On Path-Lifting Mechanisms and Unwinding in Quaternion-Based Attitude Control

The unit quaternion is a pervasive representation of rigid-body attitude used for the design and analysis of feedback control laws. Because the space of unit quaternions constitutes a double cover of the rigid-body-attitude space, quaternion-based control laws are often—by design—inconsistent, i.e., they do not have a unique value for each rigid-body attitude. Inconsistent quaternion-based control laws require an additional mechanism that uniquely converts an attitude estimate into its quaternion representation; however, conversion mechanisms that are memoryless—e.g., selecting the quaternion having positive scalar component—have a limited domain where they remain injective and, when used globally, introduce discontinuities into the closed-loop system. We show—through an explicit construction and Lyapunov analysis—that such discontinuities can be hijacked by arbitrarily small measurement disturbances to stabilize attitudes far from the desired attitude. To remedy this limitation, we propose a hybrid-dynamic algorithm for smoothly lifting an attitude path to the unit-quaternion space. We show that this hybrid-dynamic mechanism allows us to directly translate quaternion-based controllers and their asymptotic stability properties (obtained in the unit-quaternion space) to the actual rigid-body-attitude space. We also show that when quaternion-based controllers are not designed to account for the double covering of the rigid-body-attitude space by a unit-quaternion parameterization, they can give rise to the unwinding phenomenon, which we characterize in terms of the projection of asymptotically stable sets. Finally, we employ the main results to show that certain hybrid feedbacks can globally asymptotically stabilize the attitude of a rigid body.

fascinating difficulty is the topological complexity of the underlying state space of rotation matrices, : a boundaryless compact manifold that is not diffeomorphic to any vector space.This property of precludes the existence of a continuous time-invariant state-feedback control law that globally asymptotically stabilizes a particular attitude [6], [7].For the same reason, no periodic or discontinuous feedback can robustly globally asymptotically stabilize a particular attitude [8].
Often, unit quaternions are used to parametrize .While this parametrization yields the minimal globally nonsingular1 parametrization of rigid-body attitude [9], its state space, (the set of unit-magnitude vectors in ) is a double cover of .That is, there are two (antipodal) unit quaternions corresponding to every rigid-body attitude.This creates the need to stabilize a disconnected set in the covering space [5], which has its own topological obstructions [10].As discussed in [6], these topological subtleties can cause confusion and sometimes, lead to dubious claims regarding the globality of asymptotic stability (see e.g., [1], [11]).Nevertheless, unit quaternions are still used by many authors (including the authors of this paper) today to design feedback control algorithms for attitude control.
A feedback controller designed using a quaternion representation of attitude may not be consistent with a control law defined on .That is, for every rigid-body attitude, the quaternion-based feedback may take on one of two possible values.When this is the case, analysis for quaternion-based feedback is often carried out in with a lifted dynamic equation.However, such analyses are not directly related to a feedback system defined on .This obviously begs the following questions.How is a unit quaternion representation obtained from available measurements?On what state-space is an inconsistent quaternion-based feedback defined?How is stability analysis done in the covering space related to a stability result for the actual system?
Given an estimated attitude, it is a fairly simple operation to compute the corresponding set of unit quaternions (see, e.g., [12], [13]); however, the process of selecting which quaternion to use for feedback is a less obvious operation.As noted in [4], it is often the case that the quaternion with positive "scalar" component is used for feedback.This operation is discontinuous and not defined globally.As we show in this work, the act of paring such a discontinuous quaternion-selection scheme with a widely used inconsistent quaternion-based feedback opens the door for an undesirable chattering effect.In fact, we construct an explicit disturbance-defined on -that exploits the discontinuity to stabilize a region about the manifold of 180 rotations with zero angular velocity.
To remedy this behavior, we propose a hybrid-dynamic algorithm for smoothly lifting path from onto .Our approach allows us to make an equivalence between any asymptotic (in)stability result for a closed-loop system in the covering space and a corresponding (in)stability result for the actual plant.This justifies carrying out stability analysis in a unitquaternion setting; however, when a quaternion-based feedback does not respect the two-to-one covering of , this translated stability result may not be desirable.
Often, quaternion-based feedbacks are designed to stabilize only one of two quaternions corresponding to the desired attitude.When these inconsistent feedbacks are paired with a pathlifting algorithm, they cause the so-called "unwinding phenomenon," where the feedback can unnecessarily rotate the rigid body through a full rotation.This behavior was discussed at length in [6] in terms of lifts of paths and vector fields from to .In this paper, we characterize unwinding in terms of asymptotically (un)stable sets in an extended state space projected onto the plant state space.
In practice, an explicit measurement of attitude is not available.Instead, the attitude must be reconstructed from measurements of known inertial-frame vectors expressed in body-frame coordinates [14].With measurements of at least two such linearly independent vectors, the attitude can be algebraically reconstructed by in various ways, such as solving a least-squares problem (often called "Wahba's problem" [15]) [16], [17].When using only a static attitude-reconstruction algorithm, a path-lifting mechanism (like the one herein proposed) is necessary to choose the quaternion consistently if an inconsistent feedback is used.Alternatively, dynamical filters can be used to estimate the attitude from vector observations (or IMU measurements) or from the results of static attitude attitude-reconstruction algorithms [18], [19].Regardless of the process that ultimately forms an estimate of attitude, the message of this work is clear: when an inconsistent quaternion-based feedback is used, a dynamic mechanism is needed to resolve the ambiguity in which quaternion is used for feedback.Furthermore, regardless of the mechanism that fills this role (e.g., the hybrid algorithm proposed herein or a dynamic filter as in [18] and [19]), the additional state(s) of the mechanism should be considered to correctly assess the stability properties of the closed-loop system and to rule out any possibility of unwinding.
This paper is organized as follows.Section II provides the background material for attitude control and hybrid systems used in this paper.Section III reconstructs the "select-the-quaternion-with-positive-scalar-component" mechanism in terms of a static map that selects a quaternion according to a semimetric. 2 In Section IV we show by Lyapunov analysis that, when composed with a widely used inconsistent feedback, the aforementioned quaternion-selection scheme makes the closed-loop system susceptible to arbitrarily small measurement disturbances that can act to stabilize attitudes far from the desired attitude.Section V constructs a hybrid-dynamic system that smoothly lifts paths from to .We couple this system with a quaternion-based feedback in Section VI and establish an equivalence of stability between two closed systems: one is defined in the unit-quaternion space and the other one is defined in the rigid-body-attitude space extended by a unit-quaternion memory state.Section VII discusses the unwinding phenomenon in terms of the projection of asymptotically stable sets and suggests how to avoid the behavior.Finally, we present conclusions in Section VIII.The attitude of a rigid body is defined as the relative rotation of a body-fixed frame to an inertial frame and is represented by a 3 3 orthogonal matrix with unitary determinant: an element of the special orthogonal group of order three:

