Dynamic Modeling and Noncollocated Control of a Flexible Planar Cable-Driven Manipulator
Dynamic Modeling and Noncollocated Control of a Flexible Planar Cable-Driven Manipulator
Ryan James Caverly, Student Member, IEEE, and James Richard Forbes, Member, IEEE
Abstract—This paper investigates the dynamic modeling and
cables, this may be a valid assumption, which simplifies the
passivity-based control of a planar cable-actuated system. This
dynamic model of the system, as well as any controller synthesis.
system is modeled using a lumped-mass method that explicitly considers the change in cable stiffness and winch inertia that oc-
Unfortunately, in large-scale cable-actuated systems, such as
curs when the cables are wound around their respective winches.
cable-tethered aerostats [6], this flexibility cannot be ignored.
In order to simplify the modeling process, each cable is modeled in-
The dynamic modeling of cable-actuated systems considering
dividually and then constrained to the other cables. Exploiting the
cable flexibility has been developed in [7]–[10], using variations
fact that the payload is much more massive than the cables allows
of a lumped-mass method. This modeling method lumps the the definition of a modified output called the µ-tip rate. Coupling
the µ-tip rate with a modified input realizes the definition of a pas- mass of the continuous cable into discrete locations and models
sive input–output map. The two degrees of freedom of the system
the cable flexibility and natural damping by placing springs and
are controlled by four winches. This overactuation is simplified
dampers between the lumped masses.
by employing a set of load-sharing parameters that effectively re-
The flexibility associated with cables leads to the issue of
duce four inputs to two. The performance and robustness of the
noncollocation. The goal of cable-actuated systems is to control
controllers are evaluated in the simulation.
the position and velocity of a payload located at the opposite
Index Terms—Cable-actuated systems, dynamics, motion con-
end of many flexible cables actuated by winches. Due to this
trol, parallel robots, passivity-based control.
noncollocation, a passive map cannot be established from the in- put torque of the winches to the output velocity of the payload,
NTRODUCTION I. I
and hence, passivity-based control techniques are not readily applicable. When controlling flexible robotic manipulators, the
HE use of cable-actuated systems is widespread; elevators, use of a modified input and a modified output was shown to towing systems, cranes, aerostats, and cable robots are ex-
provide a passive input–output map in [10]–[18]. In particular, amples of systems actuated by cables. Cable-actuated systems in [14]–[18], the µ-tip rate, which scales the effect of the elastic typically feature stationary actuating winches, which reduces coordinates when calculating the payload velocity, is defined.
the moving mass of the system compared with standard serial The µ-tip rate was implemented on a single-degree-of-freedom (rigid-link) manipulators whose actuators are embedded in the (DOF) cable-actuated system in [10], proving that multiple pas- manipulator, allowing for higher payload accelerations. Cable- sive mappings can be established. actuated systems also allow for relatively large workspaces and
This paper extends the study presented in [10] by deriving high maximum payload to weight ratios. Unfortunately, these the nonlinear dynamic model of a planar cable-actuated system
advantages can be overshadowed by the fact that the cables and implementing passivity-based control on the planar system. used in these systems are relatively flexible and only provide This is accomplished by first fully developing the lumped-mass an actuating force in tension, not in compression. These short- model of a single cable and constraining two of these cables comings are often addressed by reducing the acceleration of together to create a two-cable system. Two of these two-cable the desired trajectory, so that cable vibrations are not excited, systems are then constrained together to capture the dynamics or using shorter cables [1]. Although this may solve the issue of the complete planar system, which consists of four cables and of cable vibration, the performance capability of cable-actuated one payload moving in the plane. The mass and stiffness param- systems is not fully exploited.
eters of the cable change as the cable is wound in or let out. The Cable-actuated systems differ from rigid manipulators by the rigid and elastic dynamics of the complete system are then ap-
flexibility present in the cables. In cable robot applications, this proximately decoupled by making the assumption of a massive flexibility is often ignored and the cables are assumed to be payload [14]. This, along with the definition of the µ-tip rate, straight and massless [1]–[5]. For systems with relatively short allows the establishment of various passive input–output maps.
Passivity-based control is then implemented in the simulation,
Manuscript received January 14, 2014; revised May 16, 2014; accepted Au- gust 8, 2014. Date of publication September 4, 2014; date of current version
which illustrates the system’s performance. The novel contri-
December 3, 2014. This paper was recommended for publication by Associate
bution of this study is the development of the dynamic model
Editor P. R. Giordano and Editor T. Murphey upon evaluation of the reviewers’
using a Lagrangian-based lumped-mass method, the establish-
comments. The authors are with the Department of Aerospace Engineering, Univer-
ment of multiple passive mappings, and the proof of closed-loop
sity of Michigan, Ann Arbor, MI 48109 USA (e-mail: caverly@umich.edu;
stability of a planar cable-actuated system. The dynamic model
forbesrj@umich.edu).
of the planar cable-actuated system is presented with a large
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
amount of detail in an effort to avoid any ambiguity in the
Digital Object Identifier 10.1109/TRO.2014.2347573
derivation.
CAVERLY AND FORBES: DYNAMIC MODELING AND NONCOLLOCATED CONTROL 1387
In this section, the lumped-mass dynamic model of each ca- ble is derived individually and, then, constrained together. The lumped-mass method used derives from [7]–[10], [21], and [22]. As in [22], the dynamics of the winches are considered, which takes into account the additional inertia added to the winches because of the wrapping of the cable. In systems with very large cables, this additional winch inertia can be significant. The assumptions made in the following derivation of the dy- namic model include that the payload moves in the plane, the tensions in the cables are large enough that gravitational forces on the cable lumped masses can be neglected, and the mass of the payload is much greater than the mass of the cables, which allows for an approximate decoupling of the dynamics in Section III-D. Further assumptions could be made to simplify the dynamic model, but it was decided to maintain a certain
Fig. 1. (a) Schematic of a planar cable-actuated system with four cables and
amount of fidelity in the model. For instance, many dynamic
(b) lumped-mass model of a single cable.
models [2], [7] do not include winch dynamics, which simpli- fies the model. In practice, a winch is needed to wind the cable, and as such, a realistic and practical dynamic model should
RELIMINARIES II. P
include this effect. In addition, the passive maps presented in
A. Passive Systems this paper rely on the fact that the energy of the undamped and unforced system is constant, which would not be true if mass
The mapping u 2e → were added and removed representing cable lengthening and L 2e , where y = Gu, is input strictly passive (ISP) if ∃ β and shortening. For this reason, the modeling of winch dynamics
0 < δ < ∞, such that [19]
and cable wrap is of great use.
2T + β, ∀u ∈ L 2e ,T∈R + .
0 A. Modeling a Single-Cable System
When δ = 0, the mapping is passive. The weak form of the
A schematic of a single winch-driven cable modeled with n Passivity Theorem states that a passive system connected in a lumped masses attached to a payload is shown in Fig. 1(b). Let
negative feedback loop with an ISP system is L 2 stable [19].
