Control of Underactuated Driftless Nonlinear Systems
3.1 Control of Driftless Systems
3.1.1 Averaging Theory for Control
In Equation (3.1), the functionsua(q, t)are control functions, which decompose into state feedback terms and time-periodic terms: ua(q, t) = fa(q) +va(t/). The functions va(t) areT-periodic functions and
the functions fa(q)are feedback terms for stabilization of the directly controlled states. Substitution into equation (3.1) gives
˙
q =Ya(q)fa(q) +Ya(q)va(t/) =XS(q) +Ya(q)va(t/), (3.7) whereXS(q) = Ya(q)fa(q). A transformation of time,t 7→ τ, takes (3.7) into a form compatible with averaging theory,
dq
dτ =XS(q) +Ya(q)va(t). (3.8)
The average of (3.8) is
q(τ) = Truncm−1(P(τ)) (z(τ)) +O(m), dz
dτ = Truncm(Z). Transformating back to time,t,
q(t) = Truncm−1(P(t/)) (z(t/)) +O m−1 ,
˙ z= 1
Truncm(Z).
Recall that P(t) and Z can be expressed as power series in . The time transformation lowers the power series order by one. ForP(t), this might pose a problem at lower orders of averaging sinceP(t)will go from being O(m)-close toO m−1
-close. Choosing the improved mth-order average, withTruncm(P(t)), will resolve the problem,
q(t) = Truncm(P(t/)) (z(t/)) +O(m),
˙ z= 1
Truncm(Z). Averaged Coefficients
The averaged version of (3.8) will contain vector fields that have combinations of time integrals and Jacobi- Lie brackets, see Section 2.4. Since the periodic inputs act as coefficients to the input vector fields, and iterated Jacobi-Lie brackets are multi-linear, the integrals can be factored. The integral terms represent the net effect of the inputs on the Jacobi-Lie bracket terms, and will be called averaging coefficients.
Define the following notation for the averaging coefficients:
V(a)(n)(t)≡ Z t
t0
Z sn−1 . . .
Z s2
va(s1)ds1· · ·dsn−1. (3.12) For the purposes of this paper, the initial time will always bet0 = 0. Cases of multiple upper and lower indices denote products of this type of integral. An example isV(a,b)(1,1)(t),
V(a,b)(1,1)(t) = V(a)(1) V(b)(1) = Z t
0
va(s1)ds1 Z t
0
vb(s1)ds1
.
Time-averaged terms are called averaged coefficients. The single index averaged coefficients are V(a)(n)(τ) = 1
T Z T
0
V(n)(a)(τ) dτ = 1
TV(n+1)(a) (T).
Additionally define the following,
Ve(a)(n)≡V(n)(a) −V(a)(n), and for the multi-index version,
Ve(A)(N)≡V(A)(N)−V(A)(N),
where(A) = (a1, a2, ..., a|A|)and (N) = (n1, n2, ..., n|N|). Theb·symbol will denote integrals within the product structure. For example,
V(ca,b,c)
(0,0,1)c (t) = Z t
0
V(a,b)(0,0)(τ) dτ V(c)(1)(t)
= Z t
0
(va(τ)vb(τ)) dτ
Z t 0
vc(τ) dτ
.
Another example is the averaged coefficient V(1,0)(a,b)(τ) = 1 T
Z T 0
V(a,b)(1,0)(τ) dτ = 1 TV(ca,b)
(1,0)c (T).
The notation will simplify the expressions for the averaged expansions of (3.7).
Comment. Those familiar with approximate inversion techniques for open-loop approximate tracking of nonholonomic systems will notice the indefinite nature of the integrals in Equation (3.12). The generalized averaging theory does lead to definite integrals. However, the lower integral limit can be shown to corre- spond to integration of the initial conditions of dynamical system (3.7). Although the definite integrals do predict the transient response, under state feedback the transient effects of the initial conditions die out over time. Our generalized averaging theory leads to the use of indefinite integrals, as does classical averaging theory. In the motion control algorithms of Bullo, and Mart´inez and Cort´es [21, 100], the integrals are also indefinite. The examples section, Section 3.3, will more explicitly cover this difference between open-loop approximate inversion and closed-loop approximate inversion.
Averaged Expansions
First-order. The first-order averaged version of system (3.7) is
˙
z=XS+ V(a)(0)(t)Ya(z), (3.13)
where the upper and lower indicies indicate multiplication and summation. The Floquet mapping is approx- imated by
Trunc0(P(t/)) = Id. (3.14)
An improved approximation is
Trunc1(P(t/)) = Id + Z t/
0
Ve(a)(0)(τ) dτ Ya(·). (3.15) First-order averaging may be useful when the inputs are constrained to lie in an admissable set that is restric- tive. In [46], Gallivan et al. achieve a set of transition rates for thin film growth that is unachievable using the inputs that the process is normally restricted to. Although the inputs are time-periodic, the inputs for film-growth are not in control affine form, and the averaging process highlights the nonlinear effect of time- periodic inputs. For systems in the affine control form, a non-vanishing average is equivalent to constant actuation, to first-order. It is the higher-order nonlinear effect of periodic inputs that will be important.