Notation
The kinematic and dynamic equations of a rigid body are (2a) (2b) respectively, where is the attitude, is the angular velocity given in the body-fixed frame, is the inertia matrix, is an external torque, and the cross product between vectors , , is defined by a matrix multiplication: , where Members of are often parametrized in terms of a rotation about a fixed axis by the so-called Rodrigues formula: the map defined as The unit-quaternion parametrization of associates every element of with two elements of .In the sense of (3), a unit quaternion is defined as (4) and represents an element of through the map defined as (5) Note the important property that for , if and only if .We denote the double-valued inverse map as Conveniently, we will often write a quaternion as a pair , rather than as a vector.With the identity element , each unit quaternion has an inverse under the quaternion multiplication rule where and .Then, the map is a group homomorphism satisfying (7) The manifold is a covering space for and is the covering map.Precisely, for every , there exists an open neighborhood of such that , where , are open, , and for each , the restriction of to is a diffeomorphism.In particular, is everywhere a local diffeomorphism.
A fundamental property of a covering space is that a continuous path in the base space can be uniquely "lifted" to a continuous path in the covering space once a base point is specified.In terms of and , for every continuous path and for every , there exists a unique continuous path satisfying and for every [21,Th. 54.1].We call any such path a lift of over .We refer the reader to see [21], [22] for general information about covering spaces.
In addition to paths, vector fields defined on can be lifted onto as well [6].In this direction, given a Lebesgue-measurable function and an absolutely continuous path satisfying (2a) for almost all , any that is a lift of over satisfies the quaternion kinematic equation (8) for almost all , where the maps and are defined as

