Dynamic Modeling of Bellows Actuated Continuum Robots Using the Euler–Lagrange Formalism07339740

Robots Using the Euler–Lagrange Formalism

Valentin Falkenhahn, Tobias Mahl, Alexander Hildebrandt, R¨udiger Neumann, and Oliver Sawodny

Abstract—In the previous decade, multiple useful approaches for kinematic models of continuum manipulators were successfully de- veloped. However, dynamic modeling approaches needed for fast simulations and the development of model-based controller design are not powerful enough yet—especially for spatial manipulators with multiple sections. Therefore, a practicable lumped mass model similar to common dynamic models of rigid-link manipulators is needed, which can be used for simulations and model-based con- trol design. The model incorporates mechanical interconnections of parallel and serially connected bellows and uses constant cur- vature kinematics and its analytical derivatives to balance forces and energies in a global reference frame. The parameters of the resulting model are identified with measurements before the simu- lation results are experimentally validated. The obtained dynamic model can be used to both simulate the manipulator dynamics and calculate the inverse dynamics needed for model-based controller design or path planning.

Fig. 1.

Festo’s Bionic Handling Assistant is a continuum manipulator with three sections and a gripper. This continuum manipulator changes its pose by

Index Terms—Continuum kinematics, dynamic model, inverse

deflating or inflating its nine parallel and serial arranged bellows. (Source:

dynamics, lumped model, redundant robots.

Festo AG).

However, due to the complexity, all these mechanical models ONTINUUM robots and hyperredundant robots inspired were considered in static equilibrium only. by snakes, octopus arms, eels, or trunks have gathered

NTRODUCTION I. I

C increasing attention in the past decade of robotics research. hyperredundant manipulators were investigated already in the

Dynamic models for continuum manipulators or the related

With their continuous backbones and their inherent flexibility, early stages to find a closed-form description [11], but compared new fields of application are being developed, where rigid-link with kinematic models, the progress of dynamic models is still robots with joints are not beneficial, i.e., in environments with lacking [9]. multiple barriers and edges. On the other hand, industrial rigid-

Only a few are using Cosserat rod models for dynamic sim- link robotics have been developed for a long period of time, ulations of single-section manipulators [12] and manipulators and its methods for modeling, control, and path planning enable with multiple sections [13], but the resulting partial differential multiple applications [1].

equations are not suitable for fast online simulations or con- Useful kinematic models for continuum robots were devel- troller design but rather for general simulations of soft robotics

oped almost ten years ago, especially the constant curvature ap- under friction and external loads. proach [2] has been commonly used [3]. In order to investigate

A close approximation of geometrically exact solutions of hy- and derive kinematics of continuum robots, also static mechan- perflexible and hyperredundant manipulators can be achieved by

ical models in both planar and spatial manner were developed, discretizing continuous models resulting multiple discs or slices. using beam mechanics [4], [5], Cosserat rod [6], [7], elliptic in- Using mass and stiffness matrices of the slices, the dynamic tegrals [8], modal approaches [9], or even a lumped model [10]. equations of the manipulator are obtained by integrating over

multiple discs. Early works cover the spatial dynamic modeling

Manuscript received October 23, 2014; revised July 10, 2015; accepted Oc- tober 26, 2015. Date of publication December 2, 2015; date of current version

of single-section manipulators [14] and the planar modeling of

December 2, 2015. This paper was recommended for publication by Associate

multisectional manipulators [15], [16]. For manipulators with

Editor R. J. Webster III and Editor A. Kheddar upon evaluation of the reviewers’

many modules, also recursive algorithms for spatial modeling

comments. V. Falkenhahn and T. Mahl are with the Institute for System Dynamics,

[17] were developed. More recent works obtain the dynamic

University of Stuttgart, D-70569 Stuttgart, Germany (e-mail: falkenhahn@

models of slice approximated manipulators using modal trans-

isys.uni-stuttgart.de; mahl@isys.uni-stuttgart.de).

formations and Lagrangian [18] or the principle of virtual work

A. Hildebrandt and R. Neumann are with Research Mechatronics, Festo AG & Co. KG, D-73726 Esslingen, Germany (e-mail: hild@de.festo.com;

[19]. Although these models might be well suited for simula-

nea@de.festo.com).

tions, they are not applicable to model-based control design.

O. Sawodny is with the Institute for System Dynamics (ISYS), University of

Lumped models are common for rigid-link robots as they

Stuttgart, D-70569 Stuttgart, Germany (e-mail: sawodny@isys.uni-stuttgart.de). Color versions of one or more of the figures in this paper are available online

provide rather simple but powerful models suitable for fast

at http://ieeexplore.ieee.org.

simulations and model-based control. However, lumped mass

Digital Object Identifier 10.1109/TRO.2015.2496826

approaches for continuum robots are still hard to find. On the

1484 IEEE TRANSACTIONS ON ROBOTICS, VOL. 31, NO. 6, DECEMBER 2015

TABLE I

Even less approaches for bellows or muscle-driven continuum

N OMENCLATURE

manipulators exist. For a single section, a planar model with three masses along the backbone and spring–damper–actuator

Symbol Variable name

systems along the actuators was presented using the Euler–

n number of sections

Lagrange formalism [25]. However, no dynamic results were

q ik actuator state of section

i and bellows k

provided, and the static simulations were not convincing.

q i actuator states of section i q j

However, the advantages of lumped manipulator models still

actuator state j

k i configuration state of section i

remain: They are computationally cheap and the methods al-

φ i orientation of section

i (config. state)

ready developed for flexible multibody systems [26] and rigid-

κ i curvature of section

i (config. state)

link manipulators [1] can be easily transferred and applied to

ℓ i elongation of section

i (config. state)

