17.3.7 Passivity-Based Approaches One may also exploit the passivity of the rigid robot dynamics to derive robust and adaptive control algorithms for manipulators.. and Grimm, W.M., On L
Trang 117-18 Robotics and Automation Handbook
the latter term following sinceθ is constant, i.e., ˜θ = ˆθ Using the parameter update law (17.102) gives
˙
From this it follows that the position tracking errors converge to zero asymptotically and the parameter estimation errors remain bounded Furthermore, it can be shown that the estimated parameters converge
to the true parameters provided the reference trajectory satisfies the condition of persistency of excitation,
αI ≤
t0+T
t0
Y T (q d , ˙q d, ¨q d )Y (q d , ˙q d, ¨q d )dt ≤ β I (17.106)
for all t0, whereα, β, and T are positive constants.
In order to implement this adaptive feedback linearization scheme, however, one notes that the accel-eration ¨q is needed in the parameter update law and that ˆ M must be guaranteed to be invertible, possibly
by the use of projection in the parameter space Later work was devoted to overcome these two drawbacks
to this scheme by using so-called indirect approaches based on a (filtered) prediction error.
17.3.7 Passivity-Based Approaches
One may also exploit the passivity of the rigid robot dynamics to derive robust and adaptive control algorithms for manipulators These methods are qualitatively different from the previous methods which were based on feedback linearization In the passivity-based approach, we modify the inner loop control as
where v, a, and r are given as
v = ˙q d − q
a = ˙v = ¨q d − ˙ q
r = ˙q d − v = ˙ q + q with K , diagonal matrices of positive gains In terms of the linear parametrization of the robot dynamics,
the control (17.107) becomes
and the combination of Equation (17.107) with Equation (17.41) yields
Note that, unlike the inverse dynamics control, the modified inner loop control (17.41) does not achieve
a linear, decoupled system, even in the known parameter case ˆθ = θ However, in this formulation the
regressor Y in Equation (17.109) does not contain the acceleration ¨ q nor is the inverse of the estimated
inertia matrix required These represent distinct advantages over the feedback linearization based schemes
17.3.8 Passivity-Based Robust Control
In the robust passivity-based approach of [36], the term ˆθ in Equation (17.108) is chosen as
ˆ
whereθ0is a fixed nominal parameter vector and u is an additional control term The system (17.109)
then becomes
M(q ) ˙r + C(q, ˙q)r + K r = Y(a, v, q, ˙q)(˜θ + u) (17.111)
Trang 2Robust and Adaptive Motion Control of Manipulators 17-19
where ˜θ = θ0− θ is a constant vector and represents the parametric uncertainty in the system If the
uncertainty can be bounded by finding a nonnegative constant,ρ ≥ 0, such that
then the additional term u can be designed according to the expression,
u=
−ρ Y T r
||Y T r||, if ||Y T r || >
−ρ Y T r, if ||Y T r || ≤
(17.113)
The Lyapunov function
V = 1
2r
is used to show uniform ultimate boundedness of the tracking error Note that ˜θ is constant and so is not
a state vector as in adaptive control Calculating ˙V yields
˙
V = r T M ˙r+1
2r
= −r T K r + 2qT K ˙ q + 1
2r
T( ˙M − 2C)r + r T Y (˜ θ + u) (17.116)
Using the passivity property and the definition of r , this reduces to
˙
V= −qT T K q − ˙q T K ˙ q + r T Y (˜ θ + u) (17.117)
Uniform ultimate boundedness of the tracking error follows with the control u from (17.113) See [36]
for details
Comparing this approach with the approach in the section (17.3.5), we see that finding a constant bound
ρ for the constant vector ˜θ is much simpler than finding a time-varying bound for η in Equation (17.44).
The boundρ in this case depends only on the inertia parameters of the manipulator, while ρ(x, t) in
Equation (17.69) depends on the manipulator state vector and the reference trajectory and, in addition, requires some assumptions on the estimated inertia matrix ˆM(q ).
17.3.9 Passivity-Based Adaptive Control
In the adaptive approach the vector ˆθ in Equation (17.109) is now taken to be a time-varying estimate of
the true parameter vectorθ Combining the control law (17.107) with (17.41) yields
The parameter estimate ˆθ may be computed using standard methods such as gradient or least squares For
example, using the gradient update law
together with the Lyapunov function
V =1
2r
T M(q )r+q T K q +1
results in global convergence of the tracking errors to zero and boundedness of the parameter estimates since
˙
V= −qT T K q − ˙q T
K ˙ q + ˜θ T { ˙ˆθ + Y T r} (17.121) See [38] for details
Trang 317-20 Robotics and Automation Handbook
PERFORMANCE
+
+
MODEL N
MODEL 2
MODEL 1
– –
–
min Ji
J1
J2
JN
et2
etN
et1
t^N
t^2
t^1
(q, q.)