B. Hybrid Systems Framework
In this work, we appeal to the hybrid systems framework [23], [24].This is in part due to the fact that the authors have developed quaternion-based hybrid feedback controllers that achieve global asymptotic stabilization of rigid-body attitude in [5], [25], and [26] and also because the path-lifting algorithm presented here is hybrid.A hybrid system allows for both continuous and discrete evolution of the state.A hybrid system with state is defined by four objects: a flow map, , governing continuous evolution of the state by a differential inclusion, a jump map, , governing discrete evolution of the state by a difference inclusion, a flow set, , dictating where continuous state evolution is allowed, and a jump set, , dictating where discrete state evolution is allowed.We write a hybrid system in the compact form Often, we will refer to a hybrid system by its data as .Solutions to hybrid systems are defined on hybrid time domains and are parametrized by , the amount of time spent flowing and , the number of jumps that have occurred.A compact hybrid time domain is a set of the form (9) where is a nonnegative integer, .We say that is a hybrid time domain if, for each , the set is a compact hybrid time domain.On every hybrid time domain, points are naturally ordered as if and if .A hybrid arc is a function , where is a hybrid time domain and, for each fixed , the map is a locally absolutely continuous function on the interval (10) When a hybrid arc has several components, we adopt the economical notation A hybrid arc is a solution to the hybrid system if and 1) for each such that has nonempty interior, for almost all and for all ; 2) for each such that , and .Solutions are not unique if is multi-valued for some , there is more than one flowing solution from some , or it is possible to flow from some point .A solution to is maximal if it is not a truncation of another solution and it is complete if is unbounded.Given a hybrid arc , let and let .Then, the time projection of is the function defined as (11) In this work, we assume that the hybrid system satisfies the hybrid basic conditions: 1) and are closed sets in .2) is an outer semicontinuous3 set-valued mapping, locally bounded on , and such that is nonempty and convex for each .3) is an outer semicontinuous set-valued mapping, locally bounded on , and such that is nonempty for each .These properties ensure, among other things, that asymptotic stability is nominally robust [24].
A , there exists such that for all satisfying .The set of points in from which each solution is complete, bounded, and converges to is called the basin of attraction of .Note that each point in belongs to the basin of attraction of any set , since no solutions exist from these points.A compact set is asymptotically stable if it is stable and attractive from an open neighborhood of and is globally asymptotically stable if its basin of attraction is .Finally, we remark that while the above definitions are written in terms of , they equally apply to manifolds embedded in .In particular, they apply to the state spaces that we will be using in this paper: , , and discrete sets of logic variables.

III. INCONSISTENT QUATERNION-BASED FEEDBACK AND MEMORYLESS PATH LIFTING
It is quite commonplace to design an attitude control law based upon a quaternion representation.That is, the control designer creates a continuous function and closes a feedback loop around (2) by setting , where is selected to satisfy , for each .When the feedback satisfies (12) we say that is consistent.Smooth and consistent feedback control algorithms are investigated in [27] for adaptive attitude control without angular velocity measurements and recently in [28] for attitude synchronization of a formation of spacecraft.In such cases, there is little need for a quaternion representation for analysis, as could be defined in terms of .When a quaternion-based feedback is inconsistent, that is, (13) the resulting feedback does not define a unique vector field on because for satisfying , the feedback is a two-element set [6].At this point, the control designer must, for every , choose which to use for feedback.In this direction, we provide a quote from the seminal paper [4]: "In many quaternion extraction algorithms, the sign of [the 'scalar' part of the quaternion] is arbitrarily chosen positive.This approach is not used here, instead, the sign ambiguity is resolved by choosing the one that satisfies the associated kinematic differential equation.In implementation, this would probably imply keeping some immediate past values of the quaternion." There is much to be gleaned from this quotation.In particular, it suggests that inconsistent quaternion-based control laws require an extra memory state to lift a trajectory from to a trajectory in .In what follows, we reconstruct the discontinuous quaternion "extraction" algorithm mentioned in the quotation above in terms of a semimetric and use the ensuing discussion to motivate a hybrid algorithm for online lifting of an attitude trajectory from to .We define a semimetric and a related distance function from to a set as (14) From a geometric viewpoint, is the height of "above" the plane orthogonal to the vector at .When the set in (14) takes the form of for some , the distance function also takes a special form.In particular, let .Then, .One possible method to lift a path from to is to simply pick the quaternion representation of that is closest to a specific quaternion in terms of the semimetric .In particular, let us define the map as (15) The map has some useful properties, which we summarize in the following lemmas.Lemma 1: Let and .The following are equivalent: 1) is single-valued and ; 2) ; 3) for all ; 4) for any , where the map was defined in (3).Proof: For the remainder of this proof, we let . By the definition of in (15), we see that is single-valued if and only if .This provides an equivalence between 1), 2), and 3), above.Now, let and be such that .Since , the fact that satisfies (7) provides the following equivalent series of expressions .Now, since , the form of in (5)  Proof: Without loss of generality, let and .Then, so that .Since a goal of attitude control is to regulate to (or, in general, an error attitude to ), one might choose as a point of reference (since ) and use the map defined as (17) Now, following 3) from Lemma 1 we see that , that is, always chooses the quaternion with positive scalar component, so long as it is single-valued.Further, Lemma 2 allows one to lift curves with so long as does not cross the manifold of 180 rotations where is multi-valued, or else will produce a quaternion trajectory that is discontinuous.As we now show, this leads to an undesirable chattering effect when is composed with an inconsistent feedback.

