ACSE 05 Conference, 19-21 December 2005, CICC, Cairo, Egypt
Predictive functional control of a PUMA robot A. Vivas, V. Mosquera Department of Electronic, Instrumentation and Control, University of Cauca Calle 5 No. 4-70, Popayán, Colombia
[email protected] http://www.ai.ucauca.edu.co
Abstract This paper describes an efficient approach for nonlinear model predictive control, applied to a 6-DOF arm robot. The model is first linearized and decoupled by feedback, secondly a model predictive control scheme, implemented with an optimized dynamic model and running within small sampling period, is exhibited. Major simulation results performed using numerical values of an industrial PUMA 560 robot prove the effectiveness of the proposed approach. The nonlinear model-based predictive control and the widely used computed torque control are compared. Tracking performance and robustness with respect to external disturbances or errors in the model are enlightened. Keywords: Robot control, dynamic modelisation, computed torque control, predictive control.
1. Introduction Many strategies have been used in the last years for robot control [1], [2], [3]. In industrial and commercial robot arms, controllers are still usually PID. But robotic manipulators have many serially linked components, the manipulator dynamics is highly nonlinear with strong couplings existing between joints, that complicate the task of a simple PID, especially if the velocities and accelerations of the robot are high. Better solutions are then implemented with model based controllers, that use a mathematical model of the robot to explicitly compensate the dynamic terms. The most common strategy of model based control is the computed torque control [4], [5], widely used for industrial robot arms. This strategy is relatively easy to implement but the uncertainties presents in the model difficult to design an effective control algorithm based on an exact mathematical model. Other solutions may be then implemented to give to the system the robustness needed [6], [7].
In the past decade, model predictive control (MPC) has become a good control strategy for a large number of process [8], [9]. Several works [10], [11], [12] have shown that model based predictive control could be of a great interest for handling constrained processes by optimizing, over some manipulated inputs, forecasts of process behavior. It provides good performances in term of rapidity, disturbances and errors cancellations. However, a significant number of industrial applications can be found, for instance, in chemicals industries or food processing that are processes with relatively slow dynamics. But very few results concern the computation of the model predictive control with nonlinear and high dynamic process such as robot links [13], [14], where the model is highly nonlinear and the digital control has to be performed within only few milliseconds. An special predictive strategy is presented in this study (predictive functional control [15], [16]), and it will be compared with the computed torque control. The robot platform used to test these controllers is a PUMA 560 robot, widely used in robotics research, for which there is a substantial literature. Test will remark the performances in tracking and robustness behavior. This paper is organized as follows: Section (2) resumes the dynamic model formulation of a robot, Section (3) details the model predictive functional control, Section (4) is dedicated to define the model based control strategies of this study, Section (5) list major simulation results in terms of tracking performance and robustness. Finally, conclusions are given in Section (6).
2. Dynamic model formulation It is well known that the behavior of an n-DOF rigidlink arm is governed by the following equation: Γ = A(q )q+ C (q, q )q + g (q )
(1)
ACSE 05 Conference, 19-21 December 2005, CICC, Cairo, Egypt
Where Γ is the actuation torque, A(q ) is a symmetric and positive definite inertia matrix, C (q, q ) is the matrix of the Coriolis and centrifugal forces, and g(q) is the gravity force. This robot dynamic equation can also be written in a compact form:
Internal Modeling: The model used is a linear one given by:
Γ = A(q ) q+ H (q , q )
where xM is the state, u is the input, yM is the measured model output, FM, GM and CM are respectively matrices or vectors of the right dimension.
(2)
where H includes centrifugal, Coriolis and gravitational forces. The dynamic terms in Eq.(2) are highly nonlinear and coupled. The first step is to design a linearizer to dynamically linearize and decouple the robot dynamic model. The possibility of finding such a linearizing controller is guaranteed by the particular form of system dynamic. In fact, Eq.(2) is linear in the control Γ and has a full-rank matrix A(q) which can be inverted for any manipulator configuration. Thus, the following linearization is proposed: Γ = Aˆ (q ) u + Hˆ (q , q )
(3)
where u represents a new input vector. Aˆ and Hˆ are estimations of the real manipulator dynamics. In the absence of disturbance and when the dynamic model is perfectly known, Aˆ = A and Hˆ = H . In this case u has the form of a joint acceleration: u = q
(4)
This is a decoupled system with simple second order linear differential dynamics. In other words, the component ui influences, with a double integrator relationship, only the joint variable qi, independently of the motion of the other joints. This technique of nonlinear compensation and decoupling is very attractive from a control viewpoint since the nonlinear and coupled manipulator dynamics is replaced by a linear and decoupled second-order system. Nonetheless, this technique is based on the assumption of perfect cancellation of dynamic terms. Compensation may be imperfect for model uncertainty and for the approximations made in on-line computation of inverse dynamics. In the next section, a model based technique is presented which is aimed at counteracting the effects of imperfect compensation.
3. Predictive functional control This section is dedicated to briefly recall the main steps of the model predictive functional control scheme used hereafter. This predictive technique has been developed by Richalet and complete details of the computation may be found in [15], [16], [17].
xM (n) = FM xM (n −1) + GM u(n −1) yM ( n) = C M T x M ( n)
(5)
If the system is unstable, the problem of robustness caused by the controller's cancellation of the poles is usually solved by a model decomposition [15]. Reference trajectory: The predictive control strategy of the MPC is summarized in Figure 1. Given the set point trajectory over a receding horizon [0, h], the predicted process output yˆ P will reach the future set point following a reference trajectory yR . In Figure 1, ε (n) = c(n) − yP (n) is the position tracking error at time n, c is the set point trajectory, yP is the process output, and CLTR is the closed loop time response. Over the receding horizon, the reference trajectory yR , which is the path towards the future set point, is given by: c(n + i ) − yR (n + i ) = α i (c(n) − yP ( n)) for 0 ≤ i ≤ h (6)
where α (0< α
