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A Case Study in Approximate Linearization: The Acrobot Example
Richard M. Murray Electronics Research Laboratory University of California Berkeley, CA 94720 John Hauser Department of EE-Systems University of Southern California Los Angeles, CA 90089–0781

Memorandum No. UCB/ERL M91/46 29 April 1991

Electronics Research Laboratory
College of Engineering University of California, Berkeley 94720

A Case Study in Approximate Linearization: The Acrobot Example
Richard M. Murray∗ Electronics Research Laboratory University of California Berkeley, CA 94720 murray@united.berkeley.edu John Hauser† Department of EE-Systems University of Southern California Los Angeles, CA 90089–0781 hauser@nyquist.usc.edu

29 April 1991

Abstract
The acrobot is a simple mechanical system patterned after a gymnast performing on a single parallel bar. By swinging her legs, a gymnast is able to bring herself into an inverted position with her center of mass above the part and is able to perform manuevers about this configuration. This report studies the use of nonlinear control techniques for designing a controller to operate in a neighborhood of the manifold of inverted equilibrium points. The techniques described here are of particular interest because the dynamic model of the acrobot violates many of the necessary conditions required to apply current methods in linear and nonlinear control theory. The approach used in this report is to approximate the system in such a way that the behavior of the system about the manifold of equilibrium points is correctly captured. In particular, we construct an approximating system which agrees with the linearization of the original system on the equilibrium manifold and is full state linearizable. For this class of approximations, controllers can be constructed using recent techniques from differential geometric control theory. We show that application of control laws derived in this manner results in approximate trajectory tracking for the system under certain restrictions on the class of desired trajectories. Simulation results based on a simplified model of the acrobot are included.
Research supported in part by an IBM Manufacturing fellowship and the National Science Foundation, under grant IRI-90-14490. † Fred O’Green Assistant Professor of Engineering


i

1 INTRODUCTION

1

1

Introduction

Recent developments in the theory of geometric nonlinear control provide powerful methods for controller design for a large class of nonlinear systems. Many systems, however, do not satisfy the restrictive conditions necessary for either full state linearization [7, 5] or input-output linearization with internal stability [2]. In this paper, we present an approach to controller design based on finding a linearizable nonlinear system that well approximates the true system over a desirable region. We outline an engineering procedure for constructing the approximating nonlinear system given the true system. We demonstrate this approach by designing a nonlinear controller for a simple mechanical system patterned after a gymnast performing on a single parallel bar. There has been considerable work in the area of system approximation including Jacobian linearization, pseudo-linearization [10, 12], approximation with a nonlinear system [8], and extended linearization [1]. Much of the work on system approximation has been directed toward analysis and the development of conditions that must be satisfied by the approximate systems rather than on the explicit construction of such approximations. Notable exceptions include the standard Jacobian approximation and the recent work of Krener using polynomial system approximations [9]. Wang and Rugh [12] also provide an approach for constructing configuration scheduled linear transformations to pseudo-linearize the system (note that this approach provides a family of approximations rather that a single system approximation). Rather than using polynomial systems or families of linear systems to approximate the given system, we approximate the given nonlinear system with a single nonlinear system that is full state linearizable. We use as a guiding example the problem of controlling the acrobot (for acrobatic-robot) shown in Figure 1. The acrobot is a highly simplified model of a human gymnast performing on a single parallel bar. By swinging her legs (a rotation at the hip) the gymnast is able to bring herself into a completely inverted position with her straightened legs pointing upwards and her center of mass above the bar. The acrobot consists of a simple two link manipulator operating in a vertical plane. The first joint (corresponding to the gymnast’s hand sliding freely on the bar) is free to rotate. A motor is mounted at the second joint (between the links) to provide a torque input to the system (corresponding to the gymnast’s ability to generate torques at the hip). A life size acrobot is currently being instrumented for experimentation at U.C. Berkeley. The eventual goal in controlling this system is to precisely execute realis-

1 INTRODUCTION
~ m 2 d d d d d d d l d 2 d d m2 g d θ2 d d d d~ m1      l1  θ1     m1 g  ¡e ¡ e

2

Figure 1: Acrobot: an acrobatic robot. Patterned after a gymnast on a parallel bar, the acrobot is only actuated at the middle (hip) joint; the first joint, corresponding to the gymnast’s hands on the bar, is free to spin about its axis. tic gymnastic routines. Our modest initial goal is to understand and design controllers capable of system control in a neighborhood of the manifold of inverted equilibrium positions. That is, we would like to have the acrobot follow a smooth trajectory while inverted, such as that shown in Figure 2. This report presents a detailed study of the stabilization and tracking for the acrobot. We begin with a complete, mathematical description of the system in Section 2. The application of standard control techniques to the acrobot is studied in Section 3. Section 4 briefly introduces the theory of approximate linearization and develops a family of nonlinear controllers using this theory. A comparison of these controllers against a standard linear controller is given in Section 5. Finally, we discuss more general nonlinear control problems and how our results for the acrobot can be applied to them.

1 INTRODUCTION

3

Figure 2: Motion of the acrobot along the manifold of inverted equilibrium positions. The application of the methods presented here require substantial algebraic computation. We have used Mathematica [13] to perform much of our computation for us. We list in the body of this paper the specific Mathematica files which were used to obtain or check indicated results. The listings for these files can be found in the appendix.

