**2. Mathematical model**

(7)

(8)

leads to the nonlinear modeling of the system, as follows:

246 Nonlinear Systems - Design, Analysis, Estimation and Control

**Figure 3.** Mechanical system with a viscous damper and a nonlinear spring [3].

unactuated submodels.

The nonlinear feedback technique, also called feedback linearization, is a representative method for controlling nonlinear systems. The main concept of feedback linearization is to transfer the original nonlinear system algebraically into the linear system by inserting equivalent inputs to suppress the nonlinearities of the former. The feedback linearization control of fully actuated systems has been discussed in several well-known textbooks [4, 5] in which this theory has been completely developed. Previous studies have pointed out that fully actuated systems are feedback linearizable through nonlinear feedback [6, 7]. In this chapter, we introduce the feedback linearization control for a class of multiple-input and multipleoutput (MIMO) underactuated systems. The analysis process is conducted using an algebra

foundation in which the mathematical model is simplified through matrix equations.

First, the mathematical model of underactuated mechanical systems is separated into two subsystems: actuated states and unactuated states. Then, we design a controller in which nonlinear feedback is partly applied to both actuated and unactuated dynamics. Subsequently, actuated submodel is "linearized" using a nonlinear feedback method; thus, the unactuated dynamics is regarded as internal model. Seeing actuated states as system outputs, a nonlinear control law is designed to drive state trajectories to the references. However, this controller does not promise the stability of unactuated states. Therefore, its structure should be adjusted to guarantee the stability of both actuated and unactuated states based on the nonlinear feedback of all system states. The control scheme now exhibits the linear combination of two components that are distinctly acquired from the nonlinear feedback of both the actuated and

In comparison with traditional controllers, such as the proportional-integral-derivative (PID) controller, partial feedback linearization (PFL) exhibits several advantages. In the PID conIn general, the physical behavior of a MIMO mechanical system is governed by a set of differential equations of motion. Consider an underactuated system with *n* degrees of freedom driven by *m* actuators (*m*<*n*). The mathematical model, which is composed of *n* ordinary differential equations, is simplified in matrix form as follows:

$$\mathbf{M(q)}\ddot{\mathbf{q}} + \mathbf{C(q, \dot{q})}\dot{\mathbf{q}} + \mathbf{G(q)} = \mathbf{F} \tag{9}$$

where = 1 2 ⋯ <sup>∈</sup>*Rn* is the vector of the generalized coordinates, and **<sup>F</sup>** <sup>∈</sup> *Rn* denotes the vector of the control inputs. Given that the system has more control signals than actuators, **F** has only *m* nonzero components as = × 1 , with <sup>=</sup> 1 2 ⋯ <sup>∈</sup>*Rm* being a vector of nonzero input forces. **M**(**q**) = **M***<sup>T</sup>*(**q**) = [*mij*]*<sup>n</sup>* × *<sup>n</sup>* ∈ *Rn* × *<sup>n</sup>* is the symmetric mass matrix, , ˙ <sup>=</sup> × ∈ × is the Coriolis and centrifugal matrix, and <sup>=</sup> 1 2 ⋯ ∈ indicates the gravity vector.

As an underactuated system, its *n* output signals are driven by *m* actuators. Meanwhile, its mathematical model is divided into two auxiliary dynamics, namely, actuated and unactuated systems. Correspondingly, <sup>=</sup> 1 2 ⋯ <sup>∈</sup>*Rm* for actuated states and <sup>=</sup> + 1 ⋯ <sup>∈</sup>*Rn*−*<sup>m</sup>* for unactuated states are defined. The matrix differential equation (9) can then be divided into two equations as follows:

$$\mathbf{M}\_{11}\left(\mathbf{q}\right)\ddot{\mathbf{q}}\_a + \mathbf{M}\_{12}\left(\mathbf{q}\right)\ddot{\mathbf{q}}\_u + \mathbf{C}\_{11}\left(\mathbf{q}, \dot{\mathbf{q}}\right)\dot{\mathbf{q}}\_a + \mathbf{C}\_{12}\left(\mathbf{q}, \dot{\mathbf{q}}\right)\dot{\mathbf{q}}\_u + \mathbf{G}\_1\left(\mathbf{q}\right) = \mathbf{U} \tag{10}$$

