Sliding Mode Control Part 9 pdf

In the first phase of the manipulator motion the classical controller CL gives smaller orientation error thanthe GVC controller but after about 1 s the GVC controller gives better perform

represents a rotational inertia about joint axis arising from the motion of other manipulator

links Figure 2(h) compares the kinetic energy for the whole manipulator K and for all joints The great value of K3 can be related to the dominant values of N3 (the two informations can be

obtained only for the GVC controller) From Figure 2(i) it is observable that the kinetic energy

is reduced faster using GVC controller than using the CL controller

4.2.2 Yasukawa-like manipulator

The manipulator is depicted in Figure 1(b) The given below results are based on Herman(2009b)

The polynomial trajectories were described with initial pointsθ i1 = (1/3)π rad, θ i2 = π

rad, θi3 = (−1/2)π rad, final points θ f 1 = (−2/3)π rad, θ f 2 = 0 rad, θ f 3 = (1/2)π rad, and the time duration t f = 1 s The starting points were different from the initialpoints Δ = +0.2,+0.2,+0.2 rad It was assumed the following control coefficients set:

k D = diag{10, 10, 10}, Λ = diag{15, 15, 15}, k P = diag{150, 150, 150} for the GVC

controller (30) For the classical controller (28) we assumed the set k D = diag{10, 10, 10},

Λ=diag{30, 30, 30} Diagonal elements values of the matrixΛ are two times smaller for theGVC controller than for the classical one For the same set of coefficients performance of theclassical controller are worse than for the considered case

Profiles of the desired joint position and velocity trajectories are shown in Figure 3(a) The jointposition errors for the GVC and the classical (CL) controller are shown in Figures 3(b) and 3(c),respectively It is observable that the errors for the GVC controller tend very fast to zero andthe manipulator works correctly But for the CL controller the joint position errors tend to

zero more slowly Increasing the gain coefficients kD orΛ could lead to better performanceobviously under condition avoidance undesirable over-regulation This observation confirmsFigure 3(d) because the error norm (in logarithmic scale) has distinctly smaller values forthe GVC controller than for the CL one Figure 3(e) presents the joint torques obtained usingthe GVC controller (for the classical one they have almost the same values) In Figure 3(f)

elements of the matrix N are shown (such information gives only for the GVC controller).

These quantities represents some rotational inertia along each axis which arise from otherlinks motion They are characteristic for the tested manipulator and for the desired joint

velocity set Values N3 are dominant almost all time what says that the third joint is the most laden Figure 3(g) a kinetic energy time history for the total manipulator K and for all joints

is presented Most of the kinetic energy is related to the second joint (K2) (and also to the same link) This fact may be associated with the dominant values N2 in the time interval

0.4÷0.6 s Figure 3(h) compares the kinetic energy reduction for the manipulator if bothcontrol algorithms are used It can be noticed that after some time this energy is reducedmuch faster using the GVC controller than using the classical one

4.3 NQV - joint space

Simulations were done for the DDArm manipulator with the parameters given in Table 1 and

under the same conditions The assumed gain coefficients set was kD=diag{10, 10, 10}, Λ=diag{15, 15, 15}, kP =diag{150, 150, 150}for the NQV controller and k D=diag{10, 10, 10},

Λ = diag{15, 15, 15} for the CL one This means also that the desired joint position andvelocity trajectories are assumed as in Figure 2(a)

Simulation results obtained from the NQV controller (42) and the CL controller (28) arepresented in Figure 4 The joint position errors for the NQV and the CL controller, arepresented in Figure 4(a) and 4(b) One can observe that for the NQV controller all position

269Non-Adaptive Sliding Mode Controllers in Terms of Inertial Quasi-Velocities

Q [Nm]

t [s]

GVC Q1

Q2 Q3

(e)

0 2 4 6 8

N2 N3

(f)

0 50 100

150

200K [J]

t [s]

