Adaptive backstepping sliding mode control for n-DOF serial hydraulic manipulator 23

CHAPTER 3 Dynamics of n-DOF Serial Hydraulic Manipulator Including Actuator Dynamics

3.5 Adaptive backstepping sliding mode control for n-DOF serial hydraulic manipulator 23

Regarding the system dynamics derived in the previous sub-section, the backstepping sliding mode control for an n-DOF system is designed. First, define state error as e₁x₁x_1dRⁿ, and

2 2 2

n

d R

  

e x x where x_1d,x_2dRⁿ are the vector of desired joint angles and angular velocities, respectively.

Step 1: the sliding surface and its derivative are designed as

1 2 1 1

1 2 1 2

n n

R R

   



  



s e λ e

s e λ e (3.25)

with λ1diag



11 12 ... 1_i ... 1_n



R^{n n} is the positive definite matrix.

A new equivalent variable is introduced as

2 2 1 2 1 1

2 2 1 2 1 2

n

s d

n

s d

R R

     



    



x x s x λ e

x x s x λ e (3.26)

Define ˆΔ is an estimation of mismatched lumped uncertainties, and Δ Δ Δ ˆ is an estimated error between the actual and estimated lumped uncertainties. The estimation of the lumped uncertainties satisfies:

ˆ   1

Δ s (3.27)

Choose a Lyapunov candidate V₁ as

1 1 1

1 1

2 2

T T

V  s Ms  Δ Δ (3.28)

Taking derivative the Lyapunov candidate V₁, with Property 4-1 and substituting Equations (3.25) and (3.26) yields:

  ^{   }

 

1 1^T 3 ˆ 1 2_s ( ,1 2) 2_s ( )1 1 ^T _ext ^T

V s x  Δ Δ M x x C x x x G x J x F Δ Δ (3.29) Define an error between an actual virtual torque x₃ and desired torque x_3d as e₃x₃x_3dRⁿ. Then the desired torque is designed as

   

¹

3_d 1 2_s ( ,1 2) 2_s ( )1 1 ^T _ext ˆ 1 1 1tanh

s

 

        

 

x M y x C x x x G x J x F Δ K s η s (3.30)

where K1diag K



11 K12 ... K1_i ... K1_n



and η1diag



11 12 ... 1_i ... 1_n



are n-by-n positive definite matrices. The term K s_{1 1} is added to enhance the convergence rate when s₁ is large;

1

1 11 12

tanh tanh tanh ... tanh

T n n

s s s s

s

s s

    R

 

       

  

        

        

s with s_1i is the sliding surface of joint i^th

and _s is an arbitrarily small value.

Remark 3-2: The term η1tanh



s1 _s



is employed instead of using signum function, i.e.,

1sign

 

1

η s because this discontinuous function is un-differentiable. Therefore, to closely achieve the robustness, small value of gain _s is chosen.

Substituting Equation (3.30) into Equation (3.29) yields:

1

1 1^{T T}_a 3 1 1 1tanh ^T

s

V 

   

      

s J e K s η s Δ Δ Δ (3.31)

Substituting the adaptive law in Equation (3.27) yields:

1

1 1^{T T}_a 3 1 1 1tanh _mismatched

s

V 



   

      

s J e K s η s (3.32)

Remark 3-3: The aim of using the adaptive law in Equation (3.27) is to cancel out influences of the lumped uncertainties to force the model as same as an ideal model. In practical simulation, to enhance the convergence rate of estimation, this equation is modified as

 

1 1 1 1

ˆ   s  ssign

Δ K s η s (3.33)

where K_s1,η_s1R^{n n} are diagonal positive definite matrices of learning rate to increase the convergence rate of the estimated lumped uncertainties.

Remark 3-4: Regarding Assumption 3-3, the lumped uncertainties Δ is unknown but bounded and its estimated error is also bounded by ˆ

mismatched



  

Δ Δ Δ , then the term, η₁ is chosen to be greater than the upper bound of the estimated error, i.e., _1imismatched i_, . This means the controller gains are much smaller instead of designing with large value enough to cover the upper bound of uncertainties in the conventional design.

From Equation (3.32), the derivative of the Lyapunov function V₁ is semi-negative if the error e₃ converges to zero. Therefore, this error should be compensated so as to keep the torque error as small as possible.

Step 2: Define the sliding surface as

2 3 2 3

Rn

 





s e λ e (3.34)

where λ2 diag



21 22 ... 2_i ... 2_n



R^{n n} is a positive matrix definite.

