Adaptive λ -tracking for locomotion systems

Robotics and Autonomous Systems 54 (2006) 529–545 www.elsevier.com/locate/robot

Adaptive λ-tracking for locomotion systems Carsten Behn ∗ , Klaus Zimmermann Faculty of Mechanical Engineering, Department of Technical Mechanics, Technische Universit¨at Ilmenau, PF 100565, 98684 Ilmenau, Germany Received 24 June 2004; received in revised form 23 March 2006; accepted 7 April 2006 Available online 6 June 2006

Abstract This paper deals with the (adaptive) control of mechanical systems, which are inspired by biological ideas. We introduce a certain type of mathematical models of worm-like locomotion systems and present some theoretical control investigations. Only discrete straight worms will be considered in this paper: chains of point masses moving along a straight line. We introduce locomotion systems in the form of a straight chain of k = 3 interconnected point masses, where we focus on interaction which emerges from a surface texture as asymmetric Coulomb friction. We consider two different types of drives: (i) The point masses are under the action of external forces, which can be regarded as external force control inputs. (ii) We deal with massless linear springs of fixed stiffnesses and controllable original spring lengths, which can be regarded as internal control inputs. The locomotion systems with these two types of drive mechanisms are described by mathematical models, which fall into the category of nonlinearly perturbed, multi-input, multi-output systems (MIMO-systems), where the outputs of the system are, for instance, the positions of the point masses or the displacements of the point masses. The goal is to simply control these systems in order to track given reference trajectories to achieve movement of the system. Because one cannot expect to have complete information about a sophisticated mechanical or biological system, but instead only structural properties are known, we deal with uncertain systems. Therefore, the method of adaptive control is chosen in this paper. Since we deal with nonlinearly perturbed MIMOsystems, we focus on the adaptive λ-tracking control objective to achieve our goal. This means tracking of a given reference signal for any pre-specified accuracy λ > 0. The objective is not to obtain information about the characteristics of the system or about system parameters, but simply to control the unknown system. This control objective allows us to design simple adaptive controllers, which achieve λ-tracking. Numerical simulations of tracking different reference signals, for an arbitrary choice of the system parameters, will demonstrate and illustrate, that the introduced, simple adaptive controller works successfully and effectively. c 2006 Elsevier B.V. All rights reserved.

Keywords: Worm-like locomotion systems; Scales and spikes; Asymmetric Coulomb friction; Adaptive control; λ-tracking; Relative degree two systems

1. Introduction 1.1. General aspects The considered locomotion systems are inspired by biological ideas. Most such inspired systems are dominated by walking machines—pedal locomotion. A lot of biological models (bipedal up to octopedal) are studied in the literature and their constructions were transferred by engineers in different forms of realization. A-pedal forms of locomotion, respectively, show their advantages in inspection techniques ∗ Corresponding author. Fax: +49 3677 691823.

E-mail addresses: [email protected] (C. Behn), [email protected] (K. Zimmermann). c 2006 Elsevier B.V. All rights reserved. 0921-8890/$ - see front matter doi:10.1016/j.robot.2006.04.005

or in applications to medical technology for diagnostic systems and minimally invasive surgery. Hence, this type of locomotion and its drive mechanisms are current topics of main focus, see for instance the PADeMIS-research-group (http://www.tu-ilmenau.de/PADeMIS) and the BIOLOCHresearch-group (http://www.ics.forth.gr/BIOLOCH). We consider a-pedal locomotion systems, which have the earthworm as a living biological prototype. A worm is, in our theory, a terrestrial locomotion system of one dominant linear dimension with no active legs or wheels. The model of a worm is a one-dimensional continuum that serves as the support of various fields [31]. We will not consider a continuous distribution of mass (with a density function), but rather a discrete one. Therefore, we deal with discrete straight worms: chains of point masses moving along a straight line, Fig. 1.

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C. Behn, K. Zimmermann / Robotics and Autonomous Systems 54 (2006) 529–545

Fig. 1. A discrete worm—a chain of point masses.

Our main goal is to track given, prescribed trajectories (for example optimal gaits which are derived from the kinematics in [28]) to realize locomotion of the worm system. Of course we know that locomotion is more than a problem of tracking trajectories, but our later presented controller will realize movement of the worm-like systems. The main problem is that one cannot expect to have complete information about a sophisticated mechanical or biological system, but instead only structural properties (e.g., minimum phase condition, strict relative degree) are known. Therefore, how do we have to design a promising control input if we have only incomplete knowledge of the system parameters? These uncertainties occur, for example, through the interaction of the worm with its environment. Observing the locomotion of worms one can recognize a conversion of internal motions into a change of external position (undulatory locomotion) [30]. The interaction could emerge from a surface texture as asymmetric Coulomb friction, see [10,4], or from a surface endowed with scales or bristles (or passive wheels with a ratchet, [31]) preventing backward displacements—in the literature they speak of spikes for short, see [28]. We focus on the motion of the worm-like system under the action of asymmetric Coulomb friction forces. But, in a rough terrain the friction coefficients might be unknown and randomly changing. This fact and supposed unknown system parameters lead to uncertain systems, mentioned above, and we have to chose the method of adaptive control. We consider two different types of adaptive control inputs (drives): (i) devices which produce internal displacements or forces, thus mimicking muscles (internal control inputs), and (ii) external forces which could be realized by magnetic fields ([34,35]) (external control inputs). By means of these two types of actuators we achieve our main goal: adaptive λ-tracking. The λ-tracking control objective is to determine a universal λ-servomechanism (a control strategy y 7→ u) to track a given reference signal with a prescribed accuracy λ > 0. This means, that a small tracking error of size λ is tolerated, see Fig. 2. This λ-tracking control objective allows us to design simple adaptive controllers, which can be easily implemented. 1.2. Aspects from control theory A wide range of control theory deals with the problem that, for a known system, a controller has to be designed in order that the feedback system achieves the pre-specified control objective [15]. The fundamental difference between this approach and that of adaptive control (in this paper) is that the system is not known exactly, only structural information about the system, like the relative degree or the minimum phase

Fig. 2. The λ-radius around a reference signal.

