**2.1 Deflection**

340 Serial and Parallel Robot Manipulators – Kinematics, Dynamics, Control and Optimization

applications. Shi *et al.* (1998) discussed some key issues in the dynamic control of

As a consequence of the interest in using flexible structures in robotics, several papers regarding the design of controllers for the manipulation task of flexible manipulators are found in the literature (Latornell *et al.*, 1998), (Choi and Krishnamurthy, 1994) and (Chang

In Tsujita *et al.* (2004), the trajectory and force controller of a flexible manipulator is proposed. From the point of view of structural dynamics, the trajectory control for a flexible manipulator is dedicated to the control of the global elastic deformation of the system, and the force control is dedicated to the control of the local deformation at the tip of the endeffector. Thus, preferably trajectory and force controls are separated in the control strategy. Static and dynamic hybrid position/force control algorithms have been developed for flexible-macro/rigid-micro manipulator systems (Yoshikawa *et al.*, 1996). The robust cooperative control scheme of two flexible manipulators in the horizontal workspace is presented in Matsuno and Hatayama (1999). A passive controller has been developed for the

In Miyabe *et al*. (2004), the automated object capture with a two-arm flexible manipulator is addressed, which is a basic technology for a number of services in space. This object capturing strategy includes symmetric cooperative control, visual servoing, the resolution of the inverse kinematics problem, and the optimization of the configuration of a two-arm

The effective use of flexible robotic manipulators in industrial environment is still a challenge for modern engineering. Usually there are several possible trajectories to perform a given task. A question that arises when programming robots is which is the best trajectory. There is no definitive solution, since the answer to this question depends on the selected performance index. Focused on industrial applications, the optimal path planning of a flexible manipulator is addressed in the present chapter. The manipulator is requested to perform a task in a vertical plane. Under this condition the gravitational effects are taken into account. Energy consumption is minimized when the movement is conducted through a suitable path. Energy is calculated by means of the evaluation of the joint torque along the path. End-effector accuracy is improved by reducing the vibration effect and increasing manipulability. The determination of the position takes into account the influence of structural flexibility. Weighting parameters are used to set the importance of each objective. The optimization scenario is composed by an optimal control formulation, solved by means of a nonlinear programming algorithm. The improvement obtained through a global optimization procedure is discussed. Numerical results demonstrate the viability of the

A control formulation to determine the optimal torque profile is proposed. The optimal manipulability is also taken into account. The effect of using end-effector positioning error as performance index is discussed. As a result, the contribution of the present work is the proposition of a methodology to evaluate the influence of different performance indexes in a

The paper is organized as follows. In Section 2, model of deflection, torque and manipulability are presented. Section 3 recalls the general optimal control formulation and the performance indexes are defined. Geometrical insight about the design variables is given. Multi-criteria programming aspects such as Pareto-optimality and objective weighting are presented in section 4. The global optimization strategy is outlined in section

payload manipulation with two planar flexible arms (Damaren 2000).

lightweight robots for several applications.

redundant flexible manipulator.

proposed methodology.

multi-criteria optimization environment.

and Chen, 1998).

Different schemes for modeling of the manipulators have been studied by a number of researchers. The mathematical model of the manipulator is generally derived from energy principles and, for a simple rigid manipulator, the rigid arm stores kinetic energy due to its moving inertia, and stores potential energy due its position in the gravitational field.

A flexible link also stores deformation energy by virtue of its deflection, joint and drive flexibility. Joints have concentrated compliance that may often be modeled as a pure spring storing only strain energy. Drive components such as shafts and belts may appear distributed. They store kinetic energy due to their low inertia, and a lumped parameter spring model often succeeds well to consider such an effect.

The most important modeling techniques for single flexible link manipulators can be grouped under the following categories: assumed modes method, finite element method and lumped parameters technique.

In the assumed modes approach, the link flexibility is usually represented by a truncated finite modal series, in terms of spatial mode eigenfunctions and time-varying mode amplitudes. Although this method has been widely used, there are several ways to choose link boundary conditions and mode eigenfunctions. Some contributions in this field were presented by Cannon and Schmitz (1984), Sakawa *et al.* (1985), Bayo (1986), Tomei and Tornambe (1988), among others. Nagaraj *et al.* (2001), Martins *et al.* (2002) and Tso *et al.* (2003) studied single-link flexible manipulators by using Lagrange's equation and the assumed modes method.

Regarding the finite element formulation, Nagarajan and Turcic (1990) derived elemental and system equations for systems with both elastic and rigid links. Bricout *et al.* (1990) studied elastic manipulators. Moulin and Bayo (1991) also used finite element discretization to study the end-point trajectory tracking for flexible arms and showed that a non-causal solution for the actuating torque enables tracking of an arbitrary tip displacement with any desired accuracy.

