Consider the PD control law with gravity compensation for n-DOF robots and assume that the desired position q d is constant.. The analysis of the PD control with gravity compensation for
Trang 17.2.2 Time Derivative of the Lyapunov Function
The time derivative of the Lyapunov function candidate (7.8) along the jectories of the closed-loop system (7.2) may be written as
Using Property 4.2, which establishes that ˙q T
1
2M˙ − C q = 0 and ˙˙ M (q) = C(q, ˙q) + C(q, ˙q) T, the time derivative of the Lyapunov function candidateyields
˙
V (˜ q, ˙q) = − ˙q T K v q + γ ˙q˙ TSech2(˜q) T M (q) ˙q − γtanh(˜q) T K p q˜
+ γtanh(˜ q) T K v q − γtanh(˜q)˙ T C(q, ˙q) T q ˙ (7.11)
We now proceed to upper-bound ˙V (˜ q, ˙q) by a negative definite function
of the states ˜q and ˙q To that end, it is convenient to find upper-bounds for
On the other hand, note that in view of (7.7), the following inequality also
holds true since K is a diagonal positive definite matrix,
Trang 2166 7 PD Control with Gravity Compensation
γtanh(˜ q) T K p˜q ≥ γλmin{K p } tanh(˜q)2
which in turn, implies the key inequality
−γtanh(˜q) T K p q ≤ −γλ˜ min{K p } tanh(˜q)2.
A bound on γtanh(˜ q) T K v q that is obtained directly is˙
γtanh(˜ q) T K v q ≤ γλ˙ Max{K v } ˙q tanh(˜q)
The upper-bound on the term −γtanh(˜q) T C(q, ˙q) T q must be carefully˙selected Notice that
−γtanh(˜q) T C(q, ˙q) T q = −γ ˙q˙ T C(q, ˙q)tanh(˜ q)
≤ γ ˙q C(q, ˙q)tanh(˜q)
Then, considering Property 4.2 but in its variant that establishes the existence
of a constant k C1 such that C(q, x)y ≤ k C1x y for all q, x, y ∈ IR n,
The two following conditions guarantee that the matrix Q is positive
defi-nite, hence, these conditions are sufficient to ensure that ˙V (˜ q, ˙q) is a negative
Trang 3The first condition is trivially satisfied since K pis assumed to be diagonalpositive definite The second condition also holds due to the upper-bound
(7.10) imposed on γ.
According to the arguments above, there always exists a strictly positive
constant γ such that the function V (˜ q, ˙q), given by (7.8) is positive definite,
while ˙V (˜ q, ˙q) expressed as (7.12), is negative definite For this reason, V (˜q, ˙q)
is a strict Lyapunov function
Finally, Theorem 2.4 allows one to establish global asymptotic stability ofthe origin It is important to underline that it is not necessary to know the
value of γ but only to know that it exists This has been done to validate the
result on global asymptotic stability that was stated
7.3 Conclusions
Let us restate the most important conclusion from the analyses done in thischapter
Consider the PD control law with gravity compensation for n-DOF robots
and assume that the desired position q d is constant
• If the symmetric matrices K p and K v of the PD control law with ity compensation are positive definite, then the origin of the closed-loopequation, expressed in terms of the state vector
grav-˜
q T q˙TT
, is a globallyasymptotically stable equilibrium Consequently, for any initial condition
con-Measurement and Control, Vol 103, pp 119–125
The following texts present also the proof of global asymptotic stabilityfor the PD control law with gravity compensation of robot manipulators
• Spong M., Vidyasagar M., 1989, “Robot dynamics and control”, John
Wi-ley and Sons
• Yoshikawa T., 1990, “Foundations of robotics: Analysis and control”, The
MIT Press
Trang 4168 7 PD Control with Gravity Compensation
A particularly simple proof of stability for the PD controller with gravitycompensation which makes use of La Salle’s theorem is presented in
• Paden B., Panja R., 1988, “Globally asymptotically stable PD+ controller for robot manipulators”, International Journal of Control, Vol 47, No 6,
pp 1697–1712
The analysis of the PD control with gravity compensation for the case in
which the desired joint position q d is time-varying is presented in
• Kawamura S., Miyazaki F., Arimoto S., 1988, “Is a local linear PD feedback control law effective for trajectory tracking of robot motion?”, in Proceed-
ings of the 1988 IEEE International Conference on Robotics and tion, Philadelphia, PA., pp 1335–1340, April
Automa-Problems
1 Consider the PD control with gravity compensation for robots Let q d (t)
be the desired joint position
Assume that there exists a constant vector x ∈ IR n such that
where k p and k v are positive constants
a) Obtain the closed-loop equation in terms of the state vector
˜
q ˙˜ qT
Is this equation linear in the state ?
