A flexible manipulator consists of rigid components and flexible beams connected by joints in a serial chain topology as shown in Figure 3.2. The body at one of the extremities of the chain is designated as the base bodyB1and the body at the other end of the chain is designated as the tip bodyBN. The in-between bodies of the chain are connected to only two joints. Body zero,B0, is the designation reserved for the inertial body.
The joints are numbered in a similar fashion as the bodies. For the nth body,Bn, the inboard joint is designated asJn, and the outboard joint is designated asJn+1. The designation of inboard (outboard) refers to the topological position of the body or joint in the manipulator, which is closer to/further from the inertial body. The first joint,J1, is the joint connecting the chain to the inertial body,B0.
The joints may have from 0 to 6 degrees of freedom (DOF). If the manipulator is floating in space for example, then J1is set to allow the 6-DOF. Similarly, if the system consists of a mobile robot to which a flexible manipulator is attached, then J1 must possess translation degrees of freedom. For a typical case as in industrial manipulators,J1only includes rotational degrees of freedom.
A joint with 0-DOF serves the purpose of rigidly connecting flexible beams with flexible beams, as inJN, for example, and flexible beams with rigid bodies, as inJ2,
66 Flexible robot manipulators
3
3 4
2 2
1
0
1 4
N – 2
N – 1
N N – 1
N
Y1
Z1 O1 X1
Figure 3.2 Flexible manipulator
J3andJN−1, for example. By rigidly connecting flexible beams, flexible bodies of greater complexity may be modelled; higher displacements may be achieved and the complexity of the shape functions reduced by assuming a flexible beam composed of several flexible beams connected base to tip. The dimension of the model, however, is increased.
3.5.1 Joint kinematics
JointJnconnecting rigid bodiesBn−1andBnis represented in Figure 3.3. Here a joint representation similar to that of Rodriguez et al. (1992), but with the increasing body numbering from base to tip as in Hardt (1999) is used. The reference position of JnonBn−1is designated as the inboard point ofJn, On−1ˆ . Similarly, the reference position of JnonBnis designated as the outboard point ofJn, On, the origin of the body reference frame{On, XnYnZn}. The inter-body position vector, which describes the position of Onrelative to On−1expressed in{On−1, Xn−1Yn−1Zn−1}, is defined as rn−1, n.
The rotation ofBn relative toBn−1 is defined through the orthogonal rotation matrix Rn/n−1. The columns of Rn/n−1 are the projections of the basis vectors of reference frame{On, XnYnZn} on {On−1, Xn−1Yn−1Zn−1}. Similarly to equation (3.36), the angular velocity of Bn relative to Bn−1 expressed in{On, XnYnZn} is obtained from
ωn/n−1= Rn/n−1T ˙Rn/n−1 (3.84)
n – 1
The relative angular velocity between the two rigid bodies may be expressed in terms of the time derivative of Euler angles (Sincarsin et al., 1993) (the case of typical robotic manipulators), therefore,ωn/n−1may be written as a product of a rotational joint matrix, Hnω(qnω) (rotation Jacobian matrix of joint n), multiplied by the vector of angular velocity parameters,˙qnω;
ωn/n−1= Hnω(qnω)˙qnω (3.86)
where Hnv(qnω) is the translation joint matrix (translation Jacobian matrix of joint n), and ˙qnvis the vector of linear velocity parameters. Hnv(qnω) is dependent on qnω
if joint translation occurs inBn−1. If translation occurs inBnit is a constant matrix.