By using a lumped parameter model, Zhu *et al.* (1999) simulated the tip position tracking of a single-link flexible manipulator. Khalil and Gautier (2000) used a lumped elasticity model for flexible mechanical systems. Megahed and Hamza (2004) used a variation of the finite segment multi-body dynamics approach to model and simulate planar flexible link manipulator with rigid tip connections to revolute joints.

Santos *et al.* (2007) proposed the computation of flexibility by means of a spring-massdamper system. According to this analogy, the first spring and damper constants are related to the joint behavior, and the following sets of spring and damper represent link flexibility. The variables and the parameters of the model are interpreted as angular quantities.

In this work the description of the deflection related to a rigid link is proposed. It is achieved by means of an Euler-Bernoulli beam formulation and covers the case for small deflections of a beam subject to lateral loads.

The bending moment *M*, shear forces *Q* and deflections *w* for a cantilever beam subjected to a point load *P* at the free end are given by

$$M(\mathbf{x}) = P(\mathbf{x} - L) \tag{1}$$

$$Q(\mathbf{x}) = \mathbf{P} \tag{2}$$

$$\text{cov}(\mathbf{x}) = \frac{P\mathbf{x}^2 \text{(3L - x)}}{6EI} \tag{3}$$

$$\text{cov}(L) = \frac{PL^3}{3EI}.\tag{4}$$

$$
\Delta\theta = \arcsin\left(\frac{\mathcal{W}}{L}\right). \tag{5}
$$

$$\mathcal{S} = \sum\_{j=1}^{2} T\_j(\theta) - T\_j(\theta + \Delta\theta) \tag{6}$$


$$
\pi\_f = \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta}\_f} \right) - \frac{\partial L}{\partial \theta\_f} \tag{7}
$$

$$Q(q)\ddot{q} + \mathcal{L}(q,\dot{q}) + \mathcal{G}(q) = \text{ }\tau\text{ }\tag{8}$$

$$Q\_{11} = \left(\frac{1}{3}\right)m\_1\,a\_1^2 + \left(\frac{1}{3}\right)m\_2\,a\_2^2 + m\_2\,a\_1^2 + m\_2\,a\_1\,a\_2\cos(\theta\_2) \tag{9}$$

$$Q\_{12} = \left(\frac{1}{3}\right) m\_2 \ a\_2^2 + 0.5 \, m\_2 \ a\_1 \ a\_2 \cos(\theta\_2) \tag{10}$$

$$Q\_{21} = \left(\frac{1}{3}\right) m\_2 \ a\_2^2 + 0.5 \, m\_2 \ a\_1 \ a\_2 \cos(\theta\_2) \tag{11}$$

$$Q\_{22} = \left(\frac{1}{3}\right) m\_2 \, a\_2^2 \tag{12}$$

$$\mathcal{L}\_{11} = -m\_2 \, a\_1 \, a\_2 \, \sin(\theta\_2) \dot{\theta}\_1 \, \dot{\theta}\_2 - 0.5 \, m\_2 \, a\_1 \, a\_2 \, \sin(\theta\_2) \, \dot{\theta}\_2 \tag{13}$$

$$\mathcal{L}\_{21} = \begin{array}{c} 0.5 \ m\_2 \ a\_1 \ a\_2 \ \sin \{\theta\_2\} \ \dot{\theta\_2}^{'} \end{array} \tag{14}$$

$$\mathcal{G}\_{11} = -0.5 \, m\_1 \, g \, \, a\_1 \cos(\theta\_1) - 0.5 \, m\_2 \, g \, \, a\_2 \cos(\theta\_1 + \theta\_2) - m\_2 \, g \, \, a\_1 \cos(\theta\_1) \tag{15}$$

$$G\_{21} = -0.5 \, m\_2 \, g \, a\_2 \cos(\theta\_1 + \theta\_2) \tag{16}$$

Multi-Criteria Optimal Path Planning of Flexible Robots 345

reference nodes, presented by circles in Figure 1. The abscissa presents the time instants (�� = �� �� = 1) while the ordinate is the joint angle. The value of intermediate joint angles

This approach ensures a smooth transition on the joint angles, velocities and accelerations, which are positive aspects from the mechanical perspective. A smooth movement preserves

0.0 0.2 0.4 0.6 0.8 1.0

Time (s)

Optimal programming problems for continuous systems are included in the field of the calculus of variations. A continuous-step dynamic system is described by an *n*dimensional state vector **x**(*t*) at time *t*. The choice of an *m*-dimensional control vector **u**(*t*) determines the time rate of change of the state vector through the dynamics given by the