$$\mathbf{M}\_{21}(\mathbf{q})\ddot{\mathbf{q}}\_a + \mathbf{M}\_{22}\left(\mathbf{q}\right)\ddot{\mathbf{q}}\_u + \mathbf{C}\_{21}\left(\mathbf{q}, \dot{\mathbf{q}}\right)\dot{\mathbf{q}}\_a + \mathbf{C}\_{22}\left(\mathbf{q}, \dot{\mathbf{q}}\right)\dot{\mathbf{q}}\_u + \mathbf{G}\_2\left(\mathbf{q}\right) = \mathbf{0},\tag{11}$$

where **M**11(**q**), **M**12(**q**), **M**21(**q**), **M**22(**q**) are the submatrices of **M**(**q**); and 11 , ˙ , 12 , ˙ , 21 , ˙ , 22 , ˙ are the submatrices of 11 , ˙ . Therefore, matrices **M**(**q**), , ˙ , and **G**(**q**) of Equation (9) exhibit the following form:

$$\mathbf{M(q)} = \begin{bmatrix} \mathbf{M\_{11}(q)} & \mathbf{M\_{12}(q)} \\ \mathbf{M\_{21}(q)} & \mathbf{M\_{22}(q)} \end{bmatrix}, \mathbf{C(q, \dot{q})} = \begin{bmatrix} \mathbf{C\_{11}(q, \dot{q})} & \mathbf{C\_{12}(q, \dot{q})} \\ \mathbf{C\_{21}(q, \dot{q})} & \mathbf{C\_{22}(q, \dot{q})} \end{bmatrix}, \mathbf{G(q)} = \begin{bmatrix} \mathbf{G\_{1}(q)} \\ \mathbf{G\_{2}(q)} \end{bmatrix}.$$

Notably, matrix **M**(**q**) is symmetric positive definite, 12 = 21 . The actuated equation (10) shows direct relationship between the actuated states **q***a* and the actuators **U**. By contrast, the unactuated equation (11) does not display the constraint between the unactuated states **q***<sup>u</sup>* and the inputs **U**. Physically, input signals **U** drive the actuated states **q***a* directly and the unactuated states **q***u* indirectly.

## **3. Nonlinear feedback control**

System dynamics, which is composed of Equations (10) and (11), is transformed into a simpler model with an equivalent linear form based on the nonlinear feedback method [7]. Note that **M**22(**q**) is a positive definite matrix. The unactuated states **q***u* can be determined from Equation (11) as

$$\ddot{\mathbf{q}}\_{u} = -\mathbf{M}\_{22}^{-1}\left(\mathbf{q}\right)\left\{\mathbf{M}\_{21}\left(\mathbf{q}\right)\ddot{\mathbf{q}}\_{a} + \mathbf{C}\_{21}\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}\_{a} + \mathbf{C}\_{22}\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}\_{u} + \mathbf{G}\_{2}\left(\mathbf{q}\right)\right\}.\tag{12}$$

In underactuated mechanical systems, the unactuated state **q***u* has a geometric relationship with the actuated state **q***a*. Therefore, control input **U** indirectly acts on **q***u* through **q***a*. Substituting Equation (12) into Equation (10) and simplifying the equation yield the following:

$$
\bar{\mathbf{M}}(\mathbf{q})\ddot{\mathbf{q}}\_a + \overline{\mathbf{C}}\_1(\mathbf{q}, \dot{\mathbf{q}})\dot{\mathbf{q}}\_a + \overline{\mathbf{C}}\_2(\mathbf{q}, \dot{\mathbf{q}})\dot{\mathbf{q}}\_u + \overline{\mathbf{G}}\_1(\mathbf{q}) = \mathbf{U} \tag{13}
$$

where

$$
\overline{\mathbf{M}}(\mathbf{q}) = \mathbf{M}\_{11}(\mathbf{q}) - \mathbf{M}\_{12}(\mathbf{q})\mathbf{M}\_{22}^{-1}(\mathbf{q})\mathbf{M}\_{21}(\mathbf{q}).
$$

