Inertial Measurements Based Velocity-free Attitude Stabilization
The existing attitude controllers (without angular velocity measurements) involve explicitly the orientation (e.g., the unit-quaternion) in the feedback. Unfortunately, there does not exist any sensor that directly measures the orientation of a rigid body, and hence, the attitude must be reconstructed using a set of inertial vector measurements as well as the angular velocity (which is assumed to be unavailable in velocity-free control schemes). To overcome this circular reasoning-like problem, we propose a velocity-free attitude stabilization control scheme relying solely on inertial vector measurements. The originality of this control strategy stems from the fact that the reconstruction of the attitude as well as the angular velocity measurements are not required at all. Moreover, as a byproduct of our design approach, the proposed controller does not lead to the unwinding phenomenon encountered in unit-quaternion based attitude controllers.
The attitude control problem of rigid bodies has been widely studied over the last decades. The interest devoted to this problem is motivated by its technical challenges as well as its practical implications in many areas such aerospace and marine engineering. The main technical difficulty encountered in this type of mechanical systems may be attributed to the fact that the orientation (angular position) of the rigid body is not a straightforward integration of the angular velocity. Several interesting solutions to the attitude control problem have been proposed in the literature (see, for instance, , , , ).
As it is customary in the position control of mechanical systems, the majority of the control schemes developed for rigid bodies are (roughly speaking) of Proportional-Derivative (PD) type, where the proportional action is in terms of the orientation and the derivative action (generating the necessary damping) is in terms of the angular velocity. The requirement of the angular velocity can be removed through an appropriate design, usually based on the passivity properties of the system, as done in , , , ,  and , for instance. Since there is no sensor that directly measures the orientation, the explicit use of the attitude (e.g., the unit-quaternion) in the control law calls for the development of suitable attitude estimation algorithms that reconstruct the attitude from the measurements provided by some appropriate sensors (such as inertial measurements units (IMUs)). Therefore, the integration of the attitude estimation algorithm with the attitude controller has to be taken with extra care as the separation principle does not generally hold for non-linear systems.
Initially, the attitude determination from vector observations, has been tackled as a static optimization problem for which several solutions, based on Wahba’s problem, have been proposed . These algorithms have been refined, later on, incorporating filtering techniques of Kalman-type to handle the measurement noise (see, for instance, [3, 20] and the survey paper ). On the other hand, probably the most simple and yet practical dynamic attitude estimation approach is the linear complementary filtering (see, for instance, ), where the angular velocity and the inertial vector measurements are fused (via some appropriate linear filters) to recover the orientation of the rigid body for small angular movements. This approach has been extended to nonlinear complementary filtering for the attitude estimation from vector measurements in [7, 12, 13, 14, 16].
In fact, the existing dynamic estimation algorithms (involving signals filtering) make use of the angular velocity information to reconstruct the orientation of the rigid body. Therefore, a natural question that may arise is whether it makes sense to use velocity-free attitude controllers such as those proposed, for instance, in ,  and , since the angular velocity will be used (anyways) to recover the attitude via dynamic estimation algorithms. In this context, the main contribution of the present paper, is the development of a new attitude stabilization control scheme that uses explicitly inertial vector measurements without requiring (either directly or indirectly) the velocity measurement. This controller is a true velocity-free scheme since neither the angular velocity nor the orientation are used in the control law. Moreover, as it will be shown later, the unwinding phenomenon111Using the unit-quaternion representation, the same attitude can be represented by two different unit-quaternion, namely and . This fact, often leads to the so-called unwinding phenomenon, which is undesirable since some closed-loop trajectories, for some initial conditions close to the desired attitude equilibrium, can undergo an unnecessary homoclinic-like orbit [1, 2]. (inherent to the quaternion representation) is avoided in our approach.
Ii Rigid-body attitude representation and equations of motion
The attitude (orientation) of a rigid body can be represented in several ways, among which, one can recall the rotation matrix evolving in the special orthogonal group of degree three,
where is a 3-by-3 identity matrix.
Another globally non-singular representation of the attitude consists of using four-dimensional vectors , called unit-quaternion, evolving in the three-sphere , embedded in , .
A unit-quaternion is composed of a scalar component and a vector component , such that . A rotation matrix describing a rotation by an angle about the unit-vector , can be represented by the unit-quaternion or such that and . Note that the mapping from to is not a one-to-one mapping as there are two unit quaternion that represent the rotation matrix . The rotation matrix (that describes the orientation of the body-attached frame with respect to the inertial frame) is related to the unit-quaternion through the Rodriguez formula (see, for instance, [8, 18]). The mapping is given by
where , such that
with , and is the set of 3-by-3 skew symmetric matrices. Given a rotation matrix and two vectors , we have the following useful properties: , , and , where denotes the vector cross product.