2 SYSTEM DESCRIPTION

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2

System description

Considered as a mechanical system, the acrobot has unforced dynamics identical to those of a two link robot. Using a Lagrangian analysis (see for example [11]), the dynamics of the acrobot can be written as ¨ ˙ ˙ M (θ)θ + C(θ, θ)θ + G(θ) = 0 τ

where θ = (θ1 , θ2 ) is the vector of relative joint angles as shown in Figure 1, M is the (uniformly positive definite) inertia tensor, C contains the Coriolis and centrifugal forces, G contains the effects of gravity, and τ is the torque applied between the first and second links. Using point mass approximations, a simple analysis yields (acrobot.m) a + b + 2c cos θ2 b + c cos θ2 b + c cos θ2 b ˙ ˙ ˙ −c sin θ2 θ2 −c sin θ2 (θ1 + θ2 ) ˙1 c sin θ2 θ 0 −d sin θ1 − e sin(θ1 + θ2 ) e sin(θ1 + θ2 )

M (θ) = ˙ C(θ, θ) = G(θ) = where

(1) (2) (3)

2 2 a = m1 l1 + m2 l1 2 b = m2 l2 c = m2 l1 l2

d = gm1 l1 + gm2 l1 e = gm2 l2

Due to the presence of rotary joints, these dynamics are highly nonlinear and contain many important trigonometric terms. Defining x := θ ˙ θ

we can write the system as a standard nonlinear system, affine in the control u := τ , x = f (x) + g(x)u ˙ where the system vector fields, f and g, are given by f (x) := ˙ θ ˙ −M −1 (C θ + G) and g(x) := 0 M −1 0 1 (4)

2 SYSTEM DESCRIPTION Parameter l1 l2 m1 m2 g Units m m kg kg m/s2 Balanced Value 1/2 1 8 8 10 Actual Value 1/2 3/4 7 8 9.8

5

Table 1: Acrobot parameters. The balanced values correspond to a version of the acrobot which has a connected equilibrium set. Since the system has a single input, we can find a one-dimensional set of equilibrium points (e.g., inverted positions) that the system can achieve. This set consists of all states where f (x0 ) + g(x0 )u0 = 0 for some input u0 . In particular, this is only true if ˙ θ = 0 G(x0 ) = and it follows from equation (4) that u0 = e sin(θ1 + θ2 ) d sin θ2 = −e sin(θ1 + θ2 ) We will refer to the input u0 associated with an equilibrium point x0 as the trim. It is the DC offset needed to counteract the drift vector field, f , at x0 . The equilibrium set consists of one or more connected components. In particular, if d = e, then we have one connected component, otherwise we have two connected components. These two components consist of equilibrium points where the center of mass of the system is above and below the axis of the first joint, respectively. It is easy to see that if (θ1 , θ2 ) is an equilibrium point, then (−θ1 , −θ2 ) and (π ± θ1 , π ± θ2 ) are also equlibrium points (see Figure 3). The kinematic and dynamic parameters for acrobot are given in Table 1. Two sets of values are given. The first corresponds to an acrobot which has an equilibrium set which is a single connected component (i.e., d = e). The second set of values is the approximate parameter values for the physical system at U.C. Berkeley. We have rounded units to rational numbers to ease the computational burden. We shall use the former (“balanced”) values unless otherwise noted. 0 u 1 0

2 SYSTEM DESCRIPTION
 

6

             v v v v v ƒ  ƒ

v  v v v v v v v  v v       ƒ ƒ  ƒ

Non-inverted positions
4 ˜ 4 ˜ 4 ˜ 4 ˜ 4 ˜ ƒ   ƒ     v  v v v v v v v  v v  v ƒ  vƒ v v  v             

l D l D l D l D l D

Inverted positions

Figure 3: Equilibrium points for θ2 = α. In general the inverted equilibrium points are in a separate component of the equilibrium set from the noninverted ones. The equilibrium points for the two sets of parameters are shown in Figure 4. For the “balanced” parameter values, the equilibrium set consists 1 of all θ1 + 1 θ2 = 0, θ1 + 2 θ2 = π, and θ2 = ±π. This last set of points 2 corresponds to the case where the center of mass of the system is coincident with the axis of the first joint, and hence every value of θ1 corresponds to an equilibrium configuration. Note also that there is a gap in the range of θ1 for which the “actual” system may be balanced.

2 SYSTEM DESCRIPTION

7

Figure 4: Equilibrium points for acrobot. The left figure is the equilibrium set using the balanced parameter values, the right plot using the actual parameter values.

3 LINEARIZATION TECHNIQUES

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3

Linearization techniques

In this section we explore the application of linearization techniques to the control of the acrobot. We distinguish between two different linearization methods. The first is linearization about a point, in which we approximate the vector fields f and g by their linearizations about an equilibrium point. If the linearization is stabilizable to that equilibrium point, then in a suitably small neighborhood the nonlinear system can also be stabilized (by linear feedback). A more recent technique is feedback linearization (see, for example, Isidori [6]). This method uses a change of coordinates and nonlinear state feedback to transform the nonlinear system description to a linear one (in the new coordinates).

3.1

Linearization about an equilibrium point

If we let (x0 , u0 ) ∈ R4 × R denote an equilibrium point for the acrobot, the linearization about (x0 , u0 ) is given by z = Az + bv ˙ where z = x − x0 v = u − u0 ∂ A = ∂x (f (x) + g(x)u)|(x0 ,u0 ) b = g(x0 )

It is straightforward to check that the acrobot linearization is completely ˙ ˙ controllable in a neighborhood of θ1 = θ2 = 0, θ1 = θ2 = 0 (straight up). At this point     0 0 0 1 0         0  0   0 0 1   A =  b =   g     l   − ll1l+l2 − lgm2 0 0 2 m 2 1 1 1 m1     1 2 m +(l +l )2 m l1 1 1 2 2 − lg g(l1 m1 +ll1 m2 +l2 m2 ) 0 0 l1 2 m1 l2 l2 m m 1
1 2 1 2

We refer to this method of linearization as Jacobian linearization since it replaces the system vector fields by their Jacobians with respect to x and u evaluated at a point. The linearized system is completely controllable if and only if det b Ab · · · An−1 b = 0 (5)

By smoothness, it follows that the system is controllable in a neighborhood of the origin. We defer the analysis of points where controllability is lost until later in this section.

