Robot Manipulators 2011 Part 10 docx

Moreover, in the position loop P controller design, the controller parameters at each axis of robot manipulator are selected with identical values by P-gain with 2.4.. The results indica

Finally, the FPGA utility of the motion control IC for robot manipulator in Fig 4 is evaluated and the result is listed in Table 3 The overall circuits included a Nios II embedded processor (5.1%, 2,468 ALUTs) and a motion control IP (14.4%, 6,948 ALUTs) in Fig 4, use 19.5% (9,416 ALUTs) utility of the Stratix II EP2S60 Nevertheless, for the cost consideration, the less expensive device - Stratix II EP2S15 (12,480 ALUTs and 419,329 RAM bits) is a better choice In Fig 4, the software/hardware program in parallel processing enhances the controller performance of the motion system for the robot manipulator

(0 degree)

(0 degree)

(51.25 degree) (116.87 degree)

(26.56 degree)

(26.56 degree)

(56.19 degree) (114.31 degree) (-58.13 degree)

(45.0 degree)

(45 degree)

(-34.88 degree) (66.81 degree) (-31.94 degree)

(c) Figure 11 Simulation result of Cartesian space (a) (0,0,0) (b) (200,100,300) (c) (300,300,300) to join space

Resource usage in FPGA (ALUTs) Module Circuit

71 Others circuit

316 Five sets of PMW module

206 Position controller module

748 Speed controller module

5,322 Inverse kinematics module

9,416 Total

285 Five sets of QEP module

2,468 Nios II embedded processor IP

Resource usage in FPGA (ALUTs) Module Circuit

71 Others circuit

316 Five sets of PMW module

206 Position controller module

748 Speed controller module

5,322 Inverse kinematics module

9,416 Total

285 Five sets of QEP module

2,468 Nios II embedded processor IP

Table 3 Utility evaluation of the motion control IC for robot manipulator in FPGA

5 Experimental system and results

Figure 12 presents the overall experimental system which includes an FPGA experimental board, five sets of inverter, five sets rectifier, a power supplier system and a Mitsubishi Movemaster RV-M1 micro articulated robot The micro articulated robot has five servo axes

FPGA-Realization of a Motion Control IC for Robot Manipulator 307 (excluding the hand) and its specification is shown in Fig.13 Each axis is driven by a 24V

DC servo motor with a reduction gear The operation ranges of the articulated robot are wrist roll ±180 degrees (J5-axis), wrist pitch ±90 degrees (J4-axis), elbow rotation 110 degrees (J3-axis), shoulder rotation 130 degrees (J2-axis) and waist rotation 300 degrees (J1-axis) The gear ratios for J1 to J5 axis of the robot are 1:100, 1:170, 1:110, 1:180 and 1:110, respectively Each DC motor is attached an optical encoder Through four times frequency circuit, the encoder pulses generate 800pulses/cycle at J1 to J3 axis and 380pulses/cycle at J4 and J5 axis The maximum path velocity is 1000mm/s and the lifting capacity is 1.2kg including the hand The total weight of this robot is 19 kg The inverter has 4 sets of IGBT type power transistors The collector-emitter voltage of the IGBT is rating 600V, the gate-emitter voltage

is rating ±12V, and the collector current in DC is rating 25A and in short time (1ms) is 50A The photo-IC, Toshiba TLP250, is used for gate driving circuit of IGBT Input signals of the inverter are PWM signals from FPGA chip The FPGA-Altera Stratix II EP2S60F672C5ES in Fig 1(a) is used to develop a full digital motion controller for robot manipulator A Nios II embedded processor can be download to this FPGA chip

Robot manipulator

FPGA board

(5) Inverter for DC motor

(5) Rectifier Power

Figure 12 Experimental system

In Fig.12 or Fig.4, the realization in PWM switching frequency, dead-band of inverter, position and speed control sampling frequency are set at 18k Hz, 1.28μs, 762 Hz and 1525 Hz, respectively Moreover, in the position loop P controller design, the controller parameters at each axis of robot manipulator are selected with identical values by P-gain with 2.4 However,

in the speed loop PI controller design, the controller parameters at each J1~J5 axis are selected with different values by [3.17, 0.05], [3.05, 0.12], [2.68, 0.07], [2.68, 0.12] and [2.44, 0.06], respectively