The three-sphere (where the unit-quaternion evolve) has a natural Lie group structure given by the quaternion multiplication (which is distributive and associative but not commutative) denoted by “”. The multiplication of two quaternion and is defined as
and has the quaternion as the identity element. Given a unit-quaternion , we have , where and is 3-dimensional zero vector.
Throughout this paper, will denote the quaternion associated to the three-dimensional vector . A vector expressed in the inertial frame can be expressed in the body frame by or equivalently in terms of unit-quaternion as , where , , and is the unit-quaternion associated to as per (1).
The rigid body attitude kinematics, in terms of the rotation matrix, are given by , and, in terms of the unit-quaternion, by
with being the angular velocity of the rigid body expressed in the body-attached frame
The rigid body rotational dynamics are governed by
were is a symmetric positive definite constant inertia matrix of rigid body with respect to the body attached frame , and is the external torque applied to the system expressed in .
Iii Problem Statement
We assume that the rigid body is equipped with at least two inertial sensors that provide measurements (in the body-attached frame) of constant and known inertial vectors , , satisfying the following assumption:
At least two vectors, among the inertial vectors, are not collinear.
The vector measurements in the body-attached frame are denoted by , . The vectors and are related by . We assume that the angular velocity of rigid body as well as the attitude ( or ) are unknown (unavailable for feedback). The only variables available for feedback are the inertial vector measurements. Our objective is to design a control input (using only the vector measurements) guaranteeing almost global asymptotic stability of the equilibrium points characterized by or, equivalently, . Since the motion on (which is not a contractible space), is hampered by some well known topological obstructions that preclude global asymptotic stability results via time-invariant continuous state feedback (see, for instance, [1, 10]), the notion of almost global asymptotic stability is commonly used in this context and is defined as follows:
An equilibrium point of a dynamical system is said to be almost globally asymptotically stable if it is stable and all the trajectories starting in some open dense subset of the state space converge to .
Iv Main results
Iv-a Preliminary results
Let us define the following dynamic auxiliary system:
with an arbitrary initial condition , where with being a design variable to be defined later in Theorem 1.
Let us define the vectors , , as or, equivalently, , corresponding to the vector in the frame attached to the auxiliary system (5).
Let denote the discrepancy between the actual rigid-body orientation and the orientation of the auxiliary system (5), which corresponds to the unit quaternion error whose dynamics is governed by
In the sequel, for the sake of simplicity, the arguments associated to the rotation matrices will be omitted. It is understood that , and correspond, respectively, to the rotation matrices , and , associated, respectively, to the unit quaternion , and .
Before stating our main results, let us define the following variables: , , and , where and for , and let us state the following useful Lemmas:
The following equalities hold:
Using the facts that , and , we have
Rewriting in terms of the unit-quaternion , i.e., , and using the facts that and , one gets
Under Assumption 1, the matrices and are positive definite.
For any we have
which shows that . Now, without loss of generality, assuming that and are the two non-collinear vectors, one has
where . It is clear that is equivalent to , which is satisfied, for some , if and only if and are collinear. Similarly, for , if and only if and are collinear. Since and are non-collinear, they cannot be both collinear to the same , and hence, for all . Consequently, from (11), one can conclude that for all , which ends the proof.
Under Assumption 1, the following statements hold:
is equivalent to or , were are the unit eigenvectors of .
is equivalent to or , were are the unit eigenvectors of .
Using Lemma 1, is equivalent to
from which one can see that is a trivial solution. The other solutions can be obtained by assuming and multiplying (12) by to obtain , which shows that is a solution since is positive definite (as per Lemma 2). Now, we need to find the values of for which . Using (12) with , we obtain . Since is non-singular, the last equality is satisfied if and only if for some constant scalar , which shows that , where are the unit eigenvectors of . Similar steps can be used to prove and hence omitted here. This completes the proof.
It worth pointing out that the unit-eigenvectors of are also the unit-eigenvectors of . This can be easily deduced from the fact that , with . Likewise, the unit-eigenvectors of are also the unit-eigenvectors of . Note also that the eigenvalues of (resp. ) can be arbitrarily increased by increasing the gains (resp. ).
Iv-B Vector measurements based attitude stabilization
We propose the following control law:
where and are defined in subsection IV-A, and the following input for the auxiliary system (5):
Under the proposed control law, the closed loop dynamics are given by
where , and
and . Note that the scalar parts of and that appear in (15) are related to the vector parts as follows: and . It is clear that the closed loop dynamics (15) are autonomous.
