# Valve Deadzone/Backlash Compensation for Lifting Motion Control of Hydraulic Manipulators

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Model and Problem Formulation

#### 2.1. Manipulator Dynamics

_{1}and O

_{2}represent the rotation centers of the upper and lower ears of the hydraulic cylinder, respectively; O

_{3}is the center of gravity of the manipulator, and let OO

_{2}= L

_{1}, O

_{1}O

_{2}= L

_{2}, OO

_{1}= L

_{3}, OO

_{3}= L

_{4}, O

_{1}′O

_{2}= L, ∠O

_{1}OO

_{3}= β

_{0}, ∠O

_{1}OO

_{2}= q

_{0}, ∠O

_{1}′OO

_{1}= q, ∠OO

_{1}′O

_{2}= α

_{0}.

_{1}A

_{1}− P

_{2}A

_{2}is the output force of the cylinder, P

_{1}and P

_{2}are the pressures in the cylinder forward and return chamber, respectively, A

_{1}and A

_{2}are the ram areas of the forward and return chambers, respectively; m is the manipulator mass; f(t) represents the nonlinear friction torque and its expression will be given later; d

_{1}(t) is the modeling error including external disturbances and unmodeled dynamics, etc.

_{p}= L − L

_{2}and combining Equation (2), we then have

#### 2.2. Friction Dynamics

_{0}, σ

_{1}and σ

_{2}are friction force parameters, which can be physically interpreted as the stiffness of the bristles between two contact surfaces, damping coefficient of the bristles, and viscous coefficient, respectively; the unmeasurable internal friction state z physically stands for the average deflection of the bristles between two contact surfaces, and its dynamics is given by

_{c}and f

_{s}denote the level of the normalized Coulomb friction and stiction force, respectively; c

_{1}, c

_{2}, c

_{3}are various shape coefficients to approximate various friction effects.

**Remark**

**1.**

#### 2.3. Pressure Dynamics

_{1}(q) = V

_{01}+ A

_{1}x

_{p}, V

_{2}(q) = V

_{02}− A

_{2}x

_{p}are the control volume of the forward and return chambers, respectively; V

_{01}and V

_{01}are the original total volumes of the two chambers; β is the effective oil bulk modulus; C

_{t}is the internal leakage coefficient; Q

_{1}is the supplied oil flow to the forward chamber and Q

_{2}is the return oil flow to the return chamber, and both of them are positive; d

_{21}(t) and d

_{22}(t) are the modeling errors in the dynamics of the two chambers, including complex unmodeled leakage, valve dynamics, etc.

#### 2.4. Flow Characteristics

_{s}and P

_{r}are the supplied and return pressures, respectively; k

_{q}is the flow gain; x

_{v}is the spool displacement of the proportional valve; C

_{d}is the discharge coefficient; w is the area gradient of the orifice; ρ is the density of the oil.

_{1}can be uniformly expressed as

_{2}has a similar form as:

_{v}= k

_{i}u, where k

_{i}is a positive electrical constant and u is the control input voltage. Therefore, Equations (16) and (17) can be transformed to

_{t}is the total flow gain with respect to the control input u.

**Remark**

**2.**

## 3. Nonlinear Adaptive Robust Controller Design

#### 3.1. Design Model and Issues to Be Addressed

_{0}, σ

_{1}, σ

_{2}, k

_{t}, β, C

_{t}. Define an unknown parameter vector as θ = [θ

_{1}, θ

_{2}, θ

_{3}, θ

_{4}, θ

_{5}, θ

_{6}]

^{T}, where θ

_{1}= σ

_{0}, θ

_{2}= σ

_{1}, θ

_{3}= σ

_{1}+ σ

_{2}, θ

_{4}= k

_{t}β, θ

_{5}= β, θ

_{6}= C

_{t}β. Define a set of state variables as $x={[{x}_{1},{x}_{2},{x}_{3},{x}_{4}]}^{T}=[q,\dot{q},$${P}_{1},{P}_{2}{]}^{T}$. Utilizing these state variables, the system described by Equations (5), (6), (9), (19) can be expressed as

_{d}(t) = x

_{1d}(t), design a bounded control input u such that the system output x

_{1}can track x

_{1d}as closely as possible despite of system nonlinearities, various modeling uncertainties as well as valve deadzone/backlash nonlinearities.