GVC K1

K2

K3 K

desired joint position thd and joint velocity vd trajectory for all joints of manipulator; b) joint position errors e for GVC controller; c) joint position errors e for classical (CL) controller; d)

comparison between joint position error norms||e||(in logarithmic scale) for both

controllers; e) joint torques Q obtained using GVC controller; f) elements of matrix N

obtained from GVC controller; g) kinetic energy reduced by each joints and by the totalmanipulator (GVC controller); h) comparison between kinetic energy (in logarithmic scale)

for classical (K CL ), and GVC (K GVC) controller

manipulator joints and entire kinetic energy K; f) comparison between kinetic energy

reduction for NQV and CL controller (in logarithmic scale)

errors tend to zero after about 1.6 s For the CL controller errors e1, e2tend very fast to zero but

e3tends to zero more slowly than for the NQV controller Figure 3(c) shows the joint torques

obtained from the NQV controller The big initial value of the joint torque Q3 arises fromthe fact that we feed back some quantity including the kinematic and dynamical parameters

of the manipulator instead of the joint velocity only However, for the tested manipulator

this value is allowed as results from reference An et al (1988) The articulated inertia D kfor

each joint (Figure 4(d)) can be obtained only using the NQV controller Each value D ksays

how much inertia rotates about the k-th joint axis Most of the rotational inertia is transfered

by the third joint axis which means that dynamical interactions are great for the third jointand the third link Figure 4(e) gives a time history of the kinetic energy for each joint andfor the manipulator Most of the energy is related to the third link which can be explained by

great values of D3 Next Figure 4(f) compares the kinetic energy (in logarithmic scale) which is

reduced by the manipulator After about 1.6 s the kinetic energy is canceled for NQV controllermuch faster than for CL controller

4.4 GVC - operational space

The simulation results are obtained for a 3 D.O.F Yasukawa-like manipulator Herman (2009a).The first objective is to show performance of the GVC controller (55) in the manipulatoroperational space The following parameters are different than in Table 2:

• link masses: m1=5 kg, m3=60 kg;

271Non-Adaptive Sliding Mode Controllers in Terms of Inertial Quasi-Velocities

• link inertias: J xx2 = 0.6 kgm2, J xz1 = 0.02 kgm2, J yy1 = 0.05 kgm2, J yy2 = 0.8 kgm2,

1 1.5

The desired position and orientation described by the vector x d = p dx p dy p dz o dφ o dϑ

Tare shown in Figures 5(a) and 5(b) Herman (2009c)

The simulations results realized in MATLAB/SIMULINK (Figure 6) come from referenceHerman (2009a).) The control gain matrices were assumed for all controllers as follows:

kD = diag{20, 20, 20},Λ = diag{20, 20, 20, 20, 20, 20}, kP = diag{20, 20, 20, 20, 20}, ρ = 1

Viscous damping coefficients were the same for all joints F=diag{2, 2, 2}

Figures 6(a) and 6(b) show the position and the orientation error for the GVC controller (55)

in the operational space, respectively One can observe that both errors converge to zero afterabout 2 s Next, in Figures 6(c) and 6(d) the same errors for the classical controller (50) arepresented As arises from both figures in order to achieve the steady-state the controller needsmore than 3 s At the same time the orientation errors are only close to zero In the first phase

of the manipulator motion the classical controller (CL) gives smaller orientation error thanthe GVC controller but after about 1 s the GVC controller gives better performance Thisphenomenon results from the fact that the dynamical parameters set in the controller (55)

is used From Figure 6(e) one can observe that after 1 s the kinetic energy K GVC (for the

GVC controller) is reduced faster than for the classical controller K CL(results are presented

on logarithmic scale)

