Control of Underactuated 1-Homogeneous Systems with Drift
5.3 Examples
5.3.1 Nonholonomic Integrator
oscillatory terms will require averaging, whereas the states that have no oscillatory terms, or ignorable, perturbative oscillatory terms, inP(t)will not require averaging.
Trajectory Tracking To track a trajectory, replace x(t) withx(t)−xd(t); the system must be locally controllable along the trajectory. For the discretized feedback, the Nyquist criteria is a limiting factor in tracking a trajectory for the indirectly controlled states. As the trajectory evolves, the Jacobi-Lie brackets contributing to small-time local controllability may vary. The trajectory tracking feedback strategy is more complicated since multiple parametrizations wil be needed. If a system is well-controllable, then a single parametrization will suffice to track trajectories. Systems with group symmetris are well-controllable if and only if they are small-time locally controllable at the identity.
Configuration Control via Kinematic Motions. To recover the stabilization via kinematics motions from Bullo [21], only theα-parameters are used for feedback. The system must be initially at rest. Kinematic motions are those that return the sysstem to rest and the end of each actuation period.
symmetric product,
hY1 : Y2iZ =
0 0 1
,
span the vertical subbundle ofT E. Here, the symmetric product operates on unlifted vector fields due to the identification of vertical vector fields with sections on the base,XV(E)7→Γ(E) =X(Q). Decompose the control inputs into state-feedback and oscillatory terms,
ua(x, t) =fa(x) +1
va(t/) =−(kpqa+kvq˙a) +1
va(t/).
The equations of motion become
˙
x=XS+1
Yaliftva(t/), (5.50)
whereXS =Z+Yaliftfa(x). With the feedback,XS ∈ M0, and the system (5.50) is still a 1-homogeneous control system. The feedback does not affect the symmetric product calculation at all, consequently we use Γ =XSfor all symmetric product calculations, c.f. Definition 37. Since the symmetric product can be used to add a third vector to complete the basis for the tangent space, and there are no bad symmetric products up to this level, the system should be locally controllable. Using oscillatory time dependent inputs forva(τ), the averaged form of the equations of motion is
˙
z=XS(q) + V(a)(1)(t)h
Yalift, XSi
−1
2V(a,b)(1,1)(t)hYa : Ybilift. The corresponding partial Floquet map is
Trunc0(P(t)) = Id, with an improved version being
Trunc1(P(t/)) = Id + Z t/
0
Ve(a)(1)(τ) dτ h
Yalift, XSi
−1 2
Z t/
0
Ve(a,b)(1,1)(τ) dτ D
Yalift : YbliftE .
Recall that the actual Floquet map also incorporates the flowΦY0,t/aliftva, which is
ΦY0,t/aliftva = Id + V(a)(1)(t/)Yalift. (5.51) Choosing the inputs
v1(t) =α112sin(t), v2(t) =α212sin(t) the equations of motion become
˙
z=Z(q)−α12hY1 : Y2ilift, whereα12=α112α212. Explicitly written, the equations of motion are
d dt
z1 z2 z3
˙ z1
˙ z2
˙ z3
=
˙ z1
˙ z2
˙ z3
−(kpz1+kvz˙1)
−(kpz2+kvz˙2)
−(kpz1+kvz˙1)z2
−
0 0 0 0 0 1
α12,
whose linearization about the origin is
˙ z=
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
−kp 0 0 −kv 0 0 0 −kp 0 0 −kv 0
0 0 0 0 0 0
z−
0 0 0 0 0 1
α12. (5.52)
System (5.52) is stabilizable. As, the first two directly controlled states are exponentially stable, the primary concern is the last state. There are several options regarding stabilization of control system.
Discrete Orbit Stabilization The first method will stabilize the system to an orbit using Theorem 37.
To determine the range of feedback gains that will stabilize the system, it is necessary to work out the derivations of the feedback law in the proof of Theorem 37. The continuous time evolution is
d dt
z3
˙ z3
= 0 1
0 0 z3
˙ z3
− 0
1
α12.