Let be the state vector belonging to the state space , with . Now, one can state our first theorem.
Consider system (3)-(4) under the control law (13). Let (14) be the input of (5). Assume that vector measurements , corresponding to the inertial vectors , are available, and Assumption 1 holds. Then,
The equilibria of the closed-loop system (15) are given by222Note that is not a single equilibrium point as we have multiple values for (unit eigenvectors of ). The same remark goes for and . , , and , where and are, respectively, the unit eigenvectors of and .
The equilibrium point is asymptotically stable with the domain of attraction containing the following domain333Note that the domain of attraction includes the domain }, with , , where denotes the maximum eigenvalue of . If the control gains are sufficiently large such that , the domain approaches }.:
where , and is the minimum eigenvalue of .
There exist and such that if and , then , and are unstable, and is almost globally asymptotically stable.
Consider the positive definite, radially unbounded, function such that
where , are defined in subsection IV-A. Using (1) and the properties of the skew symmetric matrix and the unit-quaternion, one can show that and , where and . One can also show that the dynamics of and are governed by
where we used the fact that and .
Consequently, one can rewrite (16) as follows:
were we used Lemma 1 and the properties of the unit-quaternion and the skew-symmetric matrix to obtain the second equality. Since the closed-loop dynamics are autonomous, we proceed with LaSalle’s invariance theorem. From (20), setting leads to , which in view of Lemma 3, implies that or . Since is constant, one can conclude from (6) that . Since , it follows from (14) that , and consequently since . Since , from (4), it follows that . Using this last fact, together with the fact that , one can conclude from (13) that . Again, invoking Lemma 3, one has , or . It is clear that the largest invariant set in , for the closed loop system, characterized by is given by .
Since we showed that , one has , for all , which shows that is a positively invariant sublevel set. Since and , it is clear that . Consequently, . Therefore, , and do not belong to . Finally, since the largest invariant set in , corresponding to , is nothing else but , the second claim of the theorem is proved.
Now, we need to show that , and are unstable. First, let us focus our attention on and that belong to the manifold characterized by . In fact, the closed loop dynamics of are given by , which can be written, around , as follows:
where . Consider the following positive definite function: , whose time derivative along the trajectories of (21) is given by
Since and are bounded (as per (16) and (20)), and is not an invariant manifold, it is clear that, in the neighborhood of , there exists a finite parameter such that . Consequently, one has .
Since is an eigenvalue of , . Hence, as long as and , which shows that and are unstable.
Now, let us show that is unstable, using Chetaev arguments444The reader is referred to  for more details about Chetaev Theorem.. To this end, we will show the instability of the manifold characterized by . Let us define , and consider the dynamics of and in the neighborhood of
where . Let us consider the following Chetaev function: . Define the set where . Let us also define a subset of where , that is, . Note that is non-empty for all , and . The time derivative of , in view of (23), is given by
where we used the fact that , with , and the fact that, around the manifold characterized by , with , there exists a positive parameter such that . Since is an eigenvalue of , . Therefore, picking , it follows that for all .
Picking the initial conditions , must leave since is bounded on and everywhere in . Since , must leave through the circle and not through the edges (i.e., or ). Since this can happen for arbitrarily small , the equilibrium is unstable and so is .
Finally, since the possible stable manifolds associated with the unstable equilibria belong to a set of Lebesgue measure zero in the state space, it is clear that is almost globally attractive.
For the sake of simplicity, the desired attitude has been taken as . However, it is straightforward to extend the proposed control law to the case of an arbitrary constant desired attitude by taking .
Iv-C Attitude stabilization using preconditioned vector measurements
In this section, we will show almost global asymptotic stability of the desired equilibrium point without any conditions on the minimum eigenvalues of and . This results are made possible through an appropriate preconditioning of the inertial measurement vectors.
Let us assume that we have only two vector measurements corresponding, respectively, to two inertial vectors (which are assumed non-collinear). Let us define the normalized (unit) vectors , and . It is clear that forms a unit orthonormal basis. Let us define , and . Using the fact that and , one can easily show that , for . Let us define , , , and , with , .
The choice of , and leads to and . This can be shown as follows: Since is a unit orthonormal basis, one has since , . Furthermore, since with , it is clear that .
In this case, using Lemma 1 and the new values of and , it follows that
Now, one can state our second theorem:
As in the proof of Theorem 1, one can show that the closed-loop system is autonomous. Consider the positive definite function (16) which, due to the fact that and , can be written as