Perfect position force tracking of robot

The controller renders a dynamic sliding mode for all time and since the equilibrium of the dynamic sliding surface is driven by terminal attractors in the position and force controlled

with dynamical terminal sliding mode

control

Article in Journal of Robotic Systems · September 2001

DOI: 10.1002/rob.1041

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Tracking of Robots

with Dynamical Terminal

Sliding Mode Control

V Parra-Vega∗

Sección de Mecatrónica Depto de Ing Eléctrica CINVESTAV

A.P 14-740,México,D.F.

07000 México e-mail: vparra@mail.cinvestav.mx

A Rodríguez-Angeles

Systems Signals and Control Group Faculty of Applied Mathematics University of Twente

P.O Box 217

7500 AE Enschede The Netherlands

G Hirzinger

Institute of Robotics and Mechatronics German Aerospace Center–DLR P.O Box 1116

82230 Wessling,Germany

Received 28 May 2000; accepted 7 March 2001

According to a given performance criteria, perfect tracking is defined as the perfor-mance of zero tracking error in finite time It is evident that robotic systems, in par-ticular those that carry out compliant task, can benefit from this performance since perfect tracking of contact forces endows one or many constrained robot manipula-tors to interact dexterously with the environment In this article, a dynamical terminal

Journal of Robotic Systems 18(9), 517–532 (2001)

© 2001 by John Wiley & Sons, Inc

sliding mode controller that guarantees tracking in finite-time of position and force

errors is proposed The controller renders a dynamic sliding mode for all time and

since the equilibrium of the dynamic sliding surface is driven by terminal attractors

in the position and force controlled subspaces, robust finite-time convergence for both

tracking errors arises The controller is continuous; thus chattering is not an issue and

the sliding mode condition as well the invariance property are explicitly verified

Sur-prisingly, the structure of the controller is similar with respect to the infinite-time

track-ing case, i.e., the asymptotic stability case, and the advantage becomes more evident

because terminal stability properties are obtained with the same Lyapunov function of

the asymptotic stability case by using more elaborate error manifolds instead of a more

complicated control structure A simulation study shows the expected perfect tracking

and a discussion is presented © 2001 John Wiley & Sons, Inc.

1 INTRODUCTION

In constrained motion tasks, the end-effector moves

in compliant direction so as to exert a desired profile

of force in the constrained force degree of freedom

(FDoF) while moving along the unconstrained

posi-tion degree of freedom (PDoF) To achieve this goal,

a combination of position and force control loops

are required to drive simultaneously the

manipu-lator along each DoF and to keep the end-effector

in contact to the environment Over the last decade

numerous contributions have proposed alternative

approaches for what was considered for long time

an open problem in robot control.1–4 Among all

these approaches, we focus on the explicit force

feedback control algorithms for rigid, fully actuated

robot manipulators in contact to known (infinitely)

stiff environments, and with known upper bound of

physical parameters.a The system is modeled using

differential algebraic equations (DAE)5, and hence

the contact force stands for the Lagragian of the

con-strained system The control objective is to achieve

simultaneously finite-time convergence of force and

position tracking errors under parametric

uncer-tainty, with a continuous controller Now, we discuss

the background and address the contribution of this

article

1.1 Explicit Force Control

Two basic approaches have prevailed over the years

in explicit force feedback robot control research

A1 The first one proposed5 exploits the

par-tition coordinates of the solution of the

a We study only the stage of constrained motion, leaving out the

impact and transition phases.

implicit equation that models the constraint

to obtain a decoupled dynamics for the open-loop PDoF and the FDoF Thus, robot dynamics are explicitly obtained for each DoF in terms of a unique set of indepen-dent generalized coordinates Though this set always exists, it is not evident how to handle it in a large workspace.6 Control structure is rather involved, though sim-ple stability arguments are used to prove the global stability and several control tech-niques have been proposed.7–12

A2 In the second approach,13 a passivity-based

algorithm that does not use any

coordi-nate partition of system dynamics, but introduces two orthogonal projections to construct an orthogonalized error coor-dinate system, leads to a local asymp-totic stability result This approach is computationally more efficient since the solution of the implicit equation is not required at all; besides that this scheme exploits effectively some fundamental phys-ical properties of robot dynamics The con-trol objective is translated into reshaping the desired closed-loop mechanical energy

of the system such that a local minimum arises on desired trajectories The control structure is simpler with respect to that in ref 5, though involved stability arguments are used to prove the local asymptotic convergence of force and position tracking errors.13
14