$$
\overline{\mathbf{C}}\_{1}(\mathbf{q}, \dot{\mathbf{q}}) = \mathbf{C}\_{11}(\mathbf{q}, \dot{\mathbf{q}}) - \mathbf{M}\_{12}(\mathbf{q})\mathbf{M}\_{22}^{-1}(\mathbf{q})\mathbf{C}\_{21}(\mathbf{q}, \dot{\mathbf{q}})
$$

$$
\overline{\mathbf{C}}\_{2}(\mathbf{q}, \dot{\mathbf{q}}) = \mathbf{C}\_{12}(\mathbf{q}, \dot{\mathbf{q}}) - \mathbf{M}\_{12}(\mathbf{q})\mathbf{M}\_{22}^{-1}(\mathbf{q})\mathbf{C}\_{22}(\mathbf{q}, \dot{\mathbf{q}}) \text{ and}
$$

$$
\overline{\mathbf{G}}\_{1}(\mathbf{q}) = \mathbf{G}\_{1}(\mathbf{q}) - \mathbf{M}\_{12}(\mathbf{q})\mathbf{M}\_{22}^{-1}(\mathbf{q})\mathbf{G}\_{2}(\mathbf{q}).
$$

is a positive definite matrix for every = <sup>∈</sup>*Rn*. Equation (13) is transformed into

$$
\ddot{\mathbf{q}}\_a = \overline{\mathbf{M}}^{-1}(\mathbf{q}) \left\{ \mathbf{U} - \overline{\mathbf{C}}\_1(\mathbf{q}, \dot{\mathbf{q}}) \dot{\mathbf{q}}\_a - \overline{\mathbf{C}}\_2(\mathbf{q}, \dot{\mathbf{q}}) \dot{\mathbf{q}}\_u - \overline{\mathbf{G}}\_1(\mathbf{q}) \right\}. \tag{14}
$$

By inserting Equation (14) into Equation (12), we obtain

$$\ddot{\mathbf{q}}\_{\nu} = -\mathbf{M}\_{22}^{-1}(\mathbf{q}) \left\{ \mathbf{M}\_{21}(\mathbf{q}) \overline{\mathbf{M}}^{-1}(\mathbf{q}) \mathbf{U} + \overline{\mathbf{C}}\_{3} \left( \mathbf{q}, \dot{\mathbf{q}} \right) \dot{\mathbf{q}}\_{u} + \overline{\mathbf{C}}\_{4} \left( \mathbf{q}, \dot{\mathbf{q}} \right) \dot{\mathbf{q}}\_{u} + \overline{\mathbf{G}}\_{2} \left( \mathbf{q} \right) \right\}, \tag{15}$$

where

(11)

(12)

(13)

( )

. The actuated equation

where **M**11(**q**), **M**12(**q**), **M**21(**q**), **M**22(**q**) are the submatrices of **M**(**q**); and 11 , ˙ , 12 , ˙ , 21 , ˙ , 22 , ˙ are the submatrices of 11 , ˙ . Therefore, matrices

> 11 12 11 12 1 21 22 21 22 2

& & & & &

é ùé ù éù == = ê úê ú êú ë ûë û ëû **Mq Mq C qq C qq G q M q C qq G q M q M q C qq C qq G q**

(10) shows direct relationship between the actuated states **q***a* and the actuators **U**. By contrast, the unactuated equation (11) does not display the constraint between the unactuated states **q***<sup>u</sup>* and the inputs **U**. Physically, input signals **U** drive the actuated states **q***a* directly and the

System dynamics, which is composed of Equations (10) and (11), is transformed into a simpler model with an equivalent linear form based on the nonlinear feedback method [7]. Note that **M**22(**q**) is a positive definite matrix. The unactuated states **q***u* can be determined from Equation

In underactuated mechanical systems, the unactuated state **q***u* has a geometric relationship with the actuated state **q***a*. Therefore, control input **U** indirectly acts on **q***u* through **q***a*. Substituting Equation (12) into Equation (10) and simplifying the equation yield the following:

() () ( ) ( )

, , , , , . , ,

() () ( ) () ()

**M**(**q**), , ˙ , and **G**(**q**) of Equation (9) exhibit the following form:

Notably, matrix **M**(**q**) is symmetric positive definite, 12 = 21

−1 21 ,

−1 2 .