condition, is available (since the parameters of the worm-like motion system are not known precisely, the adaptive control law has to be designed so that the controller learns from the behaviour of the system, and based on this information, adjusts its parameters). This area has been intensively studied over the last 50 years. ˚ om ([3], 1987) for a survey article. See the summary of Astr¨ Up to the beginning of the 1980s, most adaptive control mechanisms would attempt to identify or to estimate certain parameters of the system, and then design a feedback controller on the basis of this information. In this paper, we design adaptive controllers, which are not based on any parameter identification or estimation algorithms. The objective is not to obtain information about the system, but simply to control the unknown system, where we use the high-gain property of this system, [16]. This approach is called non-identifier-based highgain adaptive control in the literature, see [15] for a survey article. The structure of the feedback law and the adaptation law of the controller in this paper are not novel in the literature, but they were only applied to systems with relative degree one. The considered mechanical systems have relative degree two. Therefore, the novelty in this paper is the application of the controller to systems with relative degree two to prove λ-tracking. There are only a few papers which focus the adaptive λ-tracking problem for system with relative degree two, but the feedback law in this paper is simpler than the introduced ones in [12,16,33,24]. More precisely comparisons to these controllers are done in Section 5 and in [6]. 1.3. Arrangement of the paper Based on these introducing remarks and considerations, Section 2 deals with two models of a worm-like locomotion system with asymmetric Coulomb friction and their mathematical descriptions. In Section 3 we present the locomotion systems with different numbers of control inputs, which fall into the category of nonlinearly perturbed, multi-input, multi-output systems. Therefore, Section 4 deals with the presentation of the general system class and its properties. Sections 5 and 6 are the main theoretical sections with the description of the control objective, the adaptive controllers and the main theorems. Section 7 shows some numerical simulations. Finally, Sections 8 and 9 illustrate experiments and prototypes, and

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some conclusions, respectively. The proof of the presented theorem can be found in Appendix. 1.4. Notation R≥0 , R>0 := [0, ∞), (0, ∞), respectively; C+ , C− the open right-, left-half complex plane, respectively; √ kxk := x T x, x ∈ Rn ; C(I ; Rn ) the set of continuous functions I → Rn , I ⊂ R an interval; L∞ (I ; Rn ) the space of measurable essentially bounded functions I → Rn , I ⊂ R an interval, with norm k · k∞ ; kxk∞ := ess supt∈I kx(t)k; R the set of all differentiable functions yref (·) : R≥0 → Rm with y˙ref absolutely continuous on compact intervals and yref (·), y˙ref (·), y¨ref (·) ∈ L∞ (R≥0 ; Rm ).

Fig. 3. The earthworm, [5].

2. Models of a worm-like motion system with asymmetric Coulomb friction As mentioned in the introduction, we deal with worm-like locomotion systems, which have the earthworm as a living biological prototype. Only discrete straight worms will be considered: chains of point masses moving along a straight line. We consider two locomotion systems in the form of a chain of k = 31 point masses in a common straight line which are interconnected consecutively by linear viscoelastic elements like springs and viscous damping elements, [6,7,14,29,30]. Global displacement can be achieved by (periodic) change of shape and interaction with the environment (undulatory locomotion). We focus on interaction which emerges from a surface texture as asymmetric Coulomb friction. Additionally, we consider two different types of drive mechanisms in the following: (i) The point masses are under the action of external forces, which can be regarded as external force control inputs. (ii) We deal with massless linear springs of fixed stiffnesses and controllable original spring lengths, which can be regarded as internal control inputs. The next three subsections present the asymmetric Coulomb friction force, and the two different worm-like systems. 2.1. The asymmetric Coulomb friction force The earthworm has an internal drive to change the distance between its segments. But this internal motion is insufficient to obtain a change of external position. The interaction of the worm and the ground is very important to achieve global displacement. The biological paradigm earthworm offers bristles, see Fig. 3 and [23]. These bristles cause friction. It is necessary to equip each point mass with scales or spikes of common orientation to 1 We point out, that the later presented theory does not have to be restricted to the case k = 3. Later results show their validity for fixed, but arbitrary k ∈ N.

Fig. 4. The asymmetric dry friction force.

realize this friction in a mechanical model. These scales have to provide motion in only one (positive) direction. In [28], this ground interaction is modelled by differential constraints like x˙i ≥ 0, i = 1, . . . , 3. Hence, the scales in [28] prevent velocities from being negative. Due to [6,7] we do not want to deal with reactive forces. In this paper, we model this ground interaction as impressed forces. More precisely, we deal with an asymmetric (orientation dependent—anisotropic) dry friction force as a Coulomb friction force, see [6,7,29,30]. The asymmetric dry friction force is taken to be different in the friction magnitude depending on the direction of each point mass motion, see Fig. 4, where we set vi := x˙i for i = 1, . . . , 3. The dry friction force function is  + −Fi , vi > 0 Fi (·) : R → R mit vi 7→ Fi0 , vi = 0 , i = 1, . . . , 3,  − Fi , vi < 0 (2.1) where Fi+ , Fi− > 0 are fixed with Fi−  Fi+ and Fi0 ∈
−Fi+ , Fi− is arbitrary. In the case of symmetric friction we have Fi− = Fi+ . With regard to later numerical calculations we approximate this dry friction function by means of the tanh(·)-function as

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Fig. 6. Model of a worm-like motion system with internal control inputs. Fig. 5. Model of a worm-like motion system with external control inputs.

follows   f i (·) : R → −Fi+ , Fi− vi 7→ −

Fi−

+ 2

Fi+

with

tanh(avi ) +

Fi− − Fi+ , 2

i = 1, . . . , 3, (2.2)

where the constant a ∈ R is sufficiently large, for example a = 40, [6,7]. The main advantage of this approximation for later calculations and proofs is that there holds: f i (·) ∈ C ∞ (R; R)

and

f i (·) ∈ L∞ (R; R) .

(2.3)

Remark 2.1. We remark that the static friction force Fi0 (for F − −F +

vi = 0) takes the value i 2 i in the case of this approximation of the dry friction force. This fact does not match reality, because the value of the static friction force depends on the equilibrium of the forces. There only holds the inequality −Fi+ < Fi0 < Fi− . The stick–slip effect (see [4]) is not characterized in this approximation, because this effect is not the object of main focus in this paper.  2.2. Model with external control inputs

with x1 (0) = x10 , x˙1 (0) = x11 , x2 (0) = x20 , x˙2 (0) = x21 , x3 (0) = x30 and x˙3 (0) = x31 (all initial values are real numbers). Because we measure the whole position, the output equations are y2 (t) = x2 (t),

y3 (t) = x3 (t)

v3 (t) 

for all t ∈ R.