A third alternative that appears as an efficient combination of the first and second approaches has been proposed:

A3 The partition coordinates of A1 are pro-posed in order to obtain globally the orthog-onal projections of A2; robot dynamics are

not embedded in the solution of the implicit

function, which leads to the control

struc-ture of the second approach to yield global

asymptotic stability with simple stability

arguments.15
16 In ref 17 closed-loop error

dynamics are partially decoupled, and in

refs 18 and 19 overcompensated controllers

are proposed along this line In any case,

the literature available up to date, not only

for explicit force control but also for all

the other force control strategies,1
2 does

not assure finite-time convergence (FTC) of

tracking errors (FTT)

1.2 The Paradigm of FTT and Applications

FTT for physical systems such as robot

manipula-tors has attracted little attention; therefore it is

con-venient now to define the FTT paradigm: “Design

a control system that yields FTC of tracking errors with

a real-time compliant control input; that is,the

con-troller that renders FTT should be realizable with

cur-rent software and hardware technology.” To this end,

we impose some constraint on the control design,

such as the controller should be: (1) continuous;

(2) with no unbounded effort; (3) with no high

frequency; (4) causal; and (5) robust to

paramet-ric uncertainty and initial conditions The paradigm

FTT is not only fundamental to yield perfect

track-ing and great performance, but for many problems

in robotics it is a desirable property To name a

few, we review briefly the following tasks that can

benefit if FTT is implemented: (a) walking robots

where it is needed to assure that the state of the

leg is in the given desired trajectory before other

leg deattaches from ground; (b) event-based

algo-rithms where the discrete states are assumed to

belong to given compact sets at given time; (c)

con-tact transition tasks where concon-tact detection depends

on complex algorithms to detect the exact state of

the system at given instant; (d) dynamic simulation

systems where complex and high order models are

used to render realistic motion of articulated

bod-ies (however the complexity of the system requires

that stringent assumptions are imposed on the

model and important information is neglected, and

thus FTC can yield better realistic simulators by

relaxing such assumptions since convergence time

could be set arbitrarily); (e) closed-loop

identifica-tion of robot parameters, where weaker condiidentifica-tions

on the regressor can be imposed since real

tra-jectories can be substituted by desired tratra-jectories

at given time and the persistent excitation condi-tion can be designed beforehand; (f) obstacle avoid-ance methods, wherein consider that real trajectories follow exactly the planned trajectories (otherwise overdesigned desired trajectories are given); (g) opti-mal path planning usually considers that the sys-tem follows exactly the optimal path, which is not always the case, and then optimality is not achieved; (h) object manipulation and multirobot coordination usually require perfect timing for all finger robots

to release and grasp the object, and during (i) strained motion, where the end-effector is in con-tact to the environment and thus real position and force trajectories follow the trajectories that satisfy the constraint (otherwise the system may damage the constrained object) In this article we focus on the FTT paradigm for constrained motion using second order sliding mode control with terminal attractors20

in the sliding surface

1.3 Terminal Attractors

In order to design a control system that fully com-plies with the FTT paradigm, we exploit the

little-used technique called terminal sliding mode control.

Although terminal attractors have been subject of intensive research in the numerical and neural net-works research community, where the application is mainly bound to computer computations,21 that is not the case for physical systems The philosophy of design of terminal sliding mode control is basically

a conventional static sliding mode controller with a nonlinear sliding surface of tracking errors, where the dynamics of this surface exhibits an attractor with FTC (called the terminal attractor20), and thus tracking errors converge in finite time The termi-nal attractor is modeled as a first order differen-tial equation that violates the Lipschitz condition, and the attractive singularity located precisely in zero position error renders unbounded control in the bounded domain with internal instability of the dif-ferential equation In the second order derivative of this differential equation appears the singularity in zero position tracking error, as is the case for second order systems such as robot arms

