## 1 Introduction

Many control systems are defined on manifolds that are not homeomorphic to Euclidean space, where we use the term ‘Euclidean space’ to mean some space, not imposing any metric on it. The geometric, or coordinate-free, approach has been developed to deal with those systems without being dependent on the choice of coordinates.^{1, 4, 23} However, a state-space manifold often appears as an embedded manifold in Euclidean space and the control system naturally extends from the manifold to the ambient Euclidean space: one example is the free rigid body system on which naturally extends to . In such a case, it might be advantageous to use one single global Cartesian coordinate system in the ambient Euclidean space to design controllers for the original system on the manifold, eliminating the necessity to use rather complex tools from differential geometry or multiple local coordinate systems. For example, in the case of the free rigid body system, neither adding nor subtracting two rotation matrices is allowed in the geometric approach partly because the result does not lie on , which may be mathematically orthodox, but would discourage control engineers from understanding or applying the geometric results. Since any two rotation matrices, as matrices, can be conveniently added or subtracted in , there is no reason to refrain from carrying out such basic and convenient operations as additions and subtractions. Moreover, since one can utilize one single global Cartesian coordinate system in the ambient Euclidean space , he is free from such discontinuities as those that often occur due to the switching of local coordinate systems and chart-wise designed control laws. As such, in this paper we propose a new method that is an alternative to both the geometric approach, which adheres to differential geometric tools, and the classical approach, which employs local coordinates such as Euler angles for rigid bodies.

A brief summary of the proposed method is provided as follows. Given a control system whose dynamics evolve on a manifold , we embed into some Euclidean space and extend the system to a system whose dynamics evolve in or conservatively in a neighborhood of in . We then legitimately modify the extended system outside to add transversal stability to while the original dynamics on are kept intact. It follows that becomes an attractive invariant manifold of the resulting system denoted . We apply any controller design method available in Euclidean space to design controllers for in for stabilization of a point on or tracking of a reference trajectory on , and then restrict the controllers to which yield controllers for the original system on for the stabilization or tracking on . To showcase this method, the linearization technique in is chosen in this paper to design tracking controllers although we could alternatively apply other techniques available in such as homogeneous approximation,^{10} model predictive control,^{3} iterative learning control,^{24} differential flatness,^{12} etc.

The theory of embedding of manifolds in Euclidean space has a long history in mathematics, including several famous theorems such as the Nash embedding theorems^{18, 19} and the Whitney embedding theorem.^{2} The embedding technique has been also applied in control theory. For example, it was used to produce a simple proof of the Pontryagin maximum principle on manifolds,^{5} and was combined with the transversal stabilization technique to yield feedback-based structure-preserving numerical integrators for simulation of dynamical systems.^{6}
A series of relevant works have been made by Maggiore and his collaborators on local transverse feedback linearizability of control-invariant submanifolds and virtual holonomic constraints.^{17, 20, 21} The focus of Maggiore is placed on creation of a submanifold for a given system and its transversal stabilization via feedback for path-following controller synthesis, whereas our work in this paper is focused on embedding and extending a state space manifold of a given system into Euclidean space and its transversal stabilization for tracking controller synthesis. Moreover, our method has the merit to use one single global Euclidean coordinate system whereas the method by Maggiore does not. Another merit of our method is its openness to accommodate any existing control method developed in Euclidean space.

The paper is organized as follows. Section 2 is devoted to embedding into Euclidean space, transversal stabilization, tracking controller design via linearization, and their application to the rigid body system and the quadcopter drone system. Several tracking controllers are proposed for the two systems, and the exponential convergence of their tracking error dynamics is rigorously proven and numerical simulations are carried out to demonstrate the controllers’ good tracking ability and robustness to unknown disturbances. The paper is concluded in Section 3. The contributions of the paper are summarized as follows: 1. the development of a new controller design methodology with the embedding and transversal stabilization technique which allows to convert difficult control problems on a manifold to tractable control problem in Euclidean space and to use one single global Euclidean coordinate system in controller synthesis; and 2. the design of exponentially tracking controllers with the developed method for the rigid body system and the quadcopter system which are designed via linearization in ambient Euclidean space but are still expressed geometrically, i.e. in a coordinate-free manner. It is noted that a presentation of preliminary results was given at the 56th IEEE Conference on Decision and Control.