The transformation vi := x˙i yields the following system in matrix–vector representation

0

 0    0   c12  = − m 1   c  12   m2  0

0

0

1

0

0

0

0

1

0



0    0 0 0 0 1    c12 d12 d12 0 − 0   m1 m1 m1  c12 + c23 c23 d12 d12 + d23 d23   − −  m2 m2 m2 m2 m2   c23 c23 d23 d23 − 0 − m3 m3 m3 m3     0 0 0 0     0   x1 (t)   0 0   0    x (t)  0      2   0 0 0    u 1 (t)      1  x3 (t)  1       F1 (v1 (t))  ×  , 0 0  u 2 (t) +   + m m   1 v (t)   1   1    u 3 (t)    1 1    v2 (t)  0 0  F2 (v2 (t))    m2   m2     1 v3 (t) 1 0 0 F3 (v3 (t)) m3 m3     y1 (t) 1 0 0 0 0 0      y2 (t) = 0 1 0 0 0 0 y3 (t)

In this subsection, we consider the case where the point masses are under the action of external forces, hence these forces can be regarded as control inputs u i (·), i = 1, . . . , 3, see Fig. 5. We derive the differential equations of motion of the wormlike locomotion system by using Newton’s second law:  m 1 x¨1 (t) = c12 (x2 (t) − x1 (t)) + d12 (x˙2 (t) − x˙1 (t))    + u 1 (t) + F1 (x˙1 (t)) ,     m 2 x¨2 (t) = c23 (x3 (t) − x2 (t)) − c12 (x2 (t) − x1 (t))  + d23 (x˙3 (t) − x˙2 (t)) − d12 (x˙2 (t) − x˙1 (t))   + u 2 (t) + F2 (x˙2 (t)) ,    m 3 x¨3 (t) = −c23 (x3 (t) − x2 (t)) − d23 (x˙3 (t) − x˙2 (t))    + u 3 (t) + F3 (x˙3 (t)) , (2.4)

y1 (t) = x1 (t),

• x1 (t) x (t)  2    x3 (t)   v (t)  1    v2 (t) 

0 0 1 0 0 0  × x1 (t) x2 (t) x3 (t)

v1 (t)

v2 (t)

T v3 (t) ,

                                                                                                                          

(2.5) with x1 (0) = x10 , x2 (0) = x20 , x3 (0) = x30 , v1 (0) = v10 , v2 (0) = v20 and v3 (0) = v30 . 2.3. Model with internal control inputs For technical realization we consider a different worm-like locomotion system. The fundamental difference from the first model (Fig. 5) is that we do not assume the existence of external force inputs. We rather consider the case of massless linear springs of fixed stiffness c12 , c23 and controllable original spring lengths l12 (·), l23 (·). Hence, these elongations of the springs can be regarded as the new control inputs of the system, see Fig. 6. Again, we derive the differential equations of motion of this system by using Newton’s second law:  m 1 x¨1 (t) = c12 (x2 (t) − x1 (t)) + d12 (x˙2 (t) − x˙1 (t))    − u 12 (t) + F1 (x˙1 (t))     m 2 x¨2 (t) = c23 (x3 (t) − x2 (t)) − c12 (x2 (t) − x1 (t))  + d23 (x˙3 (t) − x˙2 (t)) − d12 (x˙2 (t) − x˙1 (t)) (2.6)   + u 12 (t) − u 23 (t) + F2 (x˙2 (t))    m 3 x¨3 (t) = −c23 (x3 (t) − x2 (t)) − d23 (x˙3 (t) − x˙2 (t))    + u 23 (t) + F3 (x˙3 (t))

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with x1 (0) = x10 , x˙1 (0) = x11 , x2 (0) = x20 , x˙2 (0) = x21 , x3 (0) = x30 and x˙3 (0) = x31 (all initial values are real numbers). Putting u 12 (·) := c12l12 (·)

and

u 23 (·) := c23l23 (·),

(2.7)

then u i j is in fact a control of the original spring length. Therefore, we have internal inputs and no longer external force inputs, as in [6] and [7]. New outputs of this new system could be the actual distances of the point masses y1 := x2 − x1

and

y2 := x3 − x2 .

(2.8)

Then we have the following equations of motion of this system in matrix–vector presentation • x1 (t) x (t)  2    x3 (t)   v (t)  1    v2 (t) 

v3 (t) 

0

 0    0   c12  = − m 1   c  12   m2  0

0

0

1

0

0

0

0

1

0

0 d12 − m1 d12 m2

0 c12 m1 c12 + c23 − m2 c23 m3  0    0 x1 (t)  x (t)   2   0    x3 (t)  c12  − ×  v (t) +  m 1  1      c12 v2 (t)   m2  v3 (t) 0

   x10 x1 (0) x (0) x   2   20      x3 (0) x30      v (0) = v  ,  1   10      v2 (0) v20 

0 c23 m2 c23 − m3 0



0    0    0    c23   − m2   c23 m3

0



0    1    0    d23    m2   d23 − m3 

0 d12 m1 d12 + d23 − m2 d23 0 m  3 0  0    0 !   1 l12 (t)  +  m F1 (v1 (t))  1 l23 (t)  1  F2 (v2 (t))   m2  1 F3 (v3 (t)) m3

       ,      



v30 v3 (0) ! " # −1 1 0 0 0 0 y1 (t) = y2 (t) 0 −1 1 0 0 0  × x1 (t) x2 (t) x3 (t) v1 (t) v2 (t)

(2.5) and (2.9). It is possible to distinguish these systems depending on the number of acting control inputs. We present these different cases for the two systems in the following two subsections, where we also present one member of each case.

T v3 (t) .

                                                                                                                                                  

(2.9)

3. Systems with different numbers of control inputs

In this section we consider the presented worm-like locomotion systems in their mathematical presentation, see

3.1. Cases for system (2.5) We have for system (2.5): Case 1: Three control inputs are acting on the system. Since the degree of freedom of this system is equal to the number of available control inputs, the system is called fully actuated. Case 2: Two control inputs are acting, hence the system is called underactuated. Case 3: Only one control input is acting on the system. This system is underactuated, too. This classification is also made in [7]. For an example of Case 3, we exemplarily choose u 1 (·) ≡ 0 and u 2 (·) ≡ 0. From (2.5) we arrive at the following system (already in normalized form) by means of the only output equation y3 (t) = x3 (t) for all t ∈ R: • x3 (t) x˙ (t)  3    x1 (t)   x (t)  2    x˙1 (t) 

x˙2 (t) 

0  c − 23   m3   0  =  0     0   c23 m2

−

1

0

0

0

0



d23 m3

0

c23 m3

0

d23 m3

0

0

0

1

0

0

0

0

0

1

              

c12 d12 d12 − 0 m1 m1 m1 d23 c12 + c23 d12 d12 + d23 − − m2 m m2 m2  2   0 0    1  x3 (t)      m 3 F3 ( y˙3 (t))   x˙ (t)  1      3  m      0 3   x1 (t)     + u (t) + ×  0    , 0 3 x (t)      2         0  1  x˙1 (t)  F1 (x˙1 (t))      m1   0    x˙2 (t) 1 0 F2 (x˙2 (t)) m2     x3 (0) x30 x˙ (0) x   3   31      x1 (0) x10      x (0) = x  ,  2   20      x˙1 (0) x11  x21 x˙2 (0) h y3 (t) = 1 0 0 0  × x3 (t) x˙3 (t)

c12 − m1 c12 m2 

0

0

x1 (t)

i x2 (t)

x˙1 (t)

T x˙2 (t) .