b) Assume that the desired position is q d (t) = αt where α is any real
constant Show that
lim
t →∞ q(t) = 0 ˜
3 Verify the expression of ˙V (˜ q, ˙q) obtained in (7.4).
Trang 54 Consider the 3-DOF Cartesian robot studied in Example 3.4 (see page 69)and shown in Figure 3.5 Its dynamic model is given by
where K p , K v are positive definite matrices
a) Obtain g(q) Verify that g(q) = g is a constant vector.
b) Define ˜q = [˜ q1 ˜2 ˜3]T Obtain the closed-loop equation Is the
closed-loop equation linear in the state ?
c) Is the origin the unique equilibrium of the closed-loop equation?d) Show that the origin is a globally asymptotically stable equilibriumpoint
5 Consider the following variant of PD control with gravity compensation2
c) Show that the origin is a globally asymptotically stable equilibriumpoint
6 Consider the PD control law with gravity compensation where the matrix
K v is a function of time, i.e.
τ = K p˜q − K v (t) ˙q + g(q) and where q d is constant, K p is a positive definite matrix and K v (t) is also positive definite for all t ≥ 0.
a) Obtain the closed-loop equation in terms of the state vector
˜
q T q˙TT
Is the closed-loop equation autonomous?
b) Verify that the origin is the only equilibrium point
c) Show that the origin is a stable equilibrium
7 Is the matrix Sech2(x) positive definite?
2This problem is taken from Craig J J., 1989, “ Introduction to robotics: Mechanics and control”, Second edition, Addison–Wesley.
Trang 6PD Control with Desired Gravity
Compensation
We have seen that the position control objective for robot manipulators
(whose dynamic model includes the gravitational torques vector g(q)), may be
achieved globally by PD control with gravity compensation The ing control law given by Equation (7.1) requires that its design symmetric
correspond-matrices K p and K v be positive definite On the other hand, this controller
uses explicitly in its control law the gravitational torques vector g(q) of the
dynamic robot model to be controlled
Nevertheless, it is worth remarking that even in the scenario of position
control, where the desired joint position q d ∈ IR n is constant, in the mentation of the PD control law with gravity compensation it is necessary to
imple-evaluate, on-line, the vector g(q(t)) In general, the elements of the vector g(q) involve trigonometric functions of the joint positions q, whose evaluations, re-
alized mostly by digital equipment (e.g ordinary personal computers) take a
longer time than the evaluation of the ‘PD-part’ of the control law In certainapplications, the (high) sampling frequency specified may not allow one to
evaluate g(q(t)) permanently Naturally, an ad hoc solution to this situation
is to implement the control law at two sampling frequencies: a high frequencyfor the evaluation of the PD-part, and a low frequency for the evaluation of
g(q(t)) An alternative solution consists in using a variant of this controller,
the so-called PD control with desired gravity compensation The study of this
controller is precisely the subject of the present chapter
The PD control law with desired gravity compensation is given by
τ = K p q + K˜ v ˙˜q + g(q d) (8.1)
where K p , K v ∈ IR n ×nare symmetric positive definite matrices chosen by the
designer As is customary, the position error is denoted by ˜q = q d − q ∈ IR n,
where q d stands for the desired joint position Figure 8.1 presents the diagram of the PD control law with desired gravity compensation for robotmanipulators Notice that the only difference with respect to the PD controller
Trang 7block-with gravity compensation (7.1) is that the term g(q d ) replaces g(q) The
practical convenience of this controller is evident when the desired position
q d (t) is periodic or constant Indeed, the vector g(q d ), which depends on q d and not on q, may be evaluated off-line once q dhas been defined and therefore,
it is not necessary to evaluate g(q) in real time.
Figure 8.1.Block-diagram: PD control with desired gravity compensation
The closed-loop equation we get by combining the equation of the robotmodel (II.1) and the equation of the controller (8.1) is
Trang 88 PD Control with Desired Gravity Compensation 173
for any initial condition
q d(0)T q˙d(0)TT
∈ IR 2n.