## 2 Main Results

### 2.1 Mathematical Preliminaries

The usual Euclidean inner product is exclusively used for vectors and matrices in this paper, i.e.

for any two matrices of equal size. The norm induced from this inner product, which is called the Frobenius or Euclidean norm, is exclusively used for vectors and matrices. Let and

denote the symmetrization operator and the skew-symmetrization operator, respectively, on square matrices, which are defined by

for any square matrix . Then,

Namely,

with respect to the Euclidean inner product. Let denote the usual matrix commutator that is defined by for any pair of square matrices and of equal size. It is easy to show that

In other words, for all and ; for all and ; and for all and . Let denote the set of all rotation matrices, which is defined as . Let denote the set of all skew symmetric matrices, which is defined as . The hat map is defined by

for . The inverse map of the hat map is called the vee map and denoted such that for all and for all . 1. for all and .

2. .

3. for all .

4. and for all .
Given a function and a subset of , the set is defined as . In particular, when consists of a single point, say , we just write to mean . Every function and manifold is assumed to be smooth in this paper unless stated otherwise. Stability, stabilization and tracking are all understood to be local unless globality is stated explicitly. The reader is referred to the book by Bloch^{1} for more information on manifolds.

### 2.2 Embedding in Euclidean Space and Transversal Stabilization

#### 2.2.1 Theory

Let be an -dimensional regular manifold in , where . Consider a control system on given by

(1) |

Notice that

(2) |

where denotes the tangent space to at . Suppose that there is a control system on given by

(3) |

that satisfies

(4) |

In other words, is an extension of to and becomes a restriction of to . By (2) and (4), is an invariant manifold of .

Suppose that there is a function such that

(5) |

and

(6) |

for all and . With this function, construct a system in as

(7) |

where the vector field is defined by

(8) |

Since every point in is a minimum point of , vanishes on identically. Hence, by (4) and (8)

(9) |

In other words, the system coincides with the original system on . Hence, is an invariant manifold of as well. Along any flow of

(10) |

by (6).

If there are positive numbers and such that

(11) |

for all , then is positively invariant for and every flow of starting in converges to as . In particular, for all and .

###### Proof.

The following corollary shows a typical situation in which to construct such a function that satisfies (5), (6) and the hypothesis of Theorem 2.2.1. Suppose that there is a function such that ; that there is an open set such that and every point in is a regular point of ; that for all ; and that there is a number

such that the smallest singular value of

is larger than for every . Suppose also that is used to define the system in (7) and (8), where is an positive definite symmetric matrix. Then, there is an open set in with such that every trajectory of starting in remains in for all future time and exponentially converges to as .###### Proof.

Let , where is an positive definite symmetric matrix. Then, in column vector form. It is easy to show that this function satisfies (5) and (6) for all . By hypothesis, for all . Hence, for any , . Let and choose a number such that which is possible due to continuity of the function . With these numbers and , the hypothesis of Theorem 2.2.1 holds true. Hence, by Theorem 2.2.1, is a positively invariant region of attraction for , and for all and . This inequality implies that

for all and all , where . Since every point of is a regular point of , can be used as part of local coordinates such that . Hence, the above inequality shows that the convergence of to is exponential. ∎

Our goal is to design controllers for the system whose dynamics evolve on the manifold . Since the system in coincides with on , and is an invariant manifold of , we can first design controllers for in one single global Cartesian coordinate system for and then restrict them to to come up with controllers for the original system . This method becomes much more tractable when is an attractive invariant manifold of , which is guaranteed by the hypothesis in Theorem 2.2.1. Notice that the size of the region of attraction of for the dynamics is immaterial since the set is not a region of interest but only an auxiliary ambient region in which we take full advantage of the Euclidean structure of .

#### 2.2.2 Application to the Rigid Body System

As a main example throughout the paper, we use the free rigid body system with full actuation whose equations of motion are given by

(12a) | ||||

(12b) |

where is the state vector consisting of a rotation matrix and a body angular velocity vector ; is the control torque; and

is the moment of inertial matrix of the rigid body. From here on, we regard the system (

12) as a system defined on , treating as a matrix. It is then easy to verify that is an invariant set of (12), i.e. every flow starting in remains in for all . Assume that the full state of the system is available, which allows us to apply the following controller(13) |

to transform the above system to

(14a) | ||||

(14b) |

where is the new control vector. Note that is an invariant set of (14). Let and define a function by

(15) |

where is a constant. It is easy to verify that and

(16) |

With this function , the modified rigid body system corresponding to (7) and (8) is computed as

(17a) | ||||

(17b) |

where .

We now show that Theorem 2.2.1 holds in the rigid body case. There are numbers and such that

for all .