This puzzling behavior seems then disastrous, and not surprisingly, the consequence of this unusual and unconventional formulation is that few control algorithms based on terminal

attrac-tors are available for robot manipulaattrac-tors in free

motion,22–25 though these controllers are not fully compliant with the FTT paradigm because they may

violate (2), (3) above These control schemes need

a discontinuous control input to achieve FTC, and

since it is virtually impossible to reproduce the

the-oretically infinite bandwidth of a signum function

in real time, a saturation function is proposed to

realize the controller Thus, a sliding mode does not

strictly arise and the singularity induces internal

instability all the time, which eventually may

ren-der unbounded control input and instability, while

in ref 26 a complex procedure is proposed to achieve

finite-time convergence with high frequency control

inputs

In ref 27 an algorithm is proposed to

sequen-tially induce sliding modes to avoid singularity;

however, a discontinuous controller is needed, and

again in order to realize the controller a

satura-tion funcsatura-tion is implemented In this condisatura-tion, the

sequence to avoid singularity cannot be guaranteed

To induce well-posed terminal attractors, it is

fundamental then to induce a sliding mode for all

time, and in order to realize this controller in real

time, the control action must be continuous,

other-wise the sliding mode condition cannot be strictly

verified

1.4 Sliding Modes with Continuous Control

Chattering is a problem that arises in variable

struc-ture control systems due to the finite bandwidth of

the software and hardware, and many techniques

have been proposed to attenuate to some extent

this phenomena.28
29 However, boundary-layer-like

methods30 do not strictly verify the sliding mode

condition, while numerical solutions31 or second

order sliding surfaces32 are not well developed

yet for mechatronic systems On the other hand

dynamic sliding modes33–35 seem to be a promising

technique; however, backsteeping methods are really

complicated in comparison to the class of controller

obtained in passivity-based robot control.36Thus, we

develop further the passivity-based dynamic sliding

mode control proposed in ref 16 to obtain a

slid-ing mode regime with continuous control for DAE

systems

1.5 Contribution

We propose a solution for the FTT paradigm

using a continuous controller for robot

manipula-tors that are subject to known holonomic constraint

and known upper bound of physical parameters

To this end, we further elaborate on the third alternative16 outlined in A3 to introduce terminal attractors in orthogonal position and force dynamic sliding surfaces The controller compensates the parametric uncertainty, and terminal attractors show

their implosive attractiveness to induce FTC of both

position and force tracking errors The closed-loop system is free of singularity and preserves the passivity of the open-loop system, and thus this algorithm might be extended to other classes of mechanical systems that are passive in the open loop Computer simulation shows the performance

of a 2 DoF rigid arm

The article has been organized as follows Section 2 presents the formulation of the problem Section 3 shows the robot dynamics in the error space In Section 4 the controller and its stability proof is presented, while in Section 5 some remarks are presented A simulation study is discussed in Section 6, and conclusions are presented in Section 7

2 PROBLEM FORMULATION

When the robot end-effector is in contact to a smooth surface, a geometric (holonomic) constraint

is imposed by the forward kinematic equation

3Cartesian coordinates and the three Euler angles,

q∈
n stands for the joint generalized coordinates,

the holonomic constraint is formulated in X

coordi-= 0, where it is assumed that the con-straint is twice differentiable Using the concon-straint

can be expressed in q coordinates as 

= 0 A classical mechanics formulation37 has been used in ref 13 to yield a model of a

rigid serial n-link robot manipulator with all

revo-lute joints as

+ − f (1)

× n symmetric positive definite inertial matrix, B0 is an n × n positive

def-˙q represents the n × n Cori-olis matrix, U stands for the n torque inputs, and

 f represents a model of the sliding friction force at the contact point For simplicity we assume a

vis-cous model such that  f is linear in terms of the velocity ˙q by  f = f ˙q, where  f T

x J x and

˙X < 1 It is assumed that the constraint

 ∈ C2
n→
m denotes m−smooth surfaces and

it is consistent and independent in a sense that J T

 has full column rank m and a unique and analytic

solution of (1) exists if initial conditions are chosen

to satisfy (2) and its derivatives up to order two.5
38

J T

+= J T

+ = J T

   T −1models the

normal-ized matrix that points outward and therefore the

Lagragian ∈
m physically stands for the

magni-tude of the force applied at the contact point (2) with

J  ≡ J  = !X J x∈
n ×m , and J x= !q stands as the

direct Jacobian

In order to obtain the representation of the

sys-tem in error coordinates, we use the standard linear

parametrization Y r $ of robot dynamics in terms of

the nominal reference ˙q r and its derivative ¨q r, where

arguments are omitted from now on when no

con-fusion arises, as40

Y r $ = Y r
q
˙q
˙q r
¨q r$

= H ¨q r+C + B0+  f˙q r + G (3)

