Grammatikopoulos, L., Karras, G. & Petsa, E. (2004). Camera calibration combining images

Hashimoto, K., Ebine, T. & Kimura, H. (1996). Visual servoing with hand-eye

Hutchinson, S., Hager, G. D. & Corke, P. I. (1996). A tutorial on visual servo control, *IEEE*

Karagiannis, D. & Astolfi, A. (2005). A new solution to the problem of range identification in

Kelly, R. (1999). Regulation of manipulators in generic task space: an energy

a nonlinear benchmarksystem using a saturated passivity-based controller, *IEEE*

for Three-Dimensional Target Tracking: Stability and *L*2-Gain Performance Analysis,

with two vanishing points, *International Archives of the Photogrammetry, Remote Sensing*

manipulator-optimal control approach, *IEEE Transactions on Robotics and Automation*

perspective vision systems, *IEEE Transactions on Automatic Control* 50(12): 2074–2077.

shapingplus damping injection approach, *IEEE Transactions on Robotics and*

systems, *IEEE Transactions on Automatic Control* 48(12): 2232–2238.

*Transactions on Control Systems Technology* 7(2): 289–293.

*IEEE Trans. Robotics and Automation* 8(3): 313–326.

*IEEE Transactions on Control Tech.* 15(1): 40.

*& Spatial Information Sciences* 35(5): 99–104.

*Trans. Robot. Autom.* 12(9): 651.

*Automation* 15(2): 381–386.

to be done. Further works in these directions are undergoing.

129800 and PAPIIT-UNAM Project IN120009.

*Applications* 1(4): 223–240.

12(5): 766–774.

*on Automatic Control* 47(4): 658–663.

Fig. 9. Experiments of the visual servoing: the joint image trajectory and the desired trajectory for joint 1 (up), 2 (middle) and 3 (down).

### **5. Conclusions**

12 Will-be-set-by-IN-TECH

Fig. 9. Experiments of the visual servoing: the joint image trajectory and the desired

trajectory for joint 1 (up), 2 (middle) and 3 (down).

This chapter has presented an IBVS based on passivity of the robot image dynamics obtained by lumping together the manipulator dynamics, forward kinematics and perspective projection the camera. Using the passivity, controller design was considerably simplified. Regulation and tracking performances were illustrated through laboratory experiments.

Although the general basic idea presented in this chapter is applicable to *n*-DOF scenarios, much research works related to more precise state observation of the robot image dynamics in the image space using its perspective projection, and analysis of the robustness issues remain to be done. Further works in these directions are undergoing.

### **6. Acknowledgement**

The authors acknowledge the financial support from CONACyT-Mexico through Project 129800 and PAPIIT-UNAM Project IN120009.

### **7. References**

Kelly, R., Bugarin, E., Cervantes, I. & Alvarez-Ramirez, J. (2006). Monocular direct visual servoing for regulation of manipulators moving in the 3d cartesian space, *Decision and Control, 2006 45th IEEE Conference on*, pp. 1782–1787.

**10**

Recep Burkan

*Turkey*

*Istanbul University, Faculty of Engineering, Mechanical Engineering Department, Istanbul*

**Modelling of Bound Estimation Laws and**

**Robust Controllers for Robot Manipulators**

**Using Functions and Integration Techniques**

Some robust control methods have been developed in the past in order to increase tracking performance in the presence of parametric uncertainties. In the presence of parametric uncertainty, unmodelled dynamics and other sources of uncertainties, robust control laws are used. Corless-Leitmann [1] approach is a popular approach used for designing robust controllers for robot manipulators. In early application of Corless-Leitmann [1] approach to robot manipulators [2, 3], it is difficult to compute uncertainty bound precisely. Because, uncertainty bound on parameters depends on the inertia parameters, the reference trajectory and manipulator state vector. Spong [4] proposed a new robust controller for robot manipulators using the Lyapunov theory that guaranties stability of uncertain systems. In this approach, Leithmann [5] or Corless-Leithman [1] approach is used for designing the robust controller. One of the advantage of Spong's approach [4] is that uncertainty on parameter is needed to derive robust controller and uncertainty bound parameters depends only on the inertia parameters of the robots. Yaz [6] proposed a robust control law based on Spong's study [4] and global exponential stability of uncertain system is guaranteed. However, disturbance and unmodelled dynamics are not considered in algorithm of [4, 6]. Danesh at al [7] develop Spong's approach [4] in such a manner that control scheme is made robust not only to uncertain inertia parameters but also to robust unmodelled dynamics and disturbances. Koo and Kim [8] introduce adaptive scheme of uncertainty bound on parameters for robust control of robot manipulators. In [8], upper uncertainty bound is not known as would be in robust controller [4] and uncertainty bound is estimated with estimation law in order to control the uncertain system. A new robust control approach is proposed by Liu and Goldenerg [9] for robot manipulators based on a decomposition of model uncertainty. Parameterized uncertainty is distinguished from unparameterized uncertainty and a compensator is designed for parameterized and unparameterized uncertainty. A decomposition-based control design framework for mechanical systems with

In order to increases tracking performance of uncertain systems, design of uncertainty bound estimation functions are considered. For this purpose, some uncertainty bound estimation functions are developed [11-15] based on a Lyapunov function, thus, stability of

**1. Introduction**

model uncertainties is proposed by Liu [10].

## **Modelling of Bound Estimation Laws and Robust Controllers for Robot Manipulators Using Functions and Integration Techniques**

Recep Burkan

*Istanbul University, Faculty of Engineering, Mechanical Engineering Department, Istanbul Turkey*

### **1. Introduction**

14 Will-be-set-by-IN-TECH

162 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Kelly, R., Bugarin, E., Cervantes, I. & Alvarez-Ramirez, J. (2006). Monocular direct visual

Kelly, R., Carelli, R., Nasisi, O., Kuchen, B. & Reyes, F. (2000). Stable visual servoing

Ma, Y., Kosecka, J. & Sastry, S. (1999). Vision guided navigation for a nonholonomic mobile

Ortega, R. & Spong, M. (1989). Adaptive motion control of rigid robots: A tutorial, *Automatica*

Slotine, J. & Li, W. (1987). On the Adaptive Control of Robot Manipulators, *The International*

Spong, M., Hutchinson, S. & Vidyasagar, M. (2006). *Robot modeling and control*, John Wiley &

Takegaki, M. & Arimoto, S. (1981). A new feedback method for dynamic control of

Tomei, P. (1991). Adaptive PD controller for robot manipulators, *IEEE Transactions on Robotics*

Weiss, L., Sanderson, A. & Neuman, C. (1987). Dynamic sensor-based control of robots with visual feedback, *Robotics and Automation, IEEE Journal of* 3(5): 404–417. Yuan, J. (1989). A general photogrammetric method for determining object positionand orientation, *IEEE Transactions on Robotics and Automation* 5(2): 129–142.

manipulators, *ASME, Transactions, Journal of Dynamic Systems, Measurement and*

robot, *Robotics and Automation, IEEE Transactions on* 15(3): 521–536.

*and Control, 2006 45th IEEE Conference on*, pp. 1782–1787.

Khalil, H. K. (2002). *Nonlinear Systems*, Prentice-Hall, New Jersey.

*Journal of Robotics Research* 6(3): 49.

5(1): 39–48.

25(6): 877–888.

*Control* 103: 119–125.

*and Automation* 7(4): 565–570.

Sons.

servoing for regulation of manipulators moving in the 3d cartesian space, *Decision*

of camera-in-hand robotic systems, *Mechatronics, IEEE/ASME Transactions on*

Some robust control methods have been developed in the past in order to increase tracking performance in the presence of parametric uncertainties. In the presence of parametric uncertainty, unmodelled dynamics and other sources of uncertainties, robust control laws are used. Corless-Leitmann [1] approach is a popular approach used for designing robust controllers for robot manipulators. In early application of Corless-Leitmann [1] approach to robot manipulators [2, 3], it is difficult to compute uncertainty bound precisely. Because, uncertainty bound on parameters depends on the inertia parameters, the reference trajectory and manipulator state vector. Spong [4] proposed a new robust controller for robot manipulators using the Lyapunov theory that guaranties stability of uncertain systems. In this approach, Leithmann [5] or Corless-Leithman [1] approach is used for designing the robust controller. One of the advantage of Spong's approach [4] is that uncertainty on parameter is needed to derive robust controller and uncertainty bound parameters depends only on the inertia parameters of the robots. Yaz [6] proposed a robust control law based on Spong's study [4] and global exponential stability of uncertain system is guaranteed. However, disturbance and unmodelled dynamics are not considered in algorithm of [4, 6]. Danesh at al [7] develop Spong's approach [4] in such a manner that control scheme is made robust not only to uncertain inertia parameters but also to robust unmodelled dynamics and disturbances. Koo and Kim [8] introduce adaptive scheme of uncertainty bound on parameters for robust control of robot manipulators. In [8], upper uncertainty bound is not known as would be in robust controller [4] and uncertainty bound is estimated with estimation law in order to control the uncertain system. A new robust control approach is proposed by Liu and Goldenerg [9] for robot manipulators based on a decomposition of model uncertainty. Parameterized uncertainty is distinguished from unparameterized uncertainty and a compensator is designed for parameterized and unparameterized uncertainty. A decomposition-based control design framework for mechanical systems with model uncertainties is proposed by Liu [10].

In order to increases tracking performance of uncertain systems, design of uncertainty bound estimation functions are considered. For this purpose, some uncertainty bound estimation functions are developed [11-15] based on a Lyapunov function, thus, stability of

Modelling of Bound Estimation Laws and

0 0r 0 r0

action. The corrected velocity error is given as

bound on parametric uncertainty such that

candidate is defined as [15, 16].

time derivative of V of system (9) is

Equation (12) is arranged as

Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 165

where is a positive definite matrix. Then the following nominal control law is considered:

τ M (q)q C (q,q)q G (q) Kσ

where *π*0Rp represents the fixed parameters in dynamic model and Kσ is the vector of PD

Where u(t) is the additional robust control input. It is assumed that there exists an unknown

Since ρR+p is assumed to be unknown, ρ should be estimated with the estimation law to

 M(q)σ C(q,q)σ Kσ Y(q,q,q ,q )( r r π u(t)) (9) By taking into account above parameters and control algorithm, the Lyapunov function

where BRnxn is a positive diagonal matrix, *(t)* is chosen as a pxp dimensional diagonal

(11)

TT T T T T 2 V -q Kq-q ΛKΛq σ Yu(t) σ Yπ ρ(t) (t) (t)ρ(t) ρ(t) (t) ρ(t) (12)

T T 2

(ρ ρ*ˆ ˆˆˆ* (t)) (t) (t)(ρ ρ(t)) (ρ ρ(t)) (t) ρ(t) 0

 *[ M(q) C(q,q)] n*

(13)

(4)

σ *q qr* q Λq (5)

π πρ 0 (7)

ρ(t) ρ ρ *ˆ* (t) (8)

*R* [17, 18], and taking

0 rr rr 0 τ τ Y(q,q,q ,q )u(t) Y(q,q,q ,q )( π u(t))-Kσ (6)

(10)

rr 0

Y(q,q,q ,q )π Kσ

The control input τ is defined in terms of the nominal control vector 0 τ as

control the system properly. ρ*ˆ* (t) shows the estimate of ρ and ρ(t) is

1 11 T T T2 V( ,q,ρ(t)) σ M(q)σ q Bq ρ(t) (t) ρ(t) 2 22

matrix changes in time. The time derivative of V along the trajectories is

 T 1 T TT T 2 V σ M(q) σ M(q)σ q Bq ρ(t) (t) (t)ρ(t) ρ(t) (t) (t) 2 *= + ++ +*

TT T T

V -q Kq-q ΛKΛ q σ Yu(t) σ Yπ

Substituting (6) into (1) and after some algebra yields

Taking *B K* 2 , using the property - 2 0 *T*

uncertain system is guaranteed. In early derivation of uncertainty bound estimation laws [11-13], only a single derivation is possible because selection of variable function is difficult for other derivation and first order differential equation is used. Only exponential function and logarithmic functions are used for derivations because it is diffucut to define variable functions for other derivations.

In previous studies, some robust control laws are introduced, however, a method for derivation of adaptive bound estimation law for robust controllers is not proposed. Recently, a new approach for derivation of bound estimation laws for robust control of robot manipulators is proposed [14, 15]. A general equation is developed based on the Lyapunov theory in order to derive adaptive bound estimation laws and stability of uncertain system is guaranteed. In the approach [15], some functions depending on robot kinematics and control parameters and proper integration techniques can be used for derivation of new bound estimation laws. Then, new bound estimation laws are derived and this derivations also show how the general rule can be used for derivation of different bound estimation laws. After that, four new robust controllers are designed based on each bound estimation law. Lyapunov theory based on Corless-Leitmann [1] approach is used and uniform boundedness error convergence is achieved. This study also shows that bound estimation laws for robust control input do not only include these derivations but also allows derivation of other bound estimation laws for robust controllers provided that appropriate function and proper integration techniques are chosen. In this work, based on the study [15], some appropriate functions are developed and proper integration techniques are chosen. As results, new uncertainty bound estimation laws for robust control input are developed and new robust controllers are proposed. In derivations, some functions and integration techniques are used.

### **2. A method for derivation of bound estimation laws**

In the absence of friction or other disturbances, the dynamic model of an n-link manipulator can be written as [16]

$$\mathbf{M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) = \tau} \tag{1}$$

where q denotes generalised coordinates, τ is the n-dimensional vector of applied torques (or forces), *M(q)* is a positive definite mass matrix, C(q,q)q is the n-dimensional vector of centripetal and Coriolis terms and G(q) is the n-dimensional vector of gravitational terms. Equation (1) can also be expressed in the following form.

$$
\pi = \text{Y}(\mathbf{q}\_l \dot{\mathbf{q}}\_l \ddot{\mathbf{q}}\_l) \text{n} \tag{2}
$$

where *π* is a p-dimensional vector of robot inertia parameters and Y is an nxp matrix which is a function of joint position, velocity and acceleration. For any specific trajectory, the desired position, velocity and acceleration vectors are qd, *d q* and *d q* . The measured actual position and velocity errors are *d qqq* , and *d qqq* . Using the above information, the corrected desired velocity and acceleration vectors for nonlinearities and decoupling effects are proposed as:

$$
\dot{\mathbf{q}}\_r = \dot{\mathbf{q}}\_d - \Lambda \tilde{\mathbf{q}} \; ; \; \ddot{\mathbf{q}}\_r = \ddot{\mathbf{q}}\_d - \Lambda \dot{\tilde{\mathbf{q}}} \tag{3}
$$

164 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

uncertain system is guaranteed. In early derivation of uncertainty bound estimation laws [11-13], only a single derivation is possible because selection of variable function is difficult for other derivation and first order differential equation is used. Only exponential function and logarithmic functions are used for derivations because it is diffucut to define variable

In previous studies, some robust control laws are introduced, however, a method for derivation of adaptive bound estimation law for robust controllers is not proposed. Recently, a new approach for derivation of bound estimation laws for robust control of robot manipulators is proposed [14, 15]. A general equation is developed based on the Lyapunov theory in order to derive adaptive bound estimation laws and stability of uncertain system is guaranteed. In the approach [15], some functions depending on robot kinematics and control parameters and proper integration techniques can be used for derivation of new bound estimation laws. Then, new bound estimation laws are derived and this derivations also show how the general rule can be used for derivation of different bound estimation laws. After that, four new robust controllers are designed based on each bound estimation law. Lyapunov theory based on Corless-Leitmann [1] approach is used and uniform boundedness error convergence is achieved. This study also shows that bound estimation laws for robust control input do not only include these derivations but also allows derivation of other bound estimation laws for robust controllers provided that appropriate function and proper integration techniques are chosen. In this work, based on the study [15], some appropriate functions are developed and proper integration techniques are chosen. As results, new uncertainty bound estimation laws for robust control input are developed and new robust controllers are proposed. In derivations, some functions and

In the absence of friction or other disturbances, the dynamic model of an n-link manipulator

where q denotes generalised coordinates, τ is the n-dimensional vector of applied torques (or forces), *M(q)* is a positive definite mass matrix, C(q,q)q is the n-dimensional vector of centripetal and Coriolis terms and G(q) is the n-dimensional vector of gravitational terms.

where *π* is a p-dimensional vector of robot inertia parameters and Y is an nxp matrix which is a function of joint position, velocity and acceleration. For any specific trajectory, the

position and velocity errors are *d qqq* , and *d qqq* . Using the above information, the corrected desired velocity and acceleration vectors for nonlinearities and decoupling effects

*r d qq q* ; q q *r d q*

M(q)q C(q,q)q G(q) *τ* (1)

Y(q,q,q ) π (2)

(3)

*q* . The measured actual

functions for other derivations.

integration techniques are used.

can be written as [16]

are proposed as:

**2. A method for derivation of bound estimation laws**

Equation (1) can also be expressed in the following form.

desired position, velocity and acceleration vectors are qd, *d q* and *d*

where is a positive definite matrix. Then the following nominal control law is considered:

$$\begin{aligned} \mathbf{r}\_0 &= \mathbf{M}\_0(\mathbf{q})\ddot{\mathbf{q}}\_\mathbf{r} + \mathbf{C}\_0(\mathbf{q}, \dot{\mathbf{q}})\dot{\mathbf{q}}\_\mathbf{r} + \mathbf{G}\_0(\mathbf{q}) - \mathbf{K}\sigma\\ &= \mathbf{Y}(\mathbf{q}, \dot{\mathbf{q}}\_\mathbf{r}\dot{\mathbf{q}}\_\mathbf{r}, \ddot{\mathbf{q}}\_\mathbf{r})\mathbf{r}\_0 - \mathbf{K}\sigma \end{aligned} \tag{4}$$

where *π*0Rp represents the fixed parameters in dynamic model and Kσ is the vector of PD action. The corrected velocity error is given as

$$
\sigma \mathbf{o} = \mathbf{q} - \dot{\mathbf{q}}\_r = \dot{\mathbf{\tilde{q}}} + \Lambda \mathbf{\tilde{q}} \tag{5}
$$

