Control of Underactuated 1-Homogeneous Systems with Drift
5.3 Examples
5.3.2 Vibrational Control of a Passive Arm
0 5 10 15 20 25 30 35
−1 0 1 x3
0 5 10 15 20 25 30 35
−0.5 0 0.5 1
x1
0 5 10 15 20 25 30 35
−0.5 0 0.5 1
x2
Figure 5.4: Improved Continuous Point Stabilization for Nonholonomic Integrator.
The gradient corresponding to the kinetic energy metric is gradϕ(q) =Mij∂ϕ
∂qj. The potential term consists of integrable control inputs,
Vcontrol(q) = 1
2kθ12+1
√Kθ1v(t/),
wherev(t) =v(t+T). The equations of motion are
¨
qi= Γijkq˙jq˙k+Rij(q) ˙qj+grad(Vcontrol), (5.56) where the Γijk are the Christoffel symbols corresponding to the kinetic energy metric (5.55), R(q) is Rayleigh dissipation represented by the matrix,
R(q) =
−102 0 0 102
.
Although the equations of motion in Equation (5.56) exhibit periodic behavior, additional work is needed to transform them into standard form for averaging. To describe the dynamics of the system in first order form, usex= (q,q),˙
˙
x=Γ +e Dlift(x) + (gradVcontrol)lift, where
Γ =e
q˙ Γq( ˙q,q)˙
. With the definition,
Ya≡gradqa, the equations of motion can be expressed as
˙
x=eΓ +Dlift(x)−kY1liftθ1+1
Y1liftv1(t/).
A transformation of time,t7→τ, leads to dx
dτ =
eΓ +Dlift(x)−kY1liftθ1
+Y1liftv1(τ) =Xe+X.
The high-amplitude, high-frequency control inputs are the primary vector field, with the remaining dynamics acting as a perturbation. The variation of constants may be used to render the system in the required form for averaging. The vector fieldY from Equation (5.17) is computed using the series representation in Equation (5.18),
dy
dτ =Y =Xe + V(1)(1)(τ)h
Y1lift,Xei
−1
2V(1,1)(1,1)(τ)D
Y1lift : Y1liftE ,
where theΓused in the symmetric product isXe, c.f. Definition 37. The Lagrangian structure of the system implies that it is a 1-homogeneous control system. First order averaging results in
dy
dτ =Y =Xe + V(1)(1)(τ)h
Y1lift,Xei
−1
2V(1,1)(1,1)(τ)D
Y1lift : Y1liftE .
To demonstrate stability of equilibria, the Lagrange-Dirichlet criterion will be used. In order to utilize the Lagrange-Dirichlet criterion, the Lagrangian system must be transformed to a Hamiltonian system using the Legendre transformation [98]. The tangent bundle coordinates (q,q)˙ are transformed to cotangent bundle coordinates(q, p).
Theorem 41 (Lagrange-Dirichlet criterion) [98] If the matrix of the second variationδ2His either posi- tive or negative definite at the equiblirium(qe, pe), then it is a stable fixed point.
For Hamiltonian systems defined on a cotangent bundle, the equilibrium subspace is a subspace of the zero section, c.f. Definition 44. Equilibria will have vanishing momenta, (qe, pe) = (qe,0). The Lagrange- Dirichlet criterion simplifies to a non-degenerate minimum of the potential energy of the Hamiltonian.
Actuation with Cosine. Using the input function v(τ) = 2√
K cos(τ), the averaged coefficients evaluate to
V(1)(1)(τ) = 0 and V(1,1)(1,1)(τ) = 2K.
The averaged equations of motion are dy
dτ =Y =Xe−KD
Y1lift : Y1liftE .
The symmetric product term is integrable and may be incorporated into the Lagrangian as an additional potential. The averaged potential is
Vavg = 1
2kθ12+K 20
2313−1152 cos(2θ2), (5.57)
and the averaged equations of motion may be derived from the Lagrangian, Lavg(q,q) =˙ 1
2q˙TM(q) ˙q−Vavg. (5.58) When the system is transformed into Hamiltonian form, the Lagrange-Dirichlet criteria can be used to determine stability of the average, from which orbital stability of the original system (5.56) may hold.
Figure 5.6 shows minima at±π2, which together with dissipation leads to the conclusion of stability. The actual system has an asymptotically stable orbit. The results up to here are known.
Actuation with Sine. What is interesting about this example is that the vibrational control design of [23]
restricts the oscillatory actuation to be cosine. Intuitively sine should function equally well, however in developing the theory, the following integral constraints were imposed,
Z T
0
va(t) dt = 0 and Z T
0
Z t
0
va(τ) dτ dt = 0. (5.59)
Both cosine and sine satisfy the first integral constraint, however, only cosine satisfies the second constraint.
