CRC Press - Robotics and Automation Handbook Episode 1 Part 10 pdf

This ad-vanced model should include the discrete-time controller, quantized position feedback and control effort, higher order dynamics in the amplifier, higher-frequency mechanical mode

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Precision Positioning of Rotary and Linear Systems 13-19

–20 0

20

Sensor Noise at 1000 Hz Bandwidth, 3–sigma = 10.0 nm

–20 0

20

–20 0

20

Time, s

FIGURE 13.8 Effect of analog filtering on the resolution of an analog sensor.

controller loop bandwidth If the noise present in the sensor measurement is essentially random, the sensor resolution can be expressed as a spectral density and quoted as nm/√Hz The designer can then match the appropriate filter bandwidth to his resolution requirements However, a caution here is that many sensors have inherently higher levels of noise at the lower frequencies The amount of filtering required to achieve

a resolution may be higher than expected Analog sensors are usually supplied with signal conditioning electronics that convert the natural output of the sensor to a standard form (typically±10 V) These electronics should be placed as close as possible to the sensor because the low-level signals in the raw sensor output are often highly susceptible to electronic noise The conditioned signal is usually somewhat less sensitive but should still be shielded and routed as far as possible from high-power conductors Analog sensor accuracy and linearity is entirely distinct from the resolution specification Sensors may be highly linear about an operating point but often become increasingly nonlinear over their full range of travel This nonlinearity can be compensated for with a table or curve fit in software Sensor manufacturers may also provide a linearized output from their electronics package This is very useful for metrology, but a designer should confirm the update rate of this output before using for feedback Often this additional output is no longer a native analog signal, but it may have been quantized, corrected, and re-synthesized through a digital-to-analog converter All of these processes add time and phase lag to the measurement making it less suitable for use as a feedback device

13.3.6 Control Algorithms

The application of control algorithms separates mechatronics systems and mechatronic design from tradi-tional electromechanical systems Modern controllers do more than simply close a PID loop They provide additional options on feedback filtering and feedforward control, allow for coordination between multiple axes, and should provide an array of tuning and data collection utilities An overview of control algorithms used with precision machines is provided here

13.3.6.1 System Modeling

Dynamic models of the system should be developed for creating the controller and verifying its per-formance Each controller design should begin with a simple rigid-body model of the overall structure

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13-20 Robotics and Automation Handbook

1 Volts Current, A

Control

Effort Amplifier Gain

(Amps/Volt)

Forcer Gain (Newtons/Amp)

Damping (N/(m/s))

Km Force, N

Accn m/s ^ 2

Mass (kg)

Encoder Gain (counts/meter) Int 1 Int 2

Vel m/s Position, m

Position

Position, cnts 1/J

B

FIGURE 13.9 Simplified plant model used for initial single-axis tuning and motor sizing.

This simplified model is justified when, at a typical closed-loop bandwidth for mechanical systems of 10–

100 Hz, friction is low and any structural resonances are located at significantly higher frequencies While not complete, the simplified model shows the bulk rigid-body motion that can be expected given the mass

of the stage, the selection of motors and amplifiers, and the types of commanded moves It can also be used

to generate a more-accurate estimation of motor currents (and power requirements) for typical machine moves Figure 13.9 shows a block diagram of a simplified linear-motor-driven system The voltage from the analog output of the controller is converted into current by the amplifiers and into a force by the linear motors In this first model, we assume no dynamics in the amplifier and that commutation is implemented properly to eliminate force ripple The motor force accelerates the stage mass, and an encoder measures the resulting stage position The continuous-time transfer function from control-effort to stage position (in counts) is therefore

and this transfer function can be used to generate controller gains Scaling factors will always be specific

to the supplier of the control electronics, and it may take some research to find these exactly

The simplified model is adequate for the initial generation of gain parameters, but determining perfor-mance at the nanometer level always involves augmenting the model with additional information This ad-vanced model should include the discrete-time controller, quantized position feedback and control effort, higher order dynamics in the amplifier, higher-frequency mechanical modes (obtained either from a finite-element study or experimentally through a modal analysis), force ripple in the motors, error in the feedback device, and friction and process-force effects Some can be predicted analytically before building the system, but all should be measured analytically as the parts of the system come together to refine the model