The control input τ is defined in terms of the nominal control vector 0 τ as

$$\mathbf{r} = \mathbf{r}\_0 + \mathbf{Y} (\mathbf{q}\_l \dot{\mathbf{q}}\_l \dot{\mathbf{q}}\_r, \ddot{\mathbf{q}}\_r) \mathbf{u}(\mathbf{t}) = \mathbf{Y} (\mathbf{q}\_l \dot{\mathbf{q}}\_l \dot{\mathbf{q}}\_r, \ddot{\mathbf{q}}\_r) (\mathbf{n}\_0 + \mathbf{u}(\mathbf{t})) \text{-K}\mathbf{o} \tag{6}$$

Where u(t) is the additional robust control input. It is assumed that there exists an unknown bound on parametric uncertainty such that

$$
\tilde{\pi} = \mathbf{n}\_0 - \mathbf{n} \le \mathbf{p} \tag{7}
$$

Since ρR+p is assumed to be unknown, ρ should be estimated with the estimation law to control the system properly. ρ*ˆ* (t) shows the estimate of ρ and ρ(t) is

$$
\tilde{\boldsymbol{\phi}}(\mathbf{t}) = \mathbf{p} - \hat{\boldsymbol{\phi}}(\mathbf{t}) \tag{8}
$$

Substituting (6) into (1) and after some algebra yields

$$\mathbf{M(q)\dot{\sigma} + C(q,\dot{q})\sigma + K\sigma = Y(q,\dot{q},\dot{q}\_r,\ddot{q}\_r)(\tilde{\mathbf{u}} + \mathbf{u}(t))}\tag{9}$$

By taking into account above parameters and control algorithm, the Lyapunov function candidate is defined as [15, 16].

$$\mathbf{V}(\sigma, \mathbf{q}, \tilde{\mathbf{p}}(\mathbf{t})) = \frac{1}{2} \sigma^{\mathrm{T}} \mathbf{M}(\mathbf{q}) \sigma + \frac{1}{2} \tilde{\mathbf{q}}^{\mathrm{T}} \mathbf{B} \tilde{\mathbf{q}} + \frac{1}{2} \tilde{\mathbf{p}}(\mathbf{t})^{\mathrm{T}} \Phi(\mathbf{t})^{2} \tilde{\boldsymbol{\rho}}(\mathbf{t}) \tag{10}$$

where BRnxn is a positive diagonal matrix, *(t)* is chosen as a pxp dimensional diagonal matrix changes in time. The time derivative of V along the trajectories is

$$\dot{\mathbf{V}} = \mathbf{o}^{\mathrm{T}} \mathbf{M}(\mathbf{q}) \dot{\boldsymbol{\sigma}} + \frac{1}{2} \mathbf{o}^{\mathrm{T}} \dot{\mathbf{M}}(\mathbf{q}) \mathbf{o} + \tilde{\mathbf{q}}^{\mathrm{T}} \mathbf{B} \dot{\tilde{\mathbf{q}}} + \tilde{\mathbf{p}}(\mathbf{t})^{\mathrm{T}} \boldsymbol{\Phi}(\mathbf{t}) \dot{\boldsymbol{\Phi}}(\mathbf{t}) \tilde{\boldsymbol{\rho}}(\mathbf{t}) + \tilde{\boldsymbol{\rho}}(\mathbf{t})^{\mathrm{T}} \boldsymbol{\Phi}(\mathbf{t})^{2} \dot{\tilde{\boldsymbol{\rho}}}(\mathbf{t}) \tag{11}$$

Taking *B K* 2 , using the property - 2 0 *T [ M(q) C(q,q)] n R* [17, 18], and taking time derivative of V of system (9) is

$$\dot{\mathbf{V}} = \dot{\mathbf{q}}^{\mathrm{T}} \mathbf{K} \dot{\mathbf{q}} \cdot \mathbf{\tilde{q}}^{\mathrm{T}} \Lambda \mathbf{K} \Lambda \mathbf{\tilde{q}} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \mathbf{u}(\mathbf{t}) + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \mathbf{\tilde{u}} + \tilde{\rho}(\mathbf{t})^{\mathrm{T}} \Phi(\mathbf{t}) \dot{\Phi}(\mathbf{t}) \tilde{\rho}(\mathbf{t}) + \tilde{\rho}(\mathbf{t})^{\mathrm{T}} \Phi(\mathbf{t})^{2} \dot{\tilde{\rho}}(\mathbf{t}) \tag{12}$$

Equation (12) is arranged as

$$\begin{split} \dot{\mathbf{V}} &= -\dot{\mathbf{q}}^{\mathrm{T}} \mathbf{K} \dot{\mathbf{q}} \cdot \mathbf{\tilde{q}}^{\mathrm{T}} \Lambda \mathbf{K} \Lambda \ \ddot{\mathbf{q}} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \mathbf{u}(\mathbf{t}) + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \tilde{\mathbf{n}} \\ &+ \left( \boldsymbol{\rho} - \dot{\boldsymbol{\rho}}(\mathbf{t}) \right)^{\mathrm{T}} \boldsymbol{\Phi}(\mathbf{t}) \dot{\boldsymbol{\Phi}}(\mathbf{t}) (\boldsymbol{\rho} - \dot{\boldsymbol{\rho}}(\mathbf{t})) - \left( \boldsymbol{\rho} - \dot{\boldsymbol{\rho}}(\mathbf{t}) \right)^{\mathrm{T}} \boldsymbol{\Phi}(\mathbf{t})^{2} \dot{\boldsymbol{\hat{\rho}}}(\mathbf{t}) \le 0 \end{split} \tag{13}$$

*(t) (t) ˆ* (since is a constant). Remembering that and if u(t) is taken as the estimated term of uncertainty bound, that is u(t) ρ*ˆ* (t) then Equation (13 is written as

$$\begin{aligned} \dot{\mathbf{V}} &= -\dot{\mathbf{q}}^{\mathrm{T}} \mathbf{K} \dot{\mathbf{q}} \cdot \dot{\mathbf{q}}^{\mathrm{T}} \Lambda \mathbf{K} \Lambda \mathbf{q} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} (-\hat{\rho}(\mathbf{t})) \; \mathbf{+} \mathbf{o}^{\mathrm{T}} \mathbf{Y} \boldsymbol{\rho} \\ &+ [\boldsymbol{\rho} - \hat{\rho}(\mathbf{t})]^{\mathrm{T}} \boldsymbol{\Phi}(\mathbf{t}) \dot{\boldsymbol{\Phi}}(\mathbf{t}) [\boldsymbol{\rho} - \hat{\rho}(\mathbf{t})] - (\boldsymbol{\rho} - \hat{\rho}(\mathbf{t}))^{\mathrm{T}} \boldsymbol{\Phi}(\mathbf{t})^{\mathrm{2}} \; \dot{\boldsymbol{\rho}}(\mathbf{t})] \leq 0 \end{aligned} \tag{14}$$

Modelling of Bound Estimation Laws and

**2.1 First choice of (t)-1** For the first derivation of

Then

After integration, the result is

derived from Lyapunov function. As a result,

1 1 1

1

2

e e

p

Substituting Equation (24) into (23) yields

guarantess stability of uncertain system. However, (t)-1 and

search for appropriate function of (t)-1 as a solution of the Equation (23).

Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 167

The Equation (23) is a general equation for derivation of the bound estimation law and it is

derived depending on the function (t)-1. For derivation, selection of (t)-1 and integration techniques are very important. There is no certain rule for selection of (t)-1 and integration techniques for this systems. System state parameters and mathematical insight are used to

> T T 1 Y σdt)i i Y σdt) e e *i i ( ( i (t) diag( sin( ))*

T T 11 11 T T 22 22

e e e e

*sin( )(Y ) ˆ(t)*

e e *p p*

*ˆ(t) sin( )(Y ) (t) [ dt] ..... ... ................*

*p ( ( T*

*( / )cos( ) ˆ(t)*

*ˆ(t) ( / )cos( )*

*ˆ(t) sin( )(Y )*

Y σdt) Y σdt) 1 1 1 1 Y σdt) Y σdt) 2 1 2 2 2

*( ( T ( ( T*

T T p p

*p p*

Y σdt) 2 1 1 2 2 2

*ˆ(t) ( / )cos( ) (t) (t)C ..... ..... ..... ................*

*( (*

e *p*

T T p p

2 Y σdt) Y σdt) Y σdt)

*(( ( (( (*

Y σdt) Y σdt)

1 1 1 1

*p ( p p p*

2 Y σdt) Y σdt) Y σdt) 2 2 2

/ )e e e / )e e e

2 Y σdt) Y σdt)

*ˆ(t) ( sin(* T p

*sin( )*

*( sin( )cos( ) ˆ(t) ˆ(t) ( sin( )cos( )*

/ )e e

*p p*

*sin( ) C .....*

T T 11 11 T T 22 22

Y σdt) Y σdt)

Y σdt) Y σdt)

T T p p

*sin( )*

Y σdt) Y σdt)

*p p*

*p ( ( p p*

e e

*( ( ( (*

e e

*( (*

*..... ................*

*ˆ(t)* , 1 *(t)* is chosen as a time varying function such that

1

T 1 1 T 2 2

Y σdt)

T p

TT T 11 11 11 TT T 22 22 22

Y σdt)

e 1 e 1

>

e

*p*

*)cos( )*

Y σdt)

*( p*

(24)

*ˆ(t)* all derived from Equation (23)

*ˆ(t)* are unknown and

*ˆ(t)* is

1 1

(25)

(26)

(27)

*p* 1 *(t) .. .....*

1

> 1 2

*.....*

Equation (14) can be arranged as

$$\begin{aligned} \dot{\mathbf{V}} &= -\dot{\mathbf{q}}^{\mathrm{T}} \mathbf{K} \dot{\mathbf{q}} \cdot \dot{\mathbf{q}}^{\mathrm{T}} \Lambda \mathbf{K} \Lambda \, \mathbf{q} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} [\mathbf{p} \cdot \dot{\mathbf{p}}(t)] \\ &+ [\mathbf{b} \cdot \dot{\mathbf{p}}(t)]^{\mathrm{T}} \Phi(t) \dot{\Phi}(t) [\mathbf{b} \cdot \dot{\mathbf{p}}(t)] - (\mathbf{p} \cdot \dot{\mathbf{p}}(t))^{\mathrm{T}} \Phi(t)^{2} \dot{\hat{\rho}}(t)] \le 0 \end{aligned} \tag{15}$$

Consequently, a suitable expression for the time derivative of V is obtained.

$$\dot{\mathbf{V}} = \dot{\mathbf{q}}^{\mathrm{T}} \mathbf{K} \dot{\mathbf{q}} - \dot{\mathbf{q}}^{\mathrm{T}} \Lambda \mathbf{K} \Lambda \mathbf{\tilde{q}} + \left[ (\boldsymbol{\rho} - \dot{\boldsymbol{\rho}}(\mathbf{t}))^{\mathrm{T}} \right] \mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma} + \Phi(\mathbf{t}) \dot{\Phi}(\mathbf{t}) (\boldsymbol{\rho} - \dot{\boldsymbol{\rho}}(\mathbf{t})) - \Phi(\mathbf{t})^{2} \dot{\boldsymbol{\rho}}(\mathbf{t}) \Big] \leq 0 \tag{16}$$

where T T -q Kq-q ΛKΛq 0 . If the rest of Equation (16) is zero, system will be stable. Remaining terms in Equation (16) are

$$\left[ \left[ (\rho \cdot \dot{\rho}(t)) \right]^T \right] \mathbf{Y}^T \mathbf{o} + \Phi(t) \dot{\Phi}(t) (\rho - \dot{\rho}(t)) - \Phi(t)^2 \dot{\rho}(t) \right] = 0 \tag{17}$$

[(ρ-ρ*ˆ* (t)]is considered as a common multiplier then

$$\mathbf{Y}^T \mathbf{o} + \Phi(\mathbf{t})\dot{\Phi}(\mathbf{t})(\mathbf{o} - \hat{\mathbf{p}}(\mathbf{t})) - \Phi(\mathbf{t})^2 \dot{\hat{\mathbf{p}}}(\mathbf{t}) = \mathbf{0} \tag{18}$$

Hence, we look for the conditions for which the equation

T 2 Y σ (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) ρ(t) 0

is satisfied. Equation (18) can be written as

$$
\Phi(\mathbf{t})\dot{\Phi}(\mathbf{t})(\mathbf{b}-\dot{\mathbf{p}}(\mathbf{t}))-\Phi(\mathbf{t})\Phi(\mathbf{t})\dot{\hat{\mathbf{p}}}(\mathbf{t})=-\mathbf{Y}^{\mathrm{T}}\mathbf{a}\tag{19}
$$

Then

$$
\Phi(\mathbf{t})\dot{\hat{\boldsymbol{\phi}}}(\mathbf{t}) + \dot{\Phi}(\mathbf{t})\dot{\hat{\boldsymbol{\phi}}}(\mathbf{t}) = \Phi(\mathbf{t})^{-1}\mathbf{Y}^{\mathrm{T}}\mathbf{o} + \dot{\Phi}(\mathbf{t})\mathbf{b} \tag{20}
$$

Equation (20) is arranged as

$$\frac{d}{dt}(\Phi(\mathbf{t})\hat{\boldsymbol{\phi}}(\mathbf{t})) = \Phi(\mathbf{t})^{-1}\mathbf{Y}^{\mathrm{T}}\boldsymbol{\alpha} + \dot{\Phi}(\mathbf{t})\boldsymbol{\phi} \tag{21}$$

Integration both side of Equation (21) yields

$$\Phi(\mathbf{t})\hat{\phi}(\mathbf{t}) = \int \Phi(\mathbf{t})^{-1} \mathbf{Y}^{\mathrm{T}} \mathbf{c} dt + \int \dot{\Phi}(\mathbf{t}) \mathbf{p} d\mathbf{t} + \mathbf{C} \tag{22}$$

Then, a general equation for derivation derivation of bound estimation law is developed as [14, 15]

$$\hat{\phi}(\mathbf{t}) = \Phi(\mathbf{t})^{-1} \{ \left[ \Phi(\mathbf{t})^{\cdot 1} \mathbf{Y}^{\Gamma} \mathbf{c} dt \right] + \rho + \Phi(\mathbf{t})^{\cdot 1} \mathbf{C} \tag{23}$$

The Equation (23) is a general equation for derivation of the bound estimation law and it is derived from Lyapunov function. As a result, *ˆ(t)* all derived from Equation (23) guarantess stability of uncertain system. However, (t)-1 and *ˆ(t)* are unknown and *ˆ(t)* is derived depending on the function (t)-1. For derivation, selection of (t)-1 and integration techniques are very important. There is no certain rule for selection of (t)-1 and integration techniques for this systems. System state parameters and mathematical insight are used to search for appropriate function of (t)-1 as a solution of the Equation (23).

### **2.1 First choice of (t)-1**

166 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

T T 2

T T 2

*ˆ ˆ ˆˆ ˆ*

T T T T 2 V -q Kq-q ΛKΛq [( ρ ρ*ˆ*(t)] [Y σ (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) ρ(t)] 0 (16)

T T 2 [(ρ-ρ*ˆ* (t)] [Y σ (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) ρ(t)] 0 (17)

T 2 Y σ (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) ρ(t) 0 (18)

T (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) (t)ρ(t) Y σ (19)

*ˆ ˆ*ρ(t) (t) Y σ (t)ρ (20)

*ˆ* (21)

(t) C (23)

*ˆ*

[ρ ρ(t)] (t) (t)[ρ ρ(t)] (ρ ρ(t)) (t) ρ(t)] 0

[ρ-ρ(t)] (t) (t)[ρ-ρ(t)] (ρ -ρ(t)) (t) (t)] 0

where T T -q Kq-q ΛKΛq 0 . If the rest of Equation (16) is zero, system will be stable.

T 2 Y σ (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) ρ(t) 0

d -1 T (t) (t)) (t) Y σ (t)ρ dt*(*

Then, a general equation for derivation derivation of bound estimation law is developed as

*ˆ ˆˆ ˆ*

estimated term of uncertainty bound, that is u(t) ρ*ˆ* (t) then Equation (13 is written as

TT T T

V -q Kq-q ΛKΛq σ Y( ρ(t)) σ Yρ

TT T

V -q Kq-q ΛKΛq σ Y[ρ-ρ(t)]

Consequently, a suitable expression for the time derivative of V is obtained.

