• To clarify the statements, we present the following simple, closed-loop system

where x(t) is a tracking error signal, is an unknown nonlinear function, and is a learning-based estimate

• It is assumed that is periodic with a known period

• For the above system, the standard repetitive update rule is given by

• With regard to the above error system, Messner et al. noted that the techniques they presented, could not be used to show that is bounded if it is generated using the standard repetitive update rule

Introduction - Previous Research

_x = ¡ x + ' (t) ¡ ' (t)

' (t)' (t)

T' (t)

' (t) = ' (t ¡ T ) + x

' (t)

' (t ¡ T) = ' (t)

• To address the boundedness problem associated with the standard repetitive update rule, Sadegh et al. proposed the following update rule

and hence, guarantee that is bounded for all time

Introduction - Previous Research

It is well known how one can apply a projection algorithm to the adaptive estimates of a gradient adaptive update law and still accommodate the Lyapunov-based stability analysis

Unfortunately, it is not clear from the analysis by Sadegh et al. how the Lyapunov-based stability analysis accommodates the saturation of the standard repetitive update rule

' (t)

• To address the boundedness problem, we propose the following update rule

' (t) = sat (' (t ¡ T ) +x)

' (t) = sat (' (t ¡ T )) + x

• Consider the following error dynamicsGeneral Problem

_e = f (t; e) + B (t; e) [w(t) ¡ w (t)]

e(t) 2 Rn

w(t) 2 Rmw(t) 2 Rm

f (t; e) 2 RnB (t e) 2 Rn £ m andwhere are bounded provided the errorsystem is bounded, is a learning-based estimateof

_e = f (t; e)Assumption 1: The origin of the error system is uniformly asymptotically stable for and there exists a positive-definite function , a symmetric matrix , and a known matrix such that

Assumption 2: The unknown periodic function has a period of

and we assume that where is a vector of known, positive bounding constants

e(t) = 0

V1(e; t) 2 R1 Q (t) 2 Rn £ n

R (t) 2 Rn £ m

_V1 · ¡ eT Q e + eT R [w ¡ w]

w(t ¡ T ) = w(t)jwi (t)j · ¯ i for i = 1; 2; :::; m

¯ =£

¯ 1 ¯ 2 ::: ¯ m

¤2 Rm

Tw(t) 2 Rm

• The learning-based estimate is designed as follows

• Based on the definition of , we can prove the following inequality

• To facilitate subsequent analysis, we develop the following relationship

where we utilized the fact that

Learning-Based Estimate Formulation

w(t) = sat¯ (w (t ¡ T )) + keR T e

sat¯ i (»i ) =½

»i for j»i j · ¯ i

sgn (»i ) ¯ i for j»i j > ¯ i8»i 2 R1; i = 1; 2; :::; m

~w = sat¯ (w(t ¡ T ) ) ¡ sat¯ (w (t ¡ T )) ¡ keR T e

w(t) = sat¯ (w(t)) = sat¯ (w(t ¡ T ))

sat¯ (¢)

(»1i ¡ »2i)2 ¸ (sat i (»1i) ¡ sat i (»2i))

2 8j»1i j · ¯ i ;»2i 2 R1; i = 1;2;:::;m

Stability Analysis • Theorem 1: The learning based estimate designed previously, ensures that

Proof: To prove Theorem 1, we define the following non-negative function

where was defined in Assumption 1 _V2 · ¡ eT Q e + eT R ~w

+1

2ke[sat¯ (w(t)) ¡ sat¯ (w (t))]T [sat¯ (w(t)) ¡ sat¯ (w (t))]

¡1

2ke[sat¯ (w(t ¡ T )) ¡ sat¯ (w (t ¡ T ))]T [sat¯ (w(t ¡ T )) ¡ sat¯ (w (t ¡ T ))]

_V2 · ¡ eT Q e

e(t) 2 L 2 \ L 1

w(t), ~w(t), f (t; e), B (t; e) 2 L 1

_e(t) 2 L 1

Signal Chasing Arguments

limt! 1

e(t) = 0

Barbalat’s Lemma

limt! 1

e(t) = 0

V1(e; t) 2 R1

V2 = V1 +1

2ke

Z t

t¡ T[sat¯ (w(¿)) ¡ sat¯ (w(¿))]T [sat¯ (w(¿)) ¡ sat¯ (w(¿))]d¿

Dynamic Model• Dynamic equation for an n-DOF revolute robot

: link position, velocity, and acceleration

: inertia matrix : centripetal-Coriolis matrix : gravity vector

: viscous friction coefficient matrix: constant, diagonal, static friction

matrix: torque control input

M (q)Äq+ Vm(q; _q) _q+G(q) + Fd _q+ Fssgn(_q) = ¿

q(t), _q(t), Äq(t) 2 Rn

M (q) 2 Rn £ n

Vm (q; _q) 2 Rn £ n

G (q) 2 Rn

F d 2 Rn £ n

F s 2 Rn£ n

Non-periodic Effects

¿(t) 2 Rn

Dynamic Model Properties• Inertia matrix is positive-definite and symmetric

where are known positive constants

• Skew Symmetry Property

• Linearity in the parameters

• The centripetal-Coriolis matrix, gravity vector, and dynamic friction matrices can be upper bounded as follows

m1, m2

kVm (q; _q)ki 1 · ³ c1 k _qk ; kG (q)k · ³ g; kF dki 1 · ³ f d

Ys( _q)µs = Fssgn( _q)

m1 k»k2 · »T M(q)» · m2k»k2 8» 2 Rn

»T

µ12

_M(q) ¡ Vm(q; _q)¶

» = 0 8» 2 Rn

parameter estimation vector

• To quantify our objective to design a global link position tracking controller, we define the link position tracking error as follows

where the desired trajectory and its first two time derivatives are assumed to be bounded, periodic functions of time with a known period such that