###### Proof.

Define an auxiliary function by

for . Take any sufficiently small such that every satisfying is invertible. Let . Then, if , , so is invertible, which implies that is also invertible. Hence, . For each and any ,

which implies

(18) |

for any . Hence for any ,

which implies by (18) that for all . It follows that is compact in , being closed and bounded. Since the matrix inversion operation is continuous, the image of under matrix inversion is also compact. Hence, there is a number such that for all . Hence, for any

which implies for all by (15) and (16), where . This completes the proof. ∎

There is a number such that every trajectory of (17) starting in remains in for all future time and converges exponentially to as .

###### Proof.

Pick such numbers and as in the statement of Lemma 2.2.2. By Lemma 2.2.2 and Theorem 2.2.1, every trajectory of (17) starting in remains in for all future time and converges to as . Let be an arbitrary trajectory staring in at . Then, by Theorem 2.2.1, it satisfies

for all . It follows that the convergence of to is exponential since the zero matrix is a regular value of the map defined by such that ; refer to pp.22–23 of Guillemin and Pollack^{9} to see why the zero matrix is a regular value of .
∎

The technique of embedding into ambient Euclidean space and transversal stabilization was successfully tested in creating feedback integrators for structure-preserving numerical integration^{6} of the dynamics of uncontrolled dynamical systems. This technique is extended to control systems in this paper. In particular, Theorem 2.2.1, Corollary 2.2.1, Lemma 2.2.2 and Theorem 2.2.2 in this paper are new and powerful so as to guarantee exponential stability of in the transversal direction.

### 2.3 Tracking via Linearization in Ambient Euclidean Space

#### 2.3.1 Theory

Consider again the system given in (7) and its restriction to given in (1). Choose a reference trajectory for on driven by a control signal so that

We can then linearize the ambient system along the trajectory in as follows:

(19) |

where

and

Refer to Section 4.6 of Khalil^{11} about the linearization technique. Notice that the above linearization does not require any use of local charts on the state-space manifold . In that sense the above linearization is conducted globally along the reference trajectory in one global coordinate system in . Also, in comparison with such a geometric linearization method as variational linearization in Lee et al.^{14} our Jacobian linearization is straightforward and simple to carry out. The following lemma is trivial but useful:
If is an exponentially tracking controller for the ambient system for the reference trajectory , then it is also an exponentially tracking controller for the system on for the same reference trajectory.
The following theorem is an adaptation of Theorem 4.13 from the textbook by Khalil^{11} in combination with Lemma 2.3.1 above.
Suppose that a linear feedback controller exponentially stabilizes the origin for the linearized system in . Let for some and be a function defined by

If the derivative is bounded and Lipschitz on uniformly in , then the controller

enables the system on to track the reference trajectory exponentially.

Notice that the key point in the above theorem is that the controller for the system on is designed in the ambient Euclidean space .

#### 2.3.2 Application to the Rigid Body System

We here apply Theorem 2.3.1 to the free rigid body system (17). Take a reference trajectory and the corresponding control signal such that

(20) |

which can be also understood as equations that define and in terms of and its time derivatives. Assume that and are bounded over the time interval . The linearization of (17) along the reference trajectory and the reference control signal is given by

(21a) | ||||

(21b) |

where

###### Proof.

We now introduce a new matrix variable replacing as follows:

(22) |

Let

(23) |

such that

(24) |

The system (21) is transformed to

(25a) | ||||

(25b) | ||||

(25c) |

###### Proof.

For any two matrices such that the matrix

(26) |

is Hurwitz, the controller

(27) |

exponentially stabilizes the origin for the system (25).

###### Proof.

Let us first show the exponential stability of the subsystem (25a) that is decoupled from the rest of the system. Let . Along the trajectory of (25), , where it is easy to show . Hence, for all , or

(28) |

for all and , which proves exponential stability of for (25a).

Differentiating (25b) and substituting (25c) transforms the subsystem (25b) and (25c) to the following second-order system:

since . This second-order system is exponentially stabilized by the controller

(29) |

where the matrices are any matrices such that the matrix in (26) becomes Hurwitz. So, there are positive constants and such that

for all and . Since is bounded by assumption, there is a constant such that for all . By (25b) and the triangle inequality,

and

for all . It is then easy to show that

(30) |

for all and , where . Notice that the controller given in (29) is the same as the one in (27). It follows from (28) and (30) that the controller (27) exponentially stabilizes the origin for the system (25). ∎

The exponential stability of the subsystem (25a) is a consequence of adding the term in (17a), and it is consistent with Theorem 2.2.2.

The following proposition produces time-varying PID-like tracking controllers. For any three matrices such that the polynomial

(31) |

is Hurwitz, the controller

(32) |

exponentially stabilizes the origin for the system (25).

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