F i , F w ,1 , and F 1 denote the inertial frame, the frame attached to the first winch, and the frame attached to the first cable,
B. Lagrangian Dynamics respectively. The F w ,1 frame is located at the center of the first The equations of motion of a rigid body can be found by winch, and the F 1 frame is located at the payload. solving Lagrange’s equation
The kinetic energy, T 1 , potential energy, V 1 , and Rayleigh dissipation function, R 1 , of the single-cable system are
T 1 = 1 ˙q T M 1 ˙q 1 , V 1 = 1 q T K 1 q 1 , and R = 1 1 ˙q T D 1 ˙q 1 , re- −
=f+Ξ T λ
spectively, where q
1 =[θ 1 q e1 α 1 ] are the general- ized coordinates of the first cable, θ
1 L = T − V is the Lagrangian, R is the Rayleigh dissi- is the rigid coor- dinate representing the rotation of the winch drum, q
where
pation function, q are the generalized coordinates, f are the e1 =
generalized forces, Ξ is the Jacobian matrix associated with [x 1,1 x 2,1 ... x n ,1 x p,1 ] are the relative flexible co- the system’s constraints, and λ are the Lagrange multipliers. A ordinates of the cable, α 1 is the angle of rotation between frame n transpose is applied to each term on the left-hand side of (1) to F i and F 1 , M 1 =M T 1 =M p,1 +M w ,1 + i= 1 M i,1 > T keep the equation consistent with the definition of a Jacobian
0 is the mass matrix, K 1 =K 1 = diag{0, k 1,1 ,k 2,1 ,...,k n ,1 , [20].
k p,1 , 0} ≥ 0 is the stiffness matrix, D 1 =D T 1 = diag{0,
c 1,1 ,c 2,1 ,...,c n ,1 ,c p,1 , 0} ≥ 0 is the damping matrix, and
YSTEM III. S D YNAMICS
r 2 It is characterized by a payload of mass 4m p , four cables, and
Consider the planar cable-actuated system shown in Fig. 1(a). ⎤ r 2 −r1 T
1:n + 1
⎣ −r1 1:n + 1 1 1:n + 1 1 1:n + 1 −r1 1:n + 1 four actuating winches. At least three cables are required to ⎥ ⎦
M ⎥ p,1 =m p,1
provide two DOFs, because of the unidirectional capability of r 2 −r1 T 1:n + 1 J p,1 /m p,1 cables. In this paper, four cables will be used, realizing a larger workspace and making for a more intuitive derivation of the
M w ,1 =
system’s dynamics.
1388 IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 6, DECEMBER 2014
r ⎤ 2 −r1 T r 2
terms are given by
1,1 r1 T 1:1 J ˙ p,1 + j=1 J ˙ j,1 The matrix 1 denotes the identity matrix, and 1 i:j = [0 1 ...
m ˙ 1,1 r 2 −˙ m
x 2 0 and ending at index j, with every other entry zero.
1 0] T is a column vector composed of ones starting at index i
1 EAr
1 2 2 (L 1 /(n + 1) − rθ 1 ) The mass of each lumped mass is calculated as m 1,1 =
∂q ˜
σA(L 1 /n − rθ 1 ) and m i,1 = σAL 1 /n, i = 2, 3, . . . , n, where
A is the cross-sectional area of the cable, L 1 is the nominal
˙q T 1 M 1,1 ˙q 1 =
1 , ˙q 1 ) 2rc 1 (q 1 ) ˙α 2 0
2 2 1 length of the cable, σ is the density of the cable, and r is the
∂q ˜
radius of the winch. The first lumped mass changes its value as the cable is wrapped around the winch drum, decreasing
˙q T 1 M j,1 ˙q 1 = rc j (q 1 ) ˙α 2 1 T 1:j /r the total mass of the cable. The stiffness of each cable seg-
∂q ˜ 1 2
ment is calculated as k 1,1 = EA/(L 1 /(n + 1) − rθ 1 ), k i,1 =
EA(n + 1)/L 1 , i = 2, 3, . . . , n, and k p,1 = EA(n + 1)/L 1 ,
where
E is the modulus of elasticity of the cable. The longitu- ˙q T 1 M p,1 ˙q 1 = rc p (q 1 ) ˙α 2 T dinal stiffness of each cable is clearly a function of the cable’s
2 1 1:n + 1 ∂q /r 1 length, which is why the stiffness of the first cable segment
changes as the cable is wrapped around the winch drum. The
second moment of mass of the winch about the winch’s axis
of rotation is J 1 =J w ,1 + σAr θ 1 , where J w ,1 is the nominal
second moment of mass of the winch, and σAr 3 θ 1 is an ad- where g(q 1 , ˙q 1 ) = −σAr(r ˙θ 1 + r ˙α 1 − ˙x 1 ) 2 − σArc 2 1 (q 1 ) ditional second moment of mass term resulting from the cable
˙α 2 2 1 j /m 1,1 − 2rc 1 (q 1 ) ˙α 1 , c j (q 1 )=m j,1 (jL 1 /(n + 1) + k=1
c p (q 1 )=m p,1 (L 1 + lumped mass about the i axis is J
winding around the winch. The second moment of mass of each x k ,1 −θ 1 r), j = 1, 2, . . . , n, and
j,1 =m j,1 (r 2 + (jL 1 /(n +
k=1 x k ,1 +x p,1 −θ 1 r). The nonlinear terms presented
1) + j k=1 x k ,1 −θ is the axis 1 above are significantly more complicated than the ones r) 2 ), j = 1, 2, . . . , n, where i
presented in [10] because of the additional DOF. perpendicular to the plane of motion. Similarly, the second
moment of mass of the payload mass about the i axis is
B. Modeling a Two-Cable System
J p,1 =m p,1 (r 2 + (L
1 r) k=1 2 x k ,1 +x p,1 −θ ).
The null-space method [23] will be employed to constrain
Using a Lagrangian formulation, the equations of motion of two single cables together to form a two-cable system. If cables the single-cable system are
could operate in tension and compression, two cables would
be sufficient to fully actuate the system in two DOFs. Due to
M 1 ¨ q 1 +D 1 ˙q 1 +K 1 q 1 =ˆ b 1 τ 1 +f non,1
(2) the unidirectionality of cables, the two-cable system, shown in Fig. 2, does not provide two DOFs. The dynamic model of the
two-cable system will be developed as a stepping stone toward where ˆ b 1 =[10] , and the nonlinear terms are defined as
the complete system.