(15)

and if u(t) is taken as the

(14)

*ˆ* (since is a constant). Remembering that

Then

[14, 15]

*(t) (t)*

Equation (14) can be arranged as

Remaining terms in Equation (16) are

[(ρ-ρ*ˆ* (t)]is considered as a common multiplier then

is satisfied. Equation (18) can be written as

Integration both side of Equation (21) yields

Equation (20) is arranged as

Hence, we look for the conditions for which the equation

For the first derivation of *ˆ(t)* , 1 *(t)* is chosen as a time varying function such that

$$\Phi(\mathbf{t})^{-1} = \text{diag}(\mathcal{J}\_{\mathbf{i}} \mathbf{e}^{(a\_i \int \mathbf{Y}^{\mathbb{T}} \text{odt} \mathbf{t})\_i} \sin(\mathbf{e}^{(a\_i \int \mathbf{Y}^{\mathbb{T}} \text{odt} \mathbf{t})\_i})) \tag{24}$$

Substituting Equation (24) into (23) yields

$$
\begin{bmatrix}
\hat{\rho}(t)\_{1} \\
\hat{\rho}(t)\_{2} \\
\dots \\
\hat{\rho}(t)\_{r}
\end{bmatrix} = \Phi(t)^{-1} \left| \begin{bmatrix}
\rho\_{1}\mathbf{e}^{(a\_{1}\big[\mathbf{Y}^{\top}\mathbf{c}\mathbf{d}\mathbf{t})\_{1}} \sin(\mathbf{e}^{(a\_{1}\big[\mathbf{Y}^{\top}\mathbf{c}\mathbf{d}\mathbf{t})\_{1}}) \mathbf{Y}^{T}\sigma \\
\rho\_{2}\mathbf{e}^{(a\_{2}\big[\mathbf{Y}^{\top}\mathbf{c}\mathbf{d}\mathbf{t})\_{2}} \sin(\mathbf{e}^{(a\_{2}\big[\mathbf{Y}^{\top}\mathbf{c}\mathbf{d}\mathbf{t})\_{2}}) \mathbf{Y}^{T}\sigma \\
\dots \\
\dots \\
\end{bmatrix} \begin{bmatrix}
\rho\_{1} \\
\rho\_{2} \\
\dots \\
\dots \\
\rho\_{p}
\end{bmatrix} + \Phi(t)^{-1} \begin{bmatrix} 1 \\ 1 \\ \dots \\
1 \end{bmatrix} \right| \tag{25}
$$

After integration, the result is

$$
\begin{bmatrix}
\hat{\rho}(t)\_1\\ \hat{\rho}(t)\_2\\ \dots\\ \hat{\rho}(t)\_p
\end{bmatrix} = \Phi(t)^{-1} \begin{bmatrix}
\end{bmatrix} + \begin{bmatrix}
\rho\_1\\ \rho\_2\\ \dots\\ \rho\_p
\end{bmatrix} + \Phi(t) \mathbf{C}^{-1} \begin{bmatrix} 1\\ 1\\ 1\\ \dots\\ 1 \end{bmatrix} \tag{26}
$$

Then

TT T 11 11 11 TT T 22 22 22 T T p p 2 Y σdt) Y σdt) Y σdt) 1 1 1 2 Y σdt) Y σdt) Y σdt) 2 2 2 2 Y σdt) Y σdt) / )e e e / )e e e / )e e *p p (( ( (( ( p ( ( p p ( sin( )cos( ) ˆ(t) ˆ(t) ( sin( )cos( ) ..... ................ ˆ(t) ( sin(* T p T T 11 11 T T 22 22 T T p p 1 2 Y σdt) Y σdt) Y σdt) 1 Y σdt) Y σdt) 2 Y σdt) Y σdt) p e e e e e e e *p p p ( p ( ( ( ( ( ( ..... )cos( ) sin( ) sin( ) C ..... sin( )* (27)

If *ˆ( )* 0 is taken as initial condition, constant C is equivalent to Cos(1). So, the estimation law for the uncertainty bound is derived as.

TT T 11 11 11 TT T 22 22 22 T T p p 2 Y σdt) Y σdt) Y σdt) 1 1 1 2 Y σdt) Y σdt) Y σdt) 2 2 2 2 Y σdt) Y σdt) / )e e e / )e e e / )e e *p p (( ( (( ( p ( ( p p ( sin( )cos( ) ˆ(t) ˆ(t) ( sin( )cos( ) ..... ................ ˆ(t) ( sin(* T p T T 11 11 T T 22 22 T T p p 1 2 Y σdt) Y σdt) Y σdt) 1 Y σdt) Y σdt) 2 Y σdt) Y σdt) p e e e e e 1 e e *p p p ( p ( ( ( ( ( ( ..... )cos( ) sin( ) sin( ) Cos( ) ..... sin( )* (28)

### **2.2 Second choice of (t)-1**

For the second derivation of *ˆ(t)* , 1 *(t)* is defined as

$$\Phi(t)^{-1} = \operatorname{diag}(\beta\_i \frac{\mathbf{e}^{\left[\left(a\_i \mathbf{Y}^\top \text{cdt}\right)\_i\right]}}{1 + \mathbf{e}^{\left(\left[\frac{2a\_i \mathbf{Y}^\top \text{cdt}}{\mathbf{c}}\right]\_i\right)}}\tag{29}$$

Modelling of Bound Estimation Laws and

2

*p*

*ˆ(t)*

If 0 *ˆ( )*

*.....*

Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 169

e

( Y σdt) 1 2 Y σdt)

1 ( Y σdt) 2 2 2 Y σdt)

1 ( Y σdt) 2 2 2 Y σdt)

1 1 e

e 1 e e 1 e

*(*

*(*

*.....*

e 1 e *p*

*p (*

T p

(32)

2 Y σdt)

( Y σdt) 1 2 Y σdt)

e 1 e e

*(*

*(*

*.....*

(35)

e 1 e *p*

*p (*

T

2 Y σdt)

T p

(33)

T p p T p p

( Y σdt) ( Y σdt)

p p T p p

*T T*

>

> >

*p*

( Y σdt) ( Y σdt)

*arctan( ) .....*

*p*

 i i ii i Sin ( Y σdt) Cos( Y σdt) *)* (34)

 *p*

*)*

*C .....*

1 1

is taken as initial condition, constant C is equivalent to -arctan(1). So, the

*)*

e

e

T 1 1 T 1 1

T 2 2 T 2 2

T 1 1

*( /) arctan( )*

( Y σdt) 2 ( Y σdt)

2 ( Y σdt) 2 2 2 2 Y σdt)

*ˆ(t) (/) arctan( )*

1 1 2 Y σdt)

*(*

*(*

1 e e

1 e

e

1 e *p*

estimation law for the uncertainty bound is derived as.

1 1 2 Y σdt)

*(*

*(*

1 e e

1 e

e

1 e *p*

e

1 ( Y σdt)

*......... ˆ(t)*

*p p (*

2

**2.3 Third choice of (t)-1** For the third derivation of

After integration, the result is

*p*

*ˆ(t)*

*.....*

e

1 ( Y σdt)

*......... ˆ(t)*

*p p (*

T 2 2

T p

T 1 1 T

T 2 2 T 2 2

e

e

e

*ˆ(t)* , 1 *(t)* is defined as

*(Y ) ˆ(t)*

1

1

*ˆ(t) (Y )*

1 2T T *(t) diag(*

2 T T 1 1 11 11 1 1 2 T T 2 1 2 2 22 22 2

Sin ( Y σdt) Cos( Y σdt) Sin ( Y σdt) Cos( Y σdt)

2 T T p p pp p

*ˆ(t) (Y ) (t) [ dt] ..... ..... .........*

Sin ( Y σdt) Cos( Y σdt)

*p T p*

T p p

2 Y σdt)

( Y σdt)

*(/) arctan(*

T 1 1

*( /) arctan( )*

( Y σdt) 2 ( Y σdt)

T 2 2

T p p T p

2 Y σdt)

( Y σdt)

*(/) arctan(*

Substitution of Equation (34) into Equation (23) yields

1

*(t) C .....*

1

2 ( Y σdt) 2 2 2 2 Y σdt)

*ˆ(t) (/) arctan( )*

Substituting Equation (29) into (23) yields

$$
\begin{bmatrix}
\hat{\rho}(t)\_1\\\hat{\rho}(t)\_2\\\dots\\\hat{\rho}(t)\_p
\end{bmatrix} = \Phi(t)^{-1} \begin{bmatrix}
\boldsymbol{\theta} \\
1 + \frac{\mathbf{e}^{\left\|(\boldsymbol{\alpha}\_1 \mathbf{Y}^\top \boldsymbol{\mathrm{ad}})\right\|\_1}{1 + \mathbf{e}^{\left\|(\boldsymbol{\alpha}\_2 \mathbf{Y}^\top \boldsymbol{\mathrm{ad}})\right\|\_2}}(\boldsymbol{Y}^T \boldsymbol{\sigma})\_1\\\boldsymbol{\theta} \frac{\mathbf{e}^{\left\|(\boldsymbol{\alpha}\_2 \mathbf{Y}^\top \boldsymbol{\mathrm{ad}})\right\|\_2}{1 + \mathbf{e}^{\left\|(\boldsymbol{\alpha}\_2 \mathbf{Y}^\top \boldsymbol{\mathrm{ad}})\right\|\_2}}(\boldsymbol{Y}^T \boldsymbol{\sigma})\_2\\\dots}
\dots
\dots
\end{bmatrix} \begin{bmatrix}
\boldsymbol{\rho}\_1\\\boldsymbol{\rho}\_2\\\vdots\\\boldsymbol{\rho}\_p\\\vdots\\\boldsymbol{\rho}\_p
\end{bmatrix} + \Phi(t)^{-1} \mathbf{C} \begin{bmatrix}
1\\1\\\vdots\\\ldots\\1
\end{bmatrix} \tag{30}
$$

After integration, the result is

$$
\begin{bmatrix}
\hat{\rho}(t)\_1\\ \hat{\rho}(t)\_2\\ \dots\\ \hat{\rho}(t)\_p
\end{bmatrix} = \Phi(t)^{-1} \begin{bmatrix}
(\beta\_1 / \alpha\_1) \arctan(\mathbf{e}^{\left[ (a\_1 \mathbf{Y}^\top \text{cdt})\_1 \right]} \\
(\beta\_2 / \alpha\_2) \arctan(\mathbf{e}^{\left[ (a\_2 \mathbf{Y}^\top \text{cdt})\_2 \right]} \\
\dots \\
(\beta\_p / \alpha\_p) \arctan(\mathbf{e}^{\left[ (a\_p \mathbf{Y}^\top \text{cdt})\_p \right]}
\end{bmatrix} + \Phi(t) \begin{bmatrix}
\rho\_1 \\ \rho\_2 \\ \dots \\ \rho\_p
\end{bmatrix} + C \begin{bmatrix} 1 \\ 1 \\ \dots \\ 1 \end{bmatrix} \tag{31}
$$

After multiplication by 1 *(t)* , the result will be

T 1 1 T 1 1 T 1 1 T 2 2 T 2 2 T 2 2 T p p T p ( Y σdt) 2 ( Y σdt) 1 1 2 Y σdt) 1 ( Y σdt) 2 ( Y σdt) 2 2 2 2 Y σdt) ( Y σdt) 2 2 Y σdt) e e 1 e e e 1 e e e 1 e *p ( ( p p p ( ( /) arctan( ) ˆ(t) ˆ(t) (/) arctan( ) ..... .........  )* (32)

If 0 *ˆ( )* is taken as initial condition, constant C is equivalent to -arctan(1). So, the estimation law for the uncertainty bound is derived as.

T 1 1 T 1 1 T 1 1 T 2 2 T 2 2 T 2 2 T p p T p ( Y σdt) 2 ( Y σdt) 1 1 2 Y σdt) 1 ( Y σdt) 2 ( Y σdt) 2 2 2 2 Y σdt) ( Y σdt) 2 2 Y σdt) e e 1 e e e 1 e e e 1 e *p ( ( p p p ( ( /) arctan( ) ˆ(t) ˆ(t) (/) arctan( ) ..... ......... ˆ(t) (/) arctan(* T 1 1 T 1 1 T 2 2 T 2 2 T p p T p p T p ( Y σdt) 1 2 Y σdt) 1 ( Y σdt) 2 2 2 Y σdt) ( Y σdt) ( Y σdt) 2 Y σdt) e 1 e e 1 1 e e 1 e *p ( ( p p ( arctan( ) ..... ..... )* (33)

### **2.3 Third choice of (t)-1**

168 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

T T p p

T T p p

*sin( )*

1 e

*T*

*(Y )*

*T*

1

*T*

*(Y )*

( Y σdt)

T p p

( Y σdt)

*i i ( (t) diag( )*

Y σdt) Y σdt)

T T 11 11 T T 22 22

*p p*

*ˆ(t)* , 1 *(t)* is defined as

( Y σdt) 1 1 2 Y σdt)

e 1 e e 1 e

*(*

*(*

*ˆ(t) (Y ) (t) [ dt] ..... .....*

e 1 e *p*

*p p*

*ˆ(t) ( / )arctan( )*

*( / )arctan( ) ˆ(t)*

After multiplication by 1 *(t)* , the result will be

*......... ˆ(t)*

1 1 ( Y σdt) 2 1 2 2 2 2 Y σdt)

T p p

( Y σdt) 2 Y σdt)

T p

*p p (*

1 1 1 1 ( Y σdt) 2 1 2 2 2

*ˆ(t) ( / )arctan( ) (t) (t) C ..... ..... ..... .........*

e *p p*

Y σdt) Y σdt)

Y σdt) Y σdt)

*sin( )*

2 Y σdt) Y σdt) Y σdt) 2 2 2

/ )e e e / )e e e

2 Y σdt) Y σdt)

e e

*sin( ) Cos( ) .....*

*( ( ( (*

e e

1

e

*( (*

*ˆ(t) ( sin(* T p

*( sin( )cos( ) ˆ(t) ˆ(t) ( sin( )cos( )*

/ )e e

*p p*

1

2

e e 1

p

*p ( ( p p*

*..... ................*

2 Y σdt) Y σdt) Y σdt)

*(( ( (( (*

is taken as initial condition, constant C is equivalent to Cos(1). So, the estimation

TT T 11 11 11 TT T 22 22 22

e

T i T i

2 Y σdt)

e 1 e 1

*p* 1

*(t) C .....*

1

( Y σdt)

*i*

*p*

*)cos( )*

Y σdt)

*( p*

1 2

*.....*

(29)

1 1

(30)

(31)

(28)

If *ˆ( )* 0

law for the uncertainty bound is derived as.

**2.2 Second choice of (t)-1** For the second derivation of

1 1 1

Substituting Equation (29) into (23) yields

*p*

*ˆ(t)*

After integration, the result is

 For the third derivation of *ˆ(t)* , 1 *(t)* is defined as

$$\left(\Phi(\mathbf{t})\right)^{-1} = \text{diag}\left(\boldsymbol{\beta}\_{\text{i}}\text{Sin}^{2}\left(\mathbf{a}\_{\text{i}}\right)\mathbf{Y}^{\text{T}}\mathbf{c}\mathbf{d}\mathbf{t}\right)\text{Cov}\left(\mathbf{a}\_{\text{i}}\right)\mathbf{Y}^{\text{T}}\mathbf{c}\mathbf{d}\mathbf{t}\mathbf{t}\mathbf{j}\_{\text{i}}\right) \tag{34}$$

Substitution of Equation (34) into Equation (23) yields

$$
\begin{bmatrix}
\hat{\rho}(t)\_1 \\
\hat{\rho}(t)\_2 \\
\dots \\
\hat{\rho}(t)\_p
\end{bmatrix} = \Phi(t)^{-1} \begin{bmatrix}
\beta\_1 \text{\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\text{\\\\\\\\\\\\\\\\\\\\\\\}\ \mathbf{C}^{\text{\text}} \text{\text{\textdth}} \mathbf{t}\_1 (\mathbf{Y}^T \sigma \mathbf{d})\_1 (\mathbf{Y}^T \sigma)\_1 \\
\beta\_2 \text{\\\\\\} \mathbf{C}^{\text{\text\textquag}} \begin{bmatrix}
\mathbf{y}^T \mathbf{d} \mathbf{t} \end{bmatrix} \mathbf{C} \text{\textquag} \begin{bmatrix}
\mathbf{a}\_1 \end{bmatrix} \mathbf{Y}^T \mathbf{d} \mathbf{t} \end{bmatrix} \mathbf{e}\_1 \begin{bmatrix}
\rho\_1 \\
\rho\_2 \\
\vdots \\
\rho\_p \end{bmatrix} + \begin{bmatrix}
\rho\_1 \\
\rho\_2 \\
\vdots \\
\rho\_p \\
\end{bmatrix} \\
\begin{bmatrix}
1 \\
1 \\
\dots \\
1
\end{bmatrix}
\end{bmatrix} + \Phi(t)^{-1} \mathbf{C} \begin{bmatrix}
1 \\
\vdots \\
1 \\
\dots \\
1
\end{bmatrix}
\tag{35}
$$

After integration, the result is

$$
\begin{bmatrix}
\hat{\rho}(t)\_1\\ \hat{\rho}(t)\_2\\ \dots\\ \hat{\rho}(t)\_p
\end{bmatrix} = \Phi(t)^{-1} \begin{bmatrix}
(\beta\_1/a\_1)\frac{\mathrm{Sin}^3(a\_1)\mathrm{Y}^1\mathrm{odt})\_1\\ 3\\ (\beta\_2/a\_2)\frac{\mathrm{Sin}^3(a\_2)\mathrm{Y}^1\mathrm{odt})\_2}{3}\\ \dots \\ \cdots \\ (\beta\_p/a\_p)\frac{\mathrm{Sin}^3(a\_p)\mathrm{Y}^1\mathrm{odt})\_p
\end{bmatrix} + \begin{bmatrix}
\rho\_1\\ \rho\_2\\ \dots\\ \dots\\ \rho\_p
\end{bmatrix} + \Phi(t)^{-1}\mathrm{C}\begin{bmatrix}1\\ 1\\ \vdots\\ 1\end{bmatrix} \tag{36}
$$

Modelling of Bound Estimation Laws and

uniformly ultimate bounded.

*ˆ(t)* , and

*V* 0 for ||x||>w where

Note that

function (13) is written as [14, 15].