13.3.6.2 Feedback Compensation

Commercially available controllers generally use some form of classical proportional-integral-derivative (PID) control as their primary control architecture This PID controller is usually coupled with one or more second-order digital filters to allow the user to design notch and low-pass filters for loop-shaping requirements In continuous-time notation, the preferred form of the PID compensator is

C (s ) = K p

1+ T D s+ 1

T I s

= K p

T D T I s2+ T I s+ 1

Separating the proportional term by placing it in front of the compensator allows the control engineer to compensate for the most common system changes (payload, actuator, and sensor resolution) by changing a

single gain term (K p) Techniques for designing continuous-time PID loops are well documented in almost any undergraduate-level feedback controls textbook The preferred method for designing, analyzing, and

characterizing servo system performance is by measuring loop-transmissions, otherwise known as

open-loop transfer functions This measure quantitatively expresses the bandwidth of the system in terms of the

crossover frequency and the damping in terms of the phase margin.

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13.3.6.3 Feedforward Compensation

Feedforward compensation uses a quantitative model of the plant dynamics to prefilter a command to improve tracking of a reference signal Feedforward control is typically not studied as often as feedback control, but it can cause tremendous differences in the ability of a system to follow a position command Feedforward control differs from feedback in that feedforward control does not affect system stability If

a perfect model of the plant were available with precisely known initial conditions, then the feedforward filter would conceptually consist of an inverse model of the plant Preshaping the reference signal with this inverse model before commanding it to the plant will effectively lead to unity gain (prefect tracking)

at all frequencies If properly tuned, the feedback system is required only to compensate for modeling inaccuracies in the feedforward filter and to respond to external disturbances In practical implementations, the plant cannot be exactly inverted over all frequencies, but the feedforward gain will instead boost tracking ability at low frequencies

Formal techniques exist for creating feedforward filters for generalized discrete-time plants Tomizuka [37] developed the zero phase error tracking controller (ZPETC) that inverts the minimum-phase zeroes

of a plant model and cancels the phase shift of the nonminimum phase components Nonminimum phase zeroes are a common occurrence in discrete-time systems The ZPETC technique begins with a discrete-time model of the closed-loop system (plant and controller),

G p (z−1)= z −d B c (z−1)B c+(z−1)

in which the numerator is factored in minimum-phase zeros B c+and non-minimum-phase zeros B c− The

z −d term is the d-step delay incurred due to computation time or any phase lag The ZPETC filter for this

system becomes

G ZPETC (z−1)= z d A c (z−1)B c−(z)

which cancels the poles and minimum-phase zeros of the system, and cancels the phase of the nonminimum phase zeros This is a noncausal filter, but it is made causal with a look ahead from the trajectory generator

In multi-axis machine, it is generally allowable to delay the position command by a few servo samples provided the commands to all axes are delayed equally and the motions remain synchronized

Feedforward control techniques are also available that modify the input command to cancel trajectory-induced vibrations One such technique, trademarked under the name Input Shaping was developed by Singer and Seering [31] at MIT Input Shaping,TMand related techniques, convolve the input trajectory with a series of carefully timed and scaled impulses These impulses are timed to be spaced by a half-period

of the unwanted oscillation and are scaled so that the oscillation produced by the first impulse is cancelled

by the second impulse Much of the research work in this area has involved making the filter robust to changes in the oscillation frequency The downside to this technique, in very general terms, is that the time allowed for the rigid-body move must be lengthened by one-half period of the oscillation to be cancelled Specialized feedforward techniques can be applied when the commanded move is cyclic and repetitive

In this case, the move command can be expressed as a Fourier series and decomposed into a summation

of sinusoidal components [19] The steady-state response of a linear system to a sinusoid is well defined as

a magnitude and phase shift of the input signal, and so each sinusoidal component of the command can

be pre-shifted by the required amount such that the actual movement matches the required trajectory

13.4 Conclusions

Precision positioning of rotary and linear systems requires a mechatronic approach to the overall system design The philosophy of determinism in machine design holds that the performance of the system can be predicted using familiar engineering principles In this chapter, we have presented some of the components that enter into a deterministic precision machine design Concepts such as alignment errors,