                                                                                                                                                

(3.1)

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3.2. Cases for system (2.9) For system (2.9) we only have two cases: one control input or two control inputs. For an example of action of one control input we choose, for example, l12 (·) ≡ 0. The equations of motion in the matrix–vector representation (already in normalized form) are • x1 (t) x (t)  2    x3 (t)   v (t)  1    v2 (t) 

v3 (t) 

0  0    0   c12  = − m 1   c  12   m2  0

0

0

1

0

0

0

0

1

0 0 0 0 c12 d12 d12 0 − m1 m1 m1 c12 + c23 c23 d12 d12 + d23 − − m2 m2 m2 m2 c23 c23 d23 − 0 m3 m3 m3   0   0     x1 (t) 0       x (t)  0    0   2        0    1  x3 (t)    F (t)) (v     1 1 ×  +  0  (l23 (t)) +  m ,    1 v1 (t)  c23    1    −   v2 (t)  m 2  F (t)) (v   2 2   m   2 c 23   v3 (t) 1 m3 F3 (v3 (t)) m3     x1 (0) x10 x (0) x   2   20      x3 (0) x30      v (0) = v  ,  1   10      v2 (0) v20  v30 v3 (0) h i y2 (t) = 0 −1 1 0 0 0  × x1 (t) x2 (t) x3 (t) v1 (t)

v2 (t)

T v3 (t) .

0



0    1    0    d23    m2   d23 − m3

                                                                                                                                              

(3.2) 4. General system class All mathematical models of the presented systems ((2.5), (2.9), (3.1) and (3.2)) and of all other, not presented systems, which can arise in the other cases of Section 3, fall into the category of nonlinearly perturbed, multi-input u(·), multioutput y(·) control systems (MIMO-systems): • control system, because we have internal or external drives to control the system; • MIMO-system, because we have up to three drives and outputs; • nonlinearly perturbed, because the modelled ground interaction through the asymmetric Coulomb friction force (impressed force) is a nonlinear part of the system.

Therefore we focus on the following general form of a nonlinearly perturbed, multi-input u(·), multi-output y(·) control system for theoretical investigations: • y(t)  y˙ (t) z(t) 

y(0)

    y(t) 0 0 0   y˙ (t) + G  u(t) 0 z(t) A5  0 + g1 (s1 (t), y(t), z(t))  , g2 (s2 (t), y(t)) y0 , y˙ (0) = y1 , z(0) = z 0 ,

 =

=

0 Im 0 A 2 0 A0 

         

(4.1)

        

with y(t), y˙ (t), u(t) ∈ Rm , z(t) ∈ Rn−2m , real matrices A2 , G ∈ Rm×m , A0 ∈ R(n−2m)×m , A5 ∈ R(n−2m)×(n−2m) and n ≥ 2m, see [6]. 4.1. Remark This special structure of the linear part of system (4.1) is based on the following transformation: we consider the finitedimensional, linear, minimum-phase, m-input u(·), m-output y(·) systems of strict relative degree two and with spectrum of the high-frequency gain in the open right-half complex plane; more precisely, the system x(t) ˙ = Ax(t) + Bu(t), y(t) = C x(t)

x(0) = x0

(4.2)

with x(t), x0 ∈ Rn , y(t), u(t) ∈ Rm (hence, this system is square), 2m ≤ n, A ∈ Rn×n , B, C T ∈ Rn×m . The system (4.2) has strict relative degree r if, and only if, there exists an integer r ∈ N such that  = 0, k = 0, . . . , r − 2 k CA B ∈ G L m (R), k = r − 1. The invariant zeros of the system (4.2) are the elements of the set     s In − A B s ∈ C det =0 . C 0 The system is called a minimum phase system, if all invariant zeros have strictly negative real parts. To look up these definitions the reader’s attention is invited to [12]. If we suppose that the system (4.2) satisfies the following conditions (i) C B = 0 and det(C AB) 6= 0, i.e. strict relative degree 2; (ii) σ (C AB) ⊂ C+ , i.e. the spectrum of the “high-frequency gain” lies in the open right-half complex plane; (iii) the system is minimum phase; then there exists a suitable coordinate transformation S −1 x(t) = (y(t)T , y˙ (t)T , z(t)T )T (see, for example [12, lemma 4.5.3]) which converts (4.2) into the equivalent form

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•      y(t) 0 Im 0 y(t) 0  y˙ (t) =  A1 A2 A3   y˙ (t) + C AB  u(t), z(t) A4 0 A5 z(t) 0 (4.3)     y(0) y0  y˙ (0) =  y1  := S −1 x0 z0 z(0) 

with z(t) ∈ Rn−2m and real matrices A1 , . . . , A5 of conforming formats. It is readily verified that the minimum phase condition (iii) in this remark yields σ (A5 ) ⊂ C− . In the case of defining g1 := A1 y + A3 z, g2 := A4 y and G := C AB system (4.3) coincides with system (4.1). 4.2. Definition of the system class We suppose that the system (4.1) has the following properties, see [6], (i) σ (G) ⊂ C+ , i.e. the spectrum of the ‘high-frequency gain’ G lies in the open right-half
complex plane.
(ii) s1 (·) ∈ L∞ R≥0 ; Rq1 and s2 (·) ∈ L∞ R≥0 ; Rq2 , i.e. they could be regarded as (bounded) disturbance terms, where a dependence like s1 (t) = ψ1 (t, y(t), y˙ (t), z(t)) and s2 (t) = ψ2 (t, y(t), y˙ (t), z(t)) is possible. (iii) The functions g1 : Rq1 × Rm × Rn−2m → Rm and g2 : Rq2 × Rm → Rn−2m are continuous, and, for compact sets C1 ⊂ Rq1 and C2 ⊂ Rq2 , there exist two constants c1 , c2 ≥ 0 such that kg1 (s, y, z)k ≤ c1 [1 + kyk + kzk] for all (s, y, z) ∈ C1 × Rm × Rn−2m , kg2 (s, y)k ≤ c2 [1 + kyk] for all (s, y) ∈ C2 × Rm . (iv) The unperturbed system (provided gi (·) ≡ 0, i = 1, 2) is (in the case of n > 2m) minimum phase (stable zero dynamic), hence σ (A5 ) ⊂ C− . In the case of n = 2m there exists no zero dynamic. It is easy to prove that every system of this system class has the strict relative degree two.