This step is designed to deal with the actuator dynamics. Regarding the third equation of (3.16), the internal leakage term, Qleak i, Cleak i,



Pai1Pai2



, cannot be measured; then, this term should be estimated using adaptive law. Define the ˆ

leak  leak leak

Δ Q Q as the estimated error between the estimated and actual internal leakage.

Taking derivative the sliding surface s₂, one obtains:

 

2  3 2 3 f 1, 2   _leak 3_d  2 3

s e λ e x x gu ψQ x λ e (3.35)

Choosing the second Lyapunov function V₂ as

2 1 2 2

1 1

2 2

T T

leak leak

V  V s s  Δ Δ (3.36)

Taking derivative of the Lyapunov function V₂ and substituting Equation (3.35) yields:

 

 

1

2 1 3 1 1 1

2 1 2 3 2 3

tanh ,

T T

a mismatched

s

T T

leak d leak leak

V 

   

      

     

s J e K s η s δ

s f x x gu ψQ x λ e Δ Δ

(3.37)

To guarantee the system robustness and global stability, the control input signal u is designed as

 

1 2

1, 2 ˆ_leak 3_d 2 2 2tanh ^T_a 1 2 3

s

    

         

u g f x x ψQ x K s η s J s λ e (3.38)

where K2 diag K



21 K22 ... K2_i ... K2_n



and η2 diag



 21 22 ... 2_i ... 2_n



are n-by-n positive definite matrices. The term K s_{2 2} is added to enhance the convergence rate when s₂ is large.

A function tanh tanh ¹ tanh ² ... tanh

T n

i i i in

s s s s

s s s

    R

 

       

  

        

        

s (i=1,2) in both Equations

(3.30) and (3.38) are used to replace a conventional signum function to achieve chattering free.

Hence, the derivative of the Lyapunov function V₂ becomes:

1 2

2 1 1 1 1 1 2 2 2 2 2

2 1 2 3

tanh tanh

T T T T

ext

s s

T T T T

leak leak leak a

V  

     

         

  



s s

s K s s η F s K s s η

s ψΔ Δ Δ s J λ e

(3.39)

To achieve system performance as an ideal model, the term s ψΔ^T_{2 leak} Δ Δ^T_{leak leak} should be cancelled out. Therefore, the following adaptive law is satisfied:



²



, 1 2

1 2

,

, if

ˆ

0 otherwise

i i

leak i ai ai

ai ai

leak i

s P P

P P

C

 

 

 

 

(3.40)

Remark 3-5: The adaptive law, presented in Equation (3.40), is considered in the steady state, i.e., all parameters are in equilibrium points. In other words, the constant values of pressures and actual leakage coefficient are exhibited; therefore, their derivative are zero, i.e., Cˆ_{leak i,} P_ai1P_ai20. This assumption is to facilitate the control strategy design and verify the effectiveness of using BSMC. Indeed, applying this adaptive law is difficult to accurately estimate the matched uncertainties since these parameters are time-variant. Therefore, the use of an observer, will be presented in the later chapters, is preferable to obtain accurate estimations.

Remark 3-6: The aim of using the adaptive law in Equation (3.40) is to cancel out influences of the leakage to achieve the ideal model with no uncertainties or disturbances. In practical simulation, to enhance the convergence rate of estimation, this equation is modified as

 



¹

  ^{ }

¹

 ^{ }

, 1 2 2 , 1 2 2

ˆleak i  leak diag a  a ^  leak s diag a  a ^ sign

C η ψ P P s η ψ P P s (3.41)

where η_leak and η_{leak s,} are n-by-n diagonal positive definite matrices to adjust the convergence rate of the estimation; sign

 

s2 



sign s

 

21 sign s

 

22 ... sign s

 

2_n



^TRⁿ.

Remark 3-7: Regarding Assumption 3-5, the estimated error of the leakage coefficient is bounded by

leak  ˆleak  leak  matched _matched

Δ Q Q δ . Then the robust matrix gain η₂ is chosen to be greater than the error of estimated leakage coefficient, i.e., _2i_matched. This means the controller gains are much smaller instead of designing with large value enough to cover the upper bound of uncertainties in conventional design.