Writing equation (3.85) together with equation (3.87) yields (Jain and Rodriguez, 1992)
The absolute linear and angular acceleration vectors of Bn, expressed in {On, XnYnZn}, may be obtained as in equations (3.41) and (3.43), or by analogy
68 Flexible robot manipulators to equations (3.34) and (3.43), as
an= Rn/n−1T
Writing the above two equations together, similar to equation (3.88), yields +
In the case of a rigid connection whereBn−1is a flexible beam andBnis either a flexible beam or a rigid-body, the reference frames {O ∧
n−1, X ∧
n−1Y ∧
n−1Z ∧
n−1} and {On, XnYnZn} coincide, and equations (3.88) and (3.91) are obtained by evaluating vk,ωk, akandαkat O ∧
and
3.5.2 Dynamics of a rigid body
The dynamic equations of a rigid body expressed in the body reference frame may be obtained through the Principle of Virtual Powers, similar to the case of a flexible beam. Considering bodyBnof Figure 3.3 one has
Applying the variational operator,δ, to equation (3.95), and solving equation (3.94) yields which may be written in compact form as
MrnAn+ Nrn(ωn) = Fn− ΦTr n+1, nFn+1 (3.98) The control force and the control torque at the joints may be obtained by per-forming a virtual power balance across joint n. Since a joint has no mass, the linear
70 Flexible robot manipulators relative motion is expressed as
δ (Hnv˙qnv)TFn= δ ˙qTnvHTnvFn (3.99) and the angular relative motion as
δ (Hnω˙qnω)TMn= δ ˙qnωT HTnωMn (3.100) The externally applied generalized force vector (torques and forces) is then written as
Tnv
3.5.3 Dynamics of a rigid–flexible–rigid body
A flexible beam attached to two rigid bodies, one at each end, may be seen as a building block for a flexible manipulator. The rigid bodies are the supporting structures for actuators and sensors at the joints, and are typically of significant mass. However, if that is not the case, their mass may be easily neglected, and what remains are their geometric properties. To this end, consider the following:
Rigid body:Bn, from equation (3.97), and setting ˙qn = 0 and ¨qn = 0 in equations Flexible beam:Bn+1, from equations (3.83), (3.92) and (3.93),
Me n+1, rr Me n+1, re
Rigid body:Bn+2, from equation (3.97),
Mr n+2An+2+ Nr n+2= Fn+2− ΦTrn+3, n+2Fn+3 (3.109) The generalized coordinates of the rigid–flexible–rigid (RFR) body are the absolute linear and angular velocity vectorsBn, vnandωnrespectively, and the elastic bending displacements, torsion angle and shear angles of bodyBn+1, qKn+1and qϕn+1. The dynamic model of the RFR body may then be obtained by reapplying the Principle of Virtual Powers. The generalized virtual velocities areδVnandδ ˙qe n+1, therefore,
δVn+1= Φr n+1, nδVn (3.110)
and
δVn+2=
Φr n+2, n+1Φr n+1, n--Φe n+2, n+1 δVn
δ ˙qe n+1
(3.111) The acceleration of the flexible beam is given in relation to the acceleration ofBn
by equation (3.104), and the acceleration ofBn+2is obtained from equations (3.104) and (3.108) as
An+2=
Φr n+2, n+1Φr n+1, n--Φe n+2, n+1 An
¨qe n+1
+ Φr n+2, n+1Nn+1+ Ne n+2
(3.112) Adding the contributions of the three bodies yields
and adding the coefficients ofδVTn andδ ˙qe nT +1and writing in matrix form results in the following system of dynamic equations:
Mrfrn
An
¨qe n+1
+ Nrfrn =
I 0
Fn−
+ Φrn+3, n+2Φr n+2, n+1Φr n+1, nT
Φrn+3, n+2Φe n+2, n+1T ,
Fn+3
(3.114)
72 Flexible robot manipulators
3.5.4 Dynamics of a serial multi-RFR body system
Consider an articulated chain of RFR bodies representing a flexible manipulator. It follows from equations (3.88), (3.107) and (3.103) that
Vn+3=
Φrn+3, n+2Φr n+2, n+1Φr n+1, n-- Φrn+3, n+2Φe n+2, n+1 Vn
˙qe n+1
+ Hn+3˙qn+3 (3.117)
and from equations (3.91) and (3.112) that An+3= Renumbering equations (3.114), (3.117) and (3.118) in terms of the nth RFR body, and rewriting the generalized velocity, acceleration and force vectors in order to include the elastic contributions results in the dynamic equation,
Mrfrn
the velocity equation,
Furthermore, the externally applied generalized force vector in equation (3.101) may be rewritten as
Equations (3.119), (3.121) and (3.122) are in a form compatible with the problem formulation and solution methods for rigid manipulators presented in (Ascher et al.