−1 21 , ˙

−1 22 , ˙ and

( ) () ()

248 Nonlinear Systems - Design, Analysis, Estimation and Control

unactuated states **q***u* indirectly.

(11) as

where

= 11 − 12 22

1 , ˙ = 11 , ˙ − 12 22

2 , ˙ = 12 , ˙ − 12 22

1 = 1 − 12 22

**3. Nonlinear feedback control**

$$
\overline{\mathbf{C}}\_{3}(\mathbf{q},\dot{\mathbf{q}}) = \mathbf{C}\_{21}(\mathbf{q},\dot{\mathbf{q}}) - \mathbf{M}\_{21}(\mathbf{q})\overline{\mathbf{M}}^{-1}(\mathbf{q})\overline{\mathbf{C}}\_{1}(\mathbf{q},\dot{\mathbf{q}}).
$$

$$
\overline{\mathbf{C}}\_{4}(\mathbf{q},\dot{\mathbf{q}}) = \mathbf{C}\_{22}(\mathbf{q},\dot{\mathbf{q}}) - \mathbf{M}\_{21}(\mathbf{q})\overline{\mathbf{M}}^{-1}(\mathbf{q})\overline{\mathbf{C}}\_{2}(\mathbf{q},\dot{\mathbf{q}}).
$$

$$
\text{and } \overline{\mathbf{G}}\_{2}(\mathbf{q}) = \mathbf{G}\_{2}(\mathbf{q}) - \mathbf{M}\_{21}(\mathbf{q})\overline{\mathbf{M}}^{-1}(\mathbf{q})\overline{\mathbf{G}}\_{1}(\mathbf{q}).
$$

Therefore, the dynamic behavior of a mechanical underactuated system can be described by actuated dynamics (14) and unactuated dynamics (15) in which the mathematical relationships among **q***a*, **q***u*, and **U** can be observed clearly.

Considering the actuated states **q***a* as the system outputs, actuated dynamics (14) can be "linearized" by defining

$$
\ddot{\mathbf{q}}\_a = \mathbf{V}\_{a'} \tag{16}
$$

with **V***<sup>a</sup>* ∈ *Rm* as the equivalent control inputs. Then, the control signals **U** become

$$\mathbf{U} = \overline{\mathbf{M}}(\mathbf{q})\mathbf{V}\_a + \overline{\mathbf{C}}\_1(\mathbf{q}, \dot{\mathbf{q}})\dot{\mathbf{q}}\_a + \overline{\mathbf{C}}\_2(\mathbf{q}, \dot{\mathbf{q}})\dot{\mathbf{q}}\_a + \overline{\mathbf{G}}\_1(\mathbf{q})\,. \tag{17}$$

Controller **U** is designed to drive the actuated states **q***a* to the desired values **q***ad*. To track the given state trajectories, the following equivalent control inputs are selected:

$$\mathbf{V}\_{a} = \ddot{\mathbf{q}}\_{\alpha l} - \mathbf{K}\_{\alpha l} \left( \dot{\mathbf{q}}\_{a} - \dot{\mathbf{q}}\_{\alpha l} \right) - \mathbf{K}\_{\alpha p} \left( \mathbf{q}\_{a} - \mathbf{q}\_{\alpha l} \right). \tag{18}$$

Given that **q***ad* = **const**, Equation (18) can be reduced into

$$\mathbf{V}\_a = -\mathbf{K}\_{ad}\dot{\mathbf{q}}\_a - \mathbf{K}\_{ap}\left(\mathbf{q}\_a - \mathbf{q}\_{ad}\right),\tag{19}$$

with **K***ad* = diag(*Kad*1, *Kad*2, …, *Kadm*) ∈ *Rm* × *<sup>m</sup>*, **K***ap* = diag(*Kap*1, *Kap*2, …, *Kapm*) ∈ *Rm* × *<sup>m</sup>* as positive diagonal matrices.