**Proof**

If Φ,

Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 171

Where ε>0. If the control input (39) is substituted into the control law (6) for the control of the model manipulator, then, the control law (6) is continuous and the closed-loop system is

It is assumed that there exists an unknown bound on parametric uncertainty such that

π πρ 0 and π π 0

TT T T T TT T T T

Where *T TT x [ q ,q ]* and Q=diag[TK, K]. Based on the Leitman [1], we can show that

*Qx ˆ ................*

Where λmin(Q) denotes the minimum eigenvalue of Q. Second term in Equation (41), if

*u*

V -q Kq-q ΛKΛq σ Y (t) σ Y σ Y

TT T

*Qx ˆ ˆ*

From the Cauchy-Schawartz inequality and our assumption on . If ||YT σ||<ε then

T

Y Y Y (t) - (t) Y

T V - (t) 4 *x Qx /*

*ˆ(t)* is bounded. The rest of the proof can be seen in [4, 8].

*T*

*T*

*ˆ ˆ () u*

σ Yu(t) σ Y( (t)) Y (t) (t) Y

T T

TT T

This last term achieves a maximum value of

TT T

*Qx u ˆ*

2 w = (t)/2 (Q) min

>

*ˆ* (42)

*ˆ(t)* are substituded into (13), the time derivative of the Lyapunov

T 1 1 T 1

(Y (t) (Y

*) ˆ )*

T

T

Y

T

T

*( ) ˆ ˆ*

T T

*ˆ* (45)

ˆ(t) / 4 when ||YT σ||=ε/2. We have that

p T

*) ˆ )*

(Y (t) (Y

*p p*

*u ˆ*

(41)

 (40)

(43)

(44)

If *ˆ( )* 0 is taken as initial condition, constant C is equivalent to zero. So, the estimation law for the uncertainty bound is derived as .

$$\begin{bmatrix} \hat{\rho}(t)\_1\\ \hat{\rho}(t)\_2\\ \dots\\ \hat{\rho}(t)\_p \end{bmatrix} = \begin{bmatrix} (\boldsymbol{\rho}\_1^2 / \boldsymbol{a}\_1) \frac{\text{Sinc}^5(\boldsymbol{a}\_1 \big[ \mathbf{Y}^\text{-} \text{cdt} \big] \mathbf{C} \text{s}(\boldsymbol{a}\_1 \big[ \mathbf{Y}^\text{-} \text{cdt} \big])}{3} \\\ (\boldsymbol{\rho}\_2^2 / \boldsymbol{a}\_2) \frac{\text{Sinc}^5(\boldsymbol{a}\_2 \big[ \mathbf{Y}^\text{-} \text{cdt} \big] \mathbf{C} \text{s}(\boldsymbol{a}\_2 \big[ \mathbf{Y}^\text{-} \text{cdt} \big])}{3} \\\ \dots \\\ \dots \\\ (\boldsymbol{\rho}\_p^2 / \boldsymbol{a}\_p) \frac{\text{Sinc}^5(\boldsymbol{a}\_p \big[ \mathbf{Y}^\text{-} \text{cdt} \big] \mathbf{C} \text{s}(\boldsymbol{a}\_p \big[ \mathbf{Y}^\text{-} \text{cdt} \big])}{3} \end{bmatrix} + \begin{bmatrix} \rho\_1\\ \rho\_2\\ \dots\\ \dots\\ \rho\_p \end{bmatrix} \end{bmatrix} \tag{37}$$

If we substitute , , and *ˆ(t)* into Equation (16), the right terms of Equation (16) T T 2 [(ρ-ρ*ˆ*(t)] [Y σ (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) ρ(t)] 0 will be always zero and the derivation of the Lyapunov function will become a negative semidefinite function such that

$$\dot{\mathbf{V}} = \dot{\mathbf{q}}^T \mathbf{K} \dot{\mathbf{q}} \cdot \mathbf{\tilde{q}}^T \Lambda \mathbf{K} \Lambda \mathbf{\tilde{q}} \le 0 \tag{38}$$

So, the system is stable for all *ˆ(t)* derived from Equation (23).

### **3. Design of robust contol laws**

Based on the uncertainty bound estimation laws derived in section 2, and in [15], it is possible to develop robust control inputs.

### **3.1 Robust control law 1**

In order to define first robust control input, the following theorem is proposed.

#### **Theorem:**

Additional control input in control law (6) is

$$(u(t))\_i = \begin{cases} -\frac{(Y^T \sigma)\_i}{\left| (Y^T \sigma)\_i \right|} \hat{\rho}(t)\_i & \text{if } \left| (Y^T \sigma)\_i \right| > \varepsilon\_i \\\ -\frac{(Y^T \sigma)\_i}{\varepsilon\_i} \hat{\rho}(t)\_i & \text{if } \left| (Y^T \sigma)\_i \right| \le \varepsilon\_i \end{cases} \tag{39}$$

Where ε>0. If the control input (39) is substituted into the control law (6) for the control of the model manipulator, then, the control law (6) is continuous and the closed-loop system is uniformly ultimate bounded.

### **Proof**

170 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

3 T 1 1

Sin ( Y σdt)

2 1 1 2 2 2

3 T p p

*ˆ(t) (/) (t) (t) C ..... ..... ..... ......... ˆ(t)*

Sin ( Y σdt) 3

3

2 2

is taken as initial condition, constant C is equivalent to zero. So, the estimation

5 T T 1 11 1 2

Sin ( Y σdt) Cos( Y σdt) 3 Sin ( Y σdt) Cos( Y σdt) 3

*ˆ(t)* into Equation (16), the right terms of Equation (16)

T T V -q Kq-q ΛKΛq 0 (38)

5 T T 2 p pp p

Sin ( Y σdt) Cos( Y σdt) 3

1 1 5 T T 2 22 2 2 2 2 2 2

*p p*

T T 2 [(ρ-ρ*ˆ*(t)] [Y σ (t) (t)(ρ ρ*ˆ ˆ* (t)) (t) ρ(t)] 0 will be always zero and the derivation of the

*ˆ(t)* derived from Equation (23).

Based on the uncertainty bound estimation laws derived in section 2, and in [15], it is

*i T i ii T*

*i T*

*(Y ) ˆ(t) if (Y )*

*i i i*

  (39)

In order to define first robust control input, the following theorem is proposed.

*T*

*i*

*i T*

*i*

*(Y ) ˆ(t) if (Y ) (Y ) (u(t))*

*..... ..... ......... ˆ(t)*

3 1

1

(36)

(37)

Sin ( Y σdt) 1

1 1

*(/)*

*ˆ(t)*

law for the uncertainty bound is derived as .

*ˆ(t)*

If we substitute , , and

So, the system is stable for all

**3.1 Robust control law 1**

**Theorem:**

**3. Design of robust contol laws**

possible to develop robust control inputs.

Additional control input in control law (6) is

If *ˆ( )* 0  p

1 1

*( /)*

*ˆ(t) (/)*

*p p*

Lyapunov function will become a negative semidefinite function such that

*(/)*

*(/)*

1 1 3 T

*p p*

*p*

It is assumed that there exists an unknown bound on parametric uncertainty such that

$$
\|\mathbf{n}\_0 - \mathbf{n} \le \mathbf{p} \qquad \text{and} \qquad \|\mathbf{n}\_0 - \mathbf{n}\| \le \delta \tag{40}
$$

If Φ, *ˆ(t)* , and *ˆ(t)* are substituded into (13), the time derivative of the Lyapunov function (13) is written as [14, 15].

$$\begin{split} \dot{\mathbf{V}} &= -\dot{\mathbf{\tilde{q}}}^{\mathrm{T}} \mathbf{K} \dot{\mathbf{\tilde{q}}} \cdot \mathbf{\tilde{q}}^{\mathrm{T}} \mathbf{A} \mathbf{K} \mathbf{A} \tilde{\mathbf{q}} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \boldsymbol{u}(\mathbf{t}) + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \tilde{\boldsymbol{\pi}} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \tilde{\rho} \\ &= -\dot{\mathbf{\tilde{q}}}^{\mathrm{T}} \mathbf{K} \dot{\mathbf{\tilde{q}}} \cdot \mathbf{\tilde{q}}^{\mathrm{T}} \boldsymbol{\Lambda} \mathbf{K} \boldsymbol{\Lambda} \tilde{\mathbf{q}} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \boldsymbol{u}(\mathbf{t}) + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \tilde{\boldsymbol{\pi}} - \mathbf{o}^{\mathrm{T}} \mathbf{Y} (\rho - \hat{\rho}(\mathbf{t}) \\ &= \boldsymbol{\prec} \mathbf{x}^{\mathrm{T}} \mathbf{Q} \mathbf{x} + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \boldsymbol{u}(\mathbf{t}) + \mathbf{o}^{\mathrm{T}} \mathbf{Y} \hat{\rho}(\mathbf{t}) \end{split} \tag{41}$$

Where *T TT x [ q ,q ]* and Q=diag[TK, K]. Based on the Leitman [1], we can show that *V* 0 for ||x||>w where

$$\mathbf{w}^2 \hat{=} \hat{\rho}(\mathbf{t}) / 2 \boldsymbol{\lambda}\_{\text{min}}(\mathbf{Q}) \tag{42}$$

Where λmin(Q) denotes the minimum eigenvalue of Q. Second term in Equation (41), if ||YTσ||>ε then

$$\leq \mathbf{x}^{\mathsf{T}} \mathbf{Q} \mathbf{x} + \mathbf{o}^{\mathsf{T}} \mathbf{Y} \hat{\rho}(\mathbf{t}) \mathbf{o}^{\mathsf{T}} \mathbf{Y} \begin{bmatrix} \mathbf{Y}^{\mathsf{T}} \sigma \boldsymbol{\mathsf{t}}\_{1} \\ \hline \lVert \mathbf{Y}^{\mathsf{T}} \sigma \boldsymbol{\mathsf{t}}\_{1} \rVert \\ \hline \end{bmatrix} \begin{aligned} \label{t} \hat{\rho}(\mathbf{t})\_{1} \\ \hline \lVert \mathbf{Y}^{\mathsf{T}} \sigma \boldsymbol{\mathsf{t}}\_{1} \rVert \\ \hline \end{aligned} \tag{43}$$

$$\leq \mathbf{x}^{\mathsf{T}} \mathbf{Q} \mathbf{x} + \mathbf{o}^{\mathsf{T}} \mathbf{Y} (\hat{\rho}(\mathbf{t})) \cdot \hat{\rho}(\mathbf{t})) \leq 0$$

From the Cauchy-Schawartz inequality and our assumption on . If ||YT σ||<ε then

$$\begin{split} \boldsymbol{\sigma}^{\mathrm{T}} \mathbf{Y} \mathbf{u}(\mathbf{t}) + \boldsymbol{\sigma}^{\mathrm{T}} \mathbf{Y} (\hat{\boldsymbol{\rho}}(\mathbf{t})) &\leq (\mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma})^{\mathrm{T}} \left( \left\lVert \hat{\boldsymbol{\rho}}(\mathbf{t}) \right\rVert \frac{\mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma}}{\left\lVert \mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma} \right\rVert} + \boldsymbol{\mu}(\mathbf{t}) \right) \\ &\leq (\mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma})^{\mathrm{T}} \left( \left\lVert \hat{\boldsymbol{\rho}}(\mathbf{t}) \right\rVert \frac{\mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma}}{\left\lVert \mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma} \right\rVert} \frac{\mathbf{Y}^{\mathrm{T}} \boldsymbol{\sigma}}{\boldsymbol{\sigma}} \left\lVert \hat{\boldsymbol{\rho}}(\mathbf{t}) \right\rVert \right) \end{split} \tag{44}$$

This last term achieves a maximum value of ˆ(t) / 4 when ||YT σ||=ε/2. We have that

$$\dot{\mathbf{V}} \le \mathbf{-x}^T \mathbf{Q} \mathbf{x} + \varepsilon \|\hat{\boldsymbol{\rho}}(\mathbf{t})\| / \, / \, 4 \tag{45}$$

Note that *ˆ(t)* is bounded. The rest of the proof can be seen in [4, 8].

### **3.2 Robust control law 2**

Based on the bound estimation law *ˆ(t)* derived from general Equation (23), additional control input u(t) are defined [15]. The additional control input in control law (6) is defined as [15]

$$\mathbf{u}(\mathbf{t}) = \begin{bmatrix} -\hat{\rho}(t)\_1 \text{sgn}(\mathbf{Y}^\top \sigma)\_1 \\ -\hat{\rho}(t)\_2 \text{sgn}(\mathbf{Y}^\top \sigma)\_2 \\ \vdots \\ -\hat{\rho}(t)\_p \text{sgn}(\mathbf{Y}^\top \sigma)\_p \end{bmatrix} \tag{46}$$

Modelling of Bound Estimation Laws and

y

l1

lc1

Fig. 1. Two-link planar robot [4].

**4. Simulation results**

this robot is given by

Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 173

For illustration, a two-link robot manipulator is given in Figure 1 [4]. Parameterisation of

m2, I2

q2

1=m11c12+m2l12+I1, *π*2= m2lc22+I2 , *π*3=m2l1lc2 ,

q1

l2

lc2

m1, I1

With this parameterisation, the component yij of Y(q,q,q ) in Equation 2 are given as

y11 1 q ; y12 1 2 q q ; 2 y cos(q )(2q q sin(q )(q 2q q ) 13 2 1 2 2 2 12 *)* ;

y14= gccos(q1); y15= gccos(q1); y16= gccos(q1+q2) ;

y14*=*gccos(q1); y15= gccos(q1) ; y16= gccos(q1+q2)

y21=0; y22 r1 2 q q*r* ; ; y23 2 r1 2 1 r1 cos(q )q sin(q )(q q )

Y(q,q,q ,qr r *)* in Equation (4) have the components

y21=0; y22 1 2 q q ; 2 y23 2 1 2 1 cos(q )q sin(q )(q ) ;

*π*4=m1lc1, *π*5=m2l1, *π*6=m2lc2, (51)

x

y24=0 ; y25=0 ; y26= gccos(q1+q2).(52)

y11 r1 q ; y12 r1 2 q q *r* ;

y13 2 r1 2 2 1 2 1 r2 2 r2 cos(q )(2q q sin(q )(q q q q q q ) *r r )* ;

y24=0 ; y25=0 ; y26= gccos(q1+q2). (53)

### **3.3 Robust control law 3**

i denote the *i*th component of the vector T Y , i choose as the ith component of . Then, considering the *ˆ(t)* derived from Equation (23), u(t) for each *ˆ(t)* is defined as as follows: For Equation (28), u(t)i is

$$\mathbf{u}(\mathbf{t}) = \begin{bmatrix} -\frac{\mathsf{u}\_{i}}{\left|\mathsf{o}\_{i}\right|} \left[ -(\boldsymbol{\beta}\_{i}^{2}/\mathsf{a}\_{i}) \mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right| \sin(\mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right| + \mathrm{Cost}(\mathbf{1}) \mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right| \sin(\mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right|) + \boldsymbol{\rho}\_{i} \right] & \text{if} \quad \left| \mathbf{\hat{o}}\_{i} \right| > \mathbf{\hat{e}}\_{i} \\\ -\frac{\mathsf{u}\_{i}}{\left|\mathsf{o}\_{i}\right|} \left[ -(\boldsymbol{\beta}\_{i}^{2}/\mathsf{a}\_{i}) \mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right| \sin(\mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right|) \cos(\mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right| + \mathrm{Cost}(\mathbf{1}) \mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right| \sin(\mathbf{e}^{a} \left| \mathbf{\hat{J}}^{\textnormal{val}} \right|) + \boldsymbol{\rho}\_{i} \right] & \text{if} \quad \left| \mathbf{\hat{o}}\_{i} \right| \leq \mathbf{\hat{e}}\_{i} \end{bmatrix} \tag{47}$$

For Equation (33),

$$\mathbf{u}(\mathbf{t})\_{i} = \begin{bmatrix} -\frac{\mathsf{v}\_{i}}{|\mathbf{v}\_{i}|} \left[ (\boldsymbol{\beta}\_{i}^{2} \;/\ \ \boldsymbol{\alpha}\_{i}) \frac{\mathbf{e}^{\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}}{1 + \mathbf{e}^{2\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}} \arctan(\mathbf{e}^{\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}) - \arctan(1) \frac{\mathbf{e}^{\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}}{1 + \mathbf{e}^{2\left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}} + \mathbf{p}\_{i} \right] \tag{48}$$

$$\begin{bmatrix} -\frac{\mathsf{v}\_{i}}{\mathsf{e}\_{i}} \left[ (\boldsymbol{\beta}\_{i}^{2} \;/\ \boldsymbol{\alpha}\_{i}) \frac{\mathbf{e}^{\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}}{1 + \mathbf{e}^{2\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}} \arctan(\mathbf{e}^{\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}) - \arctan(1) \frac{\mathbf{e}^{\boldsymbol{\alpha}\_{i} \left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}}{1 + \mathbf{e}^{2\left\lVert \mathbf{v}\_{i} \mathbf{d} \right\rVert}} + \mathbf{p}\_{i} \right] \quad \text{if} \quad \left| \mathbf{v}\_{i} \right| \leq \mathsf{e}\_{i} \end{bmatrix}$$

