Differential Flatness Condition for a 2DOF Underactuated Mechanical System

Prof. Dr. Waladin Khairi Sa'id Dr. Shibly Ahmed HameedControl & System Eng. Dept. Control & System Eng. Dept.

University of Technology University of TechnologyBaghdad-Iraq

E-mail: waladinksy@yahoo.comBaghdad-Iraq

E-mail: shiblyhameed@yahoo.com

ABSTRACTIn this paper, a simple flatness condition for the two degree of freedom underactuated

mechanical system is derived. A differential geometry is used as a mathematical tool in our derivation of the flatness condition. The flatness condition is found as a direct inner product between the covariant derivative of a vector field annihilate the codistribution spanned by the force matrix (the force matrix that appears in the Euler-Lagrange equation), and the force matrix itself. Several applications and examples available of underactuated mechanical systems are 2DOF systems or may be underactuated mechanical systems by one control with higher configuration variables than two which reduces to 2DOF system. Systems that are classified as differentially flat have many useful properties which can be used in the design of effective controller for the nonlinear systems. Two examples of flat 2DOF underactuated mechanical systems are considered (after satisfying flatness condition) these are the TORA system and the Inertia-Wheel pendulum. The flat output is derived first then the relations between the states and inputs with the flat output and its derivatives are obtained. In addition, a nonlinear controller was designed for a TORA system based on the flatness property which enforced the state trajectory to follow a desired reference trajectory. The simulation results demonstrate the effectiveness of the proposed controller.

Keywords: flatness condition, underactuated mechanical system, trajectory generation, nonlinear controller.

الخالصة:درجتين ذات الدفع تحت الميكانيكية لألنظمة بسيط إستواء شرط إشتقاق تم البحث هذا في

بين. داخلي ضرب عن عبارة وجد وقد اإلستواء شرط إلشتقاق الهندسي التفاضل إستخدام تم للحريةفي ( تظهر التي القوة مصفوفة القوة مصفوفة بواسطة بناءه تم سطح على عمودي متجه مشتقة

. ( Euler-Lagrangeمعادالت الميكانيكية لألنظمة كثيرة تطبيقات هنالك توجد نفسها القوة وصفوفة . الميكانيكية األنظمة درجتين إلى وتخفض للحرية أعلى درجات ذات أو للحرية درجتين ذات الدفع تحت

فعال مسيطر تصميم معها يمكن خصائصمهمة ضتملك مستوية إنها على تصنف التي الدفع تحت . هما للحرية درجتين ذات مستوية ألنظمة مثاالن الدفع تحت الميكانيكية األنظمة TORA systemلقيادة

and the Inertia-Wheel pendulum .الحالة متغيرات وبين المستوي الإلخراج بين العالقات إشتقاق تم . أل لمنظومة الخطي مسيطر تصميم ذلك إلى باإلضافة المثاالن لهاذان TORA والمسيطر

للمنظومة العددي التمثيل نتيجة ذلك بينت كما فعاليته أثبت وقد اإلستواء صفة من باإلستفادة

INTRODUCTIONThe nonlinear behavior of mechanical system exited the engineers and the mathematicians

to develop theoretical tools that deal with this type of system. The differential geometry is one of the main mathematical topics tool used in the analysis and design for the controller of the nonlinear system [Isidori et. al. [1981]& Isidori [1985]]Recently a branch of mechanical system named as underactuated mechanical system is distinguished for its important application and wide appearance in real-life system. The underactuated mechanical system is a mechanical system with number of

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controllers is less than the number of the configuration variables that describes the mechanical system when using Euler-Lagrange equation to develop the system model. Underactuated system appears in a broad range of applications including Robotics, Aerospace Systems, Flexible Systems, Mobile Systems, and Locomotive Systems.

Differential flatness or shortly flatness has been introduced for about fifteen years ago by Fliess et. al..[1999]. Flatness is a property that characterizes the dynamical system behavior. If a system is flat then it is equivalent to a system without dynamics (trivial system) described by a collection of independent variables, the flat outputs, having the same number (number of outputs) as the number of controllers [Fliess et. al. [1999]]. The main benefit of flatness property is that all states and inputs can be determined from the flat outputs (flat output and its derivative) without integration (one-to-one relations). The great feature of flat system is the trajectory generation. The trajectory is first designed in flat space and then reflected to system states and inputs through a one-to-one relation.

