Adaptive position tracking control of electro-hydraulic six-degree-of-freedom driving simulator subject to perturbation

Abstract

In this paper, a new Lyapunov-based adaptive control scheme for the position tracking of an electro-hydraulic Stewart–Gough platform type six-degree-of-freedom (DOF) driving simulator is proposed. The kinematics and the dynamics of both the moving platform and the hydraulic actuator are first derived, and the linearized dynamic equation of the moving platform is developed for the controller design. Two control loops are employed, in which the platform controller is in the outer loop and the actuator controllers are in the inner loop. In the platform controller, the compound proportional-differential (PD) feedback and adaptive compensation controller is designed. The PD part is used to create the principal control signal, and the adaptive part is used to compensate for the parameter uncertainty and external disturbance. To suppose that the upper bound of the perturbation is known, a gradient descent method-based adaptive law is developed for parameter estimation. For each actuator controller a servo force control scheme is designed to ensure the actuator to accurately track the desired force output by the platform controller. All the controllers are designed together based on the Lyapunov method to guarantee the stability of the whole system. Numerical simulation on a six-DOF simulator shows that the proposed control scheme has better tracking performance than its non-adaptive counterpart.

Keywords

Driving simulator Stewart–Gough platform adaptive control electro-hydraulic servo system force control

1. Introduction

A driving simulator is an effective equipment for training car drivers and testing on-board instruments, and it usually employs a Stewart–Gough (S-G) platform type six-degree-of-freedom (DOF) parallel mechanism. Since this mechanism was put forward as a tire test machine by Gough in 1956 and proposed as a flight simulator by Stewart in 1965,^1–3 it has attracted much research interest. In recent years, the S-G platform has been widely used in flight or vehicle motion simulation, manipulating additionally complicated surface machining and multi-dimensional vibration isolation.⁴

For the driving simulation, to simulate the motion of a vehicle and provide an immersing driving sense, a S-G platform type simulator is required with a high dynamic performance, especially the position tracking accuracy. There are difficulties in improving the position tracking, as the simulator has the following characteristics. The dynamics of the S-G platform mechanism is nonlinear and highly coupled because each leg exerts some disturbance on others by the moving platform. In addition, the system also has parameter uncertainties, external disturbance and high-order unmodeled dynamics. A large number of highly nonlinear terms in the dynamic equation will cause a heavy computation burden in the model-based control strategies for the platform.⁵ Additionally, an electro-hydraulic S-G platform also introduces the nonlinear dynamics of servo-valves and hydraulic cylinders into the system. Therefore, the high-accuracy tracking control of this platform is a challenging issue. The dynamics of the S-G platform has been studied by a number of methods, such as the Newton–Euler method, virtual work principle and Lagrange equation.^6–8 Lee et al.⁹ proposed the approximate dynamics in the position control of the S-G platform and simplified the nonlinear coefficient matrices into constant ones on the small motion range condition of the platform. Chen et al.¹⁰ employed the experimental method to estimate the inertial parameters and friction coefficients for an electro-hydraulic motion simulator by means of the least squares method. Much research work has been done to improve the tracking performance, and some control strategies have been used on this platform. Kang et al.¹¹ proposed the robust tracking controllers in both Cartesian coordinates and actuator coordinates; Huang et al.¹² designed a sliding-mode controller and presented the stability analysis based on Lyapunov stability theorem to guarantee that the motion tracking error converges to zero asymptotically. Other sliding-mode control designs included the direct sliding-mode controller, integral sliding-mode controller and adaptive sliding controller.^13–15 However, the above studies did not consider the hydraulic actuator dynamics. Other control methods, such as generalized predictive control,¹⁶ H-infinity control¹⁷ and adaptive control, are also used for the S-G platform.¹⁸ The adaptive control in Huang and Fu¹⁸ supposed the overall system parameters are subject to uncertainties and did not consider the external disturbance. Adaptive control uses the on-line parameter estimation method for the real-time parameters of the control plant to ensure that the control law has satisfactory performance when the plant parameters are completely unknown and/or could change with time.¹⁹ Adaptive control has been successfully applied in many fields, such as aircraft autopilots and complicated industrial process control.^20,21

In this paper, we propose a new adaptive control strategy for the position tracking of the S-G platform with perturbation of both parameter uncertainties and external disturbance. We suppose the upper bound of the perturbation is known. We consider both the platform dynamics and the actuator dynamics, and employ two loops with the platform controller in the outer loop and the actuator controllers in the inner loop in the overall control scheme. In the platform controller, the compound proportional-differential (PD) feedback and the adaptive compensation controller is designed. The PD part is used to create the main control signal, and the adaptive part is used to compensate for the parameter uncertainty and external disturbance. For the actuator controllers in the inner loop, we designed a servo force controller to ensure the actuator accurately tracks the desired force, which is output by the platform controller. The main differences between this work and Wu and Gu¹⁵ are that the latter used an adaptive controller to identify the constant uncertain parameter, plus a sliding controller to eliminate the influence of uncertain parameters and the external disturbance, and it did not consider the dynamics of hydraulic actuator. Our method uses a PD controller plus an adaptive controller to compensate for the parameter uncertainty and external disturbance, and considers the hydraulic actuator dynamics and control. All the platform controller and the actuator controllers are designed together based on the Lyapunov method to guarantee the stability of the whole system. Finally, simulation is completed to validate the proposed method.

2. Driving simulator dynamics

The S-G type driving simulator consists of a fixed base, a moving platform and six variable-length legs to connect the fixed base and the moving platform (see Figure 1). The connection of the fixed base and each leg is through a universal joint, and that of the moving platform and each leg is through a universal joint or spherical joint. For an electro-hydraulic S-G platform, each leg employs a hydraulic servo-actuator, which can perform accurate motions controlled by the servo-valve.