In Figure 6(f) the position error norms (on logarithmic scale) measured in the manipulatortask space for the GVC controller and the classical controller (CL) are compared It can be seenthat the position error norm||ep||GVCis smaller than the error norm||ep||CL Comparisonbetween the orientation error norms for both controllers are given in Figure 6(g) In the firstphase of the manipulator motion the classical controller (CL) gives smaller orientation errorthan the GVC controller but after about 0.9 s the latter controller gives better performance.This behavior also results from the fact that the dynamical parameters set in the controller(55) is used The joint torques for the GVC controller are shown in Figure 6(h) It is observable

that at the start (before 0.2 s) the torque in the third joint Q3has great value (it is a consequence

comparison between position error norm on logarithmic scale for both controllers; g)

comparison between orientation error norm on logarithmic scale for both controllers (GVC

and CL); h) joint torques Q k for GVC controller; i) joint torques Q kfor CL controller

273Non-Adaptive Sliding Mode Controllers in Terms of Inertial Quasi-Velocities

of including dynamical parameters of the system in the GVC controller) As it is shown fromFigure 6(i) the third joint torque for the CL controller has smaller value than using the GVCone However, after about 0.3 s the torques for both controllers are comparable.

The gain matrices were chosen as (the same for both controllers, i.e the NGVC and the CL):

kD = diag{4, 4, 4},Λ =diag{20, 20, 20, 20, 20, 20}, kP =diag{5, 5, 5, 5, 5},δ =0.25 whereas

the viscous damping coefficients were F=diag{2, 2, 2}

In Figures 7(a) and 7(b) the position and the orientation error for the NGVC controller (59) inthe operational space are shown Both errors tend to zero after about 1.5 s The same errors forthe classical (CL) controller (50) are given in Figures 7(c) and 7(d) After 3 s (Figure 7(c)) theposition steady-state is not achieved As a result to ensure the satisfying error convergence,the CL controller needs more time than 3 s The same conclusion can be made about theorientation error convergence (Figure 7(d)) The joint applied torques for the NGVC controllerare shown in Figure 7(e) Comparing Figures 7(e) and 7(f) it can be observed that maximumvalues of the torques using the NGVC controller are not much larger than if the CL controller

is applied

The diagonal elements of the matrixΦ are given in Figure 8(a) whereas the off-diagonal ones

in Figure 8(b) Recall that the matricesΦTandΦ give an additional gain in the term ΦT k DΦ ofthe controller (59) It can be concluded that the NGVC controller uses small control coefficients

k D kto ensure fast position and orientation trajectory tracking Moreover, each elementΦ2

kk represents an rotational inertia corresponding to the k-th quasi-velocity, whereasΦki (for i = k)

show dynamic coupling between the joint velocities (and also between the appropriate links).Such information is available only from the NGVC controller

From Figure 8(c) it can be seen that the kinetic energy K which must be reduced by the manipulator concerns mainly the third quasi-velocity K3(and also by the 3-th link) Figure 8(d)compares the kinetic energy reduction (on logarithmic scale) for both controllers After about

1 s the kinetic energy K NGVCfor the NGVC controller decreases faster than for the classical

controller K CL Consequently, the NGVC control algorithm gives faster error convergence thanthe CL control algorithm

4.6 Discussion

From the presented simulation results arises the fact that the proposed nonlinear controllers interms of the IQV ensures faster, than the classical controller, the position and orientation errorconvergence Moreover, the kinetic energy reduction is also faster if the IQV controller is used

An disadvantage of the IQV controllers is that sometimes, at the beginning of motion, great

controller; e) joint applied torques Q for NGVC controller; f) joint applied torques Q for CL

controller

initial torque can occur The great values come from including the manipulator parameters setinto the control algorithm Note, however that the same reason causes the benefit concerningthe fast error convergence and fast kinetic energy reduction Thus, it should be verified iffor the considered manipulator the real torques are acceptable It can be done via simulation

because the expected torques are determined from the time history of Q To obtain comparable results as for the IQV controller we have to assume for the CL controller the matrix k Dwithbigger gain coefficients However, at the same time elements of the matrixΛ should be enoughgreat to ensure fast error convergence From all presented cases arise that if the IQV controller

is used then the gain matrix kDhas rather small values One can say that they serve for precisetuning because the resultant gain matrix is related to the system dynamics

