Kurfess T.R. Robotics and Automation Handbook

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Precision Positioning of Rotary and Linear Systems 13

-19

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

–20

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

Position, nm

–20

Sensor Noise at 100 Hz Bandwidth, 3–sigma = 3.2 nm

Position, nm

–20

Sensor Noise at 10 Hz Bandwidth, 3–sigma = 1.0 nm

Position, nm

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.

-20 Robotics and Automation Handbook

Ka Ke 11

Current, AVolts

Control

Effort

Amplifier Gain

(Amps/Volt)

Forcer Gain

(Newtons/Amp)

Damping

(N/(m/s))

Force, N

Accn

m/s

Mass

(kg)

Encoder Gain

(counts/meter)

Int 1 Int 2

Vel. m/s Position, m

Position

Position, cnts

1/J

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

(

)

s(ms +b)

(13.19)

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, errorin the feedback

device, and frictionand process-forceeffects. Some can be predicted analyticallybefore 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

(

)

= K



1 + T

s +



= K

+ T

s + 1

(13.20)

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

). 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.

Precision Positioning of Rotary and Linear Systems 13

-21

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

inaccuraciesin the feedforwardﬁlterand to respondto external disturbances. In practicalimplementations,

the plant cannot be exactly invertedoverall frequencies, but the feedforward gain willinsteadboost 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),

−1

) =

−d

−

−1

)

−

(13.21)

in which the numerator is factored in minimum-phase zeros B

and non-minimum-phase zeros B

−

.The

−d

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

system becomes

ZPETC

−1

) =

−1

−

(z)

−1

−

(1)

(13.22)

whichcancels the poles and minimum-phase zerosof 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,

and 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,

-22 Robotics and Automation Handbook

forceand metrology loops, and kinematic constraint guide the designof 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. Precision 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.FoundationsofUltraprecision MechanismDesign. GordonandBreach

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.

Modeling and

Identiﬁcation for

Robot Motion Control

Dragan Kosti

Technische Universiteit Eindhoven

Bram de Jager

Technische Universiteit Eindhoven

Maarten Steinbuch

Technische Universiteit Eindhoven

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

14.5 Identiﬁcation of Dynamics Not Covered with a

Rigid-Body Dynamic Model

14.6 Case-Study: Modeling and Identiﬁcation of a

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:

-2 Robotics and Automation Handbook

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 neglectinglink offsets accordingto 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 ignoredin a dynamic model. If robot joints are not equippedwithgearboxes, 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

Modeling and Identiﬁcation for Robot Motion Control 14

-3

(LS) estimation of the parameters, joint motions, speeds, and accelerations can be reconstructed via an

observer [9,10]. Care should be takenwhen designingrobot trajectories to be realized duringidentiﬁcation.

Guidelines for their design are highlighted in [11,12].

Afterthe estimation is ﬁnished, experimentalvalidation of the model has to be done. Its objective is to test

how accurately the model represents actual robot dynamics. If the accuracy is satisfactory, the usefulness of

the model for model-based control purposes is established. For validation purposes, a manipulator should

execute motions similar to those it is supposed to perform in practice. In the task space, these are the

sequences of straight-line and curved movements. Hence, a writing task deﬁned in the task space might

be used as a representative validation trajectory. The choice of such a task is intuitive because of at least

two reasons: writing letters is, indeed, a concatenation of straight-line and curved primitives, similar to

common robot movements, and such a task may strongly excite the robot dynamics due to abrupt changes

in accelerations and nonuniform speed levels. An example of a writing task applicable for model validation

can be found in [13].

A dynamic model can be further used in model-based control algorithms. The model contributes to the

performance of the applied control algorithms to the extent that it matches the real robot dynamics. In

practice, the rigid-body models may cover the real dynamics only within the low-frequency range, as the

real dynamics feature ﬂexible effects at higher frequencies. These effects are caused by elasticity in joints

and by distributed link ﬂexibilities. For high-performance motion control, the inﬂuence of ﬂexibilities

cannot be disregarded. Here we present a technique for the identiﬁcation and modeling of these effects.

The rest of the chapter is organized as follows. In Section 14.2, kinematic and rigid-body dynamic

models will be derived for a general robot manipulator. A regressor form of a dynamic model will be

introduced, and a set of often used friction models will be presented. An estimation of robot inertial and

friction parameters will be explained in Section 14.3. A way to validate a dynamic model, suitable for

motion control purposes, will be presented in Section 14.4. Identiﬁcation and modeling of the dynamics

not covered by a rigid-body model will be discussed in Section 14.5. Section 14.6 is a case-study. It

demonstrates modeling and identiﬁcation strategy for a realistic direct-drive manipulator. Conclusions

will come at the end.

14.2 Robot Modeling for Motion Control

14.2.1 Kinematic Modeling

Consider a serial robot manipulator with n-joints, shown in Figure 14.1. A minimal kinematic parameter-

isation for this manipulator can be established according to the well-known DH (Denavits-Hartenberg)

convention [1,3]. This convention systematically assigns coordinate frames to the robot joints and induces

the following DH parameters: twist angles α

, link lengths a

, joint displacements q

, and link offsets d

−1

‘0’

‘

’

‘

−1’

‘

’

FIGURE 14.1 A general serial manipulator with n joints.