2.1. Kinematics and dynamics of themoving platform

To describe the motion of the moving platform, two coordinate frames are established with one designated O₁-X₁Y₁Z₁ (hereinafter referred to as {p}) moving with the platform and another designated O-XYZ (hereinafter referred to as {b}) fixed on the ground, as shown in Figure 1. These two coordinate frames coincide when the moving platform is in its initial neutral position. A generalized coordinate vector q is used to describe the position and orientation of the moving platform, q = [x,y,z,α,β,γ]^T, where x, y and z are the coordinates of the point O₁ in {b}, and α, β and γ are the orientation angles of the moving platform relative to the fixed coordinate frame {b}. These three angles are Z-Y-X Euler angles, defining the three sequential rotations of coordinate frame {p}: first γ degrees around the Z₁ axis, next β degrees around the Y₁ axis and finally α degrees around the X₁ axis. So the total transformation is

R_{α β γ} = R_{α} R_{β} R_{γ} = [\begin{matrix} C β \cdot C γ & - C α \cdot S γ + S α \cdot S β \cdot C γ & S α \cdot S γ + C α \cdot S β \cdot C γ \\ C β \cdot S γ & C α \cdot C γ + S α \cdot S β \cdot S γ & - S α \cdot C γ + C α \cdot S β \cdot S γ \\ - S β & S α \cdot C β & C α \cdot C β \end{matrix}]

(1)

where, for notational simplicity, Cα, Cβ and Cγ represent cos(α), cos(β) and cos(γ), respectively, and similarly Sα, Sβ and Sγ represent sin(α), sin(β) and sin(γ), respectively. R _α, R _β and R _γ are the corresponding transformations for each rotation:

R_{a} = [\begin{matrix} 1 & 0 & 0 \\ 0 & C α & - S α \\ 0 & S α & C α \end{matrix}], R_{β} = [\begin{matrix} C β & 0 & S β \\ 0 & 1 & 0 \\ - S β & 0 & C β \end{matrix}] and R_{γ} = [\begin{matrix} C γ & - S γ & 0 \\ S γ & C γ & 0 \\ 0 & 0 & 1 \end{matrix}] .

Denote S _i as the hydraulic cylinder vector from point B_i to A_i, then

S_{i} = t_{i} + R_{α β γ} \cdot p_{i} - b_{i}

(2)

where t _i is the vector from point O to O₁, t _i = [x,y,z]^T, p _i is the vector from point O₁ to A_i, described in {p}, and b _i is the vector from point O to B_i.

So the length of the hydraulic cylinder can be written as

l_{i} = ‖ S_{i} ‖ = ‖ t_{i} + R_{α β γ} \cdot p_{i} - b_{i} ‖

(3)

The axial speed of the hydraulic cylinder can be written as

{\overset{\cdot}{l}}_{i} = ‖ {\overset{\cdot}{S}}_{i} ‖ = ‖ v_{p} + ω_{p} \times p_{i} ‖

(4)

where v _p is the velocity of point O₁, and ω _p is the angular velocity of the platform.

Based on the Newton–Euler method, the dynamics equations of the platform can be given as

m_{e} a_{p} = D_{e} \cdot τ + G

(5)

(I \cdot ω_{p 1}) \times ω_{p 1} - I \cdot ε_{p 1} = H \cdot τ

(6)

where m _e is a 3 $\times$ 3 dimensional mass matrix, m _e = m E, in which E is the unit matrix and m is the total mass of the platform and the load; a _p is the acceleration of the platform, $a_{p} = {[\overset{\cdot\cdot}{x}, \overset{\cdot\cdot}{y}, \overset{\cdot\cdot}{z}]}^{T}$ ; D _e is the direction cosine matrix of the six cylinders, D _e = [n ₁,n ₂,n ₃,n ₄,n ₅,n ₆]^T, in which n _i = S _i/l_i ; τ is the actuating force matrix of the six hydraulic actuators, τ = [ τ ₁, τ ₂, τ ₃, τ ₄, τ ₅, τ ₆]^T; G is gravitational force, G = [0,0, mg]^T; I is the total inertial matrix of the platform and the load in coordinate frame {p}; and ω _p1 and ε _p1 are the angular velocity and the angular acceleration, respectively, of the platform, both in coordinate frame {p}. Matrix H = [p ₁ × e ₁, p ₂ × e ₂, p ₃ × e ₃, p ₄ × e ₄, p ₅ × e ₅, p ₆ × e ₆], in which e ₁, e ₂, e ₃, e ₄, e ₅ and e ₆ are the six hydraulic cylinder vectors expressed in coordinate frame {p}.

2.2. Dynamics of the hydraulic actuator

Each hydraulic actuator is shown schematically in Figure 2. It takes the combination form of a two-stage servo-valve and a servo force cylinder. The former includes a prepositioned two-jet flapper valve and another four-way spool valve as the power stage. The flow equations of each cylinder and valve can be written as

\frac{A (l_{0} + Δ l)}{β_{e}} {\overset{\cdot}{p}}_{l} = - A \overset{\cdot}{l} - C_{tm} p_{l} + Q_{l}

(7)

Q_{l} = C_{d} w_{v} x_{v} \sqrt{\frac{1}{ρ} (p_{s} - sgn (x_{v}) p_{l})}

(8)

where $A$ is the cross-section area of the forward chamber of the cylinder; $l_{0}$ is the initial length of the cylinder when the moving platform is located at its neutral position; $Δ l$ is the displacement of the piston, $Δ l = l - l_{0}$ ; $β_{e}$ is effective bulk modulus. $C_{tm}$ is the leakage coefficient of the cylinder; $p_{l}$ is the load pressure of the cylinder; $Q_{l}$ is the load flow of the cylinder; $C_{d}$ is the discharge coefficient; $w_{v}$ is the orifice area gradient of the servo-valve; $x_{v}$ is the spool displacement; $ρ$ is fluid density; $p_{s}$ is the supply pressure to the servo-valve.

