Chapter 2 Preliminary
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 av is the acceleration of the vehicle, fˆ 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∈ ℝ12 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 γΦ 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 uM> 0 exists such that u t( ) ≤uM ∀ ≥ ; 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 +BBT +FFT +2αP+γ P ≤0 where γ2 =γL2 +uM2γH2 . 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 ≤γL zɶ and
( ( )H z −H z u( ))ˆ ≤uMγH 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 zT 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 Yi = yi yɺi … yili−1T. It is easy to verify that Yn = 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 xi( )
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 (A12 12× ,C6 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 xo. The value at xo 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,...,rm, where ri 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 pqr 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 ep, 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 errorexponentially 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 Ks, and switching function ϕo 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 so will take the following form:
[ ]
where QI =Q−1= QTI1 QTI2T. 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, xd = qdT qɺdTT and suppose (q q q q qd,ɺd,ɺɺd, , )ˆ ˆɺ ∈Qd×Qest. 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 eɺp, ep. ▓
Proof . Consider the Lyapunov function candidate
tracking and estimation errors, respectively. Next, we will assess the time derivative of V by investigating Vɺ1 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 QI2 is also bounded, satisfying QI2 ≤ ϒ for some Q ϒQ>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 = ΣzɶT −1zɶ (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 ≤ −2ξzɶTΣ−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 8Ksξσ > ϒQ2 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 8Ksξσ > ϒQ2 can be ensured. Then, applying (3.46) and satisfying the condition 8Ksξσ > ϒQ2 for (3.56) can guarantee that
(
2 2)
1 o
Vɺ≤ −κ s + zɶ (3.57)
for some κ1 > . Thus, we can readily conclude that 0 eˆɺp, eˆp
and estimation error zɶ converge to zero exponentially. Consequently, q q− converge to zero as well as ˆ eˆp, eˆɺp converge to ep, eɺp, and our control goal ep, eɺp →0 can be achieved, provided the
hypotheses of this Theorem are met. It is worthy to note that the stability result here is in fact a local result for the hypothesis requiring that (q q q q qd, ɺd,ɺɺd, , )ˆ ˆɺ needs to lie inside the
set Qd×Qest in the Theorem. ▓
The conventional control scheme for Stewart platform needs to apply some inverse kinematics solver to convert the desired posture command to the desired trajectories of leg lengths. Instead of controlling the posture directly, the 6 actuators are controlled separately.
Unlike the conventional control, here in this paper the output feedback control directly controls the platform posture via feedback with observed states. This control scheme is
shown in Figure 3.1.
In the conventional control scheme as shown in Figure 2.4, q is the desired d command of the platform posture, including both desired position and orientation. For lacking the posture information of the platform, the desired posture must be transformed into the desired length of each leg actuator h q( d). In Figure 3.1, the observer receives the
input u and the estimation error which is the difference between the output y and the estimation transformed by the inverse kinematics ˆy=h x( )ˆ , where y is the vector of 6 leg lengths measured by LVDT.
u y
ˆe
q
dˆx
−
+ˆy
Figure 3.1 Schematic of Output Feedback Control
3.3 Simulation Results 3.3.1 Effect of the Observer
The simulation is carried out to verify the feasibility of the proposed observer-based
forward kinematics and output feedback control scheme. Some important parameters used in the simulation are given in Table 3.1.
Table 3.1 Simulation data of Stewart platform
Parameter Descriptions Value
Figure 3.2 Applying the nonlinear observer to solve the forward kinematics in real time
To verify the performance of the observer in this simulation, we adopt the control loop shown by Figure 3.2, where q is the desired command of platform posture including the d
position and orientation information. For lacking actual posture measurement, the desired posture command must be transformed into the desired length r of each actuator. Here, u and y denote the input and output of the system model, and more specifically y is the vector of 6 leg lengths measured by the sensors. The observer receives the input u and the estimation error ˆe which is the difference between the output y and the estimated leg length vector ˆy that is transformed by inverse kinematics yˆ =h x( )ˆ .
Circular Motion
The first simulation to illustrate the observer effect is to let the motion platform to execute a circular motion in x-y plane. The desired posture trajectory is shown in Table 3.2.
Table 3.2 Translational motion trajectory of simulation Case 1
Sway 0.3sin(7t) m
Surge 0.3cos(7t) m
Heave 0.55m
Pitch 0 rad
Roll 0 rad
Yaw 0 rad
The simulation is carried out from the initial state q t( )0 =
[
0 0 0.55 0 0 0]
Tand with integration step size of 1.0 ms. The observer gain matrix K used here is to assign the closed-loop eigenvalues to λ= − × 15 3 32T for the six blocks (such as those described in (3.28)). In this simulation, we assume that the observed initial state is q tˆ( )0 =
[
0.2 0 0.75 0 0 0]
T, which represents that the estimated posture of the movingplatform is subjected to 20 cm translation error in both x- and z-axes at initial time t=t0. The plots shown in Figure 3.3 report that the actual and observed (dashed line) translations along x-, y-, and z-axes as well as over x-y plane. It is shown that the initial estimation errors occur on x- and z-axis, but y-axis estimation is still affected slightly. Moreover, the estimated states converge to the actual state trajectories at about t=0.57s.
(a) (b)
(c) (d)
Figure 3.3 Observed and actual translation along (a) x-axis, (b) y-axis, (c) z-axis, and over (d) x-y plane
Rotation Motion
In the second simulation, we examine the observer effects when the motion platform is exercising a rotational motion, where the reference inputs of rotation angles are chosen to be sinusoidal functions. The simulation results are shown in Figure 3.4, and the desired posture trajectory is shown in Table 3.3.
Table 3.3 Rotational motion trajectory of simulation Case 2
Sway 0 m
Surge 0 m
Heave 0.55 m
Roll 0.2sin(5t) rad Pitch 0.2cos(5t) rad
Yaw 0 rad
In this simulation case, we assume that the observed initial posture state is at q tˆ( )0 =
[
0 0 0.55 0.0873 0 0]
T, which represents that the estimated posture of the movingplatform is subjected to 5 degrees of orientation errors about x-axis (angle α ) at t=t0. The plots shown in Figure 3.4 illustrate that the actual and observed (dashed line) angles about x-, y-, and z-axes. Moreover, the observed rotation trajectories also converge to the true values at about t=0.45 s.
(a) (b)
(c) (d)
Figure 3.4 Observed and actual rotations about (a) x-axis, (b) y-axis, (c) z-axis, and (d) enlargement of (b)
3.3.2 Simulation of Output Feedback Control
The third simulation is carried out with the control scheme as shown in Figure 3.1 to
examine the feasibility of the proposed output feedback control scheme in this application.
Circular Motion
In the first case, a circular motion of the moving platform over x-y plane is to be executed. The desired trajectory for circular motion tracking is specified in Table 3.4.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Table 3.4 Circular motion trajectory with output feedback control of simulation case 3 trajectories over x-y plane and along z-axes will converge to the desired ones at about t=0.6 s.
(a) (b)
Figure 3.5 Desired (solid) and actual (dashed) circular motion over (a) x-y plane, and along (b) z-axis
output feedback control when the moving platform is to exercise a desired rotational motion. The desired rotational motion trajectory is shown in Table 3.5.
Table 3.5 Rotational motion trajectory with output feedback control of simulation case 4
Sway 0 m trajectory is in dashed curve, and both will get closer and gradually coincide at about t=0.5 s. Furthermore, in the steady state, owing to the x- and y-axis rotations, the yaw angle of the motion platform slightly oscillates.
(c)
Figure 3.6 Desired (solid) and actual (dashed) rotational motion in (a) roll angle, (b) pitch angle, and (c) yaw angle
Model with uncertainties
This simulation shows the performance of the output feedback control in the model with 10% time varying uncertainty of the weight exists on the platform. The desired trajectory is specified in Table 3.6.
Table 3.6 Desired motion trajectory for simulation of output feedback control on model comparison of the controls applied to systems with and without uncertainty in x axis. The
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
dotted line shows the result when 10% time varying uncertainty of the weight exists on the platform. The RMS of the control errors are 0.0078 m for the system without uncertainty and 0.0089 m for that with uncertainty. In the simulation, not only the observed trajectories converge to actual ones, but the actual trajectories track the desired ones well. The performance of control for system with uncertainty is also verified.
Figure 3.7 Simulation results of output feedback control on model with mass uncertainty
3.4 Experimental Results
After having performed computer simulation, we also conducted real experiments to realistically demonstrate the feasibility of the proposed output feedback control scheme based on our proposed real-time forward kinematics solution using nonlinear observers.
The experiments are also categorized into two groups. The first part is to demonstrate the
0 0.2 0.4 0.6 0.8 1
-0.1 -0.05 0 0.05 0.1 0.15
Time (sec)
Translation along x axis (m)
(d) x-axis with uncertainty
Actual Desired
with uncertainty
observer performance, whereas the second part is to validate our proposed output feedback control. In the first experimental group, we employ tilt sensors and off-line Newton-Raphson algorithm as means to extract posture state of the moving platform.
Note that the tilt sensors are used to measure the pitch and roll angles, whereas the Newton-Raphson iterative algorithm can provide a rather precise forward kinematics
Note that the tilt sensors are used to measure the pitch and roll angles, whereas the Newton-Raphson iterative algorithm can provide a rather precise forward kinematics