where $∈
lis composed of, and is possibly a

prod-uct of, physical parameters and Y r∈
n ×l stands for

the regressor If we add (3) to (1) we obtain

H ˙ S+C + B0+  fS = U + J T

+ − Y r $
(4) where

We now have the following

Statement of the problem: Design a continuous

con-troller U which guarantees trajectory tracking in finite

time of desired time-varying pose and contact force It is

assumed that: (i) the upper bound of the unknown

param-eter vector $ is known; (ii) the regressor Y r is

avail-able; (iii) the kinematic constraint  is twice differentiable

and exactly known; (iv) the state (position q,velocity ˙q,

and contact momentum F ,and thus ) is available; and

(v) desired trajectories q T

d
˙q T

d , ¨q T

d
F T

d
 T

d  T are known bounded analytical functions.

Error equation (4) and constraint (2) define a

differential algebraic system whose solution is

con-strained to evolve in an invariant manifold defined

by

0=



n × R n × R m × R+ d

i

i = 0
' ' '
2



This manifold will be exploited in the following section to synthesize a convenient orthogonalized

error coordinate system using terminal sliding modes

in order to design the controller U according to the

statement

3 ERROR DYNAMICS FOR CONSTRAINED MOTION

To design an appropriate error equation, we keep in mind at this stage that the regressor to be compen-sated must be continuous since a continuous con-troller must be designed Let us note also that we want to preserve the passivity in the closed loop and thus we are looking for a similar control structure and stability analysis of refs 13 and 16 As can be seen now the problem is translated into the

refor-mulation of a new error state S in (5), which means the design of new nominal references  ˙q T

r
¨q T

r  T ∈

2n in (5), (6) We present now the open-loop error dynamics using the partitioning method5 for the

error manifolds13with terminal attractors,20and with-out coordinate reduction of system dynamics.15
16

3.1 An Orthogonalized Terminal Sliding Surface

− m independent generalized coordinates q2∈
n −m, and

the following partition of joint space coordinate

q∈
n arises:

q = q T

1
q T

Now according to (7), the derivative of (2), that is,

d

dt  = J  ˙q ≡ 0, with its corresponding partition given

by (8), yields

J  ˙q = J1 ˙q1+ J 2 ˙q2 (9)

= J 1 J 2 



˙q1

˙q2



≡ 0

where J 1 = !/!q1 ∈
m ×m and J 2 = !/!q2 ∈

m Solving (9) for ˙q1 yields

˙q1= - ˙q2 where - = −J 1 −1J 2 (10)

and -
n −m →
m has full column rank m since

by assumption rank = m and thus J1 −1 is well posed in the finite workspace imposed by the holonomic constraint (2) Taking into account the

partition (8) and using (10), the generalized velocity

˙q =  ˙q T
˙q T  T can be written as

˙q =



- ˙q2

˙q2



(11)

=



- q

I n −m



Q

where Q ∈
n is well posed Since J  Q ≡

0m , the image of J  lies on the null space of

Q; that is, the state space is decomposed into two

orthogonal subspaces such that
n can be

writ-ten as the direct sum R n



∗ Now, with the unique

set of joint independent generalized coordinates

T

2
˙q T

2 T ∈
−m, consider the nominal reference

˙qr = Q ˙q 2d − 01q r

2+ Sdp − K14 p 

+ J T

 51F − SdF + K24 F  (13) where ˙q r∈
n and

with 1q2= q2− q 2d , 1F = t

t0 − d subscript d denotes the desired reference value.