When integrated over a period, z3(t)
˙ z3(t)
= 1 T
0 1
z3(0)
˙ z3(0)
−
T+12T2 T
α12.
The discretized version is therefore equal to z3(k+ 1)
˙
z3(k+ 1)
= 1 T
0 1
z3(k)
˙ z3(k)
−
T+12T2 T
α12.
Choosingα12to be error feedback,
α12=Kpz(k) +Kvz(k),˙ the discretized system becomes
z3(k+ 1)
˙
z3(k+ 1)
=
1−(T +12T2)Kp T−(T+12T2)Kv
−T Kp 1−T Kv
z3(k)
˙ z3(k)
.
ChooseKpandKv so that the system is stabilized. Note that the value ofis critical because it determines the period T, and consequently will affect the range ofKp and Kv that will stabilize the system. As the limit→ 0is approached, the range of feedback gains will increase. In the limit, the discretized averaged system and the continuous system coincide.
At this point, there are two options available: (1) we may feedback instantaneous values of the state, or (2) we may use the Floquet mapping to feedback averaged values of the state. Only the first option will be contemplated. For Option 1, the oscillatory inputs need to be faster then the natural dynamics of the directly controlled states, otherwise the direct state feedback will degrade the oscillatory signal, and consequently the feedback signal to the indirectly controlled states. A plot of the step response if shown in Figure 5.1.
Due to the second sinusoidal input, the origin is no longer a fixed point, and the actual time-varying system will have a stable orbit. The parameters arekp = 3, kv = 4, Kp = 0.5,Kv = 1.9, = 1/7. The period isT = 2π, which gives a frequency of just over1Hz. All of the controllers found in the remainder of this
−0.5 0 0.5 1 x1
−0.5 0 0.5 1 x2
−0.5 0 0.5 1 x3
Figure 5.1: Discretized Orbit Stabilization for Nonholonomic Integrator, Option 1.
example use these control parameters also.
Discretized Stabilization to a Point For stabilization to a point to occur, the control law must vanish at the desired equilibrium. The equivalent control law will exponentially stabilize the system:
α112(t) = sign(α)p
|α|sin(t) and α212(t) =p
|α|sin(t). (5.53) The origin is a fixed point and the resulting control law will stabilize to the origin precisely, i.e., the orbit collapses to a point. The resulting control law is shown in Figure 5.2, using the same parameters as before, kp = 3,kv = 4,Kp = 0.5,Kv = 1.9, and= 1/7.
Continuous Stabilization to a Point We may modify the control strategy to utilize continuous feedback instead of discretized feedback. The α-parametrization the same as Equation (5.53). The inverse of the Floquet mapping,P(t), will be used to translate the instantaneous state values to the averaged state values for full-state feedback. For first-order averaging,P(t)≈Id, so only the flowΦY0,t/aliftva is needed to compute the Floquet mappingPe(t). Figure 5.3 shows a simulation using continuous feedback of the average.
Improved Continuous Stabilization to a Point The Floquet mapping using the standard averaging method does an decent job, however, by using the Floquet mapping from the improved averaging method, the stabi- lization can be smoothed out and improved. The improved expansion ofP−1(t)is
Trunc1 P−1(t)
= Id− Z t
0
Ve(a)(1)(τ) dτ h
Yalift, XSi +
Z t
0
Ve(a,b)(1,1)(τ) dτ hYa : Ybilift (5.54)
−0.5 0 0.5 1 x1
−0.5 0 0.5 1 x2
−0.5 0 0.5 1 x3
Figure 5.2: Discretized Point Stabilization for Nonholonomic Integrator.
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
0 5 10 15 20 25 30 35
−0.5 0 0.5 1
x3
Figure 5.3: Continuous Point Stabilization for Nonholonomic Integrator.
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.