ψ i bending angle of section i

other lumped models, like inverse dynamics control. Further-

j H i homogen. transf. matrix of frame

i in ref. frame j

more, the control strategy of bellows actuated manipulators that

rotation matrix of frame

i in reference frame j

are limited by slow dynamics on the one hand and undesired

r position (r in homogeneous representation)

g gravity vector

interaction of the actuators during faster motions on the other

F force (F in homogeneous representation)

hand could be improved using inverse dynamics control.

bending momentum of section i m i

i , bend

Hence, in this paper, an explicit lumped dynamic manipula-

mass of section i

J S i moment of inertia of mass m i

tor model is presented that provides a useful analytical approx-

ω i rotational velocity of mass m i

imation and considers dynamic couplings of parallel and serial

T kinetic energy U

actuators using the Euler–Lagrange formalism. The presented

potential

Q generalized force

model is applicable to continuum manipulators with intrinsic

M mass matrix of dynamic equation

actuators that can be described with kinematics based on homo-

C Coriolis matrix of dynamic equation N

geneous transformations like constant curvature kinematics [3],

force vector of dynamic equation

τ actuation force vector of dynamic equation

variable curvature [27], or piecewise constant curvature [28]

Index Subscript and superscript description

kinematics. Compared with a previous published conference

paper [29], also rotational energies are considered now and an

w world

ih head of section i

outline toward identification of parameters is provided as well.

ib base of section i S

In order to derive the model, for each section of a multi-

center of mass of section

B ik center of bellows area of section

i and bellows k

sectional continuum manipulator, lumped point masses were

e active (effective) force

chosen. Based on a mass–spring–damper system of a single

a actuation force p

bellows [30], a lumped section model with spring–damper rep-

passive force

⋆ general placeholder for similar items

resentation for each actuator is developed. Using the kinematic model, local forces are transformed into the world reference frame, such that an Euler–Lagrange formalism can be used [1].

one hand, this might be caused by the idea that only continuum Then, the theoretical model is applied to Festo’s Bionic Hand- models with distributed parameters instead of concentrated pa- ling Assistant [31], a bellows-actuated continuum manipulator rameters resulting in partial differential equations can describe with three sections (see Fig. 1). the dynamic complexity of continuum manipulators. On the

After introducing the kinematic model of the multisectional other hand, most continuum manipulators are not operated close constant curvature manipulator in Section II, the dynamic model to dynamic boundaries and thus require only kinematic and static is derived in Section III: First, the general model is presented; mechanical models. However, a few lumped manipulator mod- then, the generalized forces, the potential, and the kinetic en- els for continuum manipulators were developed so far, most of ergy are derived. Furthermore, the analytical derivatives of the them in the last years.

homogeneous transformations of the kinematic model need to Lumped approximations for hyperredundant octopus-like

be computed before the resulting dynamic model with its prop- manipulators were developed, e.g., to describe the planar motion erties can be presented. In Section IV, the dynamic parameters of an arm by using decoupled actuator models with a diagonal are identified on the example manipulator before the model is mass matrix resulting in a decoupled differential equation sys- reduced in Section V. After an experimental validation of the tem obtained by Lagrange formalism [20]. Another algorithm model in Section VI, a final conclusion is drawn in Section VII. that describes the spatial net motion of a platform with multi-

ple arms was derived by recursively computing reaction forces INEMATIC II. K M ODEL of actuator segments of multiple arms taking into account the

A bellows-driven continuum manipulator in general has three dynamics of the adjacent actuators [21]–[23]. Although forces actuators per section (see Fig. 2c). By serially connecting the are exchanged, the dynamic actuator models itself are still de- sections, a manipulator with multiple sections can be con- coupled. A planar model for a tendon-driven manipulator was structed, i.e., with three sections and nine bellows in total (see obtained by defining a high number of point masses. The large Fig. 2a). In order to describe the relative and absolute posi- but sparse stiffness and mass matrices combined with a Rayleigh tions of the end effector and the actuators, a kinematic model is damping approach could be used for a modal decomposition and needed. For this purpose, different coordinate frames are intro- modal control [24].

duced: actuator space, configuration space, and task space.

FALKENHAHN et al.: DYNAMIC MODELING OF BELLOWS-ACTUATED CONTINUUM ROBOTS USING THE EULER–LAGRANGE FORMALISM 1485

Serial combination of local task space reference frame defined at each section’s head result in a Cartesian task space representation of a continuum Fig. 2. (a) Actuator states q can be measured and controlled, while (b) con-

Fig. 3.

manipulator’s end effector in a world reference frame. figuration states k describe the shape of the manipulator.

First, the different spaces are introduced before the transfor- mation of the forward kinematics is explained.