Before constraining two cables together, the equations of mo-
tion of the two cables must be expressed in a convenient form.
f non,1 =−˙ M 1 ˙q 1 +
˙q T
2 1 M 1 ˙q 1 ∂q The generalized coordinates of the two-cable system are writ- 1 T ten as p T = 1 q e1 α 1 θ 2 q e2 T α 2 . The unconstrained
equations of motion of the complete system are ∂ ˜ −
2 M p p ¨ +D p ˙p + K p p =B p τ +f non,p ∂q (3)
where M p = diag{M 1 ,M 2 }, D p = diag{D 1 ,D 2 }, K p =
These terms account for the nonlinear effects because diag {K 1 ,K 2 }, f non,p =[f non,1 T f non,2 T ] ,B p = diag{ˆ b 1 ,ˆ b 2 }, of the changing mass and stiffness of the first cable
and τ T =[τ τ ] . These equations of motion do not yet ac-
2 1 M 1 ˙q 1 )=˜ ∂/ ˜ ∂q 1 ( 2 ˙q 1 M p,1 ˙q 1 )+ count for the constraint between the two cables. The equations ∂/ ˜ ˜ ∂q ( 1 ˙q T M
segment, where ˜ ∂/ ˜ ∂q 1 ( 1 ˙q T
˙q )+ n ∂/ ˜ ˜ ∂q ( 1 1 2 1 w ,1 1 j=1 1 2 ˙q T 1 M j,1 ˙q 1 ) is the par- of motion of each individual cable system are simply assembled tial derivative of the kinetic energy with respect to q 1 , while together. Note that the subscript 2 denotes the properties of the keeping the terms ˙q T
1 and ˙q 1 constant, and ˜ ∂/ ˜ ∂q 1 ( 2 q 1 K 1 q 1 ) second cable system. The equations of motion of the second ca- is the partial derivative of the potential energy with respect to ble system are identical to the first, with the values of the second q 1 , while keeping the terms q T 1 and q 1 constant. The nonlinear system substituted in. The properties of the second cable can be
CAVERLY AND FORBES: DYNAMIC MODELING AND NONCOLLOCATED CONTROL 1389
Fig. 2. Two cables constrained together.
different from those of the first cable, including the number of is lumped masses. A transformation can be done to rearrange the generalized coordinates into a convenient form, ˙p = Φ ˙κ, where
cos(α j )
sin(α j )
. (7) κ =[θ 1 θ 2 α 1 α 2 q T
C ij =
− sin(α j ) cos(α j ) ⎡
e1 q e2 ] , and
1 ⎤ 0 0 0 0 0 The two cables can be constrained together by forcing the pay-
load velocity described by both to be identical. Equating (5) and ⎢
(6) and isolating for ˙α 1 and ˙α 2 gives
(8) This allows the rigid coordinates to be assembled together and where ˆ J α =
−1 α [−J θ1 J θ2 −J e1 the elastic coordinates to be assembled together. Applying ˙p = J ]. Equation (8) can be used to relate the unconstrained
α1 −J α2
e2
Φ ˙κ and premultiplying (3) by Φ T
gives
coordinates to the constrained coordinates, ˙κ = Γ ˙q 12 , where
12 = 1 θ 2 q T
e1 q e2 , and
κ κ ¨ +D κ ˙κ + K κ κ =B κ τ +f non,κ
where M =Φ T M Φ, D =Φ κ T p κ D p Φ, K κ =Φ T K p Φ,
κ =Φ p , and f non,κ =Φ f non,p . The payload velocity can
B ⎥ Γ . (9)
be described by either the generalized coordinates of the first cable system or the second cable system
Subtracting (6) from (5) gives the constraint equation Ψ ˙κ = 0, ˙ρ = J θ1 ˙θ 1 +J e1 ˙q e1 +J α1 ˙α 1 (5) where Ψ =
−J θ2 J α1 −J α2 J e1 −J e2 tiplying (4) by Γ T and applying the constraints gives
=J θ2 ˙θ 2 +J e2 ˙q e2 +J α2 ˙α 2 (6)
Γ M κ Γ ¨ q 12 +Γ T D κ Γ ˙q 12 +Γ T K κ κ =Γ T B κ τ where J θ1 =C i1 [ −r 0 ] and J θ2 =C i2 [ −r 0 ] are the
rigid Jacobians of the first and second cable systems, respec- +Γ T f non,κ −Γ T M κ ˙Γ ˙κ + Γ T Ψ T λ (10)
tively; J T
where λ are the Lagrange multipliers associated with the con- are the elastic Jacobians of the first- and second-cable systems,
e1 =C i1 [1 1:n + 1 0 ] and J e2 =C i2 [1 1:n + 1 0 ]
straints. This can be restated as
respectively; and J
α1 =C i1 [−r L 1 + k=1 x k ,1 +x p,1 −
M 12 ¨ q 12 +D 12 ˙q 12 +K 12 q 12 =B 12 τ +f non,12 (11) are the Jacobians associated with the angle α for the first and where M =Γ T M Γ, D =Γ T D Γ, K q =Γ second cable systems, respectively. The matrices C T i1 and C i2 12 κ 12 κ 12 12 K κ κ,
B =Γ T B , and f =Γ are rotation matrices that describe the rotation from T F 1 to F i 12 κ non,12 (f non,κ −M κ ˙Γ ˙κ). Substitut- and the rotation from F 2 to F i , respectively, where F j defines ing ˙κ = Γ ˙q 12 into Ψ ˙κ = 0 gives the result Γ T Ψ T λ = 0 by
reference frame j. The rotation matrix from F j to F i ,C ij (α j ), virtue of the null-space method [23].
1390 IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 6, DECEMBER 2014
The payload velocity can be fully described by In order to constrain the first two-cable system to the second two-cable system, the payload velocity of both systems must
˙ρ = J θ 12 ˙θ 12 +J e12 ˙q e12
be the same. To accomplish this, the null-space method will where ˙θ T
be employed [23] as was done previously for the two-cable e1 e2 system. The constraint can be written as Ξ ˙η = 0 where Ξ =
J θ 12 (θ 12 ,q e12 )=[J θ1 0 ]+[J α1 0 ]ˆ J −1 α [ −J θ1 J θ2 ]
−J θ 34 J e12 −J e34 rates ˙η can be related to the independent generalized rates ˙z by
J e12 (θ 12 ,q e12 )=[J e1 0 ]+[J α1 0 ]ˆ J −1 α [ −J e1 J e2 ].
˙η = R ˙z where
Note that the Jacobians of the single-DOF two-cable system [10] ⎡ ⎤ θ 12 are constant, and in particular, the rigid Jacobians are scalar val-
ues. In this paper, the Jacobians are clearly functions of θ 12 R
and q e12 , and the rigid Jacobian is a matrix. This adds an ad- q e34 ditional degree of complexity to the dynamic model and the The components of R are
er
control formulation in Section IV. In practice, the rigid and elastic Jacobians of the two-cable system are approximated
J e12 (θ 12 , 0) [15]. See the Appendix for an explicit derivation of the equations where R ˆ θr =J −1 θ 34 J θ 12 , R ˆ θe = R ˆ θ e,1 R ˆ θ e,2
R θ e,1 = of motion for a two-cable system with a single lumped mass in J −1 θ 34 J e12 , and ˆ R
−1 θ e,2 = −J θ 34 J e34 . After premultiplying by each cable.