**Assumption**

**1.**

_{1d}∈C

^{3}and bounded; in practical hydraulic systems under normal working conditions, P

_{1}and P

_{2}are always bounded by P

_{s}and P

_{r}(i.e., 0≤ P

_{r}< P

_{1}< P

_{s}, 0≤ P

_{r}< P

_{2}< P

_{s}).

**Assumption**

**2.**

_{max}= [θ

_{1max}, …, θ

_{6max}]

^{T},θ

_{min}= [θ

_{1min}, …, θ

_{6min}]

^{T}are the known upper and lower bounds; and the unmodeled disturbances D

_{1}(t), D

_{2}(t) are bounded by

_{1}, δ

_{2}are the upper bounds of D

_{1}(t) and D

_{2}(t), respectively; and

#### 3.2. Projection Mapping and Parameter Adaptation

_{i}denotes the ith element of the vector •, and the operation < for two vectors is performed in terms of the corresponding elements of the vectors.

_{1}f

_{3}z and θ

_{2}f

_{4}z, according to [23], the following dual state observer is constructed

_{1}and η

_{2}are two learning functions to be synthesized later; the projection mapping in Equation (27) is given as

_{max}= f

_{s}, z

_{min}= −f

_{s}).

_{1}and η

_{2}, the projection mapping in Equation (28) guarantees [23]

#### 3.3. Controller Design

_{1}= x

_{1}− x

_{1d}is the output tracking error; k

_{1}is any positive feedback gain; α

_{1}is the virtual control input of x

_{2}, e

_{2}is the deviation between them; differentiating Equation (31) and noting Equation (20), we have

_{L}can be treated as the virtual control input and simultaneously design a virtual control function α

_{2}for it. Additionally, define the input discrepancy as e

_{3}= P

_{L}− α

_{2}then the virtual control law α

_{2}can be designed as

_{2}is a positive feedback gain; α

_{2a}is the model compensation with the parameter estimates $\widehat{\theta}$; α

_{2s}is the robust feedback control consisting of the linear term α

_{2s1}and the nonlinear term α

_{2s2}. Substituting Equation (33) and the expression of e

_{3}into Equation (32), we then acquire

_{2}is written as

_{M}= θ

_{max}− θ

_{min}, θ

_{1M}= θ

_{1max}− θ

_{1min}, θ

_{2M}= θ

_{2max}− θ

_{2min}, z

_{M}= z

_{max}− z

_{min}, ε

_{2}is a positive design parameter. Then the above designed α

_{2s2}satisfies:

_{3}, its dynamics can be represented as

_{2}is assumed to be known, which means the motion acceleration of the hydraulic manipulator should be measurable. If the acceleration is unavailable, the design method and main results of this paper are still effective, with just making some modifications, as illustrated in [13]. From Equation (38), the resulting controller is given as

_{a}is the model compensation term with online parameter adaptation; u

_{s}

_{1}is the negative linear feedback term with gain k

_{3}> 0 to stabilize the nominal model of the system; u

_{s}

_{2}is the nonlinear robust feedback term to handle modeling uncertainties in Equation (38). Combining Equation (39), we can rewrite Equation (38) as

_{3}is defined as

_{s}

_{2}is designed as

_{3}> 0 is an arbitrarily small design parameter. Then u

_{s}

_{2}satisfies the following conditions:

#### 3.4. Main Results

**Theorem**

**1.**

_{1}(t) = D

_{2}(t) = 0, namely, there only exists parametric uncertainties and friction nonlinearity in the system, choosing large enough feedback gains k

_{1}, k

_{2}, k

_{3}, such that the following defined matrix Λ is positive definite:

_{2}e

_{2}+ φ

_{3}e

_{3}, giving the dual state observer in Equation (27) and learning functions as

_{1}and γ

_{2}are positive learning gains, then the proposed control method (39) can ensure that all the system signals are bounded under closed-loop operation, and asymptotic tracking performance is also obtained (i.e., t→∞, e

_{1}, e

_{2}, e

_{3}→ 0).