A. Kinematic Coordinate Frames In actuator space, the actuator states q T = q

11 q 12 ... q

Fig. 4. Kinematic spaces and their corresponding transformations.

n3

(1a)

(1b) consist of three position coordinates and two orientation angles of n sections representing the length l of each bellows can be

1 q 2 ... q {3n}

directly measured and controlled (see Fig. 2a). Two indexing x = xy z ϕ x ϕ y

(4) methods allow a bellows- and section-wise notation (1a) of which can also be expressed by a homogeneous transforma-

the actuator state q ik with section index

i ∈ {1, . . . , n} and tion matrix H including a rotation matrix R and a translation bellows index k ∈ {1, 2, 3} and a global actuator-wise notation vector o

(1b) of all vector elements q j with global actuator index j∈ {1, . . . , 3n}. j

∈R In configuration space, the configuration states k characterize 4×4 (5)

an actuator-independent model of the manipulator by describ- ing its general shape. These configuration states are provided that describes the relative pose between two Cartesian coordi- by the kinematic model. A well-suited kinematic model for nate frames i and j and allows a derivation of any task coordinate continuum manipulators with multiple sections is the constant combination, e.g., position only or complete pose. curvature approach [2], [3] that defines the configuration states (see Fig. 2b) as the section’s backbone length ℓ, its curvature κ,

B. Kinematic Transformations

and the orientation φ of the deformation for each section i by

After defining the coordinate frames, the mappings between k i = φ i κ i ℓ i .

(2) these frames need to be determined (see Fig. 4): a specific map- For highly conical sections, the model can also be extended to ping f specific that calculates configuration states k in response to

a piecewise constant curvature approach [28]. The curvature the according actuator states q and a general mapping f κ general

is reciprocal to the section’s bending radius, while the product that transforms configuration states into task states. of backbone length and curvature define the bending angle

The specific mapping f specific ψ describes the relation between

actuator states and the configuration states for each section i.

ψ i =κ i ℓ i .

(3) Using the constant curvature approach [3], the configuration states (2) are obtained from actuator states (1) by

Industrial manipulators are programmed in the end effector’s task frame. These task frames often use Cartesian coordinates

f specific : k i =k i (q i1 ,q i2 ,q i3 )=k i (q i ). (6) as they are very intuitive for humans. Therefore, Cartesian co-

ordinate frames R for the end effector R e , for the head R ih This manipulator specific mapping may differ for each manip- and base R ib of the manipulator’s serially linked sections, and ulator. For instance, applying the constant curvature approach

a world reference frame R w are introduced (see Fig. 3). As to a section of the Bionic Handling Assistant with median dis- the manipulator cannot twist itself, the task space coordinates tance r B i from section back bone to bellows back bone results

1486 IEEE TRANSACTIONS ON ROBOTICS, VOL. 31, NO. 6, DECEMBER 2015

into the reference coordinate system R w (12) is obtained by √

in the transformation equations 1 [28]

calculating the product

φ i = arctan2

(k(q)) =

1b ξb ξ ξ

(k (q )) (14)

κ i =2

i1 i2

const. (q ξ=1 i1 +q i2 +q i3 )r B i

, (7b)

ℓ of all local sectional transformations (9) and the constant initial

i = 1 3 (q i1 +q i2 +q i3 ).

(7c)

transformation w H 1b (see Fig. 3).

If all three actuators have the same length such that the sec- tion is not bent but straight, the curvature κ i is zero, and the

YNAMIC III. D M ODEL orientation φ i is not defined.

In order to derive a dynamic model of a continuum manip- The general mapping

ulator with multiple sections that can be used for model-based

(8) control and that includes all main characteristics like couplings between the bellows and is still computationally efficient, the between the base (index b) and the head element (index h) of distributed mass needs to be discretized. The dynamic model for

f general,loc : ib H ih = ib H ih (k i )

a single section i is based on the homogeneous transformation discrete masses is derived using the Euler–Lagrange equations (5) and defines the pose of each section’s head. As the configu- ration states are manipulator-independent, the definition of the

=Q j (15) general transformation is valid for all continuum manipulators

∂q j with constant curvature and is obtained by

dt ∂ ˙q j

∂q j

for the general coordinates q j with the kinetic energy T , the ⎡

(1 − c

velocity-independent potential

U , and the corresponding gen-

eralized forces Q j . Having n discrete masses m i , their total ⎢ ⎢

φ (c κℓ − 1) + 1 s φ c φ (c κℓ − 1) c φ s κℓ

s φ (1 − c κℓ ) ⎥ 2 ⎥ kinetic energy

ω i J S i ω i (16) ⎣

is the sum of their translational energies depending on the linear using the abbreviations

velocity of each mass w ˙r S i and the rotational kinetic energies

c φ = cos φ i ,

s φ = sin φ i ,

(10a) depending on the rotation rate ω i . The potential U includes gravitational potentials and rotational spring potentials due to

(10b) bending. Without loss of generality, translational spring forces If the curvature κ i is zero and the section’s orientation φ i is not due to extension are covered in the generalized forces

c κℓ = cos κ i ℓ i ,

s κℓ = sin κ i ℓ i .

defined, (9) becomes

H ih ⎥ = ⎢ ⎥ (11) that consist of the projection of n f active forces F e ⎢ at the w

⎣ 0 0 1ℓ i ⎥ ⎦

position r excluding potential forces considered in U.

A. General Model Derivation

implying an upright section that is only characterized by its The discrete masses representing the distributed mass and the length ℓ i . moments of inertia have to be chosen. Therefore, it is assumed The general mapping f general of the tool at the head of the that

upper section i = n in relation to a world reference frame with

r each section i has one concentrated mass m i located at a

general,glob :

H ih = H ih (k)

locally fixed position r S i in the section’s head reference frame (see Fig. 5), and

is obtained by sequentially stacking the serial sections on top

r the moments of inertia ih J S i of the discrete masses remain

of each other. In this case, the section’s base coordinates are constant in the local coordinate frame and are approxi- identical with the head coordinates of the section below such

mated as a hollow cylinder. 2

that Based on these assumptions, the general idea of the model {i−1}h H ib =I 4 (13) derivation is presented next.