Second-order. Second- and higher-order average approximations are usually sought when the first-order average of the time-periodic inputs vanishes.
Assumption 2 For higher-order averaging, it is assumed that the time-averages of the oscillatory control inputs vanish, i.e.,V(a)(0)(t) = 0.
The second-order average has the form,
˙
z=XS(z) +V(a)(1)(t) [Ya(z), XS(z)] + 1
2V(a,b)(1,0)(t) [Ya(z), Yb(z)], (3.16a) Trunc1(P(t/)) = Id +V(1)(a)(t/)Ya(·), (3.16b) whose derivation requires integration by parts and Assumption 2. The averaged coefficient notation allows for a quick assessment of the nature of the integral product, as the lower indices determine which terms are integrated and how many times to iterate the integral.
Third-order If the LARC is satisfied via higher levels of iterated Lie brackets, then higher-order averaging is required. The averaged vector field to third order1is
˙
z=XS+V(a)(1)(t) [Ya, XS] +1
2V(a,b)(1,0)(t) [Ya, Yb] +2
V(a)(2)(t)−1
2TV(a)(1)(t)
[XS,[XS, Ya]]
−1 32
V(ca,b)
(c1,0)(t) +1
2TV(a,b)(1,0)(t)
[XS,[Ya, Yb]]
+1 32
V(a,b)(1,1)(t) + V(a,b)c
(1,0)c (t)−TV(a,b)(1,0)(t)
[Ya,[Yb, XS]] +1
32V(a,b,c)(1,1,0)(t) [Ya,[Yb, Yc]].
(3.17)
The Floquet mapping is
Trunc2(P(t/)) = Id +V(a)(1)(t/)Ya+2 Z t/
0
Ve(a)(1)(τ) dτ [Ya, XS] +1
22 Z t/
0
Ve(a,b)(1,0)(τ) dτ [Ya, Yb] +1
22V(1,1)(a,b)(t/)Ya·Yb. (3.18) Higher-order. To compute fourth- and higher-order averages follow the averaged expansion procedure delineated in Chapter 2, taking into account the affine control decomposition. Appendix 3.A contains the lower-order averaged expansions, to give the reader a sense of the procedure.
The Averages Revisited
The existence of the vector fieldsfa(q)complicates matters because they act like drift terms. Suppose that first-order, direct state feedback was not necessary, fa(q) = 0. The simplifications afforded in this case allow for higher-order averages to be computed.
Third-order. The third-order equations simplify to
˙ z= 1
2V(a,b)(1,0)(t) [Ya, Yb] + 1
32V(a,b,c)(1,1,0)(t) [Ya,[Yb, Yc]], (3.19)
1See Appendix 3.A for the relevant calculations
and the reductions actually make a fourth-order expansion tractable.
Fourth-order. The averaged autonomous vector field to fourth-order is
˙
z = V(a,b)(1,0)(t) [Ya, Yb] +1
32V(a,b,c)(1,1,0)(t) [Ya,[Yb, Yc]] (3.20)
= 1 123
V(a,b,c,d)c
(1,0,1,0)c (t) [[Ya, Yb],[Yc, Yd]]−
V(a,b,c(1,0,cc,d)
0,1)(t) + V(a,b,c,d)(1,1,1,0)(t)
[Ya,[Yb,[Yc, Yd]]]
.
The Floquet mapping is
Trunc3(P(t/)) = Id +V(1)(a)(t/)Ya+1 22
Z t/
0
Ve(1,0)(a,b)(τ) dτ [Ya, Yb] +1
22V(a,b)(1,1)(t/)Ya·Yb
− 1
22V(a)(1)(t/)V(b,c)(1,0)(t)t Ya·[Yb, Yc] +1
63V(a,b,c)(1,1,1)(t)Ya·Yb·Yc
+ 1
43V(a)(1)(t/) Z t/
0
V(1,0)(b,c)(σ) dσ (Ya·[Yb, Yc] + [Yb, Yc]·Ya).
(3.21) For driftless systems, it is the iterated Jacobi-Lie brackets generated using the set{Ya}m1 that is impor- tant for both determination of controllability, and for realization of control. When the vector fieldXS 6= 0, the averaged expansions become more complicated, but the Jacobi-Lie brackets that are needed for control do not change. Therefore, it always useful to examine the case whereXS = 0to determine the appropriate averaged coefficients and Jacobi-Lie brackets to analyze.