IV. NON-ROBUSTNESS
Let and let be a continuous function satisfying (18) where is a strictly increasing continuous function satisfying .Now, define as (19) and consider the inconsistent feedback (20) While this control law makes the set globally attractive for the lifted closed-loop system defined by ( 8), (2b), and setting , it renders stable and unstable.When composed with , one might expect that the resulting feedback globally asymptotically stabilizes the identity element of ; however, we show that any such expected global attractivity property is not robust to arbitrarily small disturbances.
Define the function as Then, for , consider the function defined implicitly in terms of the Rodrigues formula as, for every and every satisfying : otherwise.
(22) For any , the rotation matrix constitutes an angular perturbation of about the eigenaxis .The parameter controls the size of the disturbance.We note that ( 22) is well defined on .Lemma 4: For every and , is uniquely defined.Proof: Suppose that for some and .Clearly, is uniquely defined when or , since it does not depend on or in this case.
Suppose that and .This implies that , since .Then, it follows from the Rodrigues formula that for any and such that , it must be the case that or (only when ).Moreover, since , it follows that Then, we have shown that the value of is independent of the angle-axis representation of , hence, it is uniquely defined on .Let be any single-valued selection of , that is, for all and otherwise.Now, we apply the disturbance to measurements of attitude before being converted to a quaternion for use with the inconsistent feedback (20) and analyze the resulting closed-loop system.That is, we replace with in the control law defined in (20).Because and are discontinuous, we use the notion of Krasovskii solutions for discontinuous systems [29].We note that the following definition is equally valid for product spaces such as , once is isometrically identified with by vectorization.
Definition 5: Let .The Krasovskii regularization of is the set-valued mapping (23) where denotes the closed convex hull of the set .Then, given a function , a Krasovskii solution to on an interval is an absolutely continuous function satisfying (24) for almost all .We now state the main result of this section: the discontinuity created by pairing an inconsistent quaternion-based feedback with a discontinuous quaternion selection scheme makes the closed-loop system susceptible to arbitrarily small measurement disturbances that can exploit how feedback term opposes itself about the discontinuity of .Theorem 6: Let , , and satisfy (25) and define Then, the set is stable and is invariant for the closed-loop system (26) Proof: See Appendix A. The various failures of have led several authors (e.g., [30]) to derive sufficient conditions on the initial conditions of (2) to ensure that these 180 attitudes are never approached, thus obviating the use of a globally nonsingular representation of attitude like unit quaternions.However, the issues with using as a path-lifting algorithm are not a problem with the quaternion representation-they arise because is a memoryless map from to .In particular, always chooses the closest quaternion to and in general, when one compares with for some and , is multi-valued on the 2-D manifold .However, when the reference point for choosing the closest quaternion is allowed to change, it is then possible to create a dynamic algorithm for smoothly lifting a trajectory from to .We now explore such an algorithm that is hybrid in nature.

V. HYBRID ALGORITHM FOR DYNAMIC PATH LIFTING
In this section, we present a simple dynamic algorithm for lifting a path from to .The main feature of the algorithm is a memory state that provides a reference point for choosing the closest quaternion with respect to .This memory state usually remains constant, but is updated when necessary to ensure that .The basic logic behind the algorithm is pictured in Fig. 1 as a flow chart.Given a distance threshold , we define the sets , as We analyze the properties of the hybrid path-lifting algorithm by analyzing the solutions of an autonomous system that generates a wide class of useful trajectories in as input to .Theorem 7: Let and .The hybrid system (28) and its output defined in (27c) have the following properties: 1) Closed loop system (28)  That is, should be selected so that no measurement disturbance can make the choice of quaternion ambiguous.