As the manipulator consists of multiple sections, a modular with I n being the identity matrix of dimension n. Therefore, section model is derived first that can be serially coupled later.

the transformation of a vector in head coordinates of a section i

This assumption will be discussed in the identification section (see arctan2 is defined s.t. φ = arctan2(sin φ, cos φ) with φ ∈ (−π, π].

Section IV-B).

FALKENHAHN et al.: DYNAMIC MODELING OF BELLOWS-ACTUATED CONTINUUM ROBOTS USING THE EULER–LAGRANGE FORMALISM 1487

time derivatives of position vectors that are fixed in the head coordinate system (18) are derived to

H ih (20)

r i⋆ = H ˙ ih r i⋆ + H ih ˙r i⋆ (21)

and hence only depend on the corresponding derivative of the actuator-dependent homogeneous transformation matrix.

Similarly, rotation rates also depend on homogeneous trans- formation matrices and its derivatives. Using the linear operator Γ{ ⋆ } applied to ⋆ ∈ R 3×3

Fig. 5. Approximation of distributed masses as point masses located at each

section’s head. The top most point mass also includes possible gripper and load

masses; m 0 is fixed.

Γ{ ⋆ } = diag ⎝ ⎣ 1 0 0 ⎦ ⋆ ⎣ 1 0 0 ⎦ ⎠ (22)

with the properties

dt and the corresponding homogeneous transformations (5), the

dt

rotation rate w ω of the mass i can be derived [32] by

i w =Γ R ˙

ih R ih T . (24)

B. Generalized Forces For the purpose of better interpretation, bellows forces

are divided in actuation forces F B ik ,a (index a) and passive

Fig. 6. Section model with one concentrated mass and three bellows repre- sented by a spring–damper–actuator system (note: 2 nd actuator in background).

forces F B ik ,p (index p)

F B ik ,a =F B ik ,act (p ik )=A P · (p ik −p amb ) (25) Therefore, a push–pull actuator–spring–damper model of a sin-

gle bellows [30] that characterizes only the longitudinal motion

F B ik ,p =F B ik ,spr (q ik )+F B ik ,dmp ( ˙q ik ). (26) is extended to a section model containing three parallel bel- The actuation forces result from the pressure difference inside

lows: each section consists of three mass-free bellows modeled the bellows and the ambient pressure working against the termi- as push–pull actuator–spring–damper systems and a combined nal surfaces A P . The passive forces are the sum of the length- point mass at the section’s head (see Fig. 6). The gravity force F g dependent longitudinal spring force and the velocity-dependent applies at the center of mass of the i th section r S i that has a fixed damping force. relative position within the section’s head

With the bellows being always orthogonal to the head and ih r S

i = const.

(18a) base frame, the forces F B ik ,⋆ transmitted by the bellows apply

orthogonally to head and base at r

B ik . Therefore, the forces in

(18b) the local head and base coordinate frame have only entries in as well as the relative positions of the bellow’s contact points the (z)-direction. By serially stacking multiple sections on top

ih

r B ik = const.

r B ik . Using the homogeneous transformation (14), position of each other, the actuation and passive net forces F e,B ik ,a and vectors and forces 3 are transformed

F e,B ik ,p

r i⋆ = H ih

(27) from the local head coordinate system R ih of the section i to

⎨ 00F B ik ,⋆ −F B {i+1}k,⋆ 0 i<n

F e,B ik ,⋆ =

0 i=n the global reference system R w . The partial derivatives and

00F T

B ik ,⋆

at a sections head i result from the forces of the two adjacent sec-

3 If position vectors are multiplied with a homogeneous transformation, they

tions if the section is not the final section n at the manipulator’s

are extended to [r T 1] T , while force vectors are extended to [F T 0] . T

head.

1488 IEEE TRANSACTIONS ON ROBOTICS, VOL. 31, NO. 6, DECEMBER 2015

External forces in general can apply at every point, but in order to keep the equations as simple as possible, only external forces F S i applying at the centers of mass r S i are considered in the following, knowing that extending the resulting equations is easy to perform.

The generalized forces Q j (17) depend on active forces. Ac-

tive forces w F e (index e) are both the external forces w F e,S i

Fig. 7. Introduced bending stiffness M i ,bend of section i considers the lateral

stiffness due to the nonzero width of the bellows that forces single bellows to

e,S i = H ih ih F S i,ext

(28) stand straight if no external forces are applied.

that apply at the center of mass and are zero if no external contact The sum of all forces apply and the active forces of the bellows w

n sections’ potentials derived by the generalized

F e,B ik

coordinate q ik

∂M ξ ,bend

r S ξ + (35) consisting of actuation forces and passive forces with the actu-

∂q ik ation forces resulting from the pressure forces of the bellows (25), (27) and the passive forces from the dynamic spring and describes the influence of the global potential on the generalized damper forces

coordinate that is used for Euler–Lagrange equation (15). w F w

H ih F e,B ik ,act

D. Kinetic Energy

F e,B ik ,spr + ih F e,B ik ,dmp ,

Using the homogeneous derivative (21) to describe the veloc- n point masses, the kinetic energy (16) with respect to

ities of

recalling that the active bellows forces w

F e,B ik ,⋆ are the resulting the global reference system R

w is

forces of a section pair (27) if the section is not the final section.

