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Overview of the robot identification problem
Accurate dynamic models of robots are required in most advanced control schemes formulated in recent literature (Khatib, 1987; Piltan et al., 2012; Slotine et al., 1987). The precision, performance, stability and robustness of these schemes depend on, to a large extent, the accuracy of the dynamic parameters. Adap-tive and robust control scheme can tolerate some error in the dynamic param-eters, while other schemes designed to achieve perfect feedback linearization, such as computed torque control, assuming precise knowledge of the dynamic parameters. In this sense, the precise determination of the dynamic parame-ters is useful to most schemes and is crucial to some others. Furthermore, the dynamic parameters are necessary to simulate the robot dynamics.
However, accurate values of the dynamic parameters are typically unknown, even to the robot manufactures, and the measurement for some of them are practically not accessible. Thus, the indirect identification approaches are considered through the analysis of the ’input/output’ behavior of the robot fol-lowing some planned motion and on estimating the parameters value by min-imizing the difference between a function of the real robot variables and its mathematical model. The identification problem turns out to be an optimiza-tion question which searches for the correct robot model with proper dynamic parameters.
The following parts present the principle of identification procedures and the related various techniques. According to the inputs that the identification model needs, there exists three robot identification models:
• robot dynamic model (Canudas de Wit et al., 1991; Gautier, 1986; Gau-tier et al., 2008, 2013; Gautier and Khalil, 1990; Gautier, Khalil, and Restrepo, 1995; Gautier, Vandanjon, et al., 2011; Hollerbach et al., 2008; Khalil and Dombre, 2004; Khosla et al., 1985; Lu et al., 1993);
• robot energy model (Gautier, 1996; Gautier and Khalil, 1988);
• robot power model (Gautier, 1997).
The robot dynamic model is the most widely implemented identification model. It establishes the dynamic equations at individual point along the tra-jectory. The advantages of this model include that it is easy to create the dy-namic equations and it has a good excitation in the identification regression matrix, which means the regression always has a solution. By contrast, the dy-namic model contains the acceleration variables, which are usually inaccurate computation using the position measurement. While the robot energy model and power model avoid using the acceleration data. Instead, the energy model applies an integral operation on the robot power equation and the power model make use of the differential equation of the energy part. In the following, we will present these identification process. Note that for the rest part of the thesis, we denote the bold mathematical symbols for the vectors or matrix.
Case study: the 2R scara planar prototype robot of IRC-CyN
To illustrate the construction of robot dynamic model, here we present a two joint scara planar robot, called 2R robot for short. As shown in Fig. 1.1, the robot geometry is described in table (1.1) using the modified Denavit and Harten-berg notation (DHM) method, with
• j denotes the jth joint,
• σj denotes the type of joint, 0 for revolute joint, 1 for prismatic joint,
• αj is the angle between zj−1 and zj about xj−1,
• dj is the distance between zj−1 and zj along xj−1,
• θj is the angle between xj−1 and xj about zj ,
• rj is the distance between xj−1 and xj along zj ,
• q1 and q2 are joint position for joint 1 and joint 2 respectively,
• L is the length of the first robot link, L2 is the length of the second robot link.
Before regrouped into base parameters, there exist 11 standard inertia pa-rameters for each joint, such as in table (1.2):
Notice the special geometric configuration and apply the regrouping rule (Gautier, 1990) on the 2R robot, and it can be concluded as:
• joint 1 and joint 2 are direct drive so that the link 1 and 2 are attached to the rotors of motor 1 and 2. Then, the inertia moment ZZ1 and ZZ2 are the inertia moment of links plus the inertia moment of rotors I A1 and I A2 respectively, so that in the following I Aj = 0;
• joint 2 is revolute: Y Y2, MZ2, M2 can be regrouped with other dynamic parameters of link 2 and 1;
• the axe of joint 2 is parallel to that of joint 1 : XX2, XY2, XZ2, Y Z2 can be eliminated;
• joint 1 is revolute, and the axe is along the gravity direction: only the inertia parameter ZZ1 is considered;
• for joint 1, ZZ1 is grouped with M2 using the following relation ZZ1R = ZZ1 + M2L2.
It is shown in table 1.3 that there exist five base inertia parameters for 2R scara planar robot.
Table of contents :
General Introduction
1 Overview of the robot identification problem
1.1 Inverse dynamic identification model with LS
1.1.1 Case study: the 2R scara planar prototype robot of IRCCyN
1.1.2 IDIM-LS
1.1.3 Identifiability of the dynamic parameters
1.1.4 Excitation of the trajectory
1.1.5 Data processing for the inverse dynamic identification model
1.1.6 Resolution of the inverse dynamic identification model
1.1.7 Numerical tools and evaluation
1.2 Energy model identification
1.3 Power model identification
1.4 Closed-loop output error identification (CLOE)
1.5 Payload Identification
1.6 Problems in robot identification
2 Robot identification using power model and modulating functions
2.1 Modulating functions
2.2 Studies of modulating functions
2.2.1 Frequency Analysis
2.2.2 Configuration Choice
2.2.3 An introducing example with one joint robot with gravity effect
2.2.4 An introducing example with one joint robot without gravity effect
2.2.5 General case
2.3 Identification with modulating functions and power model
2.3.1 Simulation on 2R robot
2.3.2 Identification on 2R prototype robot
2.3.3 Simulation of payload dynamic parameters on 4R robot
2.4 Conclusion
3 Differentiation in parameters identification of robots
3.1 Introduction
3.1.1 A motivating example
3.2 Causal Jacobi differentiator
3.2.1 Error Analysis in Time Domain
3.2.2 Frequency Domain Analysis
3.3 Central Jacobi Differentiator
3.3.1 Another access to central Jacobi differentiator
3.3.2 Error Analysis in Time Domain
3.3.3 Error Analysis in Frequency Domain
3.4 Dynamic parameters identification of 2R robot
3.4.1 Iterative learning identification and computed torque control
3.4.2 On-line Identification
3.4.3 Non stationary inertial parameter
3.4.4 Offline Identification
3.5 Dynamic parameters identification of EMPS
3.5.1 Presentation of EMPS
3.5.2 Inverse dynamic model of EMPS
3.5.3 Identification model using motor and load positions
3.5.4 Identification model with only motor position and torque
3.5.5 Data acquisition
3.5.6 Experimental Validation
3.5.7 Comparison between two identification model with EMPS
3.6 Conclusion
4 Comparison of different identification techniques
4.1 Tests on 2R scara planar robot
4.1.1 Simulation for 2R robot identification
4.1.2 Filtering systematic error
4.1.3 Identification results on 2R prototype robot
4.2 Conclusion
5 Simplified model with real time estimation
5.1 Introduction
5.2 Simplified model and iterative learning control
5.2.1 Controller design
5.3 Real time estimation
5.3.1 Estimation model
5.4 Simulation
5.4.1 Simulation results with real time estimation
5.4.2 Results Robust To Variation Payload
5.4.3 Simulation with noise
5.5 Conclusion
Conclusion
Prospective
Bibliography
A Appendix