For Equation (36):

$$\mathbf{u}(\mathbf{t})\_{\mathbf{i}} = \begin{bmatrix} -\frac{\mathbf{u}\_{\mathrm{i}}}{|\mathbf{u}\_{\mathrm{i}}|} \left[ (\boldsymbol{\beta}\_{\mathrm{i}}^{2} \;/\ \boldsymbol{\alpha}\_{\mathrm{i}}) \frac{\sin^{5} (\boldsymbol{\alpha}\_{\mathrm{i}} \left| \int \mathbf{v}\_{\mathrm{i}} \mathrm{d}\mathbf{t} \right|) \mathrm{C} \mathbf{s} (\boldsymbol{\alpha}\_{\mathrm{i}} \left| \int \mathbf{v}\_{\mathrm{i}} \mathrm{d}\mathbf{t} \right|)}{3} + \rho\_{\mathrm{i}} \right] \qquad \text{if} \quad \left| \boldsymbol{\alpha}\_{\mathrm{i}} \right| > \varepsilon\_{\mathrm{i}} \\\ -\frac{\mathbf{u}\_{\mathrm{i}}}{\varepsilon\_{\mathrm{i}}} \left[ (\boldsymbol{\beta}\_{\mathrm{i}}^{2} \;/\ \boldsymbol{\alpha}\_{\mathrm{i}}) \frac{\sin^{5} (\boldsymbol{\alpha}\_{\mathrm{i}} \left| \int \mathbf{v}\_{\mathrm{i}} \mathrm{d}\mathbf{t} \right|) \mathrm{C} \mathbf{s} (\boldsymbol{\alpha}\_{\mathrm{i}} \left| \int \mathbf{v}\_{\mathrm{i}} \mathrm{d}\mathbf{t} \right|)}{3} + \rho\_{\mathrm{i}} \right] \quad \text{if} \quad \left| \boldsymbol{\alpha}\_{\mathrm{i}} \right| \leq \varepsilon\_{\mathrm{i}} \end{bmatrix} \tag{49}$$

#### **3.4 Robust control law 4**

From Equations (43) and (44), it is ease to define the following control law.

$$u(t) = \begin{cases} -\frac{Y^T \sigma}{\left\|(Y^T \sigma)\right\|} \left\|\hat{\boldsymbol{\rho}}(t)\right\| & \text{if} \quad \left\|(Y^T \sigma)\right\| > \varepsilon\\ -\frac{Y^T \sigma}{\varepsilon} \left\|\hat{\boldsymbol{\rho}}(t)\right\| & \text{if} \quad \left\|(Y^T \sigma)\right\| \le \varepsilon \end{cases} \tag{50}$$

### **4. Simulation results**

172 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

control input u(t) are defined [15]. The additional control input in control law (6) is defined

sgn(Y u(t)

ν dt ν dt ν dt ν dt ν dt i 2

*ˆ(t)* derived from Equation (23), u(t) for each

T 1 1 T 2 2

sgn(Y

*ˆ(t) ) ˆ(t) ) ......... ˆ(t) )*

ii i ii

υ [ / )e e e 1 e e ρ ] if υ ε

ii i ii

i i i i i i i i i i

*i i i*

*i i i*

ν dt ν dt

υ e e [( e 1 ρ )] if υ ε

υ e e [( e 1 ρ )] if υ ε

( ν dt ( ν dt υ [( ρ ] if υ ε

*sin )Cos )*

( ν dt ( ν dt υ [( ρ ] if υ ε

*sin )Cos )*

i i ii 2 ν dt 2 ν dt

i i ii 2 ν dt 2 ν dt

i i ii

i i ii

*T*

*T*

*Y ˆ(t) if (Y )*

υ [ / )e e e 1 e e ρ ] if υ ε

i ν dt ν dt ν dt ν dt ν dt i 2 i i ii

*ii i ii*

*ii i ii*

*( sin( )cos( Cos( ) sin( )*

*( sin( )cos( Cos( ) sin( )*

   , i

T

i i ii

*p p* sgn(Y

*ˆ(t)* derived from general Equation (23), additional

(46)

*ˆ(t)* is defined as as follows:

(47)

(48)

(49)

(50)

choose as the ith component of . Then,

>

**3.2 Robust control law 2**

**3.3 Robust control law 3**

For Equation (28), u(t)i is

i

υ

For Equation (33),

For Equation (36):

  i

i

i

u(t)

**3.4 Robust control law 4**

ε

*i*

*i*

*i*

*i*

i i 5

i

considering the

u(t)

i denote the *i*th component of the vector T Y

2 ν dt i

*i*

2 ν dt i

*i*

υ 1 e 1 e u(t)

i ν dt ν dt

5 ii ii i 2

υ 3

*i*

*/ )*

*i*

*/ )*

ε 3

From Equations (43) and (44), it is ease to define the following control law.

*T*

*T T*

*Y ˆ(t) if (Y ) (Y ) u(t)*

ε 1 e 1 e

*/ ) arctan( ) arctan( )*

*/ ) arctan( ) arctan( )*

ii ii i 2

as [15]

Based on the bound estimation law

For illustration, a two-link robot manipulator is given in Figure 1 [4]. Parameterisation of this robot is given by

$$\text{Fig. 1. Two-link planar robot [4].}$$

$$
\pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{l}} \mathsf{l}\_{\mathsf{c}1} \mathsf{l}\_{\mathsf{c}1} + \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{l}} \mathsf{l} + \mathsf{l}\_{\mathsf{l}} \qquad \qquad \qquad \qquad \pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}2} \mathsf{l} + \mathsf{l}\_{\mathsf{2}} \qquad \qquad \qquad \pi\_{\mathsf{5}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{l}} \mathsf{l}\_{\mathsf{c}2} \qquad \qquad \qquad \pi\_{\mathsf{5}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{l}} \mathsf{l}\_{\mathsf{c}2} \qquad \qquad \qquad \pi\_{\mathsf{5}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}2} \qquad \tag{51}
$$

$$
\pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{i}} \mathsf{l}\_{\mathsf{c}1} \qquad \qquad \qquad \qquad \pi\_{\mathsf{5}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}2} \qquad \qquad \qquad \pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}2} \qquad \qquad \pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}2} \qquad \qquad \pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}2} \qquad \qquad \pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}} \qquad \qquad \pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{2}} \mathsf{l}\_{\mathsf{c}} \qquad \qquad \pi\_{\mathsf{i}} = \mathsf{m}\_{\mathsf{i}} \mathsf{l}\_{\mathsf{$$

x

With this parameterisation, the component yij of Y(q,q,q ) in Equation 2 are given as

$$\mathbf{y}\_{11} = \ddot{\mathbf{q}}\_1; \qquad \mathbf{y}\_{12} = \ddot{\mathbf{q}}\_1 + \ddot{\mathbf{q}}\_2; \quad \mathbf{y}\_{13} = \cos(\mathbf{q}\_2)(2\ddot{\mathbf{q}}\_1 + \ddot{\mathbf{q}}\_2) - \sin(\mathbf{q}\_2)(\dot{\mathbf{q}}\_2^2 + 2\dot{\mathbf{q}}\_1\dot{\mathbf{q}}\_2);$$

$$\mathbf{y}\_{14} = \mathbf{g}\cos(\mathbf{q}); \qquad \mathbf{y}\_{15} = \mathbf{g}\cos(\mathbf{q}); \qquad \mathbf{y}\_{16} = \mathbf{g}\cos(\mathbf{q} + \mathbf{q}\_2) \; ;$$

$$\mathbf{y}\_{21} = \mathbf{0}; \qquad \mathbf{y}\_{22} = \ddot{\mathbf{q}}\_1 + \ddot{\mathbf{q}}\_2 \; ; \qquad \mathbf{y}\_{23} = \cos(\mathbf{q}\_2)\ddot{\mathbf{q}}\_1 + \sin(\mathbf{q}\_2)(\dot{\mathbf{q}}\_1^2) \; ;$$

$$\mathbf{y}\_{24} = \mathbf{0} ; \qquad \mathbf{y}\_{25} = \mathbf{0} \; ; \qquad \mathbf{y}\_{26} = \mathbf{g}\cos(\mathbf{q}\_1 + \mathbf{q}\_2). \tag{52}$$

Y(q,q,q ,qr r *)* in Equation (4) have the components

$$\mathbf{y}\_{11} = \ddot{\mathbf{q}}\_{1r} \ \mathbf{j} \ \mathbf{y}\_{12} = \ddot{\mathbf{q}}\_{r1} + \ddot{\mathbf{q}}\_{r2} \ \ ;$$

$$\mathbf{y}\_{13} = \cos(\mathbf{q}\_{2})(2\ddot{\mathbf{q}}\_{r1} + \ddot{\mathbf{q}}\_{r2}) - \sin(\mathbf{q}\_{2})(\dot{\mathbf{q}}\_{1}\dot{\mathbf{q}}\_{r2} + \dot{\mathbf{q}}\_{1}\dot{\mathbf{q}}\_{r2}) \ ;$$

$$\mathbf{y}\_{14} = \mathbf{g}\_{c}\cos(\mathbf{q}\_{1}); \qquad \mathbf{y}\_{15} = \mathbf{g}\_{c}\cos(\mathbf{q}\_{1}) \ ; \qquad \mathbf{y}\_{16} = \mathbf{g}\_{c}\cos(\mathbf{q}\_{1} + \mathbf{q}\_{2})$$

$$\mathbf{y}\_{21} = 0; \qquad \mathbf{y}\_{22} = \ddot{\mathbf{q}}\_{r1} + \ddot{\mathbf{q}}\_{r2} \ ; \qquad \mathbf{y}\_{23} = \cos(\mathbf{q}\_{2})\ddot{\mathbf{q}}\_{r1} + \sin(\mathbf{q}\_{2})(\dot{\mathbf{q}}\_{1}\dot{\mathbf{q}}\_{r1})$$

$$\mathbf{y}\_{24} = 0 \ ; \qquad \mathbf{y}\_{25} = 0 \ ; \qquad \mathbf{y}\_{26} = \mathbf{g}\_{c}\cos(\mathbf{q}\_{1} + \mathbf{q}\_{2}). \tag{53}$$

For illustrated purposes let us assume that the parameters of the unloaded manipulator are known and are given by Table 1. Using these values in Table 1, the ith component of *π* obtained by means of Equation (51) are given in Table 2. It is assumed that the parameters m2, lc2 and I2 are changed in the intervals

$$0 \le \Delta m\_2 \le 10 \quad ; \quad 0 \le \Delta l\_{c2} \le 0.5 \; ; \quad 0 \le l\_2 \le \frac{15}{12} \tag{54}$$

Modelling of Bound Estimation Laws and

vector with components ρi [4].

Table 4. Uncertainty bound [4].

Tracking error(rad)

Tracking error(rad)

Let

that a measure of uncertainty for each parameter

i denote the *i*tch component of the vector T Y

 ii i i i i

0 1 2 3 4 5

joint 2

0 1 2 3 4 5

Time(sec)

(28) when =diag(10 10), K=diag(30 30), α=1, β=1.

x 10-3 joint 1

u

 *i i*

and consequently the *i*tch component of the control input up is defined as [4].

Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 175

Having a single number ρ to measure the parameter uncertainty may lead to overly conservative design, higher than necessary gains, etc. For this purpose, different "weights" or gains to the components of u may be assigned. This can be done as follows: Supposing

 , i

i ii i i ρ υ / υ if υ ε

(ρ /ε )υ if υ ε

Since extended algorithm (56) is used, the uncertainty bounds for each parameter are shown separately in Table 4. The uncertainty bounds i in Table 4 are simply the difference between values given in Table 3 and Table 2, and the value of ρ is the Euclidean norm of the

For computer simulation, a fifth order polynomial function is used as a reference trajectory for both joints. In order to analyse performance of the proposed controllers, each control law with the same control parameters K and Ʌ is applied to the same model system using same trajectory. The control matrices Ʌ and K are chosen to be identical as Ʌ=diag(10 10) and

Tork input: Joint 2

Fig. 2. Response using the robust control law (39) with uncertainty bound estimation law

0

200

Tork input:Joint 1

400

 ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 5 7.29 6. 25 0 5 6.25

K=diag(30 30) for all controllers. The obtained results are plotted in Figures 2-4.

*i* can be defined separately as:

0 1 2 3 4 5

Time(sec)

0 1 2 3 4 5

Time(sec)

I=1,2,……….,p (58)

choose as the itch component of ,

(59)

Choosing the mean value for the range of possible i in Equation (54) yields the nominal parameter vector and the computed values for ith component of *π*0 is shown in Table 3 [4].

Table 1. Parameters of the unloaded arm [4].

Table 2. *π*i for the unloaded arm [4]

Table 3. Nominal parameter vector *π*0 [4].

With this choice of nominal parameter vector 0 and uncertainty range given by (54), it is an easy matter to calculate the uncertainty bound as follows:

$$\left\|\tilde{\boldsymbol{\pi}}\right\|^2 = \sum\_{i=1}^6 (\pi\_{i0} - \pi\_i)^2 \le 181.26\tag{55}$$

and thus 181 26 13 46 *. .* .

For explanation, Spong's algorithms are given.

$$\mathbf{u}(\mathbf{t}) = \begin{cases} -\delta \frac{\mathbf{Y}^{\mathrm{T}}\mathbf{o}}{\left\| \mathbf{Y}^{\mathrm{T}}\mathbf{o} \right\|} & \text{if} \quad \left\| \mathbf{Y}^{\mathrm{T}}\mathbf{o} \right\| > \varepsilon \\\\ -\delta \frac{\mathbf{Y}^{\mathrm{T}}\mathbf{o}}{\varepsilon} & \text{if} \quad \left\| \mathbf{Y}^{\mathrm{T}}\mathbf{o} \right\| \le \varepsilon \end{cases} \tag{56}$$

As a measure of parameter uncertainty on which the additional control input is based, ρ can be defined as

$$\mathcal{S} = \left(\sum\_{i=1}^{\mathcal{P}} \rho\_i^{\mathcal{Q}}\right)^{1/2} \tag{57}$$

Having a single number ρ to measure the parameter uncertainty may lead to overly conservative design, higher than necessary gains, etc. For this purpose, different "weights" or gains to the components of u may be assigned. This can be done as follows: Supposing that a measure of uncertainty for each parameter *i* can be defined separately as:

$$\left\lfloor \tilde{\pi}\_{i} \right\rfloor \leq \rho\_{i} \qquad \text{I} = \text{1}, \text{2}, \dots, \text{...}, \text{p} \tag{58}$$

Let i denote the *i*tch component of the vector T Y , i choose as the itch component of , and consequently the *i*tch component of the control input up is defined as [4].

$$\mathbf{u}\_{i} = \begin{cases} -\mathfrak{p}\_{i}\mathfrak{v}\_{i}/\left|\mathfrak{v}\_{i}\right| & \text{if} \quad \left|\mathfrak{v}\_{i}\right| > \mathfrak{e}\_{i} \\ -(\mathfrak{p}\_{i}/\mathfrak{e}\_{i})\mathfrak{v}\_{i} & \text{if} \quad \left|\mathfrak{v}\_{i}\right| \le \mathfrak{e}\_{i} \end{cases} \tag{59}$$

Since extended algorithm (56) is used, the uncertainty bounds for each parameter are shown separately in Table 4. The uncertainty bounds i in Table 4 are simply the difference between values given in Table 3 and Table 2, and the value of ρ is the Euclidean norm of the vector with components ρi [4].

Table 4. Uncertainty bound [4].

174 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

For illustrated purposes let us assume that the parameters of the unloaded manipulator are known and are given by Table 1. Using these values in Table 1, the ith component of *π* obtained by means of Equation (51) are given in Table 2. It is assumed that the parameters

> 15 0 12

*I* (54)

0 10 *m*2 ; 0 05 *c*2 *l .* ; 2

Choosing the mean value for the range of possible i in Equation (54) yields the nominal parameter vector and the computed values for ith component of *π*0 is shown in Table 3 [4].

m1 m2 l1 l2 lc1 lc2 I1 I2 10 5 1 1 0.5 0.5 10/12 5/12

*π*1 *π*2 *π*3 *π*4 *π*5 *π*6 8.33 1.67 2.5 5 5 2.5

13.33 8.96 8.75 5 10 8.75

With this choice of nominal parameter vector 0 and uncertainty range given by (54), it is an

04

181 26 *i i*

Y σ if Y σ ε

1 2 p 2 i i 1 ρ

*/*

As a measure of parameter uncertainty on which the additional control input is based, ρ can

Y σ if Y σ ε

T

T

05

(57)

(55)

06

(56)

03

6 2 2 0 1

 *( ).*

T

T T

ε

*i*

Y σ u(t)

  m2, lc2 and I2 are changed in the intervals

Table 1. Parameters of the unloaded arm [4].

02

Table 3. Nominal parameter vector *π*0 [4].

 181 26 13 46 *. .* . For explanation, Spong's algorithms are given.

easy matter to calculate the uncertainty bound as follows:

Table 2. *π*i for the unloaded arm [4]

01

and thus

be defined as

For computer simulation, a fifth order polynomial function is used as a reference trajectory for both joints. In order to analyse performance of the proposed controllers, each control law with the same control parameters K and Ʌ is applied to the same model system using same trajectory. The control matrices Ʌ and K are chosen to be identical as Ʌ=diag(10 10) and K=diag(30 30) for all controllers. The obtained results are plotted in Figures 2-4.

Fig. 2. Response using the robust control law (39) with uncertainty bound estimation law (28) when =diag(10 10), K=diag(30 30), α=1, β=1.

Modelling of Bound Estimation Laws and

**5. Conclusion**

diffucult.

(t) as a derivtion of

618, 1989.

**6. References**

Control Vol*.*26, pp. 1139-1144, 1981.

according to uncertainty bound estimation laws.