Substituting Equation (7) into (8) yields the following pressure dynamics:

\frac{A (l_{0} + Δ l)}{β_{e}} {\overset{\cdot}{p}}_{l} = - A \overset{\cdot}{l} - C_{tm} p_{l} + C_{d} w_{v} x_{v} \sqrt{\frac{1}{ρ} (p_{s} - sgn (x_{v}) p_{l})}

(9)

Neglecting the leakage term $C_{tm} p_{l}$ , and considering that the valve has a much more rapid response feature relative to the cylinder, its dynamics from the input current u to the spool displacement $x_{v}$ can be modeled as a proportional gain. Equation (9) can be simplified as

{\overset{\cdot}{p}}_{l} = - \frac{β_{e} \overset{\cdot}{l}}{l_{0} + Δ l} + C_{d} w_{v} \sqrt{\frac{1}{ρ} (p_{s} - sgn (u) p_{l})} u \cdot \frac{β_{e}}{A (l_{0} + Δ l)}

(10)

2.3. Dynamics of the overall system, includingthe platform and the actuators

Suppose that vector $w \in R^{6}$ represents the perturbation from the parameter errors and external disturbances. Based on Equations (5), (6) and (10), the dynamics equations of the S-G platform type simulator, including the actuators, can be given as

M (q) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) + w = J^{T} τ

(11)

p_{l}^{d} = τ / A

(12)

{\overset{\cdot}{p}}_{li} = - \frac{β_{e} {\overset{\cdot}{l}}_{i}}{l_{0} + Δ l_{i}} + C_{d} w_{v} \sqrt{\frac{1}{ρ} (p_{s} - sgn (u_{i}) p_{li})} u_{i} \cdot \frac{β_{e}}{A (l_{0} + Δ l_{i})} i = 1, 2, \dots, 6

(13)

where $M (q)$ is a 6 $\times$ 6 inertia matrix of the moving platform, and it can be explicitly written as

M = [\begin{matrix} m & 0 & 0 & 0 & 0 & 0 \\ 0 & m & 0 & 0 & 0 & 0 \\ 0 & 0 & m & 0 & 0 & 0 \\ 0 & 0 & 0 & M_{44} & M_{45} & M_{46} \\ 0 & 0 & 0 & M_{54} & M_{55} & 0 \\ 0 & 0 & 0 & M_{64} & 0 & M_{66} \end{matrix}]

(14)

in which $M_{44} = I_{x} C_{β}^{2} C_{γ}^{2} + I_{y} C_{β}^{2} S_{γ}^{2} + I_{z} S_{β}^{2}$ , $M_{45} = M_{54} = (I_{x} - I_{y} {) C}_{β} C_{γ} S_{γ}$ , $M_{46} = M_{64} = I_{z} S_{β}$ , $M_{55} = I_{x} S_{γ}^{2} + I_{y} C_{γ}^{2}$ and $M_{66} = I_{z}$ . $C (q, \overset{\cdot}{q})$ is the Coriolis and centrifugal matrix, and it can be explicitly written as $C = [\begin{matrix} 0 & 0 \\ 0 & C_{22} \end{matrix}]$ , in which the submatrix $C_{22}$ is given as

C_{22} = [\begin{matrix} - K_{1} \overset{\cdot}{β} - K_{2} \overset{\cdot}{γ} & - K_{1} \overset{\cdot}{α} - K_{3} \overset{\cdot}{β} + K_{4} \overset{\cdot}{γ} & - K_{2} \overset{\cdot}{α} + K_{4} \overset{\cdot}{β} \\ - K_{1} \overset{\cdot}{α} + K_{4} \overset{\cdot}{γ} & K_{5} \overset{\cdot}{γ} & K_{4} \overset{\cdot}{α} + K_{5} \overset{\cdot}{β} \\ K_{2} \overset{\cdot}{α} - K_{4} \overset{\cdot}{β} & - K_{4} \overset{\cdot}{α} - K_{5} \overset{\cdot}{β} & 0 \end{matrix}]

(15)

in which $K_{1} = C_{β} S_{β} (C_{γ}^{2} I_{x} + S_{γ}^{2} I_{y} - I_{z})$ , $K_{2} = C_{β}^{2} C_{γ} S_{γ} (I_{x} - I_{y})$ , $K_{3} = C_{γ} S_{β} S_{γ} (I_{x} - I_{y}), K_{4} = \frac{1}{2} C_{β} (C_{γ} - S_{γ})$ ,

$(C_{γ} + S_{γ}) (I_{x} - I_{y})$ and $K_{5} = C_{γ} S_{γ} (I_{x} - I_{y})$ . $G (q) \in R^{6}$ is the gravity term and $J$ is the Jacobian matrix, which can be explicitly written as

