• 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]