Diagonal feedback gains are 0
K1 ∈ R+ ,

5
K2∈ R m ×m

+ , and r is a terminal attractor

param-eter.b The passivity approach13
36
40 suggests that in

order to fully exploit the physical structure of robot

dynamics, the nominal reference ¨q r must be equal to

d

dt ˙q r, then (13) becomes

¨q r= ˙Q ˙q 2d − 01q r

2+ S dp − K14 p 

+ Q ¨q 2d − r01q r−1

2 1 ˙q+ ˙Sdp − K1 qp 

+ ˙J T

 51F − S dF

+ K24 F  + J T

stands for the signum function of X ∈
j

How-ever, Eq (16) is discontinuous and it is not allowed

because (3) would be discontinuous Then, if we add

powers y, that is, y x , such that x = x n /x d
x n
x d ∈ Z+
x n < x d
1<

x < 1
and x
x odd.

and subtract QK1 1S qp + J T

 5K2 2S qF to

(16) one obtains

¨q r = ¨qcont+ ¨qdisct
(17) where

¨qcont= ˙Q ˙q 2d − 01q r

2+ Sdp − K14 p 

+ Q ¨q 2d − r01q r−1

2 1 ˙q+ ˙Sdp − K1 1S qp 

+ ˙J T

 51F − SdF + K24 F 

+ J T

 51 − ˙S dF + K2 2S qF 
(18)

¨qdisct= QK1Z p − J T

with bounded 1∈
+ , 2∈
m ×m

+ , and

for the hyperbolic tangent function of X∈
k, and

every entry z p ∈ Z p
z F ∈ Z F are bounded by±1 Sub-stituting (13) into (5) and (17) into (6) gives rise to

S = QS vp − J T

˙S = ˙QS vp + Q ˙S qp + K1 1S qp  − ˙J T

 5S vF

− J T

 5 ˙ S qF + K2 2S qF  + ¨qdisct (23) where

with

S p = 1 ˙q2+ 01q r

S qF = S F − S dF
where S F ≡ 1F
(28)

and S dp
S dF are to be defined yet Equation (4) in terms of Eqs (22), (23) can be written as

H ˙ S+C +B0+ fS = U +J T

+ −Ycont$ −H ¨qdisct
(29)

where

Ycont$ = Yr
q
˙q
˙q r
¨qcont$

= H ¨qcont+C + B0+ f˙q r + G

is continuous On the other hand, Eq (17) allows one

to cast the discontinuous term H ¨q as bounded

disturbances into the right hand side of the

open-loop error equation (29), which in turn allows one

to derive a continuous controller since the

regres-sor Ycont is continuous In (22) we can see that due

to Q ∩ J = 0, the orthogonal complements Q and J

globally project the position–velocity and integral of

the force tracking errors onto orthogonal subspaces,

respectively These projections are instrumental in

the proof of stability, as becomes clear in the

follow-ing section

4 MAIN RESULT

Consider the controller U given by

U = −K d S c + J T

+



−  d − ˙S dF + K2 2S qF + <S d

vF



+ Ycont$
 (30)



$ i= −$ isat

n

j=1

S j Ycontji



for i = 1
' ' '
l (31)

where $ i > i

the inputwise saturation function of vector X =

x1
' ' '
x l  T , and Ycont= Ycontji The parameter > > 0

defines the width of the saturation function,

feed-back gains K d = K T

d ∈
n ×n

+ , < = < T ∈
m ×m

+ , and c

d are terminal attractor parameters The closed-loop

error equation between (29) and (30), (31) yields

H ˙ S= −C + B0+  fS + K d S c + J T

+ S vF + < S d

vF

−  d − Ycont$ − Y T

cont$∗ contT S (32)

where  d = H ¨qdisct− J T

+K2Z F is considered a

distur-bance, and $∗ $1
' ' '
 $ l ∈
l ×l We are now

in a position to state the stability properties of the

closed-loop system (32) in the next theorem

Theorem 1: Consider robot dynamics (1) in closed loop

with the controller (30), (31) Then, the global

finite-time convergence of tracking errors arises with

continu-ous control and singularity-free closed-loop dynamics.