R T , this transforms (14) to
R T MR ¯ ¨ z +R T DR ¯ ˙z + R T K ¯ η =R T B ¯ τ +R T ¯f non
C. Modeling the Complete System −R T M˙ ¯ R ˙z + R T Ξ T λ A similar process to that described in the previous section (15)
is used to derive the equations of motion of the complete four- where λ are the Lagrange multipliers associated with the con- cable system. Before constraining two of the two-cable sys- straint. The term R T Ξ T λ = 0, which can be shown by substi- tems together to obtain the complete system, the equations tuting ˙η = R ˙z into Ξ ˙η = 0. The equations of motion are now of motion of both two-cable systems will be expressed to- given as gether. The generalized coordinates of the complete system
M zz z ¨ +D zz ˙z + K zz z =B zz τ +f (16) are q =[θ 12 q T
TT
non,z z e12
θ 34 q e34 ] . After assembly, the uncon-
strained equations of motion of the complete system are where M zz =R T MR, D ¯ zz =R T DR, K ¯ zz z =R T K ¯ η,
B zz =R T ¯
B, and f non,z z =R T ¯f non −R T M˙ ¯ R ˙z. The reduced
M q ¨ + D ˙q + Kq = ˆ B τ +f non
(13) mass, damping, and stiffness matrices are now in the form where M = diag{M 12 ,M 34 }, D = diag{D 12 ,D 34 }, K = z z ,r r M z z ,r e
B M diag = {K 12 ,K 34 }, f non =[f non,12 non,34 ] ,ˆ = diag{B zz 12 , M T z z ,r e M z z ,ee
34 }, and τ = [ τ T 12 τ T 34 ] . Note that the subscript
the properties of a second two-cable system. The equations of zz = diag{0, K ee }
34 denotes
D zz = diag{0, D ee }, K
motion of the second two-cable system are identical to the first, where M z z ,r r =R T M θθ R θr ,M z z ,r e =R T θr θr [M θθ R θe +
with the values of the second system substituted in. A transfor- M θe ], and M z z ,ee =M ee .
mation can be done to rearrange the generalized coordinates,
The payload velocity is
˙q = Υ ˙η, where η = [ θ T
12 θ 34 q e12 q T e34 ] , and
˙ρ = U T ˙θ + U T θ e ˙q e (17)
12 ˙θ 34 ] T , ˙q e = [ ˙q T e12 ˙q T ] T ,U T e34 θ = [C 1 J ⎢ θ 12
where ˙θ T = [ ˙θ
C 2 J θ 34 ], and U T e = [C 1 J e12 C 2 J e34 ], provided that 0 < C 1 <
1 and C 2 =1−C 1 . The constants C 1 and C 2 are load-sharing
0 0 0 1 parameters that will be discussed further in Sections III-D and Applying ˙q = Υ ˙η and premultiplying (13) by Υ T gives
IV-A.
It can be shown analytically that the energy of the com-
M ¯ η ¨ +¯ D ˙η + ¯ K η =¯ B τ + ¯f non
(14) plete system without an input torque or damping is constant for
where ¯ M =Υ T
all time. The total energy of the system is H= ˙z ˙z + MΥ, ¯ D =Υ T
DΥ, ¯ K
=Υ zz T KΥ, ¯ B =Υ T B, ˆ 2
2 z T K and ¯ z. The temporal derivative of the total energy is f non =Υ f . The parameters of the transformed equa- non zz tions of motion found in (14) can be partitioned as
1 T ˙ M eθ M ee = ˙z (f non,z z −D zz ˙z + B zz τ )+ z K zz z + ˙z M zz ˙z 2 2
D ¯ = diag{0, D ee }, K ¯ = diag{0, K ee }.
CAVERLY AND FORBES: DYNAMIC MODELING AND NONCOLLOCATED CONTROL 1391
Without an input torque or damping, the time derivative of the the elastic coordinates total energy of the system is clearly zero.
D. Simplification Using the Massive Payload Assumption
e12 +J θ 12 ˙ˆJ e12 ple the rigid coordinate dynamics from the elastic coordinate
It will be convenient in Section IV to approximately decou-
˙q e . (22) dynamics. To accomplish this, it will be assumed that the mass
=B τ +f
zz
non,z z
+M
zz
of the payload is much greater than that of the cables. As shown
1 and premultiplying (22) by in [14], this assumption allows the kinetic energy of the system
Defining B e =
T e12 J −T
to be approximated by T≃T +T , where T is the kinetic B T e ρ gives e ρ
energy of only the payload moving, and T e is the kinetic en- M ˆ ee (q e )¨ q e +ˆ D ee ˙q e +ˆ K ee (q e )q e = −U e τ ˆ c +f non,e (q e , ˙q e ) ergy of the flexible coordinates vibrating when the payload is at
rest. When the payload is moving, the kinetic energy associated where ˆ M ee =B T
e M zz B e ,ˆ D ee =B e D zz B e ,ˆ K ee =
e K zz B e , and f non,e =B e f non,z z +B e M zz B ˙ e ˙q e . The stiff- kinetic energy associated with the motion of the elastic coordi- ness term in (23) is found by taking the stiffness term in
with the motion of the payload will be much greater than the B T
nates. This allows the total kinetic energy of the system when (22) and substituting into the form of K zz , giving K zz z = the payload is in motion to be approximated by only the kinetic
ee q e ) T . After premultiplying by B T e , this simply be- energy of the payload. When the payload is at rest, only the kinetic energy associated with the elastic coordinates is consid- comes K ee q e , which is equivalent to ˆ K ee q e . Note that the
ered. The total energy of the system is then approximated by the mass and stiffness matrices are functions of the elastic co- summation of both
e non,e T = ρ T
and . ordinates. The nonlinear term can be expressed as f
To find the kinetic energy of the payload in motion, the ap- C e (q e , ˙q e ) ˙q e +G e (q e ), where
˜ ∂ T gives the payload equation of motion
proximation ˙q e =¨ q e ≃ 0 is used. Substituting this into (16)
C , ˙q
e (q e e ) ˙q e =− M ˙ˆ ee ˙q e +
˙q e M ˜ ˆ ee ˙q e (24) ∂q e 2
M z z ,r r θ ¨ 12 =J θ 12 τ ˆ c +f non,z z θ
is a matrix containing the nonlinear terms related to the varying where f non,z z = T
f non,z z e T , and
non,z z θ
mass parameters, and
q e T e K ˆ ee q e (25)
2 θ 34 e ∂q ˜ e Note how τ ˆ c is distributed to τ via J θ 12 ,J θ 34 , and the constants is a column vector containing the nonlinear terms relating to the
C 1 and C 2 , which is why C 1 and C 2 are called load-sharing variable stiffness of the system. The matrix C e is constructed parameters. The velocity of the payload is now ˙ρ = J θ 12 ˙θ 12 . in such a way that (2C e − M ˙ˆ ee ) is skew symmetric. It will be Taking the time derivative of ˙ρ gives ¨ ρ =J θ 12 θ ¨ 12 + ˙J θ 12 ˙θ 12 convenient to write G e (q e ) in an alternate form and ¨ θ
12 =J θ 12 ρ ¨ −J −1 θ 12 ˙J θ 12 ˙θ 12 . Substituting these into (19)
and premultiplying by J −1 θ 12 gives
G e (q e )= M ρρ ¨ ρ −f non,ρ =ˆ τ c 1 −1 θ 12 ˆ
˙θ 1 ˙θ 2 ˙θ 3 ˙θ 4 M ρρ ˙J θ 12 ˙θ 12 . The nonlinear term can be expressed as f non,ρ =
where M ρρ =J θ 12 M z z ,r r J θ 12 , and f non,ρ =J θ 12 f non,z z θ +
2 J −1 θ 34 J ˆ e34
C ρ (ρ, ˙ρ) ˙ρ + G ρ (ρ), where C ρ (ρ, ˙ρ) contains the nonlinear terms relating to the varying mass and is constructed so that where ˆ J e34 =
e (q e ) in 2C ρ −˙ M ρρ is skew symmetric, and G ρ (ρ) contains the non- (26) is never actually computed and is only used to demon- linear terms relating to the varying stiffness. This gives the new strate the passivity of the system in Section IV-A. When us- equation of motion
e34
ing (25) to calculate G e (q e ), there is no singularity in ˙θ 1 , ˙θ 2 ,˙ θ 3 , or ˙ θ 4 . This is illustrated by examining the entry in
the first row of diag { 1 T 1 T 1 T 1 T ˙ˆ The kinetic energy of the system at rest is found by set-
M ρρ (ρ)¨ ρ −C ρ (ρ, ˙ρ) ˙ρ − G ρ (ρ) = ˆ τ c .