1997) and Pai et al. (2000). The solution method consists of writing a large algebraic system as in Lubich et al. (1992) or von Schwerin (1999), which is solved through elimination methods leading to either the global dynamic CI method (the CRB method for rigid multi-body systems), or the AB method. The former has complexity O(N3) due to the need for inverting the global system mass matrix, whereas the latter has complexity O(N). To illustrate, following (Ascher et al. 1997),the algebraic system
74 Flexible robot manipulators
for an unconstrained (Frfr4= 0) three-RFR body manipulator is written as
The global dynamics CI method consists of rearranging the block rows and columns of the above system into the form
and solving through block-row elimination in order to arrive at
&
HTΦTMΦH'
¨q + HTΦT(MΦN + N ) = T (3.125)
which is the joint space dynamic equation of the flexible manipulator. This is the solution that can be obtained through a reapplication of the Principle of Virtual Powers to the chain of RFR bodies. From equation (3.120) the following may be written:
⎡
⎣Vrfr3
Vrfr2
Vrfr1
⎤
⎦ =
⎡
⎣I Φrfr3,2 Φrfr3,2Φrfr2,1
0 I Φrfr2,1
0 0 I
⎤
⎦
⎡
⎣Hrfr3 0 0 0 Hrfr2 0
0 0 Hrfr1
⎤
⎦
⎡
⎣˙qrfr3
˙qrfr2
˙qrfr1
⎤
⎦
⇔ V = ΦH˙q (3.126)
thus identifying the productΦH as a Jacobian matrix.
The AB method on the other hand, solves the algebraic system in equation (3.123), taking advantage of its block diagonal structure. Each block corresponds to a RFR body, which is coupled to the next body in the chain through matrixΦTrfrn+1, n. The solution procedure consists of eliminating these matrices in order to decouple the diagonal blocks. Therefore, for n = N, N − 1, . . . , 1, eliminate the middle row of each block using the first row and then the last row of the block. Then, using the resulting nth block rows, eliminate matrixΦTrfrn, n−1, coupling block n− 1 with block n. The resulting system is
where
T.rfrn = Trfrn− HTrfrn .Nrfrn+ /MrfrnNrfrn
(3.128)
Drfrn = HTrfrnM/rfrnHrfrn (3.129)
N.rfrn = Nrfrn+ ΦTrfrn+1, n .Nrfrn+1 + /Mrfrn+1Nrfrn+1
+ ΦTrfrn+1, nM/rfrn+1Hrfrn+1Drfr−1
n+1T.rfrn+1 (3.130)
and
M/rfrn = Mrfrn+ ΦTrfrn+1, nM/rfrn+1Φrfrn+1, n
− ΦTrfrn+1, nM/rfrn+1Hrfrn+1D−1rfr
n+1HTrfrn+1M/rfrn+1Φrfrn+1, n (3.131) with /MrfrN = MrfrN. Equation (3.127) represents a linear system for¨qrfrnand Arfrn. For n = 1, . . . , N, ¨qrfrn may be calculated. Equation (3.131) is the CRB inertia introduced in Featherstone (1987) in the context of rigid multi-body dynamics.
3.6 Summary
A systematic modelling approach leading to a general modelling environment, which may be used either in analysis or in the development of real-time control schemes for flexible manipulator arms, has been presented. A flexible beam attached to two rigid bodies, one at each end, has been assumed as a building block for a flexible
76 Flexible robot manipulators
manipulator. The rigid bodies are the supporting structures for actuators and sensors at the joints, and are typically of significant mass. However, if that is not the case, their mass may be easily neglected, and what remains are their geometric properties.
The cross-section rotations due to elasticity of the flexible beam may be of higher order as needed and shear deformation according to the Timoshenko beam theory is included. The method of formulation for the dynamics of a flexible manipulator may be either the global dynamics CI method or the recursive AB method.