J = [\begin{matrix} n_{1}^{T} & n_{1}^{T} R_{1} p_{1} & n_{1}^{T} R_{2} p_{1} & n_{1}^{T} R_{3} p_{1} \\ n_{2}^{T} & n_{2}^{T} R_{1} p_{2} & n_{2}^{T} R_{2} p_{2} & n_{2}^{T} R_{3} p_{2} \\ n_{3}^{T} & n_{3}^{T} R_{1} p_{3} & n_{3}^{T} R_{2} p_{3} & n_{3}^{T} R_{3} p_{3} \\ n_{4}^{T} & n_{4}^{T} R_{1} p_{4} & n_{4}^{T} R_{2} p_{4} & n_{4}^{T} R_{3} p_{4} \\ n_{5}^{T} & n_{5}^{T} R_{1} p_{5} & n_{5}^{T} R_{2} p_{5} & n_{5}^{T} R_{3} p_{5} \\ n_{6}^{T} & n_{6}^{T} R_{1} p_{6} & n_{6}^{T} R_{2} p_{6} & n_{6}^{T} R_{3} p_{6} \end{matrix}]

(16)

in which $R_{1} = N (i) R_{α β γ}$ , $R_{2} = R_{α} N (j) R_{β} R_{γ}$ and $R_{3} = R_{α} R_{β} N (k) R_{γ}$ , and

N (i) = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & 0 \end{matrix}], N (j) = [\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ - 1 & 0 & 0 \end{matrix}], N (k) = [\begin{matrix} 0 & - 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] .

(17)

$τ$ is the force vector acting on the platform by the hydraulic cylinders, $τ = {[τ_{1}, τ_{2}, τ_{3}, τ_{4}, τ_{5}, τ_{6}]}^{T}$ , $p_{l}^{d}$ is the expected load pressure required by the force vector $τ$ , and $p_{l}^{d} = {[p_{l 1}^{d}, p_{l 2}^{d}, p_{l 3}^{d}, p_{l 4}^{d}, p_{l 5}^{d}, p_{l 6}^{d}]}^{T}$ .

3. Adaptive controller design

The control scheme takes an outer loop and six inner loops combined mode, as shown in Figure 3. A platform controller in the outer loop takes the PD control plus adaptive compensation for the position and orientation tracking control of the platform in Cartesian space. Each inner loop is a servo force control that is used to make each actuator track the desired force solved out by the platform controller. In the platform controller, the nonlinear PD feedback part plays a main role, and the adaptive compensation part is used to compensate the external disturbance and modeling error when the tracking error is large. When the tracking error becomes small, the former will take the most part in the control signal. Based on Lyapunov stability theorem, both the platform controller and the servo force controller are designed together in one process and the global asymptotic stability is ensured under bounded external disturbance. The adaptive law of the mass parameters of the platforms is composed of the regression matrix of the S-G platform dynamic equation and designed by the gradient descent method.

First considering that the S-G platform is substantially a kind of parallel robot, it satisfies the following three general properties²²:

$x^{T} (M (q) - 2 C (q, \overset{\cdot}{q})) x = 0 \forall x \in R^{n}$

The inertia matrix $M (q)$ is a symmetric positive matrix, and there exist positive numbers $m_{1}$ and $m_{2}$ to satisfy the following inequality:

m_{1} {‖ x ‖}^{2} \leq x^{T} M (q) x \leq m_{2} {‖ x ‖}^{2}

(18)

There exists a parameter vector that is dependent on the S-G platform parameters and makes matrices $M (q)$ , $C (q, \overset{\cdot}{q})$ and $G (q)$ satisfy the following linear relation:

M (q) ϑ + C (q, \overset{\cdot}{q}) ρ + G (q) = Φ (q, \overset{\cdot}{q}, ρ, ϑ) P

(19)

where $Φ (q, \overset{\cdot}{q}, ρ, ϑ) \in R^{n \times m}$ is the regression matrix of known parameters of the S-G platform. $P \in R^{n}$ is the unknown constant parameter vector to describe the mass property of the S-G platform.

Define $q_{d} \in R^{6}$ as the desired position and orientation of the S-G platform. Suppose that both the first-order and second-order differential of $q_{d}$ exist, and herein we suppose that the perturbation is with known upper bound, namely the norm of $w$ satisfies

‖ w ‖ \leq d_{1} + d_{2} ‖ e ‖ + d_{3} ‖ \overset{\cdot}{e} ‖

(20)

where $d_{1}$ , $d_{2}$ and $d_{3}$ are all positive constants, $e$ and $\overset{\cdot}{e}$ are the tracking error and its differential, respectively, namely $e = q - q_{d}$ and $\overset{\cdot}{e} = \overset{\cdot}{q} - {\overset{\cdot}{q}}_{d}$ .

Two new variables, $y$ and $q_{r}$ , are introduced; $y$ and the differential of $q_{r}$ are defined as²³

y = \overset{\cdot}{e} + γ e

(21)

{\overset{\cdot}{q}}_{r} = {\overset{\cdot}{q}}_{d} - γ e

(22)

where $γ$ is constant with $γ > 0$ , so

y = \overset{\cdot}{q} - {\overset{\cdot}{q}}_{r}

(23)

Let $ϑ = {\overset{\cdot\cdot}{q}}_{r}$ , $ρ = {\overset{\cdot}{q}}_{r}$ , and Equation (19) can be written as

M (q) {\overset{\cdot\cdot}{q}}_{r} + C (q, \overset{\cdot}{q}) {\overset{\cdot}{q}}_{r} + G (q) = Φ (q, \overset{\cdot}{q}, {\overset{\cdot}{q}}_{r}, {\overset{\cdot\cdot}{q}}_{r}) P

(24)

According to Equation (23), an equation of ${\overset{\cdot}{q}}_{r} = \overset{\cdot}{q} - y$ can be obtained and its substituting into Equation (24) yields