˙θ 1 q e1 , ˙θ 2 q e2 , ˙θ 3 q e3 , ˙θ 4 q e4 } K ee q, which is − 1 EArx 2 /(L /(n + 1) − rθ ) 2 . This is equivalent to
ting ˙ρ = ¨ ρ = 0. This implies J
when rearranged gives ˙θ 12 = −J −1 J e12 ˙q e12 . This can be − 1 θ 12 2 2 ˙k 1,1 x 1,1 / ˙θ 1 , which is why G e (q e ) can be stated as shown in (26). Premultiplying (26) by ˙q modified to include all elastic coordinates by defining an T e gives the following:
augmented elastic Jacobian ˆ J e12 =
ˆ ˙q e G e (q e )=− q e K ˙ˆ gives ˙θ q
e12
12 = −J J e12 ˙q e . Taking the time derivative of ˙θ 12 ee e .
gives ¨ θ = −J −1
12 θ 12 J ˆ e12 q ¨ e −(˙ J θ 12 ˆ J e12 +J θ 12 ˙ˆJ e12 ) ˙q e . Sub- This property will be useful in proving the passivity of the stituting this into (16) produces the equations of motion for system in Section IV-A. The equations of motion of the elastic
1392 IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 6, DECEMBER 2014
coordinates can be written as of both. The influence of the first and second two-cable sys- ˆ
tems is determined by the load-sharing parameters C 1 and C 2 ,
M ee (q e )¨ q e +ˆ D ee ˙q e +ˆ K ee (q e )q e which are embedded in U
θ and U e , where 0<C 1 < 1, and
= −U e τ ˆ c +C e (q e , ˙q e ) ˙q e +G e (q e ).
(28) C 2 =1−C 1 . Theorem 1: Consider the cable-actuated system described by
The kinetic and potential energies of the elastic coordinates are
the equations of motion of (16). The map τ ˆ µ is passive. T e =
2 ˙q e M ˆ ee ˙q e and V e = 1 2 q T e K ˆ ee q e . Their time derivatives,
Proof. Consider the nonnegative function [14] to be used later, are
1 H µ = H − µ(T e +V e )
T ˙ e = ˙q T
e M ˆ ee ¨ q e + ˙q e M ˙ˆ ee ˙q e 2 (29) =T
ρ + (1 − µ)(T e +V e ), 0 ≤ µ < 1.
e = ˙q T e K ˆ ee q e + q T e K ˙ˆ ee q e .
(30) Taking the time derivative of H µ and using (18) and (27)–(30)
2 results in
ONTROL IV. C F ORMULATION
H ˙ µ =˙ H − µ( ˙ T e +˙ V e )
As stated by the Passivity Theorem, the L 2 stability of a
passive system connected to an ISP system within a negative = − ˙z T D zz ˙z + ˙z T B zz τ −µ ˙q T e K ˆ ee q e feedback loop is assured. As such, identifying one or more pas- sive input–output maps is of significant interest. In this section,
1 1 it will be shown that several passive mappings can be estab-
+ q T e K ˙ˆ ee q e + ˙q T e M ˆ ee q ¨ e + ˙q T e M ˙ˆ ee ˙q e
2 2 lished for the planar cable-actuated system, which will be used
ee q e − µ ˙q e M ˙ˆ ee ˙q e of the closed-loop system.
in conjunction with ISP controllers to guarantee the L 2 stability
2 The mapping from input torque to payload velocity in a planar
− µ ˙q T e (− ˆ D ee ˙q e +C e ˙q e +G e −U e τ ˆ c ) cable-actuated system is not necessarily passive because of the
noncollocation of the system. A similar issue has been studied = ˙θ U θ τ ˆ c + µ ˙q T e U e τ ˆ c − ˙z T D zz ˙z + µ ˙q T e D ˆ ee ˙q e in flexible manipulators, leading to the use of a modified output
= ˙ρ T µ τ ˆ c − ˙z T D zz ˙z + µ ˙q T e D ˆ ee ˙q e . (34) tip rate, which has been shown to yield a passive map [11]–[18].
In particular, the reflected tip position is used in [11] and [12], Integrating (34) and knowing that ˙z T D zz ˙z = ˙q T e D ˆ ee ˙q e gives an arbitrary point located inboard of the tip is used [13], and a modified
µ-tip payload rate is used in [14]–[18]. T
˙ρ T
µ τ ˆ c dt = H µ (T ) − H µ (0) + (1 − µ) ˙z D zz ˙zdt.