535

5. Adaptive λ-tracking A work of Morse, Nussbaum and Willems & Byrnes at the beginning of the 1980s has initiated the study of adaptive controllers for dynamical systems in which the adaptation strategy does not invoke any identification mechanism, mentioned in the Introduction. Hence, this approach is different to other classical (adaptive) control books of Kokotovi´c, Sastry, Ioannou, Slotine, Isidori, Vidyasagar, Khalil, Nijmeijer, van der Schaft, and Marino. Over the last 20 years, this special field of adaptive control has become a major research topic. The first adaptive controller, not based on identification of the system parameters and being useful for a class of single-input, single-output systems (SISO-system), was given by Feuer and Morse in 1978 [13]. The first very simple adaptive controller, which uses the high-gain approach, goes back to Morse (1983, [25]) and Willems and Byrnes (1984, [32]). The simple gain adaptation, which is used in this paper, was introduced by Ilchmann and Ryan (1994, [17]) and extended by Allg¨ower and Ilchmann (1995, [1]) and Allg¨ower (1997, [2]). This approach was successful for many applications: • methanol synthesis in a polytropic, catalytic continuous stirred tank reactor on solid phase catalyst (Allg¨ower, [2]); • control of a reaction in an exothermic continuous stirred tank reactor (Allg¨ower and Ilchmann, [1]); • control of anaerobic digestion by micro-organisms of animal wastes (Ilchmann and Weirig, [18]); • control of a Biogas Tower Reactor for the waste water treatment from baker’s yeast production (Ilchmann and Pahl, [20]); • control of the end-tidal anesthetic concentration (Bullinger, [11]); • tracking control of exothermic chemical reaction models (Thuto, [21]); We will use the simple gain adaptation of these papers, but we apply the control strategy to systems with strict relative degree two to achieve λ-tracking (for worm-like locomotion systems, see [6]), which will be specified in the next subsection. 5.1. Control objective

4.3. Example System (2.5) is a nonlinearly perturbed MIMO-control
system u(t), y(t) ∈ R3 of first order like (4.1). The reader may also recognize quickly the matrices A1 , A2 and G of system (4.1) in this system constellation. Since m = 3 and n = 6 we do not have a zero dynamic. Hence there exist no matrices A3 , A4 and A5 . All properties of Section 4.2 are fulfilled, because the system has relative degree two, where the spectrum of the ‘high-frequency gain’ n lies in theoopen right-half complex plane, because σ (G) = m11 , m12 , m13 , where m 1 , m 2 , m 3 are (unknown) positive constants. Moreover, the dry friction force yields a bounded nonlinearity, which is bounded by a constant. Summarizing, all presented systems belong to this system class.

The λ-tracking control objective is to determine a universal λ-servomechanism to track a given reference signal with a prescribed accuracy. Precisely, given λ > 0, a control strategy y 7→ u is sought which, for every reference signal yref (·) ∈ R, when applied to the system, achieves λ-tracking, i.e. the following: (i) there exists a solution of the closed-loop system on the R≥0 , (ii) the trajectories of the closed-loop system are bounded on the R≥0 , (iii) convergence of the controller gain k(·) is ensured, and (iv) the output y(·) tracks yref (·) with asymptotic accuracy quantified by λ > 0 in the sense that max {0, ky(t) − yref (t)k − λ} → 0 as t → ∞.

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The last condition means that the error of the output of the system and the given reference signal yref (·) e(t) := y(t) − yref (t) is forced, via the simple adaptive feedback mechanism, towards a ball around zero of arbitrary small pre-specified radius λ > 0, see Fig. 2 and [19]. This λ-tracking control objective allows us to design simple adaptive controllers. Remark 5.1. Choosing yref (·) ≡ 0, λ = 0 we arrive at the stabilization control objective. If we want to stabilize systems with relative degree two, we have to design an adaptive controller, where the feedback law consists of the output y(·) of the system and of the derivative of the output y˙ (·), too, see [27] for an example. In [15] a simple adaptive controller is introduced    d  u(t) = − k(t)y(t) + (k(t)y(t)) , (5.1) dt  ˙ k(t) = ky(t)k2 , k(0) = k0 . This feedback law together with the adaptation law applied to any linear, minimum phase MIMO-system with strict relative degree and known ‘high-frequency gain’ in the open left-half complex plane yields a bounded solution on the R≥0 , and the properties limt→∞ y(t) = 0 and limt→∞ k(t) ∈ R hold. This controller (5.1) is simpler and more suitable than the one introduced in [12]. The controller in [12] is based on the complete knowledge of the ‘high-frequency gain’ C AB, since this matrix is a component of the feedback law. Here, we assume only structural information about this matrix to have a spectrum in the open right-half complex plane. Additionally, that controller works with two different high-gain parameters k0 (·) and k1 (·) in the form of matrices which, multiplied by the output y(·) and the derivative y˙ (·), are components of the feedback law, too. These matrices are calculated from adaptation rules. Since each matrix has the dimensions m × m a lot of calculation has to be done. The only common property of that controller and controller (5.1) is that both are based on using the output and its derivative for the feedback law. In this paper we cannot expect that the controller (5.1) will stabilize the considered systems, because we deal with nonlinearly perturbed systems with gi (·) ∈ L∞ (R; R). We have to pay particular attention to the λtracking objective and design controllers achieving λ-tracking, which will be done in the next subsection.  5.2. Controller and λ-tracker Let us consider the following λ-tracker, see also [6]. This one is a variation of the stabilizer (5.1) in [15]:  e(t) := y(t)    − yref (t), yref (·) ∈ R,  d u(t) = − k(t)e(t) + (k(t)e(t)) , (5.2)  dt   2 ˙ k(t) = max {0, ke(t)k − λ} , k(0) = k0 , with λ > 0, yref (·) ∈ R, u(t), e(t) ∈ Rm and k(t) ∈ R.

