In this section the control design for the 2D-plane case is directly derived from the pro-posed control approach for the 3D-space case presented previously. Quite naturally, this development consists in considering this case as a degenerate case of the 3D-space case.
2.7.1 Notation
The following notation is introduced for the 2D-plane case.
Figure 2.16: Inertial frameFo and body frame F (the 2D-space case).
• G is the vehicle’s center of mass,m ∈R its mass, andJ ∈R its inertia.
• Fo={O;−→ao,−→
bo} is a fixed frame with respect to which the vehicle’s absolute pose is measured.
• F={G;−→a ,−→
b } is a frame attached to the body, with the vector −→a parallel to the thrust force axis as illustrated in Fig. 2.16.
• The vector of coordinates of G in the basis of the fixed frame Fo is denoted as x= (x1, x2). Therefore, −→
OG=x1−→ao+x2−→
bo, i.e. −→
OG= (−→ao,−→ bo)x.
• The vehicle’s orientation is characterized by the oriented angle θ between −→ao and
−
→a. The rotation matrix associated with the angleθ is denoted asR(θ)∈R2×2, with
R(θ)
cosθ −sinθ sinθ cosθ
.
• The vector of coordinates associated with the linear velocity ofGwith respect toFo
is denoted asx˙ = ( ˙x1,x˙2)when expressed in the fixed frameFo, and asv = (v1, v2) when expressed in the basis of F, i.e. −→v = dtd−→
OG= (−→ao,−→
bo) ˙x= (−→a ,−→ b )v.
• The angular velocity vector of the body frame F relative to the fixed frame Fo, expressed in the body frame F, is denoted as ω ∈R.
Control design for the 2D-plane case 75
2.7.2 System modeling
The system modeling for the 2D-plane case proceeds analogously to Section 2.3. The thrust force−→
T is assumed to be −→
T =−T−→a. Alike the 3D-space case, the mathematical model of the 2D-plane case, derived from the Newton-Euler formalism, is given by
⎧⎪ the inertial frame Fo, Γe the external scalar torque induced by −→
Fe, i1[0,1], and subsystemΣ21d of System (2.45) can be simplified as follows
⎧⎨ input. Note that Assumptions 1–3 are still satisfied when Fe is replaced by γe and each ci is replaced by ci/m.
Alike the 3D-space case, we focus our control design studies on System (2.46) with u andω used hereafter as the control inputs, and we assume also that the apparent acceler-ationγe and its time-derivative are available for control design. Note thatγeandγ˙ecan be estimated using the estimation solution proposed in Section 2.5 modulo straightforward modifications. More precisely, one must only replace System (2.31) by
¨
x=−uR(θ)i1+γe,
and state the uniform boundedness results as in Lemma 3. The results of Proposition 8 still hold with the observer ofγe (assuming that u, x,˙ θ are measured) given by
⎧⎨
To gain control insight one may view a vehicle moving on a plane with two control inputs (one force and one torque) as a degenerate case of the 3D-space case where the vehicle moves on a plane with two control inputs and two other torque control inputs are
desactivated. For instance, consider the 3D-space case where the vehicle is constrained to
ko. Then the control law,e.g., proposed in Proposition 2 (i.e. the control expression (2.11)) can be simplified as follows
⎧⎪ provides the same asymptotic stability result as Proposition 2. In what follows we rigor-ously prove that this intuition is correct.
In this section the basics control laws developed in Section 2.4.1 for the 3D-space case will be used with adaptations for the 2D-plane case. The process of robustification given in Section 2.4.2 can be directly applied, and therefore is omitted in this section. We consider now the following control objectives: i) thrust direction control, ii) velocity control, iii) position control, and iv) the extension of these control laws by taking into account the unilaterality of the thrust direction. The control design for the 2D-plane case relies in the first place on the following lemma (see Section 2.8.15 for the proof) which is reminiscent of Lemma 1.
The objective is to stabilize the vehicle’s thrust direction −→a about a desired thrust direction −→γ. In practice this desired direction may be specified by a manual joystick. Let γ ∈ R2 denote the normalized vector (|γ| = 1) of coordinates of −→γ, expressed in the
Control design for the 2D-plane case 77 inertial frame F. Then, the control objective is equivalent to stabilizing R(θ)γ about i1. Define
γR(θ)γ, (2.49)
and let θ ∈ (−π;π] denote the angle between the two unit vectors i1 and γ, so that cosθ=γ1, the first component ofγ. More precisely,θis defined byθatan2(−γ2, γ1). The control objective is also equivalent to the asymptotic stabilization of θ= 0. The control result is stated next.
Proposition 9 Let k denote a positive constant, and apply the control law ω= kγ2
(1 +γ1)2 −γS2γ˙ (2.50)
to the system θ˙ = ω. Then the equilibrium point θ= 0 of the controlled system is expo-nentially stable with domain of attraction equal to(−π, π).
The proof is given in Section 2.8.16.
Velocity control
Let x˙r denote the reference velocity expressed in the inertial frame Fo, x¨r its time-derivative, andvR(θ)( ˙x−x˙r) the velocity error expressed in the body frame F. The problem of asymptotic stabilization of the linear velocity errorx˙−x˙r to zero is equivalent to the asymptotic stabilization of v to zero. Using System (2.46), one obtains the error system
˙
x=R(θ)v (2.51a)
˙
v =−ωS2v−ui1+R(θ)(γe( ˙x, t)−x¨r(t)) (2.51b)
θ˙=ω (2.51c)
with either x t
0( ˙x(s)−x˙r(s))ds, the integral of the velocity error, or xx−xr, the position tracking error when a reference trajectory xr is specified.
Let γe denote the measure or estimate of γe and define now γ as follows (instead of defining γ as a reference unit vector like previously)
γγe−x¨r(t) +h(|Iv|2)Iv, (2.52) whereh is a smooth bounded positive function satisfying Properties (2.13) and Eq. (2.14) for some positive constants η, β, and Iv ∈ R2 is defined by Eq. (2.12). From here the control results stated next asymptotically stabilizev to zero.
Proposition 10 Apply the control law (2.48) to System (2.51), with k1, k2, k3 some positive constants, and γ defined by Eq. (2.52). Suppose that
i) Assumptions 1, 4, and 5, with γ given by Eq. (2.52), are satisfied;
ii) the measurement (or estimation) error cγe−γe is constant;
iii) lim
s→+∞h(s2)s >|c|.
Then, for System (2.51b)–(2.51c)complemented with the equation I˙v =R(θ)v, there exists a constant vector Iv∗ ∈R2 such that equilibrium point(Iv,v,θ) = (I v∗,0,0)of the controlled system is asymptotically stable, with domain of attraction equal to R2×R2×(−π, π).
The proof is given in Section 2.8.17. Now to further comply with the constraint of positivity of the thrust control u(i.e. T) one can apply the following controller.
Proposition 11 Let k1, k2, k3 denote some positive constants, and γ as defined by Eq. (2.52). Let σ :R→Rdenote a strictly increasing smooth function such that σ(0) = 0 and σ(s)>−1/k1, ∀s ∈R. Apply the control law to System (2.51). Suppose that Assumptions i)–iii) of Proposition 10 are satisfied. Then the asymptotic stability result of Proposition 10 still holds.
The proof, similar to the proof of Proposition 10, is given in Section 2.8.18.
Position control with unidirectional thrust
The control objective is the combined stabilization of the velocity error v (or x˙ −
˙
xr) and the position error x = x − xr to zero. Similar to the design of the position control of the 3D-space case, the nonlinear integrator bounding technique proposed in Proposition 4 (i.e. System (2.16)) and the control structure proposed in Propositions 4 and 6 are reused. The control result, whose proof is given in Section 2.8.19, is stated next.
Proposition 12 Let k1, k2, k3 denote some positive constants. Let σ : R→ R denote a to System (2.51) wherev, γ, and y(which intervenes in the definition ofγ) are defined by Eqs. (2.19), (2.20), and (2.18) respectively, andz (which also intervenes in the definition of γ) is the solution to System (2.16). Suppose that
i) Assumptions 1, 4, and 5, with γ given by Eq. (2.20), are satisfied;
ii) the measurement (or estimation) error cγe−γe is constant;
iii) lim
s→+∞h(s2)s >|c|;
iv) ∆ > |z∗|, where z∗ denote the unique solution to the equation h(|z∗|2)z∗ =c and ∆ is the positive constant intervening in the function sat∆ (which involves in System (2.16)).
Then, for System (2.51) complemented with System (2.16), the equilibrium point (z,z,˙ x,
v,θ) = (z ∗,0,0,0,0) of the controlled system is asymptotically stable, with domain of attraction equal to R2×R2×R2×R2×(−π, π).
Analyses for Chapter 2 79