VI. QUATERNION FEEDBACK WITH DYNAMIC LIFTING
With a hybrid algorithm for path lifting in place, we consider the feedback interconnection of (2) with the hybrid pathlifting system and the quaternion-based hybrid controller , that takes a measurement as input, has a state , has dynamics (29) and produces a continuous torque .Often, quaternion-based controllers are analyzed using the lifted attitude dynamics, defined by ( 8) and (2b), thus neglecting any auxiliary lifting system.The next theorem essentially justifies this approach by relating solutions of the whole closed-loop system (including the hybrid path-lifting system) to a reduced system that has the quaternion-based hybrid controller in feedback with the lifted system defined by ( 8) and (2b).
Before stating the theorem, we define two closed-loop systems.The first closed-loop system is the feedback interconnection of (2) with the series interconnection of and .This yields the system with state defined as (30) In (30), we mean that flows can occur for when both the controller and lifting subsystems allow for flows.Jumps can occur when either the controller or lifting subsystems can jump.It may be possible that both and are satisfied at the same "time," i.e., , in which case, either jump is possible.That is, either or (the other states do not change).This is necessary to ensure that the closed-loop system satisfies the hybrid basic conditions.Now, we define the feedback interconnection of the lifted attitude system and the hybrid controller .This yields the reduced system with state defined as (31) Lemma 8: For every solution to of (30) such that , there exists a solution to of (31) such that for every , there exists such that and Conversely, for every solution to (31), there exists a solution to (30) such that for every , there exists such that and (32) is satisfied.Proof: See Appendix C. Now, we state one of our main results.The following theorem is a "separation principle" that allows one to design a feedback for the lifted system defined by ( 8), (2b) and then expect the results to translate directly to the actual system when the hybriddynamic path-lifting system is used to lift the trajectory in to .Theorem 9: Let .A compact set is stable (unstable) for the system of ( 31) if and only if the compact set (33) is stable (unstable) for the system of (30).Moreover, is attractive from for the system (31) if and only if is attractive from (34) for the system of (30).Proof: See Appendix D. Interestingly, the result of Theorem 9 is not always desired!When the set above is not designed correctly, the resulting closed-loop system can exhibit the symptom of unwinding.

VII. THE UNWINDING PHENOMENON
In Theorem 6, we showed how a particular class of inconsistent control laws (20) can be hijacked by small measurement disturbances when defined in ( 17) is used to lift paths from to .In light of Section V and Theorem 9, one might ask how the control law (20) behaves in feedback with the hybrid path lifting system .The answer is that it induces "unwinding." Though the behavior has been documented for decades (see, e.g., [3]), the term unwinding was perhaps first coined by [6] to describe a symptom of controllers that are designed for systems evolving on topologically complex manifolds using local coordinates in a covering space.In particular, the ambiguity arising from the quaternion representation can cause inconsistent quaternion-based controllers to unnecessarily rotate the rigid body through a full rotation.This behavior can be induced by inconsistent control laws like (20) that are designed to stabilize a single point in while leaving the antipodal point unstable, despite the fact that they both correspond to the same physical orientation.This behavior was elegantly described in [6] in terms of the lifts of paths and vector fields.We now provide a characterization in terms of projections of asymptotically stable sets onto the plant state space.Recall that for some set , its projection onto is defined as (35) Now, we characterize how a set of interest in the covering space (including extra dynamic states of the controller) appears when projected to the actual plant state space .In this direction, we define the operator as (36) Further, we define the covering projection as (37) Lemma 10: The maps and satisfy (38) that is, the diagram Fig. 2 commutes.Proof: Let and let .It is easy to see that .Similarly, for every , it follows that .Thus, , and so, (38) is satisfied.Let (39) Lemma 10 clarifies the purpose of controllers designed in the covering space.Suppose it is desired to asymptotically stabilize some set (in the sense that is the projection of an asymptotically stable set in the extended state space including controller states).If the dynamic controller ( 29) is designed to stabilize in the extended covering state space (as in Lemma 10), one would obviously desire that , but this should not be the only requirement.In fact, one should design to satisfy (40) in which case, we say that is consistent.That is, the controller should stabilize all points in the lifted state space whose projections under map to a point in .As the following lemma states, when (40) is not satisfied, there may be points in the plant state space whose stability relies on the controller's quaternion representation of attitude, which is hardly a desired quality.Lemma 11: Let .If is not consistent, that is, it does not satisfy (40), then there exists and such that for every satisfying and every , .Proof: If does not satisfy (40), then, clearly, there exists and such that .Then, by definition of the operator, for every , .Finally, Lemma 1 asserts that whenever and by definition of , it follows that for every satisfying , that .Unfortunately, many designs proposed in the literature (see, e.g., [1], [3], [4], [11], [27], [28], [30]- [36]) do not satisfy (40).Instead, many designs, like the inconsistent feedback (20) (having ), render the point a stable equilibrium, while rendering an unstable equilibrium.In this situation, .When seen through the map , this creates two distinct, disconnected equilibrium sets in the extended state space, with one set asymptotically stable and the other, unstable.However, both equilibrium sets project to .As the next result shows, the desired attitude can be stable, or unstable, depending on the controller's knowledge of the quaternion representation of the attitude.
Corollary 12: Let .Then, is asymptotically stable and is unstable for the system (41) where was defined in (20).Similarly, the compact set is asymptotically stable and the compact set is unstable for the hybrid system (42) Proof: We note that the stability and instability of for ( 41) is easily obtained by a simple Lyapunov analysis using the proper and positive definite function defined as .Instability of can be shown in numerous ways.To show that is asymptotically stable for the hybrid system (42), we apply Theorem 9. From (33) and (34), we obtain that is asymptotically stable for (42).By the properties of the maps , , , and , it follows that .Theorem 9 implies, in a similar fashion, that is unstable for (42).
Finally, we note that in recent works, the authors have presented a hybrid strategy for achieving a global result that is robust to measurement disturbances in [5].The results in [5] satisfy (40) and can be applied to 6-DOF rigid bodies [25] and synchronization of a network of rigid bodies [26].Several works also suggest the use of a memoryless (i.e., ) discontinuous quaternion-based feedback using the term .Such methods have been suggested in [3], [31], [37]- [40] and indeed avoid the unwinding phenomenon; however, these control laws are susceptible to measurement disturbances like the result in Theorem 6.
Corollary 13: Let , , and define , and as Then, is globally asymptotically stable for (43) Similarly, the compact set is globally asymptotically stable for (44) Proof: Global asymptotic stability of for (43) is obtained by using Lyapunov and invariance analysis [41] with the function defined as .See [5], for example.To show that is asymptotically stable for the hybrid system (44), we note that Theorem 7 implies that is globally attractive and then we apply Theorem 9.