5 Conclusion

In this paper, a review of a theoretical framework of non-adaptive sliding mode controllers

in terms of the inertial quasi-velocities (IQV) for rigid serial manipulators was provided.The dynamics of the system using several kind of the IQV, namely: the GVC, the NQV, andthe NGVC was presented The IQV equations of motion offer some advantages which areinaccessible if the classical second-order differential equations are used The IQV sliding modecontrol algorithms, based on the decomposition of the manipulator inertia matrix, can berealized both in the manipulator joint space and in its the operational space It was shown

275Non-Adaptive Sliding Mode Controllers in Terms of Inertial Quasi-Velocities

that the considered controllers are made the equilibrium point globally asymptotically orexponentially stable in the sense of Lyapunov Some advantages and disadvantages of theIQV controllers were also given in the work Moreover, the proposed control schemes are alsofeasible if the damping forces are taken into account Simulations results for two different 3D.O.F spatial manipulators have shown that the IQV controllers can give faster position andorientation error convergence and/or using smaller velocity gain coefficients than the relatedclassical control algorithms Faster kinetic energy reduction is also possible if the classicalcontroller is replaced by the IQV one It is worth noting that the discussed controllers canserve for dynamical coupling detection between the manipulator links via simulation whichallows one to avoid some expensive experimental tests.

Future works should concern investigation of the IQV controllers with models of friction,especially with Coulomb friction and dynamic friction models In order to show realperformance and properties of the controllers, experimental validation is expected

6 References

An, Ch.H.; Atkeson, Ch.G & Hollerbach, J.M (1988) Model-Based Control of a Robot

Manipulator, The MIT Press.

Berghuis, H & Nijmeijer, H (1993) A Passivity Approach to Controller - Observer Design for

Robots IEEE Transactions on Robotics and Automation, Vol.9, No 6: 740-754.

Herman, P (2005a) Sliding mode control of manipulators using first-order equations of

motion with diagonal mass matrix Journal of the Franklin Institute, Vol.342, No 4:

353-363

Herman, P (2005b) Normalised-generalised-velocity-component-based controller for a rigid

serial manipulator IEE Proc - Control Theory & Applications, Vol.152, No.5: 581-586.

Herman, P (2006) On using generalized velocity components for manipulator dynamics and

control Mechanics Research Communications, Vol.33, No 3: 281-291.

Herman, P (2009a) A nonlinear controller for rigid manipulators, PapersOnline: Methods and

Models in Automation and Robotics, Vol 14, Part 1, 10.3182/20090819-3-US-00105.

Herman, P (2009b) Strict Lyapunov function for sliding mode control of manipulators using

quasi-velocities Mechanics Research Communications, Vol.36, No 2: 169-174.

Herman, P (2009c) A quasi-velocity-based nonlinear controller for rigid manipulators

Mechanics Research Communications, Vol.36, No 7: 859-866.

Hurtado, J.E (2004) Hamel Coefficients for the Rotational Motion of a Rigid Body The Journal

of the Astronautical Sciences, Vol.52, Nos.1 and 2: 129-147.

Jain, A & Rodriguez, G (1995) Diagonalized Lagrangian Robot Dynamics IEEE Transactions

on Robotics and Automation, Vol.11, No.4: 571-584.

Junkins, J.L & Schaub H (1997) An Instantaneous Eigenstructure Quasivelocity Formulation

for Nonlinear Multibody Dynamics The Journal of the Astronautical Sciences, Vol.45,

No.3: 279-295

Kane, T.R & Levinson D.A (1983) The Use of Kane’s Dynamical Equations in Robotics The

International Journal of Robotics Research, Vol.2, No.3: 3-21.