3 LINEARIZATION TECHNIQUES
       

9

t  t t  t t  t t t  t 

             

ƒ  ƒ

ƒ  ƒ

ƒ ƒ  ƒ

Figure 5: Gravity coupling in the acrobot. By moving the center of mass to one side of the vertical axis, we can cause the entire mechanism to rotate. Controllability for the acrobot can be given physical interpretation. Consider the case when the mechanism is pointed straight up, with its center of mass directly above the pivot point (see Figure 5). We have direct control of the relative angle of the second link. By moving the second link to the left or right, we can force the center of mass to lie on either side of the pivot point and thus force the whole mechanism to rotate. This use of gravity is crucial in achieving control since equation (5) is not satisfied if g = 0 (Ab is zero). A second effect which occurs is inertial coupling between the first and second links. Since the motor exerts a torque on the second joint relative to the first joint, pushing the second joint in one direction causes the first joint to move in the opposite direction. This phenomenon is seen in the linear model by the presence of a right half plane zero; the transfer function between the hip torque and the angle of the first joint (using the balanced parameter values from Section 2) is: 3 s+2
5 3

4(s4 − 60s2 + 400) Solving for the poles of this transfer function verifies that the acrobot is open loop unstable. We now return to the question of controllability and investigate equilib-

s−2

5 3

3 LINEARIZATION TECHNIQUES

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3

2

1

1

2

3

3

2

1 0.01

1

2

3

0.5

0.02 0.03

1.0

0.04 0.05

1.5

0.06

Figure 6: Determinant of controllability matrix versus θ2 . The plot on the left corresponds to the balanced parameters and the plot on the right to the actual parameters. rium points at which the linearization is not controllable. Figure 6 shows a plot of the determinant of the controllability matrix in equation (5) versus the hip angle of the acrobot. We see that the system is controllable except at points where θ2 = ±π. Physically this configuration corresponds to the second link of the acrobot pointing back along the first. In this configuration, the balanced acrobot can swing freely about the axis of the first link and remain in an equilibrium position. So far our discussion has centered about using a linear controller for stabilization; our real interest is in trajectory tracking. We begin by reviewing trajectory tracking for a linear system ˙ x = Ax + bu We assume the system is completely controllable and we wish to track a desired state trajectory xd . Without loss of generality we can assume that (A, b) are in controllable canonical form, i.e. a chain of integrators. In this case the system can be written as x1 = x2 ˙ x2 = x3 ˙ . . . xn−1 = xn ˙ xn = u ˙ where we have placed all poles at the origin to simplify notation. If xd (·) is a desired trajectory which satisfies xd = Axd + bud for some ˙ d (i.e., xd is achievable) then we can follow this trajectory by using u u = xn ˙d

3 LINEARIZATION TECHNIQUES

11

when xd (0) = x(0). This choice of inputs corresponds to injecting the proper input at the end of the chain of integrators which model the system. To achieve trajectory tracking even if our initial condition does not satisfy xd (0) = x(0) we introduce the feedback control law u = xd + α1 (xd − xn ) + · · · + αn (xd − x1 ) ˙n n 1 and the error system satisfies e(n) + α1 e(n−1) + · · · + αn e = 0 e = xd − x

By choosing the α’s so that the resulting transfer function has all of its poles in the left half plane, e will be exponentially stable to 0 and the actual state will converge to the desired state. In the case of a linearized system, the linearization may not be a good approximation to the system for arbitrary configurations. Since we linearized about a single point, we can only guarantee trajectory tracking in a sufficiently small ball of states about that point. There are several methods for circumventing this problem; one of the most common is gain scheduling. To use gain scheduling, we design tracking controllers for many different equilibrium points and choose our gains based on the equilibrium point(s) to which we are nearest. In fact, this can be done in a more or less continuous fashion using a technique called extended linearization [12]. The basic restriction is that the desired reference trajectory must be slowly varying.

3.2

Feedback linearization

Given a nonlinear system x = f (x) + g(x)u ˙ (6)

it is sometimes possible to find a change of coordinates ξ = φ(x) and a control law u = α(x) + β(x)v such that the resulting dynamics are linear: ˙ ξ = Aξ + bv In such cases we can control the system by converting the desired trajectory or equilibrium point to our new coordinates, calculating the control v in the that space, and then pulling the control back to the original coordinates. If such a change of coordinates and feedback exists, we say that (6) is input/state linearizable.

3 LINEARIZATION TECHNIQUES

12

The conditions under which a general nonlinear system can be converted to a linear one as described above were formulated independently by Jakubcyzk and Respondek and Hunt, Su and Meyer. For the single input case, the conditions are given by the following theorem. Theorem 1 ([7, 5]). The system (6) is input/state linearizable in an open set U if and only if n−1 (i) dim span{g, adf g, · · · , adf g}(x) = n, ∀x ∈ U n−2 (ii) span{g, adf g, · · · , adf g} is an involutive distribution on U

where adj g is the iterated Lie bracket [f, · · · , [f, g] · · · ]. f The first condition is a controllability test and agrees with the linearization when evaluated at an equilibrium point. The importance of the second condition is more subtle. If condition (ii) is satisfied, then there exists a smooth h : Rn → R such that ∂h n−2 g adf g · · · adf g = 0 (7) ∂x This can be seen by applying Frobenius’ theorem: since the distribution is involutive, there exists a foliation such that the tangent space to each leaf of the foliation is spanned by the distribution restricted to that leaf. Since the leaves have dimension n − 1, there exists a scalar valued function h such that the leaves are defined by h−1 (a) for a ∈ R. Equation (7) is essentially saying that the gradient of h is perpendicular to the leaves. The standard approach in feedback linearization is to use h to define the required change of coordinates. For single input systems we define φ1 (x) = h(x) φi (x) = Li−1 h(x) f where Lf h = ∂h f is the Lie derivative of h in the direction f . The condition ∂x in equation (7) guarantees that the input will not appear until the nth derivative. Setting ξ = φ(x), our new equations are ˙ ξi = ξi+1 ˙ ξn = a(x) + b(x)u i = 1, · · · , n − 1

and by using u = b−1 (−a + v) we have a linear system (in Brunowsky canonical form).