Robust Controllers for Robot Manipulators Using Functions and Integration Techniques 177

As shown in Figures 2-4, tracking error is small and tracking performance changes

In the past, some robust controllers are developed for robot manipulators. Corless-Leitmann [1] approach is a popular approach used for designing robust controllers for robot manipulators. Spong [4] proposed a new robust controller for robot manipulators and Leithmann [5] or Corless-Leithman [1] approach is used for designing the robust controller. In [4], uncertainty bound on parameter is needed to derive robust controller and uncertainty bound parameters depends only on the inertia parameters of the robots. However, constant uncertainty bound parameters cause pure tracking performances. In order to increase tracking performance of the uncetain system, uncertainty bound estimation laws are developed [11-13]. Uncertainty bound estimation laws are updated as a function of exponential [11, 12], logarithmic [13] and trigonometric [14] functions depending on robot kinematics parameters and tracking error. A first order differential equation function is developed for derivation of control parameters and only a single derivation of uncertainty bound estimation law is possible. A new method for derivation of a bound estimation law is not proposed in [11-13], because, definition of a new variable function for other derivation is

In the study [14], a general equation is developed from Lyapunov function and uncertainty bound estimation laws depending on trigonometric functions are developed. However, a general method for derivation of uncertainty bound estimation laws is not proposed. In a recent study [15], a general method for derivation of bound estimation laws based on the Lyapunov theory is proposed. In this method, functions and integration techniques are used for derivation of uncertainty bound estimation laws. Then, relations between the bound estimation laws and robust control inputs are established and four new robust control inputs are designed depending on each bound estimation law. It is possible to derive other different uncertainty bound estimation laws from general equation (23) if appropriate functions and integration techniques are defined. In this work, three different variable

functions are defined and integration techniques are used in order to derive

distinct rule for definition of the (t) and integration techniques in order to derive

Equation (23) if appropriate function (t) and integration techniques are chosen.

relations between the uncertainty bouns and robust control laws are established. There is no

use system state parameters and mathematical insight to search for appropriate function of

with these derivations. It will be also possible to derive another bound estimation laws from

[1] M. Corless, G. Leitmann, "Continuous feedback guaranteeing uniform ultimate

[2] M. Corless, "Tracking controllers for uncertain systems: Applications to a manutec R3

boundedness for uncertain dynamic systems", IEEE Transactions Automatic

robot. Journal of Dynamic Systems Measurement and Control", Vol.111, pp. 609-

*ˆ(t)* . This study also shows that robust controllers are not limited

*ˆ(t)* and

*ˆ(t)* . We

Fig. 3. Response using the robust control law (39) with uncertainty bound estimation law (35) when =diag(10 10), K=diag(30 30), α=0.8, β=0.4.

Fig. 4. Response using the robust control law (39) with uncertainty bound estimation law (37) when =diag(10 10), K=diag(30 30), α=0.5, β=2.

As shown in Figures 2-4, tracking error is small and tracking performance changes according to uncertainty bound estimation laws.

### **5. Conclusion**

176 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

Tork input: Joint 2

Fig. 4. Response using the robust control law (39) with uncertainty bound estimation law

0

200

Tork input:Joint 1

400

Tork input: Joint 2

Fig. 3. Response using the robust control law (39) with uncertainty bound estimation law

0

200

Tork input:Joint 1

400

0 1 2 3 4 5

Time(sec)

0 1 2 3 4 5

Time(sec)

0 1 2 3 4 5

Time(sec)

0 1 2 3 4 5

Time(sec)

0 1 2 3 4 5

0 1 2 3 4 5

Time(sec)

0 1 2 3 4 5

0 1 2 3 4 5

Time(sec)

(37) when =diag(10 10), K=diag(30 30), α=0.5, β=2.

(35) when =diag(10 10), K=diag(30 30), α=0.8, β=0.4.

x 10-3 joint 1

x 10-3 joint 2

x 10-3 joint 1

x 10-3 joint 2

Tracking error(rad)

Tracking error(rad)

0

2

Tracking error(rad)

Tracking error(rad)

In the past, some robust controllers are developed for robot manipulators. Corless-Leitmann [1] approach is a popular approach used for designing robust controllers for robot manipulators. Spong [4] proposed a new robust controller for robot manipulators and Leithmann [5] or Corless-Leithman [1] approach is used for designing the robust controller. In [4], uncertainty bound on parameter is needed to derive robust controller and uncertainty bound parameters depends only on the inertia parameters of the robots. However, constant uncertainty bound parameters cause pure tracking performances. In order to increase tracking performance of the uncetain system, uncertainty bound estimation laws are developed [11-13]. Uncertainty bound estimation laws are updated as a function of exponential [11, 12], logarithmic [13] and trigonometric [14] functions depending on robot kinematics parameters and tracking error. A first order differential equation function is developed for derivation of control parameters and only a single derivation of uncertainty bound estimation law is possible. A new method for derivation of a bound estimation law is not proposed in [11-13], because, definition of a new variable function for other derivation is diffucult.

In the study [14], a general equation is developed from Lyapunov function and uncertainty bound estimation laws depending on trigonometric functions are developed. However, a general method for derivation of uncertainty bound estimation laws is not proposed. In a recent study [15], a general method for derivation of bound estimation laws based on the Lyapunov theory is proposed. In this method, functions and integration techniques are used for derivation of uncertainty bound estimation laws. Then, relations between the bound estimation laws and robust control inputs are established and four new robust control inputs are designed depending on each bound estimation law. It is possible to derive other different uncertainty bound estimation laws from general equation (23) if appropriate functions and integration techniques are defined. In this work, three different variable functions are defined and integration techniques are used in order to derive *ˆ(t)* and relations between the uncertainty bouns and robust control laws are established. There is no distinct rule for definition of the (t) and integration techniques in order to derive *ˆ(t)* . We use system state parameters and mathematical insight to search for appropriate function of (t) as a derivtion of *ˆ(t)* . This study also shows that robust controllers are not limited with these derivations. It will be also possible to derive another bound estimation laws from Equation (23) if appropriate function (t) and integration techniques are chosen.

### **6. References**

[3] M.W. Spong, M. Vidyasagar, "Robust linear compensator design for nonlinear robotic control", IEEE Journal of Robotic and Automation, Vol.3(4), pp. 345-350, 1987.

**11**

*1,3China*

*2Republic of Korea*

**Missile Cooperative Engagement Formation**

In traditional engagement model, there is only one single weapon or platform which resists to another single one, and the connections between members are few, so the complete and global information in engagement space can't be utilized sufficiently, which directly results in traditional model is confronting with more and more disadvantages especially as the high-tech and information-tech are developing faster and faster. In view of those issues, the concepts and technology of missile formation cooperative are presented, developed and expanded recently(Cui et al., 2009). Compared to traditional model, weapon or platform with cooperative manner manifests great advantages in aspects of ability of penetration, electronic countermeasures and ability of searching moving targets etc, furthermore the

Many new cooperative weapon systems are established and developed fast recently, such as Cooperative Engagement Capability (CEC) system, Net Fire System and LOw Cost Autonomous Attack System (LOCAAS) etc, wherein the LOCAAS is most relevant to our

In order to meet the requirements of future aerial warfare, United States Force has developed a series of high technical and high accurate airborne guided weapon systems, such as Joint Common Missile (JCM), Joint Direct Attack Missile (JDAM), Wind Corrected Munitions Dispenser (WCMD) and LOw Cost Autonomous Attack System (LOCAAS). These weapons are paid much attention because their great capabilities of high-precision, all-weather engagement and attacking beyond defence area etc. Especially, LOCAAS also has other great advantages besides those aspects mentioned above, such as low cost, general utilization and attacking multi-targets simultaneously, so it is paid more attention than other weapons, and it becomes an outstanding representative of high-tech weapon and new

LOCAAS is developed from an previous weapon named as Low Cost Anti-Armor Submunitions (also called LOCAAS for short) which was a kind of short-range unpowered airborne and air-to-ground guided weapon developed in 1998. At a later time, researchers added thrust system to this old LOCAAS, so it had capability of launching beyond defence

**1. Introduction**

engagement model.

synthetical engagement efficacy is developed greatly.

topic, so we will introduce it more detailed.

**Configuration Control Method**

Changzhu Wei1,2, Yi Shen1, Xiaoxiao Ma3,

Naigang Cui1 and Jifeng Guo1 *1Harbin Institute of Technology, Harbin, 2Yonsei University Observatory, Seoul, 3Xuzhou Air Force College, Xuzhou,*

## **Missile Cooperative Engagement Formation Configuration Control Method**

Changzhu Wei1,2, Yi Shen1, Xiaoxiao Ma3, Naigang Cui1 and Jifeng Guo1 *1Harbin Institute of Technology, Harbin, 2Yonsei University Observatory, Seoul, 3Xuzhou Air Force College, Xuzhou, 1,3China 2Republic of Korea*

### **1. Introduction**

178 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

[3] M.W. Spong, M. Vidyasagar, "Robust linear compensator design for nonlinear robotic control", IEEE Journal of Robotic and Automation, Vol.3(4), pp. 345-350, 1987. [4] M.W. Spong, "On the robust control of robot manipulators", IEEE Transactions on

[5] G. Leitmann, "On the efficiency of nonlinear control in uncertain linear system", Journal of Dynamic Systems Measurement and Control, Vol.102, pp. 95-102, 1981. [6] E. Yaz, "Comments on the robust control of robot manipulators", IEEE Transactions on

[7] M. Danesh, M. Keshmiri and F. Sheikholeslam, "Developing a robust control scheme

[8] K.M. Koo, J.H. Kim, "Robust control of robot manipulators with parametric

[9] G. Liu, A.A. Goldenberg, "Uncertainty decomposition-based robust control of robot

[10]. G. Liu, Decomposition-based control of mechanical systems Electrical and Computer

[11] R. Burkan, New approaches in controlling robot manipulators with parametric uncertainty, Ph.D. Thesis, Erciyes University, Institute of Science, Turkey, 2002. [12] R. Burkan, İ. Uzmay, "Upper bounding estimation for robustness to the parameter

[13] R. Burkan, İ. Uzmay, "Logarithmic based robust approach to parametric uncertainty for

[14] R. Burkan, "Upper bounding estimation for robustness to the parameter uncertainty

[15] R. Burkan, "Modelling of bound estimation laws and robust controllers for robustness

[16] M.W. Spong, S. Hutchinson, M. Vidyasagar, Robot Dynamics and Control, John

[17] J.J. Slotine, W. Li, "On the adaptive control of robotic manipulator", The International

[18] L. Sciavicco, B. Siciliano, Modeling and Control of Robot Manipulators, The McGraw-

Intelligent & Robotic Systems, Vol.46, pp.263-283, 2006.

Journal of Robotics Research, Vol.6(3), pp. 49-59, 1987.

and Robotic Systems, Vol.60, pp. 365-394, 2010.

robust to uncertain model parameters and unmodeled dynamics" Proceeding of IEEE Conference on Industrial Electronics and Applications (ICIEA), May 24-26,

uncertainty", IEEE Transactions Automatic Control, Vol.39(6), pp. 1230-1233, 1994.

manipulators", IEEE Transactions on Control Systems Technology, Vol.4, pp. 384-

Engineering, 2000 Canadian Conference on Volume: 2 Publication Year: 2000 ,

uncertainty in trajectory control of robot arm", Robotics and Autonomous Systems

control of robot manipulators", International Journal of Robust and Nonlinear

with trigonometric function in trajectory control of robot arms"*.* Journal of

to parametric uncertainty for control of robot manipulators", Journal of Intelligent

Automatic Control, Vol.37, pp.1782-1786, 1992.

Automatic Control, Vol. 38(38), pp. 511-512, 1993.

2006, Singapore.

Page(s): 966 - 970 vol.2

Vol.45, pp.99-110, 2003.

Willey&Sons, Inc., 2006.

Hill Companies, 1996.

Control, Vol.15, pp. 427-436, 2005.

393, 1996.

In traditional engagement model, there is only one single weapon or platform which resists to another single one, and the connections between members are few, so the complete and global information in engagement space can't be utilized sufficiently, which directly results in traditional model is confronting with more and more disadvantages especially as the high-tech and information-tech are developing faster and faster. In view of those issues, the concepts and technology of missile formation cooperative are presented, developed and expanded recently(Cui et al., 2009). Compared to traditional model, weapon or platform with cooperative manner manifests great advantages in aspects of ability of penetration, electronic countermeasures and ability of searching moving targets etc, furthermore the synthetical engagement efficacy is developed greatly.

Many new cooperative weapon systems are established and developed fast recently, such as Cooperative Engagement Capability (CEC) system, Net Fire System and LOw Cost Autonomous Attack System (LOCAAS) etc, wherein the LOCAAS is most relevant to our topic, so we will introduce it more detailed.

In order to meet the requirements of future aerial warfare, United States Force has developed a series of high technical and high accurate airborne guided weapon systems, such as Joint Common Missile (JCM), Joint Direct Attack Missile (JDAM), Wind Corrected Munitions Dispenser (WCMD) and LOw Cost Autonomous Attack System (LOCAAS). These weapons are paid much attention because their great capabilities of high-precision, all-weather engagement and attacking beyond defence area etc. Especially, LOCAAS also has other great advantages besides those aspects mentioned above, such as low cost, general utilization and attacking multi-targets simultaneously, so it is paid more attention than other weapons, and it becomes an outstanding representative of high-tech weapon and new engagement model.

LOCAAS is developed from an previous weapon named as Low Cost Anti-Armor Submunitions (also called LOCAAS for short) which was a kind of short-range unpowered airborne and air-to-ground guided weapon developed in 1998. At a later time, researchers added thrust system to this old LOCAAS, so it had capability of launching beyond defence

Missile Cooperative Engagement Formation Configuration Control Method 181

The missile formation control system mentioned in this chapter only has one leader missile (showed as Fig.2), and it mainly consists of the following subsystems: cooperative engagement mission planning subsystem, formation configuration describing subsystem, leader inner-loop control subsystem, follower inner-loop control subsystem and formation configuration control subsystem. The sketch map illustrates relationships between each

*I x*

, , *Vfi vfi fi*

*r x*

*r o*

*r z*

, , *Vfci vfci fci*

*r y* 1 *o r*

**2. Frame of missile formation control system**

*I z*

  *I o*

Fig. 2. Sketch map of missile formation consisted of two followers and only one leader.

, , , , *Vli vli li*

Fig. 3. Relationships between subsystems of missile formation control system.

The cooperative mission planning subsystem supplies real-time status of mission space and distribution situation of targets, and restricts the flight status of leader at the same time; the formation configuration describing system receives the information supplied by cooperative engagement subsystem and establishes associated function of engagement efficacy according to different missile formation configurations, and then makes the decision set about formation configurations based on the returned value of this efficacy function. After optimizing the values of efficacy function according to the specific mission space, the optimal formation configuration can be calculated within this decision set. The optimal

*Vlci vlci lci*

subsystem in one missile formation system is showed in Fig.3.

*I y*

area and searching moving targets within large scope. This weapon was added bidirectional data link once again in 2003, so it can implement Man-in-the-Loop control and command, which meant that this weapon system can attack targets autonomously and manifest a kind of smart engagement capability. After those improvements, this old anti-armor weapon was re-named to LOw Cost Autonomous Attack System (LOCAAS). LOCAAS adopts INS and mid-course guidance, and it also equips a smart fuze and a sensor that can be used to search moving targets, so LOCAAS can not only monitor targets within large range but also can relocate, recognize and aim at them autonomously. The most outstanding character is that LOCAAS can also make intelligent decision for choosing the optimal orientation and sequence to achieve optimal attack, and it also has the capability of on-line planning mission.

At present, United State Force sets the engagement schemes for LOCAAS as: there will be some LOCAASs flying in the battle field to cruise or put on standby, and they can connect with each other by data link. While one of them finding the targets and can't destroy them alone, it will send out the signals to require for cooperative attacking, however if it can destroy the targets by itself, this LOCAAS will attack the targets and the other LOCAASs will continue searching targets after receiving the instruction signals from the mentioned LOCAAS which has gone into the battle. The sketch map Fig.1 shows the main concepts of how LOCAASs take part in engagement cooperatively. Authors who are interested in LOCAAS can get more detailed information in the website of Lockheed Martin.

Fig. 1. The sketch map of LOCAAS engagement.

The background and significance were presented first, and then the technique frame of missile formation cooperative control system is showed, which can clearly elaborate how missiles in a formation work together in a cooperative manner. The specific relationships between each loop are analyzed, which are very necessary to clearly explain how to design missile formation control system. Followed are the main contents of this chapter, which are the detailed processes of establishing and designing missile formation control system. In this part, we will only consider the external loop of missile formation control system because the design method on individual missile inner controller is easy to be found in many literatures, and then we just assumed it is closed-loop and stable. The detailed process is divided into two steps, first is using proportion-differential control method to design missile formation keeping controller, the other explains how to design optimal keeping controller of missile formation. Some simulations are made to compare these two formation control systems proposed in this chapter at last.

### **2. Frame of missile formation control system**

180 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

area and searching moving targets within large scope. This weapon was added bidirectional data link once again in 2003, so it can implement Man-in-the-Loop control and command, which meant that this weapon system can attack targets autonomously and manifest a kind of smart engagement capability. After those improvements, this old anti-armor weapon was re-named to LOw Cost Autonomous Attack System (LOCAAS). LOCAAS adopts INS and mid-course guidance, and it also equips a smart fuze and a sensor that can be used to search moving targets, so LOCAAS can not only monitor targets within large range but also can relocate, recognize and aim at them autonomously. The most outstanding character is that LOCAAS can also make intelligent decision for choosing the optimal orientation and sequence to achieve optimal attack, and it also has the capability of on-line planning

At present, United State Force sets the engagement schemes for LOCAAS as: there will be some LOCAASs flying in the battle field to cruise or put on standby, and they can connect with each other by data link. While one of them finding the targets and can't destroy them alone, it will send out the signals to require for cooperative attacking, however if it can destroy the targets by itself, this LOCAAS will attack the targets and the other LOCAASs will continue searching targets after receiving the instruction signals from the mentioned LOCAAS which has gone into the battle. The sketch map Fig.1 shows the main concepts of how LOCAASs take part in engagement cooperatively. Authors who are interested in

The background and significance were presented first, and then the technique frame of missile formation cooperative control system is showed, which can clearly elaborate how missiles in a formation work together in a cooperative manner. The specific relationships between each loop are analyzed, which are very necessary to clearly explain how to design missile formation control system. Followed are the main contents of this chapter, which are the detailed processes of establishing and designing missile formation control system. In this part, we will only consider the external loop of missile formation control system because the design method on individual missile inner controller is easy to be found in many literatures, and then we just assumed it is closed-loop and stable. The detailed process is divided into two steps, first is using proportion-differential control method to design missile formation keeping controller, the other explains how to design optimal keeping controller of missile formation. Some simulations are made to compare these two formation control systems

LOCAAS can get more detailed information in the website of Lockheed Martin.