Kelly R & Moreno J (2005) Manipulator motion control in operational space using joint

velocity inner loops Automatica, Vol.41, No 8: 1423-1432.

Khalil, H (1996) Nonlinear Systems, Prentice Hall, New Jersey.

Kozlowski, K (1992) Mathematical Dynamic Robot Models and Identi cation of Their Parameters,

Poznan University of Technology Press, Poznan, (in Polish)

Kwatny, H.G & Blankenship, G.L (2000) Nonlinear Control and Analytical Mechanics,

Birkhäuser, Boston

Loduha T.A & Ravani B (1995) On First-Order Decoupling of Equations of Motion for

Constrained Dynamical Systems Journal of Applied Mechanics - Transactions of the ASME, Vol.62, March: 216-222.

(1996) MATLAB, Using Matlab, The MathWorks, Inc.

Mochiyama, H.; Shimemura, E & Kobayashi H (1999) Shape Control of Manipulator with

Hyper Degrees of Freedom The International Journal of Robotics Research, Vol.18, No.6:

584-600

Moreno, J & Kelly, R (2003) Velocity control of robot manipulators: analysis and experiments

International Journal of Control, Vol.76, No.14: 1420-1427.

Moreno, J.; Kelly, R & Campa, R (2003) Manipulator velocity control using friction

compensation IEE Proc - Control Theory and Applications, Vol.150, no.2: 119-126.

Moreno-Valenzuela J & Kelly R (2006) A Hierarchical Approach Manipulator Velocity Field

Control Considering Dynamic Friction Compensation Journal of Dynamic Systems, Measurement, and Control - Transactions of the ASME, Vol.128, September: 670-674.

Santibanez, V & Kelly R (1997) Strict Lyapunov Functions for Control of Robot Manipulators

Automatica, Vol.33, No.4: 675-682.

Sciavicco L & Siciliano B (1996) Modeling and Control of Robot Manipulators, The McGraw-Hill

Companies, Inc., New York

277Non-Adaptive Sliding Mode Controllers in Terms of Inertial Quasi-Velocities

Slotine J.-J., & Li W (1987) On the Adaptive Control of Robot Manipulators The International

Journal of Robotics Research, Vol.6, No.3: 49-59.

Slotine J.-J & Li W (1991) Applied Nonlinear Control, Prentice Hall, New Jersey.

Sovinsky, M.C.; Hurtado, J.E.; Griffith D.T & Turner, J.D (2005) The Hamel Representation:

Diagonalized Poincaré Form, Proceedings of IDETC’05 2005 ASME International Design Engineering Technical conference and Computers and Information in Engineering Conference, DETC2005-85650, Long Beach, California, USA, September 24-28, 2005 Spong, M.W (1992) On the Robust Control of Robot Manipulators IEEE Transactions on

Automatic Control, Vol.37, No.11: 1782-1786.

Part 4

Selected Applications of Sliding Mode Control

15

Force/Motion Sliding Mode Control of

Three Typical Mechanisms

Rong-Fong Fung1 and Chin-Fu Chang2

1Department of Mechanical & Automation Engineering National Kaohsiung First University of Science and Technology,

1 University Road, Yenchau, Kaohsiung 824,

2Institute of Engineering Science and Technology, National Kaohsiung First University of Science and Technology,

1 University Road, Yenchau, Kaohsiung 824,

Taiwan

1 Introduction

A number of papers [1-7] have been presented to address the issues of multi-body mechanisms Examples of their applications are found in gasoline and diesel engines, where the gas force acts on the slider and the motion is transmitted through the links Whether the connecting rod is assumed to be rigid or not, the steady-state and dynamic responses of the connecting rod of the mechanism with time-dependent boundary condition were obtained

by Fung et al [1-3] In addition, a number of controllers, for example, repetitive control [4],

adaptive control [5], computed torque control [6], and fuzzy neural network control [7] were designed for the multi-body mechanisms