3 LINEARIZATION TECHNIQUES

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Trajectory tracking for such a system is exactly as in the linear case. However, since we have converted the model to a linear one instead of approximating it, we do not need to stay close to any particular equilibrium point. Thus in an open set U in which the feedback linearizability equations are satisfied, we can achieve exponential trajectory tracking. To check the involutivity condition for the acrobot, we must verify that the vector fields (8) [g, adf g] [g, ad2 g] [adf g, ad2 g] f f lie in the distribution ∆ = span{g, adf g, ad2 g} f This can be done by checking that the determinant of a matrix (which is a function of x) is zero. It can be verified (exact.m) that the determinant obtained using ∆ and the second expression in equation (8) is nonzero. Hence the system is not input/state linearizable. A less restrictive class of systems is the class of input/output linearizable systems. A major difficulty is the possibility of introducing unstable internal dynamics, called zero dynamics. Since there is no predefined output function for acrobot, it might be possible to define an output such that the system is input/output linearizable and has stable zero dynamics. In this case we could again achieve trajectory tracking by relying on the stable zero dynamics to control unobservable states. Finding such an output function is nontrivial. Both of the obvious output functions (θ1 and θ2 ) have unstable zero dynamics. As we saw with the Jacobian linearization, if we use θ1 as the output, we obtain a right half plane zero in the linearized system. The effect of this right half plane zero is also present in the nonlinear system. The input/output linearizing feedback cancels this zero with a pole at the same location and results in unstable zero dynamics. Similar problems occur when using θ2 as the output. To summarize, we have shown that the acrobot is stabilizable about most equilibrium points (all but a set of measure 0) using static linear state feedback. This simple approach is not suitable for trajectory tracking, although gain scheduling and related approaches might be used to improve performance. The more global method of input/state linearization via state feedback cannot be applied to acrobot since the system is provably not input/state linearizable. In the next section we investigate the use of approximate linearization techniques to recover some of the desirable properties of feedback linearization for systems which do not meet the necessary restrictive conditions.

4 APPROXIMATE LINEARIZATION

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4

Approximate linearization

In the previous section we showed that the acrobot dynamics are not exactly linearizable by state feedback. In this section we apply the technique of approximate linearization to the acrobot. Briefly, we wish to find vector ˜ fields f and g which are close to our original vector fields but which satisfy ˜ the exact linearizability conditions. We then proceed to design a controller for the approximate system and apply it to the actual system. The usual method of approximate linearization is slightly complicated in the case of the acrobot for two reasons: we do not have a natural output function and we wish to track trajectories near a manifold of equilibrium points rather than near a single point. This chapter presents a methodology for designing a controller for a system of this type. Briefly, we will proceed in the following manner: 1. Parameterize the controllable equilibrium manifold, E, as (x1 , 0, · · · , 0). 2. Construct a smooth output, h(x), such that the linearized system at each equilibrium point has relative degree n. ˜ 3. Using h, construct approximate vector fields f and g such that they ˜ approximate f and g along the equilibrium manifold and the approximate system is exactly linearizable. ˜ 4. Using f and g , design a tracking controller for the approximate system ˜ and apply the resulting controller to the original system. We begin with a brief review of approximation theory using the presentation in Hauser et. al. [3] as a guide.

4.1

Review of approximation theory

We consider systems of the form x = f (x) + g(x)u ˙ y = h(x) (9)

The system is input/output linearizable with relative degree n in a neighborhood U if and only if for all x ∈ U i−1 (i) Lg Lf h(x) = 0 i = 1, · · · , r − 1 n−1 (ii) Lg Lf h(x) = 0

4 APPROXIMATE LINEARIZATION

15

where Lf h = ∂h f is the Lie derivative of h in the direction f . These ∂x conditions are equivalent to the exact linearization conditions in Theorem 1 of the previous section. That is, ∂h annihilates the distribution ∂x n−2 {g, adf g, · · · , adf g}. As before, we use the output ξ = h(x) and its first n derivatives to define a new set of coordinates. Using this new set of coordinates, the input/output map is given by the linear transfer function 1/sn . If the input/output conditions are not satisfied, then we can still use this basic construction as a method for generating approximate vector fields which do satisfy the conditions, at least in a neighborhood of a controllable equilibrium point. Since the behavior of the nonlinear system about an equilibrium point is determined by its linearization, any approximate system should agree with the linearized system at an equilibrium point (x0 , u0 ). ˜ ˜ That is, the approximate vector field f + g u should agree to first order with the original vector field f + gu, when evaluated at the equilibrium point. In particular, this implies that the relative degree of any approximate system should agree with the relative degree of the linearization. This motivates the following definition: the linearized relative degree of a nonlinear system in a neighborhood of an equilibrium point x0 is the relative degree of the linearization about x0 . We use this concept to construct an approximate system which has relative degree equal to the linearized relative degree of the original system. A key concept is that of higher order. A function ψ(x) is said to be higher order at x0 if the function and its first derivative vanish at x0 . More generally, a function is order k at x0 if the function and its first k derivatives vanish at x0 , and first order if only the function itself is zero at x0 . Let x0 be an equilibrium point of a nonlinear system with u0 the input required to hold the system at the equilibrium point. Suppose the linearized relative degree of the system about x0 is n. Then we can define an approximate system in a neighborhood of (x0 , u0 ) as follows: set φ1 (x) = h(x) − ψ0 (x) where ψ0 is any function that is higher order at x0 . For i = 2, · · · , n, set φi (x) = Lf φi−1 (x) + u0 Lg φi−1 (x) − ψi−1 (x) where ψi (x) is higher order at x0 . It can be shown that φ is a local diffeomorphism and hence defines a valid change of coordinates. If we write the system dynamics in this new set of coordinates, we get a chain of integrators with nonlinear perturbations (Figure 4.1).

