Contributions

A posture control scheme using a forward kinematics solution is proposed in this dissertation. With only the six leg lengths, we can obtain the posture’s information and further achieve the posture control. This control scheme is actually an output feedback control with a nonlinear observer. The observer is designed to estimate states of the Stewart platform, which is apparently a nonlinear MIMO affine system. The observed states include platform translations and rotations about three axes and their first-order time derivatives, and the sliding mode controller produces feedback with these observed states. The performance can be seen in the simulations and experimental results.

To integrate the electro-hydraulic actuator into the overall system, we formulate the electro-servovalve model and the hydraulic dynamics as the nonlinear system. Then, the dynamics of both the platform and the hydraulic actuator of the Stewart platform are taken into account to develop the backstepping controller. Further, to tackle the transformation between different states in the platform dynamics (task-space) and in the actuator (joint-space) dynamics, the aforementioned observer-based forward kinematics solver is applied. It is shown that the posture of the platform will follow a desired trajectory as closely as possible, while ensuring the overall system stability. The feasibility is also shown with simulation and experimental results.

An optimal washout filter is developed by applying the algorithm [35] to the human vestibular system. The objective is to minimize the sensation error produced from the comparison of the human vestibular signals about the actual vehicle and the simulator. The optimal motion cue algorithm is then featured by systematic integration of linear filters that are determined through an off-line design process. For the drawback with fixed parameters of the optimal algorithm, the fuzzy control rules are further designed to eliminate the sensation errors. With sensation error and time derivative of input feedback, the fuzzy logic can provide an adjusting term to further eliminate the perception errors. In the simulations, the sensation errors of classical, optimal, and fuzzy-based optimal washout filters are compared to show the performance of this algorithm.

Chapter 2 Preliminary

2.1 Kinematics of 6-DOF Motion Platform

The kinematics of a Stewart platform can be separated into inverse kinematics and forward kinematics. The former is to calculate each joint vector from the platform posture (translation and rotation). This solution can be obtained with simple equation of vector relation. The forward kinematics means to obtain the posture information with a given set of joint lengths. The following are the details of these relationships.

2.1.1 Inverse Kinematics

The Stewart platform is composed of top, base platforms, and 6 parallel actuators. The coordinate systems include the base frame (inertial frame) and the top frame (moving frame) which are shown in Figures. 2.1 and 2.2, respectively. The body coordinate is attached to the mass center of the moving platform and the fixed inertial coordinate is attached to the center of the base. The 6-DOF motion includes the 3-axis linear translations and rotations.

The 3-axis translations are represented as p_x p_y p_z^T and the 3-axis rotations are

[

α β γ , where

]

^T p_x, p_y, p_z

vertical (heave) motions, respectively

motions which are the rotating angle with respect to the 3 axes of the fixed frame. Each leg i links the attachment point motions which are the rotating angle with respect to the 3 axes of the fixed frame. Each leg

links the attachment points P_i and B_i on the top platform and the base, this configuration, the attachment points P_i and B_i can be expressed as

cos( )

represent the longitudinal (surge), lateral (sway), and represent the roll, pitch, and yaw

motions which are the rotating angle with respect to the 3 axes of the fixed frame. Each leg the top platform and the base, respectively. In

can be expressed as

Generally speaking, the inverse kinematics of the parallel manipulator has

Given a set of position and orientation, the leg vector, pointing from attachment p

to P , can be written as follows:_i

vectors can be seen in Figure

Figure 2.2 Base platform

e inverse kinematics of the parallel manipulator has

Given a set of position and orientation, the leg vector, pointing from attachment p

, can be written as follows:

are the translational offset vectors of leg i on the top platform and the and C_α are defined as sin( )α and cos( )α .

Figure 2.3. The length of the i-th leg is given by

e inverse kinematics of the parallel manipulator has a unique solution.

Given a set of position and orientation, the leg vector, pointing from attachment point B_i

(2.1) on the top platform and the

cos( ) . The relationship of th leg is given by l , and computing _i

1 2 3 4 5 6

Figure 2.3 Coordinate system of the Stewart platform

We can obtain the complete solution of the inverse kinematics as follows:

2 2 2 2 2 2 obtain the position and orientation from the given 6 actuator lengths.

2.1.2 Forward Kinematics

In contrast to the inverse kinematics is the forward kinematics which plays an important role in the posture control or the motion visualization of the Stewart platform.