**Theorem**

**2.**

_{1}(t), D

_{2}(t) are not zero at the same time), then the designed control law (39) can guarantee that all system signals are bounded under closed-loop operation. Define the Lyapunov function as

_{1}, it can be inferred that f

_{1}is always positive within the motion angle of the manipulator. Hence, V

_{2}is positive definite. Additionally, it is bounded by

_{1max}is the maximum of f

_{1}within the angular range of the manipulator motion. In such a case, it can also be obtained that t → ∞, e

_{1}, e

_{2}, e

_{3}will be bounded by $\sqrt{\frac{2\epsilon \prime}{\kappa}}$.

## 4. Simulation Results and Discussion

_{s}= 21 MPa, P

_{r}= 0 MPa, A

_{1}= 3.14 × 10

^{−2}m

^{2}, A

_{2}= 1.6 × 10

^{−2}m

^{2}, V

_{01}= 3.1416 × 10

^{−4}m

^{3}, V

_{02}= 3.04 × 10

^{−2}m

^{3}, J = 1.5 × 10

^{5}kg·m

^{2}, m = 10 t, L

_{1}= 1.6 m, L

_{2}= 2 m, L

_{3}= 3.5 m, L

_{4}= 3 m. The nonlinear function N(x

_{2}) is selected as N(x

_{2}) = x

_{2}/{2 × 10

^{−3}× [tanh(15 × 2) − tanh(1.5 × 2)] + 3 × 10

^{−3}× tanh(900 × 2)}. The simulation step size is set to 0.5 ms and the applied disturbance torque d

_{1}(t) = 5000sin(t) N·m. The following three controllers are compared:

_{1}= 200, k

_{2}= 5 × 10

^{7}, k

_{3}= 80. The initial estimate of θ is chosen as ${\widehat{\theta}}_{0}=[5\times {10}^{5},\hspace{0.17em}1100,\hspace{0.17em}2\times {10}^{5},60,\hspace{0.17em}$$5\times {10}^{8},0{]}^{T}$. The initial estimate of z is ${\widehat{z}}_{1}(0)={\widehat{z}}_{2}(0)$. The bounds of θ are chosen as θ

_{max}= [1 × 10

^{7}, 1 × 10

^{4}, 3 × 10

^{6}, 500, 2 × 10

^{9}, 0.01]

^{T}, θ

_{min}= [0, 900, 0, 50, 2 × 10

^{8}, 0]

^{T}. The bounds of z estimation are z

_{max}= 5 × 10

^{−3}, z

_{min}= −5 × 10

^{−3}. Parameter adaptation rates are set at Γ = diag{2 × 10

^{8}, 500, 2 × 10

^{5}, 2 × 10

^{−13}, 100, 1.5 × 10

^{−20}}. Friction state learning gains are γ

_{1}= 5 × 10

^{−3}, γ

_{2}= 5 × 10

^{−3}.

**Remark**

**3.**

_{1}, k

_{2,}and k

_{3}. We may implement the needed robust control gains in the following two ways. The first method is to pick up a set of values for k

_{1}, k

_{2,}and k

_{3}rigorously to ensure the theoretical stringency with various prerequisites. However, it increases the complexity of the resulting control law considerably since it may need a significant amount of offline investigating work, sometimes even be impossible. Alternatively, a pragmatic approach is to simply choose k

_{1}, k

_{2,}and k

_{3}large enough without worrying about the specific prerequisites. In this way, prerequisites will be satisfied for a certain set of values of k

_{1}, k

_{2,}and k

_{3}, at least locally around the desired trajectory to be tracked. In this paper, the second approach is used since it facilitates the online tuning process of gains in implementation.

_{1}= γ

_{2}= 0). Hence, it will be used to illustrate the effectiveness of the smooth dynamic LuGre friction model proposed in ARCBF. The control parameters are also selected to be consistent with those of ARCBF.