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force and metrology loops, and kinematic constraint guide the design of repeatable and analytic mechanics Often-neglected elements in the mechanical design are the cable management system, heat transfer and thermal management, environmental protection (protecting system from the environment it operates

in and vice-versa), and inevitable serviceability and maintance Mechatronic systems require precision mechanics; high-quality sensors, actuators, and amplifiers; and a supervisory controller This controller

is usually implemented with a microprocessor, and the discrete-time effects of sampling, aliasing, and quantization must be accounted for by the system design engineer Finally, the feedback control algorithms must be defined, implemented, and tested The elements used to create precision mechatronic systems are widely available, and it is left to the designer to choose among them trading complexity and cost between mechanics, electronics, and software

References

[1] The American Society for Precision Engineering, P.O Box 10826, Raleigh, NC 27605-0826 USA

[2] Auslander, D.M and Kempf, C.J Mechatronics: Mechanical System Interfacing Prentice Hall, Upper

Saddle River, NJ, 1996

[3] Blanding, D.L Exact Constraint: Machine Design Using Kinematic Principles American Society of

Mechanical Engineers, New York, NY, 1999

[4] Bryan, J.B The deterministic approach in metrology and manufacturing In: Proceedings of the ASME

1993 International Forum on Dimensional Tolerancing and Metrology, Dearborn, MI, Jun 17–19

1993

[5] Carbone, P and Petri, D Effect of additive dither on the resolution of ideal quantizers IEEE Trans.

Instrum Meas., 43(3):389–396, Jun 1994.

[6] DeBra, D.D Vibration isolation of precision machine tools and instruments Ann CIRP, 41(2):711–

718, 1992

[7] Donaldson, R.R and Patterson, S.R Design and construction of a large, vertical axis diamond

turning machine Proc SPIE – Int Soc Opt Eng., 433:62–67, 1983.

[8] Edlen, B The dispersion of standard air J Opt Soc Am., 43(5):339–344, 1953.

[9] Evans, C Precision Engineering: An Evolutionary View Cranfield Press, Bedford, UK, 1989.

[10] Evans, C.J., Hocken, R.J., and Estler, W.T Self-calibration: reversal, redundancy, error separation,

and “absolute testing.” CIRP Ann., 45(2):617–634, 1996.

[11] Franklin, G.F., Powell, J.D., and Workman, M.W Digital Control of Dynamic Systems, 3rd ed.,

Addison-Wesley, Reading, MA, 1998

[12] Fuller, C.R., Elliot, S.J., and Nelson, P.A Active Control of Vibration Academic Press, London, 1996 [13] Gordon, C.G Generic criteria for vibration-sensitive equipment In: Vibration Control in

Microelec-tronics, Optics, and Metrology Proc SPIE, 1619:71–85, 1991.

[14] Hale, L.C Friction-based design of kinematic couplings In: Proceedings of the 13th Annual Meeting

of the American Society for Precision Engineering, St Louis, MO, 18, 45–48, 1998.

[15] Hale, L.C Principles and Techniques for Designing Precision Machines Ph.D Thesis, MIT, Cambridge,

1999

[16] Jones, D.I.G Handbook of Viscoelastic Vibration Damping John Wiley & Sons, New York, 2001 [17] Kiong, T.K., Heng, L.T., Huifang, D., and Sunan, H Precision Motion Control: Design and

Imple-mentation Springer-Verlag, Heidelberg, 2001.

[18] Kurfess, T.R and Jenkins, H.E Ultra-high precision control In: William S Levine, editor, The

Control Handbook, 1386–1404 CRC Press and IEEE Press, Boca Raton, FL, 1996.

[19] Ludwick, S.J A Rotary Fast Tool Servo for Diamond Turning of Asymmetric Optics Ph.D Thesis, MIT,

Cambridge, 1999

[20] Marsh, E.R and Slocum, A.H An integrated approach to structural damping Precision Eng.,

18(2/3):103–109, 1996

[21] Nayfeh, S.A Design and Application of Damped Machine Elements Ph.D Thesis, MIT, Cambridge,

1998

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[22] Oppenheim, A.V and Shafer, R.W Discrete Time Signal Processing Prentice Hall, Englewood Cliffs,

NJ, 1989

[23] Oppenheim, A.V., Willsky, A.S., and Young, I.T Signals and Systems Prentice Hall, Englewood Cliffs,

NJ, 1983

[24] A◦strom K.J and Wittenmark, B Computer Controlled Systems: Theory and Design, 3rd ed., Prentice

Hall, 1996

[25] Riven, E.I Vibration isolation of precision equipment Precision Engineering, 17(1):41–56, 1995 [26] Riven, E.I Stiffness and Damping in Mechanical Design Marcel Dekker, New York, 1999.

[27] Riven, E.I Passive Vibration Isolation ASME Press, New York, 2003.

[28] Schmiechen, P and Slocum, A Analysis of kinematic systems: a generalized approach Precision

Engineering, 19(1):11–18, 1996.

[29] Schouten, C.H., Rosielle, P.C.J.N., and Schellekens, P.H.J Design of a kinematic coupling for

preci-sion applications Precipreci-sion Engineering, 20(1):46–52, 1997.

[30] Shigley, J.E and Mischke, C.R Standard Handbook of Machine Design McGraw-Hill, New York,

1986

[31] Singer, N.C and Seering, W.P Preshaping command inputs to reduce system vibration J Dynamic

Syst., Meas., Control, 112:76–82, March 1990.

[32] Slocum, A.H Kinematic couplings for precision fixturing — part 1: Formulation of design

param-eters Precision Engineering, 10(2), Apr 1988.

[33] Slocum, A.H Kinematic couplings for precision fixturing — part 2: Experimental determination of

repeatability and stiffness Precision Engineering, 10(3), Jul 1988.

[34] Slocum, A.H Precision Machine Design Prentice Hall, Englewood Cliffs, NJ, 1992.

[35] Smith, S.T and Chetwynd, D.G Foundations of Ultraprecision Mechanism Design Gordon and Breach

Science Publishers, Amsterdam, the Netherlands, 1992

[36] Stone, J., Phillips, S.D., and Mandolfo, G.A Corrections for wavelength variations in precision

interferometric displacement measurements J Res Natl Inst Stand Technol., 101(5):671–674,

1996

[37] Tomizuka, M Zero phase error tracking algorithm for digital control ASME J Dynamic Sys., Meas.,

Control, 109:65–68, 1987.

[38] Wirsching, P.H., Paez, T.L., and Ortiz, K Random Vibrations: Theory and Practice John Wiley &

Sons, New York, 1995

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14 Modeling and Identification for Robot Motion Control

Dragan Kosti ´c

Technische Universiteit Eindhoven

Bram de Jager

Maarten Steinbuch

14.1 Introduction 14.2 Robot Modeling for Motion Control Kinematic Modeling • Modeling of Rigid-Body Dynamics

• Friction Modeling 14.3 Estimation of Model Parameters Estimation of Friction Parameters • Estimation of BPS Elements • Design of Exciting Trajectory • Online Reconstruction of Joint Motions, Speeds, and Accelerations 14.4 Model Validation

Rigid-Body Dynamic Model

Direct-Drive Robotic Manipulator Experimental Set-Up • Kinematic and Dynamic Models in Closed Form • Establishing Correctness of the Models

• Friction Modeling and Estimation • Estimation of the BPS •

Dynamics Not Covered with the Rigid-Body Dynamic Model 14.7 Conclusions

14.1 Introduction

Accurate and fast motions of robot manipulators can be realized via model-based motion control schemes These schemes employ models of robot kinematics and dynamics This chapter discusses all the necessary steps a control engineer must take to enable high-performance model-based motion control These steps

are (i ) kinematic and rigid-body dynamic modeling of the robot, (ii) obtaining model parameters via direct measurements and/or identification, (iii) establishing the correctness of the models and validating the estimated parameters, and (iv) deducing to what extent the rigid-body model covers the real robot

dynamics and, if needed for high-performance control, the identification of the dynamics not covered by the derived model Better quality achieved in each of these steps contributes to higher performance of motion control

The robotics literature offers various tutorials on kinematic and dynamic modeling [1–4] A number

of modeling methods are available, meeting various requirements As for robot kinematics, the model is a mapping between the task space and the joint space The task space is the space of the robot-tip coordinates:

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end-effector’s Cartesian coordinates and angles defining orientation of the end-effector The joint space

is the space of joint coordinates: angles for revolute joints and linear displacements for prismatic joints

A robot configuration is defined in the joint space The mapping from the joint to the task space is called the forward kinematics (FK) The opposite mapping is the inverse kinematics (IK) The latter mapping reveals singular configurations that must be avoided during manipulator motions Both FK and IK can be represented as recursive or closed-form algebraic models The algebraic closed-form representation facili-tates the manipulation of a model, enabling its straightforward mathematical analysis As high accuracy of computation can be achieved faster with the closed-form models, they are preferable for real-time control

A dynamic model relates joint motions, speeds, and accelerations with applied control inputs: forces/ torques or currents/voltages Additionally, it gives insight how particular dynamic effects, e.g., nonlinear inertial and Coriolis/centripetal couplings, as well as gravity and friction effects, contribute to the overall robot dynamic behaviour A variety of methods are available for derivation of a dynamic model, which ad-mits various forms of representation Algebraic recursive models require less computational effort than the corresponding algebraic closed-form representations Recursive models have compact structure, requiring low storage resources Computational efficiency and low storage requirements are traditional arguments recommending the online use of recursive models [3] However, the power of modern digital computers makes such arguments less relevant, as online use of models in closed-form is also possible Furthermore,

an algebraic closed-form representation simplifies manipulation of a model, enabling an explicit analysis of each dynamic effect Direct insight into the model structure leads to more straightforward control design,

as compensation for each particular effect is facilitated For example, a closed-form model of gravity can

be used in the compensation of gravitational loads

Unfortunately, the derivation of closed-form models in compact form, in particular IK and dynamic models, is usually not easy, even if software for symbolic computation is used The derivation demands a series of operations, with permanent combining of intermediate results to enhance compactness of the final model To simplify derivation, a control designer sometimes approximates robot kinematic and/or inertial properties For example, by neglecting link offsets according to Denavits-Hartenberg’s (DH) notation [1,3],

a designer deliberately disregards dynamic terms related with these offsets, thus creating a less involved but incomplete dynamic model An approach to model manipulator links as slender beams can approximate robot inertial properties, since some link inertia or products of inertias may not be taken into account Consequently, the resulting model is just an approximation of the real dynamic behavior It is well-known that with transmissions between joint actuators and links, nonlinear coupling and gravity effects can become small [1–3] If reduction ratios are high enough, these effects can become strongly reduced and thus ignored in a dynamic model If robot joints are not equipped with gearboxes, which is characteristic for direct-drive robots [5], then the dynamic couplings and the gravity have critical influence on the quality

of motion control, and they must be compensated via control action More accurate models of these effects contribute to more effective compensation Additional methods to simplify dynamic modeling are neglecting friction effects or using conservative models for friction, e.g., adopting just viscous and ignoring Coulomb effects The approximate dynamic modeling is not preferable for high-performance motion control, as simplified models can compensate just for a limited number of actual dynamic effects The noncompensated effects have to be handled by stabilizing feedback control action The feedback control design is more involved if more dynamic effects are not compensated with the model

Obviously, when a model is derived, it is useful to establish its correctness Comparing it with some recursive representation of the same kinematics or dynamics is a straightforward procedure with available software packages For example, software routines specialized for robotic problems are presented in [6,7] With a model available, the next step is to estimate the model parameters Kinematic parameters are often known with a sufficient accuracy, as they can be obtained by direct measurements and additional kinematic calibration [8] In this chapter, emphasis will be on estimating the parameters of a robot dynamic model, i.e., the inertial and friction parameters The estimation of these parameters is facilitated if a model

of robot dynamics is represented in so-called regressor form, i.e., linearly in the parameters to be estimated The estimation itself is a process that requires identification experiments performed directly on a robot

To establish experimental conditions allowing for the simplest and the most time-efficient least-squares

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where

xi (k) = [e i (k), ˙e i (k), ν i (k)] T (14.29a)







1 T s 0.5T2

s





, gi =







T3

s /6

0.5T2

s

T s





Here, k abbreviates kT s A Kalman filter that optimally reconstructs the state xi in the presence of the model uncertaintyν iand measurement noiseη ihas the form:

ˆxi (k+ 1) = Ei (T s)¯xi (k)

¯xi (k)= ˆxi (k)+ ki [˜y i (k) − q r,i (k)− ciˆxi (k)] (14.30)

where ¯xidenotes updated state estimate and ki is a n× 1 vector of constant gains As the filter reconstructs deviations from the reference motions and speeds, together with the model uncertainty, the motion coordinates can be reconstructed as follows:

¯q i = q r,i + ¯e i

¯˙q i = ˙q r,i + ¯˙e i

To compute the vector ki, one needs covariances of the measurement noiseη iand of the process noise

ξ i Assuming white, zero mean quantization noise, uniformly distributed within a resolution increment

θ mof the position sensor, the straightforward choice forη iisθ2

m /12 (see [10] for details) The reference

[9] suggests a simple rule to chooseξ i: it should equal max ¨q2

r,i (maximum value of the square jerk of the motion reference) In practice, these rules are used as initial choices, while the final tuning is most

effectively done online However, use of Bode plots from the measured e i to the reconstructed ¯e i— an illustrative example is presented in Figure 14.2 — could be instructive for filter tuning The Bode plots

reveal which harmonics of e i will be amplified after reconstruction, as well as which harmonics will be

attenuated By analyzing the power spectrum of e i, a control engineer may locate spectral components

due to noise and other disturbing effects, that need to be filtered out The gain vector kican be retuned to reshape the Bode plot according to the needs

− 15

0 5

− 60

− 30 0

Frequency (Hz)

o )

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Modeling and Identification for Robot Motion Control 14-13

Elements of qr (t) should satisfy ˙q r,i (t) = ˙q r,i and ˙q r, j (t) ≡ 0 if i = j The specified set-points can be

achieved by applying the control law (14.34) with

u = −Kp(q − qr)− Kd( ˙q − ˙qr)+ n (14.36)

where Kp = diag[k p,1, , k p,n] and Kd = diag[k d,1, , k d,n] are matrices of positive position and

velocity gains, respectively, and the (n × 1) vector n contains a random excitation (noise) as ith element

and zeros elsewhere Identification is more reliable if the influence of disturbances (e.g., quantization noise)

affecting the manipulator motion and speed is low For that purpose, M, c, g, and τ fin Equation (14.34)

can be computed along qr (t) and ˙q r (t).

The spectrum analysis technique identifies an FRF of the plant dynamics P i ( j ω) = q i ( j ω)/u i ( j ω),

whereω denotes angular frequency A comprehensive description of techniques for FRF identification is

available in [24] If computed directly from the measured u i (t) and q i (t), the obtained FRF may appear

unreliable We characterize an FRF as unreliable if the coherence function

2

nq

 q q nn

(14.37)

is small. nnand q q denote the autopower spectra of the excitation signal and of the joint response, respectively, while nq is the cross power spectrum between the excitation and the response. becomes

small if the excitation and response data are correlated, which is usual if these data are measured in closed-loop If this is the case, one should carry out indirect, instead of the direct identification of the plant’s FRF

In the indirect identification, an FRF of the transfer from n i (t) to u i (t) is estimated first (n i is the i th

element of n in (14.36)) This estimation represents the sensitivity function S i ( j ω) = [1 + ( jωk d,i +

k p,i )P i ( j ω)]−1for the feedback loop in the joint i The coherence of this approach is normally sufficient for

reliable estimation in the frequency range beyond the bandwidth of the closed-loop system in the joint i Hence, a low bandwidth for the closed-loop in the joint i is preferable On the other hand, sufficiently high

bandwidths of the feedback loops in the remaining joints are needed to prevent any motion in these joints Therefore, separate sets of controller settings are required With the identified sensitivity, for a known setting of the PD gains, one may find the plant’s FRF:

P i ( j ω) = 1/S i ( j ω) − 1

j ωk d,i + k p,i

(14.38)

The obtained FRF represents dynamics of the joint i , which are not compensated with the rigid-body

dynamic model, as a set of data For the control design it is more convenient to represent these dynamics with some parametric model in a transfer function or state-space form There are a number of solutions

to fit a parametric model into FRF data [24] The least-square fit using an output error model structure model is one possibility [25]

The time-domain technique directly determines a parametric model of the plant dynamics Here, it is

only sketched for the case of direct identification, which is based on the measured excitation u i (t) and response q i (t) data Extension to indirect identification is straightforward If collected at discrete time instants t1, t2, , t ζ, the measurement pairs{u i (t1), q i (t1)}, {u i (t2), q i (t2)}, , {u i (t ζ ), q i (t ζ)} can be used for identification of the plant dynamics assuming a transfer function model of the form

Q i (z)

U i (z) = P i (z)= b1z−1+ b2z−2+ · · · + b n z −n

1− a1z−1− · · · − a n z −n (14.39)

This model, defined in Z-domain, is often referred to as the ARMA (autoregressive moving-average) model [26] Provided n < ζ , the least-squares estimates of the model coefficients

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Equation (14.43) and Equation (14.44) define forward kinematics (FK) of the RRR robot The inverse

kinematics (IK) is determined by solving the Equations (14.44) for q1, q2, and q3:

q1= asinx(d2+ d3)+ y

x2+ y2− (d2+ d3)2

x2+ y2

q3= atan±

1− p2

w h + p2

w v − a2− a22

/(2a2a3)2

p2

w h + p2

w v − a2− a2

/(2a2a3)

q2= atan(a2+ a3cos q3) p w v − a3sin q3p w h

(a2+ a3cos q3) p w h + a3sin q3p w v

p w h=

[x − (d2+ d3) sin q1]2+ [y + (d2+ d3) cos q1]2, p w v = z − d1 (14.45)

The closed-form representation (14.45) has three advantages: (i ) an explicit recognition of kinematic singularities, (ii) a direct selection of the actual robot posture by specifying a sign in the expression for

q3, as the arm can reach each point in the Cartesian space either with the elbow up (− sign) or with the elbow-down (+ sign), and (iii) a faster computation of a highly accurate IK solution than using any

recursive numerical method

A rigid-body dynamic model of the RRR robot was initially derived in the standard form (14.6) After that, for the sake of identification, it was transformed into regressor form (14.10) The number

of BPS elements is 15, coinciding with the number obtained using the formula derived in [14] For a given kinematics, this formula computes the minimum number of inertial parameters whose values can

determine the dynamic model uniquely The elements of p and R are expressed in closed-form:

p1= a2

2m3+ a2m2(a2+ 2x2)− Ixx,2+ Iyy,2

p2= −2(a2m2y2+ Ixy,2)

p3= m3(a3+ x3)

p4= m3y3

p5= a3m3(a3+ 2x3)− Ixx,3+ Iyy,3

p6= 2(a3m3y3+ Ixy,3)

p7= 2d2m2z2+ d2

2m2+ 2(d2+ d3)m3z3+ (d2+ d3)2m3+ Iyy,1+ Ixx,2+ Ixx,3

p8= −(d2+ d3)m3y3− Iyz,3

p9= −a3m3(z3+ d2+ d3)− (d2+ d3)m3x3− Ixz,3

p10= −d2m2y2− Iyz,2

p11= −a2[m3(z3+ d2+ d3)+ m2(d2+ z2)]− d2m2x2− Ixz,2

p12= a22+ a2

3+ 2a3x3

m3+ a2m2(a2+ 2x2)+ Izz,2+ Izz,3

p13= a3m3(a3+ 2x3)+ Izz,3

p14= m2(x2+ a2)+ m3a2

r1,1= ¨q1cos2q2− ˙q1˙q2sin(2q2)

r1,2= 0.5¨q1sin(2q2)+ ˙q1˙q2cos(2q2)

r1,3= 2a2{[¨q1cos(q2+ q3)− ˙q1˙q3sin(q2+ q3)] cos q2− ˙q1˙q2sin(2q2+ q3)}

r1,4= −2a2[( ¨q1sin(q2+ q3)+ ˙q1˙q3cos(q2+ q3)) cos q2+ ˙q1˙q2cos(2q2+ q3)]

r1,5= ¨q1cos2(q2+ q3)− ˙q1( ˙q2+ ˙q3)sin(2q2+ 2q3)

r1,6= −0.5¨q1sin(2q2+ 2q3)− ˙q1( ˙q2+ ˙q3)cos(2q2+ 2q3)

r1,7= ¨q1

14 -1 6 Robotics and Automation Handbook

Equation (14 .43) and Equation (14 .44) define forward kinematics... initially derived in the standard form (14 .6) After that, for the sake of identification, it was transformed into regressor form (14 .10 ) The number

of BPS elements is 15 , coinciding with the... b1z? ?1+ b2z−2+ · · · + b n z −n

1− a1z? ?1−

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