sections

4.1 Boundedness of State Trajectories S SvF

Consider the Lyapunov candidate function

2V S + V F 
(33)

where V S = S T HS and V F = S T

vF 5S vF The total deriva-tive of (33) along its solution (32) leads to

˙V = −S T

0+ f S − S T K d S c − S T

vF <5S d vF

− S T

r Ycont$∗ T

contS r − S T

r Y cont $ − S T  d

≤ −S T K d S c − S T

vF <5S d

vF − S T Ycont$∗ T

cont S

− S T Ycont$ − S T  d

≤ −S T K d S c − S T

vF <5S vF d − S T Ycont$∗ cont T S

+ S T Ycont$ + S T  d

≤ −S T K d S c − S T

vF <5S vF d − S T Ycont$∗ contT S

+ S T Ycont $ + S  d

≤ −S T K d S c − S T

vF <5S d

vF + >$0+ S T  d
(34) where we have used the fact that−S T Y r $∗ T

r S+

S T Y r $ ≤ >$0 since $0 ≥ $ Now, note that

S T  d is radially unbounded only for S and for bounded signals S T  d attains a unique

equilib-rium point at S = 0; see also that for

admissi-2
 we have that S T HQK1Z p ≤ S T HQK1,

S T HJ T

 5K2Z F ≤ S T HJ T

 5K2 On the other hand, according to the boundedness property of the

iner-tial matrix and the fact that Q and J  are functions

of bounded constant and trigonometric functions,

then  d is also bounded Along with the fact that

exists always a positive scalar A= sup vF limt A1+

+ M 2

such thatS T  d ≤ AS, where  M ∗ stands as the

∗ Thus, Eq (34) becomes

˙V ≤ −C1V D1

S − C2V D2

F + >$0+ AS (35)

where D1= 1 + c/2, C1≡  m d 2/ m D1
D2=

1+ d/2, and C2≡  m D2 If K d is large enough, and according to refs 23 and 38 one obtains

the terminal convergence of S → E0 and S vF → E1,

where E0 and E1 are bounded hyperballs with radii

r0> 0 and r1> 0, respectively Hence, for the region

outside the union of the boundaries of the domains

E = E0∪ E1 centered in the equilibrium S= 0 and

S = 0, we can conclude the terminal ultimate

boundedness of error dynamics within the

neighbor-hood of E such that terminal trajectories for V can

be obtained as

V <

 ¯A

C1

D−1 1

+

 ¯A

C2

D−1 2

for some S > 0

where ¯A = A+>$0 Thus, there exist bounded scalars

E i > 0, for i = 2
' ' '
5, such that

S ≤ E2
SvF ≤ E3
˙S ≤ E4

This establishes the boundedness of S
S vF and their

derivatives ˙S
˙ S vF

4.2 Sliding Mode and FTC for SqF

We now show that the properties of the dynamical

system defined by Eqs (25) and (15)

˙S qF = −K2 qF + ˙S vF (37)

yields a sliding mode at S qF= 0 To see this, consider

the following Lyapunov function:

V qF =1

2S

T

The total derivative of (38) along its solution (37)

gives rise to

˙V qF = −S T

qF K2 qF + S T

qF ˙S vF

≤ −K2 qF qF vF

≤ −K2 qF 5 qF

where we have used (36), and G2= K2− E5' Thus,

in order to prove that S qF → 0 in finite time, we can

always choose

in such a way that a G2> 0 in (40) guarantees the

existence of a sliding mode since Eq (39) is the

slid-ing mode condition.29 This indicates that a sliding

mode is established in finite time t qF qF 0 2,

and since for any initial condition S qF 0= 0, then

a sliding mode in S qF = 0 is enforced for all time

without reaching phase in the force controlled

sub-space, then t ≡ 0'

4.3 Sliding Mode and FTC for Sqp

If we multiply (22) by the pseudoinverse Q H =

T Q−1Q T ∈
−m×n one obtains

Q H S = Q H QS vp − Q H J T

 5S vF = S vp
(41)

since Q H Q = I and Q H J T

 = 0 −m×m ' Using

(14) and (24), the derivative of Eq (41) can be written

as follows:

˙S qp = −K1 qp + ˙Q H S + Q H ˙S' (42) Now, consider the following Lyapunov function

V qp=1

2S

T

Using (42) into the total derivative of (43) gives rise to

˙V qp = −S T

qp K1 qp + S T

qp Q H S + Q H ˙S' (44)

To proceed, notice that Q H is full column rank

− m and is composed of constant and trigono-metric functions, then Q H is bounded by a constant, and its derivative is also bounded by a function of