M (q) \overset{\cdot\cdot}{q} - M (q) \overset{\cdot}{y} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} - C (q, \overset{\cdot}{q}) y + G (q) = Φ (q, \overset{\cdot}{q}, {\overset{\cdot}{q}}_{r}, {\overset{\cdot\cdot}{q}}_{r}) P

(25)

The combination of Equation (25) and Equation (11) yields

M (q) \overset{\cdot}{y} + C (q, \overset{\cdot}{q}) y = J^{T} τ - Φ (q, \overset{\cdot}{q}, {\overset{\cdot}{q}}_{r}, {\overset{\cdot\cdot}{q}}_{r}) P - w

(26)

Let $\tilde{P} = \hat{P} - P$ , in which $\hat{P}$ is the estimation value of $P$ and $\tilde{P}$ is the estimation error. Let $e_{li} = p_{li} - p_{li}^{d}$ , where $p_{li}^{d}$ is the desired pressure of the hydraulic cylinder. The Lyapunov function of the whole system is denoted as

V = \frac{1}{2} (y^{T} M (q) y + e^{T} (K_{p 1} + γ K_{v 1}) e + {\tilde{P}}^{T} Γ^{- 1} \tilde{P}) + \sum_{i = 1}^{6} \frac{1}{2} ρ_{i} e_{li}^{2}

(27)

where $K_{p 1} = diag (k_{p 11}, k_{p 12}, . . ., k_{p 16})$ , $K_{v 1} = diag (k_{v 11}, k_{v 12}, . . ., k_{v 16})$ , $K_{p 11}$ , $K_{p 12}$ ,…, $K_{p 16}$ , $K_{v 11}$ , $K_{v 12}$ , …, $K_{v 16}$ are all positive numbers. $Γ$ is a symmetric positive matrix.

Considering that $M (q)$ is a symmetric positive matrix, the differential of $V$ is

\overset{\cdot}{V} = y^{T} M (q) \overset{\cdot}{y} + \frac{1}{2} y^{T} \overset{\cdot}{M} (q) y + e^{T} (K_{p 1} + γ K_{v 1}) \overset{\cdot}{e} + {\tilde{P}}^{T} Γ^{- 1} \overset{\cdot}{\tilde{P}} + \sum_{i = 1}^{6} ρ_{li} e_{li} {\overset{\cdot}{e}}_{li}

(28)

Define the control law of the platform controller as

τ = J (- K_{p} e - K_{v} \overset{\cdot}{e} + Φ (q, \overset{\cdot}{q}, {\overset{\cdot}{q}}_{r}, {\overset{\cdot\cdot}{q}}_{r}) \hat{P} + υ)

(29)

where $K_{p} = K_{p 1} + K_{p 2} B_{p} (e)$ and $K_{v} = K_{v 1} + K_{v 2} B_{v} (\overset{\cdot}{e})$ , where $K_{p 2} = diag (k_{p 21}, k_{p 22}, . . ., k_{p 26})$ , $K_{v 2} = diag (k_{v 21}, k_{v 22}, . . ., k_{v 26})$ , $K_{p 21}$ , $K_{p 22}$ ,…, $K_{p 26}$ , $K_{v 21}$ , $K_{v 22}$ , …, $K_{v 26}$ are all positive numbers.

$B_{p} (e) = diag (\frac{1}{α_{1} + | e_{1} |}, \frac{1}{α_{2} + | e_{2} |}, . . ., \frac{1}{α_{6} + | e_{6} |})$ and $B_{v} (e) = diag (\frac{1}{β_{1} + | {\overset{\cdot}{e}}_{1} |}, \frac{1}{β_{2} + | {\overset{\cdot}{e}}_{2} |}, . . ., \frac{1}{β_{6} + | {\overset{\cdot}{e}}_{6} |})$ , $α_{i}$ , $β_{i}$ are all positive numbers. $υ = {[υ_{1}, υ_{2}, . . ., υ_{6}]}^{T}$ , $υ_{i} = - (d_{1} + d_{2} ‖ e ‖ + d_{3} ‖ \overset{\cdot}{e} ‖) sgn (y_{i})$ , i = 1,2,…,6.

Based on Equations (26) and (29), Equation (28) can be written as

\begin{matrix} \overset{\cdot}{V} = y^{T} [- K_{p} e - K_{v} \overset{\cdot}{e} + Φ (q, \overset{\cdot}{q}, {\overset{\cdot}{q}}_{r}, {\overset{\cdot\cdot}{q}}_{r}) \tilde{P} + υ - w] - y^{T} C (q, \overset{\cdot}{q}) y + \frac{1}{2} y^{T} \overset{\cdot}{M} (q) y \\ + e^{T} K_{p} \overset{\cdot}{e} - e^{T} (K_{p 2} B_{p} (e) + γ K_{v 2} B_{v} (\overset{\cdot}{e})) \overset{\cdot}{e} + e^{T} γ K_{v} \overset{\cdot}{e} + {\tilde{P}}^{T} Γ^{- 1} \overset{\cdot}{\tilde{P}} + \sum_{i = 1}^{6} ρ_{li} e_{li} {\overset{\cdot}{e}}_{li} \end{matrix}

(30)

According to the property in Equation (17), Equation (30) can be further written as

\begin{matrix} \overset{\cdot}{V} = y^{T} [- K_{p} e - K_{v} \overset{\cdot}{e}] + e^{T} γ K_{v} \overset{\cdot}{e} + e^{T} K_{p} \overset{\cdot}{e} \\ - e^{T} (K_{p 2} B_{p} (e) + γ K_{v 2} B_{v} (\overset{\cdot}{e})) \overset{\cdot}{e} \\ + y^{T} [Φ (q, \overset{\cdot}{q}, {\overset{\cdot}{q}}_{r}, {\overset{\cdot\cdot}{q}}_{r}) \tilde{P} + υ - w] + {\tilde{P}}^{T} Γ^{- 1} \overset{\cdot}{\tilde{P}} \\ + \sum_{i = 1}^{6} ρ_{li} e_{li} {\overset{\cdot}{e}}_{li} \end{matrix}