In underactuated mechanical system, the flatness property is very important where flat output is function only to configuration variables. The number of flat outputs is equal to number of controllers, which completely describe the mechanical system state [Murray et. al. [1995].

Rathinam and Murray [1998]introduced a method in a Riemannian geometry, which provide a complete characterization of configuration flatness for systems with configuration variables and

controllers and the range of control forces depends only on configuration variables. This method allows us to determine if such a system is configuration flat and, if so, provides a constructive method for finding all possible configuration flat outputs. Rathinam and Sluis [1995] carried out another test for differential flatness using the exterior algebra. The method consists of making an initial guess for , ( inputs) of the flat outputs, which may involve parameters still to be determined. A choice of function of time for the output reduces the system to one with a single input.

Kiefer et. al. [2004] used the flatness property in the design of a controller for a 3DOF helicopter with a slight modification of the generalized force to impose the flatness property to the 3DOF helicopter model. For the weight handling equipments (like crane), the controller based flatness with constraint Lagrangian system was designed by Kiss et. al. [2000].

Using flatness the trajectory generation and tracking for nonlinear control system is presented by Van Nieuwstadt et. al. [1998] where a large class of industrial and military control problems consist of planning and following a trajectory in the presence of noise and uncertainty. For unconstrained system, a nonlinear trajectory generation using flatness based optimal control technique with B-spline as a bases function in the flat space is found in references [Milam [2003]& Murray [2006]]. For a trajectory generation for a constrained mechanical system, see reference [Milam et. al. [2000]].

The organization of this paper is as follows: basic concept of flat system is presented in section two. The flatness condition for 2DOF underactuated mechanical systems is derived ( proposition (2)) in section three, also for actuated and unactuated shape variable a simple flatness conditions are derived based on proposition (2). The result of section three are applied to the TORA system and the Inertia-Wheel pendulum in sections four and five respectively. In section six a trajectory generation is presented and a nonlinear controller formula is derived based on flatness property for the TORA system. The numerical simulation for the proposed controller is found in section seven and finally the conclusion is presented in section eight.

FLAT SYSTEMFlatness or differential flatness is an important property of underdetermined ordinary

differential equation, which has the following form:

, (1)

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The underdetermined ordinary differential equations with number of variables greater than the number of equations is said to be differentially flat if there is , outputs known as flat outputs, such that:

(2)

Equation (2) represents a one-to-one map between state and its derivative up to times, and the flat output and its derivatives to (l) times. In the case of control system the

following equation:

(3)

with considered as underdetermined equation with variables and therefore if there are flat outputs then the control system (3) is a flat system.

Therefore and according to the above definitions, the trivial system (the flat outputs) completely specifies the curve of the original system. The time that is regarded as an independence condition is preserved in both the trivial system, which consist of flat output, and the original system. In addition, in a control system, the dimension of the trivial system is equal to the number of the controller in the control system and there is a one-to-one relation between the variable of the system including the controller with the flat output and its derivatives. More precisely, if the system has state and input then the system is flat if we can find outputs of the form:

(4)

Such that:

(5)

FLATNESS of MECHANICAL SYSTEM UNDERACTUATED by ONE-CONTROLIn this section, we will follow the results on the testing and setting approach to the flat

outputs of the mechanical system underactuated by one-control introduced by Rathinam and Murray [1998]. Then a proposition is added in order to obtain the differential form of the flat outputs using simple inner product condition.

To state the result of Rathinam and Murray a quantity like the vector field and the distribution D is defined. The vector field is an annihilator to the codistribution spanned by the

force vector i.e. where .The distribution

associated with the mechanical system defined as:

(6)

Where is the covariant derivative and denote the covariant derivative of along .

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Theorem (1) Rathinam and Murray [1998]: Let be a point on Q and U be an open neighborhood of , and suppose is a submersion. If are configuration flat outputs, then:

(7)

Where is a bilinear map, which represents an inner product with respect to the Riemannian metric .Conversely if and if a certain regularity holds at , then

are configuration flat outputs around .The regularity condition is that the ratios of functions in the following set should not all be

the same at :

(8)

where , , are arbitrary vector fields around that are y-related to some vector field on and , are fixed non-vanishing vector fields such that and .The regularity conditions can be checked in coordinates as follows. Choose a function that completes

to a coordinate system. Then will be flat outputs if the following ratios of functions are not all identically equal in a local neighborhood:

,

(9a)

,

(9b)

(9c)

If these are all identically equal that means are differentially dependent and another one-dimensional distribution must be tried.

To complete the tools used to derive the flat outputs using known integrability theory of Frobenius [Boothby [1975]], the following proposition is presented.Proposition (1): Let be a non-trivial vector field such that , then the following conditions are equivalent:

, is an integer with (10)

Proof: The existence of vector filed depends on the dimension of D. Since for flat system, then this proves the existence of . Also from the condition, the kernel to the tangent subspace of the flat outputs is orthogonal to the distribution D with respect to metric g and that means is the kernel to the tangent space to flat outputs. The kernel to , spanned by

.These vector fields annihilate the differential form of the flat

outputs, i.e. , or

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Remark (1): The flat output is directly computed from its differential form if is exact otherwise flat output (refer to it )form the differential forms are obtained with an integration factor as

. Proof: If is an exact differential then the flat output obtained by direct integration of . Otherwise the differential form is still integrable because linearly independent differential

form are always integrable where with the use of an

integration factor to get . Therefore theorem (1) not only enable us to test the flatness of underactuated mechanical system but also guarantee getting m flat output as proved above.

2DOF Underactuated Flat Mechanical SystemSeveral applications and examples of underactuated mechanical systems available are 2DOF

systems or underactuated mechanical systems with configuration variables greater than two reduces to 2DOF system. These systems may be classified into two categories. The first category is the reducible systems to regular form (integrable differential form) and the second category frequently appears as the non-integrable differential form. For the second category, the systems can be reduced to a regular form under sliding mode control hierarchy as proposed in reference [Hameed [2007]].

The main contribution of this work is to derive a simple flatness condition for the 2DOF underactuated mechanical system as it is presented in proposition (2). Also and because of proposition (2), three corollaries are obtained for the special forms of the 2DOF underactuated system. Finally, the flat output and the relations between the states, inputs and the flat output and its derivatives are derived for two examples of underactuated mechanical system, which they are the TORA system and the Inertia-Wheel Pendulum.Proposition (2): The 2DOF underactuated mechanical system described by the following Euler-

Lagrange differential equation where and is a differentially flat

system if and only if the following condition holds:

, (11)

where is a vector field satisfy the inner product and is the covariant derivatives

with respect to , and .

Proof: The existence of is obvious since span a one-dimensional sub-space. Then following theorem (1), the dimension of the distribution D must equal to one in order for the system (1) to be

a flat system.The distribution D spanned by .Therefore and

must be collinear to in order to dimD equal to one and that means and

annihilate i.e. satisfy the inner product condition given in eq.(11).

The flat output may be computed according to the following proposition.Proposition (3): The integrable differential form of the flat output for the 2DOF underactuated mechanical system is given by:

(12)

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where and is the element of vector field and is the element of the inertia matrix.Proof: According to proposition (2), the distribution spanned by ( ).

Let be given by . The vector field is computed such that the

condition holds. As a result becomes .Also let be

given by then follow the condition given in proposition (1), , is

calculated as therefore .

Note that is integrable differential form (the one-form of dimension 2 is always integrable (see Boothby [1975])). Remark (2): 2DOF flat mechanical system is also a static state feedback linearizable [Fliess et. al..[1999]] and therefore the condition of equation (11) is a new condition (test) for linearizability of a 2DOF underactuated mechanical system.

Based on proposition (2), the following corollaries for 2DOF underactuated mechanical system with actuated and unactuated shape variables are stated (the shape variables are the variables that the inertia matrix function to it [Spong et. al..[1999]]). These types of mechanical systems are the main examples of underactuated system studied in many research works [17&14].Corollary (1): The 2DOF underactuated mechanical system with actuated shape variable is flat if and only if the mass element is constant. Accordingly the flat output is given by:

(13)

Proof: For the 2DOF, unactuated shape variable the force vector (one-form) is expressed by .Let , then: .The vector field satisfies the condition that is

. The covariant derivatives , is equal to .The

flatness condition in proposition (2) applied as .

Then to satisfy this condition the Christoffel symbol must equal to zero ( ), where the

Christoffel symbol is computed as .The mass matrix appears above is

only function to shape variable . For , we have and for

.Therefore, is equal to zero if and only if is constant which

satisfies the flatness condition (11). The flat output form in equation (13) follows directly the form given in proposition (3). To show that this condition is the only condition for flatness, we have to proof that the regularity condition is satisfied without any additional condition. To start let ,

then , .The first regularity condition is given by

and it is computed as . Also the second regularity

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condition is and is computed as

.

Since the two ratios are unequal, therefore the output is a flat output provided that is constant as a sufficient condition for flatness irrespective to the potential energy form.

The Acrobot is an example of non-flat 2DOF underactuated mechanical system with actuated shape variable because is not constant (see reference [Olfati-Saber[2001]] for inertia matrix of the Acrobot).

Coming analysis concerns the TORA system and the inertia-wheel pendulum as examples of 2DOF flat system with actuated shape variable. Corollary (2): 2DOF underactuated mechanical system with unactuated shape variable is differentially flat if and only if the following conditions hold:

(14)

The flat output then is equal to:

(15)

Proof: For unactuated shape variable case, the force vector is given by .

Accordingly, the vector field is equal to . Then follow the flatness condition (11) given in

proposition (2), we have .This leads to the following

condition: , where Christoffel symbols are equal to ,

.To satisfy the above , we get the following condition for

flatness:

and .The flat output in equation (15) obtained from the

general formula in proposition (3). To check the regularity condition, let , then we have

, . The first regularity condition is given by

, and for the second regularity condition

7

Where . This result leads to two cases, depending on . If then the system is flat,

otherwise if , then the mechanical system is flat if the following third regularity condition

differ from .

Note that for a potential energy the term is a Lie-derivative with respect to , i.e.

.

Remark (3): The Penduput, the Cart-Pole system and the double Inverted Pendulum are all non-flat underactuated mechanical system with unactuated shape variable because the element of the inertia matrix for all these systems is a function of the shape variable ( refer to reference [Olfati-Saber [2001]] for inertia matrix of these examples).Corollary (3): 2DOF underactuated mechanical system with constant inertia matrix and constant force vector , i.e.:

, , (16)

is a flat system with flat output of the form;

(17)

Proof: For a system with inertia matrix satisfying equation (16), the Christoffel symbols are zero

, .In addition, consists of constant elements then is equal to zero as can

be deduced from the covariant derivative, given by .

Hence, the flatness condition (11) is satisfied and the system is flat. Also the flat output can be derived directly using equation (12).

The inertia-wheel pendulum is an example of this type of systems as will be seen later.

THE TORA SYSTEMAs shown in Fig. 1, the TORA system consists of translational oscillating platform, which is

controlled via a rotational eccentric mass. The inertia matrix, potential energy and the force vector (one-form) assume the following form [Olfati-Saber [2001]];

Then the TORA system is a flat system (static feedback linearizable system) according to corollary (1) for actuated shape variable with equal to constant . The flat output from equation (12) is given by:

(18)

8

k

x

u

m1

r m2 ,I

Fig 1: The TORA system.

The underdetermined equation is found by twice differentiating equation (18):

(19)

Starting from equations (18) and (19) the one-to-one relation between the system state, control , , , , and the flat output and its derivatives are given by:

(20a)

(20b)

Where , and

The relations are one-to-one between the state plus the controller, and flat output and its derivatives up to fourth derivative.Remark (4): The TORA system is a flat system in the following region in state space:

(21) outside this region, the TORA system is not flat. This condition is represented in the flat space by the following differential inequality:

(22)

Accordingly, the system is not flat outside this region.

THE INERTIA-WHEEL PENDULUMThe inertia-wheel pendulum is a planar inverted pendulum with a revolving wheel at the end

as shown in Fig. 2,

9

Fig. 2: The Inertia-Wheel Pendulum. The actuation at the wheel and the pendulum is unactuated at the base. The inertia-wheel pendulum was first introduced by Spong et. al..[1999]. The inertia matrix, potential energy and the force

vector are , , . The vector

field is equal to and the system is flat according to corollary (3) with flat output given by:

(23)

The second derivative of is;

(24)

This equation is also the underdetermined differential equation for the inertia-wheel pendulum. Using equations (23) and (24), the one-to-one relation between the states, input and the flat output and it is derivatives are;

(25a)

(25b)

where , , and

For certain system these equations could be used to compute the required control effort for certain trajectory generated in flat space.

NONLINEAR CONTROLLER DESIGN for the TORA SYSTEM BASED FLATNESS PROPERTY

2

111, IL

2Iu

10

The flatness property will now be considered in the design of the sliding mode controller for the TORA system. The flat output is taken as the linearizable output of the TORA system. Hence, the TORA system is static feedback linearizable and accordingly, the linear model becomes (differentiating the flat output four times);

(26)

Where, ,

Note that the flatness property is not devoted to re-write the dynamical system in terms of flat output i.e., to linearize the dynamical system either by static state feedback (single-input case) or by dynamic state feedback (like endogenous, [Martin Ph. and Rouchon [1994]). In fact, it is used directly to calculate the control action value through its relation with flat output and its derivative after constructing a desired trajectory in flat space. Unfortunately, the TORA system is linearizable in a region defined by (21) as mensiond in remark (4).

The idea presented here is to use the flatness property to generate (construct) a state trajectory that is initiated from certain initial point and end at the final point. Then a nonlinear controller is designed to regulate the error function to zero. This error function is the difference between the flat output and the trajectory generated in flat space, which satisfies the above constrain.

To generate the trajectory let us assume a polynomial of degree nine. This is because the states and the control are related to the flat output and its differential to fourth derivative as given by equation (20). Therefore;

, , (27)

Where is a polynomial coefficient and is a selected time required to reach the final state. The

generated output in equation (27) is in terms of in order to parameterize the trajectory

generation with respect to . The first to fourth derivative of is equal to:

(28)

where . The initial and final point for the rest to rest condition by considering the state as is given by:

to (29)

According to initial and final states, the coefficients are computed to satisfy condition (22). Unfortunately imposing a constrain to the trajectory generation is not an easy problem, and that means one can not take any initial condition and generate a trajectory satisfying condition (22). Instead, only a limited range of initial conditions is considered.

Trajectory Generation

11

Starting from equation (28), ten equations (five for the initial and five for the final) are solved simultaneously to get for the following initial and final states;

and

(30)

Considering the time required to move from initial to final state as a parameter controlled in such a way that the trajectory generation will not violate condition (22). Condition (22) may be written in flat space as by

(31)

The nominal and the variation in system parameters are the same as those given in the first design. The following results are the initial condition represented by , the time and the coefficients of the generated output such that the generating trajectory satisfies condition (31): , and, , , , ,

, , , ,

Figure 3 shows that the function which is given by

satisfies condition (31) i.e., .

0 2 4 6 8 10

Tim e, s

-0.08

-0.04

0.00

0.04

0.08

H(t

)

Fig. 3: Function H(t) versus time.

Derivation of Controller FormulaTo derive a nonlinear controller that regulate the error function to zero, the error function is

defined first as;

, (32)

Then the error dynamics becomes;

12

(33)

Let a new controller defined by:

(34)

The error dynamics is designed to decay exponentially to zero with the following roots located at -1, -1.5, -2, -2.5. Accordingly the controller is equal to

(35)

Finally the proposed controller that forced the flat output to follow the generated trajectory and hence to regulate TORA system state to the origin is given by:

(36)

A umerical simulation is employed for the following parameter values and the initial conditions . Figure 4

shows the plot of displacement and the generated displacement, in terms of as given by equation (54a), with time. It may be noticed that the displacement coincides with the generated displacement after a short period of time. This time period, in fact, depends on the roots chosen in the construction of the controller . In addition the controller design and the simulation reveal the ability of freely designing a trajectory and avoiding critical region to reach a desired equilibrium point. This is the feature of the TORA system as a flat system.

0 2 4 6 8 10

T im e, s

-0.2

-0.1

0.0

0.1

0.2

0.3

Dis

plac

emen

t x, m

Displacem ent x

Generated displacem ent x

Fig. 4: Displacement versus time.

CONCLUSIONSIn the present work a simple flatness condition was derived for a class of mechanical system

known as underactuated mechanical system. For a 2DOF underactuated mechanical system the condition given in eq.(20) is proved as flatness property condition which it is also the exact linearization condition for this class of systems. Depends on the flatness property for actuated and unactuated mechanical system was derived and it is found as simple condition depends only on the elements of inertia matrix. The TORA system and the Inertia-Wheel pendulum are found flat

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systems according to condition (20), in addition the one-to-one relations between the system state and controller and the flat outputs are derived for both systems. The TORA flatness property is effectively used first to generate a desired trajectory in flat space satisfying the limitation presented in (21), and secondly to design a nonlinear controller as given in eq. (36). The proposed controller is found effective to force the system state to follow the generating trajectory in flat space which leads to regulate the state to origin as demonstrated in the numerical simulation presented in section seven.

REFERENCESIsidori A., Krener A. J., Gori-Giorgi C., and Monaco S. "Nonlinear decoupling via feedback: a differential geometric approach". IEEE Trans. Automatic Control, vol. (AC-26), No. 2, pp. 331-345, April 1981. (case of four authors)

Isidori A.. "Nonlinear Control Systems: An Introduction (Notes on Information and Control Systems, vol. 72)". New York: Springer-Verlag, 1985. (case of one author)

Fliess M., Levine J., Martin Ph., and Rouchon P. "A Lie-Backlund approach to equivalence and flatness of nonlinear systems". IEEE Trans. Automat. Control, vol. 44, No. 5, pp. 922-937, May 1999. (case of four authors)

Murray R. M., Rathinam M., and Sluis W. "Differential flatness of mechanical control systems: A catalog of prototype systems". Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, pp. 12-17, San Francisco, Nov. 1995. (case of three authors)

Rathinam M. and Murray R. M. "Configuration flatness of Lagrangian systems underactuated by one control ". Siam Journal on Control and Optimization, vol. 36, No. 1, pp. 164-179, Jan. 1998. (case of one author)

Rathinam M., and Sluis W. "A test for differential flatness by reduction to single input systems". Technical Memorandum No. CIT-CDS 95-018, June 1995. (case of two authors)

Kiefer T., Kugi A., and Kemmetmuller W. "Modeling and flatness-based control of a 3DOF helicopter laboratory experiment". 6th IFAC Symposium on Nonlinear Control Systems Design (NOLCOS), Sept. 2004. (case of three authors)

Kiss B., Levine J., and Mullhaupt Ph. "Modeling and motion planning for a class of weight handling equipments". Proceeding of the 14th International Conf. on Systems Engineering, Conventry, UK, Sept. 2000. (case of three authors)

Van Nieuwstadt M., Rathinam M., and Murray R. M. "Differential flatness and absolute equivalence of nonlinear control systems". Siam Journal on Control and Optimization, vol. 36, No. 4, pp. 1225-1239, 1998. (case of three authors)

Milam M. B. "Real-Time Optimal Trajectory Generation for Constrained Dynamical Systems". Ph.D. thesis, California Institute of Technology, Pasadena, California, 2003. (case of one author)

Murray R. M. "Real-Time Trajectory Generation". Lecture 3-1, CDS 270-2, April 2006. (case of one author)

Milam M. B., Mushambi K., and Murray R. M. "A new computational approach to real-time trajectory generation for constrained mechanical systems". Proceeding of the Conf. on Decision and Control, 2000. (case of three authors)

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Boothby. W. M. "An Introduction to Differentiable Manifolds and Riemannian Geometry ". New York, Academic, 1975. (case of one author)

Shibly A. Hameed. "Design of a Sliding Mode Controller for Underactuated Mechanical System". Ph.D.. thesis, Baghdad University, Baghdad, Iraq, 2007. (case of one author)

Spong M. W., Corke P., and Lozano R. "Nonlinear control of the inertia wheel pendulum". Automatica, Sept. 1999. (case of three authors)

Olfati-Saber R.. "Nonlinear control of underactuated mechanical systems with application to robotics and aerospace vehicles". Ph.D. dissertation, Dept. Elect. Eng. Comput. Sci., Massachusetts Inst. Technol., Cambridge, MA, 2001. (case of one author)

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NOMENCLATURE

Symbols: Annihilator to a tangent space .

: Distribution spanned by a set of vector fields.: force matrix (set of one-forms).

: Riemann metric in a Riemannian manifold.

: Acceleration due to gravitational attraction ( ).

: The kernel to a tangent space .: The Lagrangian of mechanical system.: Inertia matrix of mechanical system.: Configuration manifold of dimension n.: Configuration variable belonging to configuration manifold .

: Shape configuration variable.

: Tangent space to outputs .: Potential energy of mechanical system.

Greek Symbols Codistribution on a differentiable manifold.

: Determinant of the inertia matrix ( ).: Christoffel symbol in a given set of coordinates.

: Integration factor for non-exact one form.: Vector field annihilate the Force vector .

Mathematical Symbols

: Inner product on a Riemannian manifold with respect to Riemann metric.

: Bases of a tangent to configuration manifold.

: Covariant derivative of with respect to .

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16