The double integral constraint is to ensure thatV(a)(1)(t) = 0, whose vanishing allows the averaged equations of motion to be described by the simple averaged Lagrangian of (5.58). The simple Lagrangian transforms trivially to a Hamiltonian that can be analyzed using the Lagrange-Dirichlet criteria.
-0.75 -0.5 -0.25 0 0.25 0.5 0.75 -3
-2 -1 0 1 2 3
θ2
θ1
Figure 5.6: Contours forVavg, Equation (5.57).
Choosing the sine function will lead to the same net response: stabilization to the π2, with one caveat.
The oscillatory inputs will stabilize to one of the two equilibriaθ2 =±π2 depending on what region in phase space that the arm begins in. Under the averaging process, the two inputs, sine and cosine, begin in different regions of the phase space for the same initial conditions. This subtle distinction can be seen by examining the structure of the averaged equations. Under oscillatory forcing with cosine, the averaged equations of motion are
Zcos=Z− hY1 : Y1i, (5.60)
whereas, under actuation by sine,
Zsin=Z+ 2√
K[Y1, Z]− hY1 : Y1i. The averaged product evaluates to V(1)(1)(t) = 2√
K because definite integrals were used to compute the averaged coefficient. The two averaged equations of motion differ by one term, which is very important because it hinders the stability analysis from [23]. The averaged equations of motion are no longer in second-order form and the Lagrange-Dirichlet criterion cannot be applied. The Jacobi-Lie bracket term [Y1, Z]actually incorporates a bias in the angular velocities. By examining how the term[Y1, Z]leads to the velocity shift, we may define the following transformation of state,
Θ(z) =
θ1 θ2
ω1+ 2313+1152 cos(2θ40√K 2)
ω2− 8√K(5+6 cos(θ2)) 2313−1152 cos(2θ2)
. (5.61)
Using the transformation,Θ, it is possible to show that Zcos= Θ∗Zsin.
Therefore, the average respresented byZcoscan be used to represent the averaged evolution using oscillatory inputs with the sine function, only after changing coordinates. In fact, it is possible to show that for any choice of phase, i.e.,v(t) = cos(t−ϕ), there exists a transformation depending onϕ,Θϕ, that will transform the averaged vector fieldZϕ, corresponding to the choice ofcos(t−ϕ), toZcosfrom Equation (5.60). Recall from Section 2.3.2 that the monodramy map was used to calculate the averaged autonomous vector field.
From Equation (2.28), the monodramy map was defined to be Ψ = ΦX0,T. The same section, however,
discussed that there was a degree of freedom in calculating the monodramy map, the initial time could be shifted by any amount. Therefore, the choiceΨ = ΦXφ,φ+T for the monodramy map will give the averaged autonomous vector fieldZcos. The mappingΘabove, is precisely the mappingΘdiscussed in Section 2.3.2.
What this means is that there will typically exist an averaged vector field with a given phase that does not incorporate the effect of initial conditions on the average. Using the initial conditions,q0 = (0,0,0,0), the initial averaged conditions for forcing by sine and using the averaged vector fieldZcosare
q00 = Θ(q0) = (0,0,200√ 6
1161 ,−440√ 6 1161 ),
meaning that the ”elbow” part of the arm, θ2, will first be pushed towards the fixed point −π2, whereas the ”shoulder” part of the arm,θ1, will be pushed upwards. This behaviour is seen in the initial transient response of the passive arm, c.f. Figure 5.7. Over time, the transient effect due to the initial conditions die out, on account of the stabilizing feedback to the first joint.
In Figure 5.8 the same system is simulated without the use of stabilizing feedback to the first joint.
Notice that the transient response is attenuated due to the Rayleigh dissipation, but the angle of the first joint does not return to0degrees as in the case with feedback. In both cases, the angle of the second joint still stabilizes to one of the two equilibria,±π2.
Recall the zero-average assumption that is commonly found in classical averaging theory [161, 17].
The constraints are the same as (5.59), with the exception that the oscillatory input set is not restricted to satisfy the constraints. Instead, the terms are annihilated by introducing an integration constant. The net effect, as we have discussed, is to ignore certain parts of the system dynamics. Consequently, the average may not be a faithful approximation to the system in the open-loop. In particular annihilation of these terms won’t capture transients due to initial conditions, but is capable of obtaining the gross characteristic behavior, which might be the more important part. Especially if there exists stabilizing behavior that will attenuate this initial transient response. Under the classical averaging methods, both sine and cosine would result in the same averaged equations of motion. Stability of the θ2 = π2 position holds for both cases, but we have seen that the domain of attraction is not the same. Thus the classical averaging methods are only valid within a sufficiently small neighborhood of the fixed point. Introduction of integration constants is physically equivalent to averaging around the steady-state oscillatory behavior of the system; interpreted differently, classical averaging theory is akin to a linearization around the ideal average as the constant terms are ignored. Which approach to averaging is more useful depends on the calculation at hand, and also the needs of the problem statement.