A. µ-Tip Rate (35)
For 0 ≤ µ < 1, (35) simplifies to
Consider the µ-tip rate, ˙ρ µ , defined as
= µ ˙ρ + (1 − µ)J θ 12 ˙θ 12 (32) which proves that the map from τ ˆ
c to ˙ρ µ is passive . = ˙ρ − (1 − µ)J e12 ˙q e12
A PD control law of the form [14] τ ˆ c =− p ρ µ +K d ˙ρ µ can now be used to stabilize the system in an L 2 sense.
where
0 ≤ µ < 1 is a real number. To derive (32), add and subtract µJ θ 12 ˙θ 12 from (31); to derive (33), add and subtract
J e12 ˙q e12 from (31). Equivalently, all terms relating to the “12” B. µ-Tip Rate Tracking
system could be replaced by terms relating to the “34” system In order to track a desired trajectory, consider the PD control in the above definition. The µ-tip rate can be interpreted as the law found in [14]: τ ˆ = −(K p ρ ˜ µ +K d ˙˜ρ µ ), where ˜ ρ µ =ρ µ − velocity of the payload, with a weight µ that scales the influence ρ µ,d is the tracking error of the µ-tip position, and ˙˜ ρ µ = ˙ρ µ − of the time rate of change of the elastic coordinates. For specific
˙ρ µ,d is the tracking error of the µ-tip rate. The term ρ µ = µρ + values of µ, physically meaningful quantities are produced. The (1 − µ)F 12 (θ 12 ) is the µ-tip position, where ρ is the position
true velocity of the payload is captured when µ = 1, the veloc- of the payload and F 12 is the rigid forward kinematic map. The ity of the payload as if the system were not flexible is captured terms ρ µ,d and ˙ρ µ,d correspond to the prescribed desired µ- when µ = 0, and the reflected velocity of the payload is cap- tip payload position and µ-tip payload rate, respectively. Very tured when µ = −1. The µ-tip rate can also be calculated using often these terms are approximated as ρ
˙ρ d , which allows the desired payload position and velocity to be (1 − µ)U T θ ˙θ + µ ˙ρ = ˙ρ − (1 − µ)U T e ˙q e . Note that the µ-tip chosen directly, rather than computing their µ-tip counterparts rate can be found using only the rigid and elastic components [15]. This control law can be augmented with a feedforward of the first two-cable system, only the rigid and elastic compo- component based on the desired dynamics of the system. The nents of the second two-cable system, or a linear combination desired dynamics are found by selectively substituting in the
the payload velocity equation of (17): ˙ρ µ =U θ ˙θ + µU e ˙q e =
CAVERLY AND FORBES: DYNAMIC MODELING AND NONCOLLOCATED CONTROL 1393
desired kinematics into (21) and (28) where 0 ≤ µ < 1. Taking the time derivative of S µ and using (40), (41), and (43) gives
M ρρ (ρ)¨ ρ d −C ρ (ρ, ˙ρ) ˙ρ d −G ρ (ρ) = ˆ τ d (36)
e (ˆ M ee ¨˜q e +ˆ K ee ˜ q e ) where τ ˆ d is the desired feedforward torque, and
Integrating this relationship yields
q e4 K ee q e (38)
e D ˆ ee ˙˜q e dt. (44) where 0 q ˜ ei =q ei −q ei,d , and ˙˜θ i = ˙θ i − ˙θ i,d , i = 1, 2, 3, 4. The
+ (1 − µ) T ˙˜q
term G
e,d (q e ,q e,d , ˙q e , ˙q e,d ) is simply shortened to G e,d from
For
0 ≤ µ < 1, (44) reduces to
this point onwards in the interest of brevity. The desired payload
dynamics are then added to the PD control law as a feedforward ˙˜ρ µ (ˆ τ c −ˆ τ d )dt ≥ −S µ (0)
component 0 which establishes passivity from (ˆ τ c −ˆ τ d ) to ˙˜ ρ µ
ˆ τ c = (M ρρ (ρ)¨ ρ d −C ρ (ρ, ˙ρ) ˙ρ d −G ρ (ρ)) − (K p ρ ˜ µ +K d ˙˜ρ µ ). (39)
C. Filtered Error Tracking
Similar to (26), (38) does not need to be calculated online for the purposes of the feedforward control. Equation (39) clearly
Now consider an alternative form of tracking error known as shows only the desired payload dynamics from (36) are used. the filtered error, s µ = ˙˜ ρ µ + Λ˜ ρ
µ , where Λ =Λ > 0 [20]. In The form of G e,d in (38) is only introduced to assist later in the addition, the virtual reference trajectory and its corresponding passivity analysis of the system.
error are defined as ˙ρ r = ˙ρ d − Λ˜ ρ µ and ˙ ρ ˜ r = ˙ρ − ˙ρ r = ˙˜ ρ + It will be useful to calculate the error dynamics of the payload Λ ρ ˜ µ [15]. Combining the definition of the virtual reference tra-
and the elastic coordinates, for use in the passivity analysis. This jectory and the definition of the µ-tip rate allows for an alternate representation of the filtered error: s µ = ˙˜ ρ r − (1 − µ)U is accomplished by subtracting (36) from (21) and (37) from (28) T e ˙˜q e .
Now, consider the derivative control law [15]: τ ˆ = −K d s µ . M ρρ (ρ)¨ ρ ˜ −C ρ (ρ, ˙ρ) ˙˜ ρ =ˆ τ c −ˆ τ d (40) Once again, the control law can be augmented with a feedfor-
ward component. The desired system dynamics are found by
selectively substituting in the desired kinematics into (21) and = −U e (ˆ τ c −ˆ τ d )+C e (q e , ˙q e ) ˙˜ q e +˜ G e (41) (28)
where ˜ ρ =ρ−ρ d , q ˜ e =q e −q e,d , and
M ρρ (ρ)¨ ρ r −C ρ (ρ, ˙ρ) ˙ρ r −G ρ (ρ) = ˆ τ r (45)
= −U e τ ˆ r +C e (q e , ˙q e ) ˙q e,r +G e,r (q e ,q e,r , ˙q e , ˙q e,r ) (46)
where τ ˆ r is the desired feedforward torque, q e,r =q e,d , and ×diag
q ˜ e1 , q ˜ e2 , q ˜ e3 , q ˜ e4 K ˙ˆ ee q ˜ e . (42) G e,r is defined in the same way as G e,d in (38), with all the ˙˜θ 1 ˙˜θ 2 ˙˜θ 3 ˙˜θ 4 subscript d’s replaced by r’s. The virtual reference trajectory
Additionally, it is worth noting that payload dynamics of (45) can be added to the derivative control
˙˜q e G ˜ e =− q ˜ T
2 e K ˙ˆ
1 law to obtain
ee q ˜ e .
τ ˆ c =M ρρ (ρ)¨ ρ r −C ρ (ρ, ˙ρ) ˙ρ r −G ρ (ρ) − K d s µ . (47) Theorem 2: Consider the cable-actuated system described by
the equations of motion of (16). The map (ˆ τ c −ˆ τ d ρ µ is
It will be useful to calculate the virtual reference trajectory error
dynamics to assist in the passivity analysis of the system. The passive. error dynamics are found by subtracting (45) from (21) and (46) Proof: Consider the nonnegative function [14]
2 2 M ρρ (ρ)¨ ˜ ρ r −C ρ (ρ, ˙ρ) ˙˜ ρ r =ˆ τ c −ˆ τ r (48)
1394 IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 6, DECEMBER 2014
M ˆ ee (q e )¨ ˜ q e +ˆ D ee ˙˜q e +ˆ K ee (q e )˜ q e Although the use of the µ-tip rate as an output has many ad- vantages, there are some drawbacks and limitations to its use. = −U e (ˆ τ c −ˆ τ r )+C e (q e , ˙q e ) ˙˜ q e +˜ G e (49) For example, in order to know the values of ρ µ and ˙ρ µ , measure-
where q ˜ e =q e −q e,r ,˜ G e is defined by (42), and (43) remains ments of ρ, ˙ρ, θ 12 or θ 34 , and ˙θ 12 or ˙θ 34 are required, as in (32). unchanged.