VIII. CONCLUSION
Obtaining global asymptotic stability of rigid-body attitude is a fundamentally difficult task.Often, feedback controllers are designed and analyzed on a state space that is topologically simpler than ; however, it is not always clear how the analysis of such algorithms can be translated to .When unit quaternions are used to parametrize rigid-body attitude and design feedback control laws, their actual implementation relies on an algorithm to translate measurements from to .When a memoryless map is used for this task, the resulting quaternion trajectory may be discontinuous, creating an extreme measurement-disturbance sensitivity for a widely used class of quaternion-based feedback control laws.An alternative is to dynamically lift the paths using a hybrid mechanism.Such a hybrid algorithm allows one to translate stability results obtained in the covering space directly to the actual plant; however, such a feedback system can induce an undesirable unwinding response when the quaternion-based feedback is not designed to stabilize all quaternion representations of the desired attitude.Finally, when hybrid path-lifting mechanisms are used in conjunction with the hybrid quaternion-based feedbacks proposed in [5], [25], and [26], the result is global asymptotic stabilization of the identity element of .
APPENDIX A PROOF OF THEOREM 6 In what follows, we denote and define, for pairs , , .Finally, in accordance with (1), for some function , we denote , where denotes the matrix of partial derivatives with respect to and denotes the vector of partial derivatives of with respect to .Define the function and the vector field as (45) By some abuse of notation, the Krasovskii regularization of is , where the arguments of are perturbed as in ( 23) with respect to the norms defined on each space.That is, (46)

Let
. From the definition (23), one can show that (47) Since we are studying Krasovskii solutions to (26), we might normally need to evaluate ; however, the analysis in this proof obviates the need for calculating the Krasovskii regularization for regions where the calculation is nontrivial.By definition of and , the map is continuous on the set , so for all .Consider the Lyapunov function (48) Expressed in terms of rotation angle, we have equivalently

Since
, it follows that for all and if and only if and .Furthermore, the sublevel sets of are compact.
Define the function as (49)

Fig. 1 .
Fig. 1.Flow chart for dynamic path lifting from to .
(27a)Then, we propose the hybrid path-lifting algorithm as the system ,