•
•
•
FIGURE 17.9 Multiple-model-based hybrid control architecture.
17.3.10 Hybrid Control
A Hybrid System is one that has both continuous-time and discrete-event or logic-based dynamics Supervisory Control, Logic-Based Switching Control, and Multiple-Model Control are typical control
architectures in this context In the robotics context, hybrid schemes can be combined with robust and adaptive control methods to further improve robustness In particular, because the preceeding robust and
adaptive control methods provide only asymptotic (i.e., as t→ ∞) error bounds, the transient performance may not be acceptable Hybrid control methods have been shown to improve transient performance over fixed robust and adaptive controllers
The use of the term Hybrid Control in this context should not be confused with the notion of Hybrid
Position/Force Control [41] The latter is a familiar approach to force control of manipulators in which
the term hybrid refers to the combination of pure force control and pure motion control.
Figure 17.9 shows the Multiple-Model approach of [12], which has been applied to the adaptive control
of manipulators In this architecture, the multiple models have the same structure but may have different nominal parameters in case a robust control scheme is used, or different initial parameter estimates if
an adaptive control scheme is used Because all models have the same inputs and desired outputs, the
identification errors e I j are available at each instant for the j th model The idea is then to define a
performance measure, for example,
J (e I j (t)) = γ e2
I j (t) + β
t
0
e2
and switch into the closed loop the control input that results in the smallest value of J at each instant.
17.4 Conclusions
We have given a brief overview of the basic results in robust and adaptive control of robot manipulators
In most cases, we have given only the simplest forms of the algorithms, both for ease of exposition and for reasons of space An extensive literature is available that contains numerous extensions of these basic results The attached list of references is by no means an exhaustive one The book [10] is an excellent and
Trang 4Robust and Adaptive Motion Control of Manipulators 17-21
highly detailed treatment of the subject and a good starting point for further reading Also, the reprint book [37] contains several of the original sources of material surveyed here In addition, the two survey papers [1] and [28] provide additional details on the robust and adaptive control outlined here
Several important areas of interest have been omitted for space reasons including output feedback control, learning control, fuzzy control, neural networks, and visual servoing, and control of flexible robots The reader should consult the references at the end for background on these and other subjects.
References
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1208, June 2000
Trang 718 Sliding Mode Control of Robotic
Manipulators
Hector M Gutierrez
Florida Institute of Technology
18.1 Sliding Mode Controller Design–An Overview
Manipulator Motion Control Problem
Control
Sliding Mode Laws Sliding Mode Formulation of the Robotic Manipulator Motion Control Problem • Sliding Mode Controller Design 18.5 Conclusions
18.1 Sliding Mode Controller Design An Overview
Sliding mode design [1–6] has several features that make it an attractive technique to solve tracking problems in motion control of robotic manipulators, the most important being its robustness to parametric uncertainties and unmodelled dynamics, and the computational simplicity of the algorithm One way of looking at sliding mode controller design is to think of the design process as a two-step procedure First, a region of the state space where the system behaves as desired (sliding surface) is defined Then, a control action that takes the system into such surface and keeps it there is to be determined Robustness is usually achieved based on a switching control law The design of the control action can be based on different strategies, a straightforward one being to define a condition that makes the sliding surface an attractive region for the state vector trajectories Consider a nonlinear affine system of the form:
x (n i)
i = f i(x) +
m
j=1
b i j(x)u j, i = 1, , m, j = 1, , m (18.1)
whereu = [u1, , u m]T is the vector of m control inputs, and the state x is composed of the x icoordinates
to be tracked and their first (n i − 1) derivatives Such systems are called square systems because they
have as many control inputs u j as outputs to be controlled x i [3] The motion control problem to be addressed is the one of making the state vectorx track a desired trajectory r Consider first the time-varying
manifoldσ givenbytheintersectionofthesurfacess i(x, t) = 0, i = 1, , m,specifiedbythecomponentsof
Trang 8Sliding Mode Control of Robotic Manipulators 18-3
J1
q1
r1
J2
m1
m2
q2
r2
yc
xc
constrain surface
q
FIGURE 18.1 Rigid-link rigid-joint robot interacting with a constrain surface.
therefore either a position tracking problem in the q -state space or a force control problem or a hybrid of
both
Example 18.1
Consider the force control problem of the two-link robot depicted in Figure 18.1 [7] A control scheme to
track a desired contact force F d is shown in Figure 18.2, where J cis the robot’s Jacobian matrix relative to the coordinate system fixed at the point of contact x c , y c , T is the n × n diagonal selection matrix with elements equal to zero in the force controlled directions, F is the measured contact force, τ ff is the output
of the feed-forward controller, andτ smthe output of the sliding mode controller (Figure 18.2)
The matrices used to estimate the torque vector are
H(q )=
(m1+ m2)r2+ m2r2+ 2m2r1r2C2+ J1 m2r2+ m2r1r2C2
m2r2+ m2r1r2C2 m2r2+ J2
(18.8)
c (q , ˙q )=
−m2r1r2S2˙q2− 2m2r1r2S2˙q2˙q2
Fd F
robot and constrain surface
I-T Jc
–
τff
τsm
τes
FIGURE 18.2 Force controller with feed-forward compensation.
Trang 9Sliding Mode Control of Robotic Manipulators 18-5
where > 0 is the boundary layer thickness ε = /λ n−1is the corresponding boundary layer width,
whereλ is the design parameter of the sliding surface (18.2) for the case λ1= λ2= · · · = λ n The switching
control law (18.5) remains the same outside the boundary layer, and inside b L (t) the control action is interpolated by using, e.g., u sw = s/ , providing an overall piece-wise continuous control law This creates
an upper bound in tracking error given byλ (as opposed to perfect tracking): ∀t ≥ 0, |x (i ) (t) − r (i ) (t)| ≤ (2λ) i ε; i = 0, , n − 1, for all trajectories starting inside b L (t) The boundary layer thickness can be
made time-varying to tune up the control law to exploit the control bandwidth available, and in that case the sliding condition (18.4) becomes
|s (x, t)| ≥ ⇒ 1
2
d
dt s
A simple switching term that satisfies (18.13) is
u sw = (k(x, t) − ˙ ) sats
, sat(z)=
z, |z| ≤ 1
which replaces Equation (18.5) This method eliminates control chatter provided that high-frequency unmodelled dynamics are not excited [3] and that the corresponding trade-off in tracking accuracy is acceptable
The equivalent control (u eq ) [3, 5, 9] is the continuous control law that would maintain dS/dt = 0
if the dynamics were exactly known Consider the nonlinear affine system (18.15) and (18.16) with the associated sliding surface (18.17):
˙x2= f2(x1,x2)+ B2(x)u + B2(x) d(t) (18.16)
S = {x : ϕ(t) − s a(x) = s(x, t) = 0} (18.17) whereu is the m × 1 vector of control inputs, d is the m × 1 vector of input disturbances, x2is the vector
of m states to be controlled, x1is the (n − m) vector of autonomous states, f is the n × 1 vector of system’s equations, B2(x) is a m × m input gain matrix, s a(x) is a m × 1 continuous function of the states to be
tracked, andϕ(t) is the m × 1 vector of desired trajectories The equivalent control is obtained from
d
dt(ϕ(t) − sa(x, u = ueq))= 0 (18.18)
by calculating dS/dt= 0 from Equation (18.18), replacing the system’s Equation (18.15) and
Equa-tion (18.16) in the resulting expression, and finally solving for u eq This yields
u eq = − d + (G2B2)−1
d ϕ
dt − G2f2− G1f1
(18.19)
where d s a /dt = G1x1+ G2x2and G1, G2are defined as [∂s a /∂ x1]= G1, [∂s a /∂ x2]= G2
The continuous control (18.19) is a nonrobust control law because it assumes the plant model to be exact Several different techniques have been proposed to achieve robustness against parametric variations
and unmodelled dynamics (disturbances) [6,9] by a continuous control law (as opposed to a switching
control action such as Equation (18.5)), which is obviously essential in systems where the control inputs are continuous functions and hence Equation (18.5) cannot be realized One such technique [9] is based on
r Given the Lyapunov function candidate v = S T S/2, if the Lyapunov stability criteria are satisfied,
the solutionϕ(t) − s a(x) = 0 is stable for all possible trajectories of systems (18.15) and (18.16).
r The problem then becomes that of finding the control u that satisfies the Lyapunov condition
dv = −S T DS ≤ 0 for some positive definite matrix D.
Trang 10Sliding Mode Control of Robotic Manipulators 18-9
[12] Tso, S.K., Xu, Y., and Shum, H.Y., Variable structure model reference adaptive control of robot
manipulators, Proc 1991 IEEE International Conference on Robotics and Automation, Sacramento,
CA, Vol 3, pp 2148–2153, April 1991
[13] Arai, F., Fukuda,T., Shoi, W., and Wada, H., Models of mechanical systems for controller
design, Proc IFAC Workshop on Motion Control for Intelligent Automation, Perugia, Italy, pp 13–19,
1992
... IEEE International Conference on Robotics and Automation, Sacramento,CA, Vol 3, pp 21 48? ?21 53, April 1991
[ 13] Arai, F., Fukuda,T., Shoi, W., and Wada, H., Models of mechanical... m2< /small>)r2< /small>+ m2< /small>r2< /small>+ 2m2< /small>r1r2< /small>C2< /small>+...
−m2< /small>r1r2< /small>S2< /small>˙q2< /small>− 2m2< /small>r1r2< /small>S2< /small>˙q2< /small>˙q2< /small>