To confirm the effectiveness of the proposed motion control IC, the square-wave position command with ±3 degrees amplitude and 2 seconds period is firstly adopted to test the dynamic response performance At the beginning of the step response testing, the robot manipulator is moved to a specified attitude for joints J1-J5 rotating at the [9o, 40o, 60o, 45o, 10o] position Figure 14 shows the experimental results of the step response under these design

conditions, where the rise time of the step responses are with 124ms, 81ms, 80ms, 151ms and

127 ms for axis J1-J5, respectively The results also indicate that these step responses have almost zero steady-state error and no oscillation Next, to test the performance of a point-to-point motion control for the robot manipulator, a specified path is run where the robot moves from the point 1 position, (94.3, 303.5, 403.9) mm to the point 2 position, (299.8, 0, 199.6) mm, then back to point 1 After through inverse kinematics computation in (17) ~ (23), moving each joint rotation angle of the robot from point 1 to point 2 need rotation of -72.74o (-16,346 Pulses), 23.5o (8,916 Pulses), 32.16o (8,173 Pulses), -56.72o (-11,145 Pulses) and -71.18o (-8,173 Pulses), respectively Additionally, a point-to-point control scheme with constant acceleration/deceleration trapezoid velocity profile adopts to smooth the robot manipulator movement Applying this motion control scheme, all joins of robot rotate with simultaneous starting and simultaneous stopping time The acceleration/deceleration time and overall running time are set to 256ms and 1s, and the computation procedure in paragraph 3.3.1 is applied Figure 15 shows the tracking results of using this design condition at each link The results indicate that the motion of each robot link produces perfect tracking with the target command in the position or the velocity response Furthermore, the path trajectory among the position command, actual position trajectory and point-to-point straight line in Cartesian space R3 (x,y,z) are compared, with the results shown in Fig 16 Analytical results indicate that the actual position trajectory can precisely track the position command, but that the middle path between two points can not be specified in advanced Next, the performance of the linear trajectory tracking of the proposed motion control IC is tested, revealing that the robot can be precisely controlled at the specified path trajectory or not The linear trajectory path is specified where the robot manipulator moves from the starting position, (200, 100, 400) mm to the ending position, (300, 0, 300) mm, then back to starting position The linear trajectory command is generated 100 equidistant segmented points from the starting to the ending position Each middle position at Cartesian space R3 (x,y,z) will be transformed to joint space

(θ*,θ*,θ*,θ*,θ*), through the inverse kinematics computation, then sent to the servo controller

of the robot manipulator The tracking result of linear trajectory tracking is displayed in Fig

17 The overall running time is 1 second and the tracking errors are less than 4mm Similarly, the circular trajectory command generated by 300 equidistant segmented points with center (220,200,300) mm and radius 50 mm is tested again and its tracking result is shown in Fig 18 The overall running time is 3 second and the trajectory tracking errors at each axis are less than 2.5mm Figures 17~18 show the good motion tracking results under a prescribed planning trajectory Therefore, experimental results from Figs 14~18, demonstrate that the proposed FPGA-based motion control IC for robot manipulator ensures effectiveness and correctness

Movemaster RV-M1 micro articulated robot

J1-axis

J2-axis J3-axis

J4-axis

J5-axis

1:110 380

±180 0 ( ±20667 pulse)

J5

wrist roll

1:180 380

±90 0 ( ±17676 pulse)

J4

wrist pitch

1:110 800

110 0 (27958 pulse)

J3

elbow

1:170 800

130 0 (49311 pulse)

J2

should

1:100 800

300 0 (67416 pulse)

J1

waist

Gear ratio Encoder pulse x 4 (ppr)

Max working range (degree) No.

Name of link

1:110 380

±180 0 ( ±20667 pulse)

J5

wrist roll

1:180 380

±90 0 ( ±17676 pulse)

J4

wrist pitch

1:110 800

110 0 (27958 pulse)

J3

elbow

1:170 800

130 0 (49311 pulse)

J2

should

1:100 800

300 0 (67416 pulse)

J1

waist

Gear ratio Encoder pulse x 4 (ppr)

Max working range (degree) No.