(31)

Based on the transposing of Equation (21), namely ${\overset{\cdot}{y}}^{T} = {\overset{\cdot}{e}}^{T} + γ e^{T}$ , Equation (31) is further written as

\begin{matrix} \overset{\cdot}{V} = - {\overset{\cdot}{e}}^{T} K_{v} \overset{\cdot}{e} - γ e^{T} K_{p} e - e^{T} (K_{p 2} B_{p} (e) + γ K_{v 2} B_{v} (\overset{\cdot}{e})) \overset{\cdot}{e} + y^{T} [υ - w] + y^{T} Φ (q, \overset{\cdot}{q}, {\overset{\cdot}{q}}_{r}, {\overset{\cdot\cdot}{q}}_{r}) \tilde{P} + {\tilde{P}}^{T} Γ^{- 1} \overset{\cdot}{\tilde{P}} + \sum_{i = 1}^{6} ρ_{li} e_{li} {\overset{\cdot}{e}}_{li} \end{matrix}

(32)

Considering $y^{T} Φ (q, \overset{\cdot}{q}, {\overset{\cdot}{q}}_{r}, {\overset{\cdot\cdot}{q}}_{r}) \tilde{P} = {\tilde{P}}^{T} Φ^{T} (q, \overset{\cdot}{q}, {\overset{\cdot}{q}}_{r}, {\overset{\cdot\cdot}{q}}_{r}) y$ , $\overset{\cdot}{\tilde{P}} = \overset{\cdot}{\hat{P}}$ and according to gradient descent method the adaptive law is taken as

\overset{\cdot}{\hat{P}} = - {Γ Φ}^{T} (q, \overset{\cdot}{q}, {\overset{\cdot}{q}}_{r}, {\overset{\cdot\cdot}{q}}_{r}) y

(33)

Substituting Equation (33) into Equation (32) yields

\overset{\cdot}{V} = - {\overset{\cdot}{e}}^{T} K_{v} \overset{\cdot}{e} - γ e^{T} K_{p} e - e^{T} (K_{p 2} B_{p} (e) + γ K_{v 2} B_{v} (\overset{\cdot}{e})) \overset{\cdot}{e} + y^{T} [υ - w] + \sum_{i = 1}^{6} ρ_{li} e_{li} {\overset{\cdot}{e}}_{li}

(34)

Substituting the expression of $K_{p}$ , $K_{p 2}$ , $K_{v}$ and $K_{v 2}$ into Equation (34) yields

\begin{matrix} \overset{\cdot}{V} = - \sum_{i = 1}^{6} (γ (k_{p 1 i} + \frac{k_{p 2 i}}{α_{i} + | e_{i} |}) e_{i}^{2}) \\ - \sum_{i = 1}^{6} ((k_{v 1 i} + \frac{k_{v 2 i}}{β_{i} + | {\overset{\cdot}{e}}_{i} |}) {\overset{\cdot}{e}}_{i}^{2}) \\ - \sum_{i = 1}^{6} ((\frac{k_{p 2 i}}{α_{i} + | e_{i} |} + \frac{γ k_{v 2 i}}{β_{i} + | {\overset{\cdot}{e}}_{i} |}) e_{i} {\overset{\cdot}{e}}_{i}) + y^{T} (υ - w) \\ + \sum_{i = 1}^{6} ρ_{li} e_{li} {\overset{\cdot}{e}}_{li} \end{matrix}

(35)

Because $e_{i} {\overset{\cdot}{e}}_{i} \leq \frac{1}{2} (e_{i}^{2} + {\overset{\cdot}{e}}_{i}^{2})$ , therefore

\begin{matrix} \overset{\cdot}{V} \leq - \sum_{i = 1}^{6} (γ (k_{p 1 i} + \frac{k_{p 2 i}}{α_{i} + | e_{i} |}) - \frac{1}{2} (\frac{k_{p 2 i}}{α_{i} + | e_{i} |} + \frac{γ k_{v 2 i}}{β_{i} + | {\overset{\cdot}{e}}_{i} |})) e_{i}^{2} \\ - \sum_{i = 1}^{6} ((k_{v 1 i} + \frac{k_{v 2 i}}{β_{i} + | {\overset{\cdot}{e}}_{i} |}) - \frac{1}{2} (\frac{k_{p 2 i}}{α_{i} + | e_{i} |} + \frac{γ k_{v 2 i}}{β_{i} + | {\overset{\cdot}{e}}_{i} |})) {\overset{\cdot}{e}}_{i}^{2} + y^{T} (υ - w) + \sum_{i = 1}^{6} ρ_{li} e_{li} {\overset{\cdot}{e}}_{li} \end{matrix}

(36)

When $\frac{1}{2} \leq γ \leq 2$ , the following inequalities will be satisfied:

γ \frac{k_{p 2 i}}{α_{i} + | e_{i} |} - \frac{1}{2} \frac{k_{p 2 i}}{α_{i} + | e_{i} |} \geq 0

(37)

\frac{k_{v 2 i}}{β_{i} + | {\overset{\cdot}{e}}_{i} |} - \frac{1}{2} \frac{γ k_{v 2 i}}{β_{i} + | {\overset{\cdot}{e}}_{i} |} \geq 0

