4.5 Dynamic synchronization
4.5.1 The dynamic synchronization design
The synchronization problem of coordinating a follower to a leader suggests that the re-lationship between of the follower and the leader will be dynamic in the transient phase.
Thus, in this section we propose to describe this dynamic relationship in the form of a differential equation
ε˙= fε(ε) (4.78)
to control the behaviour of the follower while changing position relative to the leader. Note that dynamic synchronization is particularly suited for docking operations to a moving leader.
In this section, we restrict the coordination reference for the follower to a motion paral-lel to the motion of the leader (see Section 2.2.5) to simplify the presentation. Furthermore, the dynamic synchronization scheme is presented utilizing state feedback of the follower and state measurements of the leader to focus on the concept of dynamic synchronization rather than on state estimation.
In the presentation of this section, we will utilize the concepts of a reference and a vir-tual vehicle as defined in Figure 4.1 of Section 4.1. We define the dynamic synchronization errors
ε= xv− xr, ε˙= ˙xv− ˙xr, ε¨= ¨xv− ¨xr (4.79) where xr is the position of the reference vehicle, and xv the position of a virtual vehicle that will be used as a motion reference for the follower. Thus, the dynamic synchroniza-tion errorε defines the desired dynamic behaviour of the follower relative to the leader.
We propose a 1st order low-pass filter cascaded with a stable mass-damper-spring system (which is used as a reference filter in Fossen (2002)) to define the dynamic behaviour in the dynamic synchronization scheme
ε(3)+ (2∆∆∆+ I)ΩΩΩε¨+ (2∆∆∆+ I)ΩΩΩ2ε˙+ΩΩΩ3ε=ΩΩΩ3εr (4.80) for designed filter constants∆∆∆> 0 andΩΩΩ> 0, and whereεris the desired value forεsince
tlim→∞ε(t) =εr (4.81)
DYNAMIC SYNCHRONIZATION
Note that (4.80) guarantees that (4.79) are smooth signals, and that (4.80) can be written as a linear time invariant system
ε˙¯= A¯ε+ Bεr, ε¯= ε ε˙ ε¨ T (4.82) where
A=
0 I 0
0 0 I
−ΩΩΩ3 (2∆∆∆+ I)ΩΩΩ2 −(2∆∆∆+ I)ΩΩΩ
(4.83)
and
B=
0 0 ΩΩΩ3 T
(4.84) We define the synchronization control errors as
e= x − xv, ˙e= ˙x − ˙xv, ¨e= ¨x − ¨xv (4.85) where the states of the virtual vehicle are given from (4.79)
xv= xr+ε, ˙xv= ˙xr+ ˙ε, ¨xv= ¨xr+ ¨ε (4.86) Using the definition of the measure of tracking (3.4) of Section 3.1.1
s= ˙x − ˙y = ˙e +ΛΛΛe (4.87)
where
˙y= ˙xm−ΛΛΛe (4.88)
allows us to write the dynamics of (2.27)
M(x) ˙s = −C(x, ˙x)s − D(x, ˙x) ˙s +τ− M(x) ¨y − C(x, ˙x) ˙y − D(x, ˙x) ˙y − g(x) (4.89) Proposing the state feedback coordination control law
τ= M (x) ¨y + C (x, ˙x) ˙y + D (x, ˙x) ˙y + g (x) − Kds− Kpe (4.90) and constructing a Lyapunov function
V(t) =1
2sTM(x) s +1
2eTKpe, Kp= KTp> 0 (4.91) we get the derivative along the closed-loop trajectories
V˙(t) = −sT(D (x, ˙x) + Kd) s − eTΛΛΛTKpe (4.92) Since V(t) is positive definite, and ˙V(t) is negative definite it follows that the equilibrium (e, s) = (000, 000) is uniformly globally exponentially stable (UGES), and from convergence of s→ 000 and e → 000 that ˙e → 000.
0 10 20 30 40 50
Figure 4.8: Position errors e (top left), velocity errors ˙e (top right), dynamic synchroniza-tion referenceε(t) (bottom left), xy-plot of the follower x and leader xm(bottom right)
Table 4.4: Initial states and gains for the dynamic synchronization scheme Initial states and sliding surface gain Controller and reference filter gains
xm = [ 4 7 0 ]T Kp = diag[ 50 150 50 ]
The dynamic synchronization approach was simulated in the simulation setup of Appendix E for a docking situation where the follower is docking to a moving leader vessel. In this situation, the reference position xr coincides with the leader position xm, and control objective is thus to synchronize the follower states x and ˙x to the leader states xmand ˙xm. Initial states and gains for the simulation are given in Table 4.4, and the simulation results are shown in Figure 4.8.
