A distributed nonlinear control strategy for two-flexible-link manipulators is presented to track a desired trajectory in the robot’s workspace. The inverse dynamics problem is solved by transforming the desired trajectory from the workspace to the joint space using an intermediate space, called virtual space, and then using the quasi-static approach. To solve the nonminimum phase problem, an output redefinition technique is used. This output consists of the motor’s angle augmented with a weighted value of the link’s extremity. The distributed control strategy consists in controlling the last link by assuming that the first link is stable and follows its desired trajectories. The control law is developed to stabilize the error dynamics and to guarantee bounded internal dynamics such that the new output is as close as possible to the tip. The weighted parameter defining the noncollocated output is then selected. The same procedure is applied to control and stabilize the first link. The asymptotical stability is proved using Lyapunov theory. This algorithm is applied to a two-flexible-link manipulator in the horizontal plane, and simulations showed a good tracking of the desired trajectory in the workspace.
Many control strategies for manipulators have been the focus of several studies in recent years. The robot manipulators consist of a sequence of links and joints in various combinations. In industrial applications, most of the existing manipulators use rigid links and joints and are known as rigid manipulators. Rigid manipulators are generally slow, extremely rigid, and massive, and the useful load is very low compared to their weight. To improve the performance of the robot manipulators, their links must be lighter, and therefore they become more flexible. Flexible manipulators present more advantages when compared to rigid manipulators: they are faster and less massive and consume less energy. Some flexible manipulators are used in different areas, for example, the aerospace applications [
For flexible manipulators, the problem of workspace tracking trajectory is less covered so far than that of joint space tracking. There are few solutions to the workspace tracking problem, particularly for manipulators with many flexible links. The workspace tracking trajectory is very important since most of the tasks are defined in the operational space, such as painting, welding, and assembly. The flexible link manipulators are a nonminimum phase system when controlling the position of the end effector [
The nonlinear dynamics of flexible link manipulators combined with their under-actuated nature (the deflection variables are not actuated) present a challenging control problem. Multi-flexible-link manipulators can be controlled as one MIMO system so a single controller is used for all joints and links or as a set of interconnected subsystems so each pair of joint and link is controlled by its own controller. For the first case, many control schemes were used. Several studies used linearization around a nominal configuration of flexible manipulators model [
The paper is organized as follows. Section
Figure
Two-flexible-link manipulator.
Using Lagrange equations, the dynamical model of an
The dynamical model of the flexible link manipulators has the following properties that will be used in the control law development: From (P1), we can deduce that the diagonal elements of There exists a matrix The inertia-mass matrix The propriety (P4) is preserved for the diagonal elements of
The new noncollocated output of the
In the distributed control strategy, when controlling the
where (
Transform the desired trajectory from the workspace to the joint space using inverse kinematics and quasi-static approach.
Develop the control law for the second and first links to stabilize the error dynamics and to guarantee bounded internal dynamics such that the output is as close as possible to the tip.
Study the global stability.
To achieve the objective of workspace tracking trajectories, we need to transform the desired trajectories from the workspace to the joint space. The flexible manipulator is a nonminimum phase system when the end effector is used as the output. In this case, kinematic and dynamic relationships link the workspace and the joint space. To overcome this problem, an intermediate space called virtual space can be used. Then the desired workspace trajectory is transformed to the virtual space using an inverse kinematics relation as in rigid manipulators. To transform the desired trajectories from the virtual space to the joint pace, the quasi-static approach can be used to solve a nonlinear equation for the flexible part.
Using inverse kinematics as in rigid manipulators, the generalized coordinates in virtual space can be easily found. The deformation is assumed to be small. According to Figure
Virtual space.
Using the Jacobian matrix as in rigid manipulators, the velocity and acceleration in the virtual space can be deduced. Then, for a two-DOF manipulator, the inverse kinematics is given by the following equation:
The dynamical model (
For two-flexible-link manipulator, the distributed control strategy consists of controlling and stabilizing the last joint and flexible link by assuming that the first joint and flexible link are stable and follow their desired trajectories. Then, we move backward and apply the same procedure to the first joint and link. For each step, a new noncollocated output and a control law are developed. The weighting parameter characterizing the noncollocated output is calculated such that the tracking error is asymptotically stable. Thus, the system becomes a minimum phase with the selected new weighted outputs.