(38)

Combining Equations (37) and (38) with Equation (36) yields

\begin{array}{l} \dot{V} \leq - \sum_{i = 1}^{6} (γ (k_{p 1 i} - \frac{1}{2} \frac{k_{v 2 i}}{β_{i} + | {\dot{e}}_{i} |}) e_{i}^{2} + (k_{v 1 i} - \frac{1}{2} \frac{k_{p 2 i}}{α_{i} + | e_{i} |}) {\dot{e}}_{i}^{2}) + y^{T} (υ - w) + \sum_{i = 1}^{6} ρ_{l i} e_{l i} {\dot{e}}_{l i} \\ \leq - \sum_{i = 1}^{6} (γ (k_{p 1 i} - \frac{1}{2} \frac{k_{v 2 i}}{β_{i}}) e_{i}^{2} + (k_{v 1 i} - \frac{1}{2} \frac{k_{p 2 i}}{α_{i}}) {\dot{e}}_{i}^{2}) + y^{T} (υ - w) + \sum_{i = 1}^{6} ρ_{l i} e_{l i} {\dot{e}}_{l i} \end{array}

(39)

Let the values of $k_{p 1 i}$ , $k_{v 1 i}$ , $α_{i}$ and $β_{i}$ satisfy

k_{p 1 i} - \frac{1}{2} \frac{k_{v 2 i}}{β_{i}} > 0

(40)

and

k_{v 1 i} - \frac{1}{2} \frac{k_{p 2 i}}{α_{i}} > 0

(41)

Based on inequalities (40) and (41), Equation (39) can be rewritten as

\overset{\cdot}{V} \leq y^{T} (υ - w) + \sum_{i = 1}^{6} ρ_{li} e_{li} {\overset{\cdot}{e}}_{li}

(42)

According to Equation (20) and the definition of $υ$ under Equation (29), $y^{T} (υ - w)$ can be expressed as

\begin{matrix} y^{T} (υ - w) & = \sum_{i = 1}^{6} y_{i} [- (d_{1} + d_{2} ‖ e ‖ + d_{3} ‖ \overset{\cdot}{e} ‖) sgn (y_{i})] - y^{T} w \\ = \sum_{i = 1}^{6} [- (d_{1} + d_{2} ‖ e ‖ + d_{3} ‖ \overset{\cdot}{e} ‖) | y_{i} |] - y^{T} w \\ \leq \sum_{i = 1}^{6} (‖ w ‖ \cdot | y_{i} |) - y^{T} w = 0 \end{matrix}

(43)

\overset{\cdot}{V} \leq \sum_{i = 1}^{6} ρ_{li} e_{li} {\overset{\cdot}{e}}_{li}

(44)

According to Equation (13), we can get

\sum_{i = 1}^{6} ρ e_{li} {\overset{\cdot}{e}}_{li} = ρ e_{li} (- \frac{β_{e} {\overset{\cdot}{l}}_{i}}{l_{0} + Δ l_{i}} + C_{d} w_{v} \sqrt{\frac{1}{ρ} (P_{s} - sgn (u_{i}) p_{li})} u_{i} \cdot \frac{β_{e}}{A (l_{0} + Δ l_{i})} - {\overset{\cdot}{p}}_{li}^{d})

(45)

Define coefficient $λ = C_{d} w_{v} \sqrt{\frac{1}{ρ} (P_{s} - sgn (u_{i}) p_{li})} \cdot \frac{β_{e}}{A (l_{0} + Δ l_{i})}$ , and define the control law for hydraulic servo system as

u_{i} = - \frac{1}{λ} (- \frac{β_{e} {\overset{\cdot}{l}}_{i}}{l_{0} + Δ l_{i}} + λ e_{li} - {\overset{\cdot}{p}}_{li}^{d})

(46)

Substituting Equation (46) into Equation (45) yields

\sum_{i = 1}^{6} ρ_{li} e_{li} {\overset{\cdot}{e}}_{li} = - \sum_{i = 1}^{6} ρ_{li} e_{li}^{2} \leq 0

(47)

Based on Equations (44) and (47), we can get

\overset{\cdot}{V} \leq 0

(48)

which guarantees $lim_{t \to \infty} e = 0$ and $lim_{t \to \infty} \overset{\cdot}{e} = 0$ , based on Lyapunov stability theorem; thus, the controller is global asymptotic stable.

To summarize the above derivation, the control strategies of the whole system are as follows.

Platform controller:

τ = J (- K_{p} e - K_{v} \overset{\cdot}{e} + Φ (q, \overset{\cdot}{q}, {\overset{\cdot}{q}}_{r}, {\overset{\cdot\cdot}{q}}_{r}) \hat{P} + υ)

(49)

where

υ = {[υ_{1}, υ_{2}, . . ., υ_{6}]}^{T}, υ_{i} = - (d_{1} + d_{2} ‖ e ‖ + d_{3} ‖ \overset{\cdot}{e} ‖) sgn ({\overset{\cdot}{e}}_{i} + γ e_{i}) i = 1, 2, \dots, 6

(50)

Electro-hydraulic servo force controller:

u_{i} = - \frac{1}{λ} (- \frac{β_{e} {\overset{\cdot}{l}}_{i}}{l_{0} + Δ l_{i}} + λ e_{1 i} - {\overset{\cdot}{p}}_{li}^{d}) i = 1, 2, \dots, 6

(51)

Adaptive law:

\overset{\cdot}{\hat{P}} = - {Γ Φ}^{T} (q, \overset{\cdot}{q}, {\overset{\cdot}{q}}_{r}, {\overset{\cdot\cdot}{q}}_{r}) (\overset{\cdot}{e} + γ e)

(52)

4. Numerical simulation

Parameters of a six-DOF S-G platform type simulator are taken to verify the proposed algorithm. This platform is developed as a motion simulator and its structure is shown in Figure 1. All the parameters are listed in Table 1.