A generalized force (17) T=T trans +T rot (36)

T trans =

r S ξ (37)

(38) for its corresponding actuator state ξ=1 q

ik can be written as a sum

of all projected (20) generalized forces including bellows forces and the moment of inertia expressed in world coordinates (29) and forces applying at the center of mass (28). Note that

S ξ = R ξh ξh J w S ξ R T ξh . (39) gravitation and rotational spring forces due to bending are not

included in the generalized forces but in the potential U. To apply the Euler–Lagrange equations (15), the partial deriva- tives of (36) with respect to all generalized coordinates q ik are

needed. Using the chain rule and the identity C. Potential 4 The potential U i

i = −m i g Tw r S i +M i,bend (ψ i ,φ i )

the derivatives of the translational kinetic energy (37) for con- of a section i includes the gravitational potential of its mass m i

stant masses m i are

located at r S i and rotational spring momentum M i,bend depend-

r S ξ (41) entation φ i . This momentum is introduced, as the longitudinal

ing on the section’s bending angle ψ i (3) and the bending ori-

∂T trans

spring force of a bellows (26) derived from the single bellows model only covers the stiffness of its backbone against elonga- n ∂T

tion, but not its width-depending stiffness against lateral applied

r S ξ (42)

forces. The resulting net momentum of three parallel bellows

(see Fig. 7) disturbing each other’s longitudinal stiffness is taken

d n ∂T

m ξ H i,bend ¨ ξh r S ξ

into account by the rotational spring momentum M

momentum also stabilizes a section to stand straight with zero

curvature if no external forces apply.

∂T trans

The partial derivatives of the section’s potential are

∂M i,bend

A short proof can be found in Appendix A.

FALKENHAHN et al.: DYNAMIC MODELING OF BELLOWS-ACTUATED CONTINUUM ROBOTS USING THE EULER–LAGRANGE FORMALISM 1489

The derivatives of the homogeneous transformations H

used in (41)–(43) are presented in Appendix B. Similarly, the T

C (r,c) ξh = m ∂H ξ ∂q c r S ξ ∂q r r S ξ derivatives of the rotation matrix R can be implemented, as the

∂˙ H ξh

ξ =s

identity of

J S ξ Γ ∂q r R follows (40). To compute the derivatives of the rotational

kinetic energy (38) similar to (41)–(43), the properties of the linear operator 3ξ Γ (23) are used, such that the derivatives of T

∂q for constant inertia R S r

(50) ∂T rot

with r, c ∈ [1, 3n] indicating the row and column index of the

matrices and using the abbreviation H for w H ξh and its deriva-

tives (R for w R ξh respectively). The start index of the sums is dt ∂ ˙q ik

defined by

ξh J S R T

Γ ∂R T ∂q ik R s = section of(max(r, c)) . (51)

The last three rows of (50) result from nonconstant w J S due to +Γ RR ˙ T

R ξh J S

the rotation of constant ξh J S ξ in the world reference frame (39).

The homogeneous force vector N ∈R

w ∂T H ∂M r ,bend w = F T w +m ξ g T ∂ rot ξh N ξh

using the abbreviation R for 3 w R

ξh and its derivatives.

− ξh F T ∂ w H ξh

e,B ξ χ,p ∂q r

r B ξχ (52)

E. Resulting Dynamic Model contains stiffness and damping properties, as well as external By inserting the generalized forces (32) for all generalized co- and gravitational influences, while the pressure-dependent in-

ordinates q j and the derivatives of potential (35) and the kinetic homogeneous force vector τ (p) ∈ R 3n is defined as energy (41), (43), (45), (47) into the Euler–Lagrange equation (15), 3n coupled second-order differential equations are ob-

∂ w H ξh ξh

tained. By resorting, the differential equations can be written in

r B ξχ (53a) matrix form as

M (q) ¨ q + C(q, ˙q) ˙q + N (q, ˙q) = τ (p) ,

= H ξh F T w T∂ ξh ξh e,B ξ χ,a H ξh ∂q r r B ξχ (53b)

in which the active, controllable forces w F B ik ,a are part of the

inhomogeneous input force vector τ (p) of the differential equa- with forces in the global (53a) or local (53b) reference frame tion. The resulting mass matrix M ∈R 3n ×3n and the matrix of (19). As the pressures of the bellows are the inputs of the me- Coriolis and centrifugal gains C ∈R 3n ×3n are defined by the chanical model but their progressions cannot be chosen arbitrar- entry-wise formulas

ily, dynamic models for the pneumatic behavior of the individual bellows are necessary as well. The pneumatic model of a bel-

lows [30] based on models of pneumatic cylinders [33]–[35] is

m ∂H ξh r S

∂H ξh

(r,c)

∂q c ξ

∂q r

taken from existing models.

ξ =s

This analytical model can be efficiently programmed and im-

plemented by computing reoccurring terms only once, e.g., ∂H +Γ ∂R R T T w

(49) in (49) and (50). This reduces the complexity of the analytical model and allows a practicable handling of the derivatives.

1490 IEEE TRANSACTIONS ON ROBOTICS, VOL. 31, NO. 6, DECEMBER 2015

If the model is extended by adding additional external forces, these additional forces need to be separated into passive forces assigned to N , and actuated forces assigned to τ , although the latter should be rather uncommon.