On the basis of Equation (18) and active dynamics (16), the differential equation of the tracking error is obtained as described by

$$
\ddot{\tilde{\mathbf{q}}}\_a + \mathbf{K}\_{\alpha t} \dot{\tilde{\mathbf{q}}}\_a + \mathbf{K}\_{ap} \tilde{\mathbf{q}}\_a = \mathbf{0},
\tag{20}
$$

where = − is the tracking error vector of the actuated states. Evidently, the dynamics of the tracking error (20) is exponentially stable for every **K***ad* > **0** and **K***ap* > **0**. That is, the tracking errors of the actuated states approach zero (or **q***a* converges to **q***ad*) as *t* becomes infinite. In particular, the equivalent control **V***a* forces the actuated states **q***a* to reach the references **q***ad* asymptotically.

The control scheme (17), which corresponds to the equivalent input **V***a*, is used only to stabilize the actuated states **q***a* asymptotically. To stabilize the unactuated states **q***u*, the nonlinear feedback technique can be applied to subdynamics (15) as follows:

$$
\ddot{\mathbf{q}}\_{\mu} = \mathbf{V}\_{\mu} = -\mathbf{K}\_{\mu\iota}\dot{\mathbf{q}}\_{\mu} - \mathbf{K}\_{\imath\wp}\mathbf{q}\_{\mu} \,. \tag{21}
$$

where **V***<sup>u</sup>* ∈ *Rn* − *<sup>m</sup>* refers to the equivalent inputs of the unactuated states.

**K***ud* = diag (*Kud*1, *Kud*2, …, *Kud*(*n* − *m*) ) ∈ *R*(*n* − *m*) × (*n* − *m*) and **K***up* = diag (*Kup*1, *Kup*2, …, *Kup*(*n* − *m*) ) ∈ *R*(*n* − *m*) × (*n* − *m*) are positive matrices.

The control input **U** received from Equations (15) and (21) ensures the stability of the unactuated states **q***u* because the tracking error dynamics, that is,

$$
\ddot{\mathbf{q}}\_{\mu} + \mathbf{K}\_{\text{ud}} \dot{\mathbf{q}}\_{\mu} + \mathbf{K}\_{\text{up}} \mathbf{q}\_{\mu} = \mathbf{0} \tag{22}
$$

is stable for every **K***ud* > **0** and **K***up* > **0**. Hence, if **K***ud* and **K***up* are selected appropriately, then the equivalent inputs **V***u* can drive cargo swings **q***u* toward zero.

To stabilize the unactuated and actuated states, overall equivalent inputs are proposed by linearly combining **V***a* and **V***u* as follows:

$$\begin{split} \mathbf{V} &= \mathbf{V}\_{a} + \mathbf{a} \mathbf{V}\_{u} \\ &= -\mathbf{K}\_{ad} \dot{\mathbf{q}}\_{a} - \mathbf{K}\_{ap} \left( \mathbf{q}\_{a} - \mathbf{q}\_{ad} \right) - \mathbf{a} \left( \mathbf{K}\_{ud} \dot{\mathbf{q}}\_{u} + \mathbf{K}\_{up} \mathbf{q}\_{u} \right)' \end{split} \tag{23}$$

with **α** ∈ *Rm* × (*n* − *m*) being the weighting matrix and **V** ∈ *Rm*.