4 APPROXIMATE LINEARIZATION

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v

h

ξ4

h

ξ3

h

ξ2

h

ξ1

h

y

ψ(x, y) higher order nonlinear terms

Figure 7: Approximate linearization viewed as a chain of integrators with nonlinear perturbations (from [HKS89]). To see how this procedure produces an approximate system, we pull back the Brunowsky canonical vector fields through the diffeomorphism φ to produce the approximate vector fields:     φ2 (x) 0   .   .  .  ˜ g = [Dφ]−1  .  ˜ f = [Dφ]−1  .   .  0 φn (x) 1 0

By construction, the approximate vector fields are input/output linearizable with relative degree n. Furthermore, the vector fields agree with the original vector fields to first order at x0 since we only throw away higher order terms. There is a great deal of freedom in choosing the approximation; this freedom is manifested through the choice of the ψi ’s. If the system were input/output linearizable, then we could have chosen ψi to be zero at each step and we would have exactly the change of coordinates produced in the exact linearization procedure. Another interesting case is when we choose ψi to include all second order and higher terms; in this case our approximate system is equivalent to the Jacobian linearization. In general, however, it is not clear which terms to ignore in selecting coordinates. Currently the choice of approximation is a matter of engineering judgement. Using the approximate system, we can construct an exactly linearizing control law which is capable of trajectory tracking. In our new coordinates,

4 APPROXIMATE LINEARIZATION ξ = φ(x), the system has the form ˙ ξi = ξi+1 + ψi (x) + θi (x)(u − u0 ) ˙ ξn = Lf φn (x) + Lg φn (x)u + ψn (x) + θn (x)(u − u0 ) y = ξ1 + ψ0 (x)

17

where each θi is at least uniformly first order at x0 . With analogy to the exact linearization case, we choose 1 (n) (n−1) −Lf φn (x) + yd + αn−1 (yd − ξn ) + · · · + α0 (yd − ξ1 ) Lg φn (x) (10) where sn + αn−1 sn−1 + · · · + α0 has all its zeros in the open left half plane. Let i−1 d ξi (t) := yd (t) u= and define the tracking error as e(t) := ξ d (t) − ξ(t) This error vector encodes the deviation of the actual system trajectory from the desired trajectory of the approximate system. For ǫ sufficiently small and desired trajectories which are ǫ-near x0 and sufficiently slow, the control law (10) results in approximate tracking of the desired trajectory [3]. Thus we can approximately track any trajectory which remains close to the equilibrium point and is slowing varying. A more explicit (and more general) formulation is presented in Section 4.4.

4.2

The equilibrium manifold

In our application, we are not interested in motion near a single equilibrium point, but rather motion near a set of equilibrium points. Given a general single input system, the equilibrium points are those x0 for which f (x0 ) + g(x0 )u0 = 0 for some u0 ∈ R. We define E to be the set of all equilibrium points, x0 , such that the linearized system is controllable about x0 . Theorem 2. E is a manifold of dimension 1. Proof. Consider first the set Ex,u of all pairs (x0 , u0 ) such that f (x0 ) + g(x0 )u0 = 0 and the system is controllable at x0 . Controllability is determined by taking the determinant of a set of smooth functions and hence there exists and open ball N ∋ (x0 , u0 ) such that all equilibrium points (x′ , u′ ) ∈ N are also controllable. Let U be the union of all such N over

4 APPROXIMATE LINEARIZATION

18

u

          

x1

x2
Figure 8: Projection of the equilibrium points onto the state space [10]. Ex,u . Then U is open and Ex,u ⊂ U . Define the map F : U ⊂ Rn+1 → Rn given by F : (x, u) → f (x) + g(x)u. At any controllable equilibrium point, F (x0 , u0 ) = 0 and the Jacobian of F , DF (x0 , u0 ) = (Df (x0 ) + u0 Dg(x0 ), g(x0 )) = (A0 , b0 ), is full rank. Hence 0 is a regular value of F and F −1 (0) = Ex,u is a submanifold of Rn + 1 of dimension (n + 1) − n = 1. It remains to show that the projection is also a manifold. There are two things that can go wrong: the manifold can be tangent to the projection direction or the manifold can cross over itself. These situations are shown in Figure 8. These singularities can only occur if u0 cannot be written as a function of x0 . However, at any equilibrium point f (x0 ) + g(x0 )u0 = 0

4 APPROXIMATE LINEARIZATION

19

and u0 is not unique only if g(x0 ) = 0. This contradicts controllability and hence u0 is a unique function of x0 and neither of the situations in Figure 8 can occur. We call E the controllable equilibrium manifold and will often refer to it simply as the equilibrium manifold (as opposed to the set of all equilibrium or operating points). In general E consists of one or more connected components. For the acrobot there are always two components, consisting of the inverted and non-inverted equilibrium points. While motion on the controllable equilibrium manifold is not possible (since by definition x = 0 on the manifold), motion near the manifold can ˙ be achieved. In constructing an approximate system, we wish to do so in a way that keeps the approximation close at equilibrium points. Thus we want to throw away terms which are higher order on the equilibrium manifold (i.e., terms whose value and first derivative vanish on E) while keeping terms that vary along the equilibrium manifold. In order to construct such an approximation, it is convenient to change coordinates so that the equilibrium manifold has a simple form. A particularly convenient choice of coordinates is one in which points on the equilibrium manifold have the form (x1 , 0, · · · , 0). We can always find a parameterization of the equilibrium manifold which has this form in a neighborhood of a controllable equilibrium point, since E is a one dimensional manifold. For the acrobot, we have chosen to parameterize the equilibrium manifold using the hip angle. For the second configuration variable we use the angle of the center of mass of the system—this must be zero at all inverted equilibrium points since the center of mass must lie directly above the axis of the first link. We complete the state with the velocities of the two configuration variables. These calculations are contained in (equilibrium.m). The resulting change of coordinates (see Appendix A) is: x1 = θ2 x2 = θ1 + √ x3 = x1 ˙ x4 = x2 ˙ Other parameterizations are possible. For example, one might choose the x and y components of the system center of mass as the configuration variables. Unfortunately, the parameterization is singular about the straight up position, just as it is for a two-link robot manipulator. Another advantage of the parameterization we chose is that it simplifies some of the calculations. e sin θ2 d2 +e2 +2ed cos θ2