ODEL IV. M I DENTIFICATION

With the structural model derived (48)–(53), it can be pro- grammed efficiently, but in order to run simulations, parameters need to be identified. All kinematic parameters like the distance

of the bellows’ center to the section’s backbone r B i (7b), (18b) Fig. 8. Identification of the nonlinear stiffness behavior by measuring steady-

or the sizes of the terminal surfaces A P of the bellows (25) can be state pressures for several elongations.

taken from the CAD model or have to be identified beforehand in order to obtain the kinematic model. The position ih r S i of However, an approximation of the maximum moment of inertia each section’s mass point in its local head frame (18a) is placed can be obtained by assuming a hollow cylinder with maximal at the center

section length ℓ i m ax and radius r Bi +r bi ih r S = [0, 0, 0, 1] T

i,x m ax = 1 2 m i (r B +r b ) i 2 i + 1 12 m i ℓ 2 i m ax (57a) of the head’s profile. The remaining dynamic parameters that

ih J S

= 1 1 i,y m ax 2 have to be identified are

ih

2 m i (r B i +r b i ) + 12 m i ℓ i m ax (57b)

(57c) the bellows depending on its elongation, r the longitudinal damping force

the longitudinal stiffness or spring force 2 F

B ik ,spr (q ik ) of

ih

J S i,z m ax = m i (r B i +r b i )

F that is used for the calculation of a maximum bound of rotational

B ik ,dmp ( ˙q ik ) of the bel-

lows depending on its velocity,

energies.

These constant moments of inertia in the local frame R ih can

r each section’s bending momentum

be transformed into R w using (39) and the rotation matrix w on the bending angle of the section, R ih r the nominal mass m of each section, and

M i,bend (ψ i ) depending

of the corresponding homogeneous transformation (5). The im-

r the corresponding constant moments of inertial ih J

i depending each mass.

of

plementation of variable local inertia matrices

ih

on the configuration states (section length, curvature, and ori- In the following, these functions and constant parameters are entation) is not worth the effort due to its small influence at

identified on the Bionic Handling Assistant with appropriate possible motions. experiments and measurements.

C. Identification of Longitudinal Stiffness

A. Identification of the Nominal Section Mass The longitudinal stiffness of the bellows is represented by the The point mass m i of a section is an approximation of the bellows’ spring force F B ik ,spr (q ik ), which is a nonlinear function

real mass distributed over the whole section. A sufficient ap- depending on the length of each actuator (26). As a steady-state proximation of the section’s mass can be obtained by weighing force, it can be identified by elongating all bellows of each sec- the individual sections before assembling. Other masses being tion evenly such that the sections curvature stays zero (7). The moved like tubes and wire cables are below 1 % of the mass of spring forces are obtained by calculating actuation forces from

a section’s polyamide structure and therefore neglected. measured steady-state pressures (25) and subtracting gravita- tional influences (see Fig. 8).

B. Identification of Inertia

D. Identification of Bending Stiffness

The inertia of each section can obtained by a static model from CAD or approximated by a solid cylinder of average section

The bending stiffness of a section represented by the section’s height ℓ i avg and radius r Bi +r bi as sum of bellow’s backbone bending momentum M i,bend depends on the bending angle ψ i

distance plus bellow’s radius. The local inertia tensor results in (3) and influences the potential of a section (33). Similar to the ⎡ ih

identification process of the longitudinal stiffness, the bending J S i,x

stiffness can be obtained via the measurement of steady-state

0 ⎣ ⎥ ⎦ (55) pressures. For this purpose, actuators 2 and 3 of each section are

ih J S i = ⎢ 0 ih J S i,y

0 0 ih J

evenly actuated against actuator 1 such that the orientation φ is

either 0 or π (7a). Measured steady-state pressures are adjusted with the moments of inertia defined as

S i,z

by subtracting longitudinal stiffness pressures and gravitational

ih S

(56a) effects. The obtained relative bending pressures can be approx- imated by a linear function of the bending angle (see Fig. 9) ih J S = 1

(56b) although a variance of relative pressures exists caused by the

small creeping of the section’s material. With geometric param-

ih

J i,z = 1 2 m i (r B i +r S 2 b i ) .

(56c) eters, the linear momentum function can be computed.

FALKENHAHN et al.: DYNAMIC MODELING OF BELLOWS-ACTUATED CONTINUUM ROBOTS USING THE EULER–LAGRANGE FORMALISM 1491

Fig. 11.

Comparison of the measured translational energy and upper bounded rotational energy (57) of a lateral swing (see Fig. 10) with maximal rotation.

A. Neglecting Rotational Kinetic Energies Rotational kinetic energies T rot (16) are high on classic indus-

trial robots with rigid arms and rotational joints as all motions

Fig. 9. Linear fitting of the relative bending pressures versus bending angle

of bellows 1 working against bellows 2 and 3. Negative curvature indicates

are based on rotations only, especially due to the decreasing iner-

bending in the other direction φ = π.

tia of outer arms. Continuum manipulators do not have this kind of rotational joint. Rotations are comparably slow and can be imposed only by contrary extending and contracting bellows of

a single section. Furthermore, the main motions of the imposed section mass and the attached sections will be of translational manner as, for example, a bending of the first section results in a small rotation of the upper sections but also in a large translation of the upper sections due to their leverage factor. Using a maximum approximation of each section’s inertia (57), the rotational kinetic energy of a lateral swing (see Fig. 10) with maximal rotational velocities is significantly lower than the translational kinetic energy (see Fig. 11). Compared with this worst-case scenario, other motions generated by actuation

Fig. 10. Lateral swing triggered by contact loss of an external force to iden-

force instead of external contact loss contain even less rotational

tify the manipulator’s damping properties. Oscillations are visible on lateral deflection of the three sections

kinetic energies.

x and a selection of actuator lengths q.