˙q2 That is, there exists bounded positive scalars I1
I2

such that

Q H ≤ I1 ˙Q H ≤ I2 ˙q2' (45) Using (45) into (44) one obtains

˙V qp ≤ −K1 qp qp Q H S + Q H

≤ −K1 qp qp 2 ˙q2E2+ I1E4

where we have used (36), and G1= K1 2 ˙q2E2+

I1E4' Thus, in order to prove that S qF → 0 in finite time, we can always choose

K1> sup 2
˙q2 T

1
q2 1
q2
˙q2 =0

2 ˙q2E2+ I1E4 (47)

in such a way that G1 > 0 guarantees the

exis-tence of a sliding mode since Eq (46) is equivalent

to the sliding mode condition.29 This indicates that a sliding mode is established in finite time

t qp qp 0 1, and since for any initial

condi-tion S qp 0 = 0, then a sliding mode in S qp = 0 is enforced for all time without reaching phase in the

position controlled subspace; thus t ≡ 0'

4.4 FTC of Sp and SF

We have shown that S qp = 0 ∀t ≥ tdp > 0 is enforced

for all time; then from (26) we have the invariant

system S p = Sdp
and hence S dp plays the role of a

desired trajectory for S p and also as an input, and

since S dp = 0 ∀t ≥ tp , where t p is the convergence

time of S dp

S p = 0 ∀t ≥ t p '

A similar procedure can be followed for the sliding

surface S F to obtain

S F = 0 ∀t ≥ t F

where t F is the convergence time of S dF

4.5 Terminal Sliding Mode and

FTC of qT  ˙qT T

Using (26), (27) we can see that the existence of

a dynamic sliding mode in S qp = 0 for all time

implies

1 ˙q2= −01q r

Now consider the following Lyapunov function:

V tp=1

21q

T

The total derivative of (49) along its solution (48)

gives rise to

˙V tp = −01q T

21q2r + 1q T

2S dp

= −0n

i=1

2

2i J + 1q T

2S dp

= −2J 0

 1 2

n



i=1

1q 2i2

J

+ 1q T

≤ −2J  m

 1 2

n



i=1

1q2

2i

J

+ 1q T

2S dp

≤ −2J  m tp J + 1q T

2S dp

where J = 1 + r/2 Since S dp

t = t dp, and according to ref 22, then

V tp = 0 ∀t ≥ t tp > 0

where t tp ≤ t dp+ V

1−J

tp 0

2J  − J

which implies that

1q T

2 ˙q T

2 = 0 T

−m
0 T

−m  T

for ∀t ≥2t tp (51) regardless of system parameters Convergence of the

generalized coordinates 1q2 ˙q2 vergence of the dependent coordinates 1q1 ˙q1 since - establishes a diffeomorphism between ˙q1

and ˙q2 and the desired reference has been designed

˙q ∈

R 2nand with the constraints (2) Then, we can finally conclude the global finite-time convergence of the complete set of original joint position and joint velocity error trajectories,

1q T
1 ˙q T  T = 0 T

n
0 T

n  T ∀t ≥2t tp

where t tp can be fixed arbitrarily

4.6 Terminal Sliding Mode of F and

Dynamic sliding surface S qF = 0 implies

Since we design S dF such that it exhibits FTC at

Note that the derivative of (52) is

d

If we design ˙S F such that it exhibits FTC at t = t  > 0,

= t 

4.7 Singularity-Free Closed-Loop Dynamics

We exclude the trivial case when the system is

0= 0 at given initial conditions,24since any terminal-attractor-based

con-trol algorithm fails at this point at t = t0.c Then, we

analyze the case of 1q2 0= 0 Note that the equa-tion ¨q r in (17) violates the Lipschitz condition in the

open loop However, considering that for closed-loop

c Initial position tracking error must be different from zero at any

given initial conditions, as is usually the real case.

the force tracking errors onto orthogonal subspaces,

respectively These projections are instrumental in

the proof of stability,...

Statement of the problem: Design a continuous

con-troller U which guarantees trajectory tracking in finite

time of desired time-varying pose and contact force It is... obtain the representation of the

sys-tem in error coordinates, we use the standard linear

parametrization Y r $ of robot dynamics in terms of< /i>

the nominal