The controller, precisely the gain adaptation law, use a dead-zone as follows: if the norm of the tracking error ke(·)k is greater than λ, the gain k(·) has to increase with power 2; if the norm is smaller than λ, the output is close to the reference signal and the gain has not to increase and will stay constant. This controller is simple in its design, relies only on structural properties of the system (and not on the system’s parameters), and does not invoke any estimation or identification mechanism. It only consists of a feedback strategy and a simple parameter adaptation law. At present, we analyze further two λ-trackers, which do not have to depend on the derivative of the output of the system and achieve λ-tracking. One includes a dynamic compensator due to a controller of Miller and Davison in [24]. In [24], an “adaptive controller which provides an arbitrarily good transient and steady-state response” is introduced for linear single-input, single-output systems of higher relative degree. The control objective is very close to λ-tracking, but the control strategy is piecewise constant. Moreover, in [33], the λ-tracking approach is generalized to systems of higher relative degree but singleinput, single-output and with different nonlinearities. But the functions g1 and g2 are more general. Ref. [33] proposes a control strategy, which is more complicated as the analyzed controller. The other controller, which avoids the usage of the derivative too and which reduces in dimension (the number of used variables calculated by internal differential equations), is presented in [6,8], and applied in [6,8,9]. But this controller can not be under consideration in this paper. 6. Theorem Theorem 6.1. Let λ > 0. Then the adaptive λ-tracker (5.2) applied to the system (4.1), belonging to the system class yref (·) ∈ defined in Section 4.2, yields, for any reference signal
∞ R ; Rq1 and s (·) ∈ R, any disturbance terms s (·) ∈ L 1 ≥0 2
L∞ R≥0 ; Rq2 , and any initial data (y0 , y1 , z 0 , k0 ) ∈ Rm × Rm × Rn−2m × R, a closed-loop system (a feedback controlled initial-value problem)  y˙ (t) = ζ (t), y(0) = y0 ,     ζ˙ (t) = A2 ζ (t) + g1 (s1 (t), y(t), z(t))     − G k(t) (y(t) − yref (t)) + k(t) (ζ (t) − y˙ref (t)) i   + max {0, ky(t) − yref (t)k − λ}2 (y(t) − yref (t)) ,

ζ (0) = y1 , z˙ (t) = A5 z(t) + A0 ζ (t) + g2 (s2 (t), y(t)) , z(0) = z 0 , ˙ k(t) = max {0, ky(t) − yref (t)k − λ}2 , k(0) = k0 ,

         (6.1)

which has a solution. Every solution can be extended to a maximal solution and every maximal solution (y, ζ, z, k) : [0, t 0 ) → Rm × Rm × Rn−2m × R has the following properties: (i) (ii) (iii) (iv)

t 0 = ∞, i.e. there does not exist a finite escape time; limt→∞ k(t) exists and is finite; the solution, ζ˙ (·), z˙ (·) and u(·), as in (5.2), are bounded; lim supt→∞ ky(t) − yref (t)k ≤ λ. 

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Fig. 7. The outputs yi (·) (positions xi ) and the λ-strips.

The variable ζ (·) is needed to write the second order differential equation in y as a system of differential equations of first order. The proof can be found in Appendix. Remark 6.2. We have to point out, that the right hand side of the closed-loop system (6.1) is not necessarily continuous in all variables. This fact is a consequence of the bounded disturbance terms si , i = 1, 2, of the nonlinearities. Therefore, the right hand side is only measurable in t and continuous in all other variables. Hence, we have only an absolutely continuous solution, see the Carath´eodory Theory [26] and [22] for more details. If the right hand side does not depend on these disturbance terms, then we have a continuously differentiable solution according to the theory of ordinary differential equations. 

7.1. Model with external inputs 7.1.1. Tracking of a tanh(·)-reference signal We choose the following reference signal   0.5 + tanh(t) t 7→ yref (t) = 2.5 + tanh(t) 4.5 + tanh(t) with yref (·) ∈ R. Then we have: The three outputs (the positions of the point masses) approach very quickly the prescribed reference trajectories. As long as the outputs are outside the λ-strips the gain parameter is increasing, see Fig. 8. Fig. 9 shows the necessary control inputs to realize the motion presented in Fig. 7.

7. Simulations Remark 7.1. We present simulations of worm-like locomotion systems in the form of chains of point masses in a common straight line. The presented systems, see Figs. 5 and 6, consist only of k = 3 point masses, but the presented theory is valid for an arbitrary, but fixed k ∈ N. Therefore, the results of the following simulations could easily be extended to a larger k.  Now, we present some simulations of both worm-like locomotion systems: with external inputs (2.5) and with internal (2.9). We will only consider these fully actuated systems. Other cases are considered in [6,31]. We apply the presented λ-tracker (5.2) to these systems, and we exemplarily choose the following same parameters (dimensionless) for all simulations: • system: m 1 = m 2 = m 3 = 1, c12 = c23 = 10, d12 =
T
T
T d23 = 5, x10 , x20 , x30 = 0, 2, 4 , v10 , v20 , v30 =
T 0, 0, 0 ; • Coulomb friction forces: F1+ = F2+ = F3+ = 1, F1− = F2− = F3− = 10, hence we have by (2.2) vi 7→ 5.5 tanh(40 vi ) − 4.5, i = 1, . . . , 3; • λ-tracker (controller): λ = 0.2, k0 = 1.

7.1.2. Tracking of a sin(·)-reference signal The new reference signal is   0.5 + sin(t) t 7→ yref (t) = 2.5 + sin(t) 4.5 + sin(t) with yref (·) ∈ R. It is obvious that we do not analyze a locomotion system in this constellation since we want to track a sine signal, see Fig. 10, as a reference signal for the outputs (positions of the point masses). But this example demonstrates that the controller (5.2) also works in the case where we want to have negative velocities. These negative velocities should have been decreased by the scales (described by nonlinear dry friction). Hence, we will expect high control inputs u i (·), i = 1, . . . , 3, see Fig. 12, because this control forces have to work against the dry friction force of the scales to guarantee the tracking of the sine signal. The gain parameter k(·) converges very quickly to the finite limit, see Fig. 11. The simulations of these two subsections demonstrate that we are not able to track appropriate reference signals to achieve

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Fig. 8. The gain parameter k(·).

Fig. 9. The control inputs u i (·).

Fig. 10. The outputs yi (·) (positions xi ) and the λ-strips.

movement of the system. This drawback will be reduced by considering the model with internal control inputs in the following Section 7.2.

7.2. Model with internal inputs We remark, that we now consider the system with internal control inputs, see Fig. 6 and (2.9). Therefore,

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539

Fig. 11. The gain parameter k(·).

Fig. 12. The control inputs u i (·).

Fig. 13. The outputs yi (·) (distances) and the λ-strips.

we have new outputs of the system: distances between the point masses, no longer the positions of the point masses.

7.2.1. Tracking of a time-shift sin(·)-reference signal In this subsection we deal with the following reference signal

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Fig. 14. The gain parameter k(·).

Fig. 15. The control inputs li j (·).

t 7→ yref (t) =





2 + sin(t) . 2 + sin(t + 2)

The prescribed gait is tracked very quickly, see Fig. 13. Fig. 14 shows the convergence of the gain parameter, Fig. 15 the necessary control inputs. The worm system has two active spikes at many different time intervals, for example t ∈ [7, 7.5] or t ∈ [13, 13.5], see Fig. 16. Therefore, the worm moves slower than the one in the next subsection, but the motion is improved according to previous simulations. 7.2.2. Tracking of a gait presented in [28] In this subsection we want to track an ‘optimal fast’ gait, which is developed in [28] and presented in [31]. This gait is derived from the kinematical theory of a chain of point masses with spikes, links of variable lengths, [28]. In the context of spikes it is easier to develop a gait from the kinematical theory, so that we will not require dynamics from the very beginning. This is done in [28]. Steigenberger exemplarily derived two gaits for a chain of k = 3 point masses,

see Fig. 6. He constructed these two gaits to guarantee a certain number of active spikes during motion, for example, a gait with one active spike at any time (‘in-plane gait’), and a gait with two active spikes at any time (‘up-hill gait’). He expected, that the first gait will be very fast. We try to track this kinematically derived gait in our dynamical theory in this paper. The ‘fast’ gait in [28] (reference signal) is, for t ∈ [0, 1]:    1  l ε cos(3πt) + l − l ε, t ∈ 0, 0 0 0  3        2 1  l0 − 2l0 ε, , t∈  3 3      2   t 7→ yref (t) = −l0 ε cos(3πt) + l0 − l0 ε, t ∈ ,1  3     2  −l ε cos(3πt) + l + l ε, t ∈ 0, 0 0 0   3     2  l0 , t∈ ,1 3

         ,       

(7.1)

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Fig. 16. The motion of the worm.

Fig. 17. The outputs yi (·) (distances) and the λ-strips.

Fig. 18. The gain parameter k(·).

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Fig. 19. The control inputs li j (·).

Fig. 20. The motion of the worm.

where l0 is the original length (dimensionless chosen as 2 units) and 2ε = 0.3 is the elongation. This gait is periodically repeated. We receive the following simulations. The desired kinematic gait is approached very quickly, see Fig. 17. Fig. 18 presents the convergence of the gain parameter, Fig. 19 the used control inputs. Fig. 20 presents the fast worm, where the motion is much improved compared to the simulations in Sections 7.1.1 and 7.1.2. And apparently the motion exhibits one active spike at any time. This worm moves faster than the one in Section 7.2.1, where we choose the time-shift sine signal by coincidence. 8. Experiments and prototypes According to the second worm-like motion system (Fig. 6) we firstly create a prototype, which illustrates the basic conditions of a worm-like robot. Two stepping motors and a dummy are chosen to produce a three mass point worm system (Fig. 21). Each stepping motor can separately travel along a threaded rod in both directions with different controllable speeds to generate l12 (·) and l23 (·).

Fig. 21. Three mass point worm system.

Additionally, a special bristle-structure had been integrated (Fig. 22) to prevent the mass points from slipping backwards. At present, we are trying to establish an Excel program which, using the real values of the motor parameters, transforms

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543

Fig. 22. Bristle-structure (spikes), see [31].

the calculated, adaptive l12 (·) and l23 (·) into a function: time 7→ motorsteps per unit of time. Then, a laser measurement device (optoNCDT 1605 from Micro-Epsilon) (Fig. 23a) will be used by means of LabVIEW and a DAQ-device (Fig. 23b,c) to detect the real motion character. 9. Conclusion • A wide range of control theory deals with the problem that, for a known system, a controller has to be designed in order that the feedback system achieves the pre-specified control objective like stabilization or tracking. The fundamental difference between this approach and that of adaptive control (in this paper) is that the system is not known exactly, only structural information about the system, like relative degree or minimum phase property, is available. • We consider a mechanical system, which is inspired by biological ideas. We present some theoretical investigations of worm-like locomotion systems (chain of k = 3 interconnected point masses) with two different types of drives: (i) external force control inputs, and (ii) massless linear springs of fixed stiffnesses and controllable original spring lengths (internal control inputs). We focus on interaction which emerges from a surface texture as asymmetric Coulomb friction. • Both systems (either fully actuated or underactuated) are described by mathematical models that fall into the category of nonlinearly perturbed, minimum phase, multi-input, multi-output systems with strict relative degree two. • Since we deal with nonlinearly perturbed multi-input, multi-output control systems, which are not necessarily autonomous, particular attention is paid to the λ-tracking control objective: determine a universal λ-servomechanism (control strategy) to track a given reference signal with a prescribed accuracy λ > 0. • For this objective, we present an adaptive λ-tracker. We stress that the introduced adaptive controller consists of a very simple feedback mechanism and adaptation rule. This controller is only based on information about the output of the system—no system parameters are required. • Our main results is a theorem providing that the adaptive controller achieves λ-tracking. • Four simulations, for an arbitrary choice of the system parameters, will demonstrate and illustrate, that the adaptive controller works successfully and effectively. The adaptive

Fig. 23. Experimental setup.

nature of these controllers is expressed by the arbitrary choice of the system parameters. • For technical realization it turns out, that the worm-like locomotion system with internal control inputs is more suitable than the other one. This fact is also supported by the simulations of this worm-like system. We are able to track appropriate gaits to receive a faster motion of the system. • The system has not to be restricted to the case of k = 3 point masses, it can easily be extended to fixed, but arbitrary k ∈ N . The theory (theorem) is also valid for arbitrary numbers of point masses. • Currently, the simulations are generalized to k > 3. Acknowledgements We are indebted to Joachim Steigenberger (Technische Universit¨at Ilmenau) for fruitful and critical discussions. We thank an anonymous referee for his thorough review remarks, which substantially improved the presentation of the paper. Carsten Behn acknowledges discussions with Achim Ilchmann (TU Ilmenau) about adaptive control and λ-tracking in general. This work is supported by Deutsche Forschungsgemeinschaft (DFG, Zi 540/6-1). Appendix. Proof of Theorem 6.1 We proceed in several steps. Step 1: We show the existence of a solution of (6.1) on [0, t 0 ) for some maximal t 0 ∈ (0, ∞].