In the simulations, the leader ship tracks a sine wave reference trajectory sin(ϖt) with frequencyϖ = 1/45 with heading angleψm along the tangent line. We see in Figure 4.8 that the synchronization closed-loop errors e and ˙e converges smoothly to the origin through to the design of the dynamic reference systemε(t) as smooth signals.
CONCLUDING REMARKS
4.6 Concluding remarks
This chapter proposed a virtual vehicle approach to the motion coordination problems de-fined in Definition 2.7 and 2.8. The coordination approach was based on the design of a vir-tual vehicle that estimated the unknown states of a leader based on position measurements only. The closed-loop errors were shown to be globally practically asymptotically stable for the situation where the velocity of the follower was available in the control design.
For situations where the velocity of the follower was unknown, a stable first-order velocity filter was used to estimate the unknown states of the follower, and semiglobal practical asymptotic stability of the closed-loop errors was concluded. Simulations were presented for both approaches, and for the situation with unknown velocities for the follower, ex-perimental results illustrated the convergence of the proposed coordination scheme. The motion coordination scheme was furthermore applied to robot manipulators in a separate section to illustrate the application of the virtual vehicle approach to manipulating struc-tures. The stability results, simulation results and experimental results presented in this chapter suggest that the proposed virtual vehicle motion coordination approach is suitable for practical applications. Furthermore, a dynamic synchronization scheme was proposed to impose a smooth behaviour on the follower when changing position relative to the leader.
Chapter 5
Comparison of the observer-controller and virtual vehicle schemes
This chapter presents a discussion on the proposed observer-controller coordination scheme of Chapter 3 and the virtual vehicle coordination scheme of Chapter 4. The schemes are compared in terms of estimation principle; the approach taken to estimate the unknown states of the leader through the use of an observer or a controlled virtual vehicle, and in terms of performance and robustness; the ability to suppress disturbances, modelling er-rors, measurement noise and the practical bounds to which the schemes converge.
5.1 Estimation principle
The estimation principles of the observer-controller scheme and the virtual vehicle scheme are based on the notion of estimating the unknown states of the leader through a system that mimics (or simulates) the behaviour of the leader. In the virtual vehicle scheme, this system is a virtual system; a virtual vehicle that is constructed to stabilize to the output of the leader, and which in turn provides estimates of the states of the leader to the follower. For the observer-controller scheme, the mimicking system is the follower itself, and through the observers and controller the follower becomes a physical observer of the leader.
The information constraints imposed on the proposed coordination schemes by al-lowing the parameters of the mathematical model of the leader to be unknown, and also through the fact that only the position is available from the leader as output, suggest that the coordination schemes will not make the closed-loop errors converge to an equilibrium point at the origin, but rather to a bounded or practically stable solution close to the origin.
In particular, the presence of non-vanishing perturbations due to the unknown states ren-ders the schemes at best ultimately bounded or practically stable. The results proposed in this thesis are presented on the premise that for many applications this is sufficient. Physi-cal limitations such as measurement noise and the resolution of measurement instruments may suggest that an equilibrium point at the origin can not be stabilized, or energy con-siderations on the actuators may suggest that the system errors should not be controlled to exactly zero, but rather to a sufficiently small neighbourhood around zero. Friction or external disturbances may perturb the systems so that zero is an unattainable equilibrium, or neglected high-order nonlinearities in the model may cause the system to deviate from
an ideal reference. In these situations it is often enough to ensure that the region to which the solutions converge is sufficiently small to meet the performance demands.
Imposing in addition information constraints on the follower in situations where the velocity of the follower is unknown broadens the range of applications that is suitable for the motion coordination schemes of Chapters 3 and 4. In many systems, velocity sensors are expensive, not easily fitted to the application or contaminated with noise, and thus co-ordination schemes must be designed that do not rely on accurate velocity measurements.
In addition, coordination schemes that can maintain the fundamental stability properties with a reduced set of measurements during temporary loss of measurements or in case of permanent measurement failure have an increased robustness towards failures. The lack of velocity measurements of a system typically reduces stability results from global to semiglobal when using static control gains, and thus the region of attraction for the coordi-nation schemes is reduced from global to a region that can be tuned through control gains.
The region of attraction can, however, be increased as much as desired, and thus for most applications this presents no practical drawback in regards to performance. Note also that a global solution to the observer design problem for Euler-Lagrange systems has recently been presented in Børhaug and Pettersen (2006) using time-varying control gains, which suggests that the global region of attraction can be maintained even for systems without velocity measurements.