝑢𝑘 (𝜄 𝑑𝑘 -𝜄 𝑘 ) (32)
Substituting Eq. ( 31) into Eq. (12c) results in the following closed-loop dynamics (neglecting the approximation error associated with [START_REF] Panagi | Decentralized adaptive approximation based control of a class of large-scale systems[END_REF])

𝒘 ̃𝑢𝑘 𝑇 𝝋 𝑢𝑘 + 𝐾 𝑣𝑘 (𝜄 𝑘 -𝜄 𝑑𝑘 ) + ℎ 𝑘𝑘 𝑠 𝑘 = 𝐿 𝑘 (𝜄̇𝑑 𝑘 -𝜄̇𝑘) + (𝑢 𝑘 -𝑢 𝑑𝑘 ) (33)
Theorem 4. The 𝑘-th subsystem of the robotic manipulator ( 𝑘 = 1,2, … , 𝑛) described by Eqs. (12a)-(12c) combined with the desired motor torque control law of Eq. ( 7), the adaptation laws of the weighting vectors, the adaptive sliding gain of Eq. ( 10), updating law of current-to-torque constant of Eq. ( 25), The desired voltage control input of Eq. ( 31), updating adaptive law of weighting vector of Eq. ( 32), and subject to

𝑢 𝑑𝑘 = 𝑢 𝑘 (34)
k is stable in the sense of Lyapunov stability.

Proof.

To prove the validity of controller law of Eq. ( 31) associated with adaptation law of Eq. ( 32), consider the following Lyapunov-like function candidate

𝑉 𝑢𝑘 = 𝑉 𝑐𝑘 + 1 2 𝐿 𝑘 (𝜄 𝑘𝑑 -𝜄 𝑘 ) 2 + 1 2 𝒘 ̃𝑢𝑘 𝑇 𝑸 𝑢𝑘 -1 𝒘 ̃𝑢𝑘 𝑇 (35)
By taking the time derivative of Eq. ( 35) and considering Eq. ( 30) with adaptation law of Eq. ( 25), we can get

𝑉 ̇𝑢𝑘 = -𝐾 𝑘 𝑠 𝑘 2 + 𝑠 𝑘 𝜖 𝑘 -𝛿 𝑘 |𝑠 𝑘 | -𝑠 𝑘 ℎ 𝑘𝑘 (𝜄 𝑑𝑘 -𝜄 𝑘 ) + 𝐿 𝑘 (𝜄 𝑑𝑘 -𝜄 𝑘 )(𝜄̇𝑑 𝑘 -𝜄̇𝑘) -𝒘 ̃𝑢𝑘 𝑇 𝑸 𝑢𝑘 -1 𝒘 ̂̇𝑢𝑘 (36)
Substituting Eqs. ( 32) and ( 33) into Eq. ( 36) results in

𝑉 ̇𝑢𝑘 = -𝐾 𝑘 𝑠 𝑘 2 + 𝑠 𝑘 𝜖 𝑘 -𝛿 𝑘 |𝑠 𝑘 | -𝐾 𝑣𝑘 (𝜄 𝑘 -𝜄 𝑑𝑘 ) 2 - (𝜄 𝑑𝑘 -𝜄 𝑘 )(𝑢 𝑘 -𝑢 𝑑𝑘 ) (37)
In view of Theorem 4, Eq. ( 37) can be expressed as

𝑉 ̇𝑢𝑘 = -𝐾 𝑘 𝑠 𝑘 2 -𝐾 𝑣𝑘 (𝜄 𝑘 -𝜄 𝑑𝑘 ) 2 + 𝑠 𝑘 𝜖 𝑘 -𝛿 𝑘 |𝑠 𝑘 | ( 3 8)
The variables 𝑠 𝑘 , 𝑖 𝑘 and Γ ̃𝑘(𝑡) reach zero in finite time resulting in position error approaching zero provided that 𝛿 𝑘 ≥ |𝜖 𝑘 |.

Remark 4. The stability of the individual subsystems can guarantee the stability of the whole system.

Remark 5. The function 𝑠𝑔𝑛(. ) is a hard nonlinear function that can be approximated by a smoother function such as 𝑠𝑎𝑡(. ) [START_REF] Huang | Adaptive control of robot manipulators: a unified regressor-free approach[END_REF]. However, this can results in a modeling error due to this approximation. Therefore, in order to avoid the possible parameter drift, the robust adaptive laws can be used alternatively such as project modification, 𝜎 -modification, 𝜀 -modification, etc. [START_REF] Farrell | Adaptive approximation based control: unifying neural, fuzzy and traditional adaptive approximation approaches[END_REF].

Remark 6. If it is supposed that a sufficient number of basis functions (𝛽) are used, then the modeling error can ideally be neglected, and hence no adaptive sliding mode term could be required [START_REF] Huang | Adaptive control of robot manipulators: a unified regressor-free approach[END_REF].

Simulation Results and Discussions

Two case studies have been simulated using MATLAB/SIMULINK to prove the validity of the proposed controller: a 2-link manipulator and a 6-link biped robot; the details are as follows.

Two-link manipulator

A fully actuated 2-link manipulator was simulated using MATLAB/SIMULINK to investigate the performance of the proposed controller (Figs. [START_REF] Walker | Efficient dynamic computer simulation of robotic mechanisms[END_REF][START_REF] Balafoutis | Efficient computation of manipulator, inertia matrices and the direct dynamics problem[END_REF][START_REF] Ma | An algorithm for the inverse dynamics of n-axis general manipulator using Kane's formulation of dynamical equations[END_REF]. The parameters of the simulated manipulators are borrowed from [START_REF] Huang | Adaptive control of robot manipulators: a unified regressor-free approach[END_REF]. The target manipulator moves freely (without constrained motion) with the following desired trajectories [START_REF] Slotine | Applied nonlinear control[END_REF]

: t)) cos(2 (1 30 q 0 d1    and t)) cos(2 (1 45 q 0 d2    .
However, there are some assumptions are made for simulation implementation:  The current-to-torque constant ℎ 𝑘𝑘 is known,  The joint drive system is directly constructed with gear ratio equal to one. As a result, there are two-stage control modes for the actuators: the motor torque control mode (Eqs. ( 15) and ( 18)) and the motor voltage control mode (Eqs. ( 31) and ( 32)).



It is supposed that a sufficient number of orthogonal Chebyshev basis functions ( 𝛽 = 15 ) are used for approximation purposes and hence the approximation error is neglected (it is a typical case used for simplicity in some previous work, see e.g. [START_REF] Huang | Adaptive control of robot manipulators: a unified regressor-free approach[END_REF][START_REF] Panagi | Decentralized adaptive approximation based control of a class of large-scale systems[END_REF]).



The friction consists of two terms: Coulomb friction, 𝐵 𝑐 𝑠𝑖𝑔𝑛(𝜃 ̇), and viscous friction, 𝐵 𝑣 𝜃 ̇; with 𝐵 𝑐 = 𝐵 𝑣 = 0.5 . See [START_REF] Hfn | Modeling, walking pattern generators and adaptive control of biped robot[END_REF] and the references therein for other possible friction models.



The saturated input control is not considered and it is assumed that the motors are strong to generate the required voltages.

The feedback and the adaptation gains used are: One important point should be mentioned is that although the proposed controller does not track precisely the dynamic coefficients of the equation of motion for each subsystem, the controller tracks well the desired references. The objective of adaptive control is to follow the desired trajectory rather than convergence of unknown parameters of the system to their actual values. The system signals should be persistently excited in order to ensure good estimation for unknown parameters, see [START_REF] Ioannou | Adaptive control tutorial[END_REF] for more details.

Λ 𝑘 =
A special case has been applied to the 2-link manipulator in order to test the superiority of the proposed controller comparing with the decoupled PD control. The PD control law can be described as: 7 and8 show the position errors and the input control torques respectively considering both the decentralized adaptive partitioned approximation control (APAC) and PD control. The feedback and adaptive gain matrices are tuned by gradually increasing their values from zero to a limit at which the instability occurs, then the matrices should be scaled down [START_REF] Zhu | Virtual decomposition control, toward hyper degrees of freedom robots[END_REF]. Both controllers work well with small position error; however, the APAC shows more accuracy than the PD considering the same high gain.

) ( k k k k k k e e K s K        , with 100  k K . Figures

Six-link biped robot

In this subsection, a fully actuated 6-DOF biped robot is controlled by using the decentralized adaptive approximation control. There are different walking phases for biped locomotion; however, this section concentrates on the single support phase (SSP). The biped is considered an open chain during this walking phase in which the supporting foot is in full contact with the plane ground. This means that the dynamics of the support foot are neglected. This assumption coincides with the concept of zero-moment point (ZMP)based balance criterion for biped locomotion * . If the dynamics of the stance foot are considered and it is allowed to rotate, the system will be underactuated and the stability of the biped mechanism cannot be governed by the ZMP and another balance criterion, e.g., Poincaré map, is used alternatively. See [START_REF] Hfn | Multi-level control of zero moment point-based biped humanoid robots: a review[END_REF][START_REF] Hfn | Modeling, walking pattern generators and adaptive control of biped robot[END_REF] for more details.

Figure 9 shows the configuration of the biped robot during the SSP; for more details on modeling and control of biped robots see [47][START_REF] Hfn | Multi-level control of zero moment point-based biped humanoid robots: a review[END_REF][START_REF] Hfn | Modeling, walking pattern generators and adaptive control of biped robot[END_REF][START_REF] Hfn | Zeromoment point-based biped robot with different walking patterns[END_REF][START_REF] Hfn | A Simple algorithm for generating stable biped walking patterns[END_REF][START_REF] Hfn | Dynamic modeling of biped robot using Lagrangian and recursive Newton-Euler formulations[END_REF][START_REF] Hfn | Modeling, stability and walking pattern generators of biped robots: a review[END_REF][START_REF] Hfn | Function approximation technique-based adaptive virtual decomposition control for a serialchain manipulator[END_REF][START_REF] Hfn | On the walking pattern generators of biped robot[END_REF][START_REF] Hfn | On the dynamic optimization of biped robot[END_REF][START_REF] Hfn | Finite difference-based suboptimal trajectory planning of biped robot with continuous dynamic response[END_REF]. The physical parameters are borrowed from [START_REF] Vanderborght | Dynamic Stabilisation of the biped Lucy powered by actuators with controllable stiffness[END_REF]. It should be noted that this short subsection is focused on low-level control of the biped robot. Control of the biped mechanism may require multi-level control architecture: the walking pattern generator control level, the stabilization controller for compensation of the deviation of the ZMP (as an example of balance criterion), and the lowlevel control strategy for precise tracking of the desired references (position, torque, etc.). One of the possible strategies for modeling the biped mechanism is to consider the support foot as fixed base and hence the biped mechanism can be dealt as fixed base robots. Therefore, the dynamics and control of robotic manipulators can be applied efficiently. This assumption coincides with ZMP-based walking.

In similar to the previous simulation study, a Matlab/Simulink model used for implementing the simulation experiments. A similar control architecture described in Figs. 4,5 and the same assumptions are used for simulation implementation. Therefore, two-stage control modes are used for motion control of the target biped: the motor torque control mode and the motor voltage control mode. The decentralized adaptive approximation control is applied with the following feedback and adaptation gains:

Conclusion

This paper proposes a decentralized adaptive partitioned approximation control strategy for robotic manipulators considering actuator dynamics. Three actuator control modes are considered: torque control mode, current control mode, and voltage control mode. In the torque control mode, the armature current is well controlled by a current servo amplifier and the motor torque/current constant is known. Whereas in the current control mode only the armature current is well controlled but the torque/current constant is unknown. On the other hand, the voltage control mode is necessary when no current servo control is available. In general, the proposed control law consists of three terms: a feed-forward control term, a high gain feedback control term, and a robust sliding term. The proposed control architecture has been applied to fully actuated and fixed base robotic systems. Further work is required to consider underactuated and overactuated robotic systems and even the floating base robots. In addition, the design of a decentralized control strategy for a robot moving in constrained space considering hybrid motion/force control or impedance control is necessary for different applications where the computational complexity inherently exists such as cooperating manipulators, robotic arm/hand systems and so on. Despite most decentralized control strategies are designed based on the Euler-Lagrange (E-L) formulation, exploiting the current strategy with the integration of the Newton-Euler (N-E) formulation is a powerful technique for dealing with modeling and control of complex robotic systems. Despite the work of [START_REF] Zhu | Virtual decomposition control, toward hyper degrees of freedom robots[END_REF] has been focused on this point for modeling and decoupled control of N-E based robotic systems, the proposed controller for the whole system is distributed (not fully decentralized). Future work may focus on this point using a fully decentralized control approach considering free and constrained motion with different applications.

Fig. 1 .

 1 Fig. 1. Schematic diagram showing the dynamics of the k-th subsystem consisting of a link, joint, and armature controlled DC motor with 𝜄 𝑓 being denoted to the constant field current.

Fig. 2 .

 2 Fig. 2. (a)The control architecture of the whole system, (b) the k-th subsystem controller. There is an algebraic loop at the motor current control mode due to the dependence of the input signal on the output signal.

  𝐷𝑘 𝝋 𝐷𝑘 𝑣̇𝑘𝑠 𝑘 𝒘 ̂̇𝐶𝑘 = -𝑸 𝐶𝑘 𝝋 𝐶𝑘 𝑣 𝑘 𝑠 𝑘 𝒘 ̂̇∆𝑘 = -𝑸 ∆𝑘 𝝋 ∆𝑘 𝑠 𝑘 Γ

Figure ( 6 )

 6 Figure[START_REF] Saha | Dynamics of serial multibody systems using the decoupled natural orthogonal complement matrices[END_REF] show the position error and the input control voltage for the target robot. The robot tracks well the desired trajectory. One important point should be mentioned is that although the proposed controller does not track precisely the dynamic coefficients of the equation of motion for each subsystem, the controller tracks well the desired references. The objective of adaptive control is to follow the desired trajectory rather than convergence of unknown parameters of the system to their actual values. The system signals should be persistently excited in order to ensure good estimation for unknown parameters, see[START_REF] Ioannou | Adaptive control tutorial[END_REF] for more details. A special case has been applied to the 2-link manipulator in order to test the superiority of the proposed controller comparing with the decoupled PD control. The PD control law can be described as:

Fig. 3 .Fig. 5 .

 35 Fig. 3. The 2-link manipulator

Fig. 6 .Fig. 7 .Fig. 8 .

 678 Fig. 6. Simulation results for 2-R robot

Λ

  Figures 10 and 11 show the position error and the voltage input control for the simulated biped robot. The biped can track the desired references very well despite the presence of the unknown parameters of the system and the high degrees of freedom.

Fig. 9 .Fig. 10 .

 910 Fig. 9. The configuration of the biped robot during the single support phase

Fig. 11 .

 11 Fig. 11. The input voltage control for the biped robot during the SSP

  𝐿 𝑘𝑘 𝜄̇𝑘 + 𝑅 𝑘𝑘 𝜄 𝑘 + 𝐵 𝑘𝑘 𝑞̇𝑘 = 𝑢 𝑘 (10c) where 𝑑 (.) , 𝑐 (.) , 𝑔 𝑘 , 𝜏 𝑓𝑘 (𝑞, 𝑞̇), 𝜏 𝑚𝑘 , ℎ 𝑘𝑘 , 𝜄 𝑘 , 𝐿 𝑘𝑘 , 𝑅 𝑘𝑘 , 𝐵 𝑘𝑘 , 𝑞̇𝑘 , and 𝑢 𝑘 are the corresponding elements of the target matrix/vector.

					𝜏 𝑚𝑘 = ℎ 𝑘𝑘 𝜄 𝑘	(10b)
			Eq. (10a) is a highly coupled nonlinear system, whereas Eq.
			(10b,c) is fully decoupled. Accordingly, Eq. (10a) should be
			modified to meet the conditions of decomposition. As
			aforementioned, the objective of decoupled control is to
			control every DOF individually; therefore, Eq. (10a) can be
			further re-written as
			𝑑 𝑘𝑘 (𝑞)𝑞̈𝑘 + 𝑐 𝑘𝑘 (𝑞, 𝑞̇)𝑞̇𝑗 + ∑ 𝑑 𝑘𝑗 (𝑞)𝑞̈𝑗 + 𝑛 𝑗=1 ∑ 𝑐 𝑘𝑗 (𝑞, 𝑞̇)𝑞̇𝑗 𝑛 𝑗=1	+ 𝑔 𝑘 (𝑞) + 𝜏 𝑓𝑘 (𝑞, 𝑞̇) =
			𝑗≠𝑘	𝑗≠𝑘
			ℎ 𝑘𝑘 𝜄 𝑘	(11)
			or alternatively,
				𝑑 𝑘𝑘 (𝑞)𝑞̈𝑘 + 𝑐 𝑘𝑘 (𝑞, 𝑞̇)𝑞̇𝑘 + ∆ 𝑘 (𝑞, 𝑞̇) = 𝜏 𝑚𝑘	(12a)
			with	∆ 𝑘 = ∑ 𝑑 𝑘𝑗 (𝑞)𝑞̈𝑗 + 𝑛 𝑗=1	∑ 𝑐 𝑘𝑗 (𝑞, 𝑞̇)𝑞̇𝑗 𝑛 𝑗=1	+ 𝑔 𝑘 (𝑞) +
				𝑗≠𝑘	𝑗≠𝑘
					𝜏 𝑓𝑘 (𝑞, 𝑞̇)
					𝜏 𝑚𝑘 = ℎ 𝑘𝑘 𝜄 𝑘	(12b)
				𝐿
	∑ 𝑑 𝑘𝑗 (𝑞)𝑞̈𝑗 + 𝑛 𝑗=1	∑ 𝑐 𝑘𝑗 (𝑞, 𝑞̇)𝑞̇𝑗 + 𝑛 𝑗=1	𝑔 𝑘 (𝑞) + 𝜏 𝑓𝑘 (𝑞, 𝑞̇) =
			𝜏 𝑚𝑘 (10a)

𝑘𝑘 𝜄̇𝑘 + 𝑅 𝑘𝑘 𝜄 𝑘 + 𝐵 𝑘𝑘 𝑞̇𝑘 = 𝑢 𝑘

  𝑞̇𝑑 𝑘 -Λ 𝑘 𝑒 𝑘 , 𝑠 𝑘 = 𝑞̇𝑘 -𝑣 𝑘 = 𝑒̇𝑘 + Λ 𝑘 𝑒 𝑘 , 𝑒 𝑘 = 𝑞 𝑘 -𝑞 𝑑𝑘

	𝑑 ̂𝑘𝑘 (𝑞)𝑣̇𝑘 + 𝑐̂𝑘 𝑘 (𝑞)𝑣 𝑘 + ∆ ̂𝑘 -𝐾 𝑘 𝑠 𝑘 -Γ ̂𝑘(𝑡)𝑠𝑔𝑛(𝑠 𝑘 )
	(15)
	with
	𝑑 ̂𝑘𝑘 = 𝒘 ̂𝐷𝑘 𝑇 𝝋 𝐷𝑘
	𝑐̂𝑘 𝑘 = 𝒘 ̂𝐶𝑘 𝑇 𝝋 𝐶𝑘
	∆ ̂𝑘= 𝒘 ̂∆𝑘 𝑇 𝝋 ∆𝑘
	𝑣 𝑘 =

  𝑉 ̇𝑐𝑘 = -𝐾 𝑘 𝑠 𝑘 2 + 𝑠 𝑘 𝜖 𝑘 -𝛿 𝑘 |𝑠 𝑘 | -𝑠 𝑘 ℎ 𝑘𝑘 (𝜄 𝑑𝑘 -𝜄 𝑘 ) + 𝑠 𝑘 ℎ ̃𝑘𝑘 𝜄 𝑑𝑘 -𝑄 ℎ𝑘 -1 ℎ ̃𝑘𝑘 ℎ ̂̇𝑘𝑘 𝑉 ̇𝑐𝑘 = -𝐾 𝑘 𝑠 𝑘 2 + 𝑠 𝑘 𝜖 𝑘 -𝛿 𝑘 |𝑠 𝑘 | -𝑠 𝑘 ℎ 𝑘𝑘 (𝜄 𝑑𝑘 -𝜄 𝑘 ) +

	ℎ ̃𝑘𝑘 (𝑠 𝑘 𝜄 𝑑𝑘 -𝑄 ℎ𝑘 -1 ℎ ̂̇𝑘𝑘 ) (30)

* ZMP can be defined as the ground reaction force wrench due to the footground contact with zero horizontal moments. The ZMP location coincides with the location of the center of pressure for the balance walking; however, this is not the case for unbalanced walking. See [47-57] and the references therein for more details.


Benoît Thuilot
email: benoit.thuilot@lasmea.univ-bpclermont.fr

Christophe Cariou
email: christophe.cariou@clermont.cemagref.fr

Lionel Cordesses

Philippe Martinet

Automatic guidance of a farm tractor along curved paths, using a unique CP-DGPS

Keywords: mobile robots, nonlinear control systems, Kalman state reconstructor, GPS, agriculture

Precision agriculture involves very accurate farm vehicle control along recorded paths, which are not necessarily straight lines. In this paper, we investigate the possibility of achieving this task with a CP-DGPS as the unique sensor. The vehicle heading is derived according to a Kalman state reconstructor, and a nonlinear velocity independent control law is designed, relying on chained systems properties. Field experiments, demonstrating the capabilities of our guidance system, are reported and discussed.

Introduction

The development of guidance systems for agricultural vehicles receives more and more attention from researchers and manufacturers. The motivations are numerous, since automatic guidance:

reduces the work arduousness.

-insures an optimal work precision throughout the day and on the whole field. For instance, it enables the exact placement of field inputs (seeds, fertilizers, pesticides,. . .).

allows to operate wider tools at higher speeds, and therefore it can increase productivity. Many experiments have been conducted and reported in the literature. They can be classified into 2 categories, according to the kind of sensor to be used.

The first category uses relative information. Researchers have focused for instance on the video camera sensor [START_REF] Debain | An help guidance system for agricultural vehicles[END_REF], [START_REF] Khadraoui | Vision based control in driving assistance of agricultural vehicles[END_REF] and [START_REF] Ollis | Vision based perception for an automated harvester[END_REF]. Two kinds of difficulties arise: the first one is the reference extraction. Extreme conditions of dust, for example, reveal the weakness of the detection side. The other problem arises when the new reference is computed in an iterative way from the previous one. The automatic guidance system can lead to oscillations.

The second category uses absolute information. State of the art GPS (Carrier Phase Differential GPS or CP-DGPS) reach today the realtime centimeter accuracy. This allows the design and implementation of an absolute vehicle guidance system. Promising results have been reported: a GPS and a fiber optic gyroscope (FOG) have been used in [START_REF] Nagasaka | Automated operation in paddy fields with a fiber optic gyro sensor and GPS[END_REF]. [START_REF] O'connor | Automatic steering of a farm vehicle using GPS[END_REF] has described a solution with multiple GPS antennas. These multisensors equipments have the advantage to provide control designers with the whole vehicle attitude. However, they are quite expensive. Therefore, we are currently investigating the possibility to control vehicles with a unique GPS. Straight line following, applied to a combine harvester, has been demonstrated in [START_REF] Cordesses | GPS-based control of a land vehicle[END_REF]. The aim of this paper is to extend this work to curved path following applications. Our experimental vehicle, depicted on Figure 1, is now a farm tractor.

Figure 1: Experimental platform

We first derive the kinematic model of a farm tractor. Then, the design of a nonlinear, velocity independent, curved path following control law is detailed. Next, the estimation of the tractor heading from GPS information is discussed. Finally experimental results are displayed.

Modelisation

From a practical point of view, the tractor and the tool can be considered as a tricycle model of length l with a controlled front wheel, see Figure 2.

y δ θ ~O A A s M X Y A v l s=0 C Figure 2: Kinematic model of the farm tractor
The tractor is assumed to move on a horizontal ground. We propose here to describe its configuration with respect to the path C to be followed, rather than with respect to an absolute frame [A, X A , Y A ). For this purpose, let us denote by O the center of the tractor rear axle, and by M , the point on C which is the closest to O (see Figure 2). The vehicle configuration is then described without ambiguity by the state vector X = (s, y, θ) T with:

s : the curvilinear abscissa of point M along C, -y : the lateral deviation of the tractor with respect to C, -θ : the angular deviation of the tractor with respect to the tangent to C at point M .

Two control variables are available, namely U = (v, δ) T : v: linear velocity at point O (the control variable is actually the rear wheels rotation velocity, from which the value of v can be inferred).

δ: orientation of the front wheel.

We have derived the vehicle kinematic model according to the following assumptions:

-the tractor and the tool are a unique rigid body, -the tractor moves according to pure rolling and non slipping conditions.

They are quite common among mobile robots community: the former implies that an instantaneous rotation center exists. The latter imposes that the linear velocity vector at point O is directed along the vehicle axle. Under such assumptions, we obtain, see [START_REF]Theory of robot control[END_REF]:

     ṡ = v cos θ 1-c(s) y ẏ = v sin θ θ = v tan δ l -c(s) cos θ 1-c(s) y (1)
where c(s) denotes the curvature of path C at the curvilinear abscissa s (i.e. at point M ).

It can be noticed, that the model (1) becomes singular when y = 1 c(s) , i.e. when the point O is located at the center of curvature of the reference path. This problem is not encountered in practical situations: on one hand, the path curvature is small, and on the other hand, the farm tractor remains close to C.

Control law design

The control objective is to ensure the convergence of the tractor to the reference path C. In view of section 2, we want state variables y and θ to be brought and kept equal to 0. Moreover, control law performances are expected to be independent from tractor velocity v. This control variable is here considered as a parameter whose value will be managed according to the application. v may be constant or time-varying.

The tractor model ( 1) is clearly nonlinear. In [START_REF] Bell | Realistic autofarming closed-loop tractor control over irregular paths using kinematic GPS[END_REF], the authors propose to linearize it around the equilibrium y = θ = 0, in order that celebrated linear systems theory could be used. In this case, control design does not rely on the actual tractor model, but on an approximated one. Recent advances in Control theory have established that mobile robots models can be converted into almost linear models, namely chained forms, in an exact way, see [START_REF] Samson | Control of chained systems. Application to path following and time-varying point stabilization of mobile robots[END_REF] for instance. Such an approach is attracting since it allows us to use, for a large part, linear systems theory, while still relying upon the actual nonlinear tractor model. It is detailed hereafter.

The general chained form dedicated to systems with three states and two inputs is written as (see [START_REF] Samson | Control of chained systems. Application to path following and time-varying point stabilization of mobile robots[END_REF]):

   ȧ1 = m 1 ȧ2 = a 3 m 1 ȧ3 = m 2 (2)
with A = (a 1 , a 2 , a 3 ) T and M = (m 1 , m 2 ) T respectively the state and control vectors. In order to point out that a chained system is almost linear, just replace the time derivative by a derivation with respect to the state variable a 1 . Using the notations:

d d a 1 a i = a i and m 3 = m 2 m 1 (3)
the chained form (2) can be rewritten:

   a 1 = 1 a 2 = a 3 a 3 = m 3 (4)
The last 2 equations of system (4) constitute clearly a linear system. More generally, a n-dimensional chained system exhibits a n -1-dimensional linear subsystem.

Let us now convert tractor model ( 1) into chained form. Since control law performances are expected to be independent from the tractor velocity, the variable a 1 , which drives the evolution of the linear system (4), should be homogeneous at the distance covered by the tractor. A natural choice is then:

a 1 = s (5)
Consequently, variables a 2 and a 3 have to be related to y and θ in an invertible way. For the sake of simplicity, let us choose :

a 2 = y (6)
Straightforward computations show then that the nonlinear tractor model (1) can actually be converted into chained forms (2) or (4) from the starting choices ( 5)- [START_REF] Khadraoui | Vision based control in driving assistance of agricultural vehicles[END_REF]. More precisely, we can show successively that :

m 1 ∆ = ȧ1 = v cos θ 1 -c(s) y (7) a 3 ∆ = ȧ2 m 1 = (1 -c(s) y) tan θ (8) m 2 ∆ = ȧ3 = d dt ((1 -c(s) y) tan θ) (9) = -c(s) v sin θ tan θ - d c(s) d s v cos θ 1 -c(s) y tan θ y +v (1 -c(s) y) cos 2 θ tan δ l -c(s) cos θ 1 -c(s) y
In the sequel, state and control transformations are denoted respectively Θ(X) and Υ(U, X): A = Θ(X) defined by ( 5), ( 6) and ( 8) [START_REF] Samson | Control of chained systems. Application to path following and time-varying point stabilization of mobile robots[END_REF] M = Υ(U, X) defined by ( 7) and ( 9) [START_REF] Sussmann | A general result on the stabilization of linear systems using bounded controls[END_REF] These transformations are invertible as long as y = 1 c(s) (model singularity), v = 0, and θ = π 2 [π]. From a practical point of view, once properly initialized, the guided tractor respects these conditions.

Control design can now be completed in a very simple way: since chained form ( 4) is linear, we are led to choose the following virtual control law:

m 3 = -K d a 3 -K p a 2 (K p , K d ) ∈ R +2 (12)
As a matter of fact, reporting (12) in ( 4), leads to:

a 2 + K d a 2 + K p a 2 = 0 (13)
which implies that both a 2 and a 3 converge to zero. The same conclusion holds for y and θ in view of [START_REF] Samson | Control of chained systems. Application to path following and time-varying point stabilization of mobile robots[END_REF].

Reference path following is therefore achieved. Moreover, since the evolution of the error dynamics ( 13) is driven by a 1 = s, the gains (K d , K p ) impose a settling distance instead of a settling time. Consequently, for a given initial error, the tractor trajectory will be identical, whatever the value of v is, and even if v is time-varying. Control law performances are therefore velocity independent, as desired.

Ultimately, reporting (12) into [START_REF] Sussmann | A general result on the stabilization of linear systems using bounded controls[END_REF] provides us with the actual control law expression:

δ(y, θ) = arctan l cos 3 θ (1-c(s) y) 2 d c(s) d s y tan θ -K d (1 -c(s) y) tan θ -K p y + c(s) (1 -c(s) y) tan 2 θ + c(s) cos θ 1-c(s) y (14)
In many applications, the reference path C is a straight line, i.e. c(s) = 0. The expression of the control law (14) turns then simpler:

δ(y, θ) = arctan(l cos 3 θ(-K d tan θ -K p y)) (15)
Since control law (14) is not bounded, actuator saturation could occur. We are allowed to bound the virtual control law (12) to any arbitrary value: since chained form (4) consists in a double integrator, its asymptotic stability is still insured, see [START_REF] Sussmann | A general result on the stabilization of linear systems using bounded controls[END_REF]. Nevertheless, in view of (14), the boundedness of m 3 leads to those of δ only if the reference path curvature c(s) exhibits some good properties. They are of course satisfied in the straight line case, i.e. when c(s) = 0. In the forthcoming experimentations, control law (12) has been saturated by means of a sigmoid function:

m 3,sat (a 2 , a 3 ) = K 1 -e -k m3(a2,a3) 1 + e -k m3(a2,a3) (16)
Once saturated virtual control law (16) has been reported into [START_REF] Sussmann | A general result on the stabilization of linear systems using bounded controls[END_REF], parameter K can be tuned so as to ensure that max ∀y,∀ θ |δ(y, θ)| remains inferior to actuator limitation δ max . Finally, we must choose k = 2 K in order that m 3, sat (a 2 , a 3 ) ≈ m 3 (a 2 , a 3 ) when a 2 and a 3 are small.

The only exteroceptive sensor embarked on the tractor is a CP-DGPS. It has been located on the top of the cabin, straight up the point O. Absolute position and velocity of that point, which is the tractor control point (see Figure 2), are then available in a direct way. In addition, this location is on the highest part of the tractor, and therefore the suitest place to see as many satellites as possible.

Control law (14) requires realtime measurement of y and θ. The value of the former variable can be obtained straightforwardly: from the absolute position of point O, provided by the GPS, and the knowledge of the reference path C, we can infer the location of point M , and therefore the value of y.

The value of θ is more complex to derive. Under the pure rolling and non-slipping assumptions, the tractor heading with respect to the absolute frame [A, X A , Y A ), denoted θ, can be derived from the absolute velocity of point O, denoted (V 0, XA , V 0, YA ), also provided by the GPS:

θ = arctan V0 , Y A V 0, X A if V 0, XA = 0, sign(V 0, Y A ) π 2 if V 0, X A = 0 (17)
Therefore, the value of θ can also be obtained by a direct measurement:

θ = θ -θ d (s) (18)
where θ d (s) denotes the heading of the reference path C at the curvilinear abscissa s (i.e. at point M ).

Unfortunately, experiments have shown that a direct measurement (18) provides us with prohibitive noisy values of θ. A low pass filter could be considered. However, since the tractor model is available, a suitable alternative consists in using it through a Kalman state reconstructor. Discretizing the third equation of tractor model (1) leads to:

θk+1 = θk + v T l tan δ k ( 19
)
where T is the sampling period. Celebrated Kalman state reconstructor equations, when applied to system (19), are then written as (see for instance [START_REF] Gelb | Applied optimal estimation[END_REF]):

θk = θk-1 + v T l tan δ(y k-1 , θk-1 ) θk = θk + L( θm k -θk ) (20)
L is the scalar Kalman gain, to be chosen with respect to the sensor noise features. θm k denotes the direct measurement of θ, according to (18), at time t k = k T . θk is its filtered value at the same time, to be used in control law (14).

Experimentations

The farm tractor and its tool are depicted on Figure 1. The CP-DGPS receiver is a Dassault-Sercel dual frequency "Aquarius 5002" unit. Its claimed accuracy is about 1 centimeter, and its upper sampling frequency is 10 Hertz. The control law (14) has been implemented in high level language (C++) on a Pentium based computer.

Step responses

Although a step path is not common in agriculture, we have used the step response to check the behaviour of our closed loop system.

The reference path consists in a 2 meters step. Control law parameters (K d , K p ) have been tuned so as to impose that the error dynamics (13) presents a 15 meters settling distance1 , while Kalman parameter L has been adjusted with the value 0.08, in view of the sensor noise features. Three experiments have been carried out: in the two first, the tractor velocity was constant, v = 4 km.h -1 , and then v = 8 km.h -1 , while in the last one, the velocity has been increased linearly from v = 4 km.h -1 to v = 8 km.h -1 . The experimental results are depicted on Figure 3 and4. 3, the upper plot shows the three step responses, while the lower plot depicts the tractor raw velocity, as computed by the GPS receiver. One can check that the three step responses almost perfectly overlap: as expected, the trajectory is independent from tractor velocity, even when v is varying during the step. Moreover, the 15 meter settling distance, specified when tuning the control parameters, is actually achieved. If we investigate path following accuracy, the results appear also quite satisfactory: once the tractor is following a straight line (i.e. when x a > 70 meters), the bias µ y between the tractor trajectory and the reference path, as well as the tractor standard deviation from the mean trajectory σ y , are both very small: µ y < 2.7 cm and σ y < 3.1 cm. The values of state variable θ, required in control laws ( 14) or (15), can be obtained either by a direct measurement of tractor velocity, according to relations ( 17) and ( 18), or by filtering this direct measurement according to the Kalman state reconstructor (20). The values of θ returned by these two approaches, have been recorded during a step response with a constant tractor velocity v = 8 km.h -1 , and are displayed on Figure 4. It clearly highlights the need for filtering the values of θ returned by a direct measurement, and establishes the relevancy of the Kalman state reconstructor (20): the amplitude of θ oscillations, as well as theirs frequencies, are significantly reduced. For instance, once the tractor is following a straight line (i.e. when x a > 70 meters), the standard deviation from the mean decreases from 1.71 to 0.48 when the Kalman state reconstructor is used.

Curved path following

The first experiments reported below consist in a sinusoidal path following. Sine curves are definitely not common trajectories in agricultural tasks. Nevertheless, they are significantly different from straight lines, and therefore can be seen as a convincing test when investigating the performances of curved path following control law (14).

Period and peak to peak amplitude of the sinusoidal reference path are respectively 20 meters and 60 centimeters. The values of the control and Kalman pa-rameters K d , K p and L are identical to those used in step response experiments. At initial time, the tractor is 60 centimeters far from the reference path. Its velocity is constant: v = 6 km.h -1 . The evolution of the tractor lateral deviation with respect to the sinusoidal reference path is depicted on Figure 5. One can check that control law (14) leads to the same performances, irrespectively of the reference path features: the settling distance and the statistical variables µ y and σ y recorded during this curved path following experiment are of the same order of magnitude as those previously recorded during step response experiments. Curved path control law (14) can be gracefully degraded to the case of straight lines following by imposing c(s) = 0 : we then obtain the much simpler expression (15). In order to investigate the actual contribution of the reference path curvature c(s), we have also experimented sine curve following with the simplified control law (15). Reference path, tractor trajectory obtained with that law, and those obtained with the entire control law (14) are displayed on Figure 6. We observe that, without the reference path curvature information, the tractor follows the reference path with an 1.5 meter offset, and moreover presents large excursions when the values of c(s) are the highest. This clearly establishes the imperative need to take into account the curvature of the reference path when designing the control law.

Of course, we have also investigated control law (14) capabilities when the path to be followed is consistent with common agricultural tasks. The tractor has first been driven on a field by a human operator. The path was roughly made of straight lines and semicircles.This path has then been replayed several times in an automatic way. Examination of the results shows that the tractor trajectories almost overlap the reference path. A statistical analysis still provides us with quite satisfactory results: µ y = 3 cm and σ y = 5 cm. Nevertheless, it can be noticed that, on few parts of the reference path, the lateral deviation climbs up to large values (18.4 cm in the worst case). The explanation that can be advanced is that the reference path curvature, in these locations, is very high, thus leading to a control δ varying very quickly. This has led us to incriminate the low level servoing system, since it has not been optimized for fast tracking: neither the hydraulic components nor the controller -off the shelves devices -can achieve very high performances.

Finally, during Innov'Agri 2000 demonstrations (a French Show where new agricultural systems are presented), the tractor has replayed a complex curved path more than a hundred times without any driver abroad, establishing the system reliability.

Conclusion

The purpose of this work was to achieve curved path following from a unique CP-DGPS sensor, within the framework of agricultural applications. A Kalman state reconstructor has been derived in order to estimate the tractor heading, and a nonlinear velocity independent control law has been designed, relying upon recent developments in Control theory, precisely "chained systems" properties. Several experiments are reported, and display the control law capabilites.

Although results are quite satisfactory on an almost flat ground, further research has to be done to take into account for slope. Another important issue is to build a better hydraulic servoing system, in order to decrease both latency and settling time.

Figure 3 :

 3 Figure 3: Step responses at different velocities On Figure3, the upper plot shows the three step responses, while the lower plot depicts the tractor raw velocity, as computed by the GPS receiver. One can check that the three step responses almost perfectly overlap: as expected, the trajectory is independent from tractor velocity, even when v is varying during the step. Moreover, the 15 meter settling distance, specified when tuning the control parameters,

Figure 4 :

 4 Figure 4: Estimation of heading error θ

Figure 5 :

 5 Figure 5: Lateral deviation with respect to a sinusoidal reference path

  With curve line control law With straight line control law

Figure 6 :

 6 Figure 6: Tractor trajectories during a sine curve following: capabilities of control laws (14) and (15)

We have chosen (K d , Kp) = (0.6, 0.09) in order that the error dynamics (13) presents a double pole located at the value 0.3. Linear control tools ensure then that the settling distance is 15 meters, as it was specified.