Hence, considering **q***a* as the primary output, the total control scheme is determined by replacing **V***a* with **V** in Equation (17). By substituting Equation (23) into Equation (17), the nonlinear feedback control structure is obtained as

$$\begin{aligned} \mathbf{U} &= \left( \overline{\mathbf{C}}\_{1}(\mathbf{q}, \dot{\mathbf{q}}) - \overline{\mathbf{M}}(\mathbf{q}) \mathbf{K}\_{\alpha l} \right) \dot{\mathbf{q}}\_{\alpha} + \left( \overline{\mathbf{C}}\_{2}(\mathbf{q}, \dot{\mathbf{q}}) - \overline{\mathbf{M}}(\mathbf{q}) \mathbf{a} \mathbf{K}\_{\alpha l} \right) \dot{\mathbf{q}}\_{\alpha} \\ &- \overline{\mathbf{M}}(\mathbf{q}) \mathbf{K}\_{\alpha p} \left( \mathbf{q}\_{\alpha} - \mathbf{q}\_{\alpha l} \right) - \overline{\mathbf{M}}(\mathbf{q}) \mathbf{a} \mathbf{K}\_{\alpha p} \mathbf{q}\_{\alpha} + \overline{\mathbf{G}}\_{1}(\mathbf{q}) \end{aligned} \tag{24}$$

The nonlinear controller (23) asymptotically stabilizes all system state trajectories, as illustrated in an example presented in Section 5.

## **4. Analysis of system stability**

On the basis of Equation (18) and active dynamics (16), the differential equation of the tracking

where = − is the tracking error vector of the actuated states. Evidently, the dynamics of the tracking error (20) is exponentially stable for every **K***ad* > **0** and **K***ap* > **0**. That is, the tracking errors of the actuated states approach zero (or **q***a* converges to **q***ad*) as *t* becomes infinite. In particular, the equivalent control **V***a* forces the actuated states **q***a* to reach the references **q***ad*

The control scheme (17), which corresponds to the equivalent input **V***a*, is used only to stabilize the actuated states **q***a* asymptotically. To stabilize the unactuated states **q***u*, the nonlinear

are positive matrices.

The control input **U** received from Equations (15) and (21) ensures the stability of the unactu-

is stable for every **K***ud* > **0** and **K***up* > **0**. Hence, if **K***ud* and **K***up* are selected appropriately, then the

To stabilize the unactuated and actuated states, overall equivalent inputs are proposed by

Hence, considering **q***a* as the primary output, the total control scheme is determined by replacing **V***a* with **V** in Equation (17). By substituting Equation (23) into Equation (17), the

being the weighting matrix and **V** ∈ *Rm*.

) ∈ *R*(*n* − *m*) × (*n* − *m*)

feedback technique can be applied to subdynamics (15) as follows:

) ∈ *R*(*n* − *m*) × (*n* − *m*)

ated states **q***u* because the tracking error dynamics, that is,

equivalent inputs **V***u* can drive cargo swings **q***u* toward zero.

**K***ud* = diag (*Kud*1, *Kud*2, …, *Kud*(*n* − *m*)

linearly combining **V***a* and **V***u* as follows:

nonlinear feedback control structure is obtained as

where **V***<sup>u</sup>* ∈ *Rn* − *<sup>m</sup>* refers to the equivalent inputs of the unactuated states.

(20)

(21)

(22)

(23)

and **K***up* = di-

error is obtained as described by

250 Nonlinear Systems - Design, Analysis, Estimation and Control

asymptotically.

ag (*Kup*1, *Kup*2, …, *Kup*(*n* − *m*)

with **α** ∈ *Rm* × (*n* − *m*)

The control law **U** is obtained from the actuated dynamics (14). The stability of the remaining part (the unactuated dynamics) of the closed-loop system, called the internal dynamics, is analyzed. If the internal dynamics is stable, then the tracking control problem is solved. Substituting the control scheme (24) into the unactuated subsystem (15) yields the internal dynamics:

$$\ddot{\mathbf{q}}\_{\mu} = -\mathbf{M}\_{22}^{-1}\left(\mathbf{q}\right) \begin{pmatrix} \left(\mathbf{C}\_{21}\left(\mathbf{q}, \dot{\mathbf{q}}\right) - \mathbf{M}\_{21}\left(\mathbf{q}\right)\mathbf{K}\_{ad}\right)\dot{\mathbf{q}}\_{a} + \left(\mathbf{C}\_{22}\left(\mathbf{q}, \dot{\mathbf{q}}\right) - \mathbf{M}\_{21}\left(\mathbf{q}\right)\mathbf{q}\mathbf{K}\_{ud}\right)\dot{\mathbf{q}}\_{u} \\ -\mathbf{M}\_{21}\left(\mathbf{q}\right)\mathbf{K}\_{ap}\left(\mathbf{q}\_{a} - \mathbf{q}\_{ad}\right) - \mathbf{M}\_{21}\left(\mathbf{q}\right)\mathbf{q}\mathbf{K}\_{up}\mathbf{q}\_{u} + \mathbf{G}\_{2}\left(\mathbf{q}\right) \end{pmatrix} \tag{25}$$