Therefore, rotational kinetic energies can be neglected [29] for some purposes, especially for model-based control where

E. Identification of Longitudinal Damping computation time is important and small errors can be compen-

Due to small velocities, the damping forces F B ik ,dmp ( ˙q ik ) of sated by feedback control. This leads to a large simplification each bellows (26) have a small impact on dynamics if the ma- of the dynamic mass matrix M and the Coriolis matrix C such nipulator is driven only by actuator force. However, on swings that the main dynamics are reduced to and oscillations triggered by jumps in external forces, e.g., a

∂ H ξh frequency and the decay of the oscillation. Using a least-squares

loss of contact (see Fig. 10), damping forces characterize the n w H $∂ % T w

M (r,c) =

ξh

ξh

r S ξ r S ξ (58)

ξh

∂q c ∂q r fit, linear damping parameters can be identified from manipula-

ξ =s

tor oscillations (see Fig. 10).

∂ H ˙ ξh

∂ w H ξh

C (r,c) =

ξh

ξh

r S ξ (59)

∂q c ∂q r

ODEL V. M R EDUCTION

ξ =s

When running dynamic simulations, a detailed model is de- instead of (49) and (50). The homogeneous force vector N and sired. However, small errors are acceptable if the model is used the pressure-dependent inhomogeneous force vector τ (p) do for model-based control such that small errors can be com- not depend on kinetic energies and therefore remain the same. pensated by feedback control and the computation time can

be reduced significantly—e.g., for inverse dynamics control and model-based decoupling of the actuators using feedback-

B. Neglecting Coriolis and Centrifugal Forces linearization [36].

The computation of the Coriolis and centrifugal gain matrix In the following, two model reduction approaches are pre-

C in the dynamic equation (48) is computational very heavy sented: First, rotational kinetic energies are neglected for the even if rotational kinetic energies are neglected. The high com- computation of the dynamic matrices M and C. Second, the putational burden is caused by the dependence on the partial

influence of the Coriolis matrix C on the dynamic model (48) derivative ∂ w h H ˙ ξ /∂q j that is used only in C (50), (59) but takes is investigated and how the model changes if Coriolis forces are about half of the computation time. With small velocities ˙q ≈ 0, neglected.

however, this term becomes very small ((71) in Appendix) and

1492 IEEE TRANSACTIONS ON ROBOTICS, VOL. 31, NO. 6, DECEMBER 2015

TABLE II S ELECTED P ARAMETERS OF THE E XAMPLE M ANIPULATOR

Mass per section

[kg]

Pressure range (abs.)

[10 5 Pa]

Pressure force

[N]

Bellows length range

[m]

Bellows volume

[10 −3 m 3 ]

the Coriolis matrix

∂ w H ˙ ξh

∂ w H ξh

C (r,c) = m ξ

ξh r S ξ

ξh r S ξ ≈0 (60)

∂q c ∂q r

ξ =s

loses influence. If computation time is to be reduced, the Coriolis matrix of the dynamic model can be neglected

M (q) ¨ q + N (q, ˙q) = τ (p)

depending on the required accuracy of the model. Comparisons between measurements and simulation for both model reduction approaches are shown in the experimental val- idation next.

XPERIMENTAL VI. E V ALIDATION

Fig. 12.

Reference measurement data (—–) versus simulated trajectories of a

In order to test and verify the presented model and its ap- full model ( · · · · · · ) including rotational kinetic energies and Coriolis influences proximations, measurements on the Bionic Handling Assistant and a reduced model (– – –) without rotational kinetic energies and Coriolis were compared with open loop simulations of the model using influences. the parameters from Table II.

Measurements are recorded using a DSpace 1007 PPC with 1 ms sampling time. Reference measurements were obtained from

a pressure controlled manipulator. Note that negative pressures up to approximately −0.3 bar can be applied to the bellows using vacuum, in order to achieve higher curvatures—i.e., if two bellows of a section are pressurized, the third can be artificially held short.

Simulations were implemented in MATLAB/Simulink in- cluding the analytical derivatives of the homogeneous transfor- mations ((66), Appendix B). Singularities in the mathematical formulation of the kinematic model were excluded numerically to distinguish between curvature κ close to zero (11) instead of (9) using switching boundary of κ=1 ·10 −6 .

First, experiments with an actuated manipulator were con- ducted. Therefore, some bellows were actuated using a pressure trajectory, while the remaining were held at constant pressure. However, due to the mechanical coupling of the actuators, even the nonactuated bellows change their length. By changing only the pressure of the first bellows in the first section and in a second step the pressures of all bellows in the first section, two tilting motions are created (see Fig. 12). The second and third sections are extended in order to create a larger inertia.

These measurements can be compared with a simulation of the full model (49), (50) and a reduced model neglecting both rotational kinetic energies and Coriolis forces (58), Fig. 13. Reference measurement data (—–) versus simulated trajectories of a (61). During pressure-driven motions, both models are good coupled static ( · · · · · · ) and decoupled static (– – –) model of a pressure driven approximations of the measured trajectories. However, static trajectory in which section one is tilted. models cannot describe the motions that well (see Fig. 13). A

FALKENHAHN et al.: DYNAMIC MODELING OF BELLOWS-ACTUATED CONTINUUM ROBOTS USING THE EULER–LAGRANGE FORMALISM 1493

TABLE III

C OMPARISON OF THE R ELATIVE C OMPUTATION T IME

Model w/

Rotations

Coriolis

Quality Rel. Time

0 0.175 static (coupled) a –

static (decoupled)

–– 0.014 a High computational effort due to an implicit nonlinear model.

simulation errors, especially in the actuator domain, exist due to limits of the lumped mass approximation of the manipulator. All further approximations made in the model reduction section— neglecting rotational kinetic energies and Coriolis forces—lead to very small changes in the simulation results, while the com- putation time decreases by about

80 %. A comparison of relative computation times with respect to the full dynamic model (48) is presented in Table III. It provides the computational effort of the methods independently of simulation step size, computational power, and simulation time.