Consequently, the derivative of Lyapunov function V₂ becomes:

1 2

2 1^T 1 1 1^T 1tanh _ext 2^T 2 2 ^T2 2tanh 1^{T T}_a 2 3

s s

V  

     

 ^{s K s} ^s ^η  ^s ^F ^{s K s} ^{s η} ^s ^{s J λ e}



^(3.42)

1 2

2 ^T 1^T 1tanh _ext 2^T 2tanh

s s

V S S

 

     

        

s s

Ω s η F s η (3.43)

where

1 3 3

S

 

 

  

 

 



s e e

and

1 2

2

2 2 2 2 2

2 2 2

1 2 1

2

T

a n

T a

n

 

 

 

 

  

 

 

 

 

K J λ 0

Ω J λ K λ K λ

0 K λ K

.

Hence, in order to guarantee the system stability and robustness, the robust matrices K K₁, ₂ and slope λ₂ should be carefully designed such that the matrix

Ω

is semi-positive definite; therefore, the V2 becomes semi-negative definite. In the special case, when λ₂ is zero, the second sliding surface is simplified as a torque tracking error, i.e., s e . Then, V becomes:

1 2

2 1^T 1 1 1^T 1tanh _ext ^T2 2 2 ^T2 2tanh 0

s s

V  

     

         

s s

s K s s η F s K s s η (3.44)

as long as Assumptions 3-2 to 3-5 and Remarks 3-4 and 3-7 are hold.

Remark 3-8: In practical system, the position of the end-effector can be calculated from angular position of each joint. These values can be directly measured by using encoders. However, angular and acceleration are not available. Conventionally, these variables are obtained from taking derivative of joint angles and thereby, measured noises are also amplified. To effectively estimate unmeasurable variables for designing the controller, a second-order exact differentiation (SOED) is utilized as

 

 

 

1 1

2/3

1 1 1 1 2

2 2

1/ 2

2 2 2 1 2 1 3

3 3 2 1

sign

sign sign

 

    

 



    

   

 υ f

f κ υ q υ q υ

υ f

f κ υ f υ f υ

υ κ υ f

(3.45)

where υ υ υ₁, ₂, ₃ are estimated values of , ,q q q, respectively. Then the estimation of υ υ₂, ₃ can converge to true velocity and acceleration in finite time by choosing suitable values of observer gains κ κ κ₁, ₂, ₃ [29,78–80].

Remark 3-9: It should be noted that all estimated errors of the SOED can achieve finite-time convergence no matter what a control input signal is. This implies that the SOED observer can be designed separately to the main controller and thereby, a full state is available for designing control signals.

Remark 3-10: Generally, increasing those observer gains κ κ κ₁, ₂, ₃ results in faster convergence.

However, large values of observer gain also result in more noise influence. Too large or too small values of observer gains may lead to overshoot. Therefore, the values of all observer gains κ κ κ₁, ₂, ₃ should be chosen based on carefully tuning. Adaptive laws for automatically adjusting observer gains are not considered as the main scope in this scope.

As shown in Equation (3.44), V2



s s1, 2



is a negative-definite function, i.e.,

   

       

2 1 , 2 2 1 0 , 2 0

V s t s t V s s . It means that s₁ and s₂ are bounded. Let a function

2 1 1 1 2 2 2 2

T T

Pv  s K s s K s V and integrate function P_v2 with respect to time.

             

2 2 1 2 2 1 2

0

0 , 0 ,

t

Pv  d V s s V s t s t



^(3.46)

Because V2



s1

   

0 ,s2 0



is a bounded function, and V2



s1

   

t ,s2 t



is non-increasing and bounded, so:

2

 

0

lim

t

t Pv  d





  ^(3.47)

Additionally, P_v2 is also bounded, by Barbalat’s lemma [13], it can be shown that

2

 

lim 0^t _v 0

t P  d





 and the sliding variable constants are chosen to satisfy the Hurwitz theorem. They mean that the vectors s1

       

t ,s2 t ,e1 t ,e2 t will converge to zero as t.

3.6 Remained problems and discussion

Although the adaptive laws are designed to reduce influences of matched and mismatched uncertainties, they cannot track exactly the actual value since their calculations are directly based on the errors of tracking performance. Besides, the external force disturbance is supposed to be measured, then it is directly feedback to the controller design. If the external force is lumped into the mismatched uncertainties, the adaptive law, essentially, can also satisfy. However, in practical, these estimated errors are minimized such that they tend to zero under the regulation of the advanced control with adaptive laws but cannot be equal to zero at all time. Moreover, since the system state tracking errors oscillates around zero, then the tracking performance of the lumped mismatched uncertainties also oscillates around their real values. This implies that the estimated state cannot be accurately estimated even when increasing large updating gains. Therefore, another method like using specific observer should be considered to exhibit more precise estimation of the external force disturbance.

CHAPTER 4