A general optimization problem for such a system is to find the time history of the control

� = �������� � � �[�(�)� �(�)� �]�� ��

In the present context, the interest is focused on the joint angles and the reference nodes. Performance indexes such as positioning error, manipulability and mechanical power are derived from this information. The variable ���� = ��(��) is an element of the state vector **x**, which represents the joint angle at each time ��, related to the joint *j.* The control vector ����

��

vector **u**(*t*) for �� ����� to minimize a performance index given by

��(�) = �(�(�)� �(�)� �) (20)

(21)

between the references nodes are evaluated by means of a cubic spline interpolation.

Fig. 1. Reference nodes (o) and joint angle (—).

0.0

0.5

Joint angle (rad)

1.0

1.5

the mechanism from fatigue effects, for example.

subject to Equation (20) with *t0*, *tf*, and **x**(*ti*) specified.

equation below:

variations of the torque along the path. This means that an explicit constraint over the maximum torque is not required since it will be implicitly achieved by the present formulation.

## **2.3 Manipulability**

As an approach for evaluating quantitatively the ability of manipulators from the kinematics viewpoint the concepts of manipulability ellipsoid and manipulability measure are used.

The set of all end-effector velocities *v* which are realizable by joint velocities such that the Euclidean norm satisfies ����� ��1 are taken into account. This set is an ellipsoid in the *m*dimensional Euclidean space. In the direction of the major axis of the ellipsoid, the endeffector can move at high speed. On the other hand, in the direction of the ellipse minor axis the end-effector can move only at low speed. Also, the larger the ellipsoid the faster the endeffector can move. Since this ellipsoid represents an ability of manipulation it is called as the manipulability ellipsoid.

The principal axes of the manipulability ellipsoid can be found by making use of the singular-value decomposition of the Jacobian matrix��(�). The singular values of *J*, i.e.� ��� ���…���, are the *m* larger values taken from the *n* roots ����� � = 1� … � �� of the eigenvalues. Then �� are the eigenvalues of the matrix��(�)��(�).

One of the representative measures for the ability of manipulation derived from the manipulability ellipsoid is the volume of the ellipsoid. This is given by����, where

$$
\omega = \sigma\_1 \dots \sigma\_2 \dots \sigma\_m \tag{17}
$$

$$c\_m = \begin{cases} \left(2\pi\right)^{\frac{m}{2}} / [2.4.6\dots \text{(m-2)}.m] & \text{if } m \text{ is even} \\ 2(2\pi)^{\frac{(m-1)}{2}} / [1.3.5\dots \text{(m-2)}.m] & \text{if } m \text{ is odd}. \end{cases} \tag{18}$$

Since the coefficient��� is a constant value when the dimension *m* is fixed, the volume is proportional to *w*. Hence, *w* can be seen as a representative measure, which is called the manipulability measure, associated to the manipulator joint angle��.

Due to the direct relation between singular configuration and manipulability (through the Jacobian), the larger the manipulability measure the larger the singular configuration avoidance.

The distance from singular points can be obtained from the solution of an optimization problem in which the following objective function is minimized:

$$f(\theta) = \frac{1}{w} \tag{19}$$

Consequently, the minimization of Equation (19) means to increase the manipulability measure.

### **3. Optimal control formulation**

#### **3.1 Design variables**

The design variables are the key parameters that are updated during the workflow, aiming at increasing (or decreasing) the performance index. In this study, the design variables are

variations of the torque along the path. This means that an explicit constraint over the maximum torque is not required since it will be implicitly achieved by the present

As an approach for evaluating quantitatively the ability of manipulators from the kinematics viewpoint the concepts of manipulability ellipsoid and manipulability measure

The set of all end-effector velocities *v* which are realizable by joint velocities such that the Euclidean norm satisfies ����� ��1 are taken into account. This set is an ellipsoid in the *m*dimensional Euclidean space. In the direction of the major axis of the ellipsoid, the endeffector can move at high speed. On the other hand, in the direction of the ellipse minor axis the end-effector can move only at low speed. Also, the larger the ellipsoid the faster the endeffector can move. Since this ellipsoid represents an ability of manipulation it is called as the

The principal axes of the manipulability ellipsoid can be found by making use of the singular-value decomposition of the Jacobian matrix��(�). The singular values of *J*, i.e.� ��� ���…���, are the *m* larger values taken from the *n* roots ����� � = 1� … � �� of the

One of the representative measures for the ability of manipulation derived from the

Since the coefficient��� is a constant value when the dimension *m* is fixed, the volume is proportional to *w*. Hence, *w* can be seen as a representative measure, which is called the

Due to the direct relation between singular configuration and manipulability (through the Jacobian), the larger the manipulability measure the larger the singular configuration

The distance from singular points can be obtained from the solution of an optimization

�(�) <sup>=</sup> <sup>1</sup> �

Consequently, the minimization of Equation (19) means to increase the manipulability

The design variables are the key parameters that are updated during the workflow, aiming at increasing (or decreasing) the performance index. In this study, the design variables are

� /[2.4.6 … (��2). �� if � is even

� /[1.3.5 … (��2). �� if � is odd.