4 APPROXIMATE LINEARIZATION

20

In particular, for the balanced system parameters mentioned in Section 2, 1 the angle of the center of mass is simply θ1 + 2 θ2 whereas xcm and ycm involve trigonometric functions. This is the original motivation for defining the “balanced” set of parameters.

4.3

Constructing an (artificial) output function

In the approximation theory presented above, an output function was used to construct the approximate system. In some applications, the system possesses a natural output function that can be used for this purpose. However, in the case of the acrobot, no suitable output function is given so we must construct one. In this section we present a technique for doing so. As usual, we begin by considering the linear case. Suppose we are given a controllable linear system x = Ax + bu ˙ and we are asked to find an output y = cx which is suitable for stable trajectory tracking. By this we mean that it is easy to design a controller to make y(t) track a desired trajectory yd (t) while maintaining internal stability of the system. If the system is in Brunowsky form (i.e., a chain of integrators), then a natural output function is the output from the last integrator. This insures that the system has no zeros so that y(t), y(t), . . . , y (n−1) (t) can be used as the n coordinates of the system ˙ state. In particular, if the output y(t) converges to a constant value, then the system will converge to an equilibrium point. To construct this output when the system (A, b) is not in Brunowsky canonical form, we note that the relative degree of the system is given by the largest r such that cAi−1 b = 0 cAr−1 b = 0 i = 1, · · · , r − 1

Since we want the relative degree to be n (no zeros), we require that c b Ab · · · An−2 b = 0. (11)

Thus, any c = 0 in the (1-dimensional) null space indicated by equation (11) defines an output such that the system (c, A, b) has relative degree n.

4 APPROXIMATE LINEARIZATION We now return to the nonlinear system x = f (x) + g(x)u ˙ with the goal of constructing an output y = h(x)

21

to use in constructing an approximate system for control design. If the system with output is input/output linearizable with relative degree n around x0 then the system is linearly controllable and satisfies the nonlinear analog to (11) given by ∂h n−2 g adf g · · · adf g = 0 ∂x for all x in a neighborhood of x0 . In other words, the system is input/state linearizable—it satisfies the conditions of Theorem 1. Since many systems such as the acrobot are not input/state linearizable, we look to approximation. Our problem is one of finding a function h and approximate vector ˜ fields f and g such that ˜ ∂h g adf g · · · ˜ ˜˜ ∂x n−2 adf g = 0 ˜ ˜

(12)

for all x in a neighborhood of x0 or, more generally, in a neighborhood of the equilibrium manifold. Since it is extremely difficult to directly modify the vector fields f, g so that the system is exactly input/state linearizable, we will first construct the output function h and then use the approximate linearization methodology ˜ to construct f and g . The basic idea is to find a function h that satisfies ˜ equation (12) at each point on the equilibrium manifold. Provided that the original and approximate systems agree to first order on the equilibrium manifold, the ad -chains of the two systems will span the same subspace at each point on the equilibrium manifold, that is, n−2 n−2 span{g, adf g, · · · , adf g} = span{˜, adf g , · · · , adf g } g ˜˜ ˜ ˜

for x ∈ E. In fact, these calculations can be done directly with the linearization of the original system on the equilibrium manifold. This point is somewhat subtle, so we describe it in detail. We will assume that coordinates have been chosen such that the equilibrium manifold E has been straightened out so that each x ∈ E has the form

4 APPROXIMATE LINEARIZATION

22

(x1 , 0, . . . , 0). Let xe (x1 ) and ue (x1 ) denote the state and control for each equilibrium point (x1 , 0, . . . , 0) on E, that is, xe (x1 ) = (x1 , 0, . . . , 0) and ue (·) is such that f (xe (x1 )) + g(xe (x1 ))ue (x1 ) = 0 for each x1 such that xe (x1 ) ∈ E. Suppose, at first, that we trim the drift vector field ¯ f (x) := f (x) + g(x)uc (x) where uc (·) is any control satisfying uc (xe (x1 )) = ue (x1 ). The linearization of the trimmed system along the equilibrium manifold is then given by ¯ z = A(x1 )z + b(x1 )v ˙ where
∂g ¯ A(x1 ) := ∂f (xe (x1 )) + uc (xe (x1 )) ∂x (xe (x1 )) + g(xe (x1 )) ∂uc (xe (x1 )) ∂x ∂x ∂g = ∂f (xe (x1 )) + ue (x1 ) ∂x (xe (x1 )) + g(xe (x1 )) ∂uc (xe (x1 )) ∂x ∂x b(x1 ) := g(xe (x1 ))

In this case it is easy to verify that ¯ adj¯g(xe (x1 )) = (−A(x1 ))j b(x1 ) f Thus, letting c(·) be the derivative of the yet to be constructed output function h along the equilibrium manifold, c(x1 ) := ∂h (xe (x1 )), ∂x

equation (12) (for the trimmed system) evaluated along E takes the form ¯ c(x1 ) b(x1 ) A(x1 )b(x1 ) · · · ¯ A(x1 )n−2 b(x1 ) = 0 (13)

The equation has a smooth solution c(·) on E since the system is, by definition, linearly controllable at each of these points. Unfortunately, this linearization depends on the choice of the trim function uc (·). Certainly, one does not expect that the choice of the trim function can materially affect the directions in which the system can be controlled. Additionally,