Obviously, in the scenario where the desired position q d (t) does not satisfy
the established condition, the origin may not be an equilibrium point of theclosed-loop equation and therefore, it may not be expected to satisfy the mo-tion control objective, that is, to drive the position error ˜q(t) asymptotically
equi-librium point of the closed-loop equation is that the desired joint position q d
be a constant vector In what is left of this chapter we assume that this is thecase
As we show below, this controller may verify the position objective globally,that is,
lim
t →∞ q(t) = q d where q d ∈ IR n is a any constant vector and the robot may start off from anyconfiguration We emphasize that the controller “may achieve” the position
control objective under the condition that K p is chosen sufficiently ‘large’.Later on in this chapter, we quantify ‘large’
Considering the desired position q d to be constant, the closed-loop tion may be written in terms of the new state vector
= 0∈ IR 2nis an equilibrium point Nevertheless, besides the
origin, there may exist other equilibria Indeed, there are as many equilibria
as solutions in ˜q, may have the equation
K p q = g(q˜ d − ˜q) − g(q d ) (8.3)Naturally, the explicit solutions of (8.3) are hard to obtain Nevertheless,
as we show that later, if K p is taken sufficiently “large”, then ˜q = 0 ∈ IR n isthe unique solution
Example 8.1 Consider the model of the ideal pendulum studied in
Example 2.2 (see page 30)
J ¨ q + mgl sin(q) = τ where we identify g(q) = mgl sin(q).
In this case, the expression (8.3) takes the form
Trang 9k p q = mgl [sin(q˜ d − ˜q) − sin(q d )] (8.4)For the sake of illustration, consider the following numerical values,
k p = 0.25 q d = π/2
Either via a graphical method or numerical algorithms, one mayverify that Equation (8.4) possess exactly three solutions in ˜q The
approximated values of these solutions are: 0 (rad), −0.51 (rad) and
−4.57 (rad) This means that the PD control law with desired gravity
compensation in closed loop with the model of the ideal pendulumhas as equilibria,
law with desired gravity compensation in closed loop with the model
of the ideal pendulum, has the origin as its unique equilibrium, i.e.
0 ∈ IR2.
♦
The rest of the chapter focuses on:
• boundedness of solutions;
• unicity of the equilibrium;
• global asymptotic stability.
The studies presented here are limited to the case of robots whose jointsare all revolute
8.1 Boundedness of Position and Velocity Errors, ˜ q and ˙q
Assuming that the design matrices K p and K v are positive definite (without
assuming that K pis sufficiently “large”), and of course, for a desired constant
position q to this point, we only know that the closed-loop Equation (8.2) has
Trang 108.1 Boundedness of Position and Velocity Errors, ˜q and ˙ q 175
an equilibrium at the origin, but there might also be other equilibria In spite
of this, we show by using Lemma 2.2 that both, the position error ˜q(t) and the velocity error ˙q(t) remain bounded for all initial conditions
where K(q, ˙q) and U(q) denote the kinetic and potential energy functions of
the robot, and the constant k U is defined as (see Property 4.3)
k U= minq {U(q)}
The function V (˜ q, ˙q) may be written as
V (˜ q, ˙q) = ˙q T P (˜ q) ˙q + h(˜q) (8.5)where
Since we assumed that the robot has only revolute joints,U(q) − k U ≥ 0
for all q ∈ IR n On the other hand, we have
which is non-negative for all ˜q, q d ∈ IR n Therefore, the function h(˜ q) is
also non-negative Naturally, since the kinetic energy 1
2q˙T M (q) ˙q is a positive definite function of ˙q, then the function V (˜ q, ˙q) is non-negative for all ˜q, ˙q ∈
Trang 11where we used (3.20), i.e g(q) = ∂
∂ q U(q) Solving for M(q)¨q in the
closed-loop Equation (8.2) and substituting in (8.6) we get
2M˙ − C q was eliminated by virtue of Property 4.2 Re-˙
calling that the vector q d is constant and that ˜q = q d − q, then ˙˜q = − ˙q.
Incorporating this in Equation (8.7) we obtain
˙
V (˜ q, ˙q) = − ˙q T K v q ˙ (8.8)
Using V (˜ q, ˙q) and ˙V (˜q, ˙q) given by (8.5) and (8.8) respectively, and voking Lemma 2.2 (cf page 52), we conclude that both, ˙q(t) and ˜ q(t) are also bounded and that the velocities vector ˙q(t), is square integrable, i.e.
in- ∞
As a matter of fact, it may be shown that the velocity ˙q is not only
bounded, but that it also tends asymptotically to zero For this, notice from(8.2) that
Prop-from (8.10) we conclude that the accelerations vector ¨q(t) is also bounded
and therefore, from (8.9) and Lemma 2.2, we conclude that
lim
t →∞ ˙˜q(t) = lim
t →∞ q(t) = 0 ˙
For the sake of completeness we show next how to compute explicit
upper-bounds on the position and velocity errors Taking into account that V (˜ q, ˙q) is
a non-negative function that decreases along trajectories (i.e d
2q(t)˙ T
M (q(t)) ˙q(t) ≤ V (˜q(0), ˙q(0)) (8.12)
Trang 128.1 Boundedness of Position and Velocity Errors, ˜q and ˙ q 177
for all t ≥ 0, and where
t = 0, are measured by appropriate instruments physically collocated for this
purpose on the robot
We obtain next, explicit bounds on˜q and ˙q as a function of the initial
conditions We first notice that
where we used the fact that c ≥ 0 and that for all vectors x and y ∈ IR n
we have−x T y ≤ x T y ≤ xy, so− x y ≤ x T y Taking (8.11) into
been shown before in Example 2.11 (see page 45), the potential energyfunction is given by
Trang 13U(q) = mgl[1 − cos(q)]
Since minq {U(q)} = 0, the constant k U takes the value of zero.
Consider the numerical values used in Example 8.1
k p = 0.25 k v = 0.50
q d = π/2
Assume that we use PD control with desired gravity compensation
to control the ideal pendulum from the initial conditions q(0) = 0 and
˙
q(0) = 0.
0
5
10
15
20
25
30
35 q(t)˜ 2 [rad2]
t [s]
Figure 8.2. PD control with desired gravity compensation: graph of the position error ˜q(t)2
With the previous values it is easy to verify that
g(q d ) = mgl sin(π/2) = 1
V (˜ q(0), ˙ q(0)) = 1
2k p˜
2(0) + mgl ˜ q(0) + 1
2k p (mgl)
2= 3.87
According to the bounds (8.13) and (8.14) and taking into account the previous information, we get
˜2(t) ≤
⎡
⎣mgl +
[mgl + k p q(0)]˜ 2+ (mgl)2
k p
⎤
⎦
2
˙
q2(t) ≤ J2
k p
2 ˜
2(0) + mgl ˜ q(0) + 1
2k (mgl)
2
Trang 148.1 Boundedness of Position and Velocity Errors, ˜q and ˙ q 179
0
1
2
3 q(t)˙ 2 [(rad
s )2]
t [s]
Figure 8.3.PD control with desired gravity compensation: graph of velocity, ˙q(t)2
≤ 7.75
rad s
2
for all t ≥ 0 Figures 8.2 and 8.3 show the plots of ˜q(t)2 and ˙q(t)2
respectively, obtained by simulation We clearly appreciate from the plots that both variables satisfy the inequalities (8.15) and (8.16) Finally, it is interesting to observe from Figure 8.2 that limt →∞ q(t)˜ 2=
20.88 (evidence from simulation shows that lim t →∞ q(t) =˜ −4.57) and
limt →∞ q˙2(t) = 0 and therefore
lim
t →∞
˜
q(t)
˙
q(t) =
−4.57
This means that the solutions tend precisely to one among the three equilibria computed in Example 8.1, but which do not correspond to the origin The moral of this example is that PD control with desired gravity compensation may fail to meet the position control objective
♦
To summarize the developments above we make the following remarks Consider the PD control law with desired gravity compensation for robots
with revolute joints Assume that the design matrices K p and K v are positive
definite If the desired joint position q d (t) is a constant vector, then:
• the position error ˜q(t) and the velocity ˙q(t) are bounded Maximal bounds
on their norms are given by the expressions (8.13) and (8.14) respectively
• lim
t →∞ q(t) = 0 ∈ IR˙ n
Trang 158.2 Unicity of Equilibrium
For robots having only revolute joints, we show that with the choice of K p
sufficiently “large”, we can guarantee unicity of the equilibrium for the
closed-loop Equation (8.2) To that end, we use here the contraction mapping theorem (cf Theorem 2.1 on page 26).
The equilibria of the closed-loop Equation (8.2) satisfy
If the function f (˜ q, q d) satisfies the condition of the contraction mapping
theorem, that is, if f (˜ q, q d ) is Lipschitz (cf page 101) with Lipschitz constant
strictly smaller than 1, then the equation ˜q = f(˜q, q d) has a unique solution
have
f(x, q d)− f(y, q d) ≤ k g
λmin{K p } x − y ,
which, according to the contraction mapping theorem, implies that a sufficient
condition for unicity of the solution of f (˜ q, q d)− ˜q = 0 or equivalently of
K −1
p [g(q d − ˜q) − g(q d)]− ˜q = 0
and consequently, for the unicity of the equilibrium of the closed-loop
equa-tion, is that K p be chosen so as to satisfy
λmin{K p } > k g (8.17)
... 158.2 Unicity of Equilibrium
For robots having only revolute joints, we show that with the choice of K p
sufficiently... class="text_page_counter">Trang 14
8.1 Boundedness of Position and Velocity Errors, ˜q and ˙ q 179
0
1
2
3 q(t)˙... 0
and consequently, for the unicity of the equilibrium of the closed-loop
equa-tion, is that K p be chosen so as to satisfy
λmin{K