Name of link

Figure 13 Mitsubishi Movemaster RV-M1 micro articulated robot

FPGA-Realization of a Motion Control IC for Robot Manipulator 309

36 38 40 42 44 46

36 38 40 42 44 46

56 58 60 62 64 66

56 58 60 62 64 66

42 44 46 48 50

42 44 46 48 50

6 8 10 12 14 16

-18,000 -16,000 -14,000 -12,000 -10,000

-10,000 0 10,000 20,000

0.5 1.0 1.5 2.0 2.5 15,000

17,000 19,000 21,000 23,000

-12,000 -8,000 -4,000 0

0 2,000 4,000 6,000 8,000

0 100

200 300400 0

-10 0 10 20

0.5 1.0 1.5 2.0 2.5 -20

-10 0 10 20

X-ax is (m m)

X- error Z- error

-4 -2 0 2

FPGA-Realization of a Motion Control IC for Robot Manipulator 311

2 The function of inverse kinematics is successfully implemented by hardware in FPGA;

as the result, it diminishes the computation time from 5.6ms using Nios II processor to 840ns using FPGA hardware, and increases the system performance

3 The software/hardware co-design technology under SoPC environment has been successfully applied to the motion controller of robot manipulator

Finally, the experimental results by the step response, the point-to-point motion trajectory response and the linear and circular motion trajectory response, have been revealed that based on the novel FPGA technology, the software/hardware co-design method with parallel operation ensures a good performance in the motion control system of robot manipulator

Compared with DSP, using FPGA in the proposed control architecture has the following benefits

1 Inverse kinematics and servo position controllers are implemented by hardware and the trajectory planning is implemented by software, which can all be programmable design Therefore, the flexibility of designing a specified function of robot motion controller is greatly increased

2 Parallel processing in each block function of the motion controller makes the dynamic performance of the robot’s servo drive increasable

3 In the commercial DSP product, it is difficult to integrate all the functions of implementing a five-axis motion controller for robot manipulator into only one chip

7 References

Altera Corporation, (2004) SOPC World

Altera (2008): www.altera.com

Hall, T.S & Hamblen, J.O (2004) System-on-a-programmable-chip development platforms

in the classroom, IEEE Trans on Education, Vol 47, No 4, pp.502-507

Monmasson, E & Cirstea, M.N (2007) FPGA design methodology for industrial control

systems – a review, IEEE Trans Industrial Electronics, Vol 54, No 4, pp.1824-1842

Kabuka, M.; Glaskowsky, P & Miranda, J (1988) Microcontroller-based Architecture for

Control of a Six Joints Robot Arm, IEEE Trans on Industrial Electronics, Vol 35, No

2, pp 217-221

Kung, Y.S & Shu, G.S (2005) Design and Implementation of a Control IC for Vertical

Articulated Robot Arm using SOPC Technology, Proceeding of IEEE International

Conference on Mechatronics, 2005, pp 532~536

Kung, Y.S.; Tseng, K.H & Tai, F.Y (2006) FPGA-based servo control IC for X-Y table,

Proceedings of the IEEE International Conference on Industrial Technology, pp

2913-2918

Kung, Y.S & Tsai, M.H (2007) FPGA-based speed control IC for PMSM drive with adaptive

fuzzy control, IEEE Trans on Power Electronics, Vol 22, No 6, pp 2476-2486

Lewis, F.L.; Abdallah C.T & Dawson, D.M (1993) Control of Robot Manipulators, Macmillan

Publishing Company

Li, T.S.; Chang S.J & Chen, Y.X (2003) Implementation of Human-like Driving Skills by

Autonomous Fuzzy Behavior Control on an FPGA-based Car-like Mobile Robot,

IEEE Trans on Industrial Electronics, Vol 50, No.5, pp 867-880

Oh, S.N.; Kim, K.I & Lim, S (2003) Motion Control of Biped Robots using a Single-Chip

Drive, Proceeding of IEEE International Conference on Robotics & Automation, pp

2461~2469

Schilling, R (1998) Fundamentals of Robotics – Analysis and control, Prentice-Hall