• Since this objective is to be met despite parametric uncertainty in the dynamic model, we define the following parameter estimate error

Control Objective

e = qd ¡ q

Tqd(t) = qd(t ¡ T ) _qd(t) = _qd(t ¡ T ) Äqd(t) = Äqd(t ¡ T )

~µs = µs ¡ µs

unknown constant vector

desired trajectory

• To facilitate the subsequent control development and stability analysis, we reduce the order of the dynamic model by defining a filtered tracking error-like variable as follows

the following open-loop dynamics for the filtered tracking error can be obtained

where

Control Formulation

r = _e + ®e

M _r = ¡ Vm r + wr + Â + Y sµs ¡ ¿wr = M (qd) Äqd + Vm (qd; _qd) _qd + G (qd) + F d _qd

 = M (q) ( Äqd + ®_e) + Vm (q; _q) ( _qd + ®e) + G (q) + F d _q ¡ wr

kÂk · ½(kzk) kzk z(t) =£

e(t) r (t)¤T

jwr i (t)j · ¯ r i for i = 1; 2; :::; n

known, positive bounding function

known, positive bounding constant

Control Formulation• Given the previous open-loop error system, we design the following torque

control input

where are positive constant control gains, is generated on-line according to the following learning-based algorithm

is a positive, constant control gain, is defined previously, and the parameter estimate vector is generated on-line according to the following gradient-based adaptation algorithm

where is a constant, diagonal, positive-definite, adaptation gain matrix.

¿ = kr + kn ½2 (kzk) r + e + wr + Ys µs

k, kn 2 R1 wr (t) 2 Rn

wr (t) = sat¯ r(wr (t ¡ T )) + kL r

kL 2 R1 sat¯ r (¢)µs(t) 2 Rn

:

µs(t) = ¡ sY Ts r

¡ s 2 Rn£ n

Control Formulation• The following closed-loop error system can now be formulated

where is defined as follows

• Substituting the learning-based estimate in the above expression yields

where we used the fact that

M _r = ¡ Vm r ¡ kr ¡ e + Ys~µs + ~wr + Â ¡ kn ½2 (kzk) r

~wr (t)

wr (t) = sat¯ r (wr (t)) = sat¯ r (wr (t ¡ T ))

~wr = sat¯ r (wr (t ¡ T )) ¡ sat¯ r(wr (t ¡ T )) ¡ kL r

~wr = wr ¡ wr

Stability Analysis • Theorem 2: The proposed hybrid adaptive/learning controller ensures

global asymptotic link position tracking in the sense that

provided the control gains are selected as follows

Proof: To prove Theorem 2, we define the following non-negative function

limt! 1

e(t) = 0

minµ

®; k +kL

2

¶>

14kn

V3 =12eTe+

12

rT M r +12

~µTs ¡ ¡ 1

s~µs

+1

2kL

Z t

t¡ T[sat r (wr (¿)) ¡ sat¯r (wr (¿))]T

¢[sat¯ r (wr(¿)) ¡ sat¯r (wr (¿))]d¿

Signal Chasing Arguments

limt! 1

e(t) = 0

Barbalat’s Lemma

Stability Analysis • After taking the time derivative of the following expression is

obtained_V3 · ¡ ®eT e ¡ kr T r + r T ~wr +

£½(kzk) kzk kr k ¡ kn ½2 (kzk) kr k2¤

¡1

2kL( ~wr + kL r )T ( ~wr + kL r )

+1

2kL[sat¯ r (wr (t)) ¡ sat¯ r (wr (t))]T [sat¯ r (wr (t)) ¡ sat¯ r (wr (t))]

_V3 · ¡µ

minµ

®; k +kL

2

¶¡

14kn

¶kzk2

e(t), r (t) 2 L 2 \ L 1

wr (t), ~wr (t), _e(t) 2 L 1

z(t) =£

e(t) r (t)¤T

V3(t) 2 R1

Experimental Results• The following controller was implemented on a two-link direct-drive,

planar robot manipulator manufactured by Integrated Motion Inc.

• The two-link robot is directly actuated by switched-reluctance motors. • A Pentium 266 MHz PC running RT-Linux (real-time extension of

Linux OS) hosted the control algorithm.

• The Matlab/Simulink environment with Real-Time Linux Target for

RT-Linux was used to implement the controller.

• The Servo-To-Go I/O board provided for data transfer between the

computer subsystem and the robot.

¿ = kr + wr learning-based estimate

feedback term

• The two-link IMI robot has the following dynamic model

• The reference trajectory was selected as follows

Experimental Results·

¿1

¿2

¸=

·p1 +2p3c2 p2 + p3c2

p2 + p3c2 p2

¸ ·Äq1

Äq2

¸+

·¡ p3s2 _q2 ¡ p3s2( _q1 + _q2)p3s2 _q1 0

¸ ·_q1

_q2

¸

+·

f d1 00 f d2

¸ ·_q1

_q2

¸+

·f s1 00 fs2

¸ ·sgn ( _q1)sgn ( _q2)

¸

·qd1(t)qd2(t)

¸=

·(0:8+0:2sin(0:5t)) sin(0:5sin(0:5t)) (1¡ exp(¡ 0:6t3))(0:6+0:2sin(0:5t)) sin(0:5sin(0:5t)) (1¡ exp(¡ 0:6t3))

¸[rad]