The problem involved here is simply the reverse of that of inverse kinematics, which means to obtain the position and orientation from the given 6 actuator lengths. Unfortunately, it is difficult to solve the problem because of the inherent nonlinearity and complexity. The general form of the problem can be obtained from (2.2) as follows:

2 2 2 2 2

There are no closed-form solutions to express the translation and orientation variables

(

p_x^, p_y, , , , p_z α β γ

)

in general. The Newton-Raphson method is often used to solve the

problem, but the repetitive steps before solution convergence cause the real-time

application infeasible. Moreover, such method may even lead to infinite loops provided

selection of wrong initial values.

2.2 Dynamics of 6-DOF Motion Platform

The dynamics model of the Stewart platform is with high nonlinearity and system

uncertainties. Referring to [4], the general dynamics equations are as the standard form:

( ) ( , ) ( ) ^T( )

M q qɺɺ+C q q qɺ ɺ+G q =J q u (2.4)

where the state q=^p_x p_y p_z α β γ^^T∈ℝ includes the 3 translations and 3 ⁶ rotations, M q is the inertia matrix, ( ) C q q q( , )ɺ ɺ is the Coriolis and centrifugal force,

( )

G q is the gravitational force, u is the torque vector output by the 6 actuators, and J q( ) is the Jacobian matrix that transforms the joint torques to the force applied to the platform.

Some important system properties which will be used in the sequel are given below.

Property 2.1. C q qɺ( , ) and M qɺ( ) are bounded functions if q and qɺ are bounded.

3 3 3 3

nonlinear system. For the purpose of design of nonlinear observer, this form will be discussed in Chapter 3.

2.3 Conventional Control Scheme

The control goal of the motion platform is to track the desired posture to provide the realistic motion for operator. Due to the lack of posture information, the direct posture control is infeasible. Conventional control method for Stewart platform is to transform the desired posture from task space to joint space and obtain the desired leg lengths. However, this indirect control method cannot acquire the trajectories of posture but only guarantees tracking of each actuator. Figure 2.4 shows the control scheme.

u ^y

( )

_d

d

h q q

+ −

Figure 2.4 Stewart Platform Control Scheme with inverse kinematics

In Figure 2.4, q_d is the desired posture, h q( _d) is the desired leg length which is transformed from q_d by inverse kinematics, u is the voltage input of each actuator, and y

is the measurement of each leg length. The controller receives the errors of each leg length, and then controls each leg independently. Here, the PID controller is used to control individual actuator.

2.4 Vestibular Mathematical Model

The specific force should be first defined before we talk about the mathematical model of vestibular system. The specific force means the acceleration caused by non-gravitational force applied on the object. According the Newton’s second law, the acceleration is the applied force per unit mass, including the gravitational force and non-gravitational force.

So the total resulting acceleration can be expressed as:

ˆ ^{ng g}

v

F F

a f g

m m

= + = + (2.5)

where a_v is the acceleration of the vehicle, f^{ˆ Fng}

= m is the non-gravitational force per unit mass, or the specific force, and g ^Fg

= m is the gravitational acceleration of the vehicle.

The specific force ˆf is the summation of all non-gravitational forces applied on the vehicle. Since human can’t sense the gravitational acceleration, the specific force is the linear acceleration that affects human perception. So, the specific force is the input of the human vestibular model, and the other is angular velocity of the head.

Two mechanisms of the vestibular system work with receiving the specific force and angular velocity respectively. The otolith system receives the specific force and then

produces the perception of linear acceleration. Meanwhile, angular velocity goes through the semicircular cannels and creates the perception of rotation motion. The simplified sensation mechanisms of specific force and angular velocity proposed by Young and Oman are shown in Figures. 2.5 and 2.6, respectively.

Figure 2.5 Mathematical model of otolith

Figure 2.6 Mathematical model of semicircular Table 2.1 Human Vestibular Model Parameters [45]

Specific Force Sensation Model Parameter

Parameter Surge Sway Heave

τL 5.33 5.33 5.33

τs 0.66 0.66 0.66

τa 13.2 13.2 13.2

k 0.4 0.4 0.4

dTH(m/sec²) 0.17 0.17 0.28

Rotational Motion Sensation Model Parameter

Parameter Roll Pitch Yaw

T L 6.1 5.3 10.2

T s 0.1 0.1 0.1

T a 30 30 30

δTH(deg/sec) 3.0 3.6 2.6

The parameters of these mathematical models shown in Figure 2.5 and Figure 2.6 are shown in Table 2.1. Threshold values, ranging from 0.17 to 0.28 m/s² for linear accelerations and for 1.6 to 2.0 deg/s for angular velocities. The acceleration and angular velocities below the thresholds will not be sensed by human. These mechanisms eliminate redundant motion signals and reserve the necessary parts. This property provides the basic theory of design the washout filter.