ONCLUSION VII. C In this paper, a new approach for a dynamic model of bellows-

actuated spatial continuum manipulators has been presented that includes couplings and interconnections between the parallel and serial arranged bellows. Doing this, each section is modeled as a single lumped mass point actuated by three parallel spring– damper systems. Using the constant curvature kinematic model,

Fig. 14. Reference measurement data of a lateral swing (—–) versus simu-

forces between the sections are transformed into a common

lated trajectories of a full model ( · · · · · · ) including rotational kinetic energies and Coriolis influences and a reduced model (– – –) without rotational kinetic

world reference system such that the Euler–Lagrange equations

energies and Coriolis influences.

for each actuator can be computed. With analytic derivatives of kinematics, the dynamic manipulator model can be easily

decoupled actuator model taking into account only translational implemented. After a parameter identification on the example spring forces against static pressures does not describe dynamic manipulator, experiments show that the new model incorporates effects, gravity, and bending. A static model

the interconnections between the bellows and that it describes the real manipulator behavior well. In order to reduce compu-

N (q, ˙q) = τ (p) (62) tation time, the complexity of the model can be reduced even further, if the desired accuracy is not as important—e.g., for

derived from (48) that takes into account also bending and inverse dynamics control and model-based decoupling of the gravity has also large errors during motions. Furthermore, the actuators using feedback linearization. actuator lengths q cannot be computed explicitly from (62),

and computing the solution implicitly takes even longer than

A the dynamic simulation. PPENDIX A P ROVING THE I Second, lateral swing motions similar to the damping iden- DENTITY OF H OMOGENEOUS D ERIVATIVES

∂q j = ∂˙ ∂ ˙q j as stated in (40) is a common result the loss of an external contact force, the manipulator conducts from classical mechanics. However, due to its importance for

tification (see Fig. 9) experiment are investigated. Actuated by

The identity of H

∂H

a damped oscillation although the pressures in the bellows are the derivation of the dynamical model, a short proof will be not controlled (see Fig. 14). These motions cannot be simu- presented here. lated with static models, as the steady state does not change

Proof: The identity of (40) can be validated by using the as long as the pressure in the bellows does not change. The property developed dynamic models however—both full models and re-

∂r

∂ ˙r

duced models—can describe the motion quite well (see Fig. 14).

(63) The simulated frequency, amplitude, and damping of the tool

∂q

∂ ˙q

center point (TCP) matches the reference measurement during derived from the principle of virtual work [37] that is valid if r the large oscillations at the beginning of the swing. However, does not depend on ˙q. With the chain rule, the partial derivatives

1494 IEEE TRANSACTIONS ON ROBOTICS, VOL. 31, NO. 6, DECEMBER 2015

of r (19) and ˙r (21) are The resulting time derivative of the time-invariant transfor-

mation w H ˆιh

(70) With ih r i⋆ being constant (18) and its partial derivatives being

ξ=1 χ= 1 ∂q ξχ zero, the assumption (40) follows the property (63).

only depends on partial derivatives and the corresponding ac- tuator velocities, similar to the partial derivative of the time

A PPENDIX B derivative

D ERIVATIVES OF C ONCATENATED

∂ 2w H n H 3 OMOGENEOUS T RANSFORMATIONS ˆιh ∂ w H ˆιh ∂ ˙q ξχ

ˆιh

ξ=1 χ= 1 ∂q ξχ ∂q The derivatives of the homogeneous transformation of a sin- ik gle section ib H ih (k(q)) can be obtained by applying the chain

rule, using the general mapping f

i > ˆι mapping f specific (6) of the constant curvature approach. As the

general (12) and the specific

(71) homogeneous transformation of a single section

= ' ˆι H 3 ' ∂ 2w H ˆιh

i ≤ ˆι pends on the actuator states q ik of this section, the derivative

ˆιb

ˆιh only de-

˙q ξχ

ξ=1 χ= 1 ∂q ξχ ∂q ik

that equals the time derivative of the partial derivative

is zero with respect to actuator states of other sections. Hence,

∂ w ˆιh

the derivative of a global homogeneous transformation w H

The second time derivative of homogeneous transformations in ∂ w H ˆιh

⎪ 0 i >ˆι

⎨ the world reference frame w H = ¨ ib

ξ=1 χ= 1 ∂q ξχ ∂t is independent of sections i and actuators q ik that are above

ξ=1 χ= 1 ∂ ˙q ξχ

section ˆι. Similar to (66), also the second partial derivatives of

3 " w H ˆι 3 2w #

a section

∂ H ˆιh

˙q ξχ ˙q αβ (73) ⎧

∂ used in (43) is obtained by using (40) and can be computed with

partial derivatives of first (67) and second order (69) and the ∂k i ∂k i ∂q iβ ∂q ik time derivatives of the actuator states q.

, α = i =ˆι

Using the chain rule, the derivatives of the section’s specific with respect to actuator states of other sections are zero. Fol- mapping (7) as well as the partial derivatives of the sections

lowing (67) and defining without loss of generality α ≤ i, the general mapping (9) used in (66) and (68) can be computed second partial derivative

analytically [28] and then inserted into the formulas of the global ⎧

derivatives. Similarly, all derivatives of rotation matrices w R ih

i >ˆι

can be computed, too.

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