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The right-hand-side of (6.1) is measurable in t for fixed (y, ζ, z, k), and continuous in (y, ζ, z, k) for fixed t, because the functions g1 and g2 are continuous. It follows from the theory of ordinary differential equations, that, for any initial data (y0 , y1 , z 0 , k0 ) ∈ Rm × Rm × Rn−2m × R, (6.1) has an absolutely continuous solution and every solution has a maximal extension on [0, t 0 ) with t 0 ∈ (0, ∞]. Let (y, ζ, z, k) : [0, t 0 ) → Rm × Rm × Rn−2m × R be a maximal solution of (6.1). Step 2: Coordinate transformation. We transform system (6.1) with strict relative degree 2 into a system with strict relative degree 1 by means of a coordinate transformation. This idea can be found in [15, Proposition 4.1], where a clever introduction of an internal variable v is due to a trick of [24]. Writing dλ (x) := max {0, kxk − λ} , ∀ x ∈ Rm , system (6.1) is equivalent to •       y(t) 0 Im 0 y(t) 0           0   y˙ (t) +  f 1 (s1 (t), y(t), z(t))    y˙ (t) = 0 A2     z(t)  0 A0 A5 z(t) f 2 (s2 (t), y(t))     0 h i   d  − G  k(t) (y(t) − yref (t)) + dt (k(t) (y(t) − yref (t))) ,      0    2  ˙ = dλ (y(t) − yref (t)) , k(t)   (y(0), y˙ (0), z(0), k(0)) = (y0 , y1 , z 0 , k0 ) . 

The introduction of the internal variable v(t) := y˙ (t) + G (k(t) (y(t) − yref (t))) leads to the following closed-loop system •      y(t) 0 Im 0 0 y(t)        0  v(t) +  f 1 (s1 (t), y(t), z(t))  v(t) = 0 A2 z(t) z(t) 0 A A5 f 2 (s2 (t), y(t)) 0 G   − (A2 + Im )G  k(t) (y(t) − yref (t)) , A0 G ˙ = dλ (y(t) − yref (t))2 , k(t) (y(0), v(0), z(0), k(0)) = (y0 , y1 + Gk0 (y0 − yref (0)) , z 0 , k0 ) .

                        

(A.2) With y(t) C y(t) 

T

Im

= v(t)

T

z(t)

0


T T

0




y(t)T

v(t)T

z(t)T


T

=:

we have



G C (A2 + Im )G  = Im G = G mit σ (G) ⊂ C+ , A0 G i.e. system (A.2) has strict relative degree 1, and the spectrum of the ‘high-frequency gain’ lies in the open right-half complex plane. Moreover, with α ∈ N the invariant zeros of the system (see Section 4.1), provided gi (·) ≡ 0, i = 1, 2, are  s In − A det C

B 0



 s Im  0  = det  0 Im

−Im s Im − A 2 A0 0

0 0 s In−2m − A5 0

= (−1)α det(G) det(s In−2m − A5 ) det(s Im + Im ) = (−1)α det(G) det(s In−2m − A5 )(s + m)m = 0. !

Since σ (A5 ) ⊂ C− , this equation holds for s ∈ C with R(s) < 0. Therefore, system (A.2) is minimum phase. Conclusion: We deal now with a nonlinearly perturbed, minimum phase MIMO-system with strict relative degree 1, with the spectrum of the ‘high-frequency gain’ in the open right-half complex plane. Therefore, we can now complete the proof using arguments from [15,16,19], because λ-tracking of systems with strict relative degree 1 was proved in these papers. References

(A.1)



 Im 0 0 0 −(A2 + Im ) Im 0 0   = det   −A0 0 In−2m 0 0 0 0 Im   s Im −Im 0 G   0 s Im − A 2 0 (A2 + Im )G   ×    0 A0 s In−2m − A5 A0 G Im 0 0 0   s Im −Im 0 G −(A2 + Im )s Im s Im + Im 0 0  = det   −A0 s Im 2A0 s In−2m − A5 0  Im 0 0 0   s Im + Im 0 2m = (−1) det(G) det 2A0 s In−2m − A5 

 G (A2 + Im )G    A0 G 0

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[31] J. Steigenberger, C. Behn, K. Zimmermann, K. Abaza, Worm-like Locomotion Theory, Control and Application, in: Proceedings of the 3rd International Symposium on Adaptive Motion in Animals and Machines, AMAM, Ilmenau, Germany, September 25–30, 2005, electronical publication (6 pages), Ilmenau, Germany. [32] J.C. Willems, C.I. Byrnes, Global adaptive stabilization in the absence of information on the sign of the high frequency gain, in: Lecture Notes in Control and Information Sciences, vol. 62, Springer, Berlin, 1984. [33] X. Ye, Universal λ-tracking for nonlinearly-perturbed systems without restrictions on the relative degree, Automatica 35 (1999) 109–119. [34] K. Zimmermann, I. Zeidis, Mathematical models and prototypes of worm-like motion systems using magnetic materials; Theory and Practice of Robots and Manipulators, in: Proc. of the 15th CISM IFToMM Symposium, RoManSy 15, Montreal, Canada, 2004. [35] K. Zimmermann, I. Zeidis, V. Naletova, V. Turkov, Waves on the surface of a magnetic fluid layer in a travelling magnetic field, Journal of Magnetism and Magnetic Materials 268 (1–2) (2004) 227–231. Carsten Behn was born in Leer, Germany, on September 13th in 1974. He received the diploma degree in Mathematics from the Technische Universit¨at Ilmenau, Germany, in 2001, and the Ph.D. degree in Mechanical Engineering from the Technische Universit¨at Ilmenau, Germany, in 2005. At present, he holds a postdoctoral fellowship from the Technische Universit¨at Ilmenau, Germany. His interests include adaptive and robust control of biologically inspired mechanical motion systems. Klaus Zimmermann was born in Suhl, Germany, on August 9th in 1956. He has received the diploma degree in Mechanics and Applied Mathematics from the Kharkov State University, Ukraina, in 1980, and the Ph.D. degree in Mechanical Engineering from the Technische Universit¨at Ilmenau, Germany, in 1984. In 1990 he finished his habilitation (Dr.-Ing. habil.) in Mechanics at the Technische Universit¨at Ilmenau, Germany. At present, he is a university professor of Technical Mechanics at the Technische Universit¨at Ilmenau, Germany (since 1997). Univ.-Prof. Dr. Zimmermann is a member of GAMM and VDI. He has written the book “Technische Mechanik - Multimedial” (Leipzig, Germany: Carl-Hanser-Verlag, 2003) and is co-author of “Taschenbuch der Mechatronik” (with E. Hering, H. Steinhart et al.) (Leipzig, Germany: Carl-Hanser-Verlag, 2005).