As a comparison, a PD control law based on the true payload
Theorem 3: Consider the cable-actuated system described by position and velocity would only require a measurement of ρ
µ is and ˙ρ (which in practice could be accomplished using some sort passive.
the equations of motion of (16). The map (ˆ τ c −ˆ τ r
of vision or camera system). As a result, in order to implement Proof: Consider the nonnegative function [15]
µ-tip, PD control angular encoders and tachometers would be required on at least two of the four winches. In addition, the
e M ˆ ee ˙˜q e +˜ q e K ˆ ee ˜ q e ) (50) 2 stability results in this section rely on the assumption that the 2 payload is massive, which in practice may not always be true. where 0 ≤ µ < 1. Taking the time derivative of V and using Through simulation, a minimum suitable ratio of the payload
V= ˙˜ρ r M ρρ ˙˜ρ r + (1 − µ)( ˙˜ q
(43), (48), and (49) leads to mass to the mass of the cables was found to be four or five. In
cases where this assumption is not valid, a smaller value of µ
V=s ˙ µ (ˆ τ c −ˆ τ r ) − (1 − µ) ˙˜ q e D ˆ ee ˙˜q e .
can be used. For increased performance, the choice of µ should Integrating this relationship gives
be made so that it is as close to 1 as possible, while maintain- ing stability of the closed-loop system. For added robustness
to changes in the payload mass, the value of µ can be reduced s µ (ˆ τ c −ˆ τ r )dt = V(T ) − V(0) + (1 − µ)
˙˜q e D ˆ ee ˙˜q e dt
0 0 slightly. (51)
For 0 ≤ µ < 1, this yields UMERICAL V. N E XAMPLE
The control laws of (39) and (47) are implemented
µ (ˆ τ c −ˆ τ r )dt ≥ −V(0)
0 in a dynamic simulation of the complete planar cable- actuated system. In simulation, the cable winches are lo-
which establishes passivity from (ˆ τ c −ˆ τ r ) to s µ
. cated at (−0.5 (m), −0.75 (m)), (−0.25 (m), 1 (m)), (0.5 (m),
The results of Theorems 1–3 show that the planar cable- −0.5 (m)), and (0.4 (m), 0.8 (m)), while the payload is initially actuated system has three distinct passive input–output map- positioned at ρ
0 = [−0.1 0.1] (m). The cables have a mod-
pings. Employing the Passivity Theorem, the closed-loop sys-
E = 500 (MPa), a cross-sectional area of tem is L 2 stable as long as an ISP system is connected to the
ulus of elasticity of
A = 17.95 (mm 2 ), and a density of σ = 20 (kg/m 3 ) and are passive system in a negative feedback loop [19]. In this section, modeled using four lumped masses. A small amount of cable
PD and D controllers have been proposed, but in practice any damping was added with D
ee = 0.11. The cables are given ISP controller could be used to guarantee L 2 stability of the pretension values of 50 (N), 20 (N), 5.24 (N), and 35.91 (N) in
closed-loop system. order to maintain a positive tension in the cables at all time
It is of interest to note that while the result of Theorem 3 cou- and to ensure that the system remains in equilibrium. The pay- pled with the control law of (47) (or any ISP controller) and the load has a mass of 4 m p = 2 (kg) and each winch has a second Passivity Theorem guarantees that {˜ ρ µ , ˙˜ ρ µ }∈L 2 , the result of moment of mass of J 1 = 1.39 × 10 −5 (kg·m 2 ). The control pa- Theorem 2 coupled with the control law of (39) (or any ISP con- rameters used were µ = 0.8, K p = diag{6000, 6000} (N/m), troller) and the Passivity Theorem only guarantees that ˙ ρ ˜ µ ∈L 2 . K d = diag{150, 150} (N/(m/s)), Λ = 1 (1/s), C 1 = 0.5, and There is no guarantee that the tracking error will will go to zero C 2 = 0.5. The desired trajectory is a spiraling circle that begins in the case of Theorem 2. A similar attribute can be shown us- at the initial position (−0.1, 0.1), spirals out to a circle of radius ing Theorem 1. This clearly illustrates the benefit of using the 7 (cm), tracks the circle for two rotations, and then spirals back filtered error term, as it guarantees the L 2 stability of both the in to the initial payload location. This trajectory is shown in tracking error and the derivative of the tracking error, which is Fig. 3(a) and is described by explicitly shown in the following argument. Consider the cable-
actuated system described by the equations of motion of (16),
with the control law of (47) and (50) as a Lyapunov function can-
0≤t≤t 1
R sin ωt
didate
V , where ˙ V = −s T
µ K d s µ − (1 − µ) ˙˜ q e D ˆ ee ˙˜q e ≤ 0. From
⎨ ⎪ Theorem 3 and the Passivity Theorem [19], it can be concluded ⎪
+ρ 0 , t 1 ≤t≤t 2 that s µ ∈L 2 . Using the definition of the filtered error guarantees
ρ d (t) =
⎪ R sin ω(t − t 1 )
that { ˙˜ ⎪ ρ µ ,˜ ρ µ }∈L 2 , which in turn means ρ ˜ µ → 0 as t → ∞. In ⎪
t 3 −t
the interest of brevity, the proof of ˙ ρ ˜ → 0 as t → ∞ is omitted,
+ρ 0 , t 2 ≤t≤t 3
⎪ t 3 −t 2 R sin ω(t − t 2 )
but a similar proof can be found in [18]. Since it has only been
shown that ˜ ρ → 0 and ˙˜ ρ → 0 as t → ∞ and not ˜ ρ → 0 and
t≥t 3
˙˜ρ → 0 as t → ∞, it is recommended to choose a value of µ where ρ 0 is the initial and final positions of the payload, R is the close to
1, which results in ˜ ρ ≃˜ ρ µ and ˙ ρ ˜ ≃ ˙˜ ρ µ . maximum radius of the circle, ω is the frequency of the circle,
CAVERLY AND FORBES: DYNAMIC MODELING AND NONCOLLOCATED CONTROL 1395
Simulation results using the µ-tip PD controller of (a) τ 1 [blue, ap- proximately centered about 2 (N ·m)], τ 2 [green, about 0.8 (N ·m)], τ 3 [red, about Fig. 3.
Fig. 4.