Table 1.

Parameters of the motion simulator.

Subsystem	Parameter values
Platform and cabin	m = 196 kg	R = 1.2 m	r = 0.9 m	s = 0.15 m
	S = 0.45 m	I _xx = 85.52 kg.m²	I _yy = 71.84 kg.m²	I _zz = 152.22 kg.m²
	I _xy = 0 kg.m²	I _yz = 0 kg.m²	I _zx = 0 kg.m²
Servo-hydraulic system	p_s = 50 Mpa	P ₀ = 0.1 Mpa	A = 5.65 $\times$ 10⁻³ m²
Servo-hydraulic system	C_d = 0.65	ρ = 900 kg/m³	β _e = 6.9 $\times$ 10⁸ Pa
Controller	K_p ₁₁ = K_p ₁₂ = K_p ₁₃= K_p ₁₄= K_p ₁₅= K_p ₁₆ = 5000		K_v ₁₁ = K_v ₁₂ = K_v ₁₃= K_v ₁₄= K_v ₁₅= K_v ₁₆ = 5000
	K_p ₂₁ = K_p ₂₂ = K_p ₂₃ = K_p ₂₄ = K_p ₂₅ = K_p ₂₆ = 3000		K_v ₂₁ = K_v ₂₂ = K_v ₂₃ = K_v ₂₄ = K_v ₂₅ = K_v ₂₆ = 3000
	α ₁ = α ₂ = α ₃ = α ₄ = α ₅ = α ₆ = 1		β ₁ = β ₂ = β ₃ = β ₄ = β ₅ = β ₆ = 1
	Γ = diag(100,100,100,100)		γ = 5

Simulation tests are completed in Matlab/Simulink (see Figure 4). The platform controller, the adaptive law, the dynamic models of six actuators and the moving platform are all built with S-functions.²³ Each actuator controller is built in subsystem form with blocks provided by Simulink; the valve core control signal (Equation (51)) is completed with the “Matlab function” block in the Simulink library. Figure 5 gives the results when tracking the desired trajectories: $α = 0.4 \sin (6 π t)$ , $β = 0.5 \sin (4 π t)$ , $γ = 0.6 \sin (2 π t)$ , $x = 0.1 \sin (12 π t)$ , $y = 0.2 \sin (10 π t)$ and $z = 0.5 \sin (8 π t)$ . The perturbation is set as d ₁ = 2, d ₂ = 3 and d ₃ = 6. To testify the performance of the proposed algorithm, the PD control of the platform without adaptive compensation is also completed as the comparison, and its simulation results are also drawn in Figure 5.

The convergence in the six motions shown in Figure 5 testifies the global asymptotic stability of the proposed adaptive controller. Figure 6 demonstrates the rapid convergence of parameter estimation with the proposed adaptive law.

It can be seen from Figure 5 that by using only the PD control there exists great amplitude attenuation and phase lag in both the position and velocity tracking of all three translations, which results in large tracking errors. However, the adaptive controller shows much higher tracking performance, especially with very small error in the velocity tracking of all these three translations cases, and both the position and velocity errors gradually converge to zero. Without considering the pulse in the beginning, the maximum position tracking errors along the x, y and z coordinates are about 25%, 15% and 5%, respectively. However, the PD control has tracking errors of about 90% in all these three cases because of the amplitude attenuation and phase lag.

In the three rotations, both the adaptive controller and the PD controller show better control results: the maximum position tracking errors under the adaptive controller are about 5%, 8% and 6%, respectively. The PD controller still has a large tracking error with maximum values of about 38%, 20% and 16%, respectively, although it is better than the three translation cases. The adaptive controller still definitely outperforms the PD controller and its position and velocity tracking errors converge to zero with less fluctuation relative to the three translation cases. All the simulation results show that the proposed adaptive control algorithm has better robust performance when with parameter uncertainty and external disturbance.

5. Conclusion

This paper proposed a nonlinear adaptive control scheme for an electro-hydraulic S-G platform type simulator for motion simulation. The designed controller includes two parts: a nonlinear PD part and an adaptive compensation part. The adaptive compensation part is used to compensate for the modeling error and external disturbance and thus to effectively decrease the dynamic tracking error of the platform.

The dynamics of the electro-hydraulic actuator is also considered and a Lyapunov-based servo force control scheme is presented in this paper. The entire system, including electro-hydraulic actuators, is controlled with the combined inner and outer loop control scheme. All the controllers are designed together based on the Lyapunov method to ensure the global asymptotic stability.

Simulation demonstrated the feasibility and effectiveness of the proposed adaptive control scheme. It outperforms the non-adaptive PD control and shows better robust tracking performance.

Footnotes

Funding

This research is supported by Natural Science Fund of Heilongjiang Province of China (E201013) and the Project of Fundamental Research Funds for the Central Universities of China (DL09CB02).

Author biographies

Qiang Zhao is a full professor at Northeast Forestry University, Traffic College, the Department of Automobile Engineering, Harbin, China, and was also a visiting scholar of the Smart Structures Technology Laboratory (SSTL) at the University of Illinois at Urbana-Champaign.

Na Wang is a PhD student at Northeast Forestry University, Traffic College, Harbin, China.

Billie F Spencer, Jr is the Nathan M. and Anne M. Newmark Endowed Chair in Civil Engineering at the University of Illinois at Urbana-Champaign.

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