4 APPROXIMATE LINEARIZATION

23

since we plan to do symbolic calculations to construct the output function, we seek the simplest expressions for these objects. Note that the actual trim ue (x1 ) needed at an equilibrium point is uniquely defined. If, at a given equilibrium point xe (x1 ) we freeze the trimming control uc (x) ≡ ue (x1 ) then the linearization will be given by z = A(x1 )z + b(x1 )v ˙ where A(x1 ) :=
∂f ∂x (xe (x1 )) ∂g + ue (x1 ) ∂x (xe (x1 )) ∂uc ∂x

¯ Note that A(x1 ) = A(x1 ) due to the presence of the

term. In fact,

∂uc ¯ A(x1 ) = A(x1 ) + b(x1 ) (xe (x1 )) ∂x The following lemma shows that we can use the well-defined expression A(·) ¯ for our calculations in place of the somewhat arbitrary expression A(·). ¯ Lemma 1. Given A(·), b(·), and A(·) as defined above, ¯ span{b(x1 ), · · · , A(x1 )j−2 b(x1 )} = span{b(x1 ), · · · , A(x1 )j−2 b(x1 )} for j = 2, 3, . . .. Proof. The lemma is trivially true if j = 2. Suppose the lemma holds for j ≤ k. ¯ ¯ A(x1 )k+1 b = A(x1 ) + g(xe (x1 )) ∂uc (xe (x1 )) A(x1 )k b(x1 ) ∂x ¯ ¯ 1 )k b(x1 ) + b(x1 )[ ∂uc (xe (x1 ))A(x1 )k b(x1 )] = A(x1 )A(x ∂x The first term is contained in span{b(x1 ), · · · , A(x1 )k+1 b(x1 )} since ¯ A(x1 )k b(x1 ) ∈ span{b(x1 ), · · · , A(x1 )k b(x1 )} The second term is a multiple of b(x1 ) and hence it is also in span{b(x1 ), · · · , A(x1 )k+1 b(x1 )}.

Thus we see that the derivative c(·) of our output function h(·) solves the equation c(x1 ) b(x1 ) A(x1 )b0 · · · A(x1 )n−2 b(x1 ) = 0 (14)

4 APPROXIMATE LINEARIZATION

24

It is clear that c(x1 ) = (c1 (x1 ), · · · , cn (x1 )) (viewed as a differential oneform) is integrable. Indeed, we integrate dh(x) = c1 (x1 )dx1 + · · · + cn (x1 )dxn to get h(x) = c1 (x1 )dx1 + c2 (x1 )x2 + · · · + cn (x1 )xn

Further, since x1 parameterizes the equilibrium manifold, we have the following useful fact: Lemma 2. Suppose that c(x1 ) = 0 solves (14) with xe (x1 ) ∈ E. Then c1 (x1 ) = 0. Proof. By Lemma 1, we may assume that f (x) = 0 for x ∈ E. Since the system is linearly controllable on E, the vectors {b(x1 ), A(x1 )b(x1 ), · · · , A(x1 )n−2 b(x1 )} are linearly independent and c(x1 ) lies in the left null space of these vectors. It suffices to show that e1 = (1, 0, · · · , 0)T is linearly independent of these vectors since this implies c1 (x1 ) = c(x1 ) · e1 = 0. But we see that A(x1 ) · e1 = ∂f (x) ∂x1 xe (x1 )

and this last expression is zero since since f (x) ≡ 0 along the equilibrium manifold, parameterized by x1 . Hence e1 is in the null space of A0 and the n−2 vectors b0 , A0 b0 , · · · , A0 b0 are not in the null space of A0 since are also linearly independent by the controllability assumption. Therefore e1 n−2 is linearly independent of {b0 , A0 b0 , · · · , A0 b0 } and c1 (x0 ) = c0 ·e1 = 0. Given this fact, we can write dh(x) = = = h(x) = c1 (x1 )dx1 + c2 (x1 )dx2 + · · · + cn (x1 )dxn (x1 ) dx1 + c2 (x1 ) dx2 + · · · + cn(x1 ) dxn c1 (x1 ) c1 dx1 + c2 (x1 )dx2 + · · · + cn (x1 )dxn ˜ ˜ x1 + c2 (x1 )x2 + · · · + cn (x1 )xn ˜ ˜ n−1 {A0 b0 , · · · , A0 b0 }

Any h which matches this expression to first order is also a valid output function, with linear relative degree n. For the acrobot, the output function which results from the above calculation is (output.m) h(x) = x1 + (6 + 4 cos x1 )x2

4 APPROXIMATE LINEARIZATION

25

4.4

Approximate tracking near an equilibrium manifold

We can now extend the approximation procedure presented in Section 4.1 to construct a controller which tracks slowly varying trajectories near an equilibrium manifold. To do so, we extend the concept of a higher order function. We say a function is uniformly higher order on a manifold (parameterized by x1 ) if it is higher order in (x2 , · · · , xn ). Thus in the approximation procedure, we will ignore terms which are small near the equilibrium manifold, while keeping terms that vary along the manifold. This section details that procedure and concludes with a proof of approximate tracking for control laws constructed in this manner. It will be convenient at this point to assume that f (x0 ) = 0 for x0 ∈ E. Although we took pains to avoid making this assumption in the previous section, the benefit of allowing f (x0 ) = 0 is outweighed here by a tremendous increase in notation. We therefore assume that any nonlinear trim is included in the drift vector field. This can be accomplished in many ways, the simplist of which is to define ¯ f (x) = f (x) − g(x)ue (x1 ) v = u − ue (x1 ) and write our system as ¯ x = f (x) + g(x)v ˙ y = h(x) ¯ Suppose the linearized relative degree of the system (f , g) with respect to an output function h is n on an equilibrium manifold E = {(x1 , 0, · · · , 0)}. ¯ Assuming f (x0 ) = 0 for x0 ∈ E, we define a new set of coordinates ξ = n: φ(x) ∈ R φ1 (x) = h(x) − ψ0 (x) φ2 (x) = Lf φ1 (x) − ψ1 (x) ¯ . . . φn (x) = Lf φn−1 (x) − ψn−1 (x) ¯ where each ψi (x) is uniformly higher order on E. The system dynamics in ξ