Chapter 3

Output Feedback Posture Control

3.1 Design of the Nonlinear Observer

3.1.1 MIMO affine nonlinear system

The dynamic model described in (2.4) can be expressed in terms of state space representation for the purpose of designing the observer. The dynamic model is rewritten as the MIMO affine nonlinear system as

( ) ( ( )) ( ( )) ( )

The outputyis the vector composed of 6 actuator lengths, and can be expressed as

[

1 2 6

]

( ) ( ) ( ) ( )^T

y=h x = h q h q ⋯ h q (3.3) where

( ) ( ) ^p ( ) , 1, 2,..., 6. nonlinear system which can be expressed as

( ) ( )

3.1.2 The Concept and Conditions of the Observer

To acquire the full states without numerical iteration, the observer-based forward kinematics approach is applied to realize our hereby proposed output feedback control. The nonlinear observers developed for a class of nonlinear systems [23,24] are applicable to our system model, (3.5), and hence will be adopted in this paper. Specifically, the form of the nonlinear observer designed for the nonlinear system (3.5) is shown as

ˆ ( )ˆ ( )ˆ 1( ) [ˆ ( )]ˆ

observer gain, and Q∈ ℝ^{12 12×} is the observability matrix defined as linearization which uses Lie derivatives of h(x) along a vector field f(x), namely,

1 2

The linearized system in new coordinate is represented as

1 1

and naturally the observer is chosen as

1 1 reconstruction. The sketch of the proof will be discussed below, and the details can be seen in [23].

Theorem 3.1 For the system (3.1) with x ∈ ℝⁿ, y∈ ℝ, and u∈ ⊆ ℝU (SISO case), following assumptions hold:

(A1) The system is drift-observable in ^ℝn, and the map z= Φ( )x is uniformly Lipschitz

together with its inverse x= Φ⁻¹( )z in ^ℝn , with constants γ_Φ and γΦ−1

respectively;

(A2) the functions L(Φ⁻¹( ))z , H(Φ⁻¹( ))z are uniformly Lipschitz in ^ℝn, with Lipschitz constants γ and _L γ respectively; _H

(A3) a constant u_M> 0 exists such that u t( ) ≤u_M ∀ ≥ ; t 0

(A4) for a given α > 0, a vector K ∈ ℝⁿ and a symmetric positive definite matrix P∈ ℝn n^× exist that satisfy the following H_∞ Riccati-like inequality

2 2

(A−KC P) +P A( −KC)^T +BB^T +FF^T +2αP+γ P ≤0 where γ² =γ_L² +u_M²γ_H² . Then, the dynamic system (3.6) is such that x t( )−x tˆ( ) ≤µe^−αt x(0)−xˆ(0) .

Proof. Recalling the system in z-coordinates (3.9) and the observer (3.10), the error

dynamics can be obtained as

ˆ ˆ

( ) ( ( ) ( )) ( ( ) ( ))

zɺɶ= A KC z− ɶ+B L z −L z +F H z −H z u (3.11) where zɶ= −z zˆ From assumptions (A2) and (A3), we have L z( )−L z( )ˆ ≤γ_L zɶ and

( ( )H z −H z u( ))ˆ ≤u_Mγ_H zɶ , and the error dynamics can be written as

1 2

( )

zɺɶ= A KC z− ɶ+Bυ +Fυ (3.12) where υ₁ =L z( )−L z( )ˆ and υ₂ =( ( )H z −H z u( ))ˆ . To prove the estimation error approaches zero, consider the positive definite function of zɶ

( )z z P z^T 1 .

ν ɶ = ɶ ⁻ ɶ (3.13)

The derivative of ν is

1 1

minimum eigenvalue of a matrix (the property

1

the Lipshitz condition given in assumption 1), the inequality becomes

( ) ^t (0)

The properties of this observability matrix Q x( ) are shown in following definitions.

Definition 3.1 The map Φ( )x is said to be an observability map if it is a diffeomorphism from an open set Ω ⊆ ℝ , and the system is said to be drift-observable. If ⁿ Ω ≡ ℝⁿ, then

the system is said to be globally drift-observable. ▓

Consequently, the Jacobian of the observability map, Q x( ), is nonsingular in Ω, and the inverse map is denoted as x= Φ⁻¹( )z . The Jacobian can be computed as

1 1

( ) /z z Q ( )x

− −

∂Φ ∂ = , which is the inverse of the observability matrix shown in (3.6).

Let Yn be denoted as the vector of output derivatives of MIMO system (3.1)

1 2

T T T T

n m

Y = Y Y … Y  (3.18) where Y_i = y_i yɺ_i … y_i^li−1^T. It is easy to verify that Y_n = Φ( )x provided u t ≡( ) 0, whose fact implies that, if the vector Yn is known at time t, the invertibility of the mapping

( )x

3.1.3 Design the Observer for the Platform Model

To apply the technique of feedback linearization [42] for the dynamics model of the Stewart platform, we define Yn as a vector of the output channels and their first order time

derivatives, i.e.,

and U denotes the vector of inputs. Recalling from (3.4), the output can be expressed as a function of position and orientation states as

2 2 2 first order time derivative of the output function can be derived as

1 2 12

Then, the first term on the right hand side of (3.20), which is the Lie derivative of h x_i( )

and the other terms on the right hand side of (3.20) are the summation of Lie derivatives of

i( )

( ) ( )

Note that the above two functions contain nonlinear terms of the system, and h(x) and the Lie derivatives of h(x) are both smooth functions so that the aforementioned Lipschitz appropriately specified. Up to now, we can then define the full observer gain matrix K for

the full system pair (A_{12 12×} ,C_{6 12×} )as follows The next step to design the observer is to derive the form of the Jacobian matrix Q(x).

From (3.7), (3.8) and Φ( , )x U = Φ( )x , the Jacobian matrix is represented as

1 2 12

By applying the above procedure to our Stewart platform, the inputs obviously are not used in deriving the observability matrix so that the Jacobian can be represented as a simpler form. Thus, if the inputs can also be specified to be bounded, then the conditions of the observer can be readily satisfied. This will be addressed in Section 3.2.

The observability matrix ( ) ^{( )}x

Q x x

∂Φ

≜ ∂ is the Jacobian of the diffeomorphism map Φ ⋅( ) and should be nonsingular for state reconstruction. In fact, the property of the Jacobian matrix can be referred to a proposition in [42] as described below:

Proposition 3.1 Suppose an MIMO system has n states and a relative degree vector

has a Jacobian matrix which is nonsingular at x^o. The value at x^o of these additional functions can be chosen arbitrarily. Remark 3.1 will discuss the relative degree in our design case.

Remark 3.1 Owing to the conditions of existence of diffeomorphic mapping or nonsingular

Jacobian matrix, the relative degree of the dynamics model should be examined as described in section 3.1.2. The definition of relative degree r of an MIMO nonlinear system is the summation of r₁,...,r_m, where r_i in a set Ω is the relative degree associated with the ith output channel of the system if

1

original system.

Remark 3.2 In this section, the system described in (3.1)~(3.4) is shown to satisfy the

assumptions (A1)~(A4) of the observer design (3.6) to guarantee the convergence of the estimation error. These discussions can be summarized by the followings:

1) If Φ( )x is a diffeomorphism, then both Φ( )x and Φ⁻¹( )z are continuously differentiable. Since we have found the diffeomorphic mapping z= Φ( )x of the system (3.1), the Lipschitz conditions of both sides can be ensured.

2) The output function h(x) and the Lie derivatives of h(x) are both smooth functions and the nonlinear functions f x( ), g x( ) belong to C^∞ functions so that the Lipschitz conditions of L x( ) and H x( ) shown in (3.27) are seen to be satisfied.

3) In the controller design addressed in Section 3.2.2, we proposed a bounded control law to ensure the convergence of the estimation error.

4) For a given α >0 and choosing an observer gain K causing all eigenvalues of (A KC− ) to have negative real parts, the H^∞ Riccati-like inequality described in [23,24] guarantees that the solution P is a symmetric positive definite matrix. The proof can be seen in [24].

Remark 3.3 The conditions of the applied nonlinear observer described in Theorem 3.1 do

not include observability for any input but only drift-observability. The condition about the input is that the input must be bounded to exclude the presence of inputs that make some

system states indistinguishable. The conditions of observability for any input of the class of systems can be seen in [48],[49].

3.2 Output Feedback Control

where e denotes the error between the desired trajectory and the corresponding state, and _p

qr denotes a vector of auxiliary signals. By the linear parameterization property [41]

which assumes that the equation of motion is linear in terms of an appropriately selected set

of constant parameters θ , i.e.,

The sliding surface variable s∈R⁶ is a combination of e_p, eɺ and is expressed as _p

s t ≡ .The error dynamics can be written in terms of the sliding surface variable s and

be expressed as follows:

If the control law u is designed in the form

( )

feedback controller renders the entire Stewart platform system stable, and the control error

exponentially converges to zero.

Proof . Consider the Lyapunov function candidate

¹ ( ) .

3.2.2 Output Feedback Controller

We now illustrate the control law which has a similar form (3.37) with the estimated state. Instead of real posture q, the estimated posture qˆ is employed by the controller.

Similar to state feedback controller, the notations are defined as ˆ ˆ ˆ,

will be discussed later. The linear parametric model corresponding to the estimated state is

shown as

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

( ) _r ( , ) _r ( ) ( , , _d, _d, _d) .

M q qɺ +C q q qɺ +G q = Γ q q q q qɺ ɺ ɺɺ θ _(3.44) Thus, the output feedback control law is a function of the estimated posture, and hence is

rewritten as As shown in Theorem 3.1, the input u should have a bound to satisfy the condition of the observer. Referring to the designed input (3.45), due to the fact that Jacobian J ⋅( ),

controller gain K_s, and switching function ϕ_o are all bounded, the boundness of u solely

∈ × , suggesting that the stability analysis undertaken throughout the sequel will be

locally based.

Consider the observer described by (3.6) and apply the control law (3.45), then the error dynamics expressed in terms of s_o will take the following form:

[ ]

where Q_I =Q⁻¹= Q^T_I1 Q^T_I2^T. After applying the saturated control law (3.47), (3.48) locally (i.e. when the local condition above is met) becomes

[ ]

In the following, we will show that the proposed sliding mode control employing the

estimated state produced by the earlier developed nonlinear observer, which solves the

forward kinematics problem in real time, will render the entire Stewart platform system

locally stable, and in turn the control error will locally exponentially converge. In other

words, the control task is to let the posture of the motion platform track the desired posture

trajectory but without installing any posture or attitude sensors (i.e., only leg lengths

measured by LVDT are used).

Theorem 3.3 Considering the dynamic model of the Stewart platform as described in

(3.1)~(3.4) with the nonlinear observer as described by (3.6), here we estimate the posture

state, and apply the saturated sliding mode control law u=ub (3.47). Given the desired trajectory of the posture state, namely, x_d = q_d^T qɺ_d^T^T and suppose (q q q q q_d,ɺ_d,ɺɺ_d, , )ˆ ˆɺ ∈Q_d×Q_est. Then, the resulting closed-loop output feedback system, running

in a way of using leg lengths (output) to estimate the current posture state of the platform

via solving real-time forward kinematics problem, will ensure local asymptotical convergence of the state tracking errors eɺ_p, e_p. ▓

Proof . Consider the Lyapunov function candidate

tracking and estimation errors, respectively. Next, we will assess the time derivative of V by investigating Vɺ_{1 and} which then leads to (3.49). After substituting (3.49) into (3.51) and using the Property 2.2,

we obtain

The property of the drift-observability stated in Section 3.1.2 leads to the fact that the map

( )

z= Φ x is uniformly Lipschitz and so is its inverse map x= Φ⁻¹( )z , which consequently implies that Q⁻¹( )x is bounded on Φ Ω( ). In turn, this implies that the submatrix Q_I2 is also bounded, satisfying Q_I2 ≤ ϒ for some _Q ϒ_Q>0. Note that y h q− ( )ˆ = ɶCz, then from

(3.45) we have that

Second Step. The second part of the Lyapunov function candidate, V2, mainly accounts for the observation error as

V₂ = Σzɶ^{T −1}zɶ (3.54) where the positive definite symmetric matrix Σ satisfies an H^∞ Riccati-like inequality as described in [24]. Given that the drift-observability is fulfilled and the input u is bounded, then the time derivative of V2 is proven to be

Vɺ₂ ≤ −2ξzɶ^TΣ⁻¹zɶ= −2ξ⋅V₂ (3.55) where ξ > is a positive constant. Now the time derivative of the integrated Lyapunov 0 function V is given by the summation of (3.53) and (3.55) as

1 2

the condition 8K_sξσ > ϒ_Q² is satisfied. For a given ξ > and choosing an observer gain 0 K causing all eigenvalues of (A KC− ) to have negative real parts, the H^∞ Riccati-like inequality described in [23,24] guarantees that the solution Σ is a symmetric positive

definite matrix [24] such that the minimum eigenvalue of Σ satisfies ⁻¹ σ > as well as 0 ξσ > . As a result, with an appropriate controller gain Ks, the condition 0 8K_sξσ > ϒ_Q²

definite matrix [24] such that the minimum eigenvalue of Σ satisfies ⁻¹ σ > as well as 0 ξσ > . As a result, with an appropriate controller gain Ks, the condition 0 8K_sξσ > ϒ_Q²

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