**Case 1:**The desired position trajectory x

_{1d}= 5[1 − cos(0.5t)][1 − exp(−0.1t)]° is first implemented, shown in Figure 3. In this case, the three controllers are tested for two kinds of proportional valves with different backlash values (1 × 10

^{−6}m and 1 × 10

^{−5}m). It can be used to illustrate the influence of backlash/deadzone on control performance and the effectiveness of the new comprehensive pressure-flow equation proposed in ARCBF. The compared tracking errors of the three controllers controlled by the two valves are shown in Figure 4 and Figure 5, respectively. As illustrated, the designed control strategy ARCBF has the best tracking performance among the compared controllers since the backlash nonlinearity and nonlinear friction have been compensated effectively. It is interesting to note that, without using the backlash compensation, the control accuracy of ARCF gets worse sharply as the valve backlash increases. Additionally, by comparing the tracking errors between ARCBF and ARCB, it can be inferred that the proposed dynamic friction compensation scheme of ARCBF can effectively suppress the nonlinear friction effects in electrohydraulic systems. Furthermore, the simulation results when ε = 1 × 10

^{−6}are depicted in Figure 3, Figure 6, Figure 7 and Figure 8. The position tracking performance of ARCBF is shown in Figure 3. From Figure 6, the convergence of the parameter estimation of ARCBF is rather good, which can demonstrate the validity of parameter adaptive law. The estimation of unknown friction states of ARCBF are presented in Figure 7 and the control input is in Figure 8.

**Case 2:**To further verify the control capability of the proposed control scheme, a higher frequency motion trajectory x

_{1d}= [1 − cos(3.14t)][1 − exp(−0.1t)]° is tested and the valve backlash ε = 1 × 10

^{−5}. The position tracking performance of ARCBF is seen from Figure 9 and the compared tracking errors of the three controllers are shown in Figure 10. In this test stage, since the proportional valve switches more frequently, backlash has a greater impact on the tracking performance. However, even in such a high-frequency tracking test, the proposed control strategy is able to compensate for the unexpected effects and achieves the best performance among the three compared controllers, as shown in Figure 10. The parameter adaptation and the friction state estimation, as well as the control input of the proposed ARCBF, are omitted due to space limitation.

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Theorem**

**1.**

_{1}(t) = D

_{2}(t) = 0, then we can infer

_{1}, e

_{2}, e

_{3}]

^{T}. Combing the definition of the learning functions η

_{1}and η

_{2}, and the property in Equation (29), we can upper bound the above equation as

_{2}) is always positive, then

_{min}(Λ) is the minimal eigenvalue of matrix Λ. Therefore, V

_{1}∈ L

_{∞}and W ∈ L

_{2}. Based on the definition of V

_{1}in Equation (A1), it can be inferred that ${e}_{1},{e}_{2},{e}_{3},\tilde{\theta},{\tilde{z}}_{1},{\tilde{z}}_{2}$ are bounded; from assumptions 1 and 2, we can infer that x is bounded; based on Equation (34) and assumption 2, the boundness of ${\dot{e}}_{2}$ is; thus, obtained; from Equation (33), we can easily obtain that ${\dot{\alpha}}_{2}$ is bounded. Then the boundedness of the control input u can be concluded. Hence, all system signals are bounded under closed-loop operation.

_{1}, e

_{2}, e

_{3}, it is easy to check that the time derivative of W is bounded, thus W is uniformly continuous. By applying Barbalat’s lemma [28], W→0 as t→∞, which leads to the results in Theorem 1. □

## Appendix B

**Proof**

**of**

**Theorem**

**2.**

_{2}defined in Equation (46) can be presented by

_{2}is bounded (i.e., e

_{1}, e

_{2}, e

_{3}are bounded). Similar to the proof in Theorem 1, the boundness of u can also be obtained. □

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**MDPI and ACS Style**

Li, L.; Lin, Z.; Jiang, Y.; Yu, C.; Yao, J.
Valve Deadzone/Backlash Compensation for Lifting Motion Control of Hydraulic Manipulators. *Machines* **2021**, *9*, 57.
https://doi.org/10.3390/machines9030057

**AMA Style**

Li L, Lin Z, Jiang Y, Yu C, Yao J.
Valve Deadzone/Backlash Compensation for Lifting Motion Control of Hydraulic Manipulators. *Machines*. 2021; 9(3):57.
https://doi.org/10.3390/machines9030057

**Chicago/Turabian Style**

Li, Lan, Ziying Lin, Yi Jiang, Cungui Yu, and Jianyong Yao.
2021. "Valve Deadzone/Backlash Compensation for Lifting Motion Control of Hydraulic Manipulators" *Machines* 9, no. 3: 57.
https://doi.org/10.3390/machines9030057