Over the past 25 years, the SMC algorithm [8-10] has been taken into account for dynamic control problems The main feature of the SMC is to allow the sliding mode to occur on a prescribed switching surface, so that the system is only governed by the sliding equation and remains insensitive to a class of disturbances and parameter variations [8] It is noted that the SMC is a robust control method and has been well established in pure motion control [9] Afterwards, in order to eliminate the chattering phenomenon, which is commonly found in simulation of discontinuous SMC systems, and to simplify a hybrid numerical method that incorporates benefits of both SMC and differential algebraic equations, the (DAE) stabilization method was developed and successfully used to simulate constrained multi-body systems (MBS) whether under holonomic constraint or not [10] However, the development of a control law which has been induced by a constrained force

has not been adequately developed consistently in the previous studies Su et al [11]

attempted to use the SMC for simultaneous position and force control on a constrained robot manipulator They asserted that the control law, along with inclusion of the constraint force error in the definition of the sliding surface, produces an asymptotically stable force tracking error However, Grabbe and Bridges [12] addressed their formulation as being a departure from the typical definition of a sliding surface, which is a linear differential equation in one tracking error variable [13], and the errors in the separate force control law and stability analysis were presented in [11] Recently, Lian and Lin [14] have proposed a

Sliding Mode Control

282

new sliding surface in terms of motion error and force error, and claimed that the errors in

[11] are improved; therefore, the asymptotic stability of the motion-tracking error and

force-tracking error can be ensured However, Dixon and Zergeroglu [15] pointed out an error in

the sliding mode control stability analysis of [14]

In this chapter, our intent is to improve the errors in [11, 14] and simplify the control design

and stability proof for the three typical mechanisms, including the slider-crank mechanism,

the quick-return mechanism and the toggle mechanism as shown in Figs 1~3 respectively,

which are not seen in any references addressing the force/motion SMC Here, a separate

sliding surface is proposed using the measurements of the angular position and speed of the

crank, but the SMC algorithm is derived as well in a simple manner using only the force

tracking error to construct the controller In these schemes, the force tracking error is shown

to be arbitrarily small by changing the force control feedback gain Then, by exploiting the

structure of its dynamics, the fundamental properties of the dynamics are obtained to

facilitate controller design, whereby the asymptotic stability of motion tracking error in

sliding surface and force tracking error accumulated in controller can be ensured

The organization of this chapter is arranged as follows In Section 2, the kinematic and

dynamic analysis of the multi-body mechanism is investigated A number of previous

papers [4-7, 16-17] have shown the position and speed controllers for the regulation and

tracking problems of the multi-body mechanism in the theoretical analysis and experimental

results However, control of the constrained force has not been investigated The SMC laws

are designed in Section 3 The simulated examples are shown in Section 4 and, finally, some

conclusions are drawn

2 Dynamics analysis

2.1 Dynamic equation of motion

Based on the Euler-Lagrange formulation [4], the equation of motion for a mechanism can

Φ Φ Q is the partial derivative of the constraint equation with respect to the

coordinate and is called the constraint Jacobian matrix, Q A∈R n is the vector of

non-conservative forces and U∈R n is the vector of applied control efforts

In order to obtain the general form of the force/motion controller design, we rewrite the

nonlinear vector as:

N(Q,Q) N (Q,Q)Q N (Q)   (2) where N (Q,Q) C  ∈R n n× is the vector of coriolis and centrifugal forces; N (Q) G ∈R nis the

vector of gravitational force

Then, Equation (1) becomes:

M(Q)Q + N (Q,Q)Q N (Q) Q   U F (3) where = − T

Q

F Φ λ is the constraint force

Force/Motion Sliding Mode Control of Three Typical Mechanisms 283

2.2 Dynamic properties of the mechanism

Equation (3) is similar to the motion equation of an n-link rigid constrained robot [11, 15] in

the state space Two simplifying properties should be noted about this dynamic structure:

Property 1 The individual terms on the left-hand side of Equation (3) and the whole

dynamics are linear in terms of a suitably selected set of equivalent manipulator and load

parameters, i.e.,

M(Q)Q + N (Q,Q)Q N (Q) Y(Q,Q,Q)     (4) where Y(Q,Q,Q)  is a n r× matrix; α ∈R ris the vector of equivalent parameters

Property 2 From the given proper definition of the matrixN (Q,Q) C  , M(Q) 2N (Q,Q) − C  is

skew-symmetric The detailed proof can be seen in Appendix A

Due to the presence of m constraints, the degree of freedom of the mechanism is (n-m) In

this case, (n-m) linearly independent coordinates are sufficient to characterize the

constrained motion From the implicit function theorem, the constraint Equation (1) can

always be expressed as [18]:

=

p σ(q) (5)

Equation (5) is assumed that the elements of q are chosen to be the last (n-m) components of

Q If not the above case, Equation (1) still could always be reordered so that the last (n-m)

equations would correspond to q and the first m equations to p That is, = ⎣⎡ ⎤⎦

T

Q p q Then, to simplify the equation form of the dynamic model, defining

expressed in a reduced form as:

M(q)L(q)q N (q,q)q N (q) Q   U F (9) where

N (q,q) M(q)L(q) N (q,q)L(q)   (10)

By exploiting the structure of Equation (9), three properties can be obtained as follows:

Property 3 In terms of a suitably selected set of parameters, the motion equation (9) is still

linear, i.e

1)

M(q)L(q)q N (q,q)q N (q   Y (q,q,q)  (11)

Sliding Mode Control

The above three properties are basic principle in designing the force/motion SMC law

3 Design of the SMC Law

3.1 The sliding mode controller design

A number of previous papers have only shown the position and speed controller designed

for the regulation and tracking problems control of the constrained mechanisms However,

control of the constrained force has not been investigated in the previous studies In this

section, a separate sliding surface is proposed using the measurements of the angular

position and speed of the crank, but the SMC algorithm is derived as well in a simple

manner using only the force tracking error to construct the controller

Given a desired trajectory q d and a desired constrained forceF d, or identically a desired

multiplierλ d, which satisfy the imposed constraint, i.e., Φ(q ) d =0and = − T

F Φ (q )λ The control objective is to determine the SMC law such that →q q d and →λ λ d as t → ∞

From the SMC methodology, we define the tracking error e m∈R n m− and a sliding surface

where q r∈R n m− is the reference trajectory and Λ∈R(n m− × −) (n m) is a tunable matrix

The sliding controller [12] is defined as:

ϕ

U Y (q,q,q)  L(q)s Φ (q)λ Q (15) whereY 1 is a n r× matrix of known functions of q,q and q , L(q) is defined in Equation (6),

where K is a m m× constant matrix of force control feedback gains, and e λis the error

vector of the multipliers and defined as

Substituting Equation (15) into the dynamic model of Equation (9), whose order was

reduced using property 3, we have:

Force/Motion Sliding Mode Control of Three Typical Mechanisms 285

Y (q,q,q)  L(q)s Φ (q)λ Φ (q)λ Y (q,q,q)  (18) Defining α as a constant r-dimensional vector and replacing q by the reference

trajectoryq , then the linear parameterization of the dynamics (Property 3) leads to: r

M(q )L(q )q N (q ,q )q  N (q ) Y (q ,q ,q )  (19) Using the derivative of the sliding surface equation (14) and substituting into Equation (11),

L M − N L is also skew-symmetric, which is the same as those in [11, 14] Besides, the

special cases of the three typical mechanisms in this chapter, (L M T  −2N L C) is always equal

to zero for n m− = 1

To reduce the chattering phenomenon along the sliding surface s =0, we adopt the

quasi-linear mode controller [13], which replaces the discontinuous term of sign function of