�=��. �� … �� (17)

(18)

(19)

manipulability ellipsoid is the volume of the ellipsoid. This is given by����, where

eigenvalues. Then �� are the eigenvalues of the matrix��(�)��(�).

�

(���)

manipulability measure, associated to the manipulator joint angle��.

problem in which the following objective function is minimized:

�� = � (2�)

2(2�)

formulation.

are used.

avoidance.

measure.

**3. Optimal control formulation** 

**3.1 Design variables** 

**2.3 Manipulability** 

manipulability ellipsoid.

reference nodes, presented by circles in Figure 1. The abscissa presents the time instants (�� = �� �� = 1) while the ordinate is the joint angle. The value of intermediate joint angles between the references nodes are evaluated by means of a cubic spline interpolation.

This approach ensures a smooth transition on the joint angles, velocities and accelerations, which are positive aspects from the mechanical perspective. A smooth movement preserves the mechanism from fatigue effects, for example.

Optimal programming problems for continuous systems are included in the field of the calculus of variations. A continuous-step dynamic system is described by an *n*dimensional state vector **x**(*t*) at time *t*. The choice of an *m*-dimensional control vector **u**(*t*) determines the time rate of change of the state vector through the dynamics given by the equation below:

$$\dot{\mathbf{x}}(t) = f(\mathbf{x}(t), \mathbf{u}(t), t) \tag{20}$$

A general optimization problem for such a system is to find the time history of the control vector **u**(*t*) for �� ����� to minimize a performance index given by

$$J = \varphi[\mathbf{x}(t\_f)] + \int\_{t\_0}^{t\_f} L[\mathbf{x}(t), \mathbf{u}(t), t]dt \tag{21}$$

subject to Equation (20) with *t0*, *tf*, and **x**(*ti*) specified.

In the present context, the interest is focused on the joint angles and the reference nodes. Performance indexes such as positioning error, manipulability and mechanical power are derived from this information. The variable ���� = ��(��) is an element of the state vector **x**, which represents the joint angle at each time ��, related to the joint *j.* The control vector ����

$$
\varphi[\mathbf{x}(t\_f)] = \left[e\_1\left(\mathbf{x}(t\_f)\right) - p\_t\right]^2\tag{22}
$$

$$J = \left[ e\_1 \left( \mathbf{x}(t\_f) \right) - p\_t \right]^2 \tag{23}$$

$$L\_1(\mathbf{x}(t), \mathbf{u}(t), t) = \sum\_{j=1}^{2} \tau\_{l,j}^2 \tag{24}$$

$$L\_2(\mathbf{x}(t), \mathbf{u}(t), t) = \sum\_{j=1}^{2} \left(\frac{1}{\mathbf{w}\_{l,j}^2}\right) \tag{25}$$

$$L\_3(\mathbf{x}(t), \mathbf{u}(t), t) = \sum\_{l,j=1}^{2} \delta\_{l,j}^2 \tag{26}$$

$$\min\_{\mathbf{u}\sim\alpha} \{ f(\mathbf{u}) | h(\mathbf{u}) = 0, g(\mathbf{u}) \le 0 \} \tag{27}$$

$$p[f(\mathbf{u})] = \sum\_{k=1}^{m} a\_k f\_k(\mathbf{u}), \mathbf{u} \in \Omega. \tag{28}$$


Multi-Criteria Optimal Path Planning of Flexible Robots 349

suggested (Levy and Gomez, 1985) (Levy and Montalvo, 1985) for being used in Equation

Computational evaluation was performed aiming at evaluating the effectiveness of the

(29) to avoid undesirable points and prevent the search algorithm to fail.

**6. Numerical results** 

Fig. 2. Computational workflow.

proposed methodology, as outlined in Figure 2.


It follows that�� = � at the beginning of the optimization process. At the end, �����. By using this scaling procedure, the final objective function value provides a non-dimensional index that describes the percentage of improvement. As an example, the final value *J*=0.3 means that the overall objective was reduced to 30% of this initial value.