4 APPROXIMATE LINEARIZATION coordinates are ˙ ξ1 = ξ2 + ψ1 (x) + θ1 (x)v . . . ˙ ξn−1 = ξn + ψn−1 (x) + θn−1 (x)v ˙ ξn = Lf φn (x) + Lg φn (x)v + ψn (x) + θn (x)v ¯ y = ξ1 + ψ0 (x)

26

(15)

where each θi (x) is at least uniformly first order on E. As in the previous approximation procedure, the choice of ψ allows considerable freedom in constructing the approximation. Since the linearization is controllable on E and ∂h (x) satisfies (14), it follows that Lg φn (x0 ) = 0 for x0 ∈ E. ∂x Because the functions ψi are uniformly higher order on E and the functions θi are at least uniformly first order on E, the approximate system ˙ ξ1 = ξ2 . . . ˙ ξn−1 = ξn ˙ ξn = Lf φn (x) + Lg φn (x)v ¯ y = ξ1 is a uniform system approximation of (f, g) on E [4]. To provide approximate tracking control for the true system (15), we will use the exact asymptotic tracking control law for the approximate system (16), namely, v= where is a Hurwitz polynomial. As before, we define ξ d (t) to be the state trajectory for the approximate system induced by the desired output, yd (·), d ξi (t) := yd (i−1)

(16)

1 (n) −Lf φn (x) + yd (t) + ¯ Lg φn (x)

n−1 i=0

αi (yd − φi+1 (x))

(i)

(17)

sn + αn−1 sn−1 + · · · + α1 s + α0

(18)

(t)

We then expect that the tracking error e(t) := ξ d (t) − ξ(t)

4 APPROXIMATE LINEARIZATION

27

will remain bounded for reasonable trajectories. In fact, we will see that the size of the tracking error will be influenced by how far the desired trajectory strays from the equilibrium manifold. Since the approximate system (16) is a uniform system approximation of the true system (15) around E, we would expect that the approximation would be valid on, for instance, a cylindrical neighborhood of E given by Cǫ (E) := {ξ : π1 ξ ∈ E, ξ − π1 ξ < ǫ} where π1 ξ := (ξ1 , 0, . . . , 0) and ǫ is sufficiently small. We make use of the following fact: it is always possible to choose E ′ ⊂ E so that a given function λ(ξ) that is uniformly order ρ on E will satisfy |λ(ξ)| < K ξ − π1 ξ ρ 2 for all ξ ∈ Cǫ (E ′ ), 0 < ǫ < 1. For example, let λ(ξ) = ξ1 ξ2 . Choosing ′ = {ξ ∈ ℜ2 : |ξ | < K, ξ = 0} will guarantee the λ(ξ) < Kξ 2 on C (E ′ ), E 1 2 ǫ 2 0 < ǫ < 1. The following theorem shows that such a control law can indeed provide the desired result and provide stable approximate tracking in the neighborhood of the equilibrium manifold.

Theorem 3. Suppose (f, g) is linearly controllable at x0 and let E be the manifold of linearly controllable equilibrium points. Further assume that f (xe ) = 0 for xe ∈ E. Then, there exists a manifold E ′ ⊂ E, a change of coordinates ξ = φ(x), and an ǫ > 0 such that the approximate tracking control law (17) results in stable approximate tracking provided ξ d (t) ∈ Cǫ (E ′ ) (n) and |yd (t)| ≤ ǫ for t ≥ 0, and e(0) ≤ ǫ. Furthermore, the tracking error will be of order ǫ2 . Proof. Construct a system approximation as detailed above. For convenience, define ψ(ξ) = θ(ξ) = (ψ1 (x), · · · , ψn (x))|x=φ−1 (x) (θ1 (x), · · · , θn (x))|x=φ−1 (x)

The closed loop system given by (15) and (17) can be written as e = Ae + ψ(ξ) + θ(ξ)v ˙ where A is a Hurwitz matrix with characteristic polynomial (18).

4 APPROXIMATE LINEARIZATION As discussed above, we may take E ′ to be such that ψ(ξ) θ(ξ) Lf φn (ξ) ¯ ≤ k1 ξ − π1 ξ ≤ k1 ξ − π1 ξ ≤ k1 ξ − π1 ξ
2

28

for ξ ∈ Cδ (E ′ ), δ < 1, and some k1 < ∞. Since Lg φn (x) is nonzero on E, we can also require that E ′ and δ be such that 1 < k2 Lg φn (ξ) for ξ ∈ Cδ (E ′ ) and some k2 < ∞. Using these bounds plus the fact that ξ − π1 ξ ≤ e + ǫ (by choice of yd (·)), it follows that there exists k3 < ∞ such that ψ(ξ) + θ(ξ)v ≤ k3 ( e where ξ ∈ Cδ (E ′ ). Choose the Lyapunov function V = eT P e where P > 0 solves AT P +P A = −I. Differentiating V along the trajectories of the closed loop system, for ξ ∈ Cδ (E ′ ) and some k4 < ∞, ˙ V = − e 2 + 2eT P (ψ(ξ) + θ(ξ)(u − u0 (ξ)) ≤ − e 2 + k4 e ( e 2 + ǫ e + ǫ2 ) 1 1 2 ≤ − 4 e 2 − ( 2 − k4 ( e + ǫ)) e 2 − ( 1 e − k4 ǫ2 )2 + k4 ǫ4 2
1 2k4 2

+ ǫ e + ǫ2 )

If e ≤

− ǫ, we have 1 ˙ V

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