(a) Desired trajectory and response of (b) ρ y versus ρ x for the circular 0.2 (N ·m)], and τ 4 [cyan, about 1.45 (N ·m)] versus time. (b) T 1 [blue, about 50 portion of the trajectory and (c) ρ ˜ y versus ρ ˜ x for the entire trajectory.
(N)], T 2 [green, about 20 (N)], T 3 [red, about 5 (N)], and T 4 [cyan, about 36
(N)] versus time.
and t 1 , t 2 , and t 3 are trajectory switching times. The switching times are chosen to be multiples of the period of the circular tra- jectory, i.e., t i = 2πk/ω, i = 1, 2, 3, and k = 1, 2, . . .. The sim-
ulation uses values of R = 7 (cm), ω = 4π (rad/s), t 1 = 2.5 (s),
t 2 = 3.5 (s), and t 3 = 6 (s). The desired trajectory is used in (39) and (47) to calculate the feedforward control terms. In
simulation, the nonlinear feedforward term, i.e., C ρ (ρ, ˙ρ) ˙ρ d or
C ρ (ρ, ˙ρ) ˙ρ r , was ignored, and the feedforward mass parameter was perturbed to 90% of its actual value, in order to simplify the
simulation and to demonstrate robustness to modeling errors. The response of ρ y versus ρ x throughout the circular portion of the trajectory is given in Fig. 3(b), while the tracking error ρ ˜ y versus ρ ˜ x throughout the entire trajectory is given in Fig. 3(c).
The input torques τ 1 , τ 2 , τ 3 , τ 4 and the cable tensions T 1 , T 2 ,
T 3 , and T 4 are presented in Fig. 4. The results show that the payload follows the desired tra- jectory, even when faced with a significant feedforward model perturbation. The tracking errors shown in Fig. 3(c) are similar
for both control laws and are less than a couple of millimeters. Fig. 5. Simulation results using the µ-tip PD controller of (a) τ 1 [blue, ap- The root-mean-square (RMS) tracking error of the PD µ-tip proximately centered about 2 (N ·m)], τ 2 [green, about 0.8 (N ·m)], τ 3 [red, about controller is 1.91 (mm), and the RMS tracking error of the
0.2 (N ·m)], and τ 4 [cyan, about 1.45 (N ·m)] versus time. (b) T 1 [blue, about 50 (N)], T 2 derivative control filtered error is 2.02 (mm). It is interesting to [green, about 20 (N)], T 3 [red, about 5 (N)], and T 4 [cyan, about 36
(N)] versus time.
note in Fig. 4(a) the torque applied to the first winch τ 1 becomes
negative toward the middle of the trajectory, but Fig. 4(b) clearly shows that the tension remains positive in all cables. A similar effects of longitudinal cable vibration and the modeling of winch phenomenon is observed using the filtered error in Fig. 5. This dynamics. The modeling process begins with the modeling of a occurs when the torque applied by the other winches is suffi- single cable and constraining the system in steps using the null- cient for the torque applied to winch 1 to be negative, while space method to derive the model of the complete system. The maintaining positive cable tension.
novel contribution of the dynamic model presented in this paper is the Lagrangian derivation of a lumped-mass cable model in two dimensions, and the use of the null-space method to derive
ONCLUDING VI. C R EMARKS the dynamics of a planar cable-actuated system. The dynamic model of a planar cable-actuated system has
Multiple passive mappings were established for the planar been derived using a lumped-mass method, which includes the system, which along with the proposed controllers guarantees
1396 IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 6, DECEMBER 2014
the L 2 stability of the closed-loop system, contributing to the novelty of this study. The use of the filtered error as a control variable ensures the asymptotic convergence to zero of the pay- load’s µ-tip position and rate, but the use of the µ-tip rate error as
a control variable only ensures the L 2 stability of the payload’s
µ-tip rate. Simulation results demonstrate that the controllers allow for the true payload position and velocity to track a high- acceleration trajectory while in the presence of significant model perturbations.
A PPENDIX
In this Appendix, the dynamic model of a two-cable system is presented in detail. For simplicity, n = 1 lumped masses are used in this derivation. It will be assumed that the first cable has
a length L 1 , a modulus of elasticity
E, a cross-sectional area A,
a density σ, a payload mass m p,1 , and is wound around a winch with a second moment of mass of J w ,1 . The stiffness matrix of the first cable is K 1 = diag{0, k 1,1 ,k p,1 , 0}, where k p,1 =
2EA/L 1 is constant and k 1,1 = EA/(L 1 /2 − rθ 1 ) varies with
the rotation of the winch. The damping matrix is D 1 =
diag {0, c 1,1 ,c p,1 , 0}, where c 1,1 and c p,1 can be found experi-
mentally or set to an arbitrarily small value. The mass matrix is
where ˜ m 1 =m 1,1 +m p,1 , J 1 =J w ,1 + σAr 3 θ 1 . The lumped mass is given by m 1,1 = σA(L 1 − rθ 1 ), and the second moment of mass terms are given by J 1,1 =m 1,1 (r 2 + (L 1 /2 + x 1,1 −θ 1 r) 2 ), J p,1 =m p,1 (r 2 + (L 1 +x 1,1 +x p,1 −θ 1 r) 2 ). The equations of motion of the first cable are given by (2),
where ˆ b 1 =
T and the nonlinear matrices are
where ˜ g(q 1 , ˙q 1 ) = g(q 1 , ˙q 1 ) + σAr 3 ˙θ 2 1 , g(q 1 , ˙q 1 ) = −σAr (r ˙θ 1 + r ˙α 1 − ˙x 1 ) 2 − σArc 2 1 (q 1 ) ˙α 2 1 /m 1,1 − 2rc 1 (q 1 ) ˙α 2 1 ,
The equations of motion of the second cable are identical, with the subscript 1 replaced by 2. The two-cable system is
(g(q 1 , ˙q 1 )−
(g(q 2 , ˙q 2 )−
EAr
(L 2 /2 − rθ 2 ) 2
x 2 1,2 ) − rc p (q 2 ) ˙α 2 2
m ˙ 1,1 r( ˙x 1,1 − r ˙θ 1 ) − ˙α 1 (˙ J 1,1 +˙ J p,1 ) m ˙ 1,2 r( ˙x 1,2 − r ˙θ 2 ) − ˙α 2 (˙ J 1,2 +˙ J p,2 )
m ˙ 1,1 (r ˙θ 1 + r ˙α 1 −x 1,1 )+c 1 (q 1 ) ˙α 2 1 r+c p (q 1 ) ˙α 2 1
c p (q 1 ) ˙α 2 1
m ˙ 1,2 (r ˙θ 2 + r ˙α 2 −x 1,2 )+c 2 (q 2 ) ˙α 2 2 r+c p (q 2 ) ˙α 2 2
c p (q 2 ) ˙α 2 2
CAVERLY AND FORBES: DYNAMIC MODELING AND NONCOLLOCATED CONTROL 1397
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A CKNOWLEDGMENT
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vol. 3, pp. 1–8, Jan. 2008.
gestions have greatly improved the quality of this paper.