International

Shao, X & Sun, D (2005) A FPGA-based Motion Control IC Design, Proceeding of IEEE

International Conference on Industrial Technology, pp 131-136

Wei, R.; Gao, X.H.; Jin, M.H.; Liu, Y.W.; Liu, H.; Seitz, N.; Gruber, R & Hirzinger, G (2005)

FPGA based Hardware Architecture for HIT/DLR Hand, Proceeding of IEEE/RSJ

International Conference on intelligent Robots and System, pp 523~528

Xu, N.; Liu, H.; Chen, X & Zhou, Z (2003) Implementation of DVB Demultiplexer System

with System-on-a-programmable-chip FPGA, Proceeding of 5 th International Conference on ASIC, Vol 2, pp 954-957

Yang, G.; Liu, Y.; Cui, N & Zhao, P (2006) Design and Implementation of a FPGA-based AC

Servo System, Proceedings of the Sixth World Congress on the Intelligent Control and Automation, Vol 2, pp 8145-8149

Yasuda, G (2000) Microcontroller Implementation for Distributed Motion Control of

Mobile Robots, Proceeding of International workshop on Advanced Motion Control, pp

114-119

Zhou, Z.; Li, T.; Takahahi, T & Ho, E (2004) FPGA realization of a high-performance servo

controller for PMSM, Proceeding of the 9 th IEEE Application Power Electronics conference and Exposition, Vol.3, pp 1604-1609

17

Experimental Identification of the Inverse Dynamic Model: Minimal Encoder Resolution Needed Application to an Industrial Robot Arm

and a Haptic Interface

1IRCCyN & Nantes University, 2Haption S.A & CEA, LIST, Service de Robotique

Interactive, 3Laboratoire Central des Ponts et Chaussées

France

1 Introduction

To develop appropriate control laws and use fully the capacities of robots, a precise modelization is needed Classic models such as ARX or ARMAX can be used but in the robotic field the Inverse Dynamic Model (IDM) gives far better results In this model, the motor torques depend on the acceleration, speed and position of each joint, and of the physical parameters of the link of the robots (inertia, mass gravity, stiffness and friction) The parametric identification estimates the values of these last parameters These estimations can also help to improve the mechanical conception during retro-engineering steps… It comes that the identification process must be as accurate and reliable as possible The most popular identification methods are based on the least-squares (LS) regression

“because of its simplicity” (Atkeson et al., 1986), (Swevers et al., 1997), (Ha et al., 1989), (Kawasaki & Nishimura 1988), (Khosla & Kanade 1985), (Kozlowski 1998), (Prüfer et al., 1994) and (Raucent et al., 1992) In the last two decades, the IRCCyN robotic team has designed an identification process using IDM of robots and LS regressions which will be developed in the second part of this chapter This technique was applied and improved on several robots and prototypes (see Gautier et al., 1995 – Gautier & Poignet 2002 for example) More recently, this method was also successfully applied on haptic devices (Janot

et al., 2007)

However, it is very difficult to know how much these methods are dependent on the measurement accuracy, especially when the identification process takes place when the system is controlled by feedback So, we ignore the necessary resolution they require to produce good quality and reliable results

Some identification techniques seem robust with respect to measurement noises They are called “robust identification methods” But even if they give reliable results, they are only applied on linear systems and, overall, they are very time consuming as can be seen in (Hampel, 1971) and (Hubert, 1981) Finally, it seems difficult to apply them on robots and

we do not know how much they are robust with respect to these noises

Another simple and adequate way consists in derivating the CESTAC method (Contrôle et

Estimation Stochastique des Arrondis de Calculs developed in Vignes & La Porte, 1974)

which is based on a probabilistic approach of round-off errors using a random rounding

mode The third part of this chapter introduces the design and the application of a derivate

of the CESTAC method enabling us to estimate the minimal resolution needed for an

accurate parametric identification

This theoretical technique was successfully applied on a 6 degrees of freedom (DOF)

industrial arm (Marcassus et al., 2007) and a 3 DOF haptic device (Janot et al., 2007), the

major results obtained will be used to illustrate the use of this new tool of reliability

2 Inverse dynamic model and Least Squares estimation

2.1 General Inverse Dynamic Model

The IDM calculates the joint torques as a function of joint positions, velocities and

accelerations It is usually represented by the following equation:

where Γ is the torques vector of the actuators, q, and are respectively the joint positions,

velocities and accelerations vector of each links, A q is the inertia matrix of the robot, ( )

( )

H q, is the vector regrouping Coriolis, centrifugal and gravity torques applied on the

links, F and v Fs are respectively the viscous and Coulomb friction matrices

The parameters used in this model are XX , XY , XZ ,YY , YZ , ZZ the components of the j j j j j j

inertia tensor of link j, noted j

j

J , the mass of the link j called M , the inertia of the actuator j

noted Iaj, the first moments vector of link j around the origin of frame j noted

T j

j j j

MX MY MZ

j

MS , FV and j FS respectively the viscous and Coulomb friction j

coefficients and an offset of current measurement noted OFFSETj

The kinetic and potential energies being linear with respect to the inertial parameters, so is

the dynamic model (Gautier & Khalil, 1990) Equation (1) can thus be rewritten as:

To calculate the dynamic model we do not need all these parameters but only a set of base

parameters which are the ones necessary for this computation They can be deduced from

Experimental Identification of the Inverse Dynamic Model: Minimal Encoder Resolution

Needed Application to an Industrial Robot Arm and a Haptic Interface 315

the classical parameters by eliminating those which have no effect on the dynamic model

and by regrouping some others Actually, they represent the only identifiable parameters

Two main methods have been designed for calculating them: a direct and recursive method

based on calculation of the energy (Gautier & Khalil, 1990) and a method based on QR

numerical decomposition (Gautier, 1991) The numerical method is particularly useful for

robots consisting of closed loops

By considering only the b base parameters, equation (2) has to be rewritten as follows:

Generally, ordinary LS technique is used to estimate the base parameters by solving an

over-determined linear system obtained from the sampling of the dynamic model, along a

specifically dedicated trajectory (q,q,q), (Gautier et al., 1995) or (Khalil et al 2007)

X being the b minimum parameters vector to be identified, Y the torques measurements

vector, W the observation matrix and ρ the vector of errors, the system is described as

follows:

 

ˆXbeing the solution of the LS regression, it minimizes the 2-norm of the errors vector ρ W

is a r×b full rank and well conditioned matrix, obtained by tracking exciting trajectories and

by considering the base parameters, r being the number of samplings along a given

trajectory, r>>b

Hence, there is only one solution ˆX, (Gautier, 1997) :

with W+ the pseudo-invert matrix of W

Standard deviations of the identified parameters, σ

2 2

ρ

ˆY-WX

ˆX ˆX

j

σ

%σ =100

However, in practice, W is not deterministic This problem can be solved by filtering the

measurement vector Y and the columns of the observation matrix W

2.2.2 Data Filtering

Vectors Y and q are measures sampled during an experimental test We know that the LS

method is sensitive to outliers and leverage points so a median filter is applied to eliminate

them

The derivatives q and q are obtained without phase shift using a centered finite difference of

the vector q Knowing that q is perturbed by high frequency noises, which will be amplified by

the numeric derivations, a lowpass filter, whose order is greater than 2, is applied on q andq

The choice of the cut-off frequency ωf is very sensitive because the filtered data (q ,q ,q )f  f f

must be equal to the vector (q,q,q)  in the range [0, ωf] in order to avoid distortion of the

dynamic regressor So the filter must have a flat amplitude characteristic without phase shift in

the range [0, ωf], with the rule of thumb ωf>10*ωdyn, where ωdyn represents the dynamic

frequency of the system Considering an off-line identification, it is easily achieved with a

non-causal zero-phase digital filter by processing the input data through an IIR lowpass

Butterworth filter in both the forward and reverse directions

Since the measurement vector Y and matrix W have been filtered, a new filtered linear

system is defined:

Because there is no more signal in the range [ωf, ωs/2], where ωs is the sampling frequency,

the vector Yf and the columns of Wf are resampled at a lower rate after a lowpass filtering,

keeping one sample over nd samples, in order to obtain the final system to be identified

This process called parallel filtering is done thanks to the “decimate” function available in

the Signal Processing Toolbox of Matlab To have the same cut-off frequency ωf for the

lowpass filter, we choose:

Experimental Identification of the Inverse Dynamic Model: Minimal Encoder Resolution

Needed Application to an Industrial Robot Arm and a Haptic Interface 317

So the final linear system is:

fd fd fd

2.2.3 Exciting Trajectories

Knowing the base parameters, exciting trajectories must be designed In fact, they represent

the trajectories which excite well these parameters Design and calculations of these

trajectories can be found in (Gautier & Khalil, 1991)

If the trajectories are enough exciting, the conditioning number of W, (denoted cond(W)) is

close to 1 However, in practice, this conditioning number can reach few hundreds

depending of the high number of base parameters

If the trajectories are not enough exciting, the system is ill conditioned, undesirable

gatherings occur between inertial parameters and finally the results have no sense whatever

the encoder resolutions

3 Theory of the CESTAC method

From a theoretical point of view, the LS assumptions are violated in practical applications

In equation (6), the observation matrix W is built from the joint positions q which are

measured and fromq, q which are often computed numerically from q Therefore the

observation matrix is noisy Moreover identification process takes place when the robot is

controlled by feedback It is well known that these violations of assumption imply that the

LS estimator is biased

An adequate and simple way to evaluate the robustness of the LS estimator with respect to

the quantization errors (which contribute significantly to the bias of the estimator) consists

in deriving the CESTAC method which is based on a probabilistic approach of round-off

errors using a random rounding mode defined below:

Definition: each real number x, which is not a floating-point number, is bounded by two

consecutive floating-point numbers denoted respectively X−(rounded down) and X+

(rounded up) The random rounding mode defines the floating-point number X

representing x as being one of two values X−or X+with probability 1/2

Thus, with this random rounding mode, the same program run several times provides

different results, due to different round-off errors Under some assumptions, X can be

considered as a quasi-Gaussian distribution (Jezequel, 2004)

In our case, the position is perturbed by the encoder resolution This measurement can be

counted up (q+) as it can be counted down (q-) at each sample So, we can derive the

CESTAC method by building a new position:

( ) ( )

% 50 q P with Δ q q

Equation (15) defines our rounding mode Then, qCESTAC is filtered thanks to the data

filtering previously described Finally, we build a new linear regression called WCESTAC:

),

,( CESTAC CESTAC CESTAC

Each column of WCESTAC is filtered by using the decimation filter as explain in section xx

Hence, we obtain a new estimation of base parameters denoted XCESTAC:

Y W

We run this rounding mode N times and because of equation (15), we obtain different

results Hence, one obtains N estimation vectors denoted XCESTAC/k The mean value is

2 CESTAC CESTAC/k

1 N

1

X X

Then, the relative standard deviations are calculated:

CESTAC Xˆ

CESTAC j

We calculate the relative estimation error of the main parameters given by the following

equation:

j LS

j CESTAC

j 100 1 Xˆ Xˆ

where:

• XCESTACj is the jth main parameter identified by means of the CESTAC method,

• XLSj is the initial value of the jth main parameter identified through LS technique

Finally, we calculate the relative variation of the norm of the residual torque:

If the contribution of the noise from the observation matrix W is negligible compared to

modeling errors and current measurements noises, then %ej will be practically negligible

(less than 1%), as %e(|ρ|) (less than 1%) In this case, the bias of the LS estimator proves to

be negligible

Otherwise, the relative error %ej can not be neglected (greater than 5%), as %e(|ρ|)

(greater than 5%) This means that the LS estimator is biased and the results are

controversial

Experimental Identification of the Inverse Dynamic Model: Minimal Encoder Resolution

Needed Application to an Industrial Robot Arm and a Haptic Interface 319

The previous reasoning can be justified by considering the following equation:

• ∂ Yrepresents modeling errors and current measurements noises,

• X is the “true” solution

• Y is the “perfect” measurements vector

Hence, one obtains:

Y WX

This gives:

Y WX Y

WX

While a rounding mode is defined for q and if no variations are observed, then, it comes that

the solution X is weakly sensitive to quantizing noises It comes also that the norm of the

residual vector the is poorly correlated with ∂W We can write that:

4.1 6 DOF industrial robot arm

The robot to be identified is presented Figure 1, it is a Stäubli TX40 Its structure is a classical

anthropomorphic arm with a 6 DOF serial architecture and its characteristics can be found at

the Stäubli web site The initial encoder resolution is less than 2.10-4 degree per count so the

original resolution is more than enough to provide valuable measures

Figure 1 Stäubli TX40 to be identified

Table 1 Modified Dennavit Hartenberg Geometric Parameters for the Stäubli TX40

In order to establish the IDM, firstly we define the Modified Dennavit and Hartenberg (DHM) geometric parameters (Khalil & Kleinfinger, 1986), Table 1, based on the schematic

of Figure 2 Next, the linear regressor W and the base parameters are computed thanks to the software SYMORO+ (Khalil & Creusot 1997)

We notice that this robot has 60 base parameters, some inertial parameters are gathered with others, the letter “R” is added at the end of the regrouped parameters

Figure 2 DHM frames of the Stäubli TX40

The gathering rules give us the following analytic formulas:

ZZ1R=Ia1 + d3²*(M3+M4+M5+M6) + YY2 + YY3 + ZZ1 XX2R=-d3²*(M3+M4+M5+M6) + XX2 - YY2

XZ2R=-d3*MZ3 + XZ2 ZZ2R=Ia2 + d3²*(M3+M4+M5+M6) + ZZ2 MX2R=d3*(M3+M4+M5+M6) + MX2 XX3R=2*MZ4*RL4 + (M4+M5+M6)*RL4² + XX3 - YY3 + YY4

ZZ3R=2*MZ4*RL4 +(M4+M5+M6)*RL4² + YY4 + ZZ3 MY3R=MY3 - MZ4 -(M4+M5+M6)*RL4

ZZ4R=YY5 + ZZ4 MY4R=MY4 + MZ5 XX5R=XX5 - YY5 + YY6

Experimental Identification of the Inverse Dynamic Model: Minimal Encoder Resolution

Needed Application to an Industrial Robot Arm and a Haptic Interface 321

ZZ5R=YY6 + ZZ5 MY5R=MY5 - MZ6 XX6R=XX6 - YY6

4.1.1 Identification Results using the LS Technique

The exciting trajectories, illustrated on Figure 3, consist of polynomial positions which are designed to have constant velocities over a period (the gravity and friction parameters are thus well excited) and to reach maximum admissible accelerations (the inertia parameters are also well excited)

Figure 3 Typical Exciting Trajectory

In our case, W is a 42620x60 matrix and its conditioning number is close to 200, so the trajectories are enough exciting

The cut-off frequencies of the Butterworth filter and the decimate filter are close to 50Hz This value was found thanks to a spectral analysis

Finally, only 28 essential parameters are enough to characterize the dynamic model of the TX40 Our computations give us the whole 60 parameters, but at the end of our algorithm the parameter with the higher relative standard deviation is removed Then the LS Method is applied on the new dynamic model until the relative standard deviation of the error vector

is above a threshold: σρe≥1.04σρ, σρ being the initial relative standard deviation

Figure 4 Comparison between the measurement vector and the computed results

Figure 4 shows the measurement vector Yfd, the results of Wfd *Xfd and the error vector ρfd It

is obvious that the measured torques vector is matched by the estimated torques vector which validates the identification

The results of the identification are summed up in Table 2, only the most significant parameters for the purpose of this paper are written

Table 2 Reference Values of the Mechanical Parameters through the LS Technique

4.1.2 Identification with various lower resolution encoders

The identification protocol of the derivate CESTAC method is designed as explained in section 3 Considering the initial encoder resolution, for the first identification with a reduced resolution, we decide to use Δ1=2π/10000 Then the resolution is decreased by 1000 down to 2π/1000, 2π/500 is also used Only Δ2=2π/5000, Δ3=2π/2000 and Δ4=2π/1000, the more relevant, are presented in Table 3