In this paper, the two-flexible-link model given in [
Using (P3), the dynamical model of the two-flexible-link manipulator can be written as
In rigid and flexible part decomposition, the dynamical model (
To develop a control law, the dynamical model (
Equation (
The modified dynamical model (
Equation (
The first and the second subsystems can be characterized by two mass and inertia symmetric positive definite matrices:
The new generalized coordinate, used for the second joint and link while the first joint and link are assumed stable, is given as follows:
According to (
Using (
The internal dynamics of the second link is deduced from (
Inserting (
To control the second link, we propose the following control law:
Now going backward to the first subsystem and assuming that the second subsystem is stable, the new generalized coordinate associated to the second subsystem becomes
Using (
To find the internal dynamics of the first link,
Distributed control strategy.
This section presents the stability analysis of the tracking errors of the rigid and the flexible parts. First, we study the global stability of the rigid part by inserting the two control laws in the initial dynamical model. Second, the stability of the flexible part is studied to select the weighted parameters and guarantee bounded internal dynamics such that the new output is as close as possible to the tip.
In compact form, the first (
To study the global stability, we insert the control law given in (
The equation of motion of the two-flexible-link manipulator (
See Appendix
When using the control law (
See Appendix
To prove the asymptotical stability of the error dynamics, we propose the following Lyapunov function:
In this section, asymptotical stability of the flexible part is studied using the internal dynamics. From the dynamical model given in (
The quasi-static approach, using the inverse dynamics, neglects
The critical value of
The same procedure is now applied, proceeding backward, to the first joint-link subsystem. Using (
In the state space form, the tracking error of the flexible part of the first link can be written as follows:
As already shown in the stability analysis of the rigid part,
The critical value of
The two-flexible-link manipulator shown in Figure
In Sections
Table
System parameters.
Parameter | Value |
---|---|
Hub inertia ( |
0.1 kg·m2 |
Link length ( |
0.5 m |
Link linear density ( |
1 kg/m |
Link rigidity ( |
10 N·m2 |
Link mass ( |
0.5 kg |
Payload mass ( |
0.1 kg |
Payload inertia ( |
0.0005 kg·m2 |
Simulation results are given Figures
Desired workspace trajectory: (a)-(b) position trajectory, (c)-(d) velocity trajectories, and (e)-(f) acceleration trajectories.
Desired virtual space trajectories: (a)-(b) position trajectories, (c)-(d) velocity trajectories, and (e)-(f) acceleration trajectories.
Desired joint space trajectories: (a)-(b) position trajectories of rigid coordinates, (c)-(d) velocity trajectories of rigid coordinates, (e)-(f) acceleration trajectories of rigid coordinates, and (g)-(h) position trajectories of flexible coordinates.
Stability of second link. (a) Nyquist diagram and (b) eigenvalues evolution [
Stability of first link: eigenvalues versus
Tracking trajectories of noncollocated outputs.
Tracking errors in joint space. (a)-(b) tracking errors of noncollocated outputs and (c)-(d) tracking errors of the flexible coordinates.
Tracking in the workspace. (a)
Workspace tracking errors. (a)
For two-flexible-link manipulators, the desired workspace trajectory is chosen as lozenge form. The results of the inverse kinematics problem are given in Figures
A good tracking performance of the new noncollocated output was obtained, as shown in Figure
This paper presents a nonlinear distributed control for two-flexible-link manipulator. For the inverse dynamics, a virtual space, linked with the workspace by a simple kinematics relation as in rigid manipulators, and a quasi-static approach were used. Using this transformation procedure, a workspace desired trajectory (lozenge) has been successfully transformed to the joint space. The distributed control strategy presented in this paper uses the output redefinition technique and consists of stabilizing the flexible manipulators starting with the last joint and flexible link and going backward until the first joint and link. Lyapunov theory was used to prove the asymptotical stability. An adaptive version of this control strategy will be investigated in future work.
Using (
Deducing
The models using the collocated output (
Using the transformation
From (P4), by multiplying (
The Schur complement of
In the following, we study the error dynamics given by (
The first element is developed as
Using Taylor series, we can write