Fig. 1. The sketch map of LOCAAS engagement.

proposed in this chapter at last.

mission.

The missile formation control system mentioned in this chapter only has one leader missile (showed as Fig.2), and it mainly consists of the following subsystems: cooperative engagement mission planning subsystem, formation configuration describing subsystem, leader inner-loop control subsystem, follower inner-loop control subsystem and formation configuration control subsystem. The sketch map illustrates relationships between each subsystem in one missile formation system is showed in Fig.3.

Fig. 2. Sketch map of missile formation consisted of two followers and only one leader.

Fig. 3. Relationships between subsystems of missile formation control system.

The cooperative mission planning subsystem supplies real-time status of mission space and distribution situation of targets, and restricts the flight status of leader at the same time; the formation configuration describing system receives the information supplied by cooperative engagement subsystem and establishes associated function of engagement efficacy according to different missile formation configurations, and then makes the decision set about formation configurations based on the returned value of this efficacy function. After optimizing the values of efficacy function according to the specific mission space, the optimal formation configuration can be calculated within this decision set. The optimal formation configuration will be different when the missile formation is flying in different segments of mission space, that is, the optimal formation configuration will be decided by specific engagement requirement. The formation configuration describing subsystem will also restrict flight status of leader together with cooperative mission planning subsystem at the same time; The leader inner-loop control subsystem receives the information sent from the cooperative engagement planning subsystem and controls the leader to fly stably, and then the real flight states of leader can be obtained; The required states of follower missile can be calculated through the states in relative spatial dimension i.e. the distance between missiles which is obtained from formation configuration describing subsystem; The follower inner-loop control subsystem receives the command states, which is similar to that of leader mentioned above, and then control follower to fly stably. The actual flight states of follower missiles are fed back to the formation configuration control subsystem and used to achieve the expected formation configuration which is decided by the specific cooperative engagement mission.

This chapter focus on missile formation keeping control problems. First, we consider the actual flight states of leader as perturbation variables acting on the controller, and assume the follower inner-controller is closed and stable loop, that is, followers can track the required commands of velocity, flight path angle and flight deflection angle rapidly and stably. We will make further assumption that those three channels referred above are oneorder systems expressed as (Wei et al., 2010):

$$\begin{aligned} \dot{V}\_f &= -\frac{1}{\tau\_{vf}}(V\_f - V\_{fc}) \\ \dot{\theta}\_f &= -\frac{1}{\tau\_{\theta f}}(\theta\_f - \theta\_{fc}) \\ \dot{\nu}\_{vf} &= -\frac{1}{\tau\_{\nu\_{vf}}}(\nu\_{vf} - \nu\_{vf}) \end{aligned} \tag{1}$$

Missile Cooperative Engagement Formation Configuration Control Method 183

dimensional states for some periods of time. Next sections will present some further and

The main thought of the following parts is showed as below: basing on the kinematics relationships between missiles in inertial coordinate frame, the missile formation proportional-derivative (PD) controller via feeding back full states will be present firstly; Second, the relative motion will be established in relative coordinates frame, which can indicate the characteristics of relative motion directly. Based on this direct relative motion, the optimal controller that has non-zero given point and restrains slowly variant

The origin *r o* of this coordinate coincides with the mass centre of leader, and axis *r r o x* points to the velocity direction of leader, *r r o y* is perpendicular to *r r o x* and points to up direction, *r r o z* composes right-hand coordinate frame together with other two axes mentioned above.

The origin *OI* of this coordinate frame is fixed to an arbitrary point on ground, axis *O XI I* lies in horizontal plane and points to target, axis*O YI I* is perpendicular to *O XI I* and points to up direction, the last axis*O ZI I* also composes right-hand coordinate frame system together with

yr o1

*r*

The origin of this coordinate 1*o* coincides with the mass centre of missile, and axis 1 2 *o x* points to the velocity direction of missile, 1 2 *o y* is perpendicular to 1 2 *o x* and points to up

XI

r x

or r z

The relationship between these two coordinate frames is showed in Fig.4.

OI

Fig. 4. Relationship between relative and inertial coordinate frame.

YI

detailed researches on how to keep missile formation optimally and robustly.

perturbations is designed in the third part.

**3.1 Definition of coordinate frame system**  1. Relative coordinate frame *r r rr o xyz*

2. Inertial coordinate frame system *O XYZ I II I*

those two axes mentioned above.

ZI

3. Trajectory coordinate frame system 1 2 22 *o xyz*

**3. PD controller of missile formation keeping**

where , *Vf f* and *vf* is the velocity, flight path angle and flight deflection angle of follower respectively; , *Vfc fc* and *vfc* are commands of velocity, flight path angle and flight deflection angle of follower respectively; , *vf f* and *vf* are the inertial time constants of velocity, flight path angle and flight deflection angle channel of follower respectively. These values can be calculated by analyzing four-dimensional guidance and control system referrd in the literature of Cui et al., 2010, they are:

$$\begin{aligned} \tau\_v &= 3.51 \text{s} \\ \tau\_\theta &= 2.25 \text{s} \\ \tau\_{\varphi\_v} &= 2.37 \text{s} \end{aligned} \tag{2}$$

These inertial time constants will be used in the subsequent sections for designing and simulating missile formation keeping controller.

After missile formation achieving four-dimensionally rendezvous, this formation should carry out some subsequent missions, such as relative navigation and location, cooperative searching targets, locating and recognizing targets and cooperative penetration etc., however, these missions require missile formation keeping its configuration or relative dimensional states for some periods of time. Next sections will present some further and detailed researches on how to keep missile formation optimally and robustly.

The main thought of the following parts is showed as below: basing on the kinematics relationships between missiles in inertial coordinate frame, the missile formation proportional-derivative (PD) controller via feeding back full states will be present firstly; Second, the relative motion will be established in relative coordinates frame, which can indicate the characteristics of relative motion directly. Based on this direct relative motion, the optimal controller that has non-zero given point and restrains slowly variant perturbations is designed in the third part.

### **3. PD controller of missile formation keeping**

### **3.1 Definition of coordinate frame system**

1. Relative coordinate frame *r r rr o xyz*

182 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

formation configuration will be different when the missile formation is flying in different segments of mission space, that is, the optimal formation configuration will be decided by specific engagement requirement. The formation configuration describing subsystem will also restrict flight status of leader together with cooperative mission planning subsystem at the same time; The leader inner-loop control subsystem receives the information sent from the cooperative engagement planning subsystem and controls the leader to fly stably, and then the real flight states of leader can be obtained; The required states of follower missile can be calculated through the states in relative spatial dimension i.e. the distance between missiles which is obtained from formation configuration describing subsystem; The follower inner-loop control subsystem receives the command states, which is similar to that of leader mentioned above, and then control follower to fly stably. The actual flight states of follower missiles are fed back to the formation configuration control subsystem and used to achieve the expected formation configuration which is decided by the specific cooperative

This chapter focus on missile formation keeping control problems. First, we consider the actual flight states of leader as perturbation variables acting on the controller, and assume the follower inner-controller is closed and stable loop, that is, followers can track the required commands of velocity, flight path angle and flight deflection angle rapidly and stably. We will make further assumption that those three channels referred above are one-

1 ( )

1 ( )

*vf*

 *vf*

flight path angle and flight deflection angle channel of follower respectively. These values can be calculated by analyzing four-dimensional guidance and control system referrd in the

> 3.51s 2.25s 2.37s *v*

These inertial time constants will be used in the subsequent sections for designing and

After missile formation achieving four-dimensionally rendezvous, this formation should carry out some subsequent missions, such as relative navigation and location, cooperative searching targets, locating and recognizing targets and cooperative penetration etc., however, these missions require missile formation keeping its configuration or relative

*v*

*vf vf vfc*

*f f fc vf*

*V VV*

 and

*f f fc f*

1 ( )

*vf* is the velocity, flight path angle and flight deflection angle of follower

*vfc* are commands of velocity, flight path angle and flight deflection

are the inertial time constants of velocity,

(2)

(1)

engagement mission.

where , *Vf f*

respectively; , *Vfc*

 and

> *fc* and

angle of follower respectively; , *vf f*

literature of Cui et al., 2010, they are:

simulating missile formation keeping controller.

order systems expressed as (Wei et al., 2010):

The origin *r o* of this coordinate coincides with the mass centre of leader, and axis *r r o x* points to the velocity direction of leader, *r r o y* is perpendicular to *r r o x* and points to up direction, *r r o z* composes right-hand coordinate frame together with other two axes mentioned above. 2. Inertial coordinate frame system *O XYZ I II I*

The origin *OI* of this coordinate frame is fixed to an arbitrary point on ground, axis *O XI I* lies in horizontal plane and points to target, axis*O YI I* is perpendicular to *O XI I* and points to up direction, the last axis*O ZI I* also composes right-hand coordinate frame system together with those two axes mentioned above.

The relationship between these two coordinate frames is showed in Fig.4.

3. Trajectory coordinate frame system 1 2 22 *o xyz*

The origin of this coordinate 1*o* coincides with the mass centre of missile, and axis 1 2 *o x* points to the velocity direction of missile, 1 2 *o y* is perpendicular to 1 2 *o x* and points to up direction, 1 2 *o z* composes right-hand coordinate frame together with other two axes mentioned above. The spatial directions are parallel to those in relative coordinate frame system.

### **3.2 Model of relative motion between missiles**

#### **3.2.1 Basic assumptions**

It is assumed that the control system of missile self is closed-loop and stable, that is, missile can track the required velocity, flight path angle and flight deflection angle rapidly and stably. Further, we consider these tracking channels are one-order inertial loops described as section 2. In order to express the relationships conveniently, we will change the one-order inertial loops showed in Equ.(1) to the following forms:

$$\begin{cases} \dot{V}\_i = -\mathcal{\lambda}\_v (V\_i - V\_{ic}) \\ \dot{\theta}\_i = -\mathcal{\lambda}\_\theta (\theta\_i - \theta\_{ic}) \\ \dot{\nu}\_{vi} = -\mathcal{\lambda}\_{\psi\_v} (\nu\_{vi} - \nu\_{vic}) \end{cases} \tag{3}$$

Missile Cooperative Engagement Formation Configuration Control Method 185

*r o V*2

*X*2

*V*2

2

*X*2

2 1 2 111 \* 2 1

2 1 11 1 2 1 1 1 1 \* \* 2 1

*d d Y Y y y d d Z Z z z*

*X X x x*

*T T*

*X X x Y Y y Z Z z*

( )() *v*

2 1 \* \* 1 1

*e T T* (7)

( ) ( ) () ( ) *v v v*

Further, the deviations of relative positions can be expressed from Equ.(5) to:

2 1

*v*

 *y*

*v*2

2

*X*1

*z*

*X*1

Fig. 5. Relative position relations between two missiles.

2

*YI*

*Y*1

*ZI*

*Z*1

and making derivatives to last equation yields:

making further derivatives gets:

2 1

*Z*2

*Y*2

*V*1

*v*1

*XI*

*XI*

*r x*

1

*x*

*r z*

1

*r o*

*V*1

\*

\*

*e TT* (6)

\* \*

*x*

1

*r x r y*

where i is the number of missile,*Vi* ,*i* and *vi* represent the actual velocity, flight path angle and flight deflection angle of ith missile respectively;*Vic* ,*ic* and *vic* are desired velocity, desired flight path angle and desired flight deflection angle of ith missile respectively; *v* , and *v* are the reciprocals of inertial time constants *v* , and *v* in channels of veloctiy, flight path angle and flight deflection angle respectively.

### **3.2.2 Establishing the control model**

The kinematics equations of missile in inertial coordinate frame can be expressed as:

$$\begin{cases} \dot{X}\_i = V\_i \cos \theta\_i \cos \varphi\_{vi} \\ \dot{Y}\_i = V\_i \sin \theta\_i \\ \dot{Z} = -V\_i \cos \theta\_i \sin \varphi\_{vi} \end{cases} \tag{4}$$

The kinematics relationship of two missiles in inertial and relative coordinate frame is showed in Fig.5, so the relative positions of two missiles can be obtained from these sketch maps:

$$
\begin{bmatrix} X\_2 \\ Y\_2 \\ Z\_2 \end{bmatrix} = \begin{bmatrix} X\_1 \\ Y\_1 \\ Z\_1 \end{bmatrix} + \mathcal{T}\_2(\boldsymbol{\psi}\_{v1})\mathcal{T}\_1(\boldsymbol{\theta}\_1) \begin{bmatrix} \boldsymbol{x}^\* \\ \boldsymbol{y}^\* \\ \boldsymbol{z}^\* \end{bmatrix} \tag{5}
$$

where \* \* *x* ,*y* and \* *z* are the distances relative to leader in relative coordinate frame, and the transformation matrices in Equ.(5) are:

$$\begin{aligned} \mathbf{T}\_1(.) = \begin{bmatrix} \cos(.) & -\sin(.) & 0\\ \sin(.) & \cos(.) & 0\\ 0 & 0 & 1 \end{bmatrix}, \ \mathbf{T}\_2(.) = \begin{bmatrix} \cos(.) & 0 & \sin(.) \\ 0 & 1 & 0\\ -\sin(.) & 0 & \cos(.) \end{bmatrix} \end{aligned}$$

184 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

direction, 1 2 *o z* composes right-hand coordinate frame together with other two axes mentioned above. The spatial directions are parallel to those in relative coordinate frame

It is assumed that the control system of missile self is closed-loop and stable, that is, missile can track the required velocity, flight path angle and flight deflection angle rapidly and stably. Further, we consider these tracking channels are one-order inertial loops described as section 2. In order to express the relationships conveniently, we will change the one-order

> ( ) ( ) ( )

  (3)

 *v*

(4)

*vic* are desired velocity,

in channels of

*vi* represent the actual velocity, flight path angle

*ic* and

> , and

*v*

desired flight path angle and desired flight deflection angle of ith missile respectively;

cos cos

*i i i vi*

The kinematics relationship of two missiles in inertial and relative coordinate frame is showed in Fig.5, so the relative positions of two missiles can be obtained from these sketch

> 2 1 2 111 \* 2 1

*X X x Y Y y Z Z z*

where \* \* *x* ,*y* and \* *z* are the distances relative to leader in relative coordinate frame, and the

cos sin

*i i vi*

( )() *v*

\*

\*

cos(.) 0 sin(.)

sin(.) 0 cos(.) *T*

(.) 0 1 0

*T T* (5)

sin

*ii i*

*i v i ic i i ic vi vi vic*

*V VV*

*i* and

are the reciprocals of inertial time constants *v*

The kinematics equations of missile in inertial coordinate frame can be expressed as:

*X V Y V Z V*

2 1

cos(.) sin(.) 0 (.) sin(.) cos(.) 0

0 01 *T* , 2

system.

*v* , and *v*

maps:

**3.2.1 Basic assumptions**

where i is the number of missile,*Vi* ,

**3.2.2 Establishing the control model**

transformation matrices in Equ.(5) are:

1

**3.2 Model of relative motion between missiles**

inertial loops showed in Equ.(1) to the following forms:

and flight deflection angle of ith missile respectively;*Vic* ,

veloctiy, flight path angle and flight deflection angle respectively.

Fig. 5. Relative position relations between two missiles.