-4 Robotics and Automation Handbook

where i = 1, ..., n. The position and orientation of the ith coordinate frame with respect to the previous

one (i − 1) can be speciﬁed by the homogenous transformation matrix:

i−1

(q) =



i−1

(q)

i−1

(q)

1×3



(14.1)

Here, q is the vector of generalized coordinates:

q = [q

... q

]

(14.2)

The coordinates are linear and angular displacements of prismatic and revolute joints, respectively. The

skew-symmetric rotation matrix

i−1

∈ IR

3×3

deﬁnes the orientation of the ith frame with respect to

frame i − 1:

i−1

(q) =







cos q

−cos α

sin q

sin α

sin q

cos α

cos q

−sin α

cos q

0 sin α

cos α







(14.3)

The position vector

i−1

∈ IR

3×1

points from the origin of the frame i − 1 to the ith frame:

(

i−1

(q))

= [

cos q

sin q

] (14.4)

The forward kinematics (FK) can be computed as a product of homogeneous transformations between

the circumjacent coordinate frames:

(q) =

(q)

(q) ...

n−1

(q) =



(q)

1×3



(14.5)

The orientation and position of the tip coordinate frame n with respect to the base (inertial) frame 0 are

determined by

and

, respectively. In general case, both

and

nonlinearly depend on the

generalized coordinates, and thus it is not always possible to explicitly express q in terms of the tip position

and orientation coordinates. Consequently, there is no general closed-form representation of the inverse

kinematics (IK), and numerical techniques are often used to solve IK [1].

14.2.2 Modeling of Rigid-Body Dynamics

The rigid-body model of the robot dynamics can be derived using the Euler-Lagrange formulation [1–3].

The standard form of the model is as follows:

M(q(t))

q(t) + c(q(t),

q(t)) + g(q(t)) +τ

(t) = τ (t) (14.6)

where q is deﬁned by Equation (14.2),

q and

q are vectors of the joint speeds and accelerations, respectively,

M is the inertia matrix, c, g, and τ

are the vectorsof Coriolis/centripetal, gravitational, and friction effects,

respectively, and τ is the vector of joint control inputs (forces/torques). Elements of M, c, and g can be

determined using the homogenous transformation (14.1):

i,k



j=max(i,k)



∂

∂q



∂

∂q





, c



k=1



l=1

i,k,l



j=max(i,k,l)



∂

∂q



∂

∂q







j=i



−m

[0 0 − g 0]

∂

∂q



, i, k, l = 1, ..., n (14.7)

Modeling and Identiﬁcation for Robot Motion Control 14

-5

Here, m

is the mass of the jth link, g is acceleration due to gravity,

contains the homogenous

coordinates of the center of mass of link j expressed in the ith coordinate frame

= [x

(14.8)

while J

is the inertia tensor with elements:

j,11

−I

xx,j

+ I

yy,j

+ I

zz, j

; j

j,22

xx,j

− I

yy,j

+ I

zz, j

; j

j,33

xx,j

+ I

yy,j

− I

zz, j

j,12

= j

j,21

= I

xy,j

; j

j,13

= j

j,31

= I

xz, j

; j

j,23

= j

j,32

= I

yz, j

; j

j,14

= j

j,41

= m

j,24

= j

j,42

= m

; j

j,34

= j

j,43

= m

; j

j,44

= j

j,44

= m

(14.9)

xx,j

, I

yy,j

, and I

zz, j

denote principal moments of inertia for the link j, and I

xy,j

, I

yz, j

, and I

xz, j

are

products of inertia of the same link.

The dynamic model (14.6), excluding friction, can be represented linearly in elements of the so-called

base parameter set (BPS). These elements are combinations of the inertial parameters used in the equations

above, and they constitute the minimum set of parameters whose values can determine the dynamic model

uniquely [1,11,12,14]. A linear parameterization of the rigid-body dynamic model (14.6) has the form:

R(q(t),

q(t),

q(t))p + τ

(t) = τ (t)

(14.10)

where R ∈ IR

n×p

is the regression matrix and p ∈ IR

p×1

is the vector of the BPS elements. The regressor

form of the dynamic model is suitable for the estimation of inertial parameters.

14.2.3 Friction Modeling

The dynamic models (14.6) and (14.10) also feature the friction term τ

. In the real system, friction cannot

be disregarded if high-performance motion control is required. Hence, modeling and compensation of

friction effects are important issues in robot motion control. A basic friction model covers just Coulomb

and viscous effects:

1,i

sgn

+ α

2,i

= τ

(i = 1, ..., n) (14.11)

where τ

is the ith element of τ

, while α

1,i

and α

2,i

are Coulomb and viscous friction parameters,

respectively. This model admits a linear representation in the parameters α

1,i

and α

2,i

. This enables a

joined regressor form of the dynamic model (14.10), where the new regression matrix is formed by vertical

stacking R and the rows containing friction terms, while the parameter vector consists of p and friction

parameters. The joined regressor form allows simultaneous estimation of the BPS and friction parameters

[11,12].

However, frictionis a verycomplexinteraction between thecontactsurfacesofthe interacting objects and

needs a comprehensive model to be described correctly [15–19]. Presliding and sliding are two commonly

recognized friction regimes [15,16]. The level of friction prior to the sliding regime is called the static

friction, or the stiction. At the beginning of the sliding regime, the decreasing friction force for increasing

speed is known as the Stribeck effect. Beyond this stage, the frictionforceincreases as the speed is increasing,

which is caused by the viscous effect. A time lag between the speed and friction force can be detected,

which implies a dynamic, rather than just static nature of the friction. Several static and dynamic models

were proposed to represent the given properties of friction.

A frequently used static friction model has the form:

γ (

)sgn

+ α

2,i

= τ

(

) = α

0,i

exp[−(

s,i

)

] +α

1,i

(i = 1, ..., n) (14.12)

where γ (

) is the Stribeck curve for steady-state speeds, v

s,i

is the Stribeck velocity, and α

0,i

, α

1,i

, and α

2,i

are static, Coulomb, and viscous friction parameters, respectively. The given model describes the sliding