The local stability of the internal dynamics is guaranteed if the zero dynamics is exponentially stable. Setting **q***a* = **q***ad* in the internal dynamics (25), the zero dynamics of the system is obtained as

$$
\ddot{\mathbf{q}}\_{\boldsymbol{u}} + \mathbf{M}\_{22}^{-1}(\mathbf{q}) \left| \left( \mathbf{C}\_{22} \left( \mathbf{q}\_{\boldsymbol{u}}, \dot{\mathbf{q}}\_{\boldsymbol{u}} \right) - \mathbf{M}\_{21} \left( \mathbf{q}\_{\boldsymbol{u}} \right) \mathbf{c} \mathbf{K}\_{\boldsymbol{u}\boldsymbol{d}} \right) \dot{\mathbf{q}}\_{\boldsymbol{u}} - \mathbf{M}\_{21} \left( \mathbf{q}\_{\boldsymbol{u}} \right) \mathbf{c} \mathbf{K}\_{\boldsymbol{u}\boldsymbol{p}} \mathbf{q}\_{\boldsymbol{u}} + \mathbf{G}\_{2} \left( \mathbf{q}\_{\boldsymbol{u}} \right) \right| = \mathbf{0}. \tag{26}
$$

The zero dynamics is expanded into a set of (*n*–*m*) second-order nonlinear differential equations in which the (*n*–*m*) components of vector **q***u* are considered as variables. The stability of the zero dynamics (26) is analyzed using Lyapunov's linearization theorem [4]. By defining 2(*n*–*m*) state variables **z** ∈ *R*2 × (*n* − *m*) , the zero dynamics (26) is converted into state-space form as follows:

$$
\dot{\mathbf{z}} = \mathbf{f}\left(\mathbf{z}\right),
\tag{27}
$$

where **f**(**z**) is a vector of nonlinear functions, and **z** ∈ *R*2 × (*n* − *m*) is a state vector. System dynamics (27) is composed of 2(*n*–*m*) first-order nonlinear differential equations. This nonlinear zero dynamics is asymptotically stable around the equilibrium point **z** = **0** ( = ˙ = ) if the corresponding linearized system is strictly stable. Linearizing the zero dynamics around **z** = **0** yields a linearized system in the following form:

$$
\dot{\mathbf{z}} = \mathbf{A} \mathbf{z}\_r \tag{28}
$$

with

$$\mathbf{A} = \left(\frac{\partial \mathbf{f}}{\partial \mathbf{z}}\right)\_{\mathbf{x}=\mathbf{0}}\tag{29}$$

as a 2(*n*–*m*)×2(*n*–*m*) Jacobian matrix of components ∂*fi* /∂*xj* . The stability of the linear system (28) can be analyzed by considering the positions of the eigenvalues of **A** or using several traditional techniques, such as the Routh-Hurwitz criterion [3], the root locus method, and so on. Thus, by investigating the stability of the linear system (28), we can understand the dynamic behavior of the nonlinear system (27), or equivalently, zero dynamics (26), according to Lyapunov's linearization theorem [4].

*The nonlinear system* (26) *is asymptotically stable around the equilibrium point* ( = ˙ = ) *if the linearized system* (28) *is strictly stable*.

*The equilibrium point* ( = ˙ = ) *of the nonlinear system* (26) *is unstable if the linearized system* (28) *is unstable*.

*We cannot conclude the stability of the nonlinear system* (26) *if the linearized system* (28) *is marginally stable*.

As we will see in the examples provided in Section 5, the analysis of system stability using the aforementioned theorem yields the constraint equations of the controller parameters.