Further, the deviations of relative positions can be expressed from Equ.(5) to:

$$\mathbf{e} = \begin{bmatrix} X\_2 - X\_1 \\ Y\_2 - Y\_1 \\ Z\_2 - Z\_1 \end{bmatrix} - T\_2(\boldsymbol{\psi}\_{v1})T\_1(\theta\_1) \begin{bmatrix} x^\* \\ y^\* \\ z^\* \end{bmatrix} \tag{6}$$

and making derivatives to last equation yields:

$$\dot{\mathbf{e}} = \begin{bmatrix} \dot{X}\_2 - \dot{X}\_1 \\ \dot{Y}\_2 - \dot{Y}\_1 \\ \dot{Z}\_2 - \dot{Z}\_1 \end{bmatrix} - \frac{d\mathbf{T}\_2(\boldsymbol{\nu}\_{v1})}{d\boldsymbol{\nu}\_{v1}} \dot{\boldsymbol{\nu}}\_{v1} \mathbf{T}\_1(\boldsymbol{\theta}\_1) \begin{bmatrix} \mathbf{x}^\* \\ \mathbf{y}^\* \\ \mathbf{z}^\* \end{bmatrix} - \mathbf{T}\_2(\boldsymbol{\nu}\_{v1}) \frac{d\mathbf{T}\_1(\boldsymbol{\theta}\_1)}{d\boldsymbol{\theta}\_1} \dot{\boldsymbol{\theta}}\_1 \begin{bmatrix} \mathbf{x}^\* \\ \mathbf{y}^\* \\ \mathbf{z}^\* \end{bmatrix} \tag{7}$$

making further derivatives gets:

$$
\begin{split}
\dot{\boldsymbol{\theta}} &= \begin{bmatrix} \ddot{X}\_2 - \ddot{X}\_1 \\ \ddot{Y}\_2 - \ddot{Y}\_1 \\ \ddot{Z}\_2 - \ddot{Z}\_1 \end{bmatrix} - \frac{d\mathbf{T}\_2^2(\boldsymbol{\nu}\_{v1})}{d\boldsymbol{\nu}\_{v1}^2} \dot{\boldsymbol{\nu}}\_{v1}^2 \mathbf{T}\_1(\boldsymbol{\theta}\_1) \begin{bmatrix} \boldsymbol{x}^\* \\ \boldsymbol{y}^\* \\ \boldsymbol{z}^\* \end{bmatrix} - \frac{d\mathbf{T}\_2(\boldsymbol{\nu}\_{v1})}{d\boldsymbol{\nu}\_{v1}} \dot{\boldsymbol{\nu}}\_{v1} \mathbf{T}\_1(\boldsymbol{\theta}\_1) \begin{bmatrix} \boldsymbol{x}^\* \\ \boldsymbol{y}^\* \\ \boldsymbol{z}^\* \end{bmatrix} \\ &- 2\frac{d\mathbf{T}\_2(\boldsymbol{\nu}\_{v1})}{d\boldsymbol{\nu}\_{v1}} \dot{\boldsymbol{\nu}}\_{v1}^2 \frac{d\mathbf{T}\_1(\boldsymbol{\theta}\_1)}{d\boldsymbol{\theta}\_1} \dot{\boldsymbol{\theta}}\_1 \begin{bmatrix} \boldsymbol{x}^\* \\ \boldsymbol{y}^\* \\ \boldsymbol{z}^\* \end{bmatrix} - \mathbf{T}\_2(\boldsymbol{\nu}\_{v1}) \frac{d\mathbf{T}\_1^2(\boldsymbol{\theta}\_1)}{d\boldsymbol{\theta}\_1^2} \dot{\boldsymbol{\theta}}\_1^2 \begin{bmatrix} \boldsymbol{x}^\* \\ \boldsymbol{y}^\* \\ \boldsymbol{z}^\* \end{bmatrix} - \mathbf{T}\_2(\boldsymbol{\nu}\_{v1}) \frac{d\mathbf{T}\_1(\boldsymbol{\theta}\_1)}{d\boldsymbol{\theta}\_1} \dot{\boldsymbol{\theta}}\_1 \begin{bmatrix} \boldsymbol{x}^\* \\ \boldsymbol{y}^\* \\ \boldsymbol{z}^\* \end{bmatrix} \end{split} \tag{8}
$$

where

$$\begin{aligned} \ddot{\theta}\_1 &= -\dot{\lambda}\_\theta (\dot{\theta}\_1 - \dot{\theta}\_{1c}), \quad \ddot{\psi}\_{v1} = -\dot{\lambda}\_{\psi\_v} (\dot{\psi}\_{v1} - \dot{\psi}\_{v1c}) \\\\ \frac{dT\_1(.)}{d(.)} &= \begin{bmatrix} -\sin(.) & -\cos(.) & 0 \\ \cos(.) & -\sin(.) & 0 \\ 0 & 0 & 0 \end{bmatrix}, \quad \frac{dT\_2(.)}{d(.)} = \begin{bmatrix} -\sin(.) & 0 & \cos(.) \\ 0 & 0 & 0 \\ -\cos(.) & 0 & -\sin(.) \end{bmatrix} \\\\ \frac{d^2T\_1(.)}{d^2(.)} &= \begin{bmatrix} -\cos(.) & \sin(.) & 0 \\ -\sin(.) & -\cos(.) & 0 \\ 0 & 0 & 0 \end{bmatrix}, \quad \frac{d^2T\_2(.)}{d^2(.)} &= \begin{bmatrix} -\cos(.) & 0 & -\sin(.) \\ 0 & 0 & 0 \\ \sin(.) & 0 & -\cos(.) \end{bmatrix} \end{aligned}$$

besides, there is:

$$\begin{aligned} \ddot{X}\_i &= \dot{V}\_i \cos\theta\_i \cos\nu\_{vi} - V\_i \sin\theta\_i \dot{\theta}\_i \cos\nu\_{vi} - V\_i \cos\theta\_i \sin\nu\_{vi} \dot{\nu}\_{vi} \\ \ddot{Y}\_i &= \dot{V}\_i \sin\theta\_i + V\_i \cos\theta\_i \dot{\theta}\_i \\ \ddot{Z}\_i &= -\dot{V}\_i \cos\theta\_i \sin\nu\_{vi} + V\_i \sin\theta\_i \dot{\theta}\_i \sin\nu\_{vi} - V\_i \cos\theta\_i \cos\nu\_{vi} \dot{\nu}\_{vi} \end{aligned} \tag{9}$$

so we can get the following vector equation:

$$
\ddot{\mathbf{e}} = f\_1 + \mathbf{G}\_r \mu \tag{10}
$$

Missile Cooperative Engagement Formation Configuration Control Method 187

*V*

2 2 2

1 1 1 1 11

*ZV V V*

asymptotically equivalent to zero, we will choose the following PD control laws:

*Y V V*

further, the required control quantity can be expressed as below:

*X VV V*

*r v*

As for the expression of 1*f* , we also need to know:

The control variables of follower are:

*Gr* can be calculated by Equ.(11), it is:

expression (13) to be converged, which is lim*t*

**4.1 Establishing model of relative motion**

**4.  The condition is: iff matrix *Gr* is non-singular, the control quantity exists. The determinant of

> 2 det cos 2 *v Gr v*

Through checking this expression, it is obvious that *Gr* is non-singular when the missile formation is flying, that is, 2 *V* 0 , so we can choose proper coefficients 1 2 *k k*, to guarantee

*e* **0** .

Because relative coordinate frame is rotating, the relationship between relative derivative and absolute derivative should be considered during the process of establishing relative motion equations in relative coordinate frame. The relationship between these two

1

*c c v c*

2 2 22 2 2 2 2

*v v v v*

>

*v v v v*

*V V*

where

\* \* 1 2 2 2 1 2 \* 2 1 \* 11 2 2 11 1 11 1 1 1 \* \* 1 2 ( ) ( ) ( ) ( ) *v v rv v v v v X V x x d d Y y y d d Z z z T T fG T T* \* \*\* 2 2 1 1 1 \* 1 1 2 \* 1 1 \* 1 1 21 1 21 1 2 1 1 1 1 \* \*\* ( ) ( ) ( ) ( ) 2 ( ) ( ) *v vv v v x xx d dd d y yy d d d d z zz T TT T T T*

$$\mathbf{G}\_r = \begin{bmatrix} \lambda\_v \cos \theta\_2 \cos \psi\_{v2} & -\lambda\_\theta V\_2 \sin \theta\_2 \cos \psi\_{v2} & -\lambda\_{\varphi\_v} V\_2 \cos \theta\_2 \sin \psi\_{v2} \\ \lambda\_v \sin \theta\_2 & \lambda\_\theta V\_2 \cos \theta\_2 & 0 \\ -\lambda\_v \cos \theta\_2 \sin \psi\_{v2} & \lambda\_\theta V\_2 \sin \theta\_2 \sin \psi\_{v2} & -\lambda\_{\varphi\_v} V\_2 \cos \theta\_2 \cos \psi\_{v2} \end{bmatrix} \tag{11}$$

The control variables of follower are:

$$\mathbf{u} = \begin{bmatrix} V\_{2c} \\ \theta\_{2c} \\ \boldsymbol{\nu}\_{v2c} \end{bmatrix} \tag{12}$$

As for the expression of 1*f* , we also need to know:

$$\begin{bmatrix} \ddot{X}\_1\\ \ddot{Y}\_1\\ \ddot{Z}\_1 \end{bmatrix} = \begin{bmatrix} \dot{V}\_1 \cos \theta\_1 \cos \nu\_{v1} - V\_1 \sin \theta\_1 \dot{\theta}\_1 \cos \nu\_{v1} - V\_1 \cos \theta\_1 \sin \nu\_{v1} \dot{\nu}\_{v1} \\\ \dot{V}\_1 \sin \theta\_1 + V\_1 \cos \theta\_1 \dot{\theta}\_1 \\\ -\dot{V}\_1 \cos \theta\_1 \sin \nu\_{v1} + V\_1 \sin \theta\_1 \dot{\theta}\_1 \sin \nu\_{v1} - V\_1 \cos \theta\_1 \cos \nu\_{v1} \dot{\nu}\_{v1} \end{bmatrix}.$$

In order to eliminate the deviations of relative positions, that is, make the deviation *e* be asymptotically equivalent to zero, we will choose the following PD control laws:

$$
\ddot{\mathbf{e}} + k\_1 \dot{\mathbf{e}} + k\_2 \mathbf{e} = \mathbf{0} \tag{13}
$$

so there will be:

186 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

2 1 2 \* 2 1 \*

1 11 1 1 1 ( ), ( )

*c v v vc*

*d d*

> 2 2 2

*d d*

cos cos sin cos cos sin

*i i i vi i i i vi i i vi vi*

>

cos sin sin sin cos cos

*i i i vi i i i vi i i vi vi*

11 2 2 11 1 11 1

1 1 \* \* 1 2 ( ) ( ) ( ) ( ) *v v rv v v v*

*y yy d d d d*

 *T TT T T T*

2 2 1 1 1 \* 1 1 2 \* 1 1 \* 1 1 21 1 21 1 2 1 1 1 1 \* \*\*

*X V x x d d Y y y d d Z z z*

*T T fG T T*

( ) ( ) ( ) ( ) 2 ( ) ( ) *v vv v*

*d dd d*

( ) ( ) ( ) ( )

*v v v v v v*

2

*v*

2 1 2 11 1 11 1 1 1 \* \* 2 1

>

*X X x x d d Y Y y y d d Z Z z z*

\*

*x*

2 1 1 1 \* 1 1 1 1 21 2 1 1 \* 1

*T T e TT*

( ) ( ) ( ) 2 ( )

sin(.) cos(.) 0 (.) cos(.) sin(.) 0 (.) 0 00

cos(.) sin(.) 0 (.) sin(.) cos(.) 0 (.) 0 00

sin cos

*i i i i ii*

1 2 2

*v*

*YV V*

so we can get the following vector equation:

*v*

  *T* ,

*XV V V*

*ZV V V*

 *T* , 2

*v v v*

*T T T T*

*d d d y d d d z*

2 1 2

*v*

1

*d d*

2 1 2

*d d*

besides, there is:

where

where

\* \*

sin(.) 0 cos(.) (.) 000 (.) cos(.) 0 sin(.)

 *T*

cos(.) 0 sin(.) (.) 000 (.) sin(.) 0 cos(.)

>

2 1 2 \* 2 1 \*

\* \*\*

*x xx*

*z zz*

1 *r e f G u* (10)

\* \*

 *T*

\* \* 2 \* 1 1 \* 1 21 1 1 \* \* ( ) ( ) *v x x d y y d z z*

 *T T*

(9)

(8)

$$-k\_1 \dot{e} - k\_2 e = f\_1 + G\_r u$$

further, the required control quantity can be expressed as below:

$$\mathbf{u} = \mathbf{G}\_r^{-1}(-f\_1 - k\_1\dot{\mathbf{e}} - k\_2\mathbf{e}) \tag{14}$$

What should be done next is to analyze the existence conditions of expression (14). The condition is: iff matrix *Gr* is non-singular, the control quantity exists. The determinant of *Gr* can be calculated by Equ.(11), it is:

$$\det \mathbf{G}\_r = -\mathbb{A}\_\vartheta \mathbb{A}\_{\boldsymbol{\psi}\_v} \mathbb{A}\_{\boldsymbol{\psi}\_2} V\_2^2 \cos \theta \tag{15}$$

Through checking this expression, it is obvious that *Gr* is non-singular when the missile formation is flying, that is, 2 *V* 0 , so we can choose proper coefficients 1 2 *k k*, to guarantee expression (13) to be converged, which is lim*t e* **0** .

### **4. Optimal controller of missile formation keeping**

### **4.1 Establishing model of relative motion**

Because relative coordinate frame is rotating, the relationship between relative derivative and absolute derivative should be considered during the process of establishing relative motion equations in relative coordinate frame. The relationship between these two derivatives is:

$$\mathbf{V}\_{fr} - \mathbf{V}\_{lr} = \mathbf{V}\_r + \mathfrak{op} \times \mathbf{r} \tag{16}$$

Missile Cooperative Engagement Formation Configuration Control Method 189

where , *x y* and *z* are components of relative position vector *r* in relative coordinate frame, and the rotating angular velocity of relative coordinate frame with respect to inertial space

can be expressed as:

so, as for the expression

[,,]*T*

sin cos *vl l vl l l*

cos

After expanding all the terms from Equ.(17) to Equ.(20) finally, we can obtain the missile

cos cos cos sin sin cos cos sin cos sin cos sin

The formation motion equations (22) are nonlinear, we can treat them by linearized method to get linear forms that can be utilized and analyzed more conveniently.

We can get the absolute velocities of follower and leader in relative coordinate frame by the transformations as below:

$$\begin{aligned} \mathcal{V}\_{fr} &= \boldsymbol{\Phi}\_{l}^{r} \boldsymbol{\Phi}\_{2}^{l} \boldsymbol{\mathcal{V}}\_{f2} \\ \mathcal{V}\_{lr} &= \mathcal{V}\_{l2} \end{aligned}$$

where *r* **Φ***I* is the transformation matrix from inertial coordinate frame to relative coordinate frame:

$$\mathbf{d}\Phi\_l^r = \begin{bmatrix} \cos\theta\_l \cos\psi\_{vl} & \sin\theta\_l & -\cos\theta\_l \sin\psi\_{vl} \\ -\sin\theta\_l \cos\psi\_{vl} & \cos\theta\_l & \sin\theta\_l \sin\psi\_{vl} \\ \sin\psi\_{vl} & 0 & \cos\psi\_{vl} \end{bmatrix}$$

2 **Φ***I* is the transformation matrix from trajectory coordinate frame of follower *O xyz* 1 222 *f* to inertial coordinate frame:

$$\mathbf{O}\_2^I = \begin{vmatrix} \cos \theta\_f \cos \psi\_{vf} & -\sin \theta\_f \cos \psi\_{vf} & \sin \psi\_{vf} \\ \sin \theta\_f & \cos \theta\_f & 0 \\ -\cos \theta\_f \sin \psi\_{vf} & \sin \theta\_f \sin \psi\_{vf} & \cos \psi\_{vf} \end{vmatrix}$$

*Vf* 2 and*Vl*2 represent the velocity vectors of follower and leader in themselves trajectory coordinate frames respectively, and the components of them are:

$$\mathcal{V}\_{f2} = \begin{bmatrix} V\_f \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix}, \mathcal{V}\_{l2} = \begin{bmatrix} V\_l \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix}$$

Besides,*Vl* is the velocity of leader in inertial coordinate frame;*l* is the flight path angle of leader;*vl* is flight deflection angle of leader.

Further, the difference of absolute velocities between two missiles in the relative coordinate frame is:

$$\mathbf{V}\_{fr} - \mathbf{V}\_{lr} = \begin{bmatrix} V\_f \cos \theta\_f \cos \theta\_l \cos(\nu\_{vl} - \nu\_{vf}) + V\_f \sin \theta\_f \sin \theta\_l - V\_l \\ -V\_f \cos \theta\_f \sin \theta\_l \cos(\nu\_{vl} - \nu\_{vf}) + V\_f \sin \theta\_f \cos \theta\_l \\ V\_f \cos \theta\_f \sin(\nu\_{vl} - \nu\_{vf}) \end{bmatrix} \tag{17}$$

at the same time, the relative velocity between two missiles in relative coordinate frame:

$$\mathcal{V}\_r = [\dot{\mathbf{x}}, \dot{\mathbf{y}}, \dot{\mathbf{z}}]^T \tag{18}$$

where , *x y* and *z* are components of relative position vector *r* in relative coordinate frame, and the rotating angular velocity of relative coordinate frame with respect to inertial space can be expressed as:

$$\mathbf{d}\mathbf{p} = \begin{bmatrix} \dot{\psi}\_{vl}\sin\theta\_l\\ \psi\_{vl}\cos\theta\_l\\ \dot{\theta}\_l \end{bmatrix} \tag{19}$$

so, as for the expression*r* , there will be:

188 Recent Advances in Robust Control – Theory and Applications in Robotics and Electromechanics

*VVV r fr lr r*

where*Vfr* and*Vlr* are the absolute velocities of follower and leader in relative coordinate frame respectively;*Vr* is the relative velocity from follower to leader in relative coordinate

We can get the absolute velocities of follower and leader in relative coordinate frame by the

2

where *r* **Φ***I* is the transformation matrix from inertial coordinate frame to relative coordinate

cos cos sin cos sin sin cos cos sin sin sin 0 cos

 **Φ**

*I l vl l l vl*

2 **Φ***I* is the transformation matrix from trajectory coordinate frame of follower *O xyz* 1 222 *f* to

cos cos sin cos sin sin cos 0 cos sin sin sin cos

 **Φ**

*f f*

*Vf* 2 and*Vl*2 represent the velocity vectors of follower and leader in themselves trajectory

2 2 0, 0 0 0

Further, the difference of absolute velocities between two missiles in the relative coordinate

*fr lr f f l vl vf f f l*

at the same time, the relative velocity between two missiles in relative coordinate frame:

*V*

*V V*

cos cos cos( ) sin sin cos sin cos( ) sin cos cos sin( )

*f f l vl vff f l l*

*V V V*

*f f vl vf*

*V V* (17)

 *V V V V*

*f l*

*lr l*

coordinate frames respectively, and the components of them are:

Besides,*Vl* is the velocity of leader in inertial coordinate frame;

*vl* is flight deflection angle of leader.

 *V V* **Φ Φ** *V V*

*r I fr I f*

vector of follower relative to leader